Okay, so I’m a little slow! I’ve been working my way toward calculus with Leo for a long time. We’ve explored a great deal among the preliminaries, such as algebra, geometry, and trig, done a little “real life calculus” using cars and rockets, and I’ve even left various fun “calculus coloring books” around, which he’s picked up and variously paged through. We’ve even done some Minecraft-math before that verges on calculus. But until today I hadn’t really broached any real calculus.
Today it hit me that Minecraft provides a perfect setting to discuss integration via Riemann Sums (which most of you will remember as the method of rectangles). The reason is obvious: In Minecraft you (generally) only have rectangles, so if you want to know the area under something — say the roof of semi-circular building — you don’t have any choice but to build the thing out of blocks. Therefore, the actual area is actually most conveniently measured in terms of piles of blocks, i.e., rectangles, i.e., the basis of the Reimann Sums!
Once I realized this it took only a couple of minutes to come up with some easy examples. To you and me, this just looks like simple intro calculus, but the “patter” — the story I was telling all the way through — is all about Minecraft!
Next I explored the idea that if the blocks were smaller, the approximation would get better and better. Conveniently, the Minecraft blocks are 16-units wide, so you can divide them in half four times and still be working with integers, making it easy to actually work the detailed math.
What’s most interesting about this is that because of Leo’s facility with Minecraft, once he had his Minecraft thinking cap on (which, once on, is extremely hard to get off!), he was able to see right away how the simple version of the rectangle-based integration worked, and was also able to easily think through (approximately) how things would go if the blocks were divided in half, and then half again and again and again!
In the last example (second half of the page) Leo worked the Integral[X^2] by rectangles while I worked it by the usual (X^(N+1))/(N+1) method. (I had to help him a little with his.) And we both got about 42, which is about right! (Actually, it’s 125/3 = 41.666…)
I think that this (somewhat “duh”) realization of the relationship between Minecraft and Calculus has opened up our whole next level of math fun!