marvelous math writing prompts

Marvelous MathWriting Prompts300 Engaging Prompts and Reproducible Pages

That Motivate Kids to Write About Math—and Help You Meet the New NCTM Standards!

by Andrew Kaplan

N E W Y O R K • T O R O N T O • L O N D O N • A U C K L A N D • S Y D N E Y

M E X I C O C I T Y • N E W D E L H I • H O N G K O N G • B U E N O S A I R E S

S C H O L A S T I C

BPROFESSIONAL OOKS

Marvelous Math Writing Prompts © Andrew Kaplan, Scholastic Teaching Resources

Teachers may photocopy the designated reproducible pages for classroom use. No other part of this publication may be reproducedin whole or in part, or stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying,recording, or otherwise, without written permission of the publisher. For information regarding permission write to Scholastic Inc.,555 Broadway, New York, NY 10012.

Edited by Jacqueline GlasthalCover design by Josué Castilleja

Cover art by Mike MoranInterior design by Solutions by Design, Inc.

Interior illustrations by Mike Moran

ISBN: 0-439-21860-8

Copyright ©2001 by Andrew Kaplan. All rights reserved. Printed in the U.S.A.


Table of Contents

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Part 1—Math in School . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Let’s Talk Math, Class! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Numbers and Place Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Probability and Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Addition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Computation: Mixed Skills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Calculators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

Mental Math . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Problem Solving. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Part 2—Math in My World . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

Math at Home . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Math at Play . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

Money Math . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

News and Views About Math . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64


Why Write About Math?

When people think of math, they oftenthink of communication throughnumbers and symbols, rather thanwords. Yet the National Council ofTeachers of Mathematics (NCTM)considers the communication of maththrough writing so crucial that itspecified this form of communicationas one of its ten instructional“Standards” for the year 2000. Why iswriting about math so important? Theprocess of writing about math givesstudents opportunities to

find out what they know and don’tknow about mathematical topics;

articulate and clarify their thoughtsabout mathematical ideas andconcepts;

justify their problem-solvingtechniques and consider alternativeapproaches to problem solving;

integrate new mathematical ideasinto their thinking;

correct misconceptions; and

develop an appreciation of howmath is used in their everyday lives,and thus gain motivation forlearning more math.

In addition, when students write aboutmath they provide teachers withimportant information about what theyare learning, how they approach andassimilate new concepts, what conceptsthey need help with, and how they feelabout math class and math in general.

How to Use This BookMarvelous Math Writing Prompts

contains a variety of writing promptson a wide range of math-related topics. These prompts encourage students tothink and then write about these topicsin a number of different formats, fromletters and journal entries to wordproblems and stories.

The prompts in this book aregrouped into two parts. In Part 1, theprompts encourage students to write

4

Introduction

Marvelous Math Writing Prompts © Andrew Kaplan Scholastic Teaching Resources

about their math class and the topicsthey encounter in a school setting. Foryour convenience, these topics havebeen grouped into familiar strands,such as addition, subtraction,geometry, measurement, probabilityand statistics, and mental math. In Part2, the prompts concern students’ use ofmath in their everyday lives outside ofschool. Here, the prompts are groupedinto such topics as math at home, mathat play, and money math.

The topics in this book have beenchosen to accommodate a wide rangeof mathematical and writing abilitylevels. Within each topic, the promptsincluded require varying levels ofsophistication and confidence withskills. When working with youngerstudents or students with limited writing know-how, you may wish tostart with Drawing Math, a specialfeature in this book. Drawing Math

provides students with an opportunityto use both pictures and words torestate and solve problems. Anothergood way to introduce a topic tostudents of any age or ability level is touse Show What You Know…, a secondspecial feature. Show What You Know…gives students an opportunity to draw

on their prior knowledge of a giventopic.

As with all good problem-solvingtechniques, there is more than one right way to use this book and thewriting prompts contained within it.Sometimes, to stimulate interest, youmay wish to give students a choice oftopics. At other times, you may wish toassign a specific writing prompt gearedtoward a particular curriculum goal.Similarly, students may sometimesbenefit from working cooperatively on a prompt or from sharing theirresponses within a class or small-groupdiscussion. At other times, particularlywhen the prompts involve students’feelings and personal ideas, you maywish to give students the option ofkeeping their responses private.

Since the prompts in this book aredesigned to develop both mathematicaland verbal skills, the mechanics ofgood writing, such as punctuation,complete sentences, spelling, andcapitalization should be encouraged.However, to avoid inhibiting students’thought processes, you may wish toreserve a mechanics review until afterstudents have had a chance to get theirbasic ideas down on paper.

5Marvelous Math Writing Prompts © Andrew Kaplan Scholastic Teaching Resources

Part 1:

Math in School

S tudents can feel empowered if given an opportunity to expresswhat they are learning in their own words, as well as to shareany ideas and opinions they have. Since math class is the place

where students develop much of their math knowledge—from beingintroduced to new math skills and concepts to receiving practiceapplying these skills—it makes sense that this is also the place for themto ask questions about ideas they do not understand and to discuss theirfeelings about math. This process of learning, applying, and respondingto mathematics is enhanced by writing about it. It allows studentsadditional opportunities to record mathematical experiences, clarifyunderstanding of mathematical concepts, and request additionalinformation from teachers. For the math teacher, a student’s writingsabout math provide a way to assess both the student’s progress and thesuccess of the teacher’s instructional methods.

This section begins with students writing about math class. Studentsthen write about a variety of core mathematical topics, from place valueto problem solving, with a variety of other skills in between.


Do you think you make a good mathpartner? Explain. Also tell what youcan do to become a better one.

List three things you like and threethings you don’t like about working in a math group.

If you could change one thing aboutyour math class, what would it be?Explain why you think your class wouldbe better if this change were made.

Finish this sentence: “When I need helpwith math, I…”

List ten or more math facts you can use to describe your math class. Forexample, what is your room number?What shape is the room? How manydesks are there? How many studentshave brown eyes? What other math factscan you think of to describe your class?

Explain how you use math in anotherschool subject, such as history, science,music, or art.

Imagine that you are your own mathteacher. List ten things that you wantyour students to learn by the end of the year.

Describe the way you are graded inyour math class now. Would you like to be graded a different way? If so,describe it. If not, explain why you likethe system used now.

Imagine that you are your own mathteacher. Create a report card showinghow you are doing in math. Includeareas in which you are doing well, andothers where you need improvement.Be as specific as you can.

Create a list of instructions titled “Howto Get a Great Grade on Your NextMath Test.”

7

Let’s Talk Math, Class!

DRAWING MATH

Directions: Draw a picture of yourperfect math partner. Label all theskills and abilities that your perfectpartner would have.


How have math classes changed sinceyour parents or grandparents were yourage? Interview an adult to find out.Then write about it!

Write a letter to a friend describingwhat you did and what you learned inyour math class today.

Describe a time when something youlearned in your math class came inhandy somewhere else.

Describe a time when you helped afriend solve a math problem. What wasthe problem? Explain how you helped.

Make a dictionary of important mathwords that you know. Put the words inalphabetical order, and define each one.

When you do math, do you check yourwork? Why or why not?

Write a silly story called “The Fish WhoLoved Math Class.” In the story, showwhy the fish loved math so much!

Describe what you think math classeswill be like 100 years from now. Whatmay be different? What will probablybe the same?

Do you think you should be allowed to use a calculator when you do mathhomework? Why or why not?

8

Let ’s T a l k Ma t h , Clas s ! Let ’s T a l k Ma t h , C las s ! Let ’s T a l k Math , C lass ! Le t ’s Ta l k Mat h , C lass! L e t ’s Ta l k Math , C la ss! L e t ’s Ta l kt ’s T a l k Ma t h , Clas s ! Let ’s T a l k Ma t h , C las s ! Let ’s T a l k Math , C lass ! Le t ’s Ta l k Mat h , C lass! L e t ’s Ta l k M

MATH PROJECT

Help a younger child with math. Thenwrite about what happened. Usethese questions to help you:

What math skill did you explain?

How easy or difficult was it to

explain this math skill? Why do you

think that was?

Did you use any teaching tools,

such as models or drawings, to

help you teach? If so, what did

you use? Were these tools useful?

Why or why not?

Do you think you make a good

math teacher? Why or why not?


In Our Class, Math Really Adds Up!Using the space below, create an ad for your math class or a favoritemath skill. Start your ad with one of these slogans, or make up your own!

9

Name __________________________________________________________ Date __________________________________

Numbers: You Can Count on Them!

Money Math: It Makes Good Cents!

Geometry: It’s Not Just for Squares!

Fractions: They’re Part of theWhole Picture!

Let Subtraction Make a Differencein Your Life!


A unicycle is a vehicle with one wheel.What other words begin with the prefix“uni-”? How about the prefix “bi-”? The prefix “tri-”? What do you thinkeach of these prefixes mean?

Suppose you had 68 pennies. Howmany piles of 10 could you make?Explain how you know.

