We’re snowed with grading essays for the Evil Mathologer, and we’re trying very hard to get to a few larger posts. But for now, a quick WitCH. We don’t pay much attention to Foundation Mathematics, which is supposed to be the Easier Than Easy Maths Subject. Commenter Ron, however, emailed us about the following question on VCAA’s sample exam, posted last year. (See also this post.)

You have 90 seconds, and your time starts now.

Perhaps they were testing estimation…?

Were you able to even estimate it in 90 seconds?

50 cm from side to side, assume (estimate) the middle part is 25cm and the two overlapping parts are 12.5cm each gives me a decent estimate for the number of tiles horizontally which knocked out three of the options.

A guess (estimation) that the tiles were around 40cm high each and yes, I think I did.

But do I think it was fair for the intended level? Not at all.

Hi Red Five. Thanks for your input on this question which is just ONE of many concerns I had when looking over this SAMPLE exam paper. I think I am correct when I say that the information provided in the uncut hexagonal tile image proves that the we are working with regular hexagonal tiles. I then drew the tile layout on Isometric paper and came up with none of the 5 answers listed.

Either way, as Marty said, clearly more than 90 seconds worth of effort

Thanks, Rob. It had occurred to me that the precise number may differ from all suggested answers, but my 90 seconds ran out, so I didn’t bother. I don’t think the information provided guarantees that the tiles are regular hexagons.

Hi Marty. Can you please supply a picture of any non-regular hexagon which will perfectly match the given information in the image of the uncut tile. I am interested in finding out the interior angles. Good luck!

Hi Rob. I was imagining just squishing the hexagon down. Say, make the top and bottom four angles 150 degrees, and the other two middle angles both 10 degrees. I think the only thing that marginally contradicts the squished hexagon is the weird horizontal dotted line, of length 25cm. But that’s only a contradiction if we know that the dotted line ends at the centre of the hexagon, and we do not.

Of course we’re supposed to assume that the dotted line reaches the centre, but what kid’s gonna be pondering in that? They simply going to assume that the hexagon is regular, as was intended, and the dotted line is just there for ass-covering.

All they had to do, apart form not asking the question in the first place, is to declare the hexagons to be regular.

The height of a regular hexagon is not double the length of its side. It works out that each column in your diagram should have an extra hexagon.

Hi Joe. Thanks for your comment. This problem was really bugging me but now I can see clearly where the answer 59 comes from

That is quite funny (and depressing) that when you actually work through the problem, none of the answers are even close!

Again… maybe they were trying to test estimation (badly).

I failed to do it in 90 seconds

Yeah, but you’re merely an applied mathematician, right? Clearly you don’t have what it takes to do the Easier than Easy maths subject.

Good grief! This reminds me of a skit that I once saw. A well-known professor at a certain university was known for setting very difficult problems for his graduate students. At a Christmas gathering, the students had him dying and going to hell, whereupon the devil presented him with one of his problems. Maybe a similar fate awaits the setter of this problem.

The first column uses 0.5m, all columns thereafter use 0.375m. 3.5 = 0.375x + 0.125, x = 9.

The height of a hexagon is 0.25 * cos(30 degrees) * 2 = 0.433. The max hexagons per column is: 3.05 / 0.433 = 7.04. (Students have access to a calculator.)

Every even column has one less hexagon than every odd column. 4 * (7 + 6) + 7 = 59.

(This took me longer than 90 seconds.)

This would be a great competition question, but it’s ruined by being MC and being allocated merely 90 seconds in the easiest maths subject.

The fact that it’s multiple choice suggests that an approximate method is permissible and probably expected.

Hi, J.D. Of course approximation is permissible, and perhaps even expected. But you must then word the question accordingly.

I’m not sure about that. I think it’s understood that elimination is an acceptable strategy in multiple choice questions. Here we have a question in which it takes a good deal of effort to establish that B is the correct answer, but much less effort to see that A, C, D and E must be incorrect.

No. The question is stuffed.

Thanks, Joe. That’s a nice approach to it all.

Gotta love these “real world” maths problems. Where’s the gap/allowance for the grout?

As far as “real world” problems go, this is the best VCE one I’ve seen, because the answer could be useful. Most “real world” VCE problems are completely meaningless.

