The State of the 2018 #MTBoS (to me)

Every so often, I like to take reflect on where I think we are as a community of math educators who collaborate online. Usually after attending TMC — a math conference run for and by math educators. But please forgive me in advance. This is the ramblings of someone writing late into the night and someone who isn’t revising. Musings, at best.

Taking Stock Previously

May 2009: Why Twitter?
November 2009: How BlogBuddies Became Friends
July 2012: 40 Choose 2 First Dates, or Initial Impressions of TMC12
March 2013: Some New Things On the Interwebs & HOLY COW What is Happening!
June 2013: My Thoughts on the #MTBoS
August 2013: TMC13: The State of Things For Me
July 2014: Teacher Growth, the MTBoS, and TMC14
July 2015: My Thoughts about the Evolution of the #MTBoS: 2015 Edition

But I haven’t taken stock for a while. Partly it was because I didn’t see much change in 2016 from what I wrote in 2015. And then I made a decision to consciously avoid jumping in the numerous conversations that happened during and after TMC17, although I had lots of thoughts. [1] So now it’s time to take stock again.

Taking Stock and Making Predictions: Teacher Leader Edition

The biggest thing by far that I’m noticing is that people in the community are becoming (and seeing themselves) as teacher leaders.

When I started teaching a bit over a decade ago, I didn’t see many pathways for classroom teachers to have huge impacts on math education. (That wasn’t something I was looking for, to be clear, it just wasn’t something that I saw in my world.) In the past five years, as social media has fueled things, our classrooms aren’t silos anymore and our practice have become more public, and as this community has grown bigger, I’ve seen the rise of various pathways that teachers can become teacher leaders.

There are many ways people can become leaders from within the MTBoS community (for example, Annie Perkins/Megan Schmidt/Dan Anderson/Justin Aion inspire playful mathematics through art for so many math teachers. And people are planning and running mini-TMCs and tweet ups. New things like Desmos’ Computational Layer come around and people like Jay Chow become the expert teacher of us teachers who want to learn it. There are always people running and presenting at the Global Math Department.) That has actually been something I’ve found true since the early days of the MTBoS. It’s just that there are more and more of them now as the community has grown. I’ve found that the MTBoS provides numerous ways for people to gain their voice as a teacher leader. Tweeting is one way, short and sweet. Blogging is another. And TMC also gives some opportunities for honing a face-to-face voice (from the short and sweet my favorites, to the 30 or 60 minute sessions, to the six hour morning sessions). In a small community which attempts to be supportive and people try to raise each other up, it is easier to identify and amplify your voice, see that others find what you have to offer valuable, and gain confidence in yourself.

But we now have community members that have become recognized as teacher leaders outside of our MTBoS community. For a long time, there was Dan Meyer who fit this definition. But now there are a group of people who are in that category — MTBoS people who have kinda broken out of the MTBoS in terms of their impact on math education. Fawn Nguyen is always traveling to give talks! Julie Reulbach publishes a post and almost 10K people get it in their inboxes! John Stevens and Matt Vaudry published a book around their approach to the classroom that resonates with people! Heck, so many books are out now or coming out now by MTBoS people *cough* Christopher Danielson! Chris Shore! Edmund Harriss! Denis Sheeran! Wait, so many white men! In any case, I’ve compiled a list of all MTBoS books that I know about here.) [2] In fact, there are now a small group of MTBoS people who seem to be making the conference circuit giving keynotes. There are so many people who are now presenting or even keynoting regularly at NCTM and other conferences. And there are a lot of MTBoS folk who are on NCTM committees. (I don’t know if I’m in the minority or majority, but I see NCTM as acting as a pretty great partner to this online math teacher community.) And probably most exciting to me, I’m reading more and more tweets of people in the community taking on math coaching roles for districts, or becoming department chairs! Teachers who love math teaching leading math teachers! The MTBoS cultivates internal leaders who then can become leaders more broadly. A lot of that has happened in the past three years.

I have done a lot of thinking about the rise of the teacher leader, and what even a teacher leader is. These thoughts started in 2015, and became more solidified as I started talking with Julie Reulbach in 2018 about her TMC keynote address which is specifically about teacher leadership. (YOU SHOULD WATCH IT!) I came to the conclusion that I personally would define a teacher leader broadly… as someone whose work has a positive impact on kids outside of their own classroom.

