Category Archives: Calculus
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 ❤ 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.
**The next few posts are going to be spotlights of final projects that students did that I thought were cool or interesting and then a few reflections on doing final projects in general. I could picture doing a lot of my student’s projects as a whole class!**
If I had one more week in my non-AP Calculus class, we would study volumes of revolution. That’s probably the biggest weakness of my course right now, and I am trying to figure out a way to include that next year. A junior who is in my regular class and is taking AP next year was a bit lost when coming up with an idea, so he asked me for a topic that we do in AP but did not do in our class so he could be a bit prepared. I suggested volumes of revolution and after a lot more nudging and guidance and idea planting than I did for other students, we decided that a good project for him would be to recreate an interactive 3D model of a solid of revolution using GeoGebra and Winplot. (actually it works with solids of known cross section too).
Here’s how it works…
1. Upload a picture into GeoGebra (he chose a huge vase from the art room). Fit functions to the edges of the object on the part that will be revolved.
2. Recreate the same exact functions in Winplot (which has much better 3D capabilities than GeoGebra does).
3. Use Winplot’s revolving capabilities to revolve the surface around an axis (any axis!). And then, voila, you have a 3D model of your object that you can use the arrows on the keyboard to rotate in any direction. It actually ends up being really impressive – my student told me that he left the model up on his computer and every time he would turn it on he would rotate his vase a bit.
After I saw the success of this project, I suggested the same one to a few students in my AP class (who were required to do a much more low key, shorter version of a final project because of time restraints). They decided to recreate a bunch of sports equipment using the program, which I thought was a really cool idea! Their rotate-able objects:
NEXT YEAR: I made an instruction sheet for those AP kids because they had less time, but I’m glad I did because this was a really cool project and is something that I can see myself doing with a whole class next year. Here it is below. If you haven’t tried making any 3D models (not necessarily real objects) with Winplot, definitely try it out – it’s super cool!
I wrote about this project back on Sam’s blog this summer when Sam gave me reign of his kingdom for a month or so, but I wanted to share the student work that I got this year from it, because it was much better than last year, and some of the work is actually really beautiful/cool/interesting (Math Art, MArTH anyone?).
The basic premise of the project is to RECREATE A PICTURE USING INTEGRALS by doing the following:
- Upload a picture into GeoGebra.
- Place points around all the outlines making sure to hit critical points
- Fit functions to the outlines.
- Use integrals to shade in the areas between the outlines.
I initially waffled about whether this was a worthwhile problem or just an exercise in integrals, but having taught AP Calculus this year, I realize how these problems of just finding the area of a weird shape are interesting and important for deep understanding of the connection between a Riemann sum and how the integral actually calculates area. So basically, if you think that this is a worthwhile problem…
Find the Area of R and S given that f(x) is blah blah and g(x) is blah blah blah squared.
…then this project is just a glorified, more interesting, more complex version of that problem. If you don’t think that problem is worthwhile, well, then you probably wont like this either. Regardless, it was a great thing to do to hammer in ideas about finding the area between curves, and a great learning mode while AP’s were occurring because attendance did not really matter all that much. It took most students 3 and a half 45-minute class periods (so about 2.5 hours), though I think that more efficient students not freaking out about standardized tests, and consistently present in the classroom, might be able to do it a little quicker.
SOME OF MY FAVORITES:
ALL OF THE STUDENT WORK:
(the good, the bad, the ugly!)
GEOGEBRA INSTRUCTION SHEET:
For the last week and half of school, my non-AP Calculus class is embarking on a free choice final project. The only requirements are that they must use some sort of Calculus, they must use a real artifact (data, a picture, a video, history etc), they must incorporate technology, and they must find a way to present it to their peers.I have been so excited to see their creative streaks and see some of them get really excited about this, especially because I am impressed that they are still energized two weeks away from their graduation.
Here are some of my favorite ideas. Note that some are not very sophisticated, but are interesting nonetheless and I have been supportive regardless, as I want to see them really carry out something that they feel is their own. I will report back on these after a week and a half when they are done.
