My son is in 7th grade. I was checking his math homework the other day, and came upon this question:
The temperature yesterday was -5 degrees C. Today, the temperature is -20 degrees C. How many times colder is it today than it was yesterday?
AAAAAAARRRRGH!
I sent a note with the homework to the teacher saying (in essence), my son’s answer to this question was 4. That’s probably what the textbook publisher was looking for. But it’s wrong! Completely and utterly wrong! For the concept “four times as hot” (or cold) to have any meaning, it must take energy levels into consideration. If the temperatures had been listed in degrees Kelvin, I probably wouldn’t have said anything, but -20C is simply not four times as cold as -5C.
Okay–I posted this in GQ instead of the Pit for a reason. Can any of you come up with any justification for this question, or am I right that it’s simply meaningless drivel that some poor physics teacher is going to have to “un-teach” sometime down the road?
Lessee. Convert to Kelvin for an absolute zero scale, so we’ve got 268 K and 253 K. Assuming you do 253/268 (what I’m assuming is that you did 20/5), that comes out to 0.944029851 plus some more digits if you want to take it out even more absurdly far. That’s a temperature change of what, six percent?
Richard Feynman had the same reaction while reviewing textbooks – his example was asking students to calculate the average temperature of a bunch of stars. Math textbooks are written under pressure to include word problems from which students have to extract the necessary information. They don’t seem to mind if the situation is impossible, or the problem makes no physical sense.
Feynman’s wife likened it to living over a volcano – he worked in the basement, and would periodically erupt.
It’s a meaningless question. As posed, it has no answer. Temperature measurements can’t be treated like other common measurements, unless you use an absolute scale like Kelvin or Rankine.
Yeah, it’s stupid, but if you have a scale with a zero point on it, people are going to do this sort of thing, whatever the underlying physical motivation of the scale. They really mean “By what factor is today’s temperature further below zero on the Celsius scale than yesterday?”. In casual use, the arbitrary zero point winds up separating “hot” from “cold” in people’s minds, meaningless as that might be.
The textbook still shouldn’t reinforce it, and I’d say it’s a bad question.
I’m curious about something else - it’s a word problem to motivate an exercise in simple arithmetic, making sure the student knows how to handle negative numbers. 7th grade? It’s seems more like an elementary school level problem, doesn’t it?
I expect that the point is not to re-enforce basic arithmetic skills, but rather to learn how to apply those skills to real-life situations. This makes it all the more tragic that it is incorrect to apply those skills in the manner that the textbook expects.
At the end of each section, they have word problems tying what they’re doing to what they’ve already learned. Right now, they’re working on operations involving negative and positive fractions. Seems to me I got that stuff in 4th grade or so, but I can’t remember…
Or better yet, in keeping with the spirit of the question, try different numbers to see how much sense this makes: The temperature yesterday was 0 degrees C. Today, the temperature is -20 degrees C. How many times colder is it today than it was yesterday?
-20/0 = undefined.
The temperature yesterday was 5 degrees C. Today, the temperature is -20 degrees C. How many times colder is it today than it was yesterday?
-20/5 = -4. So it’s -4 times colder than it was yesterday or 4 times hotter than it was yesterday. WTF?
The temperature yesterday was 100 degrees C. Today, the temperature is 0 degrees C. How many times colder is it today than it was yesterday?
0/100 = 0. Today it’s 0 times as cold as yesterday. Huh?
Well, it could be said that Jill is four times shorter than 6ft than Jack is shorter than 6ft. If they were 6 foot yesterday and Jill had been hit by four times as many mallets as Jack since it’d even make sense. But ‘colder’ is still stupid because it encourages sloppy thinking, never does make sense, and is AT BEST very ambiguous and misleading, and IN ALL EXAMPLES I’VE EVER SEEN flat out wrong.
Nitpick: the scale doesn’t matter (so long as it starts at 0).
For those who care about this sort of thing, it’s a distinction between the vector space R and the affine space R, which most calculus books also screw up. The distinction is only clearly drawn post-calculus in most people’s experience.
Here’s a gem from my nephew’s junior high school science textbook (about 8 years ago):
Underneath a large photo of an astronaut doing a tethered spacewalk from the Shuttle was this caption: “As you get further from the center of the earth, the earth’s gravity decreases until, in outer space, you are weightless!”
WAG? The textbook publishers have to have “reasons” to revise and update textbooks every year or so and hence include stupid questions to justify the updates?
The aurthor(s) were brought up in government schools & universities?