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Instructor’s Manual and Test Bank
N
For
U
Elementary and Middle School
Mathematics:
R
SE
Teaching Developmentally
Tenth Edition
D
John A. Van de Walle,
Karen S. Karp,
O
Jennifer M. Bay-Williams,
C
S
All Chapters Included
All Answers Included
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TABLE OF CONTENTS
Introductory Thoughts................................................................................................................................................... iv
Course Designs and Options .......................................................................................................................................... v
Introducing These Manual Notes .............................................................................................................................. xxiv
Share Your Thoughts ................................................................................................................................................. xxv
Chapter by Chapter
Chapter 1: Teaching Mathematics in the 21st Century .................................................................................................. 1
N
Chapter 2: Exploring What It Means to Know and Do Mathematics ............................................................................. 4
Chapter 3: Teaching Through Problem Solving ............................................................................................................. 9
Chapter 4: Planning in the Problem-Based Classroom................................................................................................. 12
U
Chapter 5: Creating Assessments for Learning ............................................................................................................ 15
Chapter 6: Teaching Mathematics Equitably to All Students....................................................................................... 18
Chapter 7: Developing Early Number Concepts and Number Sense ........................................................................... 21
R
Chapter 8: Developing Meanings for the Operation..................................................................................................... 23
Chapter 9 Developing Basic Fact Fluency ................................................................................................................... 26
Chapter 10: Developing Whole-Number Place-Value Concepts.................................................................................. 29
SE
Chapter 11: Developing Strategies for Addition and
Subtraction Computation ................................................................................................................. 32
Chapter 12: Developing Strategies for Multiplication and
Division Computation ..................................................................................................................... 37
Chapter 13: Algebraic Thinking, Equations, and Functions......................................................................................... 39
D
Chapter 14: Developing Fraction Concepts ................................................................................................................. 42
Chapter 15: Developing Fraction Operations ............................................................................................................... 45
Chapter 16: Developing Decimal and Percent Concepts and Decimal Computation ................................................... 48
O
Chapter 17: Ratios, Proportions, and Proportional Reasoning ..................................................................................... 53
Chapter 18: Developing Measurement Concepts ......................................................................................................... 57
Chapter 19: Geometric Thinking and Geometric Concepts.......................................................................................... 62
C
Chapter 20: Developing Concepts of Data and Statistics ............................................................................................. 66
Chapter 21: Exploring Concepts of Probability............................................................................................................ 72
Chapter 22: Developing Concepts of Exponents, Integers, and
S
Real Numbers .................................................................................................................................. 75
Test .............................................................................................................................................................................. 78
Answer Key............................................................................................................................................................... 159
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CHAPTER 1
TEACHING MATHEMATICS IN THE 21st CENTURY
MAIN IDEAS/LEARNER OUTCOMES
The main purpose of this chapter is to introduce students to the changing world and ongoing revolution in school
mathematics and their role in the continuation of this transformation. The four chapters that follow this one will
create the foundational ideas for effective teaching in a problem-solving environment. Those chapters are to be seen
in the context of the revolution that has been going on since 1989.
N
For more than twenty-five years, mathematics has constantly undergone change. There are several significant factors
in this transformation. Over the years, the pressures influencing school mathematics have become much more
complex. The TIMSS data has caused much of the concern with the popular press pointing out that most
U
industrialized countries significantly outperform U.S. students in mathematics and science. The National
Assessment of Educational Progress (NAEP) offers ongoing indications of what American students are learning.
These data suggests that while we continue to show improvement, we are not near where we want to be. Both
TIMSS and NAEP data are referred to throughout the book.
R
This chapter explains briefly how the movement toward shared standards began with the Curriculum and
Evaluation Standards for School Mathematics, followed by the Professional Standards for Teaching Mathematics
and the Assessment Standards for School Mathematics. Considerable attention is given to the content of Principles
SE
and Standards for School Mathematics, Curriculum Focal Points and the Common Core State Standards. It may be
useful to use a time line to show your students the progression. The new Association of Mathematics Teacher
Educators (AMTE) Standards for Preparing Teachers of Mathematics (AMTE, 2017) along with NCTM publication
Principles to Actions (Appendix B) provide a focus on mathematical teaching practices. Perhaps a poster or bulletin
board with these teaching practices would be a powerful reminder to refer to throughout the semester
Pressures on teachers from state testing programs, the requirements of NCLB and the 2010 Common Core State
Standards set of focused mathematics content standards and practices adopted by 43 of the 50 states (at the date of
D
this publication) have a significant influence on what is happening in mathematics classrooms. You should
encourage students to examine Appendix A of the main text, which contains the Standards for Mathematical
Practice from the Common Core State Standards and the overarching domains. Although it is difficult to make
general statements concerning these influences across states and provinces, teacher candidates should recognize the
O
importance of discussing their own state/provincial standards in light of the vision of NCTM.
