Today was a pretty blah day until my last period class. My first three classes all had assessments so I had no fun conversations and I watched work pile up. As I came in to my last class of the day – my Geometry class – one of my Geometry teammates was waiting in my room to share that his students had been making some great strides in GeoGebra. He told me that a number of his students were really beginning to dig into what GeoGebra could do for them, especially now that we are talking about transformations. I used Geogebra extensively when writing my text and I borrowed from resources around the web for activities. One of them was an activity called A-Maze-Ing Vectors which had been created by the amazing Jennifer Silverman (@jensilvermath) and we used that activity the past two years. My teammate who had been waiting to share his good news had asked me this past summer about modifying this activity. We had had trouble completing the activity in one day and it did not take up enough for two solid days. He also had an idea about combining vector transformations on objects more complex than points. He created a pretty wonderful adaptation of the activity (you can find it here) and my students worked through it yesterday. I opened class today by projecting the last page on my AppleTV where we had to navigate a triangle through a maze and I invited a student to come up and draw on the TV (with a dry erase marker, don’t worry!) and I cannot tell you how great the conversation was in class. I sat down – a commitment of mine based on my #TalkLessAM session at TMC16 – and just watched the fireworks unfold. Kids were challenging each other, going up to the TV to draw their ideas, debating distances, talking about slope, worrying about vertices colliding with walls and discussing the option of rotating the triangle as it moved. I was SO thrilled with the engagement and the level of conversation. I credit this to a number of factors. The original activity was terrific and my colleague’s rewriting of it is creative and concise. Kids like drawing on a TV – it feels naughty or something. I sat down and got out of the way. Kids had worked this through the day before in their table groups and were invested in both supporting their teammates and making sure that their memory and their perspective was clearly heard. They were supportive of each other and slightly defensive if someone else had a different approach. After a pretty uneventful day at the end of the week it would have been easy to just limp tot he end of the day, but these kids brought each other to the finish line for the week sprinting. I am optimistic that we can pick up with a similar level of energy on Monday.

More Calculus Fun with Series

Many many thanks to the wonderful Mike Thayer (@mthayer_nj) who sent a link to this lovely video in response to yesterday’s post.

We started each Calc BC class today by revisiting the rational function that caused me so many problems yesterday. Innocently enough, I decided that it would be interesting to examine 1/(1-x) first by long division where the divisor is 1 – x and we got the power series 1 + x + x^2 + x^3 + … as the quotient. Then, we used -x + 1 as the divisor and got the series -1/x – 1/x^2 – 1/x^3 – 1/x^4 … as our quotient. Class ended too soon and I was not able to answer the question of how we could consider these two very different looking series as being equal to each other since they were each results of considering the rational function 1/(1-x).

So, after sending out my call for wisdom in yesterday’s post I went to GeoGebra and discovered something lovely. Something that I was sure my students would be able to discover for themselves. I created a GGB file and I planned our day today. Can I tell you how proud I am of my students for how they handles today? Well, I won’t wait for your permission, I will just come out and tell you how thrilled I am.

I started the day by quickly revisiting yesterday’s two division results and then I called up the GGB file with only the rational function showing. They saw that I had each of the other two series expressions typed in already (out to x^5 for each) and I asked them which of the two they wanted to see first. My morning class wanted to see the series with x as the ratio first, my after lunch class wanted to see the one with the ratio of 1/x first. In each case, after unveiling the function of choice and noticing the relationship between the rational function and this new series expansion my students made the following observations:

  1. The graphs only seem to match over a certain set of x values
  2. If I were to add more terms, that match would improve
  3. If we look at the graphs of both series then we will have a nearly complete match to the graph of the rational function

What my students realized – what I realized last night – is that the two series we can generate have completely opposite intervals of convergence. It was absolutely lovely to see geogebra help this intuition along and it was fantastic that they made this realization before seeing the second graph to confirm it.

After this breakthrough we watched the video together and all of our brains were a bit achy by the end. Amazing that Mike found/knew this link so quickly when I blogged last night.

