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Your Mathematics Standards Companion, Grades 6-8
What They Mean and How to Teach Them

- Ruth Harbin Miles - Mary Baldwin College, VA
- Lois A. Williams - Owner, C&I Mathematics Consulting

A Joint Publication of Corwin Mathematics and the National Council of Teachers of Mathematics, NCTM Stock Number 15463

Additional resources:

August 2017 | 336 pages | Corwin

*Transforming the standards into learning outcomes just got a lot easier*

In this resource, you can see in an instant how teaching to your state standards should look and sound in the classroom. Under the premise that math is math, the authors provide a Cross-Referencing Index for states implementing their own specific mathematics standards, allowing you to see and understand which page number to turn to for standards-based teaching ideas.

It’s all here, page by page:

- Get the inside scoop on which standards connect, what key vocabulary means, and time-saving tables showing where to focus instruction for each grade
- Write curriculum for: ratios and proportional relationships, the number system, expressions and equations, functions, geometry, and statistics & probability
- Use the What to Teach pages to deliver powerful standards-based lessons
- Learn effective techniques to create an environment where all students can experience math break-throughs
- Incorporate the Standards for Mathematical Practice to improve students’ ability to problem solve, construct viable arguments, use tools strategically, attend to precision, and more
- Cross-referenced index listing the standards in the following states, explaining what is unique to the standards of each state

*Your Mathematics Standards Companion* is your one-stop guide for teaching, planning, assessing, collaborating, and designing powerful mathematics curriculum.

Indexes Cross-Referencing Your State Standards

Quick Reference Guide

Acknowledgments

Letter to Grades 6–8 Teachers

Letter to Middle School Principals

Introduction

A Brief History of Current State Standards

College and Career Ready Standards for Mathematics

Instructional Shifts

Terminology Used in This Book

Effective Teaching Practices

How to Use This Book

Reflection Questions

Part 1. Ratios and Proportional Relationships

Domain Overview

Suggested Materials for This Domain

Key Vocabulary

Grade 6

Cluster A: Understand ratio concepts and use ratio reasoning to solve problems.

Sample Planning Page: Ratios and Proportional Relationships, Grade 6, Cluster A

Planning Page

Grade 7

Cluster A: Analyze proportional relationships and use them to solve real-world and mathematical problems.

Sample Planning Page: Ratios and Proportional Relationships, Grade 7, Cluster A

Planning Page

Reflection Questions: Ratios and Proportional Relationships

Part 2. The Number System

Domain Overview

Suggested Materials for This Domain

Key Vocabulary

Grade 6

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

Cluster B: Compute fluently with multi-digit numbers and find common factors and multiples.

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

Sample Planning Page: The Number System, Grade 6, Cluster C

Planning Pages

Grade 7

Cluster A: Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

Sample Planning Page: The Number System, Grade 7, Cluster A

Planning Page

Grade 8

Cluster A: Know that there are numbers that are not rational, and approximate them by rational numbers.

Sample Planning Page: The Number System, Grade 8, Cluster A

Planning Page

Reflection Questions: The Number System

Part 3. Expressions and Equations

Domain Overview

Suggested Materials for This Domain

Key Vocabulary

Grade 6

Cluster A: Apply and extend previous understandings of arithmetic to algebraic expressions.

Cluster B: Reason about and solve one-variable equations and inequalities.

Cluster C: Represent and analyze quantitative relationships between dependent and independent variables.

Sample Planning Page: Expressions and Equations, Grade 6, Cluster C

Planning Pages

Grade 7

Cluster A: Use properties of operations to generate equivalent expressions.

Cluster B: Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

Sample Planning Page: Expressions and Equations, Grade 7, Cluster A

Planning Pages

Grade 8

Cluster A: Work with radicals and integer exponents.

Cluster B: Understand the connections between proportional relationships, lines, and linear equations.

Cluster C: Analyze and solve linear equations and pairs of simultaneous linear equations.

Sample Planning Page: Expressions and Equations, Grade 8, Cluster A

Planning Pages

Reflection Questions: Expressions and Equations

Part 4. Functions

Domain Overview

Suggested Materials for This Domain

Key Vocabulary

Grade 8

Cluster A: Define, evaluate, and compare functions.

Cluster B: Use functions to model relationships between quantities.

Sample Planning Page: Functions, Grade 8, Cluster A

Planning Pages

Reflection Questions: Functions

Part 5. Geometry

Domain Overview

Suggested Materials for This Domain

Key Vocabulary

Grade 6

Cluster A: Solve real-world and mathematical problems involving area, surface area, and volume.

Sample Planning Page: Geometry, Grade 6, Cluster A

Planning Page

Grade 7

Cluster A: Draw, construct, and describe geometrical figures and describe the relationships between them.

Cluster B: Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.

Sample Planning Page: Geometry, Grade 7, Cluster B

Planning Pages

Grade 8

Cluster A: Understand congruence and similarity using physical models, transparencies, or geometry software.

Cluster B: Understand and apply the Pythagorean Theorem.

Cluster C: Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.

Sample Planning Page: Geometry, Grade 8, Cluster A

Planning Pages

Reflection Questions: Geometry

Part 6. Statistics and Probability

Domain Overview

Suggested Materials for This Domain

Key Vocabulary

Grade 6

Cluster A: Develop understanding of statistical variability.

Cluster B: Summarize and describe distributions.

Sample Planning Page: Statistics and Probability, Grade 6, Cluster B

Planning Pages

Grade 7

Cluster A: Use random sampling to draw inferences about a population.

Cluster B: Draw informal comparative inferences about two populations.

Cluster C: Investigate chance processes and develop, use, and evaluate probability models.

Sample Planning Page: Statistics and Probability, Grade 7, Cluster C

Planning Pages

Grade 8

Cluster A: Investigate patterns of association in bivariate data.

Sample Planning Page: Statistics and Probability, Grade 8, Cluster A

Planning Page

Reflection Questions: Statistics and Probability

Resources

Table 1. Standards for Mathematical Practice

Table 2. Effective Teaching Practices

Reproducibles

Reproducible 1. Percent Wheel

Reproducible 2. Frayer Model

Reproducible 3. Net of a Cube

Reproducible 4. Example for MAD (Mean Absolute Deviation)

Additional Resources

About the Authors