#
The Common Core Mathematics Companion: The Standards Decoded, Grades 3-5
What They Say, What They Mean, How to Teach Them

- Linda M. Gojak - Mathematics Consultant, NCTM Past-President
- Ruth Harbin Miles - Mary Baldwin College, VA

Series Creator: Jim Burke

*Your user’s guide to the mathematics standards*

In the 12 short months since the ELA versions of *The Common Core Companions, Grades K-2 and 3-5*, burst on the scene, they’ve already assisted tens of thousands of teachers with the day-to-day “what you do.” Teachers’ one big criticism: what about mathematics? Luckily NCTM past-president Linda Gojak and mathematics coach Ruth Harbin Miles stepped up to the task. The result? That version of the mathematics standards you wish you had.

Page by page, *The Common Core Mathematics Companions *clearly lay out:

- The mathematics embedded in each standard for a deeper understanding of the content
- Examples of what effective teaching and learning look like in the classroom
- Connected standards within each domain so teachers can better appreciate how they relate
- Priorities within clusters so teachers know where to focus their time
- The three components of rigor: conceptual understanding, procedural skills, and applications
- Vocabulary and suggested materials for each grade-level band with explicit connections to the standards
- Common student misconceptions around key mathematical ideas with ways to address them

Don’t spend another minute poring over the mathematics standards. Gojak and Miles have already done the heavy-lifting for you. Focus instead on how to teach them, using *The Common Core Mathematics Companion *as your one-stop guide for teaching, planning, assessing, collaborating, and designing powerful mathematics curriculum.

Winner of the Presidential Award for Excellence in Science and Mathematics Teaching, and director of John Carroll University’s Center for Mathematics and Science Education, **Linda Gojak** has spent 28 years teaching elementary and middle school mathematics and 15 years working with K–8 teachers. She has served as the president of the National Council of Teachers of Mathematics (NCTM), the National Council of Supervisors of Mathematics (NCSM), and the Ohio Council of Teachers of Mathematics.

**Ruth Harbin Miles** is a mathematics coach, with special expertise in developing teachers’ content knowledge and strategies for engaging students to achieve high standards in mathematics. She has served on the Board of Directors for the National Council of Teachers of Mathematics (NCTM), the National Council of Supervisors of Mathematics (NCSM), and both the Virginia Council of Teachers of Mathematics and the Kansas Association of Teachers of Mathematics.

**Also From Corwin**:

- Gojak & Miles/Common Corwin Mathematics Companion, Grades 3-5
- Taberski/The Common Core Companion, K-2
- Hull/The Common Core Mathematics Standards

A Brief History of the Common Core |

The Common Core State Standards for Mathematics |

Instructional Shifts |

Major Work of Grades 3¬–5 |

Common Core Word Wall |

The Common Core Standards for Mathematical Practice |

Effective Teaching Practices |

How to Use This Book |

Reflection Questions |

Cluster A: Represent and solve problems involving multiplication and division. |

Cluster B: Understand properties of multiplication and the relationship between multiplication and division. |

Cluster C: Multiply and divide within 100. |

Cluster D: Solve problems involving the four operations, and identify and explain patterns in arithmetic. |

Sample Planning Page: Operations and Algebraic Thinking, Grade 3, Cluster A |

Planning Pages |

Cluster A: Use the four operations with whole numbers to solve problems. |

Cluster B: Gain familiarity with factors and multiples. |

Cluster C: Generate and analyze patterns. |

Sample Planning Page: Operations and Algebraic Thinking, Grade 4, Cluster B |

Planning Pages |

Cluster A: Write and interpret numerical expressions. |

Cluster B: Analyze patterns and relationships. |

Sample Planning Page: Operations and Algebraic Thinking, Grade 5, Cluster A |

Planning Pages |

Cluster A: Use place value understanding and properties of operations to perform multi-digit arithmetic. |

Sample Planning Page: Number and Operations in Base Ten, Grade 3, Cluster A |

Planning Page |

Cluster A: Generalize place value understanding for multi-digit whole numbers. |

Cluster B: Use place value understanding and properties of operations to perform multi-digit arithmetic. |

Sample Planning Page: Number and Operations in Base Ten, Grade 4, Cluster B |

Planning Pages |

Cluster A: Understand the place value system. |

Cluster B: Perform operations with multi-digit whole numbers and with decimals to hundredths. |

Sample Planning Page: Number and Operations in Base Ten, Grade 5, Cluster B |

Planning Pages |

Cluster A: Develop understanding of fractions as numbers. |

Sample Planning Page: Number and Operations—Fractions, Grade 3, Cluster A |

Planning Page |

Cluster A: Extend understanding of fraction equivalence and ordering. |

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

Cluster C: Understand decimal notation for fractions, and compare decimal fractions. |

Sample Planning Page: Number and Operations—Fractions, Grade 4, Cluster C |

Planning Pages |

Cluster A: Use equivalent fractions as a strategy to add and subtract fractions. |

Cluster B: Apply and extend previous understandings of multiplication and division to multiply and divide fractions. |

Sample Planning Page: Number and Operations—Fractions, Grade 5, Cluster A |

Planning Pages |

Cluster A: Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects. |

Cluster B: Represent and interpret data. |

Cluster C: Geometric measurement: Understand concepts of area and relate area to multiplication and to addition. |

Cluster D: Geometric measurement: Recognize perimeter as an attribute of plane figures and distinguish between linear and area measures. |

Sample Planning Page: Measurement and Data, Grade 3, Cluster A |

Planning Pages |

Cluster A: Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit. |

Cluster B: Represent and interpret data. |

Cluster C: Geometric measurement: Understand concepts of angle and measure angles. |

Sample Planning Page: Measurement and Data, Grade 4, Cluster A |

Planning Pages |

Cluster A: Convert like measurement units within a given measurement system. |

Cluster B: Represent and interpret data. |

Cluster C: Geometric measurement: Understand concepts of volume and relate volume to multiplication and to addition. |

Sample Planning Page: Measurement and Data, Grade 5, Cluster C |

Planning Pages |

Cluster A: Reason with shapes and their attributes. |

Sample Planning Page: Geometry, Grade 3, Cluster A |

Planning Page |

Cluster A: Draw and identify lines and angles, and classify shapes by properties of their lines and angles. |

Sample Planning Page: Geometry, Grade 4, Cluster A |

Planning Page |

Cluster A: Graph points on the coordinate plane to solve real world and mathematical problems. |

Cluster B: Classify two-dimensional figures into categories based on their properties. |

Sample Planning Page: Geometry, Grade 5, Cluster A |

Planning Pages |

Table 1: Addition and Subtraction Situations, Grades 3–5 |

Table 2: Multiplication and Division Situations, Grades 3–5 |

Table 3: Standards for Mathematical Practice |

Table 4: Effective Teaching Practices |

CCSS Where to Focus Grade 3 Mathematics |

CCSS Where to Focus Grade 4 Mathematics |

CCSS Where to Focus Grade 5 Mathematics |

## Free resources

### Multiplying & Dividing Fractions Lesson

Students apply and extend previous understanding of multiplication and division to multiply and divide fractions using this lesson from *The Common Core Mathematics Companion, 3-5*. (Elementary)

### Gojak and Harbin-Miles' Life's Too Short for Long Division Webinar

Learn from Linda Gojak and Ruth Harbin-Miles impactful models and applications that lead students to a better and deeper understanding of division.

*"This companion supports implementation of the Common Core Mathematics Standards with attention to the instructional shifts: focus, coherence, and rigor. The standard-by-standard examples provide an image of what the standards looks like in the classroom for both teachers and students. The attention to students’ conceptions helps teachers plan with student thinking in mind and contributes to the collaborative work of grade-level teams."*

**Portland State University, Portland, OR**

*"This practical, exemplary resource supports teachers in their understanding and learning of the Common Core Mathematics Standards. The book does an excellent job of unpacking what the standards mean and makes explicit connections both to the Standards for Mathematical Practice and common student misconceptions. It is an ideal choice for a book study at either the grade or school level."*

**University of Nevada, Las Vegas**

*"This book is a ‘must-go-to’ for classroom teachers as well as coaches and leaders who support teachers in implementation of the Common Core Mathematics Standards. Connecting both the content standards and the practice standards with the effective teaching practices (*Principles to Actions*, NCTM, 2014), Linda Gojak and Ruth Harbin Miles provide the practical resources to enhance teachers’ understanding of the mathematics in the standards in ways that will bring them alive for all students. This is a book that will be the cornerstone of our professional development."*

**Denver Public Schools Denver, CO**

*"*The Common Core Companion* series for ELA became a runaway bestseller because it made instructional planning to address the standards manageable—and meaningful. Jim Burke and his co-authors put best practices first, equipping teachers with the 'what-it-looks-like' teaching ideas for each and every standard. Teachers have been clamoring for math version for more than a year, and now it's here! Linda Gojak and Ruth Harbin Miles do an amazing job demystifying the standards for grades 3-5. The authors have created a format and design that will have teachers dog-earing this book as they plan for instruction!" *

*"I would use this book as a starting place to plan lessons because it gives a teacher a good idea of activities and some examples for each standard. It can also be used as a toolkit for understanding how standards connect across clusters and grade levels."*

**Eastern Kentucky University**

**NCTM**