math curriculum k-12

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SAU #48 MATHEMATICS CURRICULUM GUIDE GRADES K – 12 ASHLAND CAMPTON, ELLSWORTH, HOLDERNESS, PEMI-BAKER REGIONAL PLYMOUTH, RUMNEY, THORNTON, WATERVILLE VALLEY, WENTWORTH Approved by School Board/Date: Campton May 10, 2005 Holderness April 4, 2005 Pemi-Baker October 18, 2005 Plymouth May 9, 2005 Rumney April 13, 2005 Thornton April 26, 2005 Waterville Valley April 12, 2005 Wentworth May 23, 2005

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Page 1: Math Curriculum K-12

SAU #48 MATHEMATICS CURRICULUM GUIDE GRADES K – 12

ASHLAND CAMPTON, ELLSWORTH, HOLDERNESS, PEMI-BAKER REGIONAL

PLYMOUTH, RUMNEY, THORNTON, WATERVILLE VALLEY, WENTWORTH

Approved by School Board/Date:

Campton May 10, 2005 Holderness April 4, 2005 Pemi-Baker October 18, 2005 Plymouth May 9, 2005 Rumney April 13, 2005 Thornton April 26, 2005 Waterville Valley April 12, 2005 Wentworth May 23, 2005

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Prepared by:

Ashland Elementary School Judy Kreamer

Campton Elementary School Sonja Anderson, Sandy Carter, Janet Prindle, Sherry Sinclar, Charlene Whitman

Holderness Central School Beth Allain, Joan Coursey, Ruth Harlow, Susan Long, Becky Wark

Plymouth Elementary School Cynthia Croasdale, Peter Hutchins, Karen McCloud, Mark McGlone, Jan Panagoulis

Plymouth Regional High School Donald Hudak , Donni Hughes, Amy Jemery, Stephanie Miller, Gareth Peters, Pat Palmer, Gail Poitrast

Russell Elementary School Cindy Campbell, Shelley Hancock

Thornton Central School Ann Knowles Waterville Valley Elementary School Gail Hannigan

Wentworth Elementary School Brooke Blake, Erin DeCotis, Phoebe Sanborn

Special Thanks to: Dr. Richard Evans, Professor of Mathematics, Plymouth State University

Mahesh Sharma, Director of the Center for Teaching/Learning Mathematics

SAU #48 would like to thank each and every person who worked so diligently on this great curriculum guide. We apologize if we have missed any names; please let the central office know of any omissions.

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Table of Contents Grades K – 12

A Word from the Committee …………………………………………………………………………………………….... 6 SAU #48 Mathematics Philosophy and Goals …………………………………………………………………………… 7 & 8 Quantitative and Qualitative Learning Personalities …………………………………………………………………….. 9 Levels of Knowing ………………………………………………………………………………………………………… 10 Levels of Thinking ………………………………………………………………………………………………………… 11 Grades K-2 Mathematics Scope and Sequence ...........…………………………………………………………………….. 12 - 15

Content: Number Sense Content: Computation and Operations Content: Data Analysis and Chance Content: Geometry and Measurement

Overview of Kindergarten ..................................................................................................................................................... 16 Overview of Grade 1............................................................................................................................................................... 17 Overview of Grade 2............................................................................................................................................................... 18

Page 4: Math Curriculum K-12

4Grades K-3............................................................................................................................................................................. 38-48

Grades 3 – 5 Mathematics Scope and Sequence ................………………………………………………………………… 19-24 Content: Number Sense Content: Computation and Operations Content: Data Analysis and Chance Content: Geometry and Measurement

Overview of Grade 3 ............................................................................................................................................................. 25 Overview of Grade 4 .............................................................................................................................................................. 26 Overview of Grade 5 ............................................................................................................................................................. 27 Grades 6 – 8 Mathematics Scope and Sequence…………………………………………………………………………. 28-33

Content: Number Sense Content: Computation and Operations Content: Data Analysis and Chance Content: Geometry and Measurement

Overview of Grade 6 ............................................................................................................................................................ 34 Overview of Grade 7 ........................................................................................................................................................... 35 Overview of Grade 8 ........................................................................................................................................................... 36 Assessment............................................................................................................................................................................ 37 Glossary

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Grades 4 – 8............................................................................................................................................................................ 49-78 Plymouth Regional High School Grades 9 – 12 Curriculum……………………………………………………………….. 79-138

Plymouth Regional High School Expectations and Indicators …………………………………………………………… 139-143 Appendix A - Mathematic GLE’s ………………………………………………………………………………………… 144-193

Appendix B – Frameworks ……………………………………………………………………………………………….. 194-228

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A Word from the Committee

The 21st century promises to employ workers that are problem solvers, critical thinkers, and cooperative team members. All of these are within the scope of mathematics which means that the reach of mathematics must go beyond the classroom to other areas such as the arts and communication. Mathematics teachers have a vital role in this new era as we shape malleable young people to become distinctive individuals well prepared to serve the needs of a technologically driven age.

In the early years (1992), Assistant Superintendent John True gave us the tasks of combining skills and the NCTM Standards, reaching consensus between K-8 and 9-12, exploring the material taught in grades 4 and above, and creating a scope and sequence or an outline of a new curriculum. We returned to our classrooms and conducted a month long search to find our mildew laden, cob-webbed covered, seal intact, curriculum guides. After dusting off our curriculum guides we examined the objectives set for each grade. Our eyes were opened to the fact that basic facts and shapes were being taught all the way from Kindergarten through grade 8. Fred Prevost, Mathematics Consultant for the State of New Hampshire, asked us, "When do we stop teaching how to add whole numbers?" In order to intelligently answer that question, the committee began its search for new knowledge, ideas, and methods to incorporate into our existing curriculum. The result is a blueprint that is unique as well as practical. We were not content to adopt the work of others, but were receptive to the ideas outlined in the New Hampshire Frameworks. This curriculum guide works within the dimension of the NCTM Standards, and illuminates the innovative thoughts of renowned math educators Mahesh Sharma1 and Dick Evans2. This document spans several years of ardent work by this committee in order to provide a quality plan that naturally progresses from Kindergarten through grade 12.

In the future, we hope to see that this document is falling apart, not from mildew, but from overuse. Please write notes in this resource as needed and share your thoughts and additions with colleagues and committee members to help improve this work-in-progress. We sincerely hope that you will see this document as a valuable addition to your teaching tools and resources.

Mahesh Sharma is Core Faculty and Dean of Professional Programs in Education at Cambridge College, Director of The Center for Teaching/Learning of Mathematics, and editor of Focus on Learning Problems in Mathematics. Professor Sharma works directly with children as well as helping teachers identify how students learn mathematics.

Dr. Richard E. Evans is Professor of Mathematics at Plymouth State College, Director of Pre-Award Grants, and Co-author of Mathlab Activities 1 and 2. Dr. Evans works directly with teachers and prospective teachers and has been an integral part of mathematics education in SAU #48.

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SAU #48 MATHEMATICS PHILOSOPHY & GOALS

Mathematics is a dynamic subject affected by the advances in research and technology, and the demands of a modern world. It is an essential piece of every aspect of human culture. In order to prepare our students for success in their world, a dynamic curriculum is mandatory. Research indicates that mathematics education will best serve societal needs when the curriculum is conceptually focused. When students learn mathematics by exploring and discussing concepts in the context of real life situations, what emerges from these experiences are skills which are anchored in understanding. The students can not only perform the basic procedures, but also know how to apply them to new situations. A curriculum that stresses the use of mathematics as a tool for engaging students in meaningful explorations and investigations promotes optimal mathematics learning. We believe that all students must develop their own mathematical understandings. A mathematical program must take into account various learning styles to promote positive student attitudes. The attitudes students form influence their thinking and performance, and later, influence their decisions about studying mathematics. Each one of us processes information and therefore mathematics information, differently and uniquely. This processing difference defines the unique way we learn mathematics – quantitatively or qualitatively. Students are active individuals who construct, modify and integrate ideas by interacting with materials, the world around them, and their peers. Thus, the learning of mathematics must be an active process: exploring, justifying, representing, solving, constructing, discussing using, investigating, describing, developing, and predicting. These actions require both the physical and mental involvement of students – both hands on and minds on. (NH Mathematics Framework p.3) Development of a mathematical way of thinking takes place when students and teachers communicate mathematics. They communicate when the use of effective questioning promotes articulation of the thinking process. In the words of Mahesh Sharma:

Questions instigate language production.

Language instigates models.

Models instigate thinking.

Thinking instigates understanding.

Understanding instigates knowledge, skills, and competence.

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Competence facilitates communication.

Building on this philosophy, and in order to ensure success for all students, principals and teachers should provide programs that:

Motivate all students to learn and maintain a positive attitude about math.

Engage and encourage all students to be active participants in their own study of mathematics.

Emphasize skills, concepts, applications, and accuracy, focusing on the relationships between the four mathematical

operations: addition, subtraction, multiplication and division.

Develop problem solving strategies through cooperative and collaborative interaction between students and teachers.

Involve students in real world problem solving where they generate multiple solutions and make connections to all other curriculum areas.

Use models and/or manipulatives to apply concepts and create procedures.

Insure that students and teachers communicate about math everyday.

Develop fluency and understanding of the language of mathematics.

Use technology appropriately.

Include varied assessment tools that are interpreted and used to guide instruction and planning.

Include opportunities for career exploration.

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QUANTITATIVE AND QUALITATIVE LEARNING PERSONALITIES

Math is the study of space and quantity. Through these domains emerge two distinct mathematics learning personalities: quantitative and qualitative. They define the way in which students learn mathematics. These personalities are on a continuum in which each student has a unique place.

“To learn mathematics, therefore, involves both approaches: quantitative and qualitative. To acquire a high level of fluency in mathematics work, the elements of both types of learning styles must be integrated, although in a particular individual, one learning style is more dominant than the other”. (From Improving Mathematics Instruction, Mahesh Sharma, page 15) The quantitative learner starts with the parts of a problem, and from those parts, the whole emerges. This student learns best when procedures, definitions, and formulas are presented FIRST and THEN supported with examples. This learner is usually very good at arithmetic, capable of solving word problems if they are the same as in-class examples. It follows that this learner does well on unit tests (which is like a piece of the picture) but may have difficulty on cumulative tests (which look at the whole.) In terms of manipulatives, the quantitative learner should use materials that are discrete and discontinuous. Any material that uses counting as the basis is appropriate, like Unifix Cubes, fingers, number lines, or beans. The most common error made by the quantitative learner is the error of commission (too much information, too many steps or procedures). This error can be corrected by directing the student to the whole picture because s/he is focusing on individual steps. The qualitative learner sees the whole picture and then breaks it into parts. This learner is usually very good at problem solving but procedures prove to be difficult. This student learns best when examples are given first, and then, through inductive reasoning, derives the procedures, definitions and formulas. It follows that this learner does well on cumulative tests (which look at the whole) but may have difficulty on a unit test, which is like a piece of the picture). The qualitative learner is better served with continuous, visual, spatial materials. The defining characteristics of these materials are color, shape or size. Cuisinaire rods and pattern blocks are appropriate tools. Omission is the most common error of the qualitative learner – forgetting details such as signs, decimal points and other symbols. To correct this problem the teacher’s questioning techniques must focus on the details because the student sees the whole and not the parts. In order to meet the needs of all learners, lessons should combine deductive and inductive reasoning. By starting with many examples, deriving the result (a definition, rule, or procedure) and then doing more examples, both learning styles are addressed. Continuous materials help children see patterns in order to use inductive logic. When they are combined with discrete materials that emphasize sequencing, transitivity, and deductive logic, both spatial perception and strategy skills are challenged.

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LEVELS OF KNOWING

“Every mathematical skill is a developmental concept. To master a concept at its fullest, a child needs to understand it at all levels … and at all of its levels of difficulty. The mastery of a given mathematical concept passes from intuitive level of understanding to the level where the child can explain how he has arrived at a particular result and can explain the intricacies of the concept. However, to reach this highest level of understanding and mastering, the child goes through several intermediate and partial levels of mastery.” (From Improving Mathematics Instruction, Mahesh Sharma, page I-8) Through understanding of a mathematical concept requires that the student progress through these four levels: Intuitive Level This beginning level prepares the child for specific learning by connecting the new concept to something the child already knows. This can be done through free play, brainstorming and effective questioning. Concrete The second stage of knowing involves the child manipulating concrete materials in order to solve a problem. If there are no materials available, the child will be unable to arrive at a solution. Mastery at this level is essential before children are able to continue on to the next Level of Knowing. This level is the anchor to all other levels; therefore, more time spent at this level will facilitate learning at the abstract level. Pictorial The pictorial level is the first level that involves symbolic representation of mathematical ideas and concepts. A student functioning at this level can solve problems only with the help of pictures or diagrams. To demonstrate their knowledge, students represent three-dimensional models in a two dimensional space. This level creates a bridge for the student as s/he moves from the concrete to the abstract. Abstract At the abstract level, students are able to envision the ideas they developed at the concrete and pictorial levels in order to solve problems without the actual manipulatives or a picture in front of them. Too often, instruction begins at this level and the student has no images to reference when solving a problem. “Abstract concepts are meaningless unless the student has many and diverse concrete experiences).” from Math Notebook: Levels of Knowing, volume 6, number 1 and 2, 1988. (I – C – P - A) are indicated on the Content pages. To the right of (I – C – P - A) you will find the Curriculum Frameworks represented by a number and lower case letter; example: (4a)

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Besides these levels, students must have pre-requisite skills such as following directions, ordering, sequencing and visualization. Learning mathematics in the absence of these pre-requisite skills and without progressing through the Levels of Knowing is “like having toy balloons; if they are not tied to something, they will fly away.” (From Improving Mathematics Instruction, Mahesh Sharma, page I-12) Once all of these levels have been mastered, students should be able to apply their knowledge in diverse situations. Applications may include transferring math knowledge to other Content areas, forming new ideas based on previously mastered concepts, and finding solutions to word and real life problems. Students should also be able to share their knowledge and thinking process through effective communication. Group projects, journals, think alouds, peer tutoring and conversations that utilize appropriate vocabulary are ways to demonstrate mastery of mathematical concepts through communication.

LEVELS OF THINKING An area of improvement in mathematics instruction in the classroom is to develop the students’ ability to think mathematically. This is accomplished through the development of the linguistic, conceptual and procedural components of each math concept. Students mastering these components understand and are able to apply the math concepts learned in the classroom. Linguistic The linguistic component is the acquisition of mathematical language necessary for the development of a specific concept. It is important here to realize that mathematics is a second or even a third language of a student. Conceptual The conceptual component is the visualization of an appropriate concrete model. It’s important that students have a picture in their mind’s eye. Procedural Only after the linguistic and conceptual components have been developed, can the procedural level be explored. This component involves the sequencing of steps needed in solving a problem or applying a skill.

“… mathematics is a bonafide second language, and if we want to help the child to think mathematically, we need to help him in the acquisition of the mathematical language: it’s vocabulary, it’s syntax, and, the translation ability from one language to the other.” from Improving Mathematics Instruction, Mahesh Sharma, Page I-8 “The mastery of mathematical concepts is a developmental process and is also a cumulative one. The complete mastery of any mathematical concept is achieved slowly and sequentially. …. mastery in mathematics means mastery in all of these components.” from Math Notebook, Volume 6, Number 1 & 2, Mahesh Sharma

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K-2 MATHEMATICS SCOPE AND SEQUENCE Content Kindergarten First Grade Second Grade

NUMBER SENSE Counting Rounding Patterns Multiples, Factors, Divisibility Money Appropriate Technology (indicated by asterisk)

A kindergartner should be able to: 1. Count orally up to 10 to determine numbers in a

given set. (I-C-P-A) (3a.) 2. Compare sets to determine more, less, and equal.

(I-C-P) (3a.) 3. Demonstrate an understanding of one more and one

less, up to 10. (I-C-P) (3a.) 4. Count backwards from 10 to 1 and by tens from

100, with the group. (I-C) (3a.) 5. Classify objects according to attributes. (I-C)

(5a.) 6. Put objects in groups of 10. (I-C) (3b.) 7. Count by tens to 100. (I-C) (3b.) 8. Discuss ½, 1/3, ¼ as they occur in daily activities

(I) (2a. 3a.) 9. Use concrete devices to name and compare simple

fractions. (I) (3a.) 10. *Explore the concept of one more and one less up

to 10 using a calculator. (A) (3a. 3c) 11. Recognize and name pennies and dimes. (I-C-P)

(4c.)

A first grader should be able to: 1. Count to 100 by ones, twos, fives, and tens frontwards

and backwards. (I-C-P-A) (3b.) 2. Determine the number in a set with less than 100

items. (I-C-P-A) (3a.) 3. Compare sets to determine more, less, and equal to.

(I-C-P-A) (3a.) 4. Match ordinal numbers to sets which contain that

number. (I-C-P-A) (3a.) 5. Identify the number which is one more or less than a

given set. (I-C-P-A) (3a.) 6. Identify and illustrate place value up to 99, using

groupings of tens and ones. (I-C-P-A) (3a.) 7. *Explore patterns and place value activities using

calculators. (A) (3a. 3c) 8. Identify and use odd and even numbers. (I-C-

P-A) (3b.) 9. Recognize simple fractions as equal shares of a whole

unit. (I-C-P) (3a.) 10. Create models of simple fractions (I-C-P) (3a.) 11. Recognize and record data using simple fractions. (I-

C-P-A) (3a. 5a.) 12. Recognize, name, and use pennies, nickels, dimes,

quarters, half dollars, and dollars. (I-C-P) (4c.) 13. Explore the relationship between pennies, nickels,

dimes, quarters, half dollars, and dollars. (I-C-P-A) (4c.)

14. Find equivalent money amounts using coins. (I-C-

P) (4c.)

A second grader should be able to: 1. Skip count using patterns of multiples of single digit numbers,

frontwards and backwards. (I-C-P-A) 3b.) 2. Given a two or three digit number, identify hundreds, tens, and

ones. (I-C-P-A) (3a.) 3. Order sets of numbers from smallest to largest and vice versa

using numbers 0-100. (I-C-P-A) (3a.) 4. Identify the number which is one more or one less than a 2-digit

number. (I-C-P-A) (3a.) 5. Identify the number which is 10 more or 10 less than a 2-digit

number. (I-C-P-A) (3a.) 6. Compare 2-digit numbers to determine more or less. (I-C-P-A)

(3a.) 7. Use place value models to explore other bases. (I-C) (3a.) 8. Use a place value model to identify and write 2-digit numbers. (I-

C-P-A) (3a.) 9. Use a place value model to regroup and rename 2-digit numbers.

(I-C-P) (3b.) 10. Identify and use odd and even numbers. (I-C-P-A) (3b.) 11. *Explore numbers and place value concepts using a calculator.

(A) (3a. 3c.) 12. Identify and find the nearest ten on a number line and other

models using a 2-digit number. (I-C-P-A) (3a.) 13. Create models to review simple fractions. (I-C-P) (3a.) 14. Name and write fractional parts of a whole and a set. (I-C-P-A)

(3a.) 15. Create and use models to explore and explain equivalent

fractions. (I-C-P) (3a. 3c.) 16. Use models to order fractions. (I-C-P) (3a. 3c.) 17. Discover the value of coins and paper money. (I-C-P-A) (4c.) 18. Recognize, use and record money amounts in decimal form. (C-

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K-2 MATHEMATICS SCOPE AND SEQUENCE Content Kindergarten First Grade Second Grade

COMPUTATION AND OPERATIONS Addition, Subtraction, Multiplication & Division

(of whole numbers, fractions & decimals)

Properties Percent Multiples, Factors, Divisibility Estimation Appropriate Technology (indicated by asterisk)

A kindergartner should be able to: 1. Model whole number addition situations by

joining sets of objects. (I-C-P) (2b. 3b.) 2. Model whole number subtraction situations by

removing a subset from a set. (I-C-P) (2b. 3b.)

A first grader should be able to: 1. Use manipulatives to develop and record the concepts of

addition and subtraction facts through 20, recognizing the relationship between the two operations. (I-C-P-A) (6a. 2b. 3b.)

2. Represent the joining of two sets as addition, using

numerals. (I-C-P-A) (2b.) 3. Represent the separation/comparison of sets or a

missing addend as subtraction, using numerals. (I-C-P-A) (2b.)

4. Automatize addition and subtraction facts through 20.

(A) (3c.) 5. *Explore patterns and sequences using calculators. (A)

(3a. 3c. 6a.) 6. Add and subtract money amounts of less than one dollar

using models. (I-C-P) (2b. 3b. 4c.) 7. Explore the commutative and associative properties and

the identify property of zero in addition. (I-C-P-A) (3b. 6b.)

A second grader should be able to: 1. Represent the joining of sets as addition. (I-C-P-A) (2b.) 2. Represent the separation/comparison of sets or a missing addend as

subtraction. (I-C-P-A) (2b.) 3. *Explore addition and subtraction using calculators. (A) (2b. 3c.) 4. *Explore addition and subtraction using computer software. (A) 5. Develop algorithms for addition and subtraction using

manipulatives. (I-C-P-A) (3c.) 6. Add and subtract any 2-digit numbers accurately. (I-C-P)

(3c.) 7. Develop the concept of subtraction (with regrouping) using

manipulatives to reinforce mastery of subtraction facts. (I-C-P-A) (2b. 3b. 3c.)

8. Explore, discuss and recognize the connection of addition and

subtraction as inverse operations. (I-C-P-A) (2b. 3b. 6a.) 9. Explore repeated addition and arrays as multiplication. (I-C-P-A)

(3b. 2b.) 10. *Explore patterns and sequences using calculators. (A) (3a. 3c. 6a.) 11. Explore the commutative and associative properties and the

identity property of zero in addition and explore why they do not hold for subtraction. (I-C) (3b. 6b.)

12. Determine the reasonableness of estimated sums and differences

using a variety of strategies. (I-C-P-A) (3c. 3d.) 13. Use estimation and approximation to solve problems where exact

answers are NOT required. (I-C-P-A) (3c. 3d.) 14. Explore addition of decimals using decimal squares and money

amounts as models. (I-C) (2b. 3c. 4c.) P-A) (4c.)

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K-2 MATHEMATICS SCOPE AND SEQUENCE

Content Kindergarten First Grade Second Grade DATA ANALYSIS AND CHANCE Data Collection and Organization Graphing Statistics Probability Appropriate Technology (indicated by asterisk)

A kindergartner should be able to: 1. Collect and organize objects and information

and discuss schemes for sorting. (I-C-P) (1a. 1b. 2a. 2b. 5a.)

2. Construct and interpret real graphs, i.e., graphs

that are made with physical objects. (I-C-P-A) (1a. 1b. 5a.)

A first grader should be able to: 1. Collect and organize objects and information and

discuss schemes for sorting. (I-C-P-A) (1a. 1b. 2a. 2b. 5a.)

2. Construct and interpret “real” and picture graphs. (I-

C-P-A) (1a. 1b. 2a. 5a.) 3. Verbalize observations to be recorded and added to

graphs. (I-C-P-A) (1a. 2a. 5a.) 4. Predict what is most likely to happen given a set of

facts, e.g., weather data. (I-C-P-A) (1a. 2a. 5a.)

A second grader should be able to: 1. Collect data; construct, interpret and discuss graphs (real, picto

and bar), tables and charts. (I-C-P-A) (1a. 1b. 2a. 2b. 5a.)

2. Write story problems using information from a graph. (I-C-P-A) (1a. 2a.)

3. Write about data collecting, graph or table construction, and

results. (I-C-P-A) (1a. 2a.) 4. Predict, test and compile data as to which event is most likely

or least likely to happen given appropriate information. (I-C-P-A) (1a. 2a. 5a.)

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K-2 MATHEMATICS SCOPE AND SEQUENCE Content Kindergarten First Grade Second Grade

GEOMETRY AND MEASUREMENT Two Dimensional Three Dimensional Congruency & Similarity Transformations Patterns Measurement Temperature Time Appropriate Technology (indicated by asterisk)

A kindergartner should be able to: 1. Sort and classify common geometric shapes

(at least circle, square, rectangle, and triangle). (I-C-P) (1a. 4a.)

2. Explore and make different 2- and 3-

dimensional shapes. (I-C) (4a. 4b.) 3. Use positional terms correctly (inside,

outside, above, under, beside…) (I-C-P-A) (4b.)

4. Use geometric shapes to create, explore, and

name patterns. (I-C-P) (6a. 7a) 5. Explore and compare objects in relation to

self by describing attributes of length, weight, area, volume, and temperature. (I-C) (4c.)

A first grader should be able to: 1. Identify and make geometric shapes (circle,

square/diamond, rectangle, triangle, and hexagon). (I-C-P) (4a. 4b.)

2. Link common objects to geometric shapes and

objects, using correct geometrical language (ball-sphere, box-rectangular solid). (I-C-P) (2a. 2b.)

3. Use geometric shapes to create, explore, and name

patterns. (I-C-P-A) (7a.) 4. *Explore making 2-dimensional figures using

computer software. (P-A) 5. Identify and model simple congruent figures in

different positions. (I-C-P) (4a. 4b.) 6. Copy and extend simple patterns and shapes. (I-C-

P) (4b.) 7. Continue language development of position

(inside, outside, in front of, in back of, above, under, etc.). (I-C-P-A) (4b.)

8. Use non-standard units to measure length, area,

weight and volume. (I-C-P) (4c.) 9. Explore and discover the need for a uniform unit of

length. (I-C-P) (4c.) 10. Read a Fahrenheit thermometer. (I) (4c.)

11. Know the order of the days of the week and

identify what day of the week it is. (A) 12. Tell time on a digital and analog clock using hours

and half hours. (I-C-P) (4c.)

A second grader should be able to: 1. Draw, make, and explore squares/diamonds, triangles,

rectangles, circles, rhombus, trapezoids, kites and parallelograms. (I-C-P-A) (4a.)

2. Use the terms point, line and line segment in describing 2-

dimensional figures. (I-C-P-A) (4a.) 3. *Explore making 2-dimensional figures and angle

measurements using computer software. (I-C-P-A) 4. Recognize and make shapes that can be created from 2 shapes

(e.g., all the shapes that can be made with two triangles). (I-C-P) (4b.)

5. Make, explore and describe cylinders, cones, cubes and

spheres. (I-C-P) (4b.) 6. Make, discuss and compare cubes and rectangular solids;

identify and discuss similarities and differences. (I-C-P) (1b. 4a.)

7. Explore and discuss figures with lines of symmetry. (I-C-P)

(4a.) 8. Identify congruent figures in different positions. (I-C-P-A)

(4a. 4b.) 9. Identify shapes from different views. (I-C-P-A) (4b.)

10. Copy & extend complex patterns & shapes composed of 3 or

more simple shapes. (I-C-P) (1b. 4b.) 11. Explore area as a “covering” process. (I-C-P) (4c.)

12. Estimate and use non-standard units to measure length, area,

weight and volume. (I-C-P-A) (3d. 4c.) 13. Explore and discover the need for uniform units of length and

weight. (I-C) (4c.) 14. Measure lengths of objects to the nearest centimeter, meter,

inch, foot, and yard to explore relationships between common measurement systems. (I-C-P-A) (4c.)

15. *Explore length using computer software. (P-A)

16. Compare capacities of containers. (I-C) (4c.)

17. Read thermometers in Fahrenheit & Celsius. (I-C-P-A)

(4c.) 18. Recognize time in five-minute intervals. (I-C-P-A)

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Overview of Kindergarten

The focus in kindergarten is to understand the concept of the numbers 1-10. This includes all the levels of knowing, from a concrete understanding of the numbers to reading them, writing the numerals, and applying the numbers concretely (as on a storyboard) through story problems applicable to their world. Kindergartners also explore the characteristics of common geometric shapes, using those shapes to create, identify, and name patterns. Patterns, counting, and place value are explored using the calendar and number line. Introductory graphing in kindergarten includes collecting and organizing objects and Information using a variety of real objects. Estimation strategies are applied to all kindergarten concepts.

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he focus in first grade is to understand the concept of adding and subtracting numbers through twenty. In doing so, students need to have e

s by the end of first grade.

First graders also focus on the language development of position (inside, outside, etc.) using identified 2-dimensional geometric shapes and patterns. There is an emphasis on understanding and using the concepts of time and money in real world situations. Every opportunity is made to express number sense knowledge through constructing and Interpreting graphs. Estimation strategies are applied to all first grade concepts

Overview of First Grade

Tan understanding of place value and regrouping. The goal is, after taking students through the levels of knowing, automatization of thesbasic addition and subtraction fact

Page 18: Math Curriculum K-12

18

Every opportunity is made to express number sense through constructing and interpreting graphs, charts, and tables. Estimation strategies are applied to all second grade concepts.

Overview of Second Grade

The focus in second grade is to understand the concept of adding and subtracting 2-digit numbers with regrouping. In doing so, second graders need to have a solid understanding of place value, including knowledge of other bases and the relationship between addition and subtraction. Second graders need to practice the automatization of their facts through 20. Second graders also explore the characteristics and relationships of 2- and 3 dimensional figures using geometric shapes. They reinforce their proficiency in addition and subtraction using perimeter problems. The area model is used to connect repeated addition, arrays, and multiplication.

(Please see document titled Scope and Sequence to review Mathemat

ics Scope and Sequence for Kindergarten - Second Grade)

Page 19: Math Curriculum K-12

19

3-5 MATHEMATICS SCOPE AND SEQUENCE Content Third Grade Fourth Grade Fifth Grade

NUMBER SENSE Counting Rounding Patterns Multiples, Factors, Divisibility Money Appropriate Technology (indicated by asterisk)

A r should be able to:

1. write a 3-digit number using a place value model. (I-C-P-A) (3a.)

2. e number of thousands, hundreds, tens, and

. Regroup and rename numbers up to and

-P-A) (3a. 3b.)

ne or other model. (I-C-P-A) (3a.)

5. ons. (I-C-P-A) (3a.)

6. d ) (3a.)

(3a. 3c.)

8. Use models to order fractions. (I-C-P) (3a. 3c.)

9. Identify, illustrate and write place value for

tenths and hundredths. (3a.) 10. Identify and compare fractions and decimals

which name tenths. (I-C-P-A) (3a.) 11. Skip count-using patterns of multiples of

single digit numbers. (I-C-P-A) (3b.) 12. Given a specified amount of money and a

price list, determine what can be bought. (I-C-P-A) (4c.)

13. Identify the value of any collection of coins

and dollars. (I-C-P-A) (4c.)

A ader should be able to:

1. ber -P-A) (3a. 6a.)

P-A) (3a.)

3. luding 1,000,000 in various combinations. (I-C-P-A)

g models. (I-C-P-A) (3a. 3c.)

6. ite

. (I-C-P-A) (3a. 3c. 6a.)

. Identify and compare fractions and decimals which name tenths and hundredths. (I-C-P-A) (3a.)

8. Identify the nearest ten, hundred or thousand place

of a number up to 6 digits. (I-C-P-A) (3a.) 9. Identify place value periods. (I-C-P-A) (3a.) 10. Explore and identify multiples and factors of whole

numbers. (I-C-P-A) (3a. 3b. 3c.) 11. Explore and identify numbers divisible by 2, 3, 5, 9,

and 10. (I-C-P-A) (3a. 3b. 3c) 12. Determine the amount of change to be received

from a purchase, using various methods (e.g., counting up, mental math…etc.). (I-C-P-A) (3a. 3b. 3c.)

f

1. e and write numbers (words and digits) from millions to thousandths and conversely. (I-C-P-A)

2. ess whole numbers and decimals in expanded form.

(I-C-P-A) (3a. 3c.)

3. e value from thousandths to millions of a particular digit given a whole number or decimal. (I-C-

. Order sequentially a set of whole numbers up through the

Identify, name and write equivalent fractions. (I-C-P-A)

. Recognize fractional names for decimals through

C-P-

ed -A) (1a.

3b.)

8. pulative representations of integers. (I-C-P) (3a.)

9. rs, and decimals through thousandths. (I-C-P-A) (2b. 3a. 8a.)

10. Round, given a variety of strategies, any given whole

number to any specified place value. (I-C-P-A) (3c.) 11. Explore other number systems and bases. (I-C-P-A) (2b.

3a. 6a.) 12. Explore and identify multiples and factors of whole

numbers. (I-C-P-A) (3a. 6a.) 13. Distinguish the difference between the multiple and

factor of a whole number. (I-C-P-A) (2b. 3a.) 14. Identify numbers divisible by 2, 3, 5, 9, and 10. (I-C-P-

A) (2b. 3a. 3c. 6a.) 15. Identify and classify a list of whole numbers 0 to 100 as

prime or composite. (I-C-P-A) (2b. 3a. 6a.)

third grade

Identify, illustrate and

Given a 3 or 5-digit number identify th

ones. (I-C-P-A) (3a.)

3including 1000 in various combinations of hundred, tens, and ones. (I-C

4. Using any 1, 2 or 3-digit number, identify to

the nearest 10 or 100 on a number li

Create models to review simple fracti

Name and write fractional parts of a whole ana set. (I-C-P-A

7. Create and use models to explore and explain

equivalent fractions. (I-C-P)

fourth gr

Identify, illustrate and write up to a 6-digit numusing a place value number. (I-C

2. Given up to a 6-digit number, identify the number

of thousands, hundreds, tens and ones. (I-C-

Regroup and rename numbers up to and inc

(3a. 3b. 6a.) 4. Develop the concept of remainder in division. (I-

C-P-A) (3a. 3c.) 5. Rename and rewrite whole numbers as fractions

with various denominators usin

Create models and illustrations to name and wrmixed numbers and their equivalent improper fractions

7

A ifth grader should be able to:

Identify, illustrat

(1b.)

Expr

Name the plac

P-A) (1a.)

4millions period. (I-C-P-A) (2a.)

5.(3a. 3b.)

6exploration with fractional and decimal models. (I-A) (2b. 3a. 3b.)

7. Create models or illustrations to name and write mix

numbers and their equivalent fractions. (I-C-P

Use visual and mani

Identify, compare and order fractions, intege

Page 20: Math Curriculum K-12

20

3-5 MATHEMATICS SCOPE AND SEQUENCE Content Third Grade Fourth Grade Fifth Grade

COMPUTATION AND OPERATIONS Addition, Subtraction, Multiplication & Division(of whole numbers, fractions & decimals) Properties Percent Multiples, Factors, Divisibility Estimation Appropriate Technology (indicated by asterisk)

A third grader should be able to: 1. Use manipulatives to represent and record

repeated addition and arrays as multiplication. (I-C-P-A) (2b. 3b.)

2. Use manipulatives to represent and record

repeated subtraction and arrays as division. (I-C-P-A) (2b. 3b.)

3. Explore inverse operations of addition and

subtraction and of multiplication and division. (I-C-P-A) (2b. 3b.)

4. Represent and record the separation/comparison of

sets or missing addends as subtraction. (I-C-P-A) (2b.)

5. *Explore relationships between operations using

calculators. (A) (2b. 3c. 6a.) 6. Develop algorithms for addition and subtraction

using appropriate manipulatives for any numbers less than 1000. (I-C-P-A) (2b. 3c.)

7. Add any two or more numbers less than 1000. (I-

C-P-A) (2b.) 8. Subtract two numbers less than 1000. (I-C-P-

A) (2b.) 9. Automatize all multiplication facts through ten.

(A) (2b. 3c.) 10. Add or subtract fractions with like or unlike

denominators using models. (I-C-P) (3a. 3c.) 11. Add or subtract decimals (.1 and .01) using

models, such as money and decimal squares. (I-C-P-A) (2b. 3c. 4c.)

12. Develop an algorithm for addition and subtraction

of decimals (.1 and .01) using models. (I-C-P-A) (2b. 3c.)

13. *Use calculators in appropriate computation

situations (A) (3c.) 14. Estimate sums and differences using a variety of

techniques. (I-C-P-A) (1a. 3c. 3d.)

A fourth grader should be able to: 1. Continue to develop multiplication as arrays and

repeated addition. (I-C-P-A) (3a. 3b. 3c. 6a. 7a. 8a.)

2. Continue to develop division as arrays and

repeated subtraction. (I-C-P-A) (3a. 3b. 3c. 6a. 8a.)

3. Explore multiplication and division as inverse

operations. (I-C-P-A) (3a. 3b. 3c.) 4. Automatize multiplication and division facts

through twelve. (I-C-P-A) (1b. 2b. 3a. 3b.) 5. Develop an algorithm for one or two digit

multipliers. (I-C-P-A) (3a. 3b. 3c. 8a.) 6. Perform the four operations with whole numbers.

(I-C-P-A) (3a. 3b. 8a.) 7. Evaluate a given expression containing variables

using the four operations. (I-C-P-A) (3a. 3b. 8a.) 8. Use patterns to represent and solve problems. (I-

C-P-A) (3a. 3b. 3c. 6a. 8a.) 9. Develop the concept of remainder in division and

apply to real world situations. (I-C-P-A) (2a. 3b. 8a.)

10. Add or subtract fractions with like or unlike

denominators using models. (I-C-P-A) (3a.) 11. Use physical models and illustrations to find the

sums and differences of decimals. (I-C-P-A) (3a.)

12. Develop and use algorithms to add and subtract

decimals using tenths and hundredths. (I-C-P-A) (3a.)

13. Explore the commutative and associative

properties, the property of one in multiplication, the zero identity of addition and the zero property of multiplication and explore why they do not hold for subtraction or division. (I-C-P-A) (3a.)

14. Explore and use the distributive property. (I-C-

P-A) (3a. 3b.)

A fifth grader should be able to: 1. Compute the sum or difference of two or more numerals

(including decimals) of four digits or less. (I-C-P-A) (2b.) 2. Develop and use a multiplication algorithm to find the

product of any two numbers of four digits or less (Including decimals as one of the factors). (I-C-P-A) (1a. 8a.)

3. Develop and use a division algorithm to find the quotient of

a 4-digit dividend (including decimals) and up to a 2-digit divisor (not including decimals). (I-C-P-A) (2b. 3b.)

4. Using manipulatives, develop and record algorithms for the

four operations with fractions. (I-C-P-A) (1b. 3c.) 5. Find the GCF and LCM in a set of natural numbers. (I-C-P-

A) (2b.) 6. Perform the four operations with whole numbers, fractions,

and decimals. (I-C-P-A) (1b. 6a.) 7. Explore and use order of operations. (PEMDAS) (I-C-P-

A) (2b.) 8. Evaluate a given expression containing variables. (I-C-P-

A) 9. Use patterns to represent and solve problems. (I-C-P-

A) (1a. 1b.) 10. Solve simple linear equations using concrete, informal

methods. (I-C) (1a.) 11. Identify and apply the commutative, associative, and

distributive properties of rational numbers. (I-C-P-A) (3b. 6a.)

12. Apply the properties of zero and one. (I-C-P-A) (2b.) 13. Find the prime factorization of a whole number. (I-C-

P-A) (3a.) 14. Explore the recording of prime factorization in exponential

form. (I-C-P-A) 15. Use a variety of estimation strategies. (I-C-P-A) (1a.

3d.)

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21

3-5 MATHEMATICS SCOPE AND SEQUENCE Content Third Grade Fourth Grade Fifth Grade

COMPUTATION AND OPERATIONS Addition, Subtraction, Multiplication & Division(of whole numbers,fractions & decimals) Properties Percent Multiples, Factors, Divisibility Estimation Appropriate Technology (indicated by asterisk)

15. Justify and record various estimation

strategies. (I-C-P-A) (2a. 3c. 3d.) 16. Determine the reasonableness of sums and

differences using estimation. (I-C-P-A) (3c. 3d.)

17. Use estimation, approximation or mental

computation to solve problems where exact answers are NOT required. (I-C-P-A) (3c. 3d.)

18. *Use calculators to examine

patterns/sequences to make conjectures about multiplication facts and more complex/sophisticated number patterns. (A) (1c. 3a. 3c. 6a.)

19. Explore addition/subtraction of decimals

and addition/subtraction of whole numbers. Discuss similarities. (I-C-P-A) (2b. 3c. 6a.)

20. Explore the commutative and associative

properties, the property of one in multiplication, and the zero identity of addition and multiplication and explore why they do not hold for subtraction. (I-C-P-A) (3b. 6b.)

15. Determine the reasonableness of sums,

differences, products and quotients using estimation. (I-C-P-A) (3c. 3d.)

16. Use estimation, approximation or mental

computation to solve problems where exact answers are NOT required. (I-C-P-A) (3c.)

17. *Use calculators in appropriate computation

situations. (I-C-P-A) (3c.)

16. Use estimation, approximation and mental

computation to solve problems where exact answers are or NOT required. (I-C-P-A) (2a. 3d.)

17. Determine the reasonableness of sums, differences,

products and quotients using estimation. (I-C-P-A) (2a. 3d.)

18. *Use calculators, mental math, pencil and paper and

computer methods in appropriate computation situations. (I-C-P-A) (2a. 3d. 6b.)

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22

3-5 MATHEMATICS SCOPE AND SEQUENCE

Content Third Grade Fourth Grade Fifth Grade DATA ANALYSIS AND CHANCE Data Collection and Organization Graphing Statistics Probability Appropriate Technology (indicated by asterisk)

A third grader should be able to: 1. Collect data, construct, interpret and discuss

graphs (picto and bar), tables and charts. (I-C-P-A) (1a. 1b. 2a. 2b. 5a.)

2. Interpret graphs made by classmates or found in

newspapers and books. (I-C-P-A) (1a. 1b. 2b. 5a.)

3. Interpret and discuss circle graphs. (I-C-P-A)

(1a. 1b. 2a. 2b. 5a.) 4. Write story problems using information from a

graph. (I-C-P-A) (1a. 2a. 2b.) 5. Predict, test and compile data as to which event is

most likely or least likely to happen given appropriate information. (I-C-P-A) (5a.)

6. *Explore the notion of probability with calculators

and computer software. (I-C-P-A) (5a.) 7. *Explore statistics, such as mean, median and

mode using manipulatives and calculators. (I-C-P-A) (5a.)

8. Keep a record of data collection, analysis and

chance. (I-C-P-A) (5a.)

A fourth grader should be able to: 1. Collect data; construct, interpret and discuss graphs (picto,

bar, and line), tables and charts. (I-C-P-A) (5a. 8a.)

2. Interpret graphs made by classmates or found in newspapers and books. (I-C-P-A) (5a. 8a.)

3. Interpret and discuss circle graphs. (I-C-P-A) (5a. 8a.)

4. Write story problems using information from a graph. (I-C-

P-A) (5a. 8a.) 5. Explore graphs with units different from one. (I-C-P-A) (4c.

5a. 8a.) 6. *Explore statistics, such as mean, median, mode and range

using manipulatives and calculators. (I-C-P-A) (5a. 8a.) 7. Investigate the probability of an event occurring. (I-C-P-A)

(5a. 7a. 8a.) 8. *Explore the notion of probability with calculators and

computer software. (I-C-P-A) (3a. 5a. 8a)

A fifth grader should be able to: 1. Interpret and analyze the data from a bar, line,

picto, and circle graph, charts and tables. (I-C-P-A) (1a. 2a. 3c. 3d. 5a. 6b. 7a.)

2. Construct a graph or diagram using an appropriate

scale given a set of numerical data. (I-C-P-A) (1a. 3b. 5a. 6b. 7a. 8a.)

3. Given a problem situation, collect, organize and

present the numerical data in a variety of forms, recognizing the most effective form for the data. (I-C-P-A) (1a. 2a. 3c. 5a. 7a. 8a)

4. Construct and read a stem and leaf plot. (I-C-P-A)

(6b.) 5. Identify which bar, line or picto graph reflects a

certain set of data. (I-C-P-A) (2a. 3b. 5a.) 6. Construct graphs and pictures using an x/y axis.

(I-C-P-A) (1a. 3b. 3c. 5a. 6b. 8a.) 7. *Explore statistics such as mean, median, mode

and range using pencil and paper, calculators and computers. (I-C-P-A) (1a. 3b. 3c. 5a.)

8. Given a problem solving situation involving the

likelihood of an event occurring, solve the problem by constructing a sample space, i.e., listing the possible combinations of a given number on a pair of dice. (I-C-P-A) (1a. 5a.)

Page 23: Math Curriculum K-12

23

3-5 MATHEMATICS SCOPE AND SEQUENCE Content Third Grade Fourth Grade Fifth Grade

GEOMETRY AND MEASUREMENT Two Dimensional Three Dimensional Congruency & Similarity Transformations Patterns Measurement Temperature Time Appropriate Technology (indicated by asterisk)

A third grader should be able to: 1. Draw, make and explore using manipulatives

squares/diamonds, triangles, rectangles, circles, rhombus, hexagons, trapezoids, kites and parallelograms. (I-C-P-A) (4a. 4b.)

2. Use the terms point, line and line segment in

describing 2-dimensional figures. (I-C-P-A) (4a.)

3. Name and identify angles as acute, obtuse,

and right and straight. (I-C-P-A) (4a.) 4. *Draw figures and angles using LOGO on

the computer and using paper and pencil. (I-C-P-A)

5. Separate a given shape into smaller shapes.

(I-C-P-A) (4b.) 6. Make a shape that can be made from 3

smaller shapes. (I-C-P-A) (4b.) 7. Name, make and describe cylinders, cones,

cubes, spheres, and pyramids. (I-C-P-A) (4a.) 8. Construct 3-dimensional figures. Discuss

edges, faces and vertices. (I-C-P-A) (4b.) 9. Match figures for congruency and explain

why they are congruent. (I-C-P-A) (2a. 4a.) 10. Discuss figures with lines of symmetry. (I-

C-P-A) (2a. 4a.) 11. Draw a congruent figure using many

methods. (I-C-P-A) (4a.) 12. Draw line segments in lines. (I-C-P-A) (4a.) 13. Measure lengths of objects to nearest half-

inch, quarter-inch and centimeter. (I-C-P-A) (4c.)