So that includes sharing materials/lessons with other teachers, supporting other teachers when they’re down, writing a book that changes someone’s practice, running (non-sucky) PD which pushes participants forward in their thinking about teaching, creating spaces and structures for teachers to reflect about their practice (ahem, shameless plug for The Virtual Conference of Mathematical Flavors that I’m hosting right now!), tweeting out answers to questions people ask, writing blogpost reflections about your classroom that inspire others, whatever. As Julie said in her talk, teacher leadership is not a ladder you climb. (I think it used to be.) You don’t have to leave the classroom to have that broader impact. And that a tweet that hits someone at the right time might be way more impactful and useful than a book. [3] This isn’t something “new” but it is something that I think people in the MTBoS are starting to be aware of.

With the smaller things we’re doing (cheerleading each other, having book chats online, pushing people to think about diversity and equity in their classrooms, etc.), we used to think of it as just helping each other out. But there’s a reframing that is happening. We’re doing the same things we’ve always been doing, but people are starting to recognize that this work is the work of a teacher leader. That, “OMG, I am actually a teacher leader… wait, no way… I am not… but… maybe I kinda am a little bit?” It might slowly be entering some MTBoS folk’s understanding of who they are. And that’s pretty awesome. There may be nothing different in what people are doing, but I sense the beginnings of a shift in how people are interpreting what they are doing. (And acknowledging and really accepting that what they are doing has real value to people.)

It’s like — I don’t know — I see the pathways for teachers to become leaders as numerous, crisscrossing, and some pathways are only just beginning to show themselves. Maybe vines on a trellis is the image that I’m trying to go for? All I know is that there are now a zillion different ways for someone to have a positive impact on kids outside of their own classroom… and the existence of the internet and the online math teacher community is making that even easier. I predict that there will be some cool ways people figure out how to have an impact outside of their classrooms that we can’t even image. Who knows? Find ways to get involved with new (math) teacher training programs — or create resources to help new math teachers? Argh. I literally can’t come up with good ideas… but that’s precisely the point. Who could have predicted 5 years ago that there would be a math space in the Minnesota State Fair?! I suspect in 5 years there will be a number of different awesome pathways that people will carve out for themselves to become a teacher leader in a way that aligns with their passion/interests that I totally wouldn’t have anticipated today. Which means we’re at a cusp. We’re still trying to figure out what precisely teacher leadership can look like. We have a number of examples that have emerged. But we’re likely going to see a bunch more.

We have people who are in math ed and also part and parcel of our MTBoS community. Specifically, I’m thinking of Lani Horn and Tracy Zager. Their books Motivated and Becoming the Math Teacher You Wish You’d Had tout the work of people in the MTBoS do and outline pedagogy that most everyone in the community can rally behind. But also their work also engages deeply with the idea of identity and status. And importantly, although in the past I saw certain beams of light here and there emerge about the idea of identity and status in the classroom (in the vein of complex instruction), this kind of thinking is becoming much more normal in the conversations that are being had in the MTBoS. More and more, I think people (myself totally included) that the basis of an awesome classroom has to be rooted in consideration of these things. Thinking about “how to teach completing the square,” for example, isn’t just about scaffolding and conceptual development of the idea. Being able to do that well is also predicated on building an atmosphere in the classroom that the scaffolding and conceptual development can happen within. I think more and more people (myself included) are starting to recognize this. Like Lani and Tracy do, I predict more people in math education are going to start engaging with our community, as they will come to see us as a powerful resource, and we will engage with them if we see them as an ally in our classroom work.

In general, the MTBoS conversations aren’t entirely consumed with content or sharing resources anymore. It hasn’t been that for a while. Broader discussions about identity, status, diversity, equity, social justice are now more commonplace. Not everyone is engaging in these conversations, but they’re happening and I bet this is going to turn into another pathway that the MTBoS will be creating teacher leaders that are recognized outside of the community. And I’ve seen that these conversations of identity, status, diversity, equity, social justice have been turned back onto us. Not just things to consider for our classroom community, but also for us to consider as an online math teacher community. Right now there is a small but growing recognition that we can’t brush these conversations under the rug — for the sake of our students, and for the sake of our own community. 

I don’t quite know what to make of this… but there are a ton of #MTBoS classroom teachers that have left the classroom to work full time at Desmos or Illustrative Mathematics. I went to dinner with a group of IM people in Cleveland recently and there were only two people at the giant table who I wasn’t friends with from the #MTBoS. Similarly, I know a lot of the people working at Desmos (and I heart them all!). Although I don’t know what to make of this, what I can say is that these people are working at places I believe in, and that will have a large-scale positive impact on math education. And I probably can say that if it wasn’t for the MTBoS, many of them might not have this opportunity that they decided to take.


