A simple test of our new skew dice

Last week we bought two packs of skew dice from the dice labs:

Today we decided to try a simple statistical test of the disc. Before we dove into that test, though, I asked the boys to tell me how they thought you might go about testing these dice to see how random they were.

We rolled the dice off camera. We rolled them in groups of 10 with each of us shaking up the dice in a container before each roll. After 12 rounds we had a total of 120 numbers -> here are the results:

Although we didn’t really have enough rolls to make a definitely statement about the dice, I think this was a nice way for the kids to see how a simple statistical test would work. I hope the kids are interested in playing around with these dice a bit more.

An equation with roots of sqrt(5) + sqrt(7)

My older son is working thorugh the Integrated CME Project Mathematics III book this summer. Last week he came across a pretty interesting problem in the first chapter of the book.

That chapter is about polynomials and the question was to find a polynomial with integer coefficients having a root of \sqrt{5} + \sqrt{7}. The follow up to that question was to find a polynomial with integer coefficients having a root of 3 + \sqrt{5} + \sqrt{7}.

His original solution to the problem as actually terrific. His first thought was to guess that the solution would be a quadratic with second root \sqrt{5} - \sqrt{7}. That didn’t work but it gave him some new ideas and he found his way to the solution.

Following his solution, we talked about several different ways to solve the problem. Earlier this week we revisited the problem – I wanted to make sure the ideas hadn’t slipped out of his mind.

Here’s how he approached the first part:

Here’s the second part:

Finally, we went to Mathematica to check that the polynomials that he found do, indeed, have the correct numbers as roots.

I like this problem a lot. It is a great way for kids learning algebra to see polynomials in a slightly different light. They also learn that solutions with square roots are not automatically associated with quadratics!