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Mine the Gap for Mathematical Understanding, Grades 6-8

Common Holes and Misconceptions and What To Do About Them
Discover the “why” behind students’ math mistakes with this bank of 180 downloadable, standards-aligned math tasks, common misunderstandings, and next instructional steps.

Full description

Product Details
  • Grade Level: PreK-12
  • ISBN: 9781506379821
  • Published By: Corwin
  • Series: Corwin Mathematics Series
  • Year: 2017
  • Page Count: 408
  • Publication date: September 18, 2017

Price: $37.95

Price: $37.95
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Being an effective math educator is one part based on the quality of the tasks we give, one part how we diagnose what we see, and one part what we do with what we find. Yet with so many students and big concepts to cover, it can be hard to slow down enough to look for those moments when students’ responses tell us what we need to know about next best steps. In this remarkable book, John SanGiovanni and Jennifer Rose Novak help us value our students’ misconceptions and incomplete understandings as much as their correct ones—because it’s the gap in their understanding today that holds the secrets to planning tomorrow’s best teaching.

The authors lay out 180 high-quality tasks aligned to the standards and big ideas of Grades 6–8 mathematics, including number systems, integers, ratio and proportion, equations and expressions, geometry, and statistics and probability. The tasks are all downloadable so you can use or modify them for instruction and assessment. Each big idea offers a starting task followed by:

  • what makes it a high-quality task
  • what you might anticipate before students work with the task
  • four student examples of the completed task showcasing a distinct “gap”
  • commentary on what precisely counts for mathematical understanding and the next instructional steps
  • commentary on the misconception or incomplete understanding so you learn why the student veered off course
  • three additional tasks aligned to the mathematics topic and ideas about what students might do with these additional tasks

It’s time to break our habit of rushing into re-teaching for correctness and instead get curious about the space between right and wrong answers. Mine the Gap for Mathematical Understanding is a book you will return to again and again to get better at selecting tasks that will uncover students’ reasoning, better at discerning the quality and clarity of students’ understanding, and better at planning teaching based on the gaps you see.

Key features

· 180 Rich Tasks teachers can choose from and implement immediately;

· Organized around math’s “big ideas” for grades 6-8 the number systems, integers, ratio and proportion, equations and expressions, geometry, and statistics and probability;

· Written by a teacher’s teacher in a friendly, accessible style;

· 180 examples of student work used to walk teachers through analysis of misconceptions and incomplete understandings;

· Highly templated and designed for easy use;

· Additional rich tasks provided as blackline masters through a companion website;

· Guidance around adapting tasks for differentiation and grade levels and avoiding common teacher misunderstandings and misinterpretations,



John J. SanGiovanni photo

John J. SanGiovanni

John J. SanGiovanni is a mathematics coordinator in Howard County, Maryland. There, he leads mathematics curriculum development, digital learning, assessment, and professional development. John is an adjunct professor and coordinator of the Elementary Mathematics Instructional Leadership graduate program at McDaniel College. In addition to this Figuring Out Fluency series, some of his many Corwin books include Daily Routines to Jump-Start Problem Solving, Grades K-8, Answers to Your Biggest Questions about Teaching Elementary Math, the Daily Routines to Jump-Start Math series, and Productive Math Struggle: A 6-Point Action Plan for Fostering Perseverance. John is a national mathematics curriculum and professional learning consultant who also speaks frequently at national conferences and institutes. He is active in state and national professional organizations, recently serving on the board of directors for the National Council of Teachers of Mathematics (NCTM) and currently on the board of directors for the National Council of Supervisors of Mathematics (NCSM).

Jennifer Rose Novak photo

Jennifer Rose Novak

Jennifer Rose Novak is a mathematics education associate for the Delaware Department of Education. She provides leadership in the implementation of standards-based curriculum, instruction, and professional development in mathematics education to the 19 school districts and the various charter schools in Delaware. Jenny has served as a supervisor and coach in Howard County, MD as well as an adjunct professor at McDaniel College and UMBC. She serves on a number of organizational boards, including MATHCOUNTS, the Delaware Mathematics Coalition, and the Maryland Council of Teachers of Mathematics, for which she is currently serving as president.

Table of Contents

Table of Contents

Big Ideas and Tasks at a Glance


About the Authors


Chapter 1. Rich Mathematics Tasks, Student Misconceptions, Using Tasks

Prompts With Purpose: Using High-Quality Tasks

Using Quality Tasks

Reflecting on Chapter 1

Chapter 2. Number Systems

Big Ideas

1. Addition and Subtraction of Fractions

2. Multiplication and Division of Fractions

3. Reasoning About Addition and Subtraction of Fractions

4. Reasoning About Multiplication and Division of Fractions

5. Problem Solving With Fractions

6. Decimals as Numbers

7. Addition and Subtraction With Decimals

8. Multiplication and Division With Decimals

Chapter 3. Integers

Big Ideas

9. Representing Integers

10. Representing Integers on Number Lines

11. More Representing Integers on Number Lines

12. Comparing Integers

13. Addition With Integers

14. Subtraction With Integers

15. Multiplication With Integers

16. Division With Integers

Chapter 4. Ratio, Proportion, and Percent

Big Ideas

17. Representing Ratios

18. Equivalent Ratios

19. Unit Rates

20. Using Ratios to Solve Problems

21. Reasoning With Percents

22. Unit Rate as Slope

Chapter 5. Expressions, Equations, and Functions

Big Ideas

23. Writing Expressions

24. Evaluating Expressions

25. Equivalent Expressions

26. Writing Equations

27. Solving Equations

28. Inequalities

29. Function Tables

30. Reasoning About Graphing

31. Comparing Functions

32. Systems of Equations

Chapter 6. Geometry

Big Ideas

33. Area of Composite Figures

34. Nets and Three-Dimensional Figures

35. Surface Area and Volume

36. Volume of Cylinders and Cones

37. Angle Relationships

38. Transformations, Similarity, and Congruence

39. Distance and Pythagorean Theorem

Chapter 7. Statistics

Big Ideas

40. Univariate Categorical Data

41. Univariate Quantitative Data

42. Displays of Univariate Quantitative Data

43. Deviations From the Mean

44. Bivariate Categorical Data

45. Bivariate Quantitative Data

Chapter 8. What Do We Do Next?

References and Additional Resources



Price: $37.95
Volume Discounts applied in Shopping Cart

For Instructors

Request Review Copy

When you select 'request review copy', you will be redirected to Sage Publishing (our parent site) to process your request.