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Deerfield Community School

Mathematics Curriculum

August 2005

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Deerfield Community SchoolMathematics Curriculum

Table of Contents

Contents Pages

Philosophy and Essentials 1 through 11

Grade Level CurriculumKindergarten K-1 through K-71st Grade 1-1 through 1-112nd Grade 2-1 through 2-133rd Grade 3-1 through 3-84th Grade 4-1 through 4-105th Grade 5-1 through 5-116th Grade 6-1 through 6-137th Grade 7-1 through 7-128th Grade 8-1 through 8-11

ResourcesNumber and Operations NO-1 through NO-33Geometry G-1 through G-58Functions and Algebra FA-1 through FA-52Data, Statistics, and Probability

DS-1 through DS-50

Appendix A M(N&O)- 2-3 A1Appendix B Measurement Benchmarks B1 through B2

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THE EDUCATION OF DEERFIELD’S CHILDRENA SCHOOL PHILOSOPHY

The purpose of the Deerfield Community School is to educate the young people of Deerfield by supporting goals which encourage growth in their personal lives, their homes, and community. Members of the staff are partners with parents and the larger community in supporting these goals. As partners we will meet, listen, and share knowledge. In this way we, as a community, share the commitment and the responsibility to provide for the growth of each child.

We will offer each child the tools with which to live a healthy life, encouraging health as a value in itself and as an important element for social and intellectual functioning.

We will foster the social development of each child, encouraging attention to others’ thoughts and a willingness to express one’s own as essential elements of learning.

We will encourage the intellectual growth of each child. Each child is a ready learner. Teachers must provide children with developmentally appropriate learning opportunities which challenge and inspire.

As adults, we are committed to preparing children for life as they become responsible and competent participants in the community. Children seek us as models and will listen to us. As we learn together, we will: read and write, work with concepts of number and shape, practice thinking scientifically, take an interest in the social goals of humanity, develop an involvement in the arts, and seek to live healthy lives. To do these things is basic to education.

If we are successful, young people of Deerfield will grow up valuing themselves and others, and will contribute harmoniously in their own communities.

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Mathematics Curriculum

In developing the mathematics curriculum, we recognize that the Deerfield Community School is committed to five primary goals: • students will develop a firm grounding in essential computational skills; • students will develop strong mathematical problem solving and reasoning abilities; • students will develop positive attitudes about mathematics; • students will develop the ability to use appropriate technology to solve mathematical problems; and • students will develop the ability to effectively communicate their understanding of mathematics.

Problem solving, while not a direct strand, 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. Students will use problem solving strategies to investigate and understand increasingly complex mathematical content.

Students will communicate their understanding of mathematics. 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.

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ESSENTIAL ENVIRONMENT

The Essential Environment for the student working with mathematics is a setting which recognizes each person's abilities and looks forward to each person's contributions to the classroom community. The community supports individuals and sets standards which define what is expected of contributing members. As learning takes place, members of the learning community communicate their questions and understandings, reason their ways to solutions, connect what they have learned within mathematics and to other areas of learning, and undergo a process of problem solving in which mathematics is applied to everyday life.

This Essential Environment includes these elements:

Validation: The student is viewed directly and responded to directly by the teacher. The teacher does not label the student, but instead values his mathematical work and his actions in order to better understand his perspective. In this way appropriate work plans are determined in consultation with the student. The teacher's behavior in this regard becomes a model for students as they generate work plans in consultation with one another. There is an atmosphere of respect for the genuine efforts of each person. Evidence of this atmosphere appears in the actions of adults and students as they listen carefully and respond honestly to one another through constructive criticism and praise and help one another to revisit, extend, or deepen work.

Involvement: Each student is encouraged to share particular mathematical interests and to develop expertise based upon these interests. Students realize that mathematics touches many places in people's lives and provides many points through which to develop interests and expertise. Students value one another's involvement in the various strands and seek each other out in order to share knowledge, understanding, and their excitement about learning.

Expectation: The teacher expects the student to follow through on planned work. The student expects to be able to use the teacher and other students as resources to be called upon at appropriate times to help in the work. The teacher is open to consulting on expectations about student work, showing flexibility when this is important for student learning. Each student expects to refine some of his mathematical work for public use or display.

Immersion: Mathematics is given real purpose. Students use calendars, clocks, tallies of lunch counts, order forms, and more as they go about the business of daily life in the classroom. Further, mathematics is linked with and used for other areas of study. Students gather data as they study the lives of others and as they investigate the natural world. Students create patterns as they undertake an art project. Students measure out quantities for a cooking project or a science experiment. Along with many other experiences, these uses of mathematics make it an essential part of everyday learning.

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Wondering: A spirit of inquiry is encouraged. The teacher models this by openly acknowledging his own questions and his own mistakes. Students are given plenty of opportunity to share their own wonderings and to learn from their mistakes. Approximation is viewed as a valid tool of thinking in some situations. Problem solving, based on wonderings and careful thinking, becomes a part of life in the classroom.

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ILLUSTRATING ESSENTIAL ENVIRONMENT

We are a part of our environment, and the environment is a part of us. Illustrations of how the different elements of the Essential Environment work will show them working together, as in the following anecdotal record:

When I arrived, the class was having a group meeting. The children had paired up and were conducting surveys among class members. Conversations I had had with the teacher prior to the visit established that the idea of surveys and data collection, along with representation of data through graphing, were understandings and actions which the teacher wanted the students to develop over a period of time. She does not want the students to merely learn how to read a graph. She wants them to develop reasoning skills related to data collection and interpretation.

Today's steps in this ongoing process include communication. The intent of the meeting includes reviewing purpose and checking details: "Nicholas, who's your partner and what's your survey question?" Issues are fleshed out as a part of problem solving: "So how will someone answer your survey question?" Connections are made: "It'll be interesting to compare your data with the data we collected the other day."

The teacher ends the group meeting by setting out the agenda for what will come next. Students who had finished compiling data based on their surveys would meet with her. Others would continue with their surveys, asking classmates the questions they had formulated.

In this brief snapshot of a class meeting we see several elements described as essential to the classroom environment. The teacher is setting out the expectation, making certain that students understand their plans and have thought about how their plans will work. There is an element of wondering here. A question is asked which might prompt the student to put himself in the place of the person answering his question. A question is implied which refers back to previous work, to the immersion which has been established around this mathematical topic. There is validation. Students are addressed directly. The teacher has knowledge of the students' work. She also knows her purposes for the students. She has expertise in the area of mathematics and is directing students as they develop their plans and their own expertise. There is involvement. While the overall mathematical understandings and actions of the data collection project fits with the teacher's knowledge of what students need to learn, the content of the project is largely student determined. The teacher has modeled the process prior to this class meeting. All of the students have witnessed the development of a class graph based on a survey they participated in together. Now, they are off on their own, working with their own survey questions, developing their own graph models - pursuing their own interests and questions, following through on expectations, and growing in expertise.

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ESSENTIAL ACTIONS

The Essential Actions of mathematics form a web of strategies which give the person power to use mathematics for his own problem solving purposes. The Actions discussed below are common to most mathematical situations and many other learning situations, as a person organizes for problem solving.

Recognizing: First, there must be recognition of a problem situation. Problem here is not meant in the negative sense. The problem to be solved may be a practical one, something in need of a solution before moving on; or it may be a source of wondering. As an example, we'll take a practical situation.

It is a dry fall. Your dug well has been known to run out of water at this time of year. This is a problem. You need to have some idea of how much water you have available so that you can plan. The mathematical recognition of the problem situation spots the application of mathematics as a help toward planning for the lack of water. You figure that you will have to call upon your understanding of how to measure volume. This is also another aspect of recognition - a kind of classification - knowing what the situation entails and the understandings upon which you will have to call as you work toward a solution.

Planning: After recognizing a problem situation and how math would apply to that situation, it becomes essential to develop a plan for working toward a solution. Planning involves mapping out the possibilities, then choosing from among them to develop a route which makes sense, given a person's mathematical understandings.

Based upon your recognition of the problem situation and your mathematical understanding, you think that you need to find the volume of water and that volume can be measured by multiplying length times width times height. Finding volume will be step one.Next, you realize that you will have to find out your rate of use of the water in your well. In order to determine this you will have to compare the water levels at different times.Finally, you will need to use this comparison to figure out when you could expect to run out of water at your current rate of use. This will help you to know how much you will need to conserve water.

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Researching: After recognizing a problem situation and developing a plan for coming to a solution there may be a need to gather more information. The initial need for research should become apparent as the problem situation is defined, but may come into play at any time during the problem solving process. To return to our example:

You aren't sure just how much water you have now. Research here takes the form of the physical act of lifting the well cover and using some kind of measuring device to determine the depth of the water. Perhaps you have an idea of your own, but this may be a time to use another of the Essential Actions which may come into play at any time during the problem solving process - consulting. Your neighbor has an idea. Use a piece of rope which is weighted on the drop end, put it into the well and see how far up the wetness comes by measuring with a tape measure after bringing the rope up out of the well. Now that you have the height, you think that you need to find length and width. Suddenly, you are hit with a new problem. Your well is roughly circular. You will need to call upon another Essential Action. You will need to rethink your strategy.

Rethinking: The need to think back upon a plan comes up frequently, the more frequently the more complicated the problem. Taking action, really working on something, may reveal gaps in planning which the initial thinking through did not reveal. Futhermore, assumptions may have been made which do not fit with the problem situation. The case of the dug well involves the latter.

You suddenly realize that there is no way to measure width and length since the structure that you are measuring is not rectangular in shape. You are tempted to bring in another Essential Action, approximation, by inscribing a square on the top of your well (using fastened down string), allowing for a measurement of width and length which will help determine a large portion of the volume of the water. [You are already approximating (estimating) somewhat since you are unsure of how high the intake pipe is from the bottom of the well.] Or you could go back to researching, checking reference books for a formula which will help you determine volume for a cylindrical object. The wondering part of you wants to do this, but the physical act of measuring with the rope over a couple of days has brought you to a new realization. All you really need to know is how much the water goes down over a period of time. Length is the crucial factor, since distance across will remain the same over time (no matter how low the water gets - until it runs out!). You return to planning, having chosen a new strategy for problem solving, and in this case, having simplified the problem greatly.

Solving: When the plan has been established, the research undertaken, and the rethinking accomplished, the problem may be solved. In the example of the dug well:

You measured the length of wetness on the rope on Monday and find you have 8 feet. On Wednesday you have 7 feet six inches. On Friday you have 7 feet two inches. On Sunday you have 6 feet 7 inches. You convert to inches: Monday=96" Wednesday=90" Friday=86" Sunday=79", a loss of 6 inches, then 4 inches, then 7 inches. Your rate of loss is uneven, probably due to wash days being on some days and not on others. You rethink, realizing that the variables of the situation, including uneven use of water and the (much hoped for!) possibility of rain, make an exact calculation of the date of no water fairly useless. You could keep measuring and averaging loss per day to calculate this, but you realize that something else may be more useful. You go public with your information.

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Communicating: When we communicate what we have discovered, we share our information and understandings. The form of expression becomes important. What is the best way to get my message across? Also important is the attention to appearance. What will someone else want to see and find easy to use? Mathematical expression may come out of the excitement of discovery and the desire to let the other person in on what has been found, seen, or understood. It may also serve a practical purpose, as in the case of the well.

You call your family together around the dining room table. You have begun a graph entitled The Height of the Water in Our Well. You invite the family to help you color in the bars which show heights for the days you've tabulated so far. There is space left for information from future days to be entered. You also have a chart to take outside for recording height. Though the children are not to measure the length of wetness of the rope on their own (fortunately, the well cover is very heavy), you invite them to accompany you as you work on this project together. The chart will go on the kitchen wall, a reminder to conserve water, but also a possible reassurance that things aren't as bad as they seem. Then your daughter makes a suggestion.

Extending: Communicating a mathematical discovery or mathematical information does bring a point of closure, but in many cases it may only be a pause along the way towards more work. Perhaps that piece of work will spark a new idea and be useful as a part of future work. Such was the case with the family facing drought.

