Another great trick is using diagonal lines to calculate multiplication [1]. It's a neat little trick that really helps drive home an understanding of what 'multiplication' actually is, since it's just like the OP's grid of boxes but turned on its side and contracted down to points. The trick is getting 45 degree lines so they line up more easily.
The advantage of lines is that you can spew out multiplication in the double-digit by double-digit range quite quickly. 51x23 for example is super easy to calculate - 10 100's + 17 10's + 3 1's. You're essentially just doing 50x20 + 503 + 201 + 1*3, but with lines to track everything.
Show it to a kid, and watch it blow their mind.
[1] http://lifehacker.com/5975917/quickly-multiply-big-numbers-t...