displaying data with graphs - everyday math · 7 math masters, p. 60 per partnership: tape measure...

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126 Unit 2 Using Numbers and Organizing Data

Advance PreparationPlace copies of Math Masters, page 60 near the Math Message. For the optional Readiness activity in Part 3, cut apart and tape together

four copies of Math Masters, page 406 for students to use as a graph mat and get the pattern blocks specified on page 131.

Teacher’s Reference Manual, Grades 4–6 pp. 161–167, 216–219

Key Concepts and Skills• Create a bar graph and line plot.

[Data and Chance Goal 1]

• Determine the maximum, minimum, range,

mode, and median of a data set.


• Ask and answer questions and draw

conclusions based on data landmarks,

a bar graph, and a line plot.


• Measure to the nearest half-centimeter.

[Measurement and Reference Frames Goal 1]

Key ActivitiesStudents measure their head sizes to the

nearest half-centimeter. They find the

median head size and make a bar graph and

line plot of the data.

Ongoing Assessment: Informing Instruction See page 127.

Ongoing Assessment: Recognizing Student Achievement Use journal page 46. [Data and Chance Goal 2]

Key Vocabularybar graph

MaterialsMath Journal 1, pp. 46, 47, 47A, and 47B

Student Reference Book, p. 71

Study Link 2�7 � Math Masters, p. 60

per partnership: tape measure � ruler �

stick-on notes � slate � computer with

Internet access (optional)

Constructing a KiteMath Journal 1, p. 48

compass � straightedge

Students construct a kite with a

compass and straightedge.

Math Boxes 2�8Math Journal 1, p. 45

Students practice and maintain skills

through Math Box problems.

Study Link 2�8Math Masters, p. 61

Students practice and maintain skills

through Study Link activities.

READINESS

Constructing a “Real Graph”Math Masters, pp. 62 and 406

pattern blocks � tape

Students use pattern blocks to construct

a real graph.

ENRICHMENTDetermining the Validity of the “One Size Fits All” ClaimMath Journal 1, pp. 46 and 47

Math Masters, p. 63

baseball caps with adjustable headbands �

tape measure

Students analyze a product claim by using

the class head-size data.

Teaching the Lesson Ongoing Learning & Practice Differentiation Options

�� Displaying Data with GraphsObjectives To provide practice measuring length to the

nearest half-centimeter; and to guide the construction and use

of graphs for a set of collected data.

eToolkitePresentations Interactive Teacher’s

Lesson Guide

Algorithms Practice

EM FactsWorkshop Game™

AssessmentManagement

Family Letters

CurriculumFocal Points

Common Core State Standards

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Adjusting the Activity

1 Teaching the Lesson

� Math Message Follow-Up INDEPENDENTACTIVITY

(Math Masters, p. 60)

As students are measuring and drawing the line segments, circulate and observe. Have them check each other’s work and remeasure line segments to resolve any disagreements.

� Collecting and Organizing PARTNER ACTIVITY

Head-Size Data(Math Journal 1, pp. 46, 47, 47A, and 47B;

Student Reference Book, p. 71)

To introduce this activity, read the first two paragraphs on journal page 46 as a class. To help solve Ms. Woods’s problem, have partners measure the distance around each other’s heads and record the measurement in Problem 1 on journal page 46. When students measure head size, the tape measure should measure the maximum distance around the skull. Then ask students to organize these measurements, using some of the techniques from the previous lessons. This activity can be done in a number of ways, one of which is described on the next page.

Discuss the different meanings and pronunciations of the word record.

For example, compare the phrase “record their head sizes” with the phrase

“holds the record for the fastest 100-meter dash.”

A U D I T O R Y � K I N E S T H E T I C � T A C T I L E � V I S U A L

PROBLEMBBBBBBBBBBBOOOOOOOOOOBBBBBBBBBBBBBBBBBBBBBBBBBB MMMMMEEEMMMMBLEBLLEBLBLEBLELLLLBLEBLEBLEBLEBLEEEEMMMMMMMMMMMMMMOOOOOOOOOOOOBBBBBBLBBLBLBLBLLBLLLLLPROPROPROPROPROPROPROPROPROPROPROPRPRPROPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPROROROROROROROOPPPPPPP MMMMMMMMMMMMMMMMMMMMMMEEEEEEEEEEEEELEEELEEEEEEEELLLLLLLLLLLLLLLLLLLLLRRRRRRRRRRRRRRRRRPROBLEMSOLVING