14. Regroup inches to feet and centimeters to

meters. (I-C-P-A) 15. *Explore perimeter of shapes using rulers

and computer software. (I-C-P-A)

A fourth grader should be able to: 1. Identify, describe, sketch and draw rays, right angles,

acute angles, obtuse angles and straight angles. (I-C-P-A) (4a.)

2. Identify, describe and draw parallel and perpendicular

lines. (I-C-P-A) (4a.) 3. Identify, describe and draw parallelograms, rhombuses

and trapezoids. (I-C-P-A) (4a.) 4. Recognize and make shapes that can be created from a set

of 4 or more simple shapes (i.e. tangrams, pattern blocks). (I-C-P-A) (4a. 6a.)

5. Draw a shape that has been turned or rotated. (I-C-P-A)

(4a. 4b.) 6. Enhance spatial sense using manipulatives and graphics.

(I-C-P-A) (4a. 4b.) 7. Construct 3-dimensional objects, discussing edges, faces

and vertices. (I-C-P-A) (4a. 4b.) 8. Discuss figures with lines of symmetry. (I-C-P-A)

(4a. 4b.) 9. Draw a congruent figure using many methods. (I-C-P-A)

(4a. 4b.) 10. Add or subtract units of length, including regrouping. (I-

C-P-A) (4a. 4c.) 11. Select an appropriate unit of measure for a given

situation. (I-C-P-A) (4c.) 12. Determine perimeters of closed shapes without formulas.

(I-C-P-A) (4a. 4c.) 13. Investigate figures with equal perimeters. (I-C-P-A)

(4a. 4c.) 14. Investigate the area of irregular shapes. (I-C-P-A)

(4a. 4b. 4c.) 15. After developing a standard unit, investigate areas of

rectangles, squares and right triangles (relate to multiplication and use of arrays). (I-C-P-A) (4a. 4b. 4c.)

16. Estimate and measure weight using pounds and/or

kilograms. (I-C-P-A) (4c. 8a.)

A fifth grader should be able to: 1. Identify and/or classify a selection of plane figures,

stating their properties. (I-C-P-A) (1b. 6a.)

2. Identify congruent and similar figures given a set of plane figures and their attributes. (I-C-P-A) (1a. 6a.)

3. Given a series of pictorial representations of a cube

in various rotational positions, identify those pictures which represent the same cube. (I-C-P-A) (1b. 4a. 4b. 4c. 6a.)

4. Explore and compare the properties of various

solids. (I-C-P-A) (4c. 6a.) 5. Produce a 3-dimensional object using a pictorial

representation. (I-C-P-A) (1a. 1b. 4a. 6a.) 6. Explore surface area. (I-C-P) (1a. 1b. 4a. 4b. 4d.) 7. Explore slides, flips and turns. (I-C-P-A) (1b. 4a.

4b.) 8. Explore tessellations on a plane. (I-C-P-A) (1b. 4a.) 9. Select an appropriate unit of measure given a

situation. (I-C-P-A) (3a. 6a.) 10. Estimate size, quantity, temperature, capacity and

the passage of time. (I-C-P-A) (1a. 1b. 4c.) 11. Measure a given item to an indicated precision.

(I-C-P-A) 12. Identify measure and construct acute, right and

obtuse angles. (I-C-P-A) (1b. 7a.) 13. Find the area and perimeter of any plane figure.

(I-C-P-A) (1b.) 14. Develop strategies to calculate the perimeter and

area of squares, rectangles, parallelograms, kites, triangles and circles. (I-C-P-A) (1a. 4b. 4c.)

15. *Use appropriate software to explore geometric

concepts (Tesselmania, computer software, Geometer’s Sketchpad, etc.) (I-C-P-A) (2a. 4a. 4b.)

Page 24: Math Curriculum K-12

24

3-5 MATHEMATICS SCOPE AND SEQUENCE

Content Third Grade Fourth Grade Fifth Grade

GEOMETRY AND MEASUREMENT Two Dimensional Three Dimensional Congruency & Similarity Transformations Patterns Measurement Temperature Time Appropriate Technology (indicated by asterisk)

16. After developing a standard unit of measure,

estimate, record and measure the area of rectangular regions. (I-C-P-A) (4c. 3d.)

17. Estimate the area of irregular shapes. (I-C-P-A)

(3d.)

18. Estimate and measure weight using pounds and/or kilograms. (I-C-P-A) (4c. 3d.)

19. Estimate and measure capacity using quarts,

gallons, and liters. (I-C-P-A) (4c. 3d.)

20. Read thermometers in Fahrenheit and Celsius. (I-C-P-A) (4c.)

21. Tell time to the nearest minute. (I-C-P-A)

(4c.)

22. Explore the addition of hour and half-hour time intervals. (I-C-P-A) (4c.)

17. Estimate and measure capacity using quarts,

gallons, and liters. (I-C-P-A) (4b. 4c. 8a.) 18. Read thermometers in Fahrenheit and

Celsius. (I-C-P-A) (4c. 8a.) 19. Add minutes to a specified time. (I-C-P-A)

(4c. 8a.) 20. *Use computer software to explore

geometric concepts and relationships. (I-C-P) (4a. 4b. 4c.)

Page 25: Math Curriculum K-12

25

Overview of Third Grade

The focus in third grade is to automatize all multiplication facts through ten and explore mathematical connections through the related operations of addition and division. The concepts previously learned in geometry are used to reinforce automatization of addition and subtraction of numbers less than 1000. Using models, third graders can add and subtract decimals and fractions, after learning the prerequisite skills. In geometry, third graders explore 2- and 3-dimenslonal shapes, the specific vocabulary Involved, and the construction of those shapes. Measurement strategies are applied to the shapes using standard and metric units. There is an emphasis on understanding and using the concepts of time and money in real world (including classroom) situations. Every opportunity is made to express number sense through constructing and interpreting graphs, charts, and tables. These recordings are then used to explore the concepts and terms of probability and statistics, such as mean, median, and mode. Estimation strategies are applied to all third grade concepts.

_

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26

Overview of Fourth Grade

The focus in 4th grade is to develop algorithms, using models, to add and subtract fractions and decimals after strengthening prerequisite skills. Emphasis on multiples, factors and divisibility rules will assist in the automatization of all multiplication and division facts through 12 and will enhance the connection of related operations. In geometry, 4th graders explore 2- and 3-dimensional shapes, the specific vocabulary involved with them, and- the construction of those shapes. They will apply measurement strategies to the shapes using standard and metric units to investigate perimeter and area. These concepts learned through geometry are used to reinforce the basic operations on whole numbers. There is a reinforcement of third grade concepts of time and money in real world (including classroom) situations, connecting them to fourth grade goals. In addition to collecting data and constructing graphs, tables, and charts, fourth graders use and interpret the data to predict outcomes, determine a pattern and solve dilemmas as they arise in daily living. These recordings can also be used to explore probability and statistics, such as mean, median and mode. Estimation strategies are applied to all fourth grade concepts.

Page 27: Math Curriculum K-12

27

mon multiples and factors, divisibility rules, prime factorization, exponential form, and mathe atical properties identified in the grade 5 curriculum. Concrete models should be used as extensively as possible. In geo etry, fifth graders explore, produce, and compare 2- and 3dimensional shapes using specified vocabulary. They will apply measurement strategies to the shapes using standard and metric units and will develop strategies to calculate area, perimeter, and surface area. These concepts learned through geometry are used to reinforce the four basic operations. Fifth graders collect, use, interpret and present data using specified graphs, tables and charts to predict outcomes, determine a pattern and solve dilemmas as they arise in daily living. Data can be used to explore statistics (mean, median, mode and range) and predict outcomes/likelihoods. Estimation strategies are applied to all fifth grade concepts.

Overview of Fifth Grade

The focus in 5th grade is to reinforce competency in the four basic operations, extending these operations from whole numbers to fractions and decimals, with an emphasis on com

m

m

Page 28: Math Curriculum K-12

28

6-8 MATHEMATICS SCOPE AND SEQUENCE Content Sixth Grade Seventh Grade Eighth Grade

NUMBER SENSE Counting Rounding Patterns Multiples, Factors, Divisibility Money Appropriate Technology (indicated by asterisk)

A sixth grader should be able to: 1. Identify and write (words and digits) any

given number…billions, trillions, etc. (A) (3a.)

2. Recognize fractional names for decimals

through fractional and decimal models. (I-C-P-A) (3a. 3b.)

3. Express commonly used fractions as

decimals and conversely. (I-C-P-A) (3a.) 4. Use models and manipulatives with

percent, connecting with fraction and decimal equivalents. (I-C-P-A) (3a. 3b.)

5. Use visual and manipulative

representations of integers. (I-C-P-A) (3a. 3b.)

6. Use models and patterns to identify

irrational numbers. (I-C-P) (3a. 3b. 6a.) 7. Identify, compare and order fractions,

integers, and irrationals using the entire number line. (I-C-P-A) (3a.)

8. Identify, name and write equivalent

fractions. (I-C-P-A) (3a.) 9. Round any decimal number to any

specified place. (I-C-P-A) (3a.) 10. Explore other number systems and

bases. (I-C-P-A) (3a.) 11. Identify multiples and factors of whole

numbers. (I-C-P-A) (3a.) 12. Identify numbers divisible by 2, 3, 5, 9,

and 10. (I-C-P-A) (3a.) 13. Identify prime and composite numbers.

(I-C-P-A) (3a.) 14. Recognize that the prime factorization of

a number is unique. (I) (3a. 3b. 6a.)

A seventh grader should be able to: 1. Express a whole number in exponential form,

expanded form and scientific notation. (I-C-P-A) (3a.)

2. Express prime factorization in exponential form.

(I-C-P-A) (3a.) 3. Recognize that the prime factorization of a

number is unique. (I-C-P-A) (3a.) 4. Use visual and manipulative representations of

perfect squares and perfect cubes. (I-C-P-A) (3b.)

5. Approximate the value of the square root of a

number. (I-C-P-A) (3b. 3c.) 6. Represent a given number as a whole number,

mixed number, fraction, decimal, or percent. (I-C-P-A) (3a. 3c. 3d)

7. Order a set of rational numbers. (I-C-P-A) (3c.) 8. Compare and order a set of real numbers.

(I-C-P-A) (3c.) 9. Round any number to any place value. (I-C-P-A)

(3c.) 10. Recognize when an estimate is appropriate.

(I-C-P-A) (3d.)

An eighth grader should be able to: 1. Using a model, show the geometric representation of

the square root of 2, 3, and 5. (I-C-P) (3a.)

2. Approximate the value of the square root of a number. (I-C-P-A) (3a.)

3. Recognize the relationship and patterns within the set

of real numbers. (I-C-P-A) (3a.) 4. Order any set of real numbers. (I-C-P-A) (3c.) 5. Represent any given number in any form (whole

number, mixed number, fraction, decimal, percent, perfect squares, perfect cubes, scientific notation, expanded form, exponential form and Swahili). (I-C-P-A) (3a. 3c. 3d.)

6. Recognize that the prime factorization of every

number is unique. (I-C-P-A) (3a.)

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29

6-8 MATHEMATICS SCOPE AND SEQUENCE

Content Sixth Grade Seventh Grade Eighth Grade COMPUTATION AND OPERATIONS Addition, Subtraction, Multiplication & Division

(of whole numbers, fractions & decimals)

Properties Percent Multiples, Factors, Divisibility Estimation Appropriate Technology (indicated by asterisk)

A sixth grader should be able to: 1. Find the GCF and LCM in a set of natural

numbers. (I-C-P-A) (3a.) 2. Using manipulatives, develop and record

algorithms for the four operations involving fractions. (I-C-P-A) (3b. 3c.)

3. Perform the four operations with whole

numbers, fractions, and decimals. (I-C-P-A) (3a.)

4. Explore and use order of operations

(PEMDAS). (I-C-P-A) (3a.) 5. Using manipulatives and models develop

and record algorithms for the four operations involving integers. (I-C-P-A) (3a. 3b. 3c. 8a.)

6. Evaluate a given expression containing

variables. (I-C-P-A) (6b.) 7. Use patterns and functions to represent and

solve problems. (I-C-P-A) (1a. 1b.) 8. Solve a simple equation using trial and

error when given a replacement set. (I-C-P-A) (1a. 3c.)

9. Solve simple linear equations using

concrete, informal methods. (I-C-P) (1a. 6a. 6b.)

10. Represent, simplify and solve ratios and

proportions derived from real life situations, using models and manipulatives. (I-C-P-A) (3a. 3b. 3c.)

11. Identify and apply commutative,

associative and distributive property to all numbers. (I-C-P-A) (1b. 3a. 3b.)

12. Apply properties of zero and one. (I-C-P-

A) (1b. 3a. 3b.) 13. Find the prime factorization of a whole

A seventh grader should be able to: 1. Use an algorithm to perform and record the

four operations with rational numbers. (I-C-P-A) (3a.)

2. Use order of operations with all rational

numbers. (PEMDAS) (I-C-P-A) (3a.) 3. Find the percent of a given number. (I-C-P-A)

(3a.) 4. Find what percent one number is of another.

(I-C-P-A) (3a.) 5. Develop an algorithm to approximate the

square root of a number. (I-C-P-A) (3b.) 6. Using models solve one-step equations using

variables. (I-C-P-A) (6b.) 7. Evaluate a given expression containing

variables. (I-C-P-A) (6b.) 8. Use patterns and functions to represent and

solve problems. (I-C-P-A) (1a. 1b.) 9. Solve a simple equation using trial and error

when given a replacement set. (I-C-P-A) (1a. 3c.)

10. Apply the properties of zero and one to solve

simple linear equations. (I-C-P-A) (1b. 3a. 3b.)

11. Explore and apply the properties of additive

and multiplicative inverses to solve simple linear equations. (I-C-P-A) (6a.)

12. Identify and apply commutative, associative

and distributive property to all numbers. (I-C-P-A) (1b. 3a. 3b.)

13. Use various estimation strategies. (I-C-P-A)

(3d.) 14. Use estimation, approximation and mental

computation to solve problems where exact

An eighth grader should be able to: 1. Use an algorithm to perform and record the four

operations with rational numbers. (I-C-P-A) (3a.)

2. Explore and use order of operations. (PEMDAS) (I-C-P-A) (3a.)

3. Explore, apply, and record the relationship between

percent, ratio and proportion. (I-C-P-A) (3a.) 4. Use proportions to solve problems. (I-C-P-A) (4b.)

5. Find the percent of a given number. (I-C-P-A) (3a.

3b.) 6. Find what percent one number is of another. (I-C-P-

A) (3a. 3b.) 7. Find the whole given the percent of the whole. (I-C-P-

A) (3a. 3b.) 8. Evaluate a given expression containing variables. (I-

C-P-A) (6b.) 9. Plot a linear equation on the coordinate plane. (I-C-P-

A) (6b.) 10. Use patterns and functions to represent and solve

problems. (I-C-P-a) (6b.) 11. Apply algebraic methods to solve a variety of

problems, based in reality. (I-C-P-A) (6a.) 12. Using models solve two-step equations using

variables. (I-C-P-A) (6b.) 13. Apply the properties of zero and one to solve two-

step linear equations. (I-C-P-A) (1b. 3a. 3b.)

14. Explore and apply the properties of additive and

multiplicative inverses. (I-C-P-A) (3b.) 15. Identify and apply commutative, associative and

distributive properties to all numbers. (I-C-P-A) (1b. 3a. 3b.)

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30

number in exponential form. (I-C-P-A) (3a. 3b.)

answers are or are NOT required. (I-C-P-A) (1b. 3c. 3d.)

6-8 MATHEMATICS SCOPE AND SEQUENCE Content Sixth Grade Seventh Grade Eighth Grade

COMPUTATION AND OPERATIONS Addition, Subtraction, Multiplication & Division

(of whole numbers, fractions & decimals)

Properties Percent Multiples, Factors, Divisibility Estimation Appropriate Technology (indicated by asterisk)

14. Use a variety of estimation strategies.

(I-C-P-A) (3d.) 15. Use estimation, approximation and mental

computation to solve problems where exact answers are or are NOT required. (I-C-P-A) (1b. 3c. 3d.)

16. Determine the reasonableness of sums,

differences, products and quotients using estimation. (I-C-P-A) (3c. 3d.)

17. *Use calculators, mental math, pencil and

paper and computer methods in appropriate computations situations. (I-C-P-A) (3c. 3d.)

15. Determine the reasonableness of sums,

differences, products and quotients using estimation. (I-C-P-A) (3c. 3d.)

16. *Use calculators, mental math, pencil and

paper, and computer methods in appropriate computation situations. (I-C-P-A) (3c. 3d.)

16. Use various estimation strategies. (I-C-P-A)

(3d.)

17. Use estimation, approximation and mental computation to solve problems where exact answers are or are NOT required. I-C-P-A) (1b. 3c. 3d.)

18. Determine the reasonableness of sums,

differences, products and quotients using estimation. (I-C-P-A) (3c. 3d.)

19. *Use calculators, mental math, pencil and paper

and computer methods in appropriate computation situations. (I-C-P-A) (3c. 3d.)

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31

6-8 MATHEMATICS SCOPE AND SEQUENCE Content Sixth Grade Seventh Grade Eighth Grade

DATA ANALYSIS AND CHANCE Data Collection and Organization Graphing Statistics Probability Appropriate Technology (indicated by asterisk)

A sixth grader should be able to: 1. Given a bar, line, picto or circle graph or

a chart or table, interpret and analyze the data. (I-C-P-A) (1a. 5a. 6a. 7a.)

2. Construct a graph or diagram using an

appropriate scale given a set of numerical data. (I-C-P-A) (1a. 5a. 6a.)

3. Given a problem situation, collect,

organize and present the numerical data in a variety of forms, recognizing the most effective form for the data. (I-C-P-A) (1a. 5a. 6a. 8a.)

4. Construct and read double stem and leaf

plots. (I-C-P-A) (5a.) 5. Interpret box-and-whisker plots. (I-C-P-

A) (5a.) 6. Construct graphs and pictures using x/y

axis. (I-C-P-A) (5a. 6b.) 7. Given graphs of data, identify which bar,

line, and picto graphs reflect the same set of data. (I-C-P-A) (5a.)

8. Given a graph, describe the data using an

appropriate, real life situation. (I-C-P-A) (1a. 5a.)

9. Find the mean, median, mode and range

given a set of data. (I-C-P-A) (5a.) 10. Given a problem solving situation

involving the likelihood of an event occurring, solve the problem by constructing a sample space, i.e., listing the possible combinations of a given number on a pair of dice. (I-C-P-A) (5a.)

11. Use a computer and other appropriate

technology as tools to analyze and present data. (I-C-P-A)* (5a.)

A seventh grader should be able to: 1. Given a bar, line, picto or circle graph or a

chart or table, interpret and analyze the data. (I-C-P-A) (1a. 5a. 6a. 7a)

2. Construct a graph or diagram using an

appropriate scale given a set of numerical data. (I-C-P-A) (1a. 5a. 6a.)

3. Given a problem situation, collect, organize

and present the numerical data in a variety of forms, recognizing the most effective form for the data. (I-C-P-A) (1a. 5a. 6a. 8a.)

4. Construct and interpret stem and leaf graphs

and box-and-whisker plots. (I-C-P-A) (5a.) 5. Plot data on the x/y axis. (I-C-P-A) (5a. 6b.) 6. Find the measures of central tendency given a

set of data. (I-C-P-A) (5a.) 7. Create a set of data to support a given mean,

median or mode. (I-C-P-A) (5a.) 8. Given a set of numerical data, identify the

ordered pairs and make a scatter plot. (I-C-P-A) (6b.)

9. Make appropriate inferences and predictions

based on analysis of given data. (I-C-P-A) (7a)

10. Identify the appropriate graph for a given set

of data. (I-C-P-A) (5a. 6b.) 11. Given a graph, describe the data using a real

life situation. (I-C-P-A) (5a.) 12. Develop and use the basic counting principle.

(I-C-P-A) (5a.) 13. Find the number of permutations and

combinations for a given set. (I-C-P-A) (5a. 8a.)

14. *Use the computer and other appropriate

technologies as tools to analyze and represent data. (I-C-P-A) (5a.)

An eighth grader should be able to: 1. Given a bar, line, picto or circle graph or a chart or

table, interpret and analyze the data. (I-C-P-A) (5a. 6a.)

2. Construct a graph or diagram using an appropriate

scale given a set of numerical data. (I-C-P-A) (1a. 5a. 6a.)

3. Given a problem situation, collect, organize and

present the numerical data in a variety of forms, recognizing the most effective form for the data. (I-C-P-A) (1a. 5a. 6a. 8a.)

4. Construct and interpret stem and leaf graphs.

(I-C-P-A) (5a.) 5. Using data, identify outliers, extremes and upper

and lower quartiles in order to construct box-and-whisker plots. (I-C-P-A) (5a.)

6. Find the measures of central tendency. (I-C-P-A)

(5a.) 7. Create a set of data to support a given mean,

median or mode. (I-C-P-A) (5a.) 8. Predict and find the probability of outcomes of a

simple probability experiment. (I-C-P-A) (5a.) 9. Determine the ordered pairs, make a scatter plot,

and determine the line of best fit to make predictions. (I-C-P-A) (6b.)

10. *Use the computer and other technologies as tools

to analyze and represent data. (I-C-P-A) (5a.)

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32

6-8 MATHEMATICS SCOPE AND SEQUENCE

Content Sixth Grade Seventh Grade Eighth Grade GEOMETRY AND MEASUREMENT Two Dimensional Three Dimensional Congruency & Similarity Transformations Patterns Measurement Temperature Time Appropriate Technology (indicated by asterisk)

A sixth grader should be able to: 1. Use inductive and deductive reasoning to explore

geometric concepts. (I-C-P-A) (1b. 4a. 4b. 8a.) 2. Appreciate point, line and plane, and models created

for them. (Models are the only way to represent these undefined terms) (I-C-P-A) (4a. 4b.)

3. Identify and classify a selection of plane figures,

stating their properties. (I-C-P-A) (4a. 4b.) 4. Identify congruent figures based on their properties.

(I-C-P-A) (4a.) 5. Identify similar figures and explore the relationship

between corresponding parts (using ratios). (I-C-P-A) (4a.)

6. Produce a three-dimensional object using pictorial

representations; name and compare its properties. (I-C-P-A) (1b. 4a.)

7. Given a problem-solving situation involving a 3-

dimensional space, draw an appropriate diagram to aid in the solution of the problem. (I-C-P-A) (4a. 4b.)

8. Explore reflections and rotations. (I-C-P-A)

(1b.) 9. Tessellate (tile) with a given figure and create a

figure that will tile the plane. (I-C-P-A) (4b.) 10. Make an appropriate estimate relating to size,

quantity, temperature, capacity, and passage of time. (I-C-P-A) (4c.)

11. Using a variety of appropriate tools, measure a given

item to a given precision and recognize that all measures are imprecise. (I-C-P-A) (4c.)

12. Measure a linear object, using the appropriate metric

or English unit to an indicated precision. (I-C-P-A) (4c.)

A seventh grader should be able to: 1. Use inductive reasoning to make conjectures

about geometric relationships. (I-C-P-A) (1b. 8a.)

2. Use deductive reasoning to validate conjectures

about geometric concepts. (I-C-P-A) (4a. 4b.) 3. Identify the properties of a plane figure or a solid

figure and classify them. (I-C-P-A) (4a. 4b.) 4. Identify a 3-dimensional object from a different

view. (I-C-P-A) (4a. 4b.) 5. Reproduce a 3-dimensional object from a

pictorial representation, using appropriate manipulatives. (I-C-P-A) (4a. 4b)

6. Given a problem-solving situation involving a 3-

dimensional space, draw an appropriate diagram to solve the problem. (I-C-P-A) (4a. 4b.)

7. Explain multiple transformations of 2

dimensional figures. (I-C-P-A) (4a. 4b.) 8. Make an appropriate estimate relating to size,

quantity, temperature, capacity, and passage of time. (I-C-P-A) (4c.)

9. Using a variety of appropriate tools including a

ruler, protractor, graduated cylinder, scale, trundle wheel, meter stick, measure a given item to a given precision and recognize that all measures are imprecise. (I-C-P-A) (4c.)

10. Calculate the perimeter, area and circumference

of plane figures using real world situations. (I-C-P-A) (4c.)

11. Find the volume of space figures and develop

strategies for calculating such. (I-C-P-A) (4c.) 12. Develop, write and apply with understanding

formulas for area and volume, using appropriate algebraic terminology. (I-C-P-A) (4c.)

13. Automatize the metric system! (I-C-P-A) (4c.)

An eighth grader should be able to: 1. Make and validate conjectures using inductive and/or

deductive reasoning. (I-C-P-A) (1b. 4a. 4b. 8a.)

2. Identify and classify a collection of plane figures and their properties. (I-C-P-A) (4a. 4b.)

3. Explore and develop the Pythagorean relationships to

determine the measure of unknown sides of triangles. (I-C-P-A) (4d.)

4. Explore and identify angles formed by parallels and

transversals. (I-C-P-A) (4a.) 5. Explore, design and create a tessellating pattern using

Escher’s techniques. (I-C-P-A) (4b.) 6. Given a problem-solving situation involving a 3-

dimensional space, draw an appropriate diagram to solve the problem. (I-C-P-A) (4a. 4b.)

7. Draw an object from various views, given a picture of

the object. (I-C-P-A) (4a. 4b.) 8. Develop, formalize and apply the properties of

similarity using ratio and proportion. (I-C-P-A) (4c.) 9. Construct congruent and similar figures using a

straight edge and compass. (I-C-P-A) (4c.) 10. Make an appropriate estimate relating to size,

quantity, temperature, capacity and passage of time. (I-C-P-A) (4c.)

11. Determine the surface area and volume of pyramids,

cones and spheres. (I-C-P-A) (4c.) 12. Apply the concepts of measurement using the

appropriate measure. (I-C-P-A) (4c.) 13. *Use appropriate software to explore geometric

concepts (Tesselmania, computer software, Geometer’s Sketchpad, etc.) (I-C-P-A) (4b.)

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6-8 MATHEMATICS SCOPE AND SEQUENCE Content Sixth Grade Seventh Grade Eighth Grade

GEOMETRY AND MEASUREMENT Two Dimensional Three Dimensional Congruency & Similarity Transformations Patterns Measurement Temperature Time Appropriate Technology (indicated by asterisk)

13. Find the area and perimeter of any plane

figure and develop strategies for calculating such. (I-C-P-A) (4c.)

14. *Use appropriate software to explore

geometric concepts (Tesselmania, computer software, Geometer’s Sketchpad, etc.) (I-C-P-A) (4b.)

14. *Use appropriate software to explore geometric

concepts (Tesselmania, computer software, Geometer’s Sketchpad, etc.) (I-C-P-A) (4b.)

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34

Overview of Sixth Grade

The focus in 6th grade is to increase competency in the four basic operations, extending these operations from whole numbers to fractions, decimals and integers, with an emphasis on common multiples and factors, divisibility rules, prime factorization, exponential form, and mathematical properties identified in the grade 6 curriculum. Concrete models should be used as extensively as possible, especially when exploring percents, integers and irrational numbers. In geometry, sixth graders identify classify and create 1-, 2-, and 3dimensional shapes with focus on reflections and rotations of those figures. They will use the appropriate tools and both measurement systems to accomplish these with accuracy. The beginnings of inductive and deductive reasoning are explored using geometric concepts and figures. Students will apply concrete variable manipulation to solve simple linear equations. Sixth graders collect, use, interpret and present data using specified graphs, tables and charts to predict outcomes, determine a pattern and -solve dilemmas as they arise in daily living. Data can be used to explore statistics (mean, median, mode and range) and predict outcome/likelihoods. Estimation strategies are applied to all sixth grade concepts.

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35

Overview of Seventh Grade The focus in 7th grade should be to reinforce and extend the four operations (addition, subtraction, multiplication and division) to rational numbers through the use of geometry. Geometry will also be used to apply previously developed knowledge so that seventh graders can develop, write and apply with understanding, formulas for area and volume, using appropriate algebraic terminology. Seventh graders should use models to solve and evaluate simple linear equations. Students will also use inductive and deductive reasoning throughout to make and validate conjectures about possible relationships. Complementing this focus, students will collect, use, interpret and present data using specified graphs, tables and charts to predict outcomes, determine a pattern, and solve dilemmas as they arise in daily living. Data can be used to explore statistics and probability. Seventh graders will also explore real life applications of percentage (and ratio and proportion.) Emphasis on order of operations, commutative, associative and distributive properties, additive and multiplicative inverse properties, as well as properties of zero and one should be extended to include expressing numbers and expressions in exponential form. Estimation strategies should be applied to all seventh grade concepts.

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36

Overview of Eighth Grade

The focus in 8th grade is to combine previously learned skills along with new algebraic and geometric concepts in order to problem solve around dilemmas as they arise in daily living. Computation skills learned in previous grades will be practiced through these real life situations, including those involving percentage.

tudents will extend their geometric knowledge of 2 dimensional objects into the three dimensional realm of volume and surface area. ighth graders will develop, write and apply with understanding, related formulas using appropriate algebraic terminology. Students will lso use inductive and deductive reasoning throughout to make and validate conjectures about possible relationships. Eighth graders will pply number and algebraic properties to all real numbers. They will solve and evaluate two step linear equations. Patterns, functions, and lgebraic methods should be used to represent and solve a variety of problems.

Connecting the geometric and algebraic foci, students will collect, us interpret and present data using specified graphs, tables and charts to predict outcomes, determine a pattern, and resolve issues. xplore statistics and probability.

SEaaa

e,This data can be used to e

Estimation strategies should be applied to all eighth grade concepts.

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37

ASSESSMENT

s stated in our philosophy, an assortment of assessment tools should be used to guide instruction and planning. It is our belief that nstruction and assessment are closely linked and that good teachers and good students are constantly performing informal assessments. owever, valid and meaningful information is needed by not only students and teachers, but parents, school administrators and

policymakers. It is therefore important to use mu d directly to our instruction if we are to ccurately evaluate student achievement, our

nts r objective tests: performance assessments, observations, interviews and questions, journals, open-ended

ies

AiH

ltiple methods of assessment that are relatemathematics program, and student progress. a

Although we feel that standardized tests have a role to play in evaluating programs, there are many other effective types of assessments that should be used to guide instruction on a daily basis. This section of the guide includes descriptions and some examples of assessmether than regular pencil and papeo

question responses, portfolios, student self-assessments. Sample questions from the NH Assessment with recommendations and strategfor classroom use can be found at www.ed.state.nh.us/Assessment/assessme(NHEIAP).htm (as of3/24/04). We plan to add to this ses teachers who use this guide develop more and better tools.

ction

nd 7-10 Mathematics Addenda for the NH K-12 Mathematics Curriculum rk at ricula

a Additional information and examples can be found in the 4-6 aFramewo www.plymouth.edu/psc/math/cur (as of 3/24/04) and in Mathematics Assessment: Myths, Models, Good Questions,

ical S . Copies are available in each school. and Pract uggestions, published by NCTM

Page 38: Math Curriculum K-12

developed in 12th century AD. It has thirteen columns of beads, divided by a crossbar with five beads below the bar and two

on the ideas they developed at the Concrete and Pictorial levels in order t problems without the actual

Addend

Algorithm A procedure used to solve a problem; a recipe

nalog Numbers represented by physical quantities such as rotations (clock hands), voltages (electric meter), distances

ns e situations; applications may include transferring math knowledge to other Content areas, problems

Area The num er a 2-dimensional figure; the amount of 2-dimensional space taken up by an object

rray A rectangular matrix; a rectangle of specified dimensions

xxxxx an array of 5 by 3 OR 3 by 5

38

GLOSSARY - GRADES K TO 3 Abacus pebble stacking devise used for performing all four operations and calculating square and cube roots; Chinese abacus was

above. Abstract A Level of Knowing where students are able to envisi

o solve manipulative or a picture in front of them.

Acute angle An angle whose measure is less than 90 degrees

One of the numbers to be added

A (odometer) Angle The space formed between two rays that are connected at a vertex Applicatio Using knowledge in divers forming new ideas based on previously mastered concepts, and finding solutions to a word and real life

ber of squares it takes to cov A Example: xxxxx

xxxxx

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39

ssociative property 1.) … Of addition: a + (b + c) = (a + b) = c the sum of any three numbers is the same, regardless of

e product of any three numbers is the same, regardless of

An inherent characteristic

utom tize To make autom

Base

um ting and answers the question “How many?”

A 2-dim

ircu erenc

pe regardless of order e same, regardless of order

which enables one to recognize derived from our sensory and

r concepts are called “seco

ulation of concrete materials in order to solve a problem

A grouping

2.) … of multiplication: a * (b * c) = (a * b) * th grouping

Attribute

a atic; to know without hesitation A Average Generalization of data; a central tendency of data; See mean, median, mode

1.) A system of counting 2.) Algebra: the number to which an exponent applies, such as “b” in “b to the second power”

3.) Geometry: a particular side or face of a geometric figure

Capacity The volume of a 3-dimensional figure given in terms of liquid measurement Cardinal n ber A number that is used in simple coun Chord A segment that connects two points of a circle Circle ensional figure in which all points are the same distance from a point called the center

mf e The distance around a circle (formula: C = pi * diameter) C Column One of two or more vertical sections Commutative pro rty 1) … of addition: a + b = b + a the sum of any two numbers is the same, 2.)... of multiplication: a * b = b * a the product of any two numbers is th concept An idea; some kind of lasting mental change, the result of becoming aware of similarities,

new experiences as having the similarities of an already formed class. Concepts that are moto experiences are called “primary concepts” (red, heavy, hot, sweet ...). Concepts abstracted from other

ndary concepts” (color, light, all mathematical concepts, beauty ...); See mathematical concept onceptual concrete A Level of Thinking that involves the visualization of an appropriate concrete model The second Level of c

Knowing involving the manip

Page 40: Math Curriculum K-12

40

oncr e ally manipulated (hands-on)

ore o “

un h now include measurements such as inch, foot, ounce,

ylind

no Representation of a fractional value using a decimal point and the digits 0 through 9

two points of a circle and the center; the longest chord of a circle

e another; the answer to a subtraction problem

ical quantities (analog); See analog

Distributive property and then adding the products; the most important mathematical property

e a number by which a dividend is divided; the factor of an integer or a polynomial; see division Th of a 3 – dimensional figure

quation A number sentence that us ality symbol (=) to show that two expressions have the same value; syn: number

et Materials that can be physicC

Congruent Two or m bjects that have the same size and shape (2- and 3-dimensional); all pieces identical; denoted by the symbol

≅ ” ustomary its Units of measure first developed by the Babylonians whicc

pound, cup, gallon, and mile. This system is used only in the United States and on the island of Brunei.

er A 3-dimensional figure with two parallel, congruent, circular faces C Decimal tation Denominator The number below the division bar of a fraction; it describes the number of equal pieces Diameter The segment containing Differenc The number obtained from subtracting one number from

Of or relating to calculation directly with digits rather than through mDigital easurable, phys Digits 1.) Symbols used to write numerals 2.) Fingers or toes

a (b + c) = ab + ac Multiplying a sum by a number is the same as multiplying each addend by the number

Dividend The area of an array; a number to be divided; See division Divisor edg The length of a side of an array; e line or line segment created by the intersection of planes or any of the faces Element A member of a set

es the equE sentence Example: 6 + 8 = 6 + 7 OR 6 + 8 = 14

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41

2

Expression face Any combination of operations with variables and numbers (constants) examples: 3+4 OR x-5 OR 35 - 8 *2 the 2-

Factor Numbe a ctangular

ed when a figure is turned to its reverse side

ule e

1.) A relationship between two numbers describing parts of a whole or of a set

3.) : a

Equivalent Name the same amount Estimate 1.) V. to find an answer that is close to the exact answer

.) N. An answer that is close to the exact answer Even number a whole number that has 0, 2, 4, 6, 8 in the ones place; it is divisible by two

One of dimensional surfaces making up a 3dimensional figure

rs that are combined in the multiplication operation to give a number called the product; the sides of rearray

Flip The new shape creat

Formula A general r xpressed by symbols

Fraction 2.) A division problem

A ratio written in the form

cy data

lationship between two sets

mac for construction directions.

rsity of attributes is represented within each group (gender, Multiple ld confidence in their own abilities; the

atics sends a powerful message to each and ev

-s

ogeneous one common attribute; this attribute describes the group example: bluebirds and

b

Fractional notation Representation of a decimal number in the form of division

Frequen Number of occurrences in a collection of

Function The rule for the continuation of a pattern; correspondence or re

Function hine A machine that inputs a number, uses a rule, and produces an output number; See Classroom Activities in Appendix

Heterogeneous grouping Organizing so that a dive Intelligences, leadership ...); this grouping helps children bui

ill be proficient in mathem expectation that all children can and w ery child

Hexagon A six ided polygon

Hom grouping Organizing a group so it sharesbuzzards

Page 42: Math Curriculum K-12

Improper fraction a fraction whose numerator is greater than or equal to its denominator

Infinity ted extent of time, space or quantity; represented by the symbol that was introduced in the 17th century

Intuitive hild for specific learning by connecting a new concept to something the child already knows

Isoscel

8

Horizontal Parallel to the horizon

Hypotenuse The side opposite of the right angle in a right triangle

Unlimi

Integer A positive or negative whole number, including zero

The beginning Level of Knowing that prepares the c

es triangle a triangle with two congruent sides

Length 1.) A measured distance or dimension (a 1-dimensional measurement) 2.) Duration or extent in time or space

42

Levels of Kno ing

Line strai

Line of symmetry

Line segment art of

i tic

puter language that can be

t terms

"Manipulatives belong in every classroom ... [K - 12]. Suggestions for manipulatives include: pattern blocks,

or and table blocks, scales and balances, rubber walk-on numbers, number lines, rulers, meter sticks,

w The levels through which one must progress to completely understand and master any concept; See Intuitive, Concrete, Pictorial, Abstract.

A ght path that is endless in both directions

A line on which a figure can be folded so that the two parts fit exactly

P a line that extends from one point to another point Lingu s A Level of thinking that involves the acquisition of mathematical language necessary for the development of a specific

concept LOGO Com used for computer graphics; a beginning programming language that utilizes directional

commands in order to move objects on the screen Lowes a form of a fraction where the numerator and denominator have no common factor greater than one Manipulatives

tangrams, cuisenaire rods, unifix cubes, relationshapes, tiles, geoboards, big book, tegos, clocks, calendars, dominoes, flo

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43

agrams, abacus, design cards, kinesthetic tamps, 100's board and pegs, parquetry

attern cards, maps, thermometer, weather instruments, trundle wheel, linear and liquid measures, flannel board materi ooks, apes, records, cooking implements,

eras, VCR, watch timer... as wide a equipment in every classroom." From MESPA'S

Recommendations for Effective Mathematics Instructional Programs

measuring devising, sets of objects, graphing materials, posters, Venn di dominoes, 3D shapes, word and number beads, number puzzles, coins, s blocks, ordinal board, multi-link boards, activity cards, p

als, b t recipes, telephone, timetables, schedule, sewing and weaving materials, cam variety of manipulatives as possible should be basic

, page 5

contains; mass is measured in grams, not in pounds; mass remains the same no matter the

Mathematical concept le collection of examples of the luding non-examples; all pre-requisite concepts must be solid

ber of addend

mi n of the mi

is the minuend (or a mathematical waltz - hee, hee, hee!!)

a set of numbers

factor

ation

r; syn: factor

Mass The amount of matter an object

affect of gravity; often confused with weight

Secondary concepts; in order to learn them, there must be a suitab concept, inc

Mean The arithmetic average; traditionally thought of as the "average"; found by dividing the sum of the numbers by the num

s

Median The ddle number when the data is ordered sequentially; if there is no one middle number, then the median is the mea two ddle numbers

Minuend In the expression a - b, a

Mode The number or numbers that occur most often in

Multiplicand The number being multiplied; in the expression a x b, a is the multiplicand; syn:

Multiple A number that is a product of a number and a whole number

Multiplic 1.) The area of a rectangle whose sides are the two multiplicands 2.) Repeated addition 3.) Possible notations: a x b, a * b, a (b), (a) b, (a) (b), ab Multiplier The number doing the multiplying; in the expression a x b, b is the multiplie

Number line a line that shows numbers in order

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44

ties e property, the distributive property, the identity properties of of multiplication

nte that uses the equality symbol (_) to show that two expressions have the same value; syn: equation ple: 6+8=7+7 OR 6+8=14

tor

d number

Oh boy

peration An action on a set including, but not limited to, addition, subtraction, multiplication, and division

Parallel lines Lines which are equidistant at all points and therefore will not intersect

Parallelogram A quadrilateral with two pairs of parallel lines

n

Perimeter The distance around a figure

Perpendicular lines Two lines or line segments that intersect at a 90 degree angle

pi The constant obtained by dividing the circumference of a circle by its diameter; development of pi was begun by the al approximation is 3.141592... ; The fractional approximation is 22

Number proper Includes the associative properly, the commutativmultiplication and addition; and the zero property

Number se nce An equation Exam Numeral A symbol for a number

Numera The number above the division bar in a fraction

Obtuse angle an angle whose measure is greater than 90 degrees and less than 180 degrees

Od A whole number that has a 1, 3, 5, 7, or 9 in the ones place

An expression of glee or distress that occurs in times of happiness or frustration

O

Ordinal number a number indicating order or rank (as 5`) in a series

Palindrome A sentence, number or word that appears the same forward as backward

Pattern A configuration that can be extended indefinitely

Pentago A 5-sided polygon

Percent Per 100; a way to compare a number with 100; denoted by the symbol "%"

Period In large numbers, a group of three digits separated from other groups by commas (for example, the thousands' period)

Babylonians before 1700 BC; the decim ; denoted by the 7 symbol “π "

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45

A graph that uses pictures or

Plane figure Any 2-dimensional figure

Point A single, exact location; often represented by a dot

egments

ber of ways one specific event can occur divided by the total number of

Procedural

Properties

Properties of property: a = a Any number is always equal to itself. 2.) Symmetric property: if a = b then b = a If the first number equals the second number, then the second

equals the first = c then a = c If 3=1+2 and1+2=4-1, then 3=4-1

n are equal

or

ces are triangles with a common vertex; a

Pictograph symbols to represent numbers

Pictorial A Level of knowing that involves symbolic representation of mathematical ideas and concepts. A student functioning at this level can solve problems only with the help of pictures or diagrams.

Polygon A closed figure formed by line s

Prism A 3-dimensional figure whose bases are congruent polygons in parallel planes, and whose faces are parallelograms; a prism is named by its bases' shape example: a triangular prism has bases that are triangles

Probability The likelihood of an event occurring; the num possible outcomes

A Level of Thinking that involves the sequencing of steps needed in solving a problem or applying a skill

Product The answer to a multiplication problem

See number properties

equality 1.) Reflexive

number 3.) Transitive property. If a = b and b

Proportio An equation showing that two ratios

Protract The tool used to measure angles

Pyramid A 3-dimensional figure whose base is any polygon and whose fapyramid is named by its base's shape; example: a pentagonal pyramid has a base that is a pentagon

Quadrilateral A four-sided polygon

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46

e whole picture and then breaks it into parts. This learner is to be difficult. This student learns best when examples are

, and THEN through inductive reasoning derives the procedures, definitions and formulas.

and from those parts, the whole

amples.