Recently, your daughter's class had set out a rain gauge at school to measure any rain that did fall during that dry season. She knows how to make a gauge and says she'd like to keep track of how much it rains at home. "Wouldn't it be interesting to see how much our well water changes after it rains?" Everyone agrees to help make and fill in a graph which will measure rainfall.

Deepening: Deepening comes through reflection. After working through problems, a person may take stock of new understandings and new strategies. The person engages in self-assessment, considering what he knows and understands, and how to use what he has gained. The new learning will become a part of what the person brings to problem solving situations in the future. In the case of the well:

You really knew it all along, but you didn't know that you knew it. Realizing that you just needed to measure the change in water height, you saw more clearly the logic of a situation involving a constant factor and a variable. You know that what really matters is how much water there is in the well, and that that has to do with a change in volume. But volume is a little harder to measure. Since length will change with volume and width will not, measuring length will be sufficient to indicate changes in the amount of water available. Hence, your new strategy has been to measure length only.

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Additional Essential Actions embedded in the problem solving process:

Consulting: Consulting may happen informally, or it may be a part of a strategy for problem solving. In some situations a person may consult the expertise of another person. Consulting is like researching, except that it implies working with another person to seek out information or ideas. It is natural to enter into consultations with others frequently during the problem solving process.

Approximating: Approximating or estimating is useful for several purposes. First, estimation will give a good idea of the reasonableness of an answer. Thus, after a formal calculation or measurement takes place, comparing that answer with an estimate will be a good check. Second, some situations really require only an approximate answer. Since approximations can be done more easily in the head, they make sense when they are all that is required to fit a situation. Third, the world is rough. Building a house will require some close to exact measurements if the house is to stand right. This is true for many human made things. But if we are measuring some phenomenon, such as the depth of a dug well, an approximation will have to do.

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ILLUSTRATING ESSENTIAL ACTIONS

The Essential Actions of mathematics form the solid bed of purpose upon which mathematical knowledge and understandings can be built. In the home, in the work place, and in the school, problem solving becomes meaningful within the context of real action. In the example of the well, the setting for problem solving was the home, though there was a home/school connection when the rain gauge built in school was brought in to solve a problem at home. Home/school connections involving real action purposes for mathematics can work the other way as well. At the Deerfield Community School, one example of this is a project in which a grade level takes the perceived need for useful and informative calendars and meets this need through creating, designing, producing, and marketing calendars. Connections here exist not only between home and school, but also through the areas of learning and within mathematics. Students employ decision making, informational research skills, artistic skills, writing, measurement, patterning, data collection, graphing, data analysis, and more.

During the fall of the year, students produce these calendars which will be sold to the general public in December. Students determine a theme for the calendar, develop a design both for the month grid and for facing pages, work through the specifics of layout, write and draw to provide content relative to the theme, then market the calendar, account for sales, and analyze sales data. All of these steps require problem solving strategies common to all learning, as well as problem solving strategies specific to mathematics. Students recognize problems, develop plans for solutions, consult with one another, research as needed, approximate on initial drafts, rethink plans, solve, then communicate solutions. Through this learning process, they extend and deepen their learning. The calendar project is an example of integrated learning in which mathematics plays a pivotal role.

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ESSENTIAL UNDERSTANDINGS

The section of the guide which follows is organized according to the state Grade Level Expectations (GLEs) which are broken down into the following four strands: Number and Operations; Geometry and Measurement; Functions and Algebra; Data, Statistics and Probability.

Number and Operations: Students will develop number sense and an understanding of our numeration system. 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 phenomenon. 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.

Geometry and Measurement: Students will name, describe, model, classify, and compare geometric shapes and their properties with an emphasis on their wide applicability in human activity. 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.

Functions and Algebra: 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. 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.

Data, Statistics, and Probability: Students will use data analysis, statistics and probability to analyze given situations and the outcomes of experiments. 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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Grade Level K Expectations by Strand

Number and OperationsGLE Number Grade Level Expectations Terms Defined (Definition Number) Notes and Resources

M(N&O)-K-1

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 or 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” (i.e., equal sized parts or sets) using models, explanations, or other representations.

o Rational number (N&O-1)o Whole number (N&O-2)o Fraction (N&O-3)o Equivalent numbers (N&O-14)o Composition of numbers

(N&O-15)o Decomposition of numbers

(N&O-16)o Area model to represent part to

whole relationship (N&O-17)o Set model (N&O-18)

Developing Number ConceptsBook 1 – chapter 1

Investigations in Number, Data and Space Series

Materials – unifix cubes, color tiles, teddy bear counters etc.

These resources apply to all GLE’s in this section

M(N&O)-K-2

Demonstrates understanding of the relative magnitude of numbers from 0 to 20 through investigations that demonstrate one-to-one correspondence.

Demonstrates understanding of the relative magnitude of numbers from 0 to 20 through investigations that compare whole numbers to each other or to benchmark whole numbers (5, 10).

Demonstrates understanding of the relative magnitude of numbers from 0 to 20 through investigations that demonstrate an understanding of the relation of inequality when comparing whole numbers by using “1 more” or “1 less”.

Demonstrates understanding of the relative magnitude of numbers from 0 to 20 through investigations that connect numbers orally and written as numerals to the quantities that they represent using models, representations, or number lines.

o Whole number (N&O-2)o Area model to represent part to

whole relationship (N&O-17)o Set model (N&O-18)o Linear model (N&O-19)o Relative magnitude (N&O-20)o Within number formats (N&O-

21)o Ordering (N&O-26)o Comparing (N&O-27)o Number line (N&O-28)

Developing Number ConceptsBook 1 – chapters 1 & 3

Investigations in Number Data and Space Series

Materials – beans, unifix cubes, etc.

These resources apply to allGLE’s in this section

K - 1


M(N&O)-K-3

Demonstrates conceptual understanding of mathematical operations through investigations involving addition and subtraction of whole numbers (from 0 to 10) by solving problems involving joining actions, separating actions, part-part whole relationships, and comparison situations.

Demonstrates conceptual understanding of mathematical operations through investigations involving addition of multiple one-digit whole numbers. (See Appendix A.)

o Whole number (N&O-2)o Composition of numbers (N&O-

15)o Decomposition of numbers

(N&O-16)

Developing Number Concepts – Book 2

Investigations in Number, Data and Space Series

Materials – teddy bear counters, color tiles etc.

These resources apply to allGLE’s in this section

M(N&O)-K-4

None

M(N&O)-K-5

Demonstrates understanding of monetary value through investigations involving knowing the names and values for coins (penny, nickel and dime).

o Decimal (N&O-7)o Equivalent numbers (N&O-14)o Composition of numbers (N&O-


(N&O-16)

None

M(N&O)-K-6

Mentally adds and subtracts whole numbers by naming the number that is one more or one less than the original number.

(IMPORTANT: The intent of this GLE is to embed mental arithmetic throughout the instructional program, not to teach it as a separate unit.)

All of this is referred to mentally!

Developing Number Concepts – Book 1 Chapter 3 and Book 2 Chapter 2

K - 2


M(N&O)-K-7

Makes estimates of the number of objects in a set (up to 20) by making and revising estimates as objects are counted (e.g., A student estimates the number of pennies in a jar as 20. Then the student counts the first 10 and makes another estimate based on those that have been counted and those that remain in the jar.).

(IMPORTANT: Estimation should be imbedded instructionally throughout all strands.)

Developing Number Concepts – Book 1 Chapters 1 & 3

Investigations in Number Data and Space Series

Materials – Pennies, beans, blocks etc.

M(N&O)-K-8

None

K - 3

Geometry and MeasurementGLE Number Grade Level Expectations Terms Defined (Definition Number) Notes and Resources

M(G&M)-K-1

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.

Uses properties, attributes, composition, or decomposition to recognize, name, and build polygons and circles in the environment.

o Attributes and properties (G&M-1)

o Uses composition and decomposition (G&M-5)

o Polygons (G&M-2)o Non-measurable attribute

(G&M-8)o Measurable attribute (G&M-9)

Investigations in Number Data and Space –“Making Shapes and Building Blocks” & “Collecting, Counting and Measuring”

Also found in software form under school wide applications

Materials – Pattern blocks, craft sticks, geoblocks etc.

M(G&M)-K-2

None

M(G&M)-K-3

None

M(G&M)-K-4

None

M(G&M)-K-5 None

M(G&M)-K-6

None

Developing Number Concepts – K - 4

Geometry and MeasurementGLE Number Grade Level Expectations Terms Defined (Definition Number) Notes and ResourcesM(G&M)-K-7

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]).

Demonstrates conceptual understanding of measurable attributes and compares objects visually and with direct comparison.

o Measures and uses units of measure appropriately and consistently (G&M-31)

o Makes conversions within and across systems (G&M-32)

Book 1 Chapters 1 & 3

Investigations in Number, Data and Space: “Collecting, Counting and Measuring”

Materials – ribbons, unifix cubes, blocks etc.

M(G&M)-K-8

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.

Identifies a clock and calendar as measurement tools (days of week, months of the year).

Investigations in Number, Data and Space: “Mathematical Thinking in Kindergarten”

Developing Number Concepts – Book 1 Chapter 2

M(G&M)-K-9

Demonstrates understanding of spatial relationships by location and position using positional words to locate and describe where an object is found in the environment.

Investigation in Number, Data and Space – Series

Developing Number Concepts – Books 1 & 2

M(G&M)-K-10

None

K - 5

Functions and AlgebraGLE Number Grade Level Expectations Terms Defined (Definition Number) Notes and Resources

M(F&A)–K–1

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 or three elements, or by translating AB patterns across formats (e.g., an abb pattern can be represented as snap, clap, clap or red, yellow, yellow) or by identifying number patterns in the environment.

o Patterns (F&A-1)o Extend a pattern (F&A-5)o Numeric patterns (F&A-3)o Non-numeric patterns (F&A-4)o Sequence (F&A-6)o Pattern Summary Table by grade

level (F&A-9)

Investigations in Number, Data and Space: “Pattern Trains and Hopscotch Paths

Developing Number Concepts – Book 1 Chapter 2

Materials – pattern blocks, unifix cubes, teddy bear counters etc.

M(F&A)–K–2

None

M(F&A)–K–3

None

M(F&A)–K–4

None o Equality (F&A-30)o Demonstrates conceptual

understanding of equality by solving equivalence (F&A-31)

o Number sentences (F&A-33)o Equation (F&A-32)o Algebraic equation notation

(F&A-34)o Examples of forms of equations

(F&A-35)

K - 6

Data, Statistics, and ProbabilityGLE Number Grade Level Expectations Terms Defined (Definition Number) Notes and Resources

M(DSP) – K-1

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.

(IMPORTANT: Analyzes data consistent with concepts and skills in M(DSP)–K–2.)

o Interprets a given representation (DSP-21)

o Representation (DSP-1)o Pictograph (DSP-2)o Line plot (DSP-5)o Tally chart (DSP-3)o Frequency table (DSP-4)

Investigations in Number, Data and Space: “Counting Ourselves and Others”

M(DSP) – K-2

Analyzes patterns, trends, or distributions in data in a variety of contexts by determining or using more, less, or equal (e.g., Have there been more, less, or the same number of cloudy days compared to sunny days this week?).

o Pattern (F&A-1) Investigations in Number, Data and Space: “Counting Ourselves and Others”

Developing Number Concepts Book 1 Chapter 2

M(DSP) – K-3

None

M(DSP) – K-4

None

M(DSP) – K-5

None o Combination (DSP-41)o Frequency table (DSP-4)o Tree diagram (DSP-28)o Solves problems using a variety

of counting strategies (DSP-39)

M(DSP) – K-6

None

Grade Level 1 Expectations by StrandK - 7


M(N&O)-1-1

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 or 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.

o Rational number (N&O-1)o Whole number (N&O-2)o Fraction (N&O-3)o Proper fraction (N&O-4)o Improper fraction (N&O-5)o Ratio (N&O-12)o Expanded Notation (N&O-13)o Equivalent numbers (N&O-14)o Composition of numbers




Developing Number Concepts, Book 1:Counting, Comparing, & Pattern

Developing Number Concepts, Book :Addition & Subtraction

Developing Number Concepts, Book 3: Place Value, Multiplication & Division

Investigations In Number, Data, Space: Mathematical Thinking At Grade 1

Investigations In Number, Data, Space: Building Number Sense

Investigations: Quilt Squares & Block Towns

Understanding Geometry (K. Richardson)

K - 8

Number and OperationsGLE Number Grade Level Expectations Terms Defined (Definition Number) Notes and ResourcesM(N&O)-1-2

Demonstrates understanding of the relative magnitude of numbers from 0 to 100 by ordering whole numbers.

