Eyes on Math
A Visual Approach to Teaching Math Concepts
2nd Edition
Published on 31. January 2025
Book
Paperback/Softback
304 pages
978-0-8077-8716-8 (ISBN)
Description
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 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 5 new Notice and Wonder activities to get mathematical conversations started and 5 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:
3 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.
This extensively updated teaching resource provides over 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 5 new Notice and Wonder activities to get mathematical conversations started and 5 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:
3 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.
More details
Edition
2nd New edition
Language
English
Place of publication
New York
United States
Target group
Professional and scholarly
Edition type
New edition
Product notice
Paperback (trade)
Unsewn / adhesive bound
Dimensions
Height: 251 mm
Width: 173 mm
Thickness: 20 mm
Weight
617 gr
ISBN-13
978-0-8077-8716-8 (9780807787168)
Copyright in bibliographic data and cover images is held by Nielsen Book Services Limited or by the publishers or by their respective licensors: all rights reserved.
Schweitzer Classification
Other editions
Additional editions
Book
01/2025
2nd Edition
Teachers' College Press
€112.50
Shipment within 10-20 days
Persons
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.
Content
Preface ?ix
Acknowledgments ?xiii
. ?Background ?
Visualization in Math ?
Focusing on the Important Math ?3
Building Mathematical Communication ?4
2. ?Grades K-2 ?7
Notice and Wonder
Strawberries ?8
Egg Cartons ?
Bracelets ? 2
Daisies ? 4
Donuts ?
Explaining Concepts
Counting Up by s ? 8
Counting Back by s ?2
Counting Up by 2s ?22
Counting Back by 2s ?24
Counting Up by 5s ?2
Counting Back by 5s ?28
Counting Up by s ?3
Counting Back by s ?32
Comparing Numbers by Matching ?34
Benchmark Numbers: All About 5 ?3
Benchmark Numbers: All About ?38
Ordinal Numbers ?4
Addition as Combining ?42
Adding and Adding ?44
Addition: Commutativity ?4
Addition: Changing Addends, But Not the Sum ?48
Adding or Subtracting ?5
Subtraction as Taking Away ?52
Subtraction to Compare ?54
Relating Addition and Subtraction ?5
Naming Two-Digit Numbers ?58
Naming Three-Digit Numbers ?
Place Value: Grouping in Tens ? 2
Place Value: Positions ? 4
Comparing Sizes of Numbers ?
Fractions: Halves ? 8
Fractions: Quarters ?7
Measurement: Meaning of Length ?72
Measurement: Effect of Unit Size ?74
Measurement: Standard Units of Length ?7
2-D Shapes Versus 3-D Shapes ?78
Comparing 2-D Shapes ?8
Comparing 3-D Shapes ?82
Composing Shapes ?84
Shape Puzzles ?8
Student Assessment: Draw or Model to Explain
Rearranging Addends ?89
Subitizing 8 ?9
Ordinal Numbers: Points of View ?9
Subtracting by Adding ?9
Nonstandard Measurement ?92
3. ?Grades 3-5 ?95
Notice and Wonder
Quilts ?9
Design ?98
Juice ?
Tennis ? 2
Pastries ? 4
Explaining Concepts
Multiplication: Equal Groups ?
Multiplication: Commutativity ? 8
Multiplication: The Distributive Principle ?
Multiplication: 2-Digit by 2-Digit ? 2
Division as Equal Groups or Sharing ? 4
Division: Remainders ?
Rounding Numbers ? 8
Place Value: Multiplying and Dividing by Powers of ? 2
Place Value: Renaming Numbers ? 22
Factors: What They Are ? 24
Factors Come in Pairs ? 2
Fractions: Representing ? 28
Fractions: Equivalence ? 3
Fractions: Comparing ? 32
Fractions: Mixed Number/Improper Fraction Relationship ? 34
Fractions: Common Denominators ? 3
Adding Fractions ? 38
Multiplying Fractions ? 4
Fractions: Multiplying as Resizing ? 44
Fractions as Division ? 4
Decimals: Relating Hundredths to Tenths ? 48
Decimals: Equivalence ? 5
Decimals: Adding and Subtracting ? 52
Measurement: Time Intervals ? 54
Measurement: Area of Rectangles ? 5
Perimeter Versus Area ? 58
Measurement Conversions ?
