Everyone loves an 'impossible maths exam' story

[noblegiraffe]

The latest country to have an exam that has left students in tears is New Zealand. www.theguardian.com/world/2017/nov/22/impossible-new-zealand-maths-exam-even-flummoxes-teachers

Here's one of the questions if anyone fancies a go. I'm pretty sure there must be a more elegant solution than the one I came up with!

What's actually the problem here?

I think the main problem would be spotting the step that leads to all the things you described. There are lots of other ways you could head that would be time-consuming and fruitless, and remember this is a timed exam where the pressure is on!

GH is half of EG so angle HGE is cos-1(1/2)=60 degrees =FGD,so the angle in the middle is also 60. The half kite is iscosceles too so HGB =30, leaving the other two angles at 75 each. The angle asked for is two of these =150 degrees.

Not sure trig is geometric reasoning though.

Not sure trig is geometric reasoning though

That was my issue with that solution. It felt like cheating.

Not sure trig is geometric reasoning though.

Trig is a branch of geometry.

I think the problem is spotting you have to

• draw in BG
• take notice it is a square with bisected sides so giving them a length
• work with x/2 rather than x (at least I had to)

I bet if they had said the square has length 2y it would have seemed a lot easier...

I suspect that any solution where the candidates have attempted to use trig to work out the angle will obtain mark(s).
If the question had said "use trig to work out the angle" suspect that would have made the question clearer.

Ok here we go (I think).

Draw line DE making a large equilateral triangle below a smaller iscosceles one. All the angles in large triangle are 60.

But angle GEB=GBE (half the angle we are looking for, call it x) as that triangle is iscosceles. This implies that angle BED =X-60. Looking in the small half triangle at the top now x+x-60+90=180. This solves to give x=75 and the required angle at 150.

No trig required.

Yeah... So this was an exam that all 15yos were expected to take? It would be perfectly in place in a UKMT IMC (and it wouldn't need to be at the end), I'd say, but I could believe those students who'd never dream of doing that might get stuck.

Very nice, larrygrylls. In general, any problem that can be solved with as little trig as we used for this one can also be solved without it, but that's cute.

So it is 150!?

My reasoning would have been:

More than 90 but less than 180, the two small angles look significantly less than 45 degrees and there's two of them so I guessed they were around 15 degrees.

I don't understand the actual answer

Oh nice, larry. Yes, that's the sort of answer I was expecting for 'geometrical reasoning'.

My brain was really struggling to see the triangle BEG as isosceles, it just looks wrong!

Yes criss, you were right! Unbelievable!

I liked it but for me, not for an exam. I’m beginning to develop a love for geometry. Make square side length 2, then BEG is an isosceles triangle with main side lengths 2.

EGH is a triangle with side lengths 1,2, root 3 so angle egh is 60 degrees. That makes EGB 30. As it’s an isosceles EBG must be 75 degrees, Double that to get DGE.

Noble,

Thanks for sharing it. Good for a ‘problem of the week’!

but yes, Looking at the thread I used trig, not geometry, so it is cheating.

Nice solution larry

Larry, I liked your solution. Very elegant :)

This page has the exam in its entirety, if anyone is interested:
www.stuff.co.nz/national/education/99106862/you-do-the-math-ncea-releases-exam-paper-teachers-students-complained-about
Students are also assessed internally throughout the year, so doesn't necessarily cover the whole year 11 curriculum.

It does seem that the maths exam gets complained about every year!

You are all ridiculously smart. If my DD brought me this I'd fake a heart attack, it's the only logical way out.

thanks pingoose!

I had the same problem seeing triangle DEG as equilateral, noble.

Info in the question makes it equilateral. Brain is seeing it as isosceles.

Why am I so utterly shit at maths? I'm not thick but I just don't get it. I dissected the kite and it gave me 2 right angles which I halved to get 45 degrees. Same on the other side so added together 90degrees. Now I know this is wrong because it looks wrong and its an obtuse angle and tbf, looks about 150. Where am I going wrong??

Bugger, I cocked up but i just showed it to DS3 in Y11 and he got it in about 3 minutes. Hope for him yet.

let side of square = 1 (so, DG=GB=EG=1 also)
sin(FDG) = FG / DG = 1/2 => FDG = 30
BGD = FDG = 30 (alternate angles)
GDB = GBD = GBE = GEB (two identical isosceles triangles)
x = GBD + GBE = 180 - 30 = 150

I think.

Ok, here's my (inspired by ds just now) contender for nice and neat:

Bottom angle (can't remember letters or type while looking at picture) is 60 because equilateral.

x is twice the side angle because isosceles.

Therefore 2x = 300, x = 150