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Mine the Gap for Mathematical Understanding, Grades 6-8


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Table of Contents

Big Ideas and Tasks at a Glance Acknowledgments About the Authors Introduction 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

About the Author

John SanGiovanni is a mathematics supervisor in Howard County, Maryland. There he leads mathematics curriculum development, digital learning, assessment, and professional development for 41 elementary schools and more than 1,500 teachers. John is an adjunct professor and coordinator of the Elementary Mathematics Instructional Leader graduate program at McDaniel College. He is an author and national mathematics curriculum and professional learning consultant. John is a frequent speaker at national conferences and institutes. He is active in state and national professional organizations and currently serves on the Board of Directors for the National Council of Teachers of Mathematics. 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.


"Wow! This book is a treasure trove for middle school teachers and those who support middle level mathematics education. The tasks presented are top notch, the student work is illustrative of classrooms, and the suggestions for how to respond are incredible! Each of these three would make a great resource-to have them all blended together is outstanding." -- Kevin Dykema
"Mine the Gap for Mathematical Understanding provides the framework and guidance teachers need to move beyond finding correct answers to dig deeply into the mine of student thinking, to analyze misconceptions and gaps in understanding, and to develop and implement specific strategies to support every student in deep mathematics learning. Every teacher of mathematics needs this amazing resource to learn to mine the gaps for each of their students!" -- Becky M. Walker
"This work does what other books only attempt to do. It combines instruction, assessment, and practice with open-ended and rich tasks that allow for teachers to not only immediately implement the ideas but also understand the content and pedagogy behind them. The tasks, which are immediately implementable and customizable, engage each and every learner. They are based on cutting-edge and research-based instructional frameworks and provide countless learning opportunities for students." -- Zachary Champagne
"Mine the Gap is a great tool for teachers to use to grow their own understanding of student misconceptions and incomplete understandings and how to address them. This is an indispensable resource for all involved in supporting students' growth in mathematics." -- Nathan Rosin
"More than just a nice collection of problems, this book shares a road map for teachers looking to enhance the quality of the math tasks they use with students. Teachers will appreciate the examples of actual student work paired with tips for analysis and instruction." -- Delise Andrews
"This book helps navigate how to use student work to drive instruction with rich engaging tasks, which will help all students become better mathematicians. The authors have done an excellent job of helping teachers to carefully look at student work to identify how students solved math problems, using this evidence to identify those students who understand the targeted skill, along with the misconceptions or misunderstandings of other students, with suggestions of how to move all students forward in their thinking." -- Cynthia Baumann

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