Are the Times Tables Turning?

Non-blog life (otherwise known as life) has eased up enough to get back to posting, and there’s quite a backlog of topical, “must do today” posts, plus a requested TNDOT. Usually, if I don’t get to a topical post quickly then I just drop the idea. Some recent media stuff has sufficiently annoyed me, however, that I’ll pretend it’s all still topical and I’ll post anyway.

First, another quick word about Tony Gardiner, who unexpectedly died last month. Tony’s death has affected me deeply, particularly given that I never met him, modulo attending a memorable lecture, and given that it was for only a couple years that we conversed, by email and through this blog. But I had begun to realise, and more so with his death, how much Tony had been guiding me, by explicit, admirably blunt, advice, and much more by example. I realised that I had started to compose blog posts with a “What would Tony think?” voice in my head. And Tony’s example is unparalleled. The amount that Tony contributed to mathematics education, for decades, is simply phenomenal, much of it dirt cheap or free. Tony had a missionary dedication to the mathematics education community, and he had a wisdom about mathematics education second to none. I know how fortunate I am to have Tony’s voice in my head. It will always be there.

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Another Fraction Question

OK, following up on our previous post, we have another fraction question. This one is not new, has appeared in the comments of a previous post, and many readers will have heard us bang on about it. Nonetheless, given the discussion on the previous question, and given the possibility that some new readers might not have yet read Marty’s Collected Sermons, it seems worthwhile giving the question its own post. Here it is:

    \[\color{Mulberry}\boldsymbol{(-1)^\frac26 \overset{?}{=} (-1)^\frac13}\]

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A Fraction Question

We have a question. One of the simplest possible fraction multiplications is

1/3 x 1/5 = 1/15

The question is, why? What is your simplest explanation of this equation?

To make the task absolutely clear, the explanation is to be suitable for an imagined average primary school student who is just being introduced to the notions of fraction multiplication. Explanations targeted at Peano or a Child Gauss or whomever are way, way off the point.

 

UPDATE (05/01/22)

Thank you to everyone for your comments and suggestions. They were very interesting to read. I’ll now give my own answer to the question, at which, of course, commenters are welcome and encouraged to hurl their objections. Just to be clear, I am, for once,* not pretending to be an expert. As with most commenters here, the teaching of such notions at the (upper?) primary level is well outside of my normal world. The question arose from my general concern with the undeniable awfulness of most primary level education, from specific and very enlightening discussions with Simon the Likeable, and from pseudo-teaching my nine year old daughter.

To be absolutely clear, my question was in no way intended as a trick, or as a surrogate for some other question, even if other questions naturally arose by implication. My question was intended exactly as stated:

What is the simplest explanation of 1/3 x 1/5 = 1/15 for a primary student just being introduced to fraction multiplication?

For this purpose, a deep-end definition of a/b x c/d is obviously inappropriate. At the other pole, the explanation of 1/3 x 1/5 of a pizza does not answer my question, even if it might be a valuable component of such an answer. I’m far from convinced even of that.

I agree with many of the commenters that the natural starting point is to interpret – in effect, define – “1/3 x” as “a third of”. All young children are introduced to fractions as “fractions of”: a third of a cake or whatever. We then use “1/3 x” to capture the notion of “a third of” in a precisely mathematical and symbolically manipulable way. This interpretation/definition is equivalent, as it must be, to defining “1/3 x” as “÷ 3”, but I don’t think that need concern us here; the “third of” expression is more intuitive for the intended audience.

Most easily, of course, children can take a third of suitable integer multiples: three cats or six apples or one Djokovic. That is, quantities that we are happy to divide into three equal groups. This easily extends to fractions of suitable natural numbers: a third of 6, and so on. As such, having students get used to 1/3 x 6 and the like is relatively straight-forward. (1/3 x 7, for example, is an important side story, but it can be left for later.)

But how, then, finally, does one explain 1/3 x 1/5? I’ll take it as agreed that it means “a third of 1/5”, but how do you take a third of 1/5? I think there is a simple and compelling answer.

The problem is that 1/5 is not “three things” (or a multiple of three things), as was the case with the cat clowder above. That leads many to replace 1/5 by a fifth of a pizza, and then out come the knives.

But wait a minute: 1/5 is three things. 1/5 is, of course, exactly three fifteenths. That is, the fraction 1/5 equals the fraction 3/15.** And, a third of 3/15 is obviously 1/15. In symbolic but easily decipherable and easily motivatable form,

1/3 x 1/5 = 1/3 x 3/15 = 1/15

This seems to me, by far, the most natural and simplest introduction to and explanation of non-trivial fraction multiplication. All that is required, which I agree is not a gimme, is a proper understanding of equal fractions and genuine automaticity of multiplication. (On the other hand, if a student does not have this background, the idea of their understanding fraction multiplication, or performing it well, is farcical.) But, except in conversation with Simon the Likeable, I have never seen such an approach clearly expressed.***

That’s it. Have at it.

 

*) Do not expect this to be repeated.

**) People who write that the two fractions are “equivalent” will be summarily shot. 

***) Definitely some commenters were suggesting similar approaches, but these approaches seemed to me clouded with extraneous ideas or activities. My apologies if I short-changed anyone, which is very possible.