Graphs With Scales ? 2
Coordinate Grids ? 4
Classification of Shapes ?
Parallel and Perpendicular Lines ? 8
Lines of Symmetry ? 7
Patterns Versus Nonpatterns ? 72
Algebraic Thinking: Growing Additively ? 74
Algebraic Thinking: Shrinking Additively ? 7
Algebraic Thinking: Growing Multiplicatively ? 78
Student Assessment: Draw or Model to Explain
Fractions of a Set ? 8
Fraction as Division ? 8
Relating Quotients ? 82
Comparing Products ? 83
Factors ? 84
4. ?Grades -8 ? 87
Notice and Wonder
Roller Coasters ? 88
A Giant Burger ? 9
Going to a Concert ? 92
Basketball ? 94
A Solar Eclipse ? 9
Explaining Concepts
Common Factors ? 98
Common Multiples ?2
Square Roots ?2 2
Fraction Division ?2 4
Ratios: Multiple Ratios Describe Any Situation ?2
Equivalent Ratios ?2 8
Equivalent Rates ?2
Solving Rate Problems ?2 2
Describing Percent ?2 4
Uses of Integers ?2
The Zero Principle ?2 8
Subtraction of Integers as a Directed Distance ?22
Multiplication and Division of Integers ?222
Area of a Parallelogram ?224
Area of a Triangle ?22
The Pythagorean Theorem ?23
Pi ? ?232
How Measurements Are and Are Not Related ?234
Mean ?23
Variability ?238
Sampling ?24
Probability: What It Means ?242
Unpredictability ?244
Rotations, Reflections, and Translations ?24
Scale Drawings ?248
Dilations ?25
Angles With Parallel Lines ?252
Equivalent Expressions ?254
Equation as a Balance ?25
Different Types of Equations ?258
What Is Linear? ?2
Role of the Slope in the Equation of a Line ?2 2
Systems of Equations ?2 4
Function Rules ?2
Student Assessment: Draw or Model to Explain
Subtracting Negatives ?2 8
Relating Percents ?2 9
Square Root Estimates ?27
Dividing Fractions ?272
Common Factors ?273
Appendix: Connections Between Topics and Common Core Standards ?275
References ?279
Index ?28
About the Authors ?285
Acknowledgments ?xiii
. ?Background ?
Visualization in Math ?
Focusing on the Important Math ?3
Building Mathematical Communication ?4
2. ?Grades K-2 ?7
Notice and Wonder
Strawberries ?8
Egg Cartons ?
Bracelets ? 2
Daisies ? 4
Donuts ?
Explaining Concepts
Counting Up by s ? 8
Counting Back by s ?2
Counting Up by 2s ?22
Counting Back by 2s ?24
Counting Up by 5s ?2
Counting Back by 5s ?28
Counting Up by s ?3
Counting Back by s ?32
Comparing Numbers by Matching ?34
Benchmark Numbers: All About 5 ?3
Benchmark Numbers: All About ?38
Ordinal Numbers ?4
Addition as Combining ?42
Adding and Adding ?44
Addition: Commutativity ?4
Addition: Changing Addends, But Not the Sum ?48
Adding or Subtracting ?5
Subtraction as Taking Away ?52
Subtraction to Compare ?54
Relating Addition and Subtraction ?5
Naming Two-Digit Numbers ?58
Naming Three-Digit Numbers ?
Place Value: Grouping in Tens ? 2
Place Value: Positions ? 4
Comparing Sizes of Numbers ?
