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.
David WeesJuly 22, 2013 - 5:25 am -
I like the basic elements of this idea, but I have a few things I think could be improved.
1. I can’t see the tire marks clearly in the first comparison video for the parts B and C, even when I blow it up. Maybe another image with the tire marks marked in a different colour after students ask to be able to see the tire marks more clearly?
2. There isn’t enough data for students to clearly see that the relationship is not linear. My observation is that students will happily “fit” data onto a line if possible, and when asked why it doesn’t fit a perfect line, they’ll say experimental error (or the equivalent).
Is it possible that instead of presenting students with the data, you could partner this lesson with a physics teacher and have the students model the relationship? (See http://tpt.aapt.org/resource/1/phteah/v51/i5/p268_s1?view=fulltext&bypassSSO=1 for an example of a table model using miniature cars – I think you could modify this to be an experiment students do).
Also, perhaps this YouTube video could be useful http://www.youtube.com/watch?v=KLVuakT2nks if there isn’t enough time for an experiment.
David WeesJuly 22, 2013 - 5:50 am -
Also, I’m sorry I didn’t get a chance to work on making a version of this task myself for Monday. Weekends are killer for me now in terms of getting work done, but this task is interesting enough for me that I’ll put in some effort to making my own version of it.
An aside: When I worked in a private school in London, one of my students did this project with an actual car (her father drove, she recorded the stopping distances). She told me she was going to be a lot more careful about how fast she drove after doing her project.
Molly OlsonJuly 22, 2013 - 6:46 am -
I had a friend who was an expert witness for accident court cases to give input about whether a person could have sustained the injuries claimed in the case. Another fascinating extension for this project – her job involved lots of specifics, but in general speed was one of the factors that influenced the results of the accident. I like this makeover, and it could be connected with the no texting law to make it particularly close to student experiences.
BethJuly 22, 2013 - 7:46 am -
I worked on the problem late last night. I decided to start with the graph. You can see my work here: http://algebrasfriend.blogspot.com/2013/07/makeovermonday-put-on-brakes.html
Looking forward to reading comments here since I am not satisfied with my revision.
Dan, is your first picture – the wreck where students estimate the speed copyrighted? I might like to use that one if I pursue this problem.
Ignacio ManceraJuly 22, 2013 - 3:08 pm -
Maybe a sequel/subsequent activity can be planned using this resource:
(or maybe help them buid a model using more data besides the three pictures)
Another interesting thing to done would be to recreate the program that runs in that web page using a spreadsheet or basic programming (depending on the programming skills of the kids).
Eric ScholzJuly 22, 2013 - 5:06 pm -
I like the idea of showing it first, though I don’t think it is a big difference. What I can’t wrap my head around is the accident. How many more feet would the car need to brake in order to stop AFTER the point of accident? It can easily be addressed during the lesson or be a stated speed like all cars were traveling at 20 mph at impact.
l hodgeJuly 22, 2013 - 9:34 pm -
One thing I like about the pictures is that you have to slow down and look at them fairly carefully to get the information you need. I wonder how many students would choose to use skid distance for the x-axis — speed seems more natural to me. Either way, those three points don’t provide any meaningful evidence of a non-linear model. Let’s not pretend that they do.
Evaluating the formula to fill out the table is boring, but not a slam dunk for many students. Solving for a variable under a radical is not easy even if students have seen it before. And, if they haven’t seen it before, would certainly qualify as “critical thinking”. The revision is an easier question — recognizing that speed is related to skid length, using sliders to fit a provided form, and solving graphically.
The Trooper Bob stuff is a bit silly. The purpose of extending the graph is to see what happens. Does it get flatter, less flat, will it ever be horizontal, will it turn around, will it rise above 100 mph? Maybe also look at whether it is a better predictor for high speeds or low speeds by plotting the actual data along with the curve.
Jim cibulkaJuly 23, 2013 - 4:41 am -
Can someone explain to me why this is not a kinetic energy equation (Ek=1/2mv^2)? Question
I’d love to work something like this in if possible.
David WeesJuly 23, 2013 - 7:46 am -
I took some time to re-frame this problem in another way. I don’t know how to produce the third act in a way that would interest kids, although maybe I could use the first three speeds as the data, and the final speed to frame the question.
Paul GormleyJuly 23, 2013 - 9:07 am -
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.
AlecJuly 24, 2013 - 10:20 am -
#8 Jim: it is.
It’s just that it’s the wrong way round with speed on the dependent axis making it the inverse of the quadratic kinetic energy equation, n’est-ce pas? That’s the one tiny thing I didn’t like about this as to me skid length should be a function of speed, the independent variable, but that’s me in full-on hair splitting mode… Great bit of work yet again Dan.