158 Unit 3 Multiplication and Division; Number Sentences and Algebra

Teaching the Lesson materials

Key ActivitiesStudents discuss problems in which one quantity depends on another. They illustrate this kind of relationship between pairs of numbers with a function machine and a “What’s My Rule?”table. They solve “What’s My Rule?” problems.

Key Concepts and Skills• Solve addition and subtraction problems.

[Operations and Computation Goals 1 and 2]• Solve multiplication and division problems.

[Operations and Computation Goals 3 and 4]• Use rules to complete “What’s My Rule?” tables.

[Patterns, Functions, and Algebra Goal 1]• Use words and symbols to describe and write rules for functions.

[Patterns, Functions, and Algebra Goal 1]

Key Vocabularyfunction machine • input • output • rule • “What’s My Rule?”

Ongoing Assessment: Recognizing Student Achievement Use journal page 53.[Patterns, Functions, and Algebra Goal 1]

Ongoing Learning & Practice materials

Students design polygon letters.

Students practice and maintain skills through Math Boxes and Study Link activities.

Differentiation Options materials

Students use pattern blocksto investigate and describefunctional relationships.

Students apply the “What’sMy Rule?” concept to solvea perimeter problem.

Students complete “What’sMy Rule?” tables.

� Teaching Masters (Math Masters,pp. 73 and 74)

� Teaching Aid Master (Math Masters,p. 407)

� pattern blocks (triangles, squares,hexagons)

� calculator

EXTRA PRACTICEENRICHMENTREADINESS

3

� Math Journal 1, pp. 54 and 55� Study Link Master (Math Masters,

p. 72)� straightedge

2

� Math Journal 1, p. 53� Transparency (Math Masters,

p. 407)� slate� calculator (optional)

1

Objective To review “What’s My Rule?” problems.

Technology Assessment Management System

Journal page 53, Problems 1 and 2See the iTLG.

� Math Message Follow-Up(Math Masters, p. 407)

Have students compare their completed tables.

Display the function machine on the transparency of MathMasters, page 407. Remind students how a function machineworks:

� A number (the input) is dropped into the machine.

� The machine changes the number according to a rule.

� A new number (the output) comes out the other end.

The rule for the Math Message problem is multiply by 50. Write “ 50” in the function machine. To support English languagelearners, discuss that the word rule has an everyday usage, suchas a classroom rule, and a mathematical usage.

Point out the “What’s My Rule?” table in the Math Messageproblem. Ask:● What do the numbers in the in column represent?

Number of days● What do the numbers in the out column represent?

Average number of gallons of water used by one person in thatmany days

● How are the 2 in the in column and the 100 in the out columnrelated? 2 50 � 100

Tell students that in this lesson they will review variations offunction machines.

WHOLE-CLASS

ACTIVITY

1 Teaching the Lesson

Lesson 3�1 159

Getting Started

Math Message Each person in the United States uses about 50 gallons of water per day. Use this informationto complete the “What’s My Rule?” table.

Mental Math and Reflexes Pose multidigit addition and subtraction problems.Suggestions:

30 � 50 � 80 60 � 40 � 10090 � 20 � 70 80 � 40 � 4042 � 20 � 62 53 � 30 � 8356 � 10 � 46 75 � 20 � 5532 � 62 � 94 98 � 22 � 12066 � 41 � 25 76 � 25 � 51

in out(days) (gallons)

2 100

6 30010 500

30 1,500

365 18,250

Adjusting the ActivityHave volunteers pose questions.

For example:• If 8 is dropped into the function machine,

which number will come out? 400• If 600 comes out of the function machine,

which number was dropped in? 12AUDITORY � KINESTHETIC � TACTILE � VISUAL

in out

30 15

90 75

65 50

110 95

Rule

Subtract 15

Type 1

in out

7 49

2 14

9 63

600 4,200

pounds cost

1 $3.75

2 $7.505 $18.75

11 $41.25

Rule

Multiply by 7

Type 3

in out

250 350

20 120565 665

321 421

Rule

Add 100

Type 2

in out

54 9

42 724 4

600 100

Rule

Divide by 6

Type 4

RuleA pound of nuts costs $3.75

Try This

53

Date Time

“What’s My Rule?”LESSON

3 �1Complete the “What’s My Rule?” tables and state the rules.

1. 2.in out

30 6080 11020 50

150 180290 320

Rule

out

in

Add 30

in out

130 50

290 210

100 20

350 270

420 340

Rule

out

in

�80

3. 4. Rule: There are 12 inches in 1 foot.

Rule

out

in

�23

in out

49 72

151 174272 295

588 611

480 503

in out

17 �8

12 �1327 2

�5 �3025 0

in out

in out

3 36

5 60

10 12022 264

60 720

� �

5. Rule: 6. Create your own.

Rule:

�25

162–166

Answers vary.

