Today's opener reviews solving one-step equations, dimensional analysis, and direct/inverse variations.

Today's lesson is looking at bicycle gears as an application problem for direct and inverse variations. I bring in my bike and put it on a stand so that it's easy to rotate the pedals and have the wheels turn. I give them the following handout (because I learned last year that it takes some students a really long time to copy down a table and that wasn't the best use of their class time even though it was more friendly to our budget).

We then begin to work through how many times the rear wheel turns for one revolution of the pedals in different gears. We start by leaving the front sprocket alone and changing just the rear sprocket. I have three students come up: one to rotate the pedals, one to catch the wheel after one revolution (this takes some practice and multiple tries), and one to write the number of revolutions (estimated to the nearest tenth of a revolution) in the table on the smart board.

This turns out to be an inverse variation. We then leave the rear sprocket alone and change the front sprocket, and then do the same thing.

This turns out to be a direct variation.We then try to come up with a proportion relating front teeth, rear teeth, wheel revolutions and 1 pedal revolution, and then try to calculate how many times we have to pedal in different gears to travel 1 km.

If we have time then I give them a couple of dimensional analysis problems involving Lance Armstrong (or possibly give them as homework on the blog).

In 2005 Lance Armstrong won the Tour de France for a record seventh time. Over the course of the race, his mean (average) speed was 41.7 km/h.

a) Find his mean (average) speed in ft/sec.

b) It took him 86 hours, 15 minutes and 2 seconds to complete the Tour de France. How many feet did he go?

If you're interested in cycling, then you might be interested in these videos - The Science of Cycling (part 1, part 2, part 3).

I really like this activity, but last year I felt like the students didn't really "get" it completely. I've simplified a few things and provided them the table (which gives us some extra time), so I'm hoping that helps, but I'm open to any brilliant suggestions on how to make sure they understand it.