Lesson9:&&Representing&Proportional&Relationships&with ...€¦ · Lesson&9:& Representing!Proportional!Relationshipswith!Equations&! Date:& 7/22/15! 76& ©!2014!Common!Core,!Inc.!Some!rights!reserved.!commoncore.org!

Lesson 9: Representing Proportional Relationships with Equations Date: 7/22/15

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NYS COMMON CORE MATHEMATICS CURRICULUM 7•1 Lesson 9

Lesson 9: Representing Proportional Relationships with

Equations

Student Outcomes

§ Students use the constant of proportionality to represent proportional relationships by equations in real-‐world contexts as they relate the equations to a corresponding ratio table.

Classwork Students will begin to write equations in two variables. They will analyze data that will help them understand the constant of proportionality and write the equation with two variables. Students will use an equation to find the value of an unknown variable given the value of the other variable.

Example 1 (18 minutes): Jackson’s Birdhouses Example 1: Jackson’s Birdhouses

Jackson and his grandfather constructed a model for a birdhouse. Many of their neighbors offered to buy the birdhouses. Jackson decided that building birdhouses could help him earn money for his summer camp, but he is not sure how long it will take him to finish all of the requests for birdhouses. If Jackson can build 𝟕 birdhouses in 𝟓 hours, write an equation that will allow Jackson to calculate the time it will take him to build any given number of birdhouses, assuming he works at a constant rate.

a. Write an equation that you could use to find out how long it will take him to build any number of birdhouses.

𝑯 = 𝟓 𝟕 𝑩

Define the variables. 𝐵 represents the number of birdhouses and 𝐻 represents the number of hours (time constructing birdhouses).

§ Does it matter which of these variables is independent or dependent? ú No. The number of birdhouses made could depend on how much time Jackson can work or the amount

of time he works could depend on how many birdhouses he needs to make.

§ If it is important to determine the number of birdhouses that can be built in one hour, what is the constant of proportionality?

ú number of birdhousesnumber of hours

is !!, or 1.4.

§ What does that mean in the context of this situation? ú It means that Jackson can build 1.4 birdhouses in one hour or one entire birdhouse and part of a second

birdhouse in one hour.

MP.2


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§ If it is important to determine the number of hours it takes to build one birdhouse, what is the constant of proportionality?

ú number of hoursnumber of birdhouses

is !!, or 0.71, which means that it takes him !

! of an hour to build one bird house or

!!

60 = 43 minutes to build one bird house.

§ This part of the problem asks you to write an equation that will let Jackson determine how long it will take him to build any number of birdhouses, so we want to know the value of 𝐻. This forces 𝐻 to be the dependent variable and 𝐵 to be the independent variable. Our constant of proportionality will be dependent

independent, which is !

! or !

!.

!! is !

!, so we will use the equation 𝐻 = !

!𝐵.

Use the equation above to determine the following:

b. How many birdhouses can Jackson build in 𝟒𝟎 hours?

If the equation for this situation is 𝑯 = 𝟓𝟕𝑩 and 𝑯 = 𝟒𝟎, then substitute 𝟒𝟎 into the equation for 𝑯 and solve for 𝑩

which represents the number or birdhouses.

𝟒𝟎 =𝟓𝟕 𝑩

Jackson can build 𝟓𝟔 birdhouses in 𝟒𝟎 hours.

c. How long will it take Jackson to build 𝟑𝟓 birdhouses? Use the equation from part (a) to solve the problem.

If 𝑯 = 𝟓𝟕𝑩 and 𝑩 = 𝟑𝟓, then substitute 𝟑𝟓 into the equation for 𝑩; 𝑯 = 𝟓

𝟕𝟑𝟓 ; 𝑯 = 𝟐𝟓. It will take Jackson 𝟐𝟓

hours to build 𝟑𝟓 birdhouses.

d. How long will it take to build 𝟕𝟏 birdhouses? Use the equation from part (a) to solve the problem.

