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Rates and Averages

One of the most important skills we can give our students is the ability to interpret data geometrically. The geometry of area and volume can help students understand concepts like rates, accumulations, and average value. Here are three simple examples that illustrate this point:

Figure 19. A bar graph geometrizes data from three similar problems and shows visually how the average corresponds to the height of a single rectangle with the same base and the same total area.

All of these problems involve the same calculation, and all can be illustrated on the same diagram (Figure 19). In each case the total accumulation can be interpreted geometrically as the area of three rectangles. The average will be the height of a single rectangle with the same base and the same total area. It is also possible to graph the accumulation in a way that indicates exactly how many miles had been covered (or how much money had been earned) by a given time (Figure 20).

Figure 20. A linear graph displays the accummulation from three problems, indicationg total miles covered or dollars earned. The relation between the corresponding bar and linear graphs is a precursor to calculus.

Each bar graph representation of rates (which mathematicians call a step function) leads to an accumulation graph formed from straight lines (i.e., a polygonal function). The process of finding the rate from the accumulation leads ultimately to the differential calculus, and finding accumulations from rates leads to the integral calculus. Although it is certainly not necessary for students to realize this connection as they develop their understanding of speeds and distances or salaries and earnings, every student can benefit from this type of mathematical experience both as preparation for calculus and as preparation for life.

Mathematics Awareness Month is sponsored each year by the Joint Policy Board for Mathematics to recognize the importance of mathematics through written materials and an accompanying poster that highlight mathematical developments and applications in a particular area.