Hi Marty,

Glad to see now you started to speculate Foundation Maths exams.

On the same (sample) paper published last year, there was also a multi-choice question which VCAA provided an incorrect answer I believe. If you (and maybe other commentators too) look at MCQ5, VCAA states the answer is B.

The question tested ‘same proportion’, but when you calculate the unit price for each item from II and IV, you will see item II will have different unit price… It is hard to figure out what was intended.

The paper can be accessed from here:

Click to access foundation-math-sample-w.pdf

Only MCQ has ‘official answers’ at the end. The short answer questions don’t have VCAA official answer.

Hi Sentinel. I’m not *really* speculating about Foundation. My attitude is the same as for Further-General: I ignore the subject unless someone raises an issue, as Ron did with the question above. But I am always happy for people to flag such issues/questions to me, about basically anything where they think my arrogant knowitallness might be of some assistance.

With MCQ 5, if I haven’t misread the question, I assume the intention was that “same”

ratio(“proportion” is wrong) was to include the possibility of rounding to the nearest cent? That strikes me as odd, but it also feels deliberate since the issue of rounding was easy to avoid in the question. So, presumably that version of “same” “proportion” is something stated or assumed (by VCAA anyway) in the subject? Again, this seems odd. But this is why I don’t look at these subjects unless asked. When confronted by such nonsense, I can only really think “Who gives a stuff?” (I wouldn’t think “stuff”, but I’m trying to tone down my language.)I’m happy for you to raise other issues with the sample Foundation exam, either here or on the original post. Ron also mentioned he had other issues, but didn’t elaborate. The purpose of the original post was to attack the Specialist sample questions, which were, and are, a disaster; why VCAA has not removed them is simply beyond me. I only included Foundation links for completeness. But if there are things you think worth discussing I’m happy for you to raise it.

Thanks Marty.

What you suggested (the possibility of rounding to the nearest cent) may well be a reasonable assumption, but student should not, 1, second guess (despite it being a ‘common sense’), 2, just rely on estimation.

My reason is as follows:

(1) students have their scientific calculator allowed during such an exam, not rough estimations by eye.

(I know you like using your hammer to smash calculators but for Foundation students…well).

(2) They may calculate the unit price for the sugar:

$5.98 per kg, vs $7.48 per 1.25 kg (the latter simplifies to $5.984 per kg, NOT exactly $5.98).

Also juice $3.25 per 250 ml = $1.30 per 100 ml. Another bottle of juice: $7.80/600 ml = $1.30 per 100 ml.

This item is the only one with two equal ratio/proportion.

I get when you just buy one pack then the unit price could’ve been rounded down (as discount or whatsoever), but…I have another very straight-forward reason against the suggested answer by VCAA (in real-life context).

Imagine you need 100 kg sugar, not just one kilo.

If you buy the 1-kg pack, you need 100 such packs, and you pay $598.00.

If you buy the 1.25-kg pack, you need 80 such packs, and you pay 80*7.48 = $598.40.

Totally different amounts of money. Therefore, two different ratios.

Unless you meet a very gregarious shop manager/owner who don’t care about the 40 cents, just giving you a tiny discount.

Well – who give a stuff about that – a tiny exam multi-choice question on just a sample exam?

I care, and I believe many other maths educators also care.

Because we want our kids/next generations to have the correct numeracy skills, based on CORRECT mathematics, not just quick pragmatic checking by eyes or by experience.

Thanks, Sentinel. I’ve fixed up the typesetting of your comment (the dollar signs were interpreted as LaTeX triggers.)

We’re not disagreeing at all. The MCQ is awful. I’m just questioning the ground rules, to determine the parameters of that awfulness.

I don’t think the rounding was a “reasonable” assumption for students to make, at least not on the basis of “common sense”. Words have meaning and “same” means same, at least on Planet Earth. The only querying I had is whether Foundation somehow expects such per unit costs to always be rounded to the nearest cent. You seem to suggest not, or at least not of which you are aware.

If such rounding is not standard and universal Foundation practice then of course the question is screwed. The question would be screwed even if VCAA had indicated the correct answer, E. That’s because, whether or not VCAA has said anything about rounding conventions, of course it is reasonable for students to notice the sugar ratios are the “same” to the nearest cent and then worry, reasonably, about what VCAA intended.

