# Some Random Things I Have Liked

## The Concept of Signed Areas

In calculus, after first introducing the concept of signed areas, I came up with the “backwards problem” which really tested what kids understood. (This was before we did any integration using calculus… I always teach integration of definite integrals first with things they draw and calculate using geometry, and then things they do using the antiderivatives.)

I made this last year, so apologies if I posted it last year too.

[.d0cx]

Some nice discussions/ideas came up. Two in particular:

(1) One student said that for the first problem, any line that goes through (-1.5,-1) would have worked. I kicking myself for not following that claim up with a good investigation.

(2) For all problems, only a couple kids did the easy way out… most didn’t even think of it… Take the total signed area and divide it over the region being integrated… That gives you the height of a horizontal line that would work. (For example, for the third problem, the line $y=\frac{2\pi+4}{7}$ would have worked.) If I taught the average value of a function in my class, I wouldn’t need to do much work. Because they would have already discovered how to find the average value of a function. And what’s nice is that it was the “shortcut”/”lazy” way to answer these questions. So being lazy but clever has tons of perks!

## Motivating that an antiderivative actually gives you a signed area

I have shown this to my class for the past couple years. It makes sense to some of them, but I lose some of them along the way. I am thinking if I have them copy the “proof” down, and then explain in their own words (a) what the area function does and (b) what is going on in each step of the “proof,” it might work better. But at least I have an elegant way to explain why the antiderivative has anything to do with the area under a curve.

Note: After showing them the area function, I shade in the region between $x=3$ and $x=4.5$ and ask them what the area of that bit is. If they understand the area function, they answer $F(4.5)-F(3)$. If they don’t, they answer “uhhhhhh (drool).” What’s good about this is that I say, in a handwaving way, that is why when we evaluate a definite integral, we evaluate the antiderivative at the top limit of integration, and then subtract off the antiderivative at the bottom limit of integration. Because you’re taking the bigger piece and subtracting off the smaller piece. It’s handwaving, but good enough.

## Polynomial Functions

In Precalculus, I’m trying to (but being less consistent) have kids investigate key questions on a topic before we formal delve into it. To let them discover some of the basic ideas on their own, being sort of guided there. This is a packet that I used before we started talking formally about polynomials. It, honestly, isn’t amazing. But it does do a few nice things.

[.docx]

Here are the benefits:

• The first question gets kids to remember/discover end behavior changes fundamentally based on even or odd powers. It also shows them that there is a difference between $x^2$ and $x^4$… the higher the degree, the more the polynomial likes to hang around the x-axis…
• The second question just has them list everything, whether it is significant seeming or not. What’s nice is that by the time we’re done with the unit, they will have a really deep understanding of this polynomial. But having them list what they know to start out with is fun, because we can go back and say “aww, shucks, at the beggining you were such neophytes!”
• It teaches kids the idea of a sign analysis without explaining it to them. They sort of figure it out on their own. (Though we do come together as a class to talk through that idea, because that technique is so fundamental to so much.)
• They discover the mean value theorem on their own. (Note: You can’t talk through the mean value theorem problem without talking about continuity and the fact that polynomials are continuous everywhere.)

## The Backwards Polynomial Puzzle

As you probably know, I really like backwards questions. I did this one after we did  So I was proud that without too much help, many of my kids were really digging into finding the equations, knowing what they know about polynomials. A few years ago, I would have done this by teaching a procedure, albeit one motivated by kids. Now I’m letting them do all the heavy lifting, and I’m just nudging here and there. I know this is nothing special, but this course is new to me, so I’m just a baby at figuring out how to teach this stuff.

[.docx]

# Related Rates, Yet Another Redux

I posted in 2008 how I didn’t actually find related rates all that interesting/important in calculus. The problems that I could find were contrived, and I didn’t quite get the “bigger picture.” In 2011, I posted again about something I found from a conference that used Logger Pro, was pretty interesting, and helped me get at something less formulaic.

I still don’t know how I feel about related rates. I’m torn. Part of me wants to totally eliminate them from the curriculum (which means I can also possibly eliminate implicit differentiation, because right now I see one of the main purposes of implicit differentiation is to prime students for related rates). Part of me feels there is something conceptually deeper that I can get at with related rates, and I’m missing it.

I still don’t have a good approach, but this year, I am starting with the premise that students need to leave with one essential truth:

Often times, as we change one thing, it affects a number of other things. However, the way that the other things are affected can vary greatly.

Right now, to me, that’s the heart of related rates. (To be honest, it took some conversation with my co-teacher before we were able to stumble upon this essential understanding.)

