Tag Archives: Cleverness

Some Fun Geometry Action

On the heels of learning some right triangle trig I am really trying to develop more proportional logic with my students. Just this week we had a really productive conversation about the following problem.

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Being a bit of a bull in a china shop sometimes, I proposed that we should find the height of each triangle, find each chord length and find the height of the trapezoid by finding the difference of the heights. Not elegant, I know. I was trying to make sure that we remembered some right triangle trig. that we remembered our area formula for a trapezoids, and that we try to develop some patience in solving multi-step problems. That was my plan, but as with many school plans, it did not quite unfold that way. One of my students who is a bright and quick problem-solver pointed out that simply finding each triangle area would be enough. I understand that his solution is pretty much the same as mine, but it certainly sounds more efficient. But as soon as I acknowledged that his idea was more efficient than mine another student trumped each of us. She pointed out that I had already asked them to consider the ratio of areas between the two triangles. So, if we know one area, we can automatically know the other area. If we know both areas, we find their difference as suggested by the first student who chimed in. I was so happy that she took my clue from within the problem and that she was clever enough to really save time and energy this way. I made sure to compliment her in class and I bragged about her work to two of my colleagues yesterday. Oddly, this morning when we were reviewing before today’s quiz I reminded her – and the rest of the class – of her clever idea. She had no memory of this conversation. Sigh…

I’m trying to process this and figure out what it might mean for my classroom practice. I understand that I should be more excited by my students’ ideas than they often are. I understand that I will remember context of conversations more easily than they will because I am not dealing with the cognitive load of trying to learn/understand the conversation. I am simply coming at it from such a different place. What I don’t understand is how a student can be so in command of an idea but then not remember the creative process that made her arrive at this clever conclusion. I discussed this in the faculty lounge right after Geometry today and one of her other teachers intimated that this might simply be modesty on her part. I am not sure how much faith I put in that reading.

So, while I am a bit frustrated and confused, I am choosing to focus instead on the positive energy of yesterday’s conversation, on the clever ideas that my students brought to the table, and on the fact that my students did a really nice job on their quiz today.

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.

When My Students are More Clever than I am

This happens often enough to keep me excited in my job. I love the feeling when I learn from my students. Nearly everyday I learn important things about people, about human interactions, about kindness and community. What happened today was a great example of when I learn some math from my kiddos. I blogged yesterday about helping a boy with a L’Hopital’s Rule assignment. What I did not mention was that there was a problem I could not solve. I’m not good enough with typesetting here on wordpress so I’ll do my best here. We were trying to evaluate the right hand limit of (x – 1) * tan (pi*x/2) as x approaches 1. Two clues made me think of L’Hopital. The first is that the student told me they were working on L’Hopital’s Rule. the second was the indeterminate form of 0 *  negative infinity.

We looked at a graph to convince ourselves that there is a limit and then we tried to make this a quotient to fit the rule. My instinct with trig functions is to avoid the cotangent, cosecant, and secant functions simply because I feel more comfortable visualizing the basic cosine, since, and tangent. So I made the product a quotient by dividing the tangent expression by 1/(x-1) creating an indeterminate form of negative infinity divided by positive infinity. A quick ratio of derivatives yielded something even word and the second time around was even scarier. We walked away from the problem – in part because I was trying to do three other things at the same time and in part because he was exhausted from thinning his way through the other examples. I asked one of my best BC students to consider it overnight and had some hope that he would. 

This morning at 8 I posed this question to my quiet BC class of 7 and they pounced on it. One student advised that I rewrite the tangent function as a quotient of sine divided by cosine so that L’Hopital is immediately satisfied. Another advised that I rewrite tangent as a cotangent so that I have a ratio as well. In each case, my students saw a way to rewrite a product in terms of 0 / 0 rather than my way of infinity / infinity. Neither format is lovely but we all seemed to agree that 0 / 0 seems less scary. One round of derivatives on each idea led to the conclusion that the limit was – 2 / pi. Geogebra agreed.

What strikes me is that, despite me showing an undesirable approach AND me asking them to recall something from the past, my morning class was perfectly willing to be flexible and to TRY something. Two sound ideas in about 90 seconds and we were off. I was so delighted to start my way this day.

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. 

Wow

So, I sort of pride myself on being the type of teacher who creates an environment in his classroom where conversations can flow. I have my kiddos at two large tables where they are elbow to elbow and talk regularly. Sometimes, of course, the conversation strays – bit there are often rich math conversations going on. I posted this quick story over at One Good Thing, but I want to share it here as well.

I presented my BC class (all in their second year of High School Calculus) with the equation of an ellipse centered at the origin and asked the following rather vague question – “Are there any two points on this curve where the lines tangent to the curve are perpendicular?” One girl, Chloe, immediately answered that the tangent ‘on top’ of the graph was horizontal and it would be perpendicular to the tangent on the ‘side of the graph’ which is vertical. I congratulated her and challenged the class to find some other more interesting points. A student asked what the slopes of these more interesting lines might be and then a boy, Sal, chimed in that any number you pick must be the slope of a line tangent to this ellipse. His argument was based on recognizing that between the two tangents that Chloe had mentioned the slopes range from 0 to positive infinity. In other quadrant the slope would range from 0 to negative infinity. If he had mentioned the intermediate value theorem I might have fainted on the spot from joy.

After I posted the story to One Good Thing I read Ben Blum-Smith’s most recent posting (http://researchinpractice.wordpress.com/2013/09/08/kids-summarizing/) and I now realize what an opportunity I missed by simply congratulating Sal instead of getting others to join in and complete the thought process. Read Ben’s post. You’ll be glad you did. I intend to try and incorporate this strategy into my daily practice.