I don’t expect subs to do great things. Hand out a worksheet, put simple directions on the board, maybe collect some work… that’s enough.

Thursday and Friday our freshmen were on an overnight trip. My school’s been open 7 years, and this was the first freshman trip I’d missed. So, I get coverages, of course. Luck of the draw, my first coverage is for a geometry class. And the teacher did what I might have done – left very simple directions for the sub – have the kids read about transformations and symmetry, then fill in a glossary with a few words and a sketch corresponding to each term. There was a worksheet for homework.

Hmm.

Notebooks out!

I would have preferred a few days, and I haven’t done this in a year, but the topics were fun, and I decided to change it up, on the fly.

Now, I’ve got no graph paper. Kids, maybe they do, maybe they don’t (and I am barking at kids to borrow paper and pencils from each other, jeez, they thought with a sub they really wouldn’t work?).

Let’s go. Topic is transformations. Covers moving points around the plane, but for us to see, let’s move figures around the plane and see what happens to the points.

I decide to go for concepts, with some vocab discussion, but to get them to participate in developing as many of the algebraic rules as possible. The textbook (Jurgensen, Brown, Jurgensen) has one set of vocabulary, NY State uses another, let’s just throw it all out there.

I sketch (sketch only) axes, put up a triangle at (1,2), (1,6), (2,6) and put up another at (7, -3), (7,1), (8,1). “How far did I push the original?” Across 6, down 5. NY State calls that a translation. I’ve heard it called a slide. I like “push.” The Regents may write $T_{(6,-5)}$, T for translation. The size changed? No. The shape changed? No. The orientation changed? What? The direction it points changed? No. What changed? Just where it is.

A transformation that does not change the size of a figure is called an isometry (detour into etymology). (I didn’t think about introducing “collineation” – but I had I, I wouldn’t have).

Rotations were next. I rotated something around an arbitrary point, told them that the algebra is simpler if we rotate around the origin. Put that same triangle up, and told them to rotate it 90° couterclockwise. Got E to give me the first point: $R_{O,90} (1,6) \rightarrow (6,-1)$, asked another to generalize: $R_{O,90} (x,y) \rightarrow (\textunderscore,\textunderscore)$ and he comes up with (-y,x) and I ask everyone to verify with the other points. Bingo. Repeat for 180° and 270°.

I briefly began composing rotations…. but there wasn’t time, wasn’t nearly time. My class? I would have used rotations as a segue into a brief detour into modular arithmetic and a tiny bit about groups… but I was filling a period, not planning a week. Anyhow, they had a 20 second informal demonstration (through their own answers) to inverses…

Side board. Put up a triangle. What happens if we double each coordinate? (they wait for me. Honestly, this got the least participation.) New triangle is similar, but the sides are doubled. If we used 3, would have been tripled. A half? shrink. There is a center to these dilations, and it is the origin. Again, algebra is trickier if we have a different center. Isometry? No.

Back up front, they fairly quickly reflect across the x-axis, the y-axis, y = x, I do y = -x for them.

One minute on rotational and line symmetry.

Completely out of breath. I wouldn’t normally do this in a class period. But it was better than them reading it and taking notes. We had flubs along the way, kids who reversed coordinates, but by and large I got hands up straight threw, digressed all over the place, and got the kiddies to drive the lesson. We had a little time for worksheets, and some discussion.

(I asked for the same class the next day. I got it. Hooray! Except they had a “review quiz” and I watched instead of taught.)