Some other notes – A student in my after lunch class made this observation about the complementary nature of the intervals of convergence before we even looked at geogebra. I gave one last example, f(x)= x^3/(x+5) and the student who was bothered by -1 = 1 + 2 + 4 + … quickly converted x + 5 into a form of 1 – r so we could interpret the rational function as an infinite geometric series. Another student converted it to x^2/(1 + 5/x) and we, once again, had two different series’ that had complementary intervals of convergence. I have taught this course five or six years before this and never had this ‘discovery’ pop up. Now, we cannot avoid it. It provides a wonderful context for our upcoming conversations about Taylor series and it gives us the opportunity to be more aware of convergence expectations. A pretty great day!


This post is a few days late due to, well, you know, life getting in the way. When I last checked in with you here I was preparing to put into action a plan to lag my homework with Geometry and have a series of HW assignments incorporating more review of past skills. My kiddos took their first quiz of 2016 yesterday and I plan on grading them tonight so I will have some data to back up (or refute) my reflections at that point.

What I have noticed so far:

  • Review assignments – at least the ones I have written – make my students pretty frustrated. I certainly do not want frustration to be the go to emotion for my students when I ask them to work, but I am willing to have that as a  stepping stone if I can help usher my students into a place where they are more comfortable with problem sets that do not depend on a small set of skills and ideas. I realize that I am combating years of habits and expectations.
  • We have more time to practice some of the new(ish) skills that I am hoping that they develop. We spent days in a row visiting some linear equation writing skills and some  ideas about linear combinations.
  • When we did  finally get to HW concentrating more on single sections of the text I was not receiving quite as many questions as I had been expecting. This I am taking as a positive sign. I will be more convinced that it is a positive sign if I see some stability on their quiz. We spoke briefly about the quiz today and I suspect that I will see some hesitant work. If the mistakes are more algebraic in nature I will feel better about the development of their Geometry skills and ideas.
  • We took a day in our lab to play with GeoGebra and I realize that I have to do something more consistent next year to encourage/require my students to engage with GeoGebra more frequently. What should have been a productive activity drawing some connections about the ‘centers’ of a triangle that we are examining, too many of my students were either distracted playing with zooming in and out on various images or they were flummoxed by some of the commands that we ‘learned’ in the fall. One of my new colleagues and I are brainstorming ways to make check-ins with GeoGebra a regular and meaningful part of our life in Geometry.


Part of the way that I am organizing out of school work right now is by asking my students to read based on class discussions while they do not practice those specific skills for a few days. I am not at all convinced that they are reading as I request, but I also do not think that they were doing to reading under our old structure either, so that is a wash. I will write again tomorrow after I grade the quizzes and I will check in to see if the data backs up my observations in any meaningful way.


Catching Up with the Past Week

So there are a couple of activities this past week that I want to write about. However, I have been swamped with meetings so I have fallen a bit behind.

In AP Stats we have finished our required curriculum as of 8 days ago. I am a big baseball fan and my favorite team is the New York Mets. They are having a pretty wonderful start to their year (or at least were until the last few days) so last Friday I posed the following question to my kiddos: Given that the most optimistic projection I saw for the Mets’ season had them pegged as an 87 win team, what is the likelihood of their current record (which, if I remember correctly) is 10 – 5? I liked this for a few reasons. First, it concerns baseball and likely would have a positive outlook for my Metropolitans. Second, it was not so focused on the most recent material at hand. My Stats students tend to know recent material well but struggle remembering other procedures that have not been practiced as recently. Third, it generated some nice thinking out loud about what approach to take. Being more of an algebra stream guy myself I immediately placed this in the context of a probability problem and was prepared to go down a Pascal’s triangle/binomial theorem path. Most of my Stats students don’t tend in this direction so their conversation focused instead on comparing proportions – the 87 – 75 projection with the 10 – 5 proportion. They suggested running a two proportion z test and looking at the corresponding p-value. This opened up the avenue for me to sneak in my approach and make a connection pretty visible to them. Turns out that we felt that we had enough evidence to reject the null hypothesis of the Mets being an 87 win team in favor of believing that they will exceed that win total. Their recent 5 – 5 run of games might adjust that but I do not want to know this – so I will not re-run the test right now! After we checked our trusty TI to find the p-value of this test I reminded them of the probability approach and we set up the appropriate term of the binomial expansion. Guess what happened? This calculation matched the p-value of the two proportion z test!!! This is one of those ideas that we discussed but somehow seeing these results side-by-side seemed eye opening to my kiddos. A triumph on a number of levels!

