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Lesson Guide

Algorithms Practice

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CurriculumFocal Points

Common Core State Standards

Lesson 1�11 71

Advance Preparation

Teacher’s Reference Manual, Grades 4–6 pp. 168, 169

Key Concepts and Skills• Apply place-value concepts to read and

round large numbers. [Number and Numeration Goal 1]

• Use a calculator to compute products and quotients of whole numbers. [Operations and Computation Goal 2]

• Interpret a pictograph and broken-line graph. [Data and Chance Goal 2]

• Analyze examples of inaccurate or misleading displays of data. [Data and Chance Goal 2]

Key ActivitiesStudents discuss how statistics can be presented in specific ways meant to astound the reader. They analyze a pictograph that displays incorrect and misleading information and compare broken-line graphs to decide which one is most persuasive.

Ongoing Assessment: Informing Instruction See page 74.

Ongoing Assessment: Recognizing Student Achievement Use journal page 38. [Data and Chance Goal 2]

MaterialsMath Journal 1, pp. 36 –38Study Link 1�10calculator

Playing High-Number Toss: Decimal VersionStudent Reference Book, p. 324Math Masters, p. 455per group: 4 each of number cards 0 –9 (from the Everything Math Deck, if available)Students read, write, and compare decimal numbers through thousandths.

Math Boxes 1�11Math Journal 1, p. 39straightedgeStudents practice and maintain skillsthrough Math Box problems.

Study Link 1�11Math Masters, p. 34Students practice and maintain skillsthrough Study Link activities.

ENRICHMENTCreating Persuasive GraphsMath Masters, p. 35Students create persuasive graphs to make a point or advance a cause.

EXTRA PRACTICE 5-Minute Math5-Minute Math™, pp. 19 and 95Students practice rounding numbers and estimating products.

Teaching the Lesson Ongoing Learning & Practice

132

4

Differentiation Options

Persuasive Dataand Graphs

Objective To analyze data displays and explain ways in which data can be presented to misrepresent or mislead.d

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Statistics Meant to Astound the ReaderLESSON

1�11

Date Time

Each day we eat the equivalent of 90 football fields covered with pizza!The National Association of Pizza Operators reported today that

p

Ninety acres may seem like a tremendous amount of pizza. The person who wrote this headline wants us to think so. However, let’s look at this statistic more closely.

There are 43,560 square feet in 90 ∗ 43 ,560 = 3, 920 , 400an acre, so 90 acres is about Round to 3 , 900 ,000 .3,900,000 square feet of pizza.

If Americans eat 3,900,000 square 3 , 900 ,000 ∗ 365 = 1 , 423 ,500 ,000feet of pizza each day for 365 days, Round to 1 , 420 ,000 ,000 .that is about 1,420,000,000 square feet of pizza per year.

If 1,420,000,000 square feet of pizza 1 , 42 0,0 00 ,000 / 270, 000 ,000 = 5 .2 œ5œ9œis divided by about 270,000,000 people Round to 5 .in the United States, then each person, on average, eats about 5 square feetof pizza per year.

Suppose an average pizza is about 1 square foot in area. Then each person in the United States eats approximately 5 pizzas per year.

Here is a new headline based on the information above.

An Average American Eats 5 Pizzas per Year!The National Association of Pizza Operators reported today that

1. Study the headline below.

An Average American Takes about 50,000 Automobile Trips in a Lifetime!

Write a new headline that gives the same information but will not astound the reader.Sample answer: An Average American Takes 2 Car Trips per Day!

EM3cuG6MJ1_U01_1-44.indd 36 1/11/11 5:30 PM

Math Journal 1, p. 36

Student Page

72 Unit 1 Collection, Display, and Interpretation of Data

Getting Started

Math MessageEstimate the following amounts before looking at journal page 36. Answers vary.

