This lesson teaches students about writing and evaluating expressions involving multiplication and division. It includes two examples - one about buying bags of apples where each bag contains 3 apples, and one about returning drink containers for a 5 cent deposit per container. Students complete tables for each example and write expressions to represent the relationships. They also evaluate the expressions for given values. The lesson reinforces understanding multiplication and division expressions and being able to set up and solve real-world word problems involving these operations.

1. Lesson 20: Writing and Evaluating Expressions—Multiplication and Division
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NYS COMMON CORE MATHEMATICS CURRICULUM 6•4Lesson 20
Lesson 20: Writing and Evaluating Expressions—
Multiplication and Division
Student Outcomes
 Students develop expressions involvingmultiplication and division fromreal-world problems.
 Students evaluate these expressions for given values.
Lesson Notes
This lesson buildson Lesson 19, extending the concepts usingmultiplication and division expressions.
Classwork
Opening(3 minutes)
Use a shorttime to make sure the answers to the problem set from the previous lesson areclear. The labels on the
tables should be complete.
Discussion(3 minutes)
 In the previous lesson,we created expressions thatused addition and subtraction to describethe relationship
between two quantities. How did usingtables help your understanding?
 Patterns were easy to see. Looking down the columns revealed a number pattern. Looking across the
rows revealed a constant difference between columns.
 In this lesson,we are goingto develop expressions involvingmultiplication and division,much likethe last
lesson. We will also evaluatethese expressions for given values.
Example 1 (5 minutes)
 The farmers’ market is sellingbags of apples. In every bag, there are 3 apples. If I buy one bag, how many
apples will I have?
 Three.
 If I buy two bags,how many apples will I have?
 Since2 ∙ 3 = 6, you will have 6 apples.
 If I buy three bags, how many apples will I have?
 Since3 ∙ 3 = 9, you will have 9 apples.
 Fill in the tablefor a purchaseof 4 bags of apples. Check your answer with a
partner.
Scaffolding:
Havinginterlockingcubes
ready in groups of three will
make a concrete visual for
students to see and hold for
Example 1. Put these in clear
plastic bags,if desired.

2. Lesson 20: Writing and Evaluating Expressions—Multiplication and Division
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NYS COMMON CORE MATHEMATICS CURRICULUM 6•4Lesson 20
 What if I bought some other number of bags? If I told you how many bags ,could you calculatethe number of
apples I would have altogether?
 Yes, multiply the number of bags by 3 to find the total number of apples.
 What if I bought 𝐵 bags of apples? Can you write an expression in the tablethat describes the total number of
apples I have purchased?
 3𝐵 or 3( 𝐵) or 3 · 𝐵
Take a moment to review the different notations used for multiplication. Students should be comfortable readingand
writingthe expressions in all three forms.
 What if the market had 25 bags of apples to sell? How many apples is thatin all?
 If 𝐵 = 25, then 3𝐵 = 3 ∙ 25 = 75 apples.
Exercise 1 (5 minutes)
Exercises1–4
1. The farmers’ market isselling bagsofapples. In every bag, thereare 𝟑apples.
a. Complete thetable.
Number ofBags Total Number ofApples
𝟏 𝟑
𝟐 𝟔
𝟑 𝟗
𝟒 𝟏𝟐
𝑩 𝟑𝑩
b. What ifthe market had 𝟐𝟓bagsof applesto sell? How many applesisthat in all?
If 𝑩 = 𝟐𝟓, then 𝟑𝑩 = 𝟑 ∙ 𝟐𝟓 = 𝟕𝟓apples.
c. If atruck arrived that had some number, 𝒂, moreappleson it, then how many bagswould the clerksuse to
bag up the apples?
𝒂 ÷ 𝟑bags areneeded. If thereare 𝟏or 𝟐 apples left over, an extra bag will beneeded (although not full).
d. If atruck arrived that had 𝟔𝟎𝟎more appleson it,how many bagswould theclerksuse to bag up the apples?
𝟔𝟎𝟎 𝒂𝒑𝒑𝒍𝒆𝒔÷
𝟑 𝒂𝒑𝒑𝒍𝒆𝒔
𝟏 𝒃𝒂𝒈
= 𝟐𝟎𝟎 𝒃𝒂𝒈𝒔.
e. How is part (d) different from part (b)?
Part (d)gives thenumber ofapples and asks for thenumber ofbags. Therefore, weneeded to divide the
number ofapples by 𝟑. Part (b)gives thenumber ofbags and asks for thenumber ofapples. Therefore, we
needed to multiply thenumber ofbags by 𝟑.

