Minilessons for math_practice_g3-5

Rusty Bresser and Caren Holtzman
Grades
3–5
Minilessons
for Math Practice
Minilessons
for
Math
Practice
3–5
inilessons for Math Practice, Grades 3–5 presents an
innovative approach to reinforcing upper-elementary
students’ math skills. Through twenty-seven quick, engaging activities, students
practice math concepts, skills, and processes by applying them in a variety of
problem-solving contexts throughout the school day.
Designed for use during transition times, minilessons require little or no
preparation and take only five to fifteen minutes to teach. These activities can
be repeated throughout the school year and used with any existing math
program to help students meet local, state, and national math standards.
The minilessons offer experiences in key content areas including number and
operations, algebra, geometry, data analysis and probability, and measurement.
Each activity includes an overview with an explanation of the mathematics
involved, a list of key questions to ask students, a brief vignette of how the
activity unfolded in a classroom, and ideas for extending the activity throughout
the year.
By broadening the notion of what it means to provide students with practice
in mathematics, Minilessons for Math Practice offers a valuable teaching
resource for enhancing students’ mathematical understanding and bolstering
their achievement.
Rusty Bresser, an elementary classroom teacher for more than twenty years,
is currently a Lecturer and a Supervisor of Teacher Education at the University
of California at San Diego. He is the author of Math and Literature, Grades 4–6,
Second Edition (Math Solutions Publications, 2004), a coauthor of Developing
Number Sense, Grades 3–6 (Math Solutions Publications, 1999), and a Math
Solutions Professional Development instructor.
Caren Holtzman, an elementary classroom teacher for ten years, is currently a
Lecturer and a Supervisor of Teacher Education at the University of California at
San Diego. She is the author of several children’s books as well as a coauthor
of Developing Number Sense, Grades 3–6
(Math Solutions Publications, 1999).
M
Grades
3–5
Bresser
and
Holtzman
Math
Solutions
P
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b
l
i
c
a
t
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Minilessons for
Math Practice
GRADES 3–5
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Minilessons for
Math Practice
GRADES 3–5
Rusty Bresser
Caren Holtzman
Math Solutions Publications
Sausalito, CA
䊝䊞䊟䊞
䊝䊞䊟䊞
•
•
•
•
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Math Solutions Publications
A division of
Marilyn Burns Education Associates
150 Gate 5 Road, Suite 101
Sausalito, CA 94965
www.mathsolutions.com
Copyright © 2006 by Math Solutions Publications
All rights reserved. Limited reproduction permission: The publisher grants permission to
individual teachers who have purchased this book to reproduce the Blackline Masters as
needed for use with their own students. Reproduction for an entire school district, or
commercial or any other use, in any form or by any means, is prohibited without written
permission from the publisher, except for the inclusion of brief quotations in a review.
Library of Congress Cataloging-in-Publication Data
Bresser, Rusty.
Minilessons for math practice : grades 3–5 / Rusty Bresser, Caren Holtzman.
p. cm.
Includes bibliographical references and index.
ISBN-13: 978-0-941355-75-9 (alk. paper)
ISBN-10: 0-941355-75-6
1. Mathematics—Problems, exercises, etc.—Juvenile literature. 2. Mathematics—Study
and teaching (Elementary)—Juvenile literature. 3. Problem solving—Study and teaching
(Elementary)—Juvenile literature. I. Holtzman, Caren, II. Title.
QA43.B729 2006
372.7—dc22
2006017050
Editor: Toby Gordon
Production: Melissa L. Inglis
Cover design: Jan Streitburger
Interior design: Catherine Hawkes/Cat & Mouse
Composition: Interactive Composition Corporation
Printed in the United States of America on acid-free paper
10 09 08 07 06 ML 1 2 3 4 5
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A Message from Marilyn Burns
We at Math Solutions Professional Development believe that
teaching math well calls for increasing our understanding of the
math we teach, seeking deeper insights into how children learn
mathematics, and refining our lessons to best promote students’
learning.
Math Solutions Publications shares classroom-tested lessons and
teaching expertise from our faculty of Math Solutions Inservice
instructors as well as from other respected math educators. Our
publications are part of the nationwide effort we’ve made since
1984 that now includes
• more than five hundred face-to-face inservice programs each
year for teachers and administrators in districts across the
country;
• annually publishing professional development books, now
totaling more than fifty titles and spanning the teaching of all
math topics in kindergarten through grade 8;
• four series of videotapes for teachers, plus a videotape for par-
ents, that show math lessons taught in actual classrooms;
• on-site visits to schools to help refine teaching strategies and
assess student learning; and
• free online support, including grade-level lessons, book
reviews, inservice information, and district feedback, all in our
quarterly Math Solutions Online Newsletter.
For information about all of the products and services we have
available, please visit our Web site at www.mathsolutions.com. You
can also contact us to discuss math professional development
needs by calling (800) 868-9092 or by sending an e-mail to
info@mathsolutions.com.
We’re always eager for your feedback and interested in learning
about your particular needs. We look forward to hearing from you.
®
MathSolutions
P U B L I C AT I O N S
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Acknowledgments ix
Introduction xi
Contents Chart xv
1. Number Talks 1
2. Ballpark Estimation 7
3. Clear the Board 13
4. Coin Riddle 18
5. Comparing Fractions 24
6. Coordinate Tic-Tac-Toe 30
7. Digit Place 35
8. Dot Clusters 41
9. Estimate and Measure 48
10. Estimation Jar 53
11. Function Machine 57
12. Guess My Number 64
13. Guess My Rule 69
14. How Far Away? 75
15. Looking at Data 81
16. Number Strings 87
17. Odd Number Wins 92
18. Over or Under 97
19. Personal Numbers 102
20. Race for Twenty 107
21. Target 100/Target 0 111
Contents
vii
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viii Minilessons for Math Practice
22. Tell Me All You Can 116
23. Twenty Questions 120
24. What Comes Next? 125
25. What Page Are We On? 130
26. Whole-Class Pig 135
27. Your Choice Tic-Tac-Toe 141
Blackline Masters 145
Coordinate Tic-Tac-Toe Grid 147
Dot Clusters Overhead Transparency 148
Dot Clusters Recording Sheet 149
Index 151
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We thank the following people for making this book possible: Marilyn
Burns, for her guidance and support; Toby Gordon and Maryann Wickett,
for their editorial expertise; and Melissa L. Inglis, for her work on the
book’s production.
Our sincerest appreciation to the educators and administrators who
opened their classrooms to us: from Florence Elementary School, San
Diego, CA: Principal Mary Estill, Olivia Gonzalez, Robin Gordon, Amy
Jackson, and Lisa Kimbrell; from Freese Elementary School, San Diego,
CA: Principal Midge Backensto, Melissa Mechtly, Danielle Pickett, and
Patty Starks; from Jackson Elementary School, San Diego, CA: Principal
Rupi Boyd and Tina Rasori.
Acknowledgments
ix
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Teachers have always known that their students require opportunities to
practice the things they have learned. Opportunities for practice are par-
ticularly important in mathematics. When children learn a new skill or
concept in math without having had the time to put that new idea into
practice, they tend to forget what they’ve learned.
Traditionally, practice has been synonymous with drill. The teacher
models or introduces a procedure or math fact and then gives students
worksheets full of problems so that they can “practice” over and over. The
objective of this form of practice has often been memorization rather than
understanding.
The purpose of Minilessons for Math Practice, Grades 3–5 is to
broaden the notion of what it means to provide students with practice in
mathematics. Instead of focusing just on facts, procedures, and memo-
rization, the goal of this book is to give students ongoing experiences that
will help them practice math concepts, skills, and processes so that they
may deepen their understanding of mathematics and apply what they’ve
learned to new problem situations.
Another goal of the book involves broadening classroom opportuni-
ties to do mathematics. There is limited time in the school day dedicated
solely to math. This book looks at ways to insert mathematics throughout
the students’ day. Minilessons for Math Practice offers ideas for quick
activities that can be used in various contexts.
In addition, a goal of this book is to broaden the mathematics cur-
riculum. Most teachers are required to use a district-adopted curriculum
and have little extra time for supplemental materials. The activities in this
book can be used with any existing math program to help students meet
local, state, and national math standards.
Introduction: Redefining Practice
xi
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Features of This Book
There are several key features in Minilessons for Math Practice. One is
that the activities in it take little or no preparation. They are easy to im-
plement. And the activities take only five to fifteen minutes to teach.
Throughout the day teachers find themselves transitioning their students
from one activity to another, from one place to another, or from one sub-
ject to another. These transitional times require teachers to focus the at-
tention of their students so they can move smoothly through shifts in the
day. The activities in this book convert these transitional times into rich
mathematical events.
Another important feature is that all of the activities can be repeated.
For example, many of the games in the book can be played throughout the
year to give students ongoing practice with numbers and operations.
Looking at Data can be repeated simply by changing the survey question
so that students can continue to practice analyzing data as the year pro-
gresses. For Guess My Rule, you need only to offer students different
numbers to compare. Or you might shift the Guess My Rule content area
by presenting polygons, in order to give students continued experience
with two-dimensional shapes, even after your unit on geometry is over.
The lessons in Minilessons for Math Practice focus on questioning and
classroom discussion. While the activities in the book are engaging and
fun, they become mathematically loaded when the teacher spotlights the
key mathematical concepts and skills. This spotlight becomes a focal point
for the students when the teacher asks challenging questions and helps stu-
dents develop their own ways to express their thinking using mathemati-
cal language.
Organization of the Activities
The twenty-seven activities in this book offer experiences in all of the con-
tent areas important to mathematics: number, measurement, geometry,
data analysis and probability, and algebra. As well, the lessons model how
to develop several important math processes: problem solving, reasoning
and proof, communication, connections, and representation.
See the contents chart following this introduction to identify activities
that fit the content area of choice. Since many of the activities address
more than one content area, the content area of focus for each vignette is
highlighted in the chart with a bold X. Other potential content connections
are noted with a small x.
xii Introduction: Redefining Practice
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The activities in the book each have eight components:
1. the content area(s)
2. materials
3. time
4. an overview of the activity and an explanation of the mathematics
involved
5. step-by-step teaching directions
6. a list of key questions to ask students during the lesson
7. a brief vignette from the classroom that describes how we taught
the activity
8. ideas for extending the activity throughout the year
Getting Started
The activities provided in this book have been field-tested in diverse class-
room settings. They typically take fifteen minutes or less. However, when
introducing the activities to your students, you might find you need more
time. In some cases it makes sense to budget a thirty- to forty-five-minute
time slot for the first presentation of an activity. Much of the decision will
depend on your students’ prior experience and your goals for the session.
Once you’ve made the initial time investment, the activities should run
smoothly in a five- to fifteen-minute time slot for the rest of the year.
The lessons in Minilessons for Math Practice are language rich, allow-
ing students to develop, organize, and explain their thinking. However,
the ability to communicate mathematical ideas is a skill that develops over
time. Your initial discussions with your students might be briefer and less
profound than you had anticipated. Don’t be discouraged. Over time,
with good questions and a safe environment, students will become more
confident and more competent in discussing their mathematical thinking.
This is especially true for English language learners.
The key questions listed before each vignette and the descriptions of
classroom interactions within the vignette give examples of activity struc-
tures that maximize participation and develop mathematical thinking and
language. Notice that we use different types of questions throughout the
activities. Some questions focus students on specific solutions, while other
questions focus students on multiple approaches, strategies, or techniques.
Since the activities are designed primarily for whole-class settings, we
gave special thought to meeting the needs of diverse learners. The activi-
ties need to be accessible to all students while also being rich enough to
engage all students at deep mathematical levels. Throughout the book, the
Introduction: Redefining Practice xiii
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vignettes model ways to encourage participation by all students and ways
to help students develop the language and communication skills necessary
for math talk.
We encourage you to use the book flexibly and adapt the activities to
best meet your instructional goals and your students’ needs. You might use
the activities to supplement your current unit of study in mathematics.
Alternatively, you might use some activities in Minilessons for Math Practice
to keep past math studies fresh in your students’ minds. Another option is
to use activities as previews or introductions to upcoming units of study.
We recognize that classroom teachers face more and more challenges
each year as they struggle to help their students meet local, state, and na-
tional standards and perform well on standardized tests. We hope that the
activities in this book will support you in these efforts and we encourage
you to use the book in ways that best meet the needs of your students.
However you choose to use the book, we hope you find effective and
fun ways to engage your students mathematically. We also hope this book
helps open your students’ minds to math throughout each day and
throughout the school year.
xiv Introduction: Redefining Practice
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Contents Chart
Number Data
and Analysis and
Activity Chapter Operations Algebra Geometry Measurement Probability
Number Talks 1 X
Ballpark Estimation 2 X x
Clear the Board 3 X
Coin Riddle 4 X x
Comparing Fractions 5 X
Coordinate Tic-Tac-Toe 6 X
Digit Place 7 X
Dot Clusters 8 X x
Estimate and Measure 9 x X X
Estimation Jar 10 X
Function Machine 11 x X
Guess My Number 12 X
Guess My Rule 13 X x
How Far Away? 14 X x
Looking at Data 15 x X
Number Strings 16 X
Odd Number Wins 17 X
Over or Under 18 X x
Personal Numbers 19 X
Race for Twenty 20 X
Target 100/Target 0 21 X
Tell Me All You Can 22 X
Twenty Questions 23 X x
What Comes Next? 24 X
What Page Are We On? 25 X x
Whole-Class Pig 26 x X
Your Choice Tic-Tac-Toe 27 X
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Overview
In this simple, yet powerful activity, students solve a computation problem
presented by the teacher. After students solve the problem mentally, a class
discussion ensues. The teacher asks questions and records students’ think-
ing on the board. This activity fuels mathematical thinking and computa-
tional fluency.
Start with simple problems that students can solve fairly easily. The
focus at the beginning is on helping students explain their thinking and
communicate clearly. Both of these tasks are challenging, especially for
English language learners. As students develop their communication skills,
you can gradually present more challenging problems. The teacher’s role
during number talks is to help students clarify their thinking and connect
their mathematical ideas to the appropriate symbols and notation. This
means that teachers need to ask probing questions and translate infor-
mal discussions into formal mathematical vocabulary and notation. These
teaching skills develop over time with practice. It’s as important for teach-
ers to practice number talks as it is for students.
Activity Directions
1. Write a computation problem horizontally on the board or an
overhead transparency (e.g., 27 ⫻ 2 ⫽).
2. Tell students to solve the problem mentally and pay attention to the
way they solved it.
3. After a few moments, count aloud to three and have the students tell
their answer in a whisper voice.
4. When the correct answer has been established, have individual
students share their solution strategies.
5. Record students’ strategies on the board using appropriate
mathematical notation.
Number Talks
1
䊝䊞䊟䊞
䊝䊞䊟䊞
•
•
•
•
CONTENT AREA
Number and Operations
MATERIALS
■ optional: overhead
projector
TIME
ten minutes
1
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Key Questions
• How did you solve the problem?
• Where did you start?
• Did anyone do it a different way?
From the Classroom
I started a number talk with Robin Gordon’s fourth graders. As this was
the first number talk I’d done with them, I decided to introduce a fairly
simple equation. This way, we’d be able to focus on the communication
aspect instead of getting too bogged down in the computation. After the
students understood the focus and mechanics of the whole-class number
talk, I planned to present them with more challenging problems to solve
and discuss.
“OK,” I told the students, “I’m going to write a problem on the
overhead. Your job is to solve the problem mentally. What do I mean by
that?”
“Do it in your head,” Gregory responded.
“Right,” I agreed. “You’re going to look at the problem and then solve
it mentally, in your brain. There are actually two things your brain will
be doing. Not only are you going to solve the problem, but you’re also
going to pay attention to how you solved it so we can have a conversation
about it.”
“Oh,” I added, “there’s one more thing. When you think you know
the answer, please don’t shout it out. We want to make sure everyone has
time to think and solve the problem without being interrupted or told the
answer. OK?”
I looked seriously at the students to convey the importance of this rule.
I waited for eye contact and nods. Then I wrote on the transparency:
27 ⫻ 2 ⫽
I gave the students a full thirty seconds to think about the problem.
After a more than ample pause, I spoke to the class. “Thanks for controlling
yourselves and not shouting out the answer,” I complimented. “Thumbs up
if you think you know the answer.”
Everyone showed me an upturned thumb. Instead of having all the stu-
dents vie to be the one to say the answer, I employed the choral response
approach. That way everyone would get to share.
“Well, it looks like a lot of you have an answer. Here’s what we’re
going to do,” I explained. “I’m going to count to three. When I say, ‘Three,’
2 Minilessons for Math Practice, Grades 3–5

you’re going to say the answer in a whisper voice. Ready? One. Two.
Three.”
To my surprise, I heard three or four different answers. I realized we
needed to resolve that dispute before moving ahead with the number talk.
“Hmm,” I told the class. “I heard several different answers. Take a
minute or two to talk to your neighbors about what you think the answer
is and why. Maybe that will help us agree on the correct answer.”
I gave the students a few minutes to talk to each other. As they were
discussing the problem, I prompted various groups with questions like
“What do you think the answer is?” “Can you explain your thinking to
your partners?” and “How can you prove that?” When I heard them
reaching a consensus, I asked for their attention.
“OK,” I said. “We’ll try the same thing again. I’ll count to three and
you say the answer in a whisper voice. One. Two. Three.”
“Fifty-four,” the class responded in unison.
“Yes,” I agreed. “Now for the interesting part. Who is willing to tell
how you solved the problem?”
Many students raised their hands. I called on Jonquil.
“I pictured it vertically,” she told us. “Twenty-seven plus twenty-seven.”
“So you turned this multiplication problem into an addition problem.
You knew twenty-seven times two is the same as twenty-seven plus
twenty-seven. Like this?” I asked as I wrote on the overhead.
Jonquil
27
⫹ 27
“Yes,” Jonquil replied. “And I did seven plus seven equals fourteen, so
I wrote down the four and put the one on top of the two.”
“Is this really a one?” I asked the class.
“No, it’s a ten,” Ricardo reminded us.
I am always a stickler about clarifying the value of digits when dis-
cussing arithmetic problems. Using proper terminology helps intermediate-
grade students solidify their place-value foundations. Students often need
to be reminded that face value is not necessarily place value. In this case,
the one Jonquil carried was not really a one, but a ten.
“So what did you do with that ten?” I asked Jonquil.
“I added it to the two twenties. That made fifty and I already had the
four, so it equaled fifty-four,” she explained.
I recorded Jonquil’s answer and asked if any students had thought of
it differently. Many hands were raised. I called on Ricky.
“I split the twenty-seven into a twenty and a seven. Then I added the
twenties and the sevens.”
Number Talks 3

I wrote on the overhead:
Ricky
27 ⫽ 20 ⫹ 7
20 ⫹ 20 ⫽ 40
7 ⫹ 7 ⫽ 14
40 ⫹ 14 ⫽ 54
Raquel was the next to explain. She had a method similar to Ricky’s
but used multiplication instead of addition.
Raquel
27 ⫽ 20 ⫹ 7
7 ⫻ 2 ⫽ 14
20 ⫻ 2 ⫽ 40
40 ⫹ 14 ⫽ 54
I saw an opportunity to help students make some connections.
“Does Raquel’s way remind you of anyone else’s way?” I asked the
class. There were many nods.
“Yeah, it’s just like Ricky’s,” Andrea noticed.
“Except that Raquel timesed the numbers and Ricky plussed them,”
Laurence added.
“Yes,” I replied, “Ricky and Raquel both broke the twenty-seven into
a twenty and a seven, but Ricky used addition to solve the problem and
Raquel used multiplication.” Again, I took advantage of an occasion to
gently reinforce proper mathematical terminology, replacing Laurence’s
words timesed and plussed with multiplication and addition.
“Who has a different way?” I asked again.
“Mine is like Jonquil’s, but I used multiplication,” Trudy told us.
“Tell us what you did,” I prompted.
“I thought of it vertically, but I did twenty-seven times two instead of
twenty-seven plus twenty-seven,” she explained. “First I did two times
seven, which is fourteen. I put down the four and carried the ten. Then I
did two times twenty, which is forty, plus ten more is fifty.”
Trudy had used the standard algorithm. I wrote on the overhead:
Trudy
1
27
⫻ 2
54

“Did anyone do it a different way?” I asked again. This time I called
on Jonathan.
“I took the sevens out of the twenty-sevens so I just had twenty. Then
I multiplied twenty times two and added back the sevens.”
I recorded his method on the overhead as he talked me through it.
Jonathan
20 ⫻ 2 ⫽ 40
40 ⫹ 7 ⫽ 47
47 ⫹ 7 ⫽ 54
“Wow,” I remarked as I looked at the screen. “There are a lot of dif-
ferent mathematical ideas up here. Take a minute to look at all of these
solutions. Then raise your hand if you have an idea that’s not already up
here.” I gave the students some time to digest the methods already
recorded. Janie raised her hand.
“I rounded the twenty-seven up to thirty because thirty is easy. Then I
had to take away six.”
“Where did the six come from?” I asked her.
“Because I added three to each twenty-seven to get it to thirty, so I had
to take the two threes away,” Janie explained.
I added her solution to the overhead transparency.
Janie
27 ⫹ 3 ⫽ 30
30 ⫻ 2 ⫽ 60
60 ⫺ 3 ⫽ 57
57 ⫺ 3 ⫽ 54
“This is all very impressive,” I commented to the class as I looked at
the overhead transparency filled with their ideas. “You have a lot of ways
to think about numbers and computation. That’s really important for
mathematicians. It’s also really helpful when you can see how different
methods have some of the same ideas in them. You did a nice job of mak-
ing connections among your ideas.”
I thanked the students for their attention and told them I would save
the transparency for us to revisit and add to in the future. The students
exhibited flexible thinking and good communication. I knew that these
features of their math learning would continue to improve as they regu-
larly engaged in number talks.
Number Talks 5

Extending the Activity
As students gain skill in communicating and connecting their ideas to
mathematical symbols, they can take on the task of recording their own
and each other’s thinking. Number talks benefit students in all areas of
number and operations. Regular use of number talks helps students solve
all types of arithmetic problems with more accuracy, efficiency, and flexi-
bility. And, while this number talk asked that students solve the problem
mentally, others may require or allow students to use paper and pencils.
Number talks also present an alternative to traditional test-preparation
activities. Write problems similar in content and format to those on stan-
dardized tests, and use number talks to demystify the various types of prob-
lems and strengthen students’ understanding of those problems.

Overview
Through Ballpark Estimation, students relate estimation to items in their
classroom. Ballpark Estimation is an easily adaptable activity that sup-
ports creative and divergent thinking. It also gives students an opportunity
to apply benchmark numbers when estimating. Unlike an estimation jar
activity, in which students focus on one particular container, Ballpark
Estimation encourages students to consider many items and opens up
multiple estimation opportunities in a single activity.
Activity Directions
1. Write a sentence frame on the board: In our classroom there are
about . . .
2. Write a series of benchmark numbers below the sentence (e.g., 10,
25, 50, 75, and 100).
3. Ask students to think of ways to complete the sentence for each
number. Let them discuss their ideas with their partners.
4. Write student responses on the board or an overhead
transparency.
Key Questions
• How did you come up with your idea?
• Is there anything you already know that can help you make an
estimate?
• Does anyone have a different way of thinking about it?
Ballpark Estimation
2
䊝䊞䊟䊞
䊝䊞䊟䊞
•
•
•
•
CONTENT AREA
Measurement
MATERIALS
projector
TIME
ten to fifteen minutes
7

From the Classroom
Lisa Kimbrell’s third graders had a few minutes before an all-school as-
sembly. I decided to introduce Ballpark Estimation to them. My meet-
ing with the class was in mid-September, so they were just getting to
know their teacher, the classroom routine, and each other. Presenting an
activity that focused specifically on their classroom served dual pur-
poses. It introduced them to mathematics in a relevant context while
simultaneously helping familiarize them with their own classroom
community.
I began the activity by writing on the whiteboard:
In our classroom there are about . . .
10 25 50 75 100
Then I began to explain the task. “This is an estimation activity. Would
someone please remind us what it means to estimate?”
A few students raised their hands.
“It’s when you make a guess,” Angela responded.
“You think about some numbers,” Scott added.
“It’s like when you think about how many there are,” Tony
contributed.
“Yes,” I agreed, “when you estimate you make a mathematical
guess. You actually do math thinking and then use your ideas to esti-
mate. When you make an estimate, are you supposed to know the exact
answer?”
“No?” the class chorused somewhat uncertainly.
“Right,” I said encouragingly. “You don’t need to know the exact an-
swer. If you knew the exact answer, you wouldn’t need to estimate. So
when you estimate, are you trying to get the exact right answer?”
“No.” The class seemed a bit more confident this time.
“Does it matter whose estimate is closest to the exact number?”
“No.”
“Do you win a new car if you’re closest?”
“No.”
“Does something terrible happen to you if your estimate is not the
closest?”
“No!”
“OK,” I confirmed. “Estimating is using your brain to think about
numbers. The important part is your thinking, not being right or
closest.”

I really wanted to hammer this point home. If I could set a nonthreat-
ening tone at the beginning of the year, it would pay off for a long time.
Students need to feel that estimating is a risk-free and interesting endeavor.
Teachers can encourage this perception by expressing interest in children’s
thought processes and de-emphasizing the actual total.
“So,” I continued, “here’s what I did. I wrote the beginning of a sen-
tence on the board and below it I wrote some benchmark numbers.”
I introduced the term benchmark in the course of the conversation, but
I didn’t want to get sidetracked with a long explanation or discussion. I
knew that we would return to the idea of benchmarks throughout the
year, so this was quick exposure to the idea. I wanted to keep the spotlight
on estimation in this short period of time.
“Let’s think about ten for a few moments. Can anyone think of some-
thing you have about ten of in this room?” I asked.
Many students looked around the room and quite a few began to
count.
“Remember,” I said, “you don’t need to count to find exactly ten of
something. You just need to find something that there are about ten of in
the room.”
I saw a few raised hands and called on Sandy.
“Those things,” Sandy reported, pointing to some student-created
graphs posted on construction paper.
“The surveys you made?” I asked. Math discussions often present oc-
casions to help build vocabulary, especially with English learners.
“Yes,” Sandy replied.
“How did you come up with that idea?” I asked.
Sandy explained, “It just sort of looks like ten.”
I wrote surveys under the 10 on the board.
“The punctuation charts,” William said, referring to a set of posters
on the back wall.
I added his idea to the list.
“Now,” I told the class, “I’m going to give you a minute to quietly talk
to someone next to you about other things that might fit under the ten col-
umn. Remember, you don’t need to count. You can look around the room,
but don’t leave your seats. Talk to each other and see what other ideas you
can come up with.”
From their brief partner talk, the students generated more items for
the list. After they shared their ideas and I recorded them, I moved on
to the bigger numbers. I had students partner up and discuss their ideas
about items that would fit the rest of the benchmark numbers. Eaves-
dropping on the conversations offered a window into students’ thinking
Ballpark Estimation 9

and their number sense. Gladys pointed to a basket of markers and told
her partner, Steven, that there were about twenty-five of them. Rico
disagreed, estimating there were about seventy-five markers in the
basket.
Some students continued to count items before offering any sugges-
tions. This behavior told me that either the students had not had much
prior experience with estimation or that they still were under the impres-
sion that it was important to be right or closest. In some cases, the items
children shared were not estimates at all. For fifty, they suggested states on
the map and stars on the flag. I chose to accept all ideas during this initial
exposure to Ballpark Estimation. One of the nice things about these brief
math activities is knowing that there will be a next time and that miscon-
ceptions can be addressed over time.
Several students had ideas that sparked a class discussion. Ruby esti-
mated that there were about one hundred fingernails in the classroom. An
issue arose as to whether toenails were part of the fingernail category. I
helped establish that everyone had ten fingers and hence ten fingernails.
Many students were still baffled by the prospect of estimating how many
fingernails there were in the room and/or how to count them. Knowing
that multiplication would be an important part of the math curriculum for
that class, I decided to take a moment to focus on multiplicative thinking
and how it might assist us in making estimates.
“How many fingernails does one person have?” I asked.
“Ten,” the class responded in unison.
“How about two people?” I continued.
“Twenty.”
“Four people?”
“Forty.”
“What if we wanted to find out how many fingernails there were in the
whole class?” I asked. “Can anyone think of a way to do that?”
“We could just count them,” Sheila offered.
“How do you mean?” I probed.
“Like one, two, three,” she explained.
“Yes,” I agreed, “we could count by ones.”
“Or we could count by tens,” Kenny added.
“Or fives,” Diana contributed.
“Great,” I replied. “So if we’re making estimates and it’s a big number,
there are some ways to think about it in small pieces. We can use numbers
we know like ones, fives, and tens to help us estimate.”
When we were adding to our lists, I challenged the class to come up
with at least three things for each number. I stopped the students when the
board looked like this:

In our classroom there are about . . .
10 25 50 75 100
Surveys Math Pattern Markers Fingernails
folders blocks in
a tub
Punctuation Rectangles States on Folders in Books
charts on the the map the box
Papers
word wall
Names on Board Stars on Dictionaries Crayons in
the board games the flag
Ceiling tiles
the crayon
box
Books in
the basket
Blue tubs
Posters
I looked at our lists and began to wrap up. I wanted to plant the seed
for a follow-up activity.
“I’m impressed with all of your ideas,” I told the class. “You really
seem to know what’s in your classroom and you have some interesting
ways to think about estimating. When I look at these lists, I see a lot of
things that you could actually count to find out the exact totals. That
might be an interesting activity to try sometime soon.”
The students seemed intrigued by this notion. Although there wasn’t
time for them to compare actual totals with their estimates at that mo-
ment, I left the lists on the board, giving their teacher the opportunity to
make these comparisons in the future. I thanked the students for their time
and efforts and they headed off to their assembly.
I did this activity with young third graders. It is easily adaptable for the
upper-elementary grades. Use different numbers with older students—
fractions, decimals, percents, or larger numbers. Ballpark Estimation easily
adjusts to whatever types of numbers the students are working with.
Another twist on the activity is to have students estimate about things
beyond the classroom. The estimates can focus on an entire grade, the
Ballpark Estimation 11

school, the neighborhood, or the city. This forces students to deal with
larger numbers. They can collect data about their own class and use that
information to make estimates about a larger group.
Ballpark Estimation can also become a measurement activity. Make a
chart that says
Here are some things that are about . . .
1 inch 6 inches 1 foot 2 feet 1 yard
Teachers can also use this version of the activity to focus on two- or three-
dimensional measurement. Ask students to consider items that measure in
square or cubic units. Often students lack experience with measurement.
This activity would give them repeated occasions to think about units of
measurement, their relative size, and visual models that represent the
approximate units.