If 𝑯 = 𝟓𝟕𝑩 and 𝑩 = 𝟕𝟏, then substitute 𝟕𝟏 for 𝑩 into the equation; 𝑯 =𝟓𝟕 (𝟕𝟏); 𝑯 =

𝟓𝟎.𝟕 (rounded to the nearest tenth). It will take Jackson 𝟓𝟎 hours and 𝟒𝟐 minutes (𝟔𝟎 (𝟎.𝟕)) to build 𝟕𝟏 birdhouses.

Remind students that someone may work for a fractional part of an hour, but a customer will not want to buy a partially built birdhouse. Tell students that some numbers can be left as non-‐integer answers (e.g., parts of an hour that can be written as minutes), but others must be rounded to whole numbers (e.g., the number of birdhouses completed or sold). All of this depends on the context. We must consider the real-‐life context before we determine if and how we round.

Example 2 (17 minutes): Al’s Produce Stand

Extension: Part (e): How many complete birdhouses can Jackson build in 3 hours? Use the equation from part (a) to solve the problem.

𝟑 = !𝟓𝟕!𝑩

!𝟕𝟓!𝟑 = !

𝟕𝟓! !

𝟓𝟕!𝑩

𝟒.𝟐 = 𝑩

Jackson can build 4 complete birdhouses in 3 hours.


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Let students select any two pairs of numbers from either Al’s Produce Stand or Barbara’s Produce Stand to calculate the constant of proportionality (𝑘 = dependent/independent). In order to determine the unit price, students need to divide the cost (dependent variable) by the number of ears of corn (independent variable). Lead them through the following questions to organize their thinking.

§ Which makes more sense: to use a unit rate of “ears of corn per dollar” or of “dollars (or cents) per ear of corn?”

ú The cost per ear of corn makes more sense because corn is sold as an entire ear of corn, not part of an ear of corn.

§ Based on the previous question, which will be the independent variable? ú The independent variable will be the number of ears of corn.

§ Which will be the dependent variable and why? ú The cost will be the dependent variable because the cost depends on the number of ears of corn

purchased.

Have students volunteer to share the pair of numbers they used to determine the unit rate, or constant of proportionality, and compare the values for Al’s Produce Stand and Barbara’s Produce Stand.

ú Al’s Produce Stand: 0.25 and 𝐵arbara’s Produce Stand: 0.24 § How do you write an equation for a proportional relationship?

ú 𝑦 = 𝑘𝑥 § Write the equation for Al’s Produce Stand:

ú 𝑦 = 0.25𝑥 § Write the equation for Barbara’s Produce Stand:

ú 𝑦 = 0.24𝑥 Example 2: Al’s Produce Stand

Al’s Produce Stand sells 𝟔 ears of corn for $𝟏.𝟓𝟎. Barbara’s Produce Stand sells 𝟏𝟑 ears of corn for $𝟑.𝟏𝟐. Write two equations, one for each produce stand, that models the relationship between the number of ears of corn sold and the cost. Then use each equation to help complete the tables below.

Al’s Produce Stand: 𝒚 = 𝟎.𝟐𝟓𝒙; where 𝒙 represents the number of ears of corn and 𝒚 represents the cost

Barbara’s Produce Stand: 𝒚 = 𝟎.𝟐𝟒𝒙; where 𝒙 represents the number of ears of corn and 𝒚 represents the cost

Al’s Produce Stand Barbara’s Produce Stand

𝐄𝐚𝐫𝐬 𝟔 𝟏𝟒 𝟐𝟏 𝟐𝟎𝟎 𝐄𝐚𝐫𝐬 𝟏𝟑 𝟏𝟒 𝟐𝟏 𝟐𝟎𝟖

𝐂𝐨𝐬𝐭 $𝟏.𝟓𝟎 $𝟑.𝟓𝟎 $𝟓.𝟐𝟓 $𝟓𝟎.𝟎𝟎 𝐂𝐨𝐬𝐭 $𝟑.𝟏𝟐 $𝟑.𝟑𝟔 $𝟓.𝟎𝟒 $𝟒𝟗.𝟗𝟐

§ If you use 𝐸 to represent the number of ears of corn and 𝐶 to represent the cost for the variables instead of 𝑥 and 𝑦, how would you write the equations?