BBBBBBBBBBBBBBBBBBBB ELEELEEMMMMMMMMMOOOOOOOOOBBBLBLBLBBBLBBLROOOORORORORORORORORORORO LELELELEEEEEELEMMMMMMMMMMMMLEMLLLLLLLLLLLLLLLLLLLLRRRRRRRRRRRGGGGGLLLLLLLLLLLLLVINVINVINVINVINNNNVINVINNVINVINVINVINVINV GGGGGGGGGGGOLOOOOOOLOLOLOO VVINVINLLLLLLLLLLVINVINVINVINVINNVINVINVINVINVINVINVINVINNGGGGGGGGGGOOOLOLOLOLOLLOOO VVVLLLLLLLLLLLVVVVVVVVVVVSOSOSOOSOSOSOSOSOSOOSOSOSOOOSOSOOOSOSOSOSOSOSOSOOSOSOSOSOSOSOSOSOSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS VVVVVVVVVVVVVVVVVVVVVLLLLLLLVVVVVVVVVLLLVVVVVVVVLLLLLLLLVVVVVLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLSSSSSSSSSSSSSSSSSSSSSSSS GGGGGGGGGGGGGGGGGOOOOOOOOOOOOOOOOO GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGNNNNNNNNNNNNNNNNNNNNNNNNNIIIIIIIIIIIIIIIIIIIISOLVING

ELL

LESSON

2�8

Name Date Time

Measuring and Drawing Line Segments

Measure the following line segments to the nearest1–2 centimeter.

1.

About cm

2.

About cm

3.

About cm

4.

About cm

Draw line segments having the following lengths:

5. 8 centimeters

6. 10 centimeters

7. 3.5 centimeters

11.5

7.5

9

7

128

Try This

8. Draw a line segment having the following length: 46 millimeters

Math Masters, p. 60

Teaching Master

Lesson 2�8 127

Getting Started

Math MessageTake out your ruler. Complete Math Masters, page 60.

Study Link 2�7 Follow-Up Have partners compare answers. Encourage students to add any interesting number facts to the Numbers and Their Uses Museum.

Mental Math and Reflexes Pose addition problems such as the following. Encourage students to share their strategies.

20 + 5 = 25

8 + 40 = 48

14 + 6 = 20

7 + 23 = 30

12 + 9 = 21

15 + 8 = 23

18 + 4 = 22

27 + 6 = 33

331 + 179 = 510

627 + 266 = 893

218 + 572 = 790

644 + 548 = 1,192

Ongoing Assessment: Informing Instruction

Watch for students who are having difficulty

measuring to the nearest half-centimeter.

Look for common errors such as measuring

from a point other than the 0-mark on the

ruler, rounding incorrectly, or failing to

recognize the millimeter mark halfway

between the two whole numbers.

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�

Head SizesLESSON

2�8

Date Time

Ms. Woods owns a clothing store. She is trying to decide how many children’s hats to

stock in each possible size. Should she stock the same number of hats in each size?

Or should she stock more hats in some sizes and fewer in others?

Help Ms. Woods decide. Pretend that she has asked each class in your school to

collect and organize data about students’ head sizes. She plans to combine the data

and then use it to figure out how many hats of each size to stock.

As a class, collect and organize data about one another’s head sizes.

1. Ask your partner to help you measure the distance around your head.

� Wrap the tape measure once around your head.

� See where the tape touches the end tip of the tape measure.

� Read the mark where the tape touches the end tip.

� Read this length to the nearest 1

__ 2 centimeter.

Answers vary. Record your head size. About cm

2. What is the median head size for the class? About cm

3. Find the following landmarks for the head-size data shown in the bar graph on journal page 47.

Minimum: Maximum: Range:

Mode: Median:

4. How would the landmarks above help Ms. Woods, a clothing store owner, decide how many

baseball caps of each size to stock?

Sample answer: Ms. Woods will want to stock a lot more

baseball caps in the median and mode sizes than in any

other size. She will also want to be sure not to stock

many baseball caps that are smaller than the minimum

size and larger than the maximum size.

