Everyone loves an impossible maths exam question

[noblegiraffe]

This one, from this year’s KS2 SATs apparently had children in tears and was described as ‘obscene’ by one teacher.

TBH it took me longer than I’d expect a one mark maths question for 10 year olds!

that probability one would be interesting to see the mark scheme.

I too dislike questions that you can just guess the right answer and work backwards, without showing the work as it doesn't mean you could solve it with more challenging numbers.

At the same time, when I look at it and see obvious denominators like 7 and 13, I know that the likely possibilities are relatively few. So I look for something that is a multiple of 7 that when you add 5, you'll get a multiple of 13. It takes seconds to find the right one, and was a lot less bother at this time of the morning than doing algebra, even though I could have. And yet, it's not totally random guessing either - you have to know some maths to be able to do that and explain why, so perhaps I'd have got a few marks if I'd written in words why I chose those numbers.

That’s the one, xsquared

Some of my class came strutting in saying ‘trial and error, Miss!’ I’ve spent 2 years telling them trial and error isn’t a thing

Why would you tell your children trial and error isn't a thing?? It's a recognised method in sats marking!

The slicing question really isn't horrific and yes the better mathematicians will have got it right. They would have realised that using only 2 lines means one square needs to be the size of the height so they draw the line to make that square and then the last line becomes very obvious.

Because grade 9 mathematicians shouldn’t be doing trial and error.

I saw the gcse answer by working out the denominators - can't remember how to work it out algebraically.

but for one with numbers that easy - and a little bit of reasoning, like I did above - it was a whole lot faster! You can just do it mentally like that. I could have shown full workings algebraically, but I didn't actually do them - I just reasoned that it would be multiples of 7 that were 5 less than a multiple of 13, and you only have to go as far as 26 to get there and you can be pretty certain it'd work.

They should make the numbers less obvious, because it would annoy me if I'd been able to do that, when I had the skills to show it algebraically, but didn't need to!

Seems pretty easy to me, but I'm an adult with:
a) post-secondary maths qualifications, and;
b) fully-developed exam technique.

I think (b) is the more vital factor there.
I've already learnt not to panic about weird one-mark questions.

Why do you say that mathematicians shouldn't be using trial and error? Are you objecting to the terminology (eg prefer to call it "trial and improvement"), because the actual technique is perfectly valid, eg Newton-Raphson (A-level) is a rigorous technique of trial and error (where the "error" [in the sense of the measured difference from the required answer] in one step is used in the next step to get closer).

D3 They wouldn't have been using Newton-Raphson though, they would have been using 'guess and correct'. They ideally 'should' be solving it properly. If they use guess and correct and get the right answer they'll be fine. However if they get the wrong answer they'll get zero. Whereas using a formal method if they make a slight error they'll still get method marks.

GCSE mathematicians won’t be using Newton Raphson and the old GCSE trial and improvement has been binned and replaced with iteration.

What my kids mean by trial and error is ‘a good old random guess, and if that doesn’t work, let’s pluck another number out of thin air’. Not a mathematical approach to encourage.

It was really frustrating to watch the children answer this question. They either got it in 10 seconds flat or they wasted a good 5 minutes because they couldn't work it out. I struggle to see which part of the maths curriculum it was testing.

Shall not/shan't is rewarding only those who read the Famous Five.

However, there is an argument for the Famous Five being mandatory reading for the under 11s!

T^3 - I know Y9 pupils wouldn't have been using NR for the problem in OP! As I said in my previous post, NR is A-level - I was giving it as an example of how trial and error in general is a perfectly respectable technique.

Clearly the problem with the giraffe's pupils is not that they shouldn't use trial and error but that they have a poor idea of what it means... (I get it - I was a maths teacher once and am all too familiar with the lazy Y9 pupil whose technique was just to guess).

nobleg - what type of questions do best at identifying good mathematical thinking in your experience? (Rather than just testing whether they can follow a memorized procedure.)

I think one reason the original question is hard is that the bits of language that serve to make the problem concrete do more than might be expected. Someone used to maths exams might naturally assume that the stuff about making cuts in a piece of card is irrelevant and it's just a question about drawing straight lines. But actually visualizing the physical setting makes it clear that the second cut only affects one of the two pieces made by the first cut, which is something lots of people here got wrong at first.

I was in a classroom watching little faces frown in confusion at this question.
I think the word ‘cut’ was meant to trip them up, if it had said “draw x lines to make x y and z shapes” it would have made it easier for them to understand. As pp mentioned, they either got it instantly or not at all.
I also saw the word shalln’t more times than I needed to.

Must ask DS how he got on with this one.

The answer was obvious to me, but I'm unusually good at maths and geometry, so that doesn't tell me anything about whether it's a fair question.

I'm not sure the word "cut" as opposed to "draw" makes that much difference (other than that it might make it easier for a child who thinks in very concrete terms to imagine what's going on).

It was interesting walking around the class to see how much trouble some children had with this question, whilst others got it straight away. A few who had answered all the rest were still trying to work this one out at the end of their time, having gone back to it.
I couldn't work out what maths skill it was supposed to test, unless it was testing if they knew what a square is.

Can someone please tell me the answer for xsquared’s puzzle with the coloured counters?

I sat GCSEs in 2005 and I’m sure I knew much more then than I do now!

@Fanciedachange1 - solution attached.

Of course, you could do it by trial and error : The initial set-up tells you the ratio of green to ball is 3:4. You can then try 3:4, 6:8, 9:12 etc to see what works with the second set-up.

*green to red is 3:4

The ‘shalln’t’ one isn’t unfair at all. Contractions are explicitly taught all the way through KS2 and ‘shan’t’ is one of them. Children need to think ‘do I ever hear shalln’t?’ instead of blindly following a perceived rule about the apostrophe representing one missing letter. Same with when they do a division problem with a remainder and blithely answer that eight and a half buses are needed to take the children to the zoo. That half a bus is a favourite of mine ...

The first one took me a few minutes. I didn't think to reason or logic about it, I just mentally tried different stuff.

An entertaining puzzle, but doesn't seem like what I'd expect on a maths test.

A question I've seen on adult numeracy tests: Given a certain size piece of paper (or carpet or whatever), how many pieces of a smaller size can you cut from it? This one can be deceptive - it's easy to find a layout that looks good, but in some cases much harder (and surely beyond basic numeracy) to prove it's the best. Maybe the tests choose dimensions so that's not an issue, I never thought to check.

Thank you Daddaddad!