That Elusive A – ha Moment

On Monday we returned from our two week spring break and we finally took the plunge into Power Series in our BC classes. Oh, by the way, we were looking at snow in our area on the weather forecasts. Great first day back after spring break!

So, on Monday and Tuesday we were dealing with defining Power Series’ and looking at the radius of convergence and the interval of convergence for these series’. My students seemed to be dealing with these problems pretty well. Some number of weeks ago – I cannot even remember right now – I introduced this last full chapter of our text by talking about our ultimate goal of developing Taylor series approximations and I used the function f(x) = sinx as my example weeks ago. I convinced my students that we could create a polynomial the behaves like sinx as long as we were willing to be patient enough. I started off (again, this was weeks ago!) with an approximation of sin(0.1) using geogebra and talking them through the idea that we wanted (more accurately, I wanted) to create a polynomial called P(x) that agreed with f(x) at x = 0, and whose first, second, and third derivatives all agreed with those of f(x) at x = 0. We chose x = 0 for relatively obvious reasons and since they had never seen this argument before they were willing to go along for the ride. So, we finally get to the point now where my students can follow along in the logic rather than simply watch and/or write down notes. They come to class yesterday and I tell them that in our 40 minute class I hope that we can finish 2 problems. This creates some visible unease as the idea of 2 problems each taking 20 minutes generates some snarky remarks about how hard this is going to be. What follows is a summary of the conversation with my second BC class of the day – my much more vocal and active group of the two.

Problem #1 – Estimate, correct to three decimal places, the value of sin(0.1) without using your calculator. I start a conversation about what we might be able to know about this value. We pretty quickly agree that it is positive and small. In my morning class I had a great estimate in degrees of what 0.1 radians might look like and I hope to prod the conversation in that direction. I start by asking what a logical upper bound for the estimate might be and I hope to hear someone say 1/2 since that is the smallest exact sine value they know in the unit circle. Instead, Jon tells me that it has to be less than 0.1 which is true and much more accurate. I ask him why this must be so while a number of his classmates are generating their own guesses. His neighbors are in a debate about why 1/2 is an upper bound for reasons that hover around the unit circle. When I question Jon he tells me that the function has a slop of 1 at the origin and that this slope decreases as x increases, so therefore when x increases by 0.1 y will increase by less than that. Wow. I was SO happy to hear this reasoning and I wanted to make sure that the rest of the class heard it as well. I should have dusted off Ben Blum-Smith’s idea of having another student try to restate but I honestly was not sure how many kids had even heard him. I was standing near him and he was speaking to me while his classmates were involved in conversations with each other. So, I took over and restated his point. I then pushed a bit and asked the class why Jon knew that the slope of f(x) = sinx was 1 when x was zero. Here, my mind was anticipating and hoping that someone would mention the limit of sinx / x as x approaches 0. I might have had to take a break at that point to calm my heart down. Instead I got another terrific answer – we know the derivative of sinx is cosx and we know that cos(0) = 1. I asked a student why we were suddenly talking about derivatives when Jon discussed slope and I was calmly told that the derivative IS the slope and we were ready to march on. The procedure for setting up the system of equations is tedious and time consuming and as I started the problem a number of students were rifling through their notes and found the example we did weeks ago when we generated a third degree polynomial to match up with f(x) = sinx. I was again delighted that they (a) remembered we had done this and (b) could find it so quickly in their notes. So we get the function we want and now substitute x = 0.1 into the polynomial. We have the fraction 599/6000 at this point and Jon is pretty pleased. We see that it is less than 0.1 but just barely. I remind them that the directions asked for an answer in decimals without their calculator so we dust off some long division skills and get to 0.0998. I ask a student to pull out his calculator and give me the four decimal answer that his calculator has for sin(0.1) when he recites the exact same decimals I can see some noticeable smiles on my students’ faces. They are pretty impressed. We are almost there, I can feel it.

Problem #2 asks for a four decimal approximation (I correct myself midstream because of the first problem and what I remember of our morning work) for ln (0.9), again without their calculator. So this problem has a different wrinkle. I have not yet introduced formal notation from their text regarding these series, so they don’t know about the center of convergence yet and we are not assigning the mystery, powerful a t this yet. I’m using the phrase ‘we are concentrating on x = ___’ and we want the blank to be a value close to our target but one where we can easily compute and exact value if we need to. We all agree pretty quickly that x = 1 is where we should concentrate and that ln (0.9) will be negative and small. I’m happy that I have enough discipline now to weave in this kind of ‘what do we know, what do we wonder, what can we guess’ kind of conversation into class regularly now. All this twitter and blog PD is taking hold!

So, we go through the tedious process AGAIN of matching a power series out to the third degree so that P(1) = f(1) [where f(x) is now lnx], P'(1) = f'(1), P”(1) = f”(1) and, finally, P”'(1) = f”'(1). However, we have an interesting decision to make here. For the first problem, with x = 0 as our focus, we all agreed that P(x) = a + bx + cx^2 + dx^3. With x = 1 as our focus now, we were a little anxious about this model. Students quickly offered two solution ideas – replace each x with an x + 1 or replace each x with an x – 1. I have to say I was pretty thrilled with how this conversation was unfolding. Agreement on x – 1 was reached. When I was asked why, I responded with the following two questions – (a) What is the simplest equation of a parabola with its vertex at the origin? (b) What is the simplest equation of a parabola with its vertex at the point (1,0)? Everyone seemed okay at this point. We get our polynomial and evaluate it at x = 0.9 and we arrived at the fraction -79/750. When I did the long division we arrived at -0.1053 and, once again, someone’s calculator matched this exactly. A wave of smiles and nods went around the room. Those elusive moments when you can actually see a group of people lock in on an idea are so exhilarating. It was so much fun to see this group of students attentive and engaged, not intimidated by two problems that each took about twenty minutes each. This class is my last class of the day and I ended the day in a very positive mood as a result of this conversation.


PS – Another problem day today. Here is my newest problem set. I borrowed problem #1 from @bretbenesh who was clear in explaining that he borrowed them from all over. Problem #2 is an old favorite and problem #3 is from a recent math league competition. 

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