You will not be able to adequately help your teacher candidates understand the standards documents in any one class
C
or by having them read any chapter in this book. The task at this early juncture in the course is to heighten students’
awareness of the standards and then to consistently refer to them as you travel through your course. It is important to
help teachers know that Common Core State Standards are not a curriculum but a statement of what is important in
school mathematics. Most likely your own state or province has standards with which prospective and practicing
S
teachers must be familiar. You may want to have students contrast the NCTM Standards and Curriculum Focal
Points with the Common Core State Standards or those at the state/province level where you are located.
Other Factors to Consider about School Mathematics
The power of the individual teacher cannot be ignored. The last section in Chapter 1, An Invitation to Learn and
Grow points out that the responsibility for excellent teaching and for significant learning by all students falls on the
teacher. We added a section on how to create a whole school agreement with a cohesive mathematics message. How
prospective and practicing teachers use this text and your course instruction to enhance their content knowledge
develops their ability to persevere and demonstrates their positive attitude and passion for teaching mathematics.
sdfghjkl
, wertyui
Their readiness for accepting change and their reflective disposition will play an important role in how their pre-K–8
students grow as mathematical thinkers.
SUGGESTED ACTIVITIES
Here are a few things that can be done to get students involved in the ideas just discussed.
Introduce Principles and Standards
Discuss the goals for students or read aloud the first paragraph of Chapter 1 of Principles and Actions (see Appendix
B) How is this similar or different from what they experienced as elementary or middle school students? Is this what
they are seeing in their field experiences? How possible do they believe this vision is for their classroom?
Point out the structure of the CCSS standards document. The standards define what students should understand and
be able to do. The Clusters summarize groups of related standards. The Domains are larger groups of related
N
standards (PPT Slide 1-5). The standards from different clusters and domains may sometimes be closely related,
because mathematics is a connected subject. It is critical that your methods students also read the six principles
listed on (PPT Slide 1-8 as well as the five process standards (PPT Slide 1-6). Although these are included in the
text, it would be much better if students read the full discussion of these in the actual document. With a student
U
membership to NCTM, your teacher candidates can access a Web version of the document or NCTM graciously
allows guests to sign up for 120 days of free access which should cover the time frame of most courses.
R
The Five Content Standards
Have students examine at least one of the five content standards or the domains from the Common Core State
Standards (downloadable at www.corestandards.org). You might want to spend some time discussing how these
standards are alike or different from the content in your state/province standards. Most importantly for pre service
SE
teachers, you want them to be aware of these documents to get an idea of what content is appropriate at each grade
level.
Another good discussion to have surrounds the issue of basic skills. What is meant when people say “basic skills” in
mathematics? Do the standards documents support students having basic skills? (Absolutely!) A discussion
concerning the pressures of high stakes testing and how this pressure influences what is taught in the classroom is
most useful at this early point in the semester. If there are sample-released test items in your state/province that are
available, compare those to the standards documents to see whether there are similarities and differences.
D
Introduce Professional Journals and Publications
Pass out back issues of Teaching Children Mathematics and/or Mathematics Teaching in the Middle School. Point
out some of the regular features of each journal that may be of interest. Encourage or require your students to
O
become e-members of NCTM. This membership allows full-time students to select an electronic subscription to one
of the journals—Teaching Children Mathematics, Mathematics Teaching in the Middle School or Mathematics
Teacher (for high school)—and enable them to download 25 journal articles from a non-subscribed journal. They
can also attend regional NCTM conferences for FREE!
Also show students the NCTM Essential Understandings series including the new Essential Understandings in
C
Practice, which provide details of how to teach the essential understandings effectively, Navigations books, and
other valuable NCTM publications that they should consider for their personal professional libraries. Several of the
publications, such as the Navigations books, each include a CD-ROM with other articles and several good applets
that teachers will find quite valuable.
S
POSSIBLE ASSIGNMENTS
1. Select one of the five process standards from Principles and Actions (see Appendix B) or one of the Standards for
Mathematical Practice from the Common Core State Standards (see Appendix A) and read it carefully. Discuss how
this standard reflects a vision of mathematics that is different from the one you experienced as a K-12 student.
Decide on two ideas that you think are most important for you as a future teacher to take from this standard. Create a
“to learn” list of the things you most need to learn in order to implement that standard as a teacher of mathematics.