Math Journal 1, p. 53

Student Page

� Reviewing Variations of the “What’s My Rule?” Routine(Math Masters, p. 407)

Demonstrate each type of “What’s My Rule?” table (see margin) onthe transparency of Math Masters, page 407.

� In Type 1, as in the Math Message problem, the rule and sample inputs are known, and the outputs must be determined.

� In Type 2, the rule and sample outputs are known, and theinputs must be determined.

� In Type 3, the inputs and outputs are known, and the rulemust be determined.

� In Type 4, some inputs and outputs are known, and themissing numbers and the rule must be determined.

To find the rule, students should use the pairs in which both thein and out numbers are given. Then students can use the rule tofill in the missing in and out numbers.

Pose problems like Type 4 to the class. Rules may be stated assimple statements, such as “Subtract 15,” or rules may be statedin a context like the problem in the margin (A pound of nuts costs$3.75) or as in the Math Message (50 gallons per day). Encouragestudents to supply both types of rules. Supplying a context for arule will be more difficult.

� Completing “What’s My Rule?” Tables(Math Journal 1, p. 53)

Students complete Problems 1 and 2 on their own. They work in partnerships to complete the remainder of the page. Have calculators on hand for students to use as necessary while solvingthe “What’s My Rule?” problems.

Ongoing Assessment:Recognizing Student Achievement

Use journal page 53, Problems 1 and 2 to assess students’ ability to use rulesto complete “What’s My Rule?” tables. Students are making adequate progress ifthey are able to correctly identify the in and out numbers when given the rule.Some students may be able to identify the rules in Problems 3 and 5 and usethese rules to complete the tables.

[Patterns, Functions, and Algebra Goal 1]

Journal

page 53 �Problems 1 and 2

PARTNER

ACTIVITY

WHOLE-CLASS

ACTIVITY

160 Unit 3 Multiplication and Division; Number Sentences and Algebra

� Identifying Polygon Properties (Math Journal 1, p. 54)

Students design polygon letters.

� Math Boxes 3�1(Math Journal 1, p. 55)

Mixed Practice Math Boxes in this lesson are linked with Math Boxes in Lessons 3-3 and 3-5. The skill inProblem 6 previews Unit 4 content.

Writing/Reasoning Have students write a response to thefollowing: Explain how you found the range of the data setin Problem 2. Sample answer: I subtracted the smallestnumber (16) from the largest number (25) to find therange (9).

� Study Link 3�1(Math Masters, p. 72)

Home Connection Students complete several types of“What’s My Rule?” problems.

INDEPENDENT

ACTIVITY

INDEPENDENT

ACTIVITY

INDEPENDENT

ACTIVITY

2 Ongoing Learning & Practice

54

A Polygon AlphabetLESSON

3�1

Date Time

Try reading this message:

1. Use a straightedge to design a polygon letter for each of the letters shown below. You’ll haveto simplify, because a polygon can’t have any curves, and it can’t have any “holes.”

For example, if you look at the letter “P,” you see that there is no opening in the upper part.Making it look like this, , would make it easier to read, but it would not be a polygon.

2. Which of the letters you drew are nonconvex (concave) polygons? B, C, F, M, and XHow do you know?

3. Do any of the letters you drew have special names as polygons? Explain.

4. On a separate sheet of paper, design polygon letters for the rest of the uppercase (capital)letters in the alphabet, the 26 lowercase (small) letters, or the 10 digits (0–9).

D is a hexagon. Sample answers: B is a pentagon. C is an octagon.

Sample answer: At least one vertex is pushed inward.

B C D

F M X

96 97

Sample answers:

Try This

Math Journal 1, p. 54

Student Page

55

Date Time

Math Boxes LESSON

3 �1

5. Complete.

a. 7, 15, 23, , ,

Rule:

b. 49, 42, , 28, ,

Rule:

c. , , 53, 59, , 71

Rule: �6654741

�7142135

�8473931

1. Write �, �, or � to make each numbersentence true.

a. 1 million 100,000

b. 73,099 71,999

c. 304,608 304,809

d. 5,682 7 hundred

e. 5,000,236 5,000,099�

�

�

�

�

3. Make a ballpark estimate. Write a numbermodel to show your strategy.

a. 3,389 � 2,712

� �

b. 3,452 � 1,147

� �2,4001,1003,500

6,0003,0003,000

4. Complete.

a. 21 ft � yd

b. 4 ft � in.

c. 16 ft � yd ft

d. 2 yd 2 ft � in.

e. ft in. � 568 in.44796

1548

7

6. Solve mentally or with a paper-and-pencilalgorithm.

a. $3.56 b. $6.25� $2.49 � $5.01

181 129

160 161 34–37

2. Number of spelling words correct for 10 students on the spelling test:

25, 19, 16, 25, 18, 19, 25, 24, 25, 23

a. What is the range for this set of

numbers?

b. What is the median? 23.59

6 149

Sample answers:

$6.05 $1.24

73

Math Journal 1, p. 55

Student Page

Lesson 3�1 161

STUDY LINK

3�1 “What’s My Rule?” 162–166

Name Date Time

Complete the “What’s My Rule?” tables and state the rules.

