Integration Drawing Projects ’12
I wrote about this project back on Sam’s blog this summer when Sam gave me the reigns 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:
And the Calculus Final Projects Begin
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.
Honing My Skills Instruction
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.
PREAMBLE:
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!!
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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…
Musings about Volume
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 
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:
The Time I Chickened Out… [dealing with fears of lessons not working]
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?
Drawing in Math Class
One of my favorite ways to start class is by putting out whiteboards with a problem paper-clipped at the top, and names of random groups. I love it most because every single person is engaged in mathematics within 30 seconds of class starting. In fact, students always ask me a minute or two before class starts “can we begin?” They can’t seem to resist the markers and the problem in front of them. Also, I found when I wanted to use whiteboards in the middle of class and put students in random groups that it just ate up a few minutes in each class, so this just feels more efficient (I’m kind of neurotic in terms of efficient use of class time).
Continuing my experiments with different modes of math whiteboarding, a great whiteboard warm up I tried was having them illustrate related rates type situations for objects that are changing in different ways. For example:
A pumpkin grows in a garden…
1. With a constant increase in the radius of the pumpkin
2. With a constant increase in the volume of the pumpkin
Then I had them describe what is happening to the rate of change of the important variables (so if dV/dt is constant, what is happening to dr/dt?). We then had a really good full class discussion where students explained their situation. I think this helped clarify for a lot of students the difference between “V” increasing and “dV/dt” increasing, or how just because “dV/dt” is decreasing it doesn’t mean the volume is decreasing.
This was part of a larger goal of mine to focus on big ideas and deep understanding this year – I’ve always asked students interpretation questions on tests (my final this past term had a crap-ton of writing) but I never felt like I actually directly taught them these sorts of things. For Related Rates, we solve all these problems and come up with all these numbers, but never actually talk about why they are interesting problems – the fact that as one aspect of a situation changes, another may change at a totally different rate, and that there is a relationship between all these rates that explain how things change the way they do. And honestly, I think this little activity made a huge difference – on the interpretation question on the Related Rates quiz, tons of students drew pictures to aid their explanations. 15 minutes well spent!
Concavity and Population – A Calculus Essay
(got a little burned out last month, but feeling back on track… so I’m trying to get on a blog schedule again because I really enjoy reflecting and have missed keeping up with the blog)
For the past two years, I have done a really awesome project with my students in my non-AP Calculus class using UN Population data. The kids all cite it as one of the most interesting things that we do, and it really helps me see who truly understands the concepts of concavity and inflection points and who can just set a second derivative equal to zero.
GOAL: Using data from the UN World Population Prospects, graph the population data for three different countries from the year 1950-2010. Then, describe how the population has changed over the past 60 years and will change for the next 90 using the Calculus concepts of increasing/decreasing, concave up/concave down, relative extrema and inflection points. If inclined, try to connect some of the trends that you see to history.
Of the so-called “four representations of Mathematics” (verbal, numerical, analytic and graphical), I think we tend to emphasize the analytic, and rightfully so. It’s the most sophisticated and the cleanest to use in a classroom. So I wanted to do a project that focused on the other three representations. Basically, I wanted students to tell me everything they could about concavity without getting caught up in the calculations.
We spent a couple days in class just making the graphs, which was a real test of my students’ limited Excel skills, and analyzing the data with Excel by calculating manual first and second derivatives (just calculating the difference between the population between successive years, and the difference between the differences, just to get an idea of how the function is changing and how the change is changing). This was really fun because students would be calling me over excitedly to show me things on their graph (like huge spikes in migration in a country going through a genocide, or the insane and sudden shift in the population of the former Soviet countries when the Soviet Union broke up). Then we spent another class period writing outlines for the analysis, mostly just picking out inflection points and where the population was concave up and concave down. Then the students wrote the analyses for homework. Most students wrote around 2-3 pages not including the graphs… about Calculus… and I really enjoyed reading them.
Things I liked about the project:
- Students picked their own countries, which meant that many became pretty invested in the topic. Talking about real countries that students really care about made things like “switching from increasing at an increasing rate to increasing at a decreasing rate” actually interesting. And every student’s project was different which made it awesome to grade.
- I think having students describe verbally what was going on with the population forced some to really think hard about the huge idea of concavity and brought out a lot of misconceptions, especially the perennial confusion between inflection points and relative extrema. The wonderful thing about using Standards Based Grading is that I gave them really detailed feedback and about half of them went back and edited their paper and rewrote parts to make the Calculus better – I never edited something that had already been graded in high school!
