Category Archives: Calculus
I like to start most new units in Calculus with an “anchor problem,” a common sense, every day problem that motivates new techniques and serves as a base that you can constantly refer back to. Some that I have used in the past, to varying degrees of success, are Infection for Inflection, Your Speedometer and the Intermediate Value Theorem, and Predicting Stock Prices with Differentials.
For Riemann Sums, and integration in general, I use the question that really inspired integration in the first place: how do you find the area of an irregular shape? I tell my students:
You work at the glass company. You are given the task of replacing all the glass on the front of this beautiful building, the Duxford Aviation Museum. How much glass do you need? All we know is that the building is 90 m long and 18.5 m tall in the very center.
(This task was partly inspired by this post from Shawn at ThinkThankThunk).
I have it printed on two sheets of printer paper for every group (so big enough to draw on and mark), and I give them 10 minutes to come up with an estimate. Every group writes it on a piece of paper, and then I put it in an envelope. About a week and a half later when we learn the definite integral, we calculate the actual area (using a parabola fitted to the top of the building) and the winner gets…. well nothing. But I announce it at least?
Most students struggle a bit at first and then eventually just start to try something. Some students try some sort of bizarre modified equation for the area of a circle (which I always find really interesting), some turn it into triangles, but most use the maybe-not-that-subtle hint that the window is broken up into square panes.
Right after they are finished making their predictions, we discuss. I ask them what their strategies were and how they could have made their predictions more accurate. I try to get them to come up a couple of points (that sounds manipulative):
We took an irregular shape that has no simple geometric area equation and turned it into a shape that does have a simple geometric equation.
We split up a larger shape into a bunch of smaller shapes to be able to do this.
The smaller our shapes are and the more of them there are, the more accurate our estimate would be. In fact, if we could use infinitesimally small shapes, we could be perfectly accurate.
I think that this activity really shows them how difficult the problem that we are trying to solve is, and primes them to know why we set up Riemann Sums the way that we do, but to be unsatisfied with this solution to the grand area problem. Prepped and primed for Riemann Sums, but with some foresight to know where we are going.
Side note: Simultaneous Show and Tell is a terrible name for this whiteboarding mode (because it kind of sounds like a lot of whiteboarding). Forgive me, I cannot think of anything better. So… propose a better name?
[update 11/25: Andrew in the comments suggested "Function Iron Chef" which is definitely the winner. That's what this whiteboarding mode is called now]
Students are in groups of two at a whiteboard with a VERY LARGE set of 3 X 3 axes drawn up on the board. They are sitting in a U shape so that if everyone put up their boards, every student could theoretically see everyone else’s. I put up a prompt like this:
Draw a function such that…
- does not exist
- does not exist
- and .
I put the timer on. Students are given a few minutes to draw a function (any function, lots of correct answers!) that fit the prompts. Then, at the end of the time, everyone puts their markers down and puts their board up. We spend a minute silently looking around the boards to look at everyone else’s work. Then, after a minute is up I allow the students to ask questions of each other (i.e. not just say “THAT ONE IS WRONG”). If they don’t ask questions about some that are suspect (or some that are totally correct), I will ask questions at the end to talk about specific boards. We then do 5 or 6 other rounds like this.
POSITIVES: We have done this so far with limits, continuity vs. differentiability and will do it in a few weeks with graph sketching – I think that making them do things the other way around, making them create (instead of just identifying limits or whether a function is continuous) really forces them to think harder. I also like this because when students have to show their work to their classmates, they often put a little bit more focus into making sure they are proud of what they have (and just about every student is engaged in the process, especially if you make them switch markers). I also love times to showcase mistakes as part of the learning process - we try to be as open and supportive as possible in correcting the boards. Lastly, having a discussion in a math class is always a really nice change of pace.
ISSUES: Students can get a little crazy during the discussion process and some can phrase things negatively. Not all students are good at following along verbally when discussing, and will wait for others to point out mistakes in the board. A few times the whole thing has taken a long time with all the transitions, but it has gotten better every time. I’m not sure how the weak students feel about this activity (having their work showcased and critiqued). Also, I’m not sure that this type of activity would be great for anything but a topic where the students already have some fluency and mastery.
