Lesson2:&&Proportional&Relationships&...Lesson&2:& Proportional!Relationships! Date:& 7/22/15! 17& ©!2014!Common!Core,!Inc.!Some!rights!reserved.!commoncore.org! This!work!is!licensed!under!a!!

Lesson 2: Proportional Relationships Date: 7/22/15

17

© 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-‐NonCommercial-‐ShareAlike 3.0 Unported License.

NYS COMMON CORE MATHEMATICS CURRICULUM 7•1 Lesson 2

Lesson 2: Proportional Relationships

Student Outcomes

§ Students understand that two quantities are proportional to each other when there exists a constant (number) such that each measure in the first quantity multiplied by this constant gives the corresponding measure in the second quantity.

§ When students identify the measures in the first quantity with 𝑥 and the measures in the second quantity with 𝑦, they will recognize that the second quantity is proportional to the first quantity if 𝑦 = 𝑘𝑥 for some positive number 𝑘. They apply this same relationship when using variable choices other than 𝑥 and 𝑦.

Classwork

Example 1 (10 minutes): Pay by the Ounce Frozen Yogurt!

The purpose of this example is for students to understand when measures of one quantity are proportional to measures of another quantity.

Example 1: Pay by the Ounce Frozen Yogurt!

A new self-‐serve frozen yogurt store opened this summer that sells its yogurt at a price based upon the total weight of the yogurt and its toppings in a dish. Each member of Isabelle’s family weighed their dish and this is what they found. Determine if the cost is proportional to the weight.

Weight (ounces) 𝟏𝟐.𝟓 𝟏𝟎 𝟓 𝟖

Cost ($) 𝟓 𝟒 𝟐 𝟑.𝟐𝟎

The cost _____________________________________ the weight.

Discussion

§ Does everyone pay the same cost per ounce? How do you know?

ú Yes, it costs $0.40 per ounce. If we divide each cost value by its corresponding weight, it will give the same unit price (or unit rate) of 0.40. Since we want to compare cost per ounce, we can use the unit (cost per ounce) to determine that we want to divide each cost value by each corresponding weight value.

§ Isabelle’s brother takes an extra-‐long time to create his dish. When he puts it on the scale, it weighs 15 ounces. If everyone pays the same rate in this store, how much will his dish cost? How did you calculate this cost?

ú $6. I determined the cost by multiplying 0.40 by 15 ounces § Since this is true, we say “the cost is proportional to the weight.” Have students complete the statement in

their materials.

Note: Guide students to use the precise language when saying that two quantities are proportional to each other. In this


18



𝟏𝟐 𝟏𝟔 𝟖 𝟒 𝟎

𝟏 𝟏𝟐 𝟐 𝟏 𝟏𝟐 𝟎

Cups

Ounces

example, it is correct to say that cost (y) depends on weight (x) because the dependent variable (y) depends on the independent variable (x). This understanding will be useful when determining the labels for the x-‐axis and y-‐axis when graphing ordered pairs in the coordinate plane in Lesson 5.

§ What happens if you don’t serve yourself any yogurt or toppings? How much do you pay?

ú $0. § Does the relationship above still hold true? In other words, if you buy 0 ounces of yogurt then multiply by the

cost per ounce do you get 0? ú Even for 0, you can still multiply by this constant value to get the cost (not that you would do this but

we can examine this situation for the sake of developing a rule that is always true).

§ You always multiply the number of ounces, 𝑥, by the constant that represents cost per ounce, to get the total cost, 𝑦. Pause with students to note that any variables could be chosen but, for the sake of this discussion, they are 𝑥 and 𝑦.

Teacher should label the table with the indicated variables and guide students to do the same.

§ For any measure 𝑥, how do we find 𝑦? ú Multiply it by 0.40 (unit price).

§ Indicate this on the given chart as done below. Be sure students do the same.

𝑥, Weight (ounces) 12.5 10 5 8

𝑦, Cost ($) 5 4 2 3.20

§ So, 𝑦 = 0.40𝑥. Note: Using the variables w for weight and c for cost might help students conceptualize the relationship as an equation. In this example, c = 0.40w represents cost = 0.40 x weight. Once this understanding is established, the variables w and c can be replaced with x and y, which will be necessary when graphing in the coordinate plane.

