My Calc BC classes had a test yesterday. We are deep in the midst of thinking about infinite series and I threw a question on the test that I thought would be a respite in the middle of some heavy lifting. I asked for my students to write a repeating decimal as a fraction in reduced terms. My two classes had slightly different numbers, I’ll concentrate my conversation here on this repeating decimal: 0.217217217217…
Many of my students remembered an approach where we called this number x and then created another number with the same repeating block. In this case, it would be 1000x as 217.217217217217…
A simple subtraction yields 999x = 217 and we have our desired fraction of x = 217 / 999
At least one student rewrote this as 0.217 + 0.000217 + 0.000000217 + … This student recognized this as an infinite geometric series whose first term is 217/1000 and whose common ratio is 1/1000. Remembering a nice formula gives us the same answer.
Some students fumbled on the problem not remembering either of these strategies. What I want to concentrate on here is the work of three students who all presented their reasoning essentially in the form of short paragraphs explaining how they zeroed in on this fraction. I won’t quote them directly and I will probably mix up their reasoning a bit after a long Friday.
One student presented the fraction 5/23 = 0.2173913 as a starting point. He admitted that this was after fumbling around with a couple of fractions all of which were ratios of primes. He reasoned that having no common factors would likely create ‘ugly’ decimals of some form. From there he mixed and matched some interesting reasoning. He showed that 2/9 = 0.222222… so he reasoned that 9s in denominators might be a nice thing. He showed something like 57/99 = 0.57575757… and this gave him confidence in targeting 217/999 for this problem. A different student started by pointing out that 1/5 < 0.217217217… < 1/4. This gave her an idea of where to start trying fractions that might zero in on the desired target. A third student presented some of these same ideas and concluded, somewhat apologetically, that he combined past knowledge and logic to arrive at his correct conclusion.
There is an awful lot to unpack in that particular explanation and it reminds me that one of my missions is to consistently champion such an approach to math. I displayed one of the papers on my document camera and made sure to publicly commend all three of these students. I pointed out that the ability to create this logic, to tie together past skills and ideas is FAR more meaningful than remembering some technique or formula. This is especially impressive in the time pressures involved in taking an in class test.
I had shared an old blog post in class recently when a student made a suggestion involving integration. The suggestion he made reminded me of a post I wrote a couple of years ago (you can find it here https://mrdardy.mtbos.org/2020/01/05/more-bragging/ ) and this prompted my students to notice that I am not blogging as much as I used to. They jokingly suggested that they were not worthy of blog posts. I need to make sure I use this space more frequently than I have been. I hope this is a kick start to more writing in 2022.
Oh yeah – a fun note about that post I linked above. I showed it to my students and one of them took a screen shot and forwarded to screen shot to the student I wrote about back in 2020. He replied while we were still in class together that he remembered that conversation from when he was in Calc BC!!!