nyc doe math core curriculum scope and sequences for

NYC DOE Math Core Curriculum

Scope and Sequences for Selected Grades

GRADE 4 – GO MATH 1 GRADE 4 – STATE SCOPE AND SEQUENCE 4 GRADE 6 – CPM3 18 GRADE 6 – STATE SCOPE AND SEQUENCE 27 GRADE 9 – STATE SCOPE AND SEQUENCE 38

NYC Go Math! Grade 4

GO Math! Scope and SequenceThis document contains a high-level scope and sequence for the GO Math! program intended to give teachers an overview of where instructional time will be spent across the year through use of GO Math!. It provides a suggested sequence of instruction and assessments, including where NYCDOE Periodic Assessments can be used to gauge students’ understanding of concepts and skills taught at benchmark moments throughout the year. Based on the Common Core Standards, Go Math! is divided into critical areas that offer a focused and coherent study of the key concepts and skills for each grade.

For each critical area, you will see the following:

• EssentialIdeas: The key topics of the unit; chapters and lessons are built around achieving understanding and mastery of these topics.

• Standards:The standards listed show the main standards covered throughout the Critical Area. Instruction is focused on achieving a thorough knowledge of these standards.

• MathematicalPractices:While all practices are integrated into each Critical Area, the practices listed are ones that receive particular emphasis.

• EssentialQuestions:The essential question for each chapter is listed, showing the goal of each chapter.

• AssessmentOpportunities:This listing highlights the assessments that ensure teachers can gauge student success on mastering the standards covered in the Critical Area.

Grade K: Suggested Sequence for the GO Math! program

Suggested Amount of Time (in days)

Critical Area 1: Place Value and Operations with Whole Numbers

53 days

NYCDOE Fall Benchmark Assessment

Critical Area 2: Fractions and Decimals 38 days

Critical Area 3: Geometry, Measurement, and Data 36 days

NYCDOE Spring Benchmark Assessment

State Examination1

1 The GO Math! program is paced to ensure that all pre-test and post-test standards are completely and fully covered prior to testing. As the transition to the PARCC assessments progresses, schools may choose to make decisions around the pacing of units that address post-test concepts prior to the state examination in consideration of the state’s testing program guidance (see http://www.p12.nysed.gov/assessment/math/math-ei.html).

NYC34 Planning Guide

DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B

1

NYC Scope and Sequence

NYC Scope and Sequence NYC35

Critical Area 1: Place Value and Operations with Whole Numbers Chapters 1–5 53 Days (Instructional Days: 43; Assessment Days: 10)

Critical Area 2: Fractions and Decimals Chapters 6–938 Days (Instructional Days: 30; Assessment Days: 8)

Focus or Main CC Standards

Use the four operations with whole numbers to solve problems.4.OA.3 Solve multistep word problems posed with whole numbers and having

whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

Generalize place value understanding for multi-digit whole numbers.4.NBT.1 Recognize that in a multi-digit whole number, a digit in one place repre-

sents ten times what it represents in the place to its right. 4.NBT.2 Read and write multi-digit whole numbers using base-ten numerals,

number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.

4.NBT.3 Use place value understanding to round multi-digit whole numbers to any place.

Use place value understanding and properties of operations to per-form multi-digit arithmetic.4.NBT.5 Multiply a whole number of up to four digits by a one-digit whole number,

and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

4.NBT.6 Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplica-tion and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

Also 4.OA.1, 4.OA.2, 4.OA.4, 4.OA.5, 4.N BT.4

Extend understanding of fraction equivalence and ordering.4.NF.1 Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by

using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.

4.NF.2 Compare two fractions with different numerators and different denomina-tors, e.g., by creating common denominators or numerators, or by compar-ing to a benchmark fraction such as ½. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclu-sions, e.g., by using a visual fraction model.

Build fractions from unit fractions by applying and extending previ-ous understandings of operations on whole numbers.4.NF.4 Apply and extend previous understandings of multiplication to multiply a

fraction by a whole number. 4.NF.4.a. Understand a fraction a/b as a multiple of 1/b. 4.NF.4.b. Understand a multiple of a/b as a multiple of 1/b, and use this

understanding to multiply a fraction by a whole number. 4.NF.4.c. Solve word problems involving multiplication of a fraction by

a whole number, e.g., by using visual fraction models and equations to represent the problem.

Also 4.NF.3, 4.NF.5, 4.NF.6, 4.NF.7, 4.MD.2

Highlighted Mathematical

Practices

MP.4 Model with mathematics.MP.5 Use appropriate tools strategically.MP.7 Look for and make use of structure.

MP.1 Make sense of problems and persevere in solving them.MP.2 Reason abstractly and quantitatively.MP.5 Use appropriate tools strategically.

Essential Questions

• How can you use place value to compare, add, subtract, and estimate with whole numbers? (Chapter 1)

• What strategies can you use to multiply by 1-digit numbers? (Chapter 2)• What strategies can you use to multiply by 2-digit numbers? (Chapter 3)• How can you divide by 1-digit numbers? (Chapter 4)• How can you find factors and multiples, and how can you generate and de-

scribe number patterns? (Chapter 5)

• What strategies can you use to compare fractions and write equivalent frac-tions? (Chapter 6)

• How do you add or subtract fractions that have the same denominator? (Chap-ter 7)

• How do you multiply fractions by whole numbers? (Chapter 8)• How can you record decimal notation for fractions and compare decimal frac-

tions? (Chapter 9)

Assessment Opportunities

Show What You Know Mid-Chapter Checkpoint Chapter Review/Test Chapter Test Chapter Performance TaskCritical Area Performance Task




2

NYC Go Math! Grade 4

NYC36 Planning Guide

Critical Area 3: Geometry, Measurement, and Data Chapters 10–1336 Days (Instructional Days: 28; Assessment Days: 8)

Focus or Main CC Standards

Draw and identify lines and angles, and classify shapes by properties of their lines and angles.4.G.1 Draw points, lines, line segments, rays, angles (right, acute, obtuse), and

perpendicular and parallel lines. Identify these in two-dimensional figures.4.G.2 Classify two-dimensional figures based on the presence or absence of

parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles.

4.G.3 Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry.

Also 4.OA.5, 4.MD.1, 4.MD.2, 4.MD.3, 4.MD.4, 4.MD.5, 4.MD.6, 4.MD.7

Highlighted Mathematical

Practices

MP.1 Make sense of problems and persevere in solving them.MP.2 Reason abstractly and quantitatively.MP.6 Attend to precision.

Essential Questions

• How can you draw and identify lines and angles, and how can you classify shapes? (Chapter 10)

• How can you measure angles and solve problems involving angle measures? (Chapter 11)

• How can you use relative sizes of measurements to solve problems and to generate measurement tables that show a relationship? (Chapter 12)

• How can you use formulas for perimeter and area to solve problems? (Chapter 13)

Assessment Opportunities


NYCDOE Spring Benchmark AssessmentState Examination


3

Math Benchmark Assessment Overview

The CCLS-aligned benchmark assessments are multi-item type (multiple choice, short response, and extended response) assessments designed to periodically measure student proficiency and progress across classes on a set of skills that align to CCLS grade-level standards. These assessments provide a lens for identifying some of the skills and concepts from the major work of the grade that may need to be reinforced in upcoming units if students are to meet the Common Core expectations for each grade.

The 4th Grade state sequence-aligned benchmark assessment: is offered twice per year: late fall and spring with flexible windows; takes two class periods to administer; is aligned to NYSED math curriculum maps; and covers one to three modules, or about 25-40% of the year’s instruction.

Grade 4: Suggested Sequence for NYS Suggested Instructional Time

Unit 1: Place Value, Rounding, Fluency with Addition and Subtraction Algorithms of Whole Numbers

25 days

Unit 2: Unit Conversions 7 days

Unit 3: Multiplication and Division of up to a 4-Digit Number by up to a 1-Digit Number Using Place Value

43 days


Unit 3 (continued): Multiplication and Division of up to a 4-Digit Number by up to a 1-Digit Number Using Place Value

43 days

Unit 4: Addition and Subtraction of Angle Measurements of Planar Figures 20 days

Unit 5: Order and Operations with Fractions 45 days


Unit 5 (continued): Order and Operations with Fractions 45 days

Unit 6: Decimal Fractions 20 days

State Examination

Unit 7: Exploring Multiplication 20 days

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A Story of Units: A Curriculum Overview for Grades P-5

Date: 11/21/12 3

© 2012 Common Core, Inc. All rights reserved. commoncore.org

A Story of Units Curriculum Overview NYS COMMON CORE MATHEMATICS CURRICULUM

Number and Geometry,

MeasurementFractionsKey: NumberGeometry

*Please refer to grade-level descriptions to identify partially labeled modules and the standards corresponding to all modules.

Pre-Kindergarten Kindergarten Grade 1 Grade 2 Grade 3 Grade 4 Grade 5

6/26/13 Note that date approximations are based on a first student day of 9/6/12 and last day of 6/26/13 with a testing date of approximately mid-late April.

M5: Numbers 10-20, Counting

to 100 by 1 and 10

(30 days)M6: Place Value,

Comparison, Addition and

Subtraction of Numbers to

100

(35 days)

M7: Recognizing Angles,

Faces, and Vertices of

Shapes, Fractions of Shapes

(20 days)

M6: Comparison, Addition

and Subtraction with Length

and Money

(30 days)

20 days

20 days

20 days

20 days

20 days

20 days

M5: Identify, Compose, and

Partition Shapes

(15 days)

M4: Place Value,


Subtraction of Numbers to 40

(35 days)

5/28/13

M4: Addition and Subtraction

of Numbers to 1000

(35 days)

M5: Preparation for

Multiplication and Division

Facts

(40 days)

M5: Write Numerals to 5,

Addition and Subtraction

Stories, Count to 20

(35 days)

M6: Analyze, Compare,

Create, and Compose Shapes

(10 days)

M4: Number Pairs, Addition

and Subtraction of Numbers

to 10

(47 days)

20 days

M4: Describe and Compare

Length, Weight, and Capacity

(35 days)

20 days

1/17/13

2/15/13

3/22/13

4/29/13

20 days

20 days

M5: Fractions as Numbers on

the Number Line

(35 days)

M4: Multiplication and Area

(20 days)

M3: Multiplication and

Division with Factors of 6, 7,

8, and 9

(25 days)


of Fractions

(22 days)


Division of Fractions and

Decimal Fractions

(38 days)

M5: Addition and

Multiplication with Volume

and Area

(25 days)

M7: Exploring Multiplication

(20 days)

M7: Word Problems with

Geometry and Measurement

(40 days)

20 days

20 days

20 days

20 days

20 days

20 days

M1: Classify and Count

Numbers to 10

(43 days)

Test Date

9/6/12

10/10/12


with Length, Weight,

Capacity, and Time

Measurements

(20 days)