73

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Math Journal 1, p. 46

Student PageFinding the Median Head SizeHave students record their head sizes on stick-on notes and line up in order from smallest to largest head size. Send students to the line, one at a time, and have each one compare the measurement he or she wrote on the stick-on note with those of students who are already in line. If two students have the same head size, they should stand next to each other.

Ask for suggestions for identifying the person in the middle of the line. Suggest one of the following methods, or encourage students to make up one of their own:

� Count off from one end of the line to the other (1, 2, 3, and so on). Find the person (or people) whose number is halfway between 1 and the number of the last person in line.

� Count off from both ends of the line (1, 1, 2, 2, 3, 3, and so on). The highest number identifies the middle person (or people).

� Have the person at each end of the line sit down. Repeat until one or two people are left standing.

Remind students that the median head size is the head size of the person in the middle. If there are two people, it is the measurement halfway between the two middle head sizes. Have students record the median in Problem 2 on journal page 46.

Making a Bar Graph of the DataYou can follow these steps to make a bar graph of the data.

1. On the board, draw a horizontal and vertical axis, and write 0 next to the point where the two axes meet.

2. Draw additional points, about 4 inches apart, on the horizontal axis, and label the first point with the smallest measurement. Label the rest of the points in half-centimeter increments. Write “Head Sizes to the Nearest 1 _ 2 cm” under the horizontal axis.

3. Have students attach their stick-on notes in the appropriate places above the horizontal axis. Each note should just touch the one below it.

4. Label the points on the vertical axis. Write “Number of Students” next to the vertical axis. Write an appropriate title for the graph at the top.

49

49

5

4

3

2

1

0Num

ber

of S

tude

nts

4912

4912 5012

5012

5012

5112

5112

5112

5012 51 511

2 5212 531

2

50

50

50

52

52

52

52

52

53

53

53

53

53

53

54

54

Head Sizes in 4th Grade Class

Head Sizes to the Nearest cm12

NOTE The median head size for fourth

graders is about 52 centimeters. Of all fourth

graders, 95 percent have head sizes between

49 and 55 centimeters.

Men’s and women’s hats are sized in different

ways. The size of a woman’s hat — 22, for

example –– is approximately the distance

around the head in inches. To understand a

man’s hat size — 6 7

_ 8 , for example, imagine

taking a sweatband inside the hat and forming

it into a circle. The size is the diameter of

the circle to the nearest 1

_ 8 inch. This system

began centuries ago, when new hats were

round. Men used a device called a hat screw

to force their hats into a more comfortable

oval shape.

NOTE If graphing software is available,

have students use it to create bar graphs.

Consider using the bar grapher at

http://illuminations.nctm.org/ActivityDetail.

aspx?id=63. Then have the class compare

these with the graphs students made on

journal page 47.

Adjusting the ActivityAsk students to discuss whether

the mean is a useful data landmark in this

situation.

AUDITORY � KINESTHETIC � TACTILE � VISUAL


Adjusting the Activity

5. Ask students to copy the bar graph onto journal page 47. Have them mark and label the axes and color the bars to represent the stick-on notes. Each column of stick-on notes represents one of the bars in the bar graph. Remind students to copy the title of the graph.

5

4

3

2

1

0Num

ber

of S

tud

ents

49 4912 50 501

2 51 51 12 52 52 1

2 53 53 12 54



Ask volunteers to pose questions to the class regarding the data

displayed. For example:

• Are the bars bunched together?

• Are there any isolated or way-out bars?

• Are there more or more ?

• How many and are there in all?

A U D I T O R Y � K I N E S T H E T I C � T A C T I L E � V I S U A L

Making a Line Plot of the DataDirect students to the essay on Student Reference Book, page 71, “Organizing Data.” Discuss features of line plots. Ask students to use the information in the original bar graph to create a line plot on journal page 47B. Have them label the axes and use an X to replace each stick-on note. Remind students to write the title of the graph.

Num

ber

of S

tude

nts

49 49 12 50 50 1

2 51 51 12 52 52 1

2 53 53 12 54



After students complete the line plot, have them answer the questions on journal page 47A.

