I left something important out!  Two weeks ago I did a rambling session with middle school mathletes, 7th graders, here in the Bronx (read about it here).  But while I was rambling off-topic, I diverted. This was early. And it helped me win the group. And I forgot to report it.

My plan was to ask them if they are taking math (of course), if they took math the previous few years (decimals, per cents, adding fractions, multiplying fractions, dividing fractions, check on long division), if they did early elementary (long multiplication, long addition, all the little facts), stop, and throw in something about learning to count (before counting exercises 32 – 37 vs walking from 32nd to 37th Street). This is pretty standard “suck them in” get some choral answers going, get them to think all my questions will be easy… We don’t do work, I just bring them to the tricky question. (here is the script, roughly, in a comment to a post about a previous time I visited this school for math team).

#### When we no longer fear division, we should still respect it

But early on, I got a response I hadn’t anticipated. One or two little voices tsked when I said it was good that they did division of fractions (every time they said they knew something, I said something positive). They made clear that dividing fractions was non-interesting, non-engaging, and, perhaps I am overstating it, trivial.

Now, with an hour and a half, I intended us to have fun. But having division of fractions tsked at? How could not say something? So I said that it was good that they found it easy. That a lot of people didn’t. But perhaps it would be more interesting to look at again.

And I wrote on the board $\frac{1}{4} \div \frac{3}{4}$ and immediately the squeals of “keep change change” and “keep change flip” began, and I did my best to look old and slow and confused (after having just been talking a mile a minute) – “Keep? Keep the change? What do we do?”

And they giggled, and told me, and I reinterpreted as they led. And when we were done they knew, most of them, that they were telling me to multiply by the reciprocal. And truly, before we started, some of them already knew that.

[An aside here. Why “keep change change” with kids who are sophisticated enough to “multiply by the reciprocal”? My turn to tsk.]

And I mumbled something about filling a container. How many two cup containers can be poured into an 8 cup container? (but I can’t remember the actual example) and then I drew a square, divided across and down into 4 equal squares, and shaded one quarter, and then drew an L-shaped 3-quarter piece, and asked how much of the L-shaped piece would fit into the 1-quarter piece, and the answer came back one-third pretty quick and clear with a little chatter about this different way of looking at division.

And then I went back to the original, and asked why we didn’t just divide across… $\frac{1}{4} \div \frac{3}{4} = \frac{1 \div 3}{4 \div 4} = \frac{\frac{1}{3}}{1} = \frac{1}{3}$ and were we allowed to do this or do we have to multiply by the reciprocal, and then I offered another example, $\frac{6}{25} \div \frac{3}{5}$ and they verified that they got the same answer each way, and then I offered a chance to prove the conjecture with $\frac{a}{b} \div \frac{c}{d}$ and I guided them to find that each way yields $\frac{a \times d}{b \times c}$. I drew the conclusion for them, that the algorithms are equivalent, I pointed out that the new algorithm could be more convenient, and I finished with one last example where they could use the new algorithm, but where it would be a silly choice: $\frac{9}{7} \div \frac{5}{8}$.

#### Afterthoughts

I thought this was a pretty good 4 – 6 minute run. I stayed goofy, but I answered any of them who may have been unsure about the math being serious. I made a nice point about using correct terms, as appropriate. I explicitly pointed out (and demonstrated) that even basic arithmetic has things within it that is well-worth investigating. We examined multiple algorithms for one operation. I touched on multiple models of division.

This was unplanned. And not a topic that I run through “unplanned” frequently. For the future, I’d like to improve a bunch of things. I would like to explicitly name several (maybe not all…) models of division. My spiel was weakened because I was not clear, and partially conflated two. I should have a ready example that works nicely.

Actually, I should have a list of examples that work progressively to where I am going (some people call this scaffolding, but honestly, I’m not a carpenter). And I could pull from this list. For the area model I might consider $2 \div \frac{1}{8}$ and $2 \div \frac{3}{8}$ as nice stepping-stones to perhaps $1\frac{1}{4} \div \frac{3}{8}$.

I should review Hung-Hsi Wu on fractions (hate me, progressive educators, but his stuff is hot). And – hate me more – on division.

In other words, if at some unspecified time(s) in the future I want to play, unplanned, with fractions, I would be better if I did some planning first. Where can I do some fun upper-elementary methods?

6 Comments leave one →
1. February 19, 2011 pm28 3:19 pm 3:19 pm

Last summer my brother sent me an article Wu had written, and I had to do some wok to articulate what bothered me about it. Good work to do.

I don’t like his policy and pedagogy thoughts, but I might like his thoughts on the math itself. I bet we’d learn more by talking through an article or two of his, than by looking together at something we both like.

You have read Liping Ma, I assume? She talks about how the Chinese teachers she interviewed had 3 models for division of fractions. I used to be proud of my one weak example. (How many half inches in a ruler? That’s 12 / 1/2.) I’m a bit better now, but mt facility isn’t as deep and wide as I’d like.

• February 19, 2011 pm28 11:02 pm 11:02 pm

I’d really like reading one of the Wu pieces together. And I liked what I read from Liping Ma (profound whatever – you know, PUMF), but I wanted many more examples.

For Wu, I knew you would not like his politics, but I did not know what you would say about his pedagogy. His math, I think, is good.

We’ll talk more about this, but when there is time. I know that no matter what Wu or anyone else says, I want to pay some attention to unit fractions in particular. I think they work a bit differently than ordinary fractions, and might be worth examining as special cases.

2. February 19, 2011 pm28 8:39 pm 8:39 pm

@Sue: Liping Ma rocks! Her 25 page explanation on dividing a fraction by a fraction was eye-opening: you realize that your knowledge of math fundamentals may not be as deep as you thought after reading it.

What I found most profound was her explanation of the rule (the “official” name escapes me) that the quotient stays the same if you multiply the dividend and divisor by the same number:

10/2 = 5

(10X10)/(2X10)
=100/20
=5

Using that rule, we can prove the validity of the shortcut of keep, change, flip.

3/4 ÷ 2/5

Multiply by the reciprocal of 2/5

(3/4 x 5/2) ÷ (2/5 x 5/2)

(3/4 x 5/2) ÷ 1

3/4 x 5/2

I highly recommend her book, even if you’re not an elementary school teacher.

@JD: Great off-the-cuff lesson! I need to work more on taking best advantage of these unplanned teachable moments in my classroom. Thanks for reminding me to be more mindful of them.

Paul Hawking
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