Thoughts on Variation

To me, variation is one of the most powerful ideas we meet in algebra.

1) The videos vividly illustrate the power of quadratic growth. Sadly, traffic collisions are the leading cause of death for our students.

2) The diversity of variation questions is astounding. Excellent problems can be chosen from the social sciences. Students can find their own and present them to the class.

3) English translation is essential to variation, as it is to many topics in essential mathematics (what others may call developmental mathematics). Too often we expect our students to learn by parroting what "varies," "directly," "jointly" and "inversely" mean. I do not see that anything is lost by simply telling them. Variation can be translated literally, so the conversation can be short and to the point. "Varies" means "is a multiple of" so we can translate it as `=k*` . This frees up time to talk about what variation is.

4) When we return to variation in other courses, like Math 120 and Math 125, I extend the topic to cover earthquakes. Since earthquakes vary over such a large scale, it reintroduces the idea of magnitude. Even variation on the Richter scale can be translated as a variation problem.

5) The algebra of solving variation problems can be made into a curve fitting model. I made a GeoGebra file to do this for the speed/skid problem. The file can be modified for natural events that vary directly with the square. If you want a particular situation, like linear variation, tell me. Some folks may want to solve all the variation problems by curve fitting the model. This may be more appropriate in courses like Math 125.

6) One of the most powerful aspects of variation is how easily units are handled. The multiple, `k` , means that we do not need to concern ourselves with the units. By comparison, the units in a distance rate time problem must all agree. I point this out to students and remind them that we have to include the units in their answer, even if `k` has nicely handled the units for us throughout the problem.

7) Has anyone used a photocell being moved from a light source to capture the inverse square law of the propagation of light?

--Ben