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!