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Our school operates on a trimester schedule. The fall term ends as the Thanksgiving vacation begins and the winter term ends when our spring break begins. Before our fall finals I have been taking some time to review and reorganize some ideas with my students. I have to say that the past two days with my Geometry students have been filled with a combination of two strong feelings.

The positive feeling is that I am really proud of how my students have been working together. My class is set up in three ‘pods’ of six desks each. I am spoiled and my largest class has 16 students. At the beginning of the week I uploaded five review problem sets for my Geometry students. My plan for this week is to have one problem set each day and to circulate through class during the week, putting some burden on my students to be actively reviewing ideas rather than sitting and listening to me. The students have really been going at it and I have enjoyed listening to them. A couple of good hearted heated debates have popped up and I get called in to moderate these debates. There was one particular problem that I asked that I am proud of.

The points M(3, -1), N (4,3), and P (0, 5) are the midpoints of the sides of a triangle. Find the coordinates of the vertices of this triangle.

I am not certain that I can remember the origins of this question. It is likely that I found this somewhere and if anyone recognizes where it came from, please remind me. What really pleased me about this problem was the response of one of my students. I overheard a number of students discussing this problem, so I took the opportunity to gather up the class and lead a conversation here. I drew a triangle on the board, not on a coordinate plane, and joined the midpoints of that triangle (rough sketches here) then I asked them to make some guesses about the relationships between this ‘midpoint triangle’ and the ‘parent triangle’. It was very quickly apparent that the students expect that the midpoint segments are parallel to sides of the original triangle and the guess that lengths are related came out as well. No questions were raised during this part of the conversation. This group chat convinced me that the question I asked is much more interesting than if I had asked the question in the opposite direction where I gave the parent triangle and asked for the midpoint one.

Even with an agreement in class about the relationships here, the work to translate this to the coordinate plane is a tough leap. Early in the year I gave a few questions where I would give a segments endpoint and a midpoint and ask for the other endpoint. Not a creative problem, but one that the students remember. Since I knew that I wanted to introduce vector notation early in the year, I would frame these midpoint questions by talking about this problem in terms of taking half of the journey here. So the movement from one endpoint to the midpoint is half of the journey and we want to repeat this path. This way, I try to get the students thinking in terms of component movement and not simply in terms of distance. So, I took this approach with the class on this midpoint problem. Time did not allow a conversation that was as deep as I wanted it to be and I plan on starting class today by revisiting this problem. I am unsure whether I will use the same coordinates or not, I am inclined to think I will not. I also know that I will start class today by simply trying to get a few students to explain the thought process we outlined, not to focus directly on the necessary calculations. I have been working hard at getting my Geometry students to frame a process of problem solving for complex problems. We recently worked on finding the distance between two parallel lines and I asked my students to outline their approach to the problem rather than calculating the distance, so framing a question like this in the terms of describing their problem-solving approach will make sense to many of my students. After we discussed this problem at the end of class yesterday, one of my students remarked out loud that this problem was really hard but really interesting. I was pretty pleased to hear her think out loud like that.

i started off by mentioning that I had two strong feelings so far this week and I started with the positive feeling. The not so positive feeling is a deep sense of frustration about the overwhelming lack of recall of some simple facts. At least five of my students were stymied by a simple question of finding the midpoint of a segment. They consistently wanted to apply a simple formula and they could not (would not?) stop and think about what kind of formula might even make sense. For any teacher who has taught this idea, it will be no surprise to you that the debate centered on whether to add endpoint coordinates or to subtract them. I reply by asking what midpoint means and every student quickly says middle. Then I ask what number is in the middle of 2 and 10. I try to convince them that they do not need a formula if they are willing to stop and think. I wish that I had gone to the coordinate plane quickly to try and tie together some physical sense of these points. Instead, I was visibly frustrated and asked a few students what their average would be if they earned a 90 on one test and a 100 on a second. Would their average be 5 or 95? However, thinking back on this exchange I realize that I was not effectively making a point here, I was simply showing my frustration. This phase of the year where students are trying to tie ideas together should be a time when I am happy to see the growth in my students understanding. More often, this ends up being a time of frustration with finding myself repeating ideas that I thought we had successfully conquered early in the year. I understand the pressure that students feel when they are faced with a week of long exams and stress related to trying to show mastery of twelve weeks of knowledge all at once. I have been at this for a long time now so I feel that I ought to have a better idea about how to navigate this challenging time of year. I have worked at four schools now and they all believe in final exams, so I have tried to work within that context. I also believe in the principal behind cumulative displays of knowledge. I just know that the way we do it creates such stress for most of our students that they do not feel like this is a show of knowledge, they feel it is simply more of a test of their endurance.