- I think students really learned a lot about academic technology. I purposefully didn’t give them guidelines on how I wanted them to present everything, and I think this struggle is something that is going to help students in the future, in a more independent environment of college or work.
Things I didn’t like about the project:
- Some students felt limited by population and wanted to examine data for other things. But I didn’t have other reliable data ready and didn’t want to waste time having them search for data and find bad data. So I kind of nixed some creativity and student independence for the sake of efficiency and logistics… which is something I do a lot. And something I want to get better at. But I don’t know how.
- Having them do three different countries was a bit overkill, as they kind of did the same thing for each one.
- Last year we came back to this project and added the fourth representation by fitting logistic curves using Geogebra to the populations and comparing our predictions with that of the UN, but I didn’t have time this year for the wrap up this year, which was sad.
The addition we made this year is that we are going to send our work over to the AP World History class and they are going to try to explain the graphs historically. I’m not sure how that is going to work out, but I’m all about trying interdisciplinary things.
Overall, great project and a great change of pace!
PART OF ONE STUDENT’S WORK:
DRC’s population has increased in the past century and is projected to significantly increase within the next century as well (Graph 1). The overall population rate (Graph 2) for DRC significantly increased in 1990-1995. This sudden population increase was caused by the Rwandan Genocide which reached a fever pitch in 1994. In 1995, the population jumps from 36.5 million in 1990 to 44 million. The migration rate in 1990-1995, mainly consistent of refugees, was the rate that contributed most to the overall population number fluctuation. Many Rwandans fled to DRC as refugees from the massacre that plagued their country, which was a destructive conflict that erupted between Rwanda’s two main tribes: the Hutus and the Tutsis. As aforementioned, this conflict was initiated by the colonizers, who set the stage for the genocide by favoring one tribe over the other. Also, when looking at Graph 1, I noticed there is an inflection point in the population of year 2040. Before 2040, the population of DRC had been increasing at an increasing rate. After 2040, it is projected to begin increasing at a decreasing rate. The occurrence of this inflection point is probably due to DRC reaching its carrying capacity, or reaching the point where it cannot support any more people due to lack of sufficient resources to support a consistently growing population.
(not perfect, but I love it!)
PROJECT DESCRIPTION AND INSTRUCTIONS:
GRADING RUBRIC:
STUDENT FEEDBACK:
- It was a nice project for math as it gave us the freedom to relate concepts we learn in class to historical measurements, keeping my interest. However, I didn’t feel as comfortable with the analysis as the graphs as it was exactly clear what I should be writing. I gave it a shot and got a 3 but I feel like the instructions on that part couldve been a bit more clear.
- I just find it interesting and important to teach students that what they learn in calculus isn’t just for the sake of being able to do math but because its essential for many people to come up with things we never really consider, like predicitions about the population of countries. The chanign world population is a big deal and without knowledge on critical points, derivatives and so on the world would not know what to expect in the future or how to react based on the change.
- I found the project interesting as I researched and got to know a little bit more about a certain historical conflict that interested me. I learned skills in Excel that I didn’t really know before, like doing certain types of graphs. I think I could have done better if I had more time to work on it.
- Of course the population project was very helpful as i stated above. It was interesting to see how some regions or countries differed in population growth and calculus terms helped us relate why each population graph looks the way it does.
- I loved the population project!!! I even sent you an anonymous feed back about it. I think it was one of the most interesting assignments I’ve ever done in my high school career at King’s, and I really enjoyed getting it done, even if we didn’t have much time to do it. I was genuinely interested in what the population trends on my graph meant, and I was amazed at how much historical and political events have an impact on the population. Super interesting. Thanks for assigning it!
- the population project was a good activity. it was helpful in terms of understading the material because we were able to explain the population/time graphs and why they curved up and down. It just took me a lot of time to do because there was so much to talk about for every country, so I ended u writing an essay-like analysis of each country. but I totally recommend doing this project next year.
- this project was very interesting and helped me learn new skills using excel but did not really help me learn anything new using calculus it just helped me cement the ideas we learnt in my head. I thought that the time we used doing the project could have been used to help us understand other topics better.
- I learned things I never knew I never knew. if that makes sense.