In order to get at this, we are starting our related rates unit with these two worksheets. A nice bonus is that it gets students to think about the shape of a graph, which is what we’ll be embarking on next.

The TD;DR for the idea behind the worksheets: Students study a circle which has it’s radius increase by 1 cm each second, and see how that changes the area and circumference. Then students study a circle which has it’s area increase by 10 cm^2 each second, and see how that changes the radius and circumference. The big idea is that even though one thing is changing, that one thing affects a number of different things, and it changes them in different ways.

[.docx] [.docx]

(A special thanks to Bowman for making the rocket and camera problem dynamic on Geogebra.)

It’s not like this is a deep investigation or they come out knowing anything super special. But the main takeaway that I want them to get from it becomes pretty apparent. And what’s really powerful (for me, as a teacher trying to illustrate this essential understanding) is seeing the graphs of how the various thing change.

***

I had students finish the first packet one night. Before we started going over it, or talking about it, I started today’s class asking for a volunteer to blow up balloons. (We got a second volunteer to tie the balloons.) While he practiced breathing even breaths, I tied and taped an empty balloon to the whiteboard.

Then I asked our esteemed volunteer to use one breath to blow up the first balloon. Taped it up. Again, for two breaths. Taped. Et cetera until we got a total of six balloons taped.

Then I asked what things are measurable in the balloons.

Bam. List.

(We should have listed more. Color. What it’s made of. Thickness of rubber.]

Then I asked what we did to the balloon.

Added volume. A constant volume (ish) in each balloon.

Which of the other things changed as a result?

How did they change?

This five minute start to class reinforced the main idea (hopefully). We changed one thing. It changed a bunch of other things. But just because one thing changed in one particular way doesn’t mean that everything changed in that same way. For example, just because the volume increased at a constant rate doesn’t mean the radius changed at a constant rate.

***

This is about all I got for now. I’m going to teach the rest of the topic the way I always do. It’s not up to my personal standards, but I still am struggling to get it there. I suppose to do that, I’ll have to see a more nuanced bigger picture with related rates, or find something that approaches what’s happening more visually, dynamically, or conceptually.

PS. The more I mull it over, the more I think that geogebra has to be central to my approach next year… teaching students to make sliders to change one parameter, and having them develop something that dynamically illustrates how a number of other things change. And then analyzing how those things change graphically and algebraically.

(A simple example: Have a rectangle where the diagonal changes length… what gets affected? The sides, the angle between the diagonal and the sides of the rectangle, the area, the perimeter, etc. How do each of these things get affected as the diagonal changes?)

# What does it mean to be going 58 mph at 2:03pm?

That’s the question I asked myself when I was trying to prepare a particular lesson in calculus. What does it mean to be going 58 mph at 2:03pm? More specifically, what does that 58 mean?

You see, here’s the issue I was having… You could talk about saying “well, if you went at that speed for an hour, you’d go 58 miles.” But that’s an if. It answers the question, but it feels like a lame answer, because I only have that information for a moment. That “if” really bothered me. Fundamentally, here’s the question: how can you even talk about a rate of change at a moment, when rate of change implies something is changing. But you have a moment. A snapshot. A photograph. Not enough to talk about rates of change.

And that, I realized, is precisely what I needed to make my lesson about. Because calculus is all about describing a rate of change at a moment. This gets to the heart of calculus.

I realized I needed to problematize something that students find familiar and understandable and obvious. I wanted to problematize that sentence “What does it mean to be going 58 mph at 2:03pm?”

And so that’s what I did. I posed the question in class, and we talked. To be clear, this is before we talked about average or instantaneous rates of change. This turned out to be just the question to prime them into thinking about these concepts.

Then after this discussion, where we didn’t really get a good answer, I gave them this sheet and had them work in their groups on it:

I have to say that this sheet generated some awesome discussions. The first question had some kids calculate the average rate of change for the trip while others were saying “you can’t know how fast the car is moving at noon! you just can’t!” I loved it, because most groups identified their own issue: they were assuming that the car was traveling at a constant speed which was not a given. (They also without much guidance from me discovered the mean value theorem which I threw in randomly for part (b) and (c)… which rocked my socks off!)

As they went along and did the back side of the sheet, they started recognizing that the average rate of change (something that wasn’t named, but that they were calculating) felt like it would be a more accurate prediction of what’s truly going on in the car when you have a shorter time period.

