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.
I really like your plan! I see a few places it could be tightened up:
ReplyDeletestep 1 and 2 kind of go together. All the groups are going to copy the class one into their personal one, and then they will want to know how many more their group should make. I predict it will be difficult to get them to understand that you want them to make decisions about how the table is organized.
"we compare conversion tables, and then talk about what we'd like to improve to make our stuff more usable" - it sounds like you have an answer in mind to this? Even if not, it will help if you anticipate likely responses and be ready for them.
The last step... is a big one (or maybe a steep ramp...) It seems like your existing whole-class chart involves only tangent... is that correct? I'd be leery about dropping the other ratios on them without motivation.
Thanks SO much. I'm, unfortunately, planning a bit quickly here, but here's my revised version of the instructions.
Delete"Right now we have a table on the wall that helps us convert the angle of a right triangle into the ratio of its height and width. It’s annoying to use the big class one, so we’re going to make our own table to use throughout this unit.
1. Copy the information from the class table into your personal table. You’re going to be using it for the next few days, so make sure it’s organized in a way that makes it useful to you.
2. Pick two new angles, and add them to your table.
3. Pick two new height/width ratios, and add them to your table."
Maybe tomorrow I'll ask them to add the complement of a few of their angles to the table, and maybe also the reciprocal of a few of their ratios?
This is going to be a bit messy. Thanks for the help!