Picturing numbers grade 6

How would it compare to the graph showing the number of fliers delivered?... One way to picture data for your class is by recording the data on a number line.. On Student Activity Sheet

Picturing Numbers

Data Analysis and Probability

Mathematics in Context is a comprehensive curriculum for the middle grades

It was developed in 1991 through 1997 in collaboration with the Wisconsin Center for Education Research, School of Education, University of Wisconsin-Madison and the Freudenthal Institute at the University of Utrecht, The Netherlands, with the support of the National Science Foundation Grant No 9054928.

The revision of the curriculum was carried out in 2003 through 2005, with the

support of the National Science Foundation Grant No ESI 0137414.

National Science Foundation

Opinions expressed are those of the authors and not necessarily those of the Foundation.

Boswinkel, N., Niehaus, J., Gravemeijer, K., Wijers, M., Dekker, T., Middleton, J A.,

Spence, M S., Burrill, G., & Milinkovic, J (2006) Picturing numbers In Wisconsin

Center for Education Research & Freudenthal Institute (Eds.), Mathematics in Context Chicago: Encyclopædia Britannica, Inc.

Copyright © 2006 Encyclopædia Britannica, Inc.

All rights reserved.

Printed in the United States of America.

This work is protected under current U.S copyright laws, and the performance, display, and other applicable uses of it are governed by those laws Any uses not

in conformity with the U.S copyright statute are prohibited without our express written permission, including but not limited to duplication, adaptation, and

transmission by television or other devices or processes For more information regarding a license, write Encyclopædia Britannica, Inc., 331 North LaSalle Street, Chicago, Illinois 60610.

ISBN 0-03-042402-X

1 2 3 4 5 6 073 09 08 07 06 05

The Mathematics in Context Development Team

Development 1991–1997

The initial version of Picturing Numbers was developed by Nina Boswinkel, Jansie Niehaus,

and Koeno Gravemeijer It was adapted for use in American schools by James A, Middleton, Mary S Spence, Gail Burrill, and Jasmina Milinkovic.

Wisconsin Center for Education Freudenthal Institute Staff

Research Staff

Thomas A Romberg Joan Daniels Pedro Jan de Lange

Gail Burrill Margaret R Meyer Els Feijs Martin van Reeuwijk

Project Staff

Jonathan Brendefur Sherian Foster Mieke Abels Jansie Niehaus Laura Brinker James A, Middleton Nina Boswinkel Nanda Querelle James Browne Jasmina Milinkovic Frans van Galen Anton Roodhardt Jack Burrill Margaret A Pligge Koeno Gravemeijer Leen Streefland Rose Byrd Mary C Shafer Marja van den Heuvel-Panhuizen

Peter Christiansen Julia A Shew Jan Auke de Jong Adri Treffers

Barbara Clarke Aaron N Simon Vincent Jonker Monica Wijers Doug Clarke Marvin Smith Ronald Keijzer Astrid de Wild Beth R Cole Stephanie Z Smith Martin Kindt

Fae Dremock Mary S Spence

Mary Ann Fix

Revision 2003–2005

The revised version of Picturing Numbers was developed by Monica Wijers and Truus Dekker

It was adapted for use in American schools by Gail Burrill.

Wisconsin Center for Education Freudenthal Institute Staff

Research Staff

Thomas A Romberg David C Webb Jan de Lange Truus Dekker

Gail Burrill Margaret A Pligge Mieke Abels Monica Wijers

Editorial Coordinator Editorial Coordinator Content Coordinator Content Coordinator

Project Staff

Sarah Ailts Margaret R Meyer Arthur Bakker Nathalie Kuijpers

Teri Hedges Kathleen A Steele Dédé de Haan Nanda Querelle Karen Hoiberg Ana C Stephens Martin Kindt Martin van Reeuwijk Carrie Johnson Candace Ulmer

Jean Krusi Jill Vettrus

Elaine McGrath

(c) 2006 Encyclopædia Britannica, Inc Mathematics in Context

and the Mathematics in Context Logo are registered trademarks

of Encyclopædia Britannica, Inc.

