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CMP3 and the Common Core State Standards for Mathematics

Introduction

Standards for Mathematical Content

This guide examines how Connected Mathematics Project 3 (CMP3) aligns with the Common Core State Standards for Mathematics (CCSSM). The instructional philosophy of CMP3 emphasizes inquiry and applications—an approach that embraces the CCSSM with an in-depth focus on the Standards for Mathematical Practice.

The CCSSM promote a more conceptual and analytical approach to the study of mathematics. Below is an example of how CMP3 follows this approach:

As students begin their study of mathematics in the middle grades, they have completed their study of operations with whole numbers and decimals and will complete their study of the four operations with fractions in Grade 6.

C5

Number Puzzles and Multiple Towers

Understanding the Order of Operations In Problems 1–4, solve the equation using the order of operations.

1. 12 + 4 × 6 ÷ 3 = 2. 15 – [(10 + 26) ÷ 4] =

3. (19 – 1) × (7 – 6) = 4. 2 × (8 – 4) × 10 ÷ 8 =

In Problems 5–8, insert parentheses, brackets, and/or braces to make each equation true.

5. 8 + 28 ÷ 4 = 15 6. 5 × 9 – 6 + 2 = 17

7. 20 – 3 + 1 × 3 = 8 8. 9 – 2 × 4 + 6 = 70

9. For the following problem, use grouping symbols to write an equation that represents the situation. Then solve the problem.

A florist planted pots of tulip bulbs, with 5 bulbs in each pot. He planted 6 pots with white tulip bulbs and 4 pots with red tulip bulbs. How many bulbs did he plant in all?

Unit 1 Session 3.6

note Students use the order of operations to solve problems.

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In Grade 7, students gain an in-depth understanding of rational number concepts, including terminating and repeating decimals, negative rational numbers, and complex fractions. They focus on fundamental algebra topics through their exploration of variables, along with proportional reasoning.

In Grade 8, students delve deeper in their development of algebraic concepts and skills. Students apply their knowledge of place value, properties of operations, and the inverse relationships between operations to explore, evaluate, and analyze algebraic expressions and functions.

1.3 1.41.21.1

Rational numbers are numbers that can be expressed as one integer a divided by another integer b, where b is not zero. You can write a rational number in the form ab or in decimal form.

For a rational number, ab, why does b have to be nonzero?

• Are integers rational numbers? Explain.

• Is zero a rational number? Explain.

Each negative number can be paired with a positive number. These two numbers are called opposites because they are the same distance from zero on the number line, but in different directions.

−5 −4 −3 −2 −1 0 +1 +2 +3 +4 +5

Opposites

To avoid confusion with operation signs, you can use raised signs to show negative rational numbers, such as -150. If a rational number does not have a sign, you can assume it is positive. For example, 150 is the same as +150.

Extending the Number Line1.2

Investigation 1 Extending the Number System 11

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Focus, Coherence, and Rigor

At the end of Grade 8, students have either completed all of the required content for Algebra 1, or they are prepared to begin a formal high school algebra course. Note that CMP3 offers two eighth- grade paths—traditional eighth grade and Algebra 1 eighth grade. Completing the Algebra 1 pathway in eighth grade puts students on the path tocollege-level courses by their senior year.

The CCSSM focus on a limited number of topics at each grade level and allow enough time for students to achieve fluency and even mastery of these concepts.

The subsequent year of study builds on the concepts of the previous year and prepares students to take the next coherent steps in a mathematical-development trajectory that deepens and builds on earlier grade topics and explorations.

CMP3 ensures that all students are well prepared by the end of Grade 8 for the rigor of high school mathematics.

S13S12

Butterflies, Pinwheels, and Wallpaper Symmetry and Transformations

1 Symmetry and Transformations 2 Transformations and Congruence 3 Transforming Coordinates 4 Dilations and Similar Figures

Say It With Symbols Making Sense of Symbols

1 Equivalent Expressions 2 Generating Expressions 3 Solving Equations 4 Looking Back at Functions 5 Reasoning With Symbols

It’s In The System Systems of Linear Equations and Inequalities

1 Linear Equations with Two Variables 2 Solving Linear Systems Algebraically 3 Systems of Functions and Inequalities* 4 Systems of Linear Inequalities*

*For use with Algebra 1 only

Thinking with Mathematical Models Linear and Inverse Variation

1 Exploring Data Patterns 2 Linear Models and Equations 3 Inverse Variation 4 Variability and Associations in Numerical Data 5 Variability and Associations in Categorical Data

