# PoSWW 39: A One-Sided Triangle

As for our most recent WitCH, this one comes from VicMaths, Nelson’s Specialist Mathematics Year 12 text. It is an exercise and solution from the Logic and Proof chapter (covering a new VCE topic).

## 29 Replies to “PoSWW 39: A One-Sided Triangle”

1. Red Five says:

You have said it before, in relation to Mathletics if I remember correctly…

…hire a mathematician!

2. Back in Black says:

(b) What axiom is used to get from line 2 to line 3? What is the rigorous logical justification? I might be wrong but isn’t it essentially assuming what has to be proved?

1. marty says:

Pretty much.

1. Red Five says:

It is worse than that though, because:

1. Assuming fact A is correct and then using this to prove A is correct proves nothing.
2. All the intermediate steps add nothing.
3. There is no justification WHY the first line is even valid.
4. This is not a difficult proof to get right and there are several FREE examples available, here is the first one that came up in my 0.5 second search: https://math.stackexchange.com/questions/307348/proof-of-triangle-inequality
5. If this were to be asked in a VCAA exam and a student were to have learned the solution presented in this text and write it on the exam paper… yes, well, we may never know!
6. The mark allocation sucks.

1. marty says:

Well, it’s not quite assuming what it is trying to prove. But it’s not dealing properly with |x|, which is of course the key point.

2. Terry Mills says:

As a matter if interest, if a student gave the answer to 21b as above, how many marks would you give to the student?

1. marty says:

I dunno. 1 mark out of 5? But I’m not sure I’d ever ask for the proof of the TE, except as a “this is a proof you must know” question in an analysis subject, in which case I’d grade it pretty tough.

1. Terry Mills says:

According to the marking scheme, the examiner must give 0,1 or 2 marks out of 2.

1. marty says:

I would shoot the examiner and give 1/5.

3. Terry Mills says:

This is a difficult question – for students who have never been exposed to proof in the context of basic analysis.

When I read the question, I wanted to know “What should students know before they come to this section of the book?” Or, at a more basic level, “What is the purpose of this topic in the curriculum?”

There are simpler introductions to logic and proof.

1. marty says:

Yes, it was an odd question to ask, which is why I looked at the solution. “Simpler introductions” than what?

4. The problem might be that, while the stated author(s) wrote the main body of the text, the publisher farmed out the writing of the solutions to a student or some other person either lacking mathematical knowledge or the incentive to make sure everything was sound.

1. marty says:

Thanks, Ed, but I don’t think so. The topic is new in the curriculum, but appears to be pretty simple and mechanical in what is expected. As Terry was suggesting (I think), the triangle inequality is an odd choice of exercise for this topic.

2. Terry Mills says:

It is a common practice I believe for publishers to pay someone other than the authors to work out the solutions.

1. marty says:

Yes, maybe that occurred here. But there is still the question of what the author intended. And, if the solutions to a logic and proof chapter are outsourced without the authors of the questions continuing to pay attention, that is a recipe for nonsense of exactly the type exhibited here.

1. Red Five says:

Terry (and others) you are correct that publishers do, frequently, pay people other than authors to write worked solutions, but I think that misses the point. The author of the exercise, I would expect, has already written a sketch solution when they decided if the question was valid or not. A second author would then have, presumably, proof-read the whole exercise.

What that says about the pool of talent from which the publisher went fishing I do not want to contemplate in case it is the same pool that VCAA farms for exam setters.

1. Anonymous says:

The fish doesnt fall far from the pond.

5. Tony Gardiner says:

Come on folks; give us a break.
It works just fine for positive reals!

(But perhaps
x + y <= x + y
isn't much of an advert for "Logic and Proof".)

1. Terry Mills says:

There are 12 chapters in the book. I assume that they must be all covered, say in about 24 weeks to allow plenty of time for revision for the final examination. This would be 2 weeks/chapter, at say 4 lessons/week. So all of logic and proof should be covered in 8 classes, starting from scratch.

1. Back in Black says:

Logic and proof is also in the Specialist Maths 1/2 study design, so you’re not quite starting from scratch.

@RF
Re: 6. I agree. In particular, (a) worth 2 marks leaves much to be desired.
Re: 4. Absolutely! (So to speak). I knew something was rotten in Denmark when this ‘proof’ was one I had never seen before …
Re: 5. *Ahem* …. Yes, well …. One of many elephants in the room.
Re: Hire a mathematician. The publisher probably thought they had. Here is the question: How does one recognise a ‘subject matter expert’? That is the question facing many organisations including textbook publishers. It is related to the question schools used to face when hiring teachers (I say “used to” because these days our policy makers have painted schools into the corner where they are forced – more or less – to hire the single applicant they might be lucky enough to get. Having a heart beat is good enough, but less may be required in the future).

@TM and Marty. Re: Odd choice. Cambridge has similar (and long standing) questions for complex numbers. And I’ll re-iterate that logic and proof is not introduced at the 3/4 level, it was introduced in 1/2.

@TM and EB. Re: Solutions. I’m inclined to agree. But for questions related to content new to the curriculum (particularly content that many teachers will have had little to no previous exposure to), one would hope the textbook writers themselves at least reviewed the solutions if not actually wrote them.

@TG:
Re: Giving a break. That would be Case 1 …
Re: Advert. One could reasonably say, without being specific, that there are many resources floating around that aren’t much of an advert. And one might again be tempted to wonder how you recognise an authentic ‘subject matter expert’ as opposed to a cheap (or maybe not so cheap) fake.