3. Lesson 20: Writing and Evaluating Expressions—Multiplication and Division
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NYS COMMON CORE MATHEMATICS CURRICULUM 6•4Lesson 20
Exercise 2 (5 minutes)
Students work on Exercise2 independently.
2. In New York State, there isafive cent deposit onall carbonated beverage cansand bottles. When you returnthe
empty can or bottle,you get thefive centsback.
a. Complete thetable.
Number of Containers Returned Refund in Dollars
𝟏 𝟎. 𝟎𝟓
𝟐 𝟎. 𝟏𝟎
𝟑 𝟎. 𝟏𝟓
𝟒 𝟎. 𝟐𝟎
𝟏𝟎 𝟎. 𝟓𝟎
𝟓𝟎 𝟐. 𝟓𝟎
𝟏𝟎𝟎 𝟓. 𝟎𝟎
𝑪 𝟎.𝟎𝟓𝑪
b. If we let 𝑪 represent the number ofcans, what isthe expressionthat showshow muchmoney isreturned?
𝟎. 𝟎𝟓𝑪
c. Use the expression to find outhow much money Brett wouldreceive ifhe returned 𝟐𝟐𝟐cans.
If 𝑪 = 𝟐𝟐𝟐, then 𝟎.𝟎𝟓𝑪 = 𝟎.𝟎𝟓∙ 𝟐𝟐𝟐= $𝟏𝟏.𝟏𝟎.
d. If Gavin needsto earn $𝟒.𝟓𝟎for returning cans, how many cans doeshe need to collect and return?
$𝟒.𝟓𝟎÷ 𝟎.𝟎𝟓 = 𝟗𝟎cans.
e. How ispart (d) different from part (c)?
Part (d)gives the amount ofmoney and asks for the number ofcans. Therefore, weneeded to dividethe
amount ofmoney by 𝟎. 𝟎𝟓. Part (c)gives thenumber of cans and asks for the amount ofmoney. Therefore,
we needed to multiply thenumber of cans by 𝟎.𝟎𝟓.
Discuss thesimilarities and differences between Examples 1 and 2. In both examples, the second quantity is a multiple
of the first. Multiplication by the constantterm is used to show the relationship between the quantities in the first
column and the quantities in the second column. Division isused to show the relationship between the quantities in the
second column and the quantities in the firstcolumn.

