pan-balance problems - everyday math · pdf filestudents practice solving open number...

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784 Unit 10 Using Data; Algebra Concepts and Skills

Advance PreparationFor Part 1, move the fine-tuning adjustment to calibrate your pan balance. You can adjust the balance by taping

paper clips to the underside of one pan. For the optional Readiness Activity in Part 3, set up a pan balance and

3 small containers labeled 1981, 1982, and 1983 with 10 pennies from the appropriate year in each container.

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


[Operations and Computation Goal 3]

• Use a pan-balance model to solve linear

equations with one unknown.

[Patterns, Functions, and Algebra Goal 2]

Key ActivitiesStudents find the weight of a given object

by adding or removing objects of known

weight from both pans of a pan balance.

They model pan-balance problems using

variables to represent the objects.

Ongoing Assessment: Informing Instruction See pages 787 and 788.

Key Vocabularypan balance

MaterialsMath Journal 2, pp. 333 and 334

Student Reference Book Glossary

Class Data Pad (optional) � for

demonstration: 100 standard 1" paper clips,

7 identical ballpoint pens, 4 trapezoid pattern

blocks (or 4 quarters), pan balance � slate

Finding the Volumes of Solid FiguresMath Journal 2, pp. 334A and 334B

Students measure volumes by

counting unit cubes.

Playing First to 100Student Reference Book, p. 308

Math Masters, pp. 456–458

per partnership: 2 six-sided dice,

calculator

Students practice solving open

number sentences.

Math Boxes 10�1Math Journal 2, p. 335

Students practice and maintain skills

through Math Box problems.

Ongoing Assessment: Recognizing Student AchievementUse Math Boxes, Problem 5.

[Measurement and Reference Frames

Goal 2]

Study Link 10�1Math Masters, p. 294

Students practice and maintain skills

through Study Link activities.

READINESS

Exploring Pan BalancesMath Masters, p. 295

per partnership: pan balance, small

objects (nickels, pennies, paper clips,

centimeter cubes)

Students use a pan balance to explore

equivalency.

ENRICHMENTSolving a Penny RiddleStudents describe how to use a pan balance

to determine which penny weighs more than

the others.

EXTRA PRACTICE

Weighing PenniesMath Masters, p. 296

per partnership: pan balance, 30 pennies

(10 each from 1981, 1982, and 1983)

Students weigh pennies on a pan balance to

identify the year the penny’s weight changed.

Teaching the Lesson Ongoing Learning & Practice

132

4

Differentiation Options

� Pan-Balance ProblemsObjective To introduce a pan-balance approach for solving

simple equations.s

eToolkitePresentations Interactive Teacher’s

Lesson Guide

Algorithms Practice

EM FactsWorkshop Game™

AssessmentManagement

Family Letters

CurriculumFocal Points

Common Core State Standards

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Pan-Balance ProblemsLESSON

10 �1

Date Time

Math Message

1. Explain how to use a pan balance to weigh an object.

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

2. One cube weighs as

much as 11 marbles.

3. One cube weighs

as much as oranges.

4. One whole orange weighs

as much as 22 grapes.

5. One block weighs

as much as 3 marbles.

Check your answers. The sum of the answers to Problems 2–5 should equal 39 1 _ 2 .

Sample answer: Place the object on one pan. Place unit weights, such as gramweights, on the other panuntil the two pans balance.

1

2orange

AAAA A

3 1 _ 2

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

Math Journal 2, p. 333

Student Page

Lesson 10�1 785

Getting Started

1 Teaching the Lesson

▶ Math Message Follow-Up WHOLE-CLASSDISCUSSION

(Math Journal 2, p. 333)

Algebraic Thinking Ask students to share their explanations. If examples are not given, ask volunteers to name and describe something else that operates like a pan balance. A playground teeter-totter—the board will balance if the weight on each end is the same.

Emphasize that, unlike a teeter-totter, the pans must balance when they are empty before the balance can be used to weigh objects. Show students the calibrated pan balance with the two pans in balance. Place an object such as a pen or several pattern blocks on one pan. Place the same object(s) on the other pan. Explain that the pans balance because the weight of the objects on each pan is the same; the weight in one pan is equivalent to the weight in the other pan.

▶ Demonstrating How to Solve

WHOLE-CLASS ACTIVITY

Pan-Balance Problems(Student Reference Book Glossary)

Demonstrate how to solve pan-balance problems by first using paper clips and either pattern blocks or quarters. Then demonstrate again using ballpoint pens and paper clips.

Math MessageAnswer the question at the top of page 333 in your journal.

