I gave a talk at ASSM back in April entitled The Slow Math Movement. The following is an excerpt from that talk that describes how the Illustrative Mathematics Placing a Fire Hydrant task played out in my classroom last year:

Towards the beginning of our geometry course, we give students a task from Illustrative Mathematics called Placing a Fire Hydrant. Where would you place a fire hydrant to serve all three buildings?

Through a Quick Poll, students drop a point at the location they think is best. Then we introduce the requirement that the fire hydrant should be equidistant from all three buildings.

They started on paper, using 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).

Next they moved to technology. I watched while they worked using the Class Capture feature of our technology, and using what I learned from Smith & Stein’s 5 Practices for Orchestrating Productive Mathematics Discussions to monitor, select, and sequence the student work for our whole class discussion.

My students didn’t come into this lesson knowing the vocabulary associated with special segments in triangles, so I purposefully included some incorrect solutions for placing the fire hydrant equidistant from the buildings to bring out that new vocabulary.

Kolton had constructed the midpoints of the sides of the triangle. I made him the Live Presenter so that he could discuss his solution and so that students could learn what a median of the triangle was. His measurements showed that his solution didn’t always work,

but the dynamic feature of our software let him move the buildings around and begin to consider when the intersection of the medians would be equidistant from the sides of the building.

Chaney had constructed a midsegment of the triangle, and so we looked at hers next to learn that new term.

Jameria had constructed the three midsegments of the triangle, creating a midsegment triangle. She was able to tell from her measurements that her solution didn’t always work, either, but we looked at anyway, and I told students that we would learn more about the midsegment triangle later in the course.

We moved next to Sawyer, who recognized that the correct placement of the fire hydrant should be the center of a circle that contained all three buildings, but we could see from his work that he hadn’t yet figured out how to get a circle through all three buildings.

Quinn had fashioned a circle through the three points, but still hadn’t actually constructed it.

Caroline had constructed the perpendicular bisectors of each side of the triangle. She had measured from their intersection, the circumcenter, to each building to show that they were equidistant.

As the Live Presenter, she started moving the buildings around to show that her solution always worked.

Then we asked her to construct the circumscribed circle to emphasize that the intersection of the perpendicular bisectors is the circumcenter.

As Caroline continued to move around the buildings, Gabe asked, “Why would we put the fire hydrant there?” Caroline stopped, and we took a good look at the setup.

She moved the buildings again, to exaggerate how ridiculous it would be to place a fire hydrant that far away. Our dynamic technology made the students realize that the circumcenter isn’t always the most efficient place for the fire hydrant, even if it is equidistant from the three buildings. And so we began to explore when it makes sense to put the fire hydrant equidistant from the buildings and when it no longer makes sense.

Take just a moment to contrast the Fire Hydrant task with how I used to teach special segments in triangles. Which one of these is the “Fast Math” option? Which one furthers the Slow Math Movement?

In his book The Falconer, Grant Licthman says, Questions are waypoints on the path of wisdom. Each question leads to one or more new questions or answers. Sometimes answers are dead ends; they don’t lead anywhere. Questions are never dead ends. Every question has the inherent potential to lead to a new level of discovery, understanding, or creation, levels that can range from the trivial to the sublime. (Lichtman, 35 pag.)

The technology that we use provide the impetus for students to ask questions, which leads to more questions and some answers, from and by the students. I get to watch and listen and push and probe my students by asking more questions.

What can you do this week to further The Slow Math Movement?

[Cross posted on The Slow Math Movement]

howardat58

September 9, 2015 at 6:46 pm

This is delightful !!!!!

jwilson828

September 10, 2015 at 5:40 am

Thank you, Howard.

Maya Quinn

September 9, 2015 at 9:41 pm

Incidentally:It may be worth noting that there other interpretations around where oneshouldplace the hydrant for three such buildings. A classic example is the task of placing the hydrant at point H so that it minimizes the sum of line segment distances AH + BH + CH. There are often student misconceptions about where this point will be; more can be found on the wikipage for theFermat Point.jwilson828

September 10, 2015 at 5:43 am

Thanks, Maya. We use an activity called The Shipwrecked Surfer from a Mathematics Teacher article from years ago that explores this for equilateral triangles, but we haven’t done so with other types of triangles.

Maya Quinn

September 11, 2015 at 12:15 am

The activity you mention seems to be about minimizing distance from an interior point to the sides of an equilateral triangle, rather than to its vertices. (And, in any event, the equilateral feature causes a lot of generally distinct points to overlap!)

Interestingly:I see it written up in your given link (http://www.larrycopes.com/papers/surfer.pdf) but I don’t see a mention there of the name for this fact:Viviani’s Theorem. Perhaps I have overlooked something.jwilson828

September 11, 2015 at 5:14 am

Thank you for the correction. I read “minimize the sum of the distances” and immediately thought of the equilateral triangle problem without paying attention to the difference. And yes, of course, the equilateral triangle is a special case of what would happen generally. I am glad to know about both Fermat’s Point and Viviani’s Theorem.