lesson 83 pan-balance equations, part 1 · 446 unit 8 algebra concepts and skills overview in this...

446 Unit 8 Algebra Concepts and Skills

Overview

In this and the following lesson,students are introduced to a pan-balance approach for solving simpleequations. The task is to find theweight of a given object by adding orremoving the same number of objectsof equal weight from both pans.

At first, the objects in the pans areshown pictorially. In follow-upexamples, the objects are replaced by expressions with variables. Withpractice, students will learn equation-solving techniques that go beyondinspection and trial-and-error, beforereaching a formal algebra course.

L E S S O N

83Pan-BalanceEquations, Part 1

T E A C H E R ’ S R E F E R E N C E M A N U A LSee “Solving Equations” under the topic Algebra and Uses of Variables.

A D V A N C E P R E P A R A T I O NInspect your pan balance. Move or slide the fine-tuning adjustment so that the two pans balance when they are empty. It may be necessary to tapepaper clips to the underside of one pan to ensure that the two pans balance.

T E A C H E R M A T E R I A L S box of 100 standard 1 " paper clips7 identical ballpoint pens4 trapezoid-shaped pattern blocks

(or 4 quarters)pan balance

S T U D E N T M A T E R I A L SJournal 2, pp. 342–344Study Links 82 and 83calculatorslate, chalk, eraser

MATH MESSAGE Solve. You may use your calculator.• 17 Ç (Ç13) = 30 • Ç(122) = Ç144• Ç(6 + 3) + 15 = 6

Getting Started■ 5-Minute Math

Tell students that for this activity, a clockwise turn (to the right) will be called positive. A counterclockwiseturn (to the left) will be called negative.Ask students to stand where they have enough room toturn. After each turn, have students go back to theirstarting position.• Make a turn of positive 90 degrees. Turn 90 degrees

clockwise, or to the right.• Make a turn of negative 270 degrees. Everyone should

end up facing in the same direction as they did after thepreceding turn.

14

Lesson 83 447

• Girls make a turn of positive 180 degrees. Boys make aturn of negative 180 degrees. Everyone should face inthe opposite direction from their starting position.

• Girls make a turn of negative 60 degrees. Boys make aturn of positive 120 degrees. Boys and girls should facein opposite directions.

• Make a turn of negative 90 degrees. Turn 90 degreescounterclockwise, or to the left. What are two differentways of getting back to your starting position? Make aturn of positive 90 degrees or negative 270 degrees.

■ Study Link 82 Follow-upGo over the answers with the class. Encourage students to share how they arrived at their estimates for Problem 4.

■ Math Message Follow-upHave partners check each other’s answers. Work throughdifferences until both partners agree.

Instruction and Activities■ Demonstrate the pan-balance approach.

Whole-Class ActivityTwo pan-balance examples are suggested below. Moststudents should catch on quickly. Continue with similarexamples if students appear to have difficulty.Example 1:Show the trapezoid pattern blocks and paper clips andpoint out that the goal will be to determine the weight of a single pattern block in terms of paper clips. The patternblock may weigh from 10 to 14 paper clips. (If patternblocks are not available, use quarters instead.)Place 3 pattern blocks in the left pan and 1 pattern block in the right pan. Add paper clips to the right pan,counting as you go, until the pans balance. Ask students how they might change the contents of thepans to end up with 1 pattern block in one pan, balancedwith paper clips in the other pan. Tell them that there isone rule to follow: Whatever you do, the pans must alwaysremain balanced. Suppose, for example, that a student suggests thisapproach: “Take the single block from the right pan.Then take two of the three blocks from the left pan. Then remove clips from the right pan until the two pansbalance.” If you carry out these instructions, the classwill observe that the rule (the pans must always remainbalanced) is repeatedly violated.

Note

In this example, half of 21 clips is 10 clips.Break one clip in half and remove 10 clipsplus a half clip. Since the pan balance is not a precision instrument, two blocks mayappear to balance against 21 clips, while oneblock against 10 clips may show a slight tilt. 1

2

12

Add clips to balance the pans. This mightrequire 21 clips.

Removing one block from each pan willkeep the pans balanced.

Removing one-half of the objects from eachpan will keep the pans balanced.

Success!

Example 1

Have students test their solutions by actually using thebalance. If necessary, guide students through themanipulations on the pans.Example 2:Show the ballpoint pens and paper clips and explain thatthe goal will be to determine the weight of a single pen interms of paper clips.Place 5 pens and 10 paper clips in the left pan. Place 2 pens in the right pan. Then add clips to the right pan,counting as you go, until the pans balance. Ask students how they might change the pans to end upwith 1 pen in one pan that balances with paper clips in theother pan. Tell them that whatever they do to the contentsof the pans, the pans must always remain balanced. Carry out students’ suggestions for using the balance.Offer ideas if they need help:• You can add the same number of pens to each pan or

remove the same number of pens from each pan. Youcan add the same number of clips to each pan or removethe same number of clips from each pan.

