Not long ago, I blogged about the conceptualization of polynomial long division and trying to find easier routes that would make division easier to understand, so students would find it easier to do.
Then I found James Tanton's Exploding Dots course.
This course is all about learning about different base systems, including "base x" where the base is unknown, which leads to how to perform all types of arithmetic in these bases with great conceptual understanding (though not necessarily with ease).
Then after completing the course, Tanton challenges the student to fuss around with different bases and understand what in the world is going on. So I did. In base 1/2.
Obviously, certain numbers have to be converted so that we can perform the operations. So here are number 1-30 in base 1/2.
Then I can show that the arithmetic works! Warning: It's a bit odd. Whatever "carrying" one must do in decimal arithmetic goes the opposite way in base 1/2.
And also, I can find out what decimal fractions look like in this base. It turns out some really cool results occur
Isn't that awesome?! Through a weird little course involving dots and boxes, I was able to show that an infinite geometric series with even powers of 1/2 adds up to 1/3. Insane!
Tanton does a great job here and I firmly believe that high school students would be able to understand these concepts and they can be moved through relatively quickly. Though it can't completely take over a polynomial chapter (where are the graphs?!), it can provide students with polynomial power that they own and shows students a whole new world of different bases! It's brilliant.
I encourage everyone to take a look and dare somebody to tell me this kind of math isn't fun!
Until next time, happy math!