Tag Archives: teaching math

Appropriate math questions

I had an interview for a middle school math position (why they called me, I dunno, I’m sure I was just filler), and had to bring a sample lesson plan.

I always create worksheets when asked to bring lesson plans, since it gives a better idea of how I teach. I read the standards and came up with this cartoon as a starting point:

I use buying decisions all the time in math classes, because the phrase “Pretend that it’s your money” has a near-magical effect on kids who would otherwise ignore word problems.

So I’d kick off a discussion by asking the kids to identify the difference between the two questions. Most kids of any ability will realize that the girl’s question has one answer, while the boy’s question has several.

From there, I drill down to the boy’s question. (click on the image for a slightly better view). Don’t get too quibbly on the definitions—math purists are insanely annoying.

Now, here’s the thing: this is not a lesson on functions. This is a lesson on variables and generating tables. It’s seventh grade math, not pre-algebra or algebra.

But in order to get kids to think about variables and tables, I find it’s helpful to start with a familiar situation and lead it back to math. In order to explain the difference between the girl’s question and the boy’s question, I have to introduce functions, which leads us to variables.

In my opinion, teachers don’t spend enough time leading kids through the process of identifying the definition of the unknowns (because trust me, if you say “identify the unknowns”, half the class says “Three Cheeseburgers!” and suddenly you’re in a Bill Murray skit.) and generating a table of solutions to the problem. Teachers see these as obvious, as mere activities that we use to solve problems. But low to mid ability kids see nothing obvious about defining variables and generating tables. I’ve been slowly realizing this over time, and it crystallized into my data modelling unit in my Algebra II class. This activity, which I drummed up for an interview (but will undoubtedly use at some point), was my attempt to move it back to a pre-algebra state.

Many teachers who only work with high ability kids would see this handout as ludicrously easy and in fact, I’d hand this worksheet to my top seventh graders and tell them to go it alone, while I talked the rest of the class through. Top kids should figure this out fairly quickly, and on day 2, when the rest of my kids are still practicing simple situations, they’d be on to more advanced scenarios (giving two points and figuring out the rule, giving them a table, and graphing).

But many teachers who work with lower ability kids work hard to find meaningful questions and yet miss the mark by making the question too complex. I was talking to one of my curriculum instructors, and he asked me about a lesson he was planning to give to middle school students in a low income district. He wanted to give kids fractions like 7/20, 8/25, 9/30 or 8/30, 9/35, 4/15 and have them just ballpark whether the fraction was greater or less than the obvious value nearby. So is 7/20 greater than or less than 1/3? and so on.

I told him that he was dramatically overestimating their ability, and recommended he start by doing the same thing with halves. So a list of numbers: 2/5, 9/17, 5/11, and so on. Greater than or less than one half? The kids who can do that quickly, move onto numbers around 1/4. Only after a few iterations should he give the strongest kids the exercise he was thinking of, which most of the kids wouldn’t be able to do, full stop. But he could build the weaker kids ability and fluency simply by getting them to think about greater or less than one half. I advised starting with the much simpler exercise. If I was wrong, he could have the more challenging activity ready to follow up, no harm done.

He took my advice and reported back. Most of the kids never got beyond the over/under 1/2, finding it challenging and meaningful. Most of the teachers he was working with had been sure the activity would be too easy.

I do think it’s important to set open questions for all kids, not just the top students, to let them tussle with before you settle into working problems for practice. However, the questions must be tailored to student ability, and math teachers dramatically overestimate the ability of their kids to work with these questions in a meaningful manner.

It gets old to be told that acknowledgement of low cognitive ability is “setting expectations low”. One of the comments on my “myth” post was “One question that I have is the extent to which students are playing Whack-a-Mole in order to get easy classes for a week or two.” My original response was rather rude, so I changed it to a simple “No”. It boggles my mind that anyone would think kids are faking it.

At this point, some teachers are remembering the time they taught their algebra intervention kids about logarithms, or complex numbers, or trigonometry cycles. I’m not saying you can’t teach it. They just won’t remember it. It will be as if they’ve never been taught.


Teaching Trig

(No, I am not in Alaska. Yes, I am overly fond of alliteration.)

My trig lessons were going pretty well, when I noticed that a good number of kids were having trouble identifying the opposite or adjacent side of the angle. This always seems trivial when you’re teaching a big, complicated subject. The kid gets SOHCAHTOA, gets the ratio, gets the concept of inverse, and just has a little bit of trouble figuring out which angle is which. Big deal, right? Lots of concepts grasped and a glitch on implementation. But from a testing and demonstrated knowledge standpoint, the implementation is the ball game.