Enjoy. I know my students will, and it will actually give us a funny and memorable way to talk about and avoid these common algebra mistakes.
The other thing that I showed them today to get them to stop just playing around with letters while doing Algebra is the following, which I believe I picked up at a summer workshop:
They literally laughed out loud at this. I said (in a funny, not mean and not sarcastic way), “You think that’s funny?!?!? This is the kind of stuff you guys do on quizzes. When I am correcting your work I sit and laugh and laugh and laugh at the crazy things that you do! No more crazy algebra!”
How do we stop/prevent crazy algebra mistakes besides carefully and repeatedly addressing them when they happen? Any ideas?
I’ve been getting pretty into cognitive science lately. I realize some of it is useless, and a lot of the rest of it is made up of kind of common sense things once you really think about it, but regardless, I have found it so helpful to put scientific names and research to intuitions I have in the classroom. One of the ideas that I have really liked (from Daniel T Willingham’s Why Don’t Students Like School?) is that we learn everything by connecting it to things we already know, and much of what we already know is concrete. Thus, the more you can teach through concrete examples, the more likely students are to learn the material.
EXAMPLE: Speed, iPhone prices and the Intermediate Value Theorem
This year, while teaching the Intermediate Value Theorem in AP Calculus, I did not start with the theorem itself, as I always find that language so intimidating for what is actually a simple idea. Instead I started with this:
I showed them a video of a speedometer that cuts out for about 10 seconds in the middle (ah, you’re dizzy and you pass out for a second at the wheel!). Before the cut out spot, the car was going 60 mph, and after it was going 100 mph. I then asked the to tell me:
- What was a speed that you are 100% sure that you must have gone in the time in between? Why?
- What was a speed that you could have gone in the time between, but you aren’t 100% sure? Why?
We talked about this for a few minutes, letting the students argue a bit about their thoughts and came to an agreement as a class. Then I put up a new picture that showed the original iPhone prices at some intervals. It started at $599, a few months later was $399, and then two years later was $99. Then I asked very similar questions:
- What was a price that you are 100% sure that the iPhone must have had in the time in between? Why?
- What was a price that the iPhone might have had in the time between, but you aren’t 100% sure? Why?
Again, I let them argue for a bit and discuss. After we had settled on answers, I asked what was different about the situation, keeping in mind that we had already discussed continuity in the class, but I had never mentioned this in this situation. Students said wonderful things like “To get from one price to another, the iPhone doesn’t have to pass through the other prices” and “Prices can can jump whereas speeds can’t” and I let them continue to do that until one student finally realized “Speed is continuous, whereas price is not!”
Prepped with the ideas of theorem, we took the speed situation and translated it into a mathematical theorem before looking at the actual Intermediate Value Theorem. It took about 10-15 minutes of class, which was well worth having a strong conceptual understanding of the theorem. Students still struggled mightily with proving anything with the theorem (as they have in proving anything mathematically both this year and in previous years – any advice there?) but the conceptual development of the idea was not only quicker, but I think stickier.
Isn’t that better than starting with this?
I have been reading and enjoying (though not fully buying everything in) Daniel T Willingham’s book Why Students Don’t Like School: A Cognitive Scientist Answers Questions About How the Mind Works and What It Means for the Classroom. One of the ideas that I think is really useful in planning instruction is that humans are wired to enjoy learning – some scientists believe that the brain releases a little bit of dopamine every time we solve a problem. We actually physically get pleasure from solving problems.
As an example, check out these two word picture puzzles (figure out the common expression indicated by the words and their placement):
Which of the two puzzles did you enjoy more? If you’re anything like me, or most human beings, you didn’t really enjoy the one that had the answer right above it. Even if you didn’t figure out the other one, you probably at least thought about it more than the other one (though Willingham points out that the physical response only occurs when a person solves a problem). How often do we give the answers to the riddles first in math instruction?
Here is a rule and here are examples of every type of problem you will have to do with it, now do problems like those even though you kind of already know the answer.