Example 2 (5 minutes): A Cooking Cheat Sheet!

Example 2: A Cooking Cheat Sheet!

In the back of a recipe book, a diagram provides easy conversions to use while cooking.

The ounces _____________________________________the cups.

• 0.40 • 0.40 • 0.40 • 0.40


19



§ What does the diagram tell us?

ú The number of ounces in a given number of cups. More specifically, each pair of numbers indicates the correct matching of ounces to cups.

§ Is the number of ounces proportional to the number of cups? How do you know?

ú Yes, you can multiply each number of cups by 8 to get the number of ounces.

Have students complete the statement on their materials, ounces is proportional to cups, and note how they can tell. It is important to acknowledge that you could also divide by 8 if you know the number ounces and are trying to find the number of cups. This discussion should lead to the importance of clearly defining the quantities.

§ How many ounces are there in 4 cups? 5 cups? 8 cups? How do you know? ú 32, 40, 64; each time the number of cups is multiplied by 8 to get the number of ounces.

§ For the sake of this discussion (and to provide continuity between examples), let’s represent the cups with 𝑥 and the ounces with 𝑦.

Teacher should label the diagram with the indicated variables and guide students to do the same.

§ For any number of cups 𝑥, how do we find the number of ounces, 𝑦? ú Multiply 𝑥 by 8.

§ So, 𝑦 = 8𝑥. § If we want to verify that our equation is 𝑦 = 8𝑥, which 𝑥 and 𝑦 values can we use to see if it is true? How do

you know?

ú We can choose any pair of given (𝑥, 𝑦) values since the equation should model the relationship for every pair of values.

It is a good idea to check more than one pair. Guide students to substitute the pairs of values into the equation to prove the equation is true for each one.

Exercise 1 (5 minutes)

Have students complete the following example independently, and then discuss responses as a class.

Exercise 1

During Jose’s physical education class today, students visited activity stations. Next to each station was a chart depicting how many calories (on average) would be burned by completing the activity.

Calories Burned while Jumping Rope

𝟑𝟑 𝟒𝟒 𝟐𝟐 𝟏𝟏 𝟎

𝟑 𝟒 𝟐 𝟏 𝟎 Time (minutes)

Calories Burned

Extension:

Because conversion tables can be used in two ways, an extension might ask students to write an equation that could convert ounces to cups where ounces are represented by 𝑥 and cups by 𝑦. In this case the equation would be 𝑦 = !

! which is

equivalent to 𝑦 = !!𝑥.


20



MP.1

a. Is the number of calories burned proportional to time? How do you know?

Yes, the time is always multiplied by the same number, 𝟏𝟏, to find the calories burned.

b. If Jose jumped rope for 𝟔.𝟓 minutes, how many calories would he expect to burn?

Jose would expect to burn 𝟕𝟏.𝟓 calories since 𝟔.𝟓 times 𝟏𝟏 is 𝟕𝟏.𝟓.

Example 3 (15 minutes): Summer Job

Read through Example 3 aloud. Allow for brief discussion (2 minutes) of summer jobs or ways students may have earned money over the summer. Pose the following questions:

§ How much do you think Alex had earned by the end of 2 weeks? ú He probably earned twice what he had earned in week 1.

§ How will a table help us to check Alex’s prediction?

ú It will help us to see how his earnings grow over time and whether he will have enough money by the end of the summer. A table may also help to check calculations for reasonableness.

§ Where did the two given pairs of data come from?

ú He earned $112 after working 4 weeks. Therefore, his rate was $28 for every 1 week. The total earnings is 28 times the week number.

§ Is this reasonable?

ú Yes. You could include a brief discussion of minimum wage for part-‐time workers or babysitting rates so that students have some sense of reasonable earning amounts.

§ What other pair could we complete fairly easily?

ú At 0 weeks, he has earned $0. § If he makes the same amount of money each week, how will we find out his earnings after 2 weeks? 3 weeks?

ú Since the rate will be the same, we could multiply each number of weeks by 28 to get the corresponding total earnings.

Allow students time (3 minutes) to answer part (a) and complete the remaining values, if needed. Give students time to share responses to part (a).

