Anna has 4 cakes. Three of them are squares with sides of length x, and one is rectangular and measures 2 by (3x+2). The total area of all the cakes is 13. What is the length of x?

This I'd set as a higher tier GCSE problem.To begin with I'd draw the shapes so that it's clear what we're working with. Then I'd ask my tutee to work out the areas of the individual shapes--these will be x² for the squares and 6x+4 for the rectangle. I'd get them to note that the total area of all four shapes is 13, so adding the areas of the individual shapes together, we must have 3x²+6x+4=13. We can then note that to find x we just have to solve this equation.We first take away 13 from both sides to give us 3x²+6x-9=0. I'd then get the student to note that 3 is a common factor of each coefficient of the equation, so we can write it as 3(x²+2x-3)=0. What does this actually mean? It means 3 times some number (x²+2x_3) is equal to 0. This only works if (x²+2x-3)=0m so we just need to solve this equation. We do this by factorising it into (x-1)(x+3)=0. We see this means x-1=0 or x+3=0, or equivalently x=1 or x=-3. The last step's to note that we can't have a negative length, so the answer must be x=1.