In my morning Geometry class we dipped our toes into an exploration of radians yesterday using the ProRadian Protractor designed by the fantastic Jennifer Silverman (@jensilvermath) and using an activity that she designed. I wrote a follow up HW assignment that my kiddos worked on last night. I also linked to a fabulous web site that allowed my students to explore radian measure and I shared these notes with my colleagues. There is also a lovely GeoGebra applet (also designed by jennifer Silverman) that is linked from the worksheet. I was totally excited to explore this activity with my students and I had a really nice chat with one of my teammates.

I handed out the radian protractors as well as our regular old angle protractors and we had a nice conversation about similarities between the two protractors. We had a lovely discussion about this but, looking back on yesterday , I think I allowed too many clues to seep into the conversation too quickly. Jennifer’s activity is a terrific one and I got in the way by loading too many conversations in at the beginning of the class. By having students come to my screen and try to identify where one radian measure would lie on the circle AND by having the protractors side-by-side I reduced the mystery element that I think should have been part of the classroom activity. I think I took away the opportunity to discover what was happening here. I did have one student give a GREAT explanation of why the quadrilateral radian measure was twice the triangles radian measure. She invoked a proportional idea and referenced our (n – 2)*180 formula. I had a number of students quickly see that the ratios we had been working with before (frac{x}{360}=frac{arc}{2pi r}=frac{sector}{pi r^{2}}) could be easily extended to add one more simple fraction of frac{x}{360}=frac{theta }{2pi }. That definitely felt like a triumph. So, the lesson I learned here – and I hope I remember it for next year – is to be a little more minimalist in front loading this conversation. I think that we can touch on all of these resources and really let the discovery sink in, but I feel I nudged them a little too much this time around. So the plan for next year is to hand out the radian protractor and work through the worksheet. Then hand out the angle protractor and talk about comparing them. Then, the next day after some time to think, show the web app and have them identify where one radian is. Let this unfold a little more slowly.

Borrowing from the MTBoS

I’m guessing that most of you reading this are familiar with the awkward acronym for the Math Twitter Blog-oSphere – one of the joys of tapping into this community is that they are remarkably generous about sharing ideas and resources. Today in our Geometry classes (I teach only one of the five sections we have at our school) we used an activity written by Kate Nowak (@k8nowak on twitter). It is an activity based in GeoGebra and allows the student to explore the ratio between lengths of legs in a right triangle. You can find the document we used here . I modified (very slightly) the document that Kate originally posted here. Next time I use it I will tweak it a bit. I have only twelve students in my class and chose not to explicitly team them up. They talked with their neighbors as they are usually encouraged to. However, the directions either need to be tweaked so that team references are excluded or I need to clearly team them up. I am also debating question 7. A number of students did not make the explicit leap from using the ratio they found on page 1 and using it here. I don’t necessarily want to give away too much but I may add a little prompt that they should consider the work that they have already completed. We set up a google spreadsheet and in the next couple of days I will refer to this repeatedly to show that different students working on different triangles were arriving at the same ratio. We make a big explicit deal about scale factors between similar figures. I do not think we spent enough time pointing out that scale factors within figures will also match up for similar figures. I will definitely make this more of a point of emphasis next time through my text.

I cannot thank Kate enough for sharing this activity. My students worked well and I am convinced that they will have a more solid grasp of trig ratios moving forward. As I plan out the rest of the unit I am also going to be borrowing from Sam Shah’s latest post about trig. You can find that over here.

Man – the benefits my students are reaping from people that they will never meet – such as Kate Nowak, Jennifer Silverman,  John Golden, Jed Butler, Sam Shah, Pamela Wilson, Meg Craig, and so many more – is just remarkable.

Back in the Groove

Our school has a two-week spring break at a silly, early time in the year. We have been back for a week now and I feel like my students and I are all getting back in the groove again. I know that the dreaded senior slump will continue to pick up momentum but at least I am still seeing some energy and engagement from most of my seniors.

