Monday, May 27, 2013

Optimizing Cans

This year, I gave my kids a project on optimizing cans. The goal is to minimize the amount of material/metal used for the can while holding the same content (minimize surface area while holding volume constant). This is the second time I gave this project. I like this project because it gives student the opportunity to make the connection between what they learn in class and the design of cans that they see and use daily.

As with any project, there are varying levels of achievement as you can see from the group picture below. For this project, the students had to create a poster with all the work (measurements, calculations, optimization, and conclusion) along with 2 cans (a replica of the original they'll use as reference and the optimized can). The cans have been paired for this picture with the replica of the original in the back row and the optimized can in the front row.

Group picture of original (back row) and optimized cans (front row)

The presentation of the project consisted of 4 parts:
  1. Introduction (group members and purpose of project)
  2. Presentation of the Math (measurement, calculations, optimization, and conclusion)
  3. Show the volume of the original and optimized cans are the same (completely fill a can with rice then pour it into the other)
  4. Q&A
Here are some questions I asked the students this year:
  1. What is the purpose of this project in real world terms and in mathematical terms?
  2. How did you find the radius of the can accurately? (I liked some of the different ways students had to determine the radius)
  3. What do the parts of the formula for the surface area represent? (More students struggled with this question than I expected)
  4. What was kept constant in this project and what was optimized?
  5. How did you optimize?
  6. What are critical numbers for and what do they mean in the context of the project?
  7. What is special about the new radius that you found as opposed to any other radius from any of the possible combinations of radius and height?
  8. What does your result mean for the company that created the product you used for this project? (Some groups had similar cans, while others had a more pronounced difference between the original and optimized cans. I loved some of the responses to this question)

After the all presentations were done I went back to their posters and checked their calculations and work. Below are pairs of cans from selected groups. A question for you is:
Just from looking at the cans, can you tell which pairs of cans were optimized correctly and which weren't?
In the pictures of pairs of cans below, the cans on the left are replicas of the original cans and the ones on the right are the optimized cans.
Pair #1
Pair #2
Pair #3
Pair #4
Pair #5
Pair #6
Pair #7
Pair #8
Pair #9
Pair #10
A good start can be with the question "which of the optimized cans is most unlike the others" and then proceed to "is there a pattern or similarities in the shape of the optimized cans" and "describe the relationship between the radius and the height mathematically?" I wish I had followed this line of thought during the year and asked better questions to summarize the entire project.

P.S. A few groups were cheering when they did the rice test and the rice filled their optimized cans without spilling or underfilling. I forgot how amazing it must feel for students to go through calculations that are challenging (to some) and have it work out perfectly when they build models using their final results. You can watch the anticipation as they slowly pour the last of the rice from one can to the other.

Saturday, May 11, 2013

xkcd: Paths

I guess I'm not the only one who worries about things like this.

Since we've been working on optimization problems in Calculus, it gave me an idea. The following is my extension to the above question.

How is this different from the problem in the previous comic? What are the assumptions, if any, that are made in the first comic? How would you solve this without Calculus?