Demonstrates understanding of the relative magnitude of numbers from 0 to 100 by comparing whole numbers to each other or to benchmark whole numbers (5, 10, 25, 50, 75, 100).

Demonstrates understanding of the relative magnitude of numbers from 0 to 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”, “10 less”.

Demonstrates understanding of the relative magnitude of numbers and by connecting number words (from 0 to 20) and numerals (from 0 to 100) to the quantities and positions that they represent using investigations, models, representations, or number lines.





Developing Number Concepts, Book 2: Addition & Subtraction



Investigations In Number, Data, Space: Number Games & Story Problems

M(N&O)-1-3

Demonstrates conceptual understanding of mathematical operations involving addition and subtraction of whole numbers (from 0 to 30) by solving problems involving joining actions, separating actions, part-part whole relationships, and comparison situations.

Demonstrates conceptual understanding of mathematical operations involving addition of multiple one-digit whole numbers.(See Appendix A.)



(N&O-16)

Developing Number Concepts, Book 2: Addition & SuUbtraction Investigations In Number, Data, Space: Mathematical Thinking At Grade 1



M(N&O)-1-4

None

K - 9


M(N&O)-1-5

Demonstrates understanding of monetary value by knowing the names and values for coins (penny, nickel, dime, and quarter).

Demonstrates understanding of monetary value by adding collections of like coins together to a sum no greater than $1.00.

o Decimal (N&O-7)o Expanded Notation (N&O-13)o Equivalent numbers (N&O-14)o Composition of numbers (N&O-


(N&O-16)


M(N&O)-1-6

Mentally adds and subtracts whole numbers by naming the number that is one or two more or less than the original number.

Mentally adds and subtracts whole number facts to ten (e.g., 5 + 3 = 8; 8 – 5 = 3).



M(N&O)-1-7

Makes estimates of the number of objects in a set (up to 30) and revises estimates as objects are counted (e.g., A student estimates the number of pennies in a jar as 28. Then the student counts the first 10 and makes another estimate based on those that have been counted and those that remain in the jar.).

(IMPORTANT: Estimation should be embedded instructionally throughout all strands.)





K - 10


M(N&O)-1-8

Applies properties of numbers (odd, even, composition, and decomposition [e.g., 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.





K - 11


M(G&M)-1-1

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 nonmeasurable or measurable attributes.

Uses properties, attributes, composition, or decomposition to recognize, name, build, and draw polygons and circles in the environment.





Understanding Geometry (K.Richardson)


Investigations In Number, Data, Space: Quilt Squares & Block Towns

M(G&M)-1-2

None

M(G&M)-1-3

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 (e.g., show a wooden cylinder and students identify common objects of the same shape).



M(G&M)-1-4

Demonstrates conceptual understanding of congruency by making mirror images and creating shapes that have line symmetry.



M(G&M)-1-5 None

M(G&M)-1-6

Demonstrates conceptual understanding of the length/height of a two-dimensional object using non-standard units (e.g. comparing objects to trains of small cubes, using iterations of a small unit to measure an object).

Investigations In Number, Data, Space: Quilt Squares & Block TownsInvestigations In Number, Data, Space: Bigger, Taller, Heavier, Smaller

K - 12


M(G&M)-1-7

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]).

Demonstrates conceptual understanding of measurable attributes and compares objects visually, with direct comparison, and using non-standard units.



Investigations In Number, Data, Space: Bigger, Taller, Heavier, Smaller

M(G&M)-1-8

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

Recognizes an hour and “on the ½ hour”.


Investigations In Number, Data, Space: Bigger, Taller, Heavier, Smaller

M(G&M)-1-9

Demonstrates understanding of spatial relationships using location and position by using positional words (e.g., 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.


Investigations In Number, Data, Space: Quilt Squares & Block TownsInvestigations In Number, Data, Space: Bigger, Taller, Heavier, Smaller

K - 13


M(G&M)-1-10

None

K - 14


M(F&A)–1–1

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, by finding a missing element (e.g., 2, 4, 6, ___, 10), or by translating repeating patterns across formats (e.g., an abb pattern can be represented as snap, clap, clap; or red, yellow, yellow; or 1,2,2).


level (F&A-9)






M(F&A)–1–2

None

M(F&A)–1–3

None

K - 15


M(F&A)–1–4

Demonstrates conceptual understanding of equality by finding the value that will make an open sentence true (e.g., 2 + � = 7) (limited to one operation and limited to use of addition or subtraction) using models, verbal explanations, or written equations.

o Equality (F&A-30)o Demonstrates conceptual




(F&A-35)






K - 16


M(DSP) – 1-1

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.

(IMPORTANT: Analyzes data consistent with concepts and skills in M( DSP)– 1– 2. )




Investigations In Number, Data, Space: Survey Questions & Secret Rules


M(DSP) – 1-2

Analyzes patterns, trends, or distributions in data in a variety of contexts by determining or using more, less, or equal.

o Pattern (F&A-1)Developing Number Concepts, Book 1:Counting, Comparing, & Pattern





Investigations In Number, Data, Space: Survey Questions & Secret Rules

M(DSP) – 1-3

None

M(DSP) – 1-4

None

K - 17


M(DSP) – 1-5

For a probability event in which the sample space may or may not contain equally likely outcomes, groups use experiments to describes the likelihood or chance of an event (using “more likely,” “less likely”, or “equally likely”).

o Combination (DSP-41)o Frequency table (DSP-4)o Tree diagram (DSP-28)o Solves problems using a variety


M(DSP) – 1-6

None

Grade Level 2 Expectations by Strand

K - 18


M(N&O)-2-1

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 or decomposing numbers (e.g., 34 = 17 + 17; 34 = 29 + 5); and in expanded notation (e.g., 141 = 1 hundred + 4 tens + 1 one or 141 = 100 + 40 + 1) 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.

o Rational number (N&O-1)o Whole number (N&O-2)o Fraction (N&O-3)o Proper fraction (N&O-4)o Improper fraction (N&O-5)o Ratio (N&O-12)o Expanded Notation (N&O-13)o Equivalent numbers (N&O-14)o Composition of numbers






Developing Number Concepts Using Unifix Cubes

Math Their Way

Investigations: Mathematical Thinking At Grade 2

Investigations: Coins, Coupons & Combinations

Investigations: Shapes, Halves & Symmetry

Investigations: Putting Together & Taking Apart

M(N&O)-2-2

Demonstrates understanding of the relative magnitude of numbers from 0 to 199 by ordering



K - 19


whole numbers.

Demonstrates understanding of the relative magnitude of numbers by comparing whole numbers to each other or to benchmark whole numbers (10, 25, 50, 75, 100, 125, 150, or 175).

Demonstrates understanding of the relative magnitude of numbers by demonstrating an understanding of the relation of inequality when comparing whole numbers by using "1 more", "1 less", "10 more", "10 less", "100 more", or "100 less".

Demonstrates understanding of the relative magnitude of numbers or by connecting number words and numerals to the quantities they represent using models, number lines, or explanations.




Math Their Way





M(N&O)-2-3

Demonstrates conceptual understanding of mathematical operations involving addition and subtraction of whole numbers by solving problems involving joining actions, separating actions, part-part




Developing Number Concepts, Book 3: Place Value,

K - 20


whole relationships, and comparison situations.

Demonstrates conceptual understanding of mathematical operations involving addition of multiple one-digit whole numbers. (See Appendix A.)

(N&O-16) Multiplication & Division

Math Their Way





M(N&O)-2-4

None

M(N&O)-2-5

Demonstrates understanding of monetary value by adding coins together to a value no greater than $1.99 and representing the result in dollar notation;

Demonstrates understanding of monetary value making change from $1.00 or less.

Demonstrates understanding of monetary value by recognizing equivalent coin representations of the same value (values up to $1.99).

o Decimal (N&O-7)o Expanded Notation (N&O-13)o Equivalent numbers (N&O-14)o Composition of numbers (N&O-


(N&O-16)

Math Their Way



M(N&O)-2-6

Mentally adds and subtracts whole number facts to a sum of 20.

Names the number that is 10 more or less than the original number, and mentally adds and subtracts two-

Developing Number Concepts, Book 1:Counting, Comparing & Pattern

Developing Number Concepts,

K - 21


digit multiplies of ten (e.g., 60 + 80, 90 – 30).


Book 2: Addition & Subtraction


Math Their Way





M(N&O)-2-7

Makes estimates of the number of objects in a set (up to 50) by selecting an appropriate method of estimation.

(IMPORTANT: The intent of this GLE is to embed estimation throughout the instructional program, not to teach it as a separate unit.)



Math Their WayK - 22





M(N&O)-2-8

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.



Math Their Way




K - 23


K - 24


M(G&M)-2-1

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.






Investigations: Does It Walk, Crawl, or Swim?

Investigations: How Long? How Far?


M(G&M)-2-2

None

M(G&M)-2-3

None

M(G&M)-2-4

Demonstrates conceptual understanding of congruency by composing and decomposing two-dimensional objects using models or explanations (e.g., using triangular pattern blocks to construct a figure congruent to the hexagonal pattern block).

Demonstrates conceptual understanding of congruency and uses line symmetry to demonstrate congruent parts within a shape.




M(G&M)-2-5 None

K - 25


M(G&M)-2-6

Demonstrates conceptual understanding of perimeter and area by using models or manipulatives to surround and cover polygons.

o Polygons (G&M-9)o Demonstrates conceptual

understanding of perimeter and area using models and manipulatives to surround and cover polygons (G&M-26)




M(G&M)-2-7

Measures and uses units of measures appropriately and consistently, and makes conversions within systems when solving problems across the content strands.

Benchmarks in Appendix B.





M(G&M)-2-8 None

M(G&M)-2-9

Demonstrates understanding of spatial relationships using location and position by using positional language in two- and three-dimensional situations to describe and interpret relative positions (e.g., above the surface of the desk, below the triangle on the paper).

Demonstrates understanding of spatial relationships using location and position by creating and interpreting simple maps and naming locations on simple coordinate grids.




M(G&M)-2-10

None

K - 26


M(F&A)–2–1

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 (e.g., 2, 4, 6, ___, 10).


level (F&A-9)



Math Their Way



M(F&A)–2–2

None

M(F&A)–2–3

None

M(F&A)–2–4

Demonstrates conceptual understanding of equality by finding the value that will make an open sentence

o Equality (F&A-30)o Demonstrates conceptual


K - 27


true (e.g. 2 + □ = 7) (limited to one operation and limited to use of addition or subtraction).




(F&A-35)


Math Their Way




K - 28


M(DSP) – 2-1

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.

(IMPORTANT: Analyzes data consistent with concepts and skills in M(DSP)-2-2.)






Math Their Way






K - 29


M(DSP) – 2-2

Analyzes patterns, trends, or distributions in data in a variety of contexts by determining or using more, less, or equal.

o Pattern (F&A-1)Developing Number Concepts, Book 2: Addition & Subtraction


Math Their Way




M(DSP) – 2-3 None

M(DSP) – 2-4

Uses counting techniques to solve problems involving combinations using a variety of strategies (e.g., student diagrams, organized lists, tables, tree diagrams, or others)(e.g., How many ways can you make 50 cents using nickels, dimes, and quarters?).

o Combination (DSP-41)o Frequency table (DSP-4)o Tree diagram (DSP-28)o Solves problems using a variety




Math Their Way


K - 30


M(DSP) – 2-5

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.


Math Their Way

Investigations: How Many Pockets? How Many Teeth?

