Has it occurred to anyone else that these WitCHes are a blogging Ponzi scheme? As long as we keep posting new WitCHes, no one bugs us about not polishing off the old WitCHes. What the hell; we’ll keep going until someone calls the Blog Cops. And, to continue with the scheme, this WitCH comes from the Cambridge text Specialist Mathematics 1 & 2, in the section titled Linear Diophantine equations. Happy hunting.
13 Replies to “WitCH 56: Fuzzy Dots”
I actually believe (rightly or wrongly, yet to be determined) there is a lot of good mathematics to be learned from the study of linear Diophantine equations.
My gripe with this particular book (which my school does use) is that this chapter is messy at best.
No mention of Euclid’s algorithm, a passing mention of GCDs and the jump to generalization is… not nice.
For example, if I were demonstrating the equation to a class, I might start with:
“Is there an obvious solution?” To which the answer is “Yes, .
OK, now, can we get from this to a second solution somehow?
So a second solution is
“now… lets try to generalize!”
Before anyone remarks that I, too, have not mentioned Euclid’s algorithm – in cases where there is an obvious solution I would probably not mention said algorithm until demonstrating a more difficult example such as
Spot on, RF.
Hi, RF. Further on your comment, I’ve never had to teach this stuff and it’s not obvious to me how I would. What *was* obvious to me was that the text’s approach is totally ludicrous, and incomprehensible.
That’s often the way with these WitCHes, and more generally with the school materials I see. Usually I have no clear idea of how I would teach such a topic, at least not in any detailed example-by-example sense. But I *do* know that the materials I’m reading are Plain Fucking Wrong. They rabbit on about trivia and simultaneously leave out critical facts or ideas, making it all incomprehensible and pointless. The text excerpt above is entirely typical.
Your suggestion of incorporating the Euclidean algorithm is of course very natural, though not necessary. But what is necessary is making the role of common factors crystal clear. On that, the text gives us swamp.
For some context, I actually do teach LDE’s as extension lessons to Year 8/9 students occasionally and, channeling my inner Marty Ross decided that the starting point needed to be “the idea”. In this case, the idea is that you have an equation in two variables which students (should) know has an infinite number of solutions and add the condition that we are only interested in integer solutions with the AIM to be (unlike what the text says) to find ONE solution and see if we can THEN find the PATTERN which gives ALL solutions.
Depending on how far you go with the idea, it is 2-3 lessons worth of content.
And, unlike this piece of writing which mentions but glosses over the importance of highest common factors, I always try to bring it up as a VITAL step in how we get from one solution to the next.
For difficult examples, where finding the first solution proves difficult (if time was allowed) I would THEN and ONLY THEN bring up Euclid’s algorithm.
If (and only if) additional time were available at an advanced Year 9 level, a really nice extension is the use of the algorithm in the Bezouit (have I spelled that correctly?) Identity for encryption algorithms.
Again, apologies if this feels like a rant… I have strong feelings about this topic in particular!
I’m honoured if you consider me worth channelling.
You bring up an important point, that this topic, and *many* of the good Specialist 12 topics, can be covered in a keen year 8 or year 9 class. Maybe leave out some theory or proofs – which the Cambrdige 11 text fucks up anyway, and anyway doesn’t expect anybody to read – but otherwise pretty much the same.
Side note, but perhaps relevant: since LDEs are NOT assessable at Units 3&4 level, very few schools will actually bother to teach this at 1&2 level.
Yeah, it’s not really a side note. It’s the same for a bunch of the topics in S12. It’s one more reason why the text doesn’t care if it does a half-assed job.
Another method is to rearrange the equation to . Since is an integer, the numerator must be divisible by 4. With a bit of work, the patterns identified in the book can be found.
I agree that the presentation of the material seems messy. There is no reasoning on how the solutions were obtained, they simply appear plotted on a graph. There’s a lot of interesting mathematical ideas and reasoning that could be developed, but the book just seems to completely avoid that.
