nyc doe math core curriculum scope and sequences for
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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
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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
Show What You Know Mid-Chapter Checkpoint Chapter Review/Test Chapter Test Chapter Performance TaskCritical Area Performance Task
NYCDOE Fall Benchmark Assessment
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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
Show What You Know Mid-Chapter Checkpoint Chapter Review/Test Chapter Test Chapter Performance TaskCritical Area Performance Task
NYCDOE Spring Benchmark AssessmentState Examination
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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
NYCDOE Fall Benchmark Assessment
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
NYCDOE Spring Benchmark Assessment
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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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,
Comparison, Addition and
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)
M3: Addition and Subtraction
of Fractions
(22 days)
M4: Multiplication and
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
M2: Addition and Subtraction
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)
M1: Addition and Subtraction
of Numbers to 10 and Fluency
(45 days)
M3: Ordering and Expressing
Length Measurements as
Numbers
(15 days)
M2: Place Value,
Comparison, Addition and
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)
M1: Multiplication and
Division with Factors of 2, 3,
4, 5, and 10
(25 days)
M3: Multiplication and
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)
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A Story of Units Curriculum Overview NYS COMMON CORE MATHEMATICS CURRICULUM
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
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A Story of Units Curriculum Overview NYS COMMON CORE MATHEMATICS CURRICULUM
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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A Story of Units Curriculum Overview NYS COMMON CORE MATHEMATICS CURRICULUM
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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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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Module and Approximate Number of Instructional Days
Common Core Learning Standards Addressed in Grade 4 Modules50
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)
Use the four operations with whole numbers to solve problems.
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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Module and Approximate Number of Instructional Days
Common Core Learning Standards Addressed in Grade 4 Modules50
drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.
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.
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.
Use place value understanding and properties of operations to perform multi-digit arithmetic.54
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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Module and Approximate Number of Instructional Days
Common Core Learning Standards Addressed in Grade 4 Modules50
equations, rectangular arrays, and/or area models.
Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.55
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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Module and Approximate Number of Instructional Days
Common Core Learning Standards Addressed in Grade 4 Modules50
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.
13
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Module and Approximate Number of Instructional Days
Common Core Learning Standards Addressed in Grade 4 Modules50
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?
Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.57
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.
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Module and Approximate Number of Instructional Days
Common Core Learning Standards Addressed in Grade 4 Modules50
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.
Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.59
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
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A Story of Units Curriculum Overview NYS COMMON CORE MATHEMATICS CURRICULUM
Module and Approximate Number of Instructional Days
Common Core Learning Standards Addressed in Grade 4 Modules50
Module 7:
Exploring Multiplication
(20 days)
Use the four operations with whole numbers to solve problems.
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.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.
Use place value understanding and properties of operations to perform multi-digit arithmetic.60
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.
Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.61
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
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A Story of Units Curriculum Overview NYS COMMON CORE MATHEMATICS CURRICULUM
Module and Approximate Number of Instructional Days
Common Core Learning Standards Addressed in Grade 4 Modules50
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
NYCDOE Fall Benchmark Assessment
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
NYCDOE Spring Benchmark Assessment
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
Teacher Implementation Toolkit70
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.
Scope and Sequence for Grade 6 71
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.
Teacher Implementation Toolkit72
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
Scope and Sequence for Grade 6 73
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
Teacher Implementation Toolkit74
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
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
zU
nit
Test
New
Yor
k Ci
ty S
cope
and
Seq
uenc
e fo
r CM
P3
cont
inued
Gra
de 6
*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.
Scope and Sequence for Grade 6 75
25
da
Ta a
Bo
uT
us
Stat
isti
cs a
nd D
ata
Ana
lysi
s
Inst
ruct
iona
l Ti
me
23 d
ays
Ess
enti
al Id
eas
• Th
e an
swer
s to
a s
tatis
tical
que
stio
n ar
e ca
lled
dat
a. D
ata
can
be
eith
er
num
eric
al o
r ca
teg
oric
al.
• Th
ere
are
seve
ral w
ays
to t
ry t
o s
ay w
hat
is t
ypic
al o
f a s
et o
f dat
a; in
ea
ch c
ase
a si
ngle
num
ber
, cal
led
a m
easu
re o
f cen
ter,
sum
mar
izes
the
d
ata.
Bec
ause
var
ious
mea
sure
s o
f cen
ter
are
calc
ulat
ed d
iffer
ently
, the
y re
spo
nd d
iffer
ently
to
cha
nges
in t
he d
ata
or
to u
nusu
al d
ata
valu
es.
• Th
e va
riab
ility
of a
set
of d
ata
can
be
mea
sure
d, i
nter
pre
ted
and
co
mp
ared
with
the
var
iab
ility
of o
ther
dat
a se
ts. M
easu
res
of v
aria
bili
ty t
ell
you
how
sp
read
out
the
dat
a ar
e in
rela
tion
to e
ach
oth
er o
r to
the
cen
ter.
• Fi
ndin
g m
easu
res
of c
ente
r o
r va
riab
ility
and
gra
phi
ng d
ata
are
usef
ul fo
r su
mm
ariz
ing
the
info
rmat
ion
in a
var
iab
le d
ata
set.