This post feels incomplete. I feel like I have so much more to think about and write. When rereading this post before I #pushsend, the ideas seem inchoate, and somewhat incoherent, to me. I can’t underline anything and say “THIS. THIS IS WHAT I’M TRYING TO GET AT.” Usually when I sit down to write something like this, all the blathering helps me refine my thinking and gets me to clarify what I really think. And at some point, I usually hit upon that thing that makes me feel excited and like I have found something TRUE and A REAL INSIGHT. But I am not leaving this post feeling any more lucid. No big insights emerged. So I suppose I leave this post as late night musings, as I started it as.


[1] If you don’t know what this is all about, but me being all cryptic has piqued your interest, you can get a flavor of it by reading these two posts: here and here.  This moment was important for me in one huge way. It highlighted how scaling up the online math teaching community without a central authority could lead to problems with intentionality/messaging/impact. How does a decentralized community with community-builders who are volunteers grow bigger with a coherent set of values and vision and positivity that are not only agreed upon but also enacted? Where intentionality in things that happen have a place? I don’t have answers.

[2] In 2015, I noticed there were “brands” (for lack of a better word). I still see that. The number of people with a narrow/specific idea/vision and a coherent voice that really resonates with people is growing. These people are gaining a lot of cache outside of the MTBoS. Often times, these “brands” are related to projects that were just gaining traction in 2015 (like Visual Patterns or Open Middle or Which One Doesn’t Belong or Clothesline Math or …). Conferences, books, at the MTBoS spread the word of these “brands.” And now from their rising popularity in 2015, they’ve become something bigger in math education. And with that, they carved out one pathway of “teacher leadership” that didn’t really exist before. But a number of people seem to be following it in some form. And that’s cool to see.

[3] But what I like about the MTBoS is that, taking my definition of teacher leader, it’s all a community of teachers who are trying to support each other. And so in the ways we all prop each other up, push each others’ thinking, share resources, etc., we all are having a positive impact on each others’ classrooms and each others’ kids. And so we all in small ways can consider ourselves teacher leaders. I love that it’s not a zero-sum game.

I wrote this and it doesn’t quite fit above, but I didn’t want to delete it. So I’m sticking it at the bottom of this post here. When I started teaching, I wanted to become a “master teacher.” That was my goal. Feel like I know I’m an expert in what I’m doing. I have since realized I will never consider myself a master teacher. That’s probably due to my own psychological makeup. But in the #MTBoS we have this idea that “everyone has valuable things to share.” And I truly believe that. But I also wonder if — now that I’m seeing so many ways people teach —  I don’t believe in the idea of a master teacher anymore? As opposed to my first few years of teaching,  I now don’t really think about it at all. Because over the years in the MTBoS I’ve see so many different ways people reach their kids, I recently realized that I don’t have a single ideal/standard to live up to. There are so many ways to be an awesome teacher. (Hence, my virtual conference of mathematical flavors was born. I wanted to showcase this diversity!) I just have to figure out who I am as a teacher as best as I can and just keep on trucking.



Senior Letter 2017-2018

Every year that I’ve been teaching, And at the end of each year, when I start growing wistful (but also a little bit glad to get them out of school because their second semester-ness starts to take over), I write them a letter which I give to them on the last day of classes. The letter usually always says the same thing, hits on similar themes, but I write it from the heart.

Sometimes I remember to post it on my blog, sometimes I don’t. This year (obvs) I remembered!

Even though they got to be a bit punchy at the end of the year, I’m kinda missing my seniors right now.


Part III: The Final Turn

I was in the middle of writing how I made the transition from Anti-Derivatives to Integrals, and then I got swept up with school things. So a month and a half later, at the start of my summer vacay (!), I’m going to write up the very last post in this installment, where I make The Final Turn.

As a recap: I taught anti-derivatives as “the backwards question” of derivatives, but didn’t give them any meaning other than a mathematical puzzle. And my first post (Part I) starts when I tell kids we’re going to put anti-derivatives aside for a while and we’re going to think about a road trip. In this, students recognize that velocity graphs can tell you about displacement, and we are forced to contend with left handed and right handed riemann sums. The conclusion is that “the displacement of an object can be calculated by estimating the area under a velocity graph.” Importantly, for kids, this has nothing to do with anti-derivatives. In my second post (Part II), students explore this idea in more detail. They are given a bunch of step functions for velocity and they come to recognize the difference between distance and displacement, and also why we need to specify a starting position to know an ending position. They also see the need for negative signed area to represent moving backwards. Again, this has nothing to do with anti-derivatives.

