MANIPULATIVE MATHEMATICS FOR STUDENTS
, Manipulative Mathematics
Using Manipulatives to Promote Understanding of Elementary Algebra
Concepts
Lynn Marecek
MaryAnne Anthony-Smith
1
This file is copyright 2017, Rice University. All Rights Reserved.
, Table of Contents
To the Student Page 3
Manipulative Activities Page
Game of 24 4
The Number Line Part 1-Counting Numbers and Whole Numbers 8
The Number Line Part 2-Integers 10
The Number Line Part 3-Fractions 12
Multiplication/Factors 17
Square Numbers 21
Model Fractions 24
Fractions Equivalent to One 26
Mixed Numbers and Improper Fractions 28
Equivalent Fractions 33
Model Fraction Multiplication 37
Model Fraction Division 41
Model Fraction Addition 46
Model Fraction Subtraction 50
Model Finding the Least Common Denominator 53
Addition of Signed Numbers 59
Subtraction of Signed Numbers 62
Multiples 65
Prime Numbers 68
Subtraction Property of Equality 70
Division Property of Equality 77
Visualizing Area and Perimeter 84
Measuring Area and Perimeter 87
Coin Lab 96
Exploring Slopes of Lines 99
Slope of Line Between Two Points 104
2
This file is copyright 2017, Rice University. All Rights Reserved.
, To the Student:
Research has shown that our brains learn best when we start with concrete objects and then move on to
abstract ideas. Manipulatives are concrete objects used to model abstract mathematical concepts.
Although the word ‘manipulatives’ may be new to you, using manipulatives is probably something you
have already done. If you learned to add numbers by counting your fingers, you were using manipulatives.
And maybe you used two-color counters or fraction tiles in a previous math class.
You will see Illustrations of manipulatives in Elementary Algebra whenever a new concept is introduced.
Then you will be encouraged to do a Manipulative Mathematics activity to help you develop a solid
understanding of the concept.
This booklet contains the Manipulative Mathematics activities that accompany Elementary Algebra.
Each activity relates to a mathematical topic covered in Elementary Algebra. There are two parts to each
activity:
The student worksheet will lead you through a series of questions to help you understand a
mathematical property or procedure through the use of manipulatives.
The extra practice problems reinforce the results of the student worksheet. Most Extra Practice
worksheets include a link to a website with online (or ‘virtual’) manipulatives.
Ask your math instructor if your college has manipulatives for you to use. You can access the virtual
manipulatives online any time, 24/7. We have sets of manipulatives, such as color chips, fraction pieces,
algebra tiles, geoboards, and more, in our classrooms. Our college students use them to model and learn
about critical mathematical concepts and procedures. We have heard many “a-ha’s” from students who
finally understand, for example, how to work with fractions or signed numbers.
We sincerely hope you will use our Manipulative Mathematics activities to help you succeed in
Elementary Algebra. We wholeheartedly believe that through the use of manipulatives, you can develop
an understanding of mathematics that translates into success throughout your mathematics courses.
Lynn Marecek MaryAnne Anthony-Smith
3
This file is copyright 2017, Rice University. All Rights Reserved.
, Manipulative Mathematics
Using Manipulatives to Promote Understanding of Elementary Algebra
Concepts
Lynn Marecek
MaryAnne Anthony-Smith
1
This file is copyright 2017, Rice University. All Rights Reserved.
, Table of Contents
To the Student Page 3
Manipulative Activities Page
Game of 24 4
The Number Line Part 1-Counting Numbers and Whole Numbers 8
The Number Line Part 2-Integers 10
The Number Line Part 3-Fractions 12
Multiplication/Factors 17
Square Numbers 21
Model Fractions 24
Fractions Equivalent to One 26
Mixed Numbers and Improper Fractions 28
Equivalent Fractions 33
Model Fraction Multiplication 37
Model Fraction Division 41
Model Fraction Addition 46
Model Fraction Subtraction 50
Model Finding the Least Common Denominator 53
Addition of Signed Numbers 59
Subtraction of Signed Numbers 62
Multiples 65
Prime Numbers 68
Subtraction Property of Equality 70
Division Property of Equality 77
Visualizing Area and Perimeter 84
Measuring Area and Perimeter 87
Coin Lab 96
Exploring Slopes of Lines 99
Slope of Line Between Two Points 104
2
This file is copyright 2017, Rice University. All Rights Reserved.
, To the Student:
Research has shown that our brains learn best when we start with concrete objects and then move on to
abstract ideas. Manipulatives are concrete objects used to model abstract mathematical concepts.
Although the word ‘manipulatives’ may be new to you, using manipulatives is probably something you
have already done. If you learned to add numbers by counting your fingers, you were using manipulatives.
And maybe you used two-color counters or fraction tiles in a previous math class.
You will see Illustrations of manipulatives in Elementary Algebra whenever a new concept is introduced.
Then you will be encouraged to do a Manipulative Mathematics activity to help you develop a solid
understanding of the concept.
This booklet contains the Manipulative Mathematics activities that accompany Elementary Algebra.
Each activity relates to a mathematical topic covered in Elementary Algebra. There are two parts to each
activity:
The student worksheet will lead you through a series of questions to help you understand a
mathematical property or procedure through the use of manipulatives.
The extra practice problems reinforce the results of the student worksheet. Most Extra Practice
worksheets include a link to a website with online (or ‘virtual’) manipulatives.
Ask your math instructor if your college has manipulatives for you to use. You can access the virtual
manipulatives online any time, 24/7. We have sets of manipulatives, such as color chips, fraction pieces,
algebra tiles, geoboards, and more, in our classrooms. Our college students use them to model and learn
about critical mathematical concepts and procedures. We have heard many “a-ha’s” from students who
finally understand, for example, how to work with fractions or signed numbers.
We sincerely hope you will use our Manipulative Mathematics activities to help you succeed in
Elementary Algebra. We wholeheartedly believe that through the use of manipulatives, you can develop
an understanding of mathematics that translates into success throughout your mathematics courses.
Lynn Marecek MaryAnne Anthony-Smith
3
This file is copyright 2017, Rice University. All Rights Reserved.