Cover photo credits: (left, right) © Getty Images; (middle) © Corbis

Illustrations

v Christine McCabe/© Encyclopædia Britannica, Inc.; 1 Holly Cooper-Olds;

6 Christine McCabe/© Encyclopædia Britannica, Inc.; 26, 28, 37, 41 Holly

Cooper-Olds

Photographs

1 © Corbis; 3 © Louis K Meisel Gallery, Inc./Corbis; 4 Sam Dudgeon/HRW;

7 © Bettmann/Corbis; 10 Victoria Smith/HRW; 11 © Corbis; 12 Victoria

Smith/HRW; 13 (left to right) © Corbis; Sam Dudgeon/HRW; 16 © Corbis;

17 (left to right) © Comstock, Inc.; Allan Munsie/Alamy; Jenny Thomas

Photography/HRW Photo; 19 © Corel; 21 © Brand X Pictures; (left to right)

© Corbis; © Joe McDonald/Corbis; 25 Fotosonline/Alamy; 29 © Corbis;

32 © Annie Griffiths Belt/Corbis; 35, 36 © Corbis; 38 Dennis MacDonald/

Contents v

Thousand Words

°C 30 20 10 0 –10 –20

Random sam ple of 686 adults.

Visit Family/Frie nds 30%

Travel/

Sightsee 26%

Stay at Home 21%

about what they represent such as fliers delivered,

fuel in a tank gauges, and TV programs preferred

You will look at bar graphs and number line plots

You will make a picture called a line graph to show

how temperatures change from one time interval

to the next

You will create charts that look like pies to

describe the results of your own surveys

You will also begin to describe a distribution

of data by some summary numbers—such

as the maximum, the minimum, the mean,

and the mode You will learn how to think

about the word average as the mean of a

set of numbers

In the end, you should understand something

about how pictures can describe numbers

and what types of pictures can help you

understand different situations

X

X X

X X X X X

The number of whales in the world isdecreasing rapidly Peter, Carmenza, and Ann want to do something about the situation They earn money by deliveringadvertising fliers to people’s houses Forevery flier they deliver, they get five cents,which they donate to the World WildlifeFund.

Section A: Using Bar Graphs 1

A

Using Bar Graphs

Fliers

Today, Peter delivered 100 fliers, Carmenza 200, and Ann 50

1 How much money did the three of them earn?

Helen, Amber, John, and Diego also delivered fliers In the table, youcan see how many each of them delivered

2 How many fliers were delivered in total?

Student Peter Carmenza Ann John Amber Helen Diego Number of Fliers

Delivered 100 200 50 50 100 250 250

Peter showed the information about the number of fliers in the graph

below He called his display a one-bar graph.

3 a Name one advantage of Peter’s one-bar graph compared to

Helen Diego Carmenza Peter Amber Ann John

You can also present the data fromthe table in a bar graph, which uses

a bar for each category rather thanone bar as Peter did In a bar graph,all bars have the same width

4 a Which way of presenting the number of fliers delivered do you

prefer, the table or the bar graph? Why?

b Reflect Think of another way to arrange the bars in the graph.What is one advantage of your new graph over the one above?

A bar graph has two axes, a vertical axisand a horizontal axis In thisgraph, the vertical axis shows you how many fliers were delivered.The horizontal axis shows you who delivered them A graph needslabels and a title in order to tell its story

You could also make a bar graph showing how much money eachstudent earned

5 a What would this graph look like?

b How would it compare to the graph showing the number of

fliers delivered?

Bar graphs can be used to compare sizes oramounts of different things On this graph,the horizontal axis identifies the things thatare being compared (students) The verticalaxis tells how many (marbles)

Section A: Using Bar Graphs 3

• Rosalie has twice as many marbles as Steve

• Mark has four times as many marbles as Rosalie

• Peter has four marbles more than Steve

• Alice has as many marbles as Steve

• Jane has five fewer marbles than Rosalie

• Helen lost all of the marbles she had

• Monica has as many marbles as Rosalie

6 Using the information on page 3, complete the bar graph on Student Activity Sheet 1.

7 a Who has the most marbles? How many is that?

b Who has the fewest? How many is that?

c Explain how you got your answers.

Jose found another way to show how many marbles each personhas Look at his drawing called a picture graph

8 Make a graph to show how many marbles each student has.

Make your graph different from José’s graph and the one on

Student Activity Sheet 1 Be sure to label everything on your

9 Reflect Do you think it makes a difference whether their eyes are open or closed? Why or why not?Steve

Rosalie

Have a classmate record how many seconds you can stand on one

foot with your eyes shut Write the number of seconds on a self-stick

note Then time how long a classmate can do this

One way to picture data for your class is by recording the data on a

number line Your teacher has made a number line on the board

Tape your self-stick note at the appropriate spot on the number line

This way of picturing numbers is called a number-line plotor

dot plot

Section A: Using Bar Graphs 5

Use the number-line plot your class made on the board to answerquestions 10–13

10 a Describe how long the students in your class can stand on one

foot with their eyes closed

b What times seem to be the most common?

c What is the range of times in your class?