• the number of people living in the United States• the number of pizzas the average person eats in one year• the number of times each day that the average person drives his or her car

Study Link 1�10 Follow-UpBriefly go over the answers. Students may have noticed in Problems 3 and 4 that the smaller the difference between the length and width of a rectangle, the smaller the perimeter of that rectangle. Ask students to name the dimensions (in whole units) of a rectangle with the smallest perimeter when A = 30 units2; A = 40 units2; A = 100 units2. 5 units by 6 units (P = 22 units); 5 units by 8 units (P = 26 units); 10 units by 10 units (P = 40 units)

Mental Math and Reflexes Dictate real-world comparisons and have students write an inequality to represent each one. Suggestions:

A $170 profit is greater than a $145 profit. $170 > $145

–23°C is colder than –11°C. –23°C < –11°C

9 feet below sea level is lower than 2 feet above sea level. –9 ft < 2 ft

A quiz score 3 points above the median is higher than a score 1 point below the median. 3 > –1

1 Teaching the Lesson

▶ Math Message Follow-Up

WHOLE-CLASS ACTIVITY

(Math Journal 1, p. 36)

The headline “Americans Consume 90 Acres of Pizza per Day!” is meant to convince the reader that Americans eat great quantities of pizza. Since some readers may not be able to visualize the size of an acre, the subhead is rephrased in terms of a football field (which has a surface area of about 1 acre) to ensure that all readers get the point.

Working as a class, read through the analysis of the headline on journal page 36. To support English language learners, write acre on the board and discuss its meaning.

1. Acres are converted to square feet. The headline can be converted to “Americans Consume 3,900,000 Square Feet of Pizza per Day!”

2. This quantity is multiplied by 365 to get a yearly consumption figure. The headline can be converted to “Americans Consume 1,420,000,000 Square Feet of Pizza per Year!”

3. This quantity is divided by the number of people in the United States (about 270,000,000) to get yearly consumption per person. The headline can be converted to “Each American Consumes about 5 Square Feet of Pizza per Year!”

4. Square feet are converted to a number of pizzas, assuming that each pizza is about 1 square foot in area. The headline can finally be converted to “Each Person Eats about 5 Pizzas per Year!”

ELL

Links to the FutureExpect that some students will recognize the bar over the decimal number (Math Journal 1, page 36) as a notation for a repeating decimal. Rational numbers are discussed in Unit 6. Although identification of numbers as rational or irrational is not a Grade 6 Goal, working with these numbers provides students a foundation for seventh- and eighth-grade Number and Numeration concepts.

Mathematical PracticesSMP2, SMP3, SMP4, SMP6Content Standards6.RP.3d, 6.NS.7b, 6.SP.5bBold SMP = Guiding Questions at everydaymathonline.com Bold = Focus of lesson

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Analyzing Persuasive GraphsLESSON

1�11

Date Time

1. What is mathematically wrong with the pictograph above?

by all the other conditions and circumstances combined.only a little greater than the total number of deaths causedtotal number of deaths caused by cardiovascular disease isall other diseases and circumstances combined; actually, theby this disease total many times more than deaths caused bycardiovascular disease gives the impression that deaths causedonly slightly taller than the Chronic icon. Also, the icon forobstructive pulmonary disease. However, the Cancer icon iscancer caused more than 5 times as many deaths as chronicproportional to the numbers of deaths they represent. Note thatSample answer: The heights of the disease icons are not

Math Journal 1, p. 37

Student Page

Lesson 1�11 73

This final headline doesn’t have the same effect as the original. Both provide the same information, but the way in which information is presented can make a big difference in how readers react.

Assign Problem 1 on the journal page. Have students work with partners or in groups for about 5 minutes. Allow them to use calculators. Regroup to discuss solution strategies and revised headlines. Possible strategies:

� Assume an average lifetime of about 70 years. Divide total lifetime trips by 70 to estimate the number of trips per year: 50,000 / 70 is about 714. Divide again by 365 to estimate the number of trips per day: 714 / 365 is about 1.96, or about 2 trips per day.

� Multiply 70 by 365 to find the number of days in a typical lifetime: 70 ∗ 365 = 25,550. Round to 25,000 days. Observe that 50,000 total trips in 25,000 days works out to be an average of 2 trips per day.