Example 3: Summer Job

Alex spent the summer helping out at his family’s business. He was hoping to earn enough money to buy a new $𝟐𝟐𝟎 gaming system by the end of the summer. Halfway through the summer, after working for 𝟒 weeks, he had earned $𝟏𝟏𝟐. Alex wonders, “If I continue to work and earn money at this rate will I have enough money to buy the gaming system by the end of the summer?”

To determine if he will earn enough money, he decided to make a table. He entered his total money earned at the end of Week 1 and his total money earned at the end of Week 4.

Week 𝟎 𝟏 𝟐 𝟑 𝟒 𝟓 𝟔 𝟕 𝟖

Total Earnings

$𝟐𝟖 $𝟏𝟏𝟐


21



Extension:

This problem might be too easy for some students. Below is an adaptation to the problem. The solution will follow the same reasoning as indicated in the original problem with changes to the total cost of the gaming system and the total number of weeks.

a. Work with a partner to answer Alex’s question.

Yes, Alex will have earned enough money to buy the $𝟐𝟐𝟎 gaming system by the end of the summer because he will have earned 𝟖 ⋅ 𝟐𝟖 or 𝟐𝟐𝟒 dollars for the 𝟖 weeks he worked. A sample table is shown below.

Week 𝟎 𝟏 𝟐 𝟑 𝟒 𝟓 𝟔 𝟕 𝟖

Total Earnings $𝟎 $𝟐𝟖 $𝟓𝟔 $𝟖𝟒 $𝟏𝟏𝟐 $𝟏𝟒𝟎 $𝟏𝟔𝟖 $𝟏𝟗𝟔 $𝟐𝟐𝟒

Allow for students to share responses with the class for part (b), and then record in their student pages.

b. Are Alex’s total earnings proportional to the number of weeks he worked? How do you know?

Alex’s total earnings are proportional to the number of weeks he worked. There exists a constant value, 𝟐𝟖, which can be multiplied by the number of weeks to determine the corresponding earnings for that week. The table shows an example of a proportional relationship.

Alex spent the summer helping out at his family’s business. He was hoping to earn enough money to buy a new $𝟐𝟓𝟎 gaming system by the end of the summer. Partway through the summer, after working for 𝟒 weeks, he had earned $𝟏𝟏𝟐. Alex wonders, “If I continue to work and earn money at this rate, will I have enough money in 5 more weeks to buy the gaming system by the end of the summer?”

To determine if he will earn enough money, he decided to make a table. He entered his total money earned at the end of Week 1 and his total money earned at the end of Week 4.

Week 𝟎 𝟏 𝟐 𝟑 𝟒 𝟓 𝟔 𝟕 𝟖 𝟗

Total Earnings

$𝟐𝟖 $𝟏𝟏𝟐

Yes, Alex will have earned enough money to buy the $𝟐𝟓𝟎 gaming system by the end of the summer because he will have earned 9⋅ 𝟐𝟖 or 𝟐𝟓𝟐 dollars for the 9 weeks he worked. A sample table is shown below.

Week 𝟎 𝟏 𝟐 𝟑 𝟒 𝟓 𝟔 𝟕 𝟖 𝟗

Total Earnings $𝟎 $𝟐𝟖 $𝟓𝟔 $𝟖𝟒 $𝟏𝟏𝟐 $𝟏𝟒𝟎 $𝟏𝟔𝟖 $𝟏𝟗𝟔 $𝟐𝟐𝟒 $𝟐𝟓𝟐

Alex’s total earnings are proportional to the number of weeks he worked. There exists a constant value, 𝟐𝟖, which can be multiplied by the number of weeks to determine the corresponding earnings for that week. The table shows an example of a proportional relationship.


22



Closing (2 minutes)

§ How do we know if two quantities are proportional to each other?

ú Two quantities are proportional to each other if there is one constant number that is multiplied by each measure in the first quantity to give the corresponding measure in the second quantity.

§ How can we recognize a proportional relationship when looking at a table or a set of ratios?

ú If each of the measures in the second quantity is divided by its corresponding measure in the first quantity and it produces the same number, called a constant, then the two quantities are proportional to each other.

Exit Ticket (8 minutes)

Lesson Summary

Measures in one quantity are proportional to measures of a second quantity if there is a positive number 𝒌 so that for every measure 𝒙 of the first quantity, the corresponding quantity 𝒚 is given by 𝒌𝒙. The equation 𝒚 = 𝒌𝒙 models this relationship.

