reading and drawing graphs - mcgraw-hill education · reading and drawing graphs ... lesson guide...

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224 Unit 3 Variables, Formulas, and Graphs

Advance Preparation

Teacher’s Reference Manual, Grades 4–6 pp. 161, 164, 278–289

��

Reading andDrawing Graphs

Objective To interpret and draw graphs that correspond to given situations.g

Key Concepts and Skills• Construct a graph that corresponds to a

given situation. [Data and Chance Goal 1]

• Write a “time story” based on the shape of a graph. [Data and Chance Goal 2]

• Use a graph to represent and interpret a function. [Patterns, Functions, and Algebra Goal 1]

Key ActivitiesStudents draw graphs to illustrate situations and write stories that explain graphs. Students match graphs with situations they might represent.

Ongoing Assessment: Recognizing Student Achievement Use journal page 115. [Data and Chance Goal 2]

Key Vocabularytime graph

MaterialsMath Journal 1, pp. 115–117Study Link 3�8transparency of Math Masters, p. 92 (optional) � straightedge � slate

Playing Scientific Notation TossStudent Reference Book, p. 331Math Masters, p. 472per partnership: 2 six-sided diceStudents practice translating between scientific and standard notations.

Math Boxes 3�9Math Journal 1, p. 118straightedgeStudents practice and maintain skillsthrough Math Box problems.

Study Link 3�9Math Masters, p. 93Students practice and maintain skillsthrough Study Link activities.

READINESS

Mystery Graph MatchMath Masters, p. 94per partnership: scissors, tape or glue, 1 or 2 sheets of paperStudents match mystery graphs with situations and the data they represent.

ENRICHMENTConstructing Mystery GraphsMath Masters, p. 95Students construct and analyze mystery graphs.

EXTRA PRACTICE 5-Minute Math5-Minute Math™, p. 198Students practice finding landmarks of a data set.

Teaching the Lesson Ongoing Learning & Practice

132

4

Differentiation Options

eToolkitePresentations Interactive Teacher’s

Lesson Guide

Algorithms Practice

EM FactsWorkshop Game™

AssessmentManagement

Family Letters

CurriculumFocal Points

Common Core State Standards

224_EMCS_T_TLG1_G6_U03_L09_576833.indd 224 2/9/11 1:15 PM

Matching Situations with GraphsLESSON

3�9

Date Time

Math Message

Match each situation with the graph that represents it.

1. Situation 1: A car is parked in a driveway. Graph

2. Situation 2: A car is traveling at a constant speed. Graph

3. Situation 3: A car leaves a highway tollbooth. Graph

4. Situation 4: A car approaches a red traffic light. Graph Graphs

Time

Spee

d

Graph A

Time

Spee

d

Graph B

Time

Spee

d

Graph C

Time

Spee

d

Graph D

5. Draw a graph to represent the following situation.

A woman walks up one side of a hill at a steady pace and runs down the other side. She then continues walking at a steady pace.

Time

Spee

d

�

A

Sample answer:

BD

C

EM3MJ1_G6_U03_82_122_new.indd 115 3/17/10 1:24 PM

Math Journal 1, p. 115

Student Page

Lesson 3�9 225

Getting Started

1 Teaching the Lesson

▶ Math Message Follow-Up WHOLE-CLASSDISCUSSION

(Math Journal 1, p. 115)

In Lessons 3-5 and 3-6, students used formulas and tables to construct graphs depicting various relationships—in particular, between time and speed. In this lesson, students use the shape of a graph to draw conclusions about the data or relationship the graph represents.

Discuss the graphs in the Math Message on journal page 115 and go over students’ answers. Have students complete the following statement about the relationship displayed by each of graphs A–D.

“As time passes, the speed of the car .” Graph A: decreases; Graph B: increases; Graph C: stays the same—it is zero; Graph D: stays the same.

Ask students to reword Problem 5 using a statement similar to what they used for Graphs A–D. For example, “As time passes, the speed at which the woman moves .” stays the same, then increases, then stays the same.

Ongoing Assessment: Journal page 115 �Recognizing Student Achievement

Use journal page 115 to assess students’ abilities to analyze the shape of a graph and draw conclusions about data trends. Students are making adequate progress if they are able to match the situations and graphs. Some students may be able to draw the graph for Problem 5. [Data and Chance Goal 2]

Mental Math and Reflexes Remind students that there are many real-world situations that can be modeled by positive and negative numbers. Give them real-world contexts that can be modeled with signed numbers. Have them write the signed numbers on their slates and order them from least to greatest. Suggestions:

-2°C, -9°C, 3°C, -1°C -9, -2, -1, 3 $10 profit, no profit or loss, $25 loss, $40 loss, $20 profit -40, -25, 0, 10, 20

8 miles per hour above the speed limit, 3 miles per hour below the speed limit, 6 miles per hour below the speed limit -6, -3, 8

Math MessageComplete the problems on journal page 115.

Study Link 3�8 Follow-UpDiscuss students’ responses. Students may describe the rule in Problem 11 as Subtract 6 from the in number to get the out number. Note that x - 6 = y is not one of the formulas given in Problem 11b. Students may recognize that subtracting 6 is the same as adding -6.

Mathematical PracticesSMP1, SMP2, SMP3, SMP4, SMP6Content Standards6.NS.7b

225-229_EMCS_T_TLG1_G6_U03_L09_576833.indd 225 3/19/12 8:50 AM

LESSON

3�9

Name Date Time

A Time Story

Satya runs water into his bathtub. He steps into the tub, sits down, and bathes. He gets out of the tub and drains the water. The graph shows the height of the water in the tub at different times.

Time

Hei

ght o

f Wat

er in

Tub

Math Masters, p. 92

Teaching Master


▶ Constructing a Graph

WHOLE-CLASS ACTIVITY

from a “Time Story”(Math Masters, p. 92)

Introduce this activity by telling a simple time story. Use the transparency of Math Masters, page 92 or draw the graph on the board. Read the story and explain that the graph shows the height of the water in the tub at different times. Students might recall this problem from their work with mystery graphs in Fifth Grade Everyday Mathematics.

Tell a new version of the story to the class: Satya runs water into his bathtub. He steps into the tub, sits down, and begins to bathe. The water in the tub begins to cool, so he drains some water and adds more hot water. Satya finishes bathing, gets out of the tub, and drains the water.

As a class, draw a time graph that illustrates the story you just told. Include as many details as possible.

A simple time-graph solution and a more complex version are shown in the margin. The graph on the left focuses on the main features of the story. The graph on the right adds details, such as changes in the height of water when Satya’s feet enter it and when he moves during the bath.

▶ Exploring Time Graphs PARTNER ACTIVITY


Give students time to complete journal page 116.

Ask students to share the stories they wrote to explain the graph in Problem 1. Sample answer: Mr. Olds begins the trip in a 20-mph zone. He stops for a stop sign at 2 minutes. He resumes driving in a 30-mph zone. At 3 1 _ 2 minutes, he begins to slow down for a curve in the road or a slower car. Six minutes into his trip, Mr. Olds stops at a traffic light. He resumes driving in a 30-mph zone. At 9 minutes, he speeds up to pass a car. He slows to 20 mph at 11 1 _ 2 minutes, and at 13 minutes he reaches the school.

For Problem 2, answers vary, depending on the amount of detail students decide to include. A simple time graph and a more complex version that illustrates Monica’s individual sips from the cup are shown in the margin.

PROBLEMBBBBBBBBBBBOOOOOOOOOOOBBBBBBBBBBBBBBBBBBBBBBBBB MMMMMEEEEMMMBLEBLLELBLLLLBLEBLEBLEBLEBLEBLEBLEEEEMMMMMMMMMMMMMMOOOOOOOOOOBBBBBBBLBLBLBBLBLLLLLLPROPROPROPROPROPROPROPROPROPRORPPRPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPROROROROROOROOPPPPPPP MMMMMMMMMMMMMMMMMMMEEEEEEEEEEEELLELEEEEEEEEEELLLLLLLLLLLLLLLLLLLLLRRRRRRRRRRRRRRRPROBLEMSOLVING