Vis
ual r
epre
sent
atio
ns
of a
dat
a se
t ca
n he
lp y
ou
to in
terp
ret
the
mea
sure
s o
f cen
ter
and
sp
read
, an
d re
late
thi
s to
the
ove
rall
shap
e o
f the
rep
rese
ntat
ion.
Go
als
• U
nder
stan
d a
nd u
se t
he p
roce
ss o
f sta
tistic
al in
vest
igat
ion:
po
se
que
stio
ns, c
olle
ct a
nd a
naly
ze d
ata,
and
mak
e in
terp
reta
tions
to
an
swer
que
stio
ns.
• U
se m
ultip
le re
pre
sent
atio
ns t
o o
rgan
ize
and
rep
rese
nt d
ata
and
dev
elo
p
und
erst
and
ing
of m
easu
res
of c
ente
r an
d m
easu
res
of v
aria
bili
ty fo
r d
ata
dis
trib
utio
ns.
Mai
n C
om
mo
n C
ore
Sta
ndar
ds
Com
mon
Cor
e C
onte
nt S
tand
ard
s6.
SP.A
.1: R
eco
gni
ze a
sta
tistic
al q
uest
ion
as o
ne t
hat
antic
ipat
es v
aria
bili
ty in
th
e d
ata
rela
ted
to
the
que
stio
n an
d a
cco
unts
for
it in
the
ans
wer
s.
6.SP
.A.2
: Und
erst
and
tha
t a
set
of d
ata
colle
cted
to
ans
wer
a s
tatis
tical
q
uest
ion
has
a d
istr
ibut
ion
whi
ch c
an b
e d
escr
ibed
by
its c
ente
r, sp
read
, and
o
vera
ll sh
ape.
6.SP
.A.3
: Rec
og
nize
tha
t a
mea
sure
of c
ente
r fo
r a
num
eric
al d
ata
set
sum
mar
izes
all
of i
ts v
alue
s w
ith a
sin
gle
num
ber
, whi
le a
mea
sure
of v
aria
tion
des
crib
es h
ow
its
valu
es v
ary
with
a s
ing
le n
umb
er.
6.SP
.B.4
: Dis
pla
y nu
mer
ical
dat
a in
plo
ts o
n a
num
ber
line
, inc
lud
ing
do
t p
lots
, his
tog
ram
s, a
nd b
ox
plo
ts.
6.SP
.B.5
: Sum
mar
ize
num
eric
al d
ata
sets
in re
latio
n to
the
ir co
ntex
t.
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
Part
ner
Qui
z A
Che
ck U
p 2
Part
ner
Qui
z B
Self-
Ass
essm
ent
Uni
t Te
st
New
Yor
k Ci
ty S
cope
and
Seq
uenc
e fo
r CM
P3
cont
inued
Gra
de 6
*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.
Teacher Implementation Toolkit76
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
NYCDOE Fall Benchmark Assessment
Unit 3: Rational Numbers 25 days
Unit 4: Expressions and Equations 45 days
NYCDOE Spring Benchmark Assessment
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
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Ratios Curriculum OverviewNYS COMMON CORE MATHEMATICS CURRICULUM
Key: Number GeometryRatios and Proportions
Expressions and Equations
Statistics and Probability
Functions
28
A Story of Ratios: A Curriculum Overview for Grades 6‐8 (DRAFT)Date: 5/11/13 4
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Ratios Curriculum OverviewNYS COMMON CORE MATHEMATICS CURRICULUM
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
A Story of Ratios: A Curriculum Overview for Grades 6‐8 (DRAFT)Date: 5/11/13 5
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Ratios Curriculum OverviewNYS COMMON CORE MATHEMATICS CURRICULUM
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
A Story of Ratios: A Curriculum Overview for Grades 6‐8 (DRAFT)Date: 5/11/13 6
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Ratios Curriculum OverviewNYS COMMON CORE MATHEMATICS CURRICULUM
Alignment Chart
Module and Approximate Number of Instructional Days
Common Core Learning Standards Addressed in Grade 6 Modules5
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.
31
A Story of Ratios: A Curriculum Overview for Grades 6‐8 (DRAFT)Date: 5/11/13 7
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Ratios Curriculum OverviewNYS COMMON CORE MATHEMATICS CURRICULUM
Module and Approximate Number of Instructional Days
Common Core Learning Standards Addressed in Grade 6 Modules5
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.
32
A Story of Ratios: A Curriculum Overview for Grades 6‐8 (DRAFT)Date: 5/11/13 8
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Ratios Curriculum OverviewNYS COMMON CORE MATHEMATICS CURRICULUM
Module and Approximate Number of Instructional Days
Common Core Learning Standards Addressed in Grade 6 Modules5
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
A Story of Ratios: A Curriculum Overview for Grades 6‐8 (DRAFT)Date: 5/11/13 9
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Ratios Curriculum OverviewNYS COMMON CORE MATHEMATICS CURRICULUM
Module and Approximate Number of Instructional Days
Common Core Learning Standards Addressed in Grade 6 Modules5
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.