I have a few posts bubbling in my brain and I suspect it’ll be a busy blogging week. Tonight I want to briefly touch on my AP Calculus BC class. We are just settling in to our last major required topic of the year, the Taylor / Maclaurin polynomials. I wrote a little GeoGebra demo (you can find it here) and I started off by showing them (without revealing the mechanics behind the scenes) a polynomial approximation of increasing degree for the trig function y = cos x. We played a little noticing and wondering and saw that at certain stages the polynomial did not change. It did not take long to deduce that this happened at the odd powers of the Taylor polynomial. This led to one student remembering something about the symmetry of cosine, another student mentioning that this was a y-axis symmetry and, finally, a third student mentioning that this is even symmetry. So the lack of development due to the odd powers of the Taylor made a little sense. We then switched to y = sin x (as in the link above) and, unsurprisingly, saw that the even powers seemed to do little or nothing here. We did a little more noticing and wondering watching the Taylor expand on GeoGebra. I should note that all of this was centered at x = 0 (or, in the Taylor notation, we had a = 0) GeoGebra’s sliders allowed us to begin shifting that value and some interesting (and ugly/scary) things started happening to the Taylor equation. My kiddos quickly saw that the equation seemed to be undergoing a simple horizontal transformation – at least in the x terms. The coefficients were changing in some mysterious ways. Finally, we looked at the Taylor series for y = e^x. One of my students asked a great question at this point. He asked – Why are there all those factorials in the bottoms? I skipped this question around the room a bit to see if anyone wanted to make a guess. They quickly observed that exponents in the numerator were clearly attached to the factorials int he denominator but – understandably – they had no solid guesses. Without giving away all the mechanics (we have plenty of time for that) I asked what the derivative of x^7/7! is. I was told it would be 7x^6/7!   Correct for sure, but unsatisfying. I must have made my unsatisfied face because one of my students offered a much cleaner version of that answer as x^6/6!  Again, I did not go into the mechanics at this point, but there did seem to be some sense that this was an interesting thing to note. I was pleased by the power of the graphics of the GeoGebra applet. I know that I could do something similar in Desmos but I don’t know the commands there as well as I do in GeoGebra. I will start class off tomorrow with the power series we derived for e ^ x and I’ll ask for derivatives and integrals of that. Should be fun to see them realize in this format why the derivative of e^x is itself.

Fun to be back and excited to unfold Taylor’s series’ with my students. This was one of the genuinely awe inspiring topics when I studied Calculus. I remember being amazed by this idea and it’s mechanics. I hope I can share that wonder.

Beginning New Habits AND a Fun Activity

This morning in Geometry I started by not talking for the first five minutes while my students shared their HW with each other. They talked about their answers, they puzzled over why/where they differed and they talked about using GeoGebra on their own to explore the intersection of perpendicular bisectors of triangles. I was SO delighted I almost wanted to call the rest of the day off.  I did not, though and I’m glad I stuck it out.

We looked together at GeoGebra, reviewed (again) how to find the intersection of lines, we let GeoGebra confirm that we were right. We remembered from yesterday that these lines coincide on the hypotenuse for a right triangle, in the interior for an acute triangle, and outside of an obtuse triangle. After playing with another GeoGebra sketch we all agreed that this behavior made this point of coincidence a pretty poor candidate for the center of a triangle. I pointed out that one of our students had suggested – on his way out of class yesterday – that we should concentrate on vertices rather than midpoints of sides. Again, we let GeoGebra take over and looked at a compromise by constructing a line through a vertex AND through the midpoint of a side. I named this for them as the median. I also displayed that these medians seem to ALWAYS intersect in the interior of a triangle and I named this point for them as the centroid. We all agreed that this name was ‘center-y’ enough. As time ran out, at the suggestion of another student, we asked GeoGebra to construct angle bisectors. It does so, but draws an exterior line as well. They did not complain when I erased them, but I want to examine what is really happening there. It felt a little too much like I was waving away a distraction. We saw that these angle bisectors intersect in the interior as well – setting up a great debate for tomorrow about which center is the center-est. Just thrilled with how they hung together during the intro time and during the quick GeoGebra exploring. Need to commit to both HW review time tomorrow and to revisiting the blur of activity on GeoGebra. I am planning on a lab day for Thursday so that they can manipulate these ideas themselves.

