Category Archives: Calculus
The Related Rates of an Automatic Pizza Saucer
I always struggle a bit with teaching related rates, because the focus of all textbook/AP problems is to calculate how fast something is changing in one single moment in time, which short shrifts the beauty of the topic. The main idea seems more to be thinking about how two rates are, well, related to each other. What must happen to one rate as time goes on if another rate stays constant? That’s what makes that classic ladder problem even remotely interesting (why the hell would one side be moving at a constant rate while the other is speeding up?).
I saw this gif of an automatic pizza saucer a while ago and immediately thought it would be a fabulous discussion piece for this very idea:
Someone who designed this used some sort of calculus, even if it was the intuitive kind! We talked about this in class for 15 minutes or so, and the students that enjoy wrestling with not-so-obvious and applied situations really enjoyed thinking about these types of questions:
- If the pizza spins at a constant rate and the sauce comes out at a constant rate, what has to happen to the speed of the arm?
- If the pizza spins at a constant rate and the arm moves at a constant rate, what has to happen to the rate at which the sauce comes out?
- If the arm of the pizza moves at a constant rate and the sauce comes out at a constant rate what has to happen to the rate at which the pizza spins?
We came to some interesting conclusions about the above, including that one of the situations above is not possible (I think!), which we only figured out because one student and I were vehemently defending two different and opposite things.
Anyone want to try to throw some numbers on this to figure out these very questions?
Building a Better Review for the AP Calc Exam
Both years I have taught AP Calculus AB, I have kind of dreaded the couple weeks of review. They are hard to plan for and probably really boring for students. On top of that, last year, I felt like I squandered the review time. I mostly gave them free time in class to do whatever they needed to do, and I am not sure how effective this was. 45 minutes straight of studying really dragged and I felt like students didn’t really even know what they needed to work on. In addition, the lack of structure I think prompted some students just to look at answer keys instead of struggling through problems themselves.
This year, I was dreading review again, but it really went much better and I think was much more engaging and effective. A few things I changed this year:
1. We started reviewing in class earlier, even before we finished all the material.
2. Review was more structured by me at first and slowly led to more independence, with opportunities for students to see which topics they needed most work on.
3. I spent literally 1% of the time explaining at the front of the classroom and 99% of the time having them to do the work.
The learning structures I used for review:
- 5 MINUTE SKILL DRILL
- TIMED FREE RESPONSE QUESTION
- MULTIPLE CHOICE JIGSAW
- MOODLE MULTIPLE CHOICE
- MOCK EXAM
- and then…. FREE TIME WITH PAST QUESTIONS
5 MINUTE SKILL DRILL
For about 3 weeks before we started review (i.e. while we were still doing DiffEQs, volume etc), we started class with 5 to 6 quick skill based questions. I tried to put a range of topics, from evaluating limits to writing a tangent line to finding the average value of a function. Students pulled out their notebooks and worked on the questions silently (or as silently as I could get my 85%-chatty-bro class to work). They did the ones they could do and tried the rest. After the timer ran out, I would scroll down and show the answers and show what Standard that the question corresponded with. Then, after explaining anything that needed explaining, we would vote as a class on whether to retire a topic if they felt confident or keep it on for the next day. This took about 10 minutes at the beginning of class.
I loved this because even though it ate up class time during the end of the year and forced the actual material to take longer, by the time we were ready to review, students had already brushed up on the skills and could focus on big ideas.
Next time, I will try to be more organized about it and have a booklet printed, or sheets for them to glue – I was improvising with this and I felt like it took students too much time to copy things down from the projector.
(this is sort of what it looked like below, but this is for integrals earlier in the year – I’m between computers right now and don’t have all my old files!)
TIMED FREE RESPONSE QUESTION
At the end of many units towards the end of the year, we would do a 12 minute timed Free Response Question, and this is something that we did almost daily during our review time. I would hand out a free response question on a little slip of paper, they would glue it into their notebooks and work on it for 12 minutes silently. If they didn’t know how to do it, they would just try as hard as they could, struggle through it and write down what they know. Then, after 12 minutes, I would hand out the answer key and they would grade themselves, AP style.
I loved that this forced them to struggle through a question and see what they actually know, and I loved that this got them used to AP grading (I had a much lower incidence of unit-forgetting and less-than-3-decimal precision). The trick for both of these benefits is in really holding out on the answer keys until the end of the time!
Next time, I will try to coordinate the 5 minute skill drill with this so that students can recall the topic before a tricky free response question, as I had some students who were so stuck that they didn’t really write anything down and got nothing out of the exercise.
MULTIPLE CHOICE JIGSAW
I find multiple choice harder to integrate into class than free response, but one learning structure I liked for multiple choice was Jigsaw. For those that don’t know this (I assume it is fairly common), there would be a set at 12 questions and groups of 3-4 would all work on a third of the questions together (1-4, 5-8, 9-12). Once every group got through theirs, I would rearrange the classroom so that each new group had one person who had worked on each of the sections. Then, they would either work on the rest of the questions individually and then check with each other when they got stuck, or they would just take turns and teach the other members of the group their questions. Some students reported to me that the process of explaining a question out loud really helped them understand what was going on.