- COMPETITIVE EATING RATES: A few students want to eat as many chicken wings as they can, but as they go, time when they finish each one. Then they are going to calculate the rate at which they are eating wings at a few points during the eating. Their prediction is that the more wings they eat, the slower they will eat them. I am hoping they will try to fit some sort of exponential function to the data (that might tell them their limit). They are going to compare their rates to that of an actual professional eater.
- ATTENDANCE TO THE HAJJ: The Hajj is the annual pilgrimage to Mecca that Muslims embark on once in their lifetime (or sometimes more). One student wants to look at aerial photographs of the Hajj to determine the area that the pilgrims fill up and compare the relative areas from different years to the relative levels of attendance. Then, she also wants to make functions for an old man, a young man and a woman doing the hajj that will give their position at any time given the size of the crowd in a given year.
- THE SPREAD OF SENIORITIS: A couple of students are collecting data from their friends about their GPA throughout the year to see how real senioritis is. Then, they are going to use the idea of differentials to expand on the data and predict students’ GPAs in future terms (college?) given their current slide.
- DESIGNING A GREENHOUSE: One girl wants to make a model of a curved-roof greenhouse and then use Calculus to find the amount of glass used and the volume. She also wants to do some sort of optimization exploration to see if the shape has to do with using the least amount of glass for the most sun exposure.
- CELEBRITY LAND AREA: One student is using Google Earth to find the area of various celebrity plots of land. Then he is going to compare the Google method to numerical methods (like Riemann sums and trapezoidal sums) and he is going to try to determine how Google’s mechanism for finding area works.
- INFECTION: A student has a game on her iPad where a disease is being spread around the world. I can’t remember if the object is to infect the world or to save it. Either way she is going to pick a few regions and track the spread of the disease through those regions to see if the curves are logistic, and to see how the curves of regions close to each other relate to each other.
- DERIVATIVE/ANTIDERIVATIVE CHECKERS: Two students are going to design a checkers board to practice derivatives and antiderivatives. The checkers will have derivatives on one side and antiderivatives on the other. When you jump a piece, you have to solve a derivative or antiderivative before you can capture the piece.
- GATSBY’S OPTIMAL PARTY: One student is going to design a prompt from Gatsby himself asking Calculus students to optimize his guest’s happiness at a party. I don’t know the details, but the sense I get is she is going to give Gatsby a limited budget and things that he could purchase for his party – I’m excited to see how this one turns out!
And there are lots of other great ideas too! I liked the ones above because they took one of my ideas for a prompt and totally made it their own, or just came up with something totally random that they wanted to do. I’m excited to hear how these turn out. I had a million other ideas too… here is the packet of ideas that I gave them to get them thinking.
Skills instruction was something that I was not good at when I first started teaching. I found it boring compared to big ideas and didn’t really understand why kids didn’t get things from me doing problems on the board and then them doing practice problems with each other or individually or for homework. For me, it was like “What is there to understand that simple practice wont solve?” Well, I have grown a lot since then. This year, I have implemented a lot of new tools (many whiteboard based) that have really helped out with my skills instruction. I feel like I really had a great sequence this year when doing the skills part of integrating with my regular class – I wanted to share and reflect, especially so it’s written down somewhere for me to use in the future, because some of the ideas will be useful when I teach other topics.
Just so you know how I lead up to this seminal topic in Calculus… First, I spent a considerable time (about five 45 minute class periods) exploring everything to do with Riemann Sums, both in terms of pure area and what area means in applied situations. I think that feels like a lot of time, but we tackled the conceptual side of integration very throughly and used that to motivate the idea of an antiderivative/integral. Once we motivated the integral, we focused on learning how to find antiderivatives, which is the part I want to talk about.
