Here’s a picture I drew to explain addition and subtraction of fractions to the sixth-grader:
We also ended up using a variant on Euclid’s algorithm for finding the GCD. It uses subtraction instead of division and remainder; it’s in general less efficient, but it’s easier to explain and can be easier to do in your head, when the numbers are small.
Construct a series whose first two terms are the inputs, and then continue as follows: each successive term is the absolute value of the difference between the preceding two terms — that is, simply subtract the smaller from the larger. If you reach one, the GCD is one; if you reach zero, the GCD is the previous term. (Or, you could also let the series peter out to zero, but the way I’ve stated it is simpler in practice.)
24 and 16: 24, 16, 8, 8, 0.
9 and 7: 7, 9, 2, 7, 5, 3, 2, 1.
12 and 9: 12, 9, 3, 6, 3, 3, 0.
35 and 28: 35, 28, 7, 21, 14, 7, 7, 0.
An added advantage is that the first step lends itself to an optimization that almost always short-circuits the whole process, at least for sixth-grade math problems. Take the difference of the two inputs and test whether that divides both of them. If it does, that’s the GCD.