In AP Calculus BC we are doing some pretty unexciting stuff right now – techniques of integration. The problems are (sort of) fun little algebraic puzzles but I find little room for conceptual conversations. Maybe I am just missing something obvious. But today was a bit of a revelation and I wish I knew better how to try and insert equations to tell the story. I’ll just have to use some tortured syntax to get my point across. I put up three pairs of integrals and told them that one in each pair was something they knew how to do before they met me (our school does BC as a second-year calculus course) while the second was one they needed my help with. I had an integration by parts example side by side with a boring old u substitution (the integrands were x cos(x^2) versus x cos x) and they knew which one they COULD do and we talked through integration by parts. I had a partial fraction problem side by side with a natural log problem (the integrands were (x – 2)/(x^2 – 4x + 3) versus (x + 1)/(x^2 – 4x + 3)) and again they knew the difference and we talked about partial fractions. I had a trig substitution problem against a boring old square root (this time it was sqrt (9 – x) versus sqrt (9 – x^2)) Then someone asked me a HW problem. They were asked to integrate the fifth power of tangent x. I took off writing and trying to get buy in at each of the many steps. I told them at the end that they knew each of the steps they just did not know which direction to move. I assured them that this was a process they would master with a bit of practice. As I was working, I made the decision to substitute for sec x and set up the answer in terms of that function. A student asked me why he could not use tangent to substitute. I did not have a bunch of time left so I asked him to hold his thought and talk to me at the end. He did. As a result, I made a document we’ll examine as a class on Monday comparing his solution and mine. You can grab that here I went through with math type to show his solution and mine. I’ll leave it to the students to determine why they look different and I hope they come to the conclusion that they are NOT different. To help push the conversation I created a Desmos graph and a GeoGebra graph to show my function (called d(x) in each case) and my students function (called j(x)) in each case, I will erase the f(x) that you can see by following these links because I don’t want to give the game away immediately. What troubled me was that each program dealt with my function and my student’s function just fine. When I combined them the graphing technology broke. I tweeted out to @desmos and received – as usual – a quick and helpful reply. In this case, the reply was simply ‘Thanks for sharing. This will help us make better graphs for the future.’ This is the second time this year that we have found a little glitch and I could not be more pleased with the response I have gotten each time. It is such a great way to emphasize to my students what a connected world we’re living in and how they can reach out and find help. My student said he spent a half an hour trying to figure out why his answer was ‘wrong’ since it disagreed with his text’s answer. I hope after Monday that he will begin to internalize the idea that he can check his answers in pretty powerful ways. Ways that I did not dream of when I was learning this stuff in 1982. What a fun fun experience seeing his work and getting the reply I did from Desmos. Add in the fact that I get a date with my wife at a local farm to table restaurant and the day could not get much better.
So, one of the benefits of creating a virtual presence has been that I have all sorts of new friends that I have never met. I look forward to thoughtful exchanges on my blog and on theirs, I chime in every once in a while to the torrent of information that is twitter and I am happy that I’ll be able to meet a bunch of these folks at twittermathcamp 2014 in OK this summer. However, another opportunity to actually meet some of the army of talented math folks on the internet has reared its head. The amazing Jen Silverman (@jensilvermath on twitter and at http://www.jensilvermath.com on the web) will be traveling to my school in Kingston, PA to host a one day Geogebra workshop on Saturday, May 3. Here are some reasons you should think about attending:
- Jen does amazing work on GeoGebra, she is sort of a GeoGebra Jedi Master. See this page for evidence.
- We are hoping to have a manageable crowd of about 12 – 15 folks here. Enough to share ideas but not enough to get in the way of some direct instruction when you need it.
- I’m working on taking care of lunch for everyone – so that is a definite plus.
- Oh yeah – it’s free!!!
Jen created a lovely flier for this event. If I was smarter about managing my blog I would display it below, but you can click the link to see the document.
I hope that many – if not all – of my colleagues from our middle school and high school can join us and I am reaching out to anyone within a reasonable drive of NE PA to come and join us for a day of learning and sharing.
