Lesson 11: Constant Rate Keep Scrolling Past Some Large White Spaces

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 11 8•4

Formatting is a little messy on this. My apologies. There are 15 pages.

Lesson 11: Constant Rate Keep scrolling past some large white spaces. Classwork Example 1

Give students the first question below and allow them time to work. Ask them to share their solutions with the class and then proceed with the discussion, table, and graph to finish Example 1.

Example 1

Pauline mows a lawn at a constant rate. Suppose she mows a ퟑퟓ square foot lawn in ퟐ. ퟓ minutes. What area, in square feet, can she mow in ퟏퟎ minutes? 풕 minutes?

. What is Pauline’s average rate in 2.5 minutes? 35  Pauline’s average rate in 2.5 minutes is square feet per minute. 2.5 . What is Pauline’s average rate in 10 minutes?  Let 퐴 represent the square feet of area mowed in 10 minutes. Pauline’s average rate in 10 minutes is 퐴 square feet per mintute. 10 35 퐴 . Let 퐶 be Pauline’s constant rate in square feet per minute; then, = 퐶, and = 퐶. Therefore, 2.5 10

ퟑퟓ 푨 = ퟐ. ퟓ ퟏퟎ ퟑퟓퟎ = ퟐ. ퟓ푨 ퟑퟓퟎ ퟐ. ퟓ = 푨 ퟐ. ퟓ ퟐ. ퟓ ퟏퟒퟎ = 푨

Pauline mows ퟏퟒퟎ square feet of lawn in ퟏퟎ minutes.

. If we let 푦 represent the number of square feet Pauline can mow in 푡 minutes, then Pauline’s average rate in 푡 푦 minutes is square feet per minute. 푡 . Write the two-variable equation that represents the area of lawn, 푦, Pauline can mow in 푡 minutes.

ퟑퟓ 풚 = ퟐ. ퟓ 풕 ퟐ. ퟓ풚 = ퟑퟓ풕 ퟐ. ퟓ ퟑퟓ 풚 = 풕 ퟐ. ퟓ ퟐ. ퟓ ퟑퟓ 풚 = 풕 ퟐ. ퟓ

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35 35 . What is the meaning of in the equation 푦 = 푡? 2.5 2.5 MP.7 35  The number represents the constant rate at which Pauline can mow a lawn. 2.5 . We can organize the information in a table.

Linear equation: 풕 (time in 풚 (area in ퟑퟓ minutes) 풚 = 풕 square feet) ퟐ. ퟓ ퟑퟓ ퟎ 풚 = (ퟎ) ퟎ ퟐ. ퟓ ퟑퟓ ퟑퟓ ퟏ 풚 = (ퟏ) = ퟏퟒ ퟐ. ퟓ ퟐ. ퟓ ퟑퟓ ퟕퟎ ퟐ 풚 = (ퟐ) = ퟐퟖ ퟐ. ퟓ ퟐ. ퟓ ퟑퟓ ퟏퟎퟓ ퟑ 풚 = (ퟑ) = ퟒퟐ ퟐ. ퟓ ퟐ. ퟓ ퟑퟓ ퟏퟒퟎ ퟒ 풚 = (ퟒ) = ퟓퟔ ퟐ. ퟓ ퟐ. ퟓ

. On a coordinate plane, we will let the 푥-axis represent time 푡, in minutes, and the 푦-axis represent the area of mowed lawn in square feet. Then we have the following graph.

55 50

45 풚 40 35

15 Lawn mowed in square feet, feet, square in mowed Lawn

0 1 2 3 4 5 6 7 Time in minutes, 풕 . Because Pauline mows at a constant rate, we would expect the square feet of mowed lawn to continue to rise as the time, in minutes, increases.

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Concept Development . In the last lesson, we learned about average speed and constant speed. Constant speed problems are just a special case of a larger variety of problems known as constant rate problems . . Suppose 퐴 square feet of lawn are mowed in a given time interval 푡 (minutes). Then, the average rate of lawn 퐴 mowing in the given time interval is square feet per minute. If we assume that the average rate of lawn 푡 mowing is the same constant, 퐶, for any given time interval, then we say that the lawn is mowed at a constant rate, 퐶. . Describe the average rate of painting a house.  Suppose 퐴 square feet of house are painted in a given time interval 푡 (minutes). Then the average rate 퐴 of house painting in the given time interval is square feet per minute. 푡 . Describe the constant rate of painting a house.  If we assume that the average rate of house painting is the same constant, 퐶, over any given time interval, then we say that the wall is painted at a constant rate, 퐶. . What is the difference between average rate and constant rate?  Average rate is the rate in which something can be done over a specific time interval. Constant rate assumes that the average rate is the same over any time interval. . As you can see, the way we define average rate and constant rate for a given situation is very similar. In each case, a transcription of the given information leads to an expression in two variables.