In my AP Stats I tried out the German Tank problem using resources found here at the Stats Monkey site. My two classes dealt with this in pretty different ways. My smaller class (12 students today) worked in 3 groups of four. I made a mistake in responding to one of the first ideas I heard. One group decided to invoke the empirical rule and guessed that the # of tanks was their sample mean plus three standard deviations. I responded positively to them but this simply steered the other two groups into following this lead. In my other class I was smarter and quieter. Here I had four groups of four. One group invoked the empirical rule but they also pooled their three samples together. One group used the inverse normal function on their calculator seeking a point where the area was 0.999. One group added their sample minimum to their sample maximum guessing that they should be (roughly) equidistant from the extremes. The final group doubled their median guessing it should be halfway to the max. I was thrilled with the level of discussion and the variety of responses. A great step forward from yesterday’s disappointment where they largely ignored my Radiolab assignment.

I’ll count this day in the victory column for sure.

Wrestling with the Modern World

Sometimes I am convinced that the universe is sending me important messages to sort out. I am not sure if I am always up to the task of making sense of these meanings. In my last post I was wondering aloud about how to incorporate technology into my assessments in a way that made sense. I asked my Calc BC kids to wrestle with a tough problem about circles. The problem made much more sense (to me, at least) when I graphed it using GeoGebra. It allowed me to lock in on a region of reasonable solutions. I asked if anyone out there has logical ways to incorporate this newer technology during assessments. For years my students have come armed with TI calculators. Sometimes they know how to unlock its powers, sometimes they do not. Somehow, the world of GeoGebra and Desmos (and Wolfram Alpha, and and and) seems more dangerous or intimidating to open up to classroom assessments. I worry about how to evaluate my students’ progress when I do not know where/how they found answers. So, that’s one part of what is in my head now. I have struggled with cell phone presence in my school. A little background might help explain where I am. Eight years ago when we moved north I became involved with our local Unitarian Universalist church and I volunteered as a youth group counselor. I attended a number of weekend ‘Cons’ with our youth. One of the persistent messages at these events was that this was an intentional community that was being created for the weekend. The youth were urged to be present to each other and to the event. They were expected to put electronics away for the weekend and they were asked not to engage in public displays of affection. For the most part, they bought into these requests and the energy was palpable. Kids were engaged with each other, they were talking, singing, laughing. It was a fantastic, but exhausting, weekend environment. Just last week I visited a school and sat in on four classes and two assemblies while I was there and did not see one student (or one faculty member, by the way) staring at a screen in their palm or in their lap. Kids were present to each other, to their classes, and to their assembly speakers. I found it refreshing. In my school there is a gathering area right outside my classroom window and I often see two or three kids on one bench all staring at their phones. I know that this is my bias (maybe this bias belongs to others as well!) but I find this dispiriting. In my class, I tend to stand near the door to greet people as they come in and some of them are trudging through the halls staring in their hands and barely aware of those around them. I used to have to spend time getting my classes to quiet down at the beginning of class because they’d be talking to each other as they sat down. Not so much anymore. Again – I know that this is my bias here, but I find this a bit depressing. I try to utilize the language from my UU experiences and since I teach in an independent school I CAN invoke the idea that there is a choice made in being at our school. The reality though is that this choice is often the choice of parents and not my students. At the youth group it was much more a matter of choice by the youth engaged. So, after my school visit I was feeling that my bias was being confirmed and supported by the environment of the school I visited. Then my brains was rocked yesterday by Justin Aion. Justin blogs over at and his posts (nearly daily ones!) are a treat. I have also had the pleasure and privilege of getting to know him in person here at a workshop we hosted (run by the wonderful Jennifer Silverman) and at twittermathcamp this summer. He is as delightful in person as he is through his blog. Yesterday Justin wrote a pretty moving post (you can find it here) about cell phones and I want to try to address his points as a way to help me clarify my own mixed feelings. His final point is the most important (by the way – read his whole post, don’t just take my highlights!):

If the answers to my tests can be looked up on Google, are they really worth asking in the first place?

I want my students to be creating, to be evaluating, to be synthesizing information.  I want them forming opinions and interpreting answers.  It would be great if they could determine the circumference of a circle from it’s diameter.

It would be better if they could tell me which of the given answers is the most reasonable estimate.

A smart phone can’t make judgement calls.  They can’t interpret answers.

If a smart phone can answer my test questions, I’m asking the wrong questions.

