The hard part in evaluating students is to figure out when they really "get it." When they actually understand. The kid who makes the mistake in the picture above, almost understands canceling in simplification. The hard part is to find out just what it means to understand.
For example, kids, and teachers, all agree that if x=y and a=b and none of those are zero, then :
1) x+a = y + b
2) x-a = y - b
3) x(a) = y(b)
and the one I used in class today to find the exponential curve through two points...
4) x/a = y/b...
If you disagree, say so now, because this is where it gets fun....
If you think your students really understand this idea, give them the following problem:
32 ounces = 2 pounds; 8 ounces = 1/2 pound
so 32*8 = 2(1/2) or
256 ounces = 1 pound....
If they can explain what happened, I think they understand.
2 comments:
Are you sure you did not mean to express 32*8 = 2*(1/2)?
In any case, I understand your point. If they can troubleshoot something that is stated incorrectly, then you know they "get it". Still, even if they cannot troubleshoot an incorrect statement does not mean they DON'T get it. It is potentially a different skill...more analysis and problem solving than strictly math. In fact, as presented this example is confusing because you do not say what you are trying to express. So they need to decipher what the point is first, and then find the answer.
What you are trying to express is the idea of a proportion. If you just came out and said "here is a proportion", then they would see that it is incorrectly presented and should be more like 8/32 = .5/2, which if you cross multiply provides the result of 16/16, showing that they are equal rates or ratios.
So if you use this as a way to show how numbers and units are different, and that mixing the two in a single statement can cause confusion, then I dig it.
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