Calculus

A calculus optimization poster project

I covered optimization very differently this year, as I started documenting here. Besides their assessments asking them to solve optimization problems both algebraically and on their calculators (and explaining how they did both), they did a poster project. Here are some of the finished products:

And here was the assignment…

[.docx version]

I never do projects, so this was new to me. But my kids really took to it in a way I really enjoyed. I had most of them pair up and find how “volume optimized” a can in. In other words, they took photos of cans, they decided how much metal was used to make the can (the surface area… we ignored thickness), and we asked if we could recast the can to hold more volume. That was our overarching question…

We started this the week before spring break. I think students had three days in class to work on it, and then it was due after spring break (many just had some gluing to do). I provided the posterboard and colored paper. They provided the rest.

An Example Close Up

Student Thoughts

I asked students to talk about the project in their third quarter reflections. Here are all the quotes from the reflections, where I asked them to talk about the quarter, and about the can project in particular, and give advice for changes I should make next year on it:

* I am particularly proud of the project that ___ and I worked on together. We worked really hard on it and stayed after school and although it was sort of confusing at first, once we got the hang of it I began to really understand optimization… I generally prefer projects because it allows me to be more creative and think more deeply than tests so I actually did enjoy the can project. I thought that that having to do the same thing for five cans got a bit repetitive so maybe if you were to do it again have the students do some different kinds of shapes or types of problems.

* The can project I was really proud of. ___ and I worked for hours and it and I think the end result was really good. Our poster was well made and looked good… I really liked the can project. I think we could have gone over the project more before starting because the goals were a little unclear.

* This may seem insignificant, but one of the most memorable things [from the quarter] for me was the way that this mountain of math for the cans project simplified into this beautiful little thing (h=2r) after doing all this calculus. It was quite cool when I saw that… [As for making changes for the project next year] Honestly, I’d ditch the poster element. It added nothing to my understanding, and ended up being more of a burden than anything… The calculus was certainly worth-while, but that was only like a quarter of the work. The rest was repeatedly plugging the numbers into a program I made (I tried writing a python script for the first time) and writing them down. So basically, make us do more complicated (and more in general) calculus warter than a wee-bit of calculus and a lot of “filler” kinda stuff.

* I did like the can project, but I was sometimes confused about the exact requirements. It was also difficult to finish everything in class, but it worked out when we had the extention until the Monday we got back [from Spring Break].

* The most memorable event from this quarter must have been the “Can Can” project. It gave the class and I time to apply our calculus knowledge to real world concepts… I thoroughly enjoyed the can project because I felt like I understood it entirely from day 1. The amount of work when done between a pair was not tedious at all as well.

* As for the can project, I did enjoy working on it but found it to be a bit repetitive and tedious. I also think had we more time to complete it I would have had more fun with it. I did feel I understood exactly what we were doing. I think if you were to do it next year you should allow more time so students can be more creative with their project.

* The can project was definitely worthwhile. The only thing I disliked about the project was that we used the same shape every time. I think we could have optimized different objects to make it more interesting, just  because the process became kind of repetitive. I think you should still do it next year if you would like but you could choose to alter it a little bit.

* I really liked the can project. For me, the can project was able to show directly the connection between what we were learning in Calculus and the real world which is something that really interests me. I felt like I understood what was being asked of us, and I think that it would be a good addition to next year’s Calculus curriculum as well.

* In general, optimization was my favorite/most memorable part of the quarter. It’s probably the only math I’ve ever done that requires logical, real world thinking at every step (for example, who cares about the optimization of the graph when it’s less than x=0, because you can’t have negative distance). In the past, I’ve felt that a lot of math does correlate closely to things in the real world, but this is the first time where it’s so clear how everything relates. That said, I felt like the can project went extremely well, considering this is the first time it was done in this class. I felt like I totally understood everything that was going on, and I enjoyed taking measurements, doing calculations, and seeing how much the lima bean companies were ripping us off (hint: they’re not! It’s the tuna companies that are evil). The only change I would suggest is allowing one or two days more of time to finish it. Although we got all our measurements and calculations done, the most difficult and lengthiest part of the project proved to be printing everything out, cutting it, and creating the poster.

