Celebrating Pi day with Evelyn Lamb’s idea

Last week Evelyn Lamb wrote a nice piece about \pi and continued fractions.

Since we’ve talked a little bit about continued fractions in the past, this seemed like a great way to celebrate \pi day. We started with a quick reminder about continued fractions:

 

After the quick introduction, we used my high school teacher’s fun continued fraction technique – Split, Flip, and Rat – to calculate the continued fraction for \sqrt{2}. This exercise gives you a great opportunity to talk with kids about fractions and decimals.

 

Next up was today’s activity – the continued fraction for \pi! Unfortunately, for this continued fraction split, flip, and rat doesn’t work so well. Nonetheless, we do get to have a good discussion about decimals while calculating the first two pieces of the continued fraction for \pi.

 

To calculate a few more parts of the continued fraction we went to Wolfram Alpha. Turned out to be a pretty neat way (and obviously a much quicker way) to see the next few numbers in the continued fraction. Again, we got to have a great discussion about decimals and reciprocals.

 

Now, having found a few terms in the continued fraction, we went and looked at what fractions other than 22/7 were good approximations to \pi. Happy 333/106 day everyone 🙂

 

Finaly (and sorry for the camera screw up on this one), I wanted to show a different continued fraction for \pi. In a previous video my younger son thought that we’d find a pattern in the continued fraction for \pi. We didn’t in the first one that we looked at, but there are indeed continued fractions for \pi that do have amazingly simple patterns.

 

So, a fun little project for pi day. A great opportunity to review lots of arithmetic in the context of learning about continued fractions and \pi.

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Comments

One Comment so far. Leave a comment below.
  1. myrtonos,

    The simple continued fraction can also be written in a linear form, for example the 4th convergent to π can be written as [3; 7, 15, 1, 292].

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