M3: Place Value, Counting,

and Comparison of Numbers

to 1000

(25 days) M2: Analyze, Compare,

Create, and Compose Shapes

(15 days)

M1: Analyze, Sort, Classify,

and Count up to 5

(45 days)

M2: Identify and Describe

Shapes (7 days)

M3: Comparison with Length,

Weight, and Numbers to 10

(43 days)


of Numbers to 10 and Fluency

(45 days)

M3: Ordering and Expressing

Length Measurements as

Numbers

(15 days)

M2: Place Value,


Subtraction of Numbers to 20

(35 days)

11/8/12

12/11/12

*M1: Sums and Differences

(10 days)

M3: Count and Answer "How

Many" Questions up to 10

(50 days)


Division with Factors of 2, 3,

4, 5, and 10

(25 days)


Division of up to a 4-Digit

Number by up to a 1-Digit

Number Using Place Value

(43 days)

*M2: Unit Conversions

(7 days)

M1: Place Value, Rounding,

Fluency with Addition and

Subtraction Algorithms of

Whole Numbers

(25 days)

M2: Problem Solving with

Mass, Time, and Capacity

(25 days)

M6: Collecting and

Displaying Data (10 days)

M1: Whole Number and

Decimal Fraction Place Value

to the One-Thousandths

(20 days)

M4: Addition and

Subtraction of Angle

Measurements of Planar

Figures

(20 days)

M5: Order and Operations

with Fractions

(45 days)

M2: Multi-Digit Whole

Number and Decimal

Fraction Operations

(35 days)

M6: Decimal Fractions

(20 days)

20 days

20 days

Approx. test

date for

grades 3-5M6: Graph Points on the

Coordinate Plane to Solve

Problems

(40 days)

5

rgill

Text Box

IMPORTANT: this curriculum map has been updated and is available on EngageNY at: http://www.engageny.org/resource/prekindergarten-grade-5-mathematics-curriculum-map


Date: 11/21/12 41



Sequence of Grade 4 Modules Aligned with the Standards

Module 1: Place Value, Rounding, Fluency with Addition and Subtraction Algorithms of Whole Numbers

Module 2: Unit Conversions: Addition and Subtraction of Length, Weight, and Capacity

Module 3: Multiplication and Division of up to a 4-Digit Number by up to a 1-Digit Number Using Place Value

Module 4: Addition and Subtraction of Angle Measurement of Planar Figures

Module 5: Order and Operations with Fractions

Module 6: Decimal Fractions

Module 7: Exploring Multiplication

Summary of Year

Fourth grade mathematics is about (1) developing understanding and fluency with multi-digit multiplication, and developing understanding of dividing to find quotients involving multi-digit dividends; (2) developing an understanding of fraction equivalence, addition and subtraction of fractions with like denominators, and multiplication of fractions by whole numbers; and (3) understanding that geometric figures can be analyzed and classified based on their properties, such as having parallel sides, perpendicular sides, particular angle measures, and symmetry.

Key Areas of Focus for 3-5: Multiplication and division of whole numbers and fractions—concepts, skills, and problem solving

Required Fluency: 4.NBT.4 Add and subtract within 1,000,000.

CCLS Major Emphasis Clusters Operations and Algebraic Thinking

Use the four operations with whole numbers to solve problems.

Number and Operations in Base Ten

Generalize place value understanding for multi-digit whole numbers.

Use place value understanding and properties of operations to perform multi-digit arithmetic.

Number and Operations – Fractions

Extend understanding of fraction equivalence and ordering.

Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.

Understand decimal notation for fractions, and compare decimal fractions.

Rationale for Module Sequence in Grade 4

Module 1 begins with a study of large numbers. Students are familiar with big units. For example, movies take about a gigabyte (1,000,000,000 bytes) to store on a computer while songs take about a megabyte (1,000,000 bytes). To understand these big numbers, the students rely upon

6


Date: 11/21/12 42



previous mastery of rounding and the addition and subtraction algorithms. In a sense, the algorithms have come full circle: In Grades 2 and 3 the algorithms were the abstract idea students were trying to learn, but by Grade 4 the algorithms have become the concrete knowledge students use to understand new ideas.

The algorithms continue to play a part in Module 2 on unit conversions. Repetitive by design, this module helps students draw similarities between:

10 ones = 1 ten

100 ones = 1 hundred 100 cm = 1 m

1000 ones = 1 thousand 1000 m = 1 km 1000 g = 1 kg 1000 mL = 1 L

Here again, measurement problems act as the glue that binds knowledge of the algorithms, mental math, place value, and real-world applications together into a coherent whole.

In Module 3, measurements provide the concrete foundation behind the distributive property in the multiplication algorithm: 4 × (1 m 2 cm) can be made physical using ribbon, where it is easy to see the 4 copies of 1 m and the 4 copies of 2 cm. Likewise, 4 × (1 ten 2 ones) = 4 tens 8 ones. Students then turn to the place value table with number disks to develop efficient procedures for multiplying and dividing one-digit whole numbers and use the table with number disks to understand and explain why the procedures work. Students also solve word problems throughout the module where they select and accurately apply appropriate methods to estimate, mentally calculate, or use the procedures they are learning to compute products and quotients.

Module 4 focuses as much on solving unknown angle problems using letters and equations as it does on building, drawing, and analyzing two-dimensional shapes in geometry. Students have already used letters and equations to solve word problems in earlier grades. They continue to do so in Grade 4, and now they also learn to solve unknown angle problems: work that challenges students to build and solve equations to find unknown angle measures. First, students learn the definition of degree and learn how to measure angles in degrees using a protractor. From the definition of degree and the fact that angle measures are additive, the following rudimentary facts about angles naturally follow:

1. Vertical angles are equal.

2. The sum of angle measurements on a line is 180 degrees.

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Date: 11/21/12 43



3. The sum of angle measurements around a point is 360 degrees.

Armed only with these three facts (and the two facts used to justify them), students are able to generate and solve equations that make sense, as in the following problem:

Unknown angle problems help to unlock algebraic concepts for students because such problems are visual. The x clearly stands for a specific number: If a student wished, he could place a protractor down on that angle and measure it to find x. But doing so destroys the joy of deducing the answer and solving the puzzle on his own.

Module 5 centers on equivalent fractions and operations with fractions. We use fractions when there is a given unit, the whole unit, but we want to measure using a smaller unit, called the fractional unit. To prepare students to explore the relationship between a fractional unit and its whole unit, examples of such relationships in different contexts were already carefully established earlier in the year:

360 degrees in 1 complete turn 100 cm in 1 meter 1000 g in 1 kilogram 1000 mL in 1 liter

The beauty of fractional units, once defined and understood, is that they behave just as all other units do:

“3 fourths + 5 fourths = 8 fourths” like “3 apples + 5 apples = 8 apples”

“3 fourths × 4 = 12 fourths” like “3 apples × 4 = 12 apples”

X + 240 + 90 = 360

X + 330 = 360

X = 30

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Date: 11/21/12 44



This module also includes measuring and plotting fractional numbers and adding/subtracting those measurements. In Grade 2, fractions were mostly used as adjectives (for example, half cup, third of an hour, etc.). As students do basic fraction arithmetic in Grade 4, they gradually come to understand fractions as numbers.

Module 6, on decimal fractions, starts with the realization that decimal place value units are simply special fractional units: 1 tenth = 1/10, 1 hundredth = 1/100, etc. Fluency plays an important role in this topic as students learn to relate 3/10 = 0.3 = 3 tenths.

The year ends with an exploratory module on multiplication. Students have been practicing the algorithm for multiplying by a one-digit number since Module 3. The goal of Module 7 is to structure opportunities for them to discover ways to multiply two-digit × two-digit numbers with their tools (such as place value tables, area models, bar diagrams, number disks, the distributive property and equations). Students also solve fraction and area problems that involve customary measurements (inches and feet, etc.).

Alignment Chart

Module and Approximate Number of Instructional Days

Common Core Learning Standards Addressed in Grade 4 Modules50

Module 1:

Place Value, Rounding, Fluency with Addition and Subtraction Algorithms of Whole Numbers

(25 days)

Use the four operations with whole numbers to solve problems.51

4.OA.3 Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

Generalize place value understanding for multi-digit whole numbers.

4.NBT.1 Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right. For example, recognize that 700 ÷ 70 = 10 by applying concepts of place value and division.

4.NBT.2 Read and write multi-digit whole numbers using base-ten numerals, number names, and

50

When a cluster is referred to in this chart without a footnote, the cluster is taught in its entirety. 51

4.OA.1 and 4.OA.2 are taught in Modules 3 and 7.

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Date: 11/21/12 45





expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.

4.NBT.3 Use place value understanding to round multi-digit whole numbers to any place.

Use place value understanding and properties of operations to perform multi-digit arithmetic.52

4.NBT.4 Fluently add and subtract multi-digit whole numbers using the standard algorithm.

Module 2:

Unit Conversions: Addition and Subtraction of Length, Weight, and Capacity

(7 days)

Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.53

4.MD.1 Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two-column table. For example, know that 1 ft is 12 times as long as 1 in. Express the length of a 4 ft snake as 48 in. Generate a conversion table for feet and inches listing the number pairs (1, 12), (2, 24), (3, 36), …

4.MD.2 Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale.

Module 3:

Multiplication and Division of up to a 4-Digit Number by up to a 1-Digit Number Using Place Value

(43 days)


4.OA.1 Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 x 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations.

4.OA.2 Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using

52

4.NBT.5 is taught in Modules 3 and 7; 4.NBT.6 is taught in Module 3. 53

The focus of this module is on the metric system to reinforce place value, mixed units, and word problems with unit conversions. Decimal and fraction word problems wait until Modules 5 and 6. 4.MD.3 is taught in Module 3.

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Date: 11/21/12 46





drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.


Gain familiarity with factors and multiplies.

4.OA.4 Find all factor pairs for a whole number in the range 1–100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1–100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1–100 is prime or composite.

Generate and analyze patterns.

4.OA.5 Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. For example, given the rule “Add 3” and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate in this way.


4.NBT.5 Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

4.NBT.6 Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using

54

Multiplying two 2-digit numbers is the focus of Module 7.

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equations, rectangular arrays, and/or area models.


4.MD.3 Apply the area and perimeter formulas for rectangles in real world and mathematical problems. For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor.

Module 4:

Addition and Subtraction of Angle Measurements of Planar Figures

(20 days)

Geometric measurement: understand concepts of angle and measure angles.

4.MD.5 Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement:

a. An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through 1/360 of a circle is called a “one-degree angle,” and can be used to measure angles.

b. An angle that turns through n one-degree angles is said to have an angle measure of n degrees.

4.MD.6 Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure.

4.MD.7 Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure.

Draw and identify lines and angles, and classify shapes by properties of their lines and angles.

4.G.1 Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and

55

4.MD.1 is taught in Modules 2 and 7. 4.MD.2 is taught in Modules 2, 5, 6, and 7.

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parallel lines. Identify these in two-dimensional figures.

4.G.2 Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles.

4.G.3 Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry.

Module 5:

Order and Operations with Fractions56

(45 days)

Extend understanding of fraction equivalence and ordering.

4.NF.1 Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.

4.NF.2 Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

Build fractions from unit fractions by applying and extending previous understanding of operations on whole numbers.

4.NF.3 Understand a fraction a/b with a > 1 as a sum of fractions 1/b.

a. Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.

b. Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a

56

Tenths and hundredths are important fractions in this module, represented in decimal form in Module 6.

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Date: 11/21/12 49





visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8.

c. Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction.

d. Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem.

4.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.

a. Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 × (1/4), recording the conclusion by the equation 5/4 = 5 × (1/4).

b. Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.)

c. Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie?


4.MD.2 Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or

57

4.MD.1 is taught in Modules 2 and 7. 4.MD.3 is taught in Module 3.

14


Date: 11/21/12 50





decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale.

Represent and interpret data.

4.MD.4 Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Solve problems involving addition and subtraction of fractions by using information presented in line plots. For example, from a line plot find and interpret the difference in length between the longest and shortest specimens in an insect collection.

Module 6:

Decimal Fractions

(20 days)

Understand decimal notations for fractions, and compare decimal fractions.58

4.NF.5 Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100. For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100.

4.NF.6 Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram.

4.NF.7 Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model.


4.MD.2 Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale.

58

In this module we continue to work with fractions, now including decimal form. 59

4.MD.1 is taught in Modules 2 and 7. 4.MD.3 is taught in Module 3.

15


Date: 11/21/12 51





Module 7:

Exploring Multiplication

(20 days)


4.OA.1 Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 x 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations.

4.OA.2 Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.



4.NBT.5 Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.


4.MD.1 Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two-column table. For example, know that 1 ft is 12 times as long as 1 in. Express the length of a 4 ft snake as 48 in. Generate a conversion table for feet and inches listing the number pairs (1, 12), (2, 24), (3, 36), …

4.MD.2 Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or

60

In Module 7, the focus is on multiplying two 2-digit numbers. 61

The focus now is on customary units in word problems for application of fraction concepts. 4.MD.3 is taught in Module 3.

16


Date: 11/21/12 52





decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale.

17

New York City Scope and Sequence for CMP3

The following pages contain a high-level scope and sequence for Connected Mathematics 3 and incorporate the State’s pre- and post- standards guidance (see http://www.p12.nysed.gov/assessment/math/math-ei.html). This scope and sequence is intended to give teachers an overview of where instructional time will be spent across the year through use of CMP3. It provides a suggested sequence of instruction and assessments, including where NYCDOE Periodic Assessments can be used to gauge students’ understanding of concepts and skills taught at benchmark moments throughout the year.

For each Unit, you will see the following:

• Essential Ideas The key topics of the Unit; Units are built around achieving understanding and mastery of these topics.

• Goals The mathematical and problem-solving goals that students should achieve for the Unit

• Main CC Standards The standards listed show the main content standards covered throughout the Unit. Instruction is focused on achieving a thorough knowledge of these content standards. In the case of the Standards for Mathematical Practice, all eight standards are listed for each unit because the Mathematical Practices are the foundation of the CMP approach. In each Unit, all eight Mathematical Practices are thoroughly integrated into the content. CMP is a problem-centered curriculum. Thus, the mathematical tasks or problems are the primary vehicle for student engagement with the mathematical concepts to be learned in class and in homework. The Mathematical Practices are a natural part of each CMP lesson as students use them to solve problems and develop mathematical understandings.

• Fluency Goals The key fluency expectations or examples of culminating standards for the grade

• Assessment Opportunities This listing highlights the assessments that ensure teachers can gauge student success on mastering the standards covered in the Unit.

Teacher Implementation Toolkit68

18

Grade 6

Grade 6: Suggested Sequence for CMP31 Suggested Instructional Time

Unit 1 Prime Time: Factors and Multiples 22 days

Unit 2 Comparing Bits and Pieces: Ratios, Rational Numbers, and Equivalence

25 days


Unit 3 Let’s Be Rational: Understanding Fraction Operations

16 days

Unit 4 Covering and Surrounding: Two-Dimensional Measurement

23 days

Unit 5 Decimal Ops: Computing With Decimals and Percents

24 days


Unit 6 Variables and Patterns: Focus on Algebra 25 days

State Examination

Unit 7 Data About Us: Statistics and Data Analysis 23 days

1This Scope and Sequence represents one way a school may teach the full year’s content and incorporates the state’s pre-post test standards. As the transition to the PARCC assessments progresses, schools may choose to make decisions around the sequence and pacing of Units that address post-test concepts prior to the state examination in consideration of the state’s testing program guidance (see http://www.p12.nysed.gov/assessment/math/math-ei.html).

Scope and Sequence for Grade 6 69

19

New

Yor

k Ci

ty S

cope

and

Seq

uenc

e fo

r CM

P3

G

rade

6

Pr

ime

Tim

e F

acto

rs a

nd M

ulti

ple

s

Inst

ruct

iona

l Ti

me

22 d

ays

Ess

enti

al Id

eas

• If

a nu

mb

er N

can

be

writ

ten

as a

pro

duc

t o

f tw

o w

hole

num

ber

s,

N =

a ×

b, t

hen

a an

d b

are

fact

ors

of N

. Mul

tiple

s o

f a c

an b

e fo

und

usi

ng

the

exp

ress

ion

a ×

(so

me

who

le n

umb

er),

such

as

2a, 3

a, 4

a et

c.

• W

hen

all f

acto

rs o

f a n

umb

er a

re b

roke

n d

ow

n in

to p

rime

num

ber

s, y

ou

have

a u

niq

ue p

rime

fact

oriz

atio

n. F

ind

ing

the

prim

e fa

cto

rizat

ion

of t

wo

nu

mb

ers

can

be

usef

ul in

find

ing

the

leas

t co

mm

on

mul

tiple

and

gre

ates

t co

mm

on

fact

or

of t

he n

umb

ers.

• W

hen

calc

ulat

ing

the

val

ue o

f an

exp

ress

ion,

the

op

erat

ions

hav

e to

be

per

form

ed in

a c

onv

entio

nal o

rder

, the

ord

er o

f op

erat

ions

.

• So

met

imes

a n

umer

ical

exp

ress

ion

can

be

writ

ten

in d

iffer

ent

way

s b

ut

the

exp

ress

ions

are

eq

uiva

lent

bec

ause

the

val

ue is

the

sam

e.

Go

als

• U

nder

stan

d re

latio

nshi

ps

amo

ng fa

cto

rs, m

ultip

les,

div

iso

rs, a

nd p

rod

ucts

.•

Und

erst

and

why

tw

o e

xpre

ssio

ns a

re e

qui

vale

nt.

Mai

n C

om

mo

n C

ore

Sta

ndar

ds

Com

mon

Cor

e C

onte

nt S

tand

ard

s6.

NS.

B.4

: Fin

d t

he g

reat

est

com

mo

n fa

cto

r o

f tw

o w

hole

num

ber

s le

ss t

han

or

equa

l to

100

and

the

leas

t co

mm

on

mul

tiple

of t

wo

who

le n

umb

ers

less

th

an o

r eq

ual t

o 1

2. U

se t

he d

istr

ibut

ive

pro

per

ty t

o e

xpre

ss a

sum

of t

wo

w

hole

num

ber

s 1–

100

with

a c

om

mo

n fa

cto

r as

a m

ultip

le o

f a s

um o

f tw

o

who

le n

umb

ers

with

no

co

mm

on

fact

or.

6.E

E.A

.1: W

rite

and

eva

luat

e nu

mer

ical

exp

ress

ions

invo

lvin

g w

hole

-num

ber

ex

po

nent

s.

Com

mon

Cor

e St

and

ard

s fo

r M

athe

mat

ical

Pra

ctic

eM

P.1:

Mak

e se

nse

of p

rob

lem

s an

d p

erse

vere

in s

olv

ing

the

m.

MP.

2: R

easo

n ab

stra

ctly

and

qua

ntita

tivel

y.M

P.3:

Co

nstr

uct

viab

le a

rgum

ents

and

crit

ique

the

reas

oni

ng o

f oth

ers.

MP.

4: M

od

el w

ith m

athe

mat

ics.

MP.

5: U

se a

pp

rop

riate

to

ols

str

ateg

ical

ly.

MP.

6: A

tten

d t

o p

reci

sio

n.M

P.7:

Lo

ok

for

and

mak

e us

e o

f str

uctu

re.

MP.

8: L

oo

k fo

r an

d e

xpre

ss re

gul

arity

in re

pea

ted

reas

oni

ng.

Flue

ncy

Go

als*

• D

ivid

e m

ultid

igit

num

ber

s flu

ently

usi

ng t

he s

tand

ard

alg

orit

hm.

• M

ultip

ly m

ulitd

igit

who

le n

umb

ers

usin

g t

he s

tand

ard

alg

orit

hm.*

*

Ass

essm

ents

Che

ck U

p 1

Part

ner

Qui

zC

heck

Up

2U

nit

Pro

ject

Self-

Ass

essm

ent

Uni

t Te

st

*CM

P3 d

evel

op

s flu

ency

in p

roce

dur

al s

kills

fro

m a

foun

dat

ion

of c

onc

eptu

al u

nder

stan

din

g, a

n ap

pro

ach

that

lead

s to

long

-ter

m re

tent

ion

of s

kills

and

ab

ility

to

ap

ply

tho

se s

kills

in p

rob

lem

so

lvin

g.

**re

info

rcin

g fl

uenc

y ex

pec

tatio

ns fr

om

pre

vio

us g

rad

es


20

New

Yor

k Ci

ty S

cope

and

Seq

uenc

e fo

r CM

P3

cont

inued

Gra

de 6

Co

mPa

rin

g B

iTs

an

d P

ieC

es

Rat

ios,

Rat

iona

l Num

ber

s, a

nd E

qui

vale

nce

Inst

ruct

iona

l Ti

me

25 d

ays

Ess

enti

al Id

eas

• R

atio

nal n

umb

ers

can

be

writ

ten

in fr

actio

n o

r d

ecim

al fo

rm a

nd c

an

be

rep

rese

nted

as

po

ints

or

dis

tanc

es o

n a

num

ber

-line

. A n

umb

er-li

ne

rep

rese

ntat

ion

is u

sefu

l fo

r o

rder

ing

and

co

mp

arin

g n

umb

ers.

• Fr

actio

ns a

nd d

ecim

als

can

be

rena

med

or

rep

artit

ione

d t

o fi

nd

equi

vale

nt fr

actio

ns o

r dec

imal

s. E

qui

vale

nce

is u

sefu

l for

mov

ing

bet

wee

n fr

actio

n an

d d

ecim

al re

pre

sent

atio

ns a

nd fo

r sol

ving

pro

ble

ms.

