We’ve used the Illustrative Mathematics task Placing a Fire Hydrant for several years now. Each year, the task plays out a bit differently because of the questions that the students ask and the mathematics that students notice. Which is, honestly, why I continue to teach.
I set up our work for the day as practicing I can make sense of problems and persevere in solving them and also I can attend to precision. If you don’t know how to start at Level 3, use Levels 1 and 2 to help you get there.
In an effort not to articulate all of the requirements ahead of time, I simply asked: where would you place a fire hydrant to serve buildings A, B, and C. Students dropped a point at the location they thought best.
It was then obvious from the students’ choices that they thought equidistant was important.
This year I didn’t put out tools that students might choose to use. Instead, I set the timer for them to work alone on paper for a few minutes and told them to ask for what they needed. Before I could get from the front of the room to the back, almost every hand was raised to request either a ruler or a protractor. (No one asked for a compass this year. Last year, when I had them out on the tables, lots of students used them.)
I gave students a few more minutes to work individually with the option, this time, of working with the TI-Nspire software to show their thinking. And at the end of that, I added a few more minutes, asking students to focus on how they could justify that their solution always works. Then I gave them a few minutes to discuss their thinking with a partner.
I watched (or monitored, according to Smith & Stein’s 5 Practices) while they worked using the Class Capture feature of TI-Nspire Navigator. During that time I also selected and sequenced for our whole class discussion. I wanted some of the vocabulary associated with special segments in triangles to come out of our discussion, so I didn’t immediately start with the correct solution.
We started with Autumn, who had constructed the midpoints of the sides and then created both a midsegment of the triangle and some medians of the triangle. She could tell that the intersection of the midsegment and medians was “too high”.
C chimed in that she had constructed lots of midsegments. In fact, she had created several midsegment triangles, one inside the other.
Next we went to Addison, who not only had created all three medians of the triangle but had also measured to show that the medians weren’t the answer.
That led to S, who had been trying to figure out when the intersection of the medians would be a good location for the fire hydrant.
Arienne told us about her approach next. She had placed a point inside of the buildings, measured from the point to each building, and she was moving the point around to a location that would be equidistant from the buildings.
Reagan talked with us about her solution next. She had constructed the perpendicular bisectors and measured from their intersection to each vertex to show that it always worked.
I wonder what that point has to do with the vertices. What do you see in the diagram? (I was expecting students to “see” a circle. But they didn’t. They saw a triangular prism.) I wasn’t ready to show them the circle, though. How could I help make the circle visible without telling them? A new question came to me: What if we had a 4th building? Where could we place the building so that the fire hydrant served it, too?
I quickly collected Reagan’s file and sent it out to all of the students so that they could create a 4th building that was the same distance from the fire hydrant as A, B, and C.
While they were working, Janie said, “I have a 4th building the same distance, but how do I place it so that it always works?” (On the inside, I was thrilled that Janie asked this question. It is exciting for students to realize this early on in the course that we are about generalizing and proving so that something always works and not just for one case.)
How do you place the 4th building so that it always works? What is significant about the location of the 3 buildings and the fire hydrant?
Sofia volunteered that her 4th building always works. (I have to admit that I was skeptical, but I made her the Live Presenter and asked how she made it.) Sofia had rotated building C about the fire hydrant to get d. (How many degrees? Does the number of degrees matter? Would rotating always work? Why would it work?) She rotated C again to get a 5th building between A and B. What is significant about the location of the 5 buildings and the fire hydrant?
And then they saw it. It wasn’t yet pictured, but it had become visible. All of the buildings would form a circle around the fire hydrant! The fire hydrant is the circumcenter of ∆ABC. The circle is circumscribed about the triangle.
And so the journey continues … every once in a while finding a more beautiful question.