A Challenge in Sphere-Packing
Since college and my first year of teaching, I’ve been on the lookout for applications that would be accessible problems for my students but would stretch them mathematically and draw on their talents and creativity. I believe that creativity is as important as having the right math tools in problem solving. And, I have found that if students don’t regularly have an opportunity to use their imaginations in combination with the mathematics that they are learning, their math isn’t enough to create problem solvers.
After students studied volume, they tackled the classic sphere packing problem. In geometry, sphere packing is finding an arrangement of non-overlapping identical spheres within a space. Companies that pack items may look at packing the maximum volume in packaging that has little wasted space and the minimum price. That is the “math problem”. But, the truth is, companies will sometimes abandon this mathematical “requirement” for packages that fit on a shelf better, are examples of sustainable packaging, are protective for fragile spheres, or packages that catch people’s eye. Think about how sphere-shaped items are packed, such as tennis balls in cylinders, oranges in crates, balls of yarn in bins, basketballs or golf balls in boxes, holiday ornaments separated by cardboard in packages, malted milk balls in milk cartons, beads in tubes, and more.
Benefits of this project
- Students are problem solving.
- Students are working in teams to problem solve.
- Students are seeing math in a business context.
- Students are having to write, showing and explaining correct mathematics.
- Students have to be creative and think beyond a text book problem.
- Students have to create a product that meets specific criteria.
Before you begin… Pick up plastic ornaments from a craft store (they could be opened and things put inside – and that makes them easier to store from year to year). Check at the end of the holidays when they can often be found on clearance. Each group had to pack 5 spheres one year, 7 the next year, 6 another year. (You’ll notice different-sized containers in the photos in this post.) It is important that all the spheres are the same size. Hint: You might want to buy a few extra, sometimes they are dropped on the floor and crack!
After students were put into groups, this is what they were given…
Design a Deluxe Container
A manufacturing company has hired the marketing firm that you work for to help them market a warehouse full of shiny “spheres”. Your boss has guaranteed the CEO of the company that his marketing firm can design an advertising campaign and product to sell all of the “spheres”. They need space in their warehouse as soon as possible.
Your boss has given the marketing/design/advertising team that you are a member of 2 weeks (3 weekends) to design a container for sets of 5 “spheres”- all the same size. However each team may only have one in their possession at any time. There should not be much wasted space in the container since efficiency of the container is important. Resources and cost need to be realistic and minimized if possible.
Your team needs to make the container in a very attractive and creative way as to entice a customer into noticing it and eventually buying it. The container cannot be rectangular, square box (prism), individual spheres or a cylindrical tube (they roll off the shelves easily). One container must contain all the spheres at the same time. If your container has multiple levels of spheres (instead of the spheres being all tangent to one plane) you can earn some extra design points. The materials that you use to create the container are to be chosen by your marketing team. If you plan to add something to the original “spheres” for marketing purposes (like donuts in the spheres in the shape above or dog bones in the spheres for the product at the right), it is very importan t to justify any additional expense that the container or additions may create – why will the additional product increase sales, etc.
When the container is complete, your team is to include a proposal to the boss and to other marketing teams to convince them that your container is the best for the product in terms of 1) package efficiency (calculated by finding the ratio of the volumes of the 5 spheres to the volume of the container) , 2) geometry to show how the efficiency was calculated, 3) geometry used to construct the container, 4) justification of any additional expense that you are expecting either the manufacturer or your firm to incur – make sure to include the cost of your container (use $0.01 per square inch for material used to make the container) and any add-ons!, 5) the proposed selling price of each container of product with the profit (markup divided by cost), and 6) why your design should be chosen – why it should do well in the market (a survey of prospective customers may be helpful here!). The boss expects proper English and complete sentences along with neat and detailed drawings and graphs in your proposal. It needs to be typed or word-processed (since he supplies them in your office) before it is submitted with your finished product.
For a slideshow of more student work, visit Deluxe Container – Student Work
In addition to the container and the written proposal, your team is to create and perform a 60 second videotaped television commercial for your product to encourage sales. The commercial must include some geometry to get full credit. To be considered in the “BEST” category by your boss, your video must be clear, contain the math, and motivate buyers! (If your group needs access to a video camera, see your boss (teacher) to make arrangements to use one!) The school’s green screen can be scheduled for the taping of your video as well. Software programs are available for you to use to edit your video.
The boss is going to choose the “BEST” design for production and a bonus (10 extra points) is awaiting members of the team whose design is chosen as the best in each shift (class)
|Container (55 points)||Points Possible||Points Awarded|
|All 5 spheres fit inside the finished container||20|
|Container is attractive, neatly constructed, and eye-catching||10|
|Container is original in design (multi-level design – extra points)||15|
|Proposal (55 points)|
|Logic displayed in justification||5|
|Costs, expenses, selling price and profit explained and documented, etc.||10|
|Geometry and efficiency are calculated correctly and explained in detail with diagrams and/or charts, using correct notation and geometry terminology||30|
|Grammar, complete sentences, and word processed||5|
|Creative and effective proposal||5|
|Television Commercial (50 points)|
|Advertisement is unique, creative, effective, informative||10|
|Geometry is used in the commercial correctly||15|
|Props used in the ad are appropriate and attractive||10|
|Props used in the ad are appropriate and attractive||10|
|Advertisement is 60 seconds long||5|
|Extra Credit – Best of Shift (10 points)||—|