Pan-Balance Problems with Two Balances - Everyday Math · 10LESSON More Pan-Balance Problems 2...

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Lesson 10�2 791

Advance PreparationFor Part 1, use two pan balances for demonstration purposes. Alternately, the lesson includes instructions for a single pan balance.

Teacher’s Reference Manual, Grades 4–6 pp. 291–294

Key Concepts and Skills• Use addition and subtraction to solve

pan-balance problems. [Operations and Computation Goal 1]

• Use multiplication and division to solve pan-balance problems. [Operations and Computation Goal 3]

• Use a pan-balance model to solve linear equations with two unknowns. [Patterns, Functions, and Algebra Goal 2]

Key ActivitiesStudents solve pan-balance problems involving two balances.

Ongoing Assessment: Recognizing Student Achievement Use the Math Message. [Patterns, Functions, and Algebra Goal 2]

MaterialsMath Journal 2, pp. 336–338Study Link 10�1slate � for demonstration purposes: 100 standard 1" paper clips, 3 identical ballpoint pens, 3 identical compasses (pencils removed), 2 identical plastic 6" half-circle protractors, 1 plastic 12" ruler, 1 or 2 pan balances

Making and Interpreting Line PlotsMath Journal 2, p. 339Students construct and interpret aline plot.

Math Boxes 10�2Math Journal 2, p. 340 Students practice and maintain skillsthrough Math Box problems.

Study Link 10�2Math Masters, p. 297 Students practice and maintain skillsthrough Study Link activities.

READINESS

Solving Logic PuzzlesClass Data Pad (optional)Students solve logic puzzles related to pan-balance problems.

ENRICHMENTMeasuring TimeMath Masters, p. 298Students explore a sandglass model to solve a problem about time intervals.

Teaching the Lesson Ongoing Learning & Practice Differentiation Options

�

Pan-Balance Problemswith Two Balances

Objective To develop a pan-balance approach for solving sets of two equations with two unknowns.o

eToolkitePresentations Interactive Teacher’s

Lesson Guide

Algorithms Practice

EM FactsWorkshop Game™

AssessmentManagement

Family Letters

CurriculumFocal Points

Common Core State Standards

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More Pan-Balance ProblemsLESSON

10 �2

Date Time

Math Message

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

1. One block weighs

as much as 2 balls.

2. One ball weighs

as much as 5 marbles.

Solve these problems using both pan balances. In each problem, the pans are in perfect balance. The weights of objects, such as blocks, balls, marbles, and coins, are consistent within each problem.

3.

One block weighs One coin weighs

as much as 13 marbles. as much as 7 marbles.

4.

One block weighs One ball weighs


5.

One block weighs One ball weighs


333-368_EMCS_S_G5_MJ2_U10_576434.indd 336 2/22/11 5:20 PM

Math Journal 2, p. 336

Student Page

792 Unit 10 Using Data; Algebra Concepts and Skills

Getting Started

Ongoing Assessment: Math Message �

Recognizing Student AchievementUse the Math Message to assess students’ ability to solve equations using a pan-balance model. Students are making adequate progress if they correctly solve both problems. [Patterns, Functions, and Algebra Goal 2]

1 Teaching the Lesson

▶ Math Message Follow-Up WHOLE-CLASSDISCUSSION

(Math Journal 2, p. 336)

Draw the Math Message pan-balance problems on the board or a transparency. Have volunteers share their answers and solution strategies. Under the pan balances, write: One block weighs as much as how many marbles? Ask: Can you solve this problem using only one of the pan balances? No Discuss how the information from both pan balances needs to be combined to solve the problem.

� One block weighs as much as 2 balls. One ball weighs as much as 5 marbles. Therefore, one block weighs as much as 10 marbles.

� Because one ball weighs as much as 5 marbles, 4 balls weigh as much as 20 marbles. Therefore, 2 blocks weigh as much as 20 marbles, and 1 block weighs as much as 10 marbles.

▶ Demonstrating How to Solve

WHOLE-CLASS ACTIVITY

More Complex Pan-Balance ProblemsExplain that the class will look at two other examples where the information from two pan balances needs to be combined to solve the problems.

Mental Math and Reflexes Write each riddle on the board or prepare a transparency. Students use their slates to write the number and then write the algebraic equation for the riddle: What number am I?