How long do you think it would takeyou to count up to a million by ones?What makes you think so?

Explain why “place holder” is a goodname for the zero in the number 406.

Do you think the following statement is true or false? It would take just as

long to count up to 100 by ones as it

would take to count up to 200 by twos.

Explain your response.

Pick a number! Any number! On a sheetof paper, describe it as many ways asyou can…without using the number orits name. Share the paper with a friend.Can he or she figure out what thenumber is?

10

Numbers and Place Value

Show What You Know About…Number Systems

Here are some ways that you canwrite the number five:

f ive 5 V

1 + 4

Can you think of others? Why do you think it is useful to be ableto write a number in more thanone way?

||||—


Write a silly story about No Number—a town where all the numbers werestolen one day. Describe how thepeople lived without numbers, and howthey finally got the numbers back.

Name one or more places where youhave seen Roman numerals used.

These number statements using Romannumerals are all true:

Using the true statements above, canyou figure out what these Romannumerals stand for? Explain why youthink so.

What does the symbol > stand for? Howabout the < symbol? How many othermath symbols can you think of? Whatare they? Explain in a sentence or twowhat each means.

V = ? X = ? IX = ?

III = 3

XI = 11

VIII = 8

IV = 4

DRAWING MATH

Background: An abacus is an oldcounting tool still used in Asia. Eachrow of beads on it stands for adifferent place value. Each bead atthe top of the abacus stands for fiveunits. The beads at the bottom eachstand for one unit. To show a number,move the beads toward the middle.

This abacus shows zero:

This abacus shows 263:

Directions: Draw an abacus showinganother number. Then swap with afriend. Can you figure out whatnumber your friend drew?

11

billi

ons

hund

red

mill

ions

ten

mill

ions

mill

ions

hund

red

thou

sand

s

ten

thou

sand

s

thou

sand

s

hund

reds

tens

ones


12

Num b e r s and P lace Va l ue Nu m bers a n d Place Va lue N umbe r s and P la c e Value Numb e r s and P l ace Va l u e Num b ers an d P lace V

Num b e r s an d P l ace Va l ue N u m bers a n d Place Va lue N umbe r s and P l a c e Valu e Numb e r s and P l ace Va l u e Num b ers an d P lace Vm b e r s and P lace Va l ue Nu m bers a n d Place Va lue N umbe r s and P la c e Value Numb e r s and P l ace Va l u e

What’s the largest number you cancreate by rearranging these digits:

8, 2, 4, 7, and 1

Explain why that is.

Explain what the 7 means in thenumber 2,743.

How would you write the following as a number: 8 thousand and 6 ones.Explain how you know.

When counting from zero, what’s thefirst two-digit number you come to?Why do you think ten is so important in the number system we use?

Count up from one until you reach anumber that has the letter “a” in itwhen it is spelled out. What is it? Howdid you figure it out? Can you think ofany shortcuts you can use so that youdon’t have to spell out every singlenumber to find out?

Imagine that you’re speaking to acreature from outer space, who wantsto know what “one hundred” means.How would you explain it?

Using words and pictures, describethree different ways you could solvethe problem 14 + 12 = ?

Pick a number and write a poem aboutit, telling why you think it’s so great.

What’s the highest you ever remembercounting? Describe what you werecounting, and about how long it took.

Write a sentence in which you use bothwords in one of these homophone wordpairs correctly:

How are the numbers 10 and 100 alike?How are they different?

In what ways is the number 10 like thenumber 16? How is it different?

eight ate

four for

two to

one won


List ten even numbers. Now list ten oddnumbers. Describe how you know if anumber is odd or even.

What is your address? What is theaddress next door? The address acrossthe street? In what ways are the numberson your side of the street all similar?How are they like the ones across thestreet? In what ways do they differ?

The numbers 1, 2, and 3 areconsecutive numbers. So are thenumbers 29, 30, and 31. Write adefinition that explains consecutivenumbers. Then write a sentence thathas the word “consecutive” in it.

A digital clock shows the time 7:30.What time will it show 30 minuteslater? Will a clock ever show the time7:60? Why or why not? Explain.

Is the number 3 more than, less than,or equal to the number 3.5? Explainhow you know.

List five or more places where youmight see decimals, such as 3.15 or0.66, used.

Pick any two consecutive numbers andadd them. Now add a different pair ofconsecutive numbers. Will the answeralways be odd or even? Explain whyyou think that is.

Pick any two even numbers and addthem. Will the answer always be odd oreven? Explain why you think that is.

Pick any two odd numbers and addthem. Will the answer always be odd oreven? Explain why you think that is.

Shown here are four palindromenumbers:

11 282 5,775 39,193

Figure out how all the numbers arealike. Then write a definition forpalindrome.

What numbers can you think of that arebetween 2 and 3? Where are you likelyto see numbers like these? When doyou think they can come in handy?

List five or more places where youmight see decimal-fractions, such as

or , used.34

12

12

13

Num b e r s and P lace Va l ue Nu m bers a n d Place Va lue N umbe r s and P la c e Value Numb e r s and P l ace Va l u e Num b ers an d P lace V

Num b e r s an d P l ace Va l ue N u m bers a n d Place Va lue N umbe r s and P l a c e Valu e Numb e r s and P l ace Va l u e Num b ers an d P lace VNum b e r s and P lace Va l ue Nu m bers a n d Place Va lue N umbe r s and P la c e Value Numb e r s and P l ace Va


All Set to Do Math!Imagine that each symbol below stands for a different number. Listsome of the number sets that would make the equation correct. Thendescribe on the back of this page any rules that you find to be true forall of these number sets.

14

Name __________________________________________________________ Date __________________________________

+ – 2=

+ – 2=

+ – 2=

+ – 2=

+ – 2=Marvelous Math Writing Prompts © Andrew Kaplan Scholastic Teaching Resources

Circles have no corners. Most coins are circle-shaped. How else can youdescribe a circle? How about arectangle? A square?

Do you have a favorite shape? What is it? Why do you think you like it somuch? Write a poem or a posterdescribing why it’s so great.

What do rhombuses, rectangles,parallelograms, and trapezoids all havein common? Name at least one othershape you know that shares the sametrait(s).

Describe how you could use a linesegment to split one rectangle into twotriangles. Then describe how youcould use two line segments to split arectangle into four triangles. Whatother shapes can you make from arectangle, using just two line segments?

15

Geometry

Show What You Know About…Geometry

In what ways is a square similar toa cube? In what ways are theydifferent? In what ways is atriangle similar to a pyramid? Inwhat ways are they different?Describe other ways that solidshapes and flat shapes are alike.Then describe other ways thatthey differ.

DRAWING MATH

Directions: Draw a scene of animaginary place called ShapeTown.In your picture, use each of theseshapes at least once:

circle square rectangle

parallelogram rhombus trapezoid


Use these clues to find the mysteryshape:

I have 2 short sides

that are the same length.

I have 2 long sides

that are the same length.

The sides that are opposite

each other are parallel.

I have no right angles.

What shape am I?

Can you draw me?

Now create your own riddle for adifferent shape. Then challenge aclassmate to figure out what shape you described.

Describe how a compass (like the oneshown here) can be used to help youdraw a circle.

In what ways are a protractor and acompass similar? In what ways are theydifferent?

Describe one or more ways that youcould use a sheet of loose-leaf paper tohelp you draw a square.

In your own words, explain thesimilarities and differences betweenarea and perimeter.

Figure out how many different waysthere are to arrange four square tiles.When you rearrange the tiles, do youchange the shape’s area? How about itsperimeter? Explain.

What shape could you trace from a cone? Explain.

If you cut a baseball down the middle,what shape would you see inside?Explain.

If you cut a cereal box down themiddle, what shape would you seeinside? Would the shape changedepending on how you cut the cerealbox? Explain.

The capital letters A, B, C, D, E, H, and I are all symmetrical. Explain what symmetry means. Then identifyother letters in the alphabet that aresymmetrical. Explain how you know.

16

Geom e t ry G e ometr y Geome t r y Geo m etry G eome t r y Geom e t ry G e o metr y Geome t r y Geo m e try G e omet r y Geom e t r y Geoeom e t ry G e ometr y Geome t r y Geo m etry G eome t r y Geom e t ry G e o metr y Geome t r y Geo m e try G e o

compass

protractor


Tell whether each of the followingstatements is true or false. Explainyour answers:

1. All rectangles are squares.

2. All squares are rectangles.

3. Some rectangles are squares.

Imagine that you had a pile of softballsand a pile of blocks, and that all theseitems were about the same size. Whichdo you think your toy chest would holdmore of: the balls or the blocks? Whydo you think that is?

Which could you fit more of in apopcorn tub: popcorn or unpoppedpopcorn kernels? Explain why youthink so.

Write a dialogue between a square, acircle, and a triangle in which eachtries to prove to the others that it is the most important shape.

Explain the similarities and differencesbetween a right triangle, an equilateraltriangle, and a scalene triangle.