11 a What times seemed to be in the middle?

b Was anyone able to stand for a really long time? How long?

A really short time? How short?

The number-line plot is a graph of the length of times all of yourclassmates can stand on one foot with their eyes closed Many of you probably have times that are pretty close together

12 a Group the times that are close together and tell what happens

to the plot

b On Student Activity Sheet 2, complete the table using the data

from your class Are the groups in the table different from theones chosen on the board?

13 Complete the graph on Student Activity Sheet 2, using the data

from your table Note that the bars in this graph should be drawnnext to each other The widths of all bars should be equal

Using Bar Graphs

15 Write a short report about the length of time students in your class

can stand on one foot with their eyes closed

Litter

Do you ever think about how long garbage lasts if left on its own todecay? Peter, Carmenza, and Ann found this information on the Internet

“Let’s draw a large bar graph for our classroom to make thestudents aware of this problem,” Peter suggests.

16 Reflect Explain why Peter’s idea is not a good one

17 How would you make a poster to help your classmates think

twice before they throw anything away? Use the informationfrom the table

18 Find two examples of data, not mentioned in this section, that can

be presented in:

i a bar graph;

ii a number line graph.

Section A: Using Bar Graphs 7

When Banneker was 21 years old, he took apart a neighbor’s pocketwatch, drew all its pieces, and put it back together He then carved alarge wooden replica of each piece and made a clock The clockworked, striking each hour for more than 40 years

In 1773, Banneker began making astronomical calculations thatenabled him to successfully forecast a 1789 solar eclipse

In 1791, Banneker was hired by PresidentGeorge Washington to assist Andrew Ellicottwith surveying the 10-mile square FederalTerritory (now Washington, D.C.) to lay outthe new capital of the United States

Banneker is best known for his six annualFarmer’s Almanacs published between 1792and 1797 The almanacs included information

on medicine and medical treatment, and listedtides, astronomical information, and eclipses.Banneker did all of the calculations in thealmanacs himself

Using Bar Graphs

Tables and graphs can be used to describe a wide variety of situations.Graphs take information and summarize it in a way that is easy toread They can show the number of marbles a person has, the length

of time students in your class can stand on one foot, or almost thing you want They can also help you see which things are alike,which things are most common, which things fall in the middle, andhow far apart things are To make a graph readable, label it carefully.Label the axes and give the graph a title

any-In this section you learned about bar graphs, picture graphs, andnumber-line plots Bar graphs are for data that you can put into categories, but the categories can be put on the graph in any order.The bars in a bar graph are all equal in width

A number-line plot shows data that can be quantified or described bynumbers on a number line A mark for each data point is made on anumber line

A

Number of Fliers Delivered

0 50 100 150 200 250 300

Peter Carmenza Ann John Amber Helen Diego

Students

Section A: Using Bar Graphs 9

1 a Find a bar graph in a newspaper or magazine and describe

what it shows

b Can the data you found be presented in a different way? How?

Neville and Sonia created this number-line plot

They forgot to label the axes and give it a title

2 a What does this graph tell you about the data Neville and Sonia

gathered?

b Copy and label the graph so that it fits a situation that makes

sense for the data

3 Use the data presented in the table to create bar graphs that

compare the favorite sports of sixth graders and seventh graders

Why do newspapers often use graphs to present information?

X X

X X X X X

X

X X

X X

Favorite Number of Sixth Grade Number of Seventh Grade

Sport Students Students

1 Gas tank a is half full How full is each of the other tanks?

2 If a full tank contains 40 liters of gasoline, how much gas is there

when the gauge has each of the readings shown? (Hint: The firsttank has 20 liters of gasoline.)

Catherine lives in Nashville, Tennessee Herfriend Monica moved to Atlanta, Georgia.One day Catherine drove to Georgia to visitMonica Unfortunately, she forgot howexpensive gas is and spent all of her moneyfilling up her gas tank! Catherine wondered

if she would make it to Atlanta from Nashville

on one tank of gas And if she could make itback home again without having to buy gas!

Section B: A Piece of the Pie 11

B

A Piece of the Pie

How People Spend Their Vacation

4 a What number represents 100% in

this survey?

b How many people would 50%

represent?

