# #MathArtChallenge 89: Voronoi diagrams

The Challenge: Throw some random points (or carefully selected ones!) on a plane. Identify the parts of the plane that are closest to each of those points.

Materials Needed: Paper, ruler, writing utensil.
Math concepts you could explore with this challenge: algebra, arithmetic, geometry, lines, proportions/ratios, slope, tessellations, vertices/intersections.

I made mine (above and below) using perpendicular bisectors between the points shown, but are there other ways? How might you reason your way into making one of these? (Hint, start with just 2 points, then 3, then…)

Part of my reason for selecting these is their versatility for the classroom. There are so many ways to use them for deep mathematical discussion that can leverage your student’s knowledge. In the map below, I marked the large Minneapolis Public High Schools with a star and then created a Voronoi diagram. THINK of the conversations you could have about this map.

FOR EXAMPLE:

• Which high school, just based on this map, do you think is the largest?
• Are the high schools reasonably distributed throughout the city?
• Which high school is easiest for the most people to get to? Which is the hardest? Just THINK about what your kids could bring to this discussion. Their knowledge of roads, bus maps (MPS students use the public bus system for high school), safety, residency.
• If we were to add a new high school, what area makes most sense to place it in?
• Where do you think students are most densely concentrated?
• What are the benefits of true community schools (defined here as “closest school to you”)? What are the drawbacks?
• If we laid a map of the elementary and middle schools over this, what would you expect to see? What would you want to see?

And naturally, there are easy ways to make these digitally. Here’s a quick version made in Geogebra:

Depending on how you use this activity, you may engage with different mathematical standards. I’ve listed possible connected math content above. Here are a few suggestions for how you might integrate the 8 mathematical practices. Feel free to add your own suggestions in the comments!

2.) Reason abstractly and quantitatively. What does it mean to be “closest” to? Can you measure it always through distance or are there other things we could be considering?

7.) Look for and make use of structure. How else might we make this map? Would it make sense to instead base the lines on a street grid map? (So you’re closest according to driving distance, not how birds fly?) Or based on population?

## Author: Ms. P

Math Teacher in Minneapolis, MN.