• R

atio

s ar

e co

mp

aris

ons

bet

wee

n tw

o n

umb

ers.

Yo

u ca

n sc

ale

ratio

s to

m

ake

equi

vale

nt r

atio

s. P

erce

nts

are

ratio

s w

here

100

par

ts re

pre

sent

s th

e w

hole

.

• A

rat

e is

a p

artic

ular

kin

d o

f rat

io, w

here

the

am

oun

ts c

om

par

ed a

re in

d

iffer

ent

units

. A u

nit

rate

is a

rat

e th

at h

as b

een

scal

ed t

o x

: 1.

Go

als

• U

nder

stan

d fr

actio

ns a

nd d

ecim

als

as n

umb

ers

that

can

be

loca

ted

on

the

num

ber

line

, co

mp

ared

, co

unte

d, p

artit

ione

d, a

nd d

eco

mp

ose

d.

• U

nder

stan

d r

atio

s as

co

mp

aris

ons

of t

wo

num

ber

s.

• U

nder

stan

d e

qui

vale

nce

of f

ract

ions

and

rat

ios,

and

use

eq

uiva

lenc

e to

so

lve

pro

ble

ms.

Mai

n C

om

mo

n C

ore

Sta

ndar

ds

Com

mon

Cor

e C

onte

nt S

tand

ard

s 6.

RP.

A.1

: Und

erst

and

the

co

ncep

t o

f a r

atio

and

use

rat

io la

ngua

ge

to

des

crib

e a

ratio

rela

tions

hip

bet

wee

n tw

o q

uant

ities

.

6.R

P.A

.2: U

nder

stan

d t

he c

onc

ept

of a

uni

t ra

te a

/b a

sso

ciat

ed w

ith a

rat

io

a:b

with

b ≠

0, a

nd u

se r

ate

lang

uag

e in

the

co

ntex

t o

f a r

atio

rela

tions

hip

.

6.R

P.A

.3: U

se r

atio

and

rat

e re

aso

ning

to

so

lve

real

-wo

rld a

nd m

athe

mat

ical

p

rob

lem

s, e

.g.,

by

reas

oni

ng a

bo

ut t

able

s o

f eq

uiva

lent

rat

ios,

tap

e d

iag

ram

s, d

oub

le n

umb

er li

ne d

iag

ram

s, o

r eq

uatio

ns.

6.N

S.C

.5: U

nder

stan

d t

hat

po

sitiv

e an

d n

egat

ive

num

ber

s ar

e us

ed t

og

ethe

r to

des

crib

e q

uant

ities

hav

ing

op

pos

ite d

irect

ions

or v

alue

s (e

.g.,

tem

per

atur

e ab

ove/

bel

ow z

ero,

ele

vatio

n ab

ove/

bel

ow s

ea le

vel,

cred

its/d

ebits

, pos

itive

/ne

gat

ive

elec

tric

cha

rge)

; use

pos

itive

and

neg

ativ

e nu

mb

ers

to re

pre

sent

q

uant

ities

in re

al-w

orld

con

text

s, e

xpla

inin

g th

e m

eani

ng o

f 0 in

eac

h si

tuat

ion.

6.N

S.C

.6: U

nder

stan

d a

rat

iona

l num

ber

as

a p

oin

t o

n th

e nu

mb

er li

ne.

Ext

end

num

ber

line

dia

gra

ms

and

co

ord

inat

e ax

es fa

mili

ar fr

om

pre

vio

us

gra

des

to

rep

rese

nt p

oin

ts o

n th

e lin

e an

d in

the

pla

ne w

ith n

egat

ive

num

ber

co

ord

inat

es.

6.N

S.C

.7: U

nder

stan

d o

rder

ing

and

ab

solu

te v

alue

of r

atio

nal n

umb

ers.

Com

mon

Cor

e St

and

ard

s fo

r M

athe

mat

ical

Pra

ctic

eM

P.1:

Mak

e se

nse

of p

rob

lem

s an

d p

erse

vere

in s

olv

ing

the

m.

MP.

2: R

easo

n ab

stra

ctly

and

qua

ntita

tivel

y.M

P.3:

Co

nstr

uct

viab

le a

rgum

ents

and

crit

ique

the

reas

oni

ng o

f oth

ers.

MP.

4: M

od

el w

ith m

athe

mat

ics.

MP.

5: U

se a

pp

rop

riate

to

ols

str

ateg

ical

ly.

MP.

6: A

tten

d t

o p

reci

sio

n.M

P.7:

Lo

ok

for

and

mak

e us

e o

f str

uctu

re.

MP.

8: L

oo

k fo

r an

d e

xpre

ss re

gul

arity

in re

pea

ted

reas

oni

ng.

Flue

ncy

Go

als*

• D

ivid

e m

ultid

igit

num

ber

s flu

ently

usi

ng t

he s

tand

ard

alg

orit

hm.

• O

per

ate

with

mul

tidig

it d

ecim

als

fluen

tly.

Ass

essm

ents

Part

ner

Qui

z A

Che

ck U

pPa

rtne

r Q

uiz

B

Self-

Ass

essm

ent

Uni

t Te

st

NY

CD

OE

Fal

l Ben

chm

ark

Ass

essm

ent

*CM

P3 d

evel

op

s flu

ency

in p

roce

dur

al s

kills

fro

m a

foun

dat

ion

of c

onc

eptu

al u

nder

stan

din

g, a

n ap

pro

ach

that

lead

s to

long

-ter

m re

tent

ion

of s

kills

and

ab

ility

to

ap

ply

tho

se s

kills

in p

rob

lem

so

lvin

g.


21

New

Yor

k Ci

ty S

cope

and

Seq

uenc

e fo

r CM

P3

cont

inued

Gra

de 6

LeT’

s B

e r

aTi

on

aL

Und

erst

and

ing

Fra

ctio

n O

per

atio

ns

Inst

ruct

iona

l Ti

me

16 d

ays

Ess

enti

al Id

eas

• E

stim

atio

n is

an

imp

ort

ant

par

t o

f rea

soni

ng q

uant

itativ

ely.

It e

nco

urag

es

mak

ing

sen

se o

f a s

ituat

ion,

allo

ws

you

to re

cog

nize

err

ors

, and

co

mp

lem

ents

oth

er p

rob

lem

so

lvin

g s

kills

.

• Fo

r ea

ch o

per

atio

n, t

here

is a

n ef

ficie

nt, g

ener

al a

lgo

rithm

for

com

put

ing

w

ith fr

actio

ns t

hat

wo

rks

in a

ll ca

ses.

• Va

riab

les

rep

rese

nt u

nkno

wn

valu

es.

• So

met

imes

rew

ritin

g a

pro

ble

m u

sing

a d

iffer

ent

op

erat

ion

can

be

help

ful

in fi

ndin

g t

he s

olu

tion.

Go

als

• U

nder

stan

d e

stim

atio

n as

a t

oo

l fo

r a

varie

ty o

f situ

atio

ns a

nd d

evel

op

st

rate

gie

s fo

r es

timat

ing

resu

lts o

f arit

hmet

ic o

per

atio

ns.

• R

evis

it an

d d

evel

op

mea

ning

s fo

r th

e fo

ur a

rithm

etic

op

erat

ions

and

ski

ll at

usi

ng a

lgo

rithm

s fo

r ea

ch.

• U

se v

aria

ble

s to

rep

rese

nt u

nkno

wn

valu

es a

nd e

qua

tions

to

rep

rese

nt

rela

tions

hip

s.

Mai

n C

om

mo

n C

ore

Sta

ndar

ds

Com

mon

Cor

e C

onte

nt S

tand

ard

s 6.

NS.

A.1

: Int

erp

ret

and

co

mp

ute

quo

tient

s o

f fra

ctio

ns, a

nd s

olv

e w

ord

p

rob

lem

s in

volv

ing

div

isio

n o

f fra

ctio

ns b

y fr

actio

ns, e

.g.,

by

usin

g v

isua

l fr

actio

n m

od

els

and

eq

uatio

ns t

o re

pre

sent

the

pro

ble

m.

6.E

E.B

.6: U

se v

aria

ble

s to

rep

rese

nt n

umb

ers

and

writ

e ex

pre

ssio

ns w

hen

solv

ing

a re

al-w

orld

or

mat

hem

atic

al p

rob

lem

; und

erst

and

tha

t a

varia

ble

can

re

pre

sent

an

unkn

ow

n nu

mb

er, o

r, d

epen

din

g o

n th

e p

urp

ose

at

hand

, any

nu

mb

er in

a s

pec

ified

set

.

6.E

E.B

.7: S

olv

e re

al-w

orld

and

mat

hem

atic

al p

rob

lem

s b

y w

ritin

g a

nd s

olv

ing

eq

uatio

ns o

f the

form

x +

p =

q a

nd p

x =

q fo

r ca

ses

in w

hich

p, q

and

x a

re

all n

onn

egat

ive

ratio

nal n

umb

ers.

Com

mon

Cor

e St

and

ard

s fo

r M

athe

mat

ical

Pra

ctic

eM

P.1:

Mak

e se

nse

of p

rob

lem

s an

d p

erse

vere

in s

olv

ing

the

m.

MP.

2: R

easo

n ab

stra

ctly

and

qua

ntita

tivel

y.M

P.3:

Co

nstr

uct

viab

le a

rgum

ents

and

crit

ique

the

reas

oni

ng o

f oth

ers.

MP.

4: M

od

el w

ith m

athe

mat

ics.

MP.

5: U

se a

pp

rop

riate

to

ols

str

ateg

ical

ly.

MP.

6: A

tten

d t

o p

reci

sio

n.M

P.7:

Lo

ok

for

and

mak

e us

e o

f str

uctu

re.

MP.

8: L

oo

k fo

r an

d e

xpre

ss re

gul

arity

in re

pea

ted

reas

oni

ng.

Flue

ncy

Go

als*

• D

ivid

e m

ultid

igit

num

ber

s flu

ently

usi

ng t

he s

tand

ard

alg

orit

hm.

• D

ivid

e fr

actio

ns.

Ass

essm

ents

Che

ck U

p 1

Part

ner

Qui

zC

heck

Up

2Se

lf-A

sses

smen

tU

nit

Test

*CM

P3 d

evel

op

s flu

ency

in p

roce

dur

al s

kills

fro

m a

foun

dat

ion

of c

onc

eptu

al u

nder

stan

din

g, a

n ap

pro

ach

that

lead

s to

long

-ter

m re

tent

ion

of s

kills

and

ab

ility

to

ap

ply

tho

se s

kills

in p

rob

lem

so

lvin

g.