Overview
In this activity the class works together to generate the numbers one through
twelve in order. The teacher begins the activity by rolling three dice. Using
the results of the roll and any operations, the students create in consecutive
order the numbers from one through twelve. Students may use the results of
one die, two dice, or all three dice. When students encounter a number they
cannot create using the results on the dice, the teacher rolls again. The ac-
tivity is done when students have made all the numbers from one to twelve.
Clear the Board offers opportunities for mental computation and
helps build number and operation sense. Students think flexibly and con-
sider many possibilities in order to find solutions to the computation chal-
lenges involved.
Activity Directions
1. List vertically on the board or an overhead the numbers 1 through 12.
2. Roll three dice and write the results of the roll on the board so
everyone can see them.
3. Ask students if they see any ways to make the number one using the
numbers rolled. One, two, or all three numbers rolled may be used in
conjunction with any operation.
4. After students have created a way to make one, repeat Step 3 for the
number two and record the solution on the board next to the 2.
5. Give students a few minutes to talk to each other about possible
ways to make the next few numbers using the results of the first roll
and any operation. Record their ideas on the board. When the next
number is not possible, roll again. Record the results of the new roll
on the board.
6. The activity is completed when the class has made every number on
the board in order.
Clear the Board
3
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•
•
•
CONTENT AREA
MATERIALS
■ 3 dice
projector
TIME
five to ten minutes
13

Key Questions
• How many rolls do you think it will take to clear the board?
• What numbers would you like to get now? Why?
From the Classroom
Amy Jackson wrote the numbers 1 through 12 vertically on the board before
she said a word to her fourth and fifth graders. With their interest piqued,
she asked them if they’d like to play a game. The response was quite positive.
“Great,” she told the class. This game is called Clear the Board. Our
goal is to systematically get rid of each number I listed. We have to get rid
of the numbers in order, beginning with one first and then moving down
to two, all the way to twelve.
“I’m going to roll three dice. I’ll write the numbers rolled on the board
so you can see them. Then you need to think about how you can use these
numbers to make the number one. You may use any operation—addition,
subtraction, multiplication, or division—or a combination of operations.
When you find a way to make one, I’ll clear the one off the board and we’ll
move on to the next number, two. Ready? Here I go.”
Amy rolled a 1, a 3, and a 5. She wrote these numbers on the board.
Amy was surprised that students appeared to be struggling. Why
didn’t they just point out the 1? She wondered. Then she realized her
directions might not have been clear enough.
“I need to clarify something,” she told the class. “Even though there
are three numbers, you don’t need to use all of them. You can use one, two,
or all three of the numbers rolled to make the number you need.”
Marcello raised his hand. “How about five minus three minus one?”
he suggested.
While Amy was expecting a simpler solution, she wrote Marcello’s
equation on the board next to the 1. Adriana suggested the need for paren-
theses around 5 3. Technically parentheses are not required in this situ-
ation; however, their use is not incorrect and adds clarity. Amy decided to
include them in the written model. She also pointed out that since they
rolled a 1, they could use it as is. She wrote that on the board too.
1 (5 3) 1 1, 1 1
“OK, now that we’ve made one, we can cross it off. We’re on our way
to clearing the board. Next up is two.”
“Three take away one is two,” Enrique announced.
“You could also do five minus three,” Jamal pointed out.

Amy wrote those two equations next to the 2 and then crossed off
the 2. Once the students understood the mechanics of the activity, she
gave them some time to talk in groups about ways to make the rest of the
numbers. Soon the board filled up with a variety of solutions for the next
few numbers. While Amy was impressed with the students’ flexible
thinking and comfort with mental computation, she was concerned that
the game would go on and on if she didn’t set a limit. The students were
more interested in finding multiple solutions for each number than mov-
ing on to the next number. To keep the game moving, Amy decided to
limit responses to one per number. In a few minutes the board looked like
this:
1 (5 3) 1 1, 1 1
2 3 1 2, 5 3 2
3 3 3, (5 1) 3 3, (5 3) 1 3
4 3 1 4, 5 1 4
5 5 5, 5 1 5,
5
1
5
6 5 1 6
7 (5 3) 1 7
8 5 3 8
9 5 3 1 9
10 (3 1) 5 10
11
12
The class was stuck on eleven, so Amy told the students it was time to
roll again. Before rolling, she asked a question to stimulate more mental
computation and to assess their comfort with the various operations.
“So,” she said, “I’m going to roll the three dice again. What numbers
would you like to get now?”
“A five, a six and a one,” Cristina offered.
“A five, a six, and anything would work,” Kendrick pointed out.
“How about a four, a four, and a three?” suggested Christine.
“Or a two, a three, and a six,” J. P. added.
This quick question showed Amy that her students were comfortable
breaking eleven apart in different ways but were not as comfortable think-
ing about operations other than addition. Perhaps with more time and
practice they would open up their thinking.
She rolled the dice and got two 5s and a 1. She wrote these numbers
on the board below where she’d written the first roll. Then she turned to
the class.
“Tough one, huh?” she teased.
“Just put five plus one plus five,” Kenyatta directed.
Clear the Board 15

Amy wrote the equation next to the 11 on the board.
5 1 5 11
“Aren’t you supposed to put parentheses, Ms. Jackson?” Jessie
challenged.
“No,” J. P. disagreed. “We don’t need them.”
“Let’s check,” Amy suggested.
She pointed to the first 5 and the 1.
“What’s five plus one?” she asked the class.
“Six.”
“Plus five more equals?”
“Eleven.”
“OK,” Amy continued, “let’s see what happens if we start with the
other two numbers.”
She pointed to the 1 and the second 5.
“What’s one plus five?” she asked.
“Six.”
She pointed to the first 5.
“Plus five more?”
“Eleven.”
“Hmm,” she concluded, “we get the same answer regardless of where
we start. So we don’t really need parentheses for this addition equation.”
She wrote on the board:
(5 1) 5 5 (1 5)
Although this wasn’t the time to go into a lengthy discussion about the
purpose of parentheses or the associative property, it was a nice opportu-
nity to model the concept and notation briefly.
Amy gave the students a minute to talk to a partner about how to use
these rolls to make twelve. After some computational gymnastics, the stu-
dents agreed that they couldn’t make twelve with those three numbers, so
Amy rolled again. This time she got a 5, a 2, and a 1. She wrote these num-
bers on the board below the previous roll. The students took a minute to
consider the possibilities. Christine raised her hand.
“I’ve got one,” she announced. “Five plus one times two.”
“Do I need to use parentheses for that equation?” Amy asked, rein-
forcing their prior discussion.
“Yes,” Christine told her. “Put them around the five plus one.”
Amy wrote the equation on the board and had the class double-check
to make sure everyone agreed.
(5 1) 2 12

Amy crossed out the 12 on the board and congratulated the students
for clearing the board. She asked how many rolls it took for them to
accomplish their task. Since she had written each roll on the board, the
students had an easy reference. Amy wondered aloud if they thought it
would take more or less than three rolls when they played next time. Leav-
ing them with something to think about set the stage for future experi-
ences with Clear the Board.
There are many possible twists to this game. Introduce the activity as a
whole-class, cooperative endeavor. Once students grasp the rules and
goals, have them play a competitive version in pairs. Each student writes
down the equations generated for each number in order to keep track of
them all. In this version, partners can be challenged to use the same rolls
to make their numbers. Students can compete to see who cleared the most
numbers with a given roll. Or, each student can roll the dice for his or her
turn and try to be the first to clear all twelve numbers.
The game can by extended by having students work their way from
one through twelve and then back from twelve to one. Also, if students are
relying too heavily on addition and subtraction, encourage or require the
use of other operations as a challenge.
Clear the Board 17

Overview
Riddles of all kinds intrigue children. Coin riddles stimulate children’s log-
ical thinking while simultaneously engaging them in arithmetic problems.
In this activity students use clues and mental computation to systemati-
cally narrow down possibilities until they can identify the teacher’s set of
mystery coins.
To begin, hide a few coins in your pocket, in a 35 mm film canister, or
in a small paper bag. Write clues to share with students to help them de-
duce the hidden coins. Give the students one clue at a time. Allow students
to discuss all clues, make guesses about the identity of the coins, and check
to make sure their guesses match the clues. Then reveal the coins. This
activity can be adapted easily to the level and instructional focus of your
class.
Activity Directions
1. Tell the students you have some coins in your pocket, a film canister,
or a paper bag, and their job is to guess which coins.
2. With the help of students, make a list of possible coins you could
have.
3. Give the clues one at a time, first recording a clue on the board or
overhead and then reading it aloud. Encourage discussion after each
new piece of information.
4. To model for students one way to organize information, draw a chart
on the board and fill in the possible combinations of coins. (See
page 21.)
5. Have students tell what coins they think you have.
6. Review each clue to make sure it fits the students’ guess.
7. Reveal the coins.
Coin Riddle
4
CONTENT AREA
Number and
Operations
MATERIALS
■ 3–5 coins of
different
denominations
■ clues about the
coins, written on
an index card
■ a pocket, 0.35 mm
film canister, or a
small paper bag
projector
TIME
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18

Key Questions
• What do you know now about my secret coins?
• What can you eliminate?
• What are some possible coin combinations I might have?
• How do you know?
From the Classroom
I jingled the coins in my pocket as an introduction to the Coin Riddle
activity.
“I have some coins in my pocket,” I told Danielle Pickett’s third
graders.
“I can hear them!” Terrell noted enthusiastically.
“Are you going to use them for the bake sale?” Inca inquired.
“I didn’t even know there was a bake sale today. Lucky me,” I told her.
“Maybe I will use them for the bake sale later, but for now I’m going to
use them for a math activity.”
Before introducing the actual riddle, I decided to tap into the students’
prior knowledge and get them thinking about coins in general.
“So, does anyone have any ideas about what coins might be in my
pocket?” I asked the class.
“Maybe pennies,” Jaime volunteered.
“I think it’s quarters,” Tracie offered.
“It could be pesos,” Josue posited, eliciting giggles from other students.
“Well, I do love to go to Mexico,” I told the class, “but today I’ve only
got U.S. coins. So if you know that the coins are American, what are the
possible coins I could have in my pocket? Talk to a partner about what
you know about coins and what my secret coins might be.”
After a minute or so I asked for the class’s attention and had volun-
teers tell me a coin. As the students told me a coin, I asked for its value. I
wrote the names and values on the board so students would have a refer-
ence. Several of the third graders in Danielle’s class were English learners
so I wanted to be sure they weren’t constrained by vocabulary.
penny ⫽ 1 cent
nickel ⫽ 5 cents
dime ⫽ 10 cents
quarter ⫽ 25 cents
Coin Riddle 19

half-dollar ⫽ 50 cents
dollar ⫽ 100 cents
I jingled the coins again to refocus the students. “There are a lot of dif-
ferent coins. I’m going to help you by narrowing the possibilities down a
bit. I’ll tell you that for this activity the coins could be pennies, nickels,
dimes, or quarters.”
I put a box around those four coins on the board to remind students
where to focus.
“So here’s what happens next,” I announced as I pulled an index
card out of a different pocket. “I have a riddle about my mystery coins.
The riddle will give you clues to help you figure out what coins I have in
my pocket. As I share the clues, you’ll be able to talk to a partner about
what you know about my coins and what you think they might be. There
also might be some clues that tell you what my coins aren’t. That infor-
mation could help you eliminate some of the possibilities. Are you
ready?”
“Yes!” the children responded.
“OK,” I said, “here’s my first clue. After I write it on the board, you
need to think about how this clue helps you think about what coins I
might have.” I wrote on the board:
I have three coins.
We read this clue aloud together. Then I had students turn to a partner
and brainstorm what the three coins might be. I let five or six students vol-
unteer their ideas and we agreed that there were a lot of possibilities. Then
I wrote another clue on the board:
They are all different.
“How does that clue help you?” I asked the class.
Nikki responded, “We know it’s not three pennies or three quarters or
anything like that.”
“Also, it can’t be, like, two dimes and a nickel because they’re all dif-
ferent,” Kendra added.
“Hmm,” I said. “I wonder if we could figure out what coin combina-
tions would fit my clues. Maybe a chart to show all the possible combina-
tions would help.”
I drew a chart on the board. While not integral to the Coin Riddle
activity, the chart provided another visual reference for students and also
offered a model of how to systematically organize information. The class
and I filled in the chart together.

Pennies Nickels Dimes Quarters Total Value
1 1 1
1 1 1
1 1 1
1 1 1
“It looks like we’ve got it narrowed down to four possibilities. Now
let’s see if the next clue can help you narrow it down even further,” I told
the class as I wrote the next clue on the board:
Their total value is more than 20 cents and less than 50 cents.
I gave the students a few minutes to talk about this new information
and how it could help them. During their discussion they did a lot of men-
tal computation to determine the value of each combination on the chart.
“Can we eliminate anything?” I asked the class.
“It’s not one penny, one nickel, and one dime,” Chastity announced.
“How do you know?” I probed.
“Because that only adds up to sixteen cents,” Alex explained. “Because
ten plus five is fifteen plus one more makes sixteen.”
“Oh, I see,” I told him. “So we can cross this off the chart.”
I crossed off the penny-nickel-dime option. I also took a moment to
have the class tell me the total value of the other coin combinations. I
wrote the values in the last column of the chart on the board. Then I wrote
the next clue:
Two of the coins are silver and one is copper.
I saw immediately that some of the students were confused by this
clue. In particular, the English learners looked quizzical.
“What does this mean?” I asked the class. “What is silver? What is
copper?”
“Your watch is silver,” Ignacio pointed out.
“Yes,” I agreed, pointing to my watch, “this is silver. It’s a kind of
color for metal. What else do you see that is silver?”
“Ms. Pickett’s ring.”
“The doorknob.”
“The shelf under the whiteboard.”
Coin Riddle 21

After it was clear the students knew what silver meant, I asked them
which coins were silver. They easily identified the nickel, the dime, and the
quarter. I told them that the penny was copper. We briefly talked about
other copper things to be sure they understood.
“So now what do you know about the secret coins?” I queried.
“One of the coins is a penny and the others are nickels, dimes, or quar-
ters,” Sally shared.
“All right, here’s the last clue,” I told the class as I wrote on the board:
I have no nickels.
“Talk to a partner about how this helps you figure out what coins I
have in my pocket. You might also want to use the chart we made to help
you,” I suggested.
Soon, the students knew which coins I had. In unison on the count of
three, I had them tell me what mystery coins I had. I referred back to the
chart and to the clues on the board to confirm with them that their solu-
tion fit all the criteria. Finally, I pulled the coins out of my pocket to show
the class.
“You’re right,” I told them. “I had a penny, a dime, and a quarter in
my pocket.” After the cheers subsided, I congratulated the students on
their excellent logical thinking. I encouraged them to try to make up their
own coin riddle challenges for the next time I visited.
Because I was introducing Coin Riddle for the first time, I used a whole-
class, teacher-directed approach. As students become familiar with the ac-
tivity, they can take on more ownership. For example, I could reveal the
clues all at once and let partners or groups discuss the possibilities and try
to solve the riddle.
Students can also write their own coin riddles that can be collected
and used for future Coin Riddle sessions. Before sending students off to
write their own riddles, make sure they understand what makes an effec-
tive clue. The clue must reveal some helpful information but not com-
pletely give the answer away. For example, “The total value is my lucky
number” is not a helpful clue. And a clue such as “I have four coins and
they equal four cents” gives too much away. The students need to under-
stand that the group of clues taken together leads to the one correct solu-
tion. It’s a good idea to start by leading the whole class in writing a riddle
together. Predetermining the coins can help as well. While it may take a

while for students to create viable riddles on their own, it’s worth the in-
vestment. Students develop their logical thinking and language through
both generating and solving coin riddles.
The riddle activity transfers nicely to a geometry context. Begin by
modeling a riddle about a shape. It could be two-dimensional or three-
dimensional, depending on students’ familiarity with various shapes and
geometry vocabulary. Have students discuss and solve your riddle. When
students are ready, assign a shape to a pair or group of students. Have
them work on writing a riddle about their assigned shape. Writing and
solving shape riddles helps students develop their geometry vocabulary
and concepts.
Coin Riddle 23

Overview
The study of fractions is often a great challenge for upper-elementary
students. Many students get bogged down in notation and computational
procedures. In order to combat this tendency, we need to give students
ongoing opportunities to make sense of fractions. Comparing Fractions
affords students a chance to connect fraction symbols with visual images
and to explore both the relationship between numerators and denomina-
tors and the relative size of different fractions.
Activity Directions
1. Have each student draw a game board on a piece of paper.
Comparing Fractions
Discard
2. Roll the die or spin the spinner and let students decide whether they
want to use the number as the numerator, use it as the denominator,
or discard it.
3. Roll two more times, each time allowing the students to place the
resulting number in the box of their choice. If a resulting roll is a
repeat number, roll again in order to have three different numbers.
4. Have each student draw a picture of the fraction he or she created.
5. Ask students to tell you what fractions they made. Record the
fractions and pictures on the board or an overhead transparency.
5
CONTENT AREA
Number and
Operations
MATERIALS
■ a die or spinner
numbered 1–6
■ optional: projector
TIME
five to fifteen minutes
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24

6. Lead a class discussion about the relative size of the fractions and
how the students know which is bigger or smaller. During the
discussion, place the fractions in the appropriate locations on a
number line.
7. Play again as a class and try to make as small a fraction as possible.
Key Questions
• Which is the smallest fraction? Which is the largest? How do you
know?
• Where does this fraction fit on the number line?
• What number would you like to get? Why?
From the Classroom
I introduced Comparing Fractions with a very open-ended, nonthreaten-
ing approach. Knowing that many students are shaky when it comes to
fractions, I wanted to make sure everyone felt comfortable and competent.
I didn’t want anyone to shut down because of a lack of confidence in his
or her understanding of fractions. Once I got the students engaged, I
planned to ramp up the difficulty level.
“I’ve got a new activity for you,” I told a group of fourth and fifth
graders. “It’s called Comparing Fractions. I have a die that I will roll. The
results of the rolls will determine the fraction you make. It will probably
make more sense once we get started. Let’s do one round together. To
begin you need to draw a recording sheet that looks like this.” I drew a
sample recording sheet on the board.
Comparing Fractions 25
Discard
I rolled a 3. I explained to the class, “Now you each need to decide if
you want to use the results of the roll as the numerator, the denominator,
or a discard. The discard box is like a trash can. You’re throwing that
number away and it’s not going to be part of your fraction. Once you put

a number in a box, you can’t change it. So go ahead and write a three in
whichever box you want.”
I rolled twice more, resulting in a 4 and a 1. After each roll I told the
students the results of the roll and gave them a few moments to write the
number in the box of their choosing. After the students had created their
fractions, I asked them to each draw a sketch of what their fraction looked
like. In this way the students connected the symbols to a visual represen-
tation. I then asked volunteers to tell what fraction they made and
describe their picture of it. Some of the students had made improper frac-
tions, so we had a brief discussion about what mathematicians call these
fractions and how to draw them. I recorded all the fractions and their pic-
tures on the board:
3
4
3
1
1
4
4
3
1
3
0 1
“So you were able to make six different fractions,” I noted. “Let’s
compare these fractions and think about where they would fit on a num-
ber line.”
I drew a long number line on the board and labeled it with a 0 on the
left end and a 1 about one-fourth of the way across.
“Talk to a partner,” I told the students. “Discuss where these fractions
belong on the number line. You can use the numbers and the pictures to
help you think about which is the biggest and smallest and where the
others fit in between.”
The students seemed confused, so I decided to break down the task
and model a bit of it with them. “What is the smallest fraction up here?”
I asked, pointing to the board.
“One-fourth,” Kalen offered.
“How do you know?” I asked.

“Because look at the pictures,” Kalen responded. “You can see it’s the
smallest.”
“Also fourths are smaller than thirds or ones,” Brittany added. “And
if you only have one fourth, you’re going to have the least.”
“OK,” I agreed. “So where would one-fourth go on this number line?”
“It goes after the zero but before the one because it’s less than one,”
Veronika shared.
“Does it go right in the middle between zero and one?” I pressed.
“No,” Michael replied. “That’s where one-half would go. One-fourth
is one-half of a half, so it goes closer to zero.”
I approximated the ___location for one-fourth and labeled it on the num-
ber line.
0 1
1
4
“The smallest fraction you made is now on the number line,” I con-
tinued. “Let’s put the largest fraction up here too. Which fraction is the
largest?”
“Four ones,” Chad responded.
“Yes,” I agreed, “I’m looking at all the pictures on the board, and four
is definitely the largest number. Where do we put it?”
“It goes way past the one,” Claire announced. “Like on a regular
number line it goes one, two, three, four.”
“This isn’t a regular number line?” I asked.
“Not really,” Claire explained, “because we’re putting fractions on it.”
“You can put fractions on a number line,” Jessica disagreed.
I took the opportunity to clarify some possible misconceptions about
fractions and number lines.
“This is an interesting discussion,” I told the class. “A lot of times
when we think about numbers, we think of whole numbers or counting
numbers. There are actually other types of numbers too.”
“Like negative numbers,” Cole volunteered.
“Right,” I agreed. “Fractions and decimals are numbers too. A num-
ber line is a tool we use to help us think about numbers. Number lines are
helpful when we want to compare numbers. They also can be helpful for
addition or subtraction. Right now we’re using a number line to compare
fractions and put them in order from smallest to largest.”
“Claire told us that four is the largest fraction and it goes to the right
of the one,” I continued. “I’m going to put it on the number line. Then
you’re going to have a few minutes to talk with a partner about where the
other fractions fit on our number line. You can also use your papers to
draw a number line and place the fractions where you think they belong.”

I wrote 4 on the number line and gave the students some time to talk
to each other. They had many tools available—the number line, the pic-
tures of the fractions, their own paper to sketch on, and each other to talk
to. Providing multiple access avenues is extremely important when stu-
dents are grappling with new or challenging mathematical ideas. I called
the class back to attention and we placed the rest of the fractions on the
number line. Then I moved the students on to a more challenging task.
“This time, let’s try to make a small fraction, as close to zero as possi-
ble.” I rolled a 2 and asked the students which box to put it in.
“Don’t put it as the denominator,” warned Demetrius.
“Why not?” I asked.
“Because then it will be a half. That’s already far away from zero,”
Demetrius explained.
“So where do you recommend we put the two?” I asked.
“I say throw it in the trash,” he suggested.
I wrote 2 in the discard box and rolled the die again. This time the
result was a 5.
“Where do you want to put it?” I asked the class.
“Put it on the bottom,” Sheila directed.
“Why do you want a five as the denominator?” I queried while using
proper terminology.
“The only number higher is six, and you probably won’t get a six. You
want a big number for the denominator because that means you sliced it
into more little pieces.”
“OK,” I continued as I wrote 5 in the denominator box. “So we’ve got
one more roll. Take a few moments to talk to a partner about what roll
you’re hoping for. What number would you like as the numerator to get a
fraction close to zero? Talk about why you’re hoping for that number.”
I gave the students some time to consider the possibilities. Many of
them drew pictures to help them. There was consensus that a 1 would be
the best possible number to use as the numerator. I rolled the die and the
result was a 3.
“Oh man!” Jerome lamented. “That’s way too big.”
“Well,” I responded, “it doesn’t seem that bad to me. You had a two,
a three, and a five to work with. What fraction could you have made that
would have been smaller than three-fifths?”
“We could have made two-fifths,” Samantha noted.
“Yes,” I replied as I gestured toward the number line, “you could have
gotten a little closer to zero with two-fifths. Are there any other fractions
you could have made that are closer to zero?”
“That’s the smallest fraction we could make,” Tony stated.
“How do you know?” I challenged.

“Because five was the biggest number and we made it the denomina-
tor, so that means we had the most little pieces possible. Then if we put
two as the numerator, it would be the least number of those little pieces.”
I restated Tony’s conjecture. “So you’re saying that if you want a small
fraction, you need a big denominator and a little numerator. Let’s think
about that for a moment. Look at the fractions we have on the board, talk
to a partner, and see what you think about Tony’s idea.”
I gave the students a bit of time to discuss the relationship between the
numerator and the denominator. Again, the pictures proved very useful in
clarifying the concepts and connecting the symbols to quantities. I left the
students with another question to ponder.
“So,” I summarized, “it seems like you’re starting to develop a theory
about how to make the smallest fraction possible with three rolls of the
die. The next time I visit, we’ll try to make the largest fraction with three
rolls. I wonder if you can come up with some ideas about how to do that.”
Partners can play Comparing Fractions, working either cooperatively or
competitively to make the smallest or largest fraction possible.
Comparing Fractions can easily be adjusted to the particular fraction
concept being studied. If students are working on computation, have them
make a game board that requires them to add, subtract, multiply, or di-
vide. For example, you might challenge them to make the largest product
using a game board like this:
ⴛ ⴝ
Discard
In addition, all versions of the activity can be used with decimals
rather than fractions. Regardless of the version of the activity, students
must have access to visual tools to help them make sense of the numbers
they make and the computation they do. Have them draw pictures and/or
use a number line.