Looking for Pythagoras The Pythagorean Theorem

1 Coordinate Grids 2 Squaring Off 3 The Pythagorean Theorem 4 Using the Pythagorean Theorem Understanding Real Numbers 5 Using the Pythagorean Theorem Analyzing Triangles and Circles

Growing, Growing, Growing Exponential Functions

1 Exponential Growth 2 Examining Growth Patterns Exponential Functions 3 Growth Factors and Growth Rates* 4 Exponential Decay Functions* 5 Patterns with Exponents

Contents

8Grade

10 Grade 6

Prime Time Factors and Multiples

1 Building on Factors and Multiples

2 Common Multiples and Common Factors

3 Factorizations: Searching for Factor Strings

4 Linking Multiplication and Addition

Comparing Bits and Pieces Ratios, Rational Numbers, and Equivalence

1 Making Comparisons

2 Connecting Ratios and Rates

3 Extending the Number Line

4 Comparing with Percents

Bits and Pieces II Understanding Fraction Operations

1 Extending Adding and Subtracting Fractions

2 Building on Multiplying With Fractions

3 Dividing With Fractions

4 Wrapping Up the Operations

Contents

6Grade


Incorporating the Standards for Mathematical Practice

CMP has always embodied the Standards for Mathematical Practice, and students continue to develop mathematical proficiency in CMP3 through engaging problem-based learning.

In CMP3 classrooms, students explore interesting mathematical situations and reflect on solution methods, examine why the methods work, compare methods, and relate methods to those used in previous situations. The Standards for Mathematical Practice are a natural part of each CMP3 lesson as students use them to solve problems and develop mathematical understandings.

1.2 1.31.1

Suppose a company called Custom Steel Products (CSP for short) supplies materials to builders. One common structure that CSP makes is called a truss, as shown in the figure below. (You might see a truss holding up the roof of a building.) It is made by fastening together steel rods 1 foot long.

1-foot steel rod 7-foot truss made from 27 rods

This truss has an overall length of 7 feet. The manager at CSP needs to know the number of rods in any length of truss a customer might order.

Custom Construction PartsFinding Patterns1.3

Study the drawing above to see if you can figure out what the manager needs to know. It might help to work out several cases and look for a pattern.

A Copy and complete the table below to show the number of rods in trusses of different overall lengths.

Number of Rods

Length of Truss (ft) 6 7

277 11

85432

■ ■ ■ ■

1. Make a graph of the data in your table.

2. Describe the pattern of change in the number of rods used as the truss length increases.

1.3Problem

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Thinking With Mathematical Models12

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Common Core State Standards

Mathematical Practices and Habits of Mind

Common Core State Standards

Mathematical Practices and Habits of Mind

In the Connected Mathematics curriculum you will develop an understanding of important mathematical ideas by solving problems and reflecting on the mathematics involved. Every day, you will use “habits of mind” to make sense of problems and apply what you learn to new situations. Some of these habits are described by the Common Core State Standards for Mathematical Practices (MP).

MP1 Make sense of problems and persevere in solving them.When using mathematics to solve a problem, it helps to think carefully about

• data and other facts you are given and what additional information you need to solve the problem;

• strategies you have used to solve similar problems and whether you could solve a related simpler problem first;

• how you could express the problem with equations, diagrams, or graphs;

• whether your answer makes sense.

MP2 Reason abstractly and quantitatively.When you are asked to solve a problem, it often helps to

• focus first on the key mathematical ideas;

• check that your answer makes sense in the problem setting;

• use what you know about the problem setting to guide your mathematical reasoning.

MP3 Construct viable arguments and critique the reasoning of others.When you are asked to explain why a conjecture is correct, you can

• show some examples that fit the claim and explain why they fit;

• show how a new result follows logically from known facts and principles.

When you believe a mathematical claim is incorrect, you can

• show one or more counterexamples—cases that don’t fit the claim;

• find steps in the argument that do not follow logically from prior claims.

5Common Core State Standards

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Make Sense of Problems and Persevere in Solving Them

Reason Abstractly and Quantitatively

To be effective, problems must not only embody critical concepts and skills but also must have the potential to engage students in making sense of problem situations and mathematics. CMP3 presents student-centered problems that require students to apply the math that they know to new problem situations. The suggested questions in the

Teacher Support provide the metacognitive scaffolding to help students monitor and refine their problem-solving strategies.