22

New

Yor

k Ci

ty S

cope

and

Seq

uenc

e fo

r CM

P3

cont

inued

Gra

de 6

Co

ve

rin

g a

nd

su

rr

ou

nd

ing

Tw

o-D

imen

sion

al M

easu

rem

ent

Inst

ruct

iona

l Ti

me

23 d

ays

Ess

enti

al Id

eas

• Po

lyg

ons

and

irre

gul

ar fi

gur

es c

an b

e d

eco

mp

ose

d in

to t

riang

les

and

re

ctan

gle

s to

find

the

are

a o

f the

fig

ures

.

• A

fixe

d n

umb

er o

f are

a un

its c

an b

e en

clo

sed

by

man

y d

iffer

ent

per

imet

ers,

and

a fi

xed

num

ber

of p

erim

eter

uni

ts c

an e

nclo

se m

any

diff

eren

t ar

eas.

• Fo

rmul

as fo

r th

e ar

ea a

nd p

erim

eter

of a

rect

ang

le c

an h

elp

yo

u so

lve

pro

ble

ms

by

reas

oni

ng a

bo

ut t

he re

latio

nshi

p b

etw

een

valu

es.

• Th

e vo

lum

e o

f a p

rism

can

be

tho

ught

of a

s m

ultip

lyin

g a

bas

e la

yer

of

unit

cub

es b

y th

e nu

mb

er o

f lay

ers

need

ed t

o fi

ll th

e p

rism

.

• Su

rfac

e ar

eas

of t

hree

-dim

ensi

ona

l so

lids

can

be

foun

d b

y ad

din

g t

he

area

s o

f the

face

s.

Go

als

• U

nder

stan

d w

hat

it m

eans

to

mea

sure

are

a an

d p

erim

eter

.•

Und

erst

and

and

use

the

rela

tions

hip

bet

wee

n fo

rmul

as fo

r ar

ea a

nd

per

imet

er o

f tria

ngle

s an

d p

aral

lelo

gra

ms

and

form

ulas

for

rect

ang

les.

• U

nder

stan

d v

olu

me

as fi

lling

a t

hree

-dim

ensi

ona

l sha

pe

and

dev

elo

p

stra

teg

ies

to fi

nd s

urfa

ce a

rea

by

find

ing

are

a o

f tw

o-d

imen

sio

nal s

hap

es.

Mai

n C

om

mo

n C

ore

Sta

ndar

ds

Com

mon

Cor

e C

onte

nt S

tand

ard

s 6.

EE

.C.9

: Use

var

iab

les

to re

pre

sent

tw

o q

uant

ities

in a

real

-wo

rld p

rob

lem

th

at c

hang

e in

rela

tions

hip

to

one

ano

ther

; writ

e an

eq

uatio

n to

exp

ress

o

ne q

uant

ity, t

houg

ht o

f as

the

dep

end

ent

varia

ble

, in

term

s o

f the

oth

er

qua

ntity

, tho

ught

of a

s th

e in

dep

end

ent

varia

ble

. Ana

lyze

the

rela

tions

hip

b

etw

een

the

dep

end

ent

and

ind

epen

den

t va

riab

les

usin

g g

rap

hs a

nd t

able

s,

and

rela

te t

hese

to

the

eq

uatio

n.

6.G

.A.1

: Fin

d t

he a

rea

of r

ight

tria

ngle

s, o

ther

tria

ngle

s, s

pec

ial

qua

dril

ater

als,

and

po

lyg

ons

by

com

po

sing

into

rect

ang

les

or

dec

om

po

sing

in

to t

riang

les

and

oth

er s

hap

es; a

pp

ly t

hese

tec

hniq

ues

in t

he c

ont

ext

of

solv

ing

real

-wo

rld a

nd m

athe

mat

ical

pro

ble

ms.

6.G

.A.2

: Fin

d t

he v

olu

me

of a

rig

ht re

ctan

gul

ar p

rism

with

frac

tiona

l ed

ge

leng

ths

by

pac

king

it w

ith u

nit

cub

es o

f the

ap

pro

pria

te u

nit

frac

tion

edg

e le

ngth

s, a

nd s

how

tha

t th

e vo

lum

e is

the

sam

e as

wo

uld

be

foun

d b

y m

ultip

lyin

g t

he e

dg

e le

ngth

s o

f the

pris

m. A

pp

ly t

he fo

rmul

as V

= lw

h an

d

V =

bh

to fi

nd v

olu

mes

of r

ight

rect

ang

ular

pris

ms

with

frac

tiona

l ed

ge

leng

ths

in t

he c

ont

ext

of s

olv

ing

real

-wo

rld a

nd m

athe

mat

ical

pro

ble

ms.

Com

mon

Cor

e St

and

ard

s fo

r M

athe

mat

ical

Pra

ctic

eM

P.1:

Mak

e se

nse

of p

rob

lem

s an

d p

erse

vere

in s

olv

ing

the

m.

MP.

2: R

easo

n ab

stra

ctly

and

qua

ntita

tivel

y.M

P.3:

Co

nstr

uct

viab

le a

rgum

ents

and

crit

ique

the

reas

oni

ng o

f oth

ers.

MP.

4: M

od

el w

ith m

athe

mat

ics.

MP.

5: U

se a

pp

rop

riate

to

ols

str

ateg

ical

ly.

MP.

6: A

tten

d t

o p

reci

sio

n.M

P.7:

Lo

ok

for

and

mak

e us

e o

f str

uctu

re.

MP.

8: L

oo

k fo

r an

d e

xpre

ss re

gul

arity

in re

pea

ted

reas

oni

ng.

Flue

ncy

Go

als*

• M

ultip

ly m

ulitd

igit

who

le n

umb

ers

usin

g t

he s

tand

ard

alg

orit

hm.*

*

Ass

essm

ents

Che

ck U

p 1

Che

ck U

p 2

Part

ner

Qui

z

Uni

t Pr

oje

ctSe

lf-A

sses

smen

tU

nit

Test

*CM

P3 d

evel

op

s flu

ency

in p

roce

dur

al s

kills

fro

m a

foun

dat

ion

of c

onc

eptu

al u

nder

stan

din

g, a

n ap

pro

ach

that

lead

s to

long

-ter

m re

tent

ion

of s

kills

and

ab

ility

to

ap

ply

tho

se s

kills

in p

rob

lem

so

lvin

g.

**re

info

rcin

g fl

uenc

y ex

pec

tatio

ns fr

om

pre

vio

us g

rad

es


23

*CM

P3 d

evel

op

s flu

ency

in p

roce

dur

al s

kills

fro

m a

foun

dat

ion

of c

onc

eptu

al u

nder

stan

din

g, a

n ap

pro

ach

that

lead

s to

long

-ter

m re

tent

ion

of s

kills

and

ab

ility

to

ap

ply

tho

se s

kills

in p

rob

lem

so

lvin

g.

de

Cim

aL

oP

s C

omp

utin

g w

ith

Dec

imal

s an

d P

erce

nts

Inst

ruct

iona

l Ti

me

24 d

ays

Ess

enti

al Id

eas

• E

stim

atio

n is

an

imp

ort

ant

par

t o

f rea

soni

ng q

uant

itativ

ely.

It h

elp

s yo

u m

ake

sens

e o

f a s

ituat

ion,

allo

ws

you

to re

cog

nize

err

ors

, and

co

mp

lem

ents

oth

er p

rob

lem

so

lvin

g s

kills

.

• Th

e st

and

ard

alg

orit

hm fo

r d

ivid

ing

dec

imal

s is

sup

po

rted

by

the

conn

ectio

ns b

etw

een

frac

tion

and

dec

imal

op

erat

ions

.

• Fl

uenc

y w

ith d

ecim

al o

per

atio

ns a

llow

yo

u to

so

lve

a va

riety

of

pro

ble

ms

invo

lvin

g r

atio

s an

d p

erce

nts.

• In

vers

e o

per

atio

ns c

an b

e us

ed t

o is

ola

te a

var

iab

le w

hen

solv

ing

eq

uatio

ns.

Go

als

• U

nder

stan

d e

stim

atio

n as

a t

oo

l fo

r a

varie

ty o

f situ

atio

ns, i

nclu

din

g

chec

king

ans

wer

s an

d m

akin

g d

ecis

ions

.•

Rev

isit

and

dev

elo

p m

eani

ngs

for

the

four

arit

hmet

ic o

per

atio

ns o

n w

hole

num

ber

s an

d d

ecim

als,

and

ski

ll at

usi

ng a

lgo

rithm

s fo

r ea

ch

dec

imal

op

erat

ion.

• U

se v

aria

ble

s to

rep

rese

nt u

nkno

wn

valu

es a

nd e

qua

tions

to

re

pre

sent

rela

tions

hip

s.•

Dev

elo

p u

nder

stan

din

g o

f var

ious

co

ntex

ts in

whi

ch p

erce

ntag

es a

re

used

, inc

lud

ing

sal

es t

ax, t

ips,

dis

coun

ts, p

erce

nt in

crea

ses.

Mai

n C

om

mo

n C

ore

Sta

ndar

ds

Com

mon

Cor

e C

onte

nt S

tand

ard

s 6.

NS.

B.3

: Flu

ently

ad

d, s

ubtr

act,

mul

tiply

, and

div

ide

mul

ti-d

igit

dec

imal

s us

ing

the

sta

ndar

d a

lgo

rithm

for

each

op

erat

ion.

6.E

E.A

.3: A

pp

ly t

he p

rop

ertie

s o

f op

erat

ions

to

gen

erat

e eq

uiva

lent

ex

pre

ssio

ns.

Com

mon

Cor

e St

and

ard

s fo

r M

athe

mat

ical

Pra

ctic

eM

P.1:

Mak

e se

nse

of p

rob

lem

s an

d p

erse

vere

in s

olv

ing

the

m.

MP.

2: R

easo

n ab

stra

ctly

and

qua

ntita

tivel

y.M

P.3:

Co

nstr

uct

viab

le a

rgum

ents

and

crit

ique

the

reas

oni

ng o

f oth

ers.

MP.

4: M

od

el w

ith m

athe

mat

ics.

MP.

5: U

se a

pp

rop

riate

to

ols

str

ateg

ical

ly.

MP.

6: A

tten

d t

o p

reci

sio

n.M

P.7:

Lo

ok

for

and

mak

e us

e o

f str

uctu

re.

MP.

8: L

oo

k fo

r an

d e

xpre

ss re

gul

arity

in re

pea

ted

reas

oni

ng.

Flue

ncy

Go

als*

• D

ivid

e m

ultid

igit

num

ber

s flu

ently

usi

ng t

he s

tand

ard

alg

orit

hm.

• O

per

ate

with

mul

tidig

it d

ecim

als

fluen

tly.