• If you double me and add 6, you get 20. 7; n ∗ 2 + 6 = 20• If you add 5 to me, you get 0. -5; n + 5 = 0• If you double me, add 4, subtract 4, and then divide by 2, you get 9. 9;

(n ∗ 2 + 4 - 4) / 2 = 9

• If you take half of me and triple the result, you get 30. 20; n _ 2 ∗ 3 = 30, or n ∗ 1 _ 2 ∗ 3 = 30• If you double me and double the result, you get 0. 0; n ∗ 2 ∗ 2 = 0• If you multiply me by 100 and add 25, you get 525. 5; (n ∗ 100) + 25 = 525• If you multiply me by 1,000 and add 3, you get a result that is 1,000 times as great as

2.003. 2; (n ∗ 1,000) + 3 = 2.003 ∗ 1,000

Math Message �Answer Problems 1 and 2 on page 336 of your journal.

Study Link 10�1 Follow-UpHave partners compare answers and resolve differences. Then have volunteers share their solution strategies.

Mathematical PracticesSMP1, SMP2, SMP3, SMP4, SMP6, SMP8Content Standards5.NF.5a, 5.MD.2

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Lesson 10�2 793

Example 1:

Show the protractor, ruler, and paper clips. Tell students their goal is to determine the weight of the ruler and the protractor in terms of paper clips. (See margin.)

Set up the first pan balance. Place a protractor in one pan. Then add paper clips to the other pan, one at a time, counting as you go, until the pans balance. If you have only one balance available, sketch the results of the first pan-balance situation on the board before setting up the second balance.

Set up the second pan balance. Place a protractor and ruler in one pan. Then add paper clips to the other pan, one at a time, counting as you go, until the pans balance.

Remind students that a protractor weighs 31 paper clips in the first pan balance. All that remains is to find the ruler’s weight in paper clips.

Ask: How can you change the pans to isolate the ruler in one pan, balanced by paper clips in the other pan? Tell students their solutions must follow one rule: Whatever you do, the pans must always remain balanced. Here are two possible approaches:

� Remove the protractor and replace it with 31 clips. Because the protractor weighs 31 clips, the pans remain balanced. Now remove 31 clips from each side. The pans remain balanced. So the ruler weighs as much as 39 paper clips.

� Remove the protractor from one pan. At the same time, remove 31 clips from the other pan. Because the protractor weighs 31 clips, the pans remain balanced. The ruler is balanced by 39 clips.

Example 2:

Show the ballpoint pens, compasses, and paper clips. Tell students their goal is to find the weight of a pen and the weight of a compass in terms of paper clips.

Set up the first pan balance. Place 2 pens in one pan. Then add clips to the other pan, one at a time, counting as you go, until the pans balance. (See margin.) Set up the second pan balance. Place 2 compasses in one pan. Place 1 compass and 1 pen in the other, and add paper clips, one at a time, counting as you go, until the pans balance. (See margin.)

Have students use the balances to test their solutions.

� Remove 1 _ 2 of the objects from each pan of the first pan balance. The pans remain balanced. So 1 pen weighs 10 clips.

balance #1

� Remove 1 pen from the second pan balance and replace it with 10 clips. The pans remain balanced.

exchange

4210

balance #2

balance #1

Example 1

Add clips to balance the pans. For the example here, add 31 clips.

70balance #2

remove 3131

70balance #2

remove 31

balance #2

Add clips to balance the pans.This might require 70 clips.

Exchange the protractor for 31 clips.

balance #270

31

exchange

balance #1

Example 2


balance #2


Then remove 31 clips from each pan.

Or remove the protractor from onepan and 31 clips from the other.

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6.

One coin weighs One block weighs

as much as clips. as much as clips.

7.

One can weighs One doughnut weighs

as much as blocks. as much as blocks.

8.

One weighs One weighs

as much as marbles. as much as marbles.

9.

Each can weighs B ounces. Each cube weighs A ounces.

B � ounces A � ounces39

122

42

More Pan-Balance Problems continuedLESSON

10 �2

Date Time

5 4

doughnut12

5 marbles 9 marbles

B B 18 ounces A A AB BB B B


Student Page

More Pan-Balance Problems continuedLESSON

10 �2

Date Time

10.



11.

y weighs x weighs


12.

If the cup is full, the coffee in the cup weighs as much as marbles.

If the cup is full, the coffee plus the cup weighs as much as marbles.

13. Two pens weigh as much as one compass. One pen and one compasstogether weigh 45 grams.

Complete the pan-balance problems below. Find the weights of one pen and one compass.

One pen weighs grams. One compass weighs grams.3015

2618

25

3 3 6 marbles

7 marblesx y 8 marbles4 x y y

cup empty cup full1

2

2 pens 1 compass 45g1 pen and1 compass

4 6


Student Page

Adjusting the Activity


� Remove 1 compass from each pan. The pans remain balanced. So 1 compass weighs 52 clips.

balance #252

▶ Solving Pan-Balance Problems PARTNER ACTIVITY

(Math Journal 2, pp. 336–338)

Algebraic Thinking Have partners complete the journal pages. Some of the problems show pictures of objects in the balance pans; some show squares and triangles; and some show expressions with letter variables.

