If it's Tuesday, this must be Math Concepts.

I didn't do all that well on the weekly quiz because I wasn't quite sure which was the union symbol and which one meant intersection in set notation. And I made a careless error. And I didn't read a graph correctly. What a dummy.

In class, we talked about the models of multiplication (repeated addition, array and Cartesian product), as well as the properties of multiplication (commutative, associative, distributive, etc).

We also went over some very odd strategies for subtraction problems that normally would require regrouping (a.k.a. borrowing ten from the next highest place). One of the useful ways is to add the same amount to each element of the problem, making it an easier one. The amount you add varies with the problem, but usually involves making the subtrahend (the thing being subtracted) end in a nice round zero. To wit:

43 - 38 can become, when you add 2 to both the minuend (the number you're starting with) and the subtrahend, 45 - 40. That's a much easier problem than borrowing ten from the tens place; you can do this in your head automatically.

Akin to this strategy is the method where you add ten to the ones place in the minuend and so accordingly add ten to the tens place in the subtrahend. Thus:

53 - 38 becomes 53 - 48 (where the 3 in the 53 is actually 13, represented as usual in borrowing, with a one at the 3's upper left corner). So you subtract easily in both places (13 - 8 in the ones and 5 - 4 in the tens), and get 15.

On the other hand, there were also some startlingly stupid strategies. I can't even coherently represent the worst one. It has the student starting at the left side of the problem, going through each place value. Then, depending on whether the next place value's answer on the right requires borrowing or not, the student writes either the answer or the answer minus one. If at any place the subtrahend is larger than the minuend, the student writes the difference between the subtrahend and ten plus the minuend. And so on.

It's hard to explain, harder to understand, and useless. Now how that helps students understand number theory or place value, I don't know.

## Tuesday, September 12, 2006

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