# Reverse Engineering the Average Rate of Change

## March 16, 2015

Fawn Nguyen shared this clever task, which I decided to copy for my Precalculus class. The calculations build up to the formula for average rate of change. I’ve noticed that if you ask students to write down a formula for the distance from say x = 3 to x = 7, they will immediately state 4. This an example of Gray and Tall’s procepts at work. The formula 7 – 3 simultaneously represents both the process of subtracting 3 from 7 and the resulting number 4. The knee-jerk reaction of most students is to carry out that process immediately without thought. They are uncomfortable with 7 – 3 representing a number.

“Reversing the question” provides a deceptively simple solution to this problem. Students are forced to accept 7 – 3 as a number since the problem only asks what question the calculation is designed to answer. For example, calculation D combines calculations A, B, and C. I wasn’t sure how students would react to this, but they were remarkably comfortable combining their interpretations of A, B, and C. One student incredulously baulked, “but you’re just adding up the change so it’s just the total change.” He seemed to think his answer was too obvious, but I usually have to work much harder to coach more poorly articulated responses from students.

Since students want to simplify answers, I also found that they readily made the connection between calculation D and the numerator in calculation F. I didn’t have to spell this out for them! That’s an amazing amount of scaffolding to pack into such an open-ended problem, and I could imagine applying this trick to so many other situations as well. Basically, if you have a formula that you want to motivate, this is a trick that will almost always work.