Tag Archives: slope

Great Moments in Teaching: When Worlds Collide

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 y1 from y2, then x1 from x2. 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>

Teaching Algebra, or Banging Your Head With a Whiteboard

The Five Big Ideas of First Year Algebra:

  1. Identifying the slope and y-intercept of a line from a linear equation, and graphing a linear equation provided in slope-intercept form.
  2. Solving multistep, single-variable equations that involve distribution and combination of like terms.
  3. Using substitution or elimination to solve a system of equations.
  4. Binomial multiplication
  5. Factoring a quadratic equation (a=1)

(You middle-school algebra teachers are saying “Wait, what about graphing a parabola? What about point-slope and standard form for linear equations? What about….” Stop right there. I teach the kids who didn’t make it through your classes. Some winnowing is necessary. Furthermore, I said the five BIG ideas, not the ONLY ideas.)

I’ve taught some form of algebra every year. From my first year on, I’ve nailed factoring quadratics. I do it with the generic rectangle, which has the added feature of helping out when a > 1. The integrated method I use to teach both binomial multiplication and factoring really seems to help the students put it all together.

Towards the end of my first year teaching algebra, I noticed a weird thing. My kids were bombing multi-step equations, and I couldn’t see why. I’d taught them distribution, combination, and isolating—and they’d all done well. Then they were utterly discombobulated when faced with an equation like 3x + 5(2x-7) = 4. They’d add the 3x and 5 to get 8x, then multiply it by 2x, get 16x….it was insane. Yet when I walked them through the problem breaking down distribution and combination, they got each step individually.

Then, early in my second year, I saw the same problem. Kids who had shown solid mastery of distribution, combination of like terms, and solving for x were crashing and burning when I gave them a multi-step equation that mixed and matched everything. I suddenly got it. Multi-steps up the cognitive load considerably. The kids had to take each step in the context of a larger task, and they were losing track. They couldn’t look at the problem and break it down into parts.

So I created the Distribute-Combine-Isolate worksheet, one of the best worksheets I’ve ever done. First distribute, then combine, then isolate. It gave them a sequence to follow. The improvement was tremendous. My first year students, who had much lower incoming test scores than other classes, topped all the other classes in a course-alike assessment on the multi-step equation. This year, I used the same worksheet with any Algebra 2 students who struggled with multi-steps. Again, working multi-step equations has been a major success area; I don’t have to review it, and I can put a tricky question on a test and know that all the students will either get it right or make a few minor mistakes. I am pleased.

Systems: This is the lowest priority of the Big Five when I’m working with struggling students, but it’s a high priority item for my stronger students. I find the challenge comes in when I want them to recognize a system problem. They get the technique, but the overall solution approach is still iffy. But then, this is tough. I don’t feel any real frustration or energy about it.

Leaving linear equations and binomial multiplication, arguably the easiest of the Big Five, as the most challenging and mindbogglingly crazy-making. They get it and forget it. Get it and forget it. Over and over and over and over………….[bam bam bam bam bam]

Slope: You teach them how to plot points. You teach them to see the line. You use manipulatives, transparencies with lines on them, that they can use to match up two points and see how the slope and y-intercept change. You show them how different types of situations map to different slopes. And of course, you give them endless practice.

And then you sketch a line, clearly mark the slope and the y-intercept, and ask any kid who isn’t acing the class, “So, is the slope of this line positive or negative?” and wait, and wait, and wait, and wait and sure as a villain in a Bond film, the kid will say “Um, negative?” when the slope is positive and “Positive” affirmatively when the slope is negative.

So you teach them how to model equations quickly, which works a charm and gives them all sorts of new skills. You see them become much more proficient at word problems, at seeing an equation like 8x +3y = 24 and thinking “Burgers for $8, hotdogs for $3, total of $24” and by god, it’s awesome. All this keeps, beautifully; months later, they are still showing increased competency at word problems and linear equations. You also give them endless practice worksheets where all they have to do is identify + or – on a slope image—nothing more, and they do it cheerfully and successfully. You give them the “N” rule (negative slopes form an N).

And then, you give them a test, in which they have to identify a simple system of inequalities, and a student, a mid-level student calls you over and says, “I have no idea how to do these problems.”

“Well,” you say, “look at one of the lines in the system.” The student points to a line. “Positive or negative slope?” and wait and wait and wait and wait and sure enough, the student says “Positive” when it’s negative and “Um, negative?” when it’s positive and you gnash your teeth and try to figure out how to help them without making them feel hopeless.

And later, when the same thing happens again during the test review, and you start beating yourself over the head with a whiteboard (they make them student-sized, did you know? Like slates in Laura Ingalls’ day) and then you get up and say, carefully,

“Look. When you see me beating myself over the head with a whiteboard, it’s because I am wondering what other way I could teach you this HUGE, SINGLE MOST IMPORTANT idea in first year algebra, something that I’ve told you fifty times, and believe me when I say that I’m not angry or disgusted when people don’t get it. I just can’t figure out how to make it clearer. And I think the real problem is NOT that I can’t make it clearer, but that I can’t get you to stop and think about the many, many many ways to determine the direction of a slope. All you need to do is stop and think about it and remember what you’ve done. And for some reason, many of you don’t. Let me say it again: I am not blaming you. I don’t think you’re dumb. I JUST WANT YOU TO STOP DOING IT SO I WON’T HAVE TO BEAT MYSELF OVER THE HEAD ANYMORE.”

And the class laughs, and you remind them again to stop the minute they see a line. What is the direction? What methods do they have for making that determination? Do NOT simply look at it and say “Heads, positive. Tails, negative” and guess. Please?

Lather, rinse, repeat.

As bad as slope is–and it’s terrible, horrible, the single most frustrating thing about teaching algebra to kids who struggle with math–it doesn’t have the short sharp shock value of the Binomial Multiplication Middle Term Miss.

Last week I gave my kids a geometry test and one of the questions was:


BOTH CLASSES. Every single kid (except the top 6 students, who took a different test) took x2 + x + 9, meaning that they squared (x + 3) and got x2 + 9. WHY? WHY? WHY?

I tell them that this makes baby Jesus cry. It’s the math equivalent of clubbing cute little seals. THEY MUST STOP. It hurts. And we review it, with the rectangle, which they use for factoring and SHOULD MAKE IT CLEARER, DAMMIT! and they learn it again. But I know, very soon, they will forget. If only to make me crazy.