ú 𝐶 = 0.25𝐸 and 𝐶 = 0.24𝐸


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Exercises 1-‐2

Have students complete the following examples independently, and then discuss results as a class.

Exercises

1. In 1999, the London Eye was the world’s tallest Ferris wheel. Its constant rate of rotation is so slow that people can walk onto and off the passenger capsules at ground level without the wheel stopping.

a. If the wheel rotates about 48 times in a 24-‐hour period, write an equation that represents the proportional relationship.

In this problem, it makes sense to find a constant of proportionality based on the number of rotations per hour: k = 𝟒𝟖

𝟐𝟒 = 2. The equation is r = 2h where r represents the number of rotations and h represents the

time in hours.

b. Use the equation you wrote in part (a) to find the number of rotations the Ferris wheel makes in 3 hours.

r = 2h

r = 2 (3)

r = 6 rotations

c. Use the equation you wrote in part (a) to find the number of hours it would take to make 5 complete rotations.

5= 2h 𝟏𝟐 𝟓 =

𝟏𝟐 𝟐𝒉

2.5 = h

5 rotations will take 2 hours and 30 minutes.

Note: The link below gives information and shows pictures of the London Eye Ferris wheel.

http://en.wikipedia.org/wiki/London_Eye

2. The length of the playing field in NFL football excluding the end zones is 𝟑𝟎𝟎 feet which is equivalent to 𝟏𝟎𝟎 yards.

a. Write an equation that can be used to represent this proportional relationship if the equation uses a constant of proportionality (unit rate) of yards per foot.

Since we are told to consider the number of yards per foot, yards is the dependent variable and feet is the independent variable.

k = 𝟏𝟎𝟎𝟑𝟎𝟎

= 𝟏𝟑

If f represents the number of feet and y to represents the number of yards, the equation is 𝒚 = 𝟏𝟑 𝒇.


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b. If the width of the football field is 160 feet, use the equation you wrote in part (a) to find the width of the playing field in yards.

Substitute 160 for f into the equation, and solve for y.

𝒚 = 𝟏𝟑 𝒇

𝒚 = 𝟏𝟑 𝟏𝟔𝟎

𝒚 = 𝟓𝟑𝟏𝟑

The field is 𝟓𝟑 𝟏𝟑 yards wide.

c. The end zones, by NFL regulations, are 10 yards deep on each end. Use the equation you wrote in part (a) to find the additional length to the playing field in feet if the end zones are included.

2 × 10 yards = 20 additional yards

𝒚 = 𝟏𝟑 𝒇

𝟐𝟎 = 𝟏𝟑 𝒇

(3) 20 = (3) 𝟏𝟑 f

𝟔𝟎 = 𝒇 60 feet are added to the length of the field by including the end zones.

Additional Problem: The link below is to the Grade 7 Mathematics Annotated 2014 State Test Questions resource document for August 2014. Page 41 models a selected response question. “This item measures 7.RP.2c because it assesses the student’s ability to represent a proportional relationship by an equation.”

https://www.engageny.org/resource/new-‐york-‐state-‐common-‐core-‐sample-‐questions

C losing (5 minutes)

§ What type of relationship can be modeled using an equation in the form 𝑦 = 𝑘𝑥, and what do you need to know to write an equation in this form?

ú A proportional relationship can be modeled using an equation in the form 𝑦 = 𝑘𝑥. You need to know the constant of proportionality, which is represented by 𝑘 in the equation.

§ Give an example of a real-‐world relationship that can be modeled using this type of equation and explain why. ú Distance equals rate multiplied by time. If the rate of a vehicle is going at an unchanging speed

(constant), then the distance will depend on time elapsed.

§ Give an example of a real-‐world relationship that cannot be modeled using this type of equation. Explain why. ú Distance is a dependent variable, and time is an independent variable because time is being multiplied

by the rate.


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Lesson Summary

How do you find the constant of proportionality? Divide to find the unit rate, 𝒚𝒙= 𝒌.

How do you write an equation for a proportional relationship? 𝒚 = 𝒌𝒙, substituting the value of the constant of proportionality in place of 𝒌.

