Big Ideas for Growing Mathematicians: Exploring Elementary Math with 20 Ready-to-Go Activities

130 Pages·2007·2.2 MB·English

by  Ann Kajander

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Educational Resource Grades 5–8 j
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Probability, fractals, and exponents . . . e
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BBiigg iiddeeAAss
in elementary school? You bet!
Elementary school mathematics Triangular Tessellations
doesn’t have Students investigate
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to be all skill and drill. Occasionally students the geometry and variations created by repeating
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need to see the big picture. Children are capable of patterns. GGrroowwiinngg

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understanding relatively advanced concepts, from
Fractional Salaries
Kids use a real-world scenario e
pre-algebra to infinity, when they are properly pre- A
to multiply and divide fractions.
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sented through engaging inquiry.

Kaleidocycles f
Students build and investigate their o
Big Ideas for Growing Mathematicians is an ideal r

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own geometric hexaflexigons. MMaatthheemmaattiicciiaannss
resource for elementary school mathematics enrich-
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ment programs, regular classroom instruction, or a Each unit includes a brief overview, objectives, w
home enrichment or home school program. Twenty materials, vocabulary, and step-by-step procedures. in
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intriguing projects cover a wide range of math con- Reproducible student activity sheets are included for

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tent and skills. instruction and assessment.
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ProjEcts includE h
Ann Kajander is the founder of the Kindermath
e Ages 10–14
One in a Million m
Children use Enrichment Project and an adjunct professor in the
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grains of rice to model the prob- Mathematics Department of Lakehead University. t
i exploring elementary
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ability of astronomical odds.
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Math with 20
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$19.95 (CAN $24.95)
Ready-to-go Activities
Ann Kajander
Big ide a s
Growing
for
Mathematicians

Big idea s
Grow ing
for
Mathematicians
exploring elementary Math with 20 Ready-to-go activities
ann Kajander
Chicago
Library of Congress Cataloging-in-Publication Data
Kajander, Ann, 1960–
Big ideas for growing mathematicians : exploring elementary
math with 20 ready-to-go activities / Ann Kajander.
p. cm.
ISBN-13: 978-1-56976-212-7
ISBN-10: 1-56976-212-0
1. Mathematics—Study and teaching (Elementary)—Activity
programs. I. Title.
QA135.6.K349 2003
372.7'2—dc22
2006100397

