This MitPY is a request from frequent commenter, Red Five:

*I’d like to ask what others think of teaching (mostly linear) Diophantine equations in early secondary school. They are nowhere in the curriculum but seem to be everywhere in competitions, including the AMC junior papers on occasion. I don’t see any reason to not teach them (even as an extension idea) but others may have some insights into why it won’t work.*

I suppose motivation is the main point for me. There are a

lotof “competition” type questions that are incredibly popular in olympiad and olympiad-lite contests but aren’t taught (at all) in high-school. There are massive books on these with enough content to fill the curriculum twice over.It’s good stuff, but from my perspective, there are much higher priorities. What about giving calculus a better treatment? I’d be happy with that, as a first step.How about talking about functions, sets, relations, logical operations? That would be nice. Can we also please have these things articulated into decent assessment questions? Thanks.

I agree with Glen.

If we’re talking early secondary school (Yrs 7-8), then I’d be focusing more on algebra, trigonometry, index laws etc.

In middle secondary school (Yrs 9-10), I’d be focusing more on algebra (linear and quadratic equations), trigonometry, index laws etc.

Because if students at those are struggling with this stuff (and they are), teaching Diophantine equations is a waste of valuable time. Examples of Diophantine can be taught naturally within contexts already taught (eg. Pythagoras’ Theorem – Pythagorean triples).

Besides, you’d want it to be much more than just a pure ‘exploration’ topic using a CAS or Excel.

I’d put Diophantine equations into Specialist Maths at the Yr 11 level (or Yr 10 extension level) and delete the boring as bat-shit descriptive statistics. I think students at this level have sufficient academic maturity and motivation to study them. There are many interesting applications – I explored one of them in a MAV Conference presentation (back in the day when you didn’t have to pay MAV to present – it’s my only presentation that’s not in the Conference Proceedings Book, due to me missing the deadline by a couple of days).

I haven’t thought much about this, and I guess I’m sceptical as well, but not as sceptical.

There’s no question that the most important task in Year 7 and 8 is getting the kids’ arithmetic and algebraic skills up to speed, or at least to stop them from speeding in the wrong direction. And, yes, there tends to be a specificity to Diophantine questions that is distracting from general principles and can be time-consuming. And, yes, those idiot CAS-Excel explorations are worse than useless (although I’m sure Red Five isn’t remotely contemplating such swill.)

But the specific properties of specific natural numbers are important and interesting. And, I think the interplay between the specific and the general can be engaging and illuminating, at least in Years 9 and 10, and probably earlier. In brief, numbers have souls.

To digress for a second, about ninety years ago I read a paper by my PhD supervisor on “mod 4 surfaces”. (It’s like modular arithmetic: if you have four layers of a surface together it cancels out to zero.) I tried to figure out why his argument didn’t work for mod 3 or mod 5, or anything else, and eventually I succeeded (which my supervisor then confirmed): the specific trick he used was that 2 x 2 = 2 + 2. That was it, that was the one special, eccentric fact that made mod 4 surfaces very simple in this particular context.

OK, so here’s a Diophantine question: for which natural numbers M do we have

M x M = M + M?

Or, for which natural numbers M and N do we have

M x N = M + N?

Or, the favourite of any AFL-fanatic kid, who notices that 7 goals, 7 behinds totals to 49 points: for which natural numbers G and B do we have

6G + B = G x B?

Of course, the interest in these questions is not only the answers but to

provethe answers, and preferably in a reasonably efficient manner. I think those efforts can be very rewarding, not only for practising skills, but in getting to the souls of numbers.One last one. This one is from a colleague, Dr. Death, who is writing (stunningly good) notes for a year 9 extension class. In his Expansion chapter, Dr. Death takes time out from the usual material to give this example:

A rectangle has sides of integer length. When 3 is added to the height and 2 is added to the width, the area is tripled. Find the possible dimensions of the rectangle.It seems to me that at least a few such examples and exercises, at all levels, is well worthwhile.

Nice questions that only need simple factorisation. Interesting that your first two examples have the same answer, I was expecting more from the second.

I like the third.

Thanks all, I was thinking of Year 7/8 advanced students in some form of withdrawal elective, assuming the school would consider such an activity.

My first thought was something along the lines of

which has two solutions for which and are positive integers. One solution is (trivially) but there is another.

Whereas has more than two solutions where and are positive integers.

I agree with all commenters who say it is not worth pursuing for students who are yet to show strength in arithmetic and algebra.

For those who like a historical example, search up the cannonball problem…

In summary: how many cannonballs can you have if they can be perfectly arranged in a square or perfectly stacked into a square-based pyramid?

The trivial answer is 1. There is another answer.

“This tomb holds Diophantus. Ah, what a marvel! And the tomb tells scientifically the measure of his life. God vouchsafed that he should be a boy for the sixth part of his life; when a twelfth was added, his cheeks acquired a beard; He kindled for him the light of marriage after a seventh, and in the fifth year after his marriage He granted him a son, Alas! Late begotten and miserable child, when he had reached the measure of half his father’s life, the chill grave took him. After consoling his grief by this science of numbers for four years, he reached the end of his life.” (Gow & Page, p. 93).

Develop a timeline of the life of Diophantus.

For more examples from the Greek Anthology, see Mills & Ratcliffe (2018).

References

Gow, A. S. F., & Page, D. L. (1965). The Greek anthology: Hellenistic epigrams. Cambridge UK: Cambridge University Press.

Mills, T.M. & Ratcliffe, J. (2018). Mathematics in the Greek Anthology,

In G. FitzSimons et al. (eds), Teachers Creating Impact, Proceedings of the 55th Annual Conference of the Mathematical Association of Victoria, 6-7 December 2018, Brunswick: MAV, 36-39.