Lesson 9: Interpreting the Mean as Fair Share
Let’s explore the mean of a data set and what it tells us.
Illustrative Math Unit 6.8, Lesson 9 (printable worksheets)
Lesson 9 Summary
The following diagram explains how the mean for a data set represents a “fair share” and how to find the mean for a numerical data set.
Lesson 9.1 Close to Four
Use the digits 0–9 to write an expression with a value as close as possible to 4. Each digit can be used only one time in the expression.
Lesson 9.2 Spread Out and Share
- The kittens in a room at an animal shelter are arranged in five crates, as shown.
a. The manager of the shelter wants the kittens distributed equally among the crates. How might that be done? How many kittens will end up in each crate?
b. The number of kittens in each crate after they are equally distributed is called the mean number of kittens per crate, or the average number of kittens per crate.
Explain how the expression 10 ÷ 5 is related to the average.
c. A different room in the shelter has 6 crates. No two crates contain the same number of kittens, and there is an average of 3 kittens per crate.
d. Draw or describe at least two different arrangements of kittens that match this description. You may choose to use the applet to help.
- Five servers were scheduled to work the number of hours shown in the table. They decided to share the workload, so each one would work equal hours.
a. On the grid on the left, draw a bar graph that represents the hours worked by Servers A, B, C, D, and E.
b. Think about how you would rearrange the hours so that each server gets a fair share. Then, on the grid on the right, draw a new graph to represent the rearranged hours. Be prepared to explain your reasoning.
c. Based on your second graph, what is the average or mean number of hours that the servers will work?
d. Explain why we can also find the mean by finding the value of 31 ÷ 5.
e. Which server will see the biggest change to work hours? Which server will see the least change?
Are you ready for more?
Server F, working 7 hours, offers to join the group of five servers, sharing their workload. If server F joins, will the mean number of hours worked increase or decrease? Explain how you know.
The mean of the five servers is 6.2.
If we add a server that is working 7 hours the mean hours will increase because 7 is bigger than the initial mean of 6.2 hours.
Lesson 9.3 Getting to School
- For the past 12 school days, Mai has recorded how long her bus rides to school take in minutes. The times she recorded are shown in the table.
a. Find the mean for Mai’s data. Show your reasoning.
b. In this situation, what does the mean tell us about Mai’s trip to school?
- For 5 days, Tyler has recorded how long his walks to school take in minutes. The mean for his data is 11 minutes.
a. Without calculating, predict if each of the data sets shown could be Tyler’s. Explain your reasoning.
b. Determine which data set is Tyler’s. Explain how you know.
The average is another name for the mean of a data set.
For the data set 3, 5, 6, 8, 11, 12, the average is 7.5.
3 + 5 + 6 + 8 + 11 + 12 = 45
45 ÷ 6 = 7.5
The mean is one way to measure the center of a data set. We can think of it as a balance point. For example, for the data set 7, 9, 12, 13, 14, the mean is 11.
To find the mean, add up all the numbers in the data set. Then, divide by how many numbers there are. 7 + 9 + 12 + 13 + 14 = 55 and 55 ÷ 5 = 11.
Lesson 9 Practice Problems
- A preschool teacher is rearranging four boxes of playing blocks so that each box contains an equal number of blocks. Currently Box 1 has 32 blocks, Box 2 has 18, Box 3 has 41, and Box 4 has 9.
Select all the ways he could make each box have the same number of blocks.
A. Remove all the blocks and make four equal piles of 25, then put each pile in one of the boxes.
B. Remove 7 blocks from Box 1 and place them in Box 2.
C. Remove 21 blocks from Box 3 and place them in Box 4.
D. Remove 7 blocks from Box 1 and place them in Box 2, and remove 21 blocks from Box 3 and place them in Box 4.
E. Remove 7 blocks from Box 1 and place them in Box 2, and remove 16 blocks from Box 3 and place them in Box 4.
- In a round of mini-golf, Clare records the number of strokes it takes to hit the ball into the hole of each green. She said that, if she redistributed the strokes on different greens, she could tell that her average number of strokes per hole is 3.
Explain how Clare is correct.
- Three sixth-grade classes raised $25.50, $49.75, and $37.25 for their classroom libraries. They agreed to share the money raised equally. What is each class’s equal share? Explain or show your reasoning.
- In her English class, Mai’s teacher gives 4 quizzes each worth 5 points. After 3 quizzes, she has the scores 4, 3, and 4. What does she need to get on the last quiz to have a mean score of 4? Explain or show your reasoning.
- An earthworm farmer examined two containers of a certain species of earthworm so that he could learn about their lengths. He measured 25 earthworms in each container and recorded their lengths in millimeters.
Here are histograms of the lengths for each container.
a. Which container tends to have longer worms than the other container?
b. For which container would 15 millimeters be a reasonable description of a typical length of the worms in the container?
c. If length is related to age, which container had the most young worms?
- Diego thinks that x = 3 is a solution to the equation x2 = 16. Do you agree? Explain or show your reasoning.
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