BBBBBBBBBBBBBBBBBBB ELEELEEMMMMMMMMMOOOOOOOOOOBBBLBLBLBBLBBBLOOORORORORORORORORORORORO LELELELEEEEEELEMMMMMMMMMMMLEMLLLLLLLLLLLLLLLLLLLLLRRRRRRRRRRGGGGLLLLLLLLLLLLLVVINVINVINVINNNNVINVINVINNVINVINVINVINV GGGGGGGGGGGOLOOOLOOLOLOLOO VVINVINLLLLLLLLVINVINVINVINVINVINNVINVINVINVINVINVINNGGGGGGGGGGOOOLOLOLOLOLLOOOO VVVLLLLLLLLLLLVVVVVVVVVOOSOSOSOOSOSOSOSOSOSOOSOSOSOSOOOOSOOSOSOSOSOSOSOSOOSOSOSOSOSOSOSOSOSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS VVVVVVVVVVVVVVVVVVVVVLLLLLLLVVVVVVVVVLVVVVVVVLLLLLLLLVVVVVLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLSSSSSSSSSSSSSSSSSSSSSS GGGGGGGGGGGGGGGGGOOOOOOOOOOOOOOOOOO GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGNNNNNNNNNNNNNNNNNNNNNNNNNIIIIIIIIIIIIIIIIIIISOLVING

PROBLEMBBBBBBBBBBBOOOOOOOOOOOBBBBBBBBBBBBBBBBBBBBBBBBBBB MMMMMBLEBLBLBLBLEBLELLLLLBLEBLEBLEBLEBLEEEMMMMMMMMMMMMMOOOOOOOOOOBBLBLBLBBLBLBLLLLLPROPROPROPROPROPROPROPROPROPROPROPRPRPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPROROROOROROROOOPPPPPPP MMMMMMMMMMMMMMMMMMMEEEEEEEEEEEEELEEEELELEEEEEELLLLLLLLLLLLLLLLLLLRRRRRRRRRRRRRRRRRPROBLEMSOLVING

BBBBBBBBBBBBBBBBBBBB EEELEMMMMMMMMOOOOOOOOOBBBLBLBBLBLBLBBOOOOROROROROROROROROROO LELELELEEEEEELEMMMMMMMMMMMMLEMLLLLLLLLLLLLLLLLLLLLRRRRRRRRRRRGGGLLLLLLLLLLLLLVINVINVINVINVINVINNNVINVINVINNVINVINVINVINV GGGGGGGGGGGOOOOLOLOOLOOLOO VINVINVVLLLLLLLLLVINVINVINVINVINVINVINVINVINVINVINVINVINV NGGGGGGGGGGOOOLOLOLOLOLOLOOO VVVVVVLLLLLLLLLLVVVVVVVVVOOSOSOOSOSOSOSOSOSOSOOSOSOSOSOOOSOOOSOSOSOSOSOSOOOSOSOOSOSOSOSOSOSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS VVVVVVVVVVVVVVVVVVVVVLLLLLLVVVVVVVVLLLLVVVVVVVLLLLLLLLVVVVVLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLSSSSSSSSSSSSSSSSSSSSSSS GGGGGGGGGGGGGGGGGGOOOOOOOOOOOOOOOOOOOO GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGNNNNNNNNNNNNNNNNNNNNNNNNNNIIIIIIIIIIIIIIIIIIISOLVINGTime

Hei

ght o

f Wat

er in

Tub

Time

Hei

ght o

f Wat

er in

Tub

Time

Leve

l of C

ocoa

in C

up

Time

Leve

l of C

ocoa

in C

up

EM3cuG6TLG1_225-229_U03L09.indd 226 12/13/10 8:05 AM

Mystery GraphsLESSON

3�9

Date Time

Each of the graphs at the right represents one of the situations described below. Match each situation with its graph.

1. The number of people at a school is best described by Graph .

2. The number of people in a restaurant is best described by Graph .

3. The number of people who are at home is best described by Graph .

4. The number of people in a hospital is best described by Graph .

5. The number of people driving a car is best described by Graph .

6. For one of the situations above, explain why you chose to match that situation with a particular graph.

Sample answer: For Problem 3, most people are at home early in the morning and in the evening. During the day, they go to work and school. Graph A shows this pattern.

D

E

A

C

B

6 A.M. noon 6 P.M. midnight

Graph E

Num

ber o

fPe

ople


Graph D

Num

ber o

fPe

ople


Graph C

Num

ber o

fPe

ople


Graph B

Num

ber o

fPe

ople


Graph A

Num

ber o

fPe

ople


Student Page

Time GraphsLESSON

3�9

Date Time

1. Mr. Olds drove his son Hank to school. The trip from home to school took 13 minutes. The graph below shows the speed that Mr. Olds traveled as he drove. Write a story that explains the shape of the graph.

Sample answer given in Teacher’s LessonGuide, page 226.

2. Monica filled a cup with cocoa. She drank almost half of it. She refilled the cup. Then she drank all of the cocoa in the cup. Draw a graph that illustrates this story.