In case this isn’t clear to you because you aren’t working on the sheet: think about if you knew the start time and stop time for a 360 mile trip that started at 2pm and ended a 8pm. Would you have confidence that at 4pm you were traveling around 60 mph? I’d say probably not. You could be stopping for gas or an early dinner, you might not be on a highway, whatever. But you don’t really have a good sense of what’s going on at any given moment between 2pm and 8pm. But if I said that if you had a 1 mile trip that started at 2pm and ended at 2:01pm, you might start to have more confidence that at around 2pm you were going about 60 mph. You wouldn’t be certain, but your gut would tell you that you might feel more confident in that estimate than in the first scenario. And finally if I said that you had a 0.2 mile trip that started at 2pm and ended at 2:01:02pm, you would feel more confident that you were going around 72mph at 2pm.

And here’s the key… Why does your confidence in the prediction you made (using the average rate of change) increase as your time interval decreases? What is the logic behind that intuition?

And almost all groups were hitting on the key point… that as your time interval goes down, the car has less time to fluctuate its speed dramatically. In six hours, a car can change up it’s speed a lot. But in a second, it is less likely to change up it’s speed a lot. Is it certain that it won’t? Absolutely not. You never have total certainty. But you are more confident in your predictions.

Conclusion: You gain more certainty about how fast the car is moving at a particular moment in time as you reduce the time interval you use to estimate it.

The more general mathematical conclusion: If you are estimating a rate of change of a function (for the general nice functions we deal with in calculus), if you decrease a time interval enough, the function will look less like a squiggly mess changing around a lot, and more and more like a line. Or another way to think about it: if you zoom into a function at a particular point enough, it will stop looking like a squiggly mess and more and more like a line. Thus your estimation is more accurate, because you are estimating how fast something is going when it’s graph is almost exactly a line (indicating a constant rate of change) rather than a squiggly mess.

I liked the first day of this. The discussions were great, kids seemed to get into it. After that, I explicitly introduced the idea of average rate of change, and had them do some more formulaic work (this sheet, book problems). And then  finally, I tried exploiting the reverse of the initial sheet. I gave students an instantaneous rate of change, and then had them make predictions in the future.

It went well, but you could tell that the kids were tired of thinking about this. The discussions lagged, even though the kids actually did see the relationships I wanted them to see.

My Concluding Thoughts: I came up with this idea of the first sheet the night before I was going to teach it. It wasn’t super well thought out — I was throwing it out there. It was a success. It got kids to think about some major ideas but I didn’t have to teach them these ideas. Heck, it totally reoriented the way I think about average and instantaneous rate of change. I usually have thought of it visually, like

But now I have a way better sense of the conceptual undergirding to this visual, and more depth/nuance. Anyway, my kids were able to start grappling with these big ideas on their own. However, I dragged out things too long. We spent too long talking about why we have to use a lot of average rates of changes of smaller and smaller time intervals to approximate the instantaneous rate of changes, instead of just one average rate of change over a super duper small time interval. The reverse sheet (given the instantaneous rate of change) felt tedious for kids, and the discussion felt very similar. It would have been way better to use it (after some tweaking) to introduce linear approximations a little bit later, after a break. There were too much concept work all at once, for too long a period of time.

The good news is that after some more work, we finally took the time to tie these ideas all together, which kids said they found super helpful.

# Advice from Calculus Students Past, Informing the Calculus Student Present

I’ve done Standards Based Grading in Calculus for two years now. This is the start of my third year.

One of the things I have my kids do at the end of each school year (not just in calculus, but in all my classes) is to write a letter to themselves. But in the past. Yes, I tell kids to compose a letter that can be sent back into time, to them, at the beginning of the year. Things they wish they had known at the start of the year that they know now that it is the end of the year. And I let them know whatever they write is up to them, and that I don’t look at this until way into the summer. We seal them up.

I usually share these letters with kids the following year. When I do, I ask kids to think about commonalities they noticed in the advice from students, and also, if anything struck them. We have a conversation about that. I definitely emphasize that what works for one person might not work for another.

Without further ado, here is the advice that my 2011-2012 calculus kids wrote to their past selves, which I will be sharing with my 2012-2013 calculus kids.

To me, the major commonalities are… advice to do their homework even though it’s not graded, not to use reassessments as a crutch because it’s to your benefit to learn things the first time around, and to ask for help from colleagues and Mr. Shah.

With that, I’m out like a light.

# Wealth Inequality! A Calculus Investigation

First off, I want to say that I took this wholesale from the North Carolina School of Science and Math.  Thank you NCSSM. They have a conference each year on high school math, and each time I’ve gone, the speakers I’ve liked best are the actual teachers at the school. So any good things you might want to say about this, please don’t say them to me. This is the product of the hardworking teachers over there. So please, please check out the NCSSM project here. All I will be doing in this post is talking about how I coopted it for my classroom.