* Volume optimization, more than any other topic, really stood out for me this quarter. When we first started doing it, I was confused and didn’t entirely understand what to do. I think I was a bit taken aback by translating words/pictures into mathematical equations, but once I worked at it and practiced a bunch I became better at making that translation. I thought that the can project was very interesting, and it helped me make the translation better, as well as illuminating an important real-world connection. I was interested to see which companies used their material properly! I did feel, though, that 5 cans was more than was needed — it was basically the same thing every time, so fewer cans could have been enough to still get the point across.

* I really actually liked the can project and got pretty into it. I liked it because it felt like we were working independently on applying what we learn in class to the real world. I think it should be done again next year.

* I liked the can project a lot. It was cool figuring out how much volume a can could hold if we changed the dimensions of it. At first I did not understand what to do after I found the things I needed to know (height, radius, etc.) — if there was a group where both partners did not know how to figure out the equations needed, then the project would be difficult for them. Maybe having a quick intro/hint class discussing the project will help. I think you should do it again.

* I thought that the can project was very effective because it took what we were learning and applied it to real life. I thought it was very good in allowing us to see how optimization works in reality. I definitely think it should be done again in alter years.

* Even thought I like the idea of the project, my experience with it was not a good one. It certainly illustrates the idea of optimization very well and it’s always nice to see a practical application of things we learn. But due to the circumstances of my partnership with […] it felt very tedious. I don’t think there is much you could do to change it if you are going to keep it, so I would recommend devoting more class time to this project.

* I liked the can project, however it was a little hard to do while also focusing on the problem set. It was also hard to focus on both of those in the week leading up to spring break, so if possible I would recommend splitting them up and doing at least one of them in weeks other than the one before the break. I did enjoy the project, though with the problems above I probably did not enjoy it as much as I could have. I would say to do it again next year because (as math classes don’t always directly relate to the real world) it was cool to apply what we have learned to something we may experience once we leave school.

* I actually really enjoyed the can project. It was a nice break from regular busy work and I definitely got a good handle on the concept it was trying to teach. I would highly recommend doing it again next year.

* I enjoyed the Can Project, making our poster, and working with my group members to find the optimized volumes. I definitely think you should do it again next year.

* I though the can project was good. I liked working with people to create something fun and pretty, and I liked the splitting up of labor rather than doing it on our own. I would say next year maybe give people a bit more time for the project — I felt very rushed doing it. Of course we ended up finishing, but kind of just barely, and so maybe a big more time would help.

* The most memorable thing from this quarter is the can project. In the beginning, I had difficulty understanding optimization but after doing the project it made a lot more sense. Applying the concepts to real life made them much more understandable. At first I had difficulty understanding the purpose of this project, however it proved to be beneficial to me.

Thoughts for Next Year

I got a lot of good feedback from the students, and I am glad that they are comfortable enough to share their thoughts as frankly as they did. Overall I think this thing, which I whipped up in a couple hours the day or two before I decided to do it, worked out as a good thing to do before spring break. It was low key, kids were working independently (with their partners), it allowed for some mindless work and some very mindful work, and kids seemed to learn from each other. I also got the sense from their responses that they really had their understanding of what is truly going on with optimization problems solidify.

I clearly have two big changes to make next year.

First, I need to give more time. I think the three class days that they had was appropriate to get the math done and the poster started, but I think that after this class time, I should give students a week to work on it at their leisure outside of class, while we forged forward with the material. That seemed to be one of the biggest problems — me thinking students could do everything in three days.

Second, I think I need to give a bit more choice and make things a bit more scaffolded. For some, doing 5 cans was tedious. For others, it felt appropriate. Ways to do this would be to require 3 cans, and then some options of other things to take their knowledge further. One question (which I almost did) I could ask them is to measure the volume of a can, and ask them if they could create a can with the same volume but smaller surface area (so it would be cheaper to produce). Or, as a student suggested, I could assign them different shapes and ask them to volume optimize it (boxes, spheres, cones, etc.).

