Simplest explanation of pendulum having a constant time period at low angles What is the simplest explanation for the pendulum having a constant time period at low angles? 
 A: The forces on a pendulum are shown in the diagram bellow:
We are going to use $F=ma$ where in this case we will use angular acceleration and something called moment of inertia to replace mass (this is just like a rotational version of mass given by $ml^2$.) We also replace force with torque (or moment about an axis) in this case it is given by $F_g l sin \theta$. Where $F_g=mg$.
Thus we use $F=ma$ replacing the normal versions of force, mass and acceleration with their angular equivalents.  
This gets us:
$$mgl sin\theta=ml^2\theta''$$
Dividing through gives us:
$$\frac{g}{l}sin\theta=\theta''$$
Now we assume that the maximum $\theta$ is small and use small angle approximations, that $\theta \approx sin \theta$. Meaning we get:
$$\frac{g}{l}\theta=\theta''$$
which is the general expression for SHM where $\omega^2=\frac{g}{l}$ and therefore since $T=\frac{2\pi}{\omega}$ we have $$T=2\pi \sqrt{\frac{l}{g}}$$
Which is independent of the amplitude of oscillation.
Now to my comment about the difference in acceleration. In our initial expression ($$mgl sin\theta=ml^2\theta''$$) the value $\theta$ depends only on $\theta''$. This means that since the time period is independent of the amplitude of oscillation that there is no other alternative that this is caused by a difference in acceleration. Otherwise if they had the same acceleration at their amplitudes say then the one with a smaller amplitude would have a smaller time period.
By the way I do agree with John's comment that this detail is a (far, far) to hard for 9 to 10 year olds.
A: The simplest explanation is that for small $\theta$ the equation of motion $\theta''= \frac{g}{\ell} \sin\theta$ can be replaced by $\theta''= \frac{g}{\ell} \theta$ since $\sin\theta\approx\theta$. One can also provide a rigorous mathematical account for the intuitive idea that "period of infinitesimal oscillations is independent of the amplitude"; see this article.