Overview
In Coordinate Tic-Tac-Toe, students practice using ordered pairs and
coordinate graphs in a game context. The game uses the strategy and
mechanics of tic-tac-toe with a few twists. Partners try to get four points in
a row horizontally, vertically, or diagonally on the coordinate graph. Also,
Xs and Os are marked on the intersections of lines rather than in spaces.
Activity Directions
1. Make a coordinate graph on the board or an overhead transparency.
Discuss with students how to use a coordinate graph.
Coordinate Tic-Tac-Toe
x
y
2. Draw a T-chart on the board and label the columns Teacher: X and
Class: O.
Teacher: X Class: O
6
CONTENT AREA
Algebra
MATERIALS
■ Coordinate
Tic-Tac-Toe grids,
several for each
pair of students
(see Blackline
Masters)
projector
TIME
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•
•
•
•
30

3. Begin the game by recording an ordered pair on the T-chart, plotting
the corresponding point on the grid, and marking it with an X (e.g.,
(4, 2)).
4. Have a student choose an ordered pair. Record it on the Class side
of the T-chart and mark the corresponding point on the graph with
an O.
5. Continue taking turns until one side has a line of four points
in a row.
6. Let students play additional games with partners.
Key Questions
• Why did you pick that point?
• What strategy did you use?
From the Classroom
“I’ve got a game for you today,” I announced to Tina Rasori’s class of
fourth graders. “It might seem familiar in some ways. It’s called Coordi-
nate Tic-Tac-Toe.”
I drew a 5-by-5 coordinate graph on the board. I knew the students
had been working on graphing activities lately, so I did a very quick re-
view. “Do you know what this type of graph is called?” I asked.
I was met with blank stares.
“It’s called a coordinate graph,” I told the class. “These graphs are
very useful in algebra. They can show how patterns grow and shrink.
They can also show how something moves or changes over time. Today
we’re going to play a game that will help you understand how to plot
points on a coordinate graph. Ordered pairs are used to name points on
the graph. For example, if I wrote the ordered pair (three, four), where do
you think I’d plot, or mark, my point?”
I wrote on the board:
(3, 4)
“Go over three and up four,” Jose instructed.
“Where do I start?” I asked.
“Where the axes cross,” Cindy explained.
“OK,” I said as I modeled on the board, “starting at the origin where
the axes cross at zero, I go over three on the x-axis and up four. The point
for this ordered pair is (three, four).” As I talked with the students, I put
Coordinate Tic-Tac-Toe 31

my finger on the origin and moved it to the right along the x-axis, count-
ing aloud, “One, two, three.” When I reached the 3, I counted up four,
moving my finger and counting aloud as I did so.
x
y
(3, 4)
x
y
X
I deliberately modeled the physical plotting of the point along with
incorporating the proper vocabulary so that students would become
comfortable and familiar with the terminology and structure.
Next I explained the rules for the games. “In Coordinate Tic-Tac-Toe,
you need to get four in a row instead of three in a row, and Xs and Os go
on the intersections of lines rather than in spaces, like regular tic-tac-toe.
I’ll be Xs and the class can be Os. I’ll start by putting an X on (three,
two).”
I drew a T-chart and a new graph on the board. I recorded my ordered
pair on the left side of the T-chart and then starting at the origin, I mod-
eled, as before, how to plot my ordered pair on the grid.
“Now it’s your turn. You need to think about where you want your O
and what ordered pair describes that point,” I told the class.
“How about (four, three)?” Martin suggested.
Teacher: X Class: O
(3, 2)

I added Martin’s ordered pair to the right side of the T-chart and
carefully plotted the point, beginning at the origin, counting aloud to
four as I moved my finger along the x-axis, counting aloud to three, as I
moved up the y-axis, and marking the O on the graph in the appropriate
place.
“Hmm,” I pondered as I looked at the graph, “I think I’ll take (three,
one).”
I recorded my ordered pair on the left side of the T-chart and then
again plotted it on the graph by beginning at the origin and counting
aloud as I moved my finger along the x-axis to 3 and then up one. Then I
called on Sofi to take a turn for the class.
“Definitely (three, three),” she stated.
“Why did you pick that point?” I asked her.
“Because I want to try to block you,” she explained.
“Aha,” I responded, “that’s interesting. There are really a couple of
things going on in this game. You want to get four in a row, but you also
have to pay attention to what your partner is doing. You might find you
develop some strategies as you play this game.”
I added Sofi’s data to the board, carefully continuing to model how to
plot points.
Coordinate Tic-Tac-Toe 33
x
y
X
O O
X
The game continued until the class had gotten four Os in a row. I de-
liberately let the students win this introductory game both to build their
confidence and to model being a good sport. Next I gave each pair of stu-
dents a Coordinate Tic-Tac-Toe grid (see Blackline Masters) and gave
them a few minutes to play the game on their own. The students enjoyed
the game immensely and I encouraged them to teach it to someone at
home. I knew that as students played Coordinate Tic-Tac-Toe throughout
the year, they would continue to develop strategies and build a solid foun-
dation with coordinate graphing.
Teacher: X Class: O
(3, 2) (4, 3)
(3, 1) (3, 3)

There are many ways to extend this activity. You can let students make
their own coordinate grids for practice with constructing the graphs, but I
usually start by providing prepared Coordinate Tic-Tac-Toe grids, because
sometimes when students draw their own coordinate graphs freehand, the
graphs are sloppy or slanted, making it difficult for students to see if they
have four in a row.
Another way to extend the activity is to have a strategy talk with the
class. Have students share what they have noticed after playing several
times with their partners. Discuss strategies they used and why they seem
to work. After students have had whole-class discussions about their
strategies, they can do some writing about them. Ask students to each
describe a strategy they use and explain why it works.
Also, as students become adept at the activity, they can try a challenge.
Have them use a four-quadrant coordinate graph that includes negative
numbers.
x
y
This version of the activity gives children exposure to negative numbers
in a game context. It also prepares them for using these graphs in future
algebra activities.

Overview
Students use a great deal of logic and language in this guessing game. As
they make guesses and receive clues, they zero in on the secret number.
Guidance from the teacher helps students see the progression of their
guesses and how to make conjectures that narrow down the possibilities
and lead to finding the mystery number.
To begin, choose a three-digit number with no two digits the same. In-
vite the class to guess your number. After each guess, tell the class how
many digits in the guess are correct and how many of the correct digits are
in the correct place.
Activity Directions
1. Think of a secret three-digit number with no repeating digits (e.g.,
332 has two 3s in it, so it’s not a good secret number for this game).
2. Make a recording chart on the board or an overhead with three
columns, labeled Guess, Digit, and Place.
Guess Digit Place
3. Have a student guess a three-digit number (with no repeating digits)
and record it in the Guess column.
4. In the Digit column, write the number of digits that match your
secret number.
5. In the Place column, write the number of correct digits that are in the
correct place in your secret number.
Digit Place
7
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䊝䊞䊟䊞
•
•
•
•
CONTENT AREA
MATERIALS
projector
TIME
35

6. Have another student use the information from the previous guess to
guess a new three-digit number. Record as described in Steps 3–5.
7. Stop periodically to have students discuss what they know about the
number and what number they’d like to guess next.
8. Continue until the class is able to guess your secret number.
Key Questions
• What do you know about my secret number?
• What next guess would you like to make? Why?
• How does this clue help you?
• What digits can you eliminate? Explain.
From the Classroom
“You might have played this game before,” I told Robin Gordon’s fourth
graders. “But it is such a fun one that you can play it over and over. It’s
called Digit Place. Has anyone played it before?”
Several students said they knew the game, while others smiled uncer-
tainly. I could tell this would be an introduction for many of the students,
so I knew I’d need to model carefully and check in with the students
frequently.
“Here’s how it works,” I explained. “I’m going to think of a three-
digit number and you’re going to try to guess it.” (Prior to starting the
game I decided my secret number would be 619.) “I’ll give you clues each
time you guess and the clues should help you narrow down the possibili-
ties and figure out my secret number. So what is the lowest three-digit
number?”
“One hundred,” Raquel blurted out.
“OK,” I said, “and what’s the highest three-digit number?”
“Nine hundred ninety-nine,” William responded.
100–999
“So my three-digit number is somewhere between one hundred and
nine hundred ninety-nine,” I told the class. “However, it can’t be one hun-
dred or nine hundred ninety-nine because when you play this game you
need to choose a number that doesn’t have any repeating digits. One hun-
dred has two zeros and nine hundred ninety-nine has three nines. The
number I picked has three different digits. So, you know my number is

somewhere between one hundred and nine hundred ninety-nine and
you know it has three different digits. Who would like to make a
guess?”
Many hands shot into the air. I called on Alexa, who guessed 413. I
drew a recording chart with three columns on the board and recorded her
number in the Guess column. Then I wrote a 1 in the Digit column and a
1 in the Place column.
Digit Place 37
Guess Digit Place
413 1 1
Guess Digit Place
413 1 1
537 0 0
“You guessed four hundred thirteen. You have one digit and one place
correct. What does that mean?” I asked the class.
“It means that your secret number has a four or a one or a three in it,”
Ricky explained.
“And it’s in the right place,” Ishmael added.
I asked for a volunteer to make another guess.
“How about five hundred thirty-seven?” Miguel suggested.
I added his guess to the board with the corresponding clues.
“Sometimes kids think a guess with no correct digits and places is a
bad guess,” I told the class, “but actually it’s very helpful. How does this
help you think about my secret number?”
Betty explained, “Zero digits tells you not to guess those numbers.”
“Yeah,” Victor added, “you know there’s not a five or three or seven
in your number.”
I wrote the digits 0 through 9 on the board and crossed out the 3, the
5, and the 7.
0 1 2 3
冫 4 5
冫 6 7
冫 8 9
“Miguel’s guess helps you eliminate some digits and focus your
guesses on the remaining possibilities,” I summarized. Writing the digits
on the board and crossing them out along the way can help students focus
and organize their thinking. It also encourages the kind of deductive rea-
soning that is essential in this game.

I took several more guesses. From these guesses, the students were able
to eliminate two more digits, which I crossed off the list on the board.
Then I had the students talk to a partner about what they thought they
knew about my number, what next guess they’d like to make, and why
that guess would be helpful.
0
冫 1 2
冫 3 4
冫 5
冫 6 7
冫 8 9
Guess Digit Place
413 1 1
537 0 0
124 1 0
125 1 0
410 1 1
403 0 0
I called for the class’s attention and shifted the discussion to the logi-
cal thinking.
“A lot of you are raising your hands right now,” I observed, “and I’m
assuming that means you’d like to make a guess. That’s great, but if you
want to make a guess, you also need to explain why you’re picking that
number and how you think it will help.”
I paused for a moment to let that sink in. Then I called on Ontario.
“We want to guess eight hundred nineteen because we know there can
only be a one, six, eight, or nine in your number. Also we think the one
goes in the tens place.”
“Why do you think that?” I prompted.
“Because when you look at one hundred twenty-four and one hundred
twenty-five, they both have one digit correct but it’s in the wrong place.
Then on four hundred ten, there’s one digit correct and it’s in the right
place. And we know it has to be the one because we already crossed off the
other digits.”
“Does that make sense?” I asked the class.
Many students nodded because they had made similar discoveries in
their small-group discussions. I knew some students might not follow
Ontario’s logic, but they would still benefit from being reminded that
math is about making sense. I added 819 to the chart and told the students
that they had two digits correct and two places correct. There was a great

deal of excitement and many hands were eagerly waving. I decided to slow
it down and have the students talk again before making more guesses.
“OK,” I acknowledged, “you’ve got a lot of ideas about my number at
this point. Here’s what you need to do. Talk to your partner again. This
time discuss what you know about my secret number and what guess
would give you the most information and help you figure out what the
secret number is. Make sure you can explain why this guess will be so
helpful.”
I gave the students about a minute to talk to each other. Then I called
on Cynthia.
“We want to guess six hundred nineteen,” she announced. “We know
the one is correct and we know either the eight or the nine is correct. So
we’re going to test out the six and see whether the eight or the nine is in
the right place.”
“How will you know?” I asked.
“Because if you say we only have one digit and one place correct, we’ll
know we need to put the eight back.”
“And if you say three digits correct, we’ll know we got your number,”
Tanner added.
I wrote 619 on the chart and put a 3 in the Digit column and a 3 in the
Place column. The students cheered. They had found my number.
“Wow,” I observed, looking at the game chart, “my number could
have been almost any number between one hundred and nine hundred
ninety-nine. It took you only eight guesses to get it. That’s the power of
logical thinking.”
I congratulated the students on a game well played and encouraged
them to continue trying their skills at Digit Place in the future.
Often, I’ll introduce Digit Place using a two-digit number. That way stu-
dents become familiar with the rules and terminology before they have to
use a great deal of logical thinking. Since a significant number of Robin’s
students already knew the game, I decided to jump right into a three-digit
version. When in doubt, opt for the easier game to ensure success and a
positive attitude about the upcoming challenge.
Once students are quite familiar with Digit Place, they can play on
their own with partners. Even better than one-on-one games are games
where pairs play against each other. This gives students someone to check
their ideas and also prompts more communication throughout the game.
Students can choose a two-, three-, or even four-digit number to guess.
Digit Place 39

Also, if students are curious about why the secret numbers need to
have three different digits, they might want to investigate. Allow them to
try using a number with repeating digits to see what happens and how
clues become confusing.
Digit Place also offers an excellent writing opportunity. After students
have played multiple games, they can write about strategies they use and
how they determine which numbers to guess.

Overview
The connections between numbers and geometry are often overlooked in
the traditional math curriculum. Visual and spatial learners benefit from
ways to see physical models of numbers rather than just think of them as
quantities or symbols. Dot Clusters provides a concrete, geometric way
for students to see numbers as groups of objects and think about compu-
tation in a visual way. Dot Clusters also gives students practice connecting
a visual model to its symbolic equation.
Activity Directions
1. Scatter ten beans on the overhead before turning it on. You might
want to start with a random scattering or you might choose to make
a deliberate configuration.
2. Tell students they need to figure out how many beans are on the
overhead without counting by ones.
3. Turn on the overhead and let students find the total number of beans.
4. Discuss the different ways students grouped the beans to find the answer.
5. Repeat the activity with twelve beans.
6. Show students the Dot Clusters overhead (see Blackline Masters). Let
them look at it for a few moments and discuss with their partners
how they might group the dots.
Dot Clusters
8
䊝䊞䊟䊞
䊝䊞䊟䊞
•
•
•
•
CONTENT AREA
Geometry
MATERIALS
■ overhead projector
■ 12 beans
■ Dot Clusters
overhead
transparency (see
Blackline Masters)
■ Dot Clusters
recording sheets,
1 per student (see
Blackline Masters)
■ overhead
transparency of
Dot Clusters
recording sheet
TIME
41

7. Pass out the Dot Clusters recording sheets (see Blackline Masters).
8. Tell students they need to use equal groupings to find different ways
to total the dot clusters on their sheets. Have students record the
corresponding equation for each dot grouping.
9. Share and discuss the students’ grouping strategies and equations.

Key Questions
• Where did you start?
• What did you see?
• Can you write an equation that describes how you did it?
From the Classroom
I greeted Amy Jackson’s fourth and fifth graders by telling them I had
brought a bag of pinto beans with me. After clarifying what a pinto bean
was and showing them a sample, I moved on to the math activity.
“So I have some pinto beans,” I explained to the class. “I’m going to
put them on the overhead. Your job is to try to figure out how many pinto
beans are on the overhead. The challenge is to try not to count by ones.
See if there are ways you can group the beans to make it easier to count.”
“When we count, can it be an estimate?” Kesha asked.
“That’s a great question,” I responded. “A lot of times an estimate is
good enough. In this activity, however, you need to figure out exactly how
many beans there are. You might want to do a quick estimate first, but
your answer should be the actual number of beans on the overhead.”
I had ten beans in my hand. I put them on the overhead in several
clusters.
Dot Clusters 43
Then I turned on the overhead and gave the students a minute or so to see
the beans and find the total. I asked for the students’ attention so we could
discuss what they saw and did.

“Nod your head if you know how many beans are up here,” I said to
the class.
There were nods all around. I had the whole group say the total in uni-
son so we could get right to the interesting part of the activity. I asked stu-
dents to share what they saw and how they had grouped the beans to find
the total. There were a variety of responses as the children shared their
thinking.
“I counted by twos,” David explained.
“I counted by threes,” Brandy shared.
Adriana told us she saw three groups of three plus one more. Kesha
saw four, four, and two. As each child shared, I tried to clarify and connect
his or her ideas to the bean configuration on the overhead. I pointed to the
clusters of beans and asked questions that helped connect the numbers to
the visual model, such as “Where did you start?” and “Is this what you
saw?”
I repeated the activity again, this time using twelve beans. I encour-
aged the students to look for equal groups that could make counting and
finding the total easier. Finding equal groups facilitates the use of multi-
plication, which is more efficient. After we had discussed ways they saw
the twelve beans, we moved on.
“So now I have a challenge for you,” I announced to the class.
“You’ve had a little practice looking at beans and grouping them in ways
that help you find the total. This time I’ve got a sheet with a bunch of dots.
You need to find ways to group these dots to help you figure out how
many there are. I’ll give you a minute or two to talk to a partner about
how you might group these dots. Then you’ll get a chance to go back to
your tables and work on your own sheet of dots.”
I put the Dot Clusters transparency on the overhead for the children to
see. They talked and pointed at the model and I knew they were ready to
begin working.
“OK,” I told them, “you’ve got some ideas about how to approach
this problem. I have a sheet for each of you. There are six pictures of this
group of dots on it. Your challenge is to see if you can find six different
ways to group the dots using equal groupings and find the total. The extra
challenge is for you to write an equation that describes each way you
grouped the dots.”
“Can we work with a partner?” William queried.
“I think it’s a great idea to work together on this. When you go back
to your tables, you’ll each have your own recording sheet, but you can talk
together and share your grouping strategies with each other.”
I passed out the recording sheets and the children got to work. As I
circulated around the tables, I noticed that some students were simply

circling groups of two or three dots and not necessarily looking at the big-
ger picture. I decided to interrupt them briefly.
“I’m noticing that you’re doing a good job of using equal groups,” I
told them. “Make sure you’re looking at the whole cluster before you
make those groups, though. There might be some ways you can chunk
bigger groups of dots together. Think about the symmetry of the design.
That might help you see groups in different ways.”
The students got back to work and I walked around with an overhead
transparency of the Dot Clusters recording sheet and pen. I copied a vari-
ety of students’ arrangements onto the overhead. Then I called the stu-
dents back to the rug and put the overhead up for them to see.
Dot Clusters 45
Figure 8–1 An over-
head showing different
ways students grouped
the dots.

“You found a lot of ways to solve this problem,” I complimented. “I
picked a few to share with the class. As you look at them, see if you can
figure out what each person did. Also, think about equations that could
describe the different groupings. I’ll let you talk to a partner for a minute
about that.”
I gave the students a minute to look at the samples on the overhead
and talk about equations.
“So let’s start with Cathy’s,” I suggested. “Can anyone describe what
Cathy did?”
“She grouped them by threes,” Blake noted.
“And there was one left over in the middle,” Kesha added.
“How many groups of three did she find?” I asked.
“Eight,” several students responded at once.
“Hmm,” I said, “so how can we write an equation to explain Cathy’s
way?”
“It’s eight times three plus one more,” Ruby answered.
“How do I write that?” I asked.
“Put parentheses around eight times three and then write plus one,”
Isaac explained.
Next to Cathy’s work on the overhead I wrote:
(8 ⫻ 3) ⫹ 1 ⫽ 25
“Does that look right?” I asked the class.
There were nods of approval. We moved on to add equations to the
rest of the examples on the overhead. When we were done, the overhead
looked like Figure 8–2 (see page 47).
I told the children I was impressed with their ability to connect pic-
tures to equations. I encouraged them to continue trying to picture what
equations might look like.
Dot clusters and bean clusters are highly adaptable. Do this as a quickie ac-
tivity and just put some beans on the overhead when there are a spare few
minutes of class time. Giving students recording sheets extends the activity
and allows opportunities for discussions about equations. Create Dot Clus-
ters sheets with different amounts of dots and different configurations. In
any incarnation, the activity encourages students to use geometric models
to solve numeric problems.

A literature connection can be made with books such as Greg Tang’s
The Grapes of Math (New York: Scholastic, 2001). You can read this
book aloud to offer students visual situations in which they will be en-
couraged to cluster objects in equal groups to more easily find the total.
Dot Clusters 47
(4 4) (3 3) 25 5 5 25
(4 4) 9 25
(8 3) 1 25

(9 2) 7 25 (4 2) (5 2) 7 25
Figure 8–2 Equations
explain how students
grouped the dots.

Overview
Measurement is an often overlooked part of the math curriculum.
Typically the topic is relegated to the last chapters of the math book, and
students merely receive a quick overview in preparation for a standardized
test. Sadly, many children today do not get measurement experiences out-
side of school. Even in the upper-elementary grades, students lack experi-
ence with rulers and are unfamiliar with the different measurement types
and purposes. Estimate and Measure gives students different measurement
experiences throughout the entire school year.
Activity Directions
1. Ask students to estimate the number of paces it will take you to cross
the length of the classroom.
2. Record the estimates on the board or an overhead.
3. With the class, determine the range of the estimates.
4. Discuss the estimates and have several volunteers explain their thinking.
5. Walk the length of the room as the students count your paces. Ask stu-
dents whether the actual measurement is within their range of estimates.
6. Repeat Steps 1 through 5, this time using the width of the classroom.
7. Introduce the term linear and its meaning.
8. Have students generate a class list of things to measure using linear
measurement.
9. Have students generate their own written questions about things to
estimate and measure.
Key Questions
• How did you decide on your estimate?
• What did you consider that helped you think about it?
• Can you use the information that you already have to help you make
another estimate?
Estimate and Measure
9
CONTENT AREA
Measurement
Number and
Operations
MATERIALS
projector
TIME
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48

From the Classroom
I wasn’t sure how much measurement experience Melissa Mechtly’s
third graders already had, so I decided to focus on linear measurement
first.
“I’ve been in your classroom a bunch of times,” I told the class. “But
I usually just stay up here by the rug. I want to spend today’s session get-
ting to know the space better. So the first thing I’m wondering about is
how many steps it would take me to walk across the room.”
“Which way?” Omar asked.
The classroom was in a portable trailer that was significantly rectan-
gular in shape. Omar wanted to know if I would be traveling the length or
width of the room. This was a perfect opportunity to introduce some
measurement vocabulary in context.
“Good question,” I responded. “I think I’ll walk the length of the
room, from the whiteboard all the way back to the sink. That’s the longer
way. Maybe I’ll walk the width of the room after. To walk the width of
the room, I’d need to go from the door all the way across to the bulletin
boards.”
I gestured with my arms to clarify which parts of the room I was
referring to. On the board I wrote:
Length of Room Estimates
“OK,” I continued, “let’s hear some of your estimates for the number
of steps I’ll take when I walk across the room. Remember, it’s an estimate,
so you’re making a thoughtful mathematical guess. I don’t even know
what the answer is yet. The important part of estimating is thinking math-
ematically. Good estimates help you decide if your answer is reasonable.
Think for a moment and then I’ll give a signal for you to just say your an-
swers aloud. I’ll record all the ideas on the board.”
I had students “popcorn” their estimates as I wrote them on the board.
The popcorn approach allows students to call out answers in rapid suc-
cession as I record them. I wanted to minimize the risk to individuals and
encourage all students to participate. This approach also quickly yielded
many data to discuss. In less than thirty seconds, the board was full of
estimates.
Length of Room Estimates
20 15 25 40 21 10
22 20 60 15 16 30
18 45 12 25 14 18
Estimate and Measure 49

I started with a few general questions and then I moved on to individ-
ual sharing. “What’s the lowest estimate and the highest estimate?” I
asked.
“Ten is the lowest and sixty is the highest,” Vincent responded.
“OK,” I said, “these two numbers can help us find the range of esti-
mates. How many numbers are there between ten and sixty?”
“Fifty,” several students answered almost immediately.
“OK,” I said, “so the range of your estimates is fifty.”
“Let’s hear about your thinking,” I moved on. “Is there anyone will-
ing to tell us how you decided on your estimate? What kinds of things did
you consider that helped you come up with a number?”
“Well,” Tyrone explained, “I thought about how big your steps are
and I pictured you walking across the room.”
“I did it a different way,” Terry offered. “I thought about how many
steps it would take you to go halfway across the room. Then I doubled it.”
After hearing from several other students, I took the actual measure-
ment. I started at one end of the room and walked across to the other end.
I had the students count each step aloud to keep them all involved and fo-
cused. We established that it took twenty-two steps for me to walk the
length of the room.
“So you know the length of the room is twenty-two of my steps,”
I noted. “Was the actual number of steps within the range of your
estimates?”
“Yes,” Antoinette responded, “because our estimates were between
ten and sixty.”
“OK,” I continued, “now you’re going to think about how wide the
room is, or its width. Can you use the information you already know
about the length of the room to estimate the width of the room? Talk to a
partner and then we’ll listen to your ideas.”
I gave the students some time to discuss this new question. Most of
them compared the length of the room with the width and adjusted their
estimates accordingly. We repeated the Estimate and Measure procedure,
this time measuring the width of the room.
“When we measure the length or the width of something, it’s called
linear measurement,” I told the class as I wrote linear on the board. “Does
anyone see a word within that word that can give you a hint about what
it means?”
“Ear!” Pilar noted enthusiastically.
“Well, yes,” I acknowledged, somewhat surprised. “I didn’t notice
that before, but there it is. Any other words you see in there?”
“In,” Kristin noticed.
“Good,” I replied. “What else?”
“Line?” John offered.

“Yes,” I responded, “Line is in linear. That tells us that when we’re
doing linear measurement we’re measuring a line. When I walked the
length of the room I measured a line across the room. Anytime we meas-
ure a line, whether it’s horizontal, vertical, or diagonal, we are doing lin-
ear measurement. What other linear measure could we do in this room?”
“We can measure the door,” Javier suggested.
“Yes,” I agreed as I walked over to the door, “we could measure the
height of the door or the width of the door.”
“But how could you walk up the door?” Terry challenged.
“Good point,” I responded. “Since I don’t have my antigravity boots
on, it would be pretty difficult for me to walk up the door. Are there any
tools I could use to measure the height of the door?”
“You could use a ruler,” Mia suggested.
“Or a tape measure,” Ignacio added.
“Or a yardstick,” Hakim pointed out.
“Great,” I responded. “You know about a lot of measurement tools.
These tools are very helpful for two reasons. First, they keep me from hav-
ing to defy gravity. Even more important, when we use these standard
tools, everyone knows what we’re talking about. If you went home and
told your family that your classroom is twenty-two Caren steps long, they
wouldn’t necessarily know how long the room is because they don’t know
how big my steps are. But if you told them the measurement in feet or
yards, they could have a pretty good sense of it.”
I refocused the discussion on linear measurement possibilities. The stu-
dents generated a list of things in the classroom that they could measure.
The list included things such as the rug, a book, a window, a desk, and the
calendar. Then we talked about different tools in the classroom that we
could use to do the measuring. This list included standard and nonstan-
dard measuring tools such as rulers, markers, pattern blocks, pencils, and
yardsticks. Satisfied that the students had enough ideas, I sent them back
to their tables to write down some Estimate and Measure ideas in their
journals. The students had many ideas, such as How many pencils long is
my desk? How many inches across is the sink? and How tall is the class-
room in feet? These student-generated suggestions would be used for fu-
ture Estimate and Measure opportunities.
A lot of measurement practice and concept development happen through
repeated Estimate and Measure experiences. I used a nonstandard meas-
urement (my footsteps) to introduce the activity. The nonstandard unit
Estimate and Measure 51

was less formal and allowed students to feel safe in their initial estimating
task. Using standard measurement units such as inches, feet, or centime-
ters helps students get a feel for the size of these units. It also provides
opportunities to use measuring tools and record measurements using
proper notation.
The focus can shift to other types of measurement as well. The same
procedure can be used to focus on area, surface area, volume and capacity,
or weight.