Finally, an observation of my students reflections. I am surprised at how many of them seem to crave or find happiness in the “real world application” activity. I just don’t find “real world” stuff that interesting, compared to the mathematical ideas themselves. And most of our real world applications/problems feel forced or fake, or too simplistic compared to what really happens. So I tend to eschew these sorts of things. But these comments remind me that even though I eschew them, my kids (for some reason) like them. It helps them to find a purpose for what we’re doing, and apparently they need that because I’m not able to totally convince them of the inherent beauty and interestingness of what we’re doing. (Something I work on every year.)

Optimization: An Introductory Activity & Project

I switched things around with optimization in calculus this year, and I realized if I had the time, I would spend a month on it. [1] I wonder if this shouldn’t be a crux of the class. Not the stupid “maximization and minimization” problems but finding some real good ones — in economics, physics, chemistry, ordinary situations. There have got to be tons of non-crappy ones!

Anyway, I wanted to share with you two things.

First, how I introduced the idea of optimization to my kids. Instead of going for the algebra/calculus approach, I wanted them to toy with the idea of maxima and minima, so I had them spend 35-40 minutes working on this in class:

[doc]

I thought it was pretty cool to see my kids engaged. I rarely do things like this, but I did it (I was being videotaped during this lesson… and I had never done it before… and I had the idea to create it the night before…). It was fun! And although I cut the debrief the next day short (ugh, why?), I enjoyed seeing kids engaged in problem solving through various strategies. And there was a healthy level of competition. (The winners for the 1st and 2nd tasks got a package of jelly beans, but they were so gross I threw them out! One student gave them to his rabbit who likes jelly beans, and even the rabbit didn’t like them!) But when it came down to it, it drove home the idea that optimization was something that trial and error is good for, sometimes we do it intuitively, sometimes our intuition is terrible and sometimes it is good, and sometimes we get an answer but we don’t know how to prove there isn’t a better answer (e.g. in problem #3). Some kids liked that this felt more “real world” than this world of algebra and graphing that we’ve been meandering in.

Second, I have allotted a few days for students to work on this project during class (it’s the week before Spring Break and kids are overburdened, so I didn’t want to have them do something which involved a lot of at-home time). They’ve been working on it this week, and I’ve heard some good conversations thus far. (They’re doing this in pairs, and I have one group of three.) The fundamental question is: with a given surface area, what are the dimensions of a cylinder with maximal volume?

[.doc]

Now I don’t quite know how their posters will turn out yet, or whether students will have truly gotten a lot of “mathematical” knowledge out of it. But each day, I’ve had a couple kids say things that indicate that this isn’t a terrible project. (I don’t do projects, so that’s why I’m very conscientious about it.) A few said something equivalent to “Wow, the companies could be giving me x% more creamed corn!” or how they like doing artsy-crafty things. At the very least, I can pretty much be assured that students — if I ask them if there is any question that calculus can answer at the grocery store — will be able to say yes.

Next year I will probably add the reverse component (for a given volume of liquid you want to contain, how can we package it in a cylinder to minimize cost… what about a rectangular prism… what about a cube… what about a sphere… etc.?).

[1] The one thing I found in this book my friend gave me (on science and calculus) was an experiment where you shoot a laser at some height at some angle into an aquarium, so that it hits a penny at the bottom (remember the laser beam will “change” angles as it hits the water) to minimize the time it takes for the photon to travel from the laser to the penny. I almost did it, but deciding to do it was too last minue.

Two crazy good Do Nows

Recently, I’ve been trying to be super duper conscientious of every part of my lesson. For example, I wrote out comprehensive solutions to some calculus homework, paired my kids up, handed each pair a single solution set, and had them discuss their own work/the places they got stuck/the solutions. I actually had made enough copies for each person, but I very intentionally gave each pair a single solution set. It got kids talking. (Afterwards, I told them I actually had copies for each of them.) That’s what I’m talking about — the craft of teaching. I don’t always think this deeply about my actions, but when I do, the classes always go so much better.

In that vein, of super thoughtful intentional stuffs, I wanted to share two crazy good “do nows” from last week. Not because they’re deep, but because they were so thought-out.

For one calculus class, I needed my kids to remember how to solve 5\ln(x)+1=0 (that equation was going to pop up later in the lesson and they were going to have to know how to solve it). I also know my kids are terrified of logs, but they actually do know how to solve them.

I threw the slide below up, I gave them 2 minutes, and by the end, all my kids knew how to solve it. I didn’t say a word to them. Most didn’t say a word to anyone else.

