11 Oct 2005 - 03:56
a way to teach unit conversions
Tonight's story is drawn, not from Ben's math class, but from his science class. The kids are doing a big unit on measurement the last few weeks, and his test is on Wednesday. Some of the material is on unit conversions, a topic relevant to both math and science.
The kids are doing tables of unit conversions, converting from meters to decimeters to centimeters to millimeters, and from meters to decameters to hectometers to kilometers (who uses hectometers, anyway?).
Well, Ben has been consistently getting his unit conversions backward
. He'll convert, say, 13 meters to .013 millimeters, or 12 meters to 12000 kilometers.
"You're going backward!" I was pleading. "Millimeters are littler than meters, so you always have more millimeters than meters."
"But millimeters are supposed to be little
," he said, "and kilometers are big
It was stuck in his head that way, all backward. I am afraid I understand this backwardness problem all too well, myself. But I was getting worried. Ben was getting lots and lots of practice doing these problems the wrong
way. He was drilling, I feared, a rut into his brain that would be hard to fill in.
I tried a visual aid. I taught Ben to draw a picture of a little short bar for a millimeter, a medium bar for a meter, and a big bar for a kilometer. I thought the visual aid would help, but it didn't; he already knew
that millimeters were little, and kilometers were big. More precisely, it didn't help him get problems consistently right
. What you really want for this sort of task is a procedure that gives a kid the right answer every time, so that he learns to trust himself to do it correctly.
So I decided to teach him unit conversions. Unit conversion is a special case of dimensional analysis. We've talked a bit about dimensional analysis at KTM, and at the bottom of this post I'll put some pointers to the previous blog posts and user posts we've had about dimensional analysis. But here, I'll just show step-by-step how I taught Ben (who is in 6th grade) the unit conversion technique.
Keep in mind as you follow that the Main Trick of dimensional analysis is to realize that units, such as feet, meters, grams, pounds, and so forth, can be manipulated just like numbers.
. I began by reminding Ben how canceling works when multiplying fractions. For example, 6/5 x 5/4 = 6/4. He already knew that -- but I wanted to convince myself he had that down before going any farther, because that's essential.
. I showed him a little bit about how units, like centimeters or grams or feet or what-have-you, are manipulated in expressions just like numbers. For example, in calculating the area of a rectangle that is x cm by y cm, you get:
A = (x cm) x (y cm) = xy cm^2
because the centimeter units multiply (that little ^2 symbol means 'squared'). Another example: if you want to calculate how fast you're driving if you drive 60 miles in an hour, then you would write:
rate = (60 miles)/(1 hour) = 60 miles/hour,
and your units would come out in miles per hour.
. I showed him the fractional expression: (1 cm)/(1 cm) = 1.
"Do you see why that's 1?" I said.
"One what?" I said. "Is it one centimeter?"
He thought for a second and said, "I don't think so."
"That's right," I said. "It's just 1, without any units, because the centimeters on the top and bottom canceled."
: I showed him the following expression:
(1 m)/(100 cm) = 1.
"Do you see why that's one?" I said. "It's one because 1 meter and 100 centimeters are exactly the same, so they cancel".
We did a few more of those. We did 1000 mm/1 m = 1, 1 km/1000 m = 1, and so forth.
: Remind your student that multiplying anything by 1 (and that's 1 by itself -- dimensionless, not 1 centimeter or 1 gram) leaves it unchanged.
So, for example, you can multiply by 15/15, because it's 1. You can also multiply by 1 cm/1 cm, because this is also just 1.
You can even multiply by 1000 mm/1 m, or 1 km/1000 m, because we showed in step 5 that these are also 1.
: This was the final step, where I showed him how to use the trick to do conversions.
"Here's an example," I said. "Suppose I want to convert 24 meters to centimeters."
I wrote down: 24 m = ____
Then I wrote: 24 m x ((____
"We're going to fill in those blanks so the expression on the right is equal to 1," I told him. "Then the meters will cancel on the bottom and the top, and we'll be left with the centimeters."
He knows the conversions for meters and centimeters, so we wrote:
(24 m) x ( 100 cm)/(1 m) = 24 x 100 cm.
"Now, the meters cancel each other," I said, "and we get left with 24x100, or 2400 centimeters."
: I did a few conversion problems with him, guiding him through the procedure. The last one he did successfully on his own.
I don't want to say that he had a Eureka moment, but this is a reliable procedure
that will get him through these problems, and hopefully work around that rut that was forming. We're going to practice it to automaticity.
And as he goes through school, tricks like this -- using dimensional analysis -- will get him through a lot more than unit conversions, too.
Other stuff about dimensional analysis
dimensional analysis: why and how to use it
Our first discussion of dimensional analysis was in the comments section in this thread
invented a game called "Dimensional Dominoes" for teaching kids and grownups dimensional analysis, and posted it here
Dan's dimensional dominoes (manipulatives)
unit conversion (in Comments thread)
Carolyn on teaching unit conversions
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I would simply add to Step 4 the reminder that multiplying a quantity by 1 doesn't change it's value. So, you can apply as many unit conversions as you like without changing the quantity--as long as you keep track of the units.
-- DanK - 11 Oct 2005
This is a classic example of why math is so confusing.
-- CatherineJohnson - 11 Oct 2005
This is incredibly cool.
I'm going to teach it to Christopher.
I think I'll print out Dan's dominoes...
OR, I might do screen grabs of them & post them on the front page first.
-- CatherineJohnson - 11 Oct 2005
"I would simply add to Step 4 the reminder that multiplying a quantity by 1 doesn't change it's value. So, you can apply as many unit conversions as you like without changing the quantity--as long as you keep track of the units."
Thanks for the reminder. I think I will break that out into a separate step.
-- CarolynJohnston - 11 Oct 2005
-- CarolynJohnston - 11 Oct 2005