Saturday, 6 June 2015

What makes a problem difficult?

It's not every day that mathematics makes the main news programs but that is what happened yesterday.  A GCSE ( UK exams taken by 16 year olds) maths exam question made it to the news as being very hard and having stumped many.  ( A quick search for "Hannah's sweets" in your search engine of choice should show just how much coverage it got …) 

Here is the main question…

There are n sweets in a bag.  Six of the sweets are orange.  The rest of the sweets are yellow.

Hannah takes a sweet from the bag.  She eats the sweet.

Hannah then takes at random another sweet from the bag.  She eats the sweet.

The probability that Hannah eats two orange sweets is 1/3.

Show that n2-n-90=0

..and my solution 

Start by considering the first sweet she takes out.  The probability that it is orange is 
(number of orange sweets)/( total number of sweets) which is 6/n

Now consider the second sweet she draws from the bag and the chance that this one is also orange.  Again this is  ( number of orange sweets)/( total number of sweets) but we need to remember that we removed 1 sweet already and it was orange so we have 5/(n-1)

To get the probability of the two events both happening we need to multiply the individual probabilities so we have the chance of drawing out 2 sweets and them both being orange is (6/n) x (5/(n-1)) and that multiplies out to (6x5)/(n(n-1) = 30/(n2-n)

In the question however we are told that he probability is 1/3rd so we get

1/3 = 30/(n2-n)

We know we can multiply both sides of an equation by the same thing so multiply by 3 and then by (n2-n) to get

n2-n = 90

Subtract 90 from both sides and we have n2-n-90=0 so that's the first part done.

The second part of the question asked how many sweets there were in the bag.

To work this out we ideally need to know that (x+a)(x-b) multiplies out to x2 +(a-b)x -ab

So what we are looking for are 2 numbers which multiply together to give -90 and add together to give -1. Considering this for a moment leads us to -10 and 9 so we now know that 

n2-n-90 = (n-10)(n+9) = 0

There are hence 2 possible solutions , either n=10 or n=-9

Here we need to intersect our maths with the real world and realise that as we have a bag with actual sweets in it n has to be positive so there must be 10 sweets in the bag.

Always good practice to check your answer so if the are 10 probability of drawing 2 orange sweets is 6/10 x 5/9 = 30/90 = 1/3 so that looks good.

So back to my subject line "what makes a question difficult". To solve the mystery of Hannah's sugar rush we needed 4 pieces of mathematical knowledge: how to calculate a probability; how to combine probabilities; how to rearranging equations and how to do a simple factorisation of a quadratic equation.     If you don't know those then clearly you will have a problem but I suspect many of the people classing this problem as hard would have known those things so what made this question hard?

I've been reflecting on this and I think it's because to solve it you need to head out like an explorer down an uncertain path.  We know where we are and we know where we are trying to get to but as we start out we have no clue about the journey.  It isn't instantly obvious what the formula we are asked to derive comes from, this only becomes clear once we make a start and see where it takes us.

This ability to make a start and see what happens strikes me as a very useful skill to have.  With the world moving at en ever faster pace I think to ability to innovate and develop ideas as you go along is an important capability to have.  Challenge is how we cultivate this confidence and ability to head out down the unclear path and see what turns up and to use that to move us to our destination.

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