Different Perspectives

A quick reflection here before I wake up my kiddos.

Yesterday in BC Calculus we had onto of our weekly problem days where we (mostly) put aside our current Calculus work and look at interesting problems that may or may not involve any Calculus at all. Here is problem #1 from yesterday (a problem I borrowed from @bretbenesh.

A mountain climber is about to climb a mountain. She starts at 8 am and reaches the summit at noon. She sleeps at the top of the mountain that night. The next morning, she leaves the summit at 8 am and descends using the same route she did the day before, reaching the bottom at noon. Why do you know that there is a time between 8 am and noon at which she was at exactly the same spot on the mountain on both days? We should not assume anything about her speed on either leg of the trip.


One of the things I enjoy most in teaching is seeing/hearing different ways of attacking a problem. When I read this problem I immediately sketched a height v time graph with the base of the mountain and 8 AM as the origin and the top of the mountain, noon as some arbitrary point in the first quadrant. A wiggly sketch connected the points. Day two has a y-intercept of top of mountain, 8 AM and an x-intercept of bottom of mountain, noon. No matter how I connect these points the sketch intersects my other sketch and I see the reason why. I’m surprised by this discovery, but I see it. In each of my 2 BC classes yesterday there was one student who saw through this problem and explained it away so quickly that I was wowed. In each case the student immediately switched to thinking of 2 people rather than one. If one person starts at the top at 8 am and walks down while the other starts at the bottom and walks up then they must pass each other at some point! Simple, clean, elegant. It’s fun to learn from your students, isn’t it?

Fantastic Morning from BC Calc

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.

Conversation Follow Up

A brief one tonight as I am solo dad while mrsdardy is off at a work event.


My AP Calculus BC class has a test tomorrow. Their last HW assignment – some review problems I put together was assigned for Monday night with the idea that it would help guide their studying and that today would be a day in class to discuss any concerns. Well, the class where I had the sprinting is similar to cramming conversation did not seem to take my hint at all. It was clear that plenty of cramming is planned for tonight. even asked the one girl directly about our conversation and she just sort of shrugged and said she’d be ‘sprinting’ tonight. I know that these sorts of habit are hard to change. I mean HARD to change.


So, tomorrow morning I teach a couple of classes then head to the airport. I am off to NCSSM for their Teaching Contemporary Mathematics conference. I went there about eight years ago and was wowed by Dan Teague, Floyd Bullard, and the whole atmosphere of the place. This time I am going to make a point of meeting Daren Starnes – the author of the Stats text I use. He’s also the Dept Chair at Lawrenceville and he’s been so helpful in setting up a professional visit for two of my colleagues to go there. They got snowed out Tuesday and have already rescheduled. It’s rewarding to start feeling connections outside of my building and it’ll be nice to  meet him this weekend.


Oh yeah, while I’m out on Friday my Calc BC kids will work on Problem Set 3


Changes for the New Year

So, I had recently blogged about some ideas to change the pace of my Calc BC class and I want to report on how it is going so far. We are one (partial) week into the new year. We lost Tuesday to extreme cold and I am losing the second of my two BC classes today because I’ll be visiting another classroom. As department chair, it is one of my obligations (and one of my real pleasures) to visit my colleagues to watch them at work.

I have two very different sections of BC this year. My morning class has seven students and they are somewhat reluctant to work together. They get along fine, they are just much more independent workers by nature. My afternoon class has seventeen students and they are much more social and collaborative.


I want to summarize the past two days by section, rather than by day.

Yesterday we were in our computer lab for both sections working on the Desmos activity I slightly modified from Sam Shah’s Virtual Filing Cabinet. By the way, if you haven’t seen this resource, click on the link. You’ll be glad you did. My 1st period class was typically quiet and worked individually with only a little bit of collaboration. I started class with a quick exploration of the polar functions of the form r = 1/(1 – kcos(theta)) and r = 1/(1-ksin(theta)). After five minutes, I left them alone for the next half an hour. I wrapped up class with a verification that, when k = -2, the graph is a hyperbola. A Desmos graph shows this quickly and some recall from precalc days allowed us to convert this to a rectangular equation. It was not a pretty one and the process required recalling the standard form of a hyperbola as well as remembering how to complete the square. I was pretty much the lone voice (unfortunately) but it sure seemed like they were all fine. Today, they worked on the problem set that I also linked to in my last post. I sat and worked myself and had all 7 of them sit at one of my two big tables together. Normally they split themselves with five at one table and two at the other. I thought that this would encourage more collaboration. Instead, I sat working quietly for 30 minutes while they all worked quietly as well. No talking, no looking over each other’s shoulders, no recognition of each other at all that I could see. I must admit that I was getting kind of frustrated. At one point, I catch the eye of one of them and his attention seems to be wandering. I ask him why he’s not talking to anyone and he says he answered them all except for the first question. This is very surprising to me on a number of levels. I think that the first question is the most straightforward (and most related to Calculus) and I thought that this was the one that would seem the least intimidating. The next fifteen minutes were spent sharing solution ideas to that problem as well as the other problems (we only made it through the first five together) and I have to admit I was knocked out by their creativity. Especially on the question involving counting digits. Three of my students actively shared their solution ideas and they just knocked it out of the park. Frustration turned to a combination of delight and confusion. I’ll ask some of my questions later.


