Yes, it happened. It was a rainy afternoon, and the kids were so bored that they pulled out school books to do school work. Kid1 really wanted to do spelling, but I was busy so she ended up doing math.

Lesson 28, Product of a Number and Two More

This is an incredibly sneaky lesson. It started out explaining that there's a relationship in multiplication between numbers like 2 times 4 and 3 times 3. This was followed by a brief discussion of "squares" and exponents.

Next, Kid1 started drawing rectangles ... first a rectangle that was 2 by 4. She then made this into a square by taking the 2 unit-squares off of one end, swinging them around the side, and adding another unit-square. Et voila, a square of 3 by 3. (Really, this is easy-peasy when you see the diagram in the book).

The worksheet involved drawing various rectangles, swinging the unit-squares around to almost-but-not-quite make a square, adding another unit-square to complete the square. Next, she wrote out the formulas for the various rectangles and squares (2x4=8, 3squared=9, 3squared-1=8). Then, a little nudge to go further ... without drawing rectangles or squares, try figuring out 6x8 and its "near square" (which is 7x7).

After a few of these problems, she was asked to put the concept she had discovered in her own words. This resulted in much moaning and gnashing of teeth. I suggested she tell me ("pretend I'm absolutely clueless -- how would you explain this to me?") then write out what she had said.

Next, more complex problems -- apply the concept to equations like 19x21, 49x51. This would've worked better if she were better at basic multiplication (sigh). She kept muddling up place value.

Then, a drum roll for the final problem: (n-1)x(n+1). More weeping and wailing. She was sure she couldn't do it. I read over what she had so far. She had been using the term "the middle number" to explain how to work the problems, so I suggested that a lazy person wouldn't want to write out or say "the middle number" each and every time they wanted to refer to the middle number ... maybe they would call it TMN. And, if that person doesn't feel like even writing out TMN they might choose to simply call it ... N. Or maybe it's too much trouble to even capitalize, so it just becomes (yes, you've got it) n.

OOOOOOH! OKAY!

When Mr.V got home from work I announced the Kid1 had figured out (n+1)(n-1)=n squared - 1 (NB: obviously I need to figure out how to do exponents in this wordprocessing program before I write much more about this math program). Mr.V was blown away, totally excited. Of course, he was assuming that she had derived it algebraically. Upon finding that she hadn't, he insisted that we all spend the evening, yes, deriving (n+1)(n-1)=n squared -1.

Kid1 commented as she was brushing her teeth at bedtime that she doubted she would remember all that they went over. A lot of it was on the order of "what is chair times 1? Chair. What is cat times 1? Cat. Okay, so what is n times 1? N." And the more complex "chair times negative 1 is negative chair. Cat time negative 1 is negative cat. So what is n times negative 1?" And the slightly loopy "chair times chair is chair squared. Cat times cat is cat squared. My foot times my foot is what?" "Are we including your big toes in this? Because your big toes squared are sort of gross with that damaged toenail that's peeling off." It was sort of like being in an Italo Calvino short story, like something out of t zero .

At least, if you're slightly warped out on math it was.

## 27 June 2006

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## 2 comments:

Hurray. So if we refuse to be driven nuts and the weather cooperates, eventually our kids will sit down and do complex math problems on their own for the pure joy of figuring them out (because the 'use' of that particular fact is probably not obvioius to her right now). I'm so glad. I suspected. Glad you held out. Congratulation.

Two hoots for warped math! HOOT! HOOT!

Congrats on the great math day :)

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