Mental Math and ReflexesStudents solve extended division facts problems. Write the problems on the board or Class Data Pad. Have students explain the patterns in the number of zeros and placement of the decimal point when a decimal is divided by a power of 10.

6 ÷ 3 = 2

60 ÷ 3 = 20

600 ÷ 3 = 200

81 ÷ 9 = 9

8.1 ÷ 9 = 0.9

8.1 ÷ (9 ∗ 10) = 0.09

5,600 ÷ 8 = 700

5.6 ÷ (8 ∗ 10) = 0.07

5.6 ÷ (8 ∗ 102) = 0.007

Interactive whiteboard-ready

ePresentations are available at

www.everydaymathonline.com to

help you teach the lesson.

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Adjusting the Activity


Example 1:

Show the trapezoid pattern blocks or quarters and the paper clips. Explain that the goal is to determine the weight of a single pattern block in terms of paper clips.

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: How could you change the contents of the pans so 1 pattern block in one pan is balanced with paper clips in the other pan? Tell students there is one rule to follow: Whatever you do, the pans must always remain balanced.

Suppose a student suggests the following incorrect approach: Take the single block from the right pan, and then take two of the three blocks from the left pan. Next remove clips from the right pan until the two pans balance. If you carry out these instructions, the class will observe the rule that the pans must always remain balanced is repeatedly violated.

Have students test their solutions by using the balance.

If necessary, guide students through the manipulations on the pans. (See margin.)

Example 2:

Show the ballpoint pens and paper clips. Explain that this time the goal is to determine the weight of a single pen in terms 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: How could you change the contents of the pans so 1 pen in one pan is balanced with paper clips in the other pan? Remind students that the pans must remain balanced after changing the contents of the pans.

Record the solutions on the board or a transparency by writing the

operation used for each change, for example, subtract 10 C, or subtract 2 P.

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

Have volunteers use the pan balance to explain their solutions. Possible approaches:

� Add the same number of clips to each pan or remove the same number of clips from each pan. You can also add the same number of pens to each pan or remove the same number of pens from each pan.

ELL

NOTE In this example, half of 21 clips is

10 1 _ 2 clips. Break one clip in half, and remove

10 clips plus a half clip.

The pan balance is not a precision

instrument. For example, two blocks might

appear to balance against 21 clips while

one block against 10 1 _ 2 clips might show a

slight tilt.

Removing 1 block from each pan will keep

the pans balanced.

Add clips to balance the pans. This might

require 21 clips.

Removing 1 _ 2 of the objects from each pan will

keep the pans balanced.

Success!


Pan-Balance Problems continuedLESSON

10 �1

Date Time

6. One weighs

as much as s.

7. One weighs

as much as marbles.

8. One x weighs

as much as balls.

9. One u weighs

as much as Vs.

Check your answers: The sum of the answers to Problems 6–9 should equal 10.

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

a. The content within a full bottle weighs

as much as marbles.

b. A full bottle weighs as much as marbles.

c. Explain your solutions.

I remove 1 bottle and the content of �12

� bottle from each side.

This leaves 1 bottle and the content of 1�12

� bottles on the left

and 18 marbles on the right. I remove the bottle from the left

and 6 marbles from the right leaving the content of 1�12

� bottles

on the left and 12 marbles on the right. I divide 12 by 3 since

there are three halves remaining. The content of �12

� bottle

weighs as much as 4 marbles, so the content of a whole

bottle weighs as much as 8 marbles. The weight of a full

bottle equals 14 marbles (8 � 6).

14

8

3

3

2

2

Try This

5

3x 1 ball x 7 balls

4u � 3V 3u � 6V

bothfull

halffull

Sample answer:


Student Page

Lesson 10�1 787

� Remove the same fraction of objects from each pan. For example, if the left pan contains 3 pens and the right pan contains 30 clips, you can remove 2 _ 3 of the objects from each pan—2 pens from the left pan and 20 clips from the right pan. (See margin.)

Explain that pan-balance problems are also models to help students learn how to solve algebraic equations. Ask students to look up the following terms in the Glossary of the Student Reference Book.

� Expression A group of mathematical symbols thatrepresents a number

� Algebraic expression An expression that contains a variable

� Equation A number sentence that contains an equal sign

Ask: What is an algebraic equation? A number sentence that contains an equal sign and algebraic expressions

Tell the class that algebraic expressions can be used to represent the weights in pan-balance problems. (See margin on next page.) Draw a pan balance on the board or a transparency using variables as shown in the illustration. Let P stand for the weight of 1 pen. Let C stand for the weight of 1 clip. Note that P and C do not represent the number of pens and paper clips; they stand for the weight of 1 pen and 1 paper clip, respectively.

