# Why a @Desmos Activity?

This post is really more a process of me thinking something through than anything else. However, if I missed something, I’d like to hear about it, so I’m going ahead and publishing this.

It’s summer, so I’m revising my courses. I’ve been looking at a lot of resources, and this morning was devoted to Desmos Activities.

I’ve used Desmos activities in the past. I’ve made some Desmos activities. There are things for which they are excellent. Having students write the equations of graphs, for instance – it’s a lot easier to hand them a Desmos activity than print out or project a bunch of graphs. It reduces the work involved in checking their work and lets them spend more time on thinking. Those are good goals, and Desmos helps me achieve them.

I’ve got a list of Desmos activities that I want to incorporate into my class this year. I’m especially impressed with activities on how to graph. Polygraph and Marbleslides both encourage good thinking and they’re fun. That’s a rare combination, and it’s important. Really well done.

As I looked through the calculus activities this morning, I noticed a few worth incorporating. The Daylight Hours is a really nice extension that covers some worthwhile calculus while making a nice connection to the physical world. The Rate of Change of the Exponential Function could be good.

But with many activities, I found myself thinking, “Why would I use this instead of what I’ve been doing?” Take the Functions Defined by Integrals activity. (You can see it here.) I could have every student pull up their laptop and work through the activity, and it would be fine. It’s a good activity. Or I could do what I normally do: throw a graph onto the board, ask students to find the area between x = 2 and a bunch of other x values, then talk about how what we really have here is a function. That would be fine, too.

The activity has students talking to their neighbors and responding to other ideas. But I can do that at the board by giving them a question and two minutes to talk to their neighbors before calling them back for a class discussion.

The activity uses different colors to emphasize the difference between positive and negative area. I can do that on the board.

The activity has students estimate. I can do that at the board.

The activity has some good questions that I hadn’t thought of. I could ask them at the board.

The activity has students figure out the notation on their own. I could have them do that on paper silently, then compare with a neighbor, then talk as a class.

The activity has students collect data about what the area is. This decreases the calculation burden and frees them up to do more thinking…but this topic falls at a time of the year where I want my kids to get some more practice with calculating areas. (Whether or not you agree with that as a good instructional goal is fine; I don’t even know if I agree. It’ll be on the AP exam, so we’re practicing it.) So while in general that’s a good goal, I’m not sure it’s a goal that I want to pursue.

The activity also draws in the graph of the function defined by in an integral, which is nice. That stands out to me as one of the nicest moments of the activity, actually. But…I could do that at the board. I could even have the kids use the data we’ve collected to sketch what they think the graph of the function will look like, which raises the cognitive level of the task. Although now that Desmos has a sketch feature, they could incorporate that into their activity.

The activity lets me see what every student has written in response. That’s nice, and something that’s hard to do in a classroom. That’s probably the biggest advantage of doing an activity – I can see everyone’s response instead of whatever percentage my teacher ears catch. But…I only have about 12 students in my class. I can hear most of what they say, and I can tell from their body language when I need to wander over and be closer to them.

The downside is that I’ll have students write “I don’t know” and move on. Yes, I can see the “I don’t knows” on the dashboard, but if my eyes are on my screen, then they aren’t on my students. That means I’ll miss the facial cues and body language that tells me how they feel about this.

The other downside is that it’s digital. My students spend a lot of time staring at screens. I don’t want to add to that time unless it accomplishes something I can’t easily do another way. Also, my students often lack the discipline to stay on the screen I want them to. As soon as I ask them to open up Desmos, I know that Twitter, Instagram, the physics textbook, or email will also be opened. So the Desmos activity really has to be special to make it worth those distractions.

There’s a lot of buzz around Desmos. And don’t get me wrong, I really like Desmos. We use it probably once a week in my precal class because it is so. much. better. than a TI-84. I am not in any way trying to bash Desmos.

But I’m not yet sold on the activities. I definitely like that they’re available. I have and will use them in class. There are other times where I’ll look at them and modify my non-digital lesson, like this one caused me to do. Seeing the thinking that the Desmos activities try to provoke, especially the ones written by the Desmos Teaching Faculty, challenges and improves my teaching. I love that there’s a library of things I can look at. Even if I don’t use them as is, they’re still useful.