Head Sizes continuedLESSON

2�8

Date Time

Make a bar graph of the head-size data for the class.

lab

el

title

lab

el

Head S

izes in 4

th G

rade C

lass

Head S

izes to the N

eare

st C

entim

ete

r1 2

Num

ber

of S

tudents

76

EM3MJ1_G4_U02_28-52.indd 47 10/28/10 10:39 AM

Date Time

Head Sizes continued LESSON

2�8 1. Count the Xs in the line plot you created on journal page 47B to answer the

questions below. Answers vary.

a. What is the largest head size?

b. What is the smallest head size?

c. What is the difference between the largest and the smallest head size? Write a number

model to show how you found your answer.

d. What is the mode of the head-size data?

e. What is the difference between your head size and the mode? Write a number model to

show how you found your answer.

2. Did anything about the data surprise you? Did you expect the head sizes to be close together

or spread out?

Sample answer: I expected the head sizes to be

clumped together because everyone in the class is

close in age.

3. The average head size for an infant at birth is 35 cm. Use a number model to show

how your head size compares to that of an infant.

Answers vary.

4. Imagine that you measured the head sizes of a room full of newborn babies and

created a line plot. What would you expect the graph to look like? Explain your

answer.

Sample answer: I would expect the middle of the data

to be around 35 cm. There might be some way-out

numbers because babies can be many different sizes

when they are born.

74

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Math Journal 1, p. 47A

Student Page

Student Page

Lesson 2�8 129

NOTE Stem-and-leaf plots

are covered in Grade 5.

If your curriculum requires

that this concept be

covered in Grade 4, see

www.everydaymathonline.com.

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Math Boxes LESSON

2 � 8

Date Time

12–15 4

148

1. Add mentally.

a. 2 � 4 �

b. 20 � 40 �

c. 200 � 400 �

d. � 8 � 6

e. � 80 � 60

f. � 800 � 6001,400

140

14

600

60

6

3. Subtract mentally or with a paper-and-

pencil algorithm.

a. b.

5. An ostrich can weigh about 345 pounds.

An emu can weigh about 88 pounds.

How much would they weigh together?

pounds 433

231� 84

147

603� 466

137

6. Tell whether each number sentence is

true or false.

a. 18 � 9 � 37

b. 29 � 17 � 12

c. 42 � 15 � 27

d. 17 � 40 � 24

e. 154 � 65 � 99 false

false

true

true

false

4. Write 8,042,176 in words.

2. Find the following landmarks for this set of

numbers: 12, 16, 23, 15, 16, 19, 18.

a. median

b. mode

c. maximum

d. minimum

e. range

f. mean 17

11

12

23

16

16

73–7510 11

eight million, forty-two

thousand, one hundred seventy-six


Student Page

Date Time

Head Sizes continued LESSON

2�8

74

Make a line plot of the head-size data for the class.

lab

el

title

lab

el

Head S

izes in 4

th G

rade C

lass

Num

ber

of S

tudents

Head S

izes to the N

eare

st C

entim

ete

r1 2

EM3MJ1_G4_U02_28-52.indd b47 10/28/10 10:39 AM

Math Journal 1, p. 47B

Student Page

� Analyzing Head-Size Data INDEPENDENTACTIVITY

(Math Journal 1, pp. 46 and 47)

Have students complete Problems 3 and 4 on journal page 46 on their own.

Ongoing Assessment: Journal

page 46 �Problem 4

Recognizing Student Achievement

Use journal page 46, Problem 4 to assess students’ ability to use data

landmarks and a bar graph to draw conclusions about a data set. Students are

making adequate progress if their responses include ideas such as the following:

� The smallest and largest sizes suggest the range of sizes Ms. Woods should

have in her inventory.

� The most frequently reported size and the median would suggest the sizes for

which Ms. Woods should have the largest inventory.

Some students may be able to explain why the mean would not be a useful data

landmark in this situation.


2 Ongoing Learning & Practice

� Constructing a Kite INDEPENDENTACTIVITY

(Math Journal 1, p. 48)

Students use a compass and a straightedge to construct a kite. Ask students to explain how they know the polygon is a kite.

� Math Boxes 2�8 INDEPENDENTACTIVITY

(Math Journal 1, p. 45)

Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 2-6. The skills in Problems 5 and 6 preview Unit 3 content.

Writing/Reasoning Have students write a response for the following: Describe the patterns in the number sentences in Problem 1. Sample answer: The numbers in

the second problem of each set are 10 times the numbers in the first problem. The numbers in the third problem of each set are 10 times the numbers in the second problem and 100 times the numbers in the first problem.