Explain that in expressions with variables, the multiplication symbol is often omitted. Omitting the symbol avoids confusion between the letter x and the multiplication symbol ×. For example, writing 10C has the same meaning as writing 10 ∗ C, or 10 × C.

Draw a new pan balance for each step. If possible, draw each step below the previous step.

▶ Solving Pan-Balance Problems PARTNER ACTIVITY

(Math Journal 2, pp. 333 and 334)

Algebraic Thinking Have partners complete the pan-balance problems on the journal pages. Problems show pictures of objects, squares and triangles, or expressions with variables in the balance pans. Circulate and assist.

Ongoing Assessment: Informing Instruction

Watch for students who have difficulty keeping track of the changes they make

in pan-balance objects. Have them cross out objects that are removed with

each change.

PROBLEMBBBBBBBBBBBOOOOOOOOOOBBBBBBBBBBBBBBBBBBBBBBBBBBB MMMMMEEEBLEBLLEBLBLEBLELLLLBLEBLEBLEBLEBLEBLEEEEMMMMMMMMMMMMMOOOOOOOOOOOOBBBBBBLBLBLBLBBLLBLLLLPROPROPROPROPROPROPROPROPROPROPROPPRPROPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPROROROROROROROOPPPPPPP MMMMMMMMMMMMMMMMMMMMMEEEEEEEEEEEEEEEEELEEEEEEEELLLLLLLLLLLLLLLLLLLLLLLRRRRRRRRRRRRRRRRPROBLEMSOLVING

BBBBBBBBBBBBBBBBBBBBB ELEELELEMMMMMMMMOOOOOOOOOBBBLBLBLBBLBBBLOOROROROORORORORORORORORO LELELELEEEEEELEEMMMMMMMMMMMMLEMLLLLLLLLLLLLLLLLLLLLLLRRRRRRRRRRRGGGGGLLLLLLLLLLLLLVINVINVINNNNVINVINVINVINNVINVINVINVINVV GGGGGGGGGGGGOLOOOLOOLOLOLOO VVINVINLLLLLLLLLLVINVINVINVINNVINVINVINVINVINVINVINVINNGGGGGGGGGGGOOOLOLOLOLOLLOOO VVVLLLLLLLLLLVVVVVVVVVVSOSOSOOSOSOSOSOSOSOOSOSOSOSOOOOSOOSOSOSOSOSOSOSOOSOSOSOSOSOSOSOSOSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS VVVVVVVVVVVVVVVVVVVVVLLLLLLLVVVVVVVVVLLLVVVVVVVVLLLLLLLLVVVVVLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLSSSSSSSSSSSSSSSSSSSSSS GGGGGGGGGGGGGGGGOOOOOOOOOOOOOOOOOOO GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGNNNNNNNNNNNNNNNNNNNNNNNNNIIIIIIIIIIIIIIIIIIIISOLVING

NOTE You can make a semipermanent set

of pan balances on a chalkboard by thoroughly

wetting the board and writing heavily with

chalk while the board is still wet. After it is dry,

draw the specific problems. Erase gently to

remove the problems.

10

Add clips to balance the pans. This might

require 40 clips.

40

remove 10

Removing 10 clips from each pan will keep

the pans balanced.

30

Removing 2 pens from each pan will keep

the pans balanced.

Removing 2 _ 3 of the objects from each

pan will keep the pans balanced.

remove or 20 clips2–3

30


Finding VolumeLESSON

10�1

Date Time

1. A cube with a side that is 1 unit long is called a unit cube. It has a volume of

one cubic unit. The figures below all represent rectangular prisms. Circle the

figures that represent 1 unit cube.

a. b. c d

2. Suppose these cubes are put together to form a solid figure.

What would be the volume of the figure?

cubic units

Use the pictures below to determine how many unit cubes can be packed

in each container. Then give the volume of the container. Both containers

have already been partially packed with unit cubes.

3. 4.

The solid holds

cubes. The solid holds

cubes.

Volume:

cubic units Volume:

cubic units

Find the volume of each solid figure. Some cubes are hidden.

