Making Connections

We normally have very few school days between AP tests and graduation. This year we have 8 days left (counting today) so I am trying to have a little exploratory fun with my AP Calculus BC kids. They’ve worked hard on Calculus for two years now and I know that there is plenty more Calculus out there. Instead, though, I’ve chosen to take them on a little tour of some topics that they normally would not see in their high school days. Today’s topic was different number bases (the link is to my classroom document for today) and we had some fun with a series of base 8 addition and multiplication facts. I presented them with a  picture of Lisa Simpson as a clue and one of my students noticed that she only has 8 fingers. I use this as my motivation for this conversation. So, I had this nice little handout prepared that I hoped would guide us through a fun conversation. What I did not anticipate was a terrific question that came up. Let me set it up. We visited a website that converts base ten numbers to base 6 while you type. We had fun playing with it typing in things like the year of my birth and a few other nuggets. I asked them whether a base 6 representation of a number would always be longer than the base 10 representation and we had a nice chat about that. Then a student suggested that I type in a decimal so I typed 9.2 and the website did not like this. It would have been very easy to just shrug it off. My kiddos did not. They pushed me a bit and someone suggested that I write the first few negative exponents of 6. One student wisely suggested that I work on the assumption that the coefficients would all be 1 since 1/5 is SO close to 1/6. Here’s where it got fun. Another student said something along these lines – ‘Isn’t this just going to be an infinite series?’ WOW! I was so so so pleased with this. Connecting to Taylor’s and other infinite series in the face of a (relatively) harmless decimal? So proud I was.

We tried this conversion and another student seemed suspicious of the assumption that every coefficient would be 1 (or 0) so I reverted to binary. Now our task was to convert 9.2 into a binary number. The whole number part of 1001 was easy to sell. Now, we had to convert 1/5 into a series of decreasing powers of 2 (or increasingly negative powers of 2). Well, there was quite a bit of computation involved, some nice guess and check strategy was employed, the TI calculator function of turning decimals into fractions was helpful and in the end we discovered (and were able to verify) that 9.2 in base 10 is 1001.0011001100110011…

I have made a few mentions recently of my battery being recharged. It was awfully nice to see that some of my students still have some juice left in their batteries as well.

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