1. 2.

3. Rule: 4. Rule: 7�46

5. Rule: There are 20 nickels in $1.00.

6. Create your own.

Rule:

Answersvary.

Try This

7. � 47 � 68 8. 359 � 253 � 9. 787 � 653 � 1,440612115Practice

in out

20 60190 23070 110

240 280330 370

in out

110 50

80 20

310 250

240 180

390 330

in out

131 177

34 80

104 150

629 67554 100

in out

70 490

9 63

50 350

20 140

60 420

dollars nickels

3 60

2 40

5 100

20 400100 2,000

in out

Rule

out

in

Add 40Rule

out

in

�60

Math Masters, p. 72

Study Link Master

162 Unit 3 Multiplication and Division; Number Sentences and Algebra

� Modeling Functional Relationships with PatternBlocks(Math Masters, p. 73)

To explore the relationships between pairs of numbers in“What’s My Rule?” tables using a concrete model, havestudents determine the relationship between the number

of squares and triangles and the number of sides they have. Askstudents to share strategies for Problem 4.

� Solving a Perimeter Problem(Math Masters, p. 74)

To apply students’ understanding of functional relationships, have them explore the perimeter of shapescreated by placing square pattern blocks side by side.

Students record their data in a table and use the relationshipsbetween pairs of numbers to generate a rule for finding theperimeter of any shape made by n number of squares placed sideby side.

Problem 5 challenges students to explain the rule for finding theperimeter of shapes created by placing hexagon pattern blocks side by side.

NOTE Perimeter is defined as the distance around a closed 2-dimensionalshape. Square and hexagon pattern blocks are prisms, not 2-dimensionalpolygons, as the names imply. For this activity, have students consider only the square or hexagonal bases of the pattern blocks.

� Completing “What’s My Rule?” Tables(Math Masters, p. 407)

To practice using words and symbols to describe and write rulesfor functions, have students solve “What’s My Rule?” problems.Use Math Masters, page 407 to create problems to meet the needsof individual students, or have students create and solve their ownproblems.

5–15 Min

INDEPENDENT

ACTIVITYEXTRA PRACTICE

5–15 Min

PARTNER

ACTIVITYENRICHMENT

5–15 Min

PARTNER

ACTIVITYREADINESS

3 Differentiation OptionsLESSON

3�1

Name Date Time

“What’s My Rule?” Polygon Sides

162–166

1. Use square pattern blocks to help you complete the table.

2. Suppose there are 12 squares. Explain how to find the number of sides without counting.

3. Use triangle pattern blocks to help you complete the table.

4 sides. This equals 48 sides. (12 4 � 48)Sample answer: Multiply 12 squares by

4. Suppose there are 30 sides. Explain how to find the number of triangles without counting.

equals 10 triangles. (30 3 � 10)Sample answer: Divide 30 sides by 3. This

Number of Number ofSquares Sides

1 42 83 125 207 288 32

Number of Number ofTriangles Sides

1 3

2 6

5 15

4 12

3 9

6 18

Math Masters, p. 73

Teaching Master

LESSON

3�1

Name Date Time

“What’s My Rule?” Perimeter

2. Complete the “What’s My Rule?” table. Use square pattern blocks to create the shapes.

3. Explain the rule for finding the perimeter of the shapes.

squares by 2, then add 2.Multiply the number of Sample answer:

The distance around a shape is called its perimeter. The perimeter of a square pattern block is 4 inches.

1 in.

1 in.1 in

.

1 in.

1. Place 2 square pattern blocks side by side. What is the perimeter of the shape?

inches6

1 � 1 � 1 � 1 � 4

Number of Square Perimeter ofPattern Blocks Shape (inches)

1 4

2 63 84 105 126 147 168 18

4. Use your rule to complete the following: 214 square pattern blocks are placed side by side. What is the perimeter of the shape? inches430

Try This

5. Use words or symbols to explain the rule for finding the perimeter of shapesmade by placing hexagon pattern blocks side by side.

add 2. (h 4) � 2 Sample answer: Multiply the number of hexagons by 4, then

162–166131

Math Masters, p. 74

Teaching Master