How I got them to remember how to solve that in 120 seconds, without any talking, when they are terrified of logarithms and haven’t seen them in a looong while?

I can’t quite articulate it, but I’m more proud of this single slide than a lot of other things I’ve made as a teacher. (Which is pretty much everything.)  Not deep, I know. It’s not teaching logs or getting at the underlying concept, I know. But for what I intended to do, recall prior knowledge, this was utter perfection. The flow from each problem to the next… it’s subtle. To me, anyway, it was a thing of perfection and beauty.

The second slide is below, and I threw it up before we started talking about absolute maximums/minimum in calculus.

As you can imagine, we had some good conversations. We talked about (again) whether 0.9999999… is equal to 1 or not (it is). We talked about a property of the real numbers that between any two numbers you can always find another number (dense!). I even mentioned the idea of nonstandard analysis and hyperreal numbers.

So I know it isn’t anything “special” but I was proud of these and wanted to share.

Infection Points: The Shape of a Graph

Everyone here knows that I think Bowman Dickson is the bee’s knees, the cat’s pajamas, ovaltine! Recently he posted about how he introduces inflections points in his calculus class… and just a couple days later, I was about to introduce how we use calculus to find out what a function looks like.

Usually, I introduce this in a really unengaging lecture-format. But he inspired me to … copy him. And so I did, extending some of his work, and I have had an amazing few days in calculus. So I thought I’d share it with you.

The Main Point of this Post: By creating the need for a word to talk about inflection points on graphs, we actually saw the math arise naturally. And through interrogating inflection points, we were able to articulate a general understanding of concavity. In other words… the activity we did motivated the need for more general mathematical concepts.

First, definitely read Bowman’s post. All I did was formalize it, and extend it in a few ways, by making a worksheet. I put my kids in pairs and I had them work on it (.docx):

What naturally will happen when students generate their graphs is they will get a logistic function. (Which has a beautiful inflection point! But they don’t know the word… they just see the graph.)

So here we are. The students have a graph, and they’ve been asked to explain their graph for (a) the layperson and (b) the mathematician. Most get some of it done with their partners, and then they take it home to finish individually.

The next day, at the start of class, I assign students to work in groups of 3 (with different people than their partners the previous day). They are asked to take a giant whiteboard and:

(Now I want to give credit where credit is due. I have really been struggling with using the giant whiteboards well, and having students present their work effectively and efficiently. My dear friend Susanna, when I told her about this activity, suggested the groups, the underlining of the mathy words, etc.)

This worked splendedly.

(click to enlarge)

And they had such great observations. Some groups picked up on that change where the function was increasing in one way to increasing a different way. Others talked about how the rate of change (of infected over time) was greatest. Others talked about how the function was “exponential” for the first thing, seemingly linear for the middle third, and “something else” for the last third.

Those gave rise to good short discussions, and we came up with the language for inflection points (which I call INFECTION POINTS!!! GET IT!?!) and concave up/down.

After they had a sense what those words meant, I had students work in partners on the following (.docx):

The point was to get students comfortable with the ideas before we delve into the heavy mathematical lifting. It was powerful. Especially the last page, which got students thinking about patterns, exceptions, and ways to generalize. Our big conclusions:

And with that, I’m too exhausted to type more. But that’s the general sense of what went on in an attempt to teach how to use calculus to analyze the shape of a function.

Implicit Differentiation

Normally, I don’t have trouble teaching implicit differentiation. However, I’m never satisfied with what I do. I’m fairly certain that I have taught it four different ways in the past four years. But what’s common is that we do a lot of algebra. By the end, they can find \frac{dy}{dx} for a relation like \sin(xy)+y^3=2x+y. Or something like that. But we lose the meaning of what we’re doing.

I realized we can do all this algebra, but it’s all procedure. And so there’s no real depth.

So today, after introducing implicit differentiation (including some visual motivation), I assigned 5 basic problems from the textbook. Each of the problems has an equation like 3y^3+x^2=5 and students are asked to find \frac{dy}{dx}. My kids are going to go home today and struggle with it. We’ll spend about 20 or 25 minutes in our next class going over their solutions, talking about things, whatever.

