Table of Contents

Big Ideas & Tasks at a Glance
Acknowledgments
Publisher’s Acknowledgments
About the Author
Introduction
CHAPTER 1: RICH MATHEMATICS TASKS, STUDENT MISCONCEPTIONS, USING TASKS
CHAPTER 2: ADDITION AND SUBTRACTION WITHIN 1,000
BIG IDEAS
1: Adding Within 1,000
2: Reasoning About Addition Within 1,000
3: Subtraction Within 1,000
4: Reasoning About Subtraction Within 1,000
5: Problem Solving With Addition and Subtraction
CHAPTER 3: MULTIPLICATION AND DIVISION
BIG IDEAS
6: Representing Multiplication
7: Reasoning About Multiplication
8: Properties of Multiplication
9: Representing Division
10: Reasoning About Division
11: Problem Solving With Multiplication and Division
12: Connecting Multiplication and Division
13: Representing Multi-Digit Multiplication
14: Reasoning About Multi-Digit Multiplication
15: Representing Multi-Digit Division
16: Reasoning About Multi-Digit Division
CHAPTER 4: FOUNDATIONAL FRACTION CONCEPTS
BIG IDEAS
17: Representing Fractions
18: Connecting Representations of Fractions
19: Fractions on a Number Line
20: Fractions Greater Than 1 on a Number Line
21: Decomposing Fractions
22: Equivalent Fractions on a Number Line
23: Comparing Fractions
24: Reasoning About Fractions
25: More Reasoning About Fractions
CHAPTER 5: DECIMAL CONCEPTS
BIG IDEAS
26: Representing Decimals
27: Representing Decimals as Numbers
28: Estimating and Rounding Decimals
29: Decomposing Decimals
30: Comparing Decimals
31: Addition With Decimals
32: Subtracting With Decimals
33: Problem Solving With Decimals
CHAPTER 6: ADDITION AND SUBTRACTION WITH FRACTIONS
BIG IDEAS
34: Addition With Fractions on Number Lines
35: Reasoning About Addition With Fractions
36: Subtraction With Fractions
37: Reasoning About Subtraction With Fractions
38: Problem Solving With Addition and Subtraction of Fractions
39: Addition and Subtraction With Mixed Numbers
CHAPTER 7: MULTIPLICATION AND DIVISION WITH FRACTIONS AND DECIMALS
BIG IDEAS
40: Multiplication of Fractions
41: More With Multiplication of Fractions
42: Division With Fractions
43: Problem Solving With Multiplication and Division of Fractions
44: Multiplication With Decimals
45: Division With Decimals
CHAPTER 8: WHAT DO WE DO NEXT?
References and Additional Resources

About the Author

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).

Reviews

"Wow! Mine the Gap for grades 3–5 teachers really unpacks! Not only does the book provide dozens
of great mathematical tasks, but—perhaps more importantly—it gives teachers the chance
to consider each task; anticipate student responses—a critical first step in formative assessment;
analyze actual student responses; and then consider what they would do in the classroom.
Thoughts about modifying each task provide another option for each task. At a time when teachers
are encouraged to regularly provide students with tasks that promote reasoning and problem
solving, this book will help teachers dig deeper as they mine for understandings."
*Francis (Skip) Fennell*

"Too often students (and parents and teachers) have the notion that the goal in math should
be to get the right answer. Yet insight into misconceptions and students’ thinking can tell us
much more about what students know (and don’t know) beyond simply a correct or incorrect
answer. SanGiovanni offers teachers a treasure trove of rich tasks and student work on those
tasks. Examples of how to analyze student thinking and next instructional steps make this a volume
that should be on every 3–5 math teacher’s desk!"
*Linda Gojak*

"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 for Mathematical Understanding is a much-needed and anticipated resource for
teachers, mathematics coaches, mathematics specialists, administrators, and other stakeholders.
The easy to follow, teacher-friendly format, the accompanying commentary for each student work
sample, along with the thoughtful reflection questions will quickly make this resource a ‘go to’
professional development tool."
*Latrenda Knighten*

"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*

"John SanGiovanni continues to provide teacher-friendly, must-have books. They empower teachers
by deepening their understanding of content and teaching."
*Megan Dooley*

"This book helps navigate how to use student work to drive instruction with rich engaging tasks,
which will help all students become better mathematicians. SanGiovanni has 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*