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# Coordinate Geometry: Problem Solving as Applied to Urban Planning

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Here’s the idea: your students are learning about coordinate geometry, so you teach them hoe to find the x and y axis, they plot a few points, maybe you play some lame games, and then they’re off and graphing some equations. Bo-ring!

These activities teach students about the conventions of coordinate graphing (they are not “rules”, they are “conventions”) and then applies them to the practice of solving actual problems, from delivering pizza to making maps to guide first responders. The activities were modeled from Eugene Krause’s book “Taxicab Geometry,” which is just a wonderful book on the subject.

The exercises on modeled on the idea that rather than “teaching as telling,” which involves standing in front of the class and declaring, “class, this is how you plot points. The X always goes first....yadda, yadda, yadda,” that your students can learn about the importance of describing coordinates in a certain order because it is necessary in order to prevent confusion about an exact location.

And so beings the adventures of our friends Obtuse Ollie and Acute Alice (who I lifted as an homage to Krause.) As residents of Euclid City, Ollie and Alice live on a set of streets, which run horizontal, and avenues, which run vertically. Alice and Ollie decide to get together somewhere, but the only problem is they have yet to agree what they mean when they say “let’s meet at 1st and 5th.” Is that 1st Avenue and 5th Street, or 5th Avenue and 1st Street? Hilarity ensues.

However, the fun does not stop once they’ve resolved their ambiguity. Now that they have this convention, there are still things that they can do. For example, Alice gives Ollie a set of clues about where to meet, which locates her at an exact spot. Ollie, on the other hand, knows where she is, and sends an ambiguous message about his location. Again, hilarity ensues.

As if this is not enough, our investigations lead to an “optimization problem.” Alice lives in one spot, and Ollie lives in another. The decide to that neither of them should have to walk further than the other to get together for a frappacino. Where are some of the places they could meet?

The next issue concerns the expansion of Euclid City, which, due to its popularity, now must expand both south and west, creating three new neighborhoods. How should we name the streets and avenues in these neighborhoods? Well, using a combination of negative and positive numbers, of course.

Finally, we get to the real tough problems in Euclid City: delivering pizza and running a fire department. In these two exercises, students learn about pizza delivery areas and, most difficult of all, how to set up a boundary map so that two fire departments will respond to the emergency closest to their building.

There is also a set of “Do It Yourself” activities where students can create problems of their own involving coordinate geometry and share them with other members of the class.

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