The problem situations in CMP3 are designed to support the development of students’ mathematical reasoning abilities. As students explore a set of connected problems within an Investigation, they look to understand the quantities in the problem and the relationship between these quantities. They translate a problem situation into an expression or equation and to then manipulate the equation to find a solution— or in other words, to “decontextualize.”


Construct Viable Arguments and Critique the Reasoning of Others

Model with Mathematics

In the CMP3 classroom, students routinely participate in mathematical discourse as they explain their thinking about a problem situation and their reasoning for a solution pathway. The problems that students encounter in the program offer opportunities to construct mathematical arguments and to critique others’ solutions and strategies.

The Teacher Support offers questions that support the development of a classroom culture that focuses on argument and critique as a part of solving mathematical problems.

Mathematical modeling begins in Grade 6 and continues to grow in sophistication throughout the program. CMP3 engages students in learning to construct and interpret concrete, symbolic, graphic, verbal, and algorithmic models of mathematical relationships in problem situations. As students develop fluency with these models, they apply them in different problem situations.

4.3 4.44.24.1

A In each diagram below, a large rectangle has been made from two smaller rectangles. In each case, show two different ways to calculate the area of the large rectangle.

1. 2.

3.

20 4

5

4.

2

3

8

B A large rectangle has an area of 28 square units. It has been divided into two smaller rectangles. One of the smaller rectangles has an area of 4 square units. What are possible whole-number dimensions of the large rectangle? Justify your reasoning.

4squareunits

4.2Problem

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In the CMP3 program, students work with a small set of tools as the primary vehicle for exploring problems. Once students gain familiarity with these tools, they make decisions about which tools are most appropriate for a given problem. For example, students use calculators to compute quantities, check their thinking, explore possibilities, and to use the graphing capability to examine the behavior of functions. They use manipulative tools such as polystrips to compare the rigidity of triangle and square forms and square tiles to explore relationships in area and perimeter.

CMP3 students solve problems that require them to estimate and to choose an appropriate measure or scale depending on the degree of accuracy needed.

CMP3 also supports students in developing precision in their use of mathematical language as they present solutions and defend their answers. The series of questions in a problem pushes students to more clearly articulate their solutions and the processes by which they have reached these solutions.

CMP3 helps students build mathematical understandings in ways that highlight and make use of mathematical structure. In Grade 6, for example, students look for patterns in the data presented in data tables, and they analyze numbers to determine their prime structure.

Use Appropriate Tools Strategically

Attend to Precision

Look for and Make Use of Structure


In Grade 7, students begin to see patterns in proportions from which they draw generalizations about strategies for solving proportions.

In Grade 8, students examine graphical representations of linear, exponential, and quadratic relationships so they can begin to see the structure of these relationships as functions.

As students investigate problems at each grade level, they look for connections to previous problems and previous solution strategies. Students discover opportunities to use strategies that they previously used to solve a problem in order to solve a new problem that may look on the surface to be very different. This kind of thinking and reasoning about solving problems promotes a view of mathematics as being connected in many different ways.

CMP3 provides all of the resources that you need to implement the CCSSM in your classroom. Goals and standards are described at the unit and Investigation level. Every instructional unit includes thorough guidance on the CCSSM and the background behind each lesson’s mathematics. CMP3 supports your understanding of the significance of each unit of instruction as it guides students and teachers to fully embrace the CCSSM.

Look for and Express Regularity in Repeated Reasoning

4.3 4.44.24.1

Note on Notation When you multiply a number by a letter variable, you can leave out the multiplication sign or parentheses. So, 3n means 3 * n or 3(n). This is also true for a product with more than one letter variable. So, ab means a * b or a(b).

The Distributive Property can be useful when solving problems. The Distributive Property states that if a, b, and c are any numbers, then

a(b + c) = a(b) + a(c)

• A number can be expressed as both a product and a sum.

• The area of a rectangle can be found in two different ways.

a

b c

The expression a(b + c) is in factored form. The expression a(b) + a(c) is in expanded form. The two expressions a(b + c) and a(b) + a(c) are equivalent expressions.

a(b + c) = a(b) + a(c) = ab + ac

factored form (product of two factors)

expanded form (sum of two terms)

In addition to using the Distributive Property, there needs to be an agreement as to which operation should be done first in an expression. In evaluating the expression 3 + 4 * 6, Mary thinks you get 42 and Hank thinks you get 27. Who is correct?

Ordering Operations 4.3

Investigation 4 Linking Multiplication and Addition: The Distributive Property 71

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