Why am I writing all this? To criticize Desmos? No! Emphatically no! Like I said at the beginning, I want to do more Desmos activities this year! I’m writing this as a reminder to myself.

I’ve noticed a tendency in my thinking to want to scrap what I’ve spent the last few years building and replace it almost exclusively with tons of Desmos Activities and 3-Acts and Underground Mathematics questions and those sorts of things.

All these things are tools. Desmos is a tool. A useful tool, a powerful tool, a helpful tool. Maybe it’s a nail gun. There are times where a nail gun really is the best tool for the job. But sometimes screws work just as well as nails, and I can use the drill I’m comfortable with – with some improved techniques by watching the nail gun – instead of switching.

So, while I’m revising my curriculum this year, I need to remember not to go overboard. Some of my lessons need to go. Some of them need to be tweaked. And some of them are good as is.

Unless, of course, there’s some major upside to this activity that I’m missing. That’s entirely possible, and if I missed something, I’d really like to know. Desmos guys? Will you let me know what I missed?

# U-Substitution is Like a Bad Rebound

We’ve been doing u-substitution for the last couple of days in Calculus. Too soon to tell, but I think it’s going ok.

Friday, we started. We had already talked about how antiderivative rules are just derivative rules, but backwards. So, we reviewed the chain rule. Then I wrote $\int f(g(x)g'(x)dx = F(g(x)) + C$ on the board, and we talked about what each part meant. Next up, examples: $\int \sqrt{3x^2-2} 6x dx$. What’s g? g’? f? Ok, now we find F, plug in g, and add C. You try some! (I like starting with square roots because it’s really obvious what the “inside” function is. Exponents are also good. I think in the past, I’ve had them integrate powers of 2x-1 until the powers were so high that expansion became impractical. I skipped that this year, but I may go back to it next year.)

I’m pretty sure this is how I was taught u-substitution at first. It’s how my textbook lays it out: do pattern matching, then get into actual substitution. I’ve never taught it that way, but it seemed like it was time. It seemed to help.

So, after they do a problem or two, I show them a trick: u-substitution. We rework the original example, and then they rework their previous examples. For the rest of the day, I throw up a bunch of problems on the board, and they work through them. That was Friday.

I find that a lot of students, in all areas of math, don’t know when they’re done. When have they answered the question? With u-substitution, once they’ve found the anti derivative, they could be done. It’s really just convention that we switch everything back to x. It is, however, the convention, so they have to do it. So, once we’ve actually found the antiderivative, I find these words coming out of my mouth:

“Ok, so we found the antiderivative. But, see, u-substitution is like a bad rebound. I got rid of my x and found u. I don’t want my x anymore; that’s all behind me – I want u! And things are going great, but suddenly I realize, I don’t actually want u. I want my x back. So I’m going to ditch u and go back to my x.”

It never gets more than a few chuckles, but they do remember.

I didn’t see them again until Wednesday, when I had them for two hours. There were a few questions on the homework, but most of them were about places where I’d messed up on the key. That’s always a good sign – they understand what’s going on, and are confused by mistakes. A better sign is when they realize that those are mistakes and email me about it, but this is still good.

We wrapped our our tale on Wednesday: dealing with situations where the integrand needs some massaging, like $\int x \sqrt{2x-1} dx$, and then dealing with definite integrals. I was pretty happy that the class figured out what to do and I didn’t have to tell them. I was even happier about what happened a few minutes later when they were trying a few problems themselves.

A tardy student walked in. Everything we’d done was still on the board. He pulled out his notes, and I walked over and explained what was going on on the board. He nodded like it all made sense, so I left him to try it on his own. Later in the hour, I walked past his table, and asked his neighbor how it was going. She said something like, “I’ve got the second and third, but I’m having trouble with the first.” I was about to open my mouth and start talking when I heard him say, “Oh, I’ve got the first. Here, do you want to see?” I walk off quietly and have a happy dance on the inside. I’ll take peer teaching over mine just about any day of the week; the fact that he got there without even being around for my original overview made it even sweeter.