This is where our story begins. The final part of this tale, where we connect signed areas (which students are comfortable with) to anti-derivatives in such a way that the connection makes sense and is actually kinda obvious. That was my goal, anyway.

At the end of the same packet students were working on in Part II, I gave the following velocity and position graphs. Like my students, you can probably see that the velocity graphs are approximations of “nice” curves (e.g. quadratic, square root, linear, sine, exponential, cubic).


After being asked to identify some specific things on each graph (find what happens to the corresponding position piece when the velocity was greatest… find what happens to the corresponding position piece when the velocity was zero… etc.), I asked students to write six different things they noticed/observed. Kids came up with great observations! Here are some of them taken from their nightly work (note: “palley” is our class terminology for peak or valley):

  • Zero on the velocity graph yields a slope of 0 on the position graph (the object didn’t move for that time period)
  • Most positive value on the velocity graph yields the largest positive slope on the position graph
  • Most positive value on the velocity graph corresponded with the largest change in position (from low to high) on the position graph
  • The position graph usually has one more palley than the velocity graph (when we’re dealing with polynomial velocity graphs)
  • When the velocity graph is negative, the position graph is decreasing
  • When the velocity graph is positive, the position graph is increasing
  • For the 4th graph, the velocity graph looks like a sine graph and the position graph looks like a cosine graph
  • Many of the relationships between graphs look similar to derivatives we’ve been studying
  • At the point of highest velocity, the position graph usually has a palley
  • The area under the velocity graph gives the height of the position graph
  • Velocity graph is the derivative of the position graph

Ooooh! I just found the smartboard page where we codified these in one class:


To debrief this, I had kids call out their observations and we talked about them/refined them using the graphs I was projecting at the front of the room. The most important thing I remember doing was making sure we started with “simple” observations and only then had the class move to more “complex” observations. That way kids who only noticed basic things would have a chance to voice their thinking.

From this we finally got to the conclusion we were looking for: the velocity graph is the derivative of the position graph. I had to nudge kids to make the counter-observation they get for free: the position graph is the anti-derivative of the velocity graph.

We saw all the things we had previously learned when studying the shape of a graph come out (e.g. when the derivative is negative, the original function is decreasing; when the derivative was zero, the original function often had a “palley”). Everything fit together!

We then carefully looked at small pieces of one of the graphs and discussed what we saw going from left to right, and also going from right to left.


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From left to right: we saw that the signed area under the curve on the left corresponded to the change in position on the right.

From right to left: we saw that the slopiness of the curve on the right corresponded with the value/height of the function on the left.

So we codified this. But importantly, we finally got to say “antiderivative” but with meaning. HERE’S THE KEY TURN: We knew that the derivative of position was velocity. That means by definition, the anti-derivative of velocity was position (ok, technically displacement, but let’s not cut hairs right now). But we already knew from our work previously that to get from velocity to position, we had to find signed areas. THUS, accumulated signed areas and anti-derivatives were one and the same!!!

In one class, we outlined this as such.


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In my other class, I had us codify things more formally.


Both ways of codifying this knowledge seemed to make sense for kids. And to be perfectly honest, the fact that antiderivatives were accumulated signed areas was always a bit magical to me. I know I truly understand something conceptually when something magical/mysterious changes and becomes a “well, duh, of course it has to be that way.” I don’t think in all my years of doing and teaching calculus that I had ever totally gotten rid of that feeling of magic/mystery until this year, when I really changed how I approached this with my kids.

Now I know you might be saying: but Sam, you never PROVED this relationship, that an antiderivative of a function and the signed area under that function were the same. Agreed. I wanted to have kids see and make sense of this relationship first. But I did do a basic proof of this (that the antiderivative of a function is the same as the accumulated signed area of that function) soon after. If you’re dying to know what came after, I had kids use a Riemann Sum applet (I slightly modified this one to not have the integral symbol on it or the “show actual area” button on it) to fill this sheet out [docx version: 2018-05-03 Riemann Sums]:

And at the very end of this packet is our “proof”:



Now this page/proof should have appeared super familiar to my kids. We actually had made this exact same argument earlier when seeing the derivative of the volume of the sphere is the surface area of the sphere, and when seeing the derivative of the area of a circle is the circumference of the circle (see post here). Although this proof did require us to work together as a class with a bit more handholding than I would have liked, it went pretty well.

And with that, I can put this three part beast to bed. If nothing else, this series just reminds me how much I really think about how students build and construct knowledge. It is something that I think about so naturally when designing new materials, but to explain all that thinking and how it unfolds in the classroom takes way more words than I anticpated! No more three part blog post series for a while!