Explain that some problems consist of two related parts and that students need to solve one of the parts before they have enough information to solve the other part. For example, in Problem 3, students need to complete the statement associated with the second pan balance before they can complete the statement associated with the first pan balance.

� Problem 3

Complete the second statement first. One coin weighs as much as 7 marbles. Since 1 coin weighs as much as 7 marbles and 1 block and 1 coin weigh as much as 20 marbles, 1 block must weigh as much as 13 marbles. Remove the coin and 7 marbles from the pans of the first pan balance.

Draw a new pan balance underneath the one on the board to illustrate each step. Refer to the Math Message Follow-Up, and replace each ball with 5 marbles.

Then draw another pan balance underneath to show each side of the balance divided in half. Cross off one half on each side.

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

Encourage students to begin by deciding which of the two statements should be completed first. For example, in Problem 3, the second statement needs to be completed first. However, in Problem 5, the first statement should be completed first.

Circulate and assist. A hint for Problem 12 on journal page 338 is to determine the weight of the coffee separate from the cup.

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ELL

Adjusting the Activity

Have students write equations with variables to represent each of the pan-balance problems or match the problems to a prepared list of equations. They can record the correct equation above the illustration for each problem.

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


01 1 _ 2

1

1

9 9 _ 16

1 0 5 _ 8 1

_ 4 1 _ 2 1

_ 2 1 7 _ 8

1 _ 4 3

_ 4 1 1 _ 4 3

_ 8 1 _ 2 1 1

_ 8

A full set of fraction cards was shuffled. Then 16 cards were dealt out showing the following fractions:

10 _ 10 0 _ 16 5 _ 8 2 _ 8 3 _ 6 5 _ 10 8 _ 8 14 _ 16

3 _ 12 9 _ 12 16 _ 16 1 _ 4 6 _ 16 6 _ 12 4 _ 4 2 _ 16

1. Write each of the 16 fractions on the blanks below. If a fraction is not in simplest form, write it in simplest form.

2. Make a line plot to display the data. Write the fractions below the number line in their approximate locations.

3. Find the following data landmarks. Then label them on the line plot.

a. minimum

b. maximum

c. mode(s)

d. median

4. a Find the sum of the 16 fractions.

b Write the mean of the 16 fractions as a fraction.

c Label the mean on the line plot.

5. Find the range of the data.

Displaying Fractions on a Line PlotLESSON

10 �2

Date Time

X X XX

X XXXX

X X X XXXX

18

14

38

12

916

58

34

78

0 1min.

median

mean

max.mode

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Student Page

Math Boxes LESSON

10 �2

Date Time

1. Make a magnitude estimate for the product. Choose the best answer. 4,246 * 2.5

tenths

ones

hundreds

thousands

ten-thousands

2. Circle the numbers below that are divisible by 6.

148 293 762 1,050 984

3. Name the number for each point marked on the number line.

-3 -2 -1 0 1 2 3

D A B F C E

A = B = C = D = E = F =

-1.5, or -1 1 _ 2 -0.1, or - 1 _ 10 2.5, or 2 1

_ 2

1.5, or 1 1 _ 2 0.5, or 1 _ 2 -2.9, or -2 9 _ 10

4. a. Which has the greater area, a 3 ft × 2 ft rectangle or a triangle with base 3 ft and height 5 ft?

triangle b. Which has the greater area, a triangle

with base 10 cm and height 4 cm or a parallelogram with base 5 cm and height 6 cm?

parallelogram

5. Find the volume of the prism. Fill in the circle next to the best answer.

Volume = length * width * height

A. 252 ft3

B. 50 ft3

C. 23 ft3

D. 225 ft3

247—250

189 192193 197

11

57

15 ft

5 ft3 f

t

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Student Page

Lesson 10�2 795

2 Ongoing Learning & Practice

▶ Making and Interpreting

INDEPENDENT ACTIVITY

Line Plots(Math Journal 2, p. 339)

Students make and interpret a line plot consisting of a data set of measurements obtained from fraction cards. Students plot fractional units on the number line from 0 through 1. Remind students that they can use benchmarks to place the fractions on the number line. Review fraction operations as needed to find the data landmarks.

▶ Math Boxes 10�2


(Math Journal 2, p. 340)

Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 10-4. The skill in Problem 5 previews Unit 11 content.

