AP Calculus Projects Day 2 – 2015

Here are the projects from Day 1. The projects from Day 2 are below.

Math Behind a Basketball Shot

This group talked about the math behind the ideal basketball shot, including the role Magnus Effect. They recorded some video of them shooting and used the program, Tracker, to analyze position, velocity, and acceleration of the basketball throughout its flight. Here’s a screenshot of the ball at its apex. You can see in the graph that the velocity is zero. (close enough)

Basketball Shot

Presentation: Basketball Project

Spread of Disease

This student talked about the Ebola scare a few months ago and how Ebola was starting to spread exponentially. He then explained the SIR model and used a hypothetical disease to talk about the infection rates in the town. He checked the formulas by hand and then used a spreadsheet to complete his calculations. The town of 27,830 would be infected by Day 6.

Spread of Disease

Presentation: Calc project

Parallel Parking

“What is the minimum space needed to parallel park?” This was the guiding question to this students project. She first derived the formula,

parallel parking

and then tested it out in her car. She took the measurements of the car, marked out where she needed to turn and videotaped the result. She only needed one attempt!

Here’s the video.

Presentation: Parallel Parking

Stereographic Projection

This group researched stereographic projection. They were inspired by this model on Thingiverse.


They printed it out and began their investigation. Their original plan was to create their own image but ran into some problems on how to map from the 2D to the 3D. This group worked really hard for the entire time and got into some pretty deep math.

Presentation: Stereographic Projection

Calculus in Economics

This student is going to study Economics in college so this project was right in her wheelhouse. She did a great job of explaining the Demand and Marginal Curves in her presentation. She did some work on the board to support her presentation which I forgot to get a picture of.

Calc Project

Launching a Falcon 9

This group worked on the calculations needed for SpaceX to send the Falcon 9 into orbit. They also got into the math pretty deep and looked for some assistance. Luckily, I attended high school with a current SpaceX employee. I was able to reach out and my students were able to email back and forth with him. He helped them out with some of the calculations and brought their attention to some things they had not thought of. One of the students is going into Aeronautical Engineering so he was very inspired by this project.

Presentation: INTERSTELLAR

Melting Ice Rates

This student compared the melting rates of ice depending on a few variables. One variable was the shape of the block of ice (cylinder, rectangular prism, etc) and the other was tap water vs salt water. She was very diligent and took measurements every 10 minutes for 5 hours!

Still awaiting an email for her presentation which showed all of her calculations and results.

Creating Solids of Revolution

This student took two curves and revolved them around the x-axis. She took these curves and produced the sketches of them in Inventor. She then 3D printed them and calculated their volumes. I will definitely use these in years to come so kids can visualize the solids created by revolutions of curves.

rev 1  rev2rev3

IMG_1199   3D Model

Presentation: Solids Of Revolution

Magnetic Acceleration

These students were inspired by videos of railguns they had seen. They created a much smaller, more school friendly version. You can watch the slo-mo video of it here. (The quality is not that great). They calculated that the ball bearings were moving at 8m/s (just under 18mph).

IMG_1197Screenshot of the video

Presentation: The Calculus behind Magnetic Acceleration

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AP Calculus Projects Day 1 – 2015

Today was the first day of the AP Calculus AB projects.  Here are the 5 presentations from Day 1. More to come tomorrow.


IMG_1190        IMG_1191

Cube octahedron                                              Butterfly

This student gave a verbal presentation on the origins of Origami. She showed us various ones she had made, including the 2 above. She described what made up the cube octahedron and other various pieces she had made. She then demonstrated how to make another piece fold by fold.

String Art


Fibonnaci sequence

Another one by hand


Presentation: Math and String Art

Math in Rainbows

Video explaining their project and how rainbows are made here.

Video of them creating rainbows here.

How to Solve a Rubik’s Cube

This student had zero experience with a Rubik’s Cube and decided to figure out the algorithm to solve a Rubik’s cube. After she learned the algorithms she recorded herself solving it and showed us every single algorithm. She then put together a nice video explaining it here: https://vimeo.com/130179285

Presentation: How to Solve a Rubik’s Cube Using Algorithms

Coke Bottle Revolution

CaptureDesmos to model the shape of the bottle.

Capture1Fitting a curve. Actual volume was 355mL

Capture2Adjusting to where coke actually starts.

Capture3Adjusting for thickness of the glass and vertically shifting curve down.

Capture4Final Calculation.

The actual volume was 355mL so this group calculated a 3.45% error.

Presentation: Solids of Revolution ft. Coke

Mathematical Programming

Inspired by Dan Anderson (2 doors down, always nice.) Some good math “art”

Open Processing Loops (Non-interactive)

Open Processing (Interactive)

Presentation: Calc final Presentation

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Continuity at a Point

This summer I attended an AP Summer Institute with Sergio Stadler at the Marist School in Atlanta, GA. I would highly recommend this APSI to anyone teaching AP Calculus for the first time. I taught AP Calculus AB last year, so some was repetitive but I still gained a lot from going. One of the specific things I brought home was an activity on continuity at a point. I was excited to teach this lesson since many students struggled with this last year. They struggled because I just presented the definition and then we tried to work some problems. They didn’t have an understanding of where it came from or why. Try understanding this as a high school student…

I posed the following scenario to my students:

Post 2  Post 3

I then showed them 8 different graphs.

Post 4Post 5Post 6Post 11Post 10Post 9Post 8Post 7

Here is a link to all of the graphs. (These were all colleges my students from last year ended up at.)

I then asked them to work with a partner and do the following:

Post 12

I immediately had students asking what continuous meant. My response was that “I wasn’t quite sure.” After a few minutes students had no problem using roads and bridges in their explanation.

I then asked them to take their non formal definition of continuous and make it formal.

Post 13

Students realized they had to use limits to deal with the roads and function values to deal with the bridge.  The final step was to write it formally with the three conditions as seen at the top of this post.

Post 14

This lesson went well because the students did the heavy lifting. I just clicked next on some slides and let them talk it out. We talked about if the roads don’t meet does it even matter with the bridge is? If there is not a bridge, does it even matter if the roads meet?

Students responded well with this lesson and continued to use the non formal notation (bridges, roads, etc) to help them write the formal notation on homework and tests. Here is an example on the test.

photo“not continuous, the roads meet but there is no bridge” (Then translated into formal notation)

Again, this was not something I created but I felt I had to share since it went so well.

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Benefits of a 2 hour snow delay

Yes, winter is still wreaking havoc in the Northeast. We had a big storm come in Wednesday and Thursday. Both days looked like possible snow days. We only ended up with a 2 hour delay on Thursday. Many parents decided to let their kids stay home so my classes were very small. Instead of continuing with my plans we got to dig a little deeper and look at some interesting math. 

In AP Calculus, we have a test coming up so we quickly talked about a couple review problems. Then on to the interesting things. We checked out Math Munch quickly, watched Numberphile’s Buffon’s Matches video (since the next day was 3/14), and then somehow our conversation turned to triple integrals. Something like this:



Students had no problem with it! I barely said a word. It was great to pause and just talk about some math.

In Geometry A, (first year of a 2 year sequence for Geometry) we attacked this problem since our next unit is triangle similarity where ratio/proportion is key. 



In a class where many students struggle with math and just shut down easily, I was relatively pleased with the math talk that was occurring. (Read: arguing over the correct answer and their method of solution.) We debriefed and showed many ways to solve the problem. This ranged from pictures, to charts, diagrams, verbal, and even some estimation. From individual time, to group work, to debrief this took almost the entire 43 minute shortened block (typically 81 mins). Students remarked how fast class went.

It was great to just do some good math and have great conversations with my students. But yes, I still would have taken the snow day!

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Fundamental Theorem of Calculus, Part 2

It’s been a while since I’ve posted but I had to make time for an activity I did last week in AP Calculus since it went so well. We have been calculating Riemann Sums (left, right, midpoint, and trapezoid) and students felt very comfortable doing these. We also used this Desmos file to show how increasing the number of rectangles makes our approximation closer to the actual area under the curve. (I took this Desmos file from someone else who blogged about it but I can’t find the original blog, only the Desmos file, so let me know if it was you!) At the end of the previous class we went through the typical way to bring in integrals. This is outlined at the bottom of ThinkThankThunk’s post here, however I did not yet connect it with the anti-derivative.

To connect it with the anti-derivative I used an activity from Mark Howell, a teacher from Washington, D.C. If you go to the College Board site you can find many of his publications there. I attended his AP Calculus training at an AP Calculus Summer Institute and learned a great deal from him. The students homework the previous night was to go for a 15 minute drive where they were the passenger. They first recorded the initial odometer reading. Then every 30 seconds they recorded the velocity of the car, making note of special conditions. For example, making a note if they were at a stop sign. They then recorded the final odometer reading.

They brought this data into class and graphed it by hand. They then used the midpoint rectangular approximation method, with a width of 1 minute, to find the area under the curve. Since they recorded the velocity every 30 seconds and used the midpoint method, the height of each rectangle was always the actual velocities they recorded. When finding the area of each rectangle they had to make sure they converted the width of each rectangle from 1 minute to 1/60th of an hour so the area under the curve would be in miles. Once students completed this, without prompting they immediately realized this was almost the same number as their final odometer reading minus their initial odometer reading.