Imagine that you could fly to the ceilingand take a look at this structure fromabove. How many blocks do you thinkyou would see? Explain why that is.

It’s easy to find the center of a straightline segment by using a ruler. But canyou think of a way to find the centerpoint of a circle? Describe one idea ofhow this might be done.

Imagine that you could earn $1.00 foreach side of a shape that you drew.Would it be worth more to draw apentagon, a hexagon, an octagon, or a rhombus? Explain how you know.

What’s the fewest number of sides thata regular shape can have? Explain whyyou think so.

List some items that you see every daythat are shaped like circles. Make aseparate list of items you see every daythat are shaped like squares. Which doyou think you see more of: circles orsquares? Why do you think that is?

Can two shapes be congruent but notsimilar? Can two shapes be similar butnot congruent? Explain.

17

Geom e t ry G e ometr y Geome t r y Geo m etry G eome t r y Geom e t ry G e o metr y Geome t r y Geo m e try G e omet r y Geom e t r y GeoGeom e t ry G e ometr y Geome t r y Geo m etry G eome t r y Geom e t ry G e o metr y Geome t r y Geo m e try


Pyramid PowerThis is a pyramid:

Which of these drawings can you fold into a pyramid? How do youknow? On a separate page, explain how the other figures would lookwhen folded.

18

Name __________________________________________________________ Date __________________________________

A.

C.

B.

D.


Building a Building1. You’ll need 8 cube-shaped items, such as number cubes, sugar

cubes, or building blocks, to complete this activity. Using the cubes,see if you can create the structure described in the drawings below.Here’s how:

a. Each drawing below shows a different side of the samecompleted structure. The lighter areas in the drawings are close to you. The darker areas are farther away.

b. Start with any one of the six drawings, and use the cubes to show how that side of the structure should look.

c. Now add the other five sides, one at a time.

d. When you’re done, check your model. Each side should match the drawings.

2. Describe the shape of the structure you made above.

_______________________________________________________________________________________________________

_______________________________________________________________________________________________________

_______________________________________________________________________________________________________

3. Now create your own set of plans for a different structure. Thenchallenge a friend to build it, using your set of plans.

19

Name __________________________________________________________ Date __________________________________

Left Side Front Right Side

Bottom Back Top


Most songs are written using patterns.Pick a song with a beat that follows apattern. Try to describe the patternwith words.

Look at these two number patterns:

4, 8, 12, 16, 20

27, 25, 23, 21, 19

Describe how they are alike and howthey are different.

Find a wallpaper or set of floor tilesthat create a pattern. Use words todescribe the pattern.

Describe the pattern that these shapesfollow:

Then describe how the next shape inthe pattern should look.

20

Patterns

Show What You Know About…Patterns

Write a definition for the word“pattern.” Then show an exampleof a pattern using

numbers;

colors; and

shapes.

Can you create a pattern thatuses numbers, colors, and shapesat the same time? Try it! Thenexplain in words what the pattern is.

DRAWING MATH

Background: Take a look aroundyou, and you’ll find plenty ofpatterns in nature. Honeycombs,pine cones, and the rows of corn ona cob all create patterns. Phases ofthe moon and seasons follow oneanother in a pattern. And birdstraveling together often fly inpatterns, too!

Directions: Look around yourneighborhood for another naturalpattern. Draw a picture to showwhat the pattern is. Or, if you prefer,draw a picture of a garden in whichthe flowers or vegetables aregrowing in a pattern that you makeup yourself!


How do spies and undercover agentscreate and break secret codes?Sometimes they rely on patterns likethis one to help them:

See the pattern? Once you find it,explain what it is and use the code towrite a secret message of your own.Then challenge a classmate to crackthe code and decipher the message thatyou wrote.

Write a few sentences explaining howpatterns can be used to create secretcodes.

What do checker, backgammon, andChinese checker boards all have incommon? They all contain patterns!Think of another game you know thathas to do with patterns, and explainhow patterns are used in the game.

Describe some of the different waysyou could organize the books on abookshelf. Which, if any, of thesearrangments would you consider to bea pattern? Explain why that is.

Does your school day follow a pattern?If so, describe it. If not, explain whyyou don’t feel it does.

What kinds of patterns can you find ona calendar? Describe them!

What pattern do you see in thesemultiplication sentences? Write thenext two equations that continue thepattern. Then explain the pattern inyour own words.

3 x 2 = 6

3 x 20 = 60

3 x 200 = 600

In what ways are the patterns found ona checkerboard similar to those foundon floor tiles? In what ways are theydifferent?


Here are three multiplicationsentences.

1 x 1 = 1

11 x 11 = 121

111 x 111 = 12,321

Describe any patterns you see. Thenpredict what 1,111,111 x 1,111,111equals. Use a calculator (or a penciland paper) to check your answer.

Imagine that your parents told you thatfor your allowance, you could have 1¢today, and then double the amount ofthe day before every day for a month.For example, you would get 2¢tomorrow, 4¢ the day after that, 8¢ theday after that, and so on. Would you behappy about this arrangment? Why orwhy not?

On a calculator, begin with a numberbetween 11 and 19. Add 9 to it andrecord the answer. Keep on adding 9and record the answers. Describe anypatterns that you see.

On a calculator, begin with any two-digit number that does not end in 0.Add 5 to it and record the answer. Keepon adding 5 and record the answers.Describe any patterns that you see.

Use the One Hundred Chart worksheeton page 23 to describe the picturepattern created when you put a markeron each of the following series ofnumbers:

All even numbers.

All odd numbers.

All multiples of 4.

All multiples of 9.

All prime numbers

(numbers that only have twofactors: the number itselfand 1).

All numbers that contain thedigit 2.

All numbers that contain thedigit 8.

All double-digit numbers(i.e., 11, 22, 33, and so on).

The sum of all consecutivenumbers found on the chart(i.e., 1 + 2, 2 + 3, 3 + 4, andso on).

Create a pattern of your own on theOne Hundred Chart worksheet. Thendescribe the number pattern that youmade.

22

Pat t e r n s P a t t e r ns P a t terns P atte r n s Pat t e r n s Pa t t e r ns P a t t erns P a tte r n s Pat t e r n s Pa t t e r ns P a t t erns P a tter n s Pat t e rt t e r n s P a t t e r ns P a t t erns Patte r n s Pat t e r n s Pa t t e r ns P a t t erns P a tte r n s Pat t e r n s Pa t t e r ns P a t t


One Hundred Chart

23

Name __________________________________________________________ Date __________________________________

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100


Square NumbersSome ancient Greeks liked to study numbers by using shapes, such asthose shown below. Note how these large squares are all made fromsmaller squares.

1. The numbers 4, 9, and 16 are called square numbers. Why do youthink that is?

_______________________________________________________________________________________________________

_______________________________________________________________________________________________________

2. Use a sheet of graph paper to draw the next large square that fitsthe pattern. Then draw the one after that. Describe any numberpatterns that you see. (Hint: Use the One Hundred Chart and yourmultiplication facts to help you.) Explain some ways that you couldfind even larger square numbers without drawing squares.

_______________________________________________________________________________________________________

_______________________________________________________________________________________________________

_______________________________________________________________________________________________________

_______________________________________________________________________________________________________

24

Name __________________________________________________________ Date __________________________________


Shifting Shapes1. Cut out these shapes:

2. Color one side of each shape red. Color the other side yellow.

3. Make a pattern from four or more of the shapes. Your pattern canuse ideas of shape, size, color, or number. Describe your pattern.

_______________________________________________________________________________________________________

_______________________________________________________________________________________________________

4. Now look at a classmate’s pattern and describe it.

_______________________________________________________________________________________________________

_______________________________________________________________________________________________________

5. Compare your pattern with your classmate’s pattern. In what waysare they similar? In what ways are they different?

_______________________________________________________________________________________________________

_______________________________________________________________________________________________________

25

Name __________________________________________________________ Date __________________________________


A decade is an amount of time equal toten years. What is a word for 100 years?1,000 years? Name some other wordsthat describe specific amounts of time.Why do you think it is helpful to haveso many different “time words” in ourlanguage?

You just found two beautiful rocks, andyou want to know which one weighsmore—but you don’t have a scale withyou. Make a plan for figuring it out.

Describe one or more ways that youcan measure the length of somethingwithout a standard measuring device,like a ruler.

How many measurements can youthink of to describe yourself—such asheight, weight, shoe size, and so on.See who in your class can create thelongest list of human measurements!

Find five or more kitchen items thatweigh exactly 1 pound. What are they?Are there other ways these items are all similar? In what ways are theydifferent?

How many measurements are there onthe packaging for your favorite foodproducts? List as many as you can thinkof. Then look at some actual food labelsand add any other measurements thatyou find. What does each of thesemeasurements describe?

What measurements can you find onthe dashboard of a car? Why do youthink it would be unsafe for adults todrive without these measuring devices?

Describe the distance from your hometo your school as specifically as youcan. Then explain some ways you couldmake these measurements even moreprecise.