Use this “to learn” list approach as a first step toward developing a personal professional growth plan.
sdfghjkl
Instructor’s Manual and Test Bank
N
For
U
Elementary and Middle School
Mathematics:
R
SE
Teaching Developmentally
Tenth Edition
D
John A. Van de Walle,
Karen S. Karp,
O
Jennifer M. Bay-Williams,
C
S
All Chapters Included
All Answers Included
, wertyui
TABLE OF CONTENTS
Introductory Thoughts................................................................................................................................................... iv
Course Designs and Options .......................................................................................................................................... v
Introducing These Manual Notes .............................................................................................................................. xxiv
Share Your Thoughts ................................................................................................................................................. xxv
Chapter by Chapter
Chapter 1: Teaching Mathematics in the 21st Century .................................................................................................. 1
N
Chapter 2: Exploring What It Means to Know and Do Mathematics ............................................................................. 4
Chapter 3: Teaching Through Problem Solving ............................................................................................................. 9
Chapter 4: Planning in the Problem-Based Classroom................................................................................................. 12
U
Chapter 5: Creating Assessments for Learning ............................................................................................................ 15
Chapter 6: Teaching Mathematics Equitably to All Students....................................................................................... 18
Chapter 7: Developing Early Number Concepts and Number Sense ........................................................................... 21
R
Chapter 8: Developing Meanings for the Operation..................................................................................................... 23
Chapter 9 Developing Basic Fact Fluency ................................................................................................................... 26
Chapter 10: Developing Whole-Number Place-Value Concepts.................................................................................. 29
SE
Chapter 11: Developing Strategies for Addition and
Subtraction Computation ................................................................................................................. 32
Chapter 12: Developing Strategies for Multiplication and
Division Computation ..................................................................................................................... 37
Chapter 13: Algebraic Thinking, Equations, and Functions......................................................................................... 39
D
Chapter 14: Developing Fraction Concepts ................................................................................................................. 42
Chapter 15: Developing Fraction Operations ............................................................................................................... 45
Chapter 16: Developing Decimal and Percent Concepts and Decimal Computation ................................................... 48
O
Chapter 17: Ratios, Proportions, and Proportional Reasoning ..................................................................................... 53
Chapter 18: Developing Measurement Concepts ......................................................................................................... 57
Chapter 19: Geometric Thinking and Geometric Concepts.......................................................................................... 62
C
Chapter 20: Developing Concepts of Data and Statistics ............................................................................................. 66
Chapter 21: Exploring Concepts of Probability............................................................................................................ 72
Chapter 22: Developing Concepts of Exponents, Integers, and
S
Real Numbers .................................................................................................................................. 75
Test .............................................................................................................................................................................. 78
Answer Key............................................................................................................................................................... 159
sdfghjkl
, wertyui
CHAPTER 1
TEACHING MATHEMATICS IN THE 21st CENTURY
MAIN IDEAS/LEARNER OUTCOMES
The main purpose of this chapter is to introduce students to the changing world and ongoing revolution in school
mathematics and their role in the continuation of this transformation. The four chapters that follow this one will
create the foundational ideas for effective teaching in a problem-solving environment. Those chapters are to be seen
in the context of the revolution that has been going on since 1989.
N
For more than twenty-five years, mathematics has constantly undergone change. There are several significant factors
in this transformation. Over the years, the pressures influencing school mathematics have become much more
complex. The TIMSS data has caused much of the concern with the popular press pointing out that most
U
industrialized countries significantly outperform U.S. students in mathematics and science. The National
Assessment of Educational Progress (NAEP) offers ongoing indications of what American students are learning.
These data suggests that while we continue to show improvement, we are not near where we want to be. Both
TIMSS and NAEP data are referred to throughout the book.
R
This chapter explains briefly how the movement toward shared standards began with the Curriculum and
Evaluation Standards for School Mathematics, followed by the Professional Standards for Teaching Mathematics
and the Assessment Standards for School Mathematics. Considerable attention is given to the content of Principles
SE
and Standards for School Mathematics, Curriculum Focal Points and the Common Core State Standards. It may be
useful to use a time line to show your students the progression. The new Association of Mathematics Teacher
Educators (AMTE) Standards for Preparing Teachers of Mathematics (AMTE, 2017) along with NCTM publication
Principles to Actions (Appendix B) provide a focus on mathematical teaching practices. Perhaps a poster or bulletin
board with these teaching practices would be a powerful reminder to refer to throughout the semester
Pressures on teachers from state testing programs, the requirements of NCLB and the 2010 Common Core State
Standards set of focused mathematics content standards and practices adopted by 43 of the 50 states (at the date of
D
this publication) have a significant influence on what is happening in mathematics classrooms. You should
encourage students to examine Appendix A of the main text, which contains the Standards for Mathematical
Practice from the Common Core State Standards and the overarching domains. Although it is difficult to make
general statements concerning these influences across states and provinces, teacher candidates should recognize the
O
importance of discussing their own state/provincial standards in light of the vision of NCTM.