Quarter 1.) One fourth of a whole ts

an

me other amount (distance per time passed, amount charged per pound, ... )

arisons; the ratio of a to b can be written: a to b, a: b, or a

Rectangle

The new shape created when a figure is flipped about a line

r after the division process is completed

ame

Rh us A quadrilateral with congruent sides

ang

Right triangle a triangle that has a right angle

Qualitative A mathematics learning personality where the learner sees thusually very good at problem solving but procedures provegiven FIRST

Quantitative A mathematics learning personality where the learner starts with the parts of a problem,

emerges. This student learns best when procedures, definitions, and formulas are presented FIRST and THEN supported with examples. This learner is usually very good at arithmetic and is capable of solving word problems if they are the same as in-class ex

2.) A monetary unit equal to 25 cen

Quotient The swer to a division problem

Radius The segment from the center of a circle to a point on the circle

Range The difference between the highest and lowest numbers in a set

Rate An amount measured by its relation to so

Ratio A pair of numbers used in making comp Ray A part of a line that has exactly one endpoint

parallelogram with right angles A

Reflection

Regroup See rename

Remainde The number less than the divisor that remains

Ren what happens to a quantity when it is placed in a different location in a number system; syn: regroup, borrow, and carry

omb

Right angle an le that measures 90 degrees

Page 47: Math Curriculum K-12

47

n numeral

Rotation The new shape created when a figure is turned about a point

s proximity to a specific point on a number line

ue life size or the relationship between distances on a map and the actual distances

mbers along either axis of a graph

e triandpoints; a straight path from one point to another

Set ; the individual objects of a set are the elements or members of the set; an element is said to

xample: the photocopier can be used to create similar figures through enlargement or reduction

Sphere be

ents contains some or all of the members of an inclusive set

is a symbol for threeness "+” is a symbol for combining sets

Roma Numerals developed by the Romans

Rounding A procedure that determines a number'

Row One of two or more horizontal sections

Sample A representative part of a larger group

Scale 1.) The ratio showing the relationship between a picture of an object and its tr

2.) A tool used to measure weight nu

Scalen ngle A triangle with one angle greater than 90 degrees Segment A piece of a line with two e

Sequence A set of items in a particular pattern or order

A collection of thingsbelong to a set example: the number 2 is a member (element) of the set of even numbers

Similar figures Figures that have the same shape but are not necessarily the same size e

Skip counting Counting by a number other than one; counting by multiples other than one

A 3-dimensional figure in which all the points are the same distance from a center point example: a basketball or a glo

Square A rectangle whose sides are congruent

Strategy A careful plan or method used to achieve an end

subset A set, each of whose elem

Subtrahend In the expression a - b, b is the subtrahend; it is the quantity being subtracted

Sum The answer to an addition problem

Symbol The notation that represents a concept; example: "3"

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48

ic s of a line; the shape can be folded in half so that

Syntax sions, sentences, or equations

Trading In the decimal system: to make a group of 10 from one of the next highest place value or one from 10 of the next lowest place value example: 100 can be traded for ten 10s - 10 ones c n be traded for one 10 this can also be applied to other bases

Translation The new shape created when a figure is slid across a plane

Trapezoid A quadrilateral with exactly one pair of parallel lines

Triangle A polygon with three sides

Turn The new shape created when a figure is turned about a point; syn. rotation

Unit value The amount or quantity used as a standard of measurement; the value represented by one element in the ones (units) place

Variable A symbol, usually a letter, used to represent a number or a range of numbers

Vertex The point that two rays of an angl

Vertical Perpendicular to the horizon

Weight s weight changes (compared to weight on Earth) but the

meas t of a figure with at least two dimensions)

Symbolic Representative of a concept

Symmetr A shape is symmetric if it has an identical, mirror image on both sidethe two halves match

The way in which words or numbers are put together to form phrases, expres

Tessellation Tiling; the representation of a 2-dimensional pattern with no gaps or overlaps

Total Making up a whole; the entire amount

a

e have in common

Volume The number of cubic units of space that a 3dimensional figure holds

The amount of matter an object contains, compounded by gravity; weight directly varies with gravitational changes; on the moon, an object'

Width a ured distance or dimension (a 1-dimensional measuremen

Zero properties 1.) Of addition: a + 0 = a When zero is added to any number, the sum is that number. See identity property of addition. 2.) Of multiplication: a · 0 = 0 the product of any number and zero is zero.

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GLOSSARY – GRADES 4 TO 8

49

Abacus Pebble stacking devise used for performing all four operations and calculating square and cube roots; Chinese abacus was developed in 12th century AD. It has thirteen columns of beads, divided by a crossbar with five beads below the bar and two above.

Absolute value the distance the number is from zero on a number line

Abstract A Level of Knowing where students are able to envision the ideas they developed at the Concrete and Pictorial levels anipulatives or a picture in front of them.

le: 1+2+3+4+6>12

Acute angle An angle whose measure is less than 90 degrees 90°

dja angles two angles are adjacent if they share a common vertex and side but have no common interior points

in order to solve problems without the actual m

Abundant number A number is abundant if the sum of its proper factors is greater than the number examp

Addend One of the numbers to be added

centA

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50

sides of the transversal

gment from one vertex of a polygon, per ontaining the opposite side

Analo Numbers represented by physical quantities such as rotations (clock hands), voltages (electric meter) distances (odometer)

Angle The space formed between two rays that are connected at a vertex

Applications Using ations; applications may include transferring math knowledge to other Content areas, forming fe prob ms.

ate of a desired number

Arc degree Equal to the degree of the central angle that intercepts the endpoints of the arc

Algorithm A procedure used to solve a problem; a recipe

Alternate exterior angles Pairs of non-adjacent exterior angles found on opposite sides of the transversal Alternate interior angles Pairs of non-adjacent interior angles found on opposite

Altitude A se pendicular to the line c

g

knowledge in diverse situ

new ideas based on previously mastered concepts, and finding solutions to a word and real lile

Approximate To find a number which serves as an estim Arc Any part of a curve, especially a circle

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51

e amount of 2-dimensional space taken up by an

Array A rectangular matrix; a rectangle of specified dimensions

ple: x x x x x

tive p on: a + (b + c) = (a + b) + c the sum of any three numbers is the same, regardless of

* (b * c) = (a * b) *c the product of any three numbers is the same,

ttribute An inherent characteristic

tize out hesitation

es lengths of bars to show how quantities compare

ow a repeating pattern. See vinculum.

as "b" in "b to the second power"; “b²” metric figure.

al

e tem

m that shows the first, second and third quartiles and the extreme values of a set of data

Area The number of squares it takes to cover a 2-dimensional figure; th object.

exam x x x x x x x x x x an array of 5 by 3 0R 3 by 5

Associa roperty 1.)Of additi

grouping. 2.) ... of multiplication: a

regardless of grouping.

A

Automa To make automatic; to know with

Average Generalization of data; a central tendency of data; See mean, median, mode

Axis (pl. axes) 1.) A line within a coordinate plane 2.) A central line around which the parts of the system are equally arranged

Bar graph a graph that us

Bar notation A bar placed over a sequence of numbers in a decimal amount to sh

Base 1.) A system of counting 2.) Algebra: the number to which an exponent applies, such

3.) Geometry: a particular side or face of a geo

Binomi A polynomial with exactly two terms

Bisect To divide in two equal parts Boiling point th perature at which water boil

Box-and-whisker plot A diagra

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52

Brack One form of grouping symbols; [ ]

nsional figure given in terms of liquid measurement

question "How many?"

ees as the freezing point and 100 degrees as the boiling point

ee Cel

in a circle, an angle with the vertex at its center and its sides are radii

Central tendency See measures of central tendency

A 2-dim

To enclose

Circumscribed circle A circle is circumscribed about a polygon if each vertex of the polygon is on

Braces One form of grouping symbol; { }

et

Capaci y The volume of a 3-dimet

Cardinal number A number that is used in simple counting and answers the

Celsius The temperature scale with zero degr

Centigrade S sius

Central angle

Chord A segment that connects two points on a circle

Circle ensional figure in which all points are the same distance from a point called the center

Circle graph a graph that represents data as parts of a circle. The whole circle represents the entire collection Circumference The distance around a circle (formula: C = pi x diameter)

Circumscribe

the circle

Cluster Isolated groups of values

Coeffic ent The numerical part of a monomial example: in 7ab², the coefficient is 7

Colu One of

ion

Common den

i

mn two or more vertical sections

Combinations A collection of a set of objects in which order is not important

Commiss The percent of money is earned on a sale

ominator the same denominator used in two or more fractions. Example: ¼ and ¾ have a common denominator of 4

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53

tor of 12 and 18 is 2

mul 42

Commutative property of any two numbers is the same regardless of order. 2.) ... of multiplication: a * b = b * a - the product of any two numbers is the same, regardless of order.

n ument used to draw circles and arcs and to transfer measurements

n

Complementary angles Two angles whose sum measures 90 degrees

2.) A number that can be represented in more than one array

an 180 degrees, then the polygon is concave

ing aware of similarities, which enables one to recognize s that are derived from our sensory and

) Concepts abstracted from other concepts are called “seco tical concept.

appropriate concrete model

rete materials in order to solve a problem

Conc ete Materials that can be physically manipulated (hands-on)

bol

Conjecture A mathematical statement which is the result of inductive reasoning which has neither been proved deductively nor

Common factor A number at divides two or more numbers evenly. Example: A common fac

Common tiple a number that is a multiple of two or more numbers. Example: A common multiple of 2, 6 and 14 is

1.) ... Of addition: a = b = b + a - the sum

Compass A instr

Compatible umbers Numbers that are easy to compute mentally

Composite number 1.) A number with more than two whole number factors

Concave polygon If one of the angles in a polygon is more th

Concept An idea; some kind of lasting mental change, the result of becomnew experiences as having the similarities of an already formed class. Conceptmotor experiences are called “primary concepts” (red, heavy, hot, sweet ...

ndary concepts” (color, light, all mathematical concepts, beauty ...); See mathema

Conceptual A Level of thinking that involves the visualization of an

Concrete The second Level of Knowing involving the manipulation of conc

r

Congruent Two or more objects that have the same size and shape (2- and 3-dimensional); all pieces identical; denoted by the sym “≅ ”

denied by counter example

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54

onsecutive Following an order without interruption. Example: consecutive integers...2, 3, 4 ...consecutive even integers... -2, 0, 2, 4

onstant le

straight edge

ction y using a compass and a straight edge

on is convex

nu

oordinate plane

Corresponding angles n relative to the transversal and the lines 1

ngruent or similar figures that match

each of two or more lists, the total number of possible combinations is the product of

ubit e elbow to the tip of the middle finger

ents such as inch, foot, ounce, mile. This system is used only in the United States and on the island of Brunei.

C consecutive odd integers... – 1, 1, 3, 5...

C a monomial that does not contain a variab

Construct

Create a figure using only a compass and a

Constru A drawing of a geometric figure made b

Convex polygon If all of the angles in a polygon are less than 180 degrees, then the polyg Coordinate A mber associated with a point on the number line

A grid on a plane with two perpendicular number lines C

Angles that are in the same positio

nd ∠ 2 are a pair of corresponding angles ∠ 1

a 2

Corresponding parts Parts of co

Counting principle When one item is selected from

the number of items in each list Cube 1.) A rectangular prism with all square faces 2.) The third power of a number

3.) To raise a number to the third power 4.) A number used as a factor three times Cubic 1.) Having three dimensions

2.) The unit used in measuring volume

a ancient measurement of length from thC Customary units Units of measure first developed by the Babylonians which now include measurem pound, cup, gallon, and

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55

ylind r

m xam

mal) fraction)

ota l point and the digits 0 through 9

e, then

) A /360 of the circumference of a circle easure for temperature

unt of stuff crammed

e A 3-dimensional figure with two parallel, congruent, circular faces C

Data Collection of unorganized numbers or facts Decim l A nu s, thousandths... a ber that uses the digits 0 – 9. It can also show tenths, hundredth

E ple: 397 4.28 (mixed deci 0.5034 (decimal Decim l n a decimaa tion Representation of a fractional value using Deductive reasoning (Or logical reasoning) the process of demonstrating that if certain statements are accepted as tru

other statements can be shown to follow from them Deficient numbers A number is deficient if the sum of its proper factors is less than the number. Example: 1 + 2 + 4 < 8 Degree . ngles and arcs; 11 unit of measure for a

2.) A unit of m 3.) The exponent of a variable Denominator The number below the division bar of a fraction; it describes the number of equal pieces Density 1.) The ratio of mass to volume 2.) Given equal sizes spaces or areas, density is determined by the amo into each space

e of the first effects the outcome of the second.

See variable

1.) The segm circle and the center

g one number from another; the answer to a subtraction problem

Density of numbers Between every two real numbers, there is another real number Dependent events Two events such that the outcom Example: Drawing a card out of the deck without replacing it Dependent variable Diameter ent containing two points of a 2.) The chord that contains the center of a sphere

The number obtained from subtractinDifference

Page 56: Math Curriculum K-12

56

(analog); See analog

Distributive property c) = ab + ac Multiplying a sum by a number is the same as multiplying each addend by the number then adding the products; the most important mathematical property

to be divided; See division

ro

eated

a number by which a dividend is divided; the factor of an integer or a polynomial; see

pair 2

wo sets of data

ractor and a ruler

The line o

em End point a ent or ray

expressions have the same value;

Digital Of or relating to calculation directly with digits rather than through measurable, physical quantities Digits 1.) Symbols used to write numerals 2.) Fingers or toes

a (b+ and Dividend The area of an array; a number Divisible A number is divisible by another if upon division, the remainder is ze

w many sets or how many are in each set; repDivision The inverse operation of multiplication; an operation that tells ho subtraction; see multiplication Divisor The length of a side of an array; division Dodecahedron A polyhedron with 12 faces Domain 1.) The set of all first coordinates from each ordered

.) syn. Replacement set Double bar graph a graph that uses lengths of bars to show how quantities compare for t Draw Create a figure using only a prot Edge r line segment created by the intersection of planes or any of the faces of a 3 – dimensional figure Element A m ber of a set

point at the end of a segm

Equally likely outcomes Outcomes that have the same probability of occurring

Equation A number sentence that uses the equality symbol (=) to show that two

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57

+ 8 = 14

Naming the same amount

Estimate 1.) v. A ns i

n

two

xpan ed fo Example: 345 = 300 + 40 + 5 = (3 · 100) + 4 · 10) = (4 · )

Expectant value ed value of a player is the product of the amount that can

winning:

al

Exponent The exponent tells the number of times the base is used in a factor

Exponential form A number written with a base and an exponent

Expression Any combination of operations with variables and numbers (constants) Examples: 3 + 4 OR x – 5 OR 35 – 8 · 2

syn: number sentence. Example: 6 + 8 = 6 + 7 OR 6 Equilateral triangle A triangle with three congruent sides Equivalent

To find an answer that is close to the exact answer 2.) n. n a wer that s close to the exact answer Evaluate Replacing the variables with numbers and finding the numerical value of the expressio Even number a whole number that has 0, 2, 4, 6, 8 in the ones place; it is divisible by

Event A set of one or more outcomes

E d rm A way to write a number to show the value of each digit.10) (5 · 1

A way of measuring the fairness of a game. The expectbe won by the probability of winning. The interpretation is that playing many games will lead to an “average

Experiment probability The probability of an event based on the results of an experiment

Exterior angle 1.) An angle formed when a side of a polygon is extended 1 2 2.) Given two lines and a transversal, see diagram ∠ 1, ∠ 2, ∠ 3, ∠ 4 are exterior angles 3 4

le re greater than the radius

polate

Exterior of a circ Points whose distances from the center of the circle a Extra To use known data points to predict data values at a later or earlier time

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58

terms m values. See box – and – whisker plot

dimensional surfaces making up a 3dimensional figure

led the product; the sides of a rectangular

Factor tree gram s f a composite number. The factors branch out from the previous factors l the fact

n factorial, written as Anno’s Mysterious Multiplying Jar

ame ctant value

e outcome nt; what you want to happen

b

ious

ent

e

Fractal A two or three dimensional object created by using a pattern that is repeated endlessly. The object is complex and is

3.) A ratio written in the form: a

Extremes 1.) In a proportion, the first and fourth 2.) In data, the maximum and minimu Face One of the 2-

Factor Numbers that are combined in the multiplication operation to give a number calarray

The dia howing the prime factorization ountil al ors are prime numbers

Factorial n, is the product of the numbers from 1 to n; See

Fair game a g in which each player has the same expe

Favorabl The desired result of a probability experime

Fibonacci number a num er which is part of the Fibonacci sequence

Fibonacci sequence a list of numbers in which the first two numbers are one, and each number that follows is the sum of the prevtwo numbers: 1, 1, 2, 3, 5, 8...

Finance charge The percent of money owed for delay in paym

Flip The new shape created when a figure is turned to its reverse side. See reflection

Formula A gen ral rule expressed by symbols

endlessly magnifiable

Fraction 1.) A relationship between two numbers describing parts of a whole or of a set 2.) A division problem

b

Fractional notation Representation of a decimal number in the form of division

Freezing point the temperature at which water freezes

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59

ncy

s it occurs

Front-end estimate An esti mputation of the front-end digits only. Then the estimate is adjusted by

espondence or relationship between two sets

Function machine number, uses a rule, and produces an output number; See Classroom Activities in uction directions.

a sequence in which the ratio between any two successive terms is the same

the nature of space and the shape, size and properties of figures. See the Librarian re

lide ee tra slation

horter side of a Golden Rectangle; often represented by the Greek letter “phi” (Φ); 1.618034 to 0.618034

olden Rectangle A Golden Rectangle satisfies this property: If you cut a square off one end of the rectangle, the remaining

rep

at contains the center of the sphere

eatest number that is a factor of two or more numbers; the greatest number that divides two or more numbers with no remainder

Grid A picture of lines that cross at right angles and regular intervals

alf turn A rotation of 180 degrees about a turn center

Freque Number of occurrences in a collection of data

Frequency distribution a listing of data that pairs each data item with the number of time

Frequency table A table used to group and summarize a collection of data, showing the number of occurrences at each interval

mate that begins with the co approximating the values of the other digits.

Function The rule for the continuation of a pattern; corr

A machine that inputs aAppendix for constr

Geometric sequence

Geometry a branch of mathematics dealing with Who Measu d the Earth

G S n

Golden Ratio The ratio of the longer to the s

Grectangle is similar to the original rectangle.

Graph A pictorial resentation of a set of data. See Appendix for Graphing

Great circle the intersection of the sphere with a plane th

Greatest common factor the gr

H

Heights T measurement of the altitude

Page 60: Math Curriculum K-12

60

emisphere Half a sphere

septagon

ltiple n abilities; the

icient in mathematics sends a powerful message to each

Homogeneous grouping describes the group example:

e horizon

The axis plane that is parallel to the horizon

n

dentit elem

prop

Imaginary numbers t of – 1

tion a fraction whose numerator is greater than or equal to its denominator

H

Heptagon A seven-sided polygon; syn. Heterogeneous grouping Organizing so that a diversity of attributes is represented within each group (gender, Mu Intelligences, leadership ...): This grouping helps children build confidence in their ow expectation that all children can and will be prof and every child. Hexagon A six-sided polygon Histogram a bar graph with columns next to each other to show frequency distribution

Organizing a group so it shares one common attribute; this attribute bluebirds and buzzards Horizontal Parallel to th Horizontal axis in a coordinate Hypotenuse The side opposite of the right angle in a right triangle Hypothesis An unproven theory; the “if” part of an if-then statement Icosah dro A polyhedron with twenty faces

e

I y ent See identity property Identity erty 1.) ... Of addition: For any number a, a + 0 = a; 0 is the identity element of addition 2.) ... of multiplication: For any number a, a x 1 = a; 1 is the identity element of multiplication Image The new figure formed by the transformation of a given figure

Numbers that have a factor of the square roo

Improper frac

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61

Inclusive Including all possibilities; when two events can happen at th s

e ame time

out of the de k and r the next draw Independent variable See variable

triangles and writing a proportion

reaas

1.) A statemplus 7 is greater than 5” or “5 is less than 7 + 3”

i the symbol “

Independent events Events that have no effect on each other; Ex: Drawing a card c then replacing it fo

Indirect measurement Finding a measurement by using similar

Inductive soning The process of collecting, organizing, and analyzing data to reach a general conclusion (the conjecture) (also known the Scientific Method)

Inequality ent that has two expressions that are not equal 2.) Possible interpretations: 3 + 7 >5 can be read, “3

Infinity Unlim ted extent of time, space or quantity; Represented by ∞ ” that was introduced in the 17th century

so that the drawing has sides that are tangent to the original figure

Interior angle 1.) An angle inside a polygon

,

Inscribe To draw within an existing a figure Integer A positive or negative whole number, including zero Interest Amount that is paid for the use of money over time

2.) Given two lines and a transversal, see diagram ∠5 6, ∠ ∠7, ∠8 are interior angl

es 5 6

7 8

Intuitive The be by connecting a new concept to something the a s

d subtraction or multiplication and division

Intersecting lines Lines that have exactly one point in common

ginning Level of Knowing that prepares the child for specific learning child lready know

Inverse operations Operations that undo each other, such as addition an

Page 62: Math Curriculum K-12

Irregular pe Any shape or figure with non-congruent sides

62

sha

soscel trap ruent

Kite A quadrilateral with two pairs of consecutive sides

Lateral surface All the surface of the figure except the base or bases

Least common denominator the lease common multiple of the denominators of two or more ample: The least

Le

Legs of a right triangle

1.) A measured distance or dimension (a 1-dimensional measurement) 2.) Duration or extent in time or space Levels of knowing the levels through which one must progress to completely understand and master any

concept: see Intuitive, Concrete, Pictorial, and Abstract

ik hood Probability

one dimension)

atioordinate grid

line t phed data

Line of symmetry A line on which a figure can be folded so that the two parts fit exactly _ _ _

I es ezoid A trapezoid whose non-parallel sides are cong

Isosceles triangle a triangle with two congruent sides

fractions; ex common denominator of 2/4 and 1/6 is 12

ast common multiple the smallest number that is a common multiple of two or more given numbers

The two perpendicular sides

Length

L e

Like terms Monomials with the same variables raised to the same power

Line 1.) Geometry: An undefined term that shows direction ( 2.) A straight path that is endless in both directions Linear equ on an equation that has two variables and many solutions, all of which lie in a straight line when they are graphed on a

c

Line graph A type of statistical graph used to show how values change over a period of time

Line of best fit A hat shows the general trend of the gra

Page 63: Math Curriculum K-12

Line plot diagr

gment

Linguistic n of mathematical language necessary for the development of a specific

LOGO Com ing language that utilizes directi

Lower extreme In a data set, the smallest data item

er quartile er half of the data set

An arc that is greater than a semi-circle and denoted by three letters; it contains more than 180 degrees

tangrams, cuisenaire rods, unifix cubes, relationshapes, tiles, geoboards, big book, tegos, clocks, calendars, on numbers, number lines, rulers, meter sticks,

measuring devising, sets of objects, graphing materials, posters, Venn diagrams, abacus, design cards, kinesthetic number puzzles, coins, stamps, 100's board and pegs,

thermometer, weather rials, books, tapes, records, cooking

___ _ __ __ __ _ _ _

63

Line of symmetry not a line of symmetry

A am that uses a number line to show frequency of data

line se Part of a line that extends from one point to another point

A Level of Thinking that involves the acquisitioconcept

puter language that can be used for computer graphics; a beginning programmonal commands in order to move objects on the screen

Low In a data set, the median of the low

Lowest terms a form of a fraction where the numerator and denominator have no common factor greater than one

Major arc

Manipulatives "Manipulatives belong in every classroom, [K - 12]. Suggestions for manipulatives include: pattern blocks,

dominoes, floor and table blocks, scales and balances, rubber walk-

dominoes, 3D shapes, word and number beads, parquetry blocks, ordinal board, multi-link boards, activity cards, pattern cards, maps, instruments, trundle wheel, linear and liquid measures, flannel board mate

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64

implements, recipes, telephone, timetables, schedule, sewing and weaving materials, cameras, VCR, watch timer... nipulatives as possible should be basic equipment in every classroom." From MESPA'S

Recommendations for Effective Mathematics Instructional Programs as wide a variety of ma

, page 5

Mass The amount of matter an object contains; ains the same no matter the affect of gravity; often confused with

Mathematical concept Secondary concepts; in o amples of the concept,

Mean The arithmetic average; traditionally thought of as the "average"; found by dividing the sum of the numbers by the number of ddend

res of c t of data. The mean, median and mode are measures of

of the two middle numbers

th

ts

grees

- hee, hee, hee!!)

Mirror image e revers

F D

mass is measured in grams, not in pounds; mass rem weight

rder to learn them, there must be a suitable collection of exincluding non-examples; all pre-requisite concepts must be solid

a s

Means In a proportion, the second and third terms Measu entral tendency A single number that is used to represent a secentral tendency. Median The middle number when the data is ordered sequentially; if there is no one middle number, then the median is the mean

Mental ma Mathematics done without the use of technology, including pencil and paper

Midpoint The point that separates a segment into two congruent segmen

Minor arc An arc that is less than a semi-circle and denoted by two letters; it contains less than 180 de

mathematical waltzMinuend In the expression a - b, a is the minuend (or a

th e image of a figure

B E

C A

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65

a

he nu rs

Modeling to represent a situation, object or an event with something that is similar; example: Writing an equation to represent a real life situation

exp number or a product of real numbers and varia es

tiple ber

Multiplicand The number being multiplied; in the expression a x b, a is the multiplicand; syn: factor

e area of a rectangle whose sides are the two multiplicands 2.) Repeated addition

3.) Pos b, a (b), (a) b, (a) (b), ab

Multiplicative inverses two numbers whose product is one; syn. Reciprocal; ex: ¾ and 4/3 are multiplicative inverses

Multiplicative properties 1.) ... Of zero: For any number a, a · 0 = 0 1 = a

Multiplier The number doing the multiplying; in the expression a x b, b is the multiplier; syn: factor

Mutually exclusive Two or more events such that no two events can happen at the same time

Napier’s bones

Mixed decim l See decimal Mixed number the sum of an integer and a proper fraction Mode T mber or numbers that occur most often in a set of numbe

Monomial An ression that is either a real bl

Mul A number that is a product of a number and a whole num

Multiplication 1.) Th

sible notations: a x b, a *

2.) ... of one: For any number a, a · See multiplicative identify

Ancient lattice rods used to perform multiplication

Natural numbers counting numbers

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66

Negative ger An integer less tinte han zero

m

tern t

cher request

but not

er properties y, the distributive property, the identity properties of ltiplication

Number sentence An equation that uses the equality symbol (_) to show that two expressions have the same value; syn: equation

n angle 90 degrees and less than 180 degrees

Negative nu ber A number less than zero

Net A flat pat hat can be folded to form a solid

NOT A negative student response to a tea

Null se A set w t ith no elements shown by the symbol or Ø Ø

Number line a line that shows numbers in order

Numb Includes the associative properly, the commutative propertmultiplication and addition; and the zero property of mu

Example: 6+8=7+7 OR 6+8=14 Numeral A symbol for a number

Numerator The number above the division bar in a fraction Numerical expression an expression that does not contain a variable

Obtuse angle a whose measure is greater than

Page 67: Math Curriculum K-12

67

a common endpoint and form a straight line

pair lane. The first element of the ordered pair ate) relates to the vertical axis.

divide in order from left to right

utcome The result of a probability experiment

Outlier

verestimate An estimate that is greater than the actual answer

backward

es

elogram

Parentheses One type of grouping symbol; ( )

e product of a single digit of the multiplier with the multiplicand

change to the original amount

Perfect numbers a number is perfect if the sum of its proper factors is exactly the number; ex: 1 + 2 + 3 = 6

Opposite rays Two rays that have

Ordered A pair of numbers such as (3, 2) that locates a point in a coordinate p (abscissa) relates to the horizontal axis and the second (ordin

Order of operations The rules followed to simplify expressions when more than one operation is involved 1.) Simplify the expressions inside grouping symbols 2.) Evaluate powers 3.) Multiply and 4.) Add and subtract in order from left to right Ordinal number a number indicating order or rank (as 5`) in a series

Origin The point where the axes of a coordinate plane intersect; (0, 0)

O

In a data set, a value whose distance from the center of the data is much greater than the distances of the other data values

O

Palindrome A sentence, number or word that appears the same forward as

Parallel lin Lines which are equidistant at all points and therefore will not intersect

Parall A quadrilateral with two pairs of parallel lines

Partial product In long multiplication, th

Pattern A configuration that can be extended indefinitely

Pentagon A 5-sided polygon

Percent Per 100; a way to compare a number with 100; denoted by the symbol "%"

Percent of change The ratio of the amount of

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68

Perimeter The distance around a figure

parated from other groups by commas (for example, the thousands’ period)

rtant

Perpendicular bisector (of a segment) is a line that is perpendicular to the segment at its midpoint

ents that intersect at a 90 degree angle

eter; development of pi was begun by the Babylonians before 1700 BC; the decimal approximation is 3.141592... ; The fractional approximation is 22/7; denoted by the sym

h resent numbers

evel

ts

as its sides

Polynomial A sum or difference of monomials

Population The set f all d

Positive integer Any in

as 10, 100, 1000...

Period In large numbers, a group of three digits se

Permutation An arrangement or listing of objects in a set in which order is impo

Perpendicular lines Two lines or line segm pi The constant obtained by dividing the circumference of a circle by its diam

bol "π" Pictograp A graph that uses pictures or symbols to rep

Pictorial A Level of knowing that involves symbolic representation of mathematical ideas and concepts. A student functioning at this lcan solve problems only with the help of pictures or diagrams.

Plane In geometry, an undefined term that denotes two dimensions, length and width

Plane figure Any 2-dimensional figure

Point A single, exact location; often represented by a dot

Polygon A closed figure formed by line segmen

Polyhedron (pl. polyhedra) a solid figure (3 dimensional) that has polygons

o ata in an experiment teger that is greater than zero

Power of ten The numbers 10¹, 10²10³, they also can be written

Prime fact A factor that is a prime number or

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69

tori

Prime number

Prism A 3-dimensional figure whose bases are congruent polygons in parallel planes, and whose faces are parallelograms; a prism ample: a triangular prism has bases that are triangles

Procedural nvolves the sequencing of steps needed in solving a problem or applying a skill

on problem

Prop factors the number itself

es of equali 2.) Symmetric property: if a = b then b = a If the first number equals the second number, then the second

st 3.) Transitive property: If a = b and b = c then a 3=1+2and1+2=4-1, then 3=4-1

Proportion An equation showing that two ratios are equal

Protractor The tool used to measure angles

A 3-dim ommon vertex; a pentagon

Pythagorean Theorem sum of the squares of the legs

rant

Quadrilateral

Prime fac zation Expression of a composite number as the unique product of its prime factors

a number, greater than one, with exactly two factors, namely one and the number Principle Amount of money borrowed or invested

is named by its bases' shape ex Probability The likelihood of an event occurring; the number of ways one specific event can occur divided by the total number of

possible outcomes

A Level of Thinking that i

Product The answer to a multiplicati

er All of the factors of a number, except

Properties See number properties

Properti ty 1.) Reflexive property: a = a Any number is always equal to itself.

number equals the fir = c If

Pyramid ensional figure whose base is any polygon and whose faces are triangles with a c

pyramid is named by its base's shape; example: a pentagonal pyramid has a base that is a

In a right triangle, the square of the hypotenuse is equal to the

Quad One of four sections into which the x – and y – axis divide the coordinate plane

A four-sided polygon

Page 70: Math Curriculum K-12

70

ath the whole erge EN supported

od at arithmetic and is capable of solving word problems if they are the same as in-class examples.

Qua er 1.) One fourth of a whole

rn

ta into four groups of the same size

swer to a

Quantitative A m ematics learning personality where the learner starts with the parts of a problem, and from those parts,em s. This student learns best when procedures, definitions, and formulas are presented FIRST and THwith examples. This learner is usually very go

rt 2.) A monetary unit equal to 25 cents

Quarter tu A 90 degree rotation

Quartiles The three numbers that divide an ordered set of da

Quotient The an division problem

“ Radical The symbol used to indicate a non-negative square root “

Radius The segment from the center of a circle to a point on the circle

other

mp

Ratio A pair of num making comparisons; the ratio of a to b can be written: a to b, a: b, or a

Random By chance, with no one outcome more likely than an

Random sa le When each member of the population is given an equal chance of being selected

Range The difference between the highest and lowest numbers in a set

Rate An amount measured by its relation to some other amount (distance per time passed, amount charged per pound ...)

bers used in b

um n the form a/b where “a” is any integer and “b” is any non-zero integer

endpoint

e set

s plicative inverse

Rectangle A parallelogram with right angles

Rational n ber A number that can be written i

Ray A part of a line that has exactly one

Real numbers th of rational and irrational numbers

Reciprocal Two numbers are reciprocals when their product is one; See multi

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71

pris

Rectangular pyramid

Region An area in space bounded by a closed curve

and angles

The num division process is completed

ry

Repl cement set The given set of numbers used to find the solutions of an equation; see domain

Rh us A quadrilateral with congruent sides

oman numeral Numerals developed by the Romans

otation The new shape created when a figure is turned about a point

otational symmetry A figure has rotational symmetry if when the figure is turned about a point, the figure fits exactly over its original position at least one during a complete rotation

Row One of two or more horizontal sections

Ruler A straight edge with measurement markings

Sample A representative part of a larger group

Sample space The set of all possible outcomes of an event

Rectangular prism A m whose bases are rectangles

A pyramid whose base is a rectangle

Reflection The new shape created when a figure is flipped about a line

Regroup See rename

Regular polygon A polygon with congruent sides

Remainder ber less than the divisor that remains after the

Rename what happens to a quantity when it is placed in a different location in a number system; syn: regroup, borrow, and car

a decimal whose digits, from some point on, repeat endlessly in groups of one or more; ex: 0.181818... Repeating decimal2.833333...

a

omb

Right angle an angle that measures 90 degrees

Right triangle a triangle that has a right angle

R

R

R

Rounding A procedure that determines a number's proximity to a specific point on a number line

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72

Scale 1.) The ratio showi e relationship betw n p r object and its true life size or the relationship between distances on ma i n s

2.) A tool used to m ure weight 3.) Numbers along e axis a ra

cale drawing A draw e proportional actual dimensions

Scale factor In two similar polygons w m r oli s, t e s ale c or is the ratio of the corresponding linear measurements

calene triangle A triangle with o

plot at represent a relationship between two data sets

Scientific notation

t a straight path from one point to another

set o

ng th

p and the actuaeas

eith

ing whose meas

ee

a ictu e of anl d sta

ce ph

r of g

S urements ar to the

or t o si ila s d h c fa t S ne angle greater than 90 degrees

Scatter A collection of points in a coordinate plane th

� � � �

RA

NK

IN LEA

GU

E

� � � � � � � � �

� � � � � � � � �

� EARNED RUN AVERAGE

A number in the form c x 10n where l < c ≥ 10 and n is the integer

Segmen A piece of a line with two endpoints;

Semi-circle An arc that is half a circle; contains 180°

Sequence A f items in a particular pattern or order

Page 73: Math Curriculum K-12

73

ed sum of the terms of a sequence

Set ents er nt is said to belong eme ) of t set o v numbers

Similar figures Figures that have the same shape but are not necessarily the same size example: the photocopier can be used duction

closed cross itself and encloses a part of the plane

r and denominator is one

form has no like terms and no parentheses

y

Sketch A free hand representation of a figure

lines e plane

Skip counting Counting by a number other than one; counting by multiples other than one

The m change in

Series The indicat

A collection of things; the individual objects of a set are the elem or memb s of the set; an elemeto a set example: the number 2 is a member (el nt he f e en

to create similar figures through enlargement or re

Simple curve any curve in a plane that does not

Simplest form 1.) A fraction is in simplest form when the only common factor of the numerato 2.) An expression in simplest

Simp if To remove parenthesis and combine like terms. The resulting expression is in simplest form. l

S two lines that do not intersect ankew d are not in the sam

Slide See translation

Slope easure of the steepness of a line given by the ratio of rise (change in vertical distance) over run (horizontal distance) for any two points on the line; formula: m = ∆ y (where ∆ means “change in”)

∆ x olid figure A 3-dimensional closed figure; example: prism, cone, and sphere

Solution A number that replaces a variable to make an open sentence true

Solve Find the solution to make an open sentence true

Space figure l.) A three dimensional geometric figure whose points do not all lie in the same plane

2.) Alan B. Shepard, Jr. or Luke Skywalker

dim

S

Sphere A 3- ensional figure in which all the points are the same distance from a center point example: a basketball or a globe

Page 74: Math Curriculum K-12

74

e .) A r

Square root One of two equal factors of a number; ex: 12 is a square root of 144 since

Squar

1 ectangle whose sides are congruent 2.) A rhombus with right angles

144 = ± 12

Standard form The usual short form of a number. The standard form of 5 hundreds, 7 tens and 3 ones is 573

Statistics The field of mathematics involving the collection, analysis and presentation of data

Stem-and-leaf plot a means of organizing data in which certain digits are used as stems and the remaining digits are used as leaves

angle

ent markings, used to create a straight line

Stem

6

7

Leaves

47 9

2 5 7 7

Straight an angle whose sides form a straight line and measures 180 degrees

Straight edge A tool without measurem

Strategy A careful plan or method used to achieve an end

8 7

9 6

10 6

0

1

2

3

1. 8

2 6 9

1 3 4 7

4 5

0 2 2 5 4

5 8

Key: 6 4 represents 64 grams per cup

Page 75: Math Curriculum K-12

another. In a , 1 is the subscript

s of an inclusive set

end

n problem

Supplementary angles

Surf ce area The number of square units that cover all the faces of a three dimensional figure

Sur y A means of collecting data by the analysis of some aspect of group or area

Symbol The notation that represents a concept example: "3" is a symbol for threeness; "+” is a symbol for combining sets

Subscript A figure, letter or symbol written below and to the side of 1

Subset A set, each of whose elements contains some or all of the member

Subtrah In the expression a - b, b is the subtrahend; it is the quantity being subtracted

Sum The answer to an additio

Two angles whose sum is 180°

a

ve

Symbolic Representative of a concept

75

sides of a line; the shape can be folded in half so that the atch

Symmetric property If a = b, then b = a where a and b are any number

y

Syntax

t to a circle

of digits

Tessellation Tiling; the representation of a 2-dimensional pattern with no gaps or overlaps

Tetrahedron A polyhedron with four faces

Theo etical probability A probability that is computed on the possible outcomes

Symmetric A shape is symmetric if it has an identical, mirror image on both two halves m

Symmetr A figure has symmetry when it can be folded so both parts match

The way in which words or numbers are put together to form phrases, expressions, sentences, or equations

Table A way to organize data

Tangen A line that intersects the circle at only one point

Terminating decimal A decimal in which contains a finite number

r

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76

ile To tessellate

0 from one of the next highest place value or one from 10 of the next lowest place value example: 100 can be traded for ten 10s - 10 ones can be traded for one 10 this can also be applied to other bases

Transformation A change in the size or position of a figure

A type of transformation where all points of a figure slide the same distance and in the same direction

Tr sal A line that intersects two or more lines at different points

diagram

r p s a triangle

T

Total Making up a whole; the entire amount

Trading In the decimal system: to make a group of 1

Translation

ansver

Trapezoid A quadrilateral with exactly one pair of parallel lines

Tree A picture showing outcomes of an activity

Triangle A polygon with three sides

Triangular prism A prism whose bases are triangles

Triangula yramid A pyramid whose base i

Trigonometry The study of triangular measurement

Page 77: Math Curriculum K-12

77

Trinomial A nomial with exactly three terms poly

ignoring the digits that follow

nder timat

Union The joining of two or more sets; denoted by AUB; traditionally the work “or” indicates a union

The rate that of one unit

Unit value The amount or quantity used as a standard of measurement; the value represented by one element in the ones (units) place

niverse The set of all individual things

set

used to represent a number or a range of numbers rforming a function on an independent variable (a given number or the replacement set) will

expressi

agra

A: Numbers Divisible by 5

lines

Truncate To chop off; an answer being cut off at a certain place value position,

Turn To move a figure about a point or line; syn. rotation

U es e An estimate that is less than the exact answer

Unique 1.) One and only one

2.) How to get your one and only one – “u-nique” up on them

Unit rate has a denominator

U

Unlike fractions Fractions with unequal denominators

Upper extreme In a data set, the largest data item

Upper quartile In a data set, the median of the upper half of the data

Variable 1.) A symbol, usually a letter, 2.) Independent/dependent: Pe result in a dependent variable (the answer or the solution set) Variable on An expression that has at least one variable

Venn di m a drawing that uses geometric shapes to show relationships among sets of objects

B: Odd Numbers

Vertical angles the two non-adjacent angles formed by two intersecting

Page 78: Math Curriculum K-12

zon

Vert (pl. vertices) the point that two rays of an angle have in common

al

at shows repeating decimals or the fraction bar

Weight The amount of matter an object contains, compounded by gravity; weight directly varies with gravitational changes; on the m will remain constant

mb

eas

X-axis The ho coor

X-coordinate The first coordinate of a point in an ordered pair; syn. Abscissa

X-intercept The x-coordinate of a point where the graph crosses the x-axis

Y-axis the vertical number line in a coordinate plane

Y-coordinate The second coordinate of a point in an ordered pair; s : ordinate

Y-intercept The y-coordinate of a point where the graph crosses th y-axis; notation: “b” in y = mx + b

Zero exponent The expression x° with exponent 0, has a value of one (provided x 0)

Zero pair The result of pairing one positive counter with one negative counter Zero properties 1.) ...Of addition: a + 0 = a When zero is added to any number, the sum is that number. See identity property of

addition. 2.) ... of multiplication: a · 0 = 0 the product of any number and zero is zero.

Vertical axis The axis in a coordinate plane that is perpendicular to the hori

ex

Verti Perpendicular to the horizon c

Vinculum A raised horizontal line used as a grouping symbol; ex: The bar th

Volume The number of cubic units of space that a 3dimensional figure holds

oon, an object's weight changes (compared to weight on Earth) but the mass

Whole nu ers the set of natural numbers and zero

Width a m ured distance or dimension (a 1-dimensional measurement of a figure with at least two dimensions)

rizontal number line in a dinate plane

yn

e

78

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79

PLYMOUTH REGIONAL HIGH SCHOOL

Page 80: Math Curriculum K-12

80

GRADES 9 – 12 DRAFT MATH CURRICULUM

atics pectatio r

CONTENT PAGE PLYMOUTH REGIONAL HIGH SCHOOL PHILOSOPHY AND GOALS …………………………………………………………………. 81 A WORD FROM THE MA ……….. 82

AND B (TWO YEAR PROGRAM) BASIC GEOMETRY / GEOMETRY / HONORS GEOMETRY FINITE MATHEMATICS CALCULUS / PRECALCULUS / PRECALCULUS HONORS

The following pages include a Mathem Course Sequence, the Process Strand for all classes and the Course Outlines and Ex ns fo

Plymouth Regional High School.

THEMATICS/COMPUTER SCIENCE DEPARTMENT ………………………………………………

COURSE LEVEL DESCRIPTIONS ……………………………………………………………………………………………………….. 82 PROCESS ……………………………………………………………………………………………………………………………………. 83 INTRODUCTORY STATEMENTS ………………………………………………………………………………………………………. 84-86 LINEAR ALGEBRA (ONE YEAR PROGRAM) / LINEAR ALGEBRA A

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81

RESOURCE ROOM MATH …………………………………………………………………………………………………………………. 86 SEQUENCE OF COURSES: 9TH GRADE: LEVEL 2 …… 87 LEVEL …… 90

……………………………………………………………………………… 96

LEVEL 2 Linear Algebra B ……………………………………………………………………………………………… 110

or ….. 120 LEVEL 3 Precalculus or Finite Math or Statistics or Math Topics ….. 124 HONORS AP Calculus or Calculus or Statistics or Finite Math …… 128

PRHS EXPECTATIONS AND INDICATORS ………………………… ……………………………………………………… 129

Linear Algebra A ………………………………………………………………………………………… 3 Linear Algebra …………………………………………………………………………………………

HONORS Non Linear Algebra Honors …………………………………………………………………………... 92 10TH GRADE LEVEL 2 Basic Geometry ……………… LEVEL 3 Geometry ……………………………………………………………………………………………… 100 HONORS Geometry Honors ……………………………………………………………………………………. 105 11th GRADE:

LEVEL 3 Non Linear Algebra ……………………………………………………………………………………. 113 HONORS Precalculus Honors ……………………………………………………………………………………. 117 12th GRADE: LEVEL 2 Non Linear Algebra Math Topics ……………………………………………………

…………

PLYMOUTH REGIONAL HIGH SCHOOL PHILOSOPHY AND GOALS

Plymouth Regional High School is a learning center in rural north central New Hampshire that meets the diverse educational needs of students in towns of Ashland, Campton, Ellsworth, Holderness, Plymouth, Rumney, Thornton, Waterville Valley (AREA agreement), and Wentworth. It offers broad, flexible curricula and teaching techniques to prepare students for further education and/or future career choices. The responsibility of educating our students is shared by the parents, faculty, administrators, School Board members, community members, and students themselves. The primary role of the public school is to pass on from generation togeneration the knowledge and cultural values vital to the sustenance of our society.

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of

ive and academically focused environment.

sequences which most students follow, there are exceptions. Movements up and d n and across levels are common. Student motivation and the completion of necessary prerequisites are the major factors in determining which sequence a stud t will follow. Students and parents are advised to consult with the teacher and

uidance counselor in the event that there are any questions regarding course information, prerequisite skills, and appropriate sequences. (Adapted from he Plymouth Regional High School Program of Studies, 2004-2005)

COURSE LEVEL DESCRIPTIONS*

The Mat atics Honors. It is important to note that all three levels are college preparat nd s s Curriculum Frameworks and the NCTM Principl d St ts and parents identify a learnin file Level 2 ses p arning of students who may not e ful eeded in the course. Mathematical concepts tend to be ntrodu at a con eveloped through student investigation with significant

rom the teacher. Students will usually learn to solve problems through repetition of routine problems. Students will receive support from the teacher in tudy y include substantial review of homework and previously covered content. Students

are expe to t the teacher and to seek help when needed. The course is designed to meet the needs of a stud who

Level 3 courses p st of the concepts covered in prior courses, but the c e wi . Mathematical concepts are introduced using a balance of abstract and con te appro ance from the teacher. Students will be expected to solve ro ine prob with teacher support. Students will be expected to use the textbook as a resour occ some review of homework and previously covered content before new material trod eir own learning and seeking help when needed. The course is designe meet

A WORD FROM THE MATHEMATICS/COMPUTER SCIENCE DEPARTMENT

We live in a data driven society that ultimately relies on a person’s ability to reason mathematically. In the world today we are bombarded with vast amountsquantitative information. Therefore, the level of mathematical thinking and problem solving required to analyze and communicate has increased dramatically. In such a world mathematical competence opens doors to productive futures. All students deserve an opportunity to understand the power and beauty of mathematics. The Mathematics Department offers sequences of college preparatory courses for students with varied learning styles and academic interest. It is our goal to provide the means to develop the habits of mind of a mathematician and to think critically. We believe all students can reach high standards of academic achievement though our support The Content Page shows the sequences of courses which students follow through four years at Plymouth Regional High School. Although these represent the

owen

g t

hem Department offers courses at three instructional levels: Level 2, Level 3, andory a hare an essential common core curriculum which is aligned with both the New Hampshire Mathematices an andards for School Mathematics. The level descriptions are not intended to be exclusive but rather are intended to help studeng pro that comes closest to that of the student and to determine which level will most likely meet the student’s learning needs.

cour rogress at a pace that allows for skill development and reinforcement of concepts. The course is designed to support the le hav ly retained the skills and concepts covered in prior courses which will be reviewed when n

ced crete level and developed with an increasing level of abstraction. New ideas are often diguidance fdeveloping s skills and using the textbook as a resource. Classes typicall

cted ake responsibility for their own learning with guidance from ent thrives in a directed learning environment.

rogress at a fast pace. Students are expected to have developed most of the skills and understood moours ll include some review of difficult topics that may not have been fully retained

cre aches. New ideas are often developed through student investigation with moderate guidut lems independently and solve open-ended and .non-routine problems

e ce and asionally to learn new material independently. Typical classes includis in uced. Students are expected to be self-motivated, taking responsibility for th

d to the needs of a student who thrives in a guided learning environment.

