Averaging Several Numbers

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We can build on the models developed in Averaging Two Numbers to explore more complex situations, and solve more complex problems. Let's begin by looking at a simple model for five numbers.

To portray this method symbolically, we can just create a chart and keep track of our 'moves', just like the numbers under the columns show.

Of course, there's more than one way to think about this process.

We have just taken the difference between the smallest column and the other columns and redistributed the cubes equally among the five columns. Symbolically, this would be:

• 2+ [(4-2) + (7-2) + (9-2) + (3-2)]÷5 = 5

One more method.

This is the traditional method, (4 + 7 + 2 + 9 + 3) ÷ 5 = 5.

Each of these models may prove more useful in certain situations than in others. The idea is to be flexible and use the most appropriate model for the situation, or the model that makes most sense to the problem solver. Here are some situations like those explored in class. How would you apply the models to them?

• The average of 7 numbers is 8.
• Five columns of cubes are arranged in order from smallest to largest. Each column is 3 cubes higher than the column before it. The average height of the columns is 14.
• The heights of three students are 62 inches, 69 inches and 61 inches.
• Karen has an average of 75 points on 3 assignments. On her fourth assignment, she wants to bring her average up to 80 points.
• After four games, the basketball team's average was 85. To get into the tournament, they must have a five-game average of 80 points.