Example 2

Water flows at a constant rate out of a faucet. Suppose the volume of water that comes out in three minutes is ퟏퟎ. ퟓ gallons. How many gallons of water comes out of the faucet in 풕 minutes?

. Write the linear equation that represents the volume of water, 푉, that comes out in 푡 minutes.

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Let 푪 represent the constant rate of water flow. ퟏퟎ.ퟓ 푽 ퟏퟎ.ퟓ 푽 = 푪, and = 푪; then, = . ퟑ 풕 ퟑ 풕 ퟏퟎ. ퟓ 푽 = ퟑ 풕 ퟑ푽 = ퟏퟎ. ퟓ풕 ퟑ ퟏퟎ. ퟓ 푽 = 풕 ퟑ ퟑ ퟏퟎ. ퟓ 푽 = 풕 ퟑ

10.5 10.5 . What is the meaning of the number in the equation 푉 = 푡? MP.7 3 3 10.5  The number represents the constant rate at which water flows from a faucet. 3 10.5 . Using the linear equation 푉 = 푡, complete the table. 3

Linear equation: 풕 (time in minutes) ퟏퟎ. ퟓ 푽 (in gallons) 푽 = 풕 ퟑ ퟏퟎ. ퟓ ퟎ 푽 = (ퟎ) ퟎ ퟑ ퟏퟎ. ퟓ ퟏퟎ. ퟓ ퟏ 푽 = (ퟏ) = ퟑ. ퟓ ퟑ ퟑ ퟏퟎ. ퟓ ퟐퟏ ퟐ 푽 = (ퟐ) = ퟕ ퟑ ퟑ ퟏퟎ. ퟓ ퟑퟏ. ퟓ ퟑ 푽 = (ퟑ) = ퟏퟎ. ퟓ ퟑ ퟑ ퟏퟎ. ퟓ ퟒퟐ ퟒ 푽 = (ퟒ) = ퟏퟒ ퟑ ퟑ

. On a coordinate plane, we will let the 푥-axis represent time 푡 in minutes and the 푦-axis represent the volume of water. Graph the data from the table.

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15 14 13

12 푉 11

7 gallons, in water of Volume 6

0 1 2 3 4 5 6 7 8 9 10 Time in minutes, 푡

1 . Using the graph, about how many gallons of water do you think would flow after 1 minutes? Explain. 2 1 1  After 1 minutes, between 3 and 7 gallons of water will flow. Since the water is flowing at a constant 2 2 rate, we can expect the volume of water to rise between 1 and 2 minutes. The number of gallons that 1 flows after 1 minutes then would have to be between the number of gallons that flows out after 1 2 minute and 2 minutes. . Using the graph, about how long would it take for 15 gallons of water to flow out of the faucet? Explain.  It would take between 4 and 5 minutes for 15 gallons of water to flow out of the faucet. It takes 4 minutes for 14 gallons to flow; therefore, it must take more than 4 minutes for 15 gallons to come out. 1 It must take less than 5 minutes because 3 gallons flow out every minute. 2 . Graphing proportional relationships like these last two constant rate problems provides us more information than simply solving an equation and calculating one value. The graph provides information that is not so obvious in an equation.

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Exercises 1–3

Exercises ퟏ ퟑ 풑 1. Juan types at a constant rate. He can type a full page of text in ퟐ minutes. We want to know how many pages, , Juan can type after 풕 minutes. a. Write the linear equation in two variables that represents the number of pages Juan types in any given time interval.

Let 푪 represent the constant rate that Juan types in pages per minute. Then, ퟏ 풑 ퟏ 풑 = 푪, and = 푪; therefore, = . ퟑ.ퟓ 풕 ퟑ.ퟓ 풕 ퟏ 풑 = ퟑ. ퟓ 풕 ퟑ. ퟓ풑 = 풕 ퟑ. ퟓ ퟏ 풑 = 풕 ퟑ. ퟓ ퟑ. ퟓ ퟏ 풑 = 풕 ퟑ. ퟓ

b. Complete the table below. Use a calculator and round your answers to the tenths place.