An example of posing math as a riddle instead:
It took a few days for students to learn the power rule this year, as opposed to me just writing so , which takes about 10 seconds (if you talk while you are doing it and write very, very slowly, and have to erase something in the middle because you forgot what you were doing). Despite the time needed, I felt that the cognitive payoffs with the progression I used were great, and students really internalized the idea because THEY FIGURED IT OUT THEMSELVES. Figuring out the Power Rule is something that is totally in their reach, and I would have been robbing them of some learning pleasure had I just given them the power rule at the beginning.
PHASE 1: What is a derivative? We started just by drawing tangent lines to at a bunch of points, estimating the slope and then making a table of values. I chose this function specifically because with the derivative of , it’s easy to see the pattern for the slopes in the numbers without graphing them (saving one level of abstraction). Yay, the slope at any point is just twice the x-value!
Then we did this two more times, once on a small sheet of paper for , and then once, in groups, on a huge sheet of butcher paper for . This was laborious and took a ton of time in class, but by the end I felt like students really understood well the idea of a derivative. More importantly, were ITCHING for an easier way to find it. They had all these great ideas that they were proposing, so it was easy to funnel their energy into the next phase.
PHASE 2: Finding the Rules. Then, I introduced the derivative tracer, a GeoGebra applet that does in seconds what they did in 15 minutes. I gave them a sheet of functions (below) for them to find the derivative of using the derivative tracer (kind of like collecting data in a typical canned high school science lab) and asked them to make conclusions about the derivatives that they found.
Though it was interesting to talk through with them the idea of a constant function’s derivative being 0, and a linear function’s derivative being a constant, the highlight of the lesson was seeing students figure out the power rule. When students got to that section, they seemed really proud that they could see the pattern. I had numerous students raise their hand to call me over to ask me if their idea worked, and then were so excited that it did that they immediately gave me a high-five. Students raised their hand so that I would come give them a high-five… in math class. I know the power rule is kind of easy, but I felt like they were so much more invested in the quest of learning mathematics because they figured something out themselves. Further, instead of trying to get ideas from my math notation, they had the ideas first and then I formalized it with math notation (though many students could do this no problem for themselves).
Long story short: The excitement in the room while the students were discovering something mathematical was palpable, even though that thing had been discovered many many many times before, including by their classmates sitting a few seats down. There was no “real world” motivation in this progression, no gimmicks – just the pure pleasure of mathematical discovery. So, to add to my ever lengthening list general goals for the year: I hope to avoid at all costs robbing students of the pleasure of figuring something out for themselves.
Here is my derivatives “lab” using the GeoGebra derivative tracer. Note that I’m not quite as adventurous as some and still want some structure in the classroom while “discovery” is happening. This is part of my controlling personality – tell me if you think this is too guided given my goals.
By the way, the answer to the other word picture puzzle is “mathematical induction.”
I’ve been toying around with my learning objectives for Standards Based Grading in Calculus for three years now, and I want to get some other people to weigh in on what I have. Please, take a look, tell me what you think!
- I love the first person language, which is an idea I think I stole from @kellyoshea.
- The physics modelers all have crazy acronyms for their standards like CVPM and UBFPM and ERMAHGERD. These seemed confusing to me at first, but then I thought that students might really benefit from this. The standards aren’t organized around chapter numbers, or something else arbitrary, but rather BIG DEEP IDEAS (models!). I wanted to do something similar for Calculus, so I organized mine around Local Linearity, Slope Functions, Proportional Rates and Accumulating Change (with short, simply worded descriptions in the document below). I don’t know how well this worked last year, but one goal for me is to try to always relate the standards back to their big ideas.
- I didn’t do the standards like this fully in order, and this year I am totally changing the order. But just to give you an idea of how I did things, I did all the IP and LL (limits) standards, then SF.a through SF.g (basic derivatives), then PR.a (optimization), then SF.h through SF.n (graph sketching), then PR.b-PR.h (exponential functions), then SF.o/PR.i (implicit and related rated), then all the AC standards. It was a bit confusing to go back and forth, but organizing the standards like that made it make so much more sense to me. Tell me what you think about that…
- I struggle with how general/specific to make the standards, and how to include both calculation and interpretation into the standards. Sometimes I split the two, sometimes I kept them together. This is the hardest thing for me!