I loved the interactions that this activity prompted and I loved how efficient it was for getting through many multiple choice questions (students could do this much faster than working on them on their own).
Next time, I will try to deal with the awkwardness of groups finishing at separate times and weak students incapable of explaining questions to their classmates, though I am not sure how.
MOODLE MULTIPLE CHOICE
I didn’t trust my students to do free response questions at home. They would just look up the answers and get NOTHING out of the process! But we did do a lot of multiple choice questions at home, through Moodle. It is super easy to set up quizzes, so I would just upload images of the questions from a multiple choice collection I had and set the correct answer. I would do 15 questions in a quiz, and it would take my students about 40 minutes to do. We started this about a month and a half before the exam, and then all the homework during the review time become these online multiple choice questions. Before the test, every single student did about 130 multiple choice questions, which amounts to about 3 full tests, and then many did more questions on their own outside of that.
I loved that the work was immediately self checked and automatically graded, as I think this did a lot for their learning from these questions.
Next time, I don’t think I would do so many of these as I think they got a bored with them. Also, I felt like some students were just clicking through the questions, so I would try to think of ways to get them to take these learning opportunities a bit more seriously.
MOCK EXAM
This is, of course, nothing original, but if you have the luxury of stealing a few hours from your students on a weekend for a Mock Exam, do it! Correct it for them, but don’t put a grade on it so that it can be a truly diagnostic tool. This was the most helpful thing for my students in prepping for the exam, because, on top of everything else, the Mock really helped them figure out their weaknesses so that they could really be productive when finally I gave them…
FREE TIME WITH PAST QUESTIONS
By the time I was giving them large chunks of time to work in class on their own, most students knew what their weaknesses were (from the Mock, timed Free Response, Moodle Multiple Choice etc). Whether they needed to improve their multiple choice or their free response, or they needed to work on specific topics (and could with a packet I gave them with AP Free Response questions split up by type), I felt like most students REALLY used the time well, to the point where a lot of students didn’t even bother studying the night before the exam. All the structure and diagnosing we did at the beginning, and all the work that THEY were doing instead of me talking helped them become far more independent and effective in the review process. I hope it worked – I will find out in a few weeks!
Any review structures you used that worked well?
Volume in Calculus: Conceptualizing before Formalizing
One of our PD sessions in the past was about how to support students with learning differences. One of the points that the presenter made was that most pedagogical tools that you would use the better serve these students are great tools to reach all learners. This struck me especially because I teach almost entirely students for whom English is their second language, and sometimes when I do something specifically to help students with the language of mathematics I come to larger and more general pedagogical understandings.
For example, this past week, I introduced solids of known cross-section in AP Calculus in a way that I thought would ease my students understanding of the tricky language involved in the problems, but what I ended up doing was really effectively let them develop their own conception of how these solids are formed and THEN interpret the AP problem language and integral notation in those terms. Conceptualize and then add mathematical formality to their own conceptual framework.
Here’s how it worked. I put 4 of these solids out around the room:
- First, I gave them 1-2 minutes to SILENTLY write down in bullet points how they would describe to someone else how the solid was formed.
- Then I gave them 2 minutes to share ideas in groups.
- Then I cold called on 7 or 8 students via a deck small cards with their names on them (which is by far my new favorite teaching tool). After I called on some students, I called for volunteers with any other ideas.
- LAST, I asked them to flip to the back of the paper and read the actual description.
During the “share” part, students said some of the craziest, random stuff, but most of the important parts of the description were said by various students. When it came time for them to read the description, at first they were like “whoa” because the language is still a bit daunting. But after a minute or so of close reading, they connected everything in that description with things that they themselves had said. So when it was time to do the actual integral, the intermediate notation I use made 100% sense:
So general pedagogical moral of the story? Letting students conceptualize something on their own before bringing in mathematical language and notation makes it more likely that the notation will aid in their understanding rather than provide another hurdle in learning.
A teaching fellow (a first year teacher) was observing my class (and has been observing periodically throughout the year). Afterwards, she remarked that she felt this was one of the most effective 10 minutes of the year, and I agree! And I think 10 minutes on this (instead of just 1 minute reading the question) will save lots of time in the future. Next week, I hope to try the same strategy with solids of revolution!!
An Anchor Problem for Riemann Sums
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).
(isn’t this building beautiful??)
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.
Whiteboarding Mode: Simultaneous Show and Tell
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.
The Dead Puppy Theorem and Its Corollaries
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?
Posted with BlogsyTeaching Through Concrete Examples: The Intermediate Value Theorem
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?
Let THEM Figure Out the Power Rule
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 
PHASE 1: What is a derivative? We started just by drawing tangent lines to 
Then we did this two more times, once on a small sheet of paper for 
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.”
Calculus Standards 2011-2012: Feedback Requested
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!
Some notes:
- 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
Project Implementation Reflections and Questions
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)
Bowman in Arabia 