Mine the Gap for Mathematical Understanding, Grades 6-8
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Description
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.
Contents
Big Ideas and Tasks at a Glance
Big Ideas and Tasks at a Glance
Acknowledgments
Acknowledgments
About the Authors
About the Authors
Introduction
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?
Chapter 8. What Do We Do Next?
References and Additional Resources
References and Additional Resources
Additional materials
Description
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.
Contents
Big Ideas and Tasks at a Glance
Big Ideas and Tasks at a Glance
Acknowledgments
Acknowledgments
About the Authors
About the Authors
Introduction
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?
Chapter 8. What Do We Do Next?
References and Additional Resources
References and Additional Resources
Additional materials
Reviews
Mine the Gap for Mathematical Understanding, Grades 6-8
Common Holes and Misconceptions and What To Do About Them
September 2017 | 408 pages | Corwin
| Format | Published Date | ISBN | Price |
|---|---|---|---|
| Paperback | 01/02/2026 | 9781506379821 | $40.95 |
| Lifetime | 01/02/2026 | 9781506379838 | $37.00 |
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.
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