5. 6. 7.

cubic units

cubic units

cubic units

1 cm

1 cm

2 cm

1 yd 1 yd

1 yd

1 m

1 m

1 m

1 m

1 m

1 unit

1 unit

1 unit

210

479

536

321

684

538

043

002

590

857

765

205

3 cubes

6 cu

bes

2 cubes

12

36

7 6 10

4036 40

334A-334B_EMCS_S_MJ2_G5_U10_576434.indd 334A 3/22/11 12:43 PM

Math Journal 2, p. 334A

Student Page


Ongoing Assessment: Informing Instruction

Watch for students who have difficulty using the squares and triangles or

variables in the journal page problems. Have them write number models for

expressions with shapes or variables. For example:

• 5 □s can be visualized in the pan balance as □ □ □ □ □. In a number model,

5 □s can be interpreted as □ + □ + □ + □ + □ or 5 ∗ □. (Problem 7)

• 3x can be visualized in the pan balance as xxx. In a number model, 3x can be

interpreted as x + x + x or 3 ∗ x. (Problem 8)

2 Ongoing Learning & Practice

▶ Finding the Volumes of PARTNER ACTIVITY

Solid Figures (Math Journal 2, pp. 334A and 334B)

Students find volumes by counting unit cubes.

▶ Playing First to 100 PARTNER ACTIVITY

(Student Reference Book, p. 308;

Math Masters, pp. 456–458)

Algebraic Thinking Students play First to 100 to practice a variety of skills involving variables. This game was first introduced in Lesson 4-7. For detailed instructions, see Student Reference Book, page 308.

▶ Math Boxes 10�1 INDEPENDENT

ACTIVITY (Math Journal 2, p. 335)

Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 10-3. The skill in Problem 4 previews Unit 11 content.

Writing/Reasoning Have students write a response to the following: Explain how you would use a number line to solve Problem 2a, -7 + (-3) = . Sample answer: I would start at -7 and move left 3 spaces to end at -10.

Writing/Reasoning Have students write a response to the following: Explain the strategy and reasoning you would use to solve Problem 3b with the standard multiplication algorithm. Answers vary.

10C + 5P 2P + 40C

remove 10C

Removing 10 clips from each pan will keep

the pans balanced.

5P 2P + 30C

remove 2P

Removing 2 pens from each pan will keep

the pans balanced.

3P 30C

remove 2P remove 20C

Removing 2 _ 3 of the objects from

each pan will keep the pans balanced.

P 10C

Success!


180 units3

Math Boxes LESSON

10 �1

Date Time

1. Write the coordinates of the points shown

on the coordinate grid.

a. A: (0,3)

b. B: (3,2)

c. C: (5,4)

d. D: (2,0) 10 2 3 4 5

1

0

2

4

3

5

y

x

A

B

C

D

208

153 197

2. Add or subtract.

a. –7 + (–3) = –10

b. 5 – (–8) = 13

c. –17 + 10 = –7

d. –15 – 15 = –30

e. 3 + (–20) = –17

3. Multiply. Use the algorithm of your choice.

Show your work.

a. 87

* 65 b. 39

* 24 c. 99

* 26

5,655 936 2,574

1992

5. Find the volume of the prism.

Write a number model for the formula.

4. Find the radius and diameter of the circle.

radius = 1 1 _ 2 cm

(unit)

diameter =

(unit)

3 cm

5 units

4 units

9 units

9 ∗ 4 ∗ 5 or 36 ∗ 5 = 180 units3

�

333-368_EMCS_S_MJ2_G5_U10_576434.indd 335 4/6/11 11:02 AM


Student Page

Date Time

Finding Volume continuedLESSON

10�1

8. The net pattern on the right can be folded to form an open box.

What are the dimensions of the box?

Volume =

cubic units

9. a. Which box below does not have the same volume as the other three?

b. The three boxes with the same volume each have a volume of

cubic units.

Find the volume of each solid figure.

10. Each cube that makes up the solid has

sides that are 1 cm long.

cm3


sides that are 1 in. long.

in3


sides that are 1 ft long.

ft3

Box A Box B Box C Box D

30

5 units × 3 units × 2 units

60

Box D

27

60

125

334A-334B_EMCS_S_MJ2_G5_U10_576434.indd 334B 3/22/11 12:43 PM

Math Journal 2, p. 334B

Student Page

Lesson 10�1 789

Ongoing Assessment: Math Boxes

Problem 5 �Recognizing Student Achievement

Use Math Boxes, Problem 5 to assess students’ ability to find the volume of a

rectangular prism. Students are making adequate progress if they are able to

calculate the correct volume and record the number model. Some students may

be able to record more than one number model for the formulas.

[Measurement and Reference Frames Goal 2]

▶ Study Link 10�1

INDEPENDENT ACTIVITY

(Math Masters, p. 294)

Home Connection Students use representations of pan balances to solve problems.