- I really enjoyed doing the population project!! As you can tell it took me a while to do. I like to use skills that I learn in class and apply them to situations like these. makes me feel that I did learn something. It was also a fun way to get us used to claiming what an inflection point is and what critical points is and how they look like on a graph.
- Although I want to recommend this for the next year I would not make the students do three different countries because I feel liek it was tedious and repetative talking about concavity, derivaties, and inflection points. Also I feel like that would solve your 16 page issue when grading
- I liked the project and learned a lot but I thought it was a lot of work.
- Although it was a bit hard at first because i wasn’t sure what I was supposed to write in the analysis, (if it was supposed to be explaining math concepts or history) I found it very interesting and I enjoyed doing it especially since I’m really interested in demographics.
- I learned so much about Excel! I think it was a very fun way of reasserting our knowledge to you and to ourselves. I didn’t know that what I wrote would come out of me, turns out I’m smarter than I think. The only problem was that I felt it was due to soon. That might have been because I had another project and a test due on the same day and the pressure was on.
- I feel like I did a good job, though after looking at the feedback I got I didn’t really understand exactly what I did wrong. I have to look over it again and put more work into it. But overall I feel it was a large project but It wasn’t that bad.
- It helped show how calculus us used in real life rather than on paper. plus it wasn’t that much work, you gave us two lessons to work on in class which was helpful because you were there to answer all the question at the beginning of the project where it’s usually most difficult.
- Even though i thought it was more about History and less about Math, I thought it was intresting
- I learned a lot of interesting things and tricks on the Excel that I didn’t know, which helped me a lot in other subjects.
- it was a bit hard and i didnt understand the concepts and the method of making it and describing the project
- i would not recommend it because I personally don’t know how to write math papers, and analyse graphs because i don’t know what answers you’re looking for.
- The project was not that hard, but what i found difficulties in was the analysis part of it.
- I think that the population project was interesting, as well as, I learned skills in Excel that I didnt know before. It really helped in raising our grades, and I recommend doing it next year.
- The population project definitley helped to improve my skills in Excel since the graphs we were required to create were for the most part new to me. Even though this project took a good amount of time and effort, it definitley helped me learn several important skills and I regret not submitting it on time because had I worked about an hour maximum each night and used my time in class effectively I would have been able to complete it on time.
- I just think that the project could be improved for the class next year so they get more into it and the due date was too soon which is why so many people didn’t hand it in on time and got a reduction in their grade
- It was really good, I think time was sort of an issue
- No comment. It was an interesting and novel way of teaching. Typical Mr. Bowman (That’s a good thing btw):-)
- The population project was very interesting. I learned a lot from it, but it was too much work, and it was due right before midterm week. I wouldn’t recommend doing the project next year. I didn’t feel I doing taking math. It felt like physics.
- I checked the last two boxes for a reason, because i think that students will definatly benefit from the project by learning more about certain countries and why and how population changes. Helps them determine how every change would look like on a graph and why it looks like that. At the same time i think its too much work and i feel 7arram for them to go through all of that when they can simply right a quiz on that topic :S
- We could have had a simple explanation of the subject in class and not do a really big project. To me, it was kind of really pointless, and I think that you should not do it again for the sake of the students.
How Much Should Students Retain from High School Math?
So, after a wild foray into differentiation and applications of differentiation, we just started to learn how to deal with exponential and logarithmic functions, before starting our equally wild foray into integration. I gave a pre-test to see what they remembered about exponential and logarithmic functions from the past.
It was less than encouraging when I think about the quality of math education we give to high school students.
I gave them a little more than 10 minutes to fill in what they knew on a two page pretest. Most were just about as blank as the one below:
Most students could not sketch anything that even looked like an exponential function (these are four different student’s answers):
Which might have also been because not even had any idea of what e was:
Okay, so I have fully embraced the whole idea of “teach the where they are.” I enjoy teaching this class because I have the liberty to slow down and fill in gaps in their math backgrounds and then can teach the Calculus material with the depth in conceptual understanding but without the depth in mechanical skills (if that makes any sense). Sure it’s harder to explain the idea of a limit as x goes to negative infinity for e^x when they don’t fully even understand what a negative exponent is, but I wouldn’t be in teaching if it was an easy job. So, I’m not complaining that I have to change my sacrosanct yearly plan and “lose time” – I’m just wondering if this is all they can remember about a major family of functions that I’m sure they learned about in both Algebra II and Precalculus, what did we accomplish in teaching them this in the first place? What do we expect our students to know coming out of high school? How big is that gap between what we teach and what they actually learn?