1. Guess and Check With a Partner
With inspiration from a great worksheet from Sam, I wanted students to rely on their intuition at first to find andiderivatives, instead of relying on formulae. I’ve tried things like this previously, but it really helped this time to explicitly explain that this is what we were doing – that maybe eventually we can rely on a rule, but we are going to discover the math first. I paired them up with whiteboards and set them out with the list of functions from Sam’s worksheet. Their goal: find the antiderivative of all the functions. The method: each person had a marker. One person would write down a guess for an antiderivative, and the other person would simply take the derivative of this to see if it went back to the original function. They would keep doing this until they got something correct, then write that answer down on the sheet. Then, after one person has been the “guesser’ four or five times, they switch. Example:
For the kids that actually did what I asked (others just kind of started solving them on their own, which is okay I guess), it was a really nice exercise. They worked together really well, and were so excited to tell me the rule that they had made up for integrating power functions. I had them even doing simple substitution, per great suggestion from Sam. They got good at just getting themselves to try something, and getting in the habit of checking all of their answers. One kid at the end of class told me “My brain hurts from thinking so much.” Then, after the students were done, the next class we started by collecting rules they had noticed, and it made a nice little automatic cheat sheet for them. –> SHEET WITH FUNCTIONS HERE
2. Power Rule Folding Game
Next was to tackle more complicated functions with which we could use our rules, mainly negative and fractional powers. I did this same exercise in the fall when learning how to differentiate these functions to much success, and then tried it again with differentiating power functions to much confusion (so I guess the activity has a specific niche). The idea is that everyone starts with a problem, does one step and folds over the sheet so that only their work is visible. Then everyone rotates their problems around. The next person does the next step, and then folds the paper so only their work is visible. The group keeps rotating the papers until they are all done, then they open them up and look for mistakes (if there are any). Example:
This was good for helping them drill some algebraic manipulation and develop the skill of checking their own work for mistakes… all while working very closely collaboratively. –> FOLDABLES HERE
4. Flip-Up Answers for Initial Conditions
After learning basic integration skills, we began to talk about how functions have more than one antiderivative, and how sometimes it is useful to find a specific one. After only one or two examples together, we immediately just started practicing this idea with an activity that I stole from Mimi where I placed problems around the room with the answers on the back, the idea being that students would go solve whatever problems they felt like they needed to. Example:
I enjoyed this for many of the same reasons that Mimi cited in her original post. Students could work at their own pace without feeling like they were falling behind, could pick their own problems, and could move around the room to interact with many different people (which are all huge advantages over just doing a worksheet). –> FLIP PROBLEMS HERE (though the formatting is a bit screwy)
5. Mistake Game
After two days of a little bit more traditional style instruction just to make the connection between the definite integral and area (a lesson that I need to make more discovery based next year), we then did the Mistake Game, an idea from Kelly, which I have described a few times now. Basically students work out problems on whiteboards and hide a mistake in their solution. They then present their work like as if they didn’t make a mistake and the other students have a discussion to try to find their error. The problems I chose for the mistake game where all functions for which you had to do some sort of simplifying before integrating (like distributing or dividing), which ended up being a great way of pushing them a little bit forward while giving them plenty of opportunity to really go in depth discussing this new mechanical process of a definite integral.
6. Substitution Marker
Then the last skills activity I did with integration was a few days later when we started doing substitution. I had them first try a bunch of substitution problems intuitively, and then showed them how to use a u-substitution. Then, we pulled out the whiteboards and I gave them all a sheet of problems and two markers each. They were to do all u-related work in red, and all original-integral related work in blue. What I wanted them to get comfortable with was envisioning the transition between the variables and helping see how the skeleton of the integral becomes the “outside function” of the backwards chain rule. Example: (actual student work)
This was, again, one of those activities where a bunch of students totally ignored my directions and just solved the problems (and again, not the worst thing), but I think some of the students that did it like this really benefited from using the different colors.
So why did I just ramble about all those activities? I guess what I loved about this whole sequence is how ridiculously much of the instruction for a good week and a half or so was collaborative and engaging, and forced them to think about what could have been routine material in different ways instead of just plowing through worksheets and drills. I feel like I never would have been able to pull something like this off even last year, so I am so grateful (especially to the online community) that I now have a toolbox full of sweet teaching methods. My goal is to try to mix these types of activities more often into various units, since most have skills based components. I would love any other modes of instruction that you use in your classroom to add to my toolbox!!
Side note for the Calculus people: There are a few antiderivative/integral related traps that my students fall into… any ideas on how I can stop these problems before they happen?
- I always start with the word “antiderivative” to emphasis that it’s the opposite process of a derivative, and then try to transition to “integral” as soon as possible, but it’s really tough for them to keep the vocabulary straight. I always correct them in class (mostly just trying to replace antiderivative with integral). How do you approach that vocabulary? I even had a hard time writing this post with the correct vocabulary.