26

Measurement

Show What You Know About…Measurement

Explain the purpose of each ofthese measuring tools:

Now list some other measurementtools that you know. Which areused to measure weight?Volume? Time? Temperature?Length? What else, beside thesethings, can be measured?

scale

thermometer

measuring cup

ruler


When describing the distance betweentwo cities, is it better to use units likemiles and kilometers, or units likeinches and centimeters? Why do youthink that is?

Ben Franklin once wrote, “An ounce ofprevention is worth a pound of cure.”What do you think this measurement-related saying means? List some othermeasurement-related phrases thatyou’ve heard and explain them. Or,make up one of your own!

Would you consider a map to be ameasuring device? Why or why not?Explain.

Describe a trip you have taken with your family using as manymeasurement words as you can. Forexample: How far did you go? Whatwas the temperature? How heavy wasyour luggage, and so on.

What are some ways you can measuretime without using a clock? List them!

Which do you like better, clocks withhands or digital clocks? Explain whythat is.

Imagine that you suddenly grew to beas tall as a giraffe. How might your lifechange? Write a story about what youwould see and do if you were that tall.

Imagine that you suddenly shrank to thesize of an ant. How might your lifechange? Write a story about what youwould see and do if you were that small.

27

metronomeegg timer

sundial


Which do you think is longer: thedistance around a soda can or thedistance from its top to its bottom?Write down your guess, then comparethese lengths using a piece of string.Describe what you find out.

Write a few sentences explaining thesimilarities and differences betweenliquid measures and dry measures.Provide a few examples of each.

Write a few sentences explaining thesimilarities and differences betweenitems measured in square feet andthose measured in cubic feet. Includeexamples of each.

In ancient Egypt, a foot was not exactly12 inches long. It was equal to the foot-length of the king then in power! Whatproblems might this have caused in thekingdom over time? List as many as youcan think of.

Create a poem titled “How High Is Up?”in which you describe ways that “up”can be measured.

Do you prefer using metric units ofmeasurement (like meters and liters) orstandard units (like inches, feet, andquarts)? Explain why that is.

28

Meas u remen t Measu r ement Measu r e ment M easure m ent M e asurem e n t Me a s ureme n t Mea s u remen t Meas u r emen t Measu reas u remen t Measu r ement Measu r e ment M easure m ent M e asurem e n t Me a s ureme n t Mea s u remen t

MEASUREMENT PROJECT #1

What do you think is the...

a) maximum amount of weightyou can lift at one time?

b) highest you can jump?

c) farthest you can run in oneminute?

Predict the answer to each questionabove. Then describe a way youcan test your theories.

MEASUREMENT PROJECT #2

Pick a day and record thetemperature outside at 9:00 a.m.and then again each hour, until 5:00p.m. Make a chart to record yourresults. When you’re done, write afew sentences describing theseresults and any patterns that yousee. Do you think the results wouldchange very much if you did it againthe next day? Find out!


Take a Hike—But Watch Your Weight!You’re about to head out on a two-hour hike. Take a look at the itemsbelow and decide which ones to bring. But watch out! You don’t wanta backache—so be sure to keep the total weight of the items down to8 pounds or less. Then write an imaginary story about your hike.

29

Name __________________________________________________________ Date __________________________________

(Remember: If the items weigh more than 8 pounds, go back and decide what you’ll have to leave home!)

Items You Plan to Take What They Weigh

_______________________________________________________________________________________________________

_______________________________________________________________________________________________________

_______________________________________________________________________________________________________

_______________________________________________________________________________________________________

_______________________________________________________________________________________________________

_______________________________________________________________________________________________________

small backpack: 1 pound, 10 ounces

sleeping bag: 2 pounds, 2 ounces

maps and compass:2 ounces

first-aid kit: 1 pound, 1 ounce

water canteen: 2 pounds, 4 ounces

CD player and CDs: 1 pound

food: 1 pound, 5 ounces

jackknife: 3 ounces

snacks: 6 ounces

flashlight: 8 ounces

camp stove and matches:1 pound, 8 ounces

overnight pack: 5 pounds

TOTAL WEIGHT _______________


How are pictographs like bar graphs?In what ways are they different?Explain your thinking.

Find an example of a line graph and a circle graph in a newspaper. Thendescribe the types of information thateach one shows.

Give an example of something that youmight be interested in surveying kids in your school about. Explain how asurvey is different from simply asking a few people what they think.

Weather forecasters try to figure outwhat the weather will be like tomorrowand the day after that. Geologists andseismologists try to figure out when anearthquake may strike next. In whatways are these scientists like fortune-tellers? In what ways are theydifferent?

Imagine that you have just created anew flavor of ice cream. But before youspend lots of money making gallonsand gallons of the stuff, you want tomake sure that people will buy it.Describe some things you can do tofind out.

You and a friend decide to flip a coin tosee who will go first in a game you’replaying. Do both of you have an equalchance of winning the coin flip? Why orwhy not?

If a weather forecaster announces that“there is a 75 percent chance of raintoday,” what do you think that means?How might you use this information tohelp plan your day?

30

Probability and Statistics

Show What You Know About…Probability and Statistics

What is the difference between agame of chance and a game ofskill? Explain what you think,giving examples of each. Canyou think of any games thatcombine luck with skill? Giveexamples of these, too.


Give an example of when knowingwhat happened in the past can help youto figure out what may happen in thefuture. Can we ever be certain of whatwill happen in the future? Why or whynot?

List five or more kinds of statistics, or number facts, that are used whentalking about baseball and baseballplayers. How else are numbers used in this sport?

Shown here is a Venn diagram thatcompares apples and oranges.

In your own words, explain what aVenn diagram shows. Then create onecomparing two different types of fruit.Now add another circle to yourdiagram, and use it to compare threedifferent types of fruit!

On TV, you may hear commercialsboast things like “four out of fivedentists surveyed recommend ourbrand of toothpaste.” Can you alwaystrust the “facts” that you learn fromadvertisements like these? Why or whynot? How do you feel about this?

31

DRAWING MATH

Background: Graphs come in lots ofdifferent shapes and sizes. There arecharts, pictographs,circle graphs,bar graphs, and line graphs to namejust a few! All of them allow you toorganize information in differentways.

Directions: Count the number of girlsand the number of boys in yourclassroom. Then create a pictographthat displays the results of yourcount. (Hint: To do it, you’ll first needto decide how many kids eachpicture on your graph will stand for.)When you’re done, display the samedata on a chart and on a bar graph.On which one do you find theinformation easiest to read? Why do you think that is?


Is it possible for both of thesestatements to be true?

Why or why not? Find or make upanother example of how statistics canbe used to make claims that seem tocontradict one another.

According to a television researchcompany, the average Americanwatches 3 hours and 46 minutes of TVdaily. Does that mean that every personin the United States watches thatamount of TV every day? Explain inyour own words what it does mean.

Make up a silly story about a superherowho uses statistics to solve crimes andother problems. In your story, describehow your hero or heroine uses numberfacts to save the day!

Make a chart that will help you keeptrack of your money-spending, eating,or TV-watching habits for a week. Then write three or more conclusionsthat you can make from your chart.

You and a friend decide to take turnsrolling a number cube.

Each time it lands on 4, you earn apoint. Each time it lands on 6, yourfriend earns a point. Is this a fairgame? Why or why not?

Now imagine playing the same gamewith two number cubes. Now, eachtime the sum of the cubes is 4, youearn a point. Each time the sum is 6your friend earns a point. Is this afair game? Why or why not?

How could you change the rules so thatboth games are fair?

You and a friend put the letters of thealphabet into a brown paper bag. Youthen take turns drawing letters fromthe bag.

Each time you draw a vowel fromthe bag, you score a point.

Each time your friend draws aconsonant, he or she scores a point.

Who has a better chance of winningthis game? Why is that? What could youdo to make this game fair?

32

Pro b a b i l i ty a n d Stat i s t i c s P ro b abi l i t y a nd St a t i s t i cs P r obabi l i t y and S t a t i s t ics P robab i l i t y and S t a t i st i c s Proba b i l i ty an d S tat i s t iob a b i l i ty a n d Stat i s t i c s P ro b abi l i t y a nd St a t i s t i cs P r obabi l i t y and S t a t i s t ics P robab i l i t y and S t a t i st i c s

STATISTICS PROJECT

What color eyes does the averagekid in your class have? What colorhair? How tall is he or she? Take asurvey to find out. Then write aparagraph comparing yourself withthis average kid. How might theaverage kid at another school bedifferent? Why do you think that is?


Let’s Go for a Spin…Think of all the ways you can roll the numbers between 2 and 12 usingtwo number cubes. Write these on Chart #1, below. (A few are alreadygiven to get you started.) Next, roll two number cubes at least 25 timesand use Chart #2 to keep track of your total on each roll. Then, on theback of this page, compare the shape of the data on the two charts.Explain why you think they look as they do.