Ass

essm

ents

Che

ck U

p 1

Che

ck U

p 2

Part

ner

Qui

z

Uni

t Pr

oje

ctSe

lf-A

sses

smen

tU

nit

Test

NY

CD

OE

Sp

ring

Ben

chm

ark

Ass

essm

ent

New

Yor

k Ci

ty S

cope

and

Seq

uenc

e fo

r CM

P3

cont

inued

Gra

de 6


24

va

ria

BLe

s a

nd

Pa

TTe

rn

s Fo

cus

on A

lgeb

ra

Inst

ruct

iona

l Ti

me

25 d

ays

Ess

enti

al Id

eas

• In

man

y re

al-w

orld

situ

atio

ns, o

ne v

aria

ble

qua

ntity

dep

end

s o

n an

oth

er. T

able

s, g

rap

hs, a

nd e

qua

tions

are

var

ious

rep

rese

ntat

ions

th

at c

an b

e us

ed t

o b

ette

r un

der

stan

d t

he p

atte

rn o

f cha

nge

bet

wee

n va

riab

le q

uant

ities

.

• N

ot

all r

elat

ions

hip

s ar

e lin

ear.

Line

ar re

latio

nshi

ps

have

a c

ons

tant

rat

e o

f ch

ang

e b

etw

een

varia

ble

s an

d a

re w

ritte

n in

the

form

y =

mx,

y =

b +

x,

and

y =

b +

mx.

• Th

ere

is m

ore

tha

n o

ne w

ay t

o w

rite

an e

xpre

ssio

n to

mo

del

a re

al

wo

rld s

ituat

ion.

Pro

per

ties

of o

per

atio

ns a

llow

yo

u to

gen

erat

e eq

uiva

lent

ex

pre

ssio

ns a

nd c

heck

eq

uiva

lenc

e.

• So

lutio

ns fo

r eq

uatio

ns a

nd in

equa

litie

s ca

n b

e fo

und

by

exam

inin

g t

he

tab

le o

r g

rap

h o

f the

eq

uatio

n o

r b

y re

writ

ing

it a

s a

rela

ted

eq

uatio

n.

Go

als

• D

evel

op

und

erst

and

ing

of v

aria

ble

s an

d h

ow

the

y ar

e re

late

d.

• D

evel

op

und

erst

and

ing

of e

xpre

ssio

ns a

nd e

qua

tions

.

Mai

n C

om

mo

n C

ore

Sta

ndar

ds

Com

mon

Cor

e C

onte

nt S

tand

ard

s 6.

RP.

A.3

: Use

rat

io a

nd r

ate

reas

oni

ng t

o s

olv

e re

al-w

orld

and

mat

hem

atic

al

pro

ble

ms,

e.g

., b

y re

aso

ning

ab

out

tab

les

of e

qui

vale

nt r

atio

s, t

ape

dia

gra

ms,

do

uble

num

ber

line

dia

gra

ms,

or

equa

tions

.

6.N

S.C

.8: S

olv

e re

al-w

orld

and

mat

hem

atic

al p

rob

lem

s b

y g

rap

hing

po

ints

in a

ll fo

ur q

uad

rant

s o

f the

co

ord

inat

e p

lane

. Inc

lud

e us

e o

f co

ord

inat

es

and

ab

solu

te v

alue

to

find

dis

tanc

es b

etw

een

po

ints

with

the

sam

e fir

st

coo

rdin

ate

or

the

sam

e se

cond

co

ord

inat

e.

6.E

E.A

.3: A

pp

ly t

he p

rop

ertie

s o

f op

erat

ions

to

gen

erat

e eq

uiva

lent

exp

ress

ions

.

6.E

E.B

.7: S

olv

e re

al-w

orld

and

mat

hem

atic

al p

rob

lem

s b

y w

ritin

g a

nd s

olv

ing

eq

uatio

ns o

f the

form

x +

p =

q a

nd p

x =

q fo

r ca

ses

in w

hich

p, q

and

x a

re

all n

onn

egat

ive

ratio

nal n

umb

ers.

Com

mon

Cor

e St

and

ard

s fo

r M

athe

mat

ical

Pra

ctic

eM

P.1:

Mak

e se

nse

of p

rob

lem

s an

d p

erse

vere

in s

olv

ing

the

m.

MP.

2: R

easo

n ab

stra

ctly

and

qua

ntita

tivel

y.M

P.3:

Co

nstr

uct

viab

le a

rgum

ents

and

crit

ique

the

reas

oni

ng o

f oth

ers.

MP.

4: M

od

el w

ith m

athe

mat

ics.

MP.

5: U

se a

pp

rop

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26

Math Benchmark Assessment Overview

The CCLS-aligned benchmark assessments are multi-item type (multiple choice, short response, and extended response) assessments designed to periodically measure student proficiency and progress across classes on a set of skills that align to CCLS grade-level standards. These assessments provide a lens for identifying some of the skills and concepts from the major work of the grade that may need to be reinforced in upcoming units if students are to meet the Common Core expectations for each grade.

The 6th Grade state sequence-aligned benchmark assessment: is offered twice per year: late fall and spring with flexible windows; takes two class periods to administer; is aligned to NYSED math curriculum maps; and covers one to three modules, or about 25-40% of the year’s instruction.

Grade 6: Suggested Sequence for NYS Suggested Instructional Time

Unit 1: Ratios and Unit Rates 35 days

Unit 2: Arithmetic Operations Including Dividing by a Fraction 25 days


Unit 3: Rational Numbers 25 days

Unit 4: Expressions and Equations 45 days


Unit 4 (continued): Expressions and Equations 45 days

Unit 5: Area, Surface Area, and Volume Problems 25 days

State Examination

Unit 6: Statistics 25 days

27

A Story of Ratios: A Curriculum Overview for Grades 6‐8 (DRAFT)Date: 5/11/13 3


A Story of Ratios Curriculum OverviewNYS COMMON CORE MATHEMATICS CURRICULUM

Key: Number GeometryRatios and Proportions

Expressions and Equations

Statistics and Probability

Functions

28




Sequence of Grade 6 Modules Aligned with the Standards Module 1: Ratios and Unit Rates Module 2: Arithmetic Operations Including Dividing by a Fraction Module 3: Rational Numbers Module 4: Expressions and Equations Module 5: Area, Surface Area, and Volume Problems Module 6: Statistics

Summary of Year

Sixth grade mathematics is about (1) connecting ratio and rate to whole number multiplication and division and using concepts of ratio and rate to solve problems; (2) completing understanding of division of fractions and extending the notion of number to the system of rational numbers, which includes negative numbers; (3) writing, interpreting, and using expressions and equations; and (4) developing understanding of statistical thinking.

Key Areas of Focus for Grade 6: Ratios and proportional reasoning; early expressions and equations

Required Fluency: 6.NS.2 Multi‐digit division 6.NS.3 Multi‐digit decimal operations

CCLS Major Emphasis Clusters Ratios and Proportional Relationships

Understand ratio concepts and use ratio reasoning to solve problems.

The Number System Apply and extend previous understandings of

multiplication and division to divide fractions by fractions. Apply and extend previous understandings of numbers to

the system of rational numbers. Expressions and Equations

Apply and extend previous understandings of arithmetic to algebraic expressions.

Reason about and solve one‐variable equations and inequalities.

Represent and analyze quantitative relationships between dependent and independent variables.

Rationale for Module Sequence in Grade 6

In Module 1, students build on their prior work in measurement and in multiplication and division as they study the concepts and language of ratios and unit rates. They use proportional reasoning to solve problems. In particular, students solve ratio and rate using tape diagrams, tables of equivalent ratios, double number line diagrams, and equations. They plot pairs of values generated from a ratio or rate on the first quadrant of the coordinate plane.

29




Students expand their understanding of the number system and build their fluency in arithmetic operations in Module 2. Students learned in Grade 5 to divide whole numbers by unit fractions and unit fractions by whole numbers. Now, they apply and extend their understanding of multiplication and division to divide fractions by fractions. The meaning of this operation is connected to real‐world problems as students are asked to create and solve fraction division word problems. Students continue (from Fifth Grade) to build fluency with adding, subtracting, multiplying, and dividing multi‐digit decimal numbers using the standard algorithms.

Major themes of Module 3 are to understand rational numbers as points on the number line and to extend previous understandings of numbers to the system of rational numbers, which now include negative numbers. Students extend coordinate axes to represent points in the plane with negative number coordinates and, as part of doing so, see that negative numbers can represent quantities in real‐world contexts. They use the number line to order numbers and to understand the absolute value of a number. They begin to solve real‐world and mathematical problems by graphing points in all four quadrants, a concept that continues throughout to be used into high school and beyond.

With their sense of number expanded to include negative numbers, in Module 4 students begin formal study of algebraic expressions and equations. Students learn equivalent expressions by continuously relating algebraic expressions back to arithmetic and the properties of arithmetic (commutative, associative, and distributive). They write, interpret, and use expressions and equations as they reason about and solve one‐variable equations and inequalities and analyze quantitative relationships between two variables.

Module 5 is an opportunity to practice the material learned in Module 4 in the context of geometry; students apply their newly acquired capabilities with expressions and equations to solve for unknowns in area, surface area, and volume problems. They find the area of triangles and other two‐dimensional figures and use the formulas to find the volumes of right rectangular prisms with fractional edge lengths. Students use negative numbers in coordinates as they draw lines and polygons in the coordinate plane. They also find the lengths of sides of figures, joining points with the same first coordinate or the same second coordinate and apply these techniques to solve real‐world and mathematical problems.

In Module 6, students develop an understanding of statistical variability and apply that understanding as they summarize, describe, and display distributions. In particular, careful attention is given to measures of center and variability.

30




Alignment Chart



Module 1: Ratios and Unit Rates (35 days)

Understand ratio concepts and use ratio reasoning to solve problems.

6.RP.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.”

6.RP.2 Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.”6

6.RP.3 Use ratio and rate reasoning to solve real‐world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

a. Make tables of equivalent ratios relating quantities with whole‐number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.

b. Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?

c. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.

d. Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.

5 When a cluster is referred to in this chart without a footnote, the cluster is taught in its entirety. 6 Expectations for unit rates in this grade are limited to non‐complex fractions.

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Module 2: Arithmetic Operations Including Dividing by a Fraction (25 days)

Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

6.NS.1 Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4‐cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?

Compute fluently with multi‐digit numbers and find common factors and multiples.

6.NS.2 Fluently divide multi‐digit numbers using the standard algorithm.7

6.NS.3 Fluently add, subtract, multiply, and divide multi‐digit decimals using the standard algorithm for each operation.8

6.NS.4 Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express 36 + 8 as 4 (9 + 2).

Module 3: Rational Numbers (25 days)

Apply and extend previous understandings of numbers to the system of rational numbers.

6.NS.5 Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real‐world contexts, explaining the meaning of 0 in each situation.

6.NS.6 Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane

7 This fluency standard begins in this module and is practiced throughout the remainder of the year. 8 This fluency standard begins in this module and is practiced throughout the remainder of the year.

