Friday, April 15, 2011

Phooey on Flats, Part III

This week I'm sharing how we teach reasoning in math class. We are using "find the best way to reduce problems caused by flat tires" as an example of a societal problem. ( Read the two earlier blogs in this series )

Here are the problem-solving techniques we teach in 4 through 6th grades:
1. Reasoning using logic
2. Reasoning using patterns
3. Reasoning by trial and error
5. Reasoning using a possibility chart
6. Reasoning by process of elimination
7. Solving problems using deductive reasoning
8. Estimating which answer is most reasonable
9. Reasoning by examining evidence and making notes
10. Reasoning by working backwards from a given solution
11. Determining if there is enough information to solve a problem
Which process might be most useful in deciding on a flat-tire solution?

Re-read the list, then keep them in mind as we review some real flat tire incidents. I will code these 4 different solutions red (useless), black (not sure) and green (helpful):

A. My vehicle has a mini-spare tire. Driving across the desert on paved roads, I hit a small rock and cracked my alloy wheel. The impact damaged the tire irreparably. I changed the wheel, put on the mini-spare and drove home. It took 10 days to get a new wheel and tire.

RUN-FLAT - wouldn't have helped if the rim was broken.
TIRE INFLATOR - wouldn't have helped if the rim was broken.
ROADSIDE SERVICE - the rim was ruined so my spare was essential. There was no mobile phone signal for 50 miles.
TPMS - it would have warned me that the tire lost air (but I learned that as soon as I hit the rock).

B. In the morning my tire was flat. I put on the spare. I could see a huge screw sticking out of the middle of the tread of the flat tire. I marked the spot, pulled out the screw and went to the store to have the flat fixed.

RUN-FLAT - the screw probably wouldn't have mattered to a run-flat tire; it would have held air.
TIRE INFLATOR - would have re-inflated the tire after removing the screw (or I could leave it in).
ROADSIDE SERVICE - someone could have changed my tire for me.
TPMS - it would have warned me that the tire lost air (but I could easily see it).

C. My wife was driving down the street. She realized a rear tire was flat when someone honked and pointed. She limped the car 2 blocks to a parking lot and called me to come and change the tire. The sidewall had been damaged by a pothole.

RUN-FLAT - probably wouldn't have prevented the sidewall damage.
TIRE INFLATOR -  it wouldn't have worked on the sidewall damage.
ROADSIDE SERVICE - someone DID change her tire. Me. This worked!
TPMS - it would have warned her when the tire was damaged.

DI was about to depart on a cross-country trip when I saw my front tire was low. I pulled out a screw sticking out from the corner of the tread, and the tire went flat. No tire stores in town had a new tire, and they wouldn't repair mine. Rather than continue without a spare, I bought an inflator and filled the tire. It's been holding air for 9 years (now as a spare).
RUN-FLAT - none available on my elderly car and wheel.
TIRE INFLATOR -  it DID work on the sidewall / tread puncture.
ROADSIDE SERVICE - someone could have changed my tire, but I would still have had to inflate it.
TPMS - it would have warned me that the tire was leaking.

Don't worry right now about choosing the best solution of the 4 solutions. The question is Which of the 11 math reasoning methods are we using?