Polar. Graph. Contest.

Here’s what I hung up last week:


Here’s a closeup of some of them…



These are polar graphs that students designed using Desmos. Then I printed them out on photopaper and hung them up.

This was something I wanted to do after introducing polar graphing. Why? Because one day during the polar unit, I started playing around with desmos and accidentally created:


… from something so simple …


(Now to be fair, desmos isn’t great with creating great complicated polar graphs… and it’s better to write them parametrically to get a bit more accuracy… so this is a bit of a lie of a graph in that it isn’t totally accurate… but it’s oh so pretty.) [1]

So after our unit on polar graphing, I took 10 minutes at the start of class to introduce this idea of a Polar. Graph. Contest. First I threw this image up:


I then pulled up desmos and asked my kids to shout out some polar function. I graphed it. Then I put in a slider or two. So for example, if they said r=\cos(\theta), I might have added the slider r=\cos(a\theta)+b. And then I started changing the sliders. Then I might have altered the function a bit more, like r=\cos(a\theta)+b\theta and we saw what happened. Then I gave everyone 7 minutes to just come up with something pretty.

It was magical.

Kids just started playing. They dug into old functions they had learned about. They got excited by what they were seeing. They gasped and turned their screens to show their friends. Some who were getting boring graphs saw the cool graphs their classmates were getting and were inspired to mix things up since they knew they could make neat things. #mathjoy in the house.

My heart was singing.

Then I showed kids a google doc which had all the info for the contest — and the link to the google form to submit their entries. There were initially two contests. Students needed to create the coolest polar graph with one equation. And students needed to create the coolest polar art using multiple equations. However, some students were animating the sliders and coming up with fun animations (like this or this… watch both for a while). So I added an optional third animated polar graph category.

I haven’t yet told my kids who the winners are. I want to just let them appreciate the work of their classmates for now.

After creating the bulletin board, I’ve seen kids look at the artwork. Kids from my class, but also kids from other grades. And what I’ve found fascinating is that so far, very few kids pick the same polar art pieces as their favorites. I expected everyone to love the same ones I do. But it just isn’t the case. I think when I announce the winners, I’ll have the class go to the board, have everyone point out a few that they like, and then I’ll make my grand pronouncement.

Student Feedback:

I asked my kids, when submitting their artwork, “This is something new I came up with this year. I want to know if you enjoyed doing it or not. No judgments if you didn’t. Y’all tend to be honest when I ask for feedback, and I appreciate it! I genuinely want to know. I also am a bit curious if you had any mathematical thought as you were playing on Desmos? You don’t have to say what thoughts you had (if any) — just if you had any.”

Every student responded positively. Some responses included:

  • I think this was so awesome! I love art and this felt like art to me. It is so fun when art and math intersect, I loved it!!!!!
  • Messing around with the graphs was actually more entertaining than I thought it would be. I spent a lot more time on this than I thought I would, and I feel like I’ll probably spend more time on this trying to find a really cool design I like (and possibly gaining a better understanding why the graphs look the way they do…).
  • I had so much more fun doing this than I thought I would, honestly. Once I finished my multi-equation graph, I looked at the clock on my computer and realized I had been working on it for nearly 20 minutes; it had seemed like maybe 5.
  • I really enjoyed doing this assignment. I felt that I learned a lot about polar through it. I didn’t think too much about math while making my graphs, however I thought about math a lot in order to observe and think about patterns I found in my graphs.

Now I want to be frank: there isn’t much “learning” that happens when kids are doing this assignment. This isn’t a way to teach polar. But it is a way to get kids to appreciate the power of polar when they are done working with polar, and what sorts of different kinds of graphs compared to the boring ol’ rectangular coordinate system. I just wanted kids to play, like I played, and get excited, like I got excited. It’s a slightly different way to appreciate the power of math, and I am good for that. Especially since it only took 10 minutes of classtime!


As an aside, I love that when I tweeted this out, a tweep said he was going to be doing this in his class after his kids learn about circles. Um, hell yeah!



[1] So there are two ways to graph polar in desmos. First is the straight up polar way, and the second is the parametric way. It turns out that the polar way is solid for most things, but it loses refinement at times. Let me show an example. If we graphed r=\cos(57\theta), we should get a flower with 57 petals.

And happily, if we graphed it in both polar and parametric, we get the same looking graph:


However if we zoom in a bunch, we can see that the red graph (the polar equation) is interesting and stunning, but just isn’t correct. While the zoomed in blue graph (the parametric equation) is more boring, but is technically correct.