00

10

20

30

40

1 2 3 4 5 6 7 8 9 10 11 12 13Time (min)

Spee

d (m

ph)

Sample answer:

Time

Leve

l of C

ocoa

in C

up


Student Page

Adjusting the Activity

Lesson 3�9 227

▶ Interpreting Mystery Graphs

INDEPENDENT ACTIVITY


Five graphs are shown on journal page 117. Ask students to match each situation described on the left side of the page with the appropriate graph on the right side of the page. In each graph, time is represented on the horizontal axis, and the number of people is represented on the vertical axis.

Ask students to complete the following statement about the relationship displayed by each graph: “As time passes, the number of people .”

For Graph D, the completed statement might be, “As time passes, the number of people briefly increases, then decreases, increases, and finally decreases again.”

After students have completed the relationship statement for each graph, they can look for the situation that would match that statement.

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

When most students have completed the page, discuss answers.

� Graph A represents Situation 3. Typically, the greatest number of people will be at home during the night. Between 6 A.M. and 12 noon, people are leaving for work, school, or errands. Between 12 noon and 6 P.M., people are arriving home.

� Graph B represents Situation 1. A typical school day goes from about 9 A.M. to 3 P.M. There is a fairly constant number of people in a school between those hours. The early arrival and late departure of the staff account for the gradual increase of people before 9 A.M. and the gradual decrease of people after 3 P.M.

� Graph C represents Situation 2. The number of people in a restaurant is greatest around 8 A.M., 12 noon, and 6 P.M.—typical times for breakfast, lunch, and dinner.

� Graph D represents Situation 5. The greatest number of people driving corresponds to the times that most people commute to and from work. Morning rush hour is a concentrated amount of time, while the evening rush hour is more spread out to account for those who work overtime.

� Graph E represents Situation 4. The number of patients in a hospital stays about the same. However, the increase in the number of people in the early evening could be due to visitors arriving. The decrease in the late evening could be due to fewer hospital employees working the night shift.


Math Boxes LESSON

3�9

Date Time

5. List the first 6 multiples of each number.

8

16

24

Name the least common multiple (LCM) of 8, 16, and 24.

48

8, 16, 24, 32, 40, 48

24, 48, 72, 96, 120, 14416, 32, 48, 64, 80, 96

1. The perimeter of a rectangle can be foundby the formula

P � 2 � (b � h).

Find the perimeter (P ) if b � 5.5 cm andh � 3.25 cm. Fill in the circle next to thebest answer.

A. 3.77 cm B. 7.60 cm

C. 8.75 cm D. 17.50 cm

2. Give 3 special cases for the generalpattern below.

f � m2 � f � (m � m)

3 � 52 � 3 � (5 � 5)

2.7 � 12 � 2.7 � (1 � 1)4 � (�

12�)2 � 4 � (�

12� � �

12�)

4. Rename each percent as a decimal.

a. 81% �

b. 7% �

c. 100% �

d. 70% � 0.701.000.070.81

b

h

212 103

7859 60

Sample answers:

254

3. Complete the table. Then graph the data and connect the plotted points.

Eddie travels about 8 miles per hour on his bike.

Rule: Distance traveled � 8 � number of hours

0

60

50

40

30

20

10

0 1 2 3 4 5 6 7Time (hr)

Dis

tanc

e (m

i)Time in Distance inhours (h) miles (8 � h)

0 0123 245

5640

816

7


Student Page

Study Link Master

STUDY LINK

3�9 Ferris Wheel Time Graph

140

Name Date Time

The time graph below shows the height of Rose’s head from the ground as she rides a Ferris wheel. Use the graph to answer the following questions.

00102030405060

Time (sec)

Hei

ght (

ft)

20 40 60 80 100

120

140

160

180

200

220

240

260

280

300

320

340

1. Explain what is happening from 0 to 95 seconds. People are getting on the Ferris wheel.

2. How long is Rose on the Ferris wheel before she is back to the position from which she started? About

(unit)

3. After the Ferris wheel has been completely loaded, abouthow long does the ride last before unloading begins?

(unit)

4. After the Ferris wheel has been loaded, how many timesdoes the wheel go around before unloading begins?

(unit)

5. When the ride is in full swing, approximately how longdoes one complete revolution of the wheel take?

(unit)

6. Rose takes another ride.After 130 seconds, the Ferris wheel comes to a complete stop because of an electrical failure. It starts moving again 2 minutes later. Complete the graph to show this event.