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with negative number coordinates.

a. Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., –(–3) = 3, and that 0 is its own opposite.

b. Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.

c. Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.

6.NS.7 Understand ordering and absolute value of rational numbers.

a. Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. For example, interpret –3 > –7 as a statement that –3 is located to the right of –7 on a number line oriented from left to right.

b. Write, interpret, and explain statements of order for rational numbers in real‐world contexts. For example, write –3°C > –7°C to express the fact that –3°C is warmer than ‐7°C.

c. Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real‐world situation. For example, for an account balance of –30 dollars, write |–30| = 30 to describe the size of the debt in dollars.

d. Distinguish comparisons of absolute value from statements about order. For example, recognize that an account balance less than –30 dollars represents a debt greater than 30 dollars.

6.NS.8 Solve real‐world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.

33






Module 4: Expressions and Equations (45 days)

Apply and extend previous understandings of arithmetic to algebraic expressions.9

6.EE.1 Write and evaluate numerical expressions involving whole‐number exponents.

6.EE.2 Write, read, and evaluate expressions in which letters stand for numbers.

a. Write expressions that record operations with numbers and with letters standing for numbers. For example, express the calculation “Subtract y from 5” as 5 – y.

b. Identify parts of an expression using mathematical terms (sum, term, product, factor quotient, coefficient); view one or more parts of an expression as a single entity. For example, describe the expression 2 (8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms.

c. Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real‐world problems. Perform arithmetic operations, including those involving whole‐number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). For example, use the formulas V = s3 and A = 6 s2 to find the volume and surface area of a cube with sides of length s = 1/2.

6.EE.3 Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.

6.EE.4 Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for.

9 6.EE.2c is also taught in Module 4 in the context of geometry.

34






Reason about and solve one‐variable equations and inequalities.10

6.EE.5 Understand solving an equation or inequality as a process of answering a question: Which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.

6.EE.6 Use variables to represent numbers and write expressions when solving a real‐world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.

6.EE.7 Solve real‐world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q, and x are all nonnegative rational numbers.

6.EE.8 Write an inequality of the form x > c or x < c to represent a constraint or condition in a real‐world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.

Represent and analyze quantitative relationships between dependent and independent variables.

6.EE.9 Use variables to represent two quantities in a real‐world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.

Module 5: Area, Surface Area, and Volume Problems (25 days)

Apply and extend previous understandings of arithmetic to algebraic expressions.11

6.EE.2 Write, read, and evaluate expressions in which letters stand for numbers.

c. Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real‐world problems. Perform arithmetic operations, including those

10 Except for 6.EE.8, this cluster is also taught in Module 4 in the context of geometry. 11 This standard, taught in Module 4, is practiced in this module in the context of geometry.

35






involving whole‐number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). For example, use the formulas V = s3 and A = 6 s2 to find the volume and surface area of a cube with sides of length s = 1/2.

Reason about and solve one‐variable equations and inequalities.12

6.EE.5 Understand solving an equation or inequality as a process of answering a question: Which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.

6.EE.6 Use variables to represent numbers and write expressions when solving a real‐world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.

6.EE.7 Solve real‐world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q, and x are all nonnegative rational numbers.

Solve real‐world and mathematical problems involving area, surface area, and volume.

6.G.1 Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real‐world and mathematical problems.

6.G.2 Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real‐world and mathematical problems.

6.G.3 Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real‐world and mathematical problems.

12 These standards, taught in Module 4, are practiced in this module in the context of geometry.

36






6.G.4 Represent three‐dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real‐world and mathematical problems.

Module 6: Statistics (25 days)

Develop understanding of statistical variability.

6.SP.1 Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. For example, “How old am I?” is not a statistical question, but “How old are the students in my school?” is a statistical question because one anticipates variability in students’ ages.

6.SP.2 Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.

6.SP.3 Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.

Summarize and describe distributions.

6.SP.4 Display numerical data in plots on a number line, including dot plots, histograms, and box plots.

6.SP.5 Summarize numerical data sets in relation to their context, such as by:

a. Reporting the number of observations.

b. Describing the nature of the attribute under investigation, including how it was measured and its units of measurement.

c. Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.

d. Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.

37

Algebra Benchmark Assessment Overview

The CCLS-aligned benchmark assessment is designed to periodically measure student progress across classes on a set of skills aligned to the Common Core standards and provide a lens for identifying some of the skills and concepts that may need to be taught or reinforced if students are to meet the Common Core expectations for a course. Below is a list of Common Core standards expected to be assessed on the fall benchmark.

The Algebra benchmark assessment: is offered twice per year: late fall and spring; is aligned to NYSED math curriculum maps; and covers one to three modules, or about 25-40% of the year’s instruction.

Algebra Fall Benchmark Standards Coverage

N-Q.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

A-SSE.1 Interpret expressions that represent a quantity in terms of its context.

A-SSE.1a Interpret parts of an expression, such as terms, factors, and coefficients.

A-SSE.1b Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)n as the product of P and a factor not depending on P.

A-CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

A-CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

A-CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.

A-CED.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.

A-REI.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

A-REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

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S-ID.1 Represent data with plots on the real number line (dot plots, histograms, and box plots)

S-ID.2 Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.

S-ID.3 Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).

S-ID.5 Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.

S-ID.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.

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A Story of Functions: A Curriculum Overview for Grades 9‐12 (DRAFT)Date: 5/11/13 4


A Story of Functions Curriculum OverviewNYS COMMON CORE MATHEMATICS CURRICULUM

Curriculum Map

Key:Number andQuantity

and Modeling

Geometryand Modeling

Algebra andModeling

Statistics and Probability

and Modeling

Functionsand Modeling

Grade 9 ‐‐ Algebra I Grade 10 ‐‐ Geometry Grade 11 ‐‐ Algebra II Grade 12 ‐‐ Precalculus

State Examinations State Examinations State Examinations State Examinations

6/26/13 Note that date approximations are based on a first student day of 9/6/12 and last day of 6/26/13.

M4:Expressions and Equations

(30 days)

M5:Quadratic Functions

(30 days)

Review and Examinations

M1:Complex Numbers and

Transformations(40 days)

M2:Vectors and Matrices

(40 days)

M3:Rational and Exponential

Functions(25 days)

M4: Trigonometry(15 days)

M5:Probabil ity and Statistics

(30 days)


M3: Functions(45 days)

Date9/6/12

20 daysM1:

Relationships Between Quantities and Reasoning

with Equations(30 days)

12/11/12

20 days

M2: Descriptive Statistics(20 days)

M3:Linear and Exponential

Relationships(40 days)

20 days

10/10/12

20 days 20 days

20 days

M1:Polynomial, Rational, and Radical Relationships

(45 days)

M2:Trigonometric Functions

(20 days)

20 days

11/8/12

20 days

M1:Congruence, Proof, and

Constructions(45 days)

M2:Similarity, Proof, and

Trigonometry(45 days)

20 days

2/15/13

20 days 20 days

1/17/13

20 days

M4:Inferences and Conclusions

from Data(40 days)

20 days

4/29/13

20 days 20 days


M3: Extending to Three Dimensions (10 days)

M4: Connecting Algebra and Geometry through Coordinates (25 days)

5/28/13

20 days 20 days

3/22/13

20 daysM5:

Circles with and Without Coordinates(25 days)


40




Sequence of Algebra I Modules Aligned with the Standards Module 1: Relationships Between Quantities and Reasoning with Equations Module 2: Descriptive Statistics Module 3: Linear and Exponential Relationships Module 4: Expressions and Equations Module 5: Quadratic Functions

Summary of Year

The fundamental purpose of this course is to formalize and extend the mathematics that students learned in the middle grades. Because it is built on the middle grades standards, this is a more ambitious version of Algebra I than has generally been offered. The modules deepen and extend understanding of linear and exponential relationships by contrasting them with each other and by applying linear models to data that exhibit a linear trend, and students engage in methods for analyzing, solving, and using quadratic functions. The Mathematical Practice Standards apply throughout each course and, together with the content standards, prescribe that students experience mathematics as a coherent, useful, and logical subject that makes use of their ability to make sense of problem situations.

Recommended Fluencies for Algebra I Solving characteristic problems involving the analytic geometry of lines,

including, writing the equation of a line given a point and a slope. Adding, subtracting and multiplying polynomials. Transforming expressions and chunking (seeing the parts of an

expression as a single object) as used in factoring, completing the square, and other algebraic calculations.

CCLS Major Emphasis Clusters Seeing Structure in Expressions

Interpret the structure of expressions Arithmetic with Polynomials and Rational Expressions

Perform arithmetic operations on polynomials Creating Equations

Create equations that describe numbers or relationships Reasoning with Equations and Inequalities

Understand solving equations as a process of reasoning and explain the reasoning

Solve equations and inequalities in one variable Represent and solve equations and inequalities graphically

Interpreting Functions Understand the concept of a function and use function

notation Interpret functions that arise in applications in terms of

the context Interpreting Categorical and Quantitative Data

Interpret linear models

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Rationale for Module Sequence in Algebra I

Module 1: By the end of eighth grade, students have learned to solve linear equations in one variable and have applied graphical and algebraic methods to analyze and solve systems of linear equations in two variables. Now, students analyze and explain precisely the process of solving an equation. Students, through reasoning, develop fluency writing, interpreting, and translating between various forms of linear equations and inequalities, and make conjectures about the form that a linear equation might take in a solution to a problem. They reason abstractly and quantitatively by choosing and interpreting units in the context of creating equations in two variables to represent relationships between quantities. They master the solution of linear equations and apply related solution techniques and the properties of exponents to the creation and solution of simple exponential equations.

Module 2: This module builds upon students’ prior experiences with data, providing students with more formal means of assessing how a model fits data. Students display and interpret graphical representations of data, and if appropriate, choose regression techniques when building a model that approximates a linear relationship between quantities. They analyze their knowledge of the context of a situation to justify their choice of a linear model. With linear models, they plot and analyze residuals to informally assess the goodness of fit.

Module 3: In earlier grades, students defined, evaluated, and compared functions in modeling relationships between quantities. In this module, students learn function notation and develop the concepts of domain and range. They explore many examples of functions, including sequences; they interpret functions given graphically, numerically, symbolically, and verbally, translate between representations, and understand the limitations of various representations. Students build on their understanding of integer exponents to consider exponential functions with integer domains. They compare and contrast linear and exponential functions, looking for structure in each and distinguishing between additive and multiplicative change. Students explore systems of equations and inequalities, and they find and interpret their solutions. They interpret arithmetic sequences as linear functions and geometric sequences as exponential functions. In building models of relationships between two quantities, students analyze the key features of a graph or table of a function.