1. marty says:

The triangle inequality is a natural choice when doing complex, because you have triangles staring at you, and you’re naturally studying and trying to figure out how distance/magnitude works. But the TE is an odd choice here because the “logic and proof” topic in VCE is in practice focussed and substantially mechanical, and the TE doesn’t contribute much to that focus/mechanics.

In terms of “subject matter experts”, I’m not looking to defend the publishers or whoever they got to write/proofread/etc.. But I think the poverty of mathematics textbooks can have as much or more to do with the nature of the industry. Yes, if you have a Gardiner or a Barbeau doing stuff for you that’s gonna make approximately an infinite difference. Even a marty will do in a pinch. But if you are given no time to write, or you are given an incomprehensible chapter to proofread, even a strong and school-alert mathematician would be pretty much screwed. Even assuming a coherent curriculum and assessment system, which, here, we do not have close to having.

I was once asked by a textbook editor to get involved in a new series of textbooks. I respected the editor, thought they were smart and knowledgable and sincere, and I declined. I declined because I figured I would have to co-work or co-author with people I would regard as not up to the job, and/or that I would have to proofread/whatever drafts that were unsalvageable. Maybe I was wrong. But I doubt it.

1. Back in Black says:

Yeah. I think publishers generally pay writers peanuts (unless your name ends in Woo). Often there’s a royalties clause in the contract that boosts the financial pot a bit (*). Undoubtedly in this instance there was limited time to get drafts, proof read, edit and publish (**). Having more writers helps in dividing the workload but it also divides the payment and, as you say, you’re possibly working with others who are not up to the job. And a textbook can only be as good as the curriculum its writing for, which means that if you’re lucky your textbook may rise to a standard of mediocrity.

Having said this, there were only a few chapters of new material required. The bulk of all the new textbooks has been recycled.

I think the main reason one would be involved in writing a textbook is either naivety or self-promotion. And I suppose if you’re semi-retired it makes for a nice side hustle. (***)

* When I was young, with hopes, dreams, ambitions and no mortgage, I’d hoped to write a hit song and be a one hit wonder and live off the royalties. Didn’t quite go to plan.

** I really hope this is the case. I’d be really upset if I found out that writers got access to syllabi and resources months before teachers did.

*** Math Goes to the Movies and its like excluded of course.

2. Back in Black says:

“The triangle inequality is a natural choice when doing complex, because you have triangles staring at you, and you’re naturally studying and trying to figure out how distance/magnitude works.”

Aha … Hence “A One-Sided Triangle”. I got it.

2. Banacek Spaces says:

Not all spesh 3/4 students would have 1/2 behind them. Most of my classes were 3/4 spesh straight from 1/2 methods. Smaller schools generally cannot afford to run 1/2 spesh.

1. marty says:

Geez. Really?

1. Red Five says:

A hangover from the days when there was no Specialist 1&2. There were subjects called General Mathematics A and General Mathematics B and schools could run them together in a single class.

With the new study design now saying that content such as sequences and series can be assessed at 3&4 level as part of the proof topic, I do genuinely worry for schools that do not have a dedicated 1&2 Specialist class.

6. I think the more important question is why one gives a formal proof of the triangle inequality at this point. One should not be in the business of proving things that students are not asking about, and that if you prove something, it should be framed as a clarification of something dubious or as a way of showing how propositions are related. On the basis of a geometric interpretation, the triangle inequality is easy for students to accept, and they may see a proof as a tedious bit of pedantry that cuts no ice. For many students, the first time they see the triangle inequality proved is in a first analysis course where they have been primed to see the need for careful definition and greater rigour. At that point, you can offer the challenge of giving a tighter argument; i have found that at that point they have considerable difficulty because they realize that their apprehension of absolute value is too loose. You have to go back to the definition: |x| = x when x >=0, -x when x < 0. Then |z| <= |x| iff -|x| <= z <= |x|. Then you get the triangle inequality by adding the inequalities -|x| <= x <= |x| and -|y| <= y <= |y|. This situation is somewhat subtle and one has to wonder where year 12 students are at to appreciate it.

1. Back in Black says:

Furthermore, once the triangle inequality has been proved, there are all sorts of other useful things that it can be USED to prove. In a first analysis course. Which gives the theorem and its proof RELEVANCE. I think relevance is important. The triangle inequality has no relevance in Specialist Maths – therefore the inequality and its proof is just another dry and boring direct proof. Its utility unappreciated.

Picking up on one of Marty’s earlier comments:
“The triangle inequality is a natural choice when doing complex, because you have triangles staring at you, and you’re naturally studying and trying to figure out how distance/magnitude works.”

I wonder what would happen if a student did this and then said that because its true for all complex numbers it must be true for real numbers …

2. marty says:

Thanks, Ed. I take your point.

Since “proof” in VCE maths barely exists outside of this new, contrived topic, I’m not sure how “formal” any proofs of TE would be (but which does not much negate your point). Nelson, the text with this exercise doesn’t appear to otherwise mention the TE. A second text has TE as an exercise (with “hint”) on vectors, as a byproduct of dot product stuff, and a third text has no mention of TE at all.

I could (and do) write generally about the Alice in Wonderland nature of teaching and writing for this atrocious, and atrociously examined, curriculum. Good teachers and textbook writers are pretty much winging it, with zero proper guidance or assistance from curriculum authorities or publishers. So, I give teachers a lot of leeway. I give textbooks much less leeway, if only (and not only) because their nonsense is published and spread wide.