Anyway, any thoughts are necessary! These are my standards from last year, the second time I taught Calculus.
Calculus Standards 2011-2012
I really enjoyed doing final projects with the kids this year (which may be patently obvious considering that this is my 6th post on the topic). It’s such a fun way to end the year, seeing them get excited about doing something interesting with Calculus and coming up with ideas about math that I never would have even dreamed of.
But projects can also be very frustrating, and hard to implement. Here are the things I struggled with this year. I’d love any feedback or tips.
- Since these projects were very open-ended, some students felt a bit lost, and I struggled a lot with how much guidance to give and similarly, how much to let them struggle. I just find it so hard when we have such a short amount of time to see them getting thrown off in a crazy direction, especially if it’s going to lead them to a lot of useless work. I tried so hard to “be less helpful” but I just couldn’t resist sometimes! Part of me feels like I am stealing a bit of a learning opportunity from them and part of me feels like I am just advising them to help guide their crazy teenage thought process. Also, some students just started working on their projects without really knowing why they were doing what they were doing (they just wanted to do “something about optimization”). I wanted to help them do something for their idea without turning it into my idea, but I’m not sure how well I did at that.
THOUGHTS: I think that I am going to try to have them submit proposals next year where they present some sort of thesis, or a guiding question they are going to answer in their project. This might get them to plan out their project a bit better before starting, give me a chance to give good feedback and also give them an overall question which will really guide their whole project.
- One thing that I was continually frustrated throughout the week in class that I gave them to work on the projects was that students did not work very efficiently, leaving much of it for the end. Part of it was that they just had so much time in class, but part of it was that I have no idea how to help them structure their own project to use class time well. I had tons of students show up without materials to work on their projects, and even some who would sit there and do nothing telling me that they were just going to finish at home.
THOUGHTS: I wanted to do a midpoint deadline of some sort, but because all the projects were so different, it seemed really weird to me to organize something like that. I might try having them make a schedule in the beginning of the project, but I’m not sure how to help them stick to that, or if that is even worth all the work that it would be.
- Similar to supporting them in organizing their time, I struggled helping them work well together with each other. I think group work like this is crucial in high school to learn how to structure time with someone else and communicate about a project, but the students were terrible at this. They would do things like not show up to class without telling their partner, even though they had all the materials. I even had to mediate an email war between two girls who were flipping out at each other about who was doing less for the project.
THOUGHTS: Maybe this isn’t something that I need to do something for, and maybe this is something they just have to learn by doing the project, but perhaps I could find ways to help them structure their roles in the project beforehand, or maybe just do more long-term projects like this over the course of the year.
- Last, I really want them to show off their work to each other, but I’m not sure how to make class presentations anything but the boring yawn fest that they tend to be. Students did some really cool things, but were really bad at explaining those things in a way that the class could understand. Also, it’s really hard to listen to two full class days of presentations, even for me, and it’s really hard for students to get anything out of the presentation when they are not really expected to engage in a meaningful way (not one of the presentations was interactive in any way).
THOUGHTS: I’m looking for some sort of other structure to make it more interesting. Maybe some sort of gallery walk type structure? And I also want some formal way to get those listening involved so that they really pay attention and learn – some sort of commenting system, or interactive component. It’s very hazy in my head, but this is something I am going to try to flesh out over the summer.
Any ideas would be greatly appreciated!
(Also, below is my rubric for grading these projects)
And the last project I am going to detail…
A few students did a pretty standard, but well done optimization project investigating different can shapes to find which one is the most efficient (Sam profiled his kids doing a very similar project, I loved reading his students’ reflections on it!). Then they redesigned the cans to help companies lower cost. The reason that I am profiling this because it made me realize what students find interesting in this whole optimization nonsense – I brought in cans in the winter when we first learned optimization, and we did something similar, but we never talked about the issue that really got other students’ attention…. money! I had been focusing on the shapes, but I should have been focusing on money! (Seems like a super “duh” in retrospect, and it’s not anything original, but helpful to realize nonetheless).