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83

Honors cour pected to have mastered the skills and thorough nder e, which will generally not be reviewed in the course. hem cal level. New ideas are often developed through student investigation with

inimal guidance from the teacher. Students will be expected to apply their knowledge to open-ended and non-routine problems. Students will sometimes be aterial by reading the textbook and/or solving problems on their own. Typical classes include minimal review of homework and previously

Students are expected to be highly self-motivated, taking the fullest responsibility for their own learning and seeking help when needed. The course i igne onment. (* Please r to the Ply e.)

Process Any st nt t

Solve

ituations. (1a, 1b, 2b, 4c, 5a, 6a) -solving mode. (1b, 2b)

Reason

A. Write a word problem given an equation. (1b, 6b) B. Translate verbal sentences into equations or formulas. (2b) C. Use models, known facts, properties and relationships to explain their thinking. (6b) D. Continue a pattern involving algebraic expressions. (1b, 6a, 2a) E. Use elementary deductive and inductive reasoning to solve problems. (1b) F. Identify one or more strategies for solving a problem. (1a) G. Use algebraic equations to express relationships. (1b, 6b) H. Make and defend conjectures with appropriate arguments. (1b) I. Compare and contrast strategies used to arrive at estimates, discussing advantage to each technique. (1b) J. Reflect and respond verbally and in writing to “what if…” questions. (2a, 6b, 5a) K. Determine the reasonableness of answers. (1a, 3d)

/AP ses progress at a very fast pace covering the greatest breadth and depth of topics. Students are exly stood the concepts covered in prior courses. They are expected to have retained this past knowledguMat atical concepts are often introduced at an abstract and theoreti

mexpected to learn movered material. c

s des d to meet the needs of a student who thrives in a more independent learning envir

efer mouth Regional High School Program of Studies, 2004-2005, for more details. Adopted from NCTM Standards and Lexington High School, MA Mathematics Department Webpag

ude aking a mathematics course at Plymouth Regional High School will… Problem

A. Formulate and solve real world problems using strategies. (1a) B. Define a variable and write an equation given a problem-solving situation. (1b, 6b) C. Write an equation to represent a relation, given a chart of values or a geometric representation. (1b, 6b) D. Generalize solutions and apply problem-solving strategies to new sE. Investigate new mathematics using previously learned knowledge in a problemF. Keep a journal of problems solving strategies, tips and techniques. (2a, 2b)

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84

ommunicate

A. Explore problem situations by asking and answering questions. (2a) B. Explain and justify thought processes for the solution(s) of a given problem in a variety of settings such as in cooperative

groups. (2a, 6b) C. Draw diagrams or use objects to illustrate an understanding of mathematical concepts. (2a)

b, 3b, 5a, 6a, 6d, 7a) -ROM, CBL, and laser disk). (2a,

tween the four operations and their use with all rational numbers. (2b, 3b)

C

D. Represent mathematical concepts through tables, charts, graphs and mathematical symbolism. (2a, 2nd by a variety of technologies (calculators, computers, video, CDE. Explain solutions fou

6b) F. Articulate the thought processes used in solving any problem. (2a, 6b) G. Share conjectures about possible relationships given a set of data or a pattern. (2a, 6b)

Connect

A. Use a graphing calculator to verify solutions. (1a, 6b) B. Discover relationships and patterns within the set of real numbers and algebraic expressions. (2b, 6a, 6b, 7a) C. Apply mathematical topics in a multidisciplinary or interdisciplinary setting. (1a, 2b, 6a, 6b) D. Apply mathematics to real world situations. (1a, 2b, 5a) E. Connect operations with real numbers to algebraic expressions. (2b, 6b)

. Explore the relationships beFG. Discover and explore the relation between an algebraic expression and its geometric representation. (2b, 4a, 6b)

____________________________________________________

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85

ROGRAM)

y linear functions, equations and graphs that model problem situations. The students atics and the connections between Algebra and Geometry. To do this, they will discover

ators and computer software will assist the explorations in this program.

ctive and deductive approaches to

tures. Students in Geometry and Honors Geometry will also be ir algebraic knowledge into the geometric realm by writing, applying, and using

etric and algebraic skills will be applied to real world problems.

LINEAR ALGEBRA (ONE YEAR PROGRAM) - LINEAR ALGEBRA A & B (TWO YEAR PINTRODUCTORY STATEMENT The focus in Linear Algebra is to explore and applwill investigate statistics, discrete mathempatterns, formulate expressions and solve equations involving algebraic terminology. Students will extend their geometric knowledge into the algebraic realm by writing, applying and graphing linear equations and inequalities. To complement this focus, students will translate collected and given data into algebraic statements in order to make predictions and arrive at conclusions. Whenever possible, Algebraic skills will be applied to real-world situations. Graphing calcul

This integrated Algebra program will complement the reasoning skills of the students by using indumake and validate conjectures about possible relationships.

A student who successfully completes Linear Algebra will be prepared for Geometry the following year. A student who successfully completes Linear Algebra A will be prepared for Basic Geometry the following year. These students will complete Linear Algebra B during their third year. BASIC GEOMETRY - GEOMETRY - HONORS GEOMETRY INTRODUCTORY STATEMENT The focus of Geometry is to explore and apply formal geometric topics. The students will investigate constructions, inductive and deductive reasoning, triangles, polygons, congruency, similarity, circles, areas, volumes, and the connections between Algebra and Geometry. To do this, they will discover patterns, formulate conjecexpected to prove theorems. Students will extend thecoordinate geometry. Whenever possible, geom A student who successfully completes Basic Geometry will be prepared for Linear Algebra B. A student who successfully completes Geometry will be prepared for Non-linear Algebra the following year. A student who successfully completes Honors Geometry will be prepared for Pre-Calculus Honors the following year. FINITE MATHEMATICS INTRODUCTORY STATEMENT

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is

iscrete mathematics and math of finance. As time allows, trigonometry will be introduced.

inite Mathematics is a senior mathematics course offered to students who have completed Non-linear Algebra and Geometry.

ALCULUS INTRODUCTORY STATEMENT

he focus of Calculus is to expose students to elementary calculus topics so that they will be more comfortable and successful in a college alculus course. Students will study limits and their properties, differentiation, applications of differentiation, integration, logarithmic and

applications of definite integration.

be taken concurrently with Statistics or Finite Mathematics.

LCULUS HONORS INTRODUCTORY STATEMENT

culus (Honors) combines knowledge of Linear Algebra, Non-Linear Algebra and Geometry. The focus will be to metric, exponential and logarithmic functions. For each of the

functio tud tions, the concepts of domain and range, the inverse rela s dents will study Greek and modern trigonometry and the

a

while taking Finite Mathematics or Statistics. A junior who successfully completes Pre-Ca us w

Resou Roo The go f th repare special needs students for future Math courses and/or technology courses in which mu ugh the goals and objectives of the students' IEPs dictate much of what is aught, cur orm calculations with whole and rational numbers and to think

ally. Students will use hands-on materials, as much as possible, to help advance their thinking from a concrete to a more l. dictions and to make and apply generalizations. Students

w rie Students who near Algebra A or one of several technology courses.

The focus of Finite Mathematics is to teach the mathematics covered in business and the social sciences in colleges and universities. Thourse applies the mathematics used in these fields. There are four independent areas of study: linear algebra, probability and statistics, c

d F C Tcexponential functions, and Calculus is a senior math course that may

PRE-CALCULUS - PRE-CA The study of Pre-Calexplore bas the ic linear, quadratic, absolute value, polynomial, trigono

ns s ents will investigate graphs, the inter-related patterns of transformation, and olve pertinent applications by a variety of methods. In addition, stu

th of finance. A graphing calculator is essential for this course. m A senior taking Pre-Calculus may do so concurrently

lcul ill be prepared to take either of the above courses or Calculus.

m) rce m Math (one or two-year progra

al o e Resource Room Math program is to pthey st apply mathematical concepts and skills. Altho the riculum is focused on improving their skills to perft

mathematicabstract leve They will discover patterns and then use them to make and test pre

solve real-world problems. ill use a va ty of materials and strategies to

successfully complete Resource Room Math will be prepared for Li

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87

LINEAR ALGEBRA A Mathematics 9th Grade Level 2

Worki th ode of a set of data. (5a)

Rates, os,

r, fraction, decimal, percent). (3a) gnitude of rational numbers. (3a)

Op it

B. Evaluate expressions with integers. (3b, 3c)

A student in this course will…

ng wi Data d mA. Find the mean, median, an

B. Read, interpret and construct bar, circle, and line graphs. (5a, 8a) ze and represent data. (6a) C. Use charts, tables and graphs to organi

Rati and Proportion 3a) A. Write fractions in lowest terms. (

B. Represent any given number in any form (whole numberence in maC. Recognize and demonstrate the diffe

D. Write a ratio in lowest terms. (3a) E. Calculate a unit rate. (4c, 7a) F. Write and solve proportions. (4c, 7a) G. Solve percent problems using proportions. (7a)

Geometry

A. Make predictions based on patterns in geometric figures. (7a, 8a) B. Define and identify polygons. C. Calculate the perimeters and areas of polygons. (4c)

and center. D. Identify parts of a circle: radius, diameter, chord, arc, E. Calculate the area and circumference of a circle. (4c)

erations w h Rational Numbers A. Add, subtract, multiply, and divide integers. (3b)

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88

C. Evaluate numerical expressions containing scientific notations. (3c)

INEAR ALGEBRA A (Continued)

student in this course will…

d inequalities to analyze meaningful data. (6a) Recognize algebraic patterns. (7a, 8a)

ograms. (8a) ons, including the use of exponents. (3b, 3c, 6b)

3c)

ariables. (6a, 7a)

ions. (6a) analyze real world data. (5a)

gebraically. (6b)

ng tables and equations. (3b, 6a) he domain and range of a function. (3b, 6a)

ns. (4a, 5a)

L A Using Algebra to Work with Data

A. Use variables anB. C. Interpret histD. Simplify an expression using the order of operati

ber. E. Find the absolute value of a numF. Add, subtract, multiply, and divide integers. (3b, G. Solve problems involving negative numbers. H. Write variable expressions. (6b) I. Solve real-world problems by using vJ. Simplify variable expressions. (6b) K. Use the distributive property. (3b)

situatL. Apply variable expressions in problem-solving M. Use graphing calculator technology to

Equatio and ns Functions

nipulative and algebraically. (6b) A. Write and solve one-step equations with maB. Write and solve two-step equations with manipulatives and al

) C. Solve real-world problems using equations. (6b, 1aD. Recognize and describe functions usiE. Identify tF. Apply functions to predict outcomes in order to make decisioG. Read and/or create coordinate graphs. (2a)

(1a, 5a,H. Graph equations using a table and/or graphing calculator.

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89

INEAR ALGEBRA A (Continued)

student in this course will…

rds and graphs. (6a, 7a) tes. (6a)

cribe direct variation. (4a, 6a) Explore relationships between real-world variables. (6a, 6b, 7a)

of a line. (4a, 7a)

Conne Alcount. (7a)

Worki ith

rs. (3a) nteger square roots. (4c)

Linear ebr

L A Graphing Linear Functions

A. Find unit rates from woB. Compare real-world raC. Recognize and desD. E. Find the slope F. Analyze real-world graphs. (4a, 7a) G. Graph equations using the slope and y-intercept. (4a, 6b) H. Write linear equations in slope-intercept form. (4a)

b) I. Solve a system of equations by graphing. (6

gctin gebra and Geometry A. Use proportions to estimate quantities that are difficult to

graphs. (4b) B. Use scales in drawings and on C. Use scale factors to compare sizes of objects and drawings.

D. Translate figures on the coordinate plane. (4a, 4b)

ng w Radicals A. Identify right triangles using the Pythagorean theorem. (4c)

solve real-world problems. (4c) B. Use the Pythagorean theorem toC. Identify and define rational and irrational numbe

-iD. Use a calculator with formulas resulting in non

Alg a A PRHS icatInd ors 1d, 1e, 1f, 2a, 2b, 2c, 2d, 2e, 3a, 3d, 3e, 3f, 4a, 4c, 4e, 4f, 4g, 4h, 5a, 5b, 5c, 5d, 6d, 6g, 7a, 7c, 8c, 8f, 8g

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LINEAR ALGEBRA

Mathe atics 9

90

m th Grade Level 3

d multi-step equations. C-P-A properties of equality and properties of zero and one. (formalized by using variables). P-A

butive property, including reversing the operation (formalize using variables). C-P-A Find the absolute value of a number. C-P-A

olute value. I-C-P-A

s. I-C-P-A ls. I-C-P-A

erations on polynomials. I-C-P-A

I-C-P-A

Determine if lines are parallel, perpendicular or neither. I-C-P-A

oots. I-C-P-A

The student in this course will… Computation and Operations

A. Solve open sentences anB. Explore and applyC. Explore and apply the distriD. E. Solve equations and inequalities involving absF. Solve inequalities. I-C-P-A G. Simplify expressions that contain rational numbers. P-A H. Multiply monomials. C-P-A

lI. Simplify expressions involving powers of monomiaJ. Simplify expressions involving quotients of monomiaK. Explore, design and apply algorithms to perform and record the four op

C-P-A L. Find the prime factorizations of a monomial. I-M. Find the Greatest Common Factor for a set of monomials.N. Find the slope of a line. C-P-A O. Find the x- and y- intercepts of a line. I-C-P-A P. Find the midpoint of a line segment. I-C-P-A Q. R. Write an equation of a line. I-C-P-A S. Solve systems of equations by graphing, substitution and/or elimination. I-C-P-A T. Simplify square roots. I-C-P-A U. Perform addition, subtraction, multiplication, and division with square rV. Recognize and apply direct and inver ariase v tion. I-C-P-A

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Mathematics 9

LINEAR ALGEBRA (Continued)

th Grade Level 3

mber lines. I-C-P-A uations on a coordinate plane. C-P-A

given relation is a function. I-C-P-A ualities on a coordinate plane. I-C-P-A

g. I-C-P-A ing. I-C-P-A

techniques to business situations. I-C-P-A

bers. I-C-P Numb

l/empty set and elements in sets. I-C-P-A o or more distinct variables, solve for one of them). I-C-P-A

the domain, range and inverse of a relation. I-C-P-A C-P-A

ce

pare data. P-A -A

lationship between the data and the equation. I-C-P-A

Linear ebr

The student in this course will… Geometry

A. Graph inequalities on nuB. Graph linear eqC. Determine whether a D. Graph ineqE. Solve systems of equations by graphin

graphF. Solve systems of inequalities byming G. Apply linear program

H. Find scale factor given similar figures. C-P-A I. Use scale drawings to find perimeter and area. I-C-P-A

e real-world problem. C-P-A J. Use Pythagorean Theorem to solvK. Use the Pythagorean Theorem to explore irrational num

er Sense A. Understand and apply set notation, subsets, nB. Solve literal equations (in an equation with tw

ul

C. Identify D. Show relations as sets of ordered pairs, mappings and charts. I-

IV. Data Analysis and ChanA. ms. C-P-A Analyze real-world data using histograB. Use measures of central tendency to comC. Use the line of best fit to derive equations. I-C-PD. Use the correlation coefficient to establish the re

Al ag

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PRHS icat

NON-LINE

Mathematics 9

Ind ors 1d, 1e, 1f, 2a, 2b, 2c, 2d, 2e, 3a, 3d, 3e, 3f, 4a, 4c, 4e, 4f, 4g, 4h, 5a, 5b, 5c, 5d, 6d, 6g, 7a, 7c, 8c, 8f, 8g

AR ALGEBRA HONORS*

th Grade Honors

A student in this course will…

hs and tables. B. Use functions to model growth.

rices. D. Model a situation with a simulation and make predictions.

and use direct variation to analyze data and make predictions.

Expon l F

rational exponents. ations.

E.

Logari c F

ctions. E. Understand logarithmic scales.

Linear Algebra Overview

A. Model growth with grap

C. Model with mat

E. Write F. Write and graph linear equations.

on notation. G. Write and apply point-slope form and functiH. Fit lines to data and make predictions. I. Recognize and interpret correlation coefficient. J. Write and graph linear parametric equations.

entia unctions A. Evaluate expressions that use negative and

B. Determine the doubling time or the half-life in real world situions. C. Draw graphs of exponential funct

D. Organize information and classify data. Write exponential functions that fit sets of data.

thmi unctions

tions. A. Find inverses of exponential funcB. Evaluate logarithmic equations.

linear functions. C. Solve problems using inverses of D. Graph and find equations for inverses of linear fun

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93

perties of logarithms. rse function.

ON-LINEAR ALGEBRA HONORS*(Continued)

Mathematics 9

F. Use proG. Restrict the domain of a function to obtain an inveH. Organize information and classify data.

Nth Grade Honors

A student in this course will…

Solve quadratic equations by using the graphing calculator to locate roots or zeroes. B. Write quadratic equations in intercept form.

ize or minimize quadratic functions. ite quadratic functions in vertex form.

ula. tions an equation has.

a, h and k effect the graph. m.

Data In tiga

B. Choose a representative sample. rd deviation of a data set.

ple proportion. System

Quadratic Functions

A.

C. MaximD. Complete the square to wrE. Solve equations using the quadratic formF. Use the discriminant to determine the nature and number of soluG. Factor quadratic expressions. H. Solve quadratic equations using factoring. I. Interpret how different values ofJ. Graph equations in a given forK. Organize information and classify data.

ves tions A. Organize information and classify data.

C. Find and interpret the range, interquartile and standaD. Determine the variability of data. E. Find the margin of error for a sam

s of Equations A. Solve systems of linear equations using matrices. B. Use technology to find points of intersection of graphs.

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94

bles.

nize information and classify data.

ON- E nued)

athematics 9

C. Write and graph inequalities with 2 variaD. Graph a system of linear inequalities. E. Orga

N LIN AR ALGEBRA HONORS* (Conti M th Grade Honors

A student in this course will… Radical Functions

A. Evaluate radical expressions. with radical expressions. ultiply and divide complex numbers.

complex solutions to equations that have no real solutions. mber. .

longs. ld for a set and an operation.

rganize information and classify data.

Polyno

al functions. gnize graphs of polynomial functions and describe their important features.

f cubic functions.

B. Solve equationsC. Add, subtract, mD. Find E. Calculate the magnitude of a complex nu

unctionsF. Graph and evaluate square root fG. Graph radical functions.

plex plane. H. Plot complex numbers in the comI. Identify the number systems to which a number beJ. Evaluate whether group properties hoK. O

mial Functions A. Recognize, evaluate, add and subtract polynomials. B. Multiply and divide polynomials. C. Solve cubic equations. D. Find zeroes of higher-degree polynomiE. RecoF. Find equations for graphs of cubic functions and find zeroes o

olynomials. G. Understand numeration systems using p H. Organize information and classify data.

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95

Ration unc

C. Identify important features of graphs of rational functions. rmation and classify data.

NON-LINEAR ALGEBRA HONORS* (Continued) Mathematics 9

al F tions A. Solve rational equations. B. Identify important features and find equations of translated hyperbolas.

D. Organize info

th Grade Honors

A student in this course will…

sation and classify data.

ple. nd interpret the range, interquartile and standard deviation of a data set.

Di h

ze situations involving direction.

fy data. Analyt eom

*Write and graph equations of a hyperbola.

Data Investigation

A. Organize informB. Choose a representative samC. Find aD. Determine the variability of data. E. Find the margin of error for a sample population.

screte Mat A. *Solve problems that require sorting item into groups. B. *AnalyC. *Count possibilities in situations (combinations and permutations).

tions to Pascal’s Triangle. D. *Apply the concept of combinaE. *Organize information and classi

ic G etry en two points on a coordinate plane. A. *Find the distance betwe

B. *Find focus and directrix of a parabola. C. *Write and graph equations of a circle. D. *Write and graph equations of an ellipse. E. F. *Find conics by taking a cross section of a double cone.

e used in applications. G. *Describe how a conic section can b

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Nonlin Alg

96

ear ebra/Honors PRHS icat

8g

asic Geometry

Mathematics 10

Ind ors 1a, 1d, 1e, 1f, 1g, 1h, 2a, 2b, 2c, 2d, 2e, 3a, 3d, 3e, 3f, 4a, 4c, 4e, 4f, 4g, 4h, 5a, 5b, 5c, 5d, 6d, 6g, 7a, 7b, 7c, 8c, 8f,

B

th Grade Level 2

g using inductive reasoning. (1a, 1b, 4a, 5a, 6a, 6b, 7a)

in a series. (1b, 5a, 6a, 6b, 7a) neralize basic number patterns to find the nth term in a series. (1b, 2b, 6a, 6b, 7a)

(1b, 6a, 6b, 7a) Introdu g ge

4a, 6a, 6b) s.(4a, 6a,b)

tions (daisy designs, line segments, and angles).

A student in this course will… nductive reasonin I

A. Identify patternsB. Find the next termC. GeD. Apply inductive reasoning to finding patterns in geometric shapes.

cin ometry A. Explore the attributes that can be used to sort a set of objects. (1b, 4a) B. Create definitions. (1b, 2a, 4a) C. Read and understand definitions. D. Communicate through the use of proper notation. (2a) E. Identify congruent and similar objects. (1b, 4a) F. Identify symmetry. (1a)

G. Use midpoint and slope formula to connect to algebra. (2a,H. Use coordinate and non-coordinate geometry to solve problem

Constructions

A. Use a straightedge and compass to create basic construcB. Define sketch, draw, and construct.C. Construct bisectors, midpoints, perpendiculars, and parallels. (4a) D. Construct triangles and their medians, altitudes and angle bisectors. (4a)

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97

ecial properties of each point). (4a)

asic Geometry (Continued)

Mathematics 10

E. Construct incenter, circumcenter, orthocenter, and centroid (include the spF. Construct a nine-point circle. (optional) (4a) G. Construct all types of quadrilaterals. (4a)

B

th Grade Level 2

ationship between special pairs of angles. (1b, 4a) mula for the midpoint of a segment in a coordinate plane. (1b)

ver the relationship between intersecting and non-intersecting lines, given a graph or system (perpendicular, oblique,

cally and algebraically. (1b, 2a, 2b, 4a, 6a)

riang op

Identify the types of triangles using inductive reasoning. (1b, 4a)

s. (1a, 1b) by SSS, SAS, ASA, AAS, and HL. (1a, 1b, 4a, 4b)

. (1a, 1b, 4a, 4b)

A student in this course will… Lines and angles

A. Discover the relB. Discover the forC. Disco

parallel, and skew). (1b, 2b, 4a, 6a, 6b) D. Discover the slope of a line in a coordinate plane. (1b, 4a)

a line. (1b, 2b, 4a, 6a, 6b) E. Discover the slope intercept form of the equation ofhiF. Discover how to find the intersection of lines grap

G. Identify angles formed by a two lines and a transversal. (1b, 4a) H. Discover the properties of parallel lines. (1b, 6a, 6b) I. Use the properties of parallel lines to find angle measures.

T le pr erties A. B. Identify congruent triangles. (4a) C. Measure angles of a triangle. (4c)

of a triangle to solve problemD. Use the measure of the angles E. Verify that triangles are congruent F. Verify that triangles are not congruent by AAA and SSA

solve problems. (4b) G. Use the congruence conjectures to

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le. (4a)

s. (4b)

roperties of midsegments of a triangle in a coordinate plane. (4a, 4b)

Bas eo

ics

98

H. Discover the properties of the isosceles triangI. Discover the properties of the right triangle. (4a)

and right triangle to solve problemJ. Use the properties of the isosceles triangleK. Identify and construct the midsegments of a triangle. (4a) L. Use the properties of midsegments of triangles to solve problems. (4b) M. Verify the pN. Order the measures of the sides and angles of a triangle. O. Use the triangle inequality to solve problems. (3a, 3c, 4b)

ic G metry (Continued)

Mathemat 10th Grade Level 2 A stude n th

Polygo rop

B. Find the measures of the interior and exterior angles of the polygons. (1b, 3c, 6a, 6b, 7a) , 4a)

eral is a parallelogram. (1b, 4a, 6a, 6b) F. special parallelograms. (1b, 4a) G. ties of trapezoids (including the midsegment), isosceles trapezoids, and kites. (1b, 4a) H. ies of quadrilaterals to solve geometric and real life problems. (1a, 2b, 4b)

Similarity

A. numbers. B. ve problems. (4c, 6a, 6b, 7a)

nt i is course will…

ns p erties A. Identify, name, and classify polygons. (1b, 4a)

C. Discover the properties of parallelograms. (1bD. Prove quadrilaterals are parallelograms. (1b, 4a) E. o prove a quadrilatUse coordinate geometry t

rties of Discover the propeproperDiscover the

Use the propert

Compute the ratio of twoUse proportions to sol

C. Use properties of proportions. D. Use a problem-solving plan. (1b) E. Measure corresponding parts of similar polygons. (4c) F. Identify similar triangles. (4a, 4c) G. Use similar triangles in a coordinate plane. H. Verify that triangles are similar by AAA, SSS, and SAS. (4a)

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99

I. Verify the proportionality theorems. (4a) ilarity and proportionality conjectures to solve problems in geometry and real life. (1a, 2b, 3d, 6a, 6b, 7a)

Right t gles

(2a, 2b, 6a, 7a)

2a, 2b, 3d, 4b, 4c)

H. Investigate indirect measurement using right triangles and similar triangles. (1a, 3b, 3d, 4a, 4c, 4d)

a, 4a) iscover the properties of the angles formed by tangents, chords, and secants. (1a, 4a)

Use the properties of circles to solve problems in geometry and real life. (2b, 4c)

er of a polygon. (3d, 4c, 7a) Find the area of a polygon.(3a, 3c, 3d, 4c, 4d, 7a)

uare and rectangle.

ular polygon.

le. (4c, 7a) , 4c, 7a)

ygons, circles, and similar polygons to solve problems in geometry and

robability. (5a)

J. Use the sim rian properties A. Verify the Pythagorean Theorem and its converse. (6a)

(3a) B. Find the lengths of the sides of special right triangles. n Theorem.C. Solve a right triangle using the Pythagorea

D. Identify a triangle as acute, right, or obtuse. (3a, 4a) real life. (E. Use the properties of right triangles to solve problems in geometry and

F. Find the sine, cosine, and tangent of an acute angle. (3d, 4d) G. Use the basic trigonometric ratios to solve problems in geometry and real life. (4c, 4d, 7a)

Circles A. Use vocabulary associated with circles. B. Discover the properties of tangents, secants, chords, central angles, inscribed angles, and arcs. (1C. DD.

Areas

A. Find the perimetB.

1. Sq2. Parallelogram. 3. Triangle. 4. Trapezoid. 5. Quadrilateral with perpendicular diagonals. 6. Equilateral triangle. 7. Reg8. Similar polygons.

ence of a circle. (4c, 7a) C. Find the circumferD. Find the length of an arc of a circE. Find the area of circles and regions of circles. (3b

area of polF. Use the perimeter, circumference, and b, 3c, 4c, 7a) real life. (1a, 2b, 3

G. Use areas of geometric shapes to calculate geometric p

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(1a, 4a) a)

, pyramids, cones, and spheres. (4c, 4d, 7a) ilar solids. (4c, 4d, 6a, 6b, 7a)

e surface area and volume of solids and similar solids to solve problems in geometry and real life. (1a, 2b, 3b, 3c, 3d,

Basic met

, 4f, 4g, 4h; 5 a, 5d, 5h; 6 d, 6g; 7 a, 7c; 8 c, 8f, 8g.

EOMETRY

100

Surface ea an ar d volume

. (1a, 4a) A. Identify solids that are polyhedronsB. Identify prisms, cylinders, pyramids, cones, and spheres. C. Find lateral area and surface area of prisms and cylinders. (4c, 4d, 7D. Find the surface area of pyramids and cones. (4c, 4d)

here. (4c, 7a) E. Find the surface area of a spF. Find the volume of prisms, cylindersG. Find the surface area and volume of simH. Use th

7a)

Geo ry: PRHS Indicators 1 a, 1d, 1e, 1f; 2 a, 2b, 2c, 2d, 2e; 3 d, 3e, 3f; 4 a, 4c

G

Mathematics 10th Grade Level 3 A student in this course will… nductive reasoning I

A. Identify patterns using inductive reasoning

ng patterns in geometric shapes Introdu ge

fy congruent and similar objects

dinate geometry to solve problems

B. Finding the next term in a series C. Generalize basic number patterns to find the nth term in a series D. Apply inductive reasoning to findi

cing ometry sort a set of objectsA. Explore the attributes that can be used to

B. Create definitions C. Read and understand definitions D. Communicate through the use of proper notation E. IdentiF. Identify symmetry

ula to connect to algebra G. Use midpoint and slope formH. Use coordinate and non-coor

Page 101: Math Curriculum K-12

signs, line segments, and angles)

d centroid (include the special properties of each point)

GEOMETRY (Continued)

101

Constructions ions (daisy deA. Use a straightedge and compass to create basic construct

B. Define sketch, draw, and construct C. Construct bisectors, midpoints, perpendiculars, parallels

and angle bisectors D. Construct triangles and their medians, altitudes r, anE. Construct incenter, circumcenter, orthocente

F. Nine point circle (optional) G. Construct all types of quadrilaterals

Deduct reasive oning

A. Define deductive reasoning B. Practice reasoning skills

s C. Discover the relationship between special pairs of angle

Mathematics 10th Grade Level 3 A student in this co rse will… u

Discover the relationship between intersecting and non-intersecting lines, given a graph or system (perpendicular, oblique,

a transversal

ongru tria

entify the types of triangles using inductive reasoning

Lines and angles

A. parallel, and skew)

B. Solve systems of linear equations C. Find the equation of a line

d D. Identify angles formed by a two lines anE. Discover the properties of parallel lines

easures F. Use the properties of parallel lines to find angle mG. Introduce the formal use of proofs to verify the parallel line properties

C ent ngles A. IdB. Identify congruent triangles C. Measure angles of a triangle D. Use the measure of the angles of a triangle to solve problems

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102

, ASA, AAS, and HL and SSA

eorems to solve problems e congruence postulates and theorems

Discover the properties of the isosceles triangle triangle riangle and right triangle to solve problems

Properties of t

le s

iangle in a coordinate plane

GEOMETRY (Continued)

E. Verify that triangles are congruent by SSS, SAS by AAA F. Verify that triangles are not congruent

G. Use the congruence postulates and thH. Plan and complete a proof using thI. J. Discover the properties of the right K. Use the properties of the isosceles t

riangles A. Identify and construct the midsegments of a triang

segments of triangles to solve problemB. Use the properties of midC. Verify the properties of midsegments of trD. Order the measures of the sides and angles of a triangle

roblems E. Use the triangle inequality to solve p

Mathematics 10th Grade Level 3 A student in this course will…

gons

parallelogram

ezoids, and kites

ransf ations

A. Identify the three basic rigid transformations ns, and translations

Polygons

A. Identify, name, and classify polyB. Find the measures of the angles of the polygons C. Discover the properties of parallelograms D. Prove quadrilaterals are parallelograms

ateral is a E. Use coordinate geometry to prove a quadrilF. Discover the properties of special parallelograms

midsegment), isosceles trapG. Discover the properties of trapezoids (including theH. Use the properties of quadrilaterals to solve geometric and real life problems

T orm

B. Use the properties of reflections, rotatioC. Relate transformations to line symmetry and point symmetry

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103

Simumbers

B. e pr lems C. e pr ortions D. e a p lving plan E. asu ding parts of similar polygons F. ntifG. e si ordinate plan H. rify imilar by AAA, SSS, and SAS I. rify theorems

ity and proportionality theorems y theorem to solve problems in geometry and real life

GEOMET Continued)

D. Use transformations to solve real life problems E. Use the properties of glide reflections F. Use composition of transformations

ilarities

A. Compute the ratio of two nUs oportions to solve probUs operties of propUs roblem soMe re corresponIde y similar triangles Us milar triangles in a coVe that triangles are sVe the proportionality

J. Complete proofs involving similarK. Use the similarity and proportionalit

RY ( Mathematics 10th Grade Level 3 A student in this course will… Right triangles

A. Prove right triangles congruent he Pythagorean theorem and its converse

al right triangles

roblems in geometry and real life of an acute angle

real life les or similar triangles

ircles

A. Use vocabulary associated with circles Discover the properties of tangents, secants, chords, central angles, inscribed angles, and arcs

B. Prove tC. Find the lengths of the sides of speciD. Solve a right triangle using the Pythagorean theorem E. Identify a triangle as acute, right, or obtuse F. Use the properties of right triangles to solve pG. Find the sine, cosine, and tangentH. Use the basic trigonometric ratios to solve problems in geometry andI. Investigate indirect measurement using right triang

C

B.

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104

cover the properties of the angles formed by tangents, chords, and secants

lanar Measurements A. Find the perimeter of a polygon B. Find the area of a polygon

1. Square and rectangle 2. Parallelogram 3. Triangle 4. Trapezoid 5. Quadrilateral with perpendicular diagonals 6. Equilateral triangle 7. Regular polygon 8. Similar polygons

C. Find the circumference of a circle D. Find the length of an arc of a circle E. Find the area of circles and regions of circles F. Use the perimeter, circumference, area of ilar polygons to solve problems in geometry and real

life GEOMETRY (Continued)

Mathematics 10

C. DisD. Use the properties of circles to solve problems in geometry and real life

P

polygons, circles, and sim

th Grade Level 3

A student in this course will…

Space Measurements Identify solids that are polyhedrons

and spheres ea of prisms and cylinders

F. Find the volume of prisms, cylinders, pyramids, cones, and spheres ace area and volume of similar solids

solve problems in geometry and real life Geome

A. B. Identify prisms, cylinders, pyramids, cones, C. Find lateral area and surface arD. Find the surface area of pyramids and cones E. Find the surface area of a sphere

G. Find the surfH. Use the surface area and volume of solids and similar solids to

try

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105

PRHS icat 1d f; 4 a, 4c, 4f, 4g, 4h; 5 a, 5b, 5c, 5d, 5h; 6 d, 6g; a, c; 8 c, 8f, 8g.

eometry Honors*

Ind ors , , 2d, 2e; 3d, 3e, 31 a , 1e, 1f; 2 a, 2b, 2c

G

Mathematics 10th Grade Honors

A student in this course will… Inductive reasoning

A. Identify patterns using inductive reasoning. d the next term in a series.

Introdu g ge

objects.

B. FinC. Generalize basic number patterns to find the nth term in a series. D. Apply inductive reasoning to finding patterns in geometric shapes.

cin ometry A. Explore the attributes that can be used to sort a set of B. Create definitions. C. Read and understand definitions. D. Communicate through the use of proper notation.

Page 106: Math Curriculum K-12

fy congruent and similar objects.

inate geometry to solve problems.

igns, line segments, and angles).

d centroid (include the special properties of each point).

Geometry Honors* (Continued)

106

E. IdentiF. Identify symmetry.

ula to connect to algebra. G. Use midpoint and slope formH. Use coordinate and non-coord

Constructions ns (daisy desA. Use a straightedge and compass to create basic constructio

B. Define sketch, draw, and construct. C. Construct bisectors, midpoints, perpendiculars, and parallels.

nd angle bisectors. D. Construct triangles and their medians, altitudes ar, anE. Construct incenter, circumcenter, orthocente

F. Construct the nine-point circle (optional). G. Construct all types of quadrilaterals.

Deduct reasive oning

A. Define deductive reasoning. B. Practice reasoning skills.

. C. Discover the relationship between special pairs of angles

Mathematics 10th Grade Honors

Discover the relationship between intersecting and non-intersecting lines, given a graph or system (perpendicular, oblique,

ransversal.

A student in this course will…

Lines and angles A.

parallel, and skew). B. Solve systems of linear equations. C. Find the equation of a line.

a tD. Identify angles formed by two lines and E. Discover the properties of parallel lines.

easures. F. Use the properties of parallel lines to find angle mG. Introduce the formal use of proofs to verify the parallel line properties.

Page 107: Math Curriculum K-12

Congruent triangles entify the types of triangles using inductive reasoning.

and HL. nd SSA.

rems to solve problems. e congruence postulates and theorems.

Discover the properties of the isosceles triangle. riangle. iangle and right triangle to solve problems.

Properties of t

e. s.

iangle in a coordinate plane.

Geometry Honors* (Continued)

107

A. IdB. Identify congruent triangles. C. Measure angles of a triangle. D. Use the measure of the angles of a triangle to solve problems.

, ASA, AAS,E. Verify that triangles are congruent by SSS, SASby AAA aF. Verify that triangles are not congruent oG. Use the congruence postulates and the

H. Plan and complete a proof using thI. J. Discover the properties of the right tK. Use the properties of the isosceles tr

riangles A. Identify and construct the midsegments of a triangl

egments of triangles to solve problemB. Use the properties of midsC. Verify the properties of midsegments of a trD. Order the measures of the sides and angles of a triangle.

roblems. E. Use the triangle inequality to solve p

Mathematics 10th Grade Honors A student in this course will…

gons.

arallelogram.

ezoids, and kites.

Polygons

A. Identify, name, and classify polyB. Find the measures of the angles of the polygons. C. Discover the properties of parallelograms. D. Prove quadrilaterals are parallelograms.

teral is a pE. Use coordinate geometry to prove a quadrilaF. Discover the properties of special parallelograms.

idsegment), isosceles trapG. Discover the properties of trapezoids (including the mH. Use the properties of quadrilaterals to solve geometric and real life problems.

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108

ransf ations A. Identify the three basic rigid transformations.

s, and translations.

F. Use compositions of transformations.

Simmbers.

B. e pr ems. C. e pr ortions. D. lving plan. E. s of similar polygons. F. ntifG. e si coordinate plane. H. rify imilar by AAA, SSS, and SAS. I. s.

ty and proportionality theorems. to solve problems in geometry and real life.

Geometry rs* (Continued)

ormT

B. Use the properties of reflections, rotationC. Relate transformations to line symmetry and point symmetry. D. Use transformations to solve real life problems. E. Use the properties of glide reflections.

ilarities

A. Compute the ratio of two nuUs oportions to solve problUs operties of propUse a problem-soMeasure corresponding partIde y similar triangles. Us milar triangles in aVe that triangles are sVerify the proportionality theorem

J. Complete proofs involving similariK. Use the similarity and proportionality theorem

Hono

Mathematics 10th Grade Honors

Right triangles les congruent.

B. Prove the Pythagorean Theorem and its converse. e lengths of the sides of special right triangles.

the Pythagorean Theorem.

in geometry and real life. gle.

problems in geometry and real life.

A student in this course will…

A. Prove right triang

C. Find thD. Solve a right triangle using E. Identify a triangle as acute, right, or obtuse. F. *Use the properties of right triangles to solve problemsG. *Find the sine, cosine, and tangent of an acute anH. *Use the basic trigonometric ratios to solve

Page 109: Math Curriculum K-12

109

gles. Circles

B. Discover the properties of tangents, secants, chords, central angles, inscribed angles, and arcs. Discover the properties of the angles formed by tangents, chords, and secants.

D. Use the properties of circles to solve problems in geometry and real life.

ts

1. Square and rectangle 2. Parallelogram 3. Triangle 4. Trapezoid 5. Quadrilateral with perpendicular diagonals 6. Equilateral triangle 7. Regular polygon 8. Similar polygons

C. Find the circumference of a circle. D. Find the length of an arc of a circle. E. Find the area of circles and regions of circles. F. Use the perimeter, circumference, area of ilar polygons to solve problems in geometry and real

life.

Geometry Honors* (Continued)

I. Investigate indirect measurement using right triangles or similar trian

A. Use vocabulary associated with circles.

C.

nPlanar Measureme

A. Find the perimeter of a polygon. B. Find the area of a polygon.

polygons, circles, and sim

Mathematics 10th Grade Honors A student in this course will… Space Measurements

A. Identify solids that are polyhedrons. ers, pyramids, cones, and spheres. B. Identify prisms, cylind

C. Find lateral area and surface area of prisms and cylinders. Find the surface area of pyramids and cones.

phere. D. E. Find the surface area of a s

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110

F. Find the volume of prisms, cylinders, pyramids, cones, and spheres. e area and volume of similar solids.

volume of solids and similar solids to solve problems in geometry and real life.

G. Find the surfacH. Use the surface area and

Geometry Honors PRHS Indicators c, 4f, 4g, 4h;5 a, 5b, 5c, 5d, 5h; 6 d, 6g; 7 a, 7b, 7c; 8c, 8f, 8g.

LINEAR ALGEBRA B Mathematics Grade 11 Level 2

1 a, 1d, 1e, 1f; 1g; 2a, 2b, 2c, 2d, 2e; 3a, 3d, 3e, 3f; 4 a, 4

A student in this course will… Working with Expressions

A. Simplify an expression using order of operations. B. Evaluate expressions.

Solving Equations

A. Solve one-step equations. B. Solve two-step equations.

Page 111: Math Curriculum K-12

111

A. Find the slope of a line.

C. Write equations of lines. ake predictions of real-world data.

nequalities

two-step equations using reciprocals.

problems.

H. Use inequalities to represent a group of solutions to real-world problems.

System

rm.

or subtracting and by using multiplication.

F. Graph systems of inequalities to model real-world situations when there are restrictions.

Linear Equations and Graphs

B. Graph linear equations.

D. Model linear data to m

Solving Equations and IA. Model situations with tables and graphs. B. Solve one-step andC. Solve problems using fractions. D. Solve multi-step equations. E. Use multi-step equations to solve real-worldF. Solve equations that involve more than one fraction or decimal. G. Use inequalities to represent intervals on a graph.

I. Solve inequalities.

s of Equations and Inequalities oA. Write and graph equations in standard f

les. B. Solve problems with two variabC. Solve systems of equations by addingD. Graph linear inequalities. E. Graph systems of inequalities.

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112

Continued) LINEAR ALGEBRA B ( Mathematics Grade 11 Level 2 A student in this course will…

al numbers.

dicals. s.

Quadra Fun ows.)

ze characteristics of parabolas.

Working with Radicals A. Identify and define rational and irrationB. Simplify radicals. C. Add, subtract, multiply, and divide raD. Use skills with radicals to find values in real-world problemE. Find products of monomials and binomials.

nomials to find areas of complex figures. F. Use products of monomials and biG. Recognize and find special products of binomials.

tic ctions (This section will be addressed as time all A. Analyze the shape of a graph. B. Decide whether a relationship is linear. C. RecogniD. Predict the shape of a parabola. E. Recognize side views of real objects.

mple quadratic equations. F. Use square roots and graphs to solve si

Linear ebrAlg a B PRHS icat

Ind ors

1d, 1e, 1f, 2a, 2b, 2c, 2d, 2e, 3a, 3d, 3e, 3f, 4a, 4c, 4e, 4f, 4g, 4h, 5a, 5b, 5c, 5d, 6d, 6g, 7a, 7c, 8c, 8f, 8g

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113

ON-LINEAR ALGEBRA N

Mathematics Grade 11 Level 3

A student in this course will… Linear Algebra Overview

A. Model growth with graphs and tables.

e predictions. ns.

ient.

Expon l F

negative and rational exponents. mine the doubling time or the half-life in real world situations.

Logari c F

e logarithmic equations.

.

in an inverse function.

B. Use functions to model growth. C. Model with matrices. D. Model a situation with a simulation and makE. Write and use direct variation to analyze data and make predictioF. Write and graph linear equations. G. Write and apply point-slope form and function notation.

dictions. H. Fit lines to data and make preI. Recognize and interpret correlation coefficJ. Write and graph linear parametric equations.

entia unctions A. Evaluate expressions that use B. DeterC. Draw graphs of exponential functions.

y data. D. Organize information and classifE. Write exponential functions that fit sets of data.

thm unctions A. Find inverses of exponential functions.

i

B. EvaluatC. Solve problems using inverses of linear functions.D. Graph and find equations for inverses of linear functionsE. Understand logarithmic scales. F. Use properties of logarithms. G. Restrict the domain of a function to obtaH. Organize information and classify data.

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114

ON-LINEAR ALGEBRA (Continued) N Mathematics Grade 11 Level 3

A student in this course will… Quadratic Functions

A. Solve quadratic equations by using the graphing calculator to locate roots or zeroes. ntercept form.

. rtex form.

er of solutions an equation has.

fect the graph.

ata In iga

e information and classify data.

on of a data set.

System

hs. les.

h a system of linear inequalities. classify data.

