**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?

To begin I had 6 different stacks with a different number of cubes in each stack. I then moved cubes around until each stack had the same number of cubes, 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 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?
To begin the task, I placed 9+5+3+4+7+8 on one side of the balance.

I then needed to find out the multiplication equation that would equal 36. I knew that one factor would be 6 because there were six people in the problem. Knowing my facts, I knew that 6 X 6 is 36 and the resulted in a balance. The missing factor is the balance point because it is the same as finding the mean using 36 divided by 6. It is like finding the mean using the inverse operation.

__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?

I collected pencils of 8 different lengths, measured in inches: 3, 4, 5, 6, 7, 8, 9, 10. When I laid all the pencils out end to end they measured a total of 52 inches. When I measured one section of the receipt tape, it measured to be 6.5 inches. When you fold the paper into 8 section, that represents the 8 different pencils. They are all folded the same size, which represents dividing them all equally. The mean is the average, or the amount of 1 section.

__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?

Initial apples

Apples all in one bucket

Apples when sorted out of the bucket

Putting all the apples into the bucket is like finding the sum of the apples, adding all the different peoples apples together to find out the total. When dividing by 6, or the number of people, that is the same as dividing the apples among the people. If they knew there were 36 apples and 6 people they would automatically know they each received 6 apples. This can also be represented by giving each person 1 apple until all the apples are gone. The number of apples each person gets the mean of the apples because it's the average. On average, each person picked 6 apples. Not everyone actually picked six apples but on average.

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