So the year came to a close in my calculus class. And in the last week, I wanted to try something new. And there was a confluence of things that led me to this.

I had students teach themselves how to find the area of two curves previously, when I was out sick, but then I didn’t do anything with it. I had also just seen an interesting piece on wealth inequality which piqued my interest. And I had heard of the Gini Index and the Lorenz curve before but had never pursued it seriously.

So here we are, the perfect time to go whole hog. And when doing my massive internet search, I came across NCSSM’s awesome activity and realized it was better than anything I could devise on my own. I really loved the scaffolding of the packet.

To start out class, I laid out the objective. I showed some photos from Occupy Wall Street. We read the protester’s posters aloud. And we focused on one of them: “This is not the world our parents wanted for us, nor the one we want for our kids.” I focused on that, because it implied that there was a difference in the world from the previous generation. The protester, and others, have been saying that the rich are getting richer while the rest of us are not. And my question to the class is: do you think this is true?

We talked about it generally, and I followed it up with a conversation about how we might decide if the distribution of wealth were different now than it was later. Students shared their thoughts in pairs, and they came up with some good ideas. Many pairs talked about making a histogram (wealth vs. number of people with that wealth). Others talked about comparing the top 5% with the bottom 5%. We shared our ideas as a class. I liked making them think about how one might decide this, because the answer is: there are many ways, but they all are going to involve math. We also talked about how we could compare one wealth distribution to another — and then we realized that it became tricky, fast.

I then had them make conjectures on the actual distribution of wealth in the US. And then I showed them the true answer. The true distribution shocked them.

The best part of the discussion was around what kids picked for “what they would like it to be.” We got to talk about capitalism and socialism and oligarchies. I made it really clear that I wasn’t here to make a case for one type of economic system or another. (Though some students had some strong opinions of their own.)

This initial prelude set up the remaining 2 days kids spent working on this. It gave them our overarching question (“Is income truly becoming more and more unequally distributed in the past 40 years? Or is it propaganda used by Occupy Wall Street protesters and sensational journalists?”) And off then went.

I basically made the most minor revisions to the NCSSM document and gave it to my class…

Each day, I had a goal that students had to reach, and if they didn’t they were asked to finish it at home. (At most, they only had 5 minutes of work each night. It was the last week of classes, and I wanted it to be more relaxed.) We talked at the start of each class, and I had them work in pairs. We had mini breaks/discussions to talk about big ideas. One of these included the trapezoidal rule. When introducing Riemann Sums a month or two prior, we only did them as left and right handed rectangles. But we saw how bad those approximations would be in this case where we only had 5 divisions… which necessitated the use of the trapezoidal rule. I didn’t teach my kids it, but they could do it. And some found a quicker formula to find the area, because they got sick of calculating all the areas of the trapezoids together. Huzzah! One student make a calculator program to calculate the Gini Index because it became tedious to do the calculations.

We ended the packet by just going through the US Gini Indexes for the last 40 years. We didn’t do the part asking for an investigation on other countries.

Results

We did this informally. I threw it together, I framed it in the context of Occupy Wall Street, and we went off. I didn’t collect formal feedback from my students on this (it was the last week), but I had a number of students individually let me know how much they liked it. A couple told me it was their favorite thing all year — and they loved that this had applications. One told me they spoke with an economist last summer and they were talking about economics and calculus, and the economist was talking about the Lorenz curve — but the student (at the time) didn’t understand it. I love that we could clear that up!

Also, I had two teachers observe my class the first day we started it, and I had them participate, and they said they enjoyed thinking about the questions and working on the packet.

Using This in the Future

I love the idea of using this in the future. I hope to do so next year, earlier in the year. I think I need to make the packet a little more conceptually deep, and ask some probing questions as we go along.

One type of probing question might be to ask students to draw figure out what a Lorenz curve looks like for the Gini Index to be 0 and what a Lorenz curve looks like for the Gini Index to be 1 (the packet just tells them that). Or to explain why the Lorenz curve cannot go above the line y=x. In other words, why it can’t look like:

I also think it could easily be extended to be a good poster project. One obvious idea is having students pick two countries, do a little research on them and come up with a hypothesis for which has more income inequality and justify it without mathematics. Then they would calculate the Gini Index for each. Finally they would make a poster showcasing their hypothesis and their findings.

Additionally, I could have each of them (after our in class work) read the section in the book on the Trapezoidal Rule, and make part of their poster explain this rule and how it works for any general function divided into N equally spaced rectangles. (Since I don’t formally teach it, nor do I think it needs to be formally taught.)

