This resource is about proportional relationships. To get started, read the following problem:

	A group of kids are making lemonade to sell at their lemonade stand. Following the recipe, they use 6 lemons and end up making 8 cups of lemonade. The next week they use the same recipe with 9 lemons and end up making 12 cups of lemonade.
Sources of images used for this section: Little boy, Microsoft clipart, office.com Lemon tree - Korfu Zitronen, Jean-Luc 2005, Wikimedia commons Glass of lemonade - Shekanjbeen, Mainsari66, Wikimedia Commons

Copy the following table into your notes, and use the information from the problem to fill in the empty cells:

Click here to see the completed table.

Lemons	Cups of Lemonade
3	4
6	8
9	12
12	16
15	20

Use the table to answer the following questions:

What happens to the amount of lemonade the recipe makes if you double the number of lemons?

Check Your Answer

You get double the amount of lemonade.

What happens to the amount of lemonade the recipe makes if you triple the number of lemons?

Check Your Answer

You get triple the amount of lemonade.

What happens to the amount of lemonade the recipe makes if you change the number of lemons by any scale factor?

Check Your Answer

The amount of lemonade changes by the same scale factor.

What rule did you use to fill out the missing information?

Check Your Answer

Every three lemons will make four cups of lemonade. Or 4 3 times the number of lemons will give you the number of cups of lemonade.

The kids decide to sell lemonade one more time. When they go to the fruit bowl they find that they have 5 lemons. How much lemonade can they make now?

Solve the problem on a separate piece of paper, then click to watch the answer.

Answer the following questions based on what you saw in the video.

What was the scale factor used in the first method of solving the problem?

Check Your Answer

The scale factor was 5 3 .

What did this scale factor mean?

Check Your Answer

The kids used 5 lemons instead of 3 lemons, that is 5 3 as many lemons – which makes 5 3 as much lemonade, or 6 2 3 cups.

What was the constant of proportionality used in the second method of solving the problem?

Check Your Answer

The constant of proportionality was 4 3 .

What did this constant of proportionality mean?

Check Your Answer

Each lemon would make 4 3 cups of lemonade. This means that 5 lemons would make 6 2 3 cups.

Source: Platonic solids Lantern,
The Playful Geometer, Flickr

This problem required proportional reasoning. Whatever factor you use to change one variable — in this case the number of lemons, you also use to change the other variable — the number of cups of lemonade.

Danny is helping to decorate the gym for a school dance. It takes Danny 10 minutes to hang 4 strings of flashing lights. If Danny continues to work at the same pace how long will it take him to hang 7 strings of lights?

Copy the following table into your notes, and fill it in with the values from the problem.

Click below to the see the completed table.

Check Your Answer

Strings	Minutes
4	10
7	T

Answer the following questions based on the information in the table:

1. What is the scale factor for the strings? (the factor used to change from 4 strings to 7 strings) Interactive popup. Assistance may be required.

   Hint

   4 x = 7 Interactive popup. Assistance may be required.

   Check Your Answer

   7 4
2. What scale factor should be applied to 10 minutes in order to determine the amount of minutes necessary to hang seven strings of lights? Interactive popup. Assistance may be required.

   Check Your Answer

   7 4
3. How long would it take Danny to finish hanging 7 strings of lights? Interactive popup. Assistance may be required.

   Check Your Answer

   10 by 7 4 = 70 4 = 17.5 minutes
4. State the constant of proportionality.Interactive popup. Assistance may be required.

   Hint

   4 x = 10 Interactive popup. Assistance may be required.

   Check Your Answer

   10 4 (or 5 2 )
5. Explain how you would find the number of minutes it would take to hang 7 strings of lights.Interactive popup. Assistance may be required.

   Check Your Answer

   Multiply the number of strings, 7, by the constant of proportionality, in order to find the number of minutes it takes to hang the lights.
6. How many minutes does it take to hang 7 strings of lights?

   Check Your Answer

   7 x 5 2 = 35 2 = 17.5 minutes

In proportional relationships, both variables change. What is true about the way that they change?

Check Your Answer

The numerical relationship (or scale factor or constant of proportionality) is constant.