# The Easy Question at the End

I gave a test this week in my pre-calculus class and I’d like to reflect on one test question, the last one of seven. The question set up a realistic problem that modeled the measurement of light intensity using a rational function, the topic of last week’s study. The rational function modeled the intensity, in lux, that a measuring device would detect when placed meters from a low-intensity bulb and, correspondingly, meters from a high-intensity bulb—the bulbs were at a fixed distance apart—ten meters. The details of the function are not important, what I’d like to reflect on was the students’ response to the question.

• Identify the practical domain of the rational function for this application.
• Use your calculator to estimate the minimum and maximum distances from the low-intensity bulb that would register less than 5 lux on the measuring device.

First some background. The week before the test I presented, at the board, as a lecture, a very similar problem. So similar that it showed two light bulbs ten meters apart and a measuring device between them that was modeled by a very similar rational function. We (at least a few students and I) discussed the practical domain of the rational function, given the setup; how it only involved positive values, and in particular, values between zero and 10 meters. I thought the class understood. I then went through an analytic solution of the rational function, which eventually involved the solution of a quartic equation. The solution was long and messy (it was supposed to be). After that, we solved, as a class, on our calculators, the same problem by using a graph of the rational function and a graph of a horizontal line at to find the two points of intersection of the two graphs that would answer the second question of the problem (for a value of 4 lux). I displayed my graphs, in the practical domain, on a screen at the front of the room, and the students, supposedly, created the same graphs, and found the same points of intersection, on their own calculators. The day before we had gone through a different problem that was difficult to solve using a graphing calculator, but straightforward to solve analytically. The two problems were meant to stand as endpoints on a spectrum of problems that were variously easier or harder to solve analytically or graphically.

So how did my students do with a problem that I thought was a gift (and therefore assigned fewer assessment points)? Here’s a summary table of the results.

 Total Answers Correct Answer Incorrect Answer Identify Practical Domain 34 15 19 Analytic Solution Attempted 11 11 Calculator Solution Attempted 11 5 No Solution Attempted 7 7

I think I now understand why a well-formed lecture and a seemingly attentive class do not necessarily mean that the students are learning what I think I am teaching.