I would love to hear how any of you navigate this challenge.

As I have mentioned before, our school has AP Calculus BC as a second year course in high school calculus. One of the main reasons we do this is that we feel it creates a space where we can explore, we can dig in to deep problems, and we can give our students the opportunity to really reflect in a way that the pace of a one-year (or 1.25 year) course does not allow. I have heard students say things to each other like ‘Last year, I knew how to solve those problems, but I really did not know why I was doing it that way.’ The pace of the AP curriculum creates such pressures that students too often can fall into that trap of just learning how to do something. One of the tangible joys of teaching the BC course under these conditions is that I can routinely take days like today where I just toss a problem set at my students and listen to them think. The problems bounce around – sometimes they are calculus based problems, but more often they are not. This morning our problem set started with this question:

Find the point of intersection of the lines tangent to the curve  if those lines are tangent at x = 1and x = 5

This is not a challenging problem in theory for my students in this course. However, the algebra is a bit snarly and unpleasant. They debated amongst themselves and seemed happy at the end about their work. I stepped up and did the problem on the board, with their guidance, and we arrived at the conclusion that the x-coordinate of the point of intersection was x = 3. Hmmm, I wondered aloud whether there is something going on here that x = 1, x = 5, and x = 3 seemed to have a nice relationship. I quickly examined the boring old standard parabola at the origin facing up and we saw the same thing. One of my students commented that this was probably generally true but these examples certainly did not prove this in any way. I sat back down and got to my work and a few minutes later a student named Richard called me over to show me that he had confirmed that this would always be true by replacing the 1 and 5 from my problem with p and q. His answer, instead of 3, was the average of p and q. 

I cannot accurately convey how pleased I am that he was persistent enough to do this. It would have been the easiest thing in the world to accept this conclusion and to move on to the next problem especially since his neighbors had moved on. So he removed himself from a stimulating conversation with his friends to satisfy his curiosity (and maybe to prove something to himself about his ability to push through) about a problem that likely will not have any future impact on his calculus grade. What will have an impact is his increased belief that he can solve a snarly problem. Pretty pleased, I must say that my Friday morning started off this way.

Note – This is being cross posted a bit belatedly from the lovely website https://betterqs.wordpress.com

You should drop by there if you have not yet

I continue to try and get a handle on the Discrete Math class I inherited mid August. One of the highlights has been the conversational nature of the class. Each section has 8 students and they all sit at one large conference style table. We have been wrestling with probability and there were three questions I asked on their latest HW assignment that I was happy with as I wrote. What I realized during class today was that I will be much happier next year when I rearrange them. The first question was in four parts. The context was that I had tossed a coin four times and I asked the following questions:

• What is the probability that all four are heads?
• What is the probability that all four are heads if I tell you that at least one is heads?
• What is the probability that all four are heads if I tell you that at least two of the tosses are heads?
• What is the probability that all four are heads if I tell you that at least three are heads?

A number of my students struggled to see why the changing information changed the probability. In my first class they dutifully followed my path of reasoning as I drew the outcomes and talked about the answer. It was not until we reached questions three and four that I saw light bulbs go off. In question three I told them that a friend of theirs reported having tossed a coin ten times and seeing a head each time. Their friend tells them that he KNOWS that the next toss will be tails. Their job was to convince their friend otherwise. In question four the conversation continues with their friend arguing that there is only a 1/2048 chance of eleven heads in a row. No way that will happen! Again, they were asked to address their friend’s misconception. While we discussed these two questions a number of students popped up and said that the first question now made sense. So in my afternoon class I swapped the order of the conversation and question one fell into place so much more easily. I am definitely changing this order for next year to help support my kids as they develop their understanding of probability.