33

Name __________________________________________________________ Date __________________________________

1 + 1

2

2 + 1

1 + 2

3 4

4 + 1

1 + 4

5

1 + 5

6 7

6 + 2

2 + 6

8 9 10

5 + 6

11 12

What similarities and differences do you see between the shape of thedata on the two charts above? Why do you think that may be?

CHART #2: Spin a pair of number cubes at least 25 times. Use “x” marks(stacked one above the other) to record what you roll each time.

CHART #1: Using two number cubes, show the ways you can roll a…

2 3 4 5 6 7 8 9 10 11 12


How can the equation 6 + 6 = 12 helpyou solve the problem 6 + 7 = ? Whatother addition problems can it help youto solve?

Explain how understanding place valuecan help you to solve an additionproblem like 36 + 25 = ?

How can the equation 3 + 5 = 8 helpyou find the answer to the problem 30 + 50 = ? Explain your thinking.

How are the addition problems 5 + 9and 9 + 5 alike? How are they different?Write a word problem to illustrate each.

Is it always possible to check youranswer to an addition problem (such as12 + 25 = 37) by using subtraction? Whyor why not? Explain what you think.

What happens when you add zero to anumber? Why is that?

Luvenia opened her math book andfound that the sum of the facing pageswas 243. What pages did she open to?Explain how you know.

What are you actually doing when you“regroup”? Explain.

Explain the steps you would take tosolve the equation 355 + 626 = ?

Addition

34

Show What You Know About…Addition

List three or more pieces ofadvice you would give to a friendwho was stuck on an additionproblem.

DRAWING MATH

Directions: Draw a picture to illustratethe following problem. Then solve it.

Problem: On her way out of Bernie’sBargain Basement, Trish counts herchange. She has 15¢ left. She alsohas a piece of candy that she’sbought for 25¢ and a pencil thatshe’s bought for 10¢. How muchmoney did she start with before sheentered the store?

DRAWING MATH

Problem: See how this stairway ismade up of cubes? How many morecubes would you need for a total of9 steps? Draw a picture to help yousolve the problem. Then explain howelse you could solve it.


What addition fact can help you tosolve the problem 17 – 8 = ? Explain.

What happens when you subtract anumber from itself? Explain why it is.

What happens when you subtract zerofrom a number? Explain why that is.

How can knowing the equation 7 – 2 = 5help you to solve the subtractionproblem 70 – 20 = ? Explain.

When you subtract 29 from 78, do youneed to regroup? Why or why not?Explain what you would do.

Write a story about a land calledTakeAway, where people know how to subtract, but not how to add.

35

Subtraction

Show What You Know About…Subtraction

List three or more pieces ofadvice you would give to a friendwho was stuck on a subtractionproblem. Then explain how yourfriend could check his or her work.

DRAWING MATH

Directions: Draw a picture to helpyou solve this problem. Then solve it.

Problem: There are 12 people in aroom. Six are wearing socks and 4 are wearing shoes. Three arewearing both socks and shoes. Howmany people in the room are in barefeet? (Hint: Be sure not to countanyone twice!)

MATH PROJECT

Stack 10 black checkers on top of ared checker. With a friend, take turnsremoving 1, 2, or 3checkers from thestack. The goal of the game is to make youropponent pick upthe last checkerevery time.

Play this gameuntil you thinkyou’ve figured outsome strategiesthat will alwaysallow you to win.Then describewhat thosestrategies are!


What happens when you multiply anumber by zero? How is this differentfrom what happens when you multiplya number by one? Why do you thinkthat is?

How is multiplying 3 x 5 like adding 5 + 5 + 5? What other addition problemis it like? Why is that?

About how many minutes old are you?Describe how you might figure it outusing multiplication and addition. Don’tforget to include leap years!

How can the multiplication problem 3 x 2 = 6 help you find the answer to 3 x 200 = ? How about 300 x 200 = ? Explain what you would do in each case.

Most guinea pigs have 4 toes on theirtwo front feet and only 3 toes on theirtwo rear feet. Using this information,figure out how many toes 4 guinea pigshave in all. Explain how you arrived atyour answer.

While setting up a lemonade stand forthe day, you figure out that you’ll need7 lemons for each pitcher of lemonadethat you make. You decide to make 5pitchers in all. How many lemons willyou need? Explain how you arrived atyour answer.

36

Multiplication

Show What You Know About…Multiplication

When can multiplication help youto solve a problem in real life? List some ideas. Then write aparagraph that begins with:

“I use multiplication when I…”

DRAWING MATH

Problem: Are there more legs on 3ducks or 2 dogs? Eight ducks or 5dogs? Now say you had a set ofducks and a set of dogs with anequal number of legs on each. Howmany of each animal might therebe? Draw some pictures to help yousolve these problems. Then explainhow you could use multiplication tosolve them instead.


For every 5 students who get an A inmath, the teacher will give the class agold star. How many gold stars will theclass get if 25 students get an A?Explain how you figured it out.

Marsha’s dad made a batch of hugechocolate chip cookies. The cookieswere so large that he cut each one inhalf. Marsha ate 7 halves. Her brotherGil ate 5 halves. How many wholecookies did they eat in all? Explain how you solved the problem.

If a number is divisible by 9, the digitsin the number add up to a numberdivisible by 9. For example, the digits inthe number 201,915 add up to 18 (2 + 0+ 1 + 9 + 1 + 5), which is divisible by 9.Thus 201,915 is, too! Test the rule usingsome other large numbers. Then write a rule that could help you determinewhether or not a number is divisible by10 or another number. What is it? Shareyour rules with the class.

Explain how you can use subtraction tohelp you solve the problem 20 ÷ 4 = ?

What happens when you divide anumber by one? Explain why.

What happens when you divide anumber by itself? Explain why.

A minivan can hold 8 people. Howmany vans do you need for 30 people?Explain why that is.

Jamal wants to find of 20. Terry saysthat Jamal can use division to find theanswer. Explain what Terry means.

14

37

Division

Show What You Know About…Division

What’s the difference betweendividing 3 into 12, and 12 into 3?Will both problems result in awhole number? Why or why not?

DRAWING MATH

Problem: How many ways can youevenly group the students in yourclass so that no one is left out? Drawpictures to show your results. Thenexplain how the results might differ ifone student were absent today.What about two students?


How many equations can you write thatdescribe the number 10? You can useaddition, subtraction, multiplication,division, fractions, and decimals—orany combination of these—in yourequations. See how many you can comeup with!

When doing word problems, what cluescan you look for to help you know thatyou may need to add? Subtract?Multiply? Divide? Why must you alsoread a problem carefully, rather thansimply look for these types of words?

Think of a favorite book or story youhave read or make up a story of yourown. Then write a word problem basedon the situation in that story.

Explain why you need addition,subtraction, multiplication, and

division to solve this problem:

There are 6 dogs and twice as many

cats in the window of Aaron’s

Animal Barn. There are half as

many dogs and 4 fewer cats in the

window of Patty’s Pet Kingdom.

How many animals in all will you

find in the window of each store?

Now try and write your own wordproblem that requires addition,subtraction, multiplication, and

division in order to solve it. Then give it to a classmate to solve!

Imagine that you’ve just started yourown lawn-mowing business. Describeone or more ways that you might need to use addition, subtraction,multiplication, and division. What other math skills might you need onyour new job?

38

Computation: Mixed Skills

Show What You Know About…Basic Computation Skills

Explain how a number line canhelp you solve an additionproblem. How about a probleminvolving subtraction? Division?Multiplication? In what ways arethese four computation skillsrelated? Use at least one of eachtype of equation as examples toexplain your ideas.


A Store Full of Stories

Write four or more math story problems based on this picture. Eachshould use a different operation—addition, subtraction, multiplication,or division—at least once.

1. _______________________________________________________________________________________________________

_______________________________________________________________________________________________________

2. _______________________________________________________________________________________________________

_______________________________________________________________________________________________________

3. _______________________________________________________________________________________________________

_______________________________________________________________________________________________________

4. _______________________________________________________________________________________________________

_______________________________________________________________________________________________________

39

Name __________________________________________________________ Date __________________________________


Do you think that you should beallowed to use a calculator in yourmath class? Explain why or why not.

Imagine that the key on yourcalculator broke. Explain how youcould still use your calculator to findout the product of 3 x 1,789.

Imagine that the key on yourcalculator broke. Explain how youcould still use your calculator to findout the quotient of 196 ÷ 49.

Jaime wants to use a calculator to findhow many seconds there are in a day.Describe what he should do.

Describe what happens when youdivide 0 by any number using acalculator. Then try to divide anynumber by 0. How do the answersdiffer? Explain why you think that is.

Using a calculator, solve each of thesemultiplication problems:

What do you notice about the productsin each case? Describe it.

Using a calculator, press the key.Then hit the key twice, followed bythe key three times. What happenedto your original number? Now clearyour calculator and start with anothernumber other than 5. Describe whathappens this time.

=

+

5

÷÷

x

40

Calculators

Show What You Know About…Calculators

List three ways that calculatorsare like—and three ways thatthey differ from—addingmachines and cash registers.Include examples of how eachone is used.

142,857 x 2 = ____

142,857 x 3 = ____

142,857 x 4 = ____

142,857 x 5 = ____

142,857 x 6 = ____

142,857 x 7 = ____

142,857 x 8 = ____

142,857 x 9 = ____

CALCULATOR PROJECT

The next time you go shopping, use acalculator to estimate what your totalbill will be. Then write a paragraphdescribing what happened.

cash register

calculator

adding machine


Tic-Tac-FifteenDon’t let this numberless calculator fool you. It’s actually a game thatyou can play with a friend. To play, take turns writing numbers from 1 to 9 on the keys. If you’re the first to get the “keys” in any row, column,or diagonal to add up to 15, you win! (Note that each number can beused only once in a game.)*

Play a few times and see if you can figure out some good strategiesthat would allow you to win every time. When you have some, writethem down to share with a friend.

* Suggestion: Make lots of copies of this page or ask your teacher to laminate it so that you canplay over and over again.

41

Name __________________________________________________________ Date __________________________________


Your cousin owes you 65¢. She givesyou a dollar bill. Describe how you canmentally figure out how much changeto give her back.

You have $3 and you want to buy abook that costs $2.97. How muchmoney will you have left? Explain howyou know. Then imagine that, instead,the book cost $1.47. How can youmentally figure out how much moneyyou would have left now?

In what way can thinking aboutquarters (25¢ coins) help you solve theequation 50 + 75 = ? mentally. Explain.

In your head, figure out how many

ears are there are on 12 bunnyrabbits.

paws 8 dogs have.

eggs there are in 3 dozen eggs.

Explain how you arrived at youranswer in each case.

What is the product of

10 x 7 = ? 100 x 7 = ?

10 x 9 = ? 100 x 9 = ?

10 x 12 = ? 100 x 12 = ?

1,000 x 9 = ? 100,000 x 12 = ?

Create a rule you can always use tohelp you multiply in your head anynumber by a number that has a 1followed by one or more zeroes.

What is the product of

200 x 2 = ? 200 x 4 = ? 200 x 8 = ?

Create a rule you can always use tohelp you multiply 200 by any othernumber in your head.

42

Mental Math

Show What You Know About…Mental Math

Explain how the equation 9 + 6 = 15 can help you to solveeach of these addition problemsin your head:

19 + 6 = ?

39 + 6 = ?

79 + 6 = ?

Now explain how this same strategycan help you solve other additionproblems in your head, too.


Suppose you found a $5 bill, 4 pennies,7 nickels, and 2 quarters in your coatpocket. Describe how you would addup this amount of money in your head.

While using a calculator to solve theequation: 16,863 x 2 = ?, you get thisresult on your calculator display:

Can this answer be correct? Explainwhy or why not.

In your head, figure out which of thesefractions does not belong:

Explain how you know. Then replacethat fraction with another that has anequal value to the others in the set.

Explain how you could multiply 6 x 99in your head.

Bob says he can multiply 8 x 12 in hishead. Explain how this can be done.

What kinds of numbers do you findeasiest to compute in your head?Explain why. Then create a list of yourfavorite mental math tips to share withyour friends.

Explain one way to solve this problemin your head:

5 + 23 + 55 + 10 = ?

(Hint: Start with the numbers that youfind easiest to add mentally. Then turnto the others.)

It’s 4:12 p.m. Your mom tells you thatyou have 25 minutes to get ready to goshopping with her. How can you figureout mentally what time you’ll beleaving?

To solve the problem 4 x 18 = ? mentally,Rebecca did this thinking in her head:

Explain in your own words what stepsRebecca went through in her mind.

In your mind, figure out how you canmake 45¢ using exactly 5 coins. Howabout 4 coins? Explain what you did.

12

24

36

48

59

612


Time for a Pack of Snacks

Lucky you! You have exactly $1 to spend on munchies to go with yourlunch. How many different combinations of items can you buy from thismachine? Note: You can buy more than one of a particular kind ofsnack, if you wish. But you have to do all the math in your head!

Write your possible snack combinations here:

___________________________________________________________________________________________________________

___________________________________________________________________________________________________________

___________________________________________________________________________________________________________

___________________________________________________________________________________________________________

44

Name __________________________________________________________ Date __________________________________


Which of these sentences contain anestimate? Which contain an exactnumber? Explain how you know.

More than 100 people watched thebaseball game.

Rain stopped the game for about 30 minutes.

Edna ate 3 hot dogs.

The school sold 26 team shirts.

The game lasted over 90 minutes.

Drop 10 paper clips into a paper cup oranother small container. Now estimateabout how many paper clips it wouldtake to fill the cup. Then fill the cupwith paper clips and compare your twonumbers. Describe how close yourestimate was, and why you think thatmay be.

About how far do you think it is fromthe floor to the ceiling of yourclassroom? Explain how you arrived at your estimate. Then describe a waythat you could check to see if you areright!

Look in a newspaper for someexamples of estimates and someexamples of exact numbers. Make achart showing these examples, whethereach is an estimate or an exact number,and why that type of number waschosen in each case.

45

Estimation

Show What You Know About…Estimation

Imagine a conversation betweenan estimate and an exactnumber in which each explains tothe other what it is, why it is ahandy number to use at times,and what some of those timesare. Describe what each wouldsay to the other!


You head to the store to do someshopping with $10 in your pocket. Youknow that the items you plan to buywill cost about that amount. Do youthink it’s good enough, in this case, tohave an estimate of the total before youhead for the cash register? Explain whyor why not.

Maria estimates that 28 + 37 is about70. Do you think that the exact answeris more or less than the estimate? Tellhow you can decide without adding.

What pairs of numbers in each setbelow add up to more than 100?Without doing any exact computation,explain how you know in each case.

Kira estimates that 7 x 52 is about 350.Do you think that the exact answer ismore or less than the estimate? Tellhow you can decide withoutmultiplying.

You want to buy a sandwich that costs$4.75 and a bottle of juice that costs$2.59. You have $8 in your pocket. Doyou have enough money on you?Explain how you can figure it out justby doing some math in your head.

Estimate about how many students arein your school. Then write a fewsentences explaining how you arrivedat your estimate and what you could doto find out how accurate it is.

Estimate the number of times you canwrite your name in one minute. Thenhave a partner time you. How close didyour estimate turn out to be? Explainwhy you think that is.

28 52 49

72 23 85

59 17 35

24 62 47

46

Est im a t i on E s t imat i o n Est i m a t i on E s t imat i o n Est i m a t i on E s t i mat io n Est im a t i on E s t i mat i on Est ima t i on Es t i mat ion Est ima tim a t i on E s t imat i o n Est i m a t i on E s t imat i o n Est i m a t i on E s t i mat io n Est im a t i on E s t i mat i on Est ima t i on

ESTIMATION PROJECT

How could you estimate about howmuch garbage you throw awayevery day? Create a plan, then putit into action! When you’re done,describe what you learned. Alsoexplain how your estimatecompared with those of the otherkids in your class.


In the Money

Jar A above has 999 nickels in it. About how many nickels do you thinkeach of the other jars have? Write a sentence or two explaining eachestimate.

___________________________________________________________________________________________________________

___________________________________________________________________________________________________________

___________________________________________________________________________________________________________

___________________________________________________________________________________________________________

BONUS: About how much money is in each of these jars?

47

Name __________________________________________________________ Date __________________________________


This problem has information in it thatyou do not need: A pet store has 5

beagles, 4 Siamese cats, and 6

dachshunds. How many dogs does

the pet store have? Which piece ofinformation is extra? Explain why.Then solve the problem.

Write a story problem that takes two ormore steps to solve.

Find a picture in a book or a magazine.Then write a story problem about it.

Mr. Cooper buys two of these toys andspends a total of $47. Which two toysdid he buy? Explain how you foundyour answer.

Figure out how many triangles in allthere are in this picture. Then write aparagraph explaining how you arrivedat your answer.

See if you can use 6 lines to draw 12triangles. Then explain how you figuredit out.

48

Problem Solving

Show What You Know About…Problem Solving

Directions: Write or find a wordproblem that is best solved byusing each of these strategies:

a) drawing a picture.

b) solving a similar but simplerproblem.

c) looking for a pattern.

d) guessing, checking, andrevising.

Explain how the strategy youchose can help you to solve theproblem.


Use two straight lines to divide thiswatch into three parts so that thenumbers in each section add up to thesame sum. Then describe the strategyyou used to arrive at your answer.

Can you find a way to arrange 8checkers on a checkerboard so that notwo checkers are in the same verticalor horizontal row, and no two squareswith checkers touch each other? Give ita try! Then describe how you wentabout figuring it out.

Find six numbers—all of which onlycontain the digit 1—that have a sum of123,456. That means the numbers youuse can range from 1 to 111,111. Whenyou’re done, explain on paper how youfigured it out.

Ronald takes up running as an exercise.He runs 4 miles the 1st week, 6 milesthe 2nd week, 8 miles the 3rd week,and 10 miles the 4th week. Whatpattern do you see in the data? If thepattern continues, how many miles willRonald have run in all after 6 weeks?Explain how you know.

Write your own story problem that hasan extra piece of information in it thatyou do not need in order to solve theproblem.

Write your own story problem thatcannot be solved because there is notenough information.

Ned gets to the park at 4:00. He meets

Ellen there at 4:30, and they play on

the swings until 5:00. Then they

head for the slides. Is there enoughinformation given in this story problemto figure out how long Ned stayed atthe park in all? Explain why or why not.

49

DRAWING MATH

Problem: How many different dinnercombinations are there on Ted’sdinner special? Explain how youarrived at your answer.

Main Course Side Dish Vegetableroast beef french fries cornbroiled fish rice peas

carrots


Izzie has exactly 46¢. What’s the fewestnumber of coins he could have? Whatare those coins?

Ben buys a book for $12.99. He givesthe clerk a $20 bill and gets $8.01 inchange. Did Ben get the correctchange? Explain why or why not.

Roll a sheet of loose leaf paper into along narrow tube and tape it in place.Roll a second sheet into a short fat tubeand tape it in place. Do you think onewould hold more jelly beans than theother? Explain your thinking. Then testyour guess!

How many different ways can twopeople shake hands? Three people?How about four people? Make a chartshowing your answers. Then describeany patterns you find in the solutionsthat you get.

At what age do you become a billionseconds old? Explain how you wouldgo about figuring it out.

Imagine that you had $1,000,000. If youdecided to give away $50 every hour,how long would it take you to giveaway the whole amount? How oldwould you be then? Explain how youcould arrive at an answer.

Explain how three men can cross ariver in a raft that holds up to 300pounds if two of the men weigh 150pounds each and the third man weighsexactly 300 pounds. (Hint: The men andthe raft can go back and forth acrossthe river as many times as you like.)

Sometimes if you look at the numberson a calculator upside down, they spellout a word. See if you can figure outhow to spell out the word “googol” on acalculator that’s turned upside down.(That’s the word for the number 1 with100 zeros after it.) Then explain howyou figured it out.

50

Pro b l e m Sol v i n g Pro b l em So l v i ng P r o b lem S o l v i ng P r o b lem S o l v ing P roblem S o lv in g P rob l e m Solv i n g Pro b l e m Sol v i n g Probob l e m Sol v i n g Pro b l em So l v i ng P r o b lem S o l v i ng P r ob lem S o l v ing P roblem S o lv ing P rob l e m Solv i n g


What’s the Question?This table shows how many visitors came to two different amusementparks during one weekend:

Put a check next to the questions listed here that you can answer byusing the table:

_______ How much money did visitors spend at Water World during thethree days?

_______ How many visitors came to Water World during the three days?

_______ Which amusement park had more visitors on Sunday?

_______ How many people work at Future City?

_______ How many more visitors did Water World have than Future City on Friday?

Now write three more questions that you can answer by using thetable. Give your questions to a friend to solve.

___________________________________________________________________________________________________________

___________________________________________________________________________________________________________

___________________________________________________________________________________________________________

___________________________________________________________________________________________________________

___________________________________________________________________________________________________________

Number of Visitors on Each Day

Amusement Park Friday Saturday Sunday

Water World 706 924 940

Future City 668 935 851

51

Name __________________________________________________________ Date __________________________________


Part 2:

Math in My World

Math class isn’t the only place students are exposed tomathematics. By the time they take their seats in theclassroom, many students have already glanced at clocks,

counted the change in their pockets, logged onto a computer, andplayed sports and other games that require at least a bit of math know-how. They’ve most likely come across math-related topics in books andmagazines and on their favorite TV shows. And, when the school day isdone, their math experiences continue as they shop, keep appointments,interact with friends and family members, practice the piano, and helpprepare dinner at home. Writing about these math topics—that is,writing about math as it is used outside of school—reinforces students’understanding of the math concepts they’ve learned in the classroom.Such writing also motivates students to spend more energy on theirmath studies because as students write about math, they gain anappreciation of how math is used in almost every aspect of their lives.


Create a time line of important eventsin your life. Be sure to include the dateon which these events occurred. Thenillustrate the events with photos orpictures.

An inventory is a complete list of itemsthat someone owns. Take an inventoryof your own possessions (toys, games,clothes, music CDs, and so on). Thenwrite a paragraph explaining what yourpossessions say about you.

Write a set of instructions that could beused to teach someone how to programa VCR, set a digital alarm clock, or useanother household appliance found inyour home.

Measure the area of each room in yourhome. Then create a chart showing theresults, listing the rooms in order fromsmallest to largest.

Choose an object in your bedroom orkitchen. Describe it using as manynumbers and math terms as you can.

Describe your dream house using asmany numbers and math terms as youcan. For example, how big would it be?How many rooms would it have? Atwhat temperature would you keep eachone?

53

Math at Home

DRAWING MATH

Directions: Draw a picture of yourfamily. Label the people in thepicture with as many math-relatedfacts and comparisons as you can.


How many clocks can you find in yourhome? What is the purpose of eachone? Do you think you and your familyreally need this many clocks? Explainwhy you do or don’t think so.

Why do you think clocks are built intodevices like computers, stove tops, andthe dashboards on cars? What are someother devices that contain clocks? Whydo you think that is?

How many different ways do you usecomputers in your life? How do youthink your life might be different if theyhad never been invented? Explain yourthinking.

Watch as someone in your familymakes dinner. Then list some ways that math is used in cooking.

How do your parents use math?Interview them about it! Then make a list of all the things your parents do that involve math.

At what temperature is your homekept? How about your refrigerator?Your freezer? Find out! Then compareyour answers with those of other kidsin class.

Turn on a radio for a moment. Then listall the ways you can think of that mathwas involved in bringing that song intoyour home.

Take a count of all the lightbulbs in yourhome. How many watts of electricitydoes each one use? Make a chartshowing where in your home bulbs ofdifferent wattages are used.

Write a set of instructions that wouldhelp someone get from your school toyour home. Make the directions asspecific as you can.

Write a set of directions that explainhow to make your favorite recipe.

Create a calendar showing all of thedates that are important to you andyour family.

Write a poem about the importance of certain numbers in your life. Forexample, the numbers might includeyour age, address, birthday, the numberof family members in your home, andso on.

What math skill do you think you usemost often during the day? Explainwhen and how you use it.

54

Math a t Hom e Math a t Home Math a t Hom e Math a t Home Math a t Home M ath a t Home M ath a t H ome M ath a t H ome Mth a t Hom e Math a t Home Math a t Hom e Math a t Home Math a t Home Math a t Home M ath a t H o


Math at Home

Write a few sentences about the different ways you use math in yourhome. The pictures on this page should give you some ideas.

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___________________________________________________________________________________________________________

___________________________________________________________________________________________________________

___________________________________________________________________________________________________________

___________________________________________________________________________________________________________

55

Name __________________________________________________________ Date __________________________________


Name a game or sport in which thehighest score wins. Write a fewsentences about it. Describe what agood score is in this game and how aplayer earns it.

Name a game or sport in which thelowest score wins. Write a fewsentences about it. Describe what agood score is in this game and how aplayer earns it.

Explain how your favorite card game isplayed. Be sure to include the rules andhow score is kept.

What’s your favorite sport? Explainhow math is used in that sport.

Name a game or sport in which a timeris used. What is the sport? How is atimer used in it?

Think of a favorite computer game thatuses math. Explain how math is used inthis game.

56

Math at Play

MATH PROJECT

Using a Scrabble® set, answer thesequestions:

How is math used in the tiles?

How is math used on the board?

What operations do you use to

keep score in this game?

What special squares on the

board help you to score

additional points? How?

What is the most common letter

found in a Scrabble® set? Why do

you think that is?

Which is the least common letter

found in a Scrabble® set? Why do

you think that is?

Which letter has the highest value

in Scrabble®? Why do you think

that is?

Now create your own pieces andboard for a version of the game inwhich you make equations insteadof words. How many equal signsshould you include in your game?How about plus, minus, division, andmultiplication signs? How many one-,two-, and three-digit numbers? Howmany points should each one beworth? When you’re done, test yourgame with a friend to see how well it works.


A professional basketball team is 6points behind. Arnie says that they cancatch up by making 3 straight baskets.Ellen says that they can catch up byscoring 2 straight baskets. Sam saysthat both Arnie and Ellen are right.Whom do you agree with? Explain why.

How long do you think it would takeyou to bike one mile? Explain how youmade this estimate.

Poll your classmates about theirfavorite games. Then chart the resultsand see what conclusions you can drawfrom the data.

Work with your classmates to create a bulletin-board display of the mostamazing sports-related number factsyou can find.