▶ Analyzing a PARTNER ACTIVITY

Persuasive Pictograph(Math Journal 1, p. 37)

Graphs can present information in misleading ways. Some graphs are deliberately designed to support a point of view or a course of action. Many times, incorrect graphs are not intended to be misleading. Nevertheless, the reader should always be on the lookout. Discuss the meanings of persuasive and misleading. Be sure that students understand what a pictograph is.

The pictograph on journal page 37 is a good example of a graph that is both mathematically incorrect and misleading. Have students work in pairs to analyze the pictograph and find its errors. Circulate and assist as needed.

Health Link Follow up with a discussion:

� The large figure in the graph represents the number of deaths from cardiovascular disease. The smaller figures that stand one on top of another represent the number of deaths from other leading causes. To support English language learners, discuss the meanings of the words describing various causes of death, such as cardiovascular disease and pneumonia.

� Deaths from cancer (about half a million) are a little more than half the number of deaths resulting from cardiovascular disease (a little less than 1 million). The figure representing deaths from cancer should be a little more than half as tall as the large figure. The figure representing cancer deaths is much too small.

� Similarly, the figures representing deaths from other causes are much too large in relation to the figure representing deaths from cancer.

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�

Analyzing Persuasive Graphs continuedLESSON

1�11

Date Time

The scales for the vertical axes are different.

same basic information.

1 2 3 4 5 6 7 8 9 10

4:00

3:55

3:50

3:45

3:40

3:35

3:30

3:25

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Graph B

1 2 3 4 5 6 7 8 9 10

7

6

54

3

2

1

0

Graph A

Sample answers:

�

convincing because it is a more dramatic Sample answer: Graph B will be more

representation of the data.

You are trying to convince your parents that you deserve an increase in your weeklyallowance. You claim that over the past 10 weeks, you have spent more time doing jobsaround the house, such as emptying the trash, mowing the lawn, and cleaning up afterdinner. You have decided to present this information to your parents in the form of agraph. You have made two versions of the graph and need to decide which one to use.

2. How are Graph A and Graph B similar?

3. How are Graph A and Graph B different?

4. Which graph, A or B, do you think will help you more as you try to convince yourparents that you deserve a raise in your allowance? Why?

Both graphs display the

Math Journal 1, p. 38

Student Page

Math Boxes LESSON

1�11

Date Time

1. From 1860 to 1861, 11 states seceded from the Union. All of them were reinstated between 1866 and 1870. Four new states joined the Union between 1860 and 1866. The table at the left shows the number of states in the United States from 1859 to 1870. Make a step graph

to display this information.

2. Use the circle graphs to answerthe following questions.

a. About what percent of thecontent of bread is water?

33% b. About what percent of the

content of a ripe tomato is water?

95%

3. Juan’s mean score on three tests was 86. Two of his scores were 92 and 82. What was his third score? Circle the best answer.

A. 76 B. 78

C. 80 D. 84

4. Write the value of the digit 3 in each numeral below.

8.43 3 hundredths 24.35 3 tenths 149.073

Number Years of States

1859 33 1860 32 1861–1862 23 1863 24 1864–1865 25 1866 26 1867 27 1868–1869 34 1870 37

1865 18701860Year

20

25

30

35

40

Num

ber o

f Sta

tes

Bread Pineapple

Percent of Water in Foods

Ripe TomatoSource: Astounding Averages

water water water

145

28137

3 thousandths

EM3cuG6MJ1_U01_1-44.indd 39 4/29/10 11:51 AM

Math Journal 1, p. 39

Student Page

74 Unit 1 Collection, Display, and Interpretation of Data

The pictograph on the journal page actually appeared in a newspaper. Take this opportunity to reinforce the fact that readers should not automatically accept everything that appears in print as true. Encourage students to analyze and question representations of data, even if they appear to be authentic.