40 sec

4 times

170 sec

125 sec

0 20

0102030405060

40 60 80 100

120

140

160

Time (sec)

Hei

ght (

ft)

180

200

220

240

260

280

300

320

340

Try This

Sample answer:

Math Masters, p. 93


2 Ongoing Learning & Practice

▶ Playing Scientific Notation Toss PARTNER ACTIVITY

(Student Reference Book, p. 331; Math Masters, p. 472)

Distribute 2 six-sided dice to each pair of students, as well as a game record sheet (Math Masters, page 472). Have students read the directions on page 331 in the Student Reference Book. Ask a volunteer to demonstrate how the game is played. Encourage students to play a practice game.

▶ Math Boxes 3�9



Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 3-6. The skills in Problems 4 and 5 preview Unit 4 content.

▶ Study Link 3�9


(Math Masters, p. 93)

Home Connection Students answer questions about a graph displaying the relationship between the distance of a Ferris wheel passenger from the ground and the time the passenger spends on the ride.

NOTE After completing Math Boxes, Problem 4, you might have students identify where their answers would appear on a number line.


LESSON

3�9

Name Date Time

Matching Events, Tables, and Graphs

Cut out the situations, tables, and graphs. After you match each situation with one table andone graph, tape or glue them onto a separate sheet of paper. When you are finished, youwill have one table and one graph left over.

A fern was growing rapidly in its pot fora while until it didn’t get enough water.The fern then stopped growing.

A fern was growing slowly in its pot dueto a lack of sunlight. When the fern wasmoved to a nearby windowsill, it beganto grow more rapidly.

A fern was growing rapidly in its pot fora while until it was knocked over and adog bit off the top. It stopped growing fora while before it eventually began togrow again.

Table 1

Week Height

1 4 in.2 6 in.3 8 in.4 9.5 in.5 7 in.6 7 in.7 8 in.

Table 3

Week Height

1 4 in.2 4.5 in.3 5 in.4 5.5 in.5 6 in.6 8 in.7 10 in.

Table 4

Week Height

1 4 in.2 6 in.3 8 in.4 10 in.5 12 in.6 12 in.7 12 in.

Table 2

Week Height

1 4 in.2 5 in.3 6 in.4 7 in.5 8 in.6 9 in.7 10 in.

Time

Hei

ght

Graph A

Time

Hei

ght

Graph B

Time

Hei

ght

Graph C

Time

Hei

ght

Graph D

Math Masters, p. 94

Teaching Master

LESSON

3�9

Name Date Time

Mystery Graphs

Make a mystery graph on the grid below. Be sure to label the horizontal and vertical axes. Describe the situation that corresponds to your graph on the lines provided.

Answers vary.

Math Masters, p. 95

Teaching Master

Lesson 3�9 229

3 Differentiation Options

READINESS PARTNER ACTIVITY

▶ Mystery Graph Match 5–15 Min


Students practice analyzing graph shapes by matching a graph with the situation and data table that it represents. Provide each pair of students with a copy of Math Masters, page 94, scissors, tape or glue, and one or two sheets of paper.

When all students have finished, review the following answers as a group.

� A fern was growing rapidly in its pot for a while until it didn’t get enough water. The fern then stopped growing. Table 4; Graph B

� A fern was growing slowly in its pot due to a lack of sunlight. When the fern was moved to a nearby windowsill, it began to grow more rapidly. Table 3; Graph D

� A fern was growing rapidly in its pot for a while until it was knocked over and a dog bit off the top. It stopped growing for a while before it eventually began to grow again. Table 1; Graph A

� Table 2 and Graph C are left over when students are finished. Encourage students to write a situation that would match Table 2; Graph C. Sample answer: A fern received the correct amount of sunlight, water, and nutrients; it grew at a constant rate.

ENRICHMENT


▶ Constructing Mystery Graphs 15–30 Min


To further explore graphing, have students construct original mystery graphs and describe the situations the

graphs represent on Math Masters, page 95. Ask students to analyze their peers’ graphs and determine the situations.

EXTRA PRACTICE

SMALL-GROUP ACTIVITY

▶ 5-Minute Math 5–15 Min

To offer more practice finding landmarks of a data set, see 5-Minute Math, page 198.


reading and drawing graphs - mcgraw-hill education · reading and drawing graphs ... lesson guide...

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