We are using both the PARCC Draft Model Content Framework for Mathematics and the high school sequence funded in part by the Bill and Melinda Gates Foundation and the Pearson Foundation to determine the order in which to present our geometry content.
One major difference for me is that these documents suggest starting geometry with constructions. I have had students do constructions at the end of the course the past few years, which I like, because they end up making a lot of connections about congruency and definitions. Starting the course by handing students a compass and a straightedge and directions to construct a perpendicular bisector seems like a more challenging way to get students engaged in learning geometry. So we are compromising. We gave students a problem Thursday that is from the Illustrative Mathematics site, about determining placement for a fire hydrant equidistant from 3 buildings (or “vertices”) that form an acute triangle. Actually, we didn’t use “equidistant” when we first presented the problem. We asked where they would place a fire hydrant, and let them come up with “equidistant” before making that part of the task.
When presenting tasks, we are trying to allow students to progress through the van Hiele model of geometry thought [1.]: visualization, analysis, abstraction (informal thought), deduction (formal thought), and rigor. So we started by giving students a paper copy of the location of the buildings and a ruler. Students played for a few minutes with their diagram. Many students used trial and error: they would place a point, measure the distant to each vertex, and erase and move as needed to make the point equidistant from the vertices. A few students began to think about the midpoint of each side of the triangle. In fact, they began to draw the medians of the triangle, found that they were concurrent, and decided that the fire hydrant should be located at the centroid. But when we asked them to verify that the centroid was equidistant to each vertex, they realized that the centroid wasn’t the point they needed.
In How to Solve It, George Polya suggests solving a simpler problem as a strategy for problem solving. In this case, that meant finding a point equidistant from two points instead of three. We had two students stand apart from each other and asked a third student to stand equidistant from those two students. In the first class, the third student stood at the midpoint of the “segment”. Then we asked a fourth student to stand equidistant from the two original “endpoints” but not on top of the midpoint. Ultimately, we had a line of students that formed the perpendicular bisector of the original segment.
At this point, we moved to TI-Nspire technology. Students used the perpendicular bisector tool on each side of the triangle and found the point of concurrency, the circumcenter. We used the circle tool to draw the circumscribed circle, and then we moved the vertices of the triangle around to determine whether the circumcenter would always be the best location for a fire hydrant (not necessarily, when the buildings are the vertices of an obtuse triangle).
Now we have a reason to construct a perpendicular bisector with a compass and straightedge. We will see how that goes today.
1. Crowley, M. “The van Hiele Model of the Development of Geometric Thought.” In M. Lindquist, ed., Learning and Teaching Geometry, K–12, 1987 Yearbook. Reston: National Council of Teachers of Mathematics, 1987.