- Many of my students had a strange barrier this year (that I have never seen before) when finding the area under a curve because they kept thinking of the function you integrate as “the derivative” and the function that you get out as “the original function.” So when we had a function they wanted to find the area underneath, they would take its derivative and then integrate, or some other strange thing like that. How do you introduce the integral as being the opposite of the derivative without getting that misconception (or rather, what did I do in my sequence to imply that)?
- I always, always, always have so much trouble convincing some students that u-substitution is only used for specific functions that are “backward chain rules.” But after we learn how to integrate normally, we spend a ton of time on u-substitution, and then some students try to solve EVERYTHING with u-substitutions (like 1/x^6 for example). I spent a lot of time doing activities where we pick out the functions that can be integrated with substitution and those that can’t, but for a lot of students, this obviously did not sink in. Any tips?
- I cannot for the life of me get students to remember to add a “dx” when differentiating a u to find a du. So if u = 2x, then du=2xdx. Granted we didn’t do differentials, but I still don’t understand why this was so difficult! I need some sort of conceptual trigger so they can understand why it’s so important…
I just taught my AP Calculus class the unit of Volumes of Revolution and Volumes of Known Cross Section. Overall, it went fairly well, mostly because I gave it a little time (I rushed through other topics that I deemed less important like inverse trig functions to be able to have a little extra time for volume). I felt like I did an excellent job with the cross sections, but not quite as good job with revolutions… so let me explain why:
1. Visualizing Volumes of Revolution
…but struggling with setting up the integrals
The two things that I focused on, and correctly so, were getting students to visualize the solids and to construct the integrals using an understanding of the accumulating process. Luckily, I read this awesome post from square root of negative one teach math about how she approached this. I read it the night before I started so I couldn’t get the awesome tool she used, but there was a great tip in the comments about attaching a pencil to a drill and then revolving a region with that. I used a motor from the science lab and a pencil:
Kind of hard to see in the picture, but it worked okay. Truth be told though, I got frustrated with it and didn’t end up using it, though I think it would work. Instead, I relied on Winplot, which is a computer program that can construct these solids, and some basic physical demonstrations with pencils and pieces of paper (with the same principle as the demo above). These visualizations helped some students (as the visualization above would have), but some students still could not figure the revolution part out, especially when there was a hole in the solid from revolving a more complicated region..
The thing that these visualizations did not particularly help with was going from the visualization of the solid to the integral. I think they had this difficulty because the region that they are focused on in the visualization is still the region being rotated, and it’s not those circular cross sections that they need to add up. I had one girl who could not figure out whether the circles were being added up in the x or the y direction and kept drawing her circles in the wrong direction (even though she had the shape of the solid drawn correctly). Most students got it after a while, but I was not really that sure if it was being able to trudge through that process after seeing a ton of examples or if it was actually deep understanding.
2. Visualizing Solids of Revolution
… and nailing the integral set up
But the cross-sectional solids were a different story. I did an AP Calculus workshop this past summer and I was far less than impressed at the time. I regret my attitude now though because I am realizing throughout the year that I actually got a lot of really useful things that have helped my teaching this year. This was a simple idea that the facilitator mentioned that I think worked really well. To construct the volumes of known cross-section, print out the base area, lay down some Play-Doh (or I used the sticky reusable poster tack), and then shove cross-sections in there to make the solid. This one is the solid with a base as the region bounded by , and the x-axis with semicircular cross sections perpendicular to the x-axis:
The idea from the facilitator was to have students construct these themselves, and I think that’s a great one – it would just take a lot of time with a whole class. Instead, I made a few of these (which only took me about 20 minutes) and then had students use the models to write down their idea of a how the solid is constructed FIRST, before ever being given a description. They then compared their descriptions to the textbook/AP problem style descriptions, and were like “duh, that’s what we wrote.” I was really surprised, because I teach all students for whom English is their second language and thus struggle with written descriptions like those of these solids, and also one of my colleagues had ranted last year about how hard this topic was to teach because the students couldn’t visualize the solids. But it seemed like letting them construct meaning directly from a hands-on visual model first was a good key to understanding the lengthy descriptions of these solids.