Because this was a Wednesday, I had two hours, which was probably about 30 minutes more than I used. That wasn’t bad – they started their homework, which meant they could work on it and ask me questions while I’m still there. But I feel like the lesson could have gone deeper. We got into some questions about what we’re actually doing when we find the new bounds with a definite integral, and I found myself talking about how when we do a u-substitution, we’re transforming the horizontal axis, and we’re finding the new coordinates on the horizontal axis. That’s true, and that makes sense to me, but I know a lot more math than these kids do. Complex analysis made me pretty comfortable with transformations of a plane, but that wasn’t until grad school, so I doubt they’re really happy with it. I’d like to find something – on Desmos maybe? – that gives a visual of what’s happening to the graph when we do u-substitution, but I’m not sure how to put that together. Any ideas, MTBoS?

# Revising the Shell Method – Is this the standard proof?

Background: I taught BC Calc for the first time last year, as an independent study – just one student. I leaned pretty heavily on the book’s presentation of the material. We use Larson and Edwards, 9th edition – pretty much your standard calculus book.

That particular student had, in a previous class, asked for more proofs. That’s a request I will leap at with both hands and latch on to with a death grip; we proved almost everything. The book has plenty of proofs, and I was scrambling to get the material re-learned and presentable, so I was quite willing to take the book’s proofs when they made sense.

The problem with that, of course, is that textbooks aren’t written for high school students. Have you ever looked at the proof of the Shell Method in Larson? Probably not, so I’ll tell you: it’s a nice idea, and they do a decent job explaining it. But it’s heavily algebraic. And I don’t know about your kids, but my kids don’t find algebra intuitive. In fact, I don’t find algebra intuitive. An algebraic proof will convince me that something is true, but it doesn’t help me understand why it’s true. I’d rather my kids understand something than just believe it as a revelation from on high.

This year, I’m teaching BC again. We start the year where we left off with AB – area and volume, which means we got to the Shell Method on day 2 of class. As I was looking over my notes, I realized just how rough that algebraic proof was going to be.

Quick recap: the shell method of volume is basically a Russian stacking doll, but with cylinders instead of dolls. You take infinitely many, infinitely thin outer slices of cylinders and add up their volume to get the volume of your solid of revolution. It turns out the that formula for the volume of a cylindrical shell is the circumference times the height times the width of the shell (which turns into dx in your integral, because it’s infinitely thin.)

The algebraic proof finds the volume of the entire cylinder, the volume of the hole you cut away to get the shell, and then subtracts to get the volume of the shell. Simple, straightforward, and totally obfuscated by the algebra.

Ten minutes before class started, it hit me that length*height*width is the volume of a rectangular prism. And if you take a cylindrical shell and slice it open on one side, it unrolls to a rectangular prism, where the circumference is the length. So instead of going through a bunch of messy algebra, I took a piece of paper and rolled it into a cylindrical shell. Then I unrolled it into a rectangular prism. Look, guys, same volume! Circumference becomes length! Volume formula that you learned in sixth grade? Great, let’s move on.

Why did I not see this before?

Is this the standard proof? Does everyone teach it this way and I’m just slow to the party?

If yes, then for goodness’s sake, why was that not in a textbook? That’s so much more intuitive then a half-page of algebraic manipulations! If no, then why not? Is there something I’m missing here?

# Local Linearity: The Big Reveal

Precal is currently doing a quick, basic intro to Calculus. We talked a little bit about limits (we can do them with graphs and with charts), we talked about continuity (three part definition, removable vs nonremovable), and today we talked about local linearity.

Disclaimer: I took this wholesale from an AP Summer Institute instructor.

I started by throwing this up on the projector:

I ask them to tell me everything they can about this function. I typically give them two or three minutes to write everything down. (If I’d been paying attention to the fundamental five, I’d probably have turned this into a small group discussion. “Make a list with your neighbor…” I didn’t, though, and since this was the very start of the lesson, I’m ok with it.)