What is the structure of proportional relationship equations, and how do we use them? 𝒙 and 𝒚 values are always left as variables, and when one of them is known, they are substituted into 𝒚 = 𝒌𝒙 to find the unknown using algebra.

Exit Ticket (5 minutes)


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Name ___________________________________________________ Date____________________

Lesson 9: Representing Proportional Relationships with

Equations

Exit Ticket Oscar and Maria each wrote an equation that they felt represented the proportional relationship between length in centimeters and length in inches. They each referred to a table that compared centimeters to inches. One entry in the table paired 127 centimeters with 50 inches. Who wrote the correct equation that would relate centimeters to inches? Explain why.

Oscar wrote the equation 𝑐 = 2.54 𝑖, and said that the unit rate 2.54 represents centimeters per inch.

Maria wrote the equation 𝑐 = 0.394 𝑖, and said that the unit rate 0.394 represents centimeters per inch.

Circle one:

I’m on my way. I’ve got it. I can teach it!


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Exit Ticket Sample Solutions

Oscar and Maria each wrote an equation that they felt represented the proportional relationship between length in centimeters and length in inches. They each referred to a table that compared centimeters to inches. One entry in the table paired 𝟏𝟐𝟕 centimeters with 𝟓𝟎 inches. Who wrote the correct equation that would relate centimeters to inches? Explain why.

Oscar wrote the equation 𝒄 = 𝟐.𝟓𝟒 𝒊, and said that the unit rate 𝟐.𝟓𝟒 represents centimeters per inch.

Maria wrote the equation 𝒄 = 𝟎.𝟑𝟗𝟒 𝒊, and said that the unit rate 𝟎.𝟑𝟗𝟒 represents centimeters per inch.

Oscar is correct. Oscar found the unit rate to be 𝟐.𝟓𝟒 by dividing centimeters by inches. The rate that Oscar used represents the number of centimeters per the number of inches. However, it should be noted that the variables were not well-‐defined. Since we do not know which values are independent or dependent, each equation should include a definition of each variable. For example, Oscar should have defined his variables so that 𝒄 represented the number of centimeters and 𝒊 represented the number of inches. For Maria’s equation to be correct, she should have stated that𝒄 represents number of inches and 𝒊 represents number of centimeters.

Model Problem Many traffic signals are programmed to turn green at a constant rate. In the United Kingdom, the maximum cycle time for a signal to turn green when pedestrian crossing signals are present is 40 times in one hour.

a. Write an equation that could be used to calculate the number of times the light will turn green for any number of minutes. Let 𝑐 represent the number of cycles when the light will turns green, and let 𝑚 represent the number of minutes.

b. How many times will the light turn green in 15 minutes?

c. How many minutes will it take for the light to turn green 30 times?

Solution:

a. 1 hour = 60 minutes

!"!" !"!#$%!"#$%&'

= !! cycles per minute

Since 𝑘 = !! , the equation is 𝑐 = !

!𝑚

b. !! 15 = 10

The light will turn green 10 times in 15 minutes.

c. 𝑐 = !!𝑚

30 = !!𝑚

32 30 =

32

23𝑚

45 = 𝑚 It will take 45 minutes for the light to turn green 30 times.


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Problem Set Sample Solutions

1. A person who weighs 𝟏𝟎𝟎 pounds on Earth weighs 𝟏𝟔.𝟔 lb. on the moon. a. Which variable is the independent variable? Explain why.

Weight on Earth is the independent variable because most people do not fly to the moon to weigh themselves first. The weight on the moon depends on a person’s weight on Earth.

b. What is an equation that relates weight on Earth to weight on the moon?

𝑴 =𝟏𝟔.𝟔𝟏𝟎𝟎 𝑬

𝑴 = 𝟎.𝟏𝟔𝟔𝑬 where E represents weight on Earth, and M represents weight on the moon.

c. How much would a 𝟏𝟖𝟓 pound astronaut weigh on the moon? Use an equation to explain how you know.

𝟑𝟎.𝟕𝟏 lb.

d. How much would a man that weighs 𝟓𝟎 pounds on the moon weigh on Earth?