All rights reserved. The purchase of this book entitles the indi-
vidual teacher to reproduce the forms for use in the classroom.
The reproduction of any part for an entire school or school system
or for commercial use is strictly prohibited. No form of this work
may be reproduced, transmitted, or recorded without written per-
mission from the publisher.
Cover and interior design: Monica Baziuk
Interior images © Ann Kajander
© 2007 by Ann Kajander
All rights reserved
Published by Zephyr Press
An imprint of Chicago Review Press, Incorporated
814 North Franklin Street
Chicago, Illinois 60610
ISBN-13: 978-1-56976-212-7
ISBN-10: 1-56976-212-0
Printed in the United States of America
5 4 3 2 1
Acknowledgments
I
am particularly indebted to Ralph Mason and David I am grateful to the students in my Kindermath
Stocker, whose ideas have directly supported chap- enrichment program who allowed me to conceive of
ters in this book. I appreciate the mathematical support these ideas, as well as my own children, Arthur, Robin,
of Miroslav Lovric. George Kondor has also been most and Maria, and my husband, Wally, for his interesting
generous with his materials. Other members of the questions. As well, many teacher candidates have exper-
Canadian Mathematics Education Study Group have imented with these activities and provided wonderful
provided deep inspiration and allowed me a glimpse feedback. It is to these teacher candidates that I dedi-
into the ideas that excite mathematicians. Peter Taylor cate this book. I hope you will continue in your efforts
has provided particular support and encouragement as to bring meaningful and engaging mathematical ideas
always. to your students.
Contents
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..viii .4 ..Lots.of.Dots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15
How to Use This Book. . . . . . . . . . . . . . . . . . . . . ..viii This is a collaborative activity stressing verbal
Content Areas in Each Activity. . . . . . . . . . . . . . . .x mathematical communication.
Process Skills Used in Each Activity. . . . . . . . . . ..xi .5 ..Average.Pentominoes. . . . . . . . . . . . . . . . . . . . .23
Prerequisite Knowledge and Skills for
Pentominoes are used to reason about possible
Particular Activities . . . . . . . . . . . . . . . . . . . . . . . .xii
patterns underlying the concept of averages.
.6 ..Distributing.Tiles. . . . . . . . . . . . . . . . . . . . . . . ..28
.1 ..One.in.a.Million.. . . . . . . . . . . . . . . . . . . . . . . . . . .1
This activity focuses on developing multiplication,
Students create a concrete model of a million the distributive property, and binomial
objects to investigate how unlikely “one in a million” multiplication with number tiles.
really is.
. 7 ..Guess.My.Rule. . . . . . . . . . . . . . . . . . . . . . . . . . . .33
.2 ..Math.Magic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5
Students investigate linear and nonlinear relations
Students work on a collection of favorite “math with a magician game.
magic” tricks, with explanations, for Halloween,
.8 ..The.Rice.Problem. . . . . . . . . . . . . . . . . . . . . . . . .37
April Fools’ Day, or just plain fun.
Students investigate the exponential function with
.3 ..Cooperative.Problems .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ...10
a concrete example.
A selection of geometry construction tasks are
provided with clues for group solving.
. 9 ..Lines,.Squares,.Cubes,.and.Hypercubes. . . .41 .17 ..The.Medicine.Experiment. . . . . . . . . . . . . . . . .81
Students investigate the first four dimensions with Students model kidney function with a simple
concrete models. dynamical systems experiment.
.10 ..Fractional.Salaries. . . . . . . . . . . . . . . . . . . . . . ..48 .18 ..Fabulous.Fractals. . . . . . . . . . . . . . . . . . . . . . . . .85.
Students work with concrete models for Students investigate and create their own fractal
multiplication and division of fractions. quilt, dream catcher, or other fractal piece.
.11 ..A.Temperature.Experiment. . . . . . . . . . . . . . ..52 .19 ..Triangular.Tessellations. . . . . . . . . . . . . . . . . ..90
Students investigate warming up and cooling down. Students extend their understanding of
tessellations to triangular shapes.
.12 ..Circular.Reasoning. . . . . . . . . . . . . . . . . . . . . . . .57
.20 ..Kaleidocycles. . . . . . . . . . . . . . . . . . . . . . . . . . . ..96
Students develop the formulas for the area and
circumference of circles by investigation. Students create their own
geometric hexaflexigon.
.13 ..Coins.for.Compassion. . . . . . . . . . . . . . . . . . . ..64
This is an investigation with nonstandard measures
and large numbers, with a social-justice theme. Glossary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..102
Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..105
.14 ..Toothpick.Rectangles. . . . . . . . . . . . . . . . . . . ..68
Students investigate the area and perimeter of
rectangles and their relationships.
.15 ..The.Stairs.Problem. . . . . . . . . . . . . . . . . . . . . . ..72
Students investigate ways to find the sum of a list of
consecutive numbers with a concrete model.
.16 ..Gumball.Boxes. . . . . . . . . . . . . . . . . . . . . . . . . . . .77
This activity requires filling boxes with gumballs
and looking at what happens when the sizes of
gumballs get smaller and smaller.
Introduction
M
