Square roots day 2 – approximating the square root of 2

One of the examples in my son’s Prealgebra book today was prove that \sqrt{2} is less than 2. We were having a pretty good discussion about the ideas in this example, so I thought it would be fun to see if we could go a little deeper. Since we just talked about continued fractions last weekend, I was hoping that end up being able to find something to say that was much more accurate than just “less than 2.”

Our initial discussion of the problem is here:


Next up was the beginning of looking at \sqrt{2} as a continued fraction. We’ve spent very little time on this subject, so it is still new to him and we had to go slowly through the process. Luckily the continued fraction starts to repeat fairly quickly.


We finished up by figuring out some of the fractions that approximate \sqrt{2}. This exercise was why I wanted to go down the path of calculating the continued fraction. First off, we’ll see some of the fractions that we saw already in part 1. Second, we’ll find a couple better approximations, which is neat. Third, we’ll get to see directly that these fractions are nearly equal to 2 when you square them. AND, we get lots of good fraction practice in the process. Yes!


A fun geometry problem with frisbees – or technically two Discraft discs for the purists :)

Nearly three years ago I ran across this problem:

Two circles of the same size are tangent to each other in a plane. One of the circles stays fixed and the other circle rolls around the first circle one time. How many times does the rolling circle turn around its center?

A super fun, and easy to state problem. Here’s our first run through it (FamilyMath 23 – sheesh – we are over 250 now!!):


Well . . . today I ran across the same problem in our Introduction to Geometry book. Fun to see the ideas the the boys have 3 years later: