# Daily Archives: June 4, 2012

## Calculus Final Project Spotlight: Packaging Consultants

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!

## Calculus Final Project Spotlight: Math of the Pilgrimage (Hajj)

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:

1. 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.
2. 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.
3. 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.
4. 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!

## Calculus Final Project Spotlight: Twitter Followers Math

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 <3 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.

## 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!