Module 4: In this module, students build on their knowledge from Module 3. Students strengthen their ability to discern structure in exponential expressions. They create and solve equations involving quadratic and cubic expressions. They understand that polynomials form a system analogous to the integers. In this module’s modeling applications, students reason abstractly and quantitatively in interpreting parts of an expression that represent a quantity in terms of its context; they also learn to make sense of problems and persevere in solving them by choosing or producing equivalent forms of an expression (e.g., completing the square in a quadratic expression to reveal a maximum value).

Mathematical Practices

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

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Module 5: In this module, students consider quadratic functions, comparing the key characteristics of quadratic functions to those of linear and exponential functions. Students learn through repeated reasoning to anticipate the graph of a quadratic function by interpreting the structure of various forms of quadratic expressions. In particular, they identify the real solutions of a quadratic equation as the zeros of a related quadratic function. Students expand their experience with functions to include more specialized functions—absolute value, step, and those that are piecewise‐defined. Students select from among these functions to model phenomena using the modeling cycle (see page 61 of the CCLS).

Alignment Chart


Common Core Learning Standards Addressed in Algebra I Modules

Module 1: Relationships Between Quantities and Reasoning with Equations (30 days)

Reason quantitatively and use units to solve problems.

N‐Q.1 Use units as a way to understand problems and to guide the solution of multi‐step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

N‐Q.24 Define appropriate quantities for the purpose of descriptive modeling.

N‐Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

Interpret the structure of expressions

A‐SSE.1 Interpret expressions that represent a quantity in terms of its context.★

a. Interpret parts of an expression, such as terms, factors, and coefficients.

b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)n as the product of P and a factor not depending on P.

4 This standard will be assessed in Algebra I by ensuring that some modeling tasks (involving Algebra I content or securely held content from grades 6‐8) require the student to create a quantity of interest in the situation being described.

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A‐SSE.2 Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as

(x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2).

Create equations that describe numbers or relationships

A‐CED.15 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.★

A‐CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.★

A‐CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non‐viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.★

A‐CED.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.★

Understand solving equations as a process of reasoning and explain the reasoning

A‐REI.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

Solve equations and inequalities in one variable

A‐REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

5 In Algebra I, tasks are limited to linear, quadratic, or exponential equations with integer exponents.

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Module 2: Descriptive Statistics (20 days)

Summarize, represent, and interpret data on a single count or measurement variable

S‐ID.1 Represent data with plots on the real number line (dot plots, histograms, and box plots).★

S‐ID.2 Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.★

S‐ID.3 Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).★

Summarize, represent, and interpret data on two categorical and quantitative variables

S‐ID.5 Summarize categorical data for two categories in two‐way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.★

S‐ID.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.★

a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.6

b. Informally assess the fit of a function by plotting and analyzing residuals.

c. Fit a linear function for a scatter plot that suggests a linear association.

Interpret linear models

S‐ID.7 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.★

S‐ID.8 Compute (using technology) and interpret the correlation coefficient of a linear fit.★

6 Tasks have a real‐world context. In Algebra I, exponential functions are limited to those with domains in the integers.

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S‐ID.9 Distinguish between correlation and causation.★

Module 3: Linear and Exponential Relationships (40 days)

Solve systems of equations

A‐REI.5 Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

A‐REI.67 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

Represent and solve equations and inequalities graphically

A‐REI.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

A‐REI.12 Graph the solutions to a linear inequality in two variables as a half‐plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half‐planes.

A‐REI.118 Explain why the x‐coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.★

Understand the concept of a function and use function notation

F‐IF.1 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The

7 Tasks have a real‐world context. In Algebra I, tasks have hallmarks of modeling as a mathematical practice (less defined tasks, more of the modeling cycle, etc.). 8 In Algebra I, tasks that assess conceptual understanding of the indicated concept may involve any of the function types mentioned in the standard except exponential and logarithmic functions. Finding the solutions approximately is limited to cases where f(x) and g(x) are polynomial functions.

46






graph of f is the graph of the equation y = f(x).

F‐IF.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

F‐IF.39 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n–1) for n ≥ 1.

Interpret functions that arise in applications in terms of the context

F‐IF.410 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.★

F‐IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person‐hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.★

F‐IF.611 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.★

Analyze functions using different representations

F‐IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple

9 This standard is part of the Major Content in Algebra I and will be assessed accordingly. 10 Tasks have a real‐world context. In Algebra I, tasks are limited to linear functions, quadratic functions, square root functions, cube root functions, piecewise‐defined functions (including step functions and absolute value functions), and exponential functions with domains in the integers. The focus in this module is on linear and exponential functions. 11 Tasks have a real‐world context. In Algebra I, tasks are limited to linear functions, quadratic functions, square root functions, cube root functions, piecewise‐defined functions (including step functions and absolute value functions), and exponential functions with domains in the integers. The focus in this module is on linear and exponential functions.

47






cases and using technology for more complicated cases.★

a. Graph linear and quadratic functions and show intercepts, maxima, and minima.

F‐IF.912 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

Build a function that models a relationship between two quantities

F‐BF.113 Write a function that describes a relationship between two quantities.★

a. Determine an explicit expression, a recursive process, or steps for calculation from a context.

Build new functions from existing functions

F‐BF.314 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

Construct and compare linear, quadratic, and exponential models and solve problems

F‐LE.1 Distinguish between situations that can be modeled with linear functions and with exponential functions.★

a. Prove that linear functions grow by equal differences over equal intervals, and that

12 In Algebra I, tasks are limited to linear functions, quadratic functions, square root functions, cube root functions, piecewise‐defined functions (including step functions and absolute value functions), and exponential functions with domains in the integers. The focus in this module is on linear and exponential functions. 13 Tasks have a real‐world context. In Algebra I, tasks are limited to linear functions, quadratic functions, and exponential functions with domains in the integers. 14 In Algebra I, identifying the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x+k) for specific values of k (both positive and negative) is limited to linear and quadratic functions. Experimenting with cases and illustrating an explanation of the effects on the graph using technology is limited to linear functions, quadratic functions, square root functions, cube root functions, piecewise‐defined functions (including step functions and absolute value functions), and exponential functions with domains in the integers. Tasks do not involve recognizing even and odd functions. The focus in this module is on linear and exponential functions.

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exponential functions grow by equal factors over equal intervals.

b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.

c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

F‐LE.215 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input‐output pairs (include reading these from a table).★

F‐LE.3 Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.★

Interpret expressions for functions in terms of the situation they model

F‐LE.516 Interpret the parameters in a linear or exponential function in terms of a context.★

Module 4: Expressions and Equations (30 days)

Use properties of rational and irrational numbers.

N‐RN.3 Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.

Interpret the structure of expressions

A‐SSE.1 Interpret expressions that represent a quantity in terms of its context.★

a. Interpret parts of an expression, such as terms, factors, and coefficients.

b. Interpret complicated expressions by viewing one or more of their parts as a single entity.

15 In Algebra I, tasks are limited to constructing linear and exponential functions in simple context (not multi‐step). 16 Tasks have a real‐world context. In Algebra I, exponential functions are limited to those with domains in the integers.

49






For example, interpret P(1+r)n as the product of P and a factor not depending on P.

A‐SSE.217 Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2).

Write expressions in equivalent forms to solve problems

A‐SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.★

a. Factor a quadratic expression to reveal the zeros of the function it defines.

b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.

c. Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15t can be rewritten as (1.15 1/12)12t ≈ 1.01212t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.18

Perform arithmetic operations on polynomials

A‐APR.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

Understand the relationship between zeros and factors of polynomials

A‐APR.319 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

17 In Algebra I, tasks are limited to numerical expressions and polynomial expressions in one variable. Examples: Recognize 532 � 472 as a difference of squares and see an opportunity to rewrite it in the easier‐to‐evaluate form (53�47)(53�47). See an opportunity to rewrite a2 � 9a � 14 as (a�7)(a�2). Can include the sum or difference of cubes (in one variable), and factoring by grouping. 18 Tasks have a real‐world context. As described in the standard, there is an interplay between the mathematical structure of the expression and the structure of the situation such that choosing and producing an equivalent form of the expression reveals something about the situation. In Algebra I, tasks are limited to exponential expressions with integer exponents. 19 In Algebra I, tasks are limited to quadratic and cubic polynomials in which linear and quadratic factors are available. For example, find the zeros of (x ‐ 2)(x2 ‐ 9).

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Create equations that describe numbers or relationships

A‐CED.120 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.★

A‐CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.★

A‐CED.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.★

Solve equations and inequalities in one variable

A‐REI.4 Solve quadratic equations in one variable.

a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p) 2 = q that has the same solutions. Derive the quadratic formula from this form.

b. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.21

Module 5: Quadratic Functions (30 days)

Reason quantitatively and use units to solve problems.

N‐Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

20 In Algebra I, tasks are limited to linear, quadratic, or exponential equations with integer exponents. 21 Tasks do not require students to write solutions for quadratic equations that have roots with nonzero imaginary parts. However, tasks can require the student to recognize cases in which a quadratic equation has no real solutions.

51






Interpret functions that arise in applications in terms of the context

F‐IF.422 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.★

F‐IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person‐hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.★

F‐IF.623 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.★

Analyze functions using different representations

F‐IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.★

a. Graph linear and quadratic functions and show intercepts, maxima, and minima.

b. Graph square root, cube root, and piecewise‐defined functions, including step functions and absolute value functions.

F‐IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

a. Use the process of factoring and completing the square in a quadratic function to show

22 Tasks have a real‐world context. In Algebra I, tasks are limited to linear functions, quadratic functions, square root functions, cube root functions, piecewise‐defined functions (including step functions and absolute value functions), and exponential functions with domains in the integers. 23 Tasks have a real‐world context. In Algebra I, tasks are limited to linear functions, quadratic functions, square root functions, cube root functions, piecewise‐defined functions (including step functions and absolute value functions), and exponential functions with domains in the integers.

52






zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.

F‐IF.924 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

Build a function that models a relationship between two quantities

F‐BF.125 Write a function that describes a relationship between two quantities.★

a. Determine an explicit expression, a recursive process, or steps for calculation from a context.

Build new functions from existing functions

F‐BF.326 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

Construct and compare linear, quadratic, and exponential models and solve problems

F‐LE.3 Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.★

24 In Algebra I, tasks are limited to linear functions, quadratic functions, square root functions, cube root functions, piecewise‐defined functions (including step functions and absolute value functions), and exponential functions with domains in the integers. 25 Tasks have a real‐world context. In Algebra I, tasks are limited to linear functions, quadratic functions, and exponential functions with domains in the integers. 26 In Algebra I, identifying the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x+k) for specific values of k (both positive and negative) is limited to linear and quadratic functions. Experimenting with cases and illustrating an explanation of the effects on the graph using technology is limited to linear functions, quadratic functions, square root functions, cube root functions, piecewise‐defined functions (including step functions and absolute value functions), and exponential functions with domains in the integers. Tasks do not involve recognizing even and odd functions.

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nyc doe math core curriculum scope and sequences for

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