B. Write quadratic equations in iC. Maximize or minimize quadratic functionsD. Complete the square to write quadratic functions in veE. Solve equations using the quadratic formula.

d numbF. Use the discriminant to determine the nature anG. Factor quadratic expressions.

s using factoring. H. Solve quadratic equationI. Interpret how different values of a, h and k efJ. Graph equations in a given form. K. Organize information and classify data.

D vest tions A. OrganizB. Choose a representative sample.

quartile and standard deviatiC. Find and interpret the range, interty of data. D. Determine the variabili

E. Find the margin of error for a sample proportion.

s of Equations A. Solve systems of linear equations using matrices.

tion of grapB. Use technology to find points of intersecC. Write and graph inequalities with 2 variabD. GrapE. Organize information and

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115

ON-LINEAR ALGEBRA (Continued) N Mathematics Grade 11 Level 3

A student in this course will… Radical Functions

A. Evaluate radical expressions. . lex numbers.

lot complex numbers in the complex plane. s. operation.

Polyno

ials. ply and divide polynomials.

ns and describe their important features. ns and find zeroes of cubic functions.

omials.

ation unc

A. Solve rational equations. rtant features and find equations of translated hyperbolas.

portant features of graphs of rational functions.

B. Solve equations with radical expressionse compC. Add, subtract, multiply and divid

D. Find complex solutions to equations that have no real solutions. plex number. E. Calculate the magnitude of a com

F. Graph and evaluate square root functions. G. Graph radical functions. H. PI. Identify the number systems to which a number belong

or a set and anJ. Evaluate whether group properties hold fK. Organize information and classify data.

mial Functions A. Recognize, evaluate, add and subtract polynomB. MultiC. Solve cubic equations.

l functions. D. Find zeroes of higher-degree polynomiaE. Recognize graphs of polynomial functioF. Find equations for graphs of cubic functioG. Understand numeration systems using polynH. Organize information and classify data.

R al F tions

B. Identify impontify imC. Ide

D. Organize information and classify data.

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116

ON-LINEAR ALGEBRA (Continued)

el 3

N

Mathematics Grade 11 Lev

A student in this course will… Data Investigations

A. Organize information and classify data.

ard deviation of a data set.

Di h

utations). he concept of combinations to Pascal’s Triangle.

Analyt eom

escribe how a conic section can be used in applications.

Nonlin Alg

B. Choose a representative sample. C. Find and interpret the range, interquartile and standD. Determine the variability of data. E. Find the margin of error for a sample population.

screte Mat A. *Solve problems that require sorting item into groups. B. *Analyze situations involving direction. C. *Count possibilities in situations (combinations and permD. *Apply tE. *Organize information and classify data.

ic G etry A. *Find the distance between two points on a coordinate plane. B. *Find focus and directrix of a parabola. C. *Write and graph equations of a circle.

equations of an ellipse. D. *Write and graphE. *Write and graph equations of a hyperbola. F. *Find conics by taking a cross section of a double cone. G. *D

ear ebra/Honors PRHS icat

8g Ind ors

1a, 1d, 1e, 1f, 1g, 1h, 2a, 2b, 2c, 2d, 2e, 3a, 3d, 3e, 3f, 4a, 4c, 4e, 4f, 4g, 4h, 5a, 5b, 5c, 5d, 6d, 6g, 7a, 7b, 7c, 8c, 8f,

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117

RE-CALCULUS HONORS* P Mathematics Grade 11 Honors

ons (Greek and Modern)

ic properties. ric functions, indirect measurement, arc length and angular speed. Cos x.

plitude, period and shift. uations with trig functions graphically.

Analyt rigohs of the trigonometric functions.

nge and asymptotes of graphs of trigonometric functions. s.

nverse functions of trigonometric graphs. lities graphically and/or algebraically.

Obliqu ang

d applications. Develop and use the Law of Cosines to solve oblique triangles in real-world applications.

A student in this course will… The trigonometric functi

A. Define angles and their measure. of an acute angle. B. Find the trigonometric functions

C. Derive the unit circle and the wrapping function. unit circle and periodD. Define trigonometric functions in terms of the

E. Solve problems using applications of trigonometf y = Sin x and y = F. Develop and sketch graphs o

G. Graph trig functions with changes in amH. Solve eq

ic T nometry

n grapA. Recognize and interpret patterns of transformations ond raB. Identify period, phase shift, amplitude, domain a

C. Apply graphs of trigonometric functions to solve problemD. *Define and transform iE. Solve trigonometric equations and inequaF. Develop and provide visual support algebraically for trigonometric identities. G. Verify identities.

ly. H. Solve trigonometric equations and inequalities analytical

e tr les A. Develop and use the Law of Sines to solve oblique triangles in real worl

i

B. C. Find the area of triangles. D. Complete a survey project.

lve magnitude and direction problems. E. *Use vector analysis to so

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RE-CALCULUS HONORS* (Continued)

118

P Mathematics Grade 11 Honors A student in this course will… Linear functions, quadratic functions, and absolute value functions

Graph the elementary function. domain and range, and sketch the graph.

gebraically. of functions.

Polyno

E. s algebraically. logarithmic equations to problem situations and solve graphically.

B. *Compute the possible outcomes using the Multiplication Principle. C. *Compute permutations. D. *Compute combinations.

A. B. List the transformations, theC. Solve problems graphically and alD. *Graph and solve the composition E. *Graph the inverse relation.

mial functions A. Identify the main characteristics of the graphs of polynomial functions including domain and range. B. Find the local extrema and zeros of a polynomial function.

C. Use real-world applications of polynomial functions. D. *Explore end behavior of polynomial functions.

Expone al annti d logarithmic functions

A. Graph the elementary functions of each. B. List the transformations, the domain and range, and sketch the graphs of each. C. Solve problems using applications of exponential growth or decay algebraically and/or graphically. D. Explore and use the properties of logarithmic functions.

Use logarithms to solve problemF. *Apply

Counting

A. *Define sets and set operations.

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119

S* (Continued)

s

PRE-CALCULUS HONOR Mathematics Grade 11 Honor A student in this course will…

*Represent sample spaces.

conditional and independent events.

Calculate the expected value and standard deviation of a binomial experiment.

Math o ancnt and savings options.

alues. enefits of varied payment plans.

ent, mortgage schedule, and a savings/investments annuity in order to

s/

Probability A. B. *Define properties of probability. C. *Find the probability of an event. D. *Define equally likely events.

endent events. E. *Explain the concepts of conditional and indepF. *Use multiplication and addition rules to compute the probability ofG. *Explain if a probability experiment meets the requirements necessary to be a binomial experiment. H. *Compute binomial probabilities using the formula and the tables. I.

f fi e A. Use terminology and formulas associated with debt, investme

n

B. Calculate simple interest, compound interest, and future annuity vngs, and explore the bC. Calculate monthly payments on debt or savi

D. Develop a personal financial plan including car paymreach a desired end at age 55.

Pre-Calculu Honors PRHS icat

Ind ors 1d, 1e, 1f, 2a, 2b, 2c, 2d, 2e 3a, 3d, 3e, 3f 4a, 4c, 4f, 4g, 4h 5a, 5c, 5d 6d, 6g 7a, 7c 8c, 8f, 8g

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120

SENIOR MATH TOPICS

Mathematics 12th Grade Level 2 Math Topics or Non-Linear Algebra (See 11th Grade, Level 3 for N L Algebra)

Statisti

B. Draw a histoC. lity.

any sample population and connect to applications in the business world.

Measu ent ecision and accuracy.

.

lculate with measurements and round the results. Probability

A. Find the proys an event can happen.

find probability. D. e a c es.

sed on calculated probability.

A student in this course will…

cs A. Calculate measures of central tendency for a set of data.

f data. gram to represent frequency distributions oDistinguish between range, trend, and standard deviation as measures of variabi

D. Interpret the characteristics of a normal curve. deviation to describe a set of data. E. Calculate the range and standard

F. Use standard deviation to draw conclusions about

remA. Distinguish between counting and measuring, and between prB. Read and write measurements to show precision and tolerance.

s. C. Compare measurements to specified toleranceD. Use significant digits to indicate the accuracy of a measurementE. Use precision tools to make measurements. F. Ca

bability of some sof wa

imple events. B. Count the number C. Draw diagrams and charts to help

Us alculator to find probabilitis. E. Find the odds of simple event

F. edictPr an expected outcome ba

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121

SENIOR MATH TOPICS (Continued)

Mathematics 12th Grade Level 2

A student in this course will…

Math of Finance A. Spreadsheets

1. Describe a computer spreadsheet and term

ll how it’s used. inology.

plates to solve practical problems.

s and accounts. of a loan, revolving credit account (credit cards) and/or savings program.

e tax forms. onthly budget.

Expressions g order of operations.

2. Define and use proper spreadsheet tespreadsheet tem3. Load and use simple

B. Explain terms used in financial transactions and planning. la. C. Substitute values into a formu

D. Use a calculator to solve problems with formulas. e. E. Calculate gross and net incom

F. Calculate future value of an investment. pound interest of various loanG. Determine com

H. Compute monthly paymentI. Complete employment and incomJ. Develop a personal financial plan, including a m

Algebra Review A. Working with

1. Simplify an expression usin2. Evaluate expressions.

B. Solving Equations 1. Solve one-step equations. 2. Solve two-step equations.

C. Linear Equations and Graphs 1. Find the slope of a line. 2. Graph linear equations.

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122

3. Write equations of lines. of real-world data. 4. Model linear data to make predictions

SENIOR MATH TOPICS (Continued) Mathematics 12th Grade Level 2

A student in this course will…

Congruence, Similarity and Transformations

lems.

r questions about real-life situations.

of rotations to answer questions about real-life situations. re in a plane.

a coordinate plane. Create fractals.

J. Recognize similar figures. K. Use properties of similar figures.

Use similar figures to solve real-life problems. pare perimeters and area of similar figures.

ight Triangle Trigonometry A. Name the parts of a right triangle. B. Use the Pythagorean formula to find a side of a right triangle. C. Use the characteristics of 3:4:5, 45°-45° and 30°-60° right triangles to solve practical problems. D. Use the ratios for the sine, cosine, and tangent of an angle to solve problems that involve triangles. E. Use a calculator to solve problems that involve right triangles. F. Use the Pythagorean Theorem and trigonometric ratios to measure objects indirectly. G. Use a calculator to find sine and cosine values. H. Draw a graph of sine and cosine waves.

A. Determine whether two figures are congruent. B. Use congruence to solve real-life probC. Reflect a figure about a line. D. Use properties of reflections to answeE. Rotate a figure about a point. F. Use propertiesG. Translate a figuH. Represent translations inI.

L. M. Com

R

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I. Find the amplitude, wavelength, period, and frequency of sine waves. J. Find the phase shift between two sine waves.

S (Continued)

123

SENIOR MATH TOPIC Mathematics 12th Grade Level 2

A student in this course will… f timI e or needs of the class allow, the following topics may be explored:

ar Programming

ogSequence anTopology

Math Topics

Coding Discrete MathematicsLineL ic

d Series

PRHS dicat

In ors 1a, 1d, 1e, 1f, 1g, 2a, 2b, 2c, 2d, 2e, 3a, 3d, 3e, 3f, 3g, 4c, 4e, 4f, 4g, 4h, 5a, 5b, 5c, 5d, 6d, 6g, 7a, 7c, 8a, 8c, 8f, 8g

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124

INITE MATHEMATICS

F Mathematics 12th Grade Level 3

Pre-Calculus (Refer to Pre-Calculus 11th Grade Honors), Finite Math, Statistics or Math Topics (Refer to Senior Math Topics, 12th

equen

equences. nd value of any given term.

of an arithmetic sequence. arithmetic sequence. etric.

ence.

ing. g time periods.

ke sound financial decisions. s.

Calculate future value and present value of an investment.

s plan.

Grade Level 2) The student in this course will... S ces

A. Define arithmetic and geometric sequences. B. Differentiate between arithmetic and geometric sC. Given an arithmetic sequence, find the common difference aD. Derive the formula for finding the nth termE. Derive the formula for finding the sum of an F. Determine if a sequence is arithmetic or geomG. Given a geometric sequence, find the common ratio and value of any given term.

H. Derive the formula for finding the nth term of a geometric sequence. sequence. I. Derive the formula for finding the sum of a geometric

sequJ. Calculate the sum of any arithmetic or geometric Math of Finance

A. Explain terms used in financial transactions and financial plannB. Calculate simple interest and simple interest rates for varyinC. Derive formulas used in compound interest and annuities. D. Substitute values into financial formulas to solve problems and to ma E. Use a calculator to solve financial problemF. G. Determine the amount of time needed to double or triple an investment. H. Calculate monthly payment on a loan and/or saving

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125

get.

INITE MATHEMATICS (Continued)

I. Given two investment plans determine which plan yields the largest amount. J. Develop an amortization schedule for a short-term loan. K. Calculate mortgage payments using a mortgage table. L. Develop a personal financial plan, including a monthly bud

F Mathematics 12th Grade Level 3 The student in this course will... Sets: Counting Techniques

l set, element of a set, subset, equality of sets and empty set. B. Use appropriate set notation.

List the subsets of a given set. ber of subsets in a given set. n sets. ams. ting techniques used in this unit. s a task can be performed.

Probability ine probability terms: sample space, experiment, event, probability, odds, relative frequency, expected value.

A. Define set, universa C.

D. Determine the numE. Perform the operations of union and intersection oF. Illustrate the operations performed on sets using Venn diagrG. Find the complement of a given set. H. Verify De Morgan’s laws using Venn diagrams. I. Distinguish between finite and infinite sets. J. Determine the number of elements in a given set. K. Develop a survey of local concerns that models the Venn diagramming and counL. Use the Multiplication Principle to find the number of wayM. Illustrate the Multiplication Principle using a tree diagram. N. Give the consecutive factorial values from 0! to 10!. O. Use the n! formula to evaluate problems involving factorials. P. Define permutations and combinations. Q. Differentiate between a permutation and a combination by example. R. Compute the number of permutations a task has using a formula. S. Compute the number of combinations a task has using a formula. T. Find permutations and combinations using the statistic mode on a graphing calculator.

U. Relate combinations to the binomial theorem.

A. DefB. Identify types of probability: experimental, theoretical.

Page 126: Math Curriculum K-12

iven experiment and find the probability of each event. nts.

bability distribution.

on to calculate probabilities of random variables.

a binomial experiment.

FIN E M

126

C. Construct a probabilistic model for a gD. Explain the concepts of conditional, independent and mutually exclusive eveE. Use multiplication and addition rules to compute compound probabilities. F. Display values of random variables in a proG. Use a probability distributiH. Compute the expected value of a discrete random variable. I. Explain if a probability experiment meets the requirements necessary to be J. Compute binomial probabilities using a formula.

IT ATHEMATICS (Continued) Mathematics 12th Grade Level 3 The student in this course will... *Statistics

A. Define statistics. Organize data using a frequency distribution.

edian, and mode. given situation.

standard deviation.

mally distributed. J. Sketch and label a normal curve identifying its properties.

pected values given data displayed in a normal curve. ine if the data from a given experiment is normally distributed.

M. Use the z-score formula to convert raw scores for comparison purposes. Calculate the probabilities for normal distribution by translating raw scores to z-scores to find the area under the curve.

O. Use statistics to give an analysis of a researched situation involving data that is + 3 standard deviations from the mean.

of graphing calculator to create histograms and calculate measures of central tendency and dispersion.

A. Define: quantity, direct variation, indirect variation, function.

B.C. Graphically represent data in the form of a histogram D. Graphically represent data using a pie chart. E. Compute the measures of central tendency: arithmetic mean, mF. Determine the best measure for a G. Compute measures of dispersion: range, variance, andH. Identify situations where standard deviation is used. I. Identify experiments that might obtain data that is considered to be nor

K. Find exL. Determ

N.

*Use statistics mode Linear Functions

Page 127: Math Curriculum K-12

B. Collect data and graph the results. C. Determine if the data collected from a given experiment is a function. D. Graph lines using charts and equations. E. Write the equation of a line. F. Examine the properties of parallel and perpendicuG. Solve systems of equations by graphing, substitution and elimination. H. Find and make predictions using the break-even point in business situations. I. Graph linear inequalities. J. Graph systems of linear inequalities. K. Use linear programming to solve problems.

HEMATICS (Continued)

127

lar lines.

FINITE MAT Mathematics 12th Grade Level 3 The student in this course will...

gmented matrix. form and label its rows and columns. mented matrix.

g calculator.

Finite Ma

Matrices

entity matrix and auA. Define matrix, idB. Represent a linear system in augmented matrix

gC. Write a system of linear equations from an auD. Give the dimensions of a matrix. E. Set up an augmented matrix in a graphing calculator. F. Use elementary row operations to solve a system of equations.

Solve a systemG. of equations using matrices and a graphin

thematics PRHS Indicators

8c, 8f, 8g. 1a, 1c, 1d, 1e, 1f, 1g, 1h, 2a, 2b, 2c, 2d, 2e, 3a, 3d, 3e, 3f, 4a, 4b, 4c, 4e, 4f, 4h, 5a, 5b, 5c, 5d, 6d, 6g, 7a, 7b, 7c, 7f,

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128

TATISTICS

S

Mathematics 12th Grade Level 3

A student in this course will… ntroduction to Statistics ~ “WI hat is Statistics?”

Define statistics

eter, statistic, population, sample, response

Misuse Stat

tistical problem, they should beware of misleading conclusions

A. B. Understand the need and purpose of statistics. C. Distinguish between population and sample.

eters and statistics. D. Differentiate between paramE. After reading a statistical problem, be able to identify the following: param

variable, and inference. blem. F. Identify and carry out the 7 steps in a statistical pro

G. Identify the different types of sampling.

of istics A. Identify the ways in which statistics may be misused.

B. Given a sta1. Unfavorable opinions and biases.

ges. 2. Aggravated avera3. Disregarded dispersions. 4. Persuasive tables, graphs, and charts. 5. Erroneous cause and effect conclusions. 6. Misinterpreted trends. 7. Incorrect base period in computing percentages

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129

ation to reduce chances of being misled. De

i raw data using arrays and frequency distributions. gram, frequency polygon, stem-and-leaf display, box plot.

c mean, median, and mode. viation, variance, interquartile range,

STATISTICS (Continued)

C. Evaluate quantitative inform

scriptive Statistics A. Organ zeB. Graphically represent data in the form of a histoC. Compute the measures of central tendency: arithmetiD. Compute the measures of dispersion: range, standard deviation, mean absolute de

quartile deviation. aximum, quartiles, percentiles. E. Identify measures of position: minimum, m

Mathematics 12th Grade Level 3

A student in this course will… Probability

A. Define probability terms: sample space, experime robability, and relative frequency.

nts.

riment.

variables to standard scores and then using the z-tables to find the area under the curve.

nt, event, pB. Identify types of probability: experimental, theoretical, and subjective. C. Assign simple probabilities. D. Use multiplication and addition rules to compute compound probabilities. E. Explain the concepts of conditional, independent, and mutually exclusive eveF. Identify discrete vs. continuous random variables. G. Display values of random variables in a probability distribution. H. Use a probability distribution to calculate probabilities of random variables. I. Compute the expected value of a discrete random variable.

Probability Distributions A. Explain if a probability experiment meets the requirements necessary to be a binomial experiment. B. Compute combinations. C. Compute binomial probabilities using the formula and the tables. D. Calculate the expected value and standard deviation of a binomial expeE. Compute Poisson probabilities using the tables. F. Calculate the probabilities for normal distribution by translating random

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130

l rule.

Sampli Con

Compute the mean and standard error of the sampling distribution of percentages.

3

G. Explain the empiricaH. Determine standard z-scores from specified probability requirements.

ng cepts A. Explain the need and advantages for sampling.

nite populations. B. Differentiate between finite and infiC. Compute the mean and standard deviation of the sampling distribution of means.

f the means. D. Know when the finite population correction factor is needed in order to calculate the standard error oE. Define and apply the central limit theorem. F.

STATISTICS (Continued) Mathematics 12th Grade Level

s: estimate, estimator, estimation, point estimate, and interval estimate.

ate probability distributions (z-, t-, chi-square) needed in the computation of b through d above. ine the appropriate sample size to use to estimate the population mean or percentage at different levels of

Explain the necessary steps in hypothesis testing within both the classical and p-value approaches. d

one and two-tailed tests. s for one and two-tailed tests.

A student in this course will…

Estimating Parameters A. Define and apply the following termB. Compute estimates of the population mean at different confidence levels when the population standard deviation is known

and unknown. C. Compute estimates of population percentages at different confidence levels. D. Compute estimates of population variances at different confidence levels. E. Use the appropri

. DetermFconfidence.

Hypothesis Testing for One-Sample Procedures

A.B. Compute hypothesis tests of means for one and two-tailed tests when the population standard deviation is known an

unknown. C. Compute hypothesis tests of percentages forD. Compute hypothesis tests of variance

Page 131: Math Curriculum K-12

Linear g on and correlation analysis. ositive correlation, negative correlation, or no correlation.

ndent variable (mean and individual) for forecasting purposes.

tatistics

131

Re ression and Correlation A. Define, differentiate between, and explain the purpose of regressiB. Given a set of bivariate data, reasonably predict if there will be a pC. Prepare a scatter diagram for a set of ordered pairs and interpret its meaning.

Compute a regression equation using the method of least squares D. E. Compute a standard error estimate. F. Run a t test for slope to see if there’s a true relationship between the x and y variables of a regression equation. G. Prepare interval estimates of the depe

S RHS Indicators a, 1d, 1e, 1f, 1g, 2a, 2b, 2c, 2d, 2e, 3a, 3d, 3e, 3f, 3g, 4c, 4e, 4f, 4g, 4h, 5a, 5b, 5c, 5d, 6d, 6g, 7a, 7b, 7c, 7d, 7e, 8a, 8c, 8f, 8g

P1AP CALCULUS

Mathematics 12th Grade Honors

AP Calculus, Calculus, Statis

tics, or Finite Math (Refer to Statistics or Finite Math 12th Grade Level)

tline for Calculus AB

Teache e the Teacher’s Guide-AP Calculus for sample syllabi.) Although the examin o rs may wish to enrich their courses with additional topics.

Limits

Analys othe geo t dict and to explain the observed local and global behavior of a functio

Limits o nding of the limiting process is sufficient for this course. its using algebra

Asymp t

Topical Ou This outline of topics is intended to indicate the scope of the course, but it is not necessarily the order in which the topics are to be taught.

rs may find that topics are best taught in different orders. (Seati n is based on the topics listed in the topical outline, teache

Functions, Graphs, and is f graphs. With the aid of technology, graphs of functions are often easy to produce. The emphasis is on the interplay between me ric and analytic information and on the use of calculus both to pren.

f Functions (including one-sided limits). An intuitive understa• Calculating lim• Estimating limits from graphs or tables of data to ic and unbounded behavior

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132

Understanding asymptotes in terms of graphical behavior. ptotic behavior in terms of limits involving infinity.

of change. al growth, and logarithmic growth.)

Continuity as a property of functions. The central idea of continuity is that close values of the domain lead to close values of the range.

ity in terms of limits.

•• Describing asym• Comparing relative magnitudes of functions and their rates

(For example, contrasting exponential growth, polynomi

• Understanding continu• Geometric understanding of graphs of continuous functions (Intermediate Value Theorem and Extreme Value Theorem)

AP Calculus (Continued) Mathematics 12th Grade Honors

A student in this course will… Derivatives Concept of the derivative. The concept of the derivative is presented geometrically, numerically, and analytically, and is interpreted as

Deriva ve at a point. Slope of a curve at a point. Examples are emphasized, including points at which there are vertical tangents and points at which

are no tangents. oximation.

• The Mean Value Theorem and its geometric consequences. • Equations involving derivatives. Verbal descriptions are translated into equations involving derivatives and vice versa.

an instantaneous rate of change. • Derivative defined as the limit of the difference quotient. • Relationship between differentiability and continuity. ti•

there • Tangent line to a curve at a point and local linear appr• Instantaneous rate of change as the limit of average rate of change. • Approximate rate of change from graphs and tables of values.

Derivative as a function. • Corresponding characteristics of graphs of f and f’ • Relationship between the increasing and decreasing behavior of f and the sign of f’

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133

Second derivatives.

• Corresponding characteristics of the graphs of f, f’, and f” • Relationship between the concavity of f and the sign f”. • laces where concavity changes.

pplications of derivatives. s, including the notions of monotonicity and concavity.

AP

Points of inflection as p

A• Analysis of curve• Optimization, both absolute (global) and relative (local) extreme. • Modeling rates of change, including related rates problems. • Use of implicit differentiation to find the derivative of an inverse function. • Interpretation of the derivative as a rate of change in varied applied contexts, including velocity, speed, and acceleration.

Calculus (Continued) Mathematics 12th Grade Honors

A student in this course will… Derivatives

Computation of derivatives. Knowledge of derivatives of basic functions, including Rx, exponential, logarithmic, trigonometric, and inverse trigonometric functions.

e of sums, products, and quotients of functions.

Integrals Riemann sums.

m over equal subdivisions.

Interpreta ions and properties of definite integrals. of Riemann sums.

antity over the interval:

• Basic rules for the derivativ• Chain rule and implicit differentiation.

• Concepts of a Riemann su• Computation of Riemann sums using left, right and midpoint evaluation points.

t• Definite integral as a limit• Definite integral of the rate of change of a quantity over an interval interpreted as the change of the qu

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134

• Basic properties of definite ity and linearity.)

Applications of integrals. priate integrals are used in a variety of applications to model physical, social or economic situations. Although only a

ling of applications can be included in any specific course, students should be able to adapt their knowledge and techniques to gral of a rate of

change to give accumulated change or using the method of setting up and approximating Riemann sum and representing its limit as a definite integral. To provide a common foundation, specific applications should include finding the area of a region, the volume of a solid with known cross sections, the average value of a function, and the distance traveled by a particle along a line.

AP Calculus (Continued)

integrals. (For example, additiv

Approsampsolve other similar application problems. Whatever applications are chosen, the emphasis is on using the inte

Mathematics 12th Grade Honors

o

Funda talUsUs ivative, and the analytical and graphical analysis of functions so

Techn s oAn tly from derivatives of basic functions.

by substitution of variables (including change of limits for definite integrals).

Applications Fin applications to motion along a line. Sol and using them in modeling. In particular, studying the equation y’= ky and exp

a

f x dx f b f ab

' ( ) ( ) ( )= −z

A student in this c urse will… Integrals (Continued)

men Theorem of Calculus. • e of the Fundamental theorem to evaluate definite integrals. • e of the Fundamental Theorem to represent a particular antider

defined.

ique f antidifferentiation. • tiderivatives following direc• Antiderivatives •

of antidifferentiation • ding specific antiderivatives using initial conditions, including• ving separable differential equations

onential growth.

Page 135: Math Curriculum K-12

umeral approximations to definite integrals. Use of Riemann sums and the trapezoidal Rule to approximate definite integrals of s represented algebraically, geometrically, and by tables of values.

AP Calculus

135

Nfunction

PRHS icata, 1d, 1f,

alculus

Ind ors 1e, 1g, 2a, 2b, 2c, 2d, 2e, 3a, 3d, 3e, 3f, 4c, 4e, 4f, 4g, 4h, 5a, 5b, 5c, 5d, 6d, 6g, 7a, 7b, 7c, 8a, 8c, 8f, 8g 1

C

Mathematics 12th Grade Honors

o

lationship is a function.

Limits the

hniques.

A student in this c urse will...

re-calculus overview PA. Determine if a reB. Sketch the graphs of algebraic and trigonometric functions. C. Check for symmetry with respect to both axes and the origin. D. Transform a graph. E. Find composites of functions. F. Find the zeroes of functions.

and ir properties phs or tables of data. A. Estimate limits intuitively from gra

ecB. Calculate limits using algebraic t C. Determine the intervals over which a function is continuous.

D. Find the vertical asymptotes of a function. E. Describe asymptotic behavior in terms of limits involving infinity.

F. Compare relative magnitudes of functions and their rates of change.

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136

on

functions.

raph of its derivative. s of composite functions.

d points.

ALCULUS (Continued)

Differentiati

A. Relate tangent lines and rates of change. icB. Find the derivative of algebraic or trigonometr

e slope at a point. C. Use the derivative to find thD. Match the graph of a function with the g

the derivativeE. Use the Chain Rule to find F. Find the derivatives of functions that are defined implicitly.

ines to given curves at specifieG. Use implicit differentiation to find the slope of tangent l

C Mathematics 12th Grade Honors A student in this course will... Applications of Differentiation

uantity is changing by relating it to other quantities whose rates of change are known (related

terval.

a of functions on the closed interval. n which a function is increasing or decreasing.

ma of a function.

nction using domain, range, symmetry, asymptotes, intercepts, relative extrema, and points of

ction to solve optimization problems.

te function values.

A. Find the rate at which a qrates).

B. Determine whether Rolle's Theorem can be applied to an indicated inem to a function on an indicated interval. C. Apply the Mean Value Theor

D. Determine the absolute extremE. Find critical numbers and the open intervals oF. Use the First and Second Derivative Tests to find all relative extreG. Determine concavity of a function. H. Find limits of rational functions. I. Sketch a graph of a fu

inflection. f a funJ. Use maximum and minimum values o

K. Use Newton's Method to approximate the real zeroes of a function. tive error and to approximaL. Use differentiation to estimate rela

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137

and trigonometric integrals. .

D. Compute areas using Limits Definition. E. Evaluate definite integrals. F. Use anitderivatives to evaluate definite integrals (Fundamental Theorem of Calculus).

Use the Fundamental Theorem of Calculus to find area. d the average value of a function over a given interval.

CALCULUS (Continued) Mathematics 12

Integration

A. Evaluate indefiniteB. Evaluate integrals through use of substitutionC. Use correct sigma notation.

G. H. FinI. Evaluate definite integrals using substitution. J. Approximate definite integrals using the Trapezoidal Rule and Simpson's Rule.

th Grade Honors A udent this c

iA. Differentiate logarithmB. Find indefinite integral

D. Find the inverse of a function. garithmic equations.

F. Differentiate and integrate exponential functions to another base. nverse trigonometric functions.

A. Calculate area between curves in the plane. nal solids.

C. Find work required in physics and engineering real-world problems.

st in ourse will...

Logarithm c and Exponential Functions ic functions. s of logarithmic and trigonometric functions.

C. Solve differential equations.

E. Solve exponential and lo

G. Differentiate and integrate i Applications of Definite Integration

B. Find volume of 3-dimensio

D. Calculate fluid pressure and force.

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138

Calcul

Integration Techniques A. Integrate by parts. B. Use partial fractions to evaluate the integral.

us PRHS Indicators 1d, 1e, 1f, 2b, 2c, 2d, 2e, 3a, 3d, 3e, 3f, 4c, 4e, 4f, 4g, 4h, 5a, 5b, 5c, 5d, 6d, 6g, 7a, 7c, 8a, 8c, 8f, 8g

PLYMOUTH REGIONAL HIGH SCHOOL

MISSION STATEMENT Together we challenge one another to develop and demonstrate the

EXPECTATIONS AND INDICATORS

VISION Growth: Every person, every day, some way.

attitudes, skills, and knowledge essential to attaining excellence in self, family, and community.

1. OUR GRADUATES WILL BE KNOWLEDGEABLE AND SKILLED COMMUNICATORS.

a. Give an oral presentation.

b. Write in a variety of styles.

c. Use computer-based technology to communicate.

d. Demonstrate listening skills.

e. Read and respond to a variety of written materials.

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139

f. Use a variety of visual media to communicate.

2. OUR ORMED DECISIONS.

ven information and parameters.

b. Generalize solutions and apply problem-solving strategies to new situations.

c. Use inductive and deductive reasoning to solve problems.

d goals.

. OU VALUATE, AND MANAGE INFORMATION RESOURCES

Acquire, organize, interpret, and use information from a variety of sources, e.g., Internet, periodicals, interviews, uals.

e. l, mixed, or multimedia format(s).

g. Follow accepted rules of grammar, usage, and spelling.

h. Use the writing process principles in all written work.

GRADUATES WILL EFFECTIVELY REASON, PROBLEM SOLVE, AND MAKE INF

a. Complete an experiment and draw a conclusion, gi

d. Solve problems using appropriate, creative problem-solving strategies.

e. Evaluate and reflect on their actions, choices, an

3 R GRADUATES WILL BE ABLE TO LOCATE, EAND TECHNOLOGY.

a.

videos, charts, diagrams, instruction man

b. Identify the essential question to guide research.

c. Evaluate sources for reliability, accuracy, and relevance.

d. Interpret and draw conclusions from data.

Present findings in written, ora

f. Summarize information related to a concept.

g. Understand, read, and respond to primary sources.

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ARY TO ADAPT TO THE CHANGING

E

of spreadsheet, Internet tools, and electronic data communication.

rket.

g upon previously learned material.

ons.

NEEDED TO SUCCEED IN OUR SOCIETY.

, investigate, research, and explain phenomena.

and communicate information for a variety of purposes.

athematical facts, models, strategies, properties, any relationships to solve and

explain a variety of problems.

e. Use knowledge and understanding of geographical, historical, social, cultural, and political events and relationships to , national, and world problems.

4. OUR GRADUATES WILL HAVE THE KNOWLEDGE AND SKILLS NECESSMPLOYMENT MARKET.

a. Compose clear and concise workplace communications.

b. Exhibit skills in the use

c. Converse in a clear, concise, and focused manner.

d. Exhibit knowledge of the future employment market.

e. Apply skills learned to the employment ma

f. Adapt to new situations by buildin

g. Prioritize time and tasks in an effective manner.

h. Follow written and verbal instructi

5. OUR GRADUATES WILL ACQUIRE A BODY OF KNOWLEDGE AND SKILLS

a. Make connections between subject areas.

b. Use scientific methods, concepts, and knowledge to describe

c. Use the tools of information technology to access, apply,

d. Use accurate calculations and appropriate m

research and develop theoretical solutions for local

f. Demonstrate understanding of the lifelong value of physical fitness and wellness.

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141

g. Use knowledge of literature to develop an understanding of the human experience.

IZE THEIR RESPONSIBILITIES TO PARTICIPATE IN THE

DEMO R

a. d nment at the national, state, and local levels.

b. n ote in primaries, elections, and town meetings.

c. ates.

d. decision.

e. Understand the process of a trial and the American legal system.

f. Understand freedom of expression.

g. Abide by established school codes of behavior as well as local, state, and national laws.

7. OUR GRADUATES WILL DEMONSTRATE THE ABILITY TO WORK COOPERATIVELY.

a. Work effectively, responsibly, and safely in teams.

b. Make group presentations or teach mini-lessons to a class.

c. Interact positively with other students.

d. Self-assess their role within a group or team.

e. Demonstrate a fair peer evaluation of their group.

f. Accept and delineate tasks/jobs within a small group project reflecting equal sharing of tasks.

h. Experience artistic expression through one form of fine or performing arts.

6. OUR GRADUATES WILL RECOGNC ATIC PROCESS.

Un erstand the functions and roles of gover

Ide tify dates, locations and procedures to register to v

Conduct a mock election, including deb

Work together with peers to reach a

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. OUR GRADUATES WILL DE OTHERS, AND THEIR ENVIRONMENT.

viewpoints.

Respect the personal well-being of their bodies by making healthy choices.

situations.

e. Demonstrate good sportsmanship.

142

8 MONSTRATE RESPECT FOR THEMSELVES,

a. Respect all

b.

c. Behave appropriately in all

d. Value a diversified student body and community.

f. Handle conflicts appropriately.

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MATHEM

TABLE OF CONTENTS Section I: Stems

Section II: State and Local Grade Specific Indicators…Kindergarten…........................................................... ............................................................................................

rade 3……………………………………………………….……… ………………………………………………………… rade 4……………………………………………………….…………………………………………………………. rade 5………………………………………………………..................................................................................................... rade 6…………………………………………………….…… rade 7………………………………………

Grade 8…………………………… Section III: Stems and Indicators (State Only)…………………….……………………………………………………….……… N&O………………………… G&M……………………… F&A……………………… DSP……………………… Our sincere appreciation to the Governor Wentworth Regional School District for sharing the Mathematics Grade Level Expectations in this format. You can find the Mathematics Grade Level Expectations online: http://www.ed.state.nh.us/Education/doe/organization/curriculum/NECAP/GLEs.htm

143

APPENDIX A:

ATICS GLEs

Only…………………………………………………………………………………………………………………………………………

……….…….……………………………………………………………………………………….... .......................................................

Grade 1…………………………………………………………………………………………………………………………………….. Grade 2………………………………………………………………………………………………………………………….... G …GGGG

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M (N&O) Number and Operation

M (G&M) Geometry and Measurement

Section I:

MATHEMATICS GLE’s (STEMS ONLY)

M (F&A) Functions and Algebra

M (DSP) Data, Statistics, and Probability

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STEMS AND GRADE SPECIFIC INDICATORS – Kindergarten

ing of rational numbers with respect to whole numbers from 0 to 12 through investigations that apply cy in composing and decomposing numbers using models, explanations, or other representations

Demonstrates understanding of the relative magnitude of numbers from 0 to 20 through investigations that demonstrate one-to-one bers to each other or to benchmark whole numbers (5, 10); that demonstrate an understanding of

traction of whole umbers (0 to 10) by solving problems involving joining actions, separating actions, part-part whole relationships, and comparison ituations; and addition of multiple one-digit whole numbers

Demonstrates understanding of monetary value through investigation involving knowing the names and values for coins (penny, nickel and dime) Mentally adds and subtracts whole numbers by naming the number that is one more or one less than the original number Makes estimates of the number of objects in a set (up to 20) by making and revising estimates as objects are counted Geometry and Measurement Uses properties, attributes, composition, or decomposition to sort or classify polygons (triangles, squares, rectangles, rhombi, trapezoids, and hexagons) or objects by using one non-measurable or measurable attribute; and recognizes, names, and builds polygons and circles in the environment

145

Section II: MATHMETICS GLEs

Number and Operation Demonstrates conceptual understandthe concepts of equivalen Demonstrates conceptual understanding of rational numbers with respect to positive fractional numbers (1/2) as “fair share” (equal sized parts or sets) using models, explanations, or other representations

correspondence that compare whole numthe relation of inequality when comparing whole numbers by using “1 more” or “1 less”; that connect numbers orally and written as numerals to the quantities that they represent using models, representations, or number lines Demonstrates conceptual understanding of mathematical operations through investigations involving addition and subns

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146

Demonstrates conceptual understanding of measurable attributes using comparative language to describe and compare attributes of objects (length [longer, shorter], height [taller, shorter], weight [heavier, lighter], temperature [warmer, cooler], and capacity [more, less]); and compares objects visually and with direct comparison Determines elapsed and accrued time as it relates to calendar patterns (days of the week, yesterday, today, and tomorrow), the sequence of events in a day; and identifies a clock and calendar as measuremDemonstrates understanding of spatial relation tional words to locate and describe where an object is found in the environment Functions and Algebra

d extends to specific cases a variety of patterns (sequences of shapes, sounds, movement, colors, and letters) by extending the

and Probability

a

ent tools dships using location an position by using posi

Identifies anpattern to the next one, two, and three elements, or by translating AB patterns across formats (ABB can be represented as snap, clap, clap or red, yellow, yellow) or by identifying number patterns in the environment Data, Statistics, Interprets a given representation created by the class (models and tally charts) to answer questions related to the data, or to analyze the datto formulate conclusions using words, diagrams, or verbal/scribed responses to express answers Analyzes patterns, trends, or distribution in data in a variety of contexts by determining or using more, less, or equal

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147

Section II: MATHEMATICS GLE’s STEMS AND GRADE SPECIFIC INDICATORS – Grade 1

g g and decomposing numbers and in expanded notation using models, explanations, or other

bers with respect to positive fractional numbers (benchmark fractions: a/2, a/3, or models

they represent using models,

atical operations through investigations involving addition and subtraction of whole n

onetary value by knowing the names and values for coins (penny, nickel, dime and quarter) and by

d adds ber facts to ten

counted

Number and Operation Demonstrates conceptual understanding of rational numbers with respect to whole numbers from 0 to 100 using place value, by applyinthe concepts of equivalency in composinrepresentations Demonstrates conceptual understanding of rational numa/4 where a is a whole number greater than 0 and less than or equal to the denominator) as a part to whole relationship in area where the denominator is equal to the number of parts in the whole using models, explanations, or other representations Demonstrates understanding of the relative magnitude of numbers from 0 to 100 by ordering whole numbers; by comparing whole numbers to each other or to benchmark whole numbers (5,10, 25, 50, 75, 100); by demonstrating an understanding of the relation of inequality when comparing whole numbers by using “1 more”, “1 less”, “5 more”, “5 less”, “10 more” or “10 less”; and by connecting number words (from 0 to 20) and numerals (from 0 to 100) to the quantities and positions thatrepresentations, or number lines Demonstrates conceptual understanding of mathemnumbers (0 to 30) by solving problems involving joining actions, separating actions, part-part whole relationships, and comparisosituations; and addition of multiple one-digit whole numbers Demonstrates understanding of madding collections of like coins together to a sum no greater than $1.00 Mentally adds and subtracts whole numbers by naming the number that is one or two more or one less than the original number; anor subtracts whole num Makes estimates of the number of objects in a set (up to 30) by making and revising estimates as objects are

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148

pplies properties of numbers (odd, even, composition, and decomposition [5 is the same as 2 + 3]) and field properties (commutative and dentity for addition) to solve problems and to simplify computations involving whole numbers

s,

of objects in the

Demonstrates conceptual understanding of the length/height of a two-dimensional object using non-standard units Demonstrates conceptual understanding of measurable attributes using comparative language to describe and compare attributes of objects (length [longer, shorter], height [taller, shorter], weight [heavier, lighter], temperature [warmer, cooler], and capacity [more, less]); and compares objects visually with direct comparison and using non-standard units Determines elapsed and accrued time as it relates to calendar patterns (days of the week, months of the year), the sequence of events in a day; and recognizes an hour and “on the half-hour” Demonstrates understanding of spatial relationships using location and position by using positional words (close by, on the right, underneath, above, beyond) to describe one location in reference to another on a map, in a diagram, and in the environment Functions and Algebra Identifies and extends to specific cases a variety of patterns (repeating and growing [numeric and non-numeric]) represented in models, tables, or sequences by extending the pattern to the next one, two, or three elements (ABB can be represented as snap, clap, clap; red, yellow, yellow; or 1,2,2)

Demonstrates conceptual understanding of equality by finding the value that will make an open sentence true (2 + = 7) (limited to one operation and limited to use addition or subtraction) using models, verbal explanations, or written equations

Ai Geometry and Measurement Uses properties, attributes, composition, or decomposition to sort or classify polygons (triangles, squares, rectangles, rhombi, trapezoidand hexagons) or objects by a combination of two non-measurable or measurable attributes; and recognizes, names, builds, and draws polygons and circles in the environment Given an example of a three-dimensional geometric shape (rectangular, prisms, cylinders, or spheres) finds examples nvironment that are of the same geometric shape e

Demonstrates conceptual understanding of congruency by making mirror images and creating shapes that have line symmetry

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Data, Statistics, and Probability Interprets a given representation created by the ne-to-one correspondence, and tables) to answer questions related to the data, o ams, or verbal/scribed responses to xpress answers

, or distribution in data in a variety of contexts by determining or using more, less, or equal

e ikely,” “less likely,” or “equally likely”)

149

class (models, tally charts, pictographs with or to analyze the data to formulate conclusions using words, diagr

e Analyzes patterns, trends For a probability event in which the sample space may or may not contain equally likely outcomes, groups use experiments to describe thlikelihood or chance of an event (using “more l

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STEMS AND GRADE SPECIFIC INDICATORS – Grade 2

valency in composing and decomposing numbers (34 = 17 +17) and in expanded notation 141 = 100 + 40 + 7) using

, a/3, or inator) as a part to whole relationship in area and set

inator is equal to the number of parts in the whole using models, explanations, or other representations

of the re”, “1 less”, “5 more”, “5 less”, “10 more”, “10 less” “100 more” or

nt

anding of mathematical operations through investigations involving addition and subtraction of whole numbers by solving problems involving joining actions, separating actions, part-part whole relationships, and comparison situations; and

e by adding coins together to a value no greater than $1.99 and representing the result in lue up to $1.99)

Makes estimates of the number of objects in a set (up to 50) by making and revising estimates as objects are counted

Section II: MATHEMATICS GLEs

Number and Operation Demonstrates conceptual understanding of rational numbers with respect to whole numbers from 0 to 199 using place value, by applying the concepts of equimodels, explanations, or other representations Demonstrates conceptual understanding of rational numbers with respect to positive fractional numbers (benchmark fractions: a/2a/4 where a is a whole number greater than 0 and less than or equal to the denommodels where the denom Demonstrates understanding of the relative magnitude of numbers from 0 to 199 by ordering whole numbers; by comparing whole numbers to each other or to benchmark whole numbers (5,10, 25, 50, 75, 100, 125, 150 or 175); by demonstrating an understandingelation of inequality when comparing whole numbers by using “1 mor

“100 less”; and by connecting number words (from 0 to 20) and numerals (from 0 to 100) to the quantities and positions that they represeusing models, representations, or number lines Demonstrates conceptual underst

addition of multiple one-digit whole numbers Demonstrates understanding of monetary valudollar notion; making change from $1.00 or less, or recognizing equivalent coin representations of the same value (va Mentally adds and subtracts whole numbers to a sum of 20; names the number that is 10 more or less than the original number; and adds orsubtracts two-digit multiples of ten

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151

emonstrates conceptual understanding of congruency by composing and decomposing two-dimensional objects using models or explanations (using triangular pattern blocks to constr gonal pattern block); and uses line symmetry to demonstrate congruent parts within a shape Demonstrates conceptual understandi rround and cover polygons

of measures appropriately and consistently, and makes conversions within systems when solving problems across

nal words in two- and three- dimensional

ven representation (pictographs with one-to-one correspondence, line plots, tally charts or tables) to answer questions related to the data, or to analyze the data to formulate conclusions

n in data in a variety of contexts by determining or using more, less, or equal

es,

Applies properties of numbers (odd and even) and field properties (commutative for addition, identity for addition, and associative for addition) to solve problems and to simplify computations involving whole numbers

Geometry and Measurement Uses properties, attributes, composition, or decomposition to sort or classify polygons or objects by a combination of two or more non-measurable or measurable attributes D

uct a figure congruent to the hexa

ng of perimeter and areas by using models or manipulatives to su Measures and uses units the content strands Demonstrates understanding of spatial relationships using location and position by using positiosituations to describe and interpret relative positions (above the surface of the desk, below the triangle on the paper); and creates and interprets simple maps and names locations on simple coordinate grids Functions and Algebra Identifies and extends to specific cases a variety of patterns (linear and non-numeric) represented in models, tables, or sequences byextending the pattern to the next element, or finding a missing element

Demonstrates conceptual understanding of equality by finding the value that will make an open sentence true (2 + = 7) (limited to one operation and limited to use addition or subtraction) Data, Statistics, and Probability Interprets a gi

Analyzes patterns, trends, or distributio Uses counting techniques to solve problems involving combinations using a variety of strategies (student diagrams, organized lists, tabltree diagrams or others) (How many ways can you make 50 cents using nickels, dimes, and quarters?)