But unlike solids of revolution, they had no trouble then converting the visualization into an integral (and this is almost definitely because the construction of the solid has more to do with the construction of the integral). I use this notation to set up the integrals:
I focus on the two things highlighted: first, write an integral ignoring the function that just shows the shape you are accumulating with its area equation, the direction it’s being accumulated, and the bounds. Then, go to your function to find out how to fill in the dummy variables in the area equation (like “r”). After that, you can just substitute everything into an integral and can stop thinking. I think the intermediate form made the connection between the integral and the solid and helped them really use the visualization they had in their mind for the math. The results proved it too – they really rocked this question on the Mock AP Exam, but didn’t do as well with the revolution.
So what was different between the two solids? The visualization for revolutions has less to do with the integral than the visualization for cross-sections. I realized that students were forced to try to picture the revolution, understand the accumulation and translate all that into an integral, which is a lot to absorb all at once, especially when not all parts are directly related to each other.
So what that I am going to change next year? When teaching both types of solids, I am going to show them first, have them come up with their own descriptions, and THEN show them how we would mathematically describe the solids. Also, I am going to try teaching the solids of known cross section first, because the construction of these (though often a bit harder to visualize) helps teach the accumulation process that is the important part of a solid of revolution. Then, instead of trying to understand all the new concepts at once, the revolution becomes just a step to create a solid that can be integrated similarly to something they have seen before. Also, I am going to supplement the visualizations I used this year with one that shows the region split into discs too. Perhaps with these changes my students will focus more on the discs/washers than on the revolution itself, and thus be able to set up the integrals more easily. And perhaps my notation like this will make more sense than it did this year:
So I get all these ideas for problem solving lessons, but a lot of times I struggle with pulling the trigger to try it out. I get excited about it, but then I hem and I haw, I get worried that it wont work, I debate about the implementation and fret about the logistics (planning class is tough when you have an intense type A personality but also a creative streak).
And then wonder if the thing I’m teaching could be better taught in a more straightforward manner.
So this is a time when I chickened out. To connect Riemann Sums with the physics of velocity and displacement so that the introduction of the integral is a meaningful and motivated as possible, I wanted them to use a video of my speedometer as I drove around a well known circle at our school to calculate the distance I drove. This is a modified idea from somewhere on Real Teaching Means Real Learning (though I can’t find the original post that inspired it, bah!). The video is not all that interesting but here it is:
I wanted to give them a video and let them just struggle through the task. They know that distance is velocity times time, so I wanted them to sense why this simple equation is far more difficult when the velocity is constantly changing as a motivation of why an integral is so important. I wanted them to get the idea that in order to calculate it we would have to split the trip up into much smaller segments, like a Riemann Sum, and that we could make it more accurate by doing the time at smaller and smaller intervals, but it would always still be an estimate until we did some sort of limiting process.
But then, I got worried. What if I asked them to bring computers and they didn’t? What if they got so hung up on the km/hr to m/s conversion that they couldn’t focus on the other stuff? What if they did something crazy instead of a Riemann Sum type thing? What if they couldn’t figure it out at all and we wasted a class? These are the questions that I get hung up on all the time with trying to implement #WCYDWT and #anyqs type instruction (though this is certainly different because I was asking a specific question).
So, instead we watched the video and talked as a class about the difficulty of the task with the constantly changing velocity (which meant I have no idea for how many students this really sunk in). Then I gave them this to help them solve the question:
We practiced drawing and calculating an applied Riemann sum with this, and used units to discuss why the area under the velocity vs. time graph. I think most students came away with an understanding of at least the idea that area under a velocity vs. time graph gives you displacement, but I don’t think they had the deep understanding I was hoping for, and especially not the deep motivation for integrals (which I could really tell when I tried to explain what the dx signified). It was more efficient, sure, but perhaps less effective…
…but importantly for my lesson planning was that I knew that with the scaffolding that it would work, but I wasn’t convinced it would work otherwise. How do I escape those thoughts, especially with 30 some odd teenagers staring at me for guidance every day and a tight yearly plan?