I’m not blaming their past teachers, and I’m especially not blaming my students (though my non-AP Calculus students are labeled “weak,” many of them are really quite talented but have yet to find that spark of loving mathematics or need a bit more time). Worse, I have no ideas for solutions, just those questions. I guess I can find comfort in knowing that other people are asking the same thing, and even better, that those are the questions that are driving (some of the) reform in mathematics education right now.
Introducing Differentials with Stock Predictions
The Space Between the Numbers commented on my last post that some people call launch problems like the infection one I did for concavity “Anchor Problems,” the idea being (and I quote from her) “you can keep referring back to the problem to help students latch onto the new learning by remembering this solid, relate-able context.” I think that this describes what I was trying to do perfectly, and I love having new language to talk about my teaching. So, thanks for that!
I wanted to share one more Anchor Problem that I am using in my AP Class tomorrow to introduce Differentials (which I used to what I thought great success in my non-AP class last year). I am relating the idea of making a prediction with differentials/a tangent line to making stock predictions. With charts like this:
… I’m going to have them predict the stocks price in the near and far future using the graph in any way they can. Most students last year figured out to draw a tangent line, use the slope of that tangent line to see how fast it is changing currently and then multiply that by the number of months to get the change in price, and then add that to the original price. It’s an intuitive idea that sounds way more complicated when you try to describe it. I added the “% confidence column” this year to try to get at the idea of a prediction being less and less accurate the further you are from known data.
Eventually, I want us to codify our process into a rough equation like this:
Which we can then use to look at the “equation” for using differentials:
(I think my face is crooked… I always write slopey)
This worked well last year because even when we were solving abstract problems that had nothing to do with stocks, I would ask questions like “well, how fast is your stock changing right now?” and “how many months in the future or past are we predicting?“ which served to connect the abstract to the intuitive situation. I also think it gave students a mental picture of what they are doing.
Hope it works this year too…
Introducing Inflection with Infection
One of the problems I have with math instruction that goes CONCEPT-PRACTICE-APPLICATION is that you miss out on some great opportunities to teach difficult concepts using application. When I introduced concavity this year, I did a great 10 minute activity that paid huge dividends when discussing what can become the somewhat tricky conceptual math of the second derivative and inflection points.
We got back from break this past Sunday, so I started just by asking what everyone had done for break. I had a secret agenda for this though, because from their answers I chose the person who went to the most exotic or most random place and chose them as Patient Zero for an infectious disease that was going to infect our classroom. I assigned everyone a number (I used a deck of cards, but I have done this before with the random number generator on the calculator too). Then, while Patient Zero stood on one side of the classroom everyone got as far away from him or her by standing on the other side of the classroom, so as not to catch their exotic disease. Then Patient Zero picked a card and whoever’s card got picked got infected with the disease and came over to the sick side of the room. I stood at the board and recorded data for how many total people had been infected up to each round. Then, all cards were replaced and all sick people picked someone to inflect. Then, again. Then again and again until the whole classroom was infected.
The disease spread exponentially at first, but once people started picking others that were already sick and the supply of healthy people dwindled, the spread of the disease slowed down, a nice beautiful logistic curve. Here is the data we collected:
After we hand-graphed the points together, I had them then write bullet point stories for why the graph looked like it did. Most students were quickly able to notice that the infection spread quicker and quicker at first and then started slowing down for the reasons I noted above. Many even picked out the inflection point (but not by name) by saying that this was the point the infection was spreading fastest. (The data weren’t as pretty in my other class, but that’s okay! The main point still worked, and it’s nice to see that models aren’t perfect).
After that, we were ready for Calculus. This was a perfect thing to do after a 3 week break, because I then had them tell me everything they could in terms of Calculus to fit with the stories they wrote. We had a very lively review of everything I had wanted to review grounded in this conversation about infection, and the conversation really primed us to talk about inflection points in a deep and meaningful way. The rest of the lesson was like cutting through sponge cake (is that a saying people say?) The idea of an inflection point possibly occurring when the second derivative is zero made far more sense than if I had tried to state this fact and then show why, or make more abstract graphical arguments.
My goal this year has been to motivate well all the math occurring in the classroom, both with application and pure mathematical ideas, and I think this is a good example of success! More to come on other applications of concavity.