Overview
Estimation Jar delivers a lot of math in a short period of time. In this
activity, students make estimates about the contents of a jar. By offering
visual models of quantities, this activity helps students build their number
sense. It is important to emphasize the thinking involved in making esti-
mates rather than which estimate is right or closest. Over time students
begin to use more sophisticated approaches to making estimates and
determining reasonableness.
Activity Directions
1. Fill a clear container with objects (e.g, counters, cubes, or cotton
balls).
2. Have students estimate how many objects they think are in the jar.
3. Record the estimates on the board or an overhead.
4. Have a few students share their thought processes in coming up with
their estimates.
5. Provide a benchmark for each group or table of students, such as a
handful of ten objects.
6. Discuss how the benchmark helps them think about their estimates.
7. Ask students to determine the number of objects you removed from
the jar. Have students share their calculation strategies.
8. Ask students how many objects they think are still in the jar and how
this information helps them with their estimate of the total number
of objects.
9. Continue distributing handfuls of ten objects to each of the groups,
each time determining the number of objects outside the jar.
10. Repeat Step 9 until you run out of handfuls of ten. Then count the
objects remaining in the jar.
11. Have students determine how many objects were in the jar altogether
and share their computation strategies.
Estimation Jar
10
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CONTENT AREA
MATERIALS
■ clear jar or
container (an
empty olive or
peanut butter jar
works well)
■ cubes, beans,
marbles, or other
objects, enough to
fill the jar
projector
TIME
53

Key Questions
• What did you do or look at that helped you make an estimate?
• How many objects are out? How many objects do you think are
still in?
• How does the new information change your ideas?
From the Classroom
I began the Estimation Jar activity by showing a group of fifth-grade stu-
dents a jar I had filled with color tiles. I asked them to make an estimate
of how many color tiles they thought were in the jar. I reminded them that
they couldn’t know exactly how many tiles there were, so their thinking
was the important part, not whether they were right or closest. This is an
important message for students. Often they experience estimation jars as a
contest rather than a math activity. I wanted to keep the focus on the stu-
dents’ thinking and off the stress of a competition.
After giving students a few moments to think, I asked for volunteers to
share their estimates. I recorded the estimates on the board.
“We’ve got quite a range of estimates here,” I told the class. “I won-
der why. Is anyone willing to talk about what you did or looked at that
helped you make an estimate?”
Several students raised their hands and I called on Nadine.
“I was looking at the jar and thinking about how much it would take
to fill it,” she explained.
“I looked at how big the jar is,” Tyler added.
“You also should think about how small the tiles are,” Quoc
contributed.
These short conversations are extremely valuable as they provide
models of thinking for the entire class.
“OK,” I told the class. “I’m going to give you a benchmark that might
help you think about your estimates. Is it OK to change your estimate
when you have more information?”
The class agreed that it was. I went around the room and put ten tiles
from the jar in the middle of each of the four tables in the room.
“Now you can see what ten tiles looks like. How does this new infor-
mation change your ideas?” I asked.
Miguel explained, “We can see how much space ten tiles fill and we
can figure out how much space there is in the jar.”
“Or we can just picture putting in bunches of ten to see how many fit,”
Rashad offered.

After allowing the students to contemplate the use of their benchmark
tiles, I shifted gears and asked them how many tiles were out of the jar at
the moment. It was fairly easy for them to tell me that there were forty
out, but I wanted to check for understanding and different thought
processes. I asked how they knew I had removed forty tiles from the
jar, and I got two different explanations. One student said that 10 ⫹ 10 ⫹
10 ⫹ 10 ⫽ 40. Another student said 10 ⫻ 4 ⫽ 40. We took a moment to
talk about how these two equations were equivalent and how they con-
nected to the tiles on the tables. Then we moved back to the estimation jar.
I held up the jar so everyone could see. “So we know there are forty
tiles out of the jar right now,” I told the class. “How many tiles do you
think are still in here and how does that help you with your estimate?”
“We took forty out and that was about twenty-five percent,” Marshall
replied.
“That’s the same as one-fourth,” Miguel commented.
Denise jumped on the percent bandwagon. “I’d say there’s about
sixty-five percent of the jar still full,” she offered.
“There’s three-quarters left,” Dolly posited, “and we have forty out,
so it’s going to be about one hundred five.”
Despite Dolly’s miscalculation, I was pleased that students were using
fractions and percents to describe the situation. I decided not to push this
aspect of the discussion at this point because I wanted to make sure all
students stayed engaged and focused on the estimating. I made a mental
note to revisit the fraction and percent ideas in future estimation tasks.
Nakesha brought the conversation back to the tiles in the jar.
“My new estimate is one hundred forty-four,” she explained, “because
if we took ten more out of the jar there would be fifty out altogether. Then
if we took another fifty out there would be about forty-four left. So that
would be one forty-four.”
“I counted the forty we have on the tables and then I looked at all the
tiles I could see on the inside of the jar,” Kevin explained. “It looks like
about seventy-six.”
At this point I put another ten tiles on each table and repeated the se-
quence of questions. I asked how many tiles were out of the jar, how they
knew, and how many they thought were left in the jar. We did two more
rounds of putting ten tiles on each table, so then each table had a total of
forty tiles. We estimated that there weren’t enough tiles to give each table
another ten. I dumped out the remaining tiles and counted them with the
students. We established that there were twelve tiles left. My final question
to the students was “How many tiles were in the jar altogether?” The stu-
dents had all the numerical information needed to figure it out. I gave
them about a minute to do the computation and then had several students
share their strategies with the class.
Estimation Jar 55

Randi pointed to two tables and said, “Forty plus forty is eighty. The
other two tables make another eighty. Eighty plus eighty is one sixty. One
hundred sixty plus twelve equals one seventy-two.” I wrote Randi’s idea
on the board.
40 ⫹ 40 ⫽ 80
80 ⫹ 80 ⫽ 160
160 ⫹ 12 ⫽ 172
Marshall multiplied the number of tables by the number of tiles at
each table. Then he added the remaining twelve tiles to find the total. I
recorded his strategy on the board.
40 ⫻ 4 ⫽ 160
160 ⫹ 12 ⫽ 172
Estimation Jar proved to be a rich experience. It gave students oppor-
tunities to speak, listen, estimate, and do mental computation. Over time
the students would become even more adept at combining their number
sense with computation in order to make reasonable estimates in a variety
of contexts.
There are many ways to modify the Estimation Jar activity. The modifica-
tions not only help keep the activity fresh, but also give students multiple
models for estimation and computation. One simple idea is to keep the
contents (in this case color tiles) the same and use a different-size or
different-shape jar. In this way students can still use the benchmark quan-
tities but apply them to a different space. Conversely, you can fill the orig-
inal jar with a different item (e.g., use marbles instead of color tiles) to
allow students to consider the relative proportions of the objects and
apply them to a constant container.
Another option is to begin with an empty jar and have students esti-
mate as you gradually fill it with benchmark quantities (e.g., add ten tiles
to the jar and have students estimate how many will fill the jar, then add
more to get to fifty, have them estimate again, and so on). Some teachers
have their students take ownership of a weekly estimation jar. Students are
assigned the task on a rotating basis. Each week the selected student is re-
sponsible for bringing in a filled estimation jar from home. This fosters
home-school connections and honors the items that students have in their
home environments.

Overview
During the Function Machine activity, students look at pairs of numbers
and try to generalize relationships and patterns. They also use mathemat-
ical language to describe the relationships and to connect the relationships
to math vocabulary and symbols. Function Machine is a fun and flexible
way to encourage communication and algebraic thinking. It also provides
a context for introducing and using some of the tools of algebra, such as
T-charts.
Activity Directions
1. Draw a “machine” on the board or an overhead. See the example
below.
Function Machine
In
Out
?
2. Tell students when a number is put into your machine, the machine
follows a certain rule to produce the Out number (e.g., add three).
Their job is to figure out the machine’s rule.
3. Remind students of the importance of not yelling out the rule the
machine is following when they think they’ve figured it out.
11
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CONTENT AREA
Algebra
MATERIALS
projector
TIME
ten minutes
57

4. Draw a T-chart and label the columns In and Out.
In Out
Explain to the students that the T-chart is a way to keep track of
what the machine is doing.
5. Have students tell you a number to put into the machine.
6. Record the number on the T-chart in the In column.
7. Tell students what number would come out. Record that number
under the Out column.
8. Continue to add students’ suggestions for the In number to the
T-chart, record the resulting Out numbers, and have students look
for patterns and discuss what the machine might be doing to the
numbers.
9. Add a twist by giving the students an Out number and asking them
to figure out what number went in.
10. Have students describe what they think the rule is.
11. Record their ideas on the board.
Key Questions
• What do you notice about the numbers on the T-chart?
• What might this function machine be doing?
From the Classroom
Danielle Pickett introduced the Function Machine activity to her third
graders by drawing an odd-looking tubular machine on the board. The
picture piqued their curiosity and she immediately had their attention.
“This,” she explained to the class, “is a function machine. A number
goes into my function machine, my function machine does something to
the In number, and a new number comes out. The function machine al-
ways does the same thing to the In number until you guess the rule. Your
job is to guess what my machine is doing to the In numbers.”

The students seemed excited to get started.
“There are a couple of things you need to know first,” Danielle told
them. “After you put some numbers into the machine and see what comes
out, you’ll start to have an idea of what my function machine is doing. It’s
going to be really important that you don’t shout out the answer. We want
to make sure everyone in the class has a chance to think as much as possi-
ble. OK?”
The students agreed.
“However,” she told the class, “if you do think you know what the
function machine is doing, you can test out your idea without shouting it
out. You can put numbers in and see if you can predict what will come
out.”
Danielle felt it was important to get these ground rules out in the open
at the beginning of the activity. In all classes there is a range of learners
and some students are quick to come up with answers or ideas they want
to share. When those students always jump in with their thoughts, it
has some negative effects on other students. First, the students who need
more time to think get discouraged. Second, some students stop thinking
altogether because they know someone else will just say the answer for
them.
The flip side of this dilemma is that students who do have answers and
ideas need to be encouraged. That’s why it’s important to give them some-
thing to do with their ideas other than shout them out. Meeting the needs
of a wide range of learners is one of the biggest challenges in teaching
whole-class math lessons. Making sure everyone has access to the activity
is crucial. For a function machine, a student just needs to be able to sug-
gest a number in order to get involved. This makes the entry point acces-
sible for all students. So two key questions to consider when planning
whole-class math lessons are What’s the baseline entry point for all stu-
dents? and What is the challenge or extension for this activity?
Danielle then drew a T-chart on the board, labeling the columns
In and Out. Danielle said to her class, “This chart will help us keep track
of the In and Out numbers and help us see what’s happening with the
numbers. Who would like to give me a number to put into the function
machine?”
Many students waved their hands in the air. Danielle called on Carol,
knowing that she was an English language learner who was being brave
enough to speak in front of the whole class. Even though Carol would be
saying only one number, it would help bolster her confidence and encour-
age her to take more risks in the future.
“Seven?” Carol tried.
“Well,” Danielle responded as she wrote on the T-chart, “if a seven
goes in, a ten comes out.”
Function Machine 59

“Ooooh!” Amir exclaimed. “I know what it’s doing.”
“Careful though,” Danielle cautioned. “Sometimes you need more in-
formation or data before you know exactly what’s going on. Also, please
don’t shout out the machine’s rule, as this will stop people’s thinking. Does
someone want to try a different In number?”
“Put a nine in,” Janetta suggested.
“Nine goes in, twelve comes out,” the teacher told Janetta while
recording the numbers in the chart. “Anyone else?”
“How about three?” John volunteered.
“When a three goes in, a six comes out,” Danielle told him.
At this point Danielle could tell many of the students had an idea of
what the function machine was doing. She decided to mix things up a bit to
keep them involved while giving others more time and more information.
“Nod your head if you think you know what my function machine is
doing,” she said to the class.
Most students nodded.
“Look at the T-chart,” she told the class. “What do you notice about
the numbers?”
The students looked a bit confused and hesitant, so Danielle asked a
more specific question. “Are the Out numbers greater or less than the In
numbers?”
“Greater,” the class responded.
“So let’s think about this function machine. What operations could it
be using that would make a number bigger?”
“Addition,” Vincent suggested.
“Or multiplication,” Claudia added.
“True,” Danielle agreed. “Do the Out numbers get a lot bigger or a
little bit bigger?”
“A little,” Josue replied.
“So does that tell you anything about the operation the function ma-
chine might be using?”
In
Out
?
In Out
7 10

“It’s addition, because when you multiply a number it usually gets a
lot bigger,” Kanani observed.
“Not always,” John objected. “Like if you multiply by one or zero, the
number stays the same or gets smaller.”
“Good point,” Danielle acknowledged. “Also, we’re using positive
whole numbers. Negative numbers or fractions or decimals don’t always
get bigger when you add or multiply with them.”
Danielle was pleased with this brief exchange. She had wanted to
bring some of the effects of operations to the surface of the discussion so
students would begin to think about them in terms of the Function
Machine. The discussion also allowed Danielle a chance to refer to nega-
tive numbers and fractions, reminding students that it’s a big world of
numbers out there and they needed to be careful not to overgeneralize. She
decided to move on, still using the same rule.
“Now,” she told the students, “I’m going to make it a little trickier.
This time I’m going to tell you what came out and you need to tell me
what went in.”
She wrote 8 in the Out column.
“Talk to a neighbor sitting near you. What went in that caused eight
to come out? Make sure you really explain your thinking so you under-
stand each other.”
Danielle gave the students some time to talk. This talk served two pur-
poses. It allowed students to clarify and verbalize their ideas about the
function machine. It also gave students who hadn’t yet figured out the
function an opportunity to hear others talk about it.
“So what do you think?” she asked the class. “What went into the
machine?”
“Five,” Kanani said with confidence.
“Rats!” Danielle said, “I thought I could trick you, but you’re right.
OK, I’m going to try an even harder one. What if I told you twenty-one
came out? Talk to a neighbor about what went in.”
Danielle deliberately picked a higher number so that the students
would do some mental computation. While it’s important to introduce
the function machine with numbers and operations that are easily
accessible, once the students catch on, there are opportunities to have
them do some more challenging computation in the context of the
machine. She called for the students’ attention and asked for a volunteer
to give her the In number. She wrote 18 on the T-chart after Terrell
shared it.
“All right,” Danielle said to the now very confident group, “I can see
you’re ready for the ultimate challenge. What if I told you that two came
out? This is super challenging, so talk to a neighbor and see what you
think.”
Function Machine 61

Although these third graders had had minimal exposure to negative
numbers, Danielle was curious to see how they might handle this prob-
lem. She eavesdropped on the students’ conversations. Some students were
stuck on zero as the only possibility, but the idea of negative one started to
circulate. She called on Sally.
“Negative one,” Sally exclaimed.
“Wow,” Danielle responded as she wrote 1 on the chart. “How did
you figure that out?”
“It’s because you have to go back three,” Sally explained. “And if
you’re at two and you go back three, you pass zero and get to negative
one.”
“Makes sense to me,” Danielle agreed. “So let’s talk about this func-
tion machine. What does it do to numbers? Let’s see how many different
ways you can find to explain it. Raise your hand if you have a way to
describe what the function machine does.” She called on Antonio.
“It adds three to the number,” he said.
Danielle wrote on the board:
Add three
“Is there another way to describe it?” Danielle asked.
“Plus three,” Marta contributed.
Danielle wrote Plus three on the board and asked for other ways.
“Take away three,” Nikki said.
“Hmm,” Danielle responded, “I guess that’s true if you look at the
Out number first. Let me think of a way to write that.” She wrote on the
board:
Out number 3 In number
“Does that work?” she asked the class.
The students nodded in agreement.
“So can I also write this?” Danielle asked.
In number 3 Out number
“Yes,” the class replied.
“So there are a lot of ways to describe what this function machine
does,” Danielle summarized. “I’m impressed with all your thinking
today. You’ve gotten so good at this, maybe you can think of some func-
tion machines and I can try to guess them next time.” The children were
excited by the idea and looked forward to creating their own function
machines.

Function Machine is very adaptable. Start with a fairly simple function
(e.g., 3). Depending on students’ levels and area of study, any numbers
and operations could be used. Also, you might choose to use a combina-
tion of numbers and operations (e.g., 2 1). As students become famil-
iar with function machines, they can start to make up their own rules. Stu-
dents can then take on the teacher role, leading the activity. It’s a good idea
to have children prepare their own T-charts for the In numbers from zero
to twenty along with the resulting Out numbers. Check the chart prior to
having a student share his or her rule. This prevents errors and embar-
rassment. It also speeds up the process, keeping all students on task. Of
course this means that students must suggest numbers from zero to twenty
when trying to guess the rule.
Function Machine can also easily be connected to coordinate graph-
ing. The coordinate pairs generated on the T-chart can be plotted on a
graph. Students can then discuss what they notice about the graph and
how it’s related to the function.
Function Machine 63

Overview
Guess My Number invites children to consider the structure of the num-
ber system while engaging in a logic game. Students try to guess a secret
number from within a given range of possibilities. Guess My Number also
presents an opportunity to reinforce mathematical symbols such as the
“greater than” and “less than” signs. Through Guess My Number, stu-
dents find the usefulness of number lines as tools for solving problems.
Activity Directions
1. Choose a number between a given range (e.g., 1–10, 1–20,
500–1,000). Draw a number line on the board or an overhead to
represent the range.
2. Tell the students, “I’m thinking of a number between __ and __. Who
has a guess?”
3. Elicit guesses from the students, responding each time with a clue.
For example, if a student guesses fifteen, tell the class whether your
number is greater than or less than fifteen.
4. Record the guesses and your responses on the board.
5. Ask students to explain why their guesses make sense.
6. When the students have narrowed down the possible secret numbers,
ask them to discuss with a partner the following: what they already
know about the secret number and what next guess they would like
to make.
7. Lead a brief discussion for students to share their thinking and then
continue to take guesses until the students discover the secret number.
Key Questions
• What do you know already about my number?
• How does that guess help you?
Guess My Number
12
CONTENT AREA
Number and
Operations
MATERIALS
projector
TIME
five to ten minutes
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64

From the Classroom
Patty Stark’s fifth graders had just started their Tuesday morning. As they
settled into their seats, I went to the board and drew a box with a question
mark inside it.
“Good morning,” I greeted the class. “I’ve got a secret number for you
to guess. Since it’s early in the morning I’m going to make it pretty easy for
you. I’ll tell you the number is somewhere between one and one hundred.”
“Could it be one hundred?” Martina asked.
“Yes,” I responded. “It could be any number between one and one
hundred, including one or one hundred. You can guess a number and I’ll
tell you if my secret number is greater than or less than your guess.”
I drew a number line on the board to help the students keep track of
their guesses.
Guess My Number 65
1 100
“How about fifty?” Latoya asked.
“The secret number is less than fifty,” I told her as I wrote this infor-
mation on the board. I also marked 50 on the number line with an arrow
indicating all the numbers fifty and above were too large.
1 50
⬍50
100
Before I took any more guesses I decided to have a brief discussion
about strategies for guessing. I had deliberately picked an easy number to
start with so we could focus on the mechanics and thinking involved with
the game.
“I’m a bit curious, Latoya,” I said. “Is there a particular reason you
chose fifty?”
“Yeah,” Latoya responded. “I knew your number was between one
and one hundred, so I picked fifty because it’s in the middle.”
“So how does that guess help you?” I pushed.
“Because it splits the numbers. Since you said the secret number is less
than fifty, I know it’s in the bottom half of the numbers.”
I summarized Latoya’s strategy with accompanying references to the
number line. “I think I get it,” I told the class. “All the possible numbers

are between one and one hundred. So if you guess a number right in the
middle, you can figure out which half the secret number is in and then you
can just throw away the other half and not have to worry about it.”
I proceeded to take some more guesses. Within a minute the number
line looked like this:
40 100
50
30
10
1
I decided to encourage some partner talk. I had students pair up and
briefly discuss two things—what they thought they already knew about
my secret number and what number they’d like to guess next. I wrote the
two prompts on the board to help them stay focused. Also, I was giving
them a preview of the discussion to come.
What do you already know about my secret number?
What next guess would you like to make? Why?
After a few minutes I called for the students’ attention. “So can any-
one tell something you know about my secret number?” I asked.
“It’s more than forty,” Hilario offered.
“It’s between forty and fifty,” David added.
“It’s in the forties,” Brenda posited.
“So you already know a lot about the secret number,” I validated. “So
with all this information, what number would you like to guess next? If
you raise your hand to guess a number, you also have to be willing to ex-
plain why you think that number is a helpful guess.”
I called on Destinee.
“Forty-five,” she volunteered.
“Why is forty-five helpful?” I asked.
“Because,” Destinee explained, “we know the secret number is in the
forties and forty-five is in the middle of the forties.”
“It’s just like Latoya’s idea,” Reynaldo chimed in. “You can get rid of
half the numbers that are left.”

“Aha,” I responded, “so you’re using your logical thinking to help
eliminate a bunch of possibilities with one guess. Well, I’ll tell you that the
secret number is less than forty-five. Talk to a partner again about what
you know about the number now and what guess you’d like to make
next.”
I let the students talk to each other as I added the new information to
the number line. Many students wanted to use the same strategy and pick
the number that was halfway between forty and forty-five. We took a brief
detour to establish that 42.5 was the midpoint between forty and forty-five,
but I explained that this Guess My Number game involved whole numbers
only, so they needed to choose either forty-two or forty-three.
Christina guessed forty-two. I told her that the secret number was
greater than forty-two and recorded this information on the board. Then
Kenny guessed forty-nine. Some students expressed frustration with his
guess since they already knew the number was less than forty-five. I
stopped briefly to have a talk about maintaining a safe environment.
“This is a new game we’re playing today,” I told the class. “Part of
learning and trying new things is making mistakes. It’s really important
that everyone in the class feels safe enough to share his or her ideas and
sometimes make mistakes. That’s how we learn. If you disagree with
someone or you have a different idea, that’s fine. Just make sure you com-
municate that in a way that won’t hurt anyone’s feelings. Do you know
what I’m talking about?” I asked the class as I looked at each student.
“Yes,” the students murmured.
I added Kenny’s guess to the board and moved on.
“How about forty-three?” Lisa suggested.
I wrote 43 on the board and circled it.
“Yes,” I congratulated. “My number is forty-three. I’m impressed
with everyone’s thinking. It could have been any of one hundred different
numbers, and it took you only eight guesses to get it. That shows you used
a lot of good mathematical thinking.”
Before I left the class, I called on a pair of students to lead the class in
another round of Guess My Number.
Guess My Number works equally well with fractions, decimals, or per-
cents. Giving the students some visual tools is essential. Using a number
line helps students compare numbers and order the numbers.
A 1–100 chart is another tool that works nicely for Guess My Number.
Tell students that the secret number is somewhere on the 1–100 chart.
Guess My Number 67

Cross numbers off the chart as they are eliminated. Familiarity with a
1–100 chart gives upper-elementary students a distinctive edge when it
comes to mental computation and understanding our number system.
When students have a visual model of the chart in their heads, they can
easily jump around using tens. They also have a useful geometric model
(the 10-by-10 square) to get a feel for how numbers are related to one an-
other. Playing Guess My Number with a 1–100 chart gives students fur-
ther exposure to the chart and pushes them to articulate some of the num-
ber relationships inherent in it.

Overview
Guess My Rule is a highly adaptable game that encourages children to
consider attributes of numbers and sort and classify them. Students make
guesses about a secret rule relating to numbers using a Venn diagram as a
visual model. The teacher puts the students’ guesses in the appropriate
section of the Venn. As the sections of the Venn fill with information,
students look for patterns and relationships to discern the rules that
might apply.
Activity Directions
1. Choose a rule (e.g., odd numbers, multiples of three, prime
numbers).
2. Draw a circle on the board or an overhead and explain the activity to
the students. Remind them not to shout out the rule when they have
figured it out.
3. Ask a student to tell you a number.
4. Put the number in the circle if it fits your rule. Put it outside the circle
if it doesn’t fit your rule.
5. Continue eliciting numbers and placing them in the appropriate area.
6. After recording several numbers, have students discuss the
information they have so far.
7. Ask students for numbers that will either go inside the circle or go
outside.
8. Ask the class to state the rule.
9. Repeat the activity with two different rules about numbers and a
two-circle Venn diagram.
Guess My Rule
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CONTENT AREA
Geometry
MATERIALS
projector
TIME
five to ten minutes
69

Key Questions
• What do you notice about the numbers inside the circle?
• What is the next number you want to try?
• Can anyone tell me a number that you think belongs inside/outside
the circle?
• If I told you a number, could you tell me where to put it?
From the Classroom
I started Guess My Rule with Patty Stark’s fifth graders in a simple format.
Prior to beginning the activity, I chose an easy secret rule—odd numbers.
To begin the activity, I drew a large box on the board and drew a circle
inside it.
“OK,” I told the class, “I’ve got a secret rule about numbers. You are
going to tell me a number. If that number fits my rule, I’ll put it inside the
circle. If it doesn’t fit the rule, I’ll write it outside the circle. After a while
there will be a bunch of numbers up here and you’ll start to notice things
about the numbers that might give you clues about my rule. Here’s some-
thing important.” I paused for emphasis.
“If you think you know the secret rule, don’t shout it out. We want to
give everyone in the class enough time to think about it. If someone shouts
out the answer, it cuts off everyone else’s thinking. Get it?”
I surveyed the room with my serious face.
“Good,” I moved on. “Raise your hand if you’d like to give me a
number.”
“Nine,” Timothy offered.
“Nine goes inside the circle,” I replied as I wrote 9 inside the Venn
diagram.

“How about three?” Aja suggested.
“Three goes inside the circle as well,” I told her.
I took several more numbers and added them to the board.
Guess My Rule 71
10
12
9
3
5
I stopped the number sharing and had the students refocus on the in-
formation they already had.
“Here’s what I’m going to do,” I told the class as I looked at my
watch. “I’m going to give you forty-seven seconds to talk to people at your
table about what you’re thinking about the secret rule and what next
number you might want to try. Ready, go.”
Using a short time limit forces students to get right to the point.
There’s some psychological component to having an odd number of sec-
onds in which to complete a task or discussion; students seem to rise to the
occasion and get to work quickly. Maybe it’s just the strangeness of the
situation, but whatever it is, it works.
I made an ostentatious show of keeping track of the time. When the
time was up, I called for the class’s attention.
“Time’s up,” I announced. “I’m curious to know what you’re thinking
at this point, but I don’t want to anyone to say the answer yet. You can
show me with your thumbs. Put you thumb up if you think you know the
rule. Put your thumb down if you don’t know the rule. Put your thumb to
the side if you’re not sure.”
Almost all the students had their thumbs up, so I tried a new twist to
keep the game interesting both for those who knew and those who needed
a little more time.
“Well,” I prompted, “if you think you know the rule, tell me a num-
ber that fits inside the circle.”
Many hands shot up. I called on Jessica.
“Eleven,” she said.
“Yes,” I agreed, “eleven fits the rule.”
“So does twenty-seven,” Victor added.
“Right again,” I responded. “Can anyone tell me a number that goes
outside the circle?”

“Fifty,” Jasmin volunteered.
“Good,” I said. “Now how about this challenge: I’m looking for a
three-digit number that goes outside the circle.”
“Four hundred fifty-two,” Alejandro suggested.
“Great,” I responded as I wrote the number on the board. “How
about a four-digit number that fits outside the circle?”
“One thousand seven hundred sixty-two,” Marvin volunteered.
“Wow,” I said, “it seems like you’ve figured out the rule. If I told you
a number, would you be able to tell me whether it goes inside or outside
the circle?”
Heads were nodding around the room. Many students had raised their
hands, each eager to be the one to say the rule. I diffused the situation by
allowing everyone to be the hero.
“Here’s what’s going to happen,” I told the class. “I’m going to count
to three. After I say ‘three,’ you say the rule in a quiet voice. Ready? One,
two, three.”
“Odd numbers,” the students chorused.
“You got it,” I confirmed. “Now I think you’re ready for the chal-
lenging level.”
I drew a large rectangle on the board and drew two overlapping circles
inside it.
“This time I’ve got two secret rules,” I explained. I pointed to the dif-
ferent sections of the Venn I had created. (I had decided to use multiples of
three for one rule and even numbers for the other rule.)
“Some numbers you say will fit one rule, some will fit the other rule,
some will fit both rules, and some will fit neither rule.”
I followed the same sequence I had done with the single-rule version.
Individuals told me numbers and I wrote them in the appropriate area.
After a few minutes, the board looked like this:

Guess My Rule 73
5
7
12
6
15
3
10
4
I had the students stop and talk at their tables for fifty-three seconds.
(I’m telling you, it works.) Then I called them back together for a brief
discussion.
“I noticed that there were all even numbers on the right and all the
numbers outside were odd,” Jessica pointed out.
“But the numbers on the left side are odd, too,” Caleb pointed out.
“The ones on the left are counting by threes,” Ana observed.
“They’re the multiples of three,” Eric elaborated.
“Yes, that was one of my rules,” I told the class as I labeled the circle
on the left.
“What about this other circle?” I asked. “What do you think the rule
is for this one?”
“Those are even numbers,” Phillip observed.
“Yup,” I agreed as I labeled the other circle. “So what about these
numbers in the middle?”
“They fit both. They’re even and they’re multiples of three,” Stacie
explained.
I confirmed Stacie’s assertion. I thanked the students for their work
and their thinking. I left feeling that Guess My Rule was both an engag-
ing and a mathematically worthwhile activity that could easily be modi-
fied and revisited throughout the year.
Guess My Rule is a very malleable activity. You might introduce the activ-
ity using only one rule, but then increase the challenge by using two or
even three rules simultaneously. Once students are familiar with the
format of the game, they can make up their own rules and lead class
games. When using more than one rule, make sure the rules have some
overlapping possibilities and are not mutually exclusive (e.g., odd numbers
and even numbers wouldn’t work because there’s nothing that would go in
the overlapping section).

Guess My Rule translates extremely well to geometry. Use rules about
shapes or other geometric attributes. The activity also works well with
concrete materials. Buttons, pattern blocks, even people can be physically
sorted. Students can use the visual models to ascertain the similarities and
discover the secret rules.
Guess My Rule can also be easily integrated into other curricular
areas. In a science unit on animals, for example, a Guess My Rule game
can feature mammals and animals that live in trees. In history or social
science, it can be used to compare states, countries, or cultures. In any
context, Guess My Rule helps students develop the problem-solving, rea-
soning, and communication skills that are integral parts of the NCTM
Standards.

Overview
In this activity students use the benchmark number of one hundred to
explore other numbers and their relation to one hundred. This activity
also gives students opportunities to employ their understanding of place
value and practice mental computation. Modeling the use of a number
line provides students with a visual tool for computation. In addition, the
explicit use of benchmark numbers (twenty-five, fifty, seventy-five, one
hundred) encourages students to connect unfamiliar problems with ones
they already know.
Activity Directions
1. Roll three dice or have students roll the dice. If more than one roll is
the same, roll a die again until you have three different numbers to
work with.
2. Record the results of the rolls on the board or an overhead.
3. Tell students to make all the possible two-digit numbers with the
rolls and record them on board.
4. Create a 0–100 number line and have students help you locate the
benchmark numbers 25, 50, and 75 on it.
5. Have students discuss in pairs where their two-digit numbers should
go on the number line. Then ask for volunteers to tell you how to
place the numbers.
6. Ask students which number is closest to 100 and which is farthest
away.
7. Have them use the number line to compute how close and how far
away each of the numbers is from 100. Ask students to share their
thinking and record their strategies on the board.
Key Questions
• How far away from one hundred is this number?
• How can you use what you know about benchmark numbers to help you?
How Far Away?
14
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CONTENT AREA
MATERIALS
projector
■ 3 dice
TIME
ten minutes
75

From the Classroom
I introduced How Far Away? to Melissa Mechtly’s third graders by telling
them I’d heard a lot about how much they liked math and what great
thinkers they were. “So I thought you’d be the perfect class to try a new
game,” I continued.
“We love math games!” Charlie informed me.
“Great,” I replied. “Then let’s get started. I brought some dice with me
today. In lots of games you roll one or two. For this game I’m going to roll
three.”
A murmur of excitement spread across the room.
“Yes,” I acknowledged, “it’s a pretty special game and that’s why I
wanted to find the right class to share it with.”
“So here’s how it works,” I continued. “I’m going to roll the three dice
and record the results on the board. You’re going to try to find all the dif-
ferent two-digit numbers you can make with the numbers I rolled. What
do I mean by a two-digit number?”
“Like twelve,” Reggie offered.
“Yes,” I agreed. “Twelve is a two-digit number. What are some other
two-digit numbers?”
The students shouted out a bunch of numbers. I settled them down
and then continued.
“You do know a lot of two-digit numbers,” I commented. “Turn to
a partner and explain how you can tell if it’s a two-digit number or
not.”
I gave the students less than a minute and then moved on. To begin, I
asked three different students to roll a die. When two students rolled 6s I
had one roll again because we needed three different numbers. I wrote the
numbers on the board.
6 2 4
“So there are your three numbers,” I told the class. “Now your job is
to make as many two-digit numbers as you can from those numbers. Can
anyone think of an example?”
“Sixty-two,” Natalie said.
“Yes,” I agreed, “we can take the six and the two and make sixty-two.
What other numbers can we make?”
It didn’t take long for the students to generate all six possibilities,
which I recorded on the board. I chose not to have a discussion about
probability and combinations at that point, but it would definitely be
worthwhile to do so in the future.

“OK,” I told the class, “now we’ve got these six numbers and we want
to think about them in terms of a benchmark number, one hundred.
Benchmark numbers are friendly numbers that we use and see a lot.”
I drew a number line on the board and labeled 0 at one end and 100
at the other. I asked the students where the benchmark numbers 25, 50,
and 75 would go and placed them in the appropriate spots.
“So, let’s think about these other numbers,” I told the students. “Turn
to a partner and quietly discuss where you think each of these numbers fits
on the number line and explain why you think that. Make sure you really
explain your ideas clearly so your partner understands. Also, make sure
you’re being a polite and active listener if your partner is talking. I’ll give
you about a minute and a half. Ready? Go.”
While the students talked to each other, I circulated and listened to
their discussions. Most of their ideas centered on the benchmark numbers
we had already labeled and the concepts of more and less. For example,
Patricia told her partner that 46 would go close to 50 but to the left of it
since it was a little less than 50. When I was satisfied that the students had
had sufficient time to accomplish the job, I called for their attention. The
students directed my placement of their numbers. I used a different-colored
marker to distinguish their numbers from the benchmark numbers. In a
few minutes, we had placed all of the numbers on the number line.
How Far Away? 77
0 50 100
24 26 46 64
62
42
25 75
I asked the students to identify the number that was farthest away
from 100.
“Twenty-four,” they shouted in unison.
“Because, look,” Jon explained. “It’s closer to zero than the other
numbers.”
“Yeah,” Dinora agreed. “Also, it’s the lowest number up there.”
“OK,” I told them, “I’m convinced. Now here’s my next question.
How far away from one hundred is it? Think for a moment on your own
and then I’m going to have you turn to a partner to talk about it. I’m hop-
ing that this number line and the benchmark numbers will help you an-
swer the question. You might want to look at it and talk about it with your
partner.”
While I wanted the students to feel free to solve the problem in a com-
fortable way, I didn’t want them to fall back on their same old methods

without considering the number line as a potential tool. There’s always a
delicate balance between supporting student autonomy and encouraging
students to try new models. A gentle reminder or suggestion can help
move children to new ways of thinking about problems without mandat-
ing. Also, carefully framing questions can help students focus on the de-
sired concept or strategy. I tried a question. “So who can tell us how you
used the number line to help you solve the problem?”
Several hands shot into the air. I called on Janekya. Her response
told me she had prior experience using number lines and increments of
ten.
“I pictured a number line and I started at twenty-four. I did eight hops
of ten.”
“Hold on one minute,” I requested. “Since our number line doesn’t
have tens, I’m going to draw a new one so we can see your thinking.”
I drew an open number line on the board.
24
24 34 54
44 104
94
84
74
64
ⴙ10 ⴙ10 ⴙ10 ⴙ10 ⴙ10 ⴙ10 ⴙ10 ⴙ10
“So you started at twenty-four and kept adding tens?” I checked.
“Yes,” Janekya responded.
I created a visual model on the number line. I kept the whole class in-
volved by having the students tell me the sum after each addition of ten.
Shortly we had a number line that looked like this:
“So what happened when you got to one hundred four?” I queried.
“Well, I knew I was over by four, so I had to take four away from the
eighty,” she explained.
I wrote 80 ⫺ 4 ⫽ on the board and asked the class for the answer. The
students agreed it was 76.
“Did anyone use a number line in a different way to figure out how far
twenty-four is from one hundred?” I asked.
Jon volunteered. “I saw that twenty-four was close to twenty-five,” he
began. “Then I knew twenty-five plus seventy-five is one hundred, so

twenty-four is one less than twenty-five, so I gave one to the seventy-five
to make it seventy-six.”
Although I understood Jon’s thinking, I wasn’t sure everyone had fol-
lowed along. “OK,” I said, “let me see if I understood you and can put
your ideas in my own words.”
I referred to our original number line and pointed to the numbers as I
talked about them. “You noticed that twenty-four was close to twenty-
five. You knew twenty-five plus seventy-five equals one hundred because
those are friendly numbers.” I wrote on the board:
25 ⫹ 75 ⫽ 100
“But,” I continued, “this question was about twenty-four, not twenty-
five. So you had to think about what number goes with twenty-four to
make an equation that totals one hundred.” I wrote another equation
under the first one.
25 ⫹ 75 ⫽ 100
24 ⫹ ⫽ 100
“Yes,” Jon jumped in. “And since twenty-four is one less than twenty-
five, I had to give the one to the seventy-five to make it seventy-six.”
I pointed to each number as I repeated Jon’s explanation of compen-
sation. Then I wrote seventy-six in the box to complete the second
equation.
“It’s interesting that we heard about two different ways to use a
number line to find the difference between two numbers. I’m sure you’ll
do this activity in the future with your teacher so you’ll have even more
chances to think about benchmark numbers and use number lines to
help you find the differences between numbers,” I said to wrap up the
activity.
Introduce this activity the same way for older students, but once they
catch on, change the benchmark to 1,000 and add one more die. Since
there are significantly more three-digit number combinations, narrow the
focus. Just call on three or four students and have them generate some
different three-digit numbers for the class to use. As an alternative, ask
the class to use three of the numbers on the dice to make a number that’s
How Far Away? 79

as far away from 1,000 as possible. They can also try to get as close
to 1,000 as possible or try to make numbers that are about 500 away,
250 away, and so on. This reinforces place-value concepts and encour-
ages rounding and estimation.
How Far Away? can also serve as a context for probability investiga-
tions. Students can explore how many two-digit numbers can be made
with three dice and how many three-digit numbers can be made with
four dice.

Overview
In this activity students use graphs to generate and analyze data related
to themselves and their surroundings. Graphing activities are excellent
routines for upper-elementary students. Regular work with graphs helps
students analyze and organize data, think critically, and internalize mathe-
matical vocabulary. A common upper-grade standardized test item involves
showing a graph or table and asking students to identify the median, mode,
or range. Looking at Data reviews these concepts.
Activity Directions
1. Ask students a question that will elicit a range of numerical responses
(e.g., How many people live in your home? or What is the last digit
of your phone number?). Estimation questions work well, too (e.g.,
About how many cars are parked outside the school? or About how
many teeth are in the classroom?).
2. Use a thick, dark marker to record each student response on its own
sticky note.
3. Post the sticky notes randomly on the board as you go.
4. Have students identify the lowest and highest estimate. Place the
sticky note with the lowest number at the left end of the board and
the sticky note with the highest number at the right end of the board.
Then write the numbers on the board as a range (e.g., 171–388).
5. Ask students to find the range by calculating the difference between
the lowest and highest numbers.
6. Let individual students share their approaches as you record on the
board using mathematical notation.
7. Have students arrange all the sticky notes in order from lowest to
highest so they can find the mode and median.
8. With the students, find the mode and the median.
Looking at Data
15
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CONTENT AREA
Data Analysis and
Probability
Measurement
MATERIALS
■ 3-inch-by-3-inch
sticky notes
■ a thick, dark
marker
TIME
81

Key Questions
• What is the difference between the highest and lowest numbers?
How do you know?
• What is the range?
• How can we find the median?
• How can we find the mode?
From the Classroom
I posed a question to the class in order to generate estimates that would
become the data for our graph. I had a mixed group of third through fifth
graders so I chose a question that would be accessible to the younger stu-
dents while delivering some intrigue to the older ones.
“If I started at the front door and walked all the way around the
perimeter of the school, about how many cars would I find parked on the
street?” I asked.
Two sides of the school were bordered by streets. From the classroom
window students could see one side of the school’s perimeter lined with
cars. Many started from there, counting the visible cars and using that
number as a benchmark. As individuals shared their estimates I used a
thick, dark marker to record them on sticky notes. I arbitrarily slapped
each sticky note on the board before taking the next estimate. (I chose to
write the numbers myself in order to save time, but I have also done this
activity by giving each student a sticky note and marker and having stu-
dents record and post their own estimates.)
When we finished recording and posting the estimates, we had a board
haphazardly filled with sticky notes. This “mess of numbers” became the
source of a rich discussion about data.
“Hmm,” I said, “there are a lot of numbers up here. Look at them
carefully and see if you can identify the lowest number and the highest
number on the board.”
I gave the students a few moments to peruse the board and waited
until many hands were raised. I called on Kellen, who told us 16 and 210.
I checked in with the rest of the students to see if they agreed. After estab-
lishing consensus, I pulled the 16 sticky off the board and put it at the far
left end. Then I put the 210 sticky at the right side of the board.
“OK,” I said, “the rest of your estimates are between sixteen and two
hundred ten. That helps us figure out the range. What is the range?”
I saw that some of the students were counting the sticky notes on the
board instead of calculating the actual difference between the two numbers,

so I clarified my question. “I’m noticing that some of you are trying to fig-
ure out how many sticky notes are on the board. You’re right that all the
sticky notes are between sixteen and two ten, but what you need to figure
out is the difference between sixteen and two hundred ten. The sticky notes
show some of the numbers, but we need to find out how many there’d be if
we had every number between sixteen and two ten.” I recorded the range
on the board to help students visualize the problem: 16–210.
This explanation helped the students get on track, but again I noticed
that some students were struggling to do the arithmetic mentally. I told the
students that they could work on the problem in their math journals if
they liked. After a couple of minutes I called for their attention and asked
who would like to tell us how he or she found the range. Here was a nice
opportunity to have a brief number talk.
“It’s one hundred ninety-four,” D’andre announced.
The class agreed, so I pushed on.
“Because I just did two ten minus sixteen and that’s one ninety-four,”
he explained.
I knew that D’andre had done the standard subtraction algorithm in
his math journal so I didn’t pursue it further.
“Did anyone do it a different way?” I asked the class.
“I added up,” Marsha announced.
“Tell us how you did that,” I encouraged.
“Well, I added four to sixteen to get to twenty. Then I added hundred,
so I was at one hundred twenty. Then I added ninety more to get to two
hundred ten.”
“So how did you know the range was one hundred ninety-four?”
I asked.
“Because I added four plus hundred plus ninety, and that’s one ninety-
four,” Marsha explained.
Since Marsha’s strategy might not make complete sense to students
who had not used it or been exposed to it before, I decided to record it on
the board to demonstrate her thinking for other students. I wrote:
16 4 20
20 100 120
120 90 210
4 100 90 194
“I added and subtracted,” Tina shared next.
“How did you do that?” I queried.
Tina explained her method. “Well I started at sixteen and added one
hundred. Then I was at one hundred sixteen. I added another one hundred
Looking at Data 83

and that made two hundred sixteen. I was past two hundred ten, so had to
subtract six.”
I realized this method called for a visual model as well. I wrote on the
board:
16 100 116
116 100 216
216 6 210
100 100 200, 200 – 6 194
“Great,” I said. “There are a lot of different ways to find the range.
The range helps you know about the data and which numbers you’re deal-
ing with. There are some other things to look at that also help you analyze
data and figure out what it all might mean. But as I’m looking at the
board, I still see a big mess. Let’s try to put the numbers in order. We al-
ready have the sixteen at one end and the two hundred ten at the other.
Who would like to help arrange the rest of the numbers?”
Many hands flew into the air. I called on Esteban and Paulina to come
forward and rearrange the sticky notes. They got a lot of help from the
rest of the class. I helped them by asking questions such as, “Which is
the next lowest number?” and “Are there any numbers between these
two numbers?” I also showed them that when two sticky notes had the
same number, they should put one above the other. It took them only
about a minute to get the sticky notes in order. Then we had an instant
graph and we were ready to look for the median and the mode.
“Now let’s find the mode. How can we tell what the mode is?” I asked
the class.
“It’s the number you see the most,” Tina reminded us.
“So what is the mode?” I asked, pointing to the graph. Since the sticky
notes were placed along the board in a horizontal line with repeated num-
bers stacked on top of one another, it was easy to see that 100 was the
mode because it had the tallest column.
“Now let’s see if we can find the median. How can we do that?”
I asked.
“The median is the one in the middle,” Chip explained.
“So how can we tell which one is in the middle?” I wondered aloud.
“Count up all the sticky notes and halve that number,” Kesha
suggested.
“That would work,” I agreed.
“Start at each end and move into the middle,” Nicole offered.
“Let’s try that,” I agreed.
I asked for two volunteers to help us. Michael and Tanisa obliged.
Michael started at 16 and Tanisa started at 210. They each put a finger on

the first sticky note on their end and then walked toward each other as we
counted each note. They were nose-to-nose when Michael had a finger on
60 and Tanisa was touching 75.
“Hmm,” I observed, “there seem to be two sticky notes in the middle,
not just one. I wonder what that’s about.”
“It’s because there’s an even number of estimates, “ Miguel explained.
“Yes,” I agreed. “Does anybody know how to handle it if there are
two numbers in the middle instead of just one?”
There were universal looks of uncertainty, so I explained the mathe-
matical convention to the students. “If you have an odd number of data,
you can find the one that’s exactly in the middle, and that’s the median. If
you have an even number, you take the two numbers in the middle and
find the number that’s between those two.”
Having told them the convention, I gave the students a few moments
to determine that the median was 67.5.
“OK,” I said, “let’s review what we know about the data.” I wrote on
the board:
Range: 194
Mode: 100
Median: 67.5
“This kind of information is very useful,” I concluded. “When we
started we just had a messy bunch of numbers on the board. Now we have
some information about the numbers that can help us think about those
numbers and see how they fit together. If we wanted to go outside and ac-
tually count the cars parked around the school, we could see if the actual
number was within the range of our estimates. We could also compare the
actual number to the mode and median of our estimates. When you know
the range, mode, and median, you can organize and analyze mathematical
information.”
There are many ways to adapt and extend this activity. By asking differ-
ent questions, you can easily manipulate the magnitude of the numbers
and the size of the range. At times you might choose to collect the data
and just focus in depth on one aspect. Asking questions such as, “Who
would need to know this information?” and “Who would care what
the mode is?” requires students to apply theses concepts to real-life
Looking at Data 85

examples. As a homework assignment students can look for graphs in
newspapers and magazines and bring them in to discuss and analyze.
Looking at Data can also be connected to different math strands. You
can adjust your questions to fit your current math focus. When studying
measurement, have students estimate things such as paces across the
room, square feet of classroom space, or volume of a container. After
making their estimates and analyzing the data, they can do the actual
measuring. During a probability unit, have students collect data about
dice or spinners and analyze the significance of the range, mode, and me-
dian in that context.

Overview
In this activity, students practice mentally combining numbers in different
ways using number strings. For this lesson, a number string is an expres-
sion, written in a horizontal format, that includes more than two ad-
dends. It’s best to introduce students to Number Strings after they have
had some experience solving problems with two addends, followed by
number talks.
With time and experience, students begin to use their understanding of
number relations to solve problems. For example, students might use their
knowledge of benchmark numbers (10, 25, 100, 1,000) to recombine
numbers and simplify the problem. Another approach used by many
students in the intermediate grades is compensation. In the number string
32 ⫹ 23 ⫹ 51, a student might take two from the 32 to make it 30 and
then give the two to the 23 to make it 25. Number strings also offer the
occasion to use mathematical notation, such as parentheses, and discuss
its meaning in the context of equations.
Activity Directions
1. Have students brainstorm ways to make one hundred. Write the
equations on the board or an overhead.
2. Record a number string on the board for students to solve
(e.g., 23 ⫹ 14 ⫹ 250 ⫹ 12 ⫹ 50 ⫽).
3. Allow a minute or so for students to solve the problem mentally.
4. During partner talk, direct the students to talk to a partner about
their answer and strategy for solving the number string.
5. During whole-class discussion, elicit answers and strategies from a
few volunteers and represent students’ strategies numerically on the
board.
6. Repeat Steps 2–5 using a new number string.
Number Strings
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CONTENT AREA
MATERIALS
projector
TIME
five to ten minutes
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Key Questions
• What did you do to solve the problem?
• Did anyone solve the problem in a different way?
• Tell us where you started and why.
• Were there any number combinations that made it easier to solve the
problem?
From the Classroom
I introduced Number Strings to a group of fourth and fifth graders by fo-
cusing on one hundred. I chose a familiar benchmark number so students
could focus on the mechanics of number strings rather than having to
strain over the mental computation.
“We’re going to do some math that focuses on one hundred,” I told
the class, “so before we begin the activity, let’s take a few minutes to think
about one hundred. What are some ways to make one hundred?”
“Fifty plus fifty,” several students responded almost immediately.
50 ⫹ 50 ⫽ 100
Then I called for other combinations. In a short time we had a nice list
on the board.
50 ⫹ 50 ⫽ 100
90 ⫹ 10 ⫽ 100
20 ⫹ 80 ⫽ 100
30 ⫹ 70 ⫽ 100
99 ⫹ 1 ⫽ 100
98 ⫹ 2 ⫽ 100
86 ⫹ 14 ⫽ 100
75 ⫹ 25 ⫽ 100
Next I asked the students to think of ways to make 100 using more
than two addends. Many of the students used the two-addend equations
already on the board and broke the numbers apart further to make three-
addend equations. For example, they moved from 50 ⫹ 50 ⫽ 100 to 50 ⫹
25 ⫹ 25 ⫽ 100. The students quickly generated a list of equations, which
I recorded on the board.
“OK,” I told the students, “you know a lot of different ways to make
one hundred. I’m going to write an expression with three numbers on the

board. You are going to find the answer mentally. You might find the job
easier if you try to use what you know about one hundred and hunt for
hundreds.” I wrote on the board:
50 ⫹ 12 ⫹ 50 ⫽
I deliberately chose a fairly obvious number string to build students’ con-
fidence and to provide scaffolding for more difficult problems. After giv-
ing the students a bit of time to solve the problem, I asked them to tell me
the answer in unison. The response was unanimous and correct. Now was
the time to push the communication.
“Yes,” I agreed, “the answer is one hundred twelve. What did you do
to solve the problem?”
“First I saw the fifty and fifty and that makes one hundred,” Anny
explained. “Then I added twelve more to it and got one hundred twelve.”
I recorded Anny’s idea on the board.
50 ⫹ 12 ⫹ 50 ⫽
50 ⫹ 50 ⫽ 100
100 ⫹ 12 ⫽ 112
“Did anyone solve it a different way?” I asked.
Most students had seen the one hundred first and then added twelve,
so I moved on to the next number string.
60 ⫹ 21 ⫹ 40 ⫽
Here again students had little problem finding the sum. Several stu-
dents explained that they had added the sixty and forty first and then added
twenty-one to the one hundred. I decided to offer a less obvious number
string to see how the students would approach it. I was especially interested
to see if they would continue to hunt for hundreds or find other ways to
deal with the problem. I wrote the third number string on the board.
76 ⫹ 24 ⫹ 51 ⫽
Knowing that this number string was more challenging, I gave the
students more time to think about it before eliciting responses. I wanted
to make sure they had enough to time to feel confident in their mental
computation. I also wanted to give them time to notice their own think-
ing so that they would be better prepared to talk to the class about it. I
modeled this by looking at the problem and appearing to solve it mentally
myself.
I had the students respond in unison when I asked for the sum. They
had the correct answer.
Number Strings 89

“Great,” I told them. “Now, who’s willing to tell us where you started
when you first approached the problem?”
Many students raised their hands. I called on Howard.
“First I did six and four make ten,” he began.
“Where did the six and four come from and why did you start there?”
I probed.
“Oh,” Howard replied, “I took the six from the seventy-six and the
four from the twenty-four. I started there because it’s easy to work with
tens. Also, it turned the seventy-six and twenty-four into easy tens, too.
Then I did seventy and twenty equals ninety. Ten and ninety make one
hundred, and then I added fifty-one to it to get one hundred fifty-one.”
I recorded Howard’s explanation on the board.
Howard’s Way
6 ⫹ 4 ⫽ 10
70 ⫹ 20 ⫽ 90
90 ⫹ 10 ⫽ 100
100 ⫹ 51 ⫽ 151
“Did anyone find other ways to get easy number combinations?”
I asked.
Several students raised their hands. I called on Katrina.
“I borrowed one from the seventy-six and gave it to the twenty-four,”
Katrina began.
“Why did you decide to do that?” I inquired.
“Because then it’s easier and I could make one hundred,” she ex-
plained. “I turned it into seventy-five plus twenty-five, and that’s an easy
one. It’s even on the board already. Then I just added the fifty-one and I
got one hundred fifty-one.”
Katrina was referring to our initial discussion about ways to make one
hundred. I had left the equations on the board and I was pleased to hear the
list was useful. I recorded Katrina’s strategy on the board next to Howard’s.
Katrina’s Way
76 ⫺ 1 ⫽ 75
24 ⫹ 1 ⫽ 25
75 ⫹ 25 ⫽ 100
100 ⫹ 51 ⫽ 151
I had a few more students share their thinking about the number
string. Several students started with the ones and then proceeded to the
tens. A few students added all the tens first and then went back to deal
with the ones.

I felt this was a fine introduction to Number Strings. Students had a
variety of ways to solve the equations. Furthermore, they were willing to
share their thinking and listen to each other’s ideas. I knew that as the year
progressed I would be able to employ Number Strings as a way to give
students practice with mental computation and to focus on specific
arithmetic topics we were studying.
Different number strings can be used in the classroom as the year pro-
gresses. For instance, if the goal is for students to practice finding combi-
nations of one hundred or one thousand, number strings like these could
be useful:
25 ⫹ 17 ⫹ 75 ⫽ 32 ⫹ 44 ⫹ 68 ⫽ 300 ⫹ 400 ⫹ 700 ⫽
Number Strings also works well with multiplication and when intro-
ducing and extending work with fractions and decimals. Number strings
provide students valuable practice with mental computation and combin-
ing numbers in efficient ways.
When teaching Number Strings, encourage the students to look at the
whole problem to see if they can find benchmark numbers or use familiar
combinations. This habit will greatly improve their facility with all types
of arithmetic.
Number Strings 91

Overview
In this logic game, students explore number combinations and patterns
and their effect on the results of simple addition problems. As partners
take turns removing counters from a common pile, they use logical think-
ing to consider their total, their partner’s total, and the amount left in the
pile. While the mechanics of the game are fairly simple, the logical think-
ing required to develop winning strategies pushes students’ mathematical
reasoning. Discussing experiences with the whole class brings the logic
and language to the surface of the activity.
Activity Directions
1. Start by placing fifteen counters on the overhead projector.
2. Invite a student to join you in modeling a game in front of the class.
3. Explain the rules of the game:
• Players can take one, two, or three counters on his or her turn.
• Players continue taking turns until all the counters are gone.
• The player with the odd number of counters at the end is the
winner.
4. Lead a brief discussion with students to determine some mathematical
rules that might help them as they play the game:
• odd ⫹ odd ⫽ even
• even ⫹ even ⫽ even
• odd ⫹ even ⫽ odd
5. Play one round of the game with the student volunteer.
6. Write on the board focus questions for students to consider as they
play the game in pairs.
• Does it matter who goes first?
• What strategies did you use?
Odd Number Wins
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CONTENT AREA
Number and
Operations
MATERIALS
■ counters, cubes, or
tiles, 15 per pair of
students
■ overhead projector
TIME
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7. Let students play with partners at their tables.
8. After students play several rounds of the game, lead a class discussion
based on the focus questions.
Key Questions
• What strategies did you use?
From the Classroom
I started at the overhead projector to model a game of Odd Number Wins
in front of a class of fifth graders. I placed fifteen counters on the overhead
and I invited Miguel to join me.
“So here’s the deal,” I explained to the class. “We’ve got fifteen chips
on the overhead. Miguel and I are going to take turns pulling some chips.
On each turn we can take one, two, or three chips. After we’ve taken all
the chips, we will check to see how many we have. How can we tell who
won?”
“Whoever has the most!” Danelle responded.
“Aha,” I said, “that’s the way to find the winner in most games, but
this game is a little different. You win if you end up with an odd number
of chips. It doesn’t matter who has the most, it only matters who has an
odd amount.”
“I wonder if we can find any mathematical rules that apply to this type
of situation,” I said to the class. “What happens when you add an odd
number plus an odd number?”
We had a brief discussion and generated some examples of various
simple addition scenarios. We concluded the following:
odd ⫹ odd ⫽ even
even ⫹ even ⫽ even
odd ⫹ even ⫽ odd
I felt it was useful to spend a few minutes making these ideas explicit.
These concepts might serve as a reference point for students while playing
the game. They also invite further investigation as an extension to the
activity.
“Let’s start a game,” I said to Miguel. “Do you want to go first or
should I go first?”
Odd Number Wins 93

“You can go first,” Miguel acquiesced.
“OK,” I began, “I’ll take two. Am I allowed to do that?”
“Yes,” the class responded.
Miguel then took three. After each turn I checked with the whole class
to make sure we were following the rules and to ask how many counters
were left and how many each of us had. This questioning kept the whole
class involved even though only two of us were playing. After we com-
pleted the game I gave the class some final instructions.
“In a minute or so you are going to go back to your tables and play
Odd Number Wins with a partner,” I explained. “The first few times you
play, you will just be getting to know how the game works. But after a few
games you’ll be able to concentrate on the mathematics. There are two
questions we’ll focus on when we get back together to discuss your expe-
riences.” I wrote on the board:
Does it matter who goes first?
What strategies did you use?
I sent the students off to their tables to play the game in pairs. The stu-
dents were very engaged, although some were overly invested in winning.
I reminded the class that winning was not the goal of the game. The goal
was to have some mathematical ideas to discuss when we talked about the
questions on the board. I gave the students enough time to play several
games. Then I called the students back together so we could discuss their
discoveries.
“So now you’ve got some experience with Odd Number Wins,” I told
the students. “Let’s think about these questions. Does it matter who goes
first?”
The class was split on this question. Many students were convinced it
made no difference, while others had strategies that required either they or
their opponents went first. I decided to let this controversy simmer. We
were not going to come to consensus in a brief period, and as winning
strategies emerged over time, students would see more clearly whether who
went first mattered. I recorded both possibilities on the list of strategies.
“Let’s talk about other strategies,” I suggested. “What types of things
were you doing that helped you end up with an odd number or helped
your partner end up with an even number?”
As students shared their strategies I recorded them on the board. In a
short time we had a nice list:
Go first.
Let your partner go first.