You will not be able to adequately help your teacher candidates understand the standards documents in any one class
C
or by having them read any chapter in this book. The task at this early juncture in the course is to heighten students’
awareness of the standards and then to consistently refer to them as you travel through your course. It is important to
help teachers know that Common Core State Standards are not a curriculum but a statement of what is important in
school mathematics. Most likely your own state or province has standards with which prospective and practicing
S
teachers must be familiar. You may want to have students contrast the NCTM Standards and Curriculum Focal
Points with the Common Core State Standards or those at the state/province level where you are located.
Other Factors to Consider about School Mathematics
The power of the individual teacher cannot be ignored. The last section in Chapter 1, An Invitation to Learn and
Grow points out that the responsibility for excellent teaching and for significant learning by all students falls on the
teacher. We added a section on how to create a whole school agreement with a cohesive mathematics message. How
prospective and practicing teachers use this text and your course instruction to enhance their content knowledge
develops their ability to persevere and demonstrates their positive attitude and passion for teaching mathematics.
sdfghjkl
, wertyui
Their readiness for accepting change and their reflective disposition will play an important role in how their pre-K–8
students grow as mathematical thinkers.
SUGGESTED ACTIVITIES
Here are a few things that can be done to get students involved in the ideas just discussed.
Introduce Principles and Standards
Discuss the goals for students or read aloud the first paragraph of Chapter 1 of Principles and Actions (see Appendix
B) How is this similar or different from what they experienced as elementary or middle school students? Is this what
they are seeing in their field experiences? How possible do they believe this vision is for their classroom?
Point out the structure of the CCSS standards document. The standards define what students should understand and
be able to do. The Clusters summarize groups of related standards. The Domains are larger groups of related
N
standards (PPT Slide 1-5). The standards from different clusters and domains may sometimes be closely related,
because mathematics is a connected subject. It is critical that your methods students also read the six principles
listed on (PPT Slide 1-8 as well as the five process standards (PPT Slide 1-6). Although these are included in the
text, it would be much better if students read the full discussion of these in the actual document. With a student
U
membership to NCTM, your teacher candidates can access a Web version of the document or NCTM graciously
allows guests to sign up for 120 days of free access which should cover the time frame of most courses.
R
The Five Content Standards
Have students examine at least one of the five content standards or the domains from the Common Core State
Standards (downloadable at www.corestandards.org). You might want to spend some time discussing how these
standards are alike or different from the content in your state/province standards. Most importantly for pre service
SE
teachers, you want them to be aware of these documents to get an idea of what content is appropriate at each grade
level.
Another good discussion to have surrounds the issue of basic skills. What is meant when people say “basic skills” in
mathematics? Do the standards documents support students having basic skills? (Absolutely!) A discussion
concerning the pressures of high stakes testing and how this pressure influences what is taught in the classroom is
most useful at this early point in the semester. If there are sample-released test items in your state/province that are
available, compare those to the standards documents to see whether there are similarities and differences.
D
Introduce Professional Journals and Publications
Pass out back issues of Teaching Children Mathematics and/or Mathematics Teaching in the Middle School. Point
out some of the regular features of each journal that may be of interest. Encourage or require your students to
O
become e-members of NCTM. This membership allows full-time students to select an electronic subscription to one
of the journals—Teaching Children Mathematics, Mathematics Teaching in the Middle School or Mathematics
Teacher (for high school)—and enable them to download 25 journal articles from a non-subscribed journal. They
can also attend regional NCTM conferences for FREE!
Also show students the NCTM Essential Understandings series including the new Essential Understandings in
C
Practice, which provide details of how to teach the essential understandings effectively, Navigations books, and
other valuable NCTM publications that they should consider for their personal professional libraries. Several of the
publications, such as the Navigations books, each include a CD-ROM with other articles and several good applets
that teachers will find quite valuable.
S
POSSIBLE ASSIGNMENTS
1. Select one of the five process standards from Principles and Actions (see Appendix B) or one of the Standards for
Mathematical Practice from the Common Core State Standards (see Appendix A) and read it carefully. Discuss how
this standard reflects a vision of mathematics that is different from the one you experienced as a K-12 student.
Decide on two ideas that you think are most important for you as a future teacher to take from this standard. Create a
“to learn” list of the things you most need to learn in order to implement that standard as a teacher of mathematics.
Use this “to learn” list approach as a first step toward developing a personal professional growth plan.
sdfghjkl
