Given two points in the plane there is exactly one line through them. Of course. But there is also exactly one exponential function, and one power function. Being a math nerd, I think this is cool. I also thought it could form the basis for a project wrapping up our unit on exponent and log functions. The assignment is to begin with two “personal points” formed from some numbers about you (like birthday, jersey number, etc.) and provide the graph and equation of the linear, exponential and power functions through those points. I assigned this as a portfolio, meaning it will be a summative grade (like a test), and gave them the scoring rubric in advance. The assignment sheet is here.

As a lead-in to the project, we did a lot of graphing on log scales this week. We also graphed on semi-log and log-log paper, and saw how they linearize an exponential/power function (respectively). This helped us discover and understand why the function through two points must be unique. It was nice to do some graphing by hand, both for the tactile experience and to see all the problem-solving opportunities that arose. Choosing an appropriate log scale was especially difficult for many kids.

To do the project, kids will need to use several mathematical tools:

- point-slope form of a linear equation (can’t hurt to review it yet again)
- point-ratio form of an exponential
- “logged” form of power function, with the slope as log ratio
- various exponent and log solving techniques

There is obviously no “real-world meaning” to the personal numbers students use during the project. During our log graphing week we’ve looked at several different data sets. I’m hoping the curve-fitting skill from this project can carry forward to future data analysis projects. At the least, it will give them some good algebra practice and might help make an interesting connection between these parent functions.

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