In Geometry, we have seen the importance of the perpendicular bisector. It’s kinda amazering! And before we delve too far into rotations of figures (which, again, have a lot to do with perpendicular bisectors), my co-teacher and I wanted to do some sort of investigation.

Now, after doing this, I can see what I did as being something that could introduce the awesomeness of the perpendicular bisector. It could be our anchor problem. However, we had already introduced it. So I thought this little aside would be a fun solidification of what we’ve already learned.

(To be clear, students have learned that the perpendicular bisector of a segment is the set of all points that are equidistant from the endpoints of the segments, that the three perpendicular bisectors of the sides of any triangle meet at one intersection point, and that one intersection point is the center of the circle containing all three vertices of the triangle)

Here’s the problem I gave them… however to turn them off the scent of the perpendicular bisector, beforehand I said “okay, class, now we’re doing to go on a little diversion…Here are 4 fire hydrants”

Let’s say there were fires at these four yellow locations. Which is the closest fire hydrant to each?

Okay! Some are easy to just “see” and some might require some calculations.

So next I asked students to try to color in the picture with all locations which are closest to Hydrant A, all locations which are closest to Hydrant B, all locations which are closest to Hydrant C, and all locations which are closest to Hydrant D.

Unsurprisingly, students filled in the edges of the diagram first (so the top left area was all A, the top right area was all B, etc.).

Unsurprisingly, eventually all students asked “what if a location is the same distance to two hydrants?” (I said you could just color that black.)

Surprisingly, since we had *just* been working on perpendicular bisectors, I was surprised that it took 7-10 minutes of working in groups before I heard the first student say perpendicular bisector. I made sure to have that word repeated and spread from group to group. I anticipated that as soon as students would think about A and C (and how there’s an infinite number of points equidistant, with that horizontal line that can be drawn)… or as soon as students would think about A and B (and how there’s an infinite number of points equidistant with that vertical line that can be drawn), it would unlock the whole problem. But my intuition led me astray. I honestly thought they’d finish this in 10 minutes, but it took around 20-25 minutes. But it was worth it. Eventually we got to this…

Now I will say that this region was tricky for kids…

(Note to reader: we talked more about this in the next class, and pointed out that the intersection point at the top of the yellow circled region was equidistant to A, B, and C (so it was like the intersection of the perpendicular bisectors of triangle ABC), and the intersection point at the bottom of the yellow circled region was equidistant to B, C, and D (so it was like the intersection point of the perpendicular bisectors of triangle BCD)).

After all this, I gave my kiddos a follow up “backwards” task that I invented but I didn’t have an elegant/simple way to solve. I just was curious what they would do…

I said “here’s the diagram… can you find out where the fire hydrants are?”

They had good discussions, but we had such little time left in class, and I wanted to get to some resolution. First, I said, “play with trial and error… all the hydrants are on lattice points.”

They had good discussions. But again, we were short on time. So I said: “okay, I’m going to give you ONE hydrant location. From that ONE location, you should be able to find all the others.”

And indeed, kids were figuring the others out. We had learned about perpendicular bisectors as lines of reflections, so some kids started folding their paper along the lines to get more hydrants…

And then you can get the very last one by reflecting the purple hydrant over the vertical perpendicular bisector. To me, I like that students could see that once you had ANY hydrant, they could get all the rest of the locations. For me, that’s a nice takeaway. As I said, I was just curious to see what they might come up with.

At the very end of class, literally the last 5 minutes (I wish I had more), I gave a little lecture on what these things were. They are called Voronoi diagrams. You can literally see the creation of them by this dynamic image (from Wikipedia):

And then I showed them how a giraffe’s spots are a Voronoi diagram!!! You can click the images to get a bit more

At the start of our next class, to see what students retained, I gave each student this diagram and told them to make a Voronoi diagram for it.

I was proud that most students could do this, fairly easily. Then when we had the solution…

… I asked what did all points on the purple ray represent, what did all points on the red ray prepresent, what did all points on the yellow ray represent, and what did the green point represent. Kids seemed great with this. And then I drew a circle (on the Smartboard, using the circle drawing tool) with a center at the green point, and made it bigger and bigger until lo and behold… it hit points A, B, and C. Huzzah!

Update: David Sabol shared this amazing Desmos activity (by Joel Bezaire) which is like what I posted above but is in many ways better. And deals with food deserts (thus social justice) naturally and seamlessly: https://teacher.desmos.com/activitybuilder/custom/5d2a410855693a4619850fd5

Update 2: And of course Bowman Dickson created a cool project out of this… link to his tweets about it here: https://twitter.com/bowmanimal/status/1589804876686675969 of course he did this recently but since I haven’t been using Twitter… I missed it.