Linear equation: 풕 (time in minutes) ퟏ 풑 (pages typed) 풑 = 풕 ퟑ. ퟓ ퟏ ퟎ 풑 = (ퟎ) ퟎ ퟑ. ퟓ ퟏ ퟓ ퟓ 풑 = (ퟓ) ≈ ퟏ. ퟒ ퟑ. ퟓ ퟑ. ퟓ ퟏ ퟏퟎ ퟏퟎ 풑 = (ퟏퟎ) ≈ ퟐ. ퟗ ퟑ. ퟓ ퟑ. ퟓ ퟏ ퟏퟓ ퟏퟓ 풑 = (ퟏퟓ) ≈ ퟒ. ퟑ ퟑ. ퟓ ퟑ. ퟓ ퟏ ퟐퟎ ퟐퟎ 풑 = (ퟐퟎ) ≈ ퟓ. ퟕ ퟑ. ퟓ ퟑ. ퟓ

c. Graph the data on a coordinate plane.

7 푝 6

4 Pagestyped, 3

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 Time in minutes, 푡

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d. About how long would it take Juan to type a ퟓ-page paper? Explain.

It would take him between ퟏퟓ and ퟐퟎ minutes. After ퟏퟓ minutes, he will have typed ퟒ. ퟑ pages. In ퟐퟎ minutes, he can type ퟓ. ퟕ pages. Since ퟓ pages is between ퟒ. ퟑ and ퟓ. ퟕ, then it will take him between ퟏퟓ and ퟐퟎ minutes.

2. Emily paints at a constant rate. She can paint ퟑퟐ square feet in ퟓ minutes. What area, 푨, in square feet, can she paint in 풕 minutes? a. Write the linear equation in two variables that represents the number of square feet Emily can paint in any given time interval.

Let 푪 be the constant rate that Emily paints in square feet per minute. Then, ퟑퟐ 푨 ퟑퟐ 푨 = 푪, and = 푪; therefore, = . ퟓ 풕 ퟓ 풕 ퟑퟐ 푨 = ퟓ 풕 ퟓ푨 = ퟑퟐ풕 ퟓ ퟑퟐ 푨 = 풕 ퟓ ퟓ ퟑퟐ 푨 = 풕 ퟓ

b. Complete the table below. Use a calculator and round answers to the tenths place.

Linear equation: 푨 (area painted in 풕 (time in minutes) ퟑퟐ 푨 = 풕 square feet) ퟓ ퟑퟐ ퟎ 푨 = (ퟎ) ퟎ ퟓ ퟑퟐ ퟑퟐ ퟏ 푨 = (ퟏ) = ퟔ. ퟒ ퟓ ퟓ ퟑퟐ ퟔퟒ ퟐ 푨 = (ퟐ) = ퟏퟐ. ퟖ ퟓ ퟓ ퟑퟐ ퟗퟔ ퟑ 푨 = (ퟑ) = ퟏퟗ. ퟐ ퟓ ퟓ ퟑퟐ ퟏퟐퟖ ퟒ 푨 = (ퟒ) = ퟐퟓ. ퟔ ퟓ ퟓ

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c. Graph the data on a coordinate plane.

24 22

퐴 18

14 12

10 Area painted painted Area squarein feet,

4 2

0 1 2 3 4 5 6 7 8 9 10 Time in minutes, 푡

ퟏ ퟐ d. About how many square feet can Emily paint in ퟐ minutes? Explain. ퟏ ퟏퟐ. ퟖ ퟏퟗ. ퟐ ퟐ ퟐ ퟏퟐ. ퟖ Emily can paint between and square feet in ퟐ minutes. After minutes, she paints square feet, and after ퟑ minutes, she will have painted ퟏퟗ. ퟐ square feet.

3. Joseph walks at a constant speed. He walked to a store that is one-half mile away in ퟔ minutes. How many miles, 풎, can he walk in 풕 minutes? a. Write the linear equation in two variables that represents the number of miles Joseph can walk in any given time interval, 풕.

Let 푪 be the constant rate that Joseph walks in miles per minute. Then, ퟎ.ퟓ 풎 ퟎ.ퟓ 풎 = 푪, and = 푪; therefore, = . ퟔ 풕 ퟔ 풕 ퟎ. ퟓ 풎 = ퟔ 풕 ퟔ풎 = ퟎ. ퟓ풕 ퟔ ퟎ. ퟓ 풎 = 풕 ퟔ ퟔ ퟎ. ퟓ 풎 = 풕 ퟔ

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b. Complete the table below. Use a calculator and round answers to the tenths place.