A proportional relationship is one in which the measures of one quantity are proportional to the measures of the second quantity.

In the example given below, the distance is proportional to time since each measure of distance, 𝒚, can be calculated by multiplying each corresponding time, 𝒕, by the same value, 𝟏𝟎. This table illustrates a proportional relationship between time, 𝒕, and distance, 𝒚.

Time (h), 𝒕 𝟎 𝟏 𝟐 𝟑 Distance (km), 𝒚 𝟎 𝟏𝟎 𝟐𝟎 𝟑𝟎


23



Name ___________________________________________________ Date____________________


Exit Ticket

Ms. Albero decided to make juice to serve along with the pizza at the Student Government party. The directions said to mix 2 scoops of powdered drink mix with a half gallon of water to make each pitcher of juice. One of Ms. Albero’s students said she will mix 8 scoops with 2 gallons of water to make 4 pitchers. How can you use the concept of proportional relationships to decide whether the student is correct?

Circle one:

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


24



Name ___________________________________________________ Date____________________


Exit Ticket #2

Ms. Albero decided to make juice to serve along with the pizza at the Student Government party. The directions said to pour !

! gallon of water into a container and add 2 scoops of powdered drink mix to make 1 pitcher of juice. One of Ms.

Albero’s students said she will pour 2 gallons of water into a large container and add 8 scoops of powdered drink mix to make 4 pitchers. How can you use the concept of proportional relationships to decide whether the student is correct?

Circle one:

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


25



Exit Ticket Sample Solutions

Ms. Albero decided to make juice to serve along with the pizza at the Student Government party. The directions said to mix 𝟐 scoops of powdered drink mix with a half gallon of water to make each pitcher of juice. One of Ms. Albero’s students said she will mix 𝟖 scoops with 𝟐 gallons of water to make 4 pitchers. How can you use the concept of proportional relationships to decide whether the student is correct?

Amount of Powdered Drink Mix (scoops) 𝟏 𝟐 𝟒 𝟖

Amount of Water (gallons) 𝟏𝟒

𝟏𝟐

𝟏 𝟐

As long as the amount of water is proportional to the number of scoops of drink mix, then the second quantity, amount of water, can be determined by multiplying the first quantity by the same constant. In this case, if the amount of powdered

drink mix is represented by 𝒙, and the gallons of water is represented by 𝒚, then 𝒚 = 𝟏𝟒𝒙. To determine any of the

measures of water, you will multiply the number of scoops by 𝟏𝟒.

Exit Ticket #2: Ms. Albero decided to make juice to serve along with the pizza at the Student Government party. The directions said to pour 𝟏

𝟐 gallon of water into a container and add 2 scoops of powdered drink mix to make 1 pitcher of juice. One of Ms. Albero’s students

said she will pour 𝟐 gallons of water into a large container and add 𝟖 scoops of powdered drink mix to make 𝟒 pitchers. How can you use the concept of proportional relationships to decide whether the student is correct?

In the second Exit Ticket, it is clear that the water is the first quantity and the powdered drink mix is the second quantity.

Amount of Water (gallons) 𝟏𝟒

𝟏𝟐

𝟏 𝟐

Amount of Powdered Drink Mix (scoops) 𝟏 𝟐 𝟒 𝟖

In this case, as long as the number of scoops of drink mix is proportional to the amount of water, then the second quantity, number of scoops of drink mix, can be determined by multiplying the first quantity by the same constant. In this case, if the amount of water is represented by 𝒙, and the number of scoops of drink mix is represented by 𝒚, then 𝒚 = 𝟒𝒙. To determine any number of scoops of drink mix, you will multiply the amount of water by 4.


26



Model Problems 1. At the Fremont Fall Festival, 2 hot dogs can be purchased for $5.00 and 5 hot dogs can be purchased for $12.50. Complete the table to show different numbers of hot dogs and their cost. Explain why cost is proportional to the number of hot dogs.

Number of Hot Dogs

1 2 3 4 5

Cost ($)

5.00 12.50

Solution:

Number of Hot Dogs

1 2

X 2.50

3 4 5

X 2.50

Cost ($)

2.50 5.00 7.50 10.00 12.50

!.!!!= 2.50 !".!"

