## A Fun Rabbit Hole

Last week – I know, it’s taken too long to write about this – my Precalculus Honors class started the day with a brief quiz. One of my PCH students named Max finished the quiz early and started sketching on his scrap paper. He showed me a diagram like this:

He described the problem this way – I have a square and a quarter circle coming across it. I also have a circle inscribed in the square. What is the area of these little regions? (I clumsily sketched in those regions on GeoGebra)

Well, it turns out the the topic of the day in AP Calculus BC that day was to be trigonometric substitution for integrals and this problem would be a lovely introduction to the need for this skill. AP BC was meeting for the 90 minute block and I decided that I would introduce Max’s problem, spend about ten minutes dissecting what we could and then hit a bit of a wall where I would introduce this new skill. I was pretty proud of myself and feeling very fortunate that Max thought of this question. Well, as we all know, life doesn’t always work out the way we want it to in school. I presented this problem and told them that it came up in Precalculus Honors. My BC kiddos started dissecting it right away. They concentrated on the lower left corner, they decided we should agree to a side length for the square and off they went. We decided the square should have a side length of 2 so the inscribed circle would have a radius of 1. Avoiding fractions until we HAVE to deal with them is a good plan in general, right? So, the lower region is 1/4 of the difference between the inscribed circle’s area of pi and the square’s area of 4. Good start. Next we convinced ourselves that the two remaining squiggly areas are congruent. It would have been nice if we could drop a line from the point of intersection to divide that region in two but it’s not symmetric. The different radii of the circles intersecting prevents that from being true. So, here is where I figured I would introduce this new technique. I mentioned this idea but the feeling in the room was that we should be able to answer this question using tools that a precalc student should be able to use. I was sitting in the back of the room at this point with my laptop on and a GeoGebra sketch projected on the front wall. Ideas and questions started flowing and students asked for a Desmos sketch like the one below:

So, this sketch is pretty impressive in its detail but, more importantly, this sketch happened about 20 minutes after lunch began and after I excused myself to run an errand during lunch. During the 90 minute class, my colleague David from across the hall wandered in a couple of times asking kids to explain what they were doing. He told me that Nancy, Andy, and Michael worked for at least a half an hour of they hour long lunch debating this problem. The other thing that happened while I was gone was that Andy, Kelly, and Michael had modified my Desmos sketch on my laptop pursuing their idea. Their modification is here –

I was feeling pretty great about their perseverance, their engagement, and the amount of geometry and trig that was being remembered in the service of this curious problem proposed by one of my students.  I was also more than happy to amend this week’s Calc test by taking off the one problem that relied on the trig substitution technique. I had one more class after lunch (one of my Honors Calculus sections) so I sadly erased some of the work on the board and I described the problem to that group. Some of them had already heard about it during lunch! My BC kiddos were still talking about it even after they left. At the end of the day one of our Differential Equations students wandered into my room. He said ‘I heard there was a good problem today.’ He, Owen, then proceeded to discuss the problem with Andy and Nancy who had come back to the room to discuss this. Owen dove in to the problem debating with Andy and Kelly and he produced these sketches – (the first one got rotated in translation)

I tweeted the problem out, like I do, and a former student jumped in and offered this sketch –

Another colleague, Adam, came by when he overheard this conversation and he attacked the problem using Google sketch up to find the ratio that Nick wanted – it was smaller than his proposed 10% neighborhood.

There is no real ending to this story, the weekend came, life moved on. On Monday my BC class was more focused on asking questions about this week’s test. My Precalc Honors kids were impressed by my enthusiasm in talking about all of this but they did not share Max’s curiosity about the question. I went home feeling pretty great about the sense of play and sense of curiosity of many of my students and my colleagues. While I cannot let everyday roll this way, I need (NEED!) to make sure to create spaces where this kind of magic can happen. I think almost all of the credit for this adventure lies with my students who are interested, motivated, curious, and persistent. I hope that I have helped them along by modeling curiosity and by being willing to let this kind of free range play happen in class.

## Debating Divisibility

In our Precalc Honors class we are discussing exponential and logarithmic functions now. I want to relate a fun observation/suggestion from a student a few days ago and a debate that fired up in class today.

Our text defines exponential functions as any function of the form y = a*b^x as long as b is positive and not equal to 1. One of my students, a girl named Shailee, suggested that it would feel more logical to simply say that b is greater than 1. This way, functions with a base between one and zero would instead have a negative exponent. This might make it more consistent to think about positive exponents representing growth and negative exponents representing decay. This also feels like a smoother definition for b instead of having two qualifiers, we’d only have one. Kind of a nice suggestion and one that I will be adopting for our class conversations this year.

Today I ran an activity that was suggested by Henri Picciotto when he came to do a workshop with my department in May of 2016. I had a couple of containers of 10 sided dice. They were numbered 0  through 9. I assigned a rule for each of my three groups. One group was to roll all the dice and count the number of evens. They then dispensed any that were not and rolled again. Lather, rinse, and repeat. The idea was that the number of dice remaining should model a half-life for them. The second group was looking for primes. Again, exponential decay with a base this time of 4/10. The last group was asked to look for multiples of 3. Someone asked if 0 counts as a multiple of 3. I reflexively said no but then paused and thought out loud about it. I threw the question out to twitter and we went on our merry way. We gathered data, plotted it on Desmos in a table and asked fro regression equations of the form y = a*b^x. Worked pretty well except one group went from 40 something dice down to something like 8 right away when they were supposed to have a 1/3 chance. We then checked in on twitter where interesting things were being shared. I’ll clip a few tweets below:

A side conversation also occurred when Christopher suggested that the 0 on the die was a 10 not actually a 0. This, of course, would have prevented this whole interesting conversation from happening in the first place. Anyways, this got a heated debate going in class where my students just felt uncomfortable about the idea of 3 being a factor of 0 since this implies, by a simple extension, that EVERY integer is a factor of 0. I guess we all accept without much debate that 1 is a factor of every integer, but this feels off somehow. I went off to lunch to bounce this idea off of some folks and I might have scared a couple of colleagues who are less comfortable with math. A lively debate/discussion at lunch led one colleague to casually say ‘So much just happened there’ When I returned to my classroom and my twitter feed the conversation had moved into a modular arithmetic mode. Here is a taste:

So, let me first say what an honor and a treat it is to share in a conversation like this with my students, my colleagues, and my virtual faculty lounge of folks spread around the globe. It is a mind-blowing thing to think about how much this world of education has changed for me since I took the plunge to going twitter. I am convinced (CONVINCED!!!) that life is better for my students since I did. I also want to say that the idea of modular arithmetic is one that I love to share with my students and I am determined to figure out how to find time to do so with my precalculus students since this debate brought up these ideas. I also have to admit that I am just a tad uncomfortable saying that every integer is a factor of 0. One of the side conversations at lunch went like this : Me – If 0 is a multiple of 3 then that means that 3 is a factor of 0. Rachel (science dept colleague) – If we say 3 is a factor  of 0 wouldn’t we say that 0 is a factor of 0? Me – Uhmmm, this would imply that zero divided by zero is a thing, right? This reminds me of debates I had with a friend from my old college town debating the physical meaning of 0 ^ 0

So, a delightful lunch time conversation, right? Fun to lift the curtain a bit and have my students see a debate unfolding. Fun to get my brain agitated thinking about all of the implications of saying something as simple to my kids as ‘Look for multiples of 3’. Probably a lesson to think a little more carefully about my directions to them!

Many many thanks to Henri, Sam, Christopher, David, and Bryan for engaging in this conversation and for giving me the idea of this experiment.