So, my afternoon crowd was not to be outdone by my morning crew. I slipped in a subtle reference early in the conversation with them so that they would not be inclined to simply introduce the phase shift idea. I wanted them to have a little practice untangling the mechanics involved in dealing with developing a Taylor series. They were very quick to recognize and agree that the coefficients were based on factorials so jumping from the 5th degree polynomial to the 7th degree was pretty easy for them. When I asked for the cosine they were confident about using even powers instead of odds and came to a conclusion pretty quickly. Where life got interesting was when I showed them Michael’s solution from the morning and discussed why i preferred the symmetry generated by an even powered series instead. I also discussed how Michael’s translation idea might give better results for approximating cos x with negative values of x. That’s when they stepped up and knocked me out. They suggested that we take the 6th degree polynomial approximation we had for cos x and do the following: phase shift by pi radians and reflect over the x axis. I am linking to a GeoGebra file that we created. If you want to dig into that file – here are the explanations of the functions.
a and b are self-explanatory
f is the 7th degree Taylor for sin x
g is the phase shift of this by pi/2 to approximate cos x
h is the 6th degree approximation of cos x
m is the crazy reflection/shift to move the cos x approximation backwards to another portion of the cosine curve.
Whew – what a day
We are starting our journey through the study of Taylor Polynomials today. I started with looking at y = sin x and asked them to find a ‘simpler’ function that behaved like sin x does around the origin. I sort of purposely asked this in a pretty vague way and we had a good chat about what I was asking for. One of my students offered up y = x as an answer. This gave us the opportunity to talk about the limit of sin x / x as x approaches 0. It also gave us the opportunity to talk about L’Hopital’s rule. A pretty good start in my mind. Then life got interesting. One student suggested a cubic function but I was able to get someone to urge an extension to a quadratic function that might match the sin x graph as the next step. I’m not sure what he (Michael) saw that made him jump to a cubic. He’s a really insightful student. So, I held that off and got to working on a quadratic. We agreed that the quadratic better agree with sin (0) and that the slopes should be the same. Someone suggested that the second derivatives should match as well. This resulted in a quadratic with a leading coefficient of 0. Not so good. It would have been easy for them to give up on this process, but Michael had already suggested the cubic. We had success in finding one and a GeoGebra graph confirmed that this worked over a larger region than the simpler linear function. We jumped into a fourth degree polynomial – again with failure due to a leading coefficient of 0. Here is where things really started getting promising. I asked why this was happening and a different student remembered something about even and odd symmetries. The precise language did not arrive right away, but we were able tp get that together as well. Pretty promising… A fifth degree polynomial was found and it graphed even better than the third degree. The students were getting a little tired of this process so I very quickly convinced them of the behavior of the 7th degree approximation. Michael (he was on fire this morning!) recognized the factorial pattern unfolding so we jumped ahead to the 9th degree polynomial. We were feeling pretty good about ourselves at this point. I asked them what function we might be interested in next and, luckily, I was told that cos x would be our next target. I told them that I would be quiet for the next few minutes while they worked this out for themselves. Normally, I am not at all interested in my students – especially ones at this level – simply mimicking my solution patterns. In this case, I thought that this new process was intimidating enough that they would just try to parrot my work. I was fine with that idea, this unit will take some time. However, my best laid plans were foiled. About a minute after I sat down dramatically Michael asked ‘Why don’t we just replace each x with x + pi/2?’ I was SO HAPPY, but i tried to hide that for a moment. Luckily, he spoke pretty quietly and his classmates were still working. I went back to GeoGebra and wrote a new function in his honor. Taking our last guess of h(x) which was our 9th degree polynomial and writing m(x) = h(x + pi/2) and I displayed this graph on top of the graph of cos x. It was a fantastic match but it did not have the symmetry that we had seen for the sin x approximations. The students who had plowed ahead with the polynomial model gave me their 9th degree solution and we looked at three graphs together. The cos x graph, the shifted sin x Taylor series and the cos x Taylor series. A really terrific conversation ensued. Today is what we call a T day where we have 50 minute classes. This felt like an enormously productive 50 minutes. I hope that the afternoon goes at least half as well.