Shape Times Shape

We had some very well-reasoned solutions to this problem - thank you to those of you who wrote a lot of detail. Unfortunately there are too many of you to mention everyone by name. Suryasnato from Limes Farm Junior School wrote:

The upside down isosceles triangle is 0 because every time it is in a question it is the answer. The only number that rule applies to is 0.
The diamond is 1 because every time it is in a question the other number in the question is the answer. This can only happen if the number is one.
The square is 2 because something cubed is equal to something. All the numbers must be under twelve so the squares must be 2 or 1 because when those numbers are cubed, the answer is under twelve. It can't be 1 because 1 cubed is one and the answer was a different shape from the shape in the question. The semi-circle was 8 because 2 cubed is 8.
The oval was 4 because the second question on the left hand row was 2 times the oval is 8 and 2 times 4 is 8.
The circle is double the triangle because the rectangle times 4 is equal to the circle while the rectangle times 2 is equal to the triangle. Now we need to find a number and its double. It can't be 7 and 14 because all the numbers must be twelve or below. It can't be five and ten because the triangle and the circle are in sums which involve the 2 times tables and the 4 times tables and 5 isn't in any of them. It isn't 4 and 8 because the semi-circle and oval are those numbers and you can't have two different symbols which are the same number. It can't be 3 and 6 because they aren't in the 2 or 4 times table. It can't be 2 and 4 or 1 and 2 because all those numbers are other symbols. It also can't be 0 and 0 because they are different symbols so they must be different numbers. The only possible pair is 6 and 12 so the triangle is 6 and the circle is 12.
The circle is 12 and the oval is 4 so 12 divided by 4 is the value of the rectangle. 12 divided by 4 is 3 so the rectangle is 3.
The second sum on the top is rectangle times rectangle is equal to something. Since the rectangle is 3, it is 3 times 3 which is equal to 9. So the star with more sides is 9.
That leaves the hexagon and the other star. It says 2 times star equals hexagon (see the second question on the right hand side) so we have to search for a pair of doubles. The only remaining pair of doubles under 12 is 5 and 10.

Alex and Bradley from Losley Fields Primary did it in a very slightly different order again. They sent in this Word document which explains their thought-processes very well.

Rhiannon from St Mary Redcliffe Primary also had a different way. She sent in this pdf to tell us how she did it.

Some of you said that you worked out the solution by trial and improvement - trying one number for a particular shape to see whether it worked. That can be a useful strategy although it might take a little longer than some of the ways already described.

Here is a summary picture of the value of each shape: