Grothendieck, Heron, and Brahmagupta

My son is taking the AMC 8 today so I thought we’d have sort of a light morning with math.  The section of the book we were meant to cover today was Heron’s formula for the area of a triangle.  Not exactly a “light” subject if you delve into the proof!  Instead, I took today as an opportunity to talk a little math history.

Quite a bit after Heron, in the 600’s actually, the Indian mathematician Brahmagupta found an amazing generalization of Heron’s formula.  Brahmagupta’s formula calculates the area of a cyclic quadrilateral and the triangle is a special case when one of the sides has length zero.

Though I know very little of the mathematics that Grothendieck actually studied, I took the example of Brahmagupta finding Heron’s formula as a special case of his formula as an example of solving a problem via generalization.   In the pieces I’ve read about Grothendieck in the last few days, his ability to find the right generalization of a given problem seemed to be one of his great gifts that the writers focused on.  The analogy with Heron’s formula and Brahmagupta’s formula  appeared to my best shot at mentioning Grothendieck’s contribution to math to my son.

The first video introduced Heron’s formula and discussed a little bit of history.

The next part of our talk introduced Brahmagupta’s formula and talked about why the triangle is (possibly!) a special case of this formula.  For it to be a special case, we just need to show that you can circumscribe a circle about any triangle.

In the next video we try to see if it is possible to always circumscribe a circle about a triangle.  This led to a discussion of a slightly easier problem – how do you find the set of points that is an equal distance from two given points?  Solving that problem leads to the idea that it may, in fact, always be possible to draw a circle that hits all three verticies of a triangle.

Finally, we jump over to the kitchen table to try to construct the circumcircle of a given triangle.  It is a neat construction and it is always a nice surprise when you get the three perpendicular bisectors to intersect in a single point!

So, a fun morning showing a little math history and a couple of really amazing geometry formulas.   Nice to have a light day every not and then.

Adding in binary with Duplo blocks

Several years ago I was talking about adding in binary with my older son.  On a whim we started using Duplo blocks to see how a “binary adding machine” would work.  It was a really fun exercise and I returned to it today with my younger son when we started the section in our book about adding in other bases.

I like talking about adding in binary with Duplo blocks for a couple of reasons.  First, it helps reinforce the idea of place value.  Second, it shows that you can add numbers in bases other than 10 without first converting them back to base 10.  Finally, both of these nice features happen in a setting that is pretty fun and surprising for the kids.

Our project from this morning went pretty well:

So well, in fact, that my son asked if we could do more tonight, so we did this second project.  This time we added 4 binary numbers, or was it 100 numbers for you binary fans 🙂

We’ll cover subtraction the same way, too, which is even more fun – you just need a new color to represent -1.  Can’t wait for that!