Another great piece of math to share with kids from Richard Green

Saw this really cool post from Richard Green over the weekend:

I love unsolved problems that kids can understand! In this case what really jumped off the page was that there were so many different directions to go when sharing this problem with kids. I picked the first three ideas that came to mind and used them for a fun little project with the boys this afternoon.

Sorry that this one goes a little longer than usual, but you’ll see the kids remain totally engaged (and fascinated) all the way through. So much fun!

I started by simply sharing Green’s google+ post with them:

The first project based on this unsolved problem that I thought would be interesting to kids was looking to see if they could find numbers that were written with just 0’s and 1’s in base 2 and in base 3. To a mathematician this probably doesn’t seem to be that interesting of a problem, but the kids found it to be pretty neat. They were really excited when they discovered the pattern!

The second project I thought would be interesting took about 10 minutes. The idea in this part of the project is to see if we can find a pattern in the way to convert numbers from base 2 to base 4. It took a while for the kids to see the pattern, but they were really happy when they found it. Again, the connection here probably isn’t really that surprising to mathematicians, but it is amazing to watch kids see it for the first time:

The last project that I thought the kids would find interesting was finding the probability that a number written in base n would have just 1’s and 0’s. To simplify the project we just looked at 3 digit numbers. The kids had some really great ideas here and we got to explore a couple of different ideas and patterns.

At the end of the second video in this part we returned to talking about the ideas in the original problem.

As I said at the beginning, I love sharing unsolved math problems that kids can understand. The really nice thing about the problem Richard Green shared is that there are lots of neat properties of base number arithmetic that are closely connected to this problem. Talking through some of these properties is a fun way for kids to explore math, and maybe even get a tiny little glimpse of mathematical research. Definitely a fun afternoon ðŸ™‚

Practicing fractions while learning some introductory geometry

Saw a nice introductory geometry problem in our Prealgebra book today. It is problem #7 on the 2005 AMC 8

2005 AMC 8 problem number 7

Here’s the problem:

Bill walks 1/2 mile south, then 3/4 mile east, and finally 1/2 mile south. How many miles is he, in a direct line, from his starting point?

I like a couple of different things about this problem. First, there are multiple ways to solve it, and each provides a little different insight into the geometric situation. Second, it provides a nice opportunity for a little fraction review, since you’ll likely encounter adding, multiplying, and maybe even dividing fractions in the solution.

Here’s my son’s approach to the problem:

His initial solution involved using two different triangles to find the length. I was interested to see if he could do it with just one triangle. This new solution involves drawing in two new lines, or maybe just rearranging the initial picture. I wanted to walk through this second solution because I think it is instructuve, but a little harder for a kid learning geometry to see.

It took a while, but we got there. One of the stumbling blocks was understanding what happened to the distances as we moved some of the triangles around.

I really love problems like this one. It gives a great opportunity to cover a new topic from a few different angles and also gives you an opportunity to sneak in a little review of an old topic. Definitely a fun morning.