Alternatively, I could have students (especially since we analyzed a program which calculated Riemann Sums) see if they could come up with a program that would calculate the Gini Index.

As a personal note for next year: Oh yeah, I have to remember to make a distinction between wealth inequality and income inequality. I kept conflating the two, but they are very different and I need to make sure I get that across.

# Algebra Bootcamp in Calculus

So it was the Old Math Dog who pointed out that I never wrote a post explaining how I deal with the issue of kids not knowing basic algebra in calculus. I started this practice two years ago (when I also started standards based grading) and I have seen a remarkable difference in how my classes go from my life pre-bootcamps to my life post-bootcamps…

An issue in any calculus course — and I don’t care if you’re talking about non-AP Calculus or AP Calculus — is the student’s algebra skills. They might see $\frac{1}{4}x+\pi x -4=0$ and have no idea how to solve that. Or they might not know how to find $\tan(\pi/6)$. Or they might cancel out the -1s in $\frac{x^2-1}{x-1}$ to get $\frac{x^2}{x}$. It depends on where they are coming from, but I can pretty much guarantee you that every calculus teacher says the same thing to their classes on the first day:

Calculus is easy. Algebra is hard.

In my first three years of teaching calculus, I started with how all the books started, and all my calculus teacher friends started: a precalculus review. Then we went into limits.

The problem with that is that we might review some basic trigonometry, and then we wouldn’t see it again for months. And by then, they had forgotten it. And who could blame them. The precalculus review unit at the beginning of the course wasn’t working.

As I transitioned into Standards Based Grading, I looked at everything I taught really closely, and I honed in on the particular skills/concepts I was going to be testing. And since I’d taught calculus for a number of years prior, I knew exactly where the algebra sticking points were. Thus was born The Algebra Bootcamp.

Before our first unit on limits, I carefully analyzed what things I needed students to know to understand limits to the depth I required. I then looked at all the skills and thought of all the algebraic things, and all the old concepts, they would need in order to understand limits. And from that, I crafted an algebra bootcamp, and I made SBG skills out of just those limited skills.

For example, here was our first bootcamp (which, admittedly, was longer than most of the others, because we were settling in and I was gauging where the kids were at):

and I did the same for other units… just the targeted prior knowledge that they tended to not know or struggle with…

Notice how they tend to be very concrete and specific? Like “rationalize the numerator” (because I knew we were going to be doing that when using the formal definition of the derivative) or “expand $(x+h)^n$ using the binomial theorem. Very specific things that they should know that they are going to be using in the following unit. It’s kind of funny because it is a hodgepodge of little (and often unconnected) things, and they have no idea why we’re doing a lot of what we’re doing (why are we rationalizing the numerator? why are we doing the binomial theorem?) and I don’t tell them. I say “it’s our bootcamp… once training is over you’ll see why these tools are useful.”

It is called “bootcamp” because I am not reteaching it from scratch. I’m reviewing it, and I go through things quickly. I only do a few of them in the first quarter and maybe the start of the second quarter. By that point, we’ve done what we needed to do, and they die off.

The reason that this has been so effective for me is because students aren’t having to relearn old topics/algebraic skills while concurrently learning the ideas of calculus. We review these very specific things beforehand so that when we approach the calculus topics, the focus is not on the algebraic manipulation or remembering how to find the trig values of special angles or what a piecewise function is… but  on the larger picture…. the calculus.

Remember: calculus is easy, it’s the algebra which is hard.

So we took care of the algebra beforehand, so we can see how easy calculus is.

My kids in the past two years have made so many fewer mistakes, and we’ve been able to really delve into the concepts more, because I’m no longer fielding questions like “could you review how to do X?” Doing this has also forced me to think about what the purpose of calculus class is. The more I teach it, the more I take the algebraic stuff out and the more I put the conceptual stuff in. For example, I don’t use $\cot(x)$, $\sec(x)$, and $\csc(x)$ in my course anymore  [1], because I wasn’t trying to test them on their knowledge of trigonometry. Doing these bootcamps coupled with standards based grading has forced me to keep my eye on what I really care about. Students deeply understanding the fundamental concepts of calculus. And I think you can do that without knowing how to integrate $\sec(x)\tan(x)$ just fine. [2]

[1] With the exception of $\sec^2(x)$ for the derivative of $\tan(x)$.

[2] I teach a non-AP calculus, so I have this luxury. But it’s nice. Each year I strip more and more stuff off the course and add in more and more depth. And I am glad that I understand depth to mean something other than “more complicated algebra in the same old calculus problems.”

# 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…

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?

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.