It suddenly dawned on me that I was constantly guiding students on one point, and one point only, despite a lot of time spent explaining different ways to find opposite or adjacent sides. And that this seemed awfully familiar, a memory from teaching this two years ago. I couldn’t assume they’d finally pick it up, or that this was non-trivial. I realized I needed a different approach.

And so, I gave them a Kuta handout, magic markers, and an explanation:


“Suppose I ask you to find the trig ratios for angle A. You guys have shown me you know the ratios of SOHCAHTOA. But how many of you are still not quite sure which is the opposite, the adjacent, or the hypotenuse?” (Half the class had their hands up before I finished the last sentence.)

“Yeah, that’s what I thought. So we’re going to try this. In your notes, create a right triangle like this one. Using a marker, trace the two sides of angle A.” (I can see all the kids, not just the usual worker bees, busily creating triangles and marking angles, which is encouraging. They want this method; it’s clearly a hitch in their understanding they want fixed.)

“Now, here is an important concept. The angle is created by the hypotenuse and the adjacent. The opposite is not a part of the angle.” (I repeat this several times. I can hear several kids already saying “Oh, I get it!”)

“So, which side is NOT a part of angle A, Corinne?”

“B and C.”

“Okay, remember that you describe segments by saying the two points together, like this: BC. And yes, you’re right. Segment BC is NOT a part of angle A. Sandy, what did I just say about a side NOT being being part of the angle?”

“The opposite. You said the opposite is NOT a part of the angle.”

“Awesome. So Marco, which side is NOT a part of the angle and so must be the OPPOSITE?”

“BC.”

“Great. So everyone mark BC as the opposite, like this:”

“Now. You’ve got a foolproof way of finding the opposite. Just trace over the angle, and then label the side you haven’t traced as the opposite. Now, what about the angle itself? What sides form the angle, class?”

“The adjacent and hypotenuse.”

“But how do you tell the difference between them?” asks a student.

“It’s opposite the right angle,” pops in another.

“True, and that’s a really helpful method. But if you’re worried about what “opposite” means, just do this. Take your pencil, start at the right angle. Draw a straight line through the interior, the inside, of the triangle, until you hit a side. That side is the hypotenuse.”

(Again, I can see that all the kids are doing this. It’s really friggin’ awesome to to realize how much the kids wanted to close this loophole and to have a method be so instantly useful.)

“And now, what’s left?”

“The adjacent!” the class choruses.

“There you go. That’s how you identify the sides. Helpful?”

“Yes!” (big response.)

We practiced it several more times and then I let them loose on the handout. I knew I was on the right track when one of my weakest students showed me his entire handout labelled correctly, but told me he wasn’t sure what to do next. We went through SOHCAHTOA and the light dawned. He worked every problem correctly.

From there, the next big teaching challenge was inverse, when I finally let them use calculators. Yes, up to now, they’d been using a trig table. I wanted them to fully internalize the fact that each trig value is a ratio. Their familiarity with the trig table really aided their comprehension of the inverse.

Next up was solving right triangles and figuring out what ratio to use. But this was much easier because they were now close to 100% accurate in identifying sides, so they weren’t compounding errors.

And then, the course-alike quiz:

This quiz is by another math teacher, and much easier than my quizzes. The kids nailed it. The mode–the MODE–was 100%. The mean was 83%. Only four kids failed the test, and one of them failed because he wasn’t trying.

To reward them for their great job, I still graded the quiz on a curve (meaning one of the Fs converted to a D), and weighted the quiz as 100 points instead of 50.

Then, the kids spent two days moving back and forth between the three methods of solving for sides: Pythagorean Theorem, special rights, and trig.

Once, when I was up front reviewing the results, a kid asked “So why can’t I just use trig instead of special rights?”

Another one chimed in. “Yes, I really don’t like the ratios.”

I said, “I’m fine with you using trig. But remember, you are expected to know the trig ratios for special rights, although we don’t call them by the same names. So if you really like using trig, take a little time to memorize certain values.” And then I went through the trig values for sin(30), cos(30), sin or cos(45).

“So just memorize these three, and you’re fine. And of course, you should never forget the tangent of a special right triangle.”

“ONE!” said a good number of the class. Yay.

We’re moving onto polygons, which will allow me to revisit trig when finding areas.

I’m psyched. If only I had a job next year!