# Blog Archives

## 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!

## Calculus Final Project Spotlight: 3D Solid Modeling

**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!

## 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.

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 $y=x^3$, $x=1$ 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: