Averaging Two Numbers

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Often, students think of averaging as a procedural rule, or algorithm - "Add the items, and divide by the number of items to find the average." This is adequate for the most simple 'find the average' problems, but when information is missing, or the problem involves more than simply finding a quantative average, the 'rule' is of little use. Let's look at a model that will make thinking about averaging more intuitive and build useful concepts for problem solving.

Suppose we build two columns of cubes to represent the numbers 9 and 5, and then level them off, keeping the same number of cubes and columns:

So, averaging can be thought of as a mathematical name for this leveling-off process. The example above can lead to some generalization about the height of the leveled-off columns:

Another way to look at the model gives rise to the more traditional view...

Here are some pairs of numbers to think about. Can you imagine how the two models above could represent the average of the pairs of numbers?

• 92 and 88
• 54 and 56
• 39 and 40
• 103 and 106
• 62 and 78

Students in class apply these models to situations like these. What other information can you discover about the situations? Give it a try...

• Rachel earned 18 points on her first test. She wants to earn enough points on her next test to have an average of 21 points.
• The Dragons have an average of 63 points for their first two basketball games. One of their scores was 76.
• The difference between Katy's weight and Li's weight is 8 pounds. The average of their weight is 73 pounds.