22 Oct 2005 - 16:30

lattice multiplication with carrying

This explanation at Math Forum is the clearest I've seen. Perfect.


I'm too tired to think it through right now, but I'm not so sure the final point is right.....

'Doctor Mason,' at Math Forum, says:

Multiplication really takes three steps: multiply, carry, add. The method we typically use does the multiply and carry steps together. The lattice method does all three steps separately, so it's really easier! Centuries ago, the Germans had a method for doing all three steps at once. That method takes a lot of concentration!

But I think the method I left in one of the Comments threads also separates the 3 steps.

Doesn't it?


I'm going to keep an eye out for Dr. Mason.

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You can deal with decimals correctly, too. Just find the position where the decimal "lines" intersect, and carry down the diagonal.

In a way, it's easier. But it doesn't reveal much about why it works -- the use of the distributive property in the standard algorithm is more transparent. And it's a pain to draw those lattices.

-- CarolynJohnston - 22 Oct 2005

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I think the fact that you still have to carry like in the standard algorithm defeats most of the ease of use. Seems more like a rearrangement of steps than anything and still -- conceptual insight free.

-- KDeRosa - 22 Oct 2005

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"But it doesn't reveal much about why it works ..."

Ironic, isn't it, but I don't think it is horrible. It's the old "Different Just To Be Different" mentality. How about "Anything But Traditional". After trashing traditional math teaching methods and algorithms, they are stuck. They have to do something different.

Now that my son is a month and a half into 4th grade Everyday Math, the biggest problem I see is the slower pace and the lack of practice and mastery. Do you know what homework sheet he came home with on Wednesday? One of those fact triangles for multiply and divide. (I hate those things.) Once more it's Different Just To Be Different. OK, fine, but the one page homework had 9 (I think) triangles and they were for things like 5 X 6 = 30. That's it. Kind of makes me look forward to the Lattice Method later in the year.

We also had our 15 minute parent-teacher conference today. "Hello - Your son is doing great - goodbye." with the next set of parents pacing outside of the open door. The teacher said that our son was doing really great in math. I felt like asking her if she had timed how long it took him to calculate 7 X 9. He still needs a lot of practice and the Singapore books are still sitting unopened on the table. I need to have him do this extra work, but we are still trying to get him to take less than half the night (it seems like this) to get his regular homework done.

-- SteveH - 22 Oct 2005

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Modified from Vince Calder's work at http://www.newton.dep.anl.gov/askasci/math99/math99199.htm I am not sure about the algorithm you were shown, but I know of one that "works" that is based on solid mathematical principles. The "method" is best shown by an example: Suppose we want to multiply: 768 x 52 = (7x10^2 + 6x10^1 + 8x10^0) * (5x10^1+ 2x10^0). Look at the exponents of "10". They are (2,1,0) and (1,0) respectively. So the "answer" is going to have a maximum exponent of (2+1=3). Now write the multiplication in "vertical" format, lining up the various exponents of "10" in the same columns. So we have: Exponent of "10" is 4. If the product of the 2-digit multiplication is greater than 10 the lead digit "spills over" into the next higher place value column.

4	3	2	1	0		10^x
	7	6	8
	5	2				multiply left to right
3	5					(5 x 7 = 35)
	3	0				(5 x 6 = 30)
		4	0			(5 x 8 = 40)
	1	4				(2 x 7 = 14)
		1	2			(2 x 6 = 12)
			1	6		(2 x 8 = 16)
3	9	9	3	6		add the columns

Sorry, it isn't very pretty.

All multiplications are single digit multiplications, the highest possible one is 9x9. There is no "carrying" to keep track of. Each multiplication step will only "spill over" to the column just to the left. If the addition of columns exceeds 10, then it is easy to add a "1" to the first column to the left. The problem is laid out for easy reality checks. In this case (~800x50=40,000) so you know what the answer should be.

-- SmartestTractor - 22 Oct 2005

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Steve

DO THE HOMEWORK FOR HIM!!!!

I'm not kidding.

First of all, any homework that's not of value is something you should do, and have him copy. (Obviously, homework that is valuable, he should do.)

Second, strip away all the 'extra' tasks, like recopying problems.

Re-copy problems for him (or whatever) and THEN have him solve them.

My 5th grade niece was collapsing under the weight of her math homework, which was zillions of problems all scrunched together on one worksheet. She was trying to figure stuff out in the margins, then go back and write in parts of the answers, etc.

I told my brother to buy her some quadrille paper, or use the Saxon quadrille paper I made.

Copying things & just writing anything is extremely time-consuming for a child.

Handwriting of any kind eats up a huge amount of time and energy. (You can certainly have him do daily handwriting practice during the summer, or even now, if you need to. But I'd pull the 'handwriting piece,' as educators now say, out of homework.)

-- CatherineJohnson - 22 Oct 2005

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Another thing to consider is to buy the Challenging Word Problems book and have him do just one bar model a day.

one caveat: Last summer I was having Christopher do just one bar model a day, and he really wasn't getting it. He was making the same mistakes over and over again.

I bumped him up to, I think, 3 bar models a day. That was enough.

Now I can squeeze in just one bar model a couple of times a week and he's still building competency.

-- CatherineJohnson - 22 Oct 2005

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It really is awful, having to spend time on EVERYDAY MATH when you could be doing SINGAPORE.

-- CatherineJohnson - 22 Oct 2005

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Smartest Tractor

wow

I'm going to have to look that one over

-- CatherineJohnson - 22 Oct 2005

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btw, my sister, who is married to an architectural engineer, says engineers often do 'column addition.'

Of course, for the moment I'm forgetting what that is, but I remember liking it when I tried it.

-- CatherineJohnson - 22 Oct 2005

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hmmm

I may have the wrong term

here's column addition in Everyday Math

-- CatherineJohnson - 22 Oct 2005

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Wait!

I misspoke.

I don't have Christopher doing the problems in Challenging Word Problems 3.

He's doing the problems in Primary Mathematics 3A, both the textbook & the workbook.

-- CatherineJohnson - 22 Oct 2005

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The BIG SECRET to learing via tiny amounts of effort each day (and I've done this myself, for myself) is pure repetition.

In other words, if your son were to do just one bar model a day, have him do extremely similar problems each day.

At first, have him keep doing 'number addition' problems.

"What number is 5 more than 6?"

"What number is 3 added to 2?"

etc.

You can make a huge amount of progress in very tiny amounts of time if you use strict repetition, advancing to the next step once the first step is mastered.

The next step, with addition, would be to add simple word problems.

Steve had two apples and his son gave him 3 more.
How many apples does Steve have now?

-- CatherineJohnson - 22 Oct 2005