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152

,

n appropriate makes predictions

respect to whole numbers from 0 to 999 through equivalency, resentations

, a/3, a/4, in area and

the denominator; and decimals (within a context of money) using models,

onstrates understanding of the relative magnitude of numbers from 0 to 999 by ordering whole numbers; by comparing whole

o the s

n s, or

lves problems involving addition and subtraction with regrouping; the concepts of multiplication; and addition or subtraction xt of money)

ee-ole number from a two-digit whole

number and subtracts two-digit whole numbers that are multiples of ten and three-digit whole numbers that are multiples of one hundred

For a probability event in which the sample space may or may not contain equally likely outcomes, uses experiments to describe the likelihood or chance of an event using “more likely,” “less likely,” “equally likely,” “certain,” or “impossible” In response to a teacher or student generated question or hypothesis, groups decide the most effective methods (survey, observationexperimentation) to collect the data (numerical or categorical) necessary to answer the question; collects, organizes, and appropriately displays the data; analyzes the data to draw conclusions about the question or hypothesis being tested, and whe

Section II: MATHEMATICS GLE’s

STEMS AND GRADE SPECIFIC INDICATORS – Grade 3 Number and Operation Demonstrates conceptual understanding of rational numbers with composition, decomposition, or place value using models, explanations, or other rep Demonstrates conceptual understanding of rational numbers with respect to positive fractional numbers (benchmark fractions: a/2a/6, or a/8 where a is a whole number greater than 0 and less than or equal to the denominator) as a part to whole relationshipet models where the number of parts in the whole is equal to s

explanations, or other representations Demnumbers to each other or to benchmark whole numbers (100, 250, 500, 750); or by comparing whole numbers to each other and comparingor identifying equivalent positive fractional numbers (a/2, a/3, a/4 where a is a whole number greater than 0 and less than or equal tdenominator) using models, number lines or explanation Demonstrates conceptual understanding of mathematical operations by describing or illustrating the inverse relationship between additioand subtraction of whole numbers; and the relationship between repeated addition and multiplication using models, number lineexplanations Accurately soof decimals (in the conte Mentally adds and subtracts whole numbers facts through 20; adds two-digit whole numbers; adds combinations of two-digit and thrdigit whole numbers that are multiples of ten (60 + 50, 300 + 400, 320 + 90); subtracts a one-digit wh

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153

on and

ation property of zero for single-digit whole numbers [6 x 0 = 0]) and field ion, associative for addition, identity for multiplication, and commutative for multiplication for single-

to solve problems and to simplify computation involving whole numbers

n of

onstrates conceptual understanding of congruency by matching congruent figures using reflections, translations, and rotations (flips,

Demonstrates conceptual understanding of similarity by identifying similar shapes

lygons and the area of rectangles on grids using a variety of models or anipulatives. Expresses all measures using appropriate units

Demonstrates understanding of spatial relationships using location and position by interpreting and giving directions from one location to

rhombi, trapezoids, hexagons, and circles; and builds models of rectangular prisms from three-dimensional epresentations rade 3, page 3

unctions and Algebra

dentifies and extends to specific cases a variety of patterns (linear and non-numeric) represented in models, tables, or sequences by xtending the pattern to the next one, two, or three elements element, or finding a missing elements

Makes estimates in a given situation by identifying when estimation is appropriate; selecting the appropriate methods of estimatievaluating the reasonableness of solutions appropriate to grade level GLEs across content strands Applies properties of numbers (odd, even and multiplicproperties (commutative for additdigit whole numbers [3 x 4 – 4 x 3]

Geometry and Measurement Uses properties or attributes of angles (number of angles) or sides (number of sides or length of sides) or composition or decompositioshapes to identify, describe, or distinguish among triangles, squares, rectangles, rhombi, trapezoids, hexagons, or circles Demslides, and turns) (recognizing when pentominoes are reflections, translations and rotations of each other); composing and decomposing two- and three-dimensional objects using models or explanations (given a cube, students use blocks to construct a congruent cube); and uses line symmetry to demonstrate congruent parts within a shape

Demonstrates conceptual understanding of perimeter of pom Measures and uses units of measures appropriately and consistently, and makes conversions within systems when solving problems across the content strands

another (classroom to gym, from school to home) using positional words; and between locations on a map or coordinate grid (first quadrant) using positional words or compass directions Demonstrates conceptual understanding of spatial reasoning and visualization by copying, comparing, and drawing models of triangles, squares, rectangles,rG F Ie

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154

emonstrates conceptual understanding of equality by showing equivalence between two expressions using models or different epresentations of the expression; or by finding the value that will make an open sentence true (2 + = 7) (limited to one operation and mited to use addition, subtraction, or multiplication)

ata, Statistics, and Probability

nterprets a given representation (line plots, tally charts, tables, or bar graphs) to answer questions related to the data, or to analyze the data o formulate conclusions, or to make predictions

Analyzes patterns, trends, or distribution in data in a variety of contexts by determining or using most frequent (mode), least frequent, largest, or smallest Identifies or describes representations or elements of representations that best display a given set of data or situation, consistent with the representations required in M(DSP)-3-1; and organizes and displays ta using tables, tally charts, and bar graphs to answer questions related to the data, to analyze the data, to form lve problems

to solve problems involving combinations and simple permutations using a variety of strategies (student

s, determines the likelihood or chance of an event (using “more likely,” “less likely,” “equally likely,”); and predicts the likelihood of an event using “more likely,” “less likely,”

priate

Drli D It

daulate conclusions, to make predictions, or to so

Uses counting techniquesdiagrams, organized lists, tables, tree diagrams or others) For a probability event in which the sample space may or may not contain equally likely outcome

“equally likely,” “certain,” or “impossible” and tests the prediction through experiments; and determines if a game is fair In response to a teacher or student generated question or hypothesis, groups decide the most effective methods (survey, observation, experimentation) to collect the data (numerical or categorical) necessary to answer the question; collects, organizes, and Appropriately displays the data; analyzes the data to draw conclusions about the question or hypothesis being tested, and when appromakes predictions

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Section II: MATHEMATICS GLE’s STEMS AND GRADE SPECIFIC INDICATORS – Grade 4

using models, explanations, or other representations

, a/3, a/4, and less than or equal to the denominator) as a part to whole is equal to the

denominator) as a part to whole relationship in area, set, or linear models where the number of parts in the whole are equal to, and a

Demonstrates conceptual understanding of mathematical operations by describing or illustrating the relationship between repeated

or explanations

tion of decimals and positive proper fractions with like denominators (Multiplication limited to 2 digits by 2 digits, and division limited to 1 digit divisors)

Number and Operation Demonstrates conceptual understanding of rational numbers with respect to whole numbers from 0 to 999,999 through equivalency, composition, decomposition, or place value Demonstrates conceptual understanding of rational numbers with respect to positive fractional numbers (benchmark fractions: a/2a/5, a/6, a/8 or a/10 where a is a whole number greater than 0

multiple or factor of the denominator; and decimals as hundredths within a context of money, or tenth within the context of metric measurement using models, explanations, or other representations Demonstrates understanding of the relative magnitude of numbers from 0 to 999,999 by ordering or comparing whole numbers; and ordering, comparing, or identifying equivalent proper positive fractional numbers; or decimals using models, number lines or explanations

subtraction and division (no remainders); the inverse relationship between multiplication and division of whole numbers, or the addition or subtraction of positive fractional numbers with like denominators using models, number lines, Accurately solves problems involving multiple operations on whole numbers or the use of the properties of factors and multiples; and addition or subtrac

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156

e multiples of ten

mbers (odd, even, multiplicative property of zero, and remainders) and field properties (commutative, associative, blems and to simplify computations

ses properties or attributes of angles (number of angles) or sides (number of sides or length of sides, parallelism, or perpendicularity) to lative

ses properties or attributes (shape of bases or number of lateral faces) to identify, compare, or describe three-dimensional shapes

s using models or explanations

anding of similarity by applying scales on maps, or applying characteristics of similar figures (same shape ut not necessarily the same size) to identify similar figures, or to solve problems involving similar figures. Describes relationships using

emonstrates conceptual understanding of perimeter of polygons and the area of rectangles, polygons or irregular shapes on grids using a

easures and uses units of measures appropriately and consistently, and makes conversions within systems when solving problems across

Mentally adds and subtracts whole numbers facts through 20; multiplies whole number facts to a product of 100 and calculates related division facts; adds two-digit whole numbers, combinations of two-digit and three-digit whole numbers that are multiples of ten and 4 digitwhole numbers that are multiples of 100 (limited to two addends); and subtracts a one-digit whole number from a two-digit whole number nd subtracts combinations of two-digit and three-digit whole numbers that ara

Makes estimates in a given situation by identifying when estimation is appropriate; selecting the appropriate methods of estimation and evaluating the reasonableness of solutions appropriate to grade level GLEs across content strands Applies properties of nund identity) to solve proa

Geometry and Measurement Uidentify, describe, or distinguish among triangles, squares, rectangles, rhombi, trapezoids, hexagons, or octagons; or classify angles reto 90° as more than, less than, or equal to U(rectangular prisms, triangular prisms, cylinders, or spheres) Demonstrates conceptual understanding of congruency by matching congruent figures using reflections, translations, and rotations (flips, slides, and turns) or as the result of composing and decomposing shape Demonstrates conceptual understbmodels or explanations Dvariety of models, manipulatives, or formulas. Expresses all measures using appropriate units Mthe content strands

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157

s on

emonstrates conceptual understanding of linear relationships (y = kx) as a constant rate of change by identifying, describing, or comparing situations that represent constant rates of change Demonstrates conceptual understanding of algebraic expressions by using letters of symbols to represent unknown quantities to write simple linear algebraic expressions involving any one of the four operations; or by evaluating simple linear algebraic expressions using whole numbers

emonstrates conceptual understanding of expressions using models or different pression; by simplifying numerical expressions where left to right computations may be modified only by the use

of parentheses 14 – (2 x 5) (expressions consistent with the parameters of M (F & A)-4-3) and by solving one-step linear equations of the form ax = c, x ± b = c, where a, b, and c are whole numbers with a ≠ 0 Data, Statistics, and Probability Interprets a given representation (line plots, tally charts, tables, bar graphs, pictographs, or circle graphs) to answer questions related to the data, to analyze the data to formulate or justify conclusions, to make predictions, or to solve problems Analyzes patterns, trends, or distribution in data in a variety of contexts by determining or using measures of central tendency (median or mode), or range Organizes and displays data using tables, line plots, bar graphs, and pictographs to answer questions related to the data, to analyze the data, to formulate or justify conclusions, to make predictions, or to solve problems

Demonstrates understanding of spatial relationships using location and position by interpreting and giving directions between locationa map or coordinate grid (first quadrant); plotting points in the first quadrant in context (games, mapping); and finding the horizontal and ertical distances between points on a coordinate grid in the first quadrant v

Demonstrates conceptual understanding of spatial reasoning and visualization by copying, comparing, and drawing models of triangles, squares, rectangles, rhombi, trapezoids, hexagons, octagons, and circles; and builds models of rectangular prisms from two-or three-imensional representations d

Functions and Algebra Identifies and extends to specific cases a variety of patterns (linear and non-numeric) represented in models, tables, or sequences; and writes a rule in words or symbols to find the next one D

D equality by showing equivalence between two representations of the ex

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158

Uses counting techniques to solve problems in context involving combinations or simple permutations (Given a map – determine the number of paths from point A to point B) using a variety of strategies (organized lists, tables, tree diagrams or others) For a probability event in which the sample space may or may not contain equally likely outcomes, determines the theoretical probability of an event and expresses the result as part to whole (two out of five); and predicts the likelihood of an event as a part to whole relationship and tests the prediction through experiments, and determines if a game is fair In response to a teacher or student generated question or hypothesis, groups decide the most effective methods (survey, observation, experimentation) to collect the data (numerical or categorical) necessary to answer the question; collects, organizes, and appropriately displays the data; analyzes the data to draw conclusions about the question or hypothesis being tested, and when appropriate makes predictions; and asks new questions and makes connections to real world situations

Section II: MATHEMATICS GLE’s STEMS AND GRADE SPECIFIC INDICATORS – Grade 5

Number and Operation Demonstrates conceptual understanding of rational numbers with respect to whole numbers from 0 to 9,999,999 through equivalency, composition, decomposition, or place value using models, explanations, or other representations* *Specifications for area, set, and linear models for grades 5 – 8: Fractions: The number of parts in the whole is equal to the denominator, a multiple of the denominator, or a factor of the denominator Percents: The number of parts in the whole is equal to 100, a multiple of 100, or a factor of 100 Decimals (including powers of ten): The number of parts in the whole is equal to the denominator of the fractional equivalent of the decimal, a multiple of the denominator of the fractional equivalent of the decimal, or a factor of the denominator of the fractional equivalent of the decimal Demonstrates conceptual understanding of rational numbers with respect to positive fractional numbers (proper, mixed number, and improper) (halves, fourths, eighths, thirds, sixths, twelfths, fifths, or powers of ten), decimals (to thousandths) or benchmark percents (10%, 25%, 50%, 75%, or 100%) as a part to whole relationship in area, set, or linear models using models, explanations, or other representations

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Demonstrates understanding of the relative magnitude of numbers by ordering, comparing, or identifying equivalent positive fractional numbers, decimals, or benchmark percents within number formats (fractions to fractions, decimals to decimals, or percents to percents); or integers in context using models or number lines Demonstrates conceptual understanding of mathematical operations by describing or illustrating the meaning of a remainder with respect to division of whole numbers using models, explanations, or solving problems; and addition and subtraction of decimals and positive proper fractions with unlike denominators Accurately solves problems involving multiple operations on whole numbers on whole numbers or the use of the properties of factors, multiples, prime, or composite numbers; and addition or subtraction of fractions (proper) and decimals to the hundredths place. (Division of whole numbers by up to a two-digit divisor) Mentally calculates change back from $1.00, $5.00, and $10.00; calculates multiplication and related division facts to a product of 144; multiplies a two-digit whole number by a one-digit whole number, two-digit whole numbers that are a multiple of ten, a three-digit whole number that is a multiple of 100 by a two- or three-digit number which is a multiple of 10 or 100, respectively (400 x 50, 400 x 600); and divides three- and four-digit multiples of powers of ten by their compatible factors Makes estimates in a given situation by identifying when estimation is appropriate; selecting the appropriate methods of estimation, determining the level of accuracy needed given the situation, analyzing the effect of the estimation method on the accuracy of results, and evaluating the reasonableness of solutions appropriate to grade level GLEs across content strands Applies properties of numbers (odd, even, and divisibility) and field properties (commutative, associative, identity, and distributive) to solve problems and to simplify computations

Geometry and Measurement Uses properties or attributes of angles (right, acute, or obtuse) or sides (number of congruent sides, parallelism, or perpendicularity) to identify, describe, classify or distinguish among different types of triangles (right, acute, obtuse, equiangular, or equilateral) or quadrilaterals (rectangles, squares, rhombi, trapezoids, or parallelograms) Uses properties or attributes (shape of bases, number of lateral faces, or number of bases) to identify, compare, or describe three-dimensional shapes (rectangular prisms, triangular prisms, cylinders, spheres, pyramids, or cones) Demonstrates conceptual understanding of congruency by matching congruent figures using reflections, translations, and rotations (flips, slides, and turns) or as the result of composing and decomposing shapes using models or explanations Demonstrates conceptual understanding of similarity by describing the proportional effect on the linear dimensions of triangles and rectangles when scaling up or down while preserving angle measures, or by solving related problems (including applying scales on maps). Describes relationships using models or explanations

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Demonstrates conceptual understanding of perimeter of polygons and the area of rectangles or right triangles though models, manipulatives, or formulas, the area of polygons or irregular figures on grids, and volume or rectangular prisms (cubes) using a variety of models, manipulatives, or formulas. Expresses all measures using appropriate units Measures and uses units of measures appropriately and consistently, and makes conversions within systems when solving problems across the content strands Demonstrates understanding of spatial relationships using location and position by interpreting and giving directions between locations on a map or coordinate grid (all four quadrants); plotting points in four quadrants in context (games, mapping, identifying the vertices of polygons as they are reflected, rotated, and translated); and determining horizontal and vertical distances between points on a coordinate grid in the first quadrant Demonstrates conceptual understanding of spatial reasoning and visualization by building models of rectangular and triangular prisms, cones, cylinders and pyramids from two- or three-dimensional representations Functions and Algebra Identifies and extends to specific cases a variety of patterns (linear and non-numeric) represented in models, tables, sequences, or in problem situations; or writes a rule in words or symbols for finding specific cases of a linear relationship Demonstrates conceptual understanding of linear relationships (y = kx) as a constant rate of change by identifying, describing, or comparing situations that represent constant rates of change (tell a story given a line graph about a trip) Demonstrates conceptual understanding of algebraic expressions by using letters or symbols to represent unknown quantities to write linear algebraic expressions involving any two of the four operations; or by evaluating linear algebraic expressions using whole numbers Demonstrates conceptual understanding of equality by showing equivalence between two expressions using models or different representations of the expression (expressions consistent with the parameters of M (F & A)-5-3), by solving one-step linear equations of the form ax = c, x ± b = c, or x/a = c, where a, b, and c are whole numbers with a ≠ 0 a true statement (2x + 3 = 11 {x : x = 2, 3, 4, 5}) Data, Statistics, and Probability Interprets a given representation (tables, bar graphs, circle graphs, or line graphs) to answer questions related to the data, to analyze the data to formulate or justify conclusions, to make predictions, or to solve problems Analyzes patterns, trends, or distribution in data in a variety of contexts by determining or using measures of central tendency (mean, median or mode), or range to analyze situations, or to solve problems Identifies or describes representations or elements of representations that best display a given set of data or situation, consistent with the representations required in M (DSP)-5-1; and organizes and displays data using tables, bar graphs, or line graphs to answer questions related to the data, to analyze the data, to formulate or justify conclusions, to make predictions, or to solve problems

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Uses counting techniques to solve problems in context involving combinations or simple permutations using a variety using a variety of strategies (organized lists, tables, tree diagrams, models or others) For a probability event in which the sample space may or may not contain equally likely outcomes, determines the experimental or theoretical probability of an event and expresses the result as fraction; and predicts the likelihood of an event as a fraction and tests the prediction through experiments, and determines if a game is fair

In response to a teacher or student generated question or hypothesis, groups decide the most effective methods (survey, observation,

experimentation) to collect the data (numerical or categorical) necessary to answer the question; collects, organizes, and appropriately displays the data; analyzes the data to draw conclusions about the question or hypothesis being tested, and when appropriate makes

predictions; and asks new questions and makes connections to real world situations

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Section II: MATHEMATICS GLE’s STEMS AND GRADE SPECIFIC INDICATORS – Grade 6

Number and Operation Demonstrates conceptual understanding of rational numbers with respect to ratios (comparison of two whole numbers by division a/b, a: b, and a ÷ b and where b ≠ 0); and rates (a out of b, 25%) using models, explanations, or other representations * *Specifications for area, set, and linear models for grades 5 – 8: Fractions: The number of parts in the whole are equal to the denominator, a multiple of the denominator, or a factor of the denominator Percents: The number of parts in the whole is a multiple or a factor of the numeric value representing the whole Decimals (including powers of ten): The number of parts in the whole is equal to the denominator of the fractional equivalent of the decimal, a multiple of the denominator of the fractional equivalent of the decimal, or a factor of the denominator of the fractional equivalent of the decimal Demonstrates understanding of the relative magnitude of numbers by ordering or comparing numbers with whole number bases and whole number exponents, integers, or rational numbers within and across number formats (fractions, decimals, or whole number percents from 1 – 100) using number lines or equality and inequality symbols Demonstrates conceptual understanding of mathematical operations by describing or illustrating the meaning of a power by representing the relationship between the base (whole number) and the exponent (whole number); and the effect on the magnitude of a whole number when multiplying or dividing it by a whole number, decimal or fraction; addition and subtraction of positive fractions and integers; and multiplication and division of fractions and decimals Accurately solves problems involving single or multiple operations on fractions) proper, improper, and mixed); or decimals; and addition or subtraction of integers; percent of a whole; or problems involving greatest common factor or least common multiple Mentally calculates change back from $5.00, $10.00, $20.00, $50.00 and $100.00; multiplies a two-digit whole number by a one-digit whole number, two-digit whole numbers that are a multiple of ten, a three-digit whole number that is a multiple of 100 by a two- or three-digit number which is a multiple of 10 or 100, respectively (400 x 50, 400 x 600); and divides three- and four-digit multiples of powers of ten by their compatible factors and determines the part of a whole number using benchmarks percents (1%, 10%, 25%, 50%, and 75%) Makes estimates in a given situation by identifying when estimation is appropriate; selecting the appropriate methods of estimation, determining the level of accuracy needed given the situation, analyzing the effect of the estimation method on the accuracy of results, and evaluating the reasonableness of solutions appropriate to grade level GLEs across content strands

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Applies properties of numbers (odd, even, remainders, divisibility, and prime factorization) and field properties (commutative, associative, identity [including the multiplicative property of one, distributive, and additive inverse) to solve problems and to simplify computations Geometry and Measurement Uses properties or attributes of angles (right, acute, or obtuse) or sides (number of congruent sides, parallelism, or perpendicularity) to identify, describe, classify or distinguish among different types of triangles (right, acute, obtuse, equiangular, scalene, isosceles, or equilateral) or quadrilaterals (rectangles, squares, rhombi, trapezoids, or parallelograms) Uses properties or attributes (shape of bases, number of lateral faces, number of bases, number of edges, or number of vertices) to identify, compare, or describe three-dimensional shapes (rectangular prisms, triangular prisms, cylinders, spheres, pyramids, or cones) Demonstrates conceptual understanding of congruency by predicting and describing the transformational steps (reflections, translations, and rotations) needed to show congruence (including the degree rotation) and as the result of composing and decomposing two- and three-dimensional objects using models or explanations; and using line and rotational symmetry to demonstrate congruent parts within a shape Demonstrates conceptual understanding of similarity by describing the proportional effect on the linear dimensions of polygons or circles when scaling up or down while preserving angles of polygons, or by solving related problems (including applying scales on maps). Describes relationships using models or explanations Demonstrates conceptual understanding of perimeter of polygons and the area of quadrilaterals, or triangles and the volume of rectangular prisms by using a variety of models, formulas, or by solving problems; and demonstrates understanding of the relationships of circle measures (radius to diameter and diameter to circumference) by solving related problems. Expresses all measures using appropriate units Measures and uses units of measures appropriately and consistently, and makes conversions within systems when solving problems across the content strands Functions and Algebra Identifies and extends to specific cases a variety of patterns (linear and non-numeric) represented in models, tables, sequences, graphs, or in problem situations; or writes a rule in words or symbols for finding specific cases of a linear relationship; or writes a rule in words or symbols for finding specific cases of a nonlinear relationship; and writes an expression or equation using words or symbols to express the generalization of a linear relationship (twice the term number plus 1 or 2n + 1) Demonstrates conceptual understanding of linear relationships (y = kx; y = mx + b) as a constant rate of change by constructing or interpreting graphs of real occurrences and describing the slope of linear relationships (faster, slower, greater, or smaller) in a variety of

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problem situations; and describes how change in the value of one variable relates to change in the value of a second variable in problem situations with constant rates of change Demonstrates conceptual understanding of algebraic expressions by using letters to represent unknown quantities to write linear algebraic expressions involving any of the four operations and consistent with order of operations expected at this grade level; or by evaluating linear algebraic expressions (including those with more than one variable); or by evaluating an expression within an equation (determine the value of y when x = 4 given y = 3x – 2) using whole numbers Demonstrates conceptual understanding of equality by showing equivalence between two expressions using models or different representations of the expression (expressions consistent with the parameters of M (F & A)-6-3), solving multi-step linear equations of the form ax = c, x ± b = c, where a, b, and c are whole numbers with a ≠ 0 Data, Statistics, and Probability Interprets a given representation (circle graphs, line graphs or stem-and-leaf plots) to answer questions related to the data, to analyze the data to formulate or justify conclusions, to make predictions, or to solve problems Analyzes patterns, trends, or distribution in data in a variety of contexts by determining or using measures of central tendency (mean, median or mode), or dispersion (range) to analyze situations, or to solve problems Organizes and displays data using tables, line graphs, or stem-and-leaf plots to answer questions related to the data, to analyze the data, to formulate or justify conclusions, to make predictions, or to solve problems Uses counting techniques to solve problems in context involving combinations or simple permutations using a variety using a variety of strategies (organized lists, tables, tree diagrams, models, Fundamental Counting Principle or others) For a probability event in which the sample space may or may not contain equally likely outcomes, determines the experimental or theoretical probability of an event in a problem-solving situation; and predicts the theoretical probability of an event and tests the prediction through experiments and simulations, and designs fair games In response to a teacher or student generated question or hypothesis, groups decide the most effective methods (survey, observation, experimentation) to collect the data (numerical or categorical) necessary to answer the question; collects, organizes, and appropriately displays the data; analyzes the data to draw conclusions about the question or hypothesis being tested, and when appropriate makes predictions; and asks new questions and makes connections to real world situations

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Section II: Mathematics GLE’s STEMS AND GRADE SPECIFIC INDICATORS – Grade 7

Number and Operation Demonstrates conceptual understanding of rational numbers with respect to percents as a means of comparing the same or different parts of the whole when the whole vary in magnitude; and percent as a way of expressing multiples of a number, using models, explanations, or other representations * and demonstrates conceptual understanding of square roots of perfect squares, rates, and proportional reasoning *Specifications for area, set, and linear models for grades 5 – 8: Fractions: The number of parts in the whole are equal to the denominator, a multiple of the denominator, or a factor of the denominator Percents: The number of parts in the whole is a multiple or a factor of the numeric value representing the whole Decimals (including powers of ten): The number of parts in the whole is equal to the denominator of the fractional equivalent of the decimal, a multiple of the denominator of the fractional equivalent of the decimal, or a factor of the denominator of the fractional equivalent of the decimal Demonstrates understanding of the relative magnitude of numbers by ordering, comparing, or identifying equivalent rational numbers across number formats, numbers with whole number bases and whole number exponents, integers, absolute values, or numbers represented in scientific notation using number lines or equality and inequality symbols Demonstrates conceptual understanding of mathematical operations with integers and whole number exponents (where the base is a whole number) using models, diagrams, or explanations Accurately solves problems involving proportional reasoning, percents involving discounts, tax, or tips; and rates; and addition or subtraction of integers, raising numbers to whole number powers, and determining square roots of perfect square numbers and non-perfect square numbers Mentally calculates benchmark perfect squares and related square roots; determines the part of a number using benchmark percents and related fractions Makes estimates in a given situation (including tips, discounts, and tax) by identifying when estimation is appropriate; selecting the appropriate methods of estimation, determining the level of accuracy needed given the situation, analyzing the effect of the estimation method on the accuracy of results, and evaluating the reasonableness of solutions appropriate to grade level GLEs across content strands Applies properties of numbers (odd, even, remainders, divisibility, and prime factorization) and field properties (commutative, associative, identity, distributive, inverse) to solve problems and to simplify computations, and demonstrate conceptual understanding of field

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properties as they apply to subsets if real numbers (the set of the whole numbers does not have additive inverse, the set of integers does not have multiplicative inverse) Geometry and Measurement Uses properties or attributes of angle relationships resulting from two or three intersecting lines (adjacent angles, vertical angles, straight angles, or angle relationships formed by two non-parallel lines cut by a transversal) or two parallel lines cut by a transversal to solve problems Applies theorems or relationships (triangle inequality or sum of the measures of interior angles of regular polygons) to solve problems Applies the concept of congruency by solving problems on a coordinate plane involving reflections, translations, or rotations Applies concepts of similarity by solving problems involving scaling up or down and their impact on angle measures, linear dimensions and areas of polygons, and circlers when the linear dimensions are multiplied by a constant factor. Describes effects using models or explanations Demonstrates conceptual understanding of the area of circles or the area or perimeter of composite figures (quadrilaterals, triangles, or parts of circles) and the surface area of rectangular prisms, or volume of rectangular prisms, triangular prisms, or cylinders using models, formulas, or by solving problems. Expresses all measures using appropriate units Measures and uses units of measures appropriately and consistently, and makes Demonstrates conceptual understanding of spatial reasoning and visualization by sketching three-dimensional solids; and draws nets of rectangular and triangular prisms, cylinders, and pyramids and uses the nets as a technique for finding surface area Functions and Algebra Identifies and extends to specific cases a variety of patterns (linear and non-numeric) represented in models, tables, sequences, graphs, or in problem situations; and generalizes a linear relationship using words and symbols; generalizes a linear relationship to find a specific case; or writes an expression or equation using words or symbols to express the generalization of a linear relationship Demonstrates conceptual understanding of linear relationships (y = kx; y = mx + b) as a constant rate of change by solving problems involving the relationship between slope and rate of change, by describing the meaning of slope in concrete situations, or informally determining the slope of a line from a table or graph; and distinguishes between constant and varying rates of change in concrete situations represented in tables or graphs; or describes how change in the value of one variable relates to change in the value of a second variable in problem situations with constant rates of change

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Demonstrates conceptual understanding of algebraic expressions by using letters to represent unknown quantities to write algebraic expressions (including those with whole number exponents or more than one variable); or by evaluating algebraic expressions (including those with whole number exponents or more than one variable); or by evaluating an expression within an equation (determine the value of y when x = 4 given y – 5x3 - 2) Demonstrates conceptual understanding of equality by showing equivalence between two expressions (expressions consistent with the parameters of the left- and right-hand sides of equations being solved at this grade level) using models or different representations of the expression, solving multi-step linear equations of the form ax = c with a ≠ 0, ax ± b = cx ± d, with a, c ≠ 0 and (x/a) ± b= c with a ≠ 0, where a, b, c, and d are whole numbers; or by translating a problem solving situation into an equation consistent with the parameters of the type of equations being solved for this grade level Data, Statistics, and Probability Interprets a given representation (circle graphs, scatter plots that represent discrete linear relationships, or histograms) to answer questions related to the data, to analyze the data to formulate or justify conclusions, to make predictions, or to solve problems Analyzes patterns, trends, or distribution in data in a variety of contexts by solving problem using measures of central tendency (mean, median or mode), or dispersion (Range or variation), or outliers to analyze situations to determine their effect on mean, median, or mode; and evaluate the sample from which the statistics were developed (bias) Identifies or describes representations or elements of representations that best display a given set of data or situation, consistent with the representations required in M (DSP)-7-1; and organizes and displays data using tables, line graphs, scatter plots, and circle graphs to answer questions related to the data, to analyze the data, to formulate or justify conclusions, to make predictions, or to solve problems Uses counting techniques to solve problems in context involving combinations or simple permutations using a variety using a variety of strategies (organized lists, tables, tree diagrams, models, Fundamental Counting Principle or others) For a probability event in which the sample space may or may not contain equally likely outcomes, determines the experimental or theoretical probability of an event in a problem-solving situation; and predicts the theoretical probability of an event and tests the prediction through experiments and simulations, and compares and contrasts theoretical and experimental probabilities In response to a teacher or student generated question or hypothesis, groups decide the most effective methods (survey, observation, experimentation) to collect the data (numerical or categorical) necessary to answer the question; collects, organizes, and appropriately displays the data; analyzes the data to draw conclusions about the question or hypothesis being tested while considering the limitations that

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could affect interpretations; and when appropriate makes predictions; and asks new questions and makes connections to real world situations

Section II: MATHEMATICS GLE’s STEMS AND GRADE SPECIFIC INDICATORS – Grade 8

Number and Operation (Local Option) Demonstrates conceptual understanding of rational numbers with respect to absolute values, perfect square and cube roots, and percents as a way of describing change (percent increase and decrease) using explanation, models, or other representations* *Specifications for area, set, and linear models for grades 5 – 8: Fractions: The number of parts in the whole are equal to the denominator, a multiple of the denominator, or a factor of the denominator Percents: The number of parts in the whole is a multiple or a factor of the numeric value representing the whole Decimals (including powers of ten): The number of parts in the whole is equal to the denominator of the fractional equivalent of the decimal, a multiple of the denominator of the fractional equivalent of the decimal, or a factor of the denominator of the fractional equivalent of the decimal Demonstrates understanding of the relative magnitude of numbers by ordering or comparing rational numbers, common irrational numbers, numbers with whole number or fractional bases and whole number exponents, square roots, absolute values, integers or numbers represented in scientific notation using number lines or equality and inequality symbols Accurately solves problems involving proportional reasoning (percent increase or decrease, interest rates, markups, or rates); multiplication or division of integers; and squares, cubes, and taking square or cube roots Mentally calculates benchmark perfect squares and related square roots; determines the part of a number using benchmark percents and related fractions Makes estimates in a given situation (including tips, discounts, tax, and the value of a non-perfect square root as between two whole numbers) by identifying when estimation is appropriate; selecting the appropriate methods of estimation, determining the level of accuracy needed given the situation, analyzing the effect of the estimation method on the accuracy of results, and evaluating the reasonableness of solutions appropriate to grade level GLEs across content strands Applies properties of numbers (odd, even, remainders, divisibility, and prime factorization) and field properties (commutative, associative, identity [including the multiplicative property of one], distributive, inverse) to solve problems and to simplify computations, and demonstrate conceptual understanding of field properties as they apply to subsets if real numbers when addition and multiplication are not defined in the traditional ways

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Geometry and Measurement Applies the Pythagorean Theorem to find a missing side of a right triangle, or in problem solving situations Applies concepts of similarity to determine the impact of scaling on the volume or surface area of three-dimensional figures when linear dimensions are multiplied by a constant factor; to determine the length of sides of similar triangles, or to solve problems involving growth and rate Demonstrates conceptual understanding of surface area or volume by solving problems involving surface area and volume of rectangular prisms, cylinders, pyramids, or cones. Expresses all measures using appropriate units Functions and Algebra (Local Option) Identifies and extends to specific cases a variety of patterns (linear and non-numeric) represented in models, tables, sequences, graphs, or in problem situations; and generalizes a linear relationship (non-recursive explicit equation); generalizes a linear relationship to find a specific case; generalizes a nonlinear relationship using words or symbols; or generalizes a common nonlinear relationship to find a specific case Demonstrates conceptual understanding of linear relationships (y = kx; y = mx + b) as a constant rate of change by solving problems involving the relationship between slope and rate of change; informally and formally determining slopes and intercepts represented in graphs, tables, or problem situations; or describing the meaning of slope and intercept in context; and distinguishes between linear relationships (constant rates of changes) and nonlinear relationships (varying rates of change) represented in tables, graphs, equations, or problem situations; or describes how change in the value of one variable relates to change in the value of a second variable in problem situations with constant and varying rates of change Demonstrates conceptual understanding of algebraic expressions by evaluating and simplifying algebraic expressions (including those with square roots, whole number exponents, or rational numbers); or by evaluating an expression within an equation (determine the value of y when x = 4 given y = 7 √x + 2x) Demonstrates conceptual understanding of equality by showing equivalence between two expressions (expressions consistent with the parameters of the left- and right-hand sides of equations being solved at this grade level) using models or different representations of the expression, solving formulas for one variable requiring one transformation (d = rt; d/r = t); by solving multi-step linear equations with integer coefficients; by showing that two expressions are or are not equivalent by applying commutative, associative, or distributive properties, order of operations, or substitution; and by informally solving problems involving systems of linear equations in a context

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Data, Statistics, and Probability (Local Option) Interprets a given representation (circle graphs, scatter plots, histograms, or box-and-whisker plots) to answer questions related to the data, to analyze the data to formulate or justify conclusions, to make predictions, or to solve problems Analyzes patterns, trends, or distribution in data in a variety of contexts by solving problem using measures of central tendency (mean, median or mode), or dispersion (range or variation), or outliers, quartile values, or estimated line of best fit to analyze situations to determine their effect on mean, median, or mode; and evaluate the sample from which the statistics were developed (bias, random, or non-random) Organizes and displays data using scatter plots to answer questions related to the data, to analyze the data, to formulate or justify conclusions, to make predictions, or to solve problems; or identifies representations or elements of representations that best display a given set of data or situation, consistent with the representations required in M(DSP)-8-1 Uses counting techniques to solve problems in context involving combinations or simple permutations using a variety using a variety of strategies (organized lists, tables, tree diagrams, models, Fundamental Counting Principle or others) For a probability event in which the sample space may or may not contain equally likely outcomes, determines the experimental or theoretical probability of an event in a problem-solving situation; and predicts the theoretical probability of an event and tests the prediction through experiments and simulations, and compares and contrasts theoretical and experimental probabilities In response to a teacher or student generated question or hypothesis, groups decide the most effective methods (survey, observation, experimentation) to collect the data (numerical or categorical) necessary to answer the question; collects, organizes, and appropriately displays the data; analyzes the data to draw conclusions about the question or hypothesis being tested while considering the limitations that could affect interpretations; and when appropriate makes predictions; and asks new questions and makes connections to real world situations

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Section III: MATHEMATICS GLE’s (State Only)

STEMS AND GRADE SPECIFIC INDICATORS Number and Operation Kindergarten - Demonstrates conceptual understanding of rational numbers with respect to whole numbers from 0 to 12 through investigations that apply the concepts of equivalency in composing and decomposing numbers using models, explanations, or other representations Demonstrates conceptual understanding of rational numbers with respect to positive fractional numbers (1/2) as “fair share” (equal sized parts or sets) using models, explanations, or other representations Demonstrates understanding of the relative magnitude of numbers from 0 to 20 through investigations that demonstrate one-to-one correspondence that compare whole numbers to each other or to benchmark whole numbers (5, 10); that demonstrate an understanding of the relation of inequality when comparing whole numbers by using “1 more” or “1 less”; that connect numbers orally and written as numerals to the quantities that they represent using models, representations, or number lines Demonstrates conceptual understanding of mathematical operations through investigations involving addition and subtraction of whole numbers (0 to 10) by solving problems involving joining actions, separating actions, part-part whole relationships, and comparison situations; and addition of multiple one-digit whole numbers Demonstrates understanding of monetary value through investigation involving knowing the names and values for coins (penny, nickel and dime) Mentally adds and subtracts whole numbers by naming the number that is one more or one less than the original number Makes estimates of the number of objects in a set (up to 20) by making and revising estimates as objects are counted Grade 1 - Demonstrates conceptual understanding of rational numbers with respect to whole numbers from 0 to 100 using place value, by applying the concepts of equivalency in composing and decomposing numbers and in expanded notation using models, explanations, or other representations Demonstrates conceptual understanding of rational numbers with respect to positive fractional numbers (benchmark fractions: a/2, a/3, or a/4 where a is a whole number greater than 0 and less than or equal to the denominator) as a part to whole relationship in area models where the denominator is equal to the number of parts in the whole using models, explanations, or other representations

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Demonstrates understanding of the relative magnitude of numbers from 0 to 100 by ordering whole numbers; by comparing whole numbers to each other or to benchmark whole numbers (5,10, 25, 50, 75, 100); by demonstrating an understanding of the relation of inequality when comparing whole numbers by using “1 more”, “1 less”, “5 more”, “5 less”, “10 more” or “10 less”; and by connecting number words (from 0 to 20) and numerals (from 0 to 100) to the quantities and positions that they represent using models, representations, or number lines Demonstrates conceptual understanding of mathematical operations through investigations involving addition and subtraction of whole numbers (0 to 30) by solving problems involving joining actions, separating actions, part-part whole relationships, and comparison situations; and addition of multiple one-digit whole numbers Demonstrates understanding of monetary value by knowing the names and values for coins (penny, nickel, dime and quarter) and by adding collections of like coins together to a sum no greater than $1.00 Mentally adds and subtracts whole numbers by naming the number that is one or two more or one less than the original number; and adds or subtracts whole number facts to ten Makes estimates of the number of objects in a set (up to 30) by making and revising estimates as objects are counted Applies properties of numbers (odd, even, composition, and decomposition [5 is the same as 2 + 3]) and field properties (commutative and identity for addition) to solve problems and to simplify computations involving whole numbers Grade 2- Demonstrates conceptual understanding of rational numbers with respect to whole numbers from 0 to 199 using place value, by applying the concepts of equivalency in composing and decomposing numbers (34 = 17 +17) and in expanded notation 141 = 100 + 40 + 7) using models, explanations, or other representations Demonstrates conceptual understanding of rational numbers with respect to positive fractional numbers (benchmark fractions: a/2, a/3, or a/4 where a is a whole number greater than 0 and less than or equal to the denominator) as a part to whole relationship in area and set models where the denominator is equal to the number of parts in the whole using models, explanations, or other representations Demonstrates understanding of the relative magnitude of numbers from 0 to 199 by ordering whole numbers; by comparing whole numbers to each other or to benchmark whole numbers (5,10, 25, 50, 75, 100, 125, 150 or 175); by demonstrating an understanding of the relation of inequality when comparing whole numbers by using “1 more”, “1 less”, “5 more”, “5 less”, “10 more”, “10 less” “100 more” or “100 less”; and by connecting number words (from 0 to 20) and numerals (from 0 to 100) to the quantities and positions that they represent using models, representations, or number lines

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Demonstrates conceptual understanding of mathematical operations through investigations involving addition and subtraction of whole numbers by solving problems involving joining actions, separating actions, part-part whole relationships, and comparison situations; and addition of multiple one-digit whole numbers Demonstrates understanding of monetary value by adding coins together to a value no greater than $1.99 and representing the result in dollar notion; making change from $1.00 or less, or recognizing equivalent coin representations of the same value (value up to $1.99) Mentally adds and subtracts whole numbers to a sum of 20; names the number that is 10 more or less than the original number; and adds or subtracts two-digit multiples of ten Makes estimates of the number of objects in a set (up to 50) by making and revising estimates as objects are counted Applies properties of numbers (odd and even) and field properties (commutative for addition, identity for addition, and associative for addition) to solve problems and to simplify computations involving whole numbers Grade 3 - Demonstrates conceptual understanding of rational numbers with respect to whole numbers from 0 to 999 through equivalency, composition, decomposition, or place value using models, explanations, or other representations Demonstrates conceptual understanding of rational numbers with respect to positive fractional numbers (benchmark fractions: a/2, a/3, a/4, a/6, or a/8 where a is a whole number greater than 0 and less than or equal to the denominator) as a part to whole relationship in area and set models where the number of parts in the whole is equal to the denominator; and decimals (within a context of money) using models, explanations, or other representations Demonstrates understanding of the relative magnitude of numbers from 0 to 999 by ordering whole numbers; by comparing whole numbers to each other or to benchmark whole numbers (100, 250, 500, 750); or by comparing whole numbers to each other and comparing or identifying equivalent positive fractional numbers (a/2, a/3, a/4 where a is a whole number greater than 0 and less than or equal to the denominator) using models, number lines or explanations Demonstrates conceptual understanding of mathematical operations by describing or illustrating the inverse relationship between addition and subtraction of whole numbers; and the relationship between repeated addition and multiplication using models, number lines, or explanations Accurately solves problems involving addition and subtraction with regrouping; the concepts of multiplication; and addition or subtraction of decimals (in the context of money) Mentally adds and subtracts whole numbers facts through 20; adds two-digit whole numbers; adds combinations of two-digit and three-digit whole numbers that are multiples of ten (60 + 50, 300 + 400, 320 + 90); subtracts a one-digit whole number from a two-digit whole number and subtracts two-digit whole numbers that are multiples of ten and three-digit whole numbers that are multiples of one hundred