Sometimes they need a little prompting with their ideas, but they come up with things you’d expect: it’s a line, it’s red, it has a slope of 1, it’s y-intercept is 0, it’s y=x, it’s increasing, it’s continuous, it’s odd. Someone asked me if each square was one. I shrugged – they’ve gotten used to that by this point in the year.

I ask them how much we can see. “Not much.”

I click on a point so that they can see the scale.

I ask again, how much can we see. “Barely anything! Oh, there are probably all of these curves, aren’t there?”

I zoom out, so they can see this:

They’re reactions were perfect. “Oh, wow!” “I knew it!” “You tricked us!” One girl was out of the classroom while we did this. When she walked back in, her friends told her, “You missed it! That was the graph! She tricked us!” I mention the phrase local linearity, and we’re off.

We start talking about slope. We don’t know how to find the slope of a function, so what can we do? One class wanted to find the average slope, so I had to nudge them towards finding the slope at a point. Can we do that? “Yes.” Cool, how? “Uhm….we could zoom in so that it looks like a line!” Yes, yes we can.

So we do. We find the slope at 0, at $\frac{\pi}{2}$, at $\pi$. I point out that it would be nice if we could just have a function whose output was the slope, so that we didn’t have to do all of these approximations over and over again. They agree – yes, that would be nice. (I do love it when they play along…which usually happens when I’ve set up the need before giving them something…but it also requires them to take the bait.)

We switch to $y = x^2$. I give each of them an x-value or two, and they have to find the slope by zooming in, finding two points near their x-value, and then finding the slope between them. Maybe it would be better to stick with $\sin x$, but you have to do a lot of points to see the shape, and I have really small classes.

After they all have their slopes, we put them in a table on Desmos. In one class, they did a great job finding their slopes, and it was a line. In the other, it didn’t work out so well, so we had to redo some points. Eventually, they got it, though. That would have been a spot where I could have gotten some really good estimation conversations going (why do you think that point is wrong? positive slope, decreasing function, etc.), but I didn’t. Maybe next year. They were able to figure out which points were wrong, though, and fix them without me, so I’ll take it.

If we have a bunch of points in a line, we can find the line of best fit. Desmos is really great for this – we typed in $y = mx + b$, created m and b sliders, and went to town. They got $y = 2x$ pretty quickly.

At this point, I mention the word derivative. In the advanced class, I throw up the limit definition of the derivative and talk about it. We don’t use it in precal; I just want it to simmer in their backbrain for a few months until we need it in August.

For homework, they repeat the process on two more polynomials. They get to choose which two from a list I give them. Then they have to write about what they notice, what they wonder, and how they could check those wonderings. I’m hoping some of them will notice the  Power Rule; the list they can choose from leads them their nicely. If they don’t get it, well, we repeat this with a few tweaks in Calculus, so they’ll get it then. My goal for the overview of Calculus is primarily to get them thinking about a few big ideas. We have all of next year to do stuff with it. We’ll probably also do a class discussion next time about what they noticed and wondered, and I might get them to the Power Rule that way.

Their exit ticket was to tell me why we could find the slope by zooming in. Most of them, with a bit of a nudge, could tell me that we zoomed in, saw a line, and from there we could find the slope. No one used the phrase locally linear, but I’m ok with that for this level.

I really, really liked the way this worked today. Their reactions when we zoomed out made my day. Also, in the past I’ve done this with the graphing calculator instead of Desmos (this is the first year we’re 1-1), and it was a pain. Desmos was perfect for this lesson – it’s easy to use, they’re familiar with it, and the sliders are a huge help.

I also liked that this only took about 25-30 minutes; we fit this in after a warmup and a quiz. With a whole hour, we could have done more with the theory, or I could have given them more time to practice. Things to think about….

# AP Review: Stages 2 and 3

AP Review Stage 1 is the review project. You can read more about that here. They make a binder with notes on all the major topics.

That was due about two weeks ago, so we’re into AP Review Stages 2 and 3. They’re concurrent, but they focus on two different things.