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A Story of Ratios: A Curriculum Overview for Grades 6‐8 (DRAFT)Date: 5/11/13 10
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A Story of Ratios Curriculum OverviewNYS COMMON CORE MATHEMATICS CURRICULUM
Module and Approximate Number of Instructional Days
Common Core Learning Standards Addressed in Grade 6 Modules5
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.
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A Story of Ratios: A Curriculum Overview for Grades 6‐8 (DRAFT)Date: 5/11/13 11
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A Story of Ratios Curriculum OverviewNYS COMMON CORE MATHEMATICS CURRICULUM
Module and Approximate Number of Instructional Days
Common Core Learning Standards Addressed in Grade 6 Modules5
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.
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A Story of Ratios: A Curriculum Overview for Grades 6‐8 (DRAFT)Date: 5/11/13 12
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A Story of Ratios Curriculum OverviewNYS COMMON CORE MATHEMATICS CURRICULUM
Module and Approximate Number of Instructional Days
Common Core Learning Standards Addressed in Grade 6 Modules5
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.
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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
© 2012 Common Core, Inc. All rights reserved. commoncore.org
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)
Review and Examinations
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
Review and Examinations
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)
Review and Examinations
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A Story of Functions: A Curriculum Overview for Grades 9‐12 (DRAFT)Date: 5/11/13 8
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A Story of Functions Curriculum OverviewNYS COMMON CORE MATHEMATICS CURRICULUM
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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A Story of Functions: A Curriculum Overview for Grades 9‐12 (DRAFT)Date: 5/11/13 9
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A Story of Functions Curriculum OverviewNYS COMMON CORE MATHEMATICS CURRICULUM
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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A Story of Functions: A Curriculum Overview for Grades 9‐12 (DRAFT)Date: 5/11/13 10
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions Curriculum OverviewNYS COMMON CORE MATHEMATICS CURRICULUM
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
Module and Approximate Number of Instructional Days
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 Story of Functions: A Curriculum Overview for Grades 9‐12 (DRAFT)Date: 5/11/13 11
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A Story of Functions Curriculum OverviewNYS COMMON CORE MATHEMATICS CURRICULUM
Module and Approximate Number of Instructional Days
Common Core Learning Standards Addressed in Algebra I Modules
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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A Story of Functions: A Curriculum Overview for Grades 9‐12 (DRAFT)Date: 5/11/13 12
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A Story of Functions Curriculum OverviewNYS COMMON CORE MATHEMATICS CURRICULUM
Module and Approximate Number of Instructional Days
Common Core Learning Standards Addressed in Algebra I Modules
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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A Story of Functions: A Curriculum Overview for Grades 9‐12 (DRAFT)Date: 5/11/13 13
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A Story of Functions Curriculum OverviewNYS COMMON CORE MATHEMATICS CURRICULUM
Module and Approximate Number of Instructional Days
Common Core Learning Standards Addressed in Algebra I Modules
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.
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A Story of Functions: A Curriculum Overview for Grades 9‐12 (DRAFT)Date: 5/11/13 14
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A Story of Functions Curriculum OverviewNYS COMMON CORE MATHEMATICS CURRICULUM
Module and Approximate Number of Instructional Days
Common Core Learning Standards Addressed in Algebra I Modules
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.
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A Story of Functions: A Curriculum Overview for Grades 9‐12 (DRAFT)Date: 5/11/13 15
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions Curriculum OverviewNYS COMMON CORE MATHEMATICS CURRICULUM
Module and Approximate Number of Instructional Days
Common Core Learning Standards Addressed in Algebra I Modules
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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A Story of Functions: A Curriculum Overview for Grades 9‐12 (DRAFT)Date: 5/11/13 16
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions Curriculum OverviewNYS COMMON CORE MATHEMATICS CURRICULUM
Module and Approximate Number of Instructional Days
Common Core Learning Standards Addressed in Algebra I Modules
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.
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A Story of Functions: A Curriculum Overview for Grades 9‐12 (DRAFT)Date: 5/11/13 17
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions Curriculum OverviewNYS COMMON CORE MATHEMATICS CURRICULUM
Module and Approximate Number of Instructional Days
Common Core Learning Standards Addressed in Algebra I Modules
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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A Story of Functions: A Curriculum Overview for Grades 9‐12 (DRAFT)Date: 5/11/13 18
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions Curriculum OverviewNYS COMMON CORE MATHEMATICS CURRICULUM
Module and Approximate Number of Instructional Days
Common Core Learning Standards Addressed in Algebra I Modules
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.
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A Story of Functions: A Curriculum Overview for Grades 9‐12 (DRAFT)Date: 5/11/13 19
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions Curriculum OverviewNYS COMMON CORE MATHEMATICS CURRICULUM
Module and Approximate Number of Instructional Days
Common Core Learning Standards Addressed in Algebra I Modules
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.
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A Story of Functions: A Curriculum Overview for Grades 9‐12 (DRAFT)Date: 5/11/13 20
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions Curriculum OverviewNYS COMMON CORE MATHEMATICS CURRICULUM
Module and Approximate Number of Instructional Days
Common Core Learning Standards Addressed in Algebra I Modules
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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