Even if such rounding is standard and universal in Foundation then the question is still screwed, because words still have meaning: “same” still means same.

As for my “Who gives a stuff?”, that wasn’t for a minute directed at you, or hypothetical students, trying to figure out this crap. It was directed at any teacher or student having to worry, ever, about the third decimal place of a sugar ratio. What kind of small-minded twat regards such matters of any importance or interest?

Regarding giving a stuff, however, of the MCQ being screwed, I’m 100% on your side. This MCQ demonstrates the worst, the most loathsome aspect of VCAA, the desire to nitpick on trivia while, at the exact same time, they are incapable of writing coherent questions or demonstrating an ounce of integrity or honesty.

You have every right and good cause to bash this idiotic MCQ.

Well said Marty.

I knew your ‘who gives a stuff’ was not directed at me. No worries. 😉

Many colleagues teaching Foundation Maths were actually astonished and agitated by the demanding exam paper last year. I am not teaching Foundation, but just like you, after we saw /attempted the paper/have been informed with these issues (implicit or explicit), we feel angry if the paper was not quite right. Let’s give VCAA some opportunities and time to improve…if they ever listened.

Hopefully any future official communications/practices from VCAA would be more consistent and more transparent in some way – fingers-crossed.

Thanks Marty for this; by the way, it’s Rob not Ron. I can outline a few other issues I had with this SAMPLE exam:

1: Clearly a 15 minute reading time is totally inadequate with the amount of text and image processing required to have a remote chance of getting anything positive out of it. On hindsight, the trick I suppose is to try to identify all the simple straightforward questions with a cursory glance over the exam.

2: The exam took me about 2 hours to do and I never finished MCQ20 and Q7 in Section B. Section A taking almost 30 minutes and Section B questions taking on average about 7 or 8 minutes. Do any other commenters have any comparative data?

3: MCQ14 Maths or Economics?

4: Section B Q3: Do the plumbers charge by the hour as implied at the start of the question, or by the minute as implied in part e?

5: Section B Q7: Is the pronoun ‘they’ used for PC? And does the designer in part c BUY rather than SELL blank T shirts as stated, or buy them and screen print and then re-sell them for $66? All this is taking time to process and just for 2 points!

6: Perhaps my favourite is Section B Q2 part d ii where clearly the question should have asked for the % error in the manager’s estimate of the number of shoes sold. Unless I suppose, the manager can’t count?

Agh! Sorry, Rob. I’ll look at the other question you’ve just noted.

Hi, Rob. On the specific questions you’ve noted, I don’t think any are worth a separate post, although they definitely could if they appeared on a genuine exam, rather than a sample. They’re bad. In brief:

*) MCQ14. I’m fine with the intent of the question but the wording is atrocious.

*) QB3(e). Yeah, the question isn’t great. I guess there’s only one thing to do, but it’s all pretty stupid.

*) QB7 I’m OK with “they” to refer to a person of unknown sex (but not to Hannah Gadsby), but I understand why others may object. But yes, the wording of (c) is insane. I assume the second interpretation is intended, that the designer purchased the blank shirts at $40 each, and sold them printed at $66 each, but I’m not sure. Simply appalling.

*) QB2(d)(ii) Yep, bizarre wording.

Hi Marty, thanks for your feedback and for all the other commenters. The main issue which needs to be addressed is the time taken to read and process the written and visual content of the exam paper; are these the main skills being tested rather than ‘Foundation mathematics’ ?

Furthermore, I earnestly feel that the person who wrote and proofread QB2(d) (ii) has made a fundamental error. It’s MORE THAN bizarre wording. You simply can not calculate the %error in an item of data obtained by counting; can you? Unless of course the manager miscounted; in which case the true data value is unknown.

Thanks, Rob. Quick replies.

1) Of course your “main issue” is spot on, and goes far beyond the nonsense questions being discussed here. The amount of reading and digesting required is astonishing, and absurd.

2) Your belief that the sample exam was proofread is charming.

3) I guess we don’t know without VCAA’s solution whether B2(d)(ii) should considered a “fundamental error” or just “bizarre wording”. I assumed they meant the percentage error in the estimate, in comparison to the actual (bizarre). But perhaps they meant what is literally written, the percentage “error” in the actual, in comparison to the estimate (error). Either way, garbage.

