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?)

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.

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.

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.)