Why is it when you have two equivalent fractions, their cross products are equal? Is there a good illustration/explanation of this that I could explain to a seventh grader?
This follows from the fact that for all rational numbers x, y and z, if x = y, then xz = yz. So, we start with $\displaystyle x=\frac{a}{b}$ and $\displaystyle y=\frac{a'}{b'}$ and multiply both sides of $\displaystyle x = y$ by $\displaystyle z = bb'$. It follows that $\displaystyle \frac{a}{b}bb'=\frac{a'}{b'}bb'$. Since $\displaystyle \frac{a}{b}b=a$ and $\displaystyle \frac{a'}{b'}b'=a'$, we get $\displaystyle ab'=a'b$.