This is a follow up to my three-variable project in Grade 8, which we're finally doing now that the semester test is out of the way. The kids are really enjoying it, and it helps to solidify the idea that multiple variables can cause the numerical output to change. (I tied this in our introductory discussion to familiar formulas like A=lw and P=2l + 2w). Prior to starting the project, I also quickly pulled up a real-world 3-variable table and asked them to tell me where the causes are and where the effects are, inside this format of the stockings sizing chart. This really helped them to understand and relate to the table setup on their given project sheet, in terms of visualizing why the table is set up that way.
They worked through the first table with guidance, and found the general formula pretty easily once I asked them "if n = 100, what is the formula for y? What if n = 372? What if n is just any n?" After this, they repeated the process for the second (and for some students, as well the third) table of values. The last two days we worked on verifying the general formulas they found, first manually -- learning to show proper work for multiple math test cases -- then with technology.
It took me a little bit to figure out how to set this up, but I made this http://bit.ly/excelTestTemplate to help my 8th-graders test their general formulas for their projects. I am very excited to see the outcome of this project, because as part of the testing procedure I also taught them how to program a very basic loop in the graphing calculator, in order to prompt for two variables, perform a basic formula, and then output the results. It looks like this for testing the first formula y = mn + n.
:Disp M*N + N
The kids took about 30 minutes today programming their graphing calcs to run their general formulas, testing them with a few entries from the table, and documenting it all using their camera phones in order to insert it into the math reports that they will be working all of next week. SWEETNESS. This'll be their second time writing a formal Grade 8 math paper for me, and since the first ones were pretty decent, I think this round will be as well.
I also made this project writeup guidelines to help them with framing their writeup. We discussed at various points of this project, how this mathematical process is very similar to the scientific process, with looking at the effect of a single variable at a time, then combining results to form a general hypothesis (ie. the general formula involving all variables), then test-test-testing your hypothesis using a variety of test cases and test methods. What a perfect age group to do this with, since their little minds are just opening up to the world of thinking logically and sequentially and doubting everything.
I hope you have a break in between heavy-duty algebra topics to do this with your kids. Highly recommended! Super duper multi-faceted project involving a variety of skills.
Addendum: My colleague recommended to me that at the end of the project, I show the kids the 3-D graph using x, y, z as the variables. Wow, that's pretty crazy. I don't know if they're quite ready to handle this!