This task is from Illustrative Mathematics. Students are asked to place a fire hydrant equidistant from three locations.
My students worked on paper first. I was impressed with their work this year. More so than last year, which could have to do with us swapping Units 1 and 2 from last year to this year. They used paper, rulers, folding, and compasses.
Several of them realized that if they could find the circle that contained all three locations, the center would be equidistant (and thus the location of the fire hydrant). However, their methods for finding a circle to contain all three points were not very precise (which meant they didn’t already know everything they needed to know about triangle centers).
After students worked on paper & a few students presented their ideas (I deliberately sequenced them, after reading Smith & Stein’s 5 Practices for Orchestrating Productive Mathematics Discussions), we tried to solve a simpler problem.
I asked two students in the room to stand up front. I asked another student to stand equidistant from the two students. The student stood at the midpoint of the segment containing student 1 and student 2 as endpoints. Is student 3 the only person who can be equidistant from students 1 and 2? A student in the back of the room said that she could be equidistant as well. She stood in the back. Another student in the middle of the room said that he could be equidistant as well. He stood in the middle. What is true about students 3, 4, and 5? They are all the same distance from student 1 and student 2. What else is true about students 3, 4, and 5? They are in a line. Yes, they are collinear. What else is true about students 3, 4, and 5? What would happen if we drew the line containing those students? What relationship would that line have to the segment containing students 1 and 2 as endpoints? Somehow, we got perpendicular bisector out of the conversation. Note: I’m pretty sure I read about this idea in a Mathematics Teacher years ago, but I do not have a reference for the article.
What happens if we add a 3rd person to our original problem? Are students 3, 4, and 5 equidistant from students 1, 2 and our 3rd person? No. Can anyone be equidistant to all three of our people? The students recognized the significance of taking the points two at a time – and then eventually determined that the perpendicular bisectors would be concurrent at our point of interest.
We moved to our dynamic geometry technology. Students constructed the perpendicular bisectors of the given triangle and moved the triangle. We created the circumscribed circle. We paid attention to what happened to the circumcenter. Is the circumcenter always a good location for the fire hydrant? I used Class Capture to monitor student progress.
Can the location of the circumenter give us information about the type of triangle? I made a student the Live Presenter to show us what she found about the type of triangle and the location of the circumcenter.
Finally, we ended with the straightedge and compass construction for the perpendicular bisector, focusing on the question “What segments, angles, arcs, and triangles are always congruent in the construction”.
For whatever reason, I made the list on the non-interactive whiteboard, but I marked the student findings on the interactive whiteboard. We got into isosceles triangles, congruent base angles, rhombi, and more.
And so the journey continues…