What is so “average” about the Average Rate of Change?

Posted: September 8, 2011 in Math Concepts
Tags: , ,

When asked what the average of a set of data means what answer do you think most people (including mathematics teachers) give? Our experience from leading many professional development workshops and in classroom instruction is to hear a response such as “you add up all of the numbers and divide by how many numbers there are”. Well, that is true if the question had been “How do you calculate the average of a data set?” but is not the answer to what the average means. The response typically given describes the process of how to find the average NOT what the numerical value represents in a real world context.

If we use as an example the average G.P.A. of the students in our classroom, most of our students know that the process for finding the average G.P.A. is to total up all of the G.P.A. points of the students in the classroom and then divide that number by the total number of students in the class.  Once we get a result like 3.12 the question becomes what does that number mean? The meaning is closely linked to the calculation process but is not the same thing. When finding the average we find the total number of G.P.A. points (envision all of the points accumulated on a desk in the classroom) and then divide them up equally (think about handing one G.P.A. point to one student, then the next student, etc. until every point has been physically handed out). We can now clearly see what the average G.P.A. of 3.12 means –  if every student in the class had the same number of G.P.A. points they would posess 3.12.

Likewise, the average shoe size of students in class would be the size of shoe every student in the class would have if every student in class had the same size shoe. Note that the average does not have the same meaning as the mode (the most frequent shoe size) or the median (the shoe size directly in the middle).

How does having a thorough understanding of average help one more clearly make meaning of the average rate of change? The average rate of change truly is an average but it is the average of a change!  We calculate the average rate of change by determining the total change in the ouput value and dividing by the total change in the input value. This results in a value that describes how the output values would change for every change of the input value if the output values changed by the same amount for each change in the input

Let’s consider calculating the average rate of change in the population of Austin, Texas. In 2009 the population was 1,705,075 and is anticipated to be 1,730,267 in the year 2015. To calculate the average rate of change we find the total change in population 1,730,267 – 1,705,075 = 25,192 and the total change in time 2015 – 2009 = 6 years. We then divide the total change in population by the total change in time which is 25,192 people/6 years =  4199 people/year. This means that the population of Austin will increase by 4199 people each year IF the population changes by the amount each year.

Understanding the concept of average is fundamental to the proper understanding of average rate of change in Algebra and Precalculus (and subsequently the notion of derivative in the Calculus series).

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