!= 2.50

Cost is proportional to the number of hot dogs since there exists a constant number, 2.50 (the unit price) such that the number of hot dogs multiplied by this constant gives the corresponding cost.


27



2. The school colors for Cardinal Middle School are red and yellow. The music club is ordering flowers in the school colors to sell at the first concert. Complete the table to show different number of red roses and yellow roses that could be ordered if the desired ratio of red to yellow roses is 5 to 4. Explain why these quantities are proportional.

Number of Red Roses

Number of Yellow Roses

Solution:

Below is one way to complete the table and determine if the quantities are proportional.

Number of Red Roses (x)

5

× 0.8

10 15 25

Number of Yellow Roses (y)

4 8 12 20

If x represents the number of red roses and y represents the number of yellow roses, then yellow roses are proportional to red roses since there exists a constant number 0.8 that when multiplied by each number of red roses gives the number of yellow roses. The equation y = 0.8x represents this relationship (yellow = 0.8 ·∙ red).

Note: The value 0.8 is determined by finding the quotient of y and x in its decimal form.

!!= 0.8 !

!"= 0.8 !"

!"= 0.8 !"

!"= 0.8

Using the fraction form of the quotient of y and x is also acceptable. In this case, the equation would be 𝑦 = !!𝑥.

If in this problem, however, x represented the number of yellow roses and y represented the number of red

roses, then red roses would be proportional to yellow roses since there would exist a constant number !! that

when multiplied by the number of yellow roses gives the corresponding number of red roses. The equation

𝑦 = !!𝑥 represents this relationship (red = !

! ·∙ yellow).

Note: The value !! is determined by finding the quotient of y to x. Leaving the quotient in its fraction form is

another way to write the constant.

!"!= !

! !"

!" = !

! !"

!"= !

!


28



Problem Set Sample Solutions

1. A cran-‐apple juice blend is mixed in a ratio of cranberry to apple of 𝟑 to 𝟓. a. Complete the table to show different amounts that are proportional.

Amount of Cranberry 𝟑 𝟔 𝟗

Amount of Apple 𝟓 𝟏𝟎 𝟏𝟓

b. Why are these quantities proportional?

The amount of apple is proportional to the amount of cranberry since there exists a constant number, 𝟓𝟑 ,that

when multiplied by any of the given measures for the amount of cranberry always produces the corresponding amount of apple. If the amount of cranberry is represented by 𝒙, and the amount of apple is

represented by 𝒚, then each pair of quantities satisfies the equation 𝒚 = 𝟓𝟑𝒙. A similar true relationship could

be derived by comparing the amount of cranberry to the amount of apple. In the case where 𝒙 is the amount

of apple and 𝒚 is the amount of cranberry, the equation would be 𝒚 = 𝟑𝟓𝒙.

2. John is filling a bathtub that is 𝟏𝟖 inches deep. He notices that it takes two minutes to fill the tub with three inches of water. He estimates it will take ten more minutes for the water to reach the top of the tub if it continues at the same rate. Is he correct? Explain.

Yes. In 𝟏𝟎 more minutes, the tub will reach 𝟏𝟖 inches. At that time, the ratio of time to height may be expressed as

𝟏𝟐 to 𝟏𝟖, which is equivalent to 𝟐 to 𝟑. The height of the water in the bathtub increases 𝟏 𝟏𝟐 inches every minute.

3. A conversion chart in a fabric store shows the number of inches that correspond to fractional parts of a foot. Determine if inches are proportional to feet.

There exists a constant number, 12, such that each quantity of feet when multiplied by 12 gives the number of inches. If x represents the number of feet and y represents the number of inches, then an equation that can represent this relationship is y = 12x.

Time (minutes) 𝟏 𝟐 𝟏𝟐

Bathtub Water Height (inches) 𝟏𝟏𝟐

𝟑 𝟏𝟖

9 12 6 3 0

34

1 12

14 0

Feet

Inches

Documents

Lesson2:&&Proportional&Relationships&...Lesson&2:& Proportional!Relationships! Date:& 7/22/15! 17& ©!2014!Common!Core,!Inc.!Some!rights!reserved.!commoncore.org! This!work!is!licensed!under!a!!