Once we started studying the steepness of ramps, I asked kids to compare the steepness of different ramps. Following the excellent CME Geometry some of the ramps that I gave my students had only the angle of inclination given, others had the heights and widths given, and others gave two other sides of the ramp.
The kids came up with three major strategies for comparing the steepness of ramps:
- Draw all the ramps and find their angles, and then just find their angles.
- Find the ratio between the height and the width of the ramp.
- Compare the ramp length to the width.
I asked students whether the first two rules agreed with each other, and they discovered that they did. Huzzah! Two confirmed ways to compare the steepness of ramps.
But we had a major annoyance: how do you find the height/width ratio if all you know is the angle? How do you know the angle if all you have is the height/width ratio?
So we started keeping a conversion chart on the side of the wall. We've been adding a few triangles a day for the last couple of days, but it's time to move this process along.
It's all a little bit half-baked at this point, but here's what I'm thinking:
- In class today, we break into groups and start to make more of the conversion table.
- We start keeping personal tables instead of the class one, because the list is getting too long. Each group decides how they'd like to organize their list. Do you order them by angles? By ratios? Both?
- Each group creates a few problems that can be solved with their tables.
- Then we swap tables and problems with each other. We try to solve your problems with your tables.
- We compare conversion tables, and then talk about what we'd like to improve to make our stuff more usable.
- Then I hand out a trigonometric values table and ask them to figure out what they're looking at.
If I had to guess, my problem will be that this lags a little bit, so I'm interested in ways of adding some structure to help kids feel like they're moving along at a decent pace. I'm very interested in your help sharpening this plan.