Sally is making spinners. As you can see from the diagram she has divided two circles into 4 equal parts. 
Now she is going to put numbers in the spaces on the circles.
Where should she put the numbers so that when she spins the two spinners and adds the numbers together each possible total comes up equally often? Do this for 1 possible total; 2 possible totals; 4 possible totals; 8 possible totals; and 16 possible totals. Why only those numbers of totals?
Solution
Note that because there are 4 spaces on one spinner and four spaces on the other, there are potentially 4 x 4 = 16 different totals. So the number of equally possible totals must be a factor of 16.
This can be done in five kinds of ways.
First if there is only one possible total: put the same number in all four spaces on each spinner.
Second if there are just two possible totals: put the same number in all of the spaces on one of the spinners; put two different numbers on the other spinner.
Third if there are just four possible totals: put the same number in all of the spaces of one of the spinners; put four different numbers on the other spinner.
Fourth if there are just eight possible totals: put two different numbers on one of the spinners; put four carefully chosen different numbers on the other. Be careful that there are no unwanted repeated totals here. For instance, this will work:
(How careful do you have to be here?)
Fifth if there are sixteen possible totals: put four different numbers on one of the spinners; put four carefully chosen different numbers on the other. (How careful do you have to be?)
Extension
Can Sally organise things so that there are four possible totals but that they don’t come up equally often?