5 What does this graph tell you?

6 About what fraction of people in the

survey travel and sightsee?

7 About how many people surveyed said

they visit friends or family while onvacation?

8 Reflect Graphs like the one shown onthis page are called pie charts Why doyou think they are called pie charts?

Visit Family/Friends 30%

No Vacation/

Don’t Know

Travel/

Sightsee 26%

6%

Stay at Home 21%

Summer/ Winter Resort 17%

Random sample of 686 adults.

3 On Student Activity Sheet 3, mark the fuel

gauges as indicated in the statements below

a As Catherine drove away from home, the

tank was almost full

b After 110 kilometers, about a quarter of the

gas had been used up

c Halfway to Atlanta, the tank was more

than half empty

d Fifty-five kilometers later, there was only

three-eighths of a tank left

e What do you think happened? Complete

the story

The data expressed in graphs are oftengathered by means of a survey In a survey,people are asked questions about an issue,and their answers are tallied A pie chartmay be used to present the results of asurvey.

Students at Lakeside Middle School surveyed a randomgroup of 60 students Here are some results

A Piece of the Pie

B

Data Collection

9 What do you think the word random means?

10 a Choose a circle on Student Activity Sheet 4 that will help you

make a pie chart for question I Use the circle to make a piechart for question 1

b Do the same for question II.

c Describe the results of the survey.

Answers

Bus 30 people Foot 14 people Bike 8 people Car 8 people

Question II

What Is Your Favorite Drink?

1 3 1 6 1 4 1 6 1 12

Like the students at Lakeside Middle School, you are going to collectdata The data you collect in this activity will be used throughout therest of the unit.

Instructions For each category on Student Activity Sheet 5, circle the word or

phrase that fits you best

When everyone in the class has finished, cut out all of the squaresand place them in ballot boxes— one ballot box for each question

When you come to appropriate places in the unit, you can then usethe data from each question

Section B: A Piece of the Pie 13

Mr Ramirez’s class at San Juan Middle School is studying transportation They had a discussion about how students travel to school

Ways of Traveling to School

Too many students have their parents drive them to school

The extra traffic makes it dangerous All of the cars also cause pollution.

Plenty of students ride the bus, and buses cause pollution, too.

Mr Ramirez decided to survey the students to find out how they get

to school He wants to know what part of the student body comes toschool in a car and what part takes the bus Here are the responses tohis survey question

11 Organize the data that Mr Ramirez collected in a new table that

makes it easier to understand

Another way to organize the data is to make a bar graph

12 Create a bar graph to make this data easier to understand Don’t

forget to label the axes and add a title to the graph

A Piece of the Pie

B

Student Transportation Student Transportation

Peter BusJean Bicycle

Paulo CarSusan CarHiroko FootBrian BusHarry CarJoe BusEsther FootAngela Foot

John FootSteve CarMartin CarGeorge FootJulia FootRoberto CarArden CarBill FootDominik CarCathy BicycleJill Bicycle

Transportation to School

Foot

Car

Bicycle Bus

The pie chart displays the information from the table inanother way

13 Look at the pie chart Does it tell you the same thing

the table does? Why or why not?

14 a Compare the bar graph you made to the pie chart.

Which would you use to settle the disagreementbetween Brian and Angela? Why?

b Could you use a number-line plot to show the

data? Why or why not?

The bus company has decided to change its school route unless atleast 25% of the students ride a bus to school.

15 Does Mr Ramirez’s class meet the 25% rule of the bus company?

Why or why not?

After the discussion in Mr Ramirez’s class, Arden, Susan, Steve, andMartin decide to take the bus to school

16 a What percentage of the class will now ride the bus to school?

Will the bus company change the school route?

b Reflect The bus company wants to increase the number of students riding the bus What are some reasons for riding thebus that the company could give to students and families?

Take out the data you collected about the ways in which students inyour class get to school

17 a Does your class data meet the 25% rule of the bus company?

b Do you think you have enough information to provide a picture of

how many students in your whole school ride the bus? Explain

18 a What number represents 25% of your class?

b Is that number equivalent to 25% of Mr Ramirez’s class? Explain.

Section B: A Piece of the Pie 15

B

A Piece of the Pie

The pie chart shows the survey results.