Review Stage 2 is to ensure that they know the things from the project. I give them a blank copy of “Stuff You Must Know COLD” and have them fill it out for homework. (You can find it here.) Then we do a unit circle quiz – they don’t realize how much they’ve forgotten. Then we do two stuff you must know cold quizzes. It’s a quiz a week from here until the AP exam, just to give them some accountability for their studying. We had a couple of quizzes yesterday, and I don’t think they were thrilled with their grades. Hopefully that’ll help them to focus. I have seen their copies of the handout a lot, though, as they get them out to review them. Which is good. And that’s Review Stage 2.

Review Stage 3 is to make sure they know how to use those things that they’ve just committed to memory. At an AP summer training, the presenter had split up the main AP topics into several different categories: area/volume, graphical analysis, contextual questions, differential equations, tabular questions, related rates, motion, and miscellaneous. (It’s entirely possible she’d taken this from Lin McMullin. See more here.) I give them a packet of 6 AP problems for each topic, and we do a topic a day. I let them choose if they want to work together or in groups, and they get as far as they can in class. The rest they finish for homework. I give them the scoring guides, too, so they can check their work and score as they go. I give keys to all of the homework throughout the year, so by know they know how to use this to study and check themselves instead of just copying.

Because this is mostly them working, I hardly need a whiteboard. Yesterday, I took them outside and let them work there. I did grab a mini whiteboard in case I needed it, but I was able to talk everyone through problems without actually writing anything down. They actually focused better outside, which surprised me. It may have just been the novelty, but I’ll have to keep that in mind for the future.

We also do a mock exam, and discuss that, which helps focus things as we get closer to the end. That was yesterday, and you can read more about it here.

How are we so close to the exam? Where did this year go?

# The Mock Exam Results and Analysis: 2015

We did a mock AP exam today. I must say, grading AP exams is sooo much easier than grading normal tests, because I don’t have to decide on partial credit. Right or wrong. Use the scoring guidelines. Done.

Note: if you mostly want to read what I learned and what we’ll be focusing on, skip towards the end.

My first year, we didn’t do a mock exam. We had two weeks of review, and that was it. That year was rough, in a lot of ways. That summer, I went to an AP Summer Institute, which helped a ton. Actually, it was while I was there that the scores came out. I must say, it was a little embarrassing to be surrounded by AP teachers and to find out that none of your students passed the exam.

I revamped the course, drawing extensively, on what I learned at the summer institute, and I redid precalculus, and things started getting better. Last year was less rough. Not good, not yet, but less rough. Six students took the class. Two passed, and with fives. Another one or two probably could have, but senioritis, video games, and basketball season got in the way. Not good, not yet, but better.

One of those two fives was a junior. He stayed and did BC as an independent study with me this year. (We typically only offer AB.)

Which brings us to this year. I’ve been feeling pretty good about this year. This was the first group of students who’d taken my advanced precal class, and they were a lot stronger. I was on the watch a lot earlier for senioritis, and stepped in to squash it before it got to be too big of a problem. I wasn’t quite sure how’d they do on the mock exam, though. We’ve been working free response problems for the last two weeks. They’ve been scoring their own work and telling me that they’ve been doing mostly 5s, 6s, 7s, some 8s. Those are good, and that’s certainly on track for passing. But when they weren’t working together, how would they do? And were they being generous? And what about the fact that this is a three hour exam? How would they handle brain fatigue?

They rocked it.

I’m so proud of these kids. The blew my socks off. The scores break down this way:

5 – 3 students

4 – 3 students

3 – 2 students

2 – 1 student

1 – 1 student

80% of them passed! And one of those 3s was within 3 points of being 4s! Y’all. These kids. They rock. The 5s were pretty comfortable margins, too. The BC student got a 5, too. And this is just the mock exam.

But the best part…a student who has struggled all year, a student who works slowly and doesn’t like writing things down got a 3! And he’s only about 3 points away from getting a 4, and just showing more work and explaining his answers would get him there. He needs a 4 for where he’s going to college. I was so happy – he’s sooo close to it – that I almost started crying in the middle of precal’s All Sins Forgiven Test.