4. Lesson 20: Writing and Evaluating Expressions—Multiplication and Division
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NYS COMMON CORE MATHEMATICS CURRICULUM 6•4Lesson 20
Exercise 3 (10 minutes)
Students work on Exercise3 independently.
3. The fare for a subway or a local busride is $𝟐.𝟓𝟎.
a. Complete thetable.
Number ofRides Cost ofRidesin Dollars
𝟏 𝟐. 𝟓𝟎
𝟐 𝟓. 𝟎𝟎
𝟑 𝟕. 𝟓𝟎
𝟒 𝟏𝟎.𝟎𝟎
𝟓 𝟏𝟐.𝟓𝟎
𝟏𝟎 𝟐𝟓.𝟎𝟎
𝟑𝟎 𝟕𝟓.𝟎𝟎
𝑹 𝟐.𝟓𝟎𝑹or 𝟐.𝟓𝑹
b. If we let 𝑹represent thenumber of rides, what isthe expression that showsthecost ofthe rides?
𝟐. 𝟓𝟎𝑹or 𝟐.𝟓𝑹
c. Use the expression to find outhow much money 𝟔𝟎rideswould cost.
If 𝑹 = 𝟔𝟎, then 𝟐.𝟓𝟎𝑹= 𝟐. 𝟓𝟎∙ 𝟔𝟎= $𝟏𝟓𝟎.𝟎𝟎.
d. If acommuter spends $𝟏𝟕𝟓.𝟎𝟎 on subway or busrides, how many tripsdid thecommuter take?
$𝟏𝟕𝟓.𝟎𝟎÷ $𝟐.𝟓𝟎 = 𝟕𝟎trips.
e. How is part (d) different from part (c)?
Part (d) gives theamount ofmoney and asks for thenumber of rides. Therefore, weneeded to dividethe
amount ofmoney by thecost ofeach ride ($𝟐.𝟓𝟎). Part (c) gives thenumber of rides andasks for the
amount ofmoney. Therefore, weneeded to multiply thenumber ofrides by $𝟐.𝟓𝟎.
Exercise 4 (10 minutes): Challenge Problem
4. A pendulum swingsthough acertain number ofcyclesin agiven time. Owen madeapendulumthat swings 𝟏𝟐
timesevery 𝟏𝟓seconds.
a. Construct atable showing thenumber ofcyclesthrough which apendulum swings. Includedatafor up to one
minute. Use the last row for 𝑪 cycles, and write an expression for the timeit takesfor the pendulum tomake
𝑪 cycles.
Number ofCycles Timein Seconds
𝟏𝟐 𝟏𝟓
𝟐𝟒 𝟑𝟎
𝟑𝟔 𝟒𝟓
𝟒𝟖 𝟔𝟎
𝑪
𝟏𝟓𝑪
𝟏𝟐

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b. Owen and hispendulum team settheir pendulum inmotion and counted 𝟏𝟔cycles. What wasthe elapsed
time?
𝟐𝟎seconds
c. Write an expression for thenumber ofcyclesapendulumswingsin 𝑺 seconds.
𝟏𝟐
𝟏𝟓
𝑺 or
𝟒
𝟓
𝑺 or 𝟎.𝟖 ∙ 𝑺
d. In a different experiment, Owen and hispendulum team counted thecyclesofthe pendulumfor 𝟑𝟓seconds.
How many cyclesdid they count?
𝟐𝟖cycles
Closing(2 minutes)
 In Example 1 and Exercise1, we looked at the relationship between the number of bags purchased atthe
farmers’ market and the total number of apples purchased. Wecreated two different expressions: 3𝐵
and 𝑎 ÷ 3. What does each variablerepresent, and why did we multiply by 3 in the firstexpression and divide
by 3 in the second?
 The variable 𝐵 represented the number of bags. We had to multiply by 3 because there were 3 apples
packaged in each bag. The variable 𝑎 represented the number of apples. These were grouped 3 per
bag, so we divided by 3.
 What would the expressions beif the farmers’ market sold bags that contained 5 apples instead of 3?
 5𝐵 and 𝑎 ÷ 5, respectively.
Exit Ticket (3 minutes)

6. Lesson 20: Writing and Evaluating Expressions—Multiplication and Division
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NYS COMMON CORE MATHEMATICS CURRICULUM 6•4Lesson 20
Name Date
Lesson 20: Writing and Evaluating Expressions—Multiplication
and Division
Exit Ticket
1. Anna charges $8.50 per hour to babysit. Complete the tableand answer the questions below.
Number of Hours Amount Anna Charges in Dollars
𝟏
𝟐
𝟓
𝟖
𝑯
a. Write an expression describingher earnings for working 𝐻 hours.
b. How much will sheearn if she works for 3
1
2
hours?
c. How longwill ittake Anna to earn $51.00?