It seems to me that on a chapter on linear Diophantine equations there are a few natural questions that arise. Are there any necessary or sufficient conditions for solutions to exist (there are but the book doesn’t cover this)? Can we use ideas from the general case (over the reals) to help us find any integer solutions? Given we have found a solution, can we deduce other solutions (of which there may be infinite) and can we find all possible solutions using these methods? How many solutions are there for a given equation and is there an easy way for us to know? Is there a general solution or method of solution for linear Diophantine equations?
The text states that there are an infinite number of Pythagorean triples, but this is left unproven. It is trivial to show if is a solution then so is .
The phrase “family of integer solutions” is not defined and they clearly have not plotted the “a family of integer solutions” on the Cartesian plane with just 6 dots.
The proof (unless I’m missing something obvious) has an issue on the first line. We are starting with the assumption we have a single solution and want to show that we can generate an infinite number (in fact all solutions) from this initial one. But the first line of the proof assumes we know two solutions exist, rather than one! This could be easily fixed by showing that given is a solution, we can construct another solution and .
Thanks, studyroom. I fixed up your comment (no “latex” needed) and deleted the duplicate.
I think you have a better idea than me how one might go about making a properly rich unit on LDE. Of course the text above makes no such attempt; *nothing* in VCE is ever investigated in sufficient depth to allow the concepts and truths and techniques to interact and gel. It’s all show.
On Pythagoras, the text is not obligated to delve further into their natural if distracting offside. But even then they screw it up. The “for example” is weird, and your noting of the triviality of the infinity of solutions is hilarious; presumably the writers were thinking something else, but didn’t think enough to figure out what they were thinking about.
I thought about “family of integer solutions”. I thought they escaped on a technicality, but I think you are correct: you cannot talk about both “A family …” and “THE family …” for the same LDE.
Finally, for the proof, I think it is correct as far as it goes, but that is not far enough. (And, it’s an appalling and appallingly written proof.) What they have proved is that IF is a solution THEN any second solution is of the stated form. What they don’t prove, only stating it in the final line, is the converse: any of the given form is indeed a solution.
OK, no one else seems to be rushing to jump in, so I will offer a few more suggestions of WITCH:
1. The opening sentence does NOT define what a Diophantine equation is, but just says (some very common equation which more often than not leads to surds) “may be considered a Diophantine equation”. Sure… *may* be considered a Diophantine equation as well… but more commonly is not.
2. When attempting to define what a Diophantine equation is, the text again makes a complete mess of what should be a simple definition by saying “the intention is to find linear solutions for “. Um…. yeah… when I draw the graph of I find integer solutions for but this is not my intention, my intention is to draw a few points so that I can draw the graph. “Intention” here is a but of a poor choice of phrasing.
3. The phrase “from the graph” makes the whole thing pointlessly messy. Simple number theory is a lot more elegant (in my opinion) and shows a lot more insight into what Diophantine equations actually are and how they are used (also my opinion).
4. The AMT, in one of their competitions, once a problem along the lines of:
The equation has the obvious solution , how many other solutions exist for where and are both positive integers?
In my opinion, this is a far more deep and meaningful exploration of Diophantine equations than attempting (and making a mess of it!) a “proof” of the general case extrapolated from a specific solution.
Thanks again, RF. You and studyroom had already nailed a lot of the crap. On your further remarks:
1. I think this is a very good point, and I’d go further. To me, the expression “linear diophantine equation” rings very strange. An equation is an equation is an equation. Then, what you might do with that equation is secondary. I’d much prefer to talk of a “linear diophantine problem”. Perhaps convention discourages this, but the text still did a bad job.
2. This seems to get to my elaboration of your point 1.
3. I agree. I can see that at some point it is natural to note that the integer solutions are a subset of the real solutions, and then point to the graph. But the text allows the graphical to swamp the numerical.
4. Nice AMT question. Yes, you and studyroom and I would be much happier with a deeper topic. I’m still fine and keen to include a characterisation of LDE solutions, and a proof. Just not a proof so incomprehensible and butt-ugly.
The graph here could be improved. If you’re going to mention that “the equation defines a straight line”, then why not just draw in said line, then emphasize that the solutions to the Diophantine equation are all integer points on that line? As it stands now, the positioning of the labels give the illusion that the points *aren’t* all on a straight line.
Very good points (no pun intended).