any people think of mathematics as boring or dif- evolve along with deeper understanding. The activities
ficult. Yet when mathematicians describe their provided foreground the important ideas in mathemat-
subject, they use words like amazing, intriguing, and ics—ideas of dimension, space, infinity, movement and
full of surprise, awe, creativity, and wonder. How can change, patterns and prediction—and some contain an
such different perspectives exist? element of surprise. They focus on the skills mathema-
It seems to me that part of the difficulty lies in how ticians often use to solve problems—skills of working
we teach children. “School mathematics,” until very collaboratively or talking mathematics together. They
recently (and still much more than it should), has con- focus on exploring, extending, reasoning, and in some
sisted almost exclusively of rote methods and procedures. cases justifying. Many require the important skills of
The perception is that these skills are needed before imagining and visualizing. In some activities the con-
students can do higher level mathematics, where the nection to the standard procedure is obvious; in others
excitement lies. This is somewhat analogous to insisting the focus is only on the conceptual idea. Some activities
children learn to spell properly and learn all the rules of are just plain fun. I hope they will provide a glimpse of
grammar before they are encouraged to write stories or “real” mathematics to you and your students.
create their own poetry. This book is for teachers, teacher educators, parents,
This book and its predecessor, Big Ideas for Small and most of all, kids.
Mathematicians, are meant to provide to children and
adolescents glimpses of a “mathematician’s” version
How.to.Use.This.Book
of mathematics. While the activities do not ignore the
typical content of elementary- and middle-school math- Each activity in this book is divided into informative and
ematics classrooms, they focus more on the understand- instructional sections to make it easy to understand the
ing of important ideas, rather than on procedural fluency. ideas behind the activity as well as guide the students
In fact, research has shown that procedural fluency will through the activity successfully.
viii
j The Big Idea: This first section of each activity sum- background for you and to assist you in following and
marizes the underlying mathematical concept, so you facilitating the children’s thinking. Often children have
can get the big picture before diving into the details. a correct but incomplete intuitive sense of a problem,
and it is helpful for the adult to have a sense of the pos-
j Content Areas in This Activity: This section lists
sible approaches to use.
the main mathematical content areas the activity involves.
This information is also summed up for all activities in j Helpful Terms: This box includes a list of terms rel-
the chart on page x. evant to the activity and their definitions to enable you
to review the basic terminology and concepts as well as
j Process Skills Used in This Activity: This section
to give you wording to use when explaining the ideas to
lists the main thinking and learning skills the students
the children. You will also find a complete glossary at
will use throughout the activity. This information is
the end of the book (page 102).
summed up for all activities in the chart on page xi.
j Making It Work: This is the nuts-and-bolts sec-
j Prerequisite Knowledge and Skills: This section
tion of the activity. Here you will find objectives, a list
lists whether the children will need particular content
of materials you will need, preparation instructions, the
knowledge before embarking on the activity. Most of
activity procedure, suggestions for making the most of
the activities will be enjoyable for children on some level
the activity and for helping the children through trou-
regardless of their previous knowledge and skills, but
ble spots, assessment ideas for determining if the activ-
having these technical skills will make it easier for them
ity was successful, and an extension activity or two for
to understand the math behind the activity. The chart on
those kids who want to explore more.
page xii lists this information for the activities as a whole.
j Activity Sheet: The activity sheet lists each step
j Age Appropriateness: Most of the activities in this
of the activity, as in the procedures section of Mak-
book are suitable for all ages, with varying degrees of
ing It Work, but this sheet is directed to the children,
adult assistance. In this section, you will find sugges-
with illustrations and hints to help them work through
tions for adapting various aspects of the activity to suit
the activity successfully. A photocopy of this activity
different age groups. For example, younger children can
sheet will assist you in taking them through each step,
often enjoy an activity without having to understand
although this sheet is not meant to replace your active
all of the mathematics behind it, but each activity also
involvement and guidance. You may decide not to use
allows for further exploration of concepts for older and
the activity sheet for younger kids who won’t be able to
more advanced students.
read it well, or if you prefer to take the class through the
j The Mathematical Idea: This section describes the activity orally, without the help of written instructions.
mathematical idea at the heart of the activity to provide
ix