It turns out desmos samples more points using parametrics than polar.

As a result, a few of the polar artworks my kids made aren’t “true.” Their pieces are a desmos quirk, like the red graph is above. But what a lovely desmos quirk.

Part II: Transition from Anti-Derivatives to Integrals

In my last post, I outlined the road trip scenario I used to prime kids to think about areas under curves. It had nothing to do with anti-derivatives. And that’s important to keep in mind. This post is going to try to outline how we made the intellectual leap from areas under curves to anti-derivatives. [Update: I wrote this during 40 minutes of free time I had in school today where I didn’t want to do other things I was supposed to be doing. So I didn’t get to writing the part of the lesson where we make The Leap. This post ends where we’re literally all primed to make the leap. But indeed! I will make the leap with you! But in Part III. Which will be written. Soon. I hope.]

The road trip introduced this idea that kids can approximate how far someone traveled using a left-right-midpoint Riemann Sum approximation (we did not give it that name…). It arose naturally from the roadtrip scenario.


We also made the conclusion that if we had more data, we could get a better approximation for how far someone traveled. To remind you, we started with this data:


and then we got more data:


That’s going to be our transition. We are now going to give infinite velocity data! [1]


Wonderfully, kids had no problem with doing this. The reason I highlighted question 1c is because I was very intentional about including that question. When students graphed 1e, they often would draw:


I didn’t correct anyone while they were working. And it was nice to hear a few groups have the requisite conversation about why we needed to connect the points. Afterwards, when we debriefed as a whole, this was something I highlighted. We knew the position at every moment in time, including at t=2.31 (as asked in 1c).

Kids continued on with Questions 2, 3, and 4. They flew through these, actually.


These were golden. Let me say that again: these were golden. [2] It was amazing to watch kids:

  • Parse the connection between velocity graphs and position graphs
  • Understand the idea of negative velocity
  • Think about the fact that we have to specify an initial position in order to create a position graph
  • Draw a connection between motion on a number line and a graph of position v. time
  • Understand what distance and displacement are, and see the difference between the two

Seriously, just watching kids work through these problems was… well, I’ll just say this feels like something I’m super proud of creating. It didn’t take much time to do but gave us so much fodder.

We didn’t need a lot of time to debrief these four questions. I had students highlight a few things, and I made sure we brought up the fact that we were drawing line segments for the position graphs, and not something curvy. Because constant velocity means position is changing at a constant rate, it’s linear. So for example, the position graph for 3c would look like what I have below. It isn’t a parabola.


But there was one huge thing we had to go over. With the roadtrip, we drew a connection between area and how far someone went. Most kids, as they were doing these problems, didn’t think about area. I wanted kids to think about area. So in our debrief, I explicitly asked them what our huge insight from the roadtrip was (area as distance travelled!), and if we could apply it to one of the problems.

So first I went back to question 2c. And I asked students how they calculated their answer:

pic6.pngKids said she went backwards a total of 24 units. So they did 100-24. And then I explicitly had them draw the connection to what we did with the roadtrip. This is when we talked about it being area, but “signed” to represent the backwardsness.


To be clear, some students had already been thinking in this way (about area/signed area) when working on these problems. But most hadn’t been, so we had to bring that idea to the forefront.

Then I had a student talk through Question 3 with areas in mind:


And finally, I asked groups to discuss how we could understand distance and displacement from the velocity graph.



There is one more part to this packet I had my kids work on that I will outline in my next blogpost in the series! But here’s an editable .docx of the file I made [2018-05-02 Velocity Graphs]. And here’s the document to view here:

Stay tuned for Part III.


[1] Both @calcdave and I stumbled upon the same approach for this!

[2] The only note is that a few students didn’t realize the time interval was 1/2 hour for Question 4. And it involves a fair amount of calculation.


Part I: Transition from Anti-Derivatives to Integrals

An Important Prelude: On one of the early days in my calculus class, I have groups imagine a roadtrip. It’s one of the very best things I do, because it problematizes the idea of how something can be going at a particular speed at a moment in time. Like a speed involves a rate of change of position in a time interval, but we don’t have a time interval at a moment in time. So saying something like “we were going at 58mph at 2:03pm” suddenly goes from a statement kids accept to a problematic statement. And at that moment, we’re ready for calculus.

