Taxicab geometry

The warmup activity was designed thus: I placed a coordinate plane on the slide before the kids arrived, and marked several points.  I then removed the coordinate plane and put the question on the slide: Which two points are farthest apart?

It’s not a bad question but it’s also not inherently interesting or natural.  Students looking at the cloud of points would not automatically want to know which ones were farthest apart.  So, the activity could be better (any suggestions?  Cities?) but it didn’t appear to demand any math knowledge to answer.  This is very appealing to some of the kids, and in fact it seems like you have to do this kind of thing to reach those kids at all.

After about a minute, I tell the kids that I’m curious how they’re putting this warmup on their paper at all.  They’re eyeballing.  If one pair of points is obviously farther apart than any other pair, eyeballing would be fine.  I have cleverly marked the points so that you do need some accuracy.

I ask for guesses on which points.  (students in bold) Kids point vaguely at the points.  Can we label them? OK what should we do to label them?  Can we just use letters? That’s a good idea and that’s how they do it in Geometry anyway, so let’s do that.  We label the points A-F and we are able to put guesses on the board.  There are 5 guesses.

But how can we know for sure?  Unless it’s a tie, but we would need a way to find that out too, right?  Can we measure them do you have a ruler? I have two rulers; do you want inches or centimeters?  Most of the yelling says inches and the kid who actually goes to the board decides to use centimeters.  Some of the students are disgusted by this.  So why centimeters?  Well, I just thought it would be easier. More groaning ensues because inches are so easy, and I say what do you mean by easier?  Well, centimeters are smaller so if the answers are close, it’ll be easier to tell with centimeters than inches because you’d have to use part of an inch. This gets some respect from the students and of course from me.  He measures his guess and hands the ruler to someone else, as is our custom.

Three of the guesses turn out to be 59 centimeters apart, which was not intentional but will be from now on.  Are they all exactly 59 centimeters apart?  No, probably not… So then what happened?  It’s hard to measure up there, and the dots are pretty big so if you measure from the center or from one side of the dots you get a different answer.  This is true and really good.

Still, we need more precision than we can get from the meter stick and eyeballing.  We’re back to how to mark the points on paper.  Let’s put a graph on it. A graph?  What do you mean by graph?  A grid. Really?  Grid?  Really?  (this continues for about 30 seconds) A coordinate plane. Aha let’s do that.  I put a coordinate plane (20×20) haphazardly on the slide and ask if that will suffice.  No?  Ok someone come up here and move the coordinate axes to a more convenient location.  Do you guys want graph paper?

I could narrate the rest of the period, but basically we develop distance formula from the Pythagorean Theorem (converse.)  This is nothing new.  I want to relate one other interesting exchange in detail.

So, what do we do with the vertical and horizontal distances now that we have them?  Add them.  Ok, so that’s 15 + 3 = 18.  Are these two points 18 spaces apart?  Yeah.  Let’s check.  I turn to the next slide and bring out another copy of the coordinate plane and draw the desired distance (AE) and another segment that we can count and see is 18 units long.  These are different, so the points aren’t really 18 spaces apart.  Maybe not, but if we do this with all the points, we can compare the results and tell which points are farthest apart.

This is really interesting, and in fact the sum of the x and y distances is called the Manhattan distance.  It’s a valid way to express the distance between two points, it’s just not what we’re doing right now.  I feel like I should have had him pursue that further, but how?  Or is that even worth anything?


One response to this post.

  1. Posted by blaise on June 26, 2010 at 5:52 pm

    You could go into how if you were in the city, you’d have to travel 18 street lengths, but if we were on the ocean or in a field, we’d 15.29… street lengths… or whatever.
    So I guess it shows that living in the city takes longer to get places, which is true. And I’ve arrived at this conclusion from the math, not the fact that I spent upwards of 2 hours in DC traffic today…
    The conversations you have with your kids really give me something to strive for, dude.


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