13 Oct 2005 - 04:13

Dr. Ian talks about fractions and units

In pursuit of more information about dimensional analysis, Catherine found a link to Dr. Ian on dimensional analysis at the Drexel Math Forum. He has some insights about fractions and units that are priceless. Here goes:


The ultimate way to think of a fraction

In the end, a fraction is just a division that you haven't bothered to do yet. -- Dr. Ian

It's true -- really, 5/7 just means '5 divided by 7' -- whatever number that is!

I remember reading that Wayne Wickelgren (I think it was, or maybe it was Hirsch -- my memory is shot) felt totally empowered and excited as a boy when he realized that a fraction was a division problem that he wouldn't have to actually do.

Dr. Ian also has a lot of cool insight into what makes a ratio a ratio (hint: it doesn't have any units), and how units can help you make sense of fractions:

But let's say we want to divide 6 pies evenly among 3 people. Now we have

      6 pies         6        pies
     ---------  =   ---  x   ------    =  2 (pies/person)
    3 persons        3        persons

That is, the units stay around instead of cancelling. We end up, not
with '2', but with '2 pies per person', in much the same way that if
we travel 45 miles in 30 minutes, we end up with

  45 miles        45         miles
  --------   =    ---   x    -----   =   90 miles/hour
  1/2 hour        1/2        hours

or 90 miles per hour, and not just '90'.  

That is, when you introduce units, you end up with two results: a
number, and some combination of units that tells you how to interpret
the number.

It's worth repeating

When you introduce units, you end up with two results: a number, and a combination of units that tells you how to interpret the number. -- Dr. Ian

You need to know how to interpret the number, because quantities with units are "slippery". By that I mean that you can change the actual number by changing the units. For example, 8 feet/sec describes the same speed as 8.78 km/hr, even though 8 and 8.78 are obviously not the same number.

That's why it's so important for the units to stay with the quantity as part of the answer to the problem. Remember those good old hardnosed math and science teachers who used to mark us wrong if our answer was "8.78" and not "8.78 kilometers per hour"?

But ratios aren't slippery, because they don't have any units that you can change. If Catherine has 2 dollars and I have 1 dollar, then the ratio of her money to my money is 2, period -- no units are involved in the ratio, because the dollars cancelled. A number without a unit (such as the result of any ratio calculation) is called a 'dimensionless quantity", and those are the numbers you can really count on, because you can't change them with a unit conversion.

Finally, it's pretty easy to calculate using units -- the words just come along for the ride.


Terminator update

KDeRosa sent me an updated version of his essay on Fear And Loathing In Engineering School. It's got links in it, so that people can resolve those confusing references to the mysterious number 7. I've replaced the original esssay with it at its permalink location here. (For those who want to see the original essay, it's still here in the comments where he originally wrote it, about the fourth one down). Here's to its long life as an internet classic!

Back to main page.

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I LOVE IT!!!!!!

A FRACTION IS A DIVISION YOU HAVEN'T BOTHERED TO DO YET!!!!

-- CatherineJohnson - 13 Oct 2005

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I have actually spent some time wondering whether a fraction is a division I haven't bothered to do yet, believe it or not.

-- CatherineJohnson - 13 Oct 2005

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I have a question about the math here.

Why is it written as 6 x pies?

In other words, why are '6' and 'pies' related by the multiplication sign?

Is this the same principle as 'times' meaning 'of'?

This, btw, is a major sticking point for me, and, I would guess, a lot of kids just learning math.

I was thrilled when I discovered that '2 x 3' means '2 of 3,' just as '1/2 x 4' means '1/2 of 4.'

But I have trouble seeing 'times' and 'of' are the same thing when you're talking about 6 pies.

[pause]

oh

no, I don't.

It's the same thing, right?

'6 pies' means 6 'of' pies (or, grammatically, '6 of these pies') which then means 6 x [these] pies.

Is that right?

Or have I veered off?

(I STILL have not read through the various dimensional analysis posts closely, so this is based on Total Skimming.)

-- CatherineJohnson - 13 Oct 2005

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Remember those good old hardnosed math and science teachers who used to mark us wrong if our answer was "8.78" and not "8.78 kilometers per hour"?

That is fabulously helpful!

I only 'know' this implicitly; I would have to have spent some time to figure out why you have to label.

Christopher's teacher marks off for forgetting labels, and I've supported that (and you know I'm more than happy not to support a teacher practice of which I disapprove).

Somehow, I 'knew' that the labels have to be there.

I conveyed that to Christopher, but in a purely 'procedural' way: YOUR TEACHER WILL MARK YOU DOWN, SO WRITE THEM IN!

Now I'll be able to tell him, explicitly, why it is those labels have to be there.

-- CatherineJohnson - 13 Oct 2005

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This brings me back to the troubles non-math parents have dealing with all this stuff.

People like Susan and me (Susan--I think you and I may be the only non-math parents here) have managed to absorb quite a lot of.....math common sense-y?

Something like that.

We can tell that radical constructivism is bad, we can tell that multiple solution problems are wrong, we can tell that YOU HAVE TO PUT THE LABELS IN......but we can't argue our position very effectively, because we don't know why we know these things, we just know them.

(I should stop putting words in Susan's mouth!)

Speaking only for myself, this happens constantly.

(It happens in every subject, of course; 'knowing stuff' you can't justify isn't unique to math.)

-- CatherineJohnson - 13 Oct 2005

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Obviously, this is why educating ourselves in cognitive science is a good idea.

It's not that cognitive science can, at this point, tell us everything we need to know about how to teach math & other subjects (though it can tell us a great deal).

It's that having a handle on the basic tenets of cognitive science gives you your talking points.

-- CatherineJohnson - 13 Oct 2005

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'6 pies' means 6 'of' pies (or, grammatically, '6 of these pies') which then means 6 x [these] pies.

Is that right?

Yes! That's a good way to think of it. 6 of pies, 3 of people.

-- CarolynJohnston - 13 Oct 2005

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Let's take another look at an example you used earlier:

3 apples + 2 oranges = ?

We can treat this as a dimensional analysis problem too. There is no conversion from "apples" to "oranges" and you can't simplify until the units are the same, so the answer is "3 apples + 2 oranges".

If you are willing to lose some information though, you can say:

apple ∈ [is an element of] fruit

orange ∈ fruit

So, 3 [fruit] + 2 [fruit] = 5 [fruit]

Losing less information:

apple ∈ [apples OR oranges]

orange ∈ [apples OR oranges]

3 [apples OR oranges] + 2 [apples OR oranges] = (3+2)[apples OR oranges]

The OR is a boolean operator meaning that there might be apples, oranges, or both apples and oranges.

Again, you can't add until the units are the same.

The [is an element of] symbol, ∈, may not show up in some browsers.

-- DougSundseth - 13 Oct 2005

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Cool!

-- CatherineJohnson - 13 Oct 2005

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That's a neat way of writing it......

-- CatherineJohnson - 13 Oct 2005