kitchen table math, the sequel: Fourth grade question on Bergen Academies math competition

## Saturday, July 28, 2007

Bergen Academies, a magnet high school in New Jersey, offers math competitions for area math whiz kids from 4th to 8th grade.

Here's a question on the 4th grade test from 2005:

Mary bakes a pie that starts off at 300 degrees, but it cools at a rate of 4 degrees per second. The room she is in is initially at 50 degrees, but heats up at a rate of 0.5 degrees per second. How long will it take for the room to reach half the temperature of the pie?

SusanJ said...

I think it's really neat that a 4th grader can set up and solve an algebraic equation in one unknown.

I had to read this question several times to understand it, however. The first sentence would be easier to understand if it said, "When Mary took a pie out of the oven, the pie was at 300 degrees but immediately began cooling at a rate of 4 degrees per second."

Also the last sentence as written might (on first read) be misinterpreted as referring to the original temperature of the pie. What about, "How long will it take the temperature of the room to become equal to half the temperature of the pie?"

If you follow the rule of parallel construction for good writing, the question would read, "How long will it take the temperature of the pie to become equal to twice the temperature of the room?"

Here's my complete rewrite.

When Mary took a pie out of the oven, the pie was at 300 degrees but immediately began cooling at a rate of 4 degrees per second. This caused the room to start heating up at a rate of 0.5 degrees per second. How long will it take the temperature of the pie to become equal to twice the temperature of the room?

Since this is a "real world problem" it would also be preferable if the numbers were more realistic.

David said...

"Half the temperature" does not make sense, because temperature is an inteval measure, not a ratio measure.

For example, one might think that a temperature of 40F is half as great as a temperature of 80F. But this depends on choosing an arbitrary zero point. The Celsius equivalents are 4C and 27C, and 4 is not half of 27.

I don't expect fourth graders to understand
levels of measurement
, but instructors should be aware of it, and they should avoid posing problems involving meaningless operations. Otherwise the students may learn by example that it is ok to multiply and divide temperatures.

SusanJ said...

David, I completely agree and I'm glad you brought this up. I had thought about mentioning this but decided against it partly because I'd already written a lot and partly because there was so much wrong with the question from a scientific level that I didn't know where to begin. (For example, how would you explain to a math student why you can't multiply by temperatures but you can multiply by temperature differentials as long as you are using a consistent degree size?)

The temperature issue is the biggest one but I've thought two others. One is that the rate of conductive cooling depends on the differential in temperature between the two bodies and is thus not a linear function of time.

Also heat capacity is a function of temperature.

(As you can see, I'm not a big fan of "general science" in the early grades.)

You could pose essentially the same problem by using the scenario that Mary has 300 dimes and John has 50 dimes and that every time Mary puts 4 dimes in a jar of coins, John takes out a nickel.