𝟑𝟎𝟏 lb.

2. Use this table to answer the following questions. Number of Gallons of Gas Number of Miles Driven

𝟎 𝟎 𝟐 𝟔𝟐 𝟒 𝟏𝟐𝟒 𝟏𝟎 𝟑𝟏𝟎

a. Which variable is the dependent variable and why?

The number of miles driven is the dependent variable because the number of miles you can drive depends on the number of gallons of gas you have in your tank.

b. Is the number of miles driven proportionally related to the number of gallons of gas? If so, what is the equation that relates the number of miles driven to the number of gallons of gas?

Yes, the number of miles driven is proportionally related to the number of gallons of gas because every measure of gallons of gas can be multiplied by 𝟑𝟏 to get every corresponding measure of miles driven. 𝑴 = 𝟑𝟏𝑮 where G is the number of gallons of gas, and M represents the miles driven.

c. In any ratio relating the number of gallons of gas and the number of miles driven, will one of the values always be larger? If so, which one?

Yes, the number of miles will be larger except for the point (𝟎,𝟎). The point (𝟎,𝟎) means 𝟎 miles driven uses 𝟎 gallons of gas.

d. If the number of gallons of gas is known, can you find the number of miles driven? Explain how this value would be calculated.

Yes, multiply the constant of proportionality (𝟑𝟏 mpg) by the number of gallons of gas.


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e. If the number of miles driven is known, can you find the number of gallons of gas used? Explain how this value would be calculated.

Yes, divide the number of miles driven by the constant of proportionality (𝟑𝟏 mpg).

f. How many miles could be driven with 𝟏𝟖 gallons of gas?

𝟓𝟓𝟖 miles

g. How many gallons are used when the car has been driven 𝟏𝟖 miles?

𝟏𝟖𝟑𝟏

gallons

h. How many miles have been driven when half of a gallon of gas is used?

𝟑𝟏𝟐= 𝟏𝟓. 𝟓 miles

i. How many gallons have been used when the car has been driven for a half mile?

𝟏𝟔𝟐

gallons

3. Suppose that the cost of renting a snowmobile is $𝟑𝟕.𝟓𝟎 for 𝟓 hours. a. If 𝒄 represents the cost and 𝒉 represents the hours, which variable is the dependent variable? Explain why.

Then write an equation to represent this relationship.

𝒄 is the dependent variable because the cost of using the snowmobile depends on the number of hours you use it.

𝒄 = 𝟕.𝟓𝒉

b. What would be the cost of renting 𝟐 snowmobiles for 𝟓 hours?

$𝟕𝟓

4. In Katya’s car, the number of miles driven is proportional to the number of gallons of gas used. Find the missing value in the table.

Number of Gallons of Gas Number of Miles Driven 𝟎 𝟎 𝟒 𝟏𝟏𝟐 𝟔 𝟏𝟔𝟖 𝟖 𝟐𝟐𝟒 𝟏𝟎 𝟐𝟖𝟎

a. Write an equation that will relate the number of miles driven to the number of gallons of gas.

𝑴 = 𝟐𝟖𝑮, where 𝑴 is the number of miles and 𝑮 is the number of gallons of gas

b. What is the constant of proportionality?

𝟐𝟖

Extension: Part (c): If the amount paid for one snowmobile is $30, for how many hours was the snowmobile rented?

4 hours


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c. How many miles could Katya go if she filled her 𝟐𝟐-‐gallon tank?

𝟔𝟏𝟔 miles

d. If Katya takes a trip of 𝟔𝟎𝟎 miles, how many gallons of gas would be needed to make the trip?

𝟐𝟏 𝟑𝟕 gallons

e. If Katya drives 𝟐𝟐𝟒 miles during one week of commuting to school and work, how many gallons of gas would she use?

𝟖 gallons

Documents

Lesson9:&&Representing&Proportional&Relationships&with ...€¦ · Lesson&9:& Representing!Proportional!Relationshipswith!Equations&! Date:& 7/22/15! 76& ©!2014!Common!Core,!Inc.!Some!rights!reserved.!commoncore.org!