Last year I opened the door to a conversation about proof by adapting an activity from Max Ray-Riek over at the Math Forum. I asked my Geometry kiddos to write out directions for how to make a Peanut butter and jelly sandwich. I swapped directions around somewhat randomly and asked the students to do their best to make a sandwich only following the directions given. It is a fun activity, the kids get a laugh out of it and we have some yummy afternoon snacks. Most importantly, I think that it makes a vivid point about how detailed you need to be at times when trying to tell someone how to do something. I will use this conversation and activity as a reference point over and over in the next few weeks as we begin to grapple with what it means to prove something and how you can explain to someone why you think that something is true. I took a few photos of the sandwich designs that I want to share here. In the first one, try to notice on the paper how scant the directions were. Last year I had such a thorough and detailed description written by one of my students that I posted her PDF document here on the blog. I have yet to dig through all of the directions that were turned in this year, so I do not know if there is a winner in that category again this year. Here are some of the fun photos from the day.

 

We had a lovely conversation in my afternoon Calculus class today. I wrote about it over at https://onegoodthingteach.wordpress.com/2015/10/08/taking-a-problem-apart/

There are always great posts over there.

Earlier today I wrote about one of my AP Calculus BC students who wowed me with his solution to a particularly thorny algebraic problem. I referred to being outclassed by him as he saw something I did not think of AT ALL in the problem. This tale is about someone seeing something I saw – but something I did not expect my students to remember. This involved verifying the derivative of an awkward trig expression. Below is a snapshot I took and then I will try to detail the work and the conversation.

The students were asked to show that the derivative of (sin x)^2 /(1 + cot x)  + (cos x)^2 /(1 + tan x) is equal to – cos 2x

I said out loud that my first instinct was that it would be nice to somehow get my Pythagorean identity in the numerator together. Boy it is nice to see both cosine squared and sine squared. Well, the hero of my first post shrugged and said he just attacked the quotient rule derivative here. Another student, named Samarth, had followed a path similar to the one I imagined. It is his work that I show above. He patiently talked the class through his thinking and his work and, in the process, showed that he had a strong memory of helpful trig definitions. This problem is a nasty one if you just lower your head and plow through. Samarth did not show interest in doing that, he wanted a more elegant touch to bring to play just as Connor had in the earlier post from today. In the same class as these two is another boy, named Daniel, who wowed me last week by applying the binomial theorem to a problem asking us to prove that numbers of the form 7^n – 1 are always multiples of 6. I have been in school for 9 days now and this class alone has knocked my socks off three times already.

I have been having great conversations in my other classes as well and I will be highlighting some of those soon.

It has been a pretty terrific start to the year and I am optimistic that more awesomeness lies ahead.

Last night I asked my AP Calculus BC students to tackle an especially challenging question. They were asked to generalize what the nth derivative would be for the function defined as f(x) = x^n/(1-x)

When n = 1 the derivative calculation was pretty straightforward and we ended up with 1/(1+-x)^2 asthe derivative. When n = 2 the work to find the second derivative was not as much fun, but it was doable. We ended up with 2/(1-x)^3 as its derivative. At this point, I wanted to predict that the derivative would be of the form n/(1-x)^n and call it a day. Unfortunately, this is not the correct answer and a quick check of the solutions manual confirmed this. This, sadly, is where my first period class ended its day. When my fourth period class came in, I was eager to start where I had left off and was prepared to wow my students with feats of Algebraic endurance by taking the third derivative of x^3/(1-x).

One of my more talented students in the class is named Connor. He told me that when he got to that point in the problem on his homework that he knew that there had to be a enter way. He changed the numerator from x^n into x^n – 1 + 1. This allowed him to write the original fraction as a sum of fractions with the first fraction transforming almost magically. I took a snapshot of his work and here it is below.

Connor recognized that the first fraction was now a geometric sum whose nth derivative would be zero. This is such a lovely example of what I hope our best students can/will do. Sure, he could have bulled his way through and seen the pattern emerge. Instead he insisted that there is a better way and he searched his mind to find a connection to something he had seen before. His classmates were taken aback, as was I. Once he rewrote the numerator, I thought of factoring and talked the students through that approach. As if that was not enough awesomeness for one class, I also had a student blow my mind with an approach to an ugly trig problem. Grading beckons me now, so I wil write again soon.

A promising new blog site was started by Sam S, Rachel K, and Bill T to share ideas about questions and questioning techniques over at https://betterqs.wordpress.com

This is a topic near and dear to my heart so I joined on and just posted my first item over there. You should head over to see some of the fantastic stuff happening already.