Linear equation: 풕 (time in minutes) ퟎ. ퟓ 풎 (distance in miles) 풎 = 풕 ퟔ

ퟎ. ퟓ ퟎ 풎 = (ퟎ) ퟎ ퟔ ퟎ. ퟓ ퟏퟓ ퟑퟎ 풎 = (ퟑퟎ) = ퟐ. ퟓ ퟔ ퟔ ퟎ. ퟓ ퟑퟎ ퟔퟎ 풎 = (ퟔퟎ) = ퟓ ퟔ ퟔ ퟎ. ퟓ ퟒퟓ ퟗퟎ 풎 = (ퟗퟎ) = ퟕ. ퟓ ퟔ ퟔ ퟎ. ퟓ ퟔퟎ ퟏퟐퟎ 풎 = (ퟏퟐퟎ) = ퟏퟎ ퟔ ퟔ

c. Graph the data on a coordinate plane.

d. Joseph’s friend lives ퟒ miles away from him. About how long would it take Joseph to walk to his friend’s house? Explain.

It will take Joseph a little less than an hour to walk to his friend’s house. Since it takes ퟑퟎ minutes for him to walk ퟐ. ퟓ miles, and ퟔퟎ minutes to walk ퟓ miles, and ퟒ is closer to ퟓ than ퟐ. ퟓ, it will take Joseph less than an hour to walk the ퟒ miles.

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Closing Summarize, or ask students to summarize, the main points from the lesson: . Constant rate problems appear in a variety of contexts like painting a house, typing, walking, water flow, etc. . We can express the constant rate as a two-variable equation representing proportional change. . We can graph the constant rate situation by completing a table to compute data points.

Lesson Summary

When constant rate is stated for a given problem, then you can express the situation as a two-variable equation. The equation can be used to complete a table of values that can then be graphed on a coordinate plane.

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Name Date

Problem Set Sample Solutions Students practice writing two-variable equations that represent a constant rate.

1. A train travels at a constant rate of ퟒퟓ miles per hour. a. What is the distance, 풅, in miles, that the train travels in 풕 hours? ퟒퟓ 풅 ퟒퟓ 풅 Let 푪 be the constant rate the train travels. Then, = 푪, and = 푪; therefore, = . ퟏ 풕 ퟏ 풕 ퟒퟓ 풅 = ퟏ 풕 풅 = ퟒퟓ풕 b. How many miles will it travel in ퟐ. ퟓ hours?

풅 = ퟒퟓ(ퟐ. ퟓ) = ퟏퟏퟐ. ퟓ

The train will travel ퟏퟏퟐ. ퟓ miles in ퟐ. ퟓ hours.

ퟏ 2. Water is leaking from a faucet at a constant rate of gallons per minute. ퟑ a. What is the amount of water, 풘, in gallons per minute, that is leaked from the faucet after 풕 minutes?

Let 푪 be the constant rate the water leaks from the faucet in gallons per minute. Then, ퟏ ퟏ

ퟑ 풘 ퟑ 풘 = 푪, and = 푪; therefore, = . ퟏ 풕 ퟏ 풕 ퟏ 풘 ퟑ = ퟏ 풕 ퟏ 풘 = 풕 ퟑ

b. How much water is leaked after an hour? ퟏ 풘 = 풕 ퟑ ퟏ = (ퟔퟎ) ퟑ = ퟐퟎ

The faucet will leak ퟐퟎ gallons in one hour.

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3. A car can be assembled on an assembly line in ퟔ hours. Assume that the cars are assembled at a constant rate. a. How many cars, 풚, can be assembled in 풕 hours?

Let 푪 be the constant rate the cars are assembled in cars per hour. Then, ퟏ 풚 ퟏ 풚 = 푪, and = 푪; therefore, = . ퟔ 풕 ퟔ 풕 ퟏ 풚 = ퟔ 풕 ퟔ풚 = 풕 ퟔ ퟏ 풚 = 풕 ퟔ ퟔ ퟏ 풚 = 풕 ퟔ

b. How many cars can be assembled in a week? ퟏ ퟐퟒ × ퟕ = ퟏퟔퟖ 풚 = (ퟏퟔퟖ) = ퟐퟖ A week is hours. So, ퟔ . Twenty-eight cars can be assembled in a week.