� Study Link 2�8 INDEPENDENTACTIVITY

(Math Masters, p. 61)

Home Connection Students answer questions based on gestation-period data for selected animals.


3 Differentiation Options

READINESS PARTNER ACTIVITY

� Constructing a “Real Graph” 15–30 Min

(Math Masters, pp. 62 and 406)

To provide experience with a bar graph using a concrete model, have students use pattern blocks (4 hexagons, 8 trapezoids, 2 triangles, 4 squares, 6 blue rhombi, and 3 tan rhombi) to construct a “real graph” and answer questions based on their display. When students have completed the graph, discuss an appropriate title and labels.

ENRICHMENT PARTNER ACTIVITY

� Determining the Validity of the 5–15 Min

“One Size Fits All” Claim(Math Journal 1, pp. 46 and 47; Math Masters, p. 63)

Consumer Link To investigate the application of data, have students determine whether an adjustable cap will fit all students in the class.

STUDY LINK

2�8 Gestation Period

61

73

Name Date Time

The period between the time an animal becomes pregnant and the time

its baby is born is called the gestation period. The table below shows the

number of days in the average gestation period for some animals.

1. For the gestation periods listed in the table ...

a. what is the maximum number of days?

days

b. what is the minimum number of days?

days

c. what is the range (the difference between

the maximum and the minimum)?

days

d. what is the median (middle) number of days?

days

2. Which animals have an average gestation period that is longer than 1 year?

3. How much longer is the average gestation period for a goat than for a dog? days

4. Which animal has an average gestation period that is about twice as long

as a rabbit’s?

5. Which animal has an average gestation period that is about half as long

as a squirrel’s? mouse

dog

90

giraffe, Asian elephant, and rhinoceros

151

626

19

645

Average Gestation Period(in days)

Animal Number of Days

dog 61

giraffe 457

goat 151

human 266

Asian elephant 645

mouse 19

squirrel 44

rhinoceros 480

rabbit 31

Source: World Almanac

6. 56 � 33 � 7. � 167 � 96

8. � 78 � 32 9. 271 � 89 � 18246

26389Practice

Math Masters, p. 61

Study Link Master

LESSON

2�8

Name Date Time

76

Do this activity with a partner.

Materials � set of pattern blocks from your teacher

� graph mat (4 copies of Math Masters, page 406 taped together)

1. Display the pattern blocks on the graph mat so that you can easily count and compare

the number of hexagons, trapezoids, triangles, squares, blue rhombi, and tan rhombi.

2. Use your display to answer the following questions.

a. Which pattern block appears the most? The least?

b. How many hexagons and triangles are there altogether?

c. How many more trapezoids are there than squares?

3. Use your display to complete the following statements. Sample answers:

a. There are fewer than .

b. There are more than .

c. There is the same number of as .

4. Write a question that can be answered by looking at your display. Answer your question.

a. Question

b. Answer 9

How many rhombi are there altogether?

squareshexagons

trianglesblue rhombi

trapezoidssquares

4

6

triangletrapezoid

tanrhombus

bluerhombussquare

triangle

trapezoid

hexagon

5. How many more quadrangles are there than nonquadrangles? 15Try This

Construct a “Real” Graph

Math Masters, p. 62

Teaching Master

Lesson 2�8 131

LESSON

2�8

Name Date Time

“One Size Fits All” Claim

Makers of adjustable baseball caps claim that “one size

fits all.” Do you think this is a true statement? Use the

head-size data you collected on journal pages 46 and 47

to help you decide.

Answers vary.1. Select a baseball cap and adjust the headband to the smallest size. Measure and record

the distance around the inside of the baseball cap to the nearest half centimeter.

Smallest size: cm

2. Now adjust the headband to the largest size. Measure and record.

Largest size: cm

3. Compare the measurements above with the head-size data you and your class collected.

Could this baseball cap be worn by everyone in the class? Explain your answer.

4. Do you think you have enough information to decide whether or not the claim “one size

fits all” is true? Explain.

Math Masters, p. 63

Teaching Master

NOTE For practice

with pictographs, see

www.everydaymathonline.com.


displaying data with graphs - everyday math · 7 math masters, p. 60 per partnership: tape measure...

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