Work with your classmates to create a “Guinness® Book of ClassroomRecords.” Poll your friends about theirmost amazing athletic feats andaccomplishments—such as how farthey can jump, how fast they can run,and so on. Then list the classroomrecord-holders in a book.

Create a game in which you need toknow something about coordinategeometry in order to play. Explain therules of the game to a friend. Then playa round of it together.

Write a report about a favorite athlete.Be sure to include as many amazingnumber facts and statistics about thisplayer as you can find.

Invent a new sport that you would loveto see included in the Olympics. Givethe sport a name, explain how it isplayed, and how players would bejudged or scored in this game.

Select a math skill that you havelearned about in school. Then create a board game that requires this skill to play it.

Watch a favorite game or sport on TV and imagine you are a reportercovering the event for a newspaper.Write an article telling what happenedduring the game, who eventually won,and what the final score was.


If you could change the sizes, designs,and values of our coins, what wouldyou change? Why do you think thiswould be better? If you wouldn’tchange anything, explain why that is.

Some people think that pennies are a hassle, and that the United Statesgovernment should stop making them.Do you agree or disagree? Explain yourthinking.

What is money? Write a definition for itin your own words.

What are some similarities anddifferences between keeping yourmoney in a bank and in a piggy bank?What are some other places wherepeople keep their savings? How dothese places compare?

On what coin and bill does GeorgeWashington’s face appear? Why do youthink the government decided to puthim on both these units of money?

How might the world be different ifmoney had never been invented? Wouldlife be easier or harder? What makesyou think so?

What’s the last thing that you bought?How much did you pay? Did you thinkit was worth it? Why or why not?

58

Money Math

Show What You Know About…Money

Write down the name of everycoin and denomination (unit) ofpaper money you can think ofand what each one is worth. Listthem in order of value, fromsmallest to largest.

DRAWING MATH

Directions: Imagine that it is your jobto create a new type of coin ordenomination (unit) of paper moneyfor the United States government.How much will your new unit ofmoney be worth? What will it looklike? Draw a picture to share with the class.


List three or more things that you canbuy for 25¢ or less.

Which is a better buy: 5 pencils for 20¢or 8 pencils for 40¢? Explain.

Make up a story from the point of viewof a penny about what happens to itduring its life.

When your grandparents were kids, itonly cost 5¢ to ride a bus or buy a soda.About how much more does it cost todo these things today? Write a storythat takes place in the future, when itcosts even more to do these things orsimilar ones.

Do you think that it is important tocheck your change after you buysomething? Why or why not?

Name some items that you think aretoo expensive and some other itemsthat you think are good buys. Explainwhy you think so in each case.

What did you eat for dinner last night?How much money do you think yourmeal cost? Explain.

You use a $10 bill to pay for a magazine.The magazine costs $1.95. How canyou tell if your change is correct?

Which of these jars of peanuts do youthink is the better buy? Explain whyyou think so.

Pick a job you might do, such asmowing a lawn, walking a dog, cleaningan apartment, or selling lemonade.Explain how you would decide howmuch money to charge.

List some ways that you can earnspending money. Of these ideas, whichones would you choose? How muchwould you want to charge yourcustomers? What would you do withthe money you earned? Write about it!

Why do you think two differentcomputers might have two differentprices? Explain.

How do you currently get yourspending money? Do you think that thisis a fair way? In a letter to your parents,or whoever gives you this money,explain why you do or do not agreewith this arrangement.


Suppose you won $10,000,000 in acontest. Describe what you would dowith it.

Imagine that you’ve just been offered$1,000,000—as long as you find a wayto spend all the money within a week.What would you buy? Use catalogs,newspapers, and magazines to price the items you would want and keep a running total. Can you find a way tospend all of the money in the giventime?

Do you think kids your age should getan allowance? Why or why not? If youdo think they should get an allowance,how much should it be? Explain.

Which do you think is a better deal:bringing lunch to school or buying it?Explain your thinking.

“Dough,” “bread,” and “loot” are allslang words for money. List as manyother money-related words as you canthink of. Why do you think so manydifferent ways of saying “money” exist?

What does the term “a rip-off” mean toyou? Give one or more examples ofsomething that you would consider “a rip-off.”

In what way is swapping sandwicheswith a friend at lunch similar to buyinga sandwich at a store? In what ways isit different?

People with coin collections know thatit is possible for a certain coin to beworth more than the money amountwritten on it. How can that be? Explain.

How are the tokens you use at video-game arcades similar to official U.S.currency? In what ways are theydifferent?

Give some examples of how peopleswapped goods and services beforemoney was invented. What problems doyou think there might have been withthis system? Explain.

60

Mon e y Math Money M a th M o ney M a t h Mon e y Math Money Math M oney M a th Mo n ey Mat h Mone y Math Money M a th Mon e y Math Money M ath M o ney M a t h Mon e y Math Money Math M oney M a th Mo n ey Mat h Mone y M


Pick an item you own that you wouldlike to sell. Write an ad in which youdescribe the item and tell how muchyou think it is worth.

Select some items from your home thatyou would be willing to sell in a garagesale. Explain how much money youwould ask for each one and how youarrived at that price.

What do you think the phrase “savingfor a rainy day” means?

Ben Franklin once said that “a pennysaved is a penny earned.” What do youthink he meant? Do you agree? Why orwhy not?

List all the coin combinations you canthink of that are worth 85¢. How manycan you come up with in all?

Create a menu for a restaurant you’dlove to open. Be sure to includeappropriate prices for all of the items!

Do you know what a debt is? Have youever heard of the national debt? Whatdo you think it is? Explain.

Who do you think should earn moremoney on the job: teachers or footballplayers? What makes you think so?Explain.

If you were to earn $6 an hour from thetime you were 25 years old till the dayyou turned 65 years old, and youworked 40 hours a week, how muchmoney will you have made in all?Describe all the steps you take to arriveat your answer.

If a one-fluid-ounce bottle of foodcoloring costs $1.59, about how muchshould a gallon of food coloring cost?(Hint: There are 128 fluid ounces to agallon.) Does this price surprise you?Why or why not?

What’s the fewest number of coinsneeded to make 32¢? What other coincombinations equal this amount?

Say you want to retile a floor that is 15feet long and 10 feet wide. The 12-inch-square tiles that you’re using cost $1.50each. Explain how you can figure outthe total cost of the tiles that you’llneed.

61

Mon e y Math Money M a th M o ney M a t h Mon e y Math Money Math M oney M a th Mo n ey Mat h Mone y Math Money M a th MMon e y Math Money M ath M o ney M a t h Mon e y Math Money Math M oney M a th Mo n ey Mat h Mone


Funny Money

What’s wrong with this picture? Find as many things as you can. For each thing that you find, write a sentence or two telling why itseems wrong.

___________________________________________________________________________________________________________

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___________________________________________________________________________________________________________

___________________________________________________________________________________________________________

___________________________________________________________________________________________________________

62

Name __________________________________________________________ Date __________________________________


My Own Spending ReportUse this chart to keep track of everything that you buy for a week. Then answer the questions below on the back of this page.

63

Name __________________________________________________________ Date __________________________________

a.Where did you spend most of your money this week?

b.How much did you spend in all?

c. Were you surprised by how much you spent? Explain why or why not.

d.Were you surprised by the amount you spent on a particular kind ofitem, such as food? Explain your answer.

e. Create a plan for saving some of your money for somethingexpensive that you want. How would you do it?

Day of the Week

Monday

Tuesday

Wednesday

Thursday

Friday

Saturday

Sunday

TOTAL

What I Bought Where I Bought It How Much I Spent


Pick an article from a newspaper andcircle all of the number facts used in it.Try to rewrite the story without thisinformation. What do you find whenyou try to do so?

Do you think that either boys or girlsare better at math? What makes youthink so?

Estimate the number of ads (on TV, innewspapers, on billboards and posters,and so on) you see every week. Why do you think companies choose toadvertise their products as much asthey do? How do you feel about this?

Keep track of how much television youwatch every day for a week—and howmany commercials you see during thattime. Select one of these ads anddescribe how it makes you feel.

You’ve just finished taking a reallytough math test and a friend of yourscomes up to you and asks, “Why do weneed to know this stuff, anyway?” Howdo you respond?

What is your favorite math skill?Explain why that is.

What is your least favorite math skill?Explain why that is.

Most countries use only metric units intheir measurements, while the UnitedStates uses both metric and standardunits. Which do you think is a betteridea? Why do you think so?

Can you think of any job or professionthat does not require math knowledgeat all? If so, what is it? What makes youthink that no math knowledge isneeded in it? If not, why do you thinkthat math is always important, nomatter what type of work people do?

Imagine a conversation between astudent who likes math and one whodoesn’t. What would each say to theother to explain their point of view?Write a short dialogue telling what eachone would say.

64

News and Views About Math

MATH PROJECT

Survey at least ten grown-ups to findout how they use various math skillson their job and in their life. Find away to graph the results. Then write a paragraph explaining what youlearned from the survey.


marvelous math writing prompts

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