18 Jan 2006 - 04:51

making peace with cross-multiplication

I've taught cross-multiplication to Ben before, apparently. I've even posted about it, way back when. But lately we've been dealing with problems like d/36 = 7/4, and whatever he learned about cross-multiplication last summer is gone.

I've wanted to teach Ben to do them using isolating-the-variable methods. It's difficult to get Ben to take this step-by-step approach. One problem is that he always wants to do the problems in Saxon 'the easy way' -- and I can't tell you what the easy way is, for Ben, because from where I sit, it looks as though the answers just come to him. It's a bit uncanny. I think he visualizes the proportions, somehow.

But the notion of being able to do the same thing to both sides of an equation is not coming easily to him. This is in spite of the fact that he was the pan-balance problem king of 5th grade. Apparently you can understand the heck out of pan balances, and still have trouble with understanding and manipulating equations.

But I've been driving him through all the steps involved in isolating the variable anyway. I do this because it's good to do it, and see it done, over and over. Still, sooner or later -- it may be this summer -- I'm going to end up doing massed practice with him, until he gets it. Massed practice means having him do the same thing, over and over, multiple times, without other distractors in between. Saxon is great for distributed practice; Saxon will teach a new concept or technique, and the kid will see it again at least once every night, for a long while, until he has seen it enough that it sticks for good. But Saxon only does massed practice once, when the skill is first introduced; and sometimes, a kid is going to get stuck on some concept or other, and need special emphasis on it.

Ben is maybe a year or two away from really needing to know how to isolate the variable. We're going to have to drill that idea until it sticks. In the meantime, though, I'm teaching him cross-multiplication for solving proportions, for the following reasons:

1. that's the way Saxon teaches it, and I want to be consistent with what he's learning at school.

2. It gives me the chance to show him why cross-multiplication works -- after all, equation manipulation is at the heart of cross-multiplication. it keeps the notion of doing the same thing to both sides of an equation in front of him.

3. It's a fast and slick trick for solving proportions.

I'm all for knowing the fast and slick tricks, as long as you know why they work.

-- CarolynJohnston - 18 Jan 2006

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What does he do with a problem like d/36 = 7/5?

-- BeckyC - 18 Jan 2006

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It's interesting that you say he might "visualize the proportions" to do the problems. I've heard the same description of how some autistic savants with abilities relating to calculating "do it": they "see" numbers in a way that's almost synesthetic.

Ben's autistic, right? If that's the case, then maybe he is doing the problems visually, which would explain why he's good at pan balance problems. I wonder if you might not be more right than you think you are.

BTW, can he do the pan balance problems almost instantly? I actually find them very difficult without writing out the equations corresponding to the problem, but my visualization ability is quite limited due to learning disability.

-- PaulMiller - 18 Jan 2006

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BeckyC, I don't know what he'd do with that problem precisely, but with other such problems he's been known to give correct answers involving decimals.

Paul, he didn't do the pan balance problems almost instantly. We did quite a bit of talking about them and drawing pictures before he got the hang of it, but once he got the hang, he had it down. Equation manipulation is less natural for him.

I think he is definitely developing a number sense that is deeper than mine.

-- CarolynJohnston - 18 Jan 2006

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I can't tell you what the easy way is, for Ben, because from where I sit, it looks as though the answers just come to him. It's a bit uncanny. I think he visualizes the proportions, somehow.

wow

I would love to know how he's doing this

it may be time to dust off Allan Snyder's integer & genius paper again....

-- CatherineJohnson - 19 Jan 2006

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Is integer arithmetic fundamental to mental processing?: the mind's secret arithmetic (pdf file)

They've also got it in html now.

-- CatherineJohnson - 19 Jan 2006

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Definitely take a look at Saxon 8/7, Investigation 7.

It's fantastic.

Beautiful.

It may be the first full lesson on 'isolating the variable,' and he does it as a 4-page Investigation.

I'm going to teach it to Christopher.

-- CatherineJohnson - 19 Jan 2006

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Apparently you can understand the heck out of pan balances, and still have trouble with understanding and manipulating equations.

Quelle surprise

-- CatherineJohnson - 19 Jan 2006

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I have to think this one over.

Christopher's having a hell of a time with it, too.

He's finally learned it, more or less, as a rule.

He has a good memory, so he may be able to retain the rule AS A RULE.

But that's it.

He has no idea that do-the-same-thing-to-both-sides is related in any way to the solve-via-inverse-operations approaches we used to do in Saxon.

-- CatherineJohnson - 19 Jan 2006

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He's been known to give correct answers involving decimals.

That's impressive. When I saw the problem you mentioned in the beginning of your post--d/36 = 7/4--I almost instantly saw 63 (36 [div. by] 9 = 4, so 7 [times] 9 = 63).

Becky C's example--d/36 = 7/5--is much harder, but not entirely out of the realm of possibility for mental math. It's simply 36 [div. by] 5 [times] 7, or 7 1\5 [times] 7, which is 49 and 7\5 = 49 and 1 2\5 = 50 2\5, or 50.4.

I would be extremely interested to know how your son thinks when he evaluates equations like these.

-- JdFisher - 20 Jan 2006

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When I asked him to show me how he did a problem -- the problem was d/2 = 3/5, solution d = 1.2 -- he wrote:

2*2.5 = 5

1.2*2.5 = 3.

Nothing magical about that, just good number sense.

-- CarolynJohnston - 20 Jan 2006

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That confirms my suspicion. He's solving proportions, not balancing equations, which I think is good.

He's also got that 5\2 is "2 twos and another half of 2 (i.e., 2 1\2) thing DOWN.

It's not magical in the sense that he pulls it out of thin air. But I wouldn't hesitate to call it "magical."

-- JdFisher - 20 Jan 2006

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True. And you are right that he is solving proportions, not balancing equations. He almost never makes a mistake on those problems, either, in spite of the fact that he can be kind of careless with other math problems.

-- CarolynJohnston - 20 Jan 2006

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I realized, with Christopher & bar models, that we don't have enough massed practice.

I was having him do one bar model a day, and it wasn't enough. Every day was like starting all over again.

I think I managed to move up to 3 bar models a day, which helped.

Of course now that we're buried in Prentice-Hall, we're doing no bar models at all - except when Ed 'invented' one on the spur of the moment to explain a problem about finding how many floor boards you would need to cover a room.

-- CatherineJohnson - 20 Jan 2006

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Paul

can he do the pan balance problems almost instantly? I actually find them very difficult without writing out the equations corresponding to the problem, but my visualization ability is quite limited due to learning disability

That's interesting.

I'm trying to remember my various 'stages' with the pan balance problems....

I do remember being utterly charmed the first time I saw a pan balance problem in ALGEBRA TO GO. And I remember loving the virtual pan balance problems I found on line.

But, at the same time, I felt a disconnect between pan balance problems and 'do the same thing to both sides' as well as 'isolate the variable' until I did Investigation 7 in Saxon 8/7.

-- CatherineJohnson - 20 Jan 2006

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"Apparently you can understand the heck out of pan balances, and still have trouble with understanding and manipulating equations."

I find that a pan balance analogy is only good as a tool for showing kids that you can do something to one side only if you do the same thing to the other side. After that, the real thing (equations) is better than the analogy. In other words, I would spend one lecture on the pan balance.

I look at an existing balance with the different shapes and it doesn't mean much to me. If I want any information about the system, I will convert it to an equation. I guess that is how I feel about bar models - a helpful analogy, but you need to move on.

-- SteveH - 20 Jan 2006