Isometries and Coordinate Geometry

Michael Pershan’s post on teaching congruence reminded me that way back in the beginning of summer, I’d been meaning to write up some of my geometry work, which I think is pretty unusual. Still on the list is the lesson sequencing, but here is some thoughts and sample problems on integrating Isometries and coordinate geometry.

To summarize my earlier work, explicated in Teaching Congruence, or Are You Happy, Professor Wu?, I was unhappy with the circular reasoning that geometry books present in congruence sections. Triangle ABC is congruent with Triangle DEF because all their sides and angles are congruent, and congruence is when the shapes have congruent sides and angles. Professor Wu’s writing taught me the link between congruence, similarities, and isometries (aka, transformations, or translations, rotations, and reflections). I’d previously skipped isometries, since the kids don’t need them much and they’re easy to figure out, but this discovery led me to use isometries as an introduction to congruence and similarity.

But all book chapters on isometries are very thin, or they rely on non-coordinate shapes, which is largely a waste of time. Was there any way I could bring back some other concept while working with isometries, particularly with my top students?

Which leads me straight to coordinate geometry. The most immediate tie-in is helping students figure out rotation, the most difficult of the transformations. A 90 degree rotation around a point involves perpendicular lines (“..and class, what is the relationship of perpendicular slopes, again? Class? Waiting!”). Moreover, the kids learn that the slope of the line connecting a point and its reflection must be perpendicular to the line of reflection. Finally, dilations involve all sorts of work with parallel lines. All of these reinforcements are excellent for weaker students, and are yet another reason to introduce transformations, even if only as a prelude to congruence.

But I wanted a meaningful connection for my top students, who usually grasp the basics quickly. What could I give them that would integrate algebra, coordinate geometry, and a better understanding of transformations?

Over the summer, I taught an enrichment geometry class to seventh graders whose parents got mad because I wasn’t assigning enough homework. My boss backed me–thanks, boss!—and the kids did, too—thanks, kids!—and not for the usual reasons (these are not kids who celebrate a lack of homework). The kids all told the boss that they were surprised that they weren’t able to just follow the pattern and churn out 50 problems of increasing difficulty in the same vein. “I have to really think about the problem,” said more than one, in some astonishment.

So, for example:

Homework: Reflect Triangle LMN [L: (-1,4), M: (0,7), N: (-4, 10) over line y=x+2. Prove it.

So we discussed the steps before they left. I actually posed it as a couple of questions.

1. If you sketched this and just estimated points the reflection, what would be the key information you’d need to pin down to go from “estimation” to “actual answer”?
2. Can you think of any coordinate geometry algorithms that might help you find these points?

And working with me, they came up with this procedure for each point:

1. Find the equation of the line perpendicular to the reflection line.
2. Find the solution to the reflection line and the perpendicular line. This solution is also the midpoint between the original point and its reflection.
3. Using the original point and the midpoint, find the reflection point.
4. Prove the reflection is accurate by establishing that the sides of the original triangle and the reflection are congruent.

And here it is, mapped out in Desmos—but honestly, it was much easier to do on graph paper. I just wanted to increase my own Desmos capability.

This is the cleaned up version. Maybe I should put the actual work product here. But I’m not very neat. Next time I’ll take pictures of some of the kids’ work; it’s gorgeous.

When we came in the next day, the kids excitedly told me they’d not only done the work, but “figured out how to do it without the work!” Sure, I said, and we then predicted what would happen with the reflection of y=x+3, y=-x + 4, and so on.

But what about reflecting it over the line y=-2x?

Gleesh. I didn’t have time during summer to investigate why the numbers are so ugly. The kids got tired after doing two points, and I told them to use calculators. But we did get it to work. We could see the fractions begin in the perpendicular line solutions, since we’re always adding .5x to 2x. But would it always be like that?

However, I’ve got one great activity for strong kids done–it reinforces knowledge of reflection, coordinate geometry, systems of equations, and some fairly messy algebra. Whoo and hoo.

Down side–for the first time in two years, I’m not teaching geometry this year!

All the more reason to document. Next up in this sequence is my teaching sequence. But if anyone has ideas about the translation that makes the second reflection have such unfriendly numbers, let me know.

Hey, under 1000!

Mapping Real Life with Coordinate Geometry

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.

1. He went due north for three miles, dropping off supplies for diNardo’s.
2. He then cut northeast along Steep Street to Patel’s Naan and Curry shop. He could have gone due east for 8 miles along Grimley, and then two miles due north along Freeman, but he wanted the shortest route.
3. 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.
4. Next stop, 17 miles due east along AutoBahn Boulevard to Andy’s Noodle Shop.
5. Then it was northwest along Bracken Drive to Tomas’s Taqueria, which was just 6 miles due north of Harley’s.
6. 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?