By Jessie Mathisen
A number of years ago, I had a student who was assigned an algebra project called "Grid Co." I don't know if her teacher invented it or got it from someone else. I don't even know the teacher's name, so I am unfortunately unable to give proper credit. Regardless of where it comes from, it is one of my favorite math projects for students studying algebra.
The premise of Grid Co. is that you (the student) are a consultant hired by Grid Co., the nation's leading supplier of fine grids for industrial and home use. (Kids always want to know what the grids are for- sometimes I have them come up with a fun use as a small creative writing extension to the project.) The people at Grid Co. want a rapid, accurate way for their sales staff to quote prices when customers call up to purchase custom grids, and it's your job to make that possible.
Of course, what I am really looking for here is a student-made formula that can correctly spit out a price when you insert information on the dimensions of a grid. It's a great way to help demystify formulas, which often seem like magic.
I break down Grid Co. into two portions: 2D grids and 3D grids. The 2D grids are considerably easier, so I always start with them. To give you a better idea of how I structure the project, I have included an example of the worksheet I give students at the end of this article.
Some students are pretty much able to take the assignment and run with it. However, most students need some guidance. One way I provide this is by giving students data collection tables. I find this provides focus and gets them thinking about what details are important and what details are not. (Sample data collection tables are also included at the end of this article.)
It may seem obvious, but it is critical that students actually build models and count the parts carefully. Some students run into trouble early by trying to cut corners when collecting data. For 2D grids, I usually have students just draw grids out on paper. Using dot paper (like graph paper, but with dots instead of lines) makes drawing grids easier, but it is a luxury, not a necessity. If kids want to, I'll let them build actual 2D grids, but it does take more time than using drawings. For 3D grids, I always have students build physical models. A very wide variety of materials can be used for building grids- use what is convenient or what will appeal to your particular students. In the past, I've had particularly good luck with toothpicks and stale prunes (fresh prunes are a little too soft). A grid built with toothpicks and large gumdrops was a little less stable, but far prettier. If they're available, commercial model building sets can be nice, and they won't attract ants.
Students can almost always come up with portions of the formula on their own. For example, it quickly becomes apparent that every 2D grid contains the same number of 2 hole connectors (i.e., four- one at each corner of the grid). From there, it's easy to see that simply multiplying the price of a 2 hole connector by 4 will obtain the total cost of the 2 hole connectors. On the other hand, figuring out how to model the number of 4 hole connectors is significantly more difficult. I let students mull it over for quite a while- usually I let them toy with the problem over the course of several days. At first, I give virtually no clues beyond the data collection chart, but after a while, I will gradually start suggesting ways to look for patterns in the data (interesting, but they don't usually get the algorithm from this). Then, I will start helping them look at the grid from a more functional point of view. In other words, I'll ask questions such as "Why is the number of 3 hole connectors on a given side always fewer than the length and width as measured in beams?" and "How does the number of 4 hole connectors in a row relate to the number of beams in that row?" After a while, this type of leading question will lead to the breakthroughs that students need to finally crack the algorithm.
Finally, a note for classroom teachers. I am a tutor and I also teach small groups of homeschoolers. My student who originally clued me into this project went to a small, exclusive independent school. My point is that I don't need to deal with classroom management issues, and neither did that teacher. If I were teaching in a traditional classroom, I would structure this project somewhat differently because in its current form, the students have to deal with a lot of frustration. In the environments in which I currently work, I can manage that frustration, and I think it is educationally valuable for students to sometimes really struggle (especially when they ultimately succeed and end up with a result that really wows the people they show it to). However, in the classrooms I used to teach in, that level of frustration could easily have led to a classroom management disaster. I would still tackle this sort of project, but to head off a crisis, I would probably break the project into smaller bites and announce in advance that clues would be given out at certain pre-determined times.
P.S.- I know I haven't given the algorithm here- I'm confident that you can figure it out if you try!
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