Once everyone had completed this I brought them make together to make the connection to the anti-derivative. From last class, students were fine with saying the integral was just the area under the curve, the Riemann sum. They just saw that the area under the curve for a velocity vs time graph was their final odometer reading minus their initial odometer reading. I then asked them, how is velocity related to position? (Pause…) Then a bunch of light bulbs went off. We had spent a week earlier in the year during derivatives working position/velocity/acceleration problems. Position is the anti-derivative of velocity!

Now, there was no problem writing this and all students agreed.


Here is some of their graphs and write ups. Almost all students concluded we could make our approximation better by increasing the frequency at which we recorded our velocities. (i.e. increasing the number of rectangles) Many also noted having a digital speedometer would be best to get more accurate velocity readings.

Table Graph1 Graph2 Write-up

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Explore #MTBoS: Hangman

I am participating in Exploring the MathTwitterBlogosphere for the next 8 weeks. Here is a description if you are not familiar. 

I am posting on what makes my classroom unique. One small thing which makes my classroom unique is my version of hangman. Anytime I am writing on the board in class and make a mistake I start a hangman. Head, body, 2 legs, 2 arms only. This includes any small mistakes like leaving off a digit because I am thinking about an engaging question to ask when I’m done with a boring algebra step! 

The rules:

  • A student must point out the mistake before I correct it.
  • Each mistake is worth 1 body part.
  • The students have a full week to hang “me” before the hangman gets reset.
  • If they hang me, everyone in the class gets a bonus point on their next test. 

You would be surprised how much students enjoy this little game. They love to point out when I make a mistake. (It also gives me an opportunity to discuss how making mistakes is okay. Everyone does it!) They really love to yell out loud that they are hanging me. I just hope the person walking by is not worried about my safety because there are shouts about “me” being close to death (shouting is directly related to number of body parts on the board). It also makes them pay attention to details and shows them if you make a small mistake it can throw off everything. 

I never talk about this game until someone corrects me for the first time. Then I just casually mention that I play this game. Everyone perks up. It’s amazing to see a Friday, last block class hang on to everything I put on the board when I’m on my last body part for the week. 

Let’s be honest, 10 years down the road kids will not remember all the math you taught them but they will remember that Friday they argued for 5 minutes about whether or not they deserve the final arm. And then how the class broke into applause when I finally drew a full hangman on the board. 

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Sunday Comics

Instead of reading the Sunday comics I want to share some funny moments from my classroom from this week. Ideally, I want to do this every week but realistically I know this won’t happen. I hope to share as often as I can and I encourage others to do so as well. I feel like this could be a “spin off” of One Good Thing with a mix of “Not Top 10” from ESPN. 

In no particular order,

1. Last year I taught Algebra 2/Trig in New York State. As Mr. Honner has chronicled here, these are just wonderful tests. Anyway, one of my students from last year who did not pass the test came to catch up with me. She is going to retake the test in January and was wondering if she should get a tutor. I suggested she should. Her response, “Well, I guess it should be someone besides you since I already failed it after you taught me.” Can’t deny she thinks logically. Might have to mention this in her college recommendation she asked me to write. She didn’t realize how this sounded and once I started laughing she realized what she said.

2. Before my first period class started I overheard a student conversation. One student was telling another how he was getting sick and right before school took NyQuil instead of DayQuil because he was out of DayQuil. He was actually excited to see if he would be able to stay up all day in school. He had no problem in my class, I wonder if he made it…

3. In AP Calc, I handed out a questionnaire about trig to see how much they remembered. One of the questions asked what trig was used for. One response was two words: “Boy Scouts.” So at some point this kid made a connection between trig and boy scouts. Don’t ask me what part of Boy Scouts. Just Boy Scouts in general? Amazing.

4. A student sent me an email this week from their phone. When they typed in my last name, their phone auto-corrected it to “Mr. Cialis.” She was quite embarrassed when she saw this. 

5. This last one is really from last year but was too good not to share. It was on a particular multiple choice question on the Algebra 2/Trig Regents. I’m not sure what the exact sequence was but it went something like this: Picked choice A, crossed it out, picked choice D, crossed it out, wanted to go back to A so drew an arrow to A, changed her mind so crossed out the arrow, picked choice C, crossed it out, went back to D by drawing an arrow. Her answer sheet looked something like this: 


Had to show some of the other proctors and we (including the student) had a good laugh about it. The best part was that she passed the Regents. Must have been a tough question!


These are just a few examples of why I enjoy teaching and what makes it fun and interesting. 

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