Pay attention to your partner’s total.
Take odd numbers.
Take one or three, then take two, then take one or three.
Pick odd, even, odd, even . . .
Start with an even number.
Have an odd number and leave one for your partner.
Pick three every turn and take whatever is left at the end.
Pay attention to how many are left.
“Wow,” I remarked, “you got a lot of ideas after playing only a short
time. I’m going to leave these strategies on the board, and the next time
you play Odd Number Wins you can test out some of the strategies. You’ll
probably find that some of the strategies work sometimes and some of
them work all of the time. When you find some that work all of the time,
the next mathematical question for you to think about is why that strat-
egy works. Also, you might develop more strategies as you play more. You
can add those to the list we started.”
I thanked the students for their work and thinking and told them I was
looking forward to hearing about the discoveries they would make as they
continued to play Odd Number Wins.
Language plays a key role in Odd Number Wins. In addition to con-
ducting oral discussions about strategies, have students incorporate writ-
ing. Use the key questions as writing prompts and have students reflect
in their math journals. As students play over time, they will begin to de-
velop winning strategies. Push the students to explain the logic of the
strategies and their connections to odd and even numbers in order to
make the game a rich mathematical experience worth revisiting over
time.
Once students develop winning strategies that they can explain, chal-
lenge them further by slightly altering the game. Have students start with
twenty-one counters instead of fifteen. Or change the amount of counters
they’re allowed to take on each turn.
Another extension involves exploring odd and even addends in more
depth. Use a T-chart to systematically consider all the two-addend combi-
nations of fifteen possible in the game.
Odd Number Wins 95

I used the list of combinations to open a discussion about whether or
not it was possible for both players to end up with an odd number of
chips. The T-chart provided a visual model and we hypothesized that
when an odd number is broken down into two addends, one of the ad-
dends will be odd and the other will be even. Students also noticed the
alternating odd-even patterns in the two columns. While a discussion
about the nature of odd and even numbers is not the focus of the activity,
follow-up sessions can take advantage of an opportunity to expose stu-
dents to some number theory and prepare them for future explorations.
Player A Player B
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3 12
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12 3
13 2
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Overview
Over or Under is a quick estimation activity that helps students quantify
objects and situations in their immediate surroundings. Students focus on
benchmark numbers in order to estimate amounts of different items in
their classroom. Easy to implement and engaging for students, this activ-
ity turns even a few minutes into a rich mathematical experience.
Activity Directions
1. Ask students to generate a list of items in the classroom that could be
counted.
2. Record their ideas on the board or on overhead.
3. Write some benchmark numbers on the board (e.g., 10, 25, 50, 75,
100, 250, 500, 750, 1,000) and review the concept of benchmark
numbers with the class.
4. Pick an item from the student-generated list.
5. Ask students if they think the actual number of items is over or under
each of the benchmark numbers on the board.
6. Have students share their thinking.
7. Repeat Steps 5 and 6 using a different item from the list.
Key Questions
• What are benchmark numbers?
• Is your estimate over or under __?
• How much over or under is your estimate?
Over or Under
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CONTENT AREA
Measurement
MATERIALS
projector
TIME
five to ten minutes
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From the Classroom
Robin Gordon asked her fourth graders to look around the room and talk
to a partner about things that they could count. After a few moments stu-
dents shared their ideas as she listed them on the board.
names on Popsicle sticks
hearts on the calendar
posters
books
numbers on the number line
letters in the alphabet
word study words
molecules
tables
chairs
watches
stars on the flag
boxes
pencils
paper
rectangles in the shelves
“There’s a lot of stuff to count in this room,” Robin remarked. “Now
I’m going to write some benchmark numbers on the board. As I write
these numbers, think about why they’re called benchmarks. Just what are
benchmark numbers?” Robin used another part of the board to write a list
of numbers.
10
25
50
75
100
250
500
750
1,000
“Those are the easy numbers,” Ricky responded.
“Like you use them for money,” Betty added.
“They’re numbers you use for rounding,” Ontario explained.

“Those are all good ways to talk about benchmark numbers,” the
teacher agreed. “These benchmark numbers are friendly numbers that
we’re familiar with, and they’re easy to use and compute with.”
Next Robin wrote on the board:
Over or Under
“What we’re going to do now is use these two lists. We’ve got a list of
things to count and we’ve got a list of benchmark numbers. I’m going to
pick something from your list and you’re going to estimate whether the ac-
tual amount is over or under these benchmark numbers. Let’s start with
the first item—names on Popsicle sticks.”
Robin had a cup with Popsicle sticks in the front of the room. Each
stick had one student’s name on it. Robin used these sticks to randomly
call on students or to pick a student for a special task. She chose to focus
first on the Popsicle sticks because it would be fairly easy for students to
know the amount.
She picked up the cup and pointed to the number 10 on the board. “So
what do you think?” she asked the class. “Is the number of names on these
sticks over or under ten?”
“Over!” was the overwhelming response from the class.
“Is it over or under twenty-five?” Robin asked, moving to the next
benchmark number on the board.
This time there was a mixed chorus of “overs” and “unders.”
“Hmm,” Robin said, “we’ve reached the point where people have dif-
ferent ideas about this question. Let’s talk about it.”
“Well,” Raquel explained, “I think it’s under because we only have
twenty-one students in the class.”
“Yeah, but we each have more than one name,” Mitchell countered.
Robin pulled one of the sticks out of the cup. She confirmed that it had
a first name and a last name on it.
“Oh,” Robin said, “this makes it trickier. I need to clarify the ques-
tion. How many names are on each stick?”
“Two,” the class responded.
“OK,” she continued, “so we’ll count all the first names and last
names on the sticks. Now talk to a partner about whether you think the
number of names is over or under twenty-five.”
Robin gave the students a few moments to discuss this and do some
mental computation. Most of them agreed that there were forty-two
names. Robin continued the questioning.
“So what’s the consensus?” she asked. “Is it over or under twenty-
five?”
Over or Under 99

“Over!” the class replied in unison.
“Moving on to the next benchmark number,” Robin continued, “is it
over or under fifty?”
The class replied, “Under!”
“Can anyone explain how you know it’s under fifty?” Robin asked.
“Because there are twenty-one kids in the class and each kid has a first
and last name,” Herman explained. “If you double twenty-one you get
forty-two, and that’s less than fifty.”
“Makes sense to me,” Robin agreed. “Let’s check that to make sure.
There are two names on each Popsicle stick, so count by twos as I pull
each stick out of the can.”
The students counted by twos as Robin removed the sticks one by one.
They ended at 42, confirming their idea. Robin then had them talk to a
partner about how much the actual total was over and under the two
nearest benchmarks. The students concluded that 42 was 17 more than 25
and 18 less than 50.
“This time let’s think about the number of pencils in the room. Talk to
a partner about your estimates and then we’ll use the benchmark numbers
to find out about your ideas.”
While the students talked, Robin wrote a sentence frame on the board:
The number of pencils in the room is over/under __.
She called the class to order and explained, “Sometimes it’s confusing
to just give a one-word answer, so I wrote a sentence up here to help us
keep track of what we’re talking about.”
Robin referred back to the benchmark numbers. She started with ten
and the students all agreed the number of pencils was over ten. Similarly,
they agreed the number was over twenty-five. When they got to fifty there
was a mix of responses. Robin knew this was the place to stop and have a
conversation.
“Some of you think the number of pencils in the room is over fifty and
some of you think it’s under. Let’s hear some of your thinking on this.”
“I think it’s over fifty,” Tula said. “Everyone has about two pencils
and there are also pencils at the supply table.”
“Also, Ms. Gordon has some pencils,” Brandy added.
“And Mrs. A. gave us each a pencil for Valentine’s Day,” William said.
“But some people took those home,” Jesse countered.
“And maybe some got lost,” furthered Khalil.
“A lot of factors to consider,” Robin validated. “Show me with your
thumbs what you’re thinking now. Put your thumb up if you think there
are over fifty pencils in the class. Put your thumb down if you think we
have under fifty pencils.”

Over or Under 101
Robin decided to have students use their thumbs so that she could see
each individual’s response. This participation structure also gives a more
quiet way for the whole class to respond at once. Most of the students had
their thumbs up, so Robin asked them to talk to a small group about
whether they thought there were over seventy-five or under seventy-five
pencils in the class. This led to another lively exchange. Students used a
combination of data gathering, mental computation, and classroom lore
to justify their estimates. Robin knew that these types of discussions
would continue in future episodes of Over or Under.
Use whole number benchmarks to introduce Over or Under to the class.
Fractions, decimals, and percents work well once students understand the
structure of the activity. The benchmarks might be ᎏ
1
4
ᎏ, ᎏ
1
2
ᎏ, and ᎏ
3
4
ᎏ or .25, .5,
and .75 or 25 percent, 50 percent, and 75 percent. To generate a list that
works well for fractions, ask students to think of things in the room that
are parts of groups (e.g., people with short-sleeve shirts, fiction books in
the class library) or ratios (e.g., boys to girls, pens to pencils).
Some students will be very interested in finding the actual number of
the items discussed. Depending on the item, this counting can be done as
part of the activity or student volunteers can count at a later point and
report back to the class. Once the actual number is known, students can
do mental computation to find the closest benchmark number. They can
also calculate how much over or under the actual number is from a par-
ticular benchmark.
Over or Under also works well for measurement. Have students esti-
mate the length, width, height, weight, or capacity of things in the room
in relation to benchmark numbers.

Overview
Personal Numbers is a fun way for students to get to know each other and
think about ways numbers are integrated into their lives. In this activity
students explore the role numbers play in their lives outside the classroom.
Often students feel alienated or intimidated by numbers because numbers
are not as obvious in their world as printed text is. Personal Numbers be-
gins to open students up to the realization that numbers are all around
them and serve a variety of purposes in their lives. Not only do numbers
represent quantities, but they also function as descriptors, markers for
comparison, milestones, locations, and labels. Personal Numbers helps
students generate examples of these uses and become aware of the func-
tion of numbers outside of math class.
Activity Directions
1. Share some personal numbers and tell students you’re going to ask
them to guess what the numbers might represent.
2. Lead a brief discussion focusing on what is reasonable and what isn’t
reasonable in regard to the numbers.
3. Have students talk in small groups about what each of your personal
numbers could reasonably represent.
4. Have students share their guesses about what each of your personal
numbers represents. Ask students to explain why their guesses are
reasonable.
5. Reveal what each of your personal numbers represents.
6. Introduce to students nominal, ordinal, and cardinal numbers.
Explain the purpose of each type of number and point out examples
of each in your list of personal numbers.
7. Ask students to brainstorm their own personal numbers, write them
down, and next to each, note what the number represents.
Personal Numbers
19
CONTENT AREA
Number and
Operations
MATERIALS
projector
TIME
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•
•
•
•
102

8. Let individuals share their personal numbers, and have classmates try
to guess the significance of each number.
Key Questions
• Is that a reasonable guess? Explain.
• What might that number mean?
• What couldn’t that number stand for?
From the Classroom
“We’ve been working together for a while now,” I told Robin Gordon’s
fourth graders. “You probably know some things about me from our time
together. There are other things you can tell about me just by looking at
me. I’m going to open up today and share even more about myself. I’m
going to share my personal numbers.”
“Huh?” Eddie blurted.
“Everyone has personal numbers. They are numbers that are special to
you because they represent something important in your life. I’m going to
show you some of my personal numbers and give you some time to think
about what they might mean to me.” I wrote on the board:
3805ᎏ
1
2
ᎏ
1962
7.5
2001
19
2
46
4
8
63ᎏ
1
2
ᎏ
“Before you start guessing what these personal numbers are about,
you need to know what we’re focusing on. We’re going to pay attention to
what’s reasonable and what’s unreasonable,” I explained. I wrote reason-
able and unreasonable on the board.
“When you look at my numbers, what would be a reasonable possi-
bility for one of them?”
“Maybe you were born in nineteen sixty-two,” Angel suggested.
Personal Numbers 103

“Is that reasonable? How old would I be if I were born in nineteen
sixty-two? Talk to a partner about that.”
I gave the students a few moments to discuss and do the mental com-
putation. Then I modeled how they could figure it out by jumping up by
decades.
“What’s nineteen sixty-two plus ten years?” I asked, holding up one
finger.
“Nineteen seventy-two,” the class responded.
“Plus ten more years?” I asked, while holding up another finger.
“Nineteen eighty-two.”
“Plus ten more?” I asked.
“Nineteen ninety-two.”
“Plus ten more?”
The students replied, “two thousand two.”
“OK,” I said, looking at the four fingers I was holding up, “that’s ten,
twenty, thirty, forty years to get to two thousand two. But it’s two thou-
sand five, so we need to add three more years since I already had my birth-
day this year.”
“That’s forty-three,” Mitchell declared.
“So is that reasonable?” I refocused. “Could I have been born in nine-
teen sixty-two?”
“Yes,” the class concurred.
“Give me another example of a reasonable guess,” I requested.
“Nineteen, two, four, and eight could be the ages of your kids,” Tula
noted.
“Is that reasonable?” I asked. “Could I have kids those ages?”
“Sure,” Ricky replied for the class.
“OK, so those are good examples of reasonable guesses. Can someone
give an example of an unreasonable guess?”
“It’s unreasonable that you learned to walk when you were nineteen
years old,” Raquel offered.
“Yes,” I agreed. “That would make me a very late bloomer.”
“It’s unreasonable that you have forty-six kids,” Khalil shared.
“Very unreasonable,” I agreed. “So I think you’re getting it. You’re
going to have a few minutes to talk about my personal numbers. Your goal
is to try to think of reasonable meanings for all of these numbers. It
doesn’t matter whether you actually guess the correct answer. The impor-
tant part is the reasonableness. I’ll tell you what all these numbers stand
for later so you don’t have to worry about getting them right.”
I let the students talk in small groups about my personal numbers.
Their discussions were animated and their ideas very creative. After a few
minutes I called them back together to hear some of their ideas.
“It’s reasonable that you have two sisters,” Brandy began.

“It’s reasonable that your son is eight years old,” William contributed.
“Well, seven point five is the same as seven and a half,” Mitchell
began. “And seven and a half could be your shoe size, so seven point five
could be your shoe size.”
“Is that reasonable?” I asked the class.
“Yes,” the students agreed and began to discuss their own shoe sizes.
I asked for their attention back and got them ready for the next part of
the activity. I wrote the actual meaning of each number on the board, then
I referred them to the list.
3805ᎏ
1
2
ᎏ house #
1962 birth year
7.5 shoe size
2001 car year
19 years teaching
2 son’s grade
46 states visited
4 TVs in house
8 son’s age
63ᎏ
1
2
ᎏ height in inches
“So now you know where my personal numbers come from and what
they mean,” I said to the class. “Now it’s your turn to think about some of
your own personal numbers. Before you do that, however, I want to point
out something to you. Numbers are used for different purposes. For exam-
ple, three eight zero five and a half is more like a name. It’s a number used
to identify my house. Those kinds of numbers are called nominal numbers.
If you think about sports uniforms, they usually have numbers on them.
Those are nominal numbers, too. They’re a way to help identify or give a
name to the player.” I wrote nominal—name next to 3805ᎏ
1
2
ᎏ on the board.
“There are two other kinds of numbers I’m going to tell you about.
My son’s grade is an ordinal number. He’s in second grade and that tells
the order of the grade he’s in. First, second, third, fourth—those are ordi-
nal numbers.” I wrote ordinal—order on the board next to 2.
“Finally, there are the numbers we use when we’re counting to find out
how many. Those are called cardinal numbers. I’ve been teaching for nine-
teen years. The nineteen tells how many years, so it’s a cardinal number.”
I wrote cardinal—how many on the board next to 19.
“So now it’s your turn to write down some of your personal numbers.
Try to write at least ten. Also, see if you can think of some nominal, ordi-
nal, and cardinal numbers to use. You can use some of my ideas from the
board to help you get started. But some of my personal number ideas won’t
work for you since you’re not adults with kids and jobs and cars yet.”
Personal Numbers 105

I solicited suggestions from the students for other personal numbers
they might come up with. Suggestions ranged from number of siblings to
countries visited to highest score on a video game. When I was satisfied
that the students had enough ideas, I sent them back to their seats to begin
writing down their own personal numbers. I encouraged them to write the
number and then write what it represented right next to it. Ironically, it’s
easy to forget what your own personal numbers stand for sometimes. I
planned to have different students share a few of their personal numbers
with the class on my next visit. The rest of the class would try to come up
with reasonable guesses about the numbers shared.
There are many versions of Personal Numbers. In order to incorporate
more writing into the activity, have students write clues for their personal
numbers and read them to the class. If the class is studying a particular
area of math (e.g., fractions, decimals, percents), teachers can request that
students incorporate those types of numbers into their lists. Modeling and
brainstorming might be necessary to help students think of categories for
personal numbers and ways to express the numbers. For example, teach-
ers might need to help students see that the same idea can be represented
by different types of numbers. A student might say she has three girls in
her family of five children. The teacher can help the student see that she
can also express that fact as a fraction—three-fifths of the children in her
family are girls.

Overview
Race for Twenty is a counting game in which partners employ strategic
and logical thinking. While the game itself is quite simple, the mathemat-
ics embedded within it makes it an exciting activity for upper-elementary
students. The teacher plays a crucial role in bringing the mathematics to
the surface of the game. Discussion questions lead the students to become
aware of their strategies and the structure of the logic inherent in the
game. Discussions about the game are far more important than the game
itself. The real learning happens as students listen to each other’s ideas and
analyze strategic thinking. Race for Twenty is from the nim family of logic
games. There are many other nim games that students enjoy. When stu-
dents experience other nim games (see “Odd Number Wins”), they can
compare them and begin to understand the logical structure of the game
strategies.
Activity Directions
1. Have students pair up.
2. Tell students that they are going to play a counting game. The goal is
to be the person who says “twenty.”
3. Explain the rules to the class: On each turn, a student may count
aloud either one or two numbers (e.g., Player A: 1, 2; Player B: 3;
Player A: 4; Player B: 5, 6; etc.). Play continues until one player
reaches 20.
4. Model a game with a student partner.
5. Write focus points on the board, explaining to students that there
will be a discussion based on the points:
• what you noticed
• your strategies
6. Let students play several times.
Race for Twenty
20
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•
•
•
CONTENT AREA
Algebra
MATERIALS
projector
TIME
five to ten minutes
107

7. Lead a discussion about the game based on the focus points listed on
the board. Record students’ ideas on the board or an overhead.
8. Have partners choose a strategy they hadn’t thought of before and
test it out as they play the game a few more times.
9. Lead a brief discussion to find out which ideas seem to work all of
the time and which seem to work only some of the time.
Key Questions
• What did you notice after you played a few times?
• What strategy did you use to try to get to twenty?
From the Classroom
Tina Rasori’s third and fourth graders were eager to learn a new game.
After having the students pair up, I quickly went over the ground rules so
we could jump right in.
“So this game is called Race for Twenty,” I explained to the class. “It’s
a counting game you play with a partner. When it’s your turn, you can
count one or two numbers. Your partner counts on one or two numbers
from where you left off. Whoever says ‘twenty’ is the winner. I’m going to
play one game in front of the whole class with a partner. That way you can
see how the game goes and ask any questions. Who would like to come up
here and be my partner?”
Many students raised their hands. I chose Antoinette.
“Do you want to go first or should I?” I asked Antoinette.
“You can,” she said.
“Let’s see,” I began, “one, two.”
“Three, four,” Antoinette responded.
“Five,” I said.
“Six, seven,” Antoinette continued.
I stopped there to check in with the class. While this was a two-person
activity, I wanted to make sure the whole class felt responsible for paying
attention and making sure we were following the rules.
“Is she allowed to do that?” I asked.
“Yes,” was the somewhat indignant response from the majority. The
students wanted the game to continue.
“OK,” I said. “Eight, nine.”
“Ten, eleven,” Antoinette said.
“Twelve,” I countered.

“Thirteen, fourteen,” Antoinette continued.
I hesitated there and closed my eyes for a few seconds. Then I
scratched my head and stroked my chin. I wanted to explicitly model for
the class that this game involved more than just rote counting. I wanted to
the students to actually see that I was thinking before taking my next turn.
“Fifteen,” I finally said with authority.
“Sixteen, seventeen,” Antoinette responded.
Again I slowed the pace and modeled some thinking time. This time
also allowed the students to catch on to my dilemma. A murmur of ex-
citement pulsed through the class as the students realized the teacher was
about to lose.
“Uh oh,” I said. “I think I’m in trouble. Eighteen?”
“Nineteen, twenty,” Antoinette announced in triumph as the students
applauded.
I was glad to serve as the foil in this introductory game. My relatively
painless loss showed students that the competitive aspect of the game was
less important than the game itself and the thinking it involved. I made a
point of mentioning this to the students before I sent them off to play Race
for Twenty with a partner.
“So now you’ve seen how to play the game and you know it’s about
thinking, not about winning or losing,” I summarized. “You next job is to
play with a partner. But when you’re playing, you’ll actually have two jobs.
First, you’ll be getting practice and learning how the game works. Second,
you’ll be thinking about a couple of things.” I went to the board and wrote:
What You Noticed
Your Strategies
I left enough room between the two so that I could fill in some student
comments later. It was important for me to preview the points that I
planned to use for discussion later. This way, students knew they were ac-
countable for more than just rotely playing the game. Also, they got some
time to prepare for the discussion in advance so it was less likely that my
summarizing questions would be met with blank stares. I gave the stu-
dents a few minutes to play the game with their partners. Then I called
them back together to summarize the experience. Several students were
eager to share, so I soon had some interesting ideas written on the board.
What You Noticed
Adelai: If your partner says “seventeen, eighteen,” you’ll win.
Delila: The person who goes first ends up saying “twenty.”
Javote: If your partner says “eighteen, nineteen,” you’ll win.
Race for Twenty 109

Jorge: Whoever says “fifteen” loses.
Randall: The person who says two numbers loses.
Your Strategies
Randall: Think before you say a number.
Ani: Make your partner go first.
I looked at all of the ideas on the board. I read each one aloud to make
sure all students had access to them. I used these ideas to send the students
back to take a deeper look at the game.
“Goodness,” I expressed with amazement, “you played for only a few
minutes and look at all the ideas you came up with. I am very impressed.”
“Now here’s your job for the last few minutes. You and your partner
are going to pick one of the ideas from up here and test it out. Find some-
thing that’s interesting or that you hadn’t thought of yourself and then
play Race for Twenty and see if it works.”
The children went back to playing Race for Twenty. Again, they had a
particular focus so that thinking was integrated into game playing. We
closed with a brief discussion about their findings. Students discovered
that some of the ideas seemed to work all of the time, but others worked
only sometimes. I stopped the discussion at that point, knowing that the
class would revisit Race for Twenty. Further discussions would allow stu-
dents to investigate which strategies worked consistently. From there, the
teacher could begin to push students to understand and explain why the
strategies worked. Clearly, Race for Twenty is a simple game on the sur-
face, but it provides rich opportunities for mathematical discussions.
As students gain experience with Race for Twenty, they begin to make dis-
coveries and develop strategies. Many of these early discoveries center
around numbers that are good or bad to land on. Sorting these numbers
can help students see patterns and begin to devise winning strategies.
If the class has come to a consensus on a winning strategy, create a
new challenge by slightly changing the game. For example, you could
change the target number from twenty to thirty. Or you could flip the
game so that the goal is to make your partner say twenty rather than to be
the one to say twenty.

Overview
In this game students apply their understanding of place value in order to
make strategic decisions. Students also practice mental addition and sub-
traction as they determine how close they are to the target number.
Activity Directions
1. Draw six boxes on the board or an overhead.
2. Introduce the game by playing the addition version—Target 100.
Explain to students that the goal of Target 100 is to get as close as
possible to one hundred without going over by adding up six rolls of
the die. Explain that they must use all six rolls of the die, and for
each roll, they can make the number a one or a ten.
3. Roll the die and announce the number.
4. Let students decide whether to make the number a ten or a one, and
record their number in the first box.
5. Repeat five more times, stopping periodically to let students discuss
their current total and what roll they would like to get next.
6. Ask students to find the sum of their six numbers.
7. Ask students to calculate how close they are to one hundred. Have a
few students show their computation strategies.
8. Reverse the game by having students start with one hundred and
subtract six rolls of the die to get as close to zero as possible without
going below.
Key Questions
• What roll would you like to get next? Why?
• How close were you to the target? How do you know?
Target 100/Target 0
21
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•
•
•
•
CONTENT AREA
MATERIALS
■ 1 die
projector
TIME
ten minutes
111

From the Classroom
Danielle Pickett’s third graders were working on subtraction. I decided
to introduce a game that would help them both with the concept of sub-
traction and subtraction computation. Target 0 gives students practice
with making decisions, estimating, and doing mental computation. Since I
wasn’t sure how much subtraction the students had done, I decided to in-
troduce the game using an addition version. This way the students could
learn the mechanics of the game without getting distracted by difficult
computation. Once they understood the workings of the game, I’d move
them to the subtraction version.
“I brought one die with me today,” I told the class. “That’s pretty
much all we need for a game called Target Zero.”
“I know this game,” Johannes announced.
“That’s great,” I responded. “Let’s make sure everyone knows how it
works.”
I drew six boxes in a row on the board.
“For a warm-up we’re going to play Target One Hundred with the
whole class,” I explained to the students. “I’m going to roll the die and
tell you what I got. Then you are going to decide if you want to make
that number a ten or a one. For example, if I roll a five, what are your
choices?”
“It could be five or fifty,” Dylan replied.
“Right,” I agreed. “When you decide which number you want, I’ll
write it in the first box. Then I’ll roll again and you’ll have the same deci-
sion. We’ll put that number in the next box. Are there any questions?”
“Are we going to add the numbers up?” Claire asked.
“Yes,” I responded.
“What are we trying to get to?” Rebecca asked.
“Good question,” I replied. “You’re trying to get to one hundred. You
might not get exactly one hundred, but you want to get as close as possi-
ble. Also, you don’t want to go over one hundred. Get it?”
The students nodded.
“One more thing,” I added. “Since I drew six boxes on the board, you
have to use six rolls. This makes it more challenging. Can you see why?”
“Because we might get fifty and fifty and already be at one hundred,”
Leah explained.