M(DSP) – 2-6

In response to a teacher or student generated question or hypothesis, groups decide the most effective method (e.g., survey, observation, experimentation) to collect the data (numerical or categorical) necessary to answer the question.

In response to a teacher or student generated question or hypothesis, groups collect, organize, and appropriately display the data.

In response to a teacher or student generated question or hypothesis, groups analyze the data to draw conclusions about the question or hypothesis being tested, and when appropriate make predictions.

(IMPORTANT: Analyzes data consistent with concepts and skills in M(DSP)–2–2.)

Math Their Way



K - 31



M(N&O)-3-1

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 using models, explanations, or other representations.

Demonstrates conceptual understanding of rational numbers with respect to:decimals (within a context of money) as a part of 100 using models, explanations, or other representations.

o Rational number (N&O-1)o Whole number (N&O-2)o Fraction (N&O-3)o Proper fraction (N&O-4)o Improper fraction (N&O-5)o Decimal (N&O-7)o Ratio (N&O-12)o Equivalent numbers (N&O-14)o Composition of numbers




M(N&O)-3-2

Demonstrates understanding of the relative magnitude of numbers from 0 to 999 by ordering whole numbers or by comparing whole numbers to benchmark whole numbers (100, 250, 500, or 750).

Demonstrates understanding of the relative magnitude of numbers from 0-999 by comparing whole numbers to each other.

Demonstrates understanding of the relative magnitude of numbers by 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.

o Whole number (N&O-2)o Fraction (N&O-3)o Proper fraction (N&O-4)o Improper fraction (N&O-5)o Equivalent numbers (N&O-14)o Area model to represent part to

whole relationship (N&O-17)o Set model (N&O-18)o Linear model (N&O-19)o Relative magnitude (N&O-20)o Within number formats

(N&O-21)o Ordering (N&O-26)o Comparing (N&O-27)o Number line (N&O-28)

K - 32


M(N&O)-3-3

Demonstrates conceptual understanding of mathematical operations by describing or illustrating the inverse relationship between addition and subtraction of whole numbers.

Demonstrates conceptual understanding of mathematical operations by describing or illustrating the relationship between repeated addition and multiplication using models, number lines, or explanations.


whole relationship (N&O-17)o Set model (N&O-18)o Number line (N&O-28)o Linear model (N&O-19)o Inverse relationships in

operations (N&O-29)o Relationship between repeated

addition and multiplication of whole numbers (N&O-30)

o Concept of multiplication (N&O-37)

M(N&O)-3-4

Accurately solves problems involving addition and subtraction with regrouping to 999.

Accurately solves problems involving the concept of multiplication.

Accurately solves problems involving addition or subtraction of decimals with regrouping in the context of money.

o Decimal (N&O-7)o Relationship between repeated


o Relationship between repeated subtraction and division of whole numbers (N&O-31)

o Accurately solves problems (N&O-35)

o In and out of context (N&O-36)o Concept of multiplication (N&O-

37)

M(N&O)-3-5 None

K - 33


M(N&O)-3-6

Mentally adds whole number facts through 20; adds two-digit and one-digit whole numbers; adds combinations of two-digit and three-digit whole numbers that are multiples of ten (e.g., 60 +50, 300 + 400, 320 + 90). Mentally subtracts whole number facts through 20; a one-digit whole number from a two-digit whole number (e.g., 37 – 5); and subtracts two-digit whole numbers that are multiples of ten and three-digit whole numbers that are multiples one hundred (e.g., 50 – 20, 500 – 200).



M(N&O)-3-7

Makes estimates in a given situation by identifying when estimation is appropriate, selecting the appropriate method of estimation, and evaluating the reasonableness of solutions appropriate to grade level GLEs across content strands.


M(N&O)-3-8

Applies properties of numbers (odd, even, and multiplicative property of zero for single-digit whole numbers [6 x 0 = 0]) to solve problems and to simplify computations involving whole numbers.

Applies field properties (commutative for addition, associative for addition, identity for multiplication, and commutative for multiplication for single digit whole numbers [e.g., 3 x 4 = 4 x 3]) to solve problems and to simplify computations involving whole numbers.

K - 34


M(G&M)-3-1

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.


o Non-measurable attribute (G&M-8)

o Measurable attribute (G&M-9)o Angles (G&M-10)o Triangle (G&M-3)o Quadrilaterals (G&M-4)o Polygons (G&M-2)o Uses composition and

decomposition (G&M-5)

M(G&M)-3-2

None

M(G&M)-3-3

None

M(G&M)-3-4

Demonstrates conceptual understanding of congruency by matching congruent figures using reflections, translations, and rotations (flips, slides, and turns) (e.g., recognizing when pentominoes are reflections, translations and rotations of each other). Demonstrates conceptual understanding of congruency by composing and decomposing two and three-dimensional objects using models or explanations (e.g., Given a cube, students use blocks to construct a congruent cube.).

Demonstrates conceptual understanding of congruency by using line symmetry to demonstrate congruent parts within a shape.

M(G&M)-3-5

Demonstrates conceptual understanding of similarity by identifying similar shapes.

K - 35


M(G&M)-3-6

Demonstrates conceptual understanding of perimeter of polygons on grids using a variety of models or manipulatives. Expresses all measures using appropriate units.

Demonstrates conceptual understanding of area of rectangles on grids using a variety of models or manipulatives. Expresses all measures using appropriate units.


understanding of perimeter and area using models and manipulatives to surround and cover polygons (G&M-26)


M(G&M)-3-7





M(G&M)-3-8 NoneM(G&M)-3-9

Demonstrates understanding of spatial relationships using location and position by interpreting and giving directions from one location to another (e.g., classroom to the gym, from school to home) using positional words.

Demonstrates understanding of spatial relationships using location and position between locations on a map or coordinate grid (first quadrant) using positional words or compass directions.

M(G&M)-3-10

Demonstrates conceptual understanding of spatial reasoning and visualization by copying, comparing, and drawing models of triangles, squares, rectangles, rhombi, trapezoids, hexagons, and circles.

Demonstrates conceptual understanding of spatial reasoning and visualization by building models of rectangular prisms from three-dimensional representations.

K - 36


M(F&A)–3–1

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, or finding missing elements.


level (F&A-9)

M(F&A)–3–2

None

M(F&A)–3–3

None

M(F&A)–3–4

Demonstrates conceptual understanding showing equivalence between two expressions using representations of the expressions; or by finding open sentence true (e.g., 2 + � = 7). (Limited to one operation and limited to use addition, subtraction, or multiplication.)

o Equality (F&A-30)o Demonstrates equality (F&A-31)o Number sentences (F&A-33)o Equation (F&A-32)o Algebraic equation notation

(F&A-34) o Examples of forms of equations

(F&A-35)

K - 37


M(DSP) - 3-1

Interprets a given representation (line plots, tally charts, tables, or bar graphs) to answer questions related to the data, to analyze the data to formulate conclusions, or to make predictions.



o Representation (DSP-1)o Line plot (DSP-5)o Tally chart (DSP-3)o Frequency table (DSP-4)o Bar graph (DSP-6)

M(DSP) - 3-2

Analyzes patterns, trends, or distributions in data in a variety of contexts by determining or using most frequent (mode), least frequent, largest, or smallest.

o Pattern (F&A-1)o Mode (DSP-17)

M(DSP) - 3-3

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.

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.


o Representation (DSP-1)o Identifies or describes

representations or elements of representations that best display a given set of data or situation (DSP-23)

M(DSP) - 3-4

Uses counting techniques to solve problems involving combinations and simple permutations using a variety of strategies (e.g., student diagrams, organized lists, tables, tree diagrams, or others).

o Combinations (DSP-41)o Simple Permutations (DSP-42)

K - 38


M(DSP) - 3-5

For a probability event in which the sample space may or may not contain equally likely outcomes, determines the likelihood of the occurrence of an event (using “more likely”, “less likely”, or “equally likely”).

For a probability event in which the sample space may or may not contain equally likely outcomes, 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.

o Sample space (DSP-30)o Event (DSP-31)

M(DSP) - 3-6

In response to a teacher or student generated question or hypothesis, groups decide the most effective method (e.g., survey, observation, experimentation) to collect the data (numerical or categorical) necessary to answer the question;


In response to a teacher or student generated question or hypothesis, groups analyze the data to draw conclusions about the question or hypothesis being tested, and when appropriate make predictions. (IMPORTANT: Analyzes data consistent with concepts and skills in M(DSP)–3–2.)

K - 39



M(N&O)-4-1

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 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 using models, explanations, or other representations.

Demonstrates conceptual understanding of rational numbers with respect to: decimals as hundredths within the context of money, or tenths within the context of metric measurements (e.g., 2.3 cm) using models, explanations, or other representations.

o Rational number (N&O-1)o Whole number (N&O-2)o Fraction (N&O-3)o Proper fraction (N&O-4)o Improper fraction (N&O-5)o Decimal (N&O-7)o Ratio (N&O-12) o Equivalent numbers (N&O-14)o Composition of numbers



whole relationship (N&O-17)o Set model (N&O-18)o Linear model (N&O-19)o In and out of context (N&O-36)o Factor (N&O-38)o Multiples (N&O-39)

Fraction Hamburger: puzzleInvestigations Fraction BarsGraph PaperGeoboards

M(N&O)-4-2

Demonstrates understanding of the relative magnitude of numbers from 0 to 999,999 by ordering or comparing whole numbers using models, number lines, or explanations.

Demonstrates understanding of the relative magnitude of numbers from 0 to 999,999 by ordering, comparing, or identifying equivalent proper positive fractional numbers; or decimals using models, number lines, or explanations.

o Fraction (N&O-3)o Proper fraction (N&O-4)o Decimal (N&O-7)o Equivalent numbers (N&O-14)o Area model to represent part to

whole relationship (N&O-17)o Set model (N&O-18)o Linear model (N&O-19)o Relative magnitude (N&O-20)o Within number formats

(N&O-21)o Ordering (N&O-26)o Comparing (N&O-27)o Number line (N&O-28)

K - 40


M(N&O)-4-3

Demonstrates conceptual understanding of mathematical operations by describing or illustrating the relationship between repeated subtraction and division (no remainders).

Demonstrates conceptual understanding of mathematical operations 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.

o Fraction (N&O-3)o Proper fraction (N&O-4)o Improper fraction (N&O-5)o Area model to represent part to

whole relationship (N&O-17)o Set model (N&O-18)o Linear model (N&O-19)o Number line (N&O-28)o Inverse relationships in

operations (N&O-29)o Relationship between repeated


o Concept of multiplication(N&O-37)

Math Their Way: Beans and CupsFamily MathUnifix CubesMath Bingo (all concepts)Power Blocks

M(N&O)-4-4

Accurately solves problems involving multiple operations on whole numbers or the use of the properties of factors and multiples.

Accurately solves problems involving 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.)

(IMPORTANT: Applies the conventions of order of operations where the left to right computations are modified only by the use of parentheses.)

o Whole number (N&O-2)o Fraction (N&O-3)o Proper fraction (N&O-4)o Decimal (N&O-7)o Accurately solves problems

(N&O-35)o Concept of multiplication

(N&O-37)o Factor (N&O-38)o Multiples (N&O-39)

M(N&O)-4-5 None

K - 41


M(N&O)-4-6

Mentally adds whole number facts through 20; adds two-digit whole numbers, combinations of two-digit and 3-digit whole numbers that are multiples of ten, and 4-digit whole numbers that are multiples of 100 (limited to two addends) (e.g., 67 + 24; 320 + 430; 320 + 90; 1,300 + 1,400)

Mentally subtracts a one-digit whole number from a two-digit whole number (e.g., 67 – 9); and subtracts combinations of two digit and three-digit whole numbers that are multiples of ten (e.g., 50 – 20, 230 – 80, 520 – 200).

Mentally multiplies whole number facts to a product of 100 and calculates related division facts.



M(N&O)-4-7

Makes estimates in a given situation by identifying when estimation is appropriate, selecting the appropriate method of estimation, and evaluating the reasonableness of solutions appropriate to grade level GLEs across content strands.