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Makes estimates in a given situation by identifying when estimation is appropriate; selecting the appropriate methods of estimation and evaluating the reasonableness of solutions appropriate to grade level GLE’s across content strands Applies properties of numbers (odd, even and multiplication property of zero for single-digit whole numbers [6 x 0 = 0]) and field properties (commutative for addition, associative for addition, identity for multiplication, and commutative for multiplication for single-digit whole numbers [3 x 4 – 4 x 3] to solve problems and to simplify computation involving whole numbers Grade 4 - Demonstrates conceptual understanding of rational numbers with respect to whole numbers from 0 to 999,999 through equivalency, composition, decomposition, or place value using models, explanations, or other representations Demonstrates conceptual understanding of rational numbers with respect to positive fractional numbers (benchmark fractions: a/2, a/3, a/4, a/5, a/6, a/8 or a/10 where a is a whole number greater than 0 and less than or equal to the denominator) as a part to whole is equal to the denominator) as a part to whole relationship in area, set, or linear models where the number of parts in the whole are equal to, and a multiple or factor of the denominator; and decimals as hundredths within a context of money, or tenth within the context of metric measurement using models, explanations, or other representations Demonstrates understanding of the relative magnitude of numbers from 0 to 999,999 by ordering or comparing whole numbers; and ordering, comparing, or identifying equivalent proper positive fractional numbers; or decimals using models, number lines or explanations Demonstrates conceptual understanding of mathematical operations by describing or illustrating the relationship between repeated subtraction and division (no remainders); the inverse relationship between multiplication and division of whole numbers, or the addition or subtraction of positive fractional numbers with like denominators using models, number lines, or explanations Accurately solves problems involving multiple operations on whole numbers or the use of the properties of factors and multiples; and addition or subtraction of decimals and positive proper fractions with like denominators (Multiplication limited to 2 digits by 2 digits, and division limited to 1 digit divisors) Mentally adds and subtracts whole numbers facts through 20; multiplies whole number facts to a product of 100 and calculates related division facts; adds two-digit whole numbers, combinations of two-digit and three-digit whole numbers that are multiples of ten and 4 digit whole numbers that are multiples of 100 (limited to two addends); and subtracts a one-digit whole number from a two-digit whole number and subtracts combinations of two-digit and three-digit whole numbers that are multiples of ten Makes estimates in a given situation by identifying when estimation is appropriate; selecting the appropriate methods of estimation and evaluating the reasonableness of solutions appropriate to grade level GLEs across content strands Applies properties of numbers (odd, even, multiplicative property of zero, and remainders) and field properties (commutative, associative, and identity) to solve problems and to simplify computations

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Grade 5 - Demonstrates conceptual understanding of rational numbers with respect to whole numbers from 0 to 9,999,999 through equivalency, composition, decomposition, or place value using models, explanations, or other representations* *Specifications for area, set, and linear models for grades 5 – 8: Fractions: The number of parts in the whole are equal to the denominator, a multiple of the denominator, or a factor of the denominator Percents: The number of parts in the whole is equal to 100, a multiple of 100, or a factor of 100 Decimals (including powers of ten): The number of parts in the whole is equal to the denominator of the fractional equivalent of the decimal, a multiple of the denominator of the fractional equivalent of the decimal, or a factor of the denominator of the fractional equivalent of the decimal Demonstrates conceptual understanding of rational numbers with respect to positive fractional numbers (proper, mixed number, and improper) (halves, fourths, eighths, thirds, sixths, twelfths, fifths, or powers of ten), decimals (to thousandths) or benchmark percents (10%, 25%, 50%, 75%, or 100%) as a part to whole relationship in area, set, or linear models using models, explanations, or other representations Demonstrates understanding of the relative magnitude of numbers by ordering, comparing, or identifying equivalent positive fractional numbers, decimals, or benchmark percents within number formats (fractions to fractions, decimals to decimals, or percents to percents); or integers in context using models or number lines Demonstrates conceptual understanding of mathematical operations by describing or illustrating the meaning of a remainder with respect to division of whole numbers using models, explanations, or solving problems; and addition and subtraction of decimals and positive proper fractions with unlike denominators Accurately solves problems involving multiple operations on whole numbers on whole numbers or the use of the properties of factors, multiples, prime, or composite numbers; and addition or subtraction of fractions (proper) and decimals to the hundredths place. (Division of whole numbers by up to a two-digit divisor) Mentally calculates change back from $1.00, $5.00, and $10.00; calculates multiplication and related division facts to a product of 144; multiplies a two-digit whole number by a one-digit whole number, two-digit whole numbers that are a multiple of ten, a three-digit whole number that is a multiple of 100 by a two- or three-digit number which is a multiple of 10 or 100, respectively (400 x 50, 400 x 600); and divides three- and four-digit multiples of powers of ten by their compatible factors Makes estimates in a given situation by identifying when estimation is appropriate; selecting the appropriate methods of estimation, determining the level of accuracy needed given the situation, analyzing the effect of the estimation method on the accuracy of results, and evaluating the reasonableness of solutions appropriate to grade level GLE’s across content strands

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Applies properties of numbers (odd, even, and divisibility) and field properties (commutative, associative, identity, and distributive) to solve problems and to simplify computations Grade 6 - Demonstrates conceptual understanding of rational numbers with respect to ratios (comparison of two whole numbers by division a/b, a: b, and a ÷ b and where b ≠ 0); and rates (a out of b, 25%) using models, explanations, or other representations * *Specifications for area, set, and linear models for grades 5 – 8: Fractions: The number of parts in the whole are equal to the denominator, a multiple of the denominator, or a factor of the denominator Percents: The number of parts in the whole is a multiple or a factor of the numeric value representing the whole Decimals (including powers of ten): The number of parts in the whole is equal to the denominator of the fractional equivalent of the decimal, a multiple of the denominator of the fractional equivalent of the decimal, or a factor of the denominator of the fractional equivalent of the decimal Demonstrates understanding of the relative magnitude of numbers by ordering or comparing numbers with whole number bases and whole number exponents, integers, or rational numbers within and across number formats (fractions, decimals, or whole number percents from 1 – 100) using number lines or equality and inequality symbols Demonstrates conceptual understanding of mathematical operations by describing or illustrating the meaning of a power by representing the relationship between the base (whole number) and the exponent (whole number); and the effect on the magnitude of a whole number when multiplying or dividing it by a whole number, decimal or fraction; addition and subtraction of positive fractions and integers; and multiplication and division of fractions and decimals Accurately solves problems involving single or multiple operations on fractions) proper, improper, and mixed); or decimals; and addition or subtraction of integers; percent of a whole; or problems involving greatest common factor or least common multiple Mentally calculates change back from $5.00, $10.00, $20.00, $50.00 and $100.00; multiplies a two-digit whole number by a one-digit whole number, two-digit whole numbers that are a multiple of ten, a three-digit whole number that is a multiple of 100 by a two- or three-digit number which is a multiple of 10 or 100, respectively (400 x 50, 400 x 600); and divides three- and four-digit multiples of powers of ten by their compatible factors and determines the part of a whole number using benchmarks percents (1%, 10%, 25%, 50%, and 75%) Makes estimates in a given situation by identifying when estimation is appropriate; selecting the appropriate methods of estimation, determining the level of accuracy needed given the situation, analyzing the effect of the estimation method on the accuracy of results, and evaluating the reasonableness of solutions appropriate to grade level GLE’s across content strands Applies properties of numbers (odd, even, remainders, divisibility, and prime factorization) and field properties (commutative, associative, identity [including the multiplicative property of one, distributive, and additive inverse) to solve problems and to simplify computations

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Grade 7 - Demonstrates conceptual understanding of rational numbers with respect to percents as a means of comparing the same or different parts of the whole when the whole vary in magnitude; and percent as a way of expressing multiples of a number, using models, explanations, or other representations * and demonstrates conceptual understanding of square roots of perfect squares, rates, and proportional reasoning *Specifications for area, set, and linear models for grades 5 – 8: Fractions: The number of parts in the whole are equal to the denominator, a multiple of the denominator, or a factor of the denominator Percents: The number of parts in the whole is a multiple or a factor of the numeric value representing the whole Decimals (including powers of ten): The number of parts in the whole is equal to the denominator of the fractional equivalent of the decimal, a multiple of the denominator of the fractional equivalent of the decimal, or a factor of the denominator of the fractional equivalent of the decimal Demonstrates understanding of the relative magnitude of numbers by ordering, comparing, or identifying equivalent rational numbers across number formats, numbers with whole number bases and whole number exponents, integers, absolute values, or numbers represented in scientific notation using number lines or equality and inequality symbols Demonstrates conceptual understanding of mathematical operations with integers and whole number exponents (where the base is a whole number) using models, diagrams, or explanations Accurately solves problems involving proportional reasoning, percents involving discounts, tax, or tips; and rates; and addition or subtraction of integers, raising numbers to whole number powers, and determining square roots of perfect square numbers and non-perfect square numbers Mentally calculates benchmark perfect squares and related square roots; determines the part of a number using benchmark percents and related fractions Makes estimates in a given situation (including tips, discounts, and tax) by identifying when estimation is appropriate; selecting the appropriate methods of estimation, determining the level of accuracy needed given the situation, analyzing the effect of the estimation method on the accuracy of results, and evaluating the reasonableness of solutions appropriate to grade level GLEs across content strands Applies properties of numbers (odd, even, remainders, divisibility, and prime factorization) and field properties (commutative, associative, identity, distributive, inverse) to solve problems and to simplify computations, and demonstrate conceptual understanding of field properties as they apply to subsets if real numbers (the set of the whole numbers does not have additive inverse, the set of integers does not have multiplicative inverse) Grade 8 - Demonstrates conceptual understanding of rational numbers with respect to absolute values, perfect square and cube roots, and percents as a way of describing change (percent increase and decrease) using explanation, models, or other representations*

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*Specifications for area, set, and linear models for grades 5 – 8: Fractions: The number of parts in the whole are equal to the denominator, a multiple of the denominator, or a factor of the denominator Percents: The number of parts in the whole is a multiple or a factor of the numeric value representing the whole Decimals (including powers of ten): The number of parts in the whole is equal to the denominator of the fractional equivalent of the decimal, a multiple of the denominator of the fractional equivalent of the decimal, or a factor of the denominator of the fractional equivalent of the decimal Demonstrates understanding of the relative magnitude of numbers by ordering or comparing rational numbers, common irrational numbers, numbers with whole number or fractional bases and whole number exponents, square roots, absolute values, integers or numbers represented in scientific notation using number lines or equality and inequality symbols Accurately solves problems involving proportional reasoning (percent increase or decrease, interest rates, markups, or rates); multiplication or division of integers; and squares, cubes, and taking square or cube roots Mentally calculates benchmark perfect squares and related square roots; determines the part of a number using benchmark percents and related fractions Makes estimates in a given situation (including tips, discounts, tax, and the value of a non-perfect square root as between two whole numbers) by identifying when estimation is appropriate; selecting the appropriate methods of estimation, determining the level of accuracy needed given the situation, analyzing the effect of the estimation method on the accuracy of results, and evaluating the reasonableness of solutions appropriate to grade level GLEs across content strands Applies properties of numbers (odd, even, remainders, divisibility, and prime factorization) and field properties (commutative, associative, identity [including the multiplicative property of one], distributive, inverse) to solve problems and to simplify computations, and demonstrate conceptual understanding of field properties as they apply to subsets if real numbers when addition and multiplication are not defined in the traditional ways Geometry and Measurement Kindergarten - Uses properties, attributes, composition, or decomposition to sort or classify polygons (triangles, squares, rectangles, rhombi, trapezoids, and hexagons) or objects by using one non-measurable or measurable attribute; and recognizes, names, and builds polygons and circles in the environment Demonstrates conceptual understanding of measurable attributes using comparative language to describe and compare attributes of objects (length [longer, shorter], height [taller, shorter], weight [heavier, lighter], temperature [warmer, cooler], and capacity [more, less]); and compares objects visually and with direct comparison

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Determines elapsed and accrued time as it relates to calendar patterns (days of the week, yesterday, today, and tomorrow), the sequence of events in a day; and identifies a clock and calendar as measurement tools Demonstrates understanding of spatial relationships using location and position by using positional words to locate and describe where an object is found in the environment Grade 1 - Uses properties, attributes, composition, or decomposition to sort or classify polygons (triangles, squares, rectangles, rhombi, trapezoids, and hexagons) or objects by a combination of two non-measurable or measurable attributes; and recognizes, names, builds, and draws polygons and circles in the environment Given an example of a three-dimensional geometric shape (rectangular, prisms, cylinders, or spheres) finds examples of objects in the environment that are of the same geometric shape Demonstrates conceptual understanding of congruency by making mirror images and creating shapes that have line symmetry Demonstrates conceptual understanding of the length/height of a two-dimensional object using non-standard units Demonstrates conceptual understanding of measurable attributes using comparative language to describe and compare attributes of objects (length [longer, shorter], height [taller, shorter], weight [heavier, lighter], temperature [warmer, cooler], and capacity [more, less]); and compares objects visually with direct comparison and using non-standard units Determines elapsed and accrued time as it relates to calendar patterns (days of the week, months of the year), the sequence of events in a day; and recognizes an hour and “on the half-hour” Demonstrates understanding of spatial relationships using location and position by using positional words (close by, on the right, underneath, above, beyond) to describe one location in reference to another on a map, in a diagram, and in the environment Grade 2 - Uses properties, attributes, composition, or decomposition to sort or classify polygons or objects by a combination of two or more non-measurable or measurable attributes Demonstrates conceptual understanding of congruency by composing and decomposing two-dimensional objects using models or explanations (using triangular pattern blocks to construct a figure congruent to the hexagonal pattern block); and uses line symmetry to demonstrate congruent parts within a shape Demonstrates conceptual understanding of perimeter and areas by using models or manipulatives to surround and cover polygons

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Measures and uses units of measures appropriately and consistently, and makes conversions within systems when solving problems across the content strands Demonstrates understanding of spatial relationships using location and position by using positional words in two- and three- dimensional situations to describe and interpret relative positions (above the surface of the desk, below the triangle on the paper); and creates and interprets simple maps and names locations on simple coordinate grids Grade 3 - Uses properties or attributes of angles (number of angles) or sides (number of sides or length of sides) or composition or decomposition of shapes to identify, describe, or distinguish among triangles, squares, rectangles, rhombi, trapezoids, hexagons, or circles Demonstrates conceptual understanding of congruency by matching congruent figures using reflections, translations, and rotations (flips, slides, and turns) (recognizing when pentominoes are reflections, translations and rotations of each other); composing and decomposing two- and three-dimensional objects using models or explanations (given a cube, students use blocks to construct a congruent cube); and uses line symmetry to demonstrate congruent parts within a shape Demonstrates conceptual understanding of similarity by identifying similar shapes Demonstrates conceptual understanding of perimeter of polygons and the area of rectangles on grids using a variety of models or manipulatives. Expresses all measures using appropriate units Measures and uses units of measures appropriately and consistently, and makes conversions within systems when solving problems across the content strands Demonstrates understanding of spatial relationships using location and position by interpreting and giving directions from one location to another (classroom to gym, from school to home) using positional words; and between locations on a map or coordinate grid (first quadrant) using positional words or compass directions Demonstrates conceptual understanding of spatial reasoning and visualization by copying, comparing, and drawing models of triangles, squares, rectangles, rhombi, trapezoids, hexagons, and circles; and builds models of rectangular prisms from three-dimensional representations Grade 4 - Uses properties or attributes of angles (number of angles) or sides (number of sides or length of sides, parallelism, or perpendicularity) to identify, describe, or distinguish among triangles, squares, rectangles, rhombi, trapezoids, hexagons, or octagons; or classify angles relative to 90° as more than, less than, or equal to Uses properties or attributes (shape of bases or number of lateral faces) to identify, compare, or describe three-dimensional shapes (rectangular prisms, triangular prisms, cylinders, or spheres)

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Demonstrates conceptual understanding of congruency by matching congruent figures using reflections, translations, and rotations (flips, slides, and turns) or as the result of composing and decomposing shapes using models or explanations Demonstrates conceptual understanding of similarity by applying scales on maps, or applying characteristics of similar figures (same shape but not necessarily the same size) to identify similar figures, or to solve problems involving similar figures. Describes relationships using models or explanations Demonstrates conceptual understanding of perimeter of polygons and the area of rectangles, polygons or irregular shapes on grids using a variety of models, manipulatives, or formulas. Expresses all measures using appropriate units Measures and uses units of measures appropriately and consistently, and makes conversions within systems when solving problems across the content strands Demonstrates understanding of spatial relationships using location and position by interpreting and giving directions between locations on a map or coordinate grid (first quadrant); plotting points in the first quadrant in context (games, mapping); and finding the horizontal and vertical distances between points on a coordinate grid in the first quadrant Demonstrates conceptual understanding of spatial reasoning and visualization by copying, comparing, and drawing models of triangles, squares, rectangles, rhombi, trapezoids, hexagons, octagons, and circles; and builds models of rectangular prisms from two-or three-dimensional representations Grade 5 - Uses properties or attributes of angles (right, acute, or obtuse) or sides (number of congruent sides, parallelism, or perpendicularity) to identify, describe, classify or distinguish among different types of triangles (right, acute, obtuse, equiangular, or equilateral) or quadrilaterals (rectangles, squares, rhombi, trapezoids, or parallelograms) Uses properties or attributes (shape of bases, number of lateral faces, or number of bases) to identify, compare, or describe three-dimensional shapes (rectangular prisms, triangular prisms, cylinders, spheres, pyramids, or cones) Demonstrates conceptual understanding of congruency by matching congruent figures using reflections, translations, and rotations (flips, slides, and turns) or as the result of composing and decomposing shapes using models or explanations Demonstrates conceptual understanding of similarity by describing the proportional effect on the linear dimensions of triangles and rectangles when scaling up or down while preserving angle measures, or by solving related problems (including applying scales on maps). Describes relationships using models or explanations

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Demonstrates conceptual understanding of perimeter of polygons and the area of rectangles or right triangles though models, manipulatives, or formulas, the area of polygons or irregular figures on grids, and volume or rectangular prisms (cubes) using a variety of models, manipulatives, or formulas. Expresses all measures using appropriate units Measures and uses units of measures appropriately and consistently, and makes conversions within systems when solving problems across the content strands Demonstrates understanding of spatial relationships using location and position by interpreting and giving directions between locations on a map or coordinate grid (all four quadrants); plotting points in four quadrants in context (games, mapping, identifying the vertices of polygons as they are reflected, rotated, and translated); and determining horizontal and vertical distances between points on a coordinate grid in the first quadrant Demonstrates conceptual understanding of spatial reasoning and visualization by building models of rectangular and triangular prisms, cones, cylinders and pyramids from two- or three-dimensional representations Grade 6 - Uses properties or attributes of angles (right, acute, or obtuse) or sides (number of congruent sides, parallelism, or perpendicularity) to identify, describe, classify or distinguish among different types of triangles (right, acute, obtuse, equiangular, scalene, isosceles, or equilateral) or quadrilaterals (rectangles, squares, rhombi, trapezoids, or parallelograms) Uses properties or attributes (shape of bases, number of lateral faces, number of bases, number of edges, or number of vertices) to identify, compare, or describe three-dimensional shapes (rectangular prisms, triangular prisms, cylinders, spheres, pyramids, or cones) Demonstrates conceptual understanding of congruency by predicting and describing the transformational steps (reflections, translations, and rotations) needed to show congruence (including the degree rotation) and as the result of composing and decomposing two- and three-dimensional objects using models or explanations; and using line and rotational symmetry to demonstrate congruent parts within a shape Demonstrates conceptual understanding of similarity by describing the proportional effect on the linear dimensions of polygons or circles when scaling up or down while preserving angles of polygons, or by solving related problems (including applying scales on maps). Describes relationships using models or explanations Demonstrates conceptual understanding of perimeter of polygons and the area of quadrilaterals, or triangles and the volume of rectangular prisms by using a variety of models, formulas, or by solving problems; and demonstrates understanding of the relationships of circle measures (radius to diameter and diameter to circumference) by solving related problems. Expresses all measures using appropriate units Measures and uses units of measures appropriately and consistently, and makes conversions within systems when solving problems across the content strands

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Grade 7 - Uses properties or attributes of angle relationships resulting from two or three intersecting lines (adjacent angles, vertical angles, straight angles, or angle relationships formed by two non-parallel lines cut by a transversal) or two parallel lines cut by a transversal to solve problems Applies theorems or relationships (triangle inequality or sum of the measures of interior angles of regular polygons) to solve problems Applies the concept of congruency by solving problems on a coordinate plane involving reflections, translations, or rotations Applies concepts of similarity by solving problems involving scaling up or down and their impact on angle measures, linear dimensions and areas of polygons, and circlers when the linear dimensions are multiplied by a constant factor. Describes effects using models or explanations Demonstrates conceptual understanding of the area of circles or the area or perimeter of composite figures (quadrilaterals, triangles, or parts of circles) and the surface area of rectangular prisms, or volume of rectangular prisms, triangular prisms, or cylinders using models, formulas, or by solving problems. Expresses all measures using appropriate units Measures and uses units of measures appropriately and consistently, and makes Demonstrates conceptual understanding of spatial reasoning and visualization by sketching three-dimensional solids; and draws nets of rectangular and triangular prisms, cylinders, and pyramids and uses the nets as a technique for finding surface area Grade 8 - Applies the Pythagorean Theorem to find a missing side of a right triangle, or in problem solving situations Applies concepts of similarity to determine the impact of scaling on the volume or surface area of three-dimensional figures when linear dimensions are multiplied by a constant factor; to determine the length of sides of similar triangles, or to solve problems involving growth and rate Demonstrates conceptual understanding of surface area or volume by solving problems involving surface area and volume of rectangular prisms, cylinders, pyramids, or cones. Expresses all measures using appropriate units Functions and Algebra Kindergarten - Identifies and extends to specific cases a variety of patterns (sequences of shapes, sounds, movement, colors, and letters) by extending the pattern to the next one, two, and three elements, or by translating AB patterns across formats (ABB can be represented as snap, clap, clap or red, yellow, yellow) or by identifying number patterns in the environment

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Grade 1 - Identifies and extends to specific cases a variety of patterns (repeating and growing [numeric and non-numeric]) represented in models, tables, or sequences by extending the pattern to the next one, two, or three elements (ABB can be represented as snap, clap, clap; red, yellow, yellow; or 1,2,2) Demonstrates conceptual understanding of equality by finding the value that will make an open sentence true (2 + = 7) (limited to one operation and limited to use addition or subtraction) using models, verbal explanations, or written equations Grade 2 - Identifies and extends to specific cases a variety of patterns (linear and non-numeric) represented in models, tables, or sequences by extending the pattern to the next element, or finding a missing element Demonstrates conceptual understanding of equality by finding the value that will make an open sentence true (2 + = 7) (limited to one operation and limited to use addition or subtraction) Grade 3 - Identifies and extends to specific cases a variety of patterns (linear and non-numeric) represented in models, tables, or sequences by extending the pattern to the next one, two, or three elements element, or finding a missing elements Demonstrates conceptual understanding of equality by showing equivalence between two expressions using models or different representations of the expression; or by finding the value that will make an open sentence true (2 + = 7) (limited to one operation and limited to use addition, subtraction, or multiplication) Grade 4 - Identifies and extends to specific cases a variety of patterns (linear and non-numeric) represented in models, tables, or sequences; and writes a rule in words or symbols to find the next one Demonstrates conceptual understanding of linear relationships (y = kx) as a constant rate of change by identifying, describing, or comparing situations that represent constant rates of change Demonstrates conceptual understanding of algebraic expressions by using letters of symbols to represent unknown quantities to write simple linear algebraic expressions involving any one of the four operations; or by evaluating simple linear algebraic expressions using whole numbers Demonstrates conceptual understanding of equality by showing equivalence between two expressions using models or different representations of the expression; by simplifying numerical expressions where left to right computations may be modified only by the use of parentheses 14 – (2 x 5) (expressions consistent with the parameters of M (F & A)-4-3) and by solving one-step linear equations of the form ax = c, x ± b = c, where a, b, and c are whole numbers with a ≠ 0 Grade 5 - Identifies and extends to specific cases a variety of patterns (linear and non-numeric) represented in models, tables, sequences, or in problem situations; or writes a rule in words or symbols for finding specific cases of a linear relationship

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Demonstrates conceptual understanding of linear relationships (y = kx) as a constant rate of change by identifying, describing, or comparing situations that represent constant rates of change (tell a story given a line graph about a trip) Demonstrates conceptual understanding of algebraic expressions by using letters or symbols to represent unknown quantities to write linear algebraic expressions involving any two of the four operations; or by evaluating linear algebraic expressions using whole numbers Demonstrates conceptual understanding of equality by showing equivalence between two expressions using models or different representations of the expression (expressions consistent with the parameters of M (F & A)-5-3), by solving one-step linear equations of the form ax = c, x ± b = c, or x/a = c, where a, b, and c are whole numbers with a ≠ 0 a true statement (2x + 3 = 11 {x : x = 2, 3, 4, 5}) Grade 6 - Identifies and extends to specific cases a variety of patterns (linear and non-numeric) represented in models, tables, sequences, graphs, or in problem situations; or writes a rule in words or symbols for finding specific cases of a linear relationship; or writes a rule in words or symbols for finding specific cases of a nonlinear relationship; and writes an expression or equation using words or symbols to express the generalization of a linear relationship (twice the term number plus 1 or 2n + 1) Demonstrates conceptual understanding of linear relationships (y = kx; y = mx + b) as a constant rate of change by constructing or interpreting graphs of real occurrences and describing the slope of linear relationships (faster, slower, greater, or smaller) in a variety of problem situations; and describes how change in the value of one variable relates to change in the value of a second variable in problem situations with constant rates of change Demonstrates conceptual understanding of algebraic expressions by using letters to represent unknown quantities to write linear algebraic expressions involving any of the four operations and consistent with order of operations expected at this grade level; or by evaluating linear algebraic expressions (including those with more than one variable); or by evaluating an expression within an equation (determine the value of y when x = 4 given y = 3x – 2) using whole numbers Demonstrates conceptual understanding of equality by showing equivalence between two expressions using models or different representations of the expression (expressions consistent with the parameters of M (F & A)-6-3), solving multi-step linear equations of the form ax = c, x ± b = c, where a, b, and c are whole numbers with a ≠ 0 Grade 7 - Identifies and extends to specific cases a variety of patterns (linear and non-numeric) represented in models, tables, sequences, graphs, or in problem situations; and generalizes a linear relationship using words and symbols; generalizes a linear relationship to find a specific case; or writes an expression or equation using words or symbols to express the generalization of a linear relationship Demonstrates conceptual understanding of linear relationships (y = kx; y = mx + b) as a constant rate of change by solving problems involving the relationship between slope and rate of change, by describing the meaning of slope in concrete situations, or informally determining the slope of a line from a table or graph; and distinguishes between constant and varying rates of change in concrete situations

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represented in tables or graphs; or describes how change in the value of one variable relates to change in the value of a second variable in problem situations with constant rates of change Demonstrates conceptual understanding of algebraic expressions by using letters to represent unknown quantities to write algebraic expressions (including those with whole number exponents or more than one variable); or by evaluating algebraic expressions (including those with whole number exponents or more than one variable); or by evaluating an expression within an equation (determine the value of y when x = 4 given y – 5x3 - 2) Demonstrates conceptual understanding of equality by showing equivalence between two expressions (expressions consistent with the parameters of the left- and right-hand sides of equations being solved at this grade level) using models or different representations of the expression, solving multi-step linear equations of the form ax = c with a ≠ 0, ax ± b = cx ± d, with a, c ≠ 0 and (x/a) ± b= c with a ≠ 0, where a, b, c, and d are whole numbers; or by translating a problem solving situation into an equation consistent with the parameters of the type of equations being solved for this grade level Grade 8 - Identifies and extends to specific cases a variety of patterns (linear and non-numeric) represented in models, tables, sequences, graphs, or in problem situations; and generalizes a linear relationship (non-recursive explicit equation); generalizes a linear relationship to find a specific case; generalizes a nonlinear relationship using words or symbols; or generalizes a common nonlinear relationship to find a specific case Demonstrates conceptual understanding of linear relationships (y = kx; y = mx + b) as a constant rate of change by solving problems involving the relationship between slope and rate of change; informally and formally determining slopes and intercepts represented in graphs, tables, or problem situations; or describing the meaning of slope and intercept in context; and distinguishes between linear relationships (constant rates of changes) and nonlinear relationships (varying rates of change) represented in tables, graphs, equations, or problem situations; or describes how change in the value of one variable relates to change in the value of a second variable in problem situations with constant and varying rates of change Demonstrates conceptual understanding of algebraic expressions by evaluating and simplifying algebraic expressions (including those with square roots, whole number exponents, or rational numbers); or by evaluating an expression within an equation (determine the value of y when x = 4 given y = 7 √x + 2x) Demonstrates conceptual understanding of equality by showing equivalence between two expressions (expressions consistent with the parameters of the left- and right-hand sides of equations being solved at this grade level) using models or different representations of the expression, solving formulas for one variable requiring one transformation (d = rt; d/r = t); by solving multi-step linear equations with integer coefficients; by showing that two expressions are or are not equivalent by applying commutative, associative, or distributive properties, order of operations, or substitution; and by informally solving problems involving systems of linear equations in a context Data, Statistics, and Probability

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Kindergarten - Interprets a given representation created by the class (models and tally charts) to answer questions related to the data, or to analyze the data to formulate conclusions using words, diagrams, or verbal/scribed responses to express answers Analyzes patterns, trends, or distribution in data in a variety of contexts by determining or using more, less, or equal Grade 1 - Interprets a given representation created by the class (models, tally charts, pictographs with one-to-one correspondence, and tables) to answer questions related to the data, or to analyze the data to formulate conclusions using words, diagrams, or verbal/scribed responses to express answers Analyzes patterns, trends, or distribution in data in a variety of contexts by determining or using more, less, or equal For a probability event in which the sample space may or may not contain equally likely outcomes, groups use experiments to describe the likelihood or chance of an event (using “more likely,” “less likely,” or “equally likely”) Grade 2 - Interprets a given representation (pictographs with one-to-one correspondence, line plots, tally charts or tables) to answer questions related to the data, or to analyze the data to formulate conclusions Analyzes patterns, trends, or distribution in data in a variety of contexts by determining or using more, less, or equal Uses counting techniques to solve problems involving combinations using a variety of strategies (student diagrams, organized lists, tables, tree diagrams or others) (How many ways can you make 50 cents using nickels, dimes, and quarters?) For a probability event in which the sample space may or may not contain equally likely outcomes, uses experiments to describe the likelihood or chance of an event using “more likely,” “less likely,” “equally likely,” “certain,” or “impossible” In response to a teacher or student generated question or hypothesis, groups decide the most effective methods (survey, observation, experimentation) to collect the data (numerical or categorical) necessary to answer the question; collects, organizes, and appropriately displays the data; analyzes the data to draw conclusions about the question or hypothesis being tested, and when appropriate makes predictions Grade 3 - Interprets a given representation (line plots, tally charts, tables, or bar graphs) to answer questions related to the data, or to analyze the data to formulate conclusions, or to make predictions Analyzes patterns, trends, or distribution in data in a variety of contexts by determining or using most frequent (mode), least frequent, largest, or smallest

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Identifies or describes representations or elements of representations that best display a given set of data or situation, consistent with the representations required in M(DSP)-3-1; and organizes and displays data using tables, tally charts, and bar graphs to answer questions related to the data, to analyze the data, to formulate conclusions, to make predictions, or to solve problems Uses counting techniques to solve problems involving combinations and simple permutations using a variety of strategies (student diagrams, organized lists, tables, tree diagrams or others) For a probability event in which the sample space may or may not contain equally likely outcomes, determines the likelihood or chance of an event (using “more likely,” “less likely,” “equally likely,”); and predicts the likelihood of an event using “more likely,” “less likely,” “equally likely,” “certain,” or “impossible” and tests the prediction through experiments; and determines if a game is fair In response to a teacher or student generated question or hypothesis, groups decide the most effective methods (survey, observation, experimentation) to collect the data (numerical or categorical) necessary to answer the question; collects, organizes, and appropriately displays the data; analyzes the data to draw conclusions about the question or hypothesis being tested, and when appropriate makes predictions Grade 4 - Interprets a given representation (line plots, tally charts, tables, bar graphs, pictographs, or circle graphs) to answer questions related to the data, to analyze the data to formulate or justify conclusions, to make predictions, or to solve problems Analyzes patterns, trends, or distribution in data in a variety of contexts by determining or using measures of central tendency (median or mode), or range Organizes and displays data using tables, line plots, bar graphs, and pictographs to answer questions related to the data, to analyze the data, to formulate or justify conclusions, to make predictions, or to solve problems Uses counting techniques to solve problems in context involving combinations or simple permutations (Given a map – determine the number of paths from point A to point B) using a variety of strategies (organized lists, tables, tree diagrams or others) For a probability event in which the sample space may or may not contain equally likely outcomes, determines the theoretical probability of an event and expresses the result as part to whole (two out of five); and predicts the likelihood of an event as a part to whole relationship and tests the prediction through experiments, and determines if a game is fair In response to a teacher or student generated question or hypothesis, groups decide the most effective methods (survey, observation, experimentation) to collect the data (numerical or categorical) necessary to answer the question; collects, organizes, and appropriately displays the data; analyzes the data to draw conclusions about the question or hypothesis being tested, and when appropriate makes predictions; and asks new questions and makes connections to real world situations

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Grade 5 - Interprets a given representation (tables, bar graphs, circle graphs, or line graphs) to answer questions related to the data, to analyze the data to formulate or justify conclusions, to make predictions, or to solve problems Analyzes patterns, trends, or distribution in data in a variety of contexts by determining or using measures of central tendency (mean, median or mode), or range to analyze situations, or to solve problems Identifies or describes representations or elements of representations that best display a given set of data or situation, consistent with the representations required in M (DSP)-5-1; and organizes and displays data using tables, bar graphs, or line graphs to answer questions related to the data, to analyze the data, to formulate or justify conclusions, to make predictions, or to solve problems Uses counting techniques to solve problems in context involving combinations or simple permutations using a variety using a variety of strategies (organized lists, tables, tree diagrams, models or others) For a probability event in which the sample space may or may not contain equally likely outcomes, determines the experimental or theoretical probability of an event and expresses the result as fraction; and predicts the likelihood of an event as a fraction and tests the prediction through experiments, and determines if a game is fair In response to a teacher or student generated question or hypothesis, groups decide the most effective methods (survey, observation, experimentation) to collect the data (numerical or categorical) necessary to answer the question; collects, organizes, and appropriately displays the data; analyzes the data to draw conclusions about the question or hypothesis being tested, and when appropriate makes predictions; and asks new questions and makes connections to real world situations Grade 6 - Interprets a given representation (circle graphs, line graphs or stem-and-leaf plots) to answer questions related to the data, to analyze the data to formulate or justify conclusions, to make predictions, or to solve problems Analyzes patterns, trends, or distribution in data in a variety of contexts by determining or using measures of central tendency (mean, median or mode), or dispersion (range) to analyze situations, or to solve problems Organizes and displays data using tables, line graphs, or stem-and-leaf plots to answer questions related to the data, to analyze the data, to formulate or justify conclusions, to make predictions, or to solve problems Uses counting techniques to solve problems in context involving combinations or simple permutations using a variety using a variety of strategies (organized lists, tables, tree diagrams, models, Fundamental Counting Principle or others) For a probability event in which the sample space may or may not contain equally likely outcomes, determines the experimental or theoretical probability of an event in a problem-solving situation; and predicts the theoretical probability of an event and tests the prediction through experiments and simulations, and designs fair games

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In response to a teacher or student generated question or hypothesis, groups decide the most effective methods (survey, observation, experimentation) to collect the data (numerical or categorical) necessary to answer the question; collects, organizes, and appropriately displays the data; analyzes the data to draw conclusions about the question or hypothesis being tested, and when appropriate makes predictions; and asks new questions and makes connections to real world situations Grade 7 - Interprets a given representation (circle graphs, scatter plots that represent discrete linear relationships, or histograms) to answer questions related to the data, to analyze the data to formulate or justify conclusions, to make predictions, or to solve problems Analyzes patterns, trends, or distribution in data in a variety of contexts by solving problem using measures of central tendency (mean, median or mode), or dispersion (range or variation), or outliers to analyze situations to determine their effect on mean, median, or mode; and evaluate the sample from which the statistics were developed (bias) Identifies or describes representations or elements of representations that best display a given set of data or situation, consistent with the representations required in M (DSP)-7-1; and organizes and displays data using tables, line graphs, scatter plots, and circle graphs to answer questions related to the data, to analyze the data, to formulate or justify conclusions, to make predictions, or to solve problems Uses counting techniques to solve problems in context involving combinations or simple permutations using a variety using a variety of strategies (organized lists, tables, tree diagrams, models, Fundamental Counting Principle or others) For a probability event in which the sample space may or may not contain equally likely outcomes, determines the experimental or theoretical probability of an event in a problem-solving situation; and predicts the theoretical probability of an event and tests the prediction through experiments and simulations, and compares and contrasts theoretical and experimental probabilities In response to a teacher or student generated question or hypothesis, groups decide the most effective methods (survey, observation, experimentation) to collect the data (numerical or categorical) necessary to answer the question; collects, organizes, and appropriately displays the data; analyzes the data to draw conclusions about the question or hypothesis being tested while considering the limitations that could affect interpretations; and when appropriate makes predictions; and asks new questions and makes connections to real world situations Grade 8 - Interprets a given representation (circle graphs, scatter plots that represent discrete linear relationships, or histograms) to answer questions related to the data, to analyze the data to formulate or justify conclusions, to make predictions, or to solve problems Analyzes patterns, trends, or distribution in data in a variety of contexts by solving problem using measures of central tendency (mean, median or mode), or dispersion (range or variation), or outliers to analyze situations to determine their effect on mean, median, or mode; and evaluate the sample from which the statistics were developed (bias)

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Identifies or describes representations or elements of representations that best display a given set of data or situation, consistent with the representations required in M (DSP)-7-1; and organizes and displays data using tables, line graphs, scatter plots, and circle graphs to answer questions related to the data, to analyze the data, to formulate or justify conclusions, to make predictions, or to solve problems Uses counting techniques to solve problems in context involving combinations or simple permutations using a variety using a variety of strategies (organized lists, tables, tree diagrams, models, Fundamental Counting Principle or others) For a probability event in which the sample space may or may not contain equally likely outcomes, determines the experimental or theoretical probability of an event in a problem-solving situation; and predicts the theoretical probability of an event and tests the prediction through experiments and simulations, and compares and contrasts theoretical and experimental probabilities In response to a teacher or student generated question or hypothesis, groups decide the most effective methods (survey, observation, experimentation) to collect the data (numerical or categorical) necessary to answer the question; collects, organizes, and appropriately displays the data; analyzes the data to draw conclusions about the question or hypothesis being tested while considering the limitations that could affect interpretations; and when appropriate makes predictions; and asks new questions and makes connections to real world situations

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Mathematics Reference Sheets - NECAP

The following mathematics reference sheets are identical to those that students will receive while participating in the NECAP. New Hampshire, Rhode Island, and Vermont are committed to ensuring that students develop strong conceptual foundations of the mathematical concepts that they are studying. It is the intent of the NECAP items to assess perimeter, area, and volume at the conceptual level and not the procedural level. Therefore, the formulas provided on these reference sheets do not conflict with the intention of the Grade-Level Expectations, but rather allow teachers to concentrate on developing conceptual understanding without having to be concerned with students remembering formulas or reconstructing them in an on-demand assessment situation. Furthermore, the three states believe that these reference sheets should be used as an everyday tool in the classroom to help students become familiar with the information contained on them. Please note: At grades 3 and 4 a committee of New Hampshire, Rhode Island, and Vermont educators made the decision to embed formulas as needed into test items to ensure that students will not have to transfer information from a reference sheet to a test item in their assessment booklets.] Please see the following grade specific reference sheets for more detail: http://www.ed.state.nh.us/Education/doe/organization/curriculum/NECAP/Math_Ref_Sheet_GR5.pdf http://www.ed.state.nh.us/Education/doe/organization/curriculum/NECAP/Math_Ref_Sheet_GR6.pdf http://www.ed.state.nh.us/Education/doe/organization/curriculum/NECAP/Math_Ref_Sheet_GR7.pdf http://www.ed.state.nh.us/Education/doe/organization/curriculum/NECAP/Math_Ref_Sheet_GR8.pdf

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APPENDIX B

How this Framework is Organized Rationale Societal Goals How Students Learn Mathematics Goals and Standards

How this Framework is Organized

The material in the K-12 Mathematics Curriculum Framework is organized around eight strands: Problem Solving and Reasoning; Communication and Connections; Numbers, Numeration, Operations, and Number Theory; Geometry, Measurement, and Trigonometry; Data Analysis, Statistics, and Probability; Functions, Relations, and Algebra; Mathematics of Change; and Discrete Mathematics. Within each of these areas, one or more K-12 Broad Goals identify general expectations of what ALL New Hampshire students are expected to know and be able to do. For example, the first Broad Goal in the Problem Solving and Reasoning strand states: Students will use problem-solving strategies to investigate and understand increasingly complex mathematical content.

Following each broad goal is a purpose statement which places the goal in context and elaborates on its role in the mathematics program. Further, in the case of iscrete Mathematics, a definition is provided in order to clarify this emerging area of the K-12 curriculum Standards are presented in two parts: Curriculum Standards and Proficiency Standards. The Curriculum Standards identify the scope of the content recommended for grades K-3, 4-6, and 7-12. The Proficiency Standards identify specific expectations for the assessment of cumulative learning. They will serve as the basis for the development and ongoing revision of the mathematics assessment instruments to be administered statewide at the end of grades three, six, and ten. All of the Grade 3 Proficiency Standards found in the New Hampshire Mathematics Curriculum Framework: End of Grade Three (1993) are incorporated into this K-12 framework.

The Curriculum Standards, particularly at the 7-12 level, identify more than what is included in the standards to be tested. The developers of this framework were sensitive to what constitutes a full 4-year program of mathematics in high school and the fact that students will be tested statewide at the end-of-grade ten. Local educators and policy leaders should note that the recommended content for all high school students is richer than the content that has traditionally been included in some general mathematics courses.

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Rationale

In the early part of this century, the needs of our society were dominated by an emerging industrial age driven by mass production. The needs of that society were served by mathematics education in which the acquisition of computational skills was the primary focus. Computational skills alone are no longer sufficient for the United States to remain competitive in the world marketplace. In the coming century the educational needs of our society will be very different. The economy is global, the economic environment is more competitive, and the workforce is more mobile. The acquisition of computational skills remains important, but more is needed today, due to rapidly changing technology.

The development of mathematical problem solving, reasoning, communication skills, and use of appropriate technology is essential so that people can skillfully address the more complex problems encountered in today's workplaces. We need individuals who can apply their understanding of mathematics to solve real-world problems for which there are no simple formulas and standard procedures. We need individuals who can use their knowledge of mathematics to make sense of complex situations and then communicate that understanding to others. We need individuals who are able to solve tomorrow's problems, as well as todays. Mathematics education for the twenty-first century must address these needs.

Societal Goals

We believe the goals for New Hampshire schools are closely aligned with those espoused by various national commissions and groups in their efforts to reshape the mathematics curriculum. We commit to five primary goals. That:

• all students will develop a firm grounding in essential computational skills; • all students will develop strong mathematical problem solving and reasoning abilities; • all students will develop positive attitudes about mathematics; • all students will develop the ability to use appropriate technology to solve mathematical problems; and • all students will develop the ability to communicate their understanding of mathematics effectively.

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How Students Learn Mathematics

"Students learn mathematics well only when they construct their own mathematical understanding." (Everybody Counts, p.58)

This view of learning, called constructivism is the premise upon which the reform movement in mathematics education is based. When students learn mathematics by doing mathematics, by exploring and discussing concepts in the context of physical situations, what emerges from these experiences are skills which are anchored in understanding and clarity. The students not only know the basic procedures, but also know how to apply them to new situations. Research supports the fact that students learn best by experiencing mathematics and thereby constructing understanding for themselves. Research also indicates that mathematics education will best serve societal needs when the curriculum is so conceptually focused.

The attitudes students form influence their thinking and performance, and, later, influence their decisions about studying mathematics. Students are active individuals who construct, modify, and integrate ideas by interacting with materials, the world around them, and their peers. Thus, the learning of mathematics must be an active process: exploring, justifying, representing, solving, constructing, discussing, using, investigating, describing, developing, and predicting. These actions require both the physical and mental involvement of students both hands on and minds on.

Such a curriculum has the following characteristics:

• students are actively involved in doing mathematics; • problem solving, thinking, reasoning, and communicating are everyday activities; • manipulatives are used to connect conceptual to procedural understanding; • calculators and computers are used in appropriate ways; • there is as much emphasis on application as on acquisition of knowledge and skills; • a broad range of content is addressed; and • central mathematical concepts are understood.

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Goals and Standards

Problem Solving and Reasoning | Communication and Connections | Numbers, Numeration, Operations, and Number Theory Geometry, Measurement, and Trigonometry |Data Analysis, Statistics, and Probability | Functions, Relations, and Algebra Mathematics of Change | Discrete Mathematics

Problem Solving and Reasoning

1a. K-12 Broad Goal: Students will use problem-solving strategies to investigate and understand increasingly complex mathematical content.

PURPOSE: Problem solving should serve as the organizing feature of the mathematics curriculum as well as other areas of study and be applied to everyday activities. Problem-solving must not be seen as a separate topic, but rather the centerpiece of the mathematics curriculum. Students should have many experiences in posing and solving problems from their world, from data that are meaningful to them, and from mathematical investigations.