Writing/Reasoning Have students write a response to the following: Explain your solution strategy for Problem 4a. Sample answer: The area of the rectangle is 3 ∗ 2 or 6 ft2.

The area of a triangle is 1 _ 2 the base times the height. Since the base for both figures is the same, and 1 _ 2 of the height of the triangle is greater than the height of the rectangle, the triangle’s area is greater because 3 ∗ 2.5 = 7.5 ft2.

▶ Study Link 10�2


(Math Masters, p. 297)

Home Connection Students solve pan-balance problems with one and two balances.

3 Differentiation Options

READINESS

SMALL-GROUP ACTIVITY

▶ Solving Logic Puzzles 5–15 Min

To provide experience with deductive reasoning required to solve linear equations, have students solve logic puzzles. Write the following statements on the board or the Class Data Pad.

1. All K are M.

2. All M are B.

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STUDY LINK

10�2 Pan-Balance Problems

222 228229

Name Date Time

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

1. 2.

One triangle weighs One pen weighs

as much as balls. as much as paper clips.

3.

M weighs N weighs


4.


as much as s. as much as marbles.

5.

One cup of juice weighs One apple weighs

as much as blocks. as much as blocks.

Fill in the missing numbers to make true sentences.

6. � (7 � 45) / 2 7. ((28 / 7) � 12) / 8 �

8. ((14 º 3) � 14) � 6 � 9. � (3 � 3) º ((34 / 2) º 115)050226

64

105

133

22

Practice

M N 15 marbles 2 N 20 marbles

5 11 8 marbles

18

Math Masters, p. 297

Study Link Master

LESSON

10�2

Name Date Time

Measuring Time

Franz buys two sandglasses from an antique dealer. However, when he gets home he realizes the sand in the sandglasses does not measure 1 hour. The first sandglass measures a nine-minute interval, and the other sandglass measures a thirteen-minute interval.

Franz wants to make a special cleaning solution to clean his new sandglasses. The solution needs to boil for 30 minutes. Can Franz use his sandglasses to measure 30 minutes from the time the solution starts to boil?

Explain your solution by describing what Franz should do.When the solution starts to boil, set the 13-minute sandglass. When it runs out, 13 minutes will have passed. Set both the 13-minute and the 9-minute sandglasses. When the 9-minute sandglass runs out, 13 � 9 minutes will have passed. Start the 9-minute sandglass again. When the 13 minute sandglass runs out, 13 � 9 � 4 minutes will have passed. The 9-minute sandglass will have 4 minutes of sand in its bottom. Start the9-minute sandglass again. Now the 4 minutes of sand are in thetop. When the 9-minute sandglass runs out, 13 � 9 � 4 � 4 or 30 minutes will have passed.

Math Masters, p. 298

Teaching Master


Ask students to make a third statement that could be added about K and B. All K are B.

Point out that to make the third statement, students had to put together the information from the first 2 statements. Write the following equations next to the statements on the board or Class Data Pad.

1. K = M

2. M = B

3. K = B

Ask students to substitute numbers for the letters so the sentences are still true, and write these next to the equations. Sample answers:

1. 18 = 2 ∗ 9

2. 2 ∗ 9 = 9 + 9

3. 18 = 9 + 9

Add the following pair of statements to the board or Class Data Pad.

1. 3 cubes and 1 paper clip weigh as much as 14 marbles.

2. 1 paper clip weighs as much as 2 marbles.

Ask students to make a third statement that could be added about cubes and marbles. 3 cubes weigh as much as 12 marbles, or 1 cube weighs as much as 4 marbles.

Ask students to make another pair of statements that follow this pattern. Use students’ responses to clarify their reasoning approaches.

ENRICHMENT PARTNER ACTIVITY

▶ Measuring Time 5–15 Min

(Math Masters, p. 298)

To further explore solving a problem with two unknowns, have students use a sandglass model to solve the problem on the Math Masters page. Students determine whether a sandglass with a9-minute interval and one with a 13-minute interval can be used to time a 30-minute period.

When partners have finished, discuss their solution strategies. One approach is to begin by setting the 13-minute sandglass. When it runs out, set the 13-minute and the 9-minute sandglasses at the same time. When the 9-minute sandglass runs out (13 + 9 minutes), start it again. After 4 more minutes, the 13-minute sandglass will run out (13 + 9 + 4), and there will be 4 minutes in the bottom of the 9-minute sandglass. Turn the 9-minute sandglass again so that the 4 minutes of sand on the bottom is now on the top. When this runs out, 30 minutes will have passed (13 + 9 + 4 + 4 = 30 minutes).


Pan-Balance Problems with Two Balances - Everyday Math · 10LESSON More Pan-Balance Problems 2...

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