Unifix Cubes

M(N&O)-4-8

Applies properties of numbers (odd, even, multiplicative property of zero, and remainders) to solve problems and to simplify computations.

Applies field properties (commutative, associative, and identity) to solve problems and to simplify computations.

K - 42


M(G&M)-4-1

Uses properties or attributes of angles (number of angles) or sides (number of sides, 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 90o as more than, less than, or equal to.


o Angles (G&M-10)o Parallel lines (G&M-7)o Perpendicular (G&M-6)o Triangle (G&M-3)o Quadrilaterals (G&M-4)o Polygons (G&M-2)

GeoboardsTangramsHuman Bodies3-D space figuresPower Blocks

M(G&M)-4-2

None

M(G&M)-4-3

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).

o Attributes and properties(G&M-3) o Three-dimensional shapes

(G&M-15)

M(G&M)-4-4

Demonstrates conceptual understanding of congruency by matching congruent figures using reflections, translations, or rotations (flips, slides, or turns), or as the result of composing or decomposing shapes using models or explanations.

o Congruent (G&M-16)o Reflection (G&M-17)o Translation (G&M-18)o Rotation (G&M-19)o Composing and decomposing

shapes (G&M-21)o Matching congruent figures using

reflections, translations, and rotations (G&M-20)

o Similar figures (G&M-23)o Describes the proportional effect

on linear dimensions of polygons or circles when scaling up or down while preserving the angles of polygons (G&M-24)

Mirrors

K - 43


M(G&M)-4-5

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.

o Similar figures (G&M-23)o Describes the proportional effect


o Proportional reasoning (N&O-44)

Maps

M(G&M)-4-6

Demonstrates conceptual understanding of perimeter of polygons, on grids using a variety of models, manipulatives, or formulas.

Demonstrates conceptual understanding of area of rectangles, polygons or irregular shapes on grids using a variety of models, manipulatives, or formulas.

Expresses all measures using appropriate units.


understanding of perimeter and area of polygons and irregular figures on grids (G&M-26)


o Whole number bases and whole number exponents, and fractional bases with whole number exponents (N&O-24)

o Concept of multiplication (N&O-7)

Tiles

M(G&M)-4-7





Tpe measureScale rulers yard sticksmeter sticks

M(G&M)-4-8 None

K - 44


M(G&M)-4-9

Demonstrates understanding of spatial relationships using location and position by interpreting and giving directions between locations on a map or coordinate grid (first quadrant).

Demonstrates understanding of spatial relationships using location and position by plotting points in the first quadrant in context (e.g., games, mapping);

Demonstrates understanding of spatial relationships using location and position by finding the horizontal and vertical distances between points on a coordinate grid in the first quadrant.

Computer CDMaps

M(G&M)-4-10

Demonstrates conceptual understanding of spatial reasoning and visualization by copying, comparing, and drawing models of triangles, squares, rectangles, rhombi, trapezoids, hexagons, octagons, and circles.

Demonstrates conceptual understanding of spatial reasoning and visualization by building models of rectangular prisms from two- or three dimensional representations. Geoboards

Blocks

K - 45


M(F&A)–4–1

Identifies and extends to specific cases a variety of patterns (linear and nonlinear) represented in models, tables or sequences; and writes a rule in words or symbols to find the next case.

o Patterns (F&A-1)o Extend a pattern (F&A-5)o Numeric patterns (F&A-3) o Sequence (F&A-6)o Expresses generalization or rule

using words or symbols (F&A-7)o Pattern Summary Table by grade

level (F&A-9)

Unifix Cubes

M(F&A)–4–2

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.

M(F&A)–4–3

Demonstrates conceptual understanding of algebraic expressions by using letters or 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.

o Whole number (N&O-2)o Algebraic expressions (F&A-25)o Evaluating algebraic expressions

(F&A-26)o Linear relationships (F&A-10)o Proportional linear relationships

(y = kx) (F&A-11)o Non-proportional linear

relationships (y = mx + b) (F&A-12)



(F&A-35)

K - 46


M(F&A)–4–4

Demonstrates conceptual understanding of equality by showing equivalence between two expressions using models or different representations of the expressions, by simplifying numerical expressions where left to right computations may be modified only by the use of parentheses [e.g., 14 - (2 × 5)]

and by solving one-step linear equations of the form ax = c and/orx ± b = cwhere a, b, and c are whole numbers with a≠ 0.

o Equality (F&A-30)o Demonstrates equality (F&A-31)o Number sentences (F&A-33)o Equation (F&A-32)o Algebraic equation notation

(F&A-34) o Examples of forms of equations

(F&A-35)

K - 47


M(DSP) - 4-1

Interprets a given representation (line plots, 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.



o Representation (DSP-1)o Line plot (DSP-5)o Frequency table (DSP-4)o Bar graph (DSP-6)o Pictograph (DSP-2)o Circle graph (DSP-8)

GraphingTime Liner

M(DSP) - 4-2

Analyzes patterns, trends, or distributions in data in a variety of contexts by determining or using measures of central tendency (median or mode), or range.

o Pattern (F&A-1)o Median (DSP-16)o Mode (DSP-17)o Range (DSP-18)

M(DSP) - 4-3

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.


M(DSP) - 4-4

Uses counting techniques to solve problems in context involving combinations or simple permutations (e.g., Given a map - Determine the number of paths from point A to point B.) using a variety of strategies (e.g., organized lists, tables, tree diagrams, or others).

o Solves problems using a variety of counting strategies (DSP-25)

o Combination (DSP-41)o Permutation (DSP-42)o Frequency table (DSP-4)o Tree diagram (DSP-28)

K - 48


M(DSP) - 4-5

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 (e.g., two out of five).

For a probability event in which the sample space may or may not contain equally likely outcomes predicts the likelihood of an event as a part to whole relationship (e.g., two out of five, zero out of five, five out of five) and tests the prediction through experiments; and determines if a game is fair.

o Sample space (DSP-30)o Event (DSP-31)o Theoretical probability (DSP-32)o Ratio (N&O-12)

Tiles – different coloredDiceRed beansPlaying cards

M(DSP) - 4-6

In response to a teacher or student generated question or hypothesis, groups decide the most effective method (e.g., survey, observation, experimentation) to collect the data (numerical or categorical) necessary to answer the question.


In response to a teacher or student generated question or hypothesis, groups analyze the data to draw conclusions about the question or hypothesis being tested, and when appropriate make predictions.

In response to a teacher or student generated question or hypothesis, groups ask new questions and make connections to real world situations.


K - 49



M(N&O)-5-1

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.

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 (10, 100, 1000)), 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*.

o Rational number (N&O-1)o Whole number (N&O-2)o Fraction (N&O-3)o Proper fraction (N&O-4)o Improper fraction (N&O-5)o Mixed number (N&O-6)o Decimal (N&O-7)o Percent (N&O-8)o Ratio (N&O-12)o Equivalent numbers (N&O-14)o Composition of numbers



whole relationship (N&O-17)o Set model (N&O-18)o Linear model (N&O-19)o * Specifications for Area, Set, and

Linear Models (Pg. NO-12)

From the series Investigations of Number, Data and Space

Mathematical Thinking at Grade 5

Factors, multiples, square number, prime number, addition and subtraction games, landmark numbers –100, 1000, 10,000, solving larger number operations using cluster patterns includes games, mental math, worksheets (tiles and graph paper)

Building on Numbers You Know

Strategies for estimation and computation, algebraic patterns includes games and worksheets

Name That Portion

Fractions, decimals and percents includes several games and worksheets

K - 50


M(N&O)-5-2

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.

o Rational number (N&O-1)o Whole number (N&O-2)o Fraction (N&O-3)o Proper fraction (N&O-4)o Improper fraction (N&O-5)o Mixed number (N&O-6)o Decimal (N&O-7)o Percent (N&O-8)o Ratio (N&O-12)o Equivalent numbers (N&O-14)o Composition of numbers



whole relationship (N&O-17)o Set model (N&O-18)o Linear model (N&O-19)


Name That Portion


Patterns of Change

Tables and graphs includes software

M(N&O)-5-3

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.

Demonstrates conceptual understanding of mathematical operations with respect to addition and subtraction of decimals and positive proper fractions with unlike denominators.

o Whole number (N&O-2)o Meaning of remainders with

respect to division of whole numbers (N&O-32)




K - 51


M(N&O)-5-4

Accurately solves problems involving multiple operations on whole numbers or the use of the properties of factors, multiples, prime, or composite numbers.

Accurately solves problems involving addition or subtraction of fractions (proper) and decimals to the hundredths place. (Division of whole numbers by a one- or two-digit divisor.)

(IMPORTANT: Applies the conventions of order of operations with and without parentheses.)

o Whole number (N&O-2)o Fraction (N&O-3)o Proper fraction (N&O-4)o Decimal (N&O-7)o Accurately solves problems

(N&O-35)o Factor (N&O-38)o Multiples (N&O-39)o Prime numbers (N&O-40)o Composite numbers (N&O-41)




Name That Portion


M(N&O)-5-5 None

M(N&O)-5-6

Mentally calculates change back from $1.00, $5.00, and $10.00;

Mentally calculates multiplication and related division facts to a product of 144; multiplies a two-digit whole number by a one-digit whole number (e.g., 45 x 5), two-digit whole numbers that are multiples of ten (e.g., 50 x 60), 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 (e.g., 400 x 50, 400 x 600).

Mentally calculates division of 3- and 4-digit multiples of powers of ten by their compatible factors (e.g., 360 ÷ 6; 360 ÷ 60; 3600 ÷ 6; 3600 ÷ 60; 3600 ÷ 600; 360 ÷ 12; 360 ÷ 120; 3600 ÷ 12; 3600 ÷ 120; 3600 ÷ 1200).





K - 52


M(N&O)-5-7

Makes estimates in a given situation by identifying when estimation is appropriate, selecting the appropriate method 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.





M(N&O)-5-8

Applies properties of numbers (odd, even, and divisibility) to solve problems and to simplify computations.

Applies field properties (commutative, associative, identity, and distributive) to solve problems and to simplify computations.





K - 53


M(G&M)-5-1

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).


o Angles (G&M-10)o Parallel lines (G&M-7)o Perpendicular (G&M-6)o Triangle (G&M-3)o Quadrilaterals (G&M-4)


Picturing Polygons

2-D geometry includes software and manipulatives (Power Polygons)

M(G&M)-5-2

None

M(G&M)-5-3

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).


o Three-dimensional shapes (G&M-15)

Containers and Cubes

3-D geometry and volume includes manipulatives for volume

M(G&M)-5-4

None

M(G&M)-5-5

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 effects using models or explanations.

Picturing Polygons


K - 54


M(G&M)-5-6

Demonstrates conceptual understanding of perimeter of polygons, and the area of rectangles or right triangles through models, manipulatives, or formulas.

Demonstrates conceptual understanding of area of polygons or irregular figures on grids, and volume of rectangular prisms (cubes) using a variety of models, manipulatives, or formulas.

Expresses all measures using appropriate units.

o Triangle (G&M-3)o Polygons (G&M-2)o Three-dimensional shapes

(G&M-15)o Demonstrates conceptual

understanding of perimeter and area of polygons and irregular figures on grids (G&M-26)

o Demonstrates conceptual understanding of perimeter, area, volume, or surface area using models and manipulatives (G&M-28)




Picturing Polygons


Containers and Cubes3-D geometry and volume includes manipulatives for volume

M(G&M)-5-7





Measurement Benchmarks

Estimating and measuring standard and metric linear, weight, capacity and time (scales, metric and standard weights, metric and standard measuring tapes, metric and standard rulers, stop watches)

K - 55

Geometry and MeasurementGLE Number Grade Level Expectations Terms Defined (Definition Number) Notes and ResourcesM(G&M)-5-8 None

M(G&M)-5-9

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).

Demonstrates understanding of spatial relationships using location and position by plotting points in four quadrants in context (e.g., games, mapping, identifying the vertices of polygons as they are reflected, rotated, and translated).

Demonstrates understanding of spatial relationships using location and position by determining horizontal and vertical distances between points on a coordinate grid in the first quadrant.