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ퟏ ퟖퟎ ퟐ 4. A copy machine makes copies at a constant rate. The machine can make copies in ퟐ minutes. a. Write an equation to represent the number of copies, 풏, that can be made over any time interval, 풕.

Let 푪 be the constant rate that copies can be made in copies per minute. Then, ퟖퟎ 풏 ퟖퟎ 풏 ퟏ = 푪, and = 푪; therefore, ퟏ = . ퟐ 풕 ퟐ 풕 ퟐ ퟐ ퟖퟎ 풏 = ퟏ ퟐ 풕 ퟐ ퟏ ퟐ 풏 = ퟖퟎ풕 ퟐ ퟓ 풏 = ퟖퟎ풕 ퟐ ퟐ ퟓ ퟐ ⋅ 풏 = ⋅ ퟖퟎ풕 ퟓ ퟐ ퟓ 풏 = ퟑퟐ풕

b. Complete the table below.

Linear equation: 풕 (time in minutes) 풏 (number of copies) 풏 = ퟑퟐ풕

ퟎ 풏 = ퟑퟐ(ퟎ) ퟎ ퟎ. ퟐퟓ 풏 = ퟑퟐ(ퟎ. ퟐퟓ) ퟖ ퟎ. ퟓ 풏 = ퟑퟐ(ퟎ. ퟓ) ퟏퟔ ퟎ. ퟕퟓ 풏 = ퟑퟐ(ퟎ. ퟕퟓ) ퟐퟒ ퟏ 풏 = ퟑퟐ(ퟏ) ퟑퟐ

c. Graph the data on a coordinate plane.

34 32

푛 , , 22

18 16

Number Number copies of 14 12

8 6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time in minutes, 푡

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d. The copy machine runs for ퟐퟎ seconds, then jams. About how many copies were made before the jam occurred? Explain.

Since ퟐퟎ seconds is ퟎ. ퟑ of a minute, then the number of copies made will be between ퟖ and ퟏퟔ because ퟎ. ퟑ is between ퟎ. ퟐퟓ and ퟎ. ퟓ.

5. Connor runs at a constant rate. It takes him ퟑퟒ minutes to run ퟒ miles. a. Write the linear equation in two variables that represents the number of miles Connor can run in any given time interval, 풕.

Let 푪 be the constant rate that Connor runs in miles per minute, and let 풎 represent the number of miles he ran in 풕 minutes. Then, ퟒ 풎 ퟒ 풎 = 푪, and = 푪; therefore, = . ퟑퟒ 풕 ퟑퟒ 풕 ퟒ 풎 = ퟑퟒ 풕 ퟑퟒ풎 = ퟒ풕 ퟑퟒ ퟒ 풎 = 풕 ퟑퟒ ퟑퟒ ퟒ 풎 = 풕 ퟑퟒ ퟐ 풎 = 풕 ퟏퟕ

b. Complete the table below. Use a calculator and round answers to the tenths place.

Linear equation: 풕 (time in minutes) ퟐ 풎 (distance in miles) 풎 = 풕 ퟏퟕ ퟐ ퟎ 풎 = (ퟎ) ퟎ ퟏퟕ ퟐ ퟑퟎ ퟏퟓ 풎 = (ퟏퟓ) ≈ ퟏ. ퟖ ퟏퟕ ퟏퟕ ퟐ ퟔퟎ ퟑퟎ 풎 = (ퟑퟎ) ≈ ퟑ. ퟓ ퟏퟕ ퟏퟕ ퟐ ퟗퟎ ퟒퟓ 풎 = (ퟒퟓ) ≈ ퟓ. ퟑ ퟏퟕ ퟏퟕ ퟐ ퟏퟐퟎ ퟔퟎ 풎 = (ퟔퟎ) ≈ ퟕ. ퟏ ퟏퟕ ퟏퟕ

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c. Graph the data on a coordinate plane.

d. Connor ran for ퟒퟎ minutes before tripping and spraining his ankle. About how many miles did he run before he had to stop? Explain.

Since Connor ran for ퟒퟎ minutes, he ran more than ퟑ. ퟓ miles, but less than ퟓ. ퟑ miles. Since ퟒퟎ is between ퟑퟎ and ퟒퟓ, then we can use those reference points to make an estimate of how many miles he ran in ퟒퟎ minutes, probably about ퟓ miles.

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