• You can remove the same fraction of objects from eachpan. For example, if the left pan contains 3 pens and theright pan contains 30 clips, you can remove 2 pens fromthe left pan and 20 clips from the right pan.

Students test their solutions by actually using the balance.If necessary, guide students through the manipulations onthe balance.Show students that the manipulations can also berepresented with variables. (See below.) Let P stand forthe weight of 1 pen. Let C stand for the weight of 1 clip.


Add clips to balance the pans. This mightrequire 40 clips.

10

40

remove 10

Removing 10 clips from each pan will keepthe pans balanced.

30

Removing 2 pens from each pan will keepthe pans balanced.

30

remove or 20 clips–23

Removing of the objects from each pan will keep the pans balanced.

23

10C + 5P 2 P + 40C

remove 10 C

Removing 10 clips from each pan willkeep the pans balanced.

3 P 30 C

remove 2 P remove 20C

Removing from each pan will keep thepans balanced.

23

5 P 2 P + 30 C

remove 2P

Removing 2 pens from each pan will keep the pansbalanced.

P 10 C

Success!

Example 2

■ Solve pan-balance equations. (Journal 2, pp. 342 and 343)Partner Activity Have students work in pairs to complete journal pages 342and 343.Problems show objects, placeholders, (such as and ),or expressions with variables in the pans.If students have difficulty using placeholders or variables,use the chalkboard or an overhead transparency to reworkone of the examples. Use a sequence of pan-balancesketches to keep track of the operations instead of workingdirectly with the balance.Then operate with the pan-balance sketches and symbolsas if you were manipulating the actual balance. Similarly, help students to visualize expressions, suchas 5 and 3x, which appear on the journal pages. For example:

5 may be visualized in the balance pan as. (Problem 6)

3x may be visualized as xxx. (Problem 7)Problem 8 is challenging. Encourage students who at firstdo not succeed to return to the problem after completingLesson 84.

Lesson 83 449

Journal 2, p. 342

Solve these pan-balance problems. In each figure, the two pans are in perfect balance.

1. One cube weighs

as much as marbles.

2. One cube weighs

as much as oranges.

3. One orange weighs

as much as grapes.

4. One block weighs

as much as marbles.

Check your answers:

The sum of the answers to Problems 1 and 2 should equal 14 1—2.

The sum of the answers to Problems 3 and 4 should equal 25.

3

22

31—2

11

Pan-Balance Equations

Housing BoomSome might say that Parker Brothers (the games company)has built more houses than any other developer in the world.Since 1935, the company has “built” more than 3 billionhouses for its Monopoly® game.Source: Games Magazine.

A A A

A A

(half orange)

Journal 2, p. 343

Pan-Balance Equations (continued)


5. One weighs

as much as ’s.

6. One weighs

as much as marbles.

7. One x weighs

as much as balls.

Check your answers:



Challenge

8. An empty bottle weighs as much as 6 marbles.

The contents of a full bottle weigh

as much as marbles.

A full bottle weighs as

much as marbles.14

8

3

2

2

5

3x 1 ball x 7 balls

bothfull

halffull


Practice and Extension■ Math Boxes 83 (Journal 2, p. 344)

Math Boxes in this lesson are paired with Math Boxes in Lesson 85. The skill in Problem 1 is a prerequisite forUnit 9.

■ Study Link 83Students use representations of pan balances to solveproblems.

Study Link 83


1. One weighs

as much as .

2. One cube weighs

as much as marbles.

3. Two cantaloupes weigh

as much as apples.

4. One X weighs

as much as Y.

5. One B weighs

as much as M.3

4

36

3

3

Study Link 83: Pan-Balance Problems

10

( cantaloupe)12

4X 15Y 6X 7Y

3B 3M 1B 9M

Journal 2, p. 344

1. Rename each fraction as a mixednumber.

a.28—6 =

b.34—5 =

c.19—4 =

d.24—9 =

e.43—10 = 4 3—

10

2 6—9 or 2 2—

3

4 3—4

6 4—5

4 4—6 or 4 2—

3

2. Multiply or divide mentally.

a. 386 ÷ 4 ∑

b. 673 ÷ 9 ∑

c. 68 º 50 =

d. 299 º 15 =

e. 295 ÷ 5 = 59

4485

3400

74 R7

96 R2

3. Complete the “What’s My Rule?”table and state the rule.

4. Circle < or > to show whether theproduct is less than or greater than10,000.

a. 93 º 95 < > 10,000

b. 5 º 751 < > 10,000

c. 885 º 15 < > 10,000

d. 200 º 800 < > 10,000

e. 300 º 300 < > 10,000

5. If the hexagon on your Geometry Template represents 3, what is the value of each shape below?

a. b.

c. d. e. 21 1—2

1—2

11 1—2

Math Boxes 83

Ruleout = in – 8

in out

8 0

10 2

2 Ç6

0 –8

–1 Ç9

lesson 83 pan-balance equations, part 1 · 446 unit 8 algebra concepts and skills overview in this...

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