Picturing Polygons


Patterns of Change


Math A Way of Thinking

Coordinate graphing section

M(G&M)-5-10

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.

Containers and Cubes

3-D geometry and volume includes manipulatives for volume

K - 56


M(F&A)–5–1

Identifies and extends to specific cases a variety of patterns (linear and nonlinear) represented in models, tables, sequences, or in problem situations; and writes a rule in words or symbols for finding specific cases of a linear relationship.

o Numeric patterns (F&A-3)o Extend a pattern (F&A-5)o Sequence (F&A-6)o Linear relationships (F&A-10)o Proportional linear relationships



o Expresses generalization or rule using words or symbols (F&A-7)

o Concrete situations (F&A-20)o Pattern Summary Table by grade

level (F&A-9)


Patterns of Change


Building Blocks for Algebra K-8

patterns and functions includes games

M(F&A)–5–2

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 (e.g., tell a story given a line graph about a trip).

Patterns of Change


K - 57


M(F&A)–5–3

Demonstrates conceptual understanding of algebraic expressions by using letters 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.

o Whole number (N&O-2)o Algebraic expression (F&A-25)o Evaluating algebraic expressions



relationships (y = mx =b) (F&A-12)

o Number sentences (F&A-33)o Equation (F&A-32)o Examples of forms of equations

(F&A-35)o Algebraic equation notation

(F&A-34)

Patterns of Change




M(F&A)–5–4

Demonstrates conceptual understanding of equality by showing equivalence between two expressions using models or different representations of the expressions, by solving one-step linear equations of the formax = c and/orx ± b = c and/orx/a = c, where a, b, and c are whole numbers with a≠ 0; or by determining which values of a replacement set make the equation (multi-step of the form ax ± b = c where a, b, and c are whole numbers with a≠ 0) a true statement (e.g., 2x + 3 = 11, {x: x = 2, 3, 4, 5}).

o Whole number (N&O-2)o Equality (F&A-30)o Demonstrates equality (F&A-31)o Number sentences (F&A-33)o Equation (F&A-32)o Algebraic equation notation




relationships (y = mx+b) (F&A-12)

Patterns of Change




K - 58


M(DSP) – 5-1

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.



o Representation (DSP-1)o Frequency table (DSP-4)o Bar graph (DSP-6)o Circle graph (DSP-8)o Line graph (DSP-12)


Data: Kids, Cats and Ads

Statistics (stop watches)

M(DSP) – 5-2

Analyzes patterns, trends, or distributions 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.

o Pattern (F&A-1)o Mean (DSP-15)o Median (DSP-16)o Mode (DSP-17)o Range (DSP-18)



M(DSP) – 5-3

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.

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.






M(DSP) – 5-4

None

K - 59


M(DSP) – 5-5

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 a fraction.

For a probability event in which the sample space may or may not contain equally likely outcomes, predicts the likelihood of an event as a fraction and tests the prediction through experiments; and determines if a game is fair.

o Sample space (DSP-30)o Experimental probability (DSP-

33)o Theoretical probability (DSP-32)o Event (DSP-31)o Fraction (N&O-3)o Ratio (N&O-12)

Between Never and Always

Probability includes games (spinners, colored cubes)

M(DSP) – 5-6

In response to a teacher or student generated question or hypothesis decides the most effective method (e.g., survey, observation, experimentation) to collect the data (numerical or categorical) necessary to answer the question. In response to a teacher or student generated question or hypothesis collects, organizes, and appropriately displays the data.

In response to a teacher or student generated question or hypothesis analyzes the data to draw conclusions about the question or hypothesis being tested, and when appropriate makes predictions.

In response to a teacher or student generated question or hypothesis asks new questions and makes connections to real world situations.


Between Never and Always

Probability includes games (spinners, colored cubes)

K - 60



M(N&O)-6-1

Demonstrates conceptual understanding of rational numbers with respect to ratios (comparison of two whole numbers by division a/b, a : b, and a ÷ b , where b ≠ 0); and rates (e.g., a out of b, 25%) using models, explanations, or other representations*.

o Rational number (N&O-1)o Whole number (N&O-2)o Proper fraction (N&O-4)o Improper fraction (N&O-5)o Mixed number (N&O-6)o Percent (N&O-8)o Ratio (N&O-12)o Equivalent numbers (N&O-14)o Area model to represent part to

whole relationship (N&O-17)o Set model (N&O-18)o Linear model (N&O-19)o Proportional reasoning (N&O-

44)o *Specifications for Area, Set, and


Glencoe Mathematics Application and Concepts course 1 2004 edition plus corresponding Resource Masters

Calculators (classroom set)

K - 61


Demonstrates understanding of the relative magnitude of numbers by ordering or comparing numbers with whole number bases and whole number exponents (e.g.,33, 43), 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.

o Rational number (N&O-1)o Fraction (N&O-3)o Proper fraction (N&O-4)o Improper fraction (N&O-5)o Mixed number (N&O-6)o Decimal (N&O-7)o Percent (N&O-8)o Integer (N&O-9)o Equivalent numbers (N&O-14)o Relative magnitude (N&O-20)o Within number formats

(N&O-21)o Across number formats (N&O-

22)o Whole number bases and whole

number exponents, and fractional bases with whole number exponents (N&O-24)

o Ordering (N&O-26)o Comparing (N&O-27)o Number line (N&O-28)o Describing or illustrating the

meaning of a power (N&O-33)



M(N&O)- Demonstrates conceptual understanding of o Proper fraction (N&O-4) Glencoe Mathematics Application K - 62

Number and OperationsGLE Number Grade Level Expectations Terms Defined (Definition Number) Notes and Resources6-3 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) (e.g.,33, 43).

Demonstrates conceptual understanding of mathematical operations and the effect on the magnitude of a whole number when multiplying or dividing it by a whole number, decimal, or fraction.

Demonstrates conceptual understanding of mathematical operations of addition and subtraction of positive fractions and integers; and multiplication and division of fractions and decimals.

o Improper fraction (N&O-5)o Decimal (N&O-7)o Whole number bases and whole


o Relationship between repeated addition and multiplication of whole numbers (N&O-30)

o Relationship between repeated subtraction and division of whole numbers (N&O-31)

o Effect on magnitude of a whole number when multiplying or dividing by a whole number, fraction, or decimal (N&O-34)

o Describing or illustrating the meaning of a power (N&O-33)

and Concepts course 1 2004 edition plus corresponding Resource Masters


Fraction Bars

M(N&O)-6-4

Accurately solves problems involving single or multiple operations on fractions (proper, improper, and mixed), or decimals;

Accurately solves problems involving addition or subtraction of integers; percent of a whole; or problems involving greatest common factor or least common multiple.

(IMPORTANT: Applies the conventions of order of operations with and without parentheses.)

o Fraction (N&O-3)o Proper fraction (N&O-4)o Improper fraction (N&O-5)o Mixed number (N&O-6)o Decimal (N&O-7)o Percent (N&O-8)o Accurately solves problems

(N&O-35)o GCF (N&O-42)o LCM (N&O-43)



Fraction Bars

Inter Chips

Connected Mathematics Series

M(N&O)-6-5

None

M(N&O)-6-6

Mentally calculates change back from $5.00, $10.00, $20.00, $50.00, and $100.00. All of this is referred to mentally!

K - 63


Mentally multiplies a two-digit whole number by a one-digit whole number (e.g., 45 x 5), two-digit whole numbers that are multiples of ten (e.g., 50 x 60), 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 (e.g., 400 x 50, 400 x 600).

Mentally divides 3- and 4-digit multiples of powers of ten by their compatible factors (e.g., 360 ÷ 6; 360 ÷ 60; 3600 ÷ 6; 3600 ÷ 60; 3600 ÷ 600; 360 ÷ 12; 360 ÷ 120; 3600 ÷ 12; 3600 ÷ 120; 3600 ÷ 1200).

Mentally determines the part of a whole number using benchmark percents (1%, 10%, 25%, 50%, and 75%).




M(N&O)-6-7

Makes estimates in a given situation by identifying when estimation is appropriate, selecting the appropriate method of estimation, determining the level

Glencoe Mathematics Application and Concepts course 1 2004 edition plus corresponding

K - 64


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.


Resource Masters


M(N&O)-6-8

Applies properties of numbers (odd, even, remainders, divisibility, and prime factorization) tosolve problems and to simplify computations.

Applies field properties (commutative,associative, identity [including the multiplicative property of one, e.g., 1 = 2/2 and 2/2 x ¾ = 6/8, so ¾ = 6/8], distributive, and additive inverses) to solve problems and to simplify computations.



K - 65


M(G&M)-6-1

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).


o Angles (G&M-10)o Parallel lines (G&M-7)o Perpendicular (G&M-6)o Triangle (G&M-3)o Quadrilaterals (G&M-4)




Geo Boards

M(G&M)-6-2

None

M(G&M)-6-3

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).






Geo Boards

M(G&M)-6-4

Demonstrates conceptual understanding of congruency by predicting and describing the transformational steps (reflections, translations, and rotations) needed to show congruence (including the degree of rotation) and as the result of composing and decomposing two- and three-dimensional objects using models or explanations.

Demonstrates conceptual understanding of congruency by using line and rotational symmetry to demonstrate congruent parts within a shape.




Geo Boards

M(G&M)- Demonstrates conceptual understanding of o Similar figures (G&M-23) Glencoe Mathematics Application

K - 66

Geometry and MeasurementGLE Number Grade Level Expectations Terms Defined (Definition Number) Notes and Resources6-5 similarity by describing the proportional effect on the

linear dimensions of polygons or circles when scaling up or down while preserving the angles of polygons, or by solving related problems (including applying scales on maps). Describes effects using models or explanations.

o Polygons (G&M-2)o Sum of the measures of interior

angles of polygons (G&M-13) o Describes the proportional effect


o Solves problems involving scaling up or down and their impact on angle measure, linear dimensions, and areas of polygons and circles when the linear dimensions are multiplied by a constant factor (G&M-25)


and Concepts course 1 2004 edition plus corresponding Resource Masters



Geo Boards

M(G&M)-6-6

Demonstrates conceptual understanding of perimeter of polygons, the area of quadrilaterals or triangles, and the volume of rectangular prisms by

o Polygons (G&M-2)o Triangle (G&M-3)o Three-dimensional shapes

(G&M-15)

Glencoe Mathematics Application and Concepts course 1 2004 edition plus corresponding

K - 67


using models, formulas, or by solving problems.

Demonstrates conceptual understanding of the relationships of circle measures (radius to diameter and diameter to circumference) by solving related problems. Expresses all measures using appropriate units.


o Demonstrates conceptual understanding of perimeter, area, volume, or surface area by solving problems (G&M-29)

o Demonstrates conceptual understanding of the relationships of circle measures by solving related problems (G&M-30)


o Irrational number (N&O-10) o Whole number bases and whole


Resource Masters



Geo Boards

M(G&M)-6-7



o Measures and uses units of measure appropriately and consistently

o Makes conversions within and across systems

o Irrational number

Glencoe Mathematics Application and Concepts course 1 2004 edition plus corresponding Resource MastersCalculators (classroom set)Connected Mathematics SeriesGeo Boards

M(G&M)-6-8 None

M(G&M)-6-9

None

M(G&M)-6-10

None

K - 68


K - 69


M(F&A)–6–1

Identifies and extends to specific cases a variety of patterns (linear and nonlinear) 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.

Identifies and extends to specific cases a variety of patterns and writes an expression or equation using words or symbols to express the generalization of a linear relationship (e.g., twice the term number plus 1 or 2n + 1).

o Patterns (F&A-1)o Numeric patterns (F&A-3)o Non-numeric patterns (F&A-4)o Extend a pattern (F&A-5)o Sequence (F&A-6)o Linear relationships (F&A-10)o Proportional linear relationships



o Non-linear relationships (F&A-18)

o Expresses generalization or rule using words or symbols (F&A-7)


level (F&A-9)



M(F&A)–6–2

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.

