I’m on vacation! I actually took a whole half day off to add to my spring break, spent a couple days with my grandkids (keep saying the phrase, it will get more real in a decade or three), then embarked on an epic road trip through the northwest. My goal to write more posts is much on my mind–despite my pledge, I’ve only written 10 posts this year. But I’ve gotten better at chunking–in years past, I would have written one “teaching oddness” post, rather than three.

So this new semester, new year, has already seen some teaching moments that are best thought of as crack cocaine, a hit of adrenaline that explodes in the psyche in that moment and every subsequent memory of it, the moments you know that all those feel-good movies about teaching aren’t a complete lie. Not all moments are big; this one would barely be noticed by an outsider.

I was explaining slope to one of my three huge algebra 2 classes, the most boisterous of them. Algebra 2 is tough when half your kids don’t remember or never learned Algebra 1, while the rest think they know all there is to know, which is y=mx+b and the quadratic formula (no understanding of what it means or how to factor). Meanwhile, my recent adventures in tutoring calculus (be sure to check out Ben Orlin’s comment) has increased my determination to improve conceptual understanding among my stronger students, even if my weaker ones get a tad bored.

“I want you to stop just thinking of slope as a number, something you can only get by looking at two points, subtracting y_{1} from y_{2}, then x_{1} from x_{2}. The simplest way to start this process is to consider the slope triangle, which I know a lot of you use to find the slope, but don’t really think about.”

“But think of slope as represented by an actual right triangle. The legs represent the relative change rates of the horizontal and vertical (the x and the y). The hypotenuse is the slope. You can *see* the rate of change. It’s not just a number. Evaluate slopes by their triangles and you can *see* the ratio in action.”

I’m skipping over some discussion, some give and take. As I drew pictures, I “activated prior knowledge“, elicited responses as to what slope was, what the slope-intercept form represented, etc. But this was pretty close to pure lecture. I can read the audience–they’re not hanging on every word, but they get it, I’m not preaching to snoozers.

“How many of you remember right triangle trigonometry last year, in geometry?” A few hands, mostly my top kids.

“Come on, SOHCATOA?”

“Oh, yeah, that stuff” and most hands go up.

“So when I teach right triangle trig, I do my best to beat into your heads that the trig identities are ratios. Trigonometry is, in fact, the study of the relationship between the ratios of triangle legs and the triangle’s angles.”

“And that means you can think of the slope of a line in terms of its trigonometric ratio. Take a look at the triangle again, but now use your geometry lens instead of algebra.”

“The slope of a line is rise over run in algebra. But in geometry, it’s opposite over adjacent. The slope of a line is identical to the slope triangle’s tangent ratio.”

“Holy SHIT.” Every head turned around to the back of the room (where the top kids sit), where Manuel, a big, rumpled, exceptionally bright sophomore was staring at my board work.

I smiled. Walked all the way to the back of the room, to Manuel’s desk, tapped it lightly. “Thanks. That means a lot.” Walked back all the way to the front.

Remy smiled knowingly. “That was like some sort of smart-people’s joke, right?”

“Naw,” I said. “His worlds just collided.”

I could do a bit more, explain how I followed up, but no. You either get why it’s great, or you don’t.

*<mic drop>*

March 28th, 2016 at 8:40 pm

[…] Source: Education Realist […]

March 29th, 2016 at 8:29 pm

The illustration would be better if the triangles’ sharp ends pointed to the left, instead of the right. (The “rise” and “opp” dimensions should also be moved, so that they are still next to the sides they describe.)

Unless explicitly shown otherwise, most people assume that the x-axis increases to the right, and the y-axis increases to the top. Thus, a slope that is downward-and-to-the-right is negative, and upward-and-to-the-right is positive.

March 29th, 2016 at 8:31 pm

This isn’t my picture. My picture was fabulous. and yes, it did look as you describe. I had them both going in the same direction.

I’ll go steal the picture when I get home and substitute it.

March 31st, 2016 at 12:38 am

I teach tangent as the “slope ratio” for several class periods before ever introducing the term. CPM curriculum gave me that. Have then graph lines and measure slope angles then make slope angles and measure slopes for several useful pairs and start building up our own trig table. The have then use that table to find missing sides and angles using similarity and logic only. Works! Sadly it doesn’t seem to stick very long: as soon as they learn SOHCAHTOA and use it for a week the slope ratio idea goes away for then and when I mention it or ask a question about it, big fat question mark.

March 31st, 2016 at 12:51 am

I got the idea from CPM, too! I spend a lot of time trying to beat the idea that it’s a ratio through their head. If you check out my earlier work, I’ve got some test questions that might help you test it in different ways. But I hear you about the forgetting.

November 23rd, 2017 at 6:35 pm

[…] were expected. The best way to do that, in my experience, is give them a situation and point out obvious things that connect in not so obvious […]

January 28th, 2018 at 4:44 pm

[…] 1 or lower in their freshman year, and took AP Calculus their senior year. (One of them was Manuel.) Eight students entered at the same level took regular calculus. So fourteen students were not […]

January 28th, 2018 at 7:11 pm

I was taught sin and cos using the unit circle. This was emphasized more than the ratios. To this day, I think of trig this way. I got a 5 on AP Calc and went on to a technical career.

Maybe it is sort of a gap, but it sure is easier to just think unit circle.

And yes, I can do manipulations if needed. But I just don’t start with or conceptualize with ratios of sides. I think unit circle

February 10th, 2018 at 9:00 am

[…] in their freshman year, and succeed in taking AP Calculus their senior year. (One of them was Manuel.) Eight students entered at the same level took regular calculus. So fourteen students were not […]

June 3rd, 2018 at 7:27 pm

[…] (Manuel‘s younger brother) and Benny look abashed, hearing the edge in my voice. I was mad at myself […]