“Yes,” I said, “you need to make sure you spread the numbers out so
you don’t end up going over one hundred. Are you ready?”
The students nodded their heads, so I rolled the die.
“It’s a four,” I announced. “Raise your hand if you have an idea about
what to do with the four.”
“Let’s make it forty,” Tristan suggested.
I wrote 40 in the first box and rolled again. Soon the board looked like
this.
Target 100/Target 0 113
40 5 20 3 5
“What’s our total now?” I asked the class. I gave the students a minute
to compute mentally and then called on Alexa.
“It’s seventy-three,” Alexa told us.
“Hmm,” I said, “you’ve got a total of seventy-three. I wonder what
roll you’d like to get next. Turn to a partner and talk about what would
be the best roll and why. Also talk about what you might do with a roll
that’s not the one you wanted.”
I gave the students a few minutes to talk to each other. After some an-
imated discussion, they were ready to share.
“The best roll would be a two,” Luke explained. “Because we could
make it a twenty and that would give us ninety-three. That’s really close to
one hundred.”
“A three would be too much,” Jana chimed in, “because that would
make it one hundred three.”
I rolled the die and got a 4. The students moaned in disappointment.
“Hold on,” I said. “The game’s not over. You want to get as close to
one hundred as you can without going over. So what do you need to do
with the four? Do you want to make it four or forty?”
“We have to make it four because forty is too much. We’ll go over one
hundred,” Otto noted.
I wrote 4 in the last box and asked the students for the total. After the
students established that the total was seventy-seven, I decided to take a
quick opportunity for them to do some mental computation and hopefully
apply some of the strategies they’d learned.
“Well,” I said to the class, “your total is seventy-seven. How close to
one hundred is seventy-seven?”
“Not that close,” Claire commented.

“I think you did pretty well for your first time ever playing Target One
Hundred,” I countered. “But how could you figure out the difference be-
tween seventy-seven and one hundred? Talk to a partner and then we’ll lis-
ten to some of your ideas.”
Volunteers shared several different strategies—counting up, subtract-
ing, and rounding to seventy-five, subtracting, and taking away two from
the answer. I told them I was sure they would have other opportunities to
play Target 100.
“Now you’re ready for the next level,” I told the class. “This time in-
stead of Target One Hundred, we’re going to play Target Zero. Instead of
starting with no points and working up to one hundred, you’re going to
start with one hundred points and try to get down to zero.” I wrote on the
board:
100
“The rest of the rules are the same. You need to use six rolls and you
decide each time whether to make it a ten or a one. But you’re subtracting
from one hundred instead of adding up to one hundred.”
I rolled the die. It was a 6.
“Let’s make it a sixty,” Jana suggested.
“What is one hundred minus sixty?” I asked the class.
The students were quick to respond.
100 60 40
I rolled several more times, each time asking a volunteer to tell me
whether to multiply the roll by one or by ten. In a few minutes the board
looked like this:
100 60 40
40 4 36
36 6 30
30 20 10
“Hmm,” I said, “you’ve got two more rolls and you want to get to
zero. What two rolls would like to get? Talk to a partner about some
possibilities.”
The students brainstormed many combinations of ten. It’s always nice
to take advantage of opportunities for students to review their single-digit
facts, especially those involving benchmark numbers. This task was a bit
tricky because while there are eleven different two-addend possibilities for

making ten, most of them are not possible when rolling a die twice. I
recorded the possible roll combinations on the board.
6 4 10
5 5 10
4 6 10
“OK,” I continued, “let’s roll and see what happens.”
I ended up rolling a 5 and then another 5, which thrilled the students
because they landed exactly at zero. While acknowledging that they would
not always be this lucky, the students agreed that Target 0 was a fun and
exciting game.
Target 100 and Target 0 offer a lot of flexibility. Change the target num-
ber and/or change the number of rolls required to reach the target. After
the games have been modeled with the whole class, students can play on
their own in pairs.
Target 100/Target 0 115

Overview
Tell Me All You Can encourages students to think and communicate about
equations before actually solving them. Students use what they know
about numbers and operations to make statements about an arithmetic
problem. In addition to developing number sense, this activity promotes
efficient test-taking skills by having students consider the reasonableness
and likelihood of different solutions.
Activity Directions
1. Put an open equation on the board or an overhead (e.g., 100 67 ).
2. Ask students to tell you what they know about the problem (not
what the answer is). Offer students support using prompts like the
following:
• Is the answer going to be more than or less than one hundred?
How do you know?
• Is the answer going to be more than or less than fifty? How do you
know?
• Make up a story that goes with the problem.
3. Have students prove their conjectures with mathematical
explanations.
4. Continue to generate different ideas from the students.
5. Have students solve the problem and explain their strategies.
Key Questions
• What do you know about this problem?
• How do you know?
• Can you tell a story that goes with this problem?
Tell Me All You Can
22
CONTENT AREA
Number and
Operations
MATERIALS
projector
TIME
five to ten minutes
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116

From the Classroom
Lisa Kimbrell’s third graders were having some trouble with multidigit
subtraction. While they had been exposed to several approaches to solv-
ing these problems, they often solved the problems rotely and didn’t seem
bothered by unreasonable answers. For example, they were unperturbed
when the difference was greater than either of the original numbers.
They were particularly confused by numbers with zeros in them. They
had learned some rules for borrowing and regrouping, but they weren’t
clear on what to do with the zeros. We decided to try Tell Me All You
Can as a diagnostic tool and to support their efficiency and accuracy.
“Here’s what we’re going to try,” I told the class. “I’m going to write
a subtraction problem on the board. But instead of solving the problem
and telling me the answer, you’re going to think about this problem in a
different way. You’re going to tell me everything you know about the
problem before you even find the answer. This activity is called Tell Me All
You Can.”
I wrote an equation on the board horizontally.
100 67
The horizontal format helps immensely in encouraging students to
look at the whole problem and think flexibly about it. When a problem is
written vertically, many students immediately jump to the standard algo-
rithm without giving the problem any thought at all. After writing the
problem on the board, I waited for some volunteers.
“You can write the problem vertically,” Marshall suggested.
“Yes,” I agreed. “I wrote it horizontally, but it could be rewritten
vertically.”
The students seemed a bit stumped by what to say next.
To further the discussion, I decided to ask some questions that would
prompt students to think about the problem in different ways. “Is the
answer going to be more or less than one hundred?” I asked.
“Less!” came the reply.
“Because you’re subtracting,” Dena explained. “Subtraction is taking
away. If you start with one hundred and take some away, you’re going to
have less.”
“That makes sense to me,” I responded. “Is the answer going to be
more or less than fifty? Turn to a partner and see if you can come up with
a way to talk about this.”
Tell Me All You Can 117

The students talked briefly. Although the question wasn’t particularly
challenging, I wanted to give them some time to talk about the problem in
a way they hadn’t before.
Arturo explained, “One hundred minus fifty is fifty. One hundred
minus sixty-seven means you’re taking away one more ten and seven more
ones. So you’re going to have less than fifty.”
“Ah,” I remarked, “you used the benchmark number of fifty. You
know what one hundred minus fifty is, so you used that information to
help you think about one hundred minus sixty-seven. Did anyone think
about it in a different way?”
“I just thought about tens,” Ronald shared. “One hundred minus
sixty is forty, so I know it’s going to be less than fifty.”
“OK,” I said to the class. “Here’s another way for you to think about
this problem. Talk to a partner about a story you can make up that goes
with the situation. So you need to create a story that goes with one hun-
dred minus sixty-seven.”
I gave the students a few minutes to make up their stories. Since they
were all seated on the rug, there was a lot of cross-pollination of ideas.
I called on Paulo to share his story.
“I had one hundred candies. I gave sixty-seven to my friends. How
many do I have left?”
“Does that story work?” I asked the class. There was consensus that it
was fine.
“Any other ideas?” I asked.
“There were one hundred birds. Sixty-seven flew away. How many
were left?” Angela offered.
“That works,” I acknowledged.
Many students had stories to share, but I didn’t want to spend more
time on this particular avenue. I had them tell their stories to a partner.
This way, all students got a chance to voice their ideas in a minimal
amount of time.
“Is there anything else you can tell me about this problem?” I asked.
“I think the answer is going to be in the thirties,” Esmirna told us.
“Why do you think that?” I probed.
“Because I know one hundred minus seventy is thirty. And sixty-seven
is close to seventy.”
“And it’s going to end with a three,” Patty posited.
“How do you know?” I questioned.
“Because ten minus seven is three,” she explained. “If you start at the
ones, it will be three.”
From here I had the students solve the problem mentally. We had a
brief number talk about their strategies. (See “Number Talks” for more

details.) I was pleased with this initial Tell Me All You Can discussion.
I had pushed the students beyond their usual problem-solving process.
I knew as this activity was repeated they would become more comfortable
with thinking beyond the answer. Ideally, they would get into the habit of
thinking about every problem as a whole before they attempted to solve it.
This would ensure they would think about reasonableness with flexibility,
which would help them both in test-taking situations and in real life.
This activity works equally well with any type of number problem. De-
pending on the current unit of study, choose any operation and whole num-
bers, fractions, decimals, or percents. The key is communication. As stu-
dents become accustomed to these types of discussions, they will increase
their ability to talk about the problems. Initially, they may get stuck on
finding the answer, so it’s important to ask leading questions to offer stu-
dents other ways to think about the equation.
Tell Me All You Can 119

Overview
The game of Twenty Questions presents surprising challenges for students.
In this guessing game, students generate questions to hone in on a secret
number. The act of generating questions requires much thinking and
pushes students to consider numbers, how they can be decomposed and
recombined, and their connections to real life. Students also develop flex-
ibility in their thinking about numbers as the teacher asks them to create
different types of questions. This activity encourages teacher questions
and student discussion throughout the experience.
Activity Directions
1. Think of a secret number between 1 and 100.
2. Explain the rules of the game to the students:
• Tell students that they need to ask yes-or-no questions and that a
guess is not considered a question (i.e., “Is the secret number
twelve?” is a guess, not a question).
• Tell students they must ask twenty questions to figure out what
the secret number is before they’ll be allowed to guess what
it is.
3. Have students begin asking questions. Use tallies on the board or an
overhead to keep track of the number of questions asked.
4. Stop occasionally to have students discuss what they already
know about the secret number and what questions they might ask
next.
5. Remind students that when they think they know your secret
number, rather than shout it out, they should test it out by asking a
good question about the number.
6. After students have asked twenty questions, let them guess the secret
number.
Twenty Questions
23
CONTENT AREA
Number and
Operations
MATERIALS
projector
■ optional: 1–100
chart
TIME
ten minutes
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120

Key Questions
• What do you know about the secret number so far?
• How many numbers have you narrowed it down to?
• What types of questions are helpful? Explain.
• What next question can you ask in order to eliminate a lot of
possibilities?
• Can you ask a question that connects the secret number to something
in real life?
From the Classroom
I introduced the game to a group of third and fourth graders by asking
if they were familiar with Twenty Questions. Although some students
claimed they were, I took a minute to explain the rules.
“I have a secret number and your job is to figure out what it is. There
are two important rules: You can only ask me questions about my num-
ber that I can answer with yes or no. They have to be real questions about
my number, not just a guess. The other rule is that you must ask all
twenty questions before you guess my number. So if you think you know
what my secret number is, don’t shout it out; test it out by asking a smart
question.”
The students seemed eager to get started. I drew a box with a question
mark on the board to give the students a visual reminder that they were
trying to solve a mystery.
“OK,” I said, “who wants to ask me a question about my secret
number?”
Many hands shot into the air. I called on Sofia.
“Is it even?” she asked.
“No,” I responded, “my number is not even.”
I put a tally mark on the board to keep track of the question and then
checked in with the group. “So what do you know about my secret num-
ber so far?” I asked.
“It’s odd,” several students stated immediately.
I concurred and called on Trista.
“Is it above or below fifty?” she asked.
“Hmm,” I said, “that’s a good question, but I can’t answer that with
yes or no. Can anyone think of a way to ask the question so I can tell
you yes or no?”
The students were stumped. I decided to model a question for them.
Twenty Questions 121

“What if you just asked me if it’s below fifty? I can tell you yes or no,
and then you’ll know from my answer whether it’s more or less than fifty.
Do you want to try that?”
Trista agreed and I told her that yes, my number was below fifty. I then
directed the students’ attention to the 1–100 chart posted on the wall. This
particular chart had even numbers posted in white and odd numbers
posted in red.
“Take a look at the one-to-one-hundred chart,” I told the students.
“Can someone go to the chart and point out where my secret number is on
the chart?”
Edward went to the chart and swept his arm across the top half of the
chart.
“Right,” I agreed, “You know my number is somewhere between one
and fifty. What else do you know about my secret number?”
Ivette reminded the group that the number was odd.
“Good,” I said. “So if I look at the chart, I see that all the odd num-
bers are red and that helps me.”
I pointed to the vertical stripes of odd numbers between 1 and 50. “It’s
one of these numbers,” I said. “Now who wants to ask me another
question?”
“Is it close to forty?” Edward asked.
Again I realized that the students needed help with framing their ques-
tions and asking them precisely. With repeated practice students will gain
this skill, but when introducing the game, anticipate that you’ll need some
time to model question forming. Edward’s question fit the yes-or-no
parameters, but the concept of close depends on context and could not be
answered definitively.
“That’s a tough question for me to answer. How close is close? One
person might think ten away is close but someone else might think one
away is close.”
“Is it in the twenties?” Adelia asked.
While Adelia’s question moved us away from the close-to-forty issue,
I decided to use it as an illustration for the class. I went to the 1–100 chart.
“No,” I replied, “it’s not in the twenties, so now when I look at the
possibilities, I can eliminate this whole row.” I pointed to the row of twen-
ties. Subsequently students asked several questions in the same vein.
“Is it in the thirties?”
“Is it in the forties?”
When we had established that my secret number was in the forties, I
redirected the students and had them talk to a partner about what types of
questions were helpful and what next question they might ask in order to
eliminate possibilities. I did this in order to avoid further unnecessary
questions about whether my number was in the teens, in the ones, and so

on. I wanted to move the students toward a new line of questions. The stu-
dents focused on odd numbers in the forties and were forced to generate
different types of questions.
Ellie asked if my number was more than forty-five. I asked her why she
chose forty-five.
“We already know it’s in the forties,” Ellie explained, “so I went to the
middle of the forties to see if it’s more or less.”
“Well,” I responded, “the answer to your question is no. My secret
number is not more than forty-five. That information should help you
a lot.”
I went to the hundreds chart to illustrate how Ellie had eliminated two
of the possibilities (47 and 49) with one question.
Trinesha then asked if my number was one away from forty-four.
After I told her it was not, Joey asked if my number had a one in it. When
I told him it did, I knew it was time to redirect again.
“Nod your head if you’re pretty sure you know what my secret num-
ber is,” I told the class. Almost everyone nodded.
“OK,” I reminded them, “the rules for this game are that you have to
ask twenty questions before you can guess, so even if you know my num-
ber, you need to think of more questions. So far you’ve asked eight ques-
tions. That means you need to ask twelve more before you can guess.
Maybe you can ask questions that will test out your idea to make sure you
really know the secret number.”
I got a new series of questions. “Is it below forty-three?” “Is it below
forty-two?” Does it have a four in it?” Clearly the students now knew that
my secret number was forty-one, but I wanted to push them to decompose
the number and apply some operation sense. I made a suggestion.
“I can tell by your questions that you’ve figured out my secret number.
You still need to ask a few more questions before you can guess. I wonder
if you can ask some questions that have addition or subtraction in them.
Maybe you can think about what plus what equals my number and ask me
questions like that.”
I gave the students a moment to think about this new type of question.
The wait time paid off. The students did some mental computation and
asked some good questions.
“Is it thirty plus eleven?”
“Is it forty plus one?”
“Is it twenty plus twenty-one?”
“Is it forty-three minus one?”
I decided to change directions one more time with the few remaining
questions.
“Well,” I told the class, “you seem to know my number, but you’ve
still got a couple of questions left. I’m going to give you a minute to quietly
Twenty Questions 123

talk to a partner about when and where you might find this number in real
life.”
The students were a bit hesitant but a few examples emerged.
“Could it be someone’s age?” Marshall asked.
“Yes,” I agreed.
“Is it the number of fingers on four peoples’ hands, plus one more fin-
ger?” Sofia proposed.
“Yes again,” I responded.
The stretching required to ask the questions forced the students to
think about numbers in many ways and to make many connections. When
the students had finally asked me twenty questions, I allowed them to
guess my secret number. Rather than call on one individual, I had the
whole class tell me in unison. This way all students got to be “right” and
had the opportunity to successfully participate in the activity.
The Twenty Questions game is really two games in one. At the beginning,
students use logical thinking to ask questions that systematically narrow
down the possibilities and zero in on the secret number. Somewhere about
halfway through the game, students usually have a pretty good idea of
what the secret number is. At this point, they need to think about the num-
ber in a variety of ways. They start to ask questions about magnitude and
begin breaking the number apart in a variety of ways. Prompting from the
teacher is essential in helping students form effective and varied questions.
Once students become familiar with the mechanics of the game, they
can become the leaders of the game. A student or pair of students can
think of a secret number and answer questions from the class. The game
can also be focused on specific areas of arithmetic that students need to
practice. You can encourage students to ask questions that involve multi-
plication or properties of the number (odd, even, prime, composite), for
example. Students can also use this activity to connect numbers and quan-
tities to real life, thus making math more relevant and accessible.
This activity can also be adapted for geometry. A secret shape can be
the focal point and students can ask yes-or-no questions about its proper-
ties and uses in real life.

Overview
What Comes Next? focuses students’ attention on pattern recognition, a
foundational idea of algebraic thinking, and mathematical communica-
tion. In this activity, the teacher begins a number pattern on the board.
The students need to predict the next three numbers in the pattern. Stu-
dents also need to verbally describe the pattern and justify their predic-
tions using mathematical reasoning. Students are further challenged to
find examples of the pattern in the real world.
Activity Directions
1. Begin a number pattern on the board or an overhead (e.g., 2 4 6
).
2. Ask students to think about what numbers would come next in the
pattern.
3. Have students explain the pattern in as many ways as possible and
record their ideas on the board.
4. Ask students to think about where this pattern occurs in real life.
5. Repeat Steps 1 through 4 with a different pattern.
Key Questions
• What is the pattern?
• Is there another way to describe the pattern?
• Can you think of real-life examples of this pattern?
From the Classroom
Melissa Mechtly introduced What Comes Next? to her third graders by
asking them what they knew about patterns. After establishing their
familiarity with the idea, she began the activity.
What Comes Next?
24
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CONTENT AREA
Algebra
MATERIALS
projector
TIME
five to ten minutes
125

“So I’m going to start a pattern,” Melissa told her class, “and your
job is to think about what numbers come next in the pattern. You also
need to think about why those numbers come next. When I call on you
I’m going to ask you to explain the thinking behind your answer, so be
prepared.”
It’s important to let students know what’s expected of them. Since
Melissa wanted her students to focus on justifying their answers, she
needed to let them know at the outset that they would be asked to do that.
Also, she didn’t want anyone to feel put on the spot.
She started with a simple pattern.
1 2 3
“That’s so easy!” Janet exclaimed.
“Well this is just the warm-up to make sure you understand how the
activity works,” Melissa responded. “Don’t worry. I’ve got some trickier
patterns for you.”
“So who thinks you know what the next number is?” Melissa
continued.
“Four comes next,” John announced.
“How do you know?” Melissa asked.
“Because it’s just counting,” John explained.
“Counting by what?” Melissa pushed.
“Counting by ones,” several students exclaimed.
Melissa wrote on the board:
Counting by 1s
“So if my pattern is counting by ones,” she said to her class, “what
would come after four?”
“Five!” the students replied with confidence.
“And then?”
“Six!”
Melissa filled in the pattern on the board.
1 2 3 4 5 6
“Is there another way to describe this pattern?” she asked.
“Plus one,” Chantelle suggested.
“Adding one,” Rachel contributed.
Melissa wrote these two descriptions on the board. She turned the stu-
dents’ attention to what they had done so far.

1 2 3 4 5 6 counting by ones
plus one
adding one
“OK,” she said, “you figured out my pattern and have several ways to
describe it. Now here’s another question for you. Can you think of any ex-
amples of this pattern in real life? Talk to a partner and see if you can
come up with some places where you find this pattern outside of math
class.”
Melissa gave the students a minute to talk to each other. She noticed
that most of the students were looking around the classroom for ideas and
weren’t really thinking outside of the school context. She realized she
would need to prompt them a bit. She started by taking some of their
classroom-based suggestions.
“The number line,” Omar pointed out.
“The calendar,” Carol added.
“Our table numbers,” Vincent said.
“Wow,” Melissa responded, “this pattern shows up a lot around here.
What about outside our classroom? Can you think of any examples?”
Seeing her students’ blank stares, Melissa gave them a hint. “Hakim
told me that tomorrow is his birthday. Do birthdays have a pattern?”
“Oh yeah,” Joshua said, “first you’re one year old, then you’re two
years old, then you’re three years old . . .”
Melissa cut him off well before he reached middle age. “Aha,” she
said, “so there’s another example of this pattern. I bet you’ll notice it a lot
in your lives once you start thinking about it. Now let’s try a trickier pat-
tern.” She wrote on the board:
2 4
“That’s so easy!” several students blurted out.
“Hang on,” Melissa cautioned. “I gave you only two numbers in the
pattern. That’s not a lot of information. There might be more than one
pattern that fits those numbers. Talk to a neighbor and see how many dif-
ferent possibilities you can come up with.”
Melissa let the students have a brief discussion. When she was satisfied
they had multiple answers and reasons, she called for their attention.
“OK,” she said, “who has an idea about what might come next in this
pattern? Remember you also need to tell why that number comes next.”
“It could be six,” Liliana suggested.
“And why could it be six?” Melissa probed.
“Because it’s counting by twos,” Liliana explained.
What Comes Next? 127

“All right,” Melissa responded. “If that’s the pattern, what would
come next?”
“Eight!” shouted the class.
“Right,” Melissa agreed. “Did anyone come up with another
possibility?”
“It could be two,” Chantelle announced.
“How would that work?” the teacher asked.
“Because,” Chantelle explained, “the pattern could be two, four, two,
four, two, four.”
“Oh,” Melissa replied. “It could be a repeating pattern instead of a
growing pattern. I’ll tell you that this particular pattern is a growth
pattern. So if you know that it’s a growth pattern, are there any other
possibilities?”
“Maybe eight,” Katy offered.
“Why eight?” Melissa asked.
“It could be like two times two is four and four times two is
eight.”
“I see,” Melissa said, “so it might be a multiply-by-two, or a doubling,
pattern. If it’s a doubling pattern, what would come next? Talk to a part-
ner about the next few numbers in Katy’s pattern.”
Melissa let the students work through the computation and then they
added the next four numbers to the pattern.
2 4 8 16 32 64
“Wow,” Melissa exclaimed. “Those numbers get big pretty fast. So
now here’s my next question: When does this pattern occur in real life?
Talk to your neighbors and see if you can come up with anything.”
Melissa gave the students some time to discuss the pattern. As she lis-
tened to their discussions, she realized they had a hard time distinguishing
this doubling pattern from a multiples-of-two pattern. This point is con-
fusing since both patterns rely on multiplying by two. However, multiples
of two increase by only two each time (e.g., two, four, six, eight, . . .). The
doubling pattern Melissa presented involves powers of two that cause
the numbers to increase exponentially. She decided she needed to show
the students an example. She took a piece of blank paper and held it up.
“How many rectangles do I have?” she asked the class.
“One,” the class responded.
“OK,” Melissa told them, “I’m going to fold the paper in half. Then
how many rectangles will I have when I open it back up?”
“Two,” the students replied.

Melissa folded the paper in half and opened it to confirm their predic-
tion. “So what will happen if I fold it in half and then in half again?” she
continued.
The students responded, “Four.”
Melissa continued the folding process, asking the students each time
how many rectangles there would be. The students soon caught on to the
connection between the paper folding and the pattern on the board. They
were fascinated and wanted to see how many times their teacher could
actually fold the paper. She indulged them, finally stopping when she had
created thirty-two little rectangles.
“Actually,” she pointed out to the class, “I think this pattern starts
with one, not two. Let me write that and see if the pattern still works.”
1 2 4 8 16 32
“Does this show what happened when I folded the paper? Is it still a
doubling pattern?”
The class agreed that it did represent the folded paper. Melissa told
them that this pattern occurred a lot in mathematical and scientific situa-
tions. She also told them it was called powers of two. She didn’t think it
was necessary to go into a long explanation of exponents and exponential
growth. She just wanted to make them aware of this important pattern.
She encouraged the children to look for this pattern in the real world and
come back to share what they discovered.
What Comes Next? works well as a review of multiplication facts. Start
with any number and give its subsequent multiples (e.g., 3 6 9 ).
The activity can also be used to introduce mathematical patterns such as
square numbers, triangle numbers (1 3 6 10 ), and a common
pattern in nature, Fibonacci numbers (0 1 1 2 3 5 8 ). Students
can also create their own number patterns and have the class guess what
comes next and how the patterns work.
What Comes Next? 129

Overview
Often, the upper-elementary day includes a time when the teacher reads a
chapter book aloud to the class. The chapter book read-aloud gives the
teacher an opportunity to engage children’s number sense with a quick
question: “What page do you think we’re on?” This question prompts
students to estimate and perform some mental computation.
Number sense plays a key role in children’s mathematical develop-
ment. Students need many opportunities with estimation and mental com-
putation. They also need much exposure to a variety of strategies and
methods. Leading short discussions with a real-world context, such as
estimating the page number the class is on in a read-aloud book, accom-
plishes two important goals. First, students become more flexible in their
thinking about numbers and their connections to the real world. Second,
over time students hear many different ideas about numbers and opera-
tions that help them build their own arsenal of problem-solving strategies.
Activity Directions
1. Hold up a book with a bookmark in it so that students can see where
the bookmark is placed relative to the entire thickness of the book.
2. Ask students to estimate the number of pages in the book. Record
their estimates on the board or an overhead.
3. Help the students determine the range of their estimates. Have
students share their computation strategies and record them on the
board.
4. Reveal the number of pages in the book.
5. Ask students to estimate what page the bookmark is on.
6. Lead a brief discussion about the students’ estimates that focuses on
number sense and different approaches to solving the problem.
7. Reveal the page the bookmark is on.
What Page Are We On?
25
CONTENT AREA
Number and
Operations
Data Analysis and
Probability
MATERIALS
■ a book
■ a bookmark
projector
TIME
five to ten minutes
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•
•
•
130