Yesterday, my afternoon class also met in the computer lab to work with Desmos. Again, I spent about three to five minutes looking at an animated drawing of the polar curves I mentioned above. For the next thirty minutes the class had a consistent hum of chatter, people arguing with each other about conclusions, kids looking at each other’s work. When I reconvened the class to focus on the same k = -2 case, they were engaged. telling me what the hyperbola equation was, catching a mistake I made in factoring, just a lively discussion. When class ended, I checked in with two students who were just packing up. One of them said something to the effect that my class made his head hurt a bit. He said it cheerfully and his neighbor said that my class was ‘interesting’ which is the word I use to describe difficult or challenging questions. He, too, said this rather cheerfully. I won’t be around to see them work on the problem sheet but I have asked the colleague who is subbing for me to collect their work so I can see what they can accomplish and how they approached these problems.

Now, I am left with these questions as I move forward.

  1. How do I create a situation so that my first period class actually talks to each other?
  2. Is it important enough to make that happen, given that they are productive workers? I have a pretty strong belief that talking about ideas is important, but I don’t know how to win this class over to that point of view. Is my personal bias important enough to try to change the nature of my learners in my 1st period class?
  3. Can I build momentum for these problem solving days if they only happen once per week?

I’ll keep reporting on progress and I’ll keep an eye on any wisdom that you can share int he comments section.




So, today we saw school cancelled due to the cold weather here. Woke up to an air temp below zero and wind chill about 20 below. Took the morning to finish the first of my weekly problem day assignments. I’m sticking to my guns and using this Thursday as our first class work day despite losing today to the weather. I sent out a parametric/polar practice sheet to my kids and asked them to spend some time with Desmos. Tomorrow we’ll be in the computer lab working with Desmos (gotta get started on that doc next) and then we’ll have our problem day. I’ll report back on how it goes. The doc I created that is linked above is a collection of stuff I’ve scoured from the web.

More Handshakes

So – as I anticipated, this problem was more challenging than the more typical direction where the number of people in the room is the given information. What completely knocked me out was that I saw two different diligent solution paths. One student convinced his neighbors that the formula to work with was x(x-1) where x is the number of people in the room. Nice thinking, not exactly right but a good start. Another group was diligently building a table of values. We plotted those values and it was pretty clear that a parabola was emerging. This was nice since the other formula was quadratic. However, the table of values did not match the formula. This led to a quick adjustment and we decided to solve the equation (1/2)(x)(x-1) = 253

I was told to multiply the 2 over to create x(x-1)=  506 and this is where class got exciting. The students, predictably, wanted to solve the quadratic. We had already concluded that we wanted numbers somewhere in the twenties based on the product. One of my students pointed out that we were looking at consecutive numbers whose product ends in a 6. The student patiently explained that it had to be 22 and 23 or 27 and 28. I LOVE this. The type of number sense at play here was so refreshing. First day of school a winner based on this exchange.


So, one of my favorite opening day problems for Algebra II is the handshake problem. You know, there are 18 of us in this room and if we were total strangers and went around and introduced ourselves to everyone in the room, then how many handshakes would occur are we go around the room meeting each other? I know, I know, this is a sort of pseudo-contexty type of problem but it always leads to interesting conversations. So, this year my lowest class is a Precalculus Honors class and I think I’ll start off by saying “There is a group of strangers in a room and they all go around and introduce themselves to everyone in the room. At the end of this process 253 handshakes have occurred. How many people are in the room?” 

Is this any more interesting/challenging than the standard version? I’m not certain, but I think it is.