Chemical equilibrium — why multiplication, rather than addition? Why do we multiply (and not take the sum) the concentration of products and reactants when more than one entity is present in either side and rise the power to the coefficient term?

Suppose either

$$\ce{aA + bB &lt;=&gt; cC +dD}$$

described by

$$K_{\mathrm{eq}} = \frac{[\ce{C}]^c \cdot [\ce{D}]^d} {[\ce{A}]^a \cdot [\ce{B}]^b}$$

where we multiply the concentrations and rise the power to the coefficient. Or the dissociation of

$$\ce{A_aB_b &lt;=&gt; aA+ + bB-}$$

described by 
$$K_{\mathrm{sp}} = [\ce{A}]^a \cdot [\ce{B}]^b$$

again based on a product, and not the sum.

This seeks a clarification of math based on chemistry. How to explain it to an 8th grader?
For the 8th grader, hmmm...  

Let suppose this funny equilibrium equation:

$$\ce{2 S &lt;=&gt; H}$$

Where $\ce{S}$ are ice skaters on a giant skating ground and $\ce{H}$ are "collision heaps" of 2 skaters laying on the ice.

Now, what is the frequency of collisions leading to a heap, if there is "skater concentration" $\ce{ [S] }$ ?  It should be $k_1 \ce{[S]}$ as frequency should be proportional to $\ce{[S]}$, correct ?

But wait ! When there is more skaters, each skater would collide with  other skaters more frequently,  proportionally to $\ce{[S]}$ as well !

So collision frequency would be like $k_1 \ce{[S]}^2$

Now the heap "recovery":  

It should be proportional to $\ce{[H]}$, as more heaps, more skaters would be  raising at the same time. So the heap recovery rate would be like $k_2 \ce{[H]}$.

Now imagine the equilibrium on the skating ground, where the rate of collisions is equal to the recovery rate
and number of collision heaps is constant.

Then $k_1 \ce{[S]}^2= k_2 \ce{ [H]}$

If we express the equilibrium constant $K=k_1/k_2$, then $$ K = \frac{\ce{[H]}}{\ce{[S]}^2}$$

Do you see ? No sum, but multiplication. Twice as many skaters leads to 4 times more heaps.

You can create similar system, where just for fun collide just boys and girls together.