I agree 100% with these sentiments. When I first visited my current school I saw a chapel presentation that completely won me over. It was one of the 4 or 5 major reasons why I am here. Our  Reverend addressed these ideas and won me over. I do not think that this is the real reason why I worry about cell phones or other connectivity issues on assessments or in my class. Justin writes passionately about students doing what he wants (needs?) them to do while still being connected electronically through their phones or their headphones. What troubles me is a persistent belief that I have that we all benefit when everyone is engaged in class. The student who is doing solid math while wearing headphones is depriving their classmates of a strong voice and they are depriving themselves of the opportunity to explain their own thinking or to hear the thoughts of their classmates. I believe SO strongly that learning ought to be social and interactive. Maybe I am just inflating any logical concerns about relating to each other but that is where my heart and my head are right now. I don’t know how to balance what I want, what my students want, what I believe is best for the group as a whole, and the needs of the individuals. I know that there is a sweet spot there and that it almost certainly varies by class – hell, even by time of day.

I have asked my students to have their phones on their desks this year. We know that they are in the classroom and I don’t want surreptitious use in their laps. I ask them to look up stuff, I recognize that some of them use their phone as a rudimentary calculator. I don’t pretend that these don’t exist and I want to encourage honesty and openness about their presence in the classroom. Some students have complied while others have not. I speak patiently (but consistently) with those who keep them in their laps and text friends during class.

I know that I want my students to interact and I believe that they do less of it when they are plugged in to their phone or their headphones. I want students to research and solve challenging problems and I know that they do less of that when they are not connected to the internet through their phones or tablets or laptops. I chaired a committee at our school that helped develop a 1 – 1 program in our middle school. That program should soon bubble up to our high school. I believe in technology. I do, I think it improves learning and depend understanding. I am jealous of my students when I get to display complex ideas with Desmos or GeoGebra because I am old and did not even have rudimentary graphing technology available when I was trying to learn trig and calculus. I cannot tell if my visceral reactions to cell phones is at all logical and I am trying to sort that out. Justin – thanks for making me think and making me uncomfortable. Anyone else out there reading this – please poke at me through comments or through twitter (I am @mrdardy) I want to sort through these conflicts. I want to create an environment that is meaningful for my students AND for me. I sometimes feel like the grumpy old man yelling at kids on the lawn (even though I don’t have my own yard!) even though I don’t want to believe that is me.

sigh… This stuff is hard.

Great Follow Up Day

On Friday I wrote about a pretty terrific conversation that came up at the end of our BC class. We had tackled a particularly gruesome integral – (tan x)^5 and I had done so by repeated patience substitution and I chose to let u = sec x and look for combinations of sec x tan x as the du piece. One of my students stayed after and showed me his work where he shoe u = tan x and du = (sec x)^2 He was frustrated and told me that he spent about a half an hour trying to figure out why his answer was ‘wrong’. So, I typed up my solution and had it on one side of a page. (This document and the graphs I created are all linked on my last post from Friday.) I typed up the solution my student arrived at and placed it on the back of the page. I had my students working in teams so they each had my solution and their classmates’ solution in front of them simultaneously. I asked them to examine each of them and explain why they did not agree. I heard some pretty good conversations, most of them simply concentrating on making sure that they even followed each of the solutions. We had talked about it on Friday so it was good to hear them reflecting clearly on that experience. After a couple of minutes, one of my students announced that he proved that both solutions worked. I played dumb and asked what he was talking about. He explained very calmly that since one answer was based on even powers of tangent and one was based on even powers of secant, we could show that they were nearly equal. They seemed to differ only by a constant. I then showed them the Desmos graph I created and the GeoGebra graph I created. Both programs were happy with my solution and with my student’s solution. Neither program was happy with the difference between them. But I showed them that every x input we could guess at in the difference function yielded either an undefined answer or an answer of -0.75

I used this conversation with a number of goals in mind. I want them to get in the habit of talking to each other. I want them to see that there is not just ONE way to do math problems – especially ones as sophisticated as the ones we talk about in BC Calculus. I want them to think about graphs. I want them to utilize resources such as Wolfram Alpha, GeoGebra, and Desmos. I want them to notice and wonder about relationships. They are not yet where I want them to be in these terms, but the more often I remind them and the more often I model this behavior, then the more likely they are to adopt these behaviors.

If I did not believe this, I might not have the energy to keep on keeping on in this job. But I do believe it and I do keep on keeping on.

Thank you world of math resources for my students! Thank you world of recourses for me!!

We Broke Graphing Technology Again – A Success Story

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.