Eyes on Math

This full-color resource will help K–8 teachers find new ways to clarify concepts that students find difficult.

This extensively updated teaching resource provides over 100 engaging, full-color visuals with explanations of how they can be used to stimulate mathematics learning, to explain mathematical concepts, and to assess students’ mathematical understanding in grades K–8.

Readers are provided with a strong mathematical background, downloadable copies of the visuals they can use directly, and helpful questions to raise with their students. Expected answers for each question and follow-up extensions are also provided. This expanded second edition includes 15 new Notice and Wonder activities to get mathematical conversations started and 15 new Student Assessment activities with suggestions for visuals that students can create to help teachers assess comprehension.

This user-friendly book will help teachers find new ways to clarify concepts that students find difficult. It will also help teachers working with students with low reading ability, including English language learners and special education students.

Book Features:

• 130 visuals, including color artwork and graphics.
• Questions and tasks to use with students in leading the instructional conversation.
• Expected answers and explanations of why each question is important.
• Prompts for students to show their understanding of a concept by using visuals.
• Vital mathematical background and context.

Marian Small is an international professional developer, conference speaker, bestselling author, former dean of education at the University of New Brunswick, and longtime professor of mathematics education. Visit Marian’s website at www.onetwoinfinity.ca for in-person and online professional development. Amy Lin is the director of teaching and learning at Seneca College. Amy has worked as a teacher in elementary, secondary, and postsecondary levels and as a mathematics coach, a curriculum consultant, administrator, and ministry researcher.

"Images can be leveraged as a gateway to students' mathematical reasoning, offering teachers valuable insights into their thinking. Eyes on Math is a must-have resource that empowers educators to engage students in meaningful ways, cultivating a rich understanding of mathematical concepts through thoughtful questions and exercises."
—Graham Fletcher, math specialist

“This inspiring new edition of Eyes on Math is chock-full of stimulating visuals that will spark students' and teachers' curiosity. Perhaps most important, Marian Small helps readers plan how and when to introduce each visual by sharing open-ended prompts, carefully constructed assessment questions and possible student responses, and connections to the essential ideas in mathematics. Her respect for students and teachers and their mathematical thinking shines through on each page—a delight!”
—Tracy Johnston Zager, math coach and author

“The visual models in Eyes on Math allow students to see the interconnectedness of mathematical ideas, and the provocative images and stimulating questions spark rich classroom conversations. This is a resource that every teacher should have in their library. Kudos to Small and Lin for making an amazing book even better!”
—Patrick Vennebush, chief learning officer, The Math Learning Center

Preface  ix

Acknowledgments  xiii

1.  Background  1
Visualization in Math  1
Focusing on the Important Math  3
Building Mathematical Communication  4

2.  Grades K–2  7

Notice and Wonder
Strawberries  8
Egg Cartons  10
Bracelets  12
Daisies  14
Donuts  16

Explaining Concepts
Counting Up by 1s  18
Counting Back by 1s  20
Counting Up by 2s  22
Counting Back by 2s  24
Counting Up by 5s  26
Counting Back by 5s  28
Counting Up by 10s  30
Counting Back by 10s  32
Comparing Numbers by Matching  34
Benchmark Numbers: All About 5  36
Benchmark Numbers: All About 10  38
Ordinal Numbers  40
Addition as Combining  42
Adding 0 and Adding 1  44
Addition: Commutativity  46
Addition: Changing Addends, But Not the Sum  48
Adding or Subtracting 10  50
Subtraction as Taking Away  52
Subtraction to Compare  54
Relating Addition and Subtraction  56
Naming Two-Digit Numbers  58
Naming Three-Digit Numbers  60
Place Value: Grouping in Tens  62
Place Value: Positions  64
Comparing Sizes of Numbers  66
Fractions: Halves  68
Fractions: Quarters  70
Measurement: Meaning of Length  72
Measurement: Effect of Unit Size  74
Measurement: Standard Units of Length  76
2-D Shapes Versus 3-D Shapes  78
Comparing 2-D Shapes  80
Comparing 3-D Shapes  82
Composing Shapes  84
Shape Puzzles  86

Student Assessment: Draw or Model to Explain
Rearranging Addends  89
Subitizing 8  90
Ordinal Numbers: Points of View  90
Subtracting by Adding  91
Nonstandard Measurement  92

3.  Grades 3–5  95

Notice and Wonder
Quilts  96
Design  98
Juice  100
Tennis  102
Pastries  104

Explaining Concepts
Multiplication: Equal Groups  106
Multiplication: Commutativity  108
Multiplication: The Distributive Principle  110
Multiplication: 2-Digit by 2-Digit  112
Division as Equal Groups or Sharing  114
Division: Remainders  116
Rounding Numbers  118
Place Value: Multiplying and Dividing by Powers of 10  120
Place Value: Renaming Numbers  122
Factors: What They Are  124
Factors Come in Pairs  126
Fractions: Representing  128
Fractions: Equivalence  130
Fractions: Comparing  132
Fractions: Mixed Number/Improper Fraction Relationship  134
Fractions: Common Denominators  136
Adding Fractions  138
Multiplying Fractions  140
Fractions: Multiplying as Resizing  144
Fractions as Division  146
Decimals: Relating Hundredths to Tenths  148
Decimals: Equivalence  150
Decimals: Adding and Subtracting  152
Measurement: Time Intervals  154
Measurement: Area of Rectangles  156
Perimeter Versus Area  158
Measurement Conversions  160
Graphs With Scales  162
Coordinate Grids  164
Classification of Shapes  166
Parallel and Perpendicular Lines  168
Lines of Symmetry  170
Patterns Versus Nonpatterns  172
Algebraic Thinking: Growing Additively  174
Algebraic Thinking: Shrinking Additively  176
Algebraic Thinking: Growing Multiplicatively  178

Student Assessment: Draw or Model to Explain
Fractions of a Set  180
Fraction as Division  181
Relating Quotients  182
Comparing Products  183
Factors  184

4.  Grades 6–8  187

Notice and Wonder
Roller Coasters  188
A Giant Burger  190
Going to a Concert  192
Basketball  194
A Solar Eclipse  196

Explaining Concepts
Common Factors  198
Common Multiples  200
Square Roots  202
Fraction Division  204
Ratios: Multiple Ratios Describe Any Situation  206
Equivalent Ratios  208
Equivalent Rates  210
Solving Rate Problems  212
Describing Percent  214
Uses of Integers  216
The Zero Principle  218
Subtraction of Integers as a Directed Distance  220
Multiplication and Division of Integers  222
Area of a Parallelogram  224
Area of a Triangle  226
The Pythagorean Theorem  230
Pi    232
How Measurements Are and Are Not Related  234
Mean  236
Variability  238
Sampling  240
Probability: What It Means  242
Unpredictability  244
Rotations, Reflections, and Translations  246
Scale Drawings  248
Dilations  250
Angles With Parallel Lines  252
Equivalent Expressions  254
Equation as a Balance  256
Different Types of Equations  258
What Is Linear?  260
Role of the Slope in the Equation of a Line  262
Systems of Equations  264
Function Rules  266

Student Assessment: Draw or Model to Explain
Subtracting Negatives  268
Relating Percents  269
Square Root Estimates  271
Dividing Fractions  272
Common Factors  273

Appendix: Connections Between Topics and Common Core Standards  275

References  279

Index  281

About the Authors  285