Best Midpoint / Nth point
We had previously looked at how to find distance in the coordinate plane, and so the next question we asked was, "how can we find the coordinate point that is exactly halfway between two other points?" We started by guessing:
We created a set of criteria to tell us what actually determined a "midpoint" to see if we could rank points in order of "closest" midpoint.
nth point
After we developed a formula for the midpoint, the next question was, "how can we find a point that is some fraction of the distance between two endpoints?" Students used many different tools to investigate this question. Many students started by writing or explaining their method for finding the "nth point" in words:
Some students used algebraic sequences to help them solve the problem:
Some used similar triangles:
We wanted a formula that would generalize the process, and students came up with several different ways of representing this idea.
Presentations
Student presentations are shown below.