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ALGEBRAIC THINKING (Strand D)
Number Sentences/Number
Sense (Standards 1 and 2)
- Merriam, Eve. 12 Ways to
Get to 11. New York: Simon & Schuster Books, 1993.
This book explores number
sentences as it investigates many different ways to reach the number
11.
- Have students pick a number.
Have them write as many different number sentences as possible. Encourage
students to use more than two numbers, more than one operation, and
to write the sentences using the correct order of operations.
- Hulme, Joy N.
Sea Sums. New York; Hyperion Books for Children, 1996.
Addition and subtraction
problems are told in an ocean theme.
- Have students create
a book of their own along the same lines, writing number sentences
to describe the pages of their book. Also, the ideas for use with
12 Ways to Get 11 are appropriate here.
- Dee, Ruby. Two
Ways to Count to Ten. New York: Henry Holt and Company, 1988.
The leopard king is seeking
a husband for his daughter. The animal who is able to toss a spear into
the air and count to ten before it hits the ground wins. This tale indicates
that sometimes the cleverest is the winner.
- Before reading the book,
have students try to count to twenty as fast as possible and record
the times. After reading the book, connect the different ways to count
to a number to the factors of the number.
Patterns and Variables
(Standard 1)
- Falconer, Elizabeth.
The House That Jack Built. Nashville, TN: Ideals Children's Books, 1990.
This book describes a growing
pattern, from the house that Jack built to the farmer who sows corn.
- Have students create repeating
and growing patterns, grouping patterns with the same basic structure
together.
- Scieszka, Jon and
Lane Smith. The Stinky Cheeseman and other Fairly Stupid Tales. New
York: Viking, 1992.
Many favorite fairy tales
are told from a more modern perspective. Particular stories that emphasize
patterns are “The Princess and the Bowling Ball,” “Jack’s
Story,” and “The Stinky Cheese Man.”
- After reading all of
these books, give children a strip of adding machine tape. Have them
draw a pattern on their strip, making sure they draw at least two
cycles of their pattern. Then have the children sort their patterns
on the classroom wall so that like patterns are grouped together.
Children need to justify why patterns in a group are alike and identify
the pattern as ABAB, etc. (This activity is described more fully in
the NCTM Addenda Series Patterns K-6.)
- Mitsumasa, Anno.
Anno’s Math Games II. New York: Putnam & Grosset, 1997.
A wide range of mathematics
topics are illustrated through diagrams. The section on The Magic Machine
deals with ideas connected to a function machine. The section on Counting
with Circles provides a nice introduction to ideas connected with variables.
- Read the pages in the
section on The Magic Machine. Have children identify what each machine
does. Then have children create their own “function” machine.
Read the section on Counting with Circles. Have students create similar
pages of their own.
Patterns, Tables, and
Rules (Standard 1)
- Pinczes, Elinor
J. One Hundred Hungry Ants. Boston: Houghton Mifflin, 1993.
One hundred ants are marching
to a picnic. They arrange themselves into rows of different sizes in
order to get to the picnic faster.
- Have children complete
the following chart.
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Number
of
Columns
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Number
of Ants
in
a Column
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1
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100
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2
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Write a rule that describes the pattern in the table. Graph the results.
- Hutchins, Pat. The
Doorbell Rang. New York: Mulberry Books, 1986.
A plate of cookies is divided
among several children. The number of children changes each time the
doorbell rings.
- Do the same things as
with A Hundred Hungry Ants.
- Giganti, Paul, Jr.
Each Orange Had 8 Slices. New York: Greenwillow Books, 1992.
Each page of this book contains
several problems, with multiplication as the focus.
- Take any page of the book.
Create a table to illustrate the relationships that are described.
Try to write one or more rules for the relationships. Graph the results.
Connect the steepness of the graphs to ideas of slope.
- Hulme, Joy N. Sea
Squares. New York: Hyperion Books, 1991.
This counting book uses an
ocean theme as it explores multiplication patterns of the form n x n,
from 1 x 1 to 10 x 10.
- Have children consider
the next few numbers in the pattern. Have children write a rule to
describe their pattern. If appropriate, introduce the notation for
exponents.
- Anno, Masaichiro
and Mitsumasa Anno. Anno's Mysterious Multiplying Jar. New York: Philomel
Books, 1983.
A jar contains an island
on which are two countries. Each country contains three mountains. The
patterns continue until there are 10! jars.
- Have children relate the
descriptions in the book to multiplication. If appropriate, introduce
factorial notation. Have children create their own multiplication
book. Relate the ideas in the book to problems connected with the
Multiplication Counting Principle. [e.g., Nine people are on a baseball
team. Without any restrictions, how many batting orders are possible?
- Goble, Paul. Her
Seven Brothers. New York: Aladdin Paperbacks, 1988.