In the context of the rest of the problems, it seems fair to assume that the expected solution would have been via estimation. Specifically, the intended approach is probably to use a calculator to divide the total area of the rectangle by the area of one hexagon, or an approximation thereof, and to give an answer that is less than the result, but not too much less.

For this kind of approach, I think 5 minutes would be a reasonable length of time, not 90 seconds.

But what is the basis for the 90-second time limit? Is there an assumption that time should be allotted proportionally to point value? If a paper has problems of varying difficulty, then the problems at the top end of the scale can distinguish between the most able candidates, who presumably won’t have needed all their time for the easier questions.

Perhaps I’m not familiar enough with the system in Victoria to comment on this, but I wonder if it’s being asserted that an exam on “easy” math shouldn’t have hard questions. One could imagine that the difference between different levels of exam would be primarily in the curricula they cover without there necessarily being a low ceiling for achievement within one of the more restricted syllabi.

J.D. I don’t imagine that you regard this question as acceptable, but I’m happy for you to play Nitwit’s Advocate.

As you suspect, the 90 seconds came from averaging: 120 minutes for 80 marks. Of course you are correct, that on any exam some marks will intentionally and reasonably require more time than others. You want hard questions. But there are limits, and this question is obviously way, way over the limit. If it takes 30 seconds to simply read the question, independent of trying to comprehend it enough to begin an approach, that is a bad MCQ.

Even if the question were properly diagrammed and properly worded, it’d be way over the line.

Whether the question is acceptable or not would depend on expectations in the context of the Victorian education system. If the expectation is that a student who is reasonably competent but not outstanding should be able to achieve a perfect score, then I agree that this is a bad question.

Still, the approximate method reduces the complexity of the problem considerably. One hexagon is inscribed in a circle of radius r = 25 cm. The area of the disk is square centimetres, so the area of a hexagon is probably 80% or 90% of that, or 1600 to 1800 square centimetres. The number of complete hexagon equivalents in the rectangle comes to about , give or take 5%. This already eliminates answers C, D and E. Now some percentage of the rectangle is covered by broken hexagons, and this percentage looks a lot more like 5% than 30%, which is what it would have to be for the answer to be A. So the answer has to be B.

Then take my word for it: even by the lunatic standards of VCAA, this question is not remotely acceptable. Even if the wording and diagram weren’t screwed.

In partial defence

– even low maths is going to have some kids in it who are anomalously good, so they need some questions in there to separate out the top. They literally can’t just write a bunch of fair questions.

– in Foundation, they can’t do this through tricky algebra or hard concepts, so they’ve got to differentiate on speed

– there would be kids in foundation who are absolutely drilling regular shape dimensions Monday through Sunday all year

– others would have “height is root 3 times side length” on their cheat sheets.

So it’s probably not quite as crazy as it seems.

[Judge calls the defence lawyer to the bench]. I just want to be clear: you’re defending your client with the claim that “They literally can’t just write a bunch of fair questions”? OK, I’ll consider that when making my ruling.

It sounds ludicrous but it’s a fact of having [i] elective streamed maths subjects and [ii] competitive, comparative study scores. There is going to be a small but significant fraction of kids inappropriately in Foundation who could have been in General or even Methods. Only a tiny sliver (about 1 in 400) of them can get the 50 score.

It sounds like a good idea to write an exam full of fair questions that can reasonably be answered in the allotted time by a conscientious student who belongs in the subject. But thats going to result in a situation where getting 100% isn’t enough to get a top score.

The differentiator is going to be either things like this or a godawful essay about ‘applications of trigonometry in the built environment’ or what have you.

Hi, Alex. I was being a little (but only a little) tongue in check.

I take your point. We want questions to differentiate at the top end. And, the weaker the subject and the more trivial the culture of testing, the more difficult that is to do. But I think you have, hilariously, hit upon exactly how VCAA handles this: just hit ’em with an

unfairquestion or two.VCAA does this with the other subjects as well, particularly Methods. They’ll have a ton of trivia and then a question or two that are batshit insane. Not hard. Insane. And this is all nuts.

If VCAA cannot properly test the students with

fairquestions then VCAA has screwed up well before the exam has begun. After which setting unfair questions is not a fix, it’s simply salt in the wounds.