What I Did
- Reduce extraneous literacy demand. A lot of visual information has been encoded in text. Let’s get that information back in its natural medium.
- Delay the abstraction. Tables and graphs and equations will eventually be useful but let’s delay their introduction until we need them.
- Get a better image. The illustration here is a member of the “job testimonial” genre. ie. “Trooper Bob uses math, so you should too.” I’m unconvinced that message will sway classroom opinion on Algebra even a little. Instead let’s put the student into Trooper Bob’s shoes, doing Trooper Bob’s work.
- Ask a better question. Neither of the two questions here addresses any of Trooper Bob’s concerns. The first has you extend a graph for no discernible purpose. (And why extend the graph from 60 feet to 100 feet. Is that just arbitrary?) The second poses the fantastic scenario where Trooper Bob comes to the scene of a wreck already aware of how fast the car was traveling and then proceeds to do math to figure out the length of the tire marks in front of him. Which he could just measure.
- Add intuition. Per usual.
So show this picture of a wreck. Ask your students to guess how fast you think that car was going when it hit the brakes. Tell them they have to figure out if it broke the law. Do they think it was speeding?
Then show them this image.
Ask your students to rank the cars from fastest to slowest. Ask them how they know. They’ve decided the variables “length of skid” and “speed” are positively related. But what kind of relationship is it? This is where a graph — a picture of a relationship — is so useful. Show them the data.
Have them graph the data. This is a little new to us. It isn’t linear. It isn’t quadratic. It isn’t exponential. Offer an explanation of the root model. It’s the inverse of a parabola. With the parabola, a little growth in the horizontal direction results in a lot of growth in the vertical direction. With the root model, a lot of growth is required in the horizontal direction before you get even a little growth in the vertical direction.
Now they can find the exact model for these data and evaluate it for 232.7 feet.
68 miles per hour in a residential zone? You won’t be needing that drivers license for a long time.
What You Did
Over on the blogs:
- Bob Lochel has students think through the root model much more comprehensively than the original textbook task.
Over on the Twitter:
- Nicholas Chan encourages modeling also, where students make predictions from data.
- Eric Scholz has the same, except where I start with the accident you’re trying to solve and then get smaller data for modeling, he starts by showing students the smaller data and then ending with the accident you’re trying to solve. Is the difference substantial?
- Matthew Jones sends along this clip, which would make for interesting watching after our math work. I’m not sure what work the students would do on the video, though.
- Kate says, “bring the cop to school,” which could be great, but again what math work do the students do?
Now this is the reason I follow this blog. I am a criminal justice instructor and many students come to my field as fugitives from math and science. Use of these materials is helping me develop criminal justice contextualized math resources for a class proposal.