Key Questions
• About how many pages do you think this book has? Can you explain
your thinking?
• About what fraction of the book have we read so far?
• Did anyone get the same answer using a different method?
• Did anyone come up with a different estimate or think about the
problem in a different way?
From the Classroom
I began by asking a mixed group of third- through fifth-grade students if
they were familiar with the book Holes, by Louis Sachar. Students told
me that they had read the book, seen the movie, and heard the rap song. I
didn’t know there was a rap song, but when they started singing it, I told
them I believed them.
I held up a copy of Holes with a bookmark marking my place. I con-
tinued the session by asking the students to estimate how many pages they
thought were in the book. I asked them to each think of their estimate and
raise their hand when they were ready to share it. Then I did a “lightning
round,” quickly pointing to a student, having the student say the estimate,
and writing the number on the board. In less than a minute the board was
full of estimates.
215 300 320
260 150 195
250 180 369
290 210 190
I decided to gather a bunch of estimates first without stopping to dis-
cuss individuals’ thinking. My goal was to encourage all students to offer
an estimate without pressure. I also wanted to collect a range of numbers
on the board for us to analyze.
“Let’s find the range of your estimates,” I suggested to the class.
Noticing the puzzled looks, I decided to scaffold this task a bit more.
I explained, “The first step to finding the range is to look at all the num-
bers up here and find the lowest and the highest. Take a minute to find
those two numbers.”
It didn’t take long for students to identify 150 as the lowest estimate
and 369 as the highest. I circled those two numbers on the board. I
What Page Are We On? 131

continued, “The next step to finding the range is to figure the difference
between one hundred fifty and three hundred sixty-nine.”
I gave the students a few minutes to discuss this with a partner. Then I
had a several students share their thinking.
“I lined up the numbers up and down in my head,” Aimee explained.
“Then I just did the subtraction in my head.”
369
⫺ 150
“So you used the mental chalkboard strategy,” I summarized. “Did
you start with the ones or the hundreds?”
Although I was pretty sure Aimee had used the traditional algorithm,
starting in the ones column, I wanted to publicize the possibility that there
were other ways to deal with the computation.
“I started with the ones,” Aimee confirmed. “I got two hundred
nineteen.”
I completed Aimee’s solution on the board:
369
⫺ 150
219
“Did anyone get the same answer using a different method?” I asked.
Marco shared his solution. “I added one hundred to one hundred fifty,
and that got me to two hundred fifty. Then I added another one hundred
to get to three fifty. Then I added a ten and a nine and that made three
sixty-nine.”
I decided an open number line would be an effective tool for helping
students visualize what Marco had done.
150 250 369
350 360
ⴙ100 ⴙ100
ⴙ10 ⴙ9
“So you added one hundred plus one hundred plus ten plus nine,
which equals two hundred nineteen,” I said, pointing to the number line.
“You did get the same answer as Aimee. So it looks like there’s agreement
that the range of your estimates is two hundred nineteen.”
I moved along. “OK, I’m going to tell you that you did a nice job
with your estimates. I’m thinking that because the actual number of

pages in the book is in the middle of your estimates. The book actually has
two hundred thirty-three pages.”
I gave the students a moment to digest this exciting information.
Then I showed them the bookmark that was sticking out of the top of the
book again. I gave the students a view of the top of the book so they
could see the bookmark and its position relative to the thickness of the
book.
“So you now know there are two hundred thirty-three pages in the
whole book,” I reminded the students. “With that in mind, what page of
the book do you think I’m on right now? You can see where the bookmark
is to help you. I’d like to hear some ideas about your estimates. Then you
can make some estimates and I’ll write them on the board.”
“It looks like the bookmark is about halfway,” Aja observed.
“More like eight-twelfths,” countered Christian.
“Fractions can help you think about your estimates,” I validated.
“What else can you do to help you estimate what page I’m on?”
“If we know there are two hundred thirty-three pages, then we know
the page we’re on is going to be about half of that or a little more than
half,” Devin offered.
“Well, I’ve read that book already,” Maddy stated. “I’m pretty sure
there are thirty-five chapters. So I can guess you’re on about Chapter
Twenty.”
“That could help,” I commented. “So talk to a partner about your es-
timate for what page we’re on. Also, make sure you discuss your reason-
ing so your partner understands how you got your idea.”
I gave the students a couple of minutes to talk. As they talked, I circu-
lated, showing them the book and the bookmark. Occasionally, I’d ask
students to remind me how many total pages there were in the book. This
question helped them focus on reasonableness. In a short while I called for
the class’s attention and then had students share their estimates and their
rationales for those estimates. Many students used the benchmark of one-
half to estimate.
“I think you’re on page one hundred sixteen because one hundred six-
teen plus one hundred sixteen is two hundred thirty-two, and that’s prac-
tically two hundred thirty-three,” Thea explained.
“My estimate is one twenty-five because it looks like it’s a little more
than half,” Reggie volunteered.
After listening to a few more students, I told the class that the book-
mark was actually on page 131. I asked students to each mentally calcu-
late the difference between their estimate and the actual page number.
They shared that information with their partners. I encouraged them to
use the strategies suggested for estimating in the future when I came back
to ask them what page they were on in another book.
What Page Are We On? 133

Using books as an estimating activity opens many possibilities. The same
question can be revisited over time during the course of a read-aloud. Flip
the question by telling students what page you’re on and asking them to
estimate how many pages are in the entire book. Also, books can be com-
pared with each other. Hold up two books and ask, “If the book in my
right hand has two hundred fifteen pages, about how many pages do you
think this other book has?” In addition, you can incorporate the element
of time by asking such questions as, “The book we’re reading has one
hundred seventy-nine pages. Today we read twenty-one pages. About how
many days do you think it will take to finish the whole book?”
Gathering the estimates on sticky notes and posting them on the board
also gives a context for data analysis. Students can find the range, mode,
and median of their estimates (see “Looking at Data” for more details).

Overview
The game of pig is a popular probability game usually played by partners
or in small groups. In this version of the game, the whole class participates
together, making it a quicker, yet equally engaging activity.
Probability games, especially those involving dice and cards, supply
absorbing occasions for strategic thinking and arithmetic practice in con-
text. This game gets students to consider likely and unlikely outcomes
while practicing addition. It’s also nice because it gets students up on their
feet and moving around a bit.
Activity Directions
1. Ask the students to stand up by their desks.
2. To begin a round of Whole-Class Pig, the teacher rolls two dice and
tells students the number on each die. The students state the sum
aloud.
3. Each student writes the sum on his or her paper.
4. Each student decides whether to save the points earned from that roll
or to keep going. If a student decides to save his or her points, he or
she circles the total and turns the paper over. The student also sits
down until the end of the round.
5. The round continues until all students have sat down and flipped
their papers over or until a 1 appears on a roll.
6. If a 1 appears on either die, students who did not save their points
lose all the points they accumulated on that round.
7. If two 1s appear on the dice, everyone goes back to zero, regardless
of how many points he or she saved on the previous rounds.
8. The goal is to get 100 points. It’s OK to go over 100. As soon as a
student reaches or surpasses 100, he or she should raise a hand.
9. Lead a discussion for students to share their strategies and to
consider if some sums are more likely than others.
Whole-Class Pig
26
䊝䊞䊟䊞
䊝䊞䊟䊞
•
•
•
•
CONTENT AREA
Data Analysis and
Probability
MATERIALS
■ 2 dice
TIME
135

Key Questions
• What strategy are you using?
• How did you decide when to stop?
• Are some rolls more likely than others?
From the Classroom
Rather than starting with a lengthy explanation, I decided to jump right
into the game Whole-Class Pig with Olivia Gonzalez’s fifth graders. The
students would need to ask me questions about the rules as they arose. I
asked all the students to stand up, push their chairs in, and stand in front
of their desks.
“OK,” I began, “each of you has a piece of paper and a pencil. I’m
going to roll these two dice and the sum of the dice is your score. Are you
ready?”
I checked and noted the nodding heads and smiles. I rolled the dice.
“It’s a five and a three. What’s the total?” I asked. I always tell the
amount on each die and have the students do the addition. I try to capi-
talize on any occasion for students to practice basic facts or do arithmetic
in context.
“Eight,” came the unanimous response.
“Good,” I said, “so now everyone has eight points. Write an eight on
your paper. If you want to save your points, circle the eight, flip your
paper over, and sit down. If you want to keep going, just remain standing
and leave the eight faceup.”
I looked around the room and noticed some puzzled faces. I plowed
on, hoping someone would stop me and ask for clarification.
“So, I’m looking around the room and some of you have flipped your
papers over and sat down, which means you’re stopping, but most of you
haven’t. I’m going to roll again and those of you who are still standing can
add the next roll to your total.”
“That’s not fair!” argued Andi, who had flipped her paper over and
was sitting down.
“Why not?” I queried.
“Because some people will keep getting points and we’ll be stuck with
eight.”
“Aha,” I replied, “I need to explain one other rule to this game. If you
decide to keep going and I roll a one on either die, you lose all the points
for this round.”
“What if we did circle our eight and sat down?” Daniel asked.

“Then your points are safe and you can’t lose them,” I explained.
“There’s one other rule you need to know. If I roll double ones, or snake
eyes, you lose all your points, even the ones you saved. Everyone goes all
the way back to zero. So let’s hope that doesn’t happen.”
It was my hope that students would take ownership of the game and
assert themselves. I wanted them to realize there was information missing
and that they needed to ask questions in order to find out the details they
lacked. I don’t always want to spoon-feed everything to the students. They
need to take charge of their learning and begin to insist that things make
sense before they proceed. Introducing the Whole-Class Pig game in gen-
eral terms forced the students to make sense of it and ask questions when
they needed more information. I had the students turn to their partners
and tell the rules as they understood them. After this brief interlude I
continued with the game. I rolled a 1 and a 3.
“Uh oh,” I told the class, “I just rolled a one and a three. So what does
that mean?”
“We’re back to zero,” Edgardo proclaimed glumly.
“No,” Keenan argued, “not if we saved our points.”
“That’s right,” I agreed. “If you flipped your paper over and sat down,
you still have eight points. If you kept going, you’re now at zero. So right
now everyone either has eight points or zero points. Everyone stand up
and flip your papers back up and we’re going to play Round Two.”
“What are we trying to do?” Tina asked.
“Another great question,” I replied. “The goal of this game is to get
one hundred points. So you always have the choice of saving the points
you have or continuing on for more points in each round. The only prob-
lem is that if you decide to be a pig and go for more points, you might end
up losing all your points. That’s why this game is called Whole-Class Pig.”
I rolled again and students recorded their new totals on their papers.
After each roll I asked students to either circle their new totals, sit down,
and flip their papers or hang in and continue. Each round ended either
with a roll of 1 or all students saving their points. Then we started a new
round. After several rounds I polled the students about their current
totals.
“Does anyone have more than twenty?” I asked.
I deliberately started with a low number that I knew everyone in the
class had surpassed. That way, all students would feel some success and be
able to raise their hands.
“How about more than thirty?” I continued.
I saw fewer hands this time.
“Wait,” exclaimed Umberto, “I’m not sure what I have.”
“Let’s see if the class can help,” I offered.
Whole-Class Pig 137

Umberto told me his total points from each round and I wrote them on
the board. I gave the students a few moments to find the sum of Umberto’s
number string (see “Number Strings” for more details).
Umberto’s Points
14 ⫹ 10 ⫹ 4 ⫹ 4 ⫹ 11 ⫽
An opening for a quick number talk arose, and I took advantage. I
had several students share their thinking. I began with Umberto since it
was his score we were talking about. He had just added the numbers
from left to right. Other students started with the tens or saw a lot of
fours and grouped them together first. After establishing Umberto’s total
and acknowledging the various ways to compute it, we got back to the
game.
“Is this like bingo?” David asked.
“I’m not sure I know what you mean,” I told him.
“Like, if we get one hundred, do we yell something out?” he clarified.
“Hmm,” I said, “I guess when you get to one hundred, you can raise
your hand.”
“Does it have to be exactly one hundred?” Shante asked.
“Another good question,” I responded. “You don’t need to get exactly
one hundred. You can go over.”
We continued the game. I rolled the two dice, told the students the re-
sults, and had them say the sum aloud each time. This game really al-
lowed the students’ individual personalities to shine. Some students
boldly remained standing through many rolls. Other students took a
conservative approach and proudly circled and saved their points after
one or two rolls. Others showed more hesitancy and looked to their
friends for cues. Throughout the game there was a good deal of talking,
laughing, and interaction. I established a signal so the students would
settle down and focus before each new roll. I told the students I would
count down from three, at which point I expected them to either be
standing or sitting. Also they all needed to be quietly paying attention to
the upcoming roll. This approach worked well. It allowed the children to
express their enthusiasm while allowing me to manage the group and
keep the activity moving.
We continued playing until several students announced they had
reached 100. At this point we stopped the game and had a brief
discussion.
“I’m curious to know if any of you had a strategy or plan while we
were playing,” I prompted the class.
“I think it’s good to take a risk and keep going,” Katy shared.

“I stopped after two or three or four rolls,” Henry stated.
“How did you decide when to stop?” I pushed.
“Well,” Henry explained, “if you rolled some high numbers like fives
or sixes, then I would stop sooner to save my points. If you rolled low
numbers I would keep going because I didn’t have much to lose.”
“Interesting strategy,” I acknowledged. “Other people might want to
try that strategy the next time we play.”
Then I shifted the focus of the discussion. “This game involves prob-
ability,” I explained to the class. “When I roll the dice we don’t know
exactly what is going to happen, but there might be some outcomes that
are more likely and some that are unlikely. Did anyone notice sums that
seemed more likely to occur? Were there certain numbers that came up
more than other numbers?”
I deliberately asked the same question in two ways to help students
connect the mathematical vocabulary with familiar words and concepts.
“It seemed like you rolled a lot of eights,” Martin observed.
“So eight is a more likely sum when I roll the dice?” I inserted appro-
priate vocabulary as I restated Martin’s idea.
“Yes,” Martin agreed.
“Can anyone tell a sum that seemed unlikely? Something that hardly
happened at all?”
“You never got double ones,” Cindy responded.
“Lucky for us!” Angela added.
“So it seems unlikely that double ones will occur when I roll the dice,”
I restated. “Is it possible to get double ones?”
“Yes!” was the resounding response.
“Sure,” I agreed, “it’s possible that it could happen, but it’s unlikely.
When I visit again we can explore some of these probability ideas further.
Maybe that will help you think about other strategies for the game of
Whole-Class Pig.”
Playing Whole-Class Pig with the class is lively and engaging. Once stu-
dents have played the game a few times and discussed likely and un-
likely sums, lead some other probability investigations. One easy way to
look more deeply at the probability aspect of the game is to graph the
sums as they are rolled. Display a class graph and add to it over time.
Simple tally marks next to each sum rolled will soon reveal likely and
unlikely sums.
Whole-Class Pig 139

If students play the game with partners, they can keep track of the
sums rolled and add the data to the class graph. With enough data the
graph will begin to resemble the famous bell-shaped curve. The graph can
be used as a jumping-off point for a discussion about game strategy and
how it relates to the data posted. A further investigation involves deter-
mining all the possible two-dice combinations and relating that informa-
tion to the graph and the game.
Another version of the game involves playing ten rounds of pig. The
goal is for students to see how many points they can get in those ten
rounds. They may reach or surpass 100, but that isn’t the focus. In some
ways this version mitigates the competitive race to be the first to get to
100. Instead all students are testing out their strategies to see how well
they work.
2
3
4
5
6
7
8
9
10
11
12

Overview
Your Choice Tic-Tac-Toe offers a new twist to the traditional tic-tac-toe
game. The goal remains the same—get three in a row—but on each turn
the student may choose to be either an X or an O. While challenging stu-
dents to break the old habit of remaining either X or O for the entire
game, Your Choice Tic-Tac-Toe also provides opportunities for spatial
reasoning and strategy development.
Activity Directions
1. Explain to students that Your Choice Tic-Tac-Toe is just like regular
tic-tac-toe with one twist: players may choose whether to be X or O
on each turn instead of keeping one symbol or the other for the
entire game.
2. To demonstrate for the entire class how to play the game, ask two
student volunteers to play a game as you guide them.
3. Draw a tic-tac-toe game board on the board or an overhead.
4. Player A puts either an X or an O anywhere on the game board.
5. Player B puts either an X or an O anywhere on the board.
6. Play continues until one player completes a line of three in a row.
7. Ask another pair to play a game in front of the class.
8. Have students play the game in pairs.
9. Lead a discussion for students to share their opinions of game and
discoveries they made about Your Choice Tic-Tac-Toe.
Key Questions
• What did you think of this game?
• What discoveries did you make?
Your Choice Tic-Tac-Toe
27
䊝䊞䊟䊞
䊝䊞䊟䊞
•
•
•
•
CONTENT AREA
Geometry
MATERIALS
projector
TIME
141

From the Classroom
“Raise your hand if you know how to play tic-tac-toe,” I said to Melissa
Mechtly’s third graders.
Hands shot into the air along with a murmur of excitement.
“Well,” I continued, “today you’re going to learn a new version of tic-
tac-toe. It’s the same as regular tic-tac-toe in a lot of ways. You use the
same board. You play with a partner. You use Xs and Os. You want to get
three in a row horizontally, vertically, or diagonally. But there’s one big
difference.” I wrote on the board:
“In Your Choice Tic-Tac-Toe, you get to decide whether you want to
use an X or an O on every turn. With regular tic-tac-toe you pick which
partner is X and which is O at the beginning of the game and you stick
with it. This version of the game gives you the choice of changing your
symbol on each turn.”
“Huh?” Charlie responded.
“Yeah,” Kayla agreed, “I don’t get it.”
I wasn’t surprised at the students’ confusion. Mere verbal directions
often leave questions and uncertainties. I wanted to give the students a
brief overview of the game and connect it to a game they already knew.
However, I would need to model a game in front of the whole class to help
the students really understand.
“How about this?” I suggested to the class. “Let’s get two volunteers
up here to play a game of Your Choice Tic-Tac-Toe. That way everyone
can see what the game looks like and ask questions about it.”
Hands flew into the air as almost all students wanted to play in front of
the class. I picked Aisha and Emilio to play a game. I drew a tic-tac-toe grid
on the whiteboard and handed each student a marker. Emilio put an O in
the upper right corner. Aisha put an X in the middle. Emilio put an O in the
upper left corner. Aisha put an O in the top middle box and had three Os
in a row. Emilio laughed and slapped his head as he realized his mistake.
“This is tricky,” I said. “Sometimes we’re so used to doing things a
certain way that we forget the rules have changed. Does anyone else want
to come up here and play a game in front of the class?”
I called on another boy and girl to play. I try to be aware of gender as
I seek volunteers for whole-class activities. Often boys dominate discus-
sions during math time, but it’s possible to equalize participation with
some deliberate awareness.
The second game was a bit longer since the students had some experi-
ence to draw from. They began to see the considerations necessary for this

new version of the game. After the game was over, I sent students off to
play in pairs. I told them we’d get back together in a few minutes to dis-
cuss what they had noticed and learned.
I circulated among the pairs as they played. Typically, students began
by forgetting the new rules and making some quick game-ending errors.
After a few tries, however, the students began to think more strategically
and plan for the possibility of their partners switching from Os to Xs or
vice versa. After a while I called the students back to the rug and we dis-
cussed their experiences. I started with a very general question to get the
conversation going.
“So what do you think of this game?” I asked.
“It’s fun,” Antoinette replied.
“It’s tricky because I kept forgetting that my partner could switch and
he kept getting three in a row,” Javier added.
“I liked it because I like tic-tac-toe,” Carol shared.
“OK,” I moved on, “I know you didn’t have a lot of time to play, but
did you discover anything in the time you had?”
“I made a discovery,” Jon announced. “It’s good to put an X or an O in
the middle. Then if your partner puts the same thing anywhere you can win.”
I drew a tic-tac-toe grid and modeled Jon’s discovery. Several other
students had used the same strategy so there was a consensus about his
approach. Students also realized that if both partners knew about this
strategy, it would be more difficult to win with it.
“I noticed something else,” Hakim added. “If you start with an O in
the middle and then you have Xs on the two diagonal corners of the O,
you will always win.”
“Hmm,” I said, “let’s take a look at Hakim’s idea.”
I drew another tic-tac-toe board and had Hakim tell me where to put
the Xs and the O.
Your Choice Tic-Tac-Toe 143

We checked each empty square to see what Hakim would do if his
partner put an X or an O in it. Sure enough, Hakim could get three in a
row regardless of what his partner did.
“Interesting,” I commented. “It seems like there are some good strate-
gies for Your Choice Tic-Tac-Toe. When you play again, try out some of
these ideas and also see if you can find some other ways to get three in
a row.”
I left the class, telling the students that I was looking forward to hear-
ing more about their discoveries when I returned.
Rather than start with verbal directions, another option is to jump in and
model a game with minimal introduction. Then have students generate a
list of how regular tic-tac-toe and Your Choice Tic-Tac-Toe are the same
and different. A Venn diagram or T-chart would serve well as a graphic or-
ganizer for the same-different comparisons.
Writing also works well as a follow-up to this activity. Challenge stu-
dents to write directions for the game. They can also write about their dis-
coveries or strategies. Sentence frames such as “I liked . . . ,” “I noticed . . . ,”
“I discovered . . . ,” and “I wonder . . .” help students verbalize their ideas
and communicate them in written form.
Your Choice Tic-Tac-Toe also makes a nice homework assignment.
Have students teach the game to people at home, play with them, and
report or write about the experience.

Coordinate Tic-Tac-Toe Grid
Dot Clusters Overhead Transparency
Dot Clusters Recording Sheet
Blackline Masters
145
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From
Minilessons
for
Math
Practice,
Grades
3–5
by
Rusty
Bresser
and
Caren
Holtzman.
©
2006
Math
Solutions
Publications.
x
y
x
y
X O
X O

Dot Clusters
From
Minilessons
for
Math
Practice,
Grades
3–5
by
Rusty
Bresser
and
Caren
Holtzman.
©
2006
Math
Solutions
Publications.

estimating with (activities),
7–12, 97–101
introducing the term, 9
biggest. See largest/smallest
Blackline Masters, 145–49
(game), 147
Dot Clusters (activity), 148–49
books, in estimating activities,
130–34
cardinal numbers, 105
chapter book read-aloud, 134
choral response approach, 2–3
classifying numbers by attributes
(activity), 69–74
Clear the Board (activity), 13–17
Coin Riddle (activity), 18–23
Comparing Fractions (activity),
24–29
comparison of fractions (activity),
24–29
compensation, 87
Contents Chart, xv
coordinate graphs, 31–32
negative numbers, activity
with, 34
in tic-tac-toe game context
(activity), 30–34
Coordinate Tic-Tac-Toe (game),
30–34
Blackline Master, 147
counting game with strategy and
logical thinking, 107–10
addition and subtraction
in benchmark number
activities, 75–80, 87–91
effect of patterns and number
combinations on (activity),
92–96
with Function Machine
(activity), 61–62
mental math dice activity of
generating numbers, 13–17
mental problem-solving and
discussion activity, 1–6
in Number Strings (activity),
87–91
probability game with addition,
135–40
target number game using,
111–15
algebra
coordinate graphs, tic-tac-toe
game with, 30–34
counting game with strategy
and logical thinking, 107–10
Function Machine (activity),
57–63
associative property, 16–17
Ballpark Estimation (activity),
7–12
benchmark numbers, 87, 118
100 (activities), exploring
numbers’ relationship to,
75–80, 88–91
defined, 77, 98–99
data analysis
graphing activity, 81–86
with probability game, 135–40
decimals, comparison activity, 29
decomposing numbers, with
Twenty Questions game,
120–24
dice
benchmark number activity
using, 75–80
fraction construction using,
24–29
mental math computation
activity with, 13–17
with probability game, 135–40
target number addition and
subtraction game using,
111–15
Digit Place (game/activity), 35–40
division, mental math dice
activity of generating
numbers, 13–17
Dot Clusters (activity), 41–47
Blackline Masters, 148–49
doubling patterns, 128
equations, activity to explore,
116–19
estimation
with benchmark numbers,
7–12, 97–101
classroom items, relating to
(activities), 7–12, 97–101
defining, 8–9
151
Index
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graphing estimates of perimeter
(activity), 81–86
linear measurement activity
with, 48–52
objects in a jar (activity),
estimating quantity of,
53–56
using books for (activity),
130–34
Estimation Jar (activity), 53–56
even/odd numbers, logic game
about, 92–96
exponents, 129
Fibonacci numbers, 129
fractions
with benchmark number
activity, 101
comparing, activity, 24–29
Function Machine (activity),
57–63
geometry
numbers as groups of objects
(activity), presenting,
41–47
pattern recognition
activity, 129
riddle activity with shapes, 23
Gonzalez, Olivia, 136
Gordon, Robin, 2, 36,
98–101, 103
Grapes of Math, The, 47
graphs
coordinate (see coordinate
graphs)
data analysis (activity), 81–86
greater than. See more than
growing patterns, 126–29
guessing games
numbers within given range
(activity), guessing, 64–68
rules relating to numbers,
guessing, 69–74
for zeroing in on numbers,
35–40
Guess My Number (activity),
64–68
Guess My Rule (activity),
69–74
Holes, 131
How Far Away? (activity), 75–80
Jackson, Amy, 14–17, 43
Kimbrell, Lisa, 8, 117
largest/smallest, comparing
fractions activity, 24–29
length of classroom, activity for
measuring and estimating,
48–51
less than, in number guessing
activity, 64–68
linear measure, in estimating and
measuring the classroom
(activity), 48–51
logical thinking
addition problems (activity),
effect of patterns and number
combinations on, 92–96
clues and mental computation
activity, 18–23
counting game with strategy
and, 107–10
and game of Twenty Questions,
120–24
guessing game for zeroing in on
numbers, 35–40
guessing numbers within given
range (activity), 64–68
Looking at Data (activity),
81–86
measurement
estimation activity with,
12, 101
estimation activity with linear,
48–52
graphing estimates of perimeter
(activity), 81–86
Mechtly, Melissa, 49, 76,
125–29, 142
median
in graphing activity, 81, 84, 85
in number sense activity, 134
mental chalkboard strategy, 132
mode
in graphing activity, 81, 84, 85
in number sense activity, 134
models of numbers, physical
(activity), 41–47
money, Coin Riddle (activity),
18–23
more than, in number guessing
activity, 64–68
multiplication
(activity), 61–62
mental problem-solving and
discussion activity, 1–6
multiples activity, 129
with pattern recognition
activity, 128–29
National Council of Teachers of
Math (NCTM), 74
negative numbers
coordinate graphs activity
with, 34
function machines with, 61–62
nim family of logic games, 107
nominal numbers, 105
number lines
in benchmark number activity,
75–79
with fraction activity, 26–29
with number guessing activity,
65–67
open, 132
using books as estimating
activities, 132–33
numbers
cardinal numbers, 105
Fibonacci, 129
numbers, 35–40
negative (see negative numbers)
nominal numbers, 105
odd/even, logic game with,
92–96
ordinal numbers, 105
personal numbers (activity),
102–6
physical models of (activity),
41–47
152 Index

number sense, in using books as
estimating activities, 130–34
number strings, defined, 87
Number Strings (activity), 87–91
Number Talks (activity), 1–6
Odd Number Wins
(game/activity), 92–96, 107
one hundred (100) as benchmark
number, activities for
exploring, 75–80, 88–91
one thousand (1000) as
benchmark number, activity
with, 79–80
1–100 chart
with game of Twenty
Questions, 122–24
guess my number activity with,
67–68
open number lines, 132
ordered pairs, in tic-tac-toe game
(activity), 30–34
ordinal numbers, 105
origin, in graphing, 31
Over or Under (activity), 97–101
parentheses, 14, 16–17
patterns
attributes of numbers, activity
for examining, 69–74
doubling, 128
effect of, on addition problems
(activity), 92–96
(activity), 57–63
growing, 126–29
pattern recognition activity,
125–29
repeating, 128
Personal Numbers (activity),
102–6
Pickett, Danielle, 19, 58–62, 112
Pig (game), 135–40
place value
numbers, 35–40
target numbers, addition and
subtraction game with,
111–15
powers of two, 129
practice, redefining, ix
prediction
with function machines activity,
59–63
in pattern recognition activity,
125–29
probability
game, 135–40
graphing activity, 81–86
Race for Twenty (game), 107–10
range
in graphing activity, 81, 82–83,
84, 85
in number sense activity,
130–32, 134
Rasori, Tina, 31, 108
reasonablness/unreasonablness
of personal numbers, 102,
103–5
of solutions to equations,
116–19
repeating patterns, 128
riddles, coin (activity), 18–23
rules relating to numbers, activity
of guessing, 69–74
Sachar, Louis, 131
sorting numbers by attributes
(activity), 69–74
spatial reasoning, in tic-tac-toe
game, 141–44
Stark, Patty, 65, 70
subtraction. See addition and
subtraction
symmetry, 45
Tang, Greg, 47
Target 100/Target 0 (game),
111–15
T-charts
with function machines activity,
57–63
with odd/even logic game,
95–96
ordered pairs with, 32–33
Tell Me All You Can (activity),
116–19
tic-tac-toe games
coordinate graphing with,
30–34
spatial reasoning with, 141–44
Twenty Questions (game), 120–24
Venn diagrams, 69–74
What Comes Next? (activity),
125–29
What Page Are We On? (activity),
130–34
Whole-Class Pig (game), 135–40
width of classroom, activity for
measuring and estimating,
48–51
writing
coin riddles, 22–23
with tic-tac-toe game, 144
x-axis, 31–32
y-axis, 32
(activity), 141–44
Index 153

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