There were a lot of common mistakes, as there always are. The main thing I want them to focus on is showing all of their work and all of their reasoning. Don’t just tell me the critical points; show me that you set the derivative equal to zero. Don’t just tell me the area; show me the integral. Two of the 3s would have been 4s if they’d shown more work. Use words instead of arrows.

We also need to pay attention to negatives and u-substitution more, and we probably need a refresher on differential equations. That’s one of the topics that we haven’t worked problems on yet, though, and we’re doing that next week, so I’m not surprised.

Oh, and the MC was kind of rough. The free response was generally better, and sometimes by a lot. I’m not sure what to do about that – suggest they get an AP review book and just work a bunch of problems? Or maybe I’ll dig out my old one and just give them a bunch of problems.

Tomorrow, we’ll talk about the exam. I think I’ll split them up into groups, give them the keys, and let them talk it through. Hearing it from a peer can be better than hearing it from a teacher, and teaching a peer is always a good thing.

This is just the mock exam. We still have a week and a half to really knuckle down and dig in. Most of those 3s could easily become 4s, and the 4s could possibly turn into 5s depending on how badly they want it. I’m a little nervous, though, that they’ll see these scores and start slacking off. I’ll have to push them a little harder this next week to make sure they finish strong.

For the first time, I’m actually looking forward to getting score reports in July.

So – the tl;dr version:

1.) Go to an AP Summer Institute. Those things are super helpful.

2.) Make sure pre-cal is where it needs to be so that they’re prepared.

3.) Mock exams help them and me both see what to focus on in the last few weeks of review.

4.) I love my kids.

# AP Review Stage 1: Project

I don’t assign a lot of math projects. Occasionally we’ll do some “math in real life” homework assignments, but projects of the major-grade variety, that take several weeks to finish, don’t feature heavily in my classes. They aren’t bad by any means, but they aren’t a good fit for my classes right now.

In fact, I only assign one.

This is a project that I did in high school. When we were reviewing for the AP Calculus exam, my teacher assigned this project to our class. Years later, I still have it. It’s survived moves to and from college, to grad school, home from grad school, to an apartment, and now to a house. I still use the thing when I’m preparing notes for class, and I’m still proud of it. I figure that any kind of project that has such a strong effect on me is probably worth assigning.

The assignment is pretty simple. I hand out a list of 31 topics that cover the essentials of what we’ve studied this past year. Some of them are big ideas, some of them are just detailed things they’ll need for the AP exam, some of them are places where I know they’re weak. For each topic, they have to create a page of notes, including examples. Then they compile them into a notebook with an interesting, mathematically-themed cover. I suggest color, but don’t require it. They have two weeks to work on it. I even break it down for them and tell them to have these pages done and ready to discuss in class on these days. We go over any questions they have on those topics and work some practice problems. It’s Stage 1 of AP review.

You can find the list of topics here:

AB Calc Review Project Topic List

In the past, this has gone over pretty well. My one BC student still has and uses his notebook from last year. Students usually tell me that it really helps them get ready for the exam, because they’re going over everything and explaining it in their own words. I especially like that they have to put it in their own words. It’s a higher order of learning than just working problems, and it makes them internalize it in a different way. Plus, it’s a fairly easy major grade. It’s time consuming, but it’s not hard, and it’s something they should be doing anyways to study. If you just do a thorough job, you’ll probably get an A.

The projects were due on Friday. I always like the day they’re due because I get to see what they’ve come up with. The ones who really put the time in are usually so proud of their work; it’s fun to see how invested they get in their work. Some of them really go to town on the covers. I love seeing their personalities come through. One girl told me that she stayed up late, markers and sharpies strewn everywhere, working on hers. The cover is only worth about 5 points on the grade, but it’s the most fun.

Grading always takes longer than I remember (I struggle with 11. How did my high school teacher handle 110?), but I get to see the students’ strengths. One girl doesn’t grasp Calculus intuitively, but she takes fantastic notes and explains things really well. Another boy has struggled all year, but put in a lot of work and is starting to pull a lot of the concepts together. Another filled his with inside-jokes from class. These make me smile.

This, of course, is just Stage 1 of AP Review. I’ll post on Stages 2 and 3 sometimes soon.