Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum

Session 5, Part A:
Fair Allocations (25 minutes)

In This Part: The Mean | The Mean and the Median

 The term average is a popular one; it is often used, and often used incorrectly. Although there are different types of averages, the typical definition of the word "average" when talking about a list of numbers is "what you get when you add all the numbers and then divide by how many numbers you have." This statement describes how you calculate the arithmetic mean, or average. But knowing how to calculate a mean doesn't necessarily tell you what it represents. Let's begin our exploration of the mean: Using your 45 coins, create 9 stacks of several sizes. You must use all 45 coins, and at least 1 coin must be in each of the 9 stacks. It's fine to have the same number of coins in multiple stacks. Here is one possible arrangement, or allocation, of the 45 coins: Problem A1 Record the number of coins in each of your 9 stacks. What is the mean number of coins in the 9 stacks?

 To find the mean, divide the number of coins (45) by the number of stacks.    Close Tip To find the mean, divide the number of coins (45) by the number of stacks.

Problem A2

Create a second allocation of the 45 coins into 9 stacks.

 a. Record the number of coins in each of your 9 stacks, and determine the mean for this new allocation. b. Why is the mean of this allocation equal to the mean of the first allocation? c. Describe two things that you could do to this allocation that would change the mean number of coins in the stacks.

Problem A3

Create a third allocation of the 45 coins into 9 stacks in a special way:

 • First take a coin from the pile of 45 and put it in the first stack. • Then take another coin from the pile and put it in the second stack. • Continue in this way until you have 9 stacks with 1 coin each, and 36 coins remaining. • Take a coin from the pile of 36 and put it in the first stack. • Then take another coin from the pile and put it in the second stack. • Continue in this fashion until all of the remaining coins have been used.
 a. Now record the number of coins in each of your 9 stacks, and determine the mean for this new allocation. b. What observations can you make about the mean in this special allocation?

This method produces what is called a fair or equal-share allocation. Each stack, in fact, contains the average number of coins. You might think of this as a fair allocation of the 45 coins among 9 people: Each person gets the same number of coins.

 Video Segment In this video segment, Professor Kader asks participants to create snap-cube representations of the number of people in their families. He then asks them to find a way of finding the mean without using calculation. Watch this segment for an exploration of the mean. How does the mean relate to the fair allocation of the data? If you're using a VCR, you can find the first part of this segment on the session video approximately 2 minutes and 37 seconds after the Annenberg Media logo. The second part of this segment begins approximately 8 minutes and 42 seconds after the Annenberg Media logo.

 Session 5: Index | Notes | Solutions | Video