3 Differentiation Options

READINESS PARTNER ACTIVITY

▶ Exploring Pan Balances 5–15 Min


To provide experience with a pan-balance model of equality, have students explore relationships between the weights of various objects. Have them find combinations of objects that balance the pans. Use small objects such as nickels, pennies, paper clips, and centimeter cubes. Emphasize that the pans must begin in balance. Show students how to adjust the pans if necessary. As they find combinations that balance, have them record the combinations on the master using pictures and words.

ENRICHMENT

SMALL-GROUP ACTIVITY

▶ Solving a Penny Riddle 5–15 Min

To apply students’ understanding of the pan-balance model of equality, pose the following problem: Suppose you have seven pennies and you know that one penny is heavier than the other six. How can you tell which penny is heavier in two weighings on a pan balance?

Have students share their strategies. Possible strategy:

1. Put 3 pennies on one pan and 3 pennies on the other. If the pans balance, the penny you are not weighing is the heavier penny. If the pans do not balance, you know that the heavy penny is in the heavier pan.

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Teaching Master

LESSON

10�1 Penny Weights

The materials used to make a penny were changed at the beginning

of one of these years: 1981, 1982, or 1983. As a result, the weight of

a penny has changed. Your task is to find the year the weight

of pennies changed.

1. Compare 1981 pennies to 1982 pennies.

Put ten 1981 pennies in one pan and ten 1982 pennies in the other pan.

Do the pans balance?

2. Return the pennies to their correct containers.

Put ten 1982 pennies in one pan and ten 1983 pennies in the other pan.

Do the pans balance?

3. I think penny weights changed beginning in the year because

4. Why do you think it is better to compare the weights using 10 pennies for each year

rather than only 1 penny for each year?

5. Why do you think they changed the materials used to make a penny?

Name Date Time

Sample answer: the 1983 pennies are lighter than

difference for comparing groups of 10 pennies is 10 times

Sample answer: The weight

the ones minted in 1981 and 1982.

greater than the weight difference for comparing 1 penny

for each year. Therefore, it will be easier to detect a difference

using the pan balance to compare groups of 10.

Sample answer: To reduce the cost of making pennies

Yes

No

1983

294-322_439_EMCS_B_MM_G5_U10_576973.indd 296 2/22/11 5:44 PM

Math Masters, p. 296

STUDY LINK

10�1 Pan-Balance Problems

Name Date Time

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


1. One triangle weighs

as much as squares.

2. One cube weighs

as much as marbles.

3. Two cantaloupes weigh

as much as apples.

4. One X weighs

as much as Ys.

5. One B weighs

as much as M s. 3 B 3 M 1 B 9 M

4 X 15 Y 6 X 7 Y

cantaloupe12

10

6. 4,217 7. 16,000

- 2,849 - 8,245

8. 11.47 - 8.896 = 9. 36 - 42 =

Practice

228 229

3

3

36

4

3

1,3682,574 -6

7,755

294-322_439_EMCS_B_MM_G5_U10_576973.indd 294 2/22/11 5:44 PM

Math Masters, p. 294

Study Link Master


2. Take the 3 pennies from the heavier pan. Put 1 in each pan. If the pans balance, you know that the penny you did not weigh is the heavier penny. If the pans do not balance, you know that the heavy penny is in the heavier pan.

NOTE The weight of the penny changed in 1983. Prior to 1983, a penny

weighed about 3.11 grams; it was made up of 95% copper and 5% zinc.

Since 1983, a penny has weighed 2.5 grams; it is made up of 97.5% zinc and

2.5% copper.

EXTRA PRACTICE PARTNER ACTIVITY

▶ Weighing Pennies 30+ Min


Social Studies Link Students try to determine the year in which the weight of a penny changed. Have students use a

pan balance to weigh pennies minted in 1981, 1982, and 1983 and record their results and conclusions on Math Masters, page 296.

When partners have finished, discuss their conclusions. If the weight has not changed from one year to the next, 10 pennies from one year should approximately balance 10 pennies from the following year. If the weight has changed, the two sets of 10 pennies will not balance, and it will be clear which year has heavier penny weights.

The U.S. Mint maintains a Web site that contains information about currency and coins: www.usmint.gov.

From the Mint home page, click the photo of a penny to find information specific to pennies.

Planning Ahead

In preparation for working with rates and ratios beginning in Lesson 10-4, assemble a collection of examples for a Rates and Ratios Museum.


pan-balance problems - everyday math · pdf filestudents practice solving open number...

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