Setting the Stage Now: In calculus, I had gotten my kids to take tons of derivatives, and then taught them about antiderivatives and how to take them. But of course the whole time they were doing antiderivatives, they were asking “but why are we doing this?” I got this question a few times, but I just said “you’ll see very soon… probably next week… I just want to get some of this algebraic thinking out of the way so we can focus on concepts.” [1]

The Problem: I looked back at what I’ve done in the past about how to draw the connection between the antiderivative and signed areas together, and had a bunch of pretty terrible methods. I didn’t like any of them. So I put out a call on twitter while brainstorming myself. [2] I got a ton of responses, three of which stood out to me. The first one is something I kinda did at the start of the year already (before we did any calculus), so I’m excited for when my kids see the connection…


The second came was from the appropriately named @calcdave:


What’s incredible is that when I tweeted out asking for help, I had already started brainstorming … and it was so amazing that we had such similar ideas that I had to take a photo of my scratch work to send David:


I also got a DM from Brett Parker (@parkermathed) with a screenshot of how he digs into the idea. And what was his idea?


Yup. Driving in a car. Which of course got me thinking of the start of the year when I basically transitioned to derivatives using a road trip.

What I did: Part I

So I told kids we were going to put a pin in anti-derivatives for a short while to go on a short detour. And I used Brett’s setup, but changed it to be a followup from the roadtrip we had talked about earlier this year. We read the setup together, and then I gave each group a few minutes to talk about part (a).

sheet1Surprisingly, this was not an easy question for kids. Many didn’t instantly think distance=(rate)(time). Additionally they didn’t know what assumption to make about the speed for the minute that passed between 7:10 and 7:11. I emphasized the approximate part of the question, and really told kids they would need to make an assumption.

When we debriefed, most kids suggested Alex was probably driving 69mph for the minute, so they did 69\cdot\frac{1}{60}.

Some suggested he was maybe going 70mph for most of the minute, so they suggested 70\cdot\frac{1}{60}. And with that, I suggested that maybe he was going 68mph for most of the minute… We did the calculations for all three and saw they were pretty similar.

notes1.png(We also had discussed that in that minute, perhaps Alex started at 70mph, went to 100mph, and then slowed down to 68mph… We just didn’t know. So we were making and using an assumption, but one that is pretty reasonable.)

Then kids in their groups went to the next three parts. And each group was assigned an assumption: average speed, left hand speed, or right hand speed.


The only errors I saw kids make in part (b) was not taking the different time intervals into account. Since we did one example which had a time interval of \frac{1}{60} hour, some groups were using that time interval for everything. But pretty rapidly most kids got there.


In our debrief, I wrote out the calculation kids did for the left hand assumption, and then asked students what I would have to change to do the calculation for the right hand assumption. (That was written in yellow.) This question was a key question to ensure kids understood the difference.

Then we talked why this was an approximation? (We had to make assumptions about the speed of the car at all the times inbetween the times we were given.) And then kids said that to get a more accurate distance for Alex, we’d need more data.

And that’s precisely what they got when they flipped the sheet over.

I assigned some groups the left hand assumption, and some groups the right hand assumption. I thought with so many data points, kids would be like “argh! I have to do all these calculations!” But no, there were no audible groans (I don’t think…) And they plotted the points and then did the calculations. I told kids to write out all their calculations instead of just saying how far Alex drove in the first minute, the second minute, the third minute, etc. No everyone listened. Shame. They lost out. Because when we debriefed, we saw:


Yup. The fact we could factor out the 1/60th is key. It made this all go so much faster. Only a few kids noticed that when doing the calculations. And then we compared that with the right hand assumption:


The same answer! Why? Kids saw that the sum inside the brakets (after factoring out the 1/60) would be the same because the starting speed and ending speed were both the same.

So we’re done, right? NO. We had two more moves to make.

First, we looked at the graph.


And I asked: “if we were doing the left hand assumption, what would our velocity graph look like?” And we concluded that for each 1 minute interval, Alex was driving at a constant rate. So it would look like this:


Second, I asked: Between 7:05 and 7:06, how much are we assuming Alex went? (Kids answered 67\cdot\frac{1}{60}) Here’s where I did some talking. I could probably have asked kids to think geometrically and had them come up with this, but I was running out of time.

I said: “where do we see 67 represented on this graph?” (Kids said the height of the velocity graph from 7:05 to 7:06.) I then said: “where do we see the \frac{1}{60} on this graph?” (Kids said the 1 minute length.) So I drew a rectangle. Slowly. And then shaded it in. Slowly. And turned around. Slowly.


Yes. Audible gasps from many. I then said: “What about 7:04 to 7:05? What was our calculation for the distance Alex traveled?” And after a few more, we saw:


Kids saw. Heck yeah. They got that the approximate distance that Alex traveled was algebraically calculated in one way. And then they saw that number had a graphical representation. It was awesome.

