I gave the dog problem from 2 posts again to my Wednesday student yesterday and he solved it in the way it was intended to be solved.
We start by choosing a letter to stand in for the answer we want. Let S be the number of small dogs. Also let L be the number of Large dogs. We have not be asked to find the number of large dogs but this is part of the situation.
So we can say S = L + 36 – because there are 36 more small dogs than large dogs. Actually L = S – 36 is the same and will lead us to the answer required more quickly.
Also L + S = 49 – the number of dogs. This ‘equation’ I think is intended – but I will return to this point.
We can solve the equations by substituting the first into the second to give S – 36 + S = 49.
Simplify by adding the S and the S and adding 36 to both sides, we get 2S = 85 and so S = 42.5
“But how can we have half a dog”, asked my Wednesday student, and a very fair question two. This is the more obvious reason why this is a bad question – interestingly bad but bad none the less. If we are going to encourage students to take ‘real life’ problem solving seriously, then the questions we ask should make sense.
But my other reason why I’d want to change this question comes back to the story about the black sheep. The point of the story is that as mathematicians we shouldn’t assume anything – or at least we should qualify any answer by stating clearly which further assumptions we have made – I claim the answer to this question is incomplete unless we also say
‘Assuming all dogs are large or small’ – i.e there are no medium sized dogs! Without that, we can’t use safely the equation S + L = 49!