Fractions: Halves ? 8
Fractions: Quarters ?7
Measurement: Meaning of Length ?72
Measurement: Effect of Unit Size ?74
Measurement: Standard Units of Length ?7
2-D Shapes Versus 3-D Shapes ?78
Comparing 2-D Shapes ?8
Comparing 3-D Shapes ?82
Composing Shapes ?84
Shape Puzzles ?8
Student Assessment: Draw or Model to Explain
Rearranging Addends ?89
Subitizing 8 ?9
Ordinal Numbers: Points of View ?9
Subtracting by Adding ?9
Nonstandard Measurement ?92
3. ?Grades 3-5 ?95
Notice and Wonder
Quilts ?9
Design ?98
Juice ?
Tennis ? 2
Pastries ? 4
Explaining Concepts
Multiplication: Equal Groups ?
Multiplication: Commutativity ? 8
Multiplication: The Distributive Principle ?
Multiplication: 2-Digit by 2-Digit ? 2
Division as Equal Groups or Sharing ? 4
Division: Remainders ?
Rounding Numbers ? 8
Place Value: Multiplying and Dividing by Powers of ? 2
Place Value: Renaming Numbers ? 22
Factors: What They Are ? 24
Factors Come in Pairs ? 2
Fractions: Representing ? 28
Fractions: Equivalence ? 3
Fractions: Comparing ? 32
Fractions: Mixed Number/Improper Fraction Relationship ? 34
Fractions: Common Denominators ? 3
Adding Fractions ? 38
Multiplying Fractions ? 4
Fractions: Multiplying as Resizing ? 44
Fractions as Division ? 4
Decimals: Relating Hundredths to Tenths ? 48
Decimals: Equivalence ? 5
Decimals: Adding and Subtracting ? 52
Measurement: Time Intervals ? 54
Measurement: Area of Rectangles ? 5
Perimeter Versus Area ? 58
Measurement Conversions ?
Graphs With Scales ? 2
Coordinate Grids ? 4
Classification of Shapes ?
Parallel and Perpendicular Lines ? 8
Lines of Symmetry ? 7
Patterns Versus Nonpatterns ? 72
Algebraic Thinking: Growing Additively ? 74
Algebraic Thinking: Shrinking Additively ? 7
Algebraic Thinking: Growing Multiplicatively ? 78
Student Assessment: Draw or Model to Explain
Fractions of a Set ? 8
Fraction as Division ? 8
Relating Quotients ? 82
Comparing Products ? 83
Factors ? 84
4. ?Grades -8 ? 87
Notice and Wonder
Roller Coasters ? 88
A Giant Burger ? 9
Going to a Concert ? 92
Basketball ? 94
A Solar Eclipse ? 9
Explaining Concepts
Common Factors ? 98
Common Multiples ?2
Square Roots ?2 2
Fraction Division ?2 4
Ratios: Multiple Ratios Describe Any Situation ?2
Equivalent Ratios ?2 8
Equivalent Rates ?2
Solving Rate Problems ?2 2
Describing Percent ?2 4
Uses of Integers ?2
The Zero Principle ?2 8
Subtraction of Integers as a Directed Distance ?22
Multiplication and Division of Integers ?222
Area of a Parallelogram ?224
Area of a Triangle ?22
The Pythagorean Theorem ?23
Pi ? ?232
How Measurements Are and Are Not Related ?234
Mean ?23
Variability ?238
Sampling ?24
Probability: What It Means ?242
Unpredictability ?244
Rotations, Reflections, and Translations ?24
Scale Drawings ?248
Dilations ?25
Angles With Parallel Lines ?252
Equivalent Expressions ?254
Equation as a Balance ?25
Different Types of Equations ?258
What Is Linear? ?2
Role of the Slope in the Equation of a Line ?2 2
Systems of Equations ?2 4
Function Rules ?2
Student Assessment: Draw or Model to Explain
Subtracting Negatives ?2 8
Relating Percents ?2 9
Square Root Estimates ?27
Dividing Fractions ?272
Common Factors ?273
Appendix: Connections Between Topics and Common Core Standards ?275
References ?279
Index ?28
About the Authors ?285