When Dan posted his Painted Cubes Makeover, it reminded me of another activity that was presented for Pythagorean Theorem work, but Students found a totally different connection once again.

The Unit starts with an activity about the City of Euclid, where roads strictly form blocks.

Instead of showing students a grid right away, I used a set-up that is natural in my room, the desks. While I normally run different configurations of desk set-up in my room, when I know this is coming up I reset my desks this way with this task in mind.

(Photo does not portray any actual students, any resemblance is purely coincidental)

I will ask students how I could get from Sam to Joe, and we talk about the different paths. One activity I like to do while exploring this is to have my RC car in the room. When students tell me a particular path, I will set the car up and “investigate ” the path. I do this mainly to reinforce that students need to be able to express their ideas clearly (and to have a bit of fun as well).

Naturally the conversation comes up about what the shortest path from Sam to Joe is. Students really get into arguments about what path is correct:

Students can come up with some really creative reasoning why one is longer than the other. One very popular concept is that longer means time, which means that the more turns you take the longer the path is. Then a discussion starts to define how long the path is, students normally revise it to how many desks you move, or how many students. They also start defining the direction that you move: “one student to the right, three students up, one more student to the right.” This all happens until that one student asks: Why we have to follow the road?

Other than physical constraints (and students can come up with many different pros and cons to both sides), students want to define the length as how far a paper airplane will travel between Sam and Joe. So we create a diagram like this.

As soon as one student looked at this diagram, they immediately said: “The distance is the slope Mr. A, it’s how far you move in the horizontal direction and vertical direction.” Another student piped in: “That’s not right, you add the directions- not put them in a fraction.”

Now is the time we nail down what observation that student saw. I will ask students to discuss in small groups what the student meant by the distance being the slope. Many groups talk about the diagram misleading them, that when they look at how you move, it is similar to what they know about slope. They see the angled line and make that connection. They then realize that while it is not the distance, the same process the use to find slope will also define distance in Taxi-Cab Geometry (with a slight change in operation). They go on to find slope of all the previous paths from Sam to Joe, for our pictures I will get:

- A slope of 2/3
- A slope of 1,1 and 0
- A slope of 2 and 0
- A slope of 1/2 and 1

When we do go on to the Euclid activity, students will tell me the slope between the buildings, and are even engaged if I ask them to try and define equations for the line between buildings. One other great thing students notice when they discover the slope connection, they are able to tell me what the Y value is when the intercept is a non-integer. For example; since the slope from the stadium to the gas station is 1/6, you cross the Y axis after moving 2 blocks (or units to the right), so it is only 2/6ths (or 1/3rd) of the way between 3 and 4-> 3 1/3.

Even though these observations side track the lesson slightly, I always take time to investigate such findings. Any way to make connections to math for my students is valuable, and they come into the next day very excited to explore the same activity from a different perspective.