K-12 Curriculum Standards (1a): K-3

• Make up problems based on everyday experiences. • Solve problems using a variety of strategies (for example: make a list, draw a picture, or guess and check). • Formulate and solve real-world problems. • Verify and interpret results with respect to the original problem. • Generalize solutions and apply strategies to new problem situations. • Solve multi-step problems. • Use problem solving approaches to investigate and understand new mathematical content, both independently and in groups. • Demonstrate that a problem may be solved in more than one way. • Exhibit confidence in their ability to solve problems independently and in groups. • Display increasing perseverance, and persistence in problem solving. • Write about problem solutions and solution processes.

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4-6 Building upon the K-3 experiences, in grades 4- 6:

• Solve problems using a variety of strategies (for example: look for a simpler problem, or working backwards). • Formulate and solve real-world problems. • Solve multi-step problems and problems with multiple solutions or no solution; and recognize problems where more information is

needed. • Use problem-solving approaches to investigate and understand mathematical content. • Verify and interpret results with respect to the original problem. • Demonstrate that a problem may be solved in more than one way. • Develop confidence, perseverance, and persistence in problem solving both independently and in groups. • Generalize solutions and apply strategies to new problem situations.

7-12 Building upon the K- 6 experiences, in grades 7-12:

• Determine, collect and organize the relevant data needed to solve real-world problems. • Determine the reasonableness of solutions to real-world problems. • Use technology whenever appropriate to solve real-world problems which require strategies previously learned. • Use technology whenever appropriate to solve problems related to basic living skills including, but not limited to, personal finance,

wages, banking and credit, home improvement problems, measurement, taxes, business situations, purchasing, and transportation. • Apply problem solving strategies to solve problems in the natural and social sciences and in pure mathematics.

Proficiency Standards (1a): End of Grade 3:

• Formulate problems from everyday and mathematical situations. • Solve problems that require the use of strategies (for example: making a list,drawing a picture, looking or a pattern, or acting out). • Solve problems with and without using manipulatives and calculators.

End of Grade 6:

• Solve problems that require the use of strategies (for example: working backwards; looking for patterns and relationships; guess and check; making tables, charts, and graphs; solving a simpler version of a problem; looking for similar problems; drawing a diagram; or creating a model).

• Formulate, solve, and verify problems from every-day and mathematical situations and interpret the results. • Solve multi-step problems, solve problems with multiple solutions, recognize when a problem has no solution, and recognize

problems where more information is needed.

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• Solve problems using manipulatives, graphs, charts, diagrams, and calculators.

End of Grade 10:

• Determine, collect, and organize the relevant data needed to solve real-world problems. • Choose the appropriate technology needed to solve a real-world problem. • Translate results of a computation into solutions that fit the real-world problem (for example, when a computation shows that one

needs 3. 2 gallons of paint to paint a room, how much paint do you buy?). • Determine if the solution of a real-world problem is reasonable. • Use technology to solve a problem from science, social science, or mathematics.

1b. K-12 Broad Goal: Students will use mathematical reasoning.

PURPOSE: Students need to recognize that memorized facts, rules, and procedures are only a part of mathematics. They need opportunities to use these facts, rules, and procedures to make conjectures, develop and refine their reasoning abilities, gather evidence, and produce valid rules and generalizations. Students need to be able to justify their thinking through examples and explanations and appreciate that how a problem is solved is as important as the answer.

K-12 Curriculum Standards (1b): K-3

• Draw conclusions using inductive reasoning. • Use models, known facts, properties, and relationships to explain their thinking. • Explain conjectures, solution processes, and answers. • Demonstrate belief that mathematics makes sense. • Demonstrate conservation of number and length by using reversibility of thought.

4-6 Building upon the K-3 experiences, in grades 4-6:

• Draw conclusions using inductive reasoning, elementary deductive reasoning, and reasoning by analogy. • Use models, known facts, properties, and relationships to explain their thinking. • Explain conjectures, solutions processes, and answers. • Appreciate the pervasive use and power of reasoning as a part of mathematics. • Show increasing ability to understand and apply reasoning processes and spatial reasoning (symmetry, reflections, motions in the

plane, and identifying three-dimensional objects from two-dimensional drawings).

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7-12 Building upon the K-6 experiences, in grades 7-12:

• Draw logical conclusions and make generalizations using deductive and inductive reasoning. • Formulate and test mathematical conjectures and arguments. • Determine the validity of an argument and/or a solution. • Apply mathematical reasoning skills, when appropriate, in other disciplines.

Proficiency Standards (1b): End of Grade 3:

• Continue a number pattern. • Identify the missing information needed to find a solution to a given story problem. • Compare and contrast geometric figures. • Verify an answer to a problem. • Continue a geometric pattern. • Defend a conjecture with an appropriate argument. • Discuss the use of a problem solving strategy. Example: "I chose this method to solve the problem because ..."

End of Grade 6:

• Continue a pattern involving integers and positive rational numbers. • Solve problems involving two-and three-dimensional geometric shapes and explain one's reasoning. • Use elementary deductive reasoning to solve word problems. • Use models, known facts, properties, and relationships to explain thinking and to justify answers and solution processes.

End of Grade 10:

• Use inductive reasoning to make generalizations from an observed pattern. • Use logical reasoning, as well as estimation and mental computations, to determine the validity of a solution. • Justify conjectures, defend generalizations, and write logical arguments.

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Communication and Connections

2a. K-12 Broad Goal: Students will communicate their understanding of mathematics.

PURPOSE: Reading, writing, talking, listening, and modeling, provide students with the opportunity to integrate the language of mathematics into their world, and help them to develop understanding. Actively exploring, investigating, describing, and explaining mathematical ideas promote communication which leads to a greater comprehension of mathematical concepts.

K-12 Curriculum Standards (2a): K-3

• Relate everyday language to mathematical language and symbols. • Discuss, illustrate, and write about mathematical concepts and relationships. • Use language to reflect on, clarify, and articulate thinking about mathematical ideas and situations. • Demonstrate mathematical communication through discussion, reading, writing, listening, and responding, individually and in

groups.

4-6 Building upon the K-3 experiences, in grades 4-6:

• Relate everyday language to mathematical language and symbols. • Discuss, illustrate, and write about mathematical concepts and relationships. • Use language to reflect on, clarify, and articulate thinking about mathematical ideas and situations. • Demonstrate mathematical communication through discussion, representation, reading, writing, listening, and responding,

individually and in groups. • Use a variety of technologies (for example: computers, calculators, video, CD-ROM, or laser disc, to represent and communicate

mathematical ideas). • Understand and appreciate the economy and power of mathematical symbolism and its role in the development of mathematics.

7-12 Building upon the K-6 experiences, in grades 7-12:

• Formulate questions, conjectures, definitions, and generalizations about, data, information, and problem situations. • Use a variety of technologies to represent and communicate mathematical ideas and determine the appropriateness of their use. • Understand, explain, analyze, and evaluate mathematical arguments and conclusions made by others. • Understand the efficiency and power of mathematical notation.\

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Proficiency Standards (2a): End of Grade 3:

• Discuss (in writing) mathematical concepts and relationships. • Draw pictures and use objects to illustrate mathematical concepts. • Write about the mathematical topics presented. • Defend conjectures and tentative generalizations.

End of Grade 6:

• Demonstrate an understanding of mathematical concepts and relationships through a variety of methods (for example: writing, graphing, charts, diagrams, number sentences, or symbols).

• Explain, analyze, and evaluate mathematical arguments and conclusions presented by others. • Explain conclusions, thought processes, and strategies in problem-solving situations. • Make conjectures and defend generalizations. • Evaluate the validity of a mathematical statement.

End of Grade 10:

• Evaluate given information and determine appropriate questions suggested by the situation. • Evaluate given information and determine appropriate generalizations suggested by the situation. • Describe orally and/or in writing how various technologies can be used to communicate about a specific situation. • Use mathematical symbols and notation to communicate mathematically. • Justify conjectures, defend generalizations and write logical arguments.

2b. K-12 Broad Goal: Students will recognize, develop, and explore mathematical connections.

PURPOSE: Mathematical topics, ideas, and procedures must be connected to each other and to the students' everyday experiences, both in and out of school. In particular, mathematics must be connected to all other curriculum areas. Mathematical connections will help students become aware of the usefulness of mathematics, serve to bridge the concrete and the abstract, and enable deeper understanding of important ideas.

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K-12 Curriculum Standards (2b):

K-3

• Understand the mathematical processes of addition, subtraction, and multiplication and relate them to one another. • Recognize different representations of concepts and procedures (for example, students should recognize the relationship among

seven counters, seven tally marks, and the symbol 7). • Translate among different representations as appropriate. • Recognize relationships among different topics in mathematics. • Recognize and use mathematics in other curriculum areas. • Recognize and use mathematics in their daily lives.

4-6 Building upon the K-3 experiences, in grades 4-6:

• Understand the mathematical processes and procedures of addition, subtraction, multiplication, and division and relate them to one another.

• Recognize equivalent representations of concepts and procedures and translate among them as appropriate (for example, understand how the addition of whole numbers, fractions, and decimals are related).

• Recognize relationships among different topics in mathematics. • Recognize and use mathematics in other curriculum areas and in their daily lives. • Link concepts and procedures (for example, know why you "invert and multiply" when dividing two fractions).

7-12 Building upon the K-6 experiences, in grades 7-12:

• View mathematics as an integrated whole. (Be able to synthesize the varied branches.) • Explain the relationship between a real-world problem and an appropriate mathematical model. • Explain in oral or written form how mathematics connects to other disciplines, to daily life, careers, and society. • Use models and calculators or other technologies to develop equivalent representations of the same mathematical concept. • Recognize the logical development of mathematics from basic assumptions and definitions, and understand that mathematics

frequently arises out of real-world applications. • Recognize that many real world applications require an understanding of use of mathematical concepts (for example: personal

finance, running a business, building a house, following a recipe, or sending a rocket to the moon).

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Proficiency Standards (2b): End of Grade 3:

• Demonstrate the relationship between addition and multiplication and between addition and subtraction. • Demonstrate the relationship between fractions and decimals. • Identify mathematical situations occurring in children's literature. • Identify mathematical applications in social studies (for example: graphs, tables, or maps). • Identify the use of mathematical skills and concepts in science (for example: measurement, graphs, or data analysis. • Identify examples of geometry in nature, art, and architecture. • Use probability and statistics to describe and predict simple events. • Use money in real-world situations. • Use geometric representations for fractions and decimals and to explain arithmetic operations.

End of Grade 6:

• Identify the relationships among the four basic operations on rational numbers. • Identify the relationship among the basic operations as applied to whole numbers and to positive rational numbers. • Use mathematical skills, concepts, and applications in other disciplines (for example: graphs in social studies, patterns in art, or

music and geometry in technology education).

End of Grade 10:

• Explain in oral or written form the relationships among various mathematical concepts (for example, the relationship between exponentiation and multiplication).

• Translate among equivalent representations of the same concept (for example, a table of values, an equation, and a graph may all be representations of the same function).

• Explain in oral or written form the relationships between a real-world problem and an appropriate mathematical model. • Explain in oral or written form how mathematics connects to other areas (for example: geometry in art and architecture, data

analysis in social studies and exponential growth in finance).

Numbers, Numeration, Operations, and Number Theory

3a. K-12 Broad Goal: Students will develop number sense and an understanding of our numeration system.

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PURPOSE: Students must understand numbers if they are to make sense of the ways numbers are used in their everyday world. Numbers are used to describe and interpret real-world phenomena. Students need to use numbers to quantify, to identify location, to identify a specific object in a collection, to name, to measure, and to model real-world situations. They need to understand relative magnitude in order to make sense of everyday situations.

K-12 Curriculum Standards (3a): K-3

• Order a set of numbers (0-99) from smallest to largest. • Name the whole number immediately before or after any 2-digit number. • Name the number that is ten units before or ten units after any 2-digit number. • Compare any two 2-digit numbers to determine which is greater or less. • Read and write whole numbers. • Show understanding of place value concepts via the use of physical models. • Recognize and demonstrate the difference in magnitude of whole numbers and fractions. • Demonstrate knowledge of differences in the use of ordinal and cardinal numbers. • Interpret the multiple uses of numbers encountered in the real-world.

4-6 Building upon the K-3 experiences, in grades 4-6:

• Read and write integers and positive rational numbers. • Represent and identify whole numbers, fractions, and decimals using physical models. • Use physical models to represent integers and positive rational numbers. • Explore the relationship of simple decimals to fractions. • Explore the operations of addition, subtraction, multiplication, and division of integers using manipulatives or representational

models. • Demonstrate an understanding of denominate numbers (numbers involving units of measure, such as 3 in.) through applications to

real-life situations. • Explore the meaning of 10%, 25%, 50%, 75%, and 100% and their fraction and decimal equivalents. • Demonstrate an understanding of prime and composite numbers. • Identify multiples and factors of whole numbers. • Identify numbers divisible by 2, 3, 5, 9, and 10. • Demonstrate an understanding of the periodicity of numbers. • Explore ancient numeration systems and the use of different bases (such as base 2 and 5).

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7-12 Building upon the K-6 experiences, in grades 7-12:

• Read and write rational numbers. • Use physical models to represent rational numbers. • Recognize and demonstrate the difference in magnitude of rational numbers. • Compare magnitudes of integers, rational, and irrational numbers. • Develop and use order relations for integers, rational and irrational numbers.

Proficiency Standards (3a): End of Grade 3

• Identify and write a 3-digit number given a physical model or an illustration of a place-value model, and given a 3-digit number, create a model.

• Read and write three-digit whole numbers. • Identify the number 1000 as a unit or in various combinations of hundreds, tens, and ones.

End of Grade 6:

• Name and identify a fraction or decimal, given a physical representation. • Given a decimal representation in tenths or hundredths, write an equivalent fraction. • Given an integer or a positive rational number, represent the number with the use of physical models or diagrams. • Explain the use of numbers in various every-day contexts (for example: calendars, clocks, signs, or literature). • Given a set of fractional models, name and write those that represent equivalent fractions. • Given a pair of fractions, determine which is larger by using physical models or illustrations. • Develop and use order relations for integers and positive rational numbers. • Apply number theory to the factoring of whole numbers and the equivalency of positive rational numbers.

End of Grade 10:

• Read and write rational numbers. • Use physical models to represent rational numbers. • Compare and order real numbers.

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3b. K-12 Broad Goal: Students will understand the concepts of number operations.

PURPOSE: Students need to build an awareness of the properties of an operation, see relationships among operations, and acquire insight into the effects of operations on real numbers. Students need to recognize conditions in real-world situations where the use of these operations is indicated and useful.

K-12 Curriculum Standards (3b): K-3

• Develop meaning for the operations of addition, subtraction, multiplication, and division by modeling and discussing a rich variety of problem situations.

• Demonstrate and explain the relationship between these operations. • Relate the mathematical language and symbols to problem situations and informal language. • Recognize that a wide range of problem situations can be represented by one expression. • Recognize the effect of performing the operations of addition, subtraction, multiplication, and division with whole numbers.

4-6 Building upon the K-3 experiences, in grades 4-6:

• Develop meaning for multiplication and division of whole numbers, fractions, and decimals by modeling and discussing a rich variety of problem situations.

• Demonstrate and explain the relationship among the four basic operations and, when appropriate, use the associative, commutative, and distributive properties to simplify computations.

• Explore and develop the concepts of addition and subtraction of fractions and decimals using manipulatives.

7-12 Building upon the K-6 experiences, in grades 7-12:

• Demonstrate an understanding of the operations of addition, subtraction, multiplication, and division of rational numbers, and the effect of performing these operations (for example, what can one say about the quotient when dividing by a fraction between 0 and 1?).

• Understand the standard algebraic order of operations. • Understand the properties of exponents. • Recognize when to use and how to apply the field properties.

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Proficiency Standards (3b): End of Grade 3:

• Count by ones, twos, fives, and tens. • Identify even and odd numbers and explain the difference. • Use manipulatives and pictures to represent multiplication as repeated addition or arrays. • Use manipulatives and pictures to represent division as the sharing of objects and as the number of groups of shared objects. • Given a word problem, choose the appropriate operation or operations to solve it. • Explain the relationship among the four basic operations. • Given an equation, such as "3 + 5 = 8," write a story problem that could be solved using the equation.

End of Grade 6:

• Apply the associative, commutative, and distributive properties in a problem solving situation. • Apply the multiplicative and additive properties of zero and the multiplicative property of one. • Demonstrate an understanding of multiplication as repeated addition and of division as repeated subtraction. • Demonstrate an understanding that the product of two whole numbers greater than 1 is greater than either of the factors. • Demonstrate an understanding that when dividing two whole numbers that are greater than one, the quotient will be smaller than

the dividend.

End of Grade 10:

• Examine the four basic operations from a functional perspective; that is, as operations on ordered pairs. • Connect the properties of operations on real numbers to common uses (for example, the distributive property is used in each of the

following cases: 2x + 3x = 5x; 2/7 + 3/7 = 5/7; and 2(3x + 4) = 6x +8).

3c. K-12 Broad Goal: Students will compute.

PURPOSE: The purpose of computation is to solve problems. While computation remains important in mathematics and in everyday life, advances of technology require us to rethink how computation is done today. Students must recognize that estimation, mental computation, use of calculators, and paper and pencil calculation are all appropriate ways to compute solutions to problems. Basic fact memorization should be incorporated into a rich curriculum rather than be its primary focus.

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K-12 Curriculum Standards (3c): K-3

• Model, explain, and develop proficiency with basic facts and algorithms. • Use a variety of mental computation and estimation techniques. • Use calculators in appropriate computational situations. • Given a problem, select appropriate computational techniques to solve the problem and determine the reasonableness of the result.

4-6 Building upon the K-3 experiences, in grades 4-6:

• Multiply and divide whole numbers and decimals. • Add integers and positive rational numbers using models or representations. • Use a variety of mental computation techniques. • Use calculators in appropriate computational situations. Given a problem, select an appropriate computational technique to solve

the problem and determine the reasonableness of the result. • Explore operations with fractions using manipulatives.

7-12 Building upon the K-6 experiences, in grades 7-12:

• Apply the standard algebraic order of operations with real numbers. • Apply the properties of exponents with real numbers. • Simplify expressions involving the use of grouping symbols such as parentheses.

Proficiency Standards (3c): End of Grade 3:

• Using physical models and illustrations, determine the sum or difference of fractions with like or unlike denominators. • Using physical models and illustrations, determine the sum or difference of decimals. • Develop and use algorithms to add and subtract decimals. • Subtract any two 2-digit numbers. • Use manipulatives to illustrate an algorithm for adding or subtracting whole numbers less than 1,000. • Add two or more whole numbers less than 1000. • Use a calculator to extend addition to include 4-digit numbers and subtraction to include 3-and 4-digit numbers. • Demonstrate mastery of the multiplication facts with factors less than or equal to 5.

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End of Grade 6:

• Demonstrate mastery of the multiplication facts with factors less than or equal to 10. • Select an appropriate computational technique in the solution of problems and check the reasonableness of results through mental

computation and estimation strategies. • Use calculators in appropriate problem solving situations. • Add integers using models or representations. • Multiply three digit whole numbers by two digit whole numbers. • Divide three digit whole numbers by two digit whole numbers. • Multiply and divide two and three digit decimals. • Using physical models and illustrations, determine the sum or difference of fractions with like and unlike denominators (using only

halves, fourths, and eighths). • Using physical models, illustrations, and calculators, determine the sum or difference of decimals.

End of Grade 10:

• Perform the four basic operations with rational numbers. • Simplify expressions containing rational numbers, integer exponents, and grouping symbols using conventional methods and

technology. • Evaluate numerical expressions containing scientific notation. • Use the order of operations to evaluate expressions. • Use the field properties to simplify expressions.

3d. K-12 Broad Goal: Students will use mental computation and estimation skills and strategies and know when it is appropriate to do so.

PURPOSE: Students should know what is meant by estimation and mental computation, when they are appropriate, and how close an estimate is required in a given situation. Students should be encouraged to estimate the solution of problems before computation or measurement is done, and to use estimation to determine the reasonableness of answers, and to recognize when an estimate is sufficient as an answer.

K-12 Curriculum Standards (3d): K-3

• Use a variety of estimation strategies when solving problems. • Determine the reasonableness of answers for problems involving arithmetic operations.

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• Recognize and use estimation and mental computation to solve problems where exact answers are not required. • Estimate appropriate units of measurement. • Estimate or predict an approximate solution to a problem. • Communicate the strategies used in estimation based upon previous experiences. • Use estimation to determine the reasonableness of a calculation done by calculator or computer.

4-6 Building upon the K-3 experiences, in grades 4-6:

• Use a variety of mental computation techniques in appropriate situations. • Demonstrate estimation skills through a variety of strategies. • Determine the reasonableness of answers. • Recognize and use estimation and mental computation to solve problems where exact answers are not required.

7-12 Building upon the K-6 experiences, in grades 7-12:

• Determine the reasonableness of answers for problems involving arithmetic operations on real numbers.

Proficiency Standards (3d): End of Grade 3:

• Use estimation and mental computation to determine the reasonableness of answers for problems involving addition and subtraction.

• Use estimation and mental computation to solve problems where exact answers are not required.

End of Grade 6:

• Use estimation and mental computation to determine the reasonableness of answers obtained from the four basic operations on rational numbers.

• Select and use appropriate mental computation and estimation strategies in problem situations when exact answers are not needed.

End of Grade 10:

• Use estimation and mental computation to determine the reasonableness of answers obtained from the four basic operations on real numbers.

• Select and use appropriate mental computation and estimation strategies in problem situations when exact answers are not needed.

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Geometry, Measurement, and Trigonometry

4a. K-12 Broad Goal: Students will name, describe, model, classify, and compare geometric shapes and their properties with an emphasis on their wide applicability in human activity.

PURPOSE: Geometry helps students represent and describe the world in which they live. Students need to investigate, experiment, and explore geometric properties using both technology and hands on materials.

K-12 Curriculum Standards (4a): K-3

• Name, model, describe, and classify cubes, spheres, cones, cylinders, pyramids, and rectangular solids. • Name, model, describe, and classify circles, rectangles, squares, triangles, trapezoids, parallelograms, kites, and rhombuses

(diamonds). • Name, model, describe, and classify right, acute, obtuse, and straight angles. • Determine when pairs of figures are congruent. • Determine the presence or absence of lines of symmetry for given figures.

4-6 Building upon the K-3 experiences, in grades 4-6:

• Explore, discuss, and describe properties of common two and three dimensional figures. • Explore congruence and similarity of two dimensional figures. • Investigate rotational and reflective symmetry. (Point and line symmetry.)

7-12 Building upon the K-6 experiences, in grades 7-12:

• Explore the relationship among definitions, postulates, and theorems. • Investigate the properties of two and three dimensional shapes. • Deduce properties of and relationships among figures from given assumptions. • Use compass and straightedge, manipulatives, and technology to explore geometric constructions. • Deduce properties and relationships among congruent figures and similar figures. • Explore basic transformations (for example: reflections, translations, rotations, or dilations). • Use basic transformations to demonstrate similarity, symmetry, and congruence of figures. • Explore relationships between synthetic and coordinate representations.

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• Apply principles of coordinate geometry, i.e., graph lines and circles and determine the slope and intercept of a line, mid-point of a line segment, the center and radius of a circle.

• Understand the interrelationships between algebraic and geometric representations of the same mathematical concept.

Proficiency Standards (4a): End of Grade 3:

• Use the terms points, lines, and line segments in describing two dimensional figures. • Draw line segments and lines. • Draw lines of symmetry. • Determine if two plane figures are congruent by matching. • Identify, describe, and draw a kite. • Identify and describe pyramids.

End of Grade 6:

• Identify, describe, and name properties of triangles, quadrilaterals, and other polygons. • Identify point and line symmetry in given polygons. • Measure and classify angles. • Identify and draw congruent and similar figures using graph paper.

End of Grade 10:

• Represent and solve problems using the properties of two and three dimensional geometric figures. • Use technology, manipulatives, and/or coordinate geometry to deduce and explain the properties of and the relationships among

geometric figures. • Translate between synthetic and coordinate representations.

4b. K-12 Broad Goal: Students will develop spatial sense.

PURPOSE: We live in a three dimensional world. To interpret, understand, and appreciate that world, students need to develop an understanding of space. Research suggests that there is a high correlation between spatial abilities and success in mathematics. Spatial skills include making and interpreting drawings, forming mental images, visualizing changes, and generalizing about perceptions in the environment.

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K-12 Curriculum Standards (4b): K-3

• Use position terms (for example: inside, outside, above, top-to-bottom, over, or under). • Copy and make shapes by drawing and using manipulatives (for example: pattern blocks, or tangrams). • Draw, compare, and visualize shapes in various positions. • Investigate and predict results of combining, subdividing, and changing shapes using manipulatives (for example: pattern blocks, or

tangrams). • Construct various three dimensional objects. • Describe and/or draw three-dimensional objects from different perspectives.

4-6 Building upon the K-3 experiences, in grades 4-6:

• Explore transformational geometry through the use of slides, flips, and turns and the relations to tessellation's. • Explore classification of two and three dimensional figures based upon properties. • Enhance spatial sense using manipulatives and computer graphics. • Explore the relationships and properties of two dimensional and three dimensional figures using manipulatives and technology.

7-12 Building upon the K-6 experiences, in grades 7-12:

• Explore two dimensional and three dimensional geometry using shadows, perspectives, projections, and maps. • Explore the effects of reflections, translations, rotations, and dilations in the study of congruent and similar geometric shapes and

tessellation's. • Understand, model, describe, analyze, and apply patterns produced by processes of geometric change, formally connecting

iteration, approximations, limits, self similarity, and fractals. • Use manipulatives and computer graphics to enhance spatial sense and to increase understanding of geometry and to explore its

connections to other parts of mathematics, science, and art. • Explore other geometries and their applications (for example, non Euclidean geometries have significant applications in science).

Proficiency Standards (4b): End of Grade 3:

• Divide and separate a shape into smaller shapes. • Recognize and make shapes that can be created from a set of three simple shapes. • Identify congruent figures.

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• Draw figures congruent to a given figure. • Construct three dimensional objects.

End of Grade 6:

• Tessellate (tile) a plane with a given figure and create a figure that will tile the plane. • Describe the shadow of certain figures.

End of Grade 10:

• Sketch specific two dimensional figures, given definitions and/or properties. • Demonstrate that the conditions necessary for congruence or the conditions necessary for similarity are met. • Use technology, manipulatives, and/or coordinate geometry to explain properties of transformations (for example: translations, line

reflections, rotations, dilations, and the composition of these transformations). • Demonstrate an understanding of properties among two and three dimensional figures.

4c. K-12 Broad Goal: Students will develop an understanding of measurement and systems of measurement through experiences which enable them to use a variety of techniques, tools, and units of measurement to describe and analyze quantifiable phenomena.

PURPOSE: Measurement is used in many ways throughout our lives. Students must be introduced to the standard units of measure used in both the metric and English Systems. Students should estimate and measure length, area, capacity, volume, weight, time and temperature, as well as discover practical uses of these skills. High school students must develop more mature insights into the essential role of measurement as a link between the abstractions of mathematics and the concreteness of the real-world. By using various techniques and tools, we describe and analyze quantifiable phenomena to understand and organize our world.

K-12 Curriculum Standards (4c): K-3

• Understand the need for a uniform unit of measure. • Develop measuring skills. • Investigate the attributes of length, area, capacity, volume, and weight using standard (metric and English) and nonstandard units of

measure. • Understand the attributes of time and temperature. • Relate measurement ideas to geometric ideas. • Develop the concepts of perimeter and area. • Make and use estimates of measurements.

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• Develop an understanding of money.

4-6 Building upon the K-3 experiences, in grades 4-6:

• Investigate and compare standard and nonstandard units of measurement for length, area volume, capacity, weight, time, and temperature.

• Explore and discover formulas for area and volume. • Explore estimation strategies for finding areas and volumes. • Explore conversion of units within a measurement system and between systems.

7-12 Building upon the K-6 experiences, in grades 7-12:

• Explore linear and area measures of two dimensional figures. • Explore the volume, surface area and linear measures of three dimensional figures. • Apply the Pythagorean Theorem to real-world situations. • Enhance, extend, apply, and formalize understandings and applications of measurement including strategies for determining

perimeters, areas, and volumes by using formulas, approximations, and computer geometry programs. • Use ratio and proportions to explore the properties of similar figures and use these properties to solve problems. • Choose appropriate techniques and tools to measure quantities in order to achieve specified degrees of precision, accuracy, and

error (or tolerance) of measurement. • Choose an appropriate unit of measure and use appropriate formulas to find perimeter and circumference, area of polygons and

circles, the volume and surface area of selected solids, and the measure of angles. • Choose appropriate units for measuring size, rates, and energy. • Select and use appropriate formulas and procedures to determine a measure when a direct measurement is not available. • Understand and apply measurement in career based contexts and in interdisciplinary situations.

Proficiency Standards (4c): End of Grade 3:

• Tell time to the nearest minute. • Measure line segments to the nearest half inch and quarter inch and to the nearest centimeter. • Investigate the measure of perimeters. • Add units of length that may or may not require regrouping of inches to feet or centimeters to meters. • Estimate weight using pounds or kilograms. • Estimate capacity using quarts, gallons, or liters. • Given a standard unit, estimate and measure the area of a rectangular region.

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• Given a standard unit, estimate the area of any region. • Investigate the addition of hour and half hour time intervals. • Given an amount of money, determine if a purchase can be made.

End of Grade 6:

• Find and/or estimate the perimeter and area of a given quadrilateral or triangle. • Demonstrate an understanding of the use of maps, scale drawings, and timelines. • Compare the relationship between similar figures and their areas.

End of Grade 10:

• Identify and use appropriate units of measurement. • Approximate areas of irregular shapes drawn on a grid. • Convert commonly used measurements to equivalent ones within a measurement system. • Apply the formulas for and choose an appropriate unit of measurement to find the linear and area measures associated with two

dimensional figures and the volume and surface area of three dimensional figures. • Apply the Pythagorean theorem to problem solving situations. • Select an appropriate procedure to determine a measure when a direct measurement cannot be made. • Use ratio and proportion to find the measure of all sides of similar figures.

4d. K-12 Broad Goal: Students will know the basic concepts of trigonometry and apply these concepts to real-world problems.

PURPOSE: all students should explore real-world phenomena which involve right triangle trigonometry. These experiences should include the use of the sine, cosine, and tangent ratios. Technology should be used to facilitate the learning of trigonometry, allowing students more time and power to explore realistic applications.

K-12 Curriculum Standards (4d): K-3

• None at this level.

4-6

• Explore similar figures and the ratios of corresponding lengths (sides) and areas.

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7-12

• Explore the sine, cosine, and tangent ratios in right triangles. • Use the sine, cosine, and tangent ratios to solve real-world problems.

Proficiency Standards (4d): End of Grade 3:

• Does not apply.

End of Grade 6:

• Make scale drawings, keeping sides in proportion. (Scale factor to be kept to a small whole number or fraction with denominator less than 6.)

End of Grade 10:

• Use technology or manipulatives to apply basic trigonometric ratios to solve practical real-world problem.

Data Analysis, Statistics, and Probability

5a. K-12 Broad Goal: Students will use data analysis, statistics and probability to analyze given situations and the outcomes of experiments.

PURPOSE: Collecting, organizing, displaying, and interpreting data, as well as using the information to make decisions and predictions, have become very important in our society. Statistical instruction should be carried out in a spirit of investigation and exploration so students can answer questions about data. Probability must be studied in familiar contexts encouraging students to model situations. Students need to investigate fairness, chances of winning, and uncertainty. Technology should be used as a tool throughout the investigation process.

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K-12 Curriculum Standards (5a): K-3

• Collect, organize, describe, and interpret data. • Formulate and solve problems that involve collecting, organizing, and analyzing data. • Predict outcomes and carry out simple activities involving probability. • Determine which event is most likely or least likely to happen, given appropriate information.

4-6 Building upon the K-3 experiences, in grades 4-6:

• Collect, organize, describe, represent, and interpret data in both simulations and real world situations. • Simulate, display, graph, and analyze data using technology and other means. • Investigate and explore mean, median, and mode. • Investigate and explore the basic elements of sampling. • Make predictions, inferences, and decisions based on interpretation of data. • Demonstrate an ability to read and interpret statistical data presented in text. • Explore situations involving probability.

7-12 Building upon the K-6 experiences, in grades 7-12:

• Collect, organize, describe and interpret data from familiar contexts. • Use a variety of techniques which include but are not limited to spreadsheets, tables, stem and leaf plots, box and whisker plots, to

analyze data and make predictions. • Understand sampling and recognize its role in statistical claims. • Understand and apply measures of central tendency, dispersion and correlation. • Design a statistical experiment to study a problem, conduct the experiment, interpret, and communicate the outcomes. • Use graphics technology to analyze real world data. • Apply the normal curve and its properties to familiar contexts. • Use relative frequency and probability, as appropriate, to represent and solve problems involving uncertainty. • Use simulations to estimate probabilities. • Create, interpret, and understand applications of discrete and continuous probability distributions. • Make predictions based on interpolation and extrapolation from data. • Make decisions based on interpretation of data.

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Proficiency Standards (5a): End of Grade 3:

• Collect data, construct, and interpret picture and bar graphs. • Interpret circle graphs. • Write a story problem using information from a graph. • Given appropriate information, determine which is most likely to happen or whether one event is more likely than another.

End of Grade 6:

• Construct and interpret line plots, stem and leaf plots, frequency distributions, and graphs. • Use multiple representations to display equivalent data. • Select appropriate data to solve simulations and real world problems. • Simulate, display, graph and analyze data in a variety of mediums. • Determine and explore various uses of mean, median, and mode. • Use sampling techniques to make predictions. • Given a sample space find probabilities of events.

End of Grade 10:

• Given a bar, line, circle or picture graph, interpret and analyze the data. • Choose an appropriate scale and construct a graph or diagram using a set of numerical data in a variety of mediums. • Calculate the common measures of central tendency: mean, median and mode. • Use appropriate measure of central tendency in problem situations. • Given a set of numerical data, determine the ordered pairs and make a scatter plot. • Use sample sets to make appropriate inferences and predictions. • Predict and find the probability of outcomes of a simple probability experiment. • Interpret probabilities in real world situations (for example: lotteries, or medical testing.

Functions, Relations and Algebra

6a. K-12 Broad Goal: Students will recognize patterns and describe and represent relations and functions with tables, graphs, equations and rules, and analyze how a change in one element results in a change in another.

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PURPOSE: One of the central themes of mathematics is the study of patterns, relations, and functions. This study requires students to recognize, describe, and generalize patterns and build mathematical models to predict the behavior of real-world phenomenon that exhibit the observed pattern. This study of patterns leads to an exploration of functions, a concept which is an important unifying idea in all aspects of mathematics.

K-12 Curriculum Standards (6a): K-3

• Use concrete models to create a pattern, describe the pattern, and represent the pattern symbolically in a table. • Recognize, describe, extend, and create a wide variety of patterns. • Represent and describe mathematical relationships. • Explore the use of variables and open sentences to express relationships. • Discover patterns or relationships from graphical representations.

4-6 Building upon the K-3 experiences, in grades 4-6:

• Recognize, describe, analyze, extend, and create a wide variety of patterns using models, tables, graphs, simple rules, and manipulatives such as pattern blocks.

• Explore functional relationships to describe how a change in one quantity results in a change in another. • Explore number patterns to discover properties and relationships. • Analyze properties and relationships related to prime numbers, composite numbers, rational numbers, multiples, factors, and

exponents. • Investigate field properties: commutative, associative, distributive, inverse and identity elements. • Explore simple linear equations.

7-12 Building upon the K-6 experiences, in grades 7-12:

• Analyze functions and relations to describe how a change in one quantity results in a change in another. • Use polynomial, rational, trigonometric, and exponential functions to model real-world relationships. • Use charts and tables to organize and represent data.

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Proficiency Standards (6a): End of Grade 3:

• Recognize and describe patterns that involve numbers and shapes. • Create patterns. • Write an open sentence (equation) to express a relationship.

End of Grade 6:

• Generalize simple patterns using words. • Extend a pattern using models. • Identify properties and relationships related to prime numbers, composite numbers, rational numbers, multiples, factors, and

exponents. • Determine how a change in length or width affects perimeter, area, and volume of two and three dimensional figures. • Solve simple linear equations by using concrete materials, tables, or graphs. • Apply the following properties when appropriate: commutative, associative, distributive, inverse, and identity elements.

End of Grade 10:

• Develop algebraic formulas to express relationships which occur in other disciplines (for example: science or economics). • Recognize and describe relationships within a set of data.

6b. K-12 Broad Goal: Students will use algebraic concepts and processes to represent situations that involve variable quantities with expressions, equations, inequalities, matrices and graphs.

PURPOSE: Algebra is the language through which much of mathematics is communicated. It provides a means of representing concepts at an abstract level and then applying those concepts. Students in grades K-6 should explore algebraic concepts in an informal way, emphasizing physical models, data, graphs and other mathematical representations. Formal algebraic manipulation may be deferred to later grades. The understanding of algebraic representation is a prerequisite to formal work in virtually all of mathematics. Algebraic processes are important tools in the study of natural sciences and social sciences.

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K-12 Curriculum Standards (6b): K-3

• Represent situations and number patterns with concrete materials, tables, graphs, verbal rules, and equations; and translate from one to another.

• Develop an understanding of commutative and associative properties. • Write and solve open sentences that describe everyday situations.

4-6 Building upon the K-3 experiences, in grades 4-6:

• Represent situations and number patterns with concrete materials, tables, graphs, verbal rules, and standard algebraic notation. • Understand the use of literal variables, expressions, equations, and inequalities. • Analyze tables and graphs to identify algebraic relationships. • Solve simple linear equations using informal, graphical, and concrete methods. • Plot points on a number line. • Plot points on a rectangular coordinate system. • Explore using equations to model word problems and how to solve the problems using the equations. • Use calculators, computers, and other technology to explore linear relationships.

7-12 Building upon the K-6 experiences, in grades 7-12:

• Simplify algebraic expressions using the standard order of operations. • Perform polynomial operations with manipulatives. • Model and solve problems that involve varying quantities with variables, expressions, equations, inequalities, absolute values,

vectors, and matrices. • Evaluate algebraic expressions for given values of the variable. • Write an equation, using one or two variables, which represents a real-world problem. • Write an inequality, using one or two variables, which represents a real-world problem. • Solve equations and inequalities in one or two variables, by informal and formal algebraic methods. • Use tables and graphs as tools to interpret expressions, equations, and inequalities. • Develop, explain, use, and analyze, procedures for operating on algebraic expressions and matrices. • Solve equations and inequalities of varying degrees using graphing calculators and computers as well as appropriate paper and

pencil techniques. • Model real-world problems with systems of equations or inequalities. • Solve systems of equations or inequalities using technology as well as pencil and paper techniques. • Use technology to explore the use of matrices in the solution of systems of equations.

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• Understand the logic and purposes of algebraic procedures. • Interpret algebraic equations and inequalities geometrically and describe geometric objects algebraically.

Proficiency Standards (6b): End of Grade 3:

• Use recall of number facts to solve simple equations (for example, 4 + _ = 9 may be solved by remembering the fact that 4 + 5 is 9).

• Write the number pattern described by a written or verbal rule. • Illustrate the commutative and associative laws of addition and the commutative law of multiplication with manipulatives.

End of Grade 6:

• Plot points on a number line or in the plane. • Use trial and error to find a solution to an equation from among a given replacement set. • Solve simple linear equations using concrete, informal methods. • Given a table or graph, select a sentence describing the underlying relationship(s).

End of Grade 10:

• Evaluate simple algebraic expressions for given values of the variable. • Perform simple operations on matrices. • Write an equation or inequality in one variable which represents a real-world problem. • Solve equations and inequalities in one variable. • Graph the solution set of equations and inequalities in one variable. • Use appropriate graphing technology (for example: a graphing calculator, or graphing software) to graph an equation or inequality

in two variables. • Use appropriate graphing technology (such as a graphing calculator or graphing software) to solve systems of linear equations in

two variables. • Solve and justify, orally or in writing, the algebraic solution to a real-world problem.

Mathematics of Change

7a. K-12 Broad Goal: Students will be able to use concepts about mathematical change in analyzing patterns, graphs, and applied situations.

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PURPOSE: All natural phenomena are characterized by change. Mathematics is a tool for representing and describing this change, and a preliminary understanding of change is an important precursor to the more formal ideas of calculus. Through explorations of patterns, tables, graphs, functions, and situations which focus on the nature of change, representation, understanding, and recognition of types of change can be promoted. Real-world examples of change can be examined. Proportional reasoning and experience with rates should be part of this process.

K-12 Curriculum Standards (6b): K-3

• Record data in situations where change is occurring. • Notice similarities and differences between patterns, in numerical and geometric situations. • Observe and describe term by term change in patterns. • Compare growth patterns.

4-6 Building upon the K-3 experiences, in grades 4-6:

• Explore sequences involving number and geometric patterns. • Explore rates of change from familiar contexts. • Explore meaning of comparisons (for example: cost per unit, miles per hour, wage rates, or batting averages). • Explore rates of change in discrete (cost per unit) and continuous (distance per unit of time) settings. • Interpret and compare rates of change by looking at graphs.

7-12 Building upon the K-6 experiences, in grades 7-12:

• Use proportional reasoning strategies to solve problems about rates. • Extend patterns and predict nth terms in number sequences. • Extend patterns and predict nth terms in sequences of geometric figures. • Examine tables of numbers from familiar contexts to determine if patterns exist. • Recognize and describe different types of change (for example: arithmetic, geometric, periodic, damped, or oscillating). • Calculate and describe change in continuous and discrete contexts which are familiar. • Interpret and analyze information about change in familiar situations (for example: percent change, average change, or rates such as

distance per unit time). • Recognize and interpret information about change in graphs of functions.

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Proficiency Standards (7a): End of Grade 3:

• Describe the term by term change in a pattern (for example, given a number pattern: describe the rule that was used, determine the next term and explain the reasoning used).

• Given two patterns, describe how they are similar. • Given two patterns, describe how they are different.

End of Grade 6:

• Recognize and extend sequences of number and geometric patterns. • Describe and interpret change from graphs and/or tables of data. • Find averages (for example: batting averages, or grade point averages) and compute rates in familiar contexts (for example: soft

drink consumption, distance per unit of time, hourly wages , or paint mixing).

End of Grade 10:

• Solve rate problems that involve proportional reasoning. • Extend patterns and predict nth terms in number sequences, using words and/or symbols. • Extend patterns and predict nth terms in sequences of geometric figures, using words and/or symbols. • Examine tables of numbers from familiar contexts to determine if patterns exist. • Differentiate among different types of change (for example: arithmetic, geometric, or periodic). • Calculate and describe change in continuous and discrete contexts which are familiar. • Interpret and analyze information about change in familiar situations (for example: percent change, average change, or rates such as

distance per unit time).

Discrete Mathematics

Discrete mathematics is defined as the study of topics which involve items that can be counted, rather than continuous amounts which can only be measured. Discrete mathematics is actually an umbrella term which includes such topics as: counting techniques, sets, relations, functions, logic and reasoning, patterning (iteration and recursion), algorithms, and induction. Probability, networks, graph theory, social decision making, and matrices should also be included in a discrete mathematics curriculum. Embedded in these areas are the three main themes of discrete mathematics: existence (Is there a solution?), counting (How many solutions are there?), and efficiency (What is the best solution?).

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8a. K-12 Broad Goal: Students will use a variety of tools from discrete mathematics to explore and model real-world situations.

PURPOSE: Information and communication continue to impact the modern world and require the understanding of discrete mathematics. Decision making involving sets and systems having a countable number of elements needs to be integrated throughout the curriculum. Students should have experiences with finite graphs, matrices, sequences, recursion and the development and testing of algorithms.

K-12 Curriculum Standards (6b): K-3

• Solve simple counting problems by making lists (for example, handshakes among 3 people). • Explore alternate routes on maps. • Use strategies in game situations.

4-6 Building upon the K-3 experiences, in grades 4-6:

• Represent data in an organized fashion so the number of items in a set can be determined. • Recognize, describe, extend, and create a wide variety of sequences. • Create simple algorithms as a more efficient way of solving problems. • Investigate the benefits of various alternatives in simple networks. • Use logic and inductive reasoning to make predictions related to a series of statements.

7-12 Building upon the K-6 experiences, in grades 7-12:

• Represent data and solve problem situations using graphs, trees and matrices. • Use algebraic and geometric iteration to explore patterns and solve problems. • Use combinations and permutations to solve a variety of problems. • Use algorithms for finding optimal paths and circuits in graphs. • Solve optimization problems. • Analyze different algorithms for efficiency. • Create and interpret discrete probability distributions using technologies whenever appropriate. • Use linear programming to solve problems. • Use the counting principle to solve problems.

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Proficiency Standards (8a): End of Grade 3:

• For a given situation, list all possible combinations (for example: list all possible clothes ensembles with two shirts and three pairs of pants; or, given a map, list all possible paths from point A to point B.

End of Grade 6:

• Use counting techniques to determine the number of outcomes for situations (for example: handshake problems, menu ordering, or clothes matching).

• Solve problems for finding efficient routes (for example: mail delivery, or snow plowing for two or three streets. • Given three statements organize their content into chart form to investigate their relationships.

End of Grade 10:

• Use combinations and permutations to solve a variety of problems.