A young girl searches for
seven brothers she does not know. Through a series of adventures, they
become the Big Dipper. The story is based on a Cheyenne legend.
- Have children create
designs using colored toothpicks. Have them describe the number of
toothpicks of a given color needed for 1, 2, 3, 4, ..., 100 of their
designs. Write a rule that tells a company how many colored toothpicks
they need to have if someone orders a given number of your design.
(Activities with sample children’s work are described in a chapter
by Thompson, Chappell, and Austin in the forthcoming Addenda series
on Changing the Faces of Mathematics: Perspectives on Indigenous Peoples.)
More Advanced Patterns
-- Exponential Growth (Standard 1)
- Hong, Lily Toy.
Two of Everything. Morton Grove, IL: Albert Whitman & Company, 1993.
A couple finds a brass pot
which doubles everything placed into it. The couple's life changes dramatically.
- Suppose you start with
5 coins and place them in the pot. Continue doubling the results and
record the values in a table. How long will it take before you have
1000 coins? What if you had a triple pot? What if you started with
1000 coins and had a half-pot? How long would it take before you have
less than 50 coins?
- Losi, Carol A.
The 512 Ants on Sullivan Street. New York: Scholastic, 1997.
Ants at a picnic keep doubling
until they steal all the food.
- If the pattern continues,
how many ants would be needed for the next three food items? Find
the total number of ants each time a new food is added to the story.
Try to write rules to describe your patterns. Graph the number of
ants with each food and the total number of ants.
- Birch, David. The
King's Chessboard. New York: Puffin Pied Piper Books, 1988.
A wise man does a service
for a king who insists on giving a reward. The wise man requests one
grain of rice for the first square of a chessboard, with the number
of grains doubling for each new square of the chessboard. The king eventually
realizes that there would not be enough rice in all the world to meet
the wise man's request.
- Barry, David. The
Rajah's Rice: A Mathematical Folktale from India. New York: W. H. Freeman
and Company, 1994.
A young girl heals the rajah's
sick elephants. She then defeats the rajah by requesting a reward of
rice in which the number of grains doubles each day until all the squares
of a chessboard are covered.
- Demi. One Grain
of Rice: A Mathematical Folktale. New York: Scholastic, Inc., 1997.
A young girl uses her wits
to help starving people and teach the wicked rajah a lesson. This is
another variant on the doubling tale.
- Pittman, Helena
Clare. A Grain of Rice. New York: Bantam Skylark Book, 1986.
A humble servant gets one
grain of rice on the first day from the Emperor, with the number of
grains of rice set to double each day for 100 days.
- For all four of these
books, have children create a table with the number of grains of rice
on each square of the chessboard or each day of the specified period.
Have children describe the patterns they see and graph the results,
if possible. Have children explore weight and space issues connected
to the quantities of rice.
- Chwast, Seymour.
The 12 Circus Rings. San Diego: Gulliver Books, 1993.
This book explores multiplication
and sequences as readers explore the objects in each of 12 circus rings.
- Record the number of new
animals, people, etc. who enter each of the circus rings. Also record
the total number of people who are in a given ring.
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Circus
ring
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Number
of new objects in this circus ring
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Total
number of objects in this circus ring
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Total
number of
objects in circus in all the circus rings
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1
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1
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1
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1
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2
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2
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3
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4
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3
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3
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6
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10
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4
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- Anno, Mitsumasa.
Anno's Magic Seeds. New York: Philomel Books, 1995.
Two magic seeds are given
to a young man. When a seed is buried, it produces two seeds the following
year. A variety of patterns are explored in this book.
- Have children keep a record
of the number of seeds Jack plants and picks each year throughout
the story. This information can be graphed.Equations (Standard 2)
- Edens, Cooper.
How Many Bears? New York: Atheneum, 1994.
The number of bears needed
to run a bakery is determined by solving mathematical clues given throughout
the book.
- Have children solve the
equations described in the hints on each page. Have children make
their own book so that not all the pages lead to the same final result.
Miscellaneous (classification)
- Lowell, Susan.
The Three Little Javelinas. Flagstaff, AZ: Northland Publishing Company,
1992.
The story of the three little
pigs is retold with a southwest flavor.
- Laird, Donivee
Martin. The Three Little Hawaiian Pigs and the Magic Shark. Honolulu,
HI: Barnaby Books, 1981.
This retelling of the classic
three little pigs story has a Hawaiian flavor.
- Scieszka, Jon.
The True Story of the 3 Little Pigs! New York: Puffin Books, 1989.
The story of the three pigs
is told from the point of view of the wolf.
- After reading the three
versions of the Three Little Pigs, have children use a Venn diagram
to explore the similarities and differences among the various versions.
Identifying attributes and characteristics is an important part of algebraic
thinking.
©2001
Denisse R. Thompson
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