Demonstrates conceptual understanding of linear relationships (y = kx; y = mx + b) as a constant rate of change 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.

o Ratio (N&O-12)o Proportional reasoning (N&O-44)o Linear relationships (F&A-10) o Proportional linear relationships



o Distinguishes between constant and varying rates (F&A-19)

o Slope (F&A-14)o Describes the meaning of slope

and intercept in concrete situations (F&A-15)

o Concrete situations (F&A-20)



K - 70


M(F&A)–6–3

Demonstrates conceptual understanding of algebraic expressions by using letters to represent unknown quantities to write linear algebraic expressions involving two or more of the four operations; or by evaluating linear algebraic expressions (including those with more than one variable); or by evaluating an expression within an equation (e.g., determine the value of y when x = 4 given y = 3x - 2).

o Algebraic expression (F&A-25)o Evaluating algebraic expressions

(F&A-26)o Number sentences (F&A-33)o Equation (F&A-32)o Algebraic equation notation


(F&A-35) o Linear relationships (F&A-10)o Proportional linear relationships





M(F&A)–6–4

Demonstrates conceptual understanding of equality by showing equivalence between two expressions using models or different representations of the expressions, solving multi-step linear equations of the form ax ± b = c, where a, b, and c are whole numbers with a≠ 0.

o Whole number (N&O-2)o Equality (F&A-30)o Demonstrates equality (F&A-

31)o Linear relationships (F&A-10) o Proportional linear relationships





(F&A-35)



K - 71


M(DSP) – 6-1

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.

(IMPORTANT: Analyzes data consistent with concepts and skills in M(DSP)-6-2).


o Representation (DSP-1)o Circle graph (DSP-8)o Line graph (DSP-12)o Stem-and-leaf plot (DSP-9)




M(DSP) – 6-2

Analyzes patterns, trends or distributions 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.

o Pattern (F&A-1)o Mean (DSP-15)o Median (DSP-16)o Mode (DSP-17)o Dispersion (DSP-19)o Range (DSP-18)




M(DSP) – 6-3

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.

(IMPORTANT: Analyzes data consistent with concepts and skills in M(DSP)–6–2).




M(DSP) – 6-4

Uses counting techniques to solve problems in context involving combinations or simple permutations using a variety of strategies (e.g., organized lists, tables, tree diagrams, models, Fundamental Counting Principle, or others).


o Combination (DSP-26)o Permutation (DSP-27)o Frequency table (DSP-4)o Tree diagram (DSP-28)o Fundamental Counting Principle

(DSP-29)




K - 72


M(DSP) – 6-5

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.

For a probability event in which the sample space may or may not contain equally likely outcomes, predicts the theoretical probability of an event and tests the prediction through experiments and simulations; and designs fair games.


33)o Theoretical probability (DSP-32)o Event (DSP-31)




M(DSP) – 6-6

In response to a teacher or student generated question or hypothesis decides the most effective method (e.g., survey, observation, experimentation) to collect the data (numerical or categorical) necessary to answer the question.

In response to a teacher or student generated question or hypothesis collects, organizes, and appropriately displays the data.

In response to a teacher or student generated question or hypothesis analyzes the data to draw conclusions about the question or hypothesis being tested, and when appropriate makes predictions.






K - 73



M(N&O)-7-1

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 wholes vary in magnitude (e. g., 8 girls in a classroom of 16 students compared to 8 girls in a classroom of 20 students, or 20% of 400 compared to 50% of 100) using models, explanations, or other representations*.

Demonstrates conceptual understanding of rational numbers with respect to percents as a way of expressing multiples of a number (e. g., 200% of 50) using models, explanations, or other representations*.

Demonstrates conceptual understanding of square roots of perfect squares, rates, and proportional reasoning.

o Rational number (N&O-1)o Percent (N&O-8)o Ratio (N&O-12)o Equivalent numbers (N&O-14)o Area model to represent part to

whole relationship (N&O-17)o Set model (N&O-18)o Linear model (N&O-19)o Whole number bases and whole


o Comparing (N&O-27)o Number line (N&O-28)o Multiples (N&O-39)o Proportional reasoning (N&O-

44)o * Specifications for Area, Set, and


Glencoe Applications and Concepts Course 2 (2004)

Glencoe Applications and Concepts Course 2 (2004)Resource Kit

Mathimagination Series(Steve and Janis Marcy)

Mathematics – A Way of Thinking(Robert Baratta-Lorton)

Guiding Children’s Learning of Mathematics (Kennedy and Tipps)

M(N&O)-7-2

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 (e.g., 33, 43), integers, absolute values, or numbers represented in scientific notation using number lines or equality and inequality symbols.

o Rational number (N&O-1)o Integer (N&O-9)o Equivalent numbers (N&O-14)o Relative magnitude (N&O-20)o Across number formats (N&O-

21)o Absolute value (N&O-23)o Scientific notation (N&O-25)o Ordering (N&O-26)o Comparing (N&O-27)o Describing or illustrating the


Pre-Algebra with Pizzazz Series(Steve and Janis Marcy)



K - 74


M(N&O)-7-3

Demonstrates conceptual understanding of operations with integers and whole number exponents (where the base is a whole number) using models, diagrams, or explanations.

M(N&O)-7-4

Accurately solves problems involving proportional reasoning, percents involving discounts, tax or tips, andrates.

Accurately solves problems involving addition or subtraction of integers, raising numbers to whole number powers, and determining square roots of perfect square numbers and non- perfect square numbers.

(IMPORTANT: Applies the conventions of order of operations, including parentheses, brackets, or exponents).

o Percent (N&O-8)o Ratio (N&O-12)o Accurately solves problems

(N&O-35)o Proportional reasoning (N&O-

44)

Glencoe Applications and Concepts Course 2


Pre-Algebra with Pizzazz Series(Steve and Janis Marcy)

M(N&O)-7-5 None

M(N&O)-7-6

Mentally calculates benchmark perfect squares and related square roots (e. g., 12, 22 … 122, 152, 202, 252, 1002, 10002).

Mentally determines the part of a number using benchmark percents and related fractions (1%, 10%, 25%, 33⅓ %, 50%, 66⅔%, 75%, and 100%) (e. g., 25% of 16; 33⅓% of 330).

(IMPORTANT: Mental arithmetic should be imbedded instructionally throughout all strands .)


K - 75


M(N&O)-7-7

Makes estimates in a given situation (including tips, discounts, and tax) by identifying when estimation is appropriate, selecting the appropriate method 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.


M(N&O)-7-8

Applies properties of numbers (odd, even, remainders, divisibility, and prime factorization) to solve problems and to simplify computations.

Applies field properties (commutative, associative, identity, distributive, inverses) to solve problems and to simplify computations.

Demonstrates conceptual understanding of field properties as they apply to subsets of the real numbers (e. g., the set of whole numbers does not have additive inverses, the set of integers does not have multiplicative inverses).



Pre-Algebra with Pizzazz

K - 76


M(G&M)-7-1

Uses properties 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.


o Angle (G&M-10)o Angle relationships formed by

two or more lines cut by a transversal (G&M-11)

o Parallel lines (G&M-7)



Mathimagination Series

Math – A Way of Thinking

Moving On with Geoboards(Creative Publications)

M(G&M)-7-2 Applies theorems or relationships (triangle inequality

or sum of the measures of interior angles of polygons) to solve problems.

o Triangle (G&M-3)o Triangle inequality (G&M-12)o Sum of the measures of interior

angles of polygons (G&M-13)o Angles (G&M-10)o Polygons (G&M-2)



M(G&M)-7-3

None

M(G&M)-7-4

Applies the concepts of congruency by solving problems on a coordinate plane involving reflections, translations, or rotations.

o Congruent (G&M-16)o Reflection (G&M-17)o Translation (G&M-18)o Rotation (G&M-19)o Solving problems on a coordinate

plane using reflections, translations, or rotations (G&M-22)



K - 77


M(G&M)-7-5

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 circles when the linear dimensions are multiplied by a constant factor. Describes effects using models or explanations.

o Similar figures (G&M-23)o Angles (G&M-10)o Polygons (G&M-2)o Sum of the measures of interior

angles of polygons (G&M-13)o Describes the proportional effect



o Factor (N&O-38)o Proportional reasoning (G&M-

44)





Geoboards

K - 78


M(G&M)-7-6

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 related problems. Expresses all measures using appropriate units.

o Quadrilaterals (G&M-4)o Triangle (G&M-3)o Three-dimensional shapes

(G&M-15)o Demonstrates conceptual

understanding of perimeter, area, volume, or surface area using models and manipulatives (G&M-28)


o Demonstrates conceptual understanding of the relationships of circle measures by solving related problems (G&M-30)


o Irrational number (N&O-10)o Whole number bases and whole






Geoboards

Tessellations – The Geometry of Patterns (Creative Publications)

M(G&M)-7-7 NoneM(G&M)-7-8 NoneM(G&M)-7-9 None

M(G&M)-7-10

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.

K - 79


M(F&A)–7–1

Identifies and extends to specific cases a variety of patterns (linear and nonlinear) represented in models, tables, sequences, graphs, or in problem situations.

Identifies and extends to specific cases a variety of patterns and generalizes a linear relationship using words and symbols.

Identifies and extends to specific cases a variety of patterns and 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 nonlinear relationship.

o Patterns (F&A-1)o Numeric patterns (F&A-3)o Non-numeric patterns (F&A-4)o Extend a pattern (F&A-5)o Sequence (F&A-6)o Linear relationships (F&A-10)o Proportional linear relationships



o Non-linear relationships (F&A-18)

o Equation (F&A-32)o Expresses generalization or rule

using words or symbols (F&A-7)

o Generalizes a pattern to find a specific case (F&A-8)


level (F&A-9)




K - 80


M(F&A)–7–2

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.

Demonstrates conceptual understanding of linear relationships 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.

o Ratio (F&A-12)o Proportional reasoning (N&O-44)o Linear relationships (F&A-10)o Proportional linear relationships



o Slope (F&A-14)o Intercept (F&A-17)o Non-linear relationships (F&A-

18)o Informally determines slope

(F&A-16)o Distinguishes between constant

and varying rates (F&A-19) o Solves problems involving linear

relationships (F&A-13)o Concrete situations (F&A-20)




M(F&A)–7–3

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 (e.g., determine the value of y when x = 4 given y = 5x3 - 2).



(F&A-26)o Number sentences (F&A-33)o Equation (F&A-32)o Examples of forms of equations


(F&A-34)




K - 81


M(F&A)–7–4

Demonstrates conceptual understanding of equality by showing equivalence between two expressions (expressions consistent with the parameters of the left- and right-hand sides of the equations being solved at this grade level) using models or different representations of the expressions, solving multi-step linear equations of the form ax ± b = 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.

o Equality (F&A-30)o Demonstrates equality




o Number sentences (F&A-33)o Equation (F&A-32)o Examples of forms of equations


(F&A-34)




K - 82


M(DSP) – 7-1

Interprets a given representation (circle graphs, scatter plots that represent discrete linear relationships, or histograms) to analyze the data to formulate or justify conclusions, to make predictions, or to solve problems.



o Representation (DSP-1)o Circle graph (DSP-8)o Scatter plot (DSP-13)o Histogram (DSP-7)o Linear relationships (F&A-10)o Proportional linear relationship

(F&A-11) o Non-proportional linear

relationship (F&A-12)



M(DSP) – 7-2

Analyzes patterns, trends, or distributions in data in a variety of contexts by solving problems using measures of central tendency (mean, median, or mode), dispersion (range or variation), or outliers to analyze situations to determine their effect on mean, median, or mode;

Analyzes patterns, trends, or distributions in data in a variety of contexts by evaluating the sample from which the statistics were developed (bias).

o Pattern (DSP-1)o Mean (DSP-15)o Median (DSP-16)o Mode (DSP-17)o Dispersion (DSP-19)o Range (DSP-18)o Outlier (DSP-20)o Analyzes the impact of outliers

on the mean, median and mode (DSP-22)

o Evaluates samples from which the statistics were developed (bias) (DSP-24)



M(DSP) – 7-3

Identifies or describes representations or elements of representations that best display a given set of data or situation, 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.