19 a Estimate the fraction of people who prefer programs

for children

b About how many people is this?

c In your notebook, copy and complete the table

below Use the pie chart to help you

A Piece of the Pie

B

Stacked bar graphsand pie charts can be used to tell the same story

in different ways Every year, Monroe Middle School organizes asports day Students can choose from six different sports: soccer, basketball, bicycling, softball, skating, and volleyball The stacked bargraphs and the pie charts show the results of a survey of four classes

Graphs Tell a Story

Type of Show Estimated Fraction Number of People

News and Information Movies and Entertainment Sports

Programs for Children

Television Programs

A television network wants to knowwhat kinds of programs people like towatch the most To find out, they hired

a company who took a survey of about1,000 people People chose the kinds

of programs shown on this page

Television Viewer Preferences

Movies and Entertainment

20 Compare the network’s survey results to your class results What

are the similarities and differences?

A Piece of the Pie

21 In your own words, explain what a stacked bar graph is.

22 Match each stacked bar graph with the pie chart that tells the

same story Be ready to explain your choices

23 Examine the data you collected in class about

favorite sports Make a pie chart, a bar graph,

or a stacked bar graph of your data

24 Reflect What are the advantages and disadvantages of each of these three kinds

of graphs? Which graph did you use? Why?

25 Tell the story that is presented in the graphs

for class 2

26 Reflect Based on your data, which sportswould you include if you were organizing asports day for your class?

A Piece of the Pie

In this section, you studied pie charts, bar graphs, and stacked bargraphs A pie chart is a circle divided into sections or “pieces of pie.”Each piece represents a category of data The size of the piece showsthe fraction or percent that category is of the whole pie Each piece ofthe pie chart is labeled, or a legend next to it shows what is meant byeach piece Don’t forget the title!

A stacked bar graph is a bar that represents the entire amount Thebar is divided into parts showing the fraction or part for the differentcategories making up the total amount

Graphs tell a story about the data they represent Different graphsrepresent data in different ways, so they tell different stories The type

of graph you use depends on what you want to say about a situation

B

Alternatives José Garcia The Smashers Valhalla

New Eastland Students’ Favorite Performers

0 25,000 15,000 20,000 10,000 70,000

Alternatives

José Garcia The Smashers

Valhalla

Alternatives 25,000 votes or 36% José Garcia 15,000 votes or 21% The Smashers 20,000 votes or 29% Valhalla 10,000 votes or 14%

New Eastland Students’ Favorite Performers

Legend

Section B: A Piece of the Pie 19

The two graphs represent the results

of a survey on the kinds of pets peopleprefer

1 Reflect Can the two graphs represent the same data? Explain

why or why not

2 a Think of a question about the data that can best be answered

using the bar graph

b Think of a question about the data that can best be answered

using the pie chart

A Piece of the Pie

B

Here is a table showing the results of a survey on computer gamesgiven to students at Western Middle School

3 a Create a graph that tells the story of these data Explain why

you used the graph you did

b Which computer game is the most popular one? How is this

shown in your graph?

Start making an overview of the different types of graphs in the unit and complete it while working through the rest of the unit Name the characteristics of each graph as you enter it in youroverview

Computer Games

Favorite Computer Game Number of Students Percentage of Students

Book of Magic Hit the Safe Tic Tac Go Tilings Island Falling Problems

129 33 190 98 73 119

642

20 5 30 15 11 19

100 Total

Raisa thinks about growing up to be a doctor.She wonders what the other students in herclass want to be.

Raisa surveyed her classmates Based on thesurvey results, she drew this pictograph

Section C: A Picture is Worth a Thousand Words 21

C

A Picture is Worth a Thousand Words

Your Profession

Teacher

Doctor Pilot

Retail Sales

Career Goals



Key

One Person

1 Why do you think this graph is called a pictograph?

2 How many students in Raisa’s class want to be teachers?

3 How many students are there in each of the other

4 Make a pictograph to represent the data.

5 a Make a pictograph of the data on professions that

you collected as a class

b How many of your classmates want to be teachers?

6 Compare the graphs you made in problems 4 and 5

How are they alike? How are they different? Explain

In the previous sections you studied bar graphs.

7 a Why would someone choose to use a pictograph instead

Cat and Mouse

Which animal has more babies per litter, a mouse or a cat? Grace and Huong wanted to find out First, they collected data They askedall the people in their school who have mice or cats whether or nottheir pets have ever had a litter and if so, how many babies were inthe litter Grace and Huong collected the following data about 18 miceand 12 cats

A Mouse

Number of Babies

in Litter

Cat Number of Babies