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Original poster

Everyone loves an 'impossible maths exam' story

84 replies

noblegiraffe · 22/11/2017 18:54

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!

Everyone loves an 'impossible maths exam' story

OP posts:

relaxitllbeok · 22/11/2017 20:20

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.

relaxitllbeok · 22/11/2017 20:16

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.

larrygrylls · 22/11/2017 20:12

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.

catslife · 22/11/2017 20:00

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.

TeenTimesTwo · 22/11/2017 19:58

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...

PurpleDaisies · 22/11/2017 19:57

Not sure trig is geometric reasoning though.

Trig is a branch of geometry.

Original poster

noblegiraffe · 22/11/2017 19:56

Not sure trig is geometric reasoning though

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

OP posts:

larrygrylls · 22/11/2017 19:54

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.

Original poster

noblegiraffe · 22/11/2017 19:52

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!

OP posts:

AtiaoftheJulii · 22/11/2017 19:48

Oh ok, I thought again. Yeah, got it. Three teachers should have got that surely! I've sort of got used to the students not being able to apply knowledge out of context, sadly.

Now to read the thread.

littlechous · 22/11/2017 19:47

D’oh.

I got 144°

PurpleDaisies · 22/11/2017 19:46

It isn’t that hard if you spot from the key that GB = EG.

I’m very surprised the teachers were flummoxed.

dootball · 22/11/2017 19:46

The key is to read the word square, and draw on the line BG.
Then you can you Pythagoras and non calculator trig to find one angle and then you are away.
Or is that the same method you had ?

EllenJanethickerknickers · 22/11/2017 19:45

I got cos x/4 = 1/2.

TeenTimesTwo · 22/11/2017 19:45

Looks like we agree.
I hate trying to write answers out when I can't see the diagram!

AtiaoftheJulii · 22/11/2017 19:44

Are we allowed a calculator? If so, fairly straightforward. If not, I'll have to go away and think again!

relaxitllbeok · 22/11/2017 19:42

OK, I'll bite. What's actually the problem here? To do this question, students have to know what a kite is, and that the sine of 60 degrees is 1/2 (to be picky, the converse, that if an acute angle has a sine of 1/2, it must be 60 degrees), that the angles at the base of an isosceles are equal, and that the angles in a triangle add to 180 degrees. They have to use each of these facts once. Is any of those facts supposed to be beyond them, or was it just that there were more steps than they were used to?

As for the idea that their teachers were flummoxed... well.

TeenTimesTwo · 22/11/2017 19:42

I think crisscross is right, though my trig is rusty.
reasoning follow below so if you still want to do it look away now:

call the 3 lengths that are the same 2y. B and G bisect the side of the square. So the bottom triangle FG=y and DG=2y.
because the sides are the same length there are 2 isosceles triangles.
base angles = so each x/2 so apex angle BGD = 180-x
so DGF is 90-(180-x) = x-90.

oh to sin and ah becos so cos(DGF)=y/2y=1/2
cos(x-90)=1/2
sin(x)=0.5 x=30/150 and its obtuse so 150.

cantkeepawayforever · 22/11/2017 19:42

I would say it is well in line with last year's GCSE practice papers, in terms of reasoning. In fact, as it only requires minimal trigonometry and otherwise only sums of angles within a triangle and on a straight line, it is easier than some of those, which required circle theorems and all sorts.

cantkeepawayforever · 22/11/2017 19:40

I did it like this:

BG = 2GH

Call angle EGH y
cos y = 1/2, therefore y = 60 degrees

Internal angle of the kite DGE = 60 degrees as well, due to the sum of angles on the straight line being 180.

If you bisect x to give two isosceles triangles, then looking just at BGE, sum of its internal angles = x (ie 2 x 1/2x) + 30 = 180 degrees.

Therefore x = 150 degrees.

catslife · 22/11/2017 19:34

Angle BDG looks as if it could be a right angle, but does trigonometry count as a valid method?

Original poster

noblegiraffe · 22/11/2017 19:32

That's basically what I did, parietal (except base 1/2 and hyp 1). Maybe that is the way you're meant to do it then!

OP posts:

Original poster

noblegiraffe · 22/11/2017 19:29

I've done it, but I don't think my solution will be the one that the kids would be expected to come up with. I did a bit of trig, but it didn't feel very nice.

OP posts:

parietal · 22/11/2017 19:29

Triangle EGH has one side of length 1 unit and the hypotenuse length 2 units. So you can get the angles in that and then work out the rest from there.

Janek · 22/11/2017 19:25

Ah yes, i was starting to realise that myself...

So can you do it noble?

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