I left by showing the original velocity points graphed. Reminded them of our left hand assumption, and that it was just an assumption, though a pretty good one. And then I drew the curve below. And dropped the microphonic device and left.


All in all, this was a pretty detailed blow-by-blow. And since I did it in two different classes, and things unfolded slightly differently in each, this is an amalgam of what happened. But it’s a pretty solid recap of the story I wanted to tell and how it was told. (And it’s a testament to the help of my (twitter) frands.) Most of the time, students were working. But a lot of great conversations happened as a whole class. It is a long post, but the question we worked on was only one page [editable version to download: 2018-04-30 A Road Trip Reprise] And it probably only took 20 minutes total from start to finish. A really exciting 20 minutes for me.

Stay tuned for how I used the idea that @calcdave and I both stumbled upon to make the connection between area and anti-derivatives. Right now kids have seen that there is a connection between area under a velocity curve and the distance someone has traveled. There is still no connection to anti-derivatives. That’s coming up.

[1] To be honest, I made the decision years ago to do tons of antiderivatives before introducing the integral. I wanted all the algebraic work done and solid before we introduced the concept of the integral. I didn’t want kids messing with the idea of signed area, new notation, and tough antiderivatives all at the same time. I still kind of think this is the right decision, but some doubts have crept in. I just really hate when I have to say “you’ll see why… promise…”

[2] Apparently I asked this on my blog ages ago in 2009.

A simple vector problem with a rich set of approaches

In precalculus, we do a little bit with vectors. And last year and this year, I gave a basic problem to my kids. (I think I found it in some standard textbook.) What I love about this problem is that there are so many ways kids approached it. All essentially the same, but all different enough. Because my kids weren’t all doing it the same way, it has shown me that we are teaching them well. And also, it reminds me that a super basic problem can be a super rich problem.

vector problem

Approach 1: Since we were in vector land, a few kids solved it like this. They found the vector from P to Q, and then set the magnitude equal to 5 and did the algebra.


Approach 2: Similarly, some students just used the distance formula that they had memorized.


I like that this kid expanded out (-3-x)^2 and then later eventually factored. Because that was slightly different than how the student in approach 1 solved it (leaving (x+3)^2 as is, and then taking the square root of both sides.

Approach 3: One student found the equation of a circle of radius 5 around the point (-3,1). Then they realized that they were looking for the solution to a system of equations for the circle and the line y=4. So they substituted y=4 into the equation of the circle and solved!


Approach 4: Most students took a geometric/visual approach. They drew the point (-3,1) and the line y=4. Then they drew these two triangles (seeing that the vertical distance from the point to the line was 3 and the diagonal distance from the point to the line was 5, since we want a distance of 5 away from (-3,1). Then they used 3-4-5 right triangles to get the horizontal distance.


All of these were lovely. I enjoyed seeing them all together and drawing some connections among them. Most kids were awed when they saw Approach 3. And since so many students didn’t take a straight up algebraic approach, they were like “ooooh” when they saw Approach 1/2. I supposed what I most like about this is that it really highlights how circles, distance, and vectors are all essentially tied together. I mean kids should know that circles and distance are fundamentally related (but of course they don’t always remember that). But this problem connects those two concepts with something new: vectors. And that the magnitude of vectors simply being an equation involving a circle, secretly.


Actually, while I’m writing this, I might as well share this other problem that had a couple of approaches. This was the basic question:


Approach 1: Most students took this approach. They drew the vector, and then drew a smaller vector with unit length, and then used similarity to find this new vector (with a scale factor of \frac{1}{\sqrt{5}}.


Related to this were students who simply saw the scalar that was multiplied by the vector <5,12> to be a “scale factor” that stretches/shrinks the vector by a particular factor. But why this works is because of this similar triangles argument.

Approach 2: A bunch of students used trig. They first found “the angle” and then realized that angle put on a unit circle would work! The fact that so many students saw the problem this way made me happy. I then asked what if we wanted the vector to have length 2 or 3 (instead of unit length), and they were able to answer it. We also talked about one huge deficit of this approach: you lose exactitude since they approximated the angle with their calculator. Even if they didn’t round, they wouldn’t “see” the square root of 5 pop out, when they would with the similarity argument.


Approach 3: Okay, so strangely this year, none of my students used this approach. But it is related to the similar triangles approach, and in years past, I’ve had students come up with it. So I showed it to them so they could see another approach. It’s an algebraic approach to find the scale factor.