# Exploring Arithmetic Mean

Another task from my Masters that I had to share on my blog ! Four tasks to explore arithmetic mean. Even if elementary students do not formally study arithmetic mean, it is still a fun concept to explore !

Exploration #1: The Leveling Model.

Joan, Jane, John, Jen, Jack, and Jill went up the hill to fetch a pail of apples. Joan found 9 apples, Jane picked 5 apples, John got 3 apples, Jen fetched 4 apples, Jack found 7 apples, and Jill got 8 apples. The six friends decide to share the apples equally among them. Use stacks of unifix cubes to model the apples that each friend picked (use a different color of cubes for each friend, if possible). Then use the cubes to distribute the apples among the friends. (That is, find the arithmetic mean number of apples for the friends.) How many cubes (apples) are in each stack initially? How many cubes are in each stack after distributing apples? What is the arithmetic mean number of apples?

Initially, there were  6 stacks of cubes, with 9 cubes, 5 cubes, 3 cubes, 4 cubes, 7 cubes, and 8 cubes. After distributing the cubes, there were 6 cubes in each stack. The arithmetic mean number of apples is 6.

Exploration #2: Balance Model.

Joan, Jane, John, Jen, Jack, and Jill went up the hill to fetch a pail of apples. Joan found 9 apples, Jane picked 5 apples, John got 3 apples, Jen fetched 4 apples, Jack found 7 apples, and Jill got 8 apples. Use a balance (here) to determine the arithmetic mean number of apples for the six friends. On one side of the balance write an addition equation to represent the number of apples. On the other side of the balance write a multiplication equation where one of your other factors represents the number of people. You need to determine what the other factor is that will make the balance level. What missing factor proved to be the one that balanced the original side? Why did the missing factor prove to be the balance point? What is the arithmetic mean number of apples?

The missing factor that balanced the original side is 6 (apples / person). When multiplied by the number of people, the missing factor balances the total amount of apples (6 apples / person x 6 people = 9 + 5 + 3 + 4 + 7 + 8 apples). The missing factor represents the arithmetic mean number of apples, which is 6.

Exploration #3:

Collect a number of pencils from students. The aim is to use eight pencils, but collect more so that from this number, eight pencils of different lengths can be used. Lay the pencils end to end (this works really well in the grooved tray of some marker boards or chalkboards. With the pencils laid end-to-end, measure the total length of the pencils with adding machine tape or similar strip of paper. Now fold the paper in half; fold this fold length a second time, and then this length a third time. Unfold the paper to observe eight sections of equal length, each the arithmetic mean length of the pencils. What mathematical process summarizes the laying of the pencils end to end? What mathematical process summarizes the folding of the adding machine tape? Why does the length of any of the eight pieces of paper represent the arithmetic mean of the length of the pencils?

The laying of the pencils end to end represents the sums of the pencil lengths. The folding of the adding machine tape represents the division of the total length of the pencils by the number of pencils. The length of any of the eight pieces represents the arithmetic mean of the length of the pencils, as it represents the total length of the pencils divided by the number of pencils.

Exploration #4:

Joan, Jane, John, Jen, Jack, and Jill went up the hill to fetch a pail of apples. Joan found 9 apples, Jane picked 5 apples, John got 3 apples, Jen fetched 4 apples, Jack found 7 apples, and Jill got 8 apples. When they got back down the hill, they wanted to share the apples equally. So they dumped their apples in a bushel basket and proceeded to distribute them fairly. Model this situation with unifix cubes and use them to find the arithmetic mean. What mathematical process summarizes the dumping of apples into the bushel basket? What mathematical process summarizes the distribution of the apples? Why does the number of apples each friend eventually gets represent the arithmetic mean of the apples?

The dumping of apples into the bushel basket represents the mathematical process of adding all the apples. The distribution of the apples represents the equal sharing of the apples among each friend. The number of apples each friend gets represents the arithmetic mean of the apples i.e. the total number of apples divided by the number of people.