And then… then… I’m going to hand out this sheet I wrote today.

[.doc, .pdf]
[if you’re wondering, the graphs were made by the fabulous winplot which I adore… it can do implicit plotting!]

My kids found \frac{dy}{dx} for homework. Now in class, my kids are going to interrogate what that means.

I am not sure yet how I’m going to structure the class. I think I might have us all work together on the first problem (#9), and then assign pairs to work on two of the remaining problems. And then I’ll pick one problem for each pair to present to the class. But what I’m truly happy about is that each problem gets kids to relate implicit differentiation to a graphical understanding of the derivative. It forces my kids to look at the derivative equation, and make connections to the original graph.

Although I’m proud of it, I’m honestly just not sure if this investigation is beyond the scope of my kids’s abilities. It pulls together a lot of concepts. I think it’ll work for them. This year I have a really really strong crew so I have faith. However, it’s an activity I’m going to have to give my kids time to do, and room to struggle. I know me, and I’m going to want to rush it, and I’m going to want to help them in ways that aren’t good for them. The struggle is where they’re going to learn in this, so I have to give it time and stay out.

I am in the middle of a hellish week, but if I have time, I’ll try to report back how it goes after we do it in class.

Taking a Moment… in Calculus

In calculus, I’ve historically asked kids to take the derivative of:

f(x)=\frac{2x^2+\sqrt{x}}{\sqrt{x}}

and students will immediately go to the quotient rule. OBVIOUSLY! There’s a numerator and denominator. Duh. So go at it!

Unfortunately, this is VERY UNWISE because it leads to a lot more work. And I was sick of my kids not taking a moment to think: what are my options, and what might be the best option available? Also, kids generally found it hard to deal when we started mixing the derivative rules up!

So I came up with a sheet to address this and paired kids to work on it.

(I’ve also had kids think they can do some crazy algebra with g(x)=\frac{x^2+1}{x+1}. This sheet also helped me talk with kids individually about that.)

For a little context, my kids have only learned the power rule, the product rule, the quotient rule, and that the derivative of e^x is e^x. They have not yet been formally exposed to the chain rule.

Without further ado…

[.pdf, .doc]

Digressions! Hints of the Chain Rule via the Power Rule.

In calculus today, I went off the beaten path a bit and it was a lovely digression. I think this works so well with my kids not only because they’re awesome, but because quite a few of them like to notice patterns and explore.

So far we’ve learned and proved the power rule for derivatives, and we’ve been practicing using it. So if students are given y=\sqrt{x}(1+x) and asked to find y', they know that they have to distribute and then take the derivative. [We don’t know the product rule yet.]

So… for their work due today, they were asked to find the derivative of y=(1+x)^3. And my kids wanted to go over this together in class. So when we worked it all out, we got y'=3x^2+6x+3. And then someone noticed that was the same as y'=3(1+x)^2 which looked like the power rule! Like if we had y=x^3, the derivative would be y'=3x^2, so similarly since we have y=(1+x)^3, it’s makes sense that the derivative was y=3(1+x)^2.

At this point, I decided I wanted to capitalize on this. So I said: okay, neat observation. Does it always hold?

And I threw this up…

and had students — using the rule they observed — make a conjecture as to what the derivative would be (without calculating things out). They got (working in pairs, and then sharing as a class):

And then they checked…

… and saw it was wrong. So based on this, I had them revise their conjecture, and take a stab at:

which they did… and they came up with (and worked out):

So they believed they had something that always worked… so I had them prove it. Which they did.

And it worked out!!!

So now we had proved something about the derivative of y=(1+ax)^2, and I asked them: would it work to the third power? would it work to the nth power? And I left it as an exercise for their home enjoyment (our corny term for homework). I’m really curious to see who gets how far on this!

It’s cool. They’re getting whiffs of the chain rule. I’m not going to give it to ’em or do anything else with this. We’ll wait a while. But I really like how this digression took 15 minutes, but it capitalized on something they were curious about. And we’ll see the connection later.

I felt strongly enough about how this worked out that I engineered this discussion to happen in my second calculus class. I treated it like a big surprise. What a strange observation. Instead of forging forward in class, let’s take a digression. I loved that it worked a second time too.