The students did tons and tons of calculations, but what I really loved is that they compared the price of producing the current can that the company produces and the price of producing the ideal can. They looked up the price of aluminum and estimated (or looked up? I’m not sure here) how many cans per day a factory would produce. After a bunch of multiplication, they showed that tiny, tiny changes in the shape would result in savings in the hundreds of thousands of dollars range for a year (see red number below), which is super cool.
Also, they had a really nice framework for their project. They pretended they were a packaging consulting company and even came up with a logo and a name that combined their names. I thought that was great!
NEXT YEAR: I am going to frame my optimization unit much more in the way these students went about it. I feel like this is a complicated mini experiment in terms of #anyqs – the students found for me what the actual interesting question is. For me, the shapes of the cans themselves is interesting (especially that it ends up being such a beautiful ratio), but I think a lot of kids were really amazed at how a small change in the size of the cans can result in huge savings and led them to wonder why all cans aren’t shaped the same way. So, thanks for helping improve my curriculum, (now former) students!
A student’s mother is completing the Hajj this year, the pilgrimage that Muslims take to Mecca. This is one of the five pillars of Islam (along with prayer, fasting, charity and testifying that there is only one God). All physically and financially capable Muslims must carry out this pilgrimage at least once in their lifetime. This student based her whole project on the Hajj and calculated many different things about it. Specifically, she calculated:
- How long it would take to complete each part of the Hajj (once you get there, there are certain rituals during which the pilgrims walk to various places). She used aerial photographs and official information to measure the distance (around 40 km!) and then used an average person’s walking speed to estimate that each pilgrims walks for around 10 hours during the Hajj.
- How many people can be expected to attend the Hajj in the future given data from the past 10 years and assuming exponential growth. She used previous data and the basic exponential growth model to make predictions for the next 30 years.
- How large the current area around the Kaaba is (the holiest site of Islam around with the Hajj is based). She used GeoGebra and Google Earth software to measure the area.
- And how much the area will have to increase in future years to accommodate the extra pilgrims. Based on her predictions of the increase in the number of pilgrims, she mapped out how big the area around the Kaaba will have to be for the pilgrims to all have the same amount of area. She thought it was cool they they would have to restrict the number of pilgrims, or knock down highways in order to keep the area per person the same.
The math wasn’t perfect and there were some crazy assumptions made, but I absolutely loved this project. It was from someone who had told me in the beginning of the year that math wasn’t her thing, and it was really cool to see her get excited about the project because it applied to something really interesting. All the math was very well motivated and taken from a wide range of things that we did this year. Great stuff!
NEXT YEAR: I could see doing some sort of city planning project involving Google Earth that somehow involves population growth. It would be really cool to look at current rates on population increases in areas and see what that would mean for the physical space. I am so happy that a lot of these final projects have translated into great teaching ideas!
For their final project, one group decided to make a twitter account and track how many followers they gained over time. The account was called “UknowURatKings” (King’s is our school… so YOU KNOW YOU’RE AT KING’S for those who hate txtspeak). They tweeted inside jokes about the school that you would only really get if you were pert of our community. I was following them, which was good because they ventured into inappropriate territory once (it was a nice mini experiment in social networking with students!). Here was my favorite tweet of theirs:
They had predicted that the followers function would follow a logistic model. Using a few data points, they created a logistic model of their own: they thought they would max out at around 100 followers (the size of the senior class population on twitter plus some extras), they originally told 13 people, and after one day they had something like 40 people (unfortunately, I can’t find where they uploaded their project ahh!). Based on that they created their logistic model. Then, they tweeted furiously for about a week and recorded how many followers they had each day. At the end, they compared their results with their model…
They were way off. Though they had chosen the right model, the number of followers increased slower than they thought and maxed out around 60, not 100. My favorite part of their project was that they didn’t try to fudge their numbers or make the data fit their model – instead, they talked about their assumptions that may have been flawed, their tweeting behavior skewing the results, and inconsistencies in data collection. I <3 data.
NEXT YEAR: I thought that this was a really fun and simple project, and it might be something that I try to do with my whole class when we study exponential models next year (I swear I could teach a whole term on just the logistic function). I think we could have an awesome discussion about modeling with all the different inconsistencies that will arise, and we could even add a competition component, to see who can get the most followers for their account under certain constraints… Too many ideas, too little time.