7. Lesson 20: Writing and Evaluating Expressions—Multiplication and Division
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NYS COMMON CORE MATHEMATICS CURRICULUM 6•4Lesson 20
Exit Ticket Sample Solutions
1. Annacharges $𝟖.𝟓𝟎 per hour to babysit. Complete thetableand answer thequestionsbelow.
Number ofHours Amount AnnaChargesin Dollars
𝟏 𝟖. 𝟓𝟎
𝟐 𝟏𝟕.𝟎𝟎
𝟓 𝟒𝟐.𝟓𝟎
𝟖 𝟔𝟖
𝑯 𝟖.𝟓𝟎𝑯or 𝟖.𝟓𝑯
a. Write an expression describing her earningsfor working 𝑯 hours.
𝟖.𝟓𝟎𝑯or 𝟖.𝟓𝑯
b. How much will she earn ifshe worksfor 𝟑
𝟏
𝟐
hours?
If 𝑯 = 𝟑.𝟓, then 𝟖. 𝟓𝑯 = 𝟖. 𝟓 ∙ 𝟑.𝟓 = 𝟐𝟗.𝟕𝟓. Sheearned $𝟐𝟗.𝟕𝟓.
c. How long will it take Annato earn $𝟓𝟏.𝟎𝟎?
𝟓𝟏÷ 𝟖. 𝟓 = 𝟔. It takes Anna 𝟔hours to earn $𝟓𝟏.𝟎𝟎.
Problem Set Sample Solutions
1. A radio station plays 𝟏𝟐songseach hour. They never stop for commercials, news, weather, or traffic reports.
a. Write an expression describing how many songsare played by theradio stationin 𝑯 hours.
𝟏𝟐 𝑯
b. How many songswill be played in an entireday (𝟐𝟒hours)?
𝟏𝟐∙ 𝟐𝟒 = 𝟐𝟖𝟖. Therewill be 𝟐𝟖𝟖songs played.
c. How long doesit take the radio stationto play 𝟔𝟎consecutive songs?
𝟔𝟎 𝒔𝒐𝒏𝒈𝒔 ÷
𝟏𝟐 𝒔𝒐𝒏𝒈𝒔
𝟏 𝒉𝒐𝒖𝒓
= 𝟓 𝒉𝒐𝒖𝒓𝒔.
2. A ski areahas a high speed lift that can move 𝟐, 𝟒𝟎𝟎skiersto the topofthe mountain each hour.
a. Write an expression describing how many skierscan be lifted each hour.
𝟐, 𝟒𝟎𝟎 𝑯
b. How many skierscan be moved to the topofthe mountain in 𝟏𝟒hours?
𝟏𝟒∙ 𝟐, 𝟒𝟎𝟎= 𝟑𝟑,𝟔𝟎𝟎. 𝟑𝟑,𝟔𝟎𝟎skiers can bemoved.