(IMPORTANT: Analyzes data consistent with concepts and skills in M( DSP)– 7– 2 .)





K - 83


M(DSP) – 7-4

Uses counting techniques to solve problems in context involving combinations or permutations (e.g., How many different ways can eight students place first, second, and third in a race?) using a variety of strategies (e.g., organized lists, tables, tree diagrams, models, Fundamental Counting Principle, or others).



What Are My Chances Book A, Book B(Creative Publications)

M(DSP) – 7-5

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.

For a probability event in which the sample space may or may not contain equally likely outcomes, predicts the theoretical probability of an event and tests the prediction through experiments and simulations

For a probability event in which the sample space may or may not contain equally likely outcomes compares and contrasts theoretical and experimental probabilities.


33)o Theoretical probability (DSP-32)o Event (DSP-31)



K - 84


M(DSP) – 7-6

In response to a teacher or student generated question or hypothesis decides the most effective method (e. g., survey, observation, experimentation) to collect the data (numerical or categorical) necessary to answer the question.


In response to a teacher or student generated question or hypothesis analyzes the data to draw conclusions about the question or hypothesis being tested while considering the limitations that could affect interpretations.

In response to a teacher or student generated question or hypothesis, makes predictions when appropriate.


(IMPORTANT: Analyzes data consistent with concepts and skills in M( DSP)– 7– 2. )



Grade Level 8 Expectations by StrandK - 85


M(N&O)-8-1

Demonstrates conceptual understanding of rational numbers with respect to percents as a way of describing change (percent increase and decrease) using explanations, models, or other representations*.

o Rational number (N&O-1)o Percent (N&O-8)o Ratio (N&O-12)o Area model to represent part to

whole relationship (N&O-17)o Set model (N&O-18)o Linear model (N&O-19)o Proportional reasoning (N&O-

44)o * Specifications for Area, Set, and


Glencoe, Pre-Algebra (2003) Textbook

Glencoe, Pre-Algebra (2003) Resource Kit

Glencoe, Applications and Connections (1995) Resource Kit

Algebra In the Real World – Futures Channel

M(N&O)-8-2

Demonstrates understanding of the relative magnitude of numbers by ordering or comparing

rational numbers, common irrational numbers (e.g. , , ), 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.

o Rational number (N&O-1)o Integer (N&O-9)o Irrational number (N&O-10)o Real numbers (N&O-11)o Equivalent numbers (N&O-14)o Relative magnitude (N&O-20)o Across number formats (N&O-

21)o Absolute value (N&O-23) o Whole number bases and whole


o Scientific notation (N&O-25)o Ordering (N&O-26)o Comparing (N&O-27)o Number line (N&O-28)o Describing or illustrating the


Glencoe, Algebra I (2003)

Creative Publications, Middle School Math with Pizzazz (1989)

M(N&O)-8-3 None

K - 86


Accurately solves problems involving proportional reasoning (percent increase or decrease, interest rates, markups, or rates).

Accurately solves problems involvingmultiplication or division of integers.

Accurately solves problems involving squares, cubes, and taking square or cube roots.

(IMPORTANT: Applies the conventions of order of operations.)

o Percent (N&O-8)o Integer (N&O-9)o Ratio (N&O-12)o Whole number bases and whole









M(N&O)-8-5 None


M(N&O)-8-6

Mentally calculates benchmark perfect squares andsquare roots (e.g., 12, 22 … 122, 152, 202, 252, 1002, 10002).

Mentally determines the part of a number using benchmark percents and related fractions (1%, 10%,25%,33⅓ %, 50%,66 ⅔%, 75%, and 100%) (e.g., 25% of 16; 33⅓ % of 330).

(IMPORTANT: Mental arithmetic should be imbedded instructionally throughout all strands.)


M(N&O)- Makes estimates in a given situation (including tips, Glencoe, Pre-Algebra (2003)

K - 87

Number and OperationsGLE Number Grade Level Expectations Terms Defined (Definition Number) Notes and Resources8-7 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 method 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 solutionsappropriate to grade level GLEs across content strands.


Textbook






M(N&O)-8-8

Applies properties of numbers (odd, even, remainders, divisibility, and prime factorization) to solve problems and to simplify computations..Applies field properties (commutative, associative, identity [including the multiplicative property of one, e.g. 20 x 23 = 20+3 = 23, so 20 = 1], distributive, inverses) to solve problems and to simplify computations.

Demonstrates conceptual understanding offield properties as they apply to subsets of real numbers when addition and multiplication are not defined in the traditional ways (e.g., If a ∆ b = a + b - 1, is ∆ a commutative operation?).





K - 88


M(G&M)-8-1

None

M(G&M)-8-2

Applies the Pythagorean Theorem to find a missing side of a right triangle, or in problem solving situations.

o Pythagorean Theorem (G&M-14)

o Triangle (G&M-3)


M(G&M)-8-3

None

M(G&M)-8-4

None

M(G&M)-8-5

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;

Applies concepts of similarity to determine the length of sides of similar triangles

Applies concepts of similarity to solve problems involving growth and rate.


o Similar figures (G&M-23)o Triangle (G&M-3)o Describes the proportional effect



o Ratio (N&O-12)o Proportional reasoning (N&O-44)


Native American Geometry Replacement Unit - http://www.earthmeasure.com

Compasses, Protractors, Straightedges


K - 89

Geometry and MeasurementGLE Number Grade Level Expectations Terms Defined (Definition Number) Notes and ResourcesM(G&M)-8-6

Demonstrates conceptual understanding of surface area or volume by solving problems involving surface area and volume of rectangular prisms, triangular prisms, cylinders, or pyramids. Expresses all measures using appropriate units.





o Irrational number (N&O-10)o Whole number bases and whole



Centimeter Cubes


M(G&M)-8-7

None

M(G&M)-8-8 None

M(G&M)-8-9

None

M(G&M)-8-10

None

K - 90


M(F&A)–8–1

Identifies and extends to specific cases a variety of patterns (linear and nonlinear) represented in models, tables, sequences, graphs, or in problem situations.

Identifies and extends to specific cases a variety of patterns and generalizes a linear relationship (non-recursive explicit equation)

Identifies and extends to specific cases a variety of patterns and generalizes a linear relationship to find a specific case.

Identifies and extends to specific cases a variety of patterns and generalizes a nonlinear relationship using words or symbols, or generalizes a common nonlinear relationship to find a specific case.

o Patterns (F&A-1)o Numeric patterns (F&A-3)o Non-numeric patterns (F&A-4)o Extend a pattern (F&A-5)o Sequence (F&A-6)o Expresses generalization or rule

using words or symbols (F&A-7)o Pattern Summary Table by grade

level (F&A-9)o Linear relationships (F&A-10)o Proportional linear relationships



o Non-linear relationships (F&A-18)o Concrete situations (F&A-20)o Generalizes a pattern to find a

specific case (F&A-8)






K - 91

Functions and AlgebraGLE Number Grade Level Expectations Terms Defined (Definition Number) Notes and ResourcesM(F&A)–8–2

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.

Demonstrates conceptual understanding of linear relationships (y = kx; y = mx + b) as a constant rate of change by informally and formally determining slopes and intercepts represented in graphs, tables, or problem situations.

Demonstrates conceptual understanding of linear relationships (y = kx; y = mx + b) as a constant rate of change by describing the meaning of slope and intercept in context

Distinguishes between linear relationships (constant rates of change) and nonlinear relationships (varying rates of change) represented in tables, graphs, equations, or problem situations.

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.

o Ratio (N&O-12)o Proportional reasoning (N&O-44)o Linear relationships (F&A-10)o Proportional linear relationships



o Non-linear relationships (F&A-18)o Intercept (F&A-17)o Slope (F&A-14)o Informally determines slope

(F&A-16)o Solves problems involving linear

relationships (F&A-13)o Describes the meaning of slope

and intercept in concrete situations (F&A-15)

o Distinguishes between constant and varying rates (F&A-19)






o Rational number (N&0-1)

K - 92

Functions and AlgebraGLE Number Grade Level Expectations Terms Defined (Definition Number) Notes and ResourcesM(F&A)–8–3

Demonstrates conceptual understanding of algebraic expressions by evaluating and simplifying algebraic expressions (including those with square roots, whole number exponents, or rational numbers); or demonstrates conceptual understanding of algebraic expressions by evaluating an expression within an equation (e.g., determine the value of y when x = 4 given ).



(F&A-26)o Simplifying algebraic equations

(F&A-27)o Number sentences (F&A-33) o Equation (F&A-32)o Examples of forms of equations


(F&A-34)





M(F&A)–8–4

Demonstrates conceptual understanding of equality by showing equivalence between two expressions (expressions consistent with the parameters of the left- and right-hand sides of the equations being solved at this grade level) using models or different representations of the expressions, solving formulas for a variable requiring one transformation (e.g., d = rt; d/r = t).

Demonstrates conceptual understanding of equality by solving multi-step linear equations with integer coefficients.

Demonstrates conceptual understanding of equality by showing that two expressions are or are not equivalent by applying commutative, associative, or distributive properties, order of operations, or substitution.

Demonstrates conceptual understanding of equality by informally solving problems involving systems of linear equations in a context.

o Equality (F&A-30)o Demonstrates equality (F&A-31)o Linear relationships (F&A-10)o Proportional linear relationships



o Number sentences (F&A-33)o Formula (F&A-28)o Write equivalent forms of

formulas (F&A-29)o Equation (F&A-32)o Examples of forms of equations

(F&A-35)o Simplifying algebraic equations


(F&A-34)






K - 93


M(DSP) – 8-1

Interprets a given representation (line graphs, scatter plots, histograms, or box-and-whisker plots) to analyze the data to formulate or justify conclusions, to make predictions, or to solve problems.



o Representation (DSP-1)o Histogram (DSP-7)o Scatter plot (DSP-13)o Box-and-whisker plot (DSP-21)




M(DSP) – 8-2

Analyzes patterns, trends, or distributions in data in a variety of contexts by determining or using measures of central tendency (mean, median, or mode), dispersion (range or variation), outliers, quartile values, or estimated line of best fit to analyze situations, or to solve problems.

Analyzes patterns, trends, or distributions in data in a variety of contexts by evaluating the sample from which the statistics were developed (bias, random, or non-random).

o Pattern (DSP-1)o Mean (DSP-15)o Median (DSP-16)o Mode (DSP-17)o Dispersion (DSP-19)o Range (DSP-19)o Outlier (DSP-20)o Quartiles (DSP-11)o Estimated line of best fit (DSP-

14)o Evaluates samples from which

the statistics were developed (bias) (DSP-24)

o Analyzes the impact of outliers on the mean, median and mode (DSP-22)





M(DSP) – 8-3

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.


o Scatter plot (DSP-13) o Representation (DSP-1)o Identifies or describes




Glencoe, Applications and Connections (1995) Resource KitGlencoe, Algebra I (2003)

K - 94


M(DSP) – 8-4

Uses counting techniques to solve problems in context involving combinations or permutations using a variety of strategies (e.g., organized lists, tables, tree diagrams, models, Fundamental Counting Principle, or others).


o Permutation (DSP-27)o Frequency table (DSP-4)o Tree diagram (DSP-28)o Fundamental Counting Principle

(DSP-29)





M(DSP) – 8-5

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.

For a probability event in which the sample space may or may not contain equally likely outcomes, predicts the theoretical probability of an event and tests the prediction through experiments and simulations.

For a probability event in which the sample space may or may not contain equally likely outcomes, compares and contrasts theoretical and experimental probabilities.





K - 95


M(DSP) – 8-6

In response to a teacher or student generated question or hypothesis decides the most effective method (e. g., survey, observation, experimentation) to collect the data (numerical or categorical) necessary to answer the question.


In response to a teacher or student generated question or hypothesis analyzes the data to draw conclusions about the question or hypothesis being tested while considering the limitations that could affect interpretations.

In response to a teacher or student generated question or hypothesis makes predictions when appropriate.







K - 96

grade level expectations by strandsakai.sau53.org/access/content/group/3c66f1dc-44a6-4a1b... · web...

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