Ongoing Assessment: Informing InstructionWatch for students who have difficulty recognizing the discrepancies in the heights of the figures shown in the pictograph. Some students may also not see that the areas of the figures are inaccurate. The area of the large figure representing cardiovascular disease deaths is more than 20 times the area of the figure representing cancer deaths. The graph gives the visual impression that the number of deaths from cardiovascular disease is far greater than the number of deaths from cancer.

▶ Selecting a Persuasive PARTNER ACTIVITY

Broken-Line Graph(Math Journal 1, p. 38)

Circulate and assist while students complete journal page 38. Discuss their responses. To support English language learners, discuss the meaning of the word allowance. Stress that both graphs on the journal page are correct, but the presentation of the data makes a significant difference.

Ongoing Assessment: Journal �page 38

Problems 2 and 3Recognizing Student AchievementUse journal page 38, Problems 2 and 3 to assess students’ ability to read and interpret a broken-line graph. Students are making adequate progress if they indicate that the graphs display the same basic information. Some students may be able to articulate that because the increments along the vertical axis of Graph B are smaller, the graph appears to rise faster.

[Data and Chance Goal 2]

2 Ongoing Learning & Practice

▶ Playing High-Number Toss: PARTNER ACTIVITY

Decimal Version(Student Reference Book, p. 324; Math Masters, p. 455)

Students play High-Number Toss: Decimal Version to preview Unit 2 skills of reading, writing, and comparing decimal numbers through thousandths.

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STUDY LINK

1�11 The Population of River City

Copyright ©

Wright G

roup/McG

raw-H

ill

Name Date Time

The way a graph is constructed affects how fairly the data are represented.

The mayor of River City is trying to convince the city council that the city needsmore schools. She claims that the city’s population has doubled since 1998. The mayor used the graph below to support her claim.

1. According to the graph, what was the population in 2000?

2. Between which 2 years was the increase in population the least?

3. Is the mayor’s claim misleading? Explain.

Practice

4. $0.05 º 100 = 5. $0.90 º 100 =

6. $139.25 º 100 = 7. º 100 = $89.00

140

155

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165

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190

1998 1999 2000 2001 2002 2003 2004 2005

Year

Population of River City

Popu

latio

n (th

ousa

nds)

165,000

2003 and 2004

Sample answer: Yes. The population in 2005 would have to be 310,000 for the claim to be true.

$0.89$90.00$5.00

$13,925.00

EM3MM_G6_U01_001-040.indd 34 3/29/10 5:13 PM

Math Masters, p. 34

Study Link Master

LESSON

1� 11

Name Date Time

Persuasive Graphs

Create a persuasive graph for one of the two situations below or make up asituation of your own. You want to present your case in a way most favorable toyou and your cause. Be sure that your graph does not contain false information.Remember that you are merely presenting the information in a way that will be ofthe greatest benefit to you.

� You have been in charge of sales at the school store this year. Each monthprofits have increased—from $10 in September to $18 in June. You would liketo have the same job again next year and want to show your principal why youare the best candidate for the position.

� For health reasons, you have been encouraging your uncle to lose weight. Overthe past 8 weeks, he has gone from 300 pounds to 291 pounds. You are proudof your uncle and want to show him how much progress he has made.

Math Masters, p. 35

Teaching Master

Lesson 1�11 75

▶ Math Boxes 1�11

INDEPENDENT ACTIVITY

(Math Journal 1, p. 39)

Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 1-9. The skills in Problem 4 preview Unit 2 content.

▶ Study Link 1�11

INDEPENDENT ACTIVITY

(Math Masters, p. 34)

Home Connection Students analyze how data can be presented in such a way as to be misleading.

3 Differentiation Options

ENRICHMENT INDEPENDENT

ACTIVITY

▶ Creating Persuasive Graphs 15–30 Min

(Math Masters, p. 35)

Students create graphs that represent a given situation in the most favorable light without presenting false information. Provide time for students to share the graphs they created.

EXTRA PRACTICE

SMALL-GROUP ACTIVITY

▶ 5-Minute Math 15–30 Min

To offer more practice with rounding numbers and estimating products, see 5-Minute Math, pages 19 and 95.

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