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c. How long will it take to move 𝟑,𝟔𝟎𝟎skiersto thetop ofthe mountain?
𝟑, 𝟔𝟎𝟎÷ 𝟐,𝟒𝟎𝟎= 𝟏.𝟓; it will takean hour and a half.
3. Polly writesamagazine column,for which sheearns $𝟑𝟓per hour. Createatable ofvaluesthat showsthe
relationship between the numberofhoursthat Polly works, 𝑯, and the amountofmoney Polly earns in dollars, 𝑬.
Answers will vary. Sampleanswers areshown.
Hours Polly Works (𝑯) Polly’s Earnings in Dollars (𝑬)
𝟏 𝟑𝟓
𝟐 𝟕𝟎
𝟑 𝟏𝟎𝟓
𝟒 𝟏𝟒𝟎
a. If you know how many hoursPolly works, can you determinehow muchmoney she earned? Write the
corresponding expression.
Multiplying thenumber ofhours that Polly works by her rate ($𝟑𝟓 per hour) will calculateher pay. 𝟑𝟓𝑯is the
expression for her pay in dollars.
b. Use your expression to determinehow much Polly earned after working for 𝟑
𝟏
𝟐
hours.
𝟑𝟓𝑯= 𝟑𝟓 ∙ 𝟑.𝟓 = 𝟏𝟐𝟐.𝟓. Polly makes $𝟏𝟐𝟐.𝟓𝟎for working 𝟑
𝟏
𝟐
hours.
c. If you know how much money Polly earned,can you determinehow long she worked? Writethe
corresponding expression.
Dividing Polly’s pay by 𝟑𝟓will calculatethenumber ofhours sheworked. 𝑬 ÷ 𝟑𝟓is theexpression for the
number ofhours sheworked.
d. Use your expression to determinehow long Polly worked ifshe earned $𝟓𝟐.𝟓𝟎.
𝟓𝟐.𝟓𝟎÷ 𝟑𝟓= 𝟏. 𝟓; Polly worked an hour and a halffor $𝟓𝟐.𝟓𝟎.
4. Mitchelldeliversnewspapersafter school, for whichhe earns $𝟎.𝟎𝟗 per paper. Createatable ofvaluesthat shows
the relationshipbetween thenumber ofpapersthat Mitchelldelivers, 𝑷, and the amount ofmoney Mitchellearns
in dollars, 𝑬.
Answers will vary. Sampleanswers areshown.
Number ofPapers Delivered (𝑷) Mitchell’s Earnings in Dollars (𝑬)
𝟏 𝟎. 𝟎𝟗
𝟏𝟎 𝟎. 𝟗𝟎
𝟏𝟎𝟎 𝟗. 𝟎𝟎
𝟏𝟎𝟎𝟎 𝟗𝟎.𝟎𝟎
a. If you know how many papersMitchell delivers, can you determinehow much money heearned? Writethe
corresponding expression.
Multiplying thenumber of papers that Mitchelldelivers by his rate ($𝟎.𝟎𝟗per paper) will calculate his pay.
𝟎. 𝟎𝟗𝑷is theexpression for his pay in dollars.

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b. Use your expression to determinehow much Mitchell earned by delivering 𝟑𝟎𝟎newspapers.
𝟎. 𝟎𝟗𝑷 = 𝟎.𝟎𝟗∙ 𝟑𝟎𝟎= 𝟐𝟕. Mitchell earned $𝟐𝟕.𝟎𝟎for delivering 𝟑𝟎𝟎newspapers.
c. If you know how much money Mitchell earned, can you determine how many papershe delivered? Write the
corresponding expression.
Dividing Mitchell’s pay by $𝟎.𝟎𝟗willcalculatethenumber of papers hedelivered. 𝑬 ÷ 𝟎. 𝟎𝟗is theexpression
for thenumber of papers hedelivered.
d. Use your expression to determinehow many papers Mitchell delivered ifhe earned $𝟓𝟖.𝟓𝟎last week.
𝟓𝟖.𝟓𝟎÷ 𝟎.𝟎𝟗 = 𝟔𝟓𝟎; therefore, Mitchell delivered 𝟔𝟓𝟎newspapers last week.
5. Randy isan art dealer who sellsreproductionsoffamouspaintings. Copiesofthe Mona Lisa sell for $𝟒𝟕𝟓.
a. Last year Randy sold $𝟗,𝟗𝟕𝟓.𝟎𝟎worth of Mona Lisa reproductions. How many did hesell?
𝟗, 𝟗𝟕𝟓÷ 𝟒𝟕𝟓 = 𝟐𝟏. Hesold 𝟐𝟏copies ofthe painting.
b. If Randy wantsto increase hissalesto at least $𝟏𝟓,𝟎𝟎𝟎thisyear, how many copieswill heneed tosell
(without changing the priceperpainting)?
𝟏𝟓,𝟎𝟎𝟎÷ 𝟒𝟕𝟓is about 𝟑𝟏.𝟔; therefore, hewill haveto sell 𝟑𝟐paintings.