Why is $\pi $ equal to $3.14159...$?

If the kids are not too old you could visually try the following which is very straight forward. Build a few models of a circle of paperboard and then take a wire and put it straigt around it. Mark the point of the line where it is meets itself again and then measure the length of it. You will get something like 3.14..

Now let them measure themselves the diameter and circumference of different circles and let them plot them into a graph. Tadaa they will see the that its proportional and this is something they (hopefully) already know.

Or use the approximation using the archimedes algorithm. Still its not really great as they will have to handle big numbers and the result is rather disappointing as it doesn't reveal the irrationality of pi and just gives them a more accurate number of $\pi$.

You can try doing what Archimedes did: using polygons inside and outside the circle.

Here is a webpage which seems to have a good explanation.

An other method one can try is to use the fact that the area of the circle is $\displaystyle \pi r^2$. Take a square, inscribe a circle. Now randomly select points in the square (perhaps by having many students throw pebbles etc at the figure or using rain or a computer etc). Compute the ratio of the points which are within the circle to the total. This ratio should be approximately the ratio of the areas $ = \displaystyle \frac{\pi}{4}$. Of course, this is susceptible to experimental errors :-)

Or maybe just have them compute the approximate area of circle using graph paper, or the approximate perimeter using a piece of string.

Not sure how convincing the physical experiments might be, though.

Given the age of the children, I think that wheeling a bicycle along the ground and measuring the distance traveled for one wheel revolution would be a good idea. This exercise can be continued by asking for suggestions on how to improve the accuracy (e.g. wheel the bike two revolutions etc)

As you obviously have children in the class who pose thoughtful questions, one can ask what do you think would happen to the distance measured on the cycle milometer if pi was suddenly smaller or larger (if one let down or pumped up the tyres). What would happen to the speed registered in such cases. I'm sure there is an interesting exercise for your pupils here. Good luck.