Yesterday, I wanted to close off the coordinate geometry section (distance, midpoint) before I moved into logic. Rather than put a few random problems on the board, I came up with a map description.

Han is a driver for Harley’s Restaurant Supplies, making his Monday morning route.

- He went due north for three miles, dropping off supplies for diNardo’s.
- He then cut northeast along Steep Street to Patel’s Naan and Curry shop. He
couldhave gone due east for 8 miles along Grimley, and then two miles due north along Freeman, but he wanted the shortest route.- He then drove southwest along Morespark, back past Harley’s, all the way down to Bob’s Burgers. Harley’s is exactly halfway between Patel’s and Bob’s.
- Next stop, 17 miles due east along AutoBahn Boulevard to Andy’s Noodle Shop.
- Then it was northwest along Bracken Drive to Tomas’s Taqueria, which was just 6 miles due north of Harley’s.
- Back to Harley’s for his lunch of his noodles, naan, tacos, and burgers before he started out for the afternoon.
A. Create a map of Han’s route, including street, restaurant names, and coordinates. Suggest using (0,0) for Harley’s.

B. How far did Han travel?

All the students had their own whiteboards, so they could sketch and erase as needed. Step A, the sketch, went really well. As I expected, step 2 gave students the most difficulty, but a third of the class understood it without assistance, and the rest had drawn the two descriptions as two different locations.

First student finished with the sketch on whiteboard:

(Yeah, no street names. He put them in after I took the picture).

As students began moving from the sketch to calculating the distance, I brought it back up front. What was the difference between finding the distance from Bob’s to Andy’s (due east) and Andy’s to Tomas’s (northwest)?

This is the final product—I forgot to take a picture mid-lesson. But see how some trips are starred, and some have a plus. The class identified the stars, which required further calculation to get the difference. I stress the “slope triangle” in all aspects of coordinate geometry (slope, midpoint, distance), and you can see my light colored sketches of the three relevant triangles.

Later, the class identified the missing distances, and then we added it all up. Final instructions: transfer all of this to a quality sketch in your notes. Use color to identify the triangles.

When I teach the Big Three of Coordinate Geometry (slope, midpoint, distance), I emphasize the triangle because for so many students, the formulas are just one more reason to get negatives and subtraction all hosed up. Sketch in the triangles, and you’ve got a backup. Does this answer make sense? Yes, it’s fine if they use the formulas. I will forgive them. Provided they don’t muck up the math. And remember, *knowing* the formulas is essential. I want them to recognize the format of each formula, even if they never use them.

This took about 45 minutes? Wrap up and transition, maybe 55 minutes.

A few days ago, an ~~a**l obsessive~~ overly rigid teacher called me lazy for not having weeks of lesson plans written in advance. I am usually pretty nice to commenters (which is, like, so not me) but while I don’t object to teachers who plan, I vehemently object to teachers who confuse planning with teaching, and this guy is a prime example of the moralizing putz who never got over his potty training and wants everyone else to suffer his pain.

But here’s the thing: I built this lesson in the fifteen minutes before the day started. I do not think my ability to do so is an essential aspect of good teaching. But it’s a part of teaching I really enjoy, the combination of a) my understanding of my kids’ immediate need and b) my strength at creating interesting lessons on the fly. Forcing me to put together a schedule weeks in advance would either make a liar of me or take away that essential piece of my teaching. I’d become a liar, of course. But why go through the farce?

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