# Radians as Decimals and a 2-minute Fix

This was a small thing that sent lightbulbs off. I should have done this years ago.

My kids are reasonably comfortable with radians, expressed as fractions of pi, and we do a lot of measuring with radian protractors, so they can even do things other than special angles. (More on that another day.) If I give them an angle, they can tell me what quadrant it is, and they can often explain their reasoning.

But then we hit inverse trig today. I don’t know what your calculator does, but our calculators spit out decimals when they evaluate inverse trig functions. And every year, my kids get confused. “Wait, but I thought this was supposed to be an angle in radians? It’s a decimal! Where’s the fraction? Where’s pi?” And then they get confused about whether they have an x-coordinate or an angle and they switch up arccosine and cosine (an admittedly easy thing to do) and the whole thing is a mess.

Using the radian protractors helped. I have one set that has fractions of pi on it and another set has decimals on it. Things got a little better.

We solved some equations like cos(theta) = 0.3 and got things like 1.26 and 5.02.

But then, right at the end of class, I decided to throw up a coordinate axis and label it: 0, pi/2, pi, 3/pi/2, 2pi.

“Guys, what’s pi as a decimal?”

“3.14!”

And I labeled the negative x-axis as 3.14.

“Uhhhhh…..”

“You have a calculator….

“Oh!” (furious typing) “1.57!”

And I labeled the positive y-axis as 1.57. We kept going and ended up with something like this:

Lightbulbs went off all across the room. “So what quadrant are our answers in?”

“I and IV!”

“And does that make sense?”

The bell rang, and we didn’t get to the last question, but it helped them so much just for me to write down what pi, pi/2, 3pi/2, and 2pi were as decimals. They could see where the answers they were getting belonged. I should have done this years ago.

# Han Solo and the Question I Didn’t Ask

I put the following problem on a recent homework assignment:If you can’t read it, it gives a table that maps the number of years Han Solo and Chewbacca* have been smuggling to the total value of goods they smuggled that year. I ask whether the inverse function exists, what it means in that context, and how you could change the table so that it doesn’t exist. Then, remembering that meaning matters, I ask the students to come up with a reason why someone (Han, Jabba, an Imperial customs officer…) might care about the inverse function. I was thinking something like, “If they know they made a certain amount at some point, but couldn’t remember which year…” with suitable dressing about why Jabba the Hutt is after them for not giving him the right cut or something.

The answers students gave me were baffling. There were, of course, the silly ones like “If a strong enough ion blast were to overload the Death Star’s main cannon as it fired, it could invert the world due to….” Well, you get the idea. But over and over again, students told me that Han would care about this function because he would want to see that he was making more money, that the amount that he smuggled was increasing, etc.

Now, it’s a lot easier to see that the smuggled amount increases as time goes on if the input is time and the output is the amount smuggled. That’s the whole point of talking about increasing functions. It’s a lot harder to tell if you’re looking at the inverse function, although it could certainly be done.

Finally, I found a student who wrote this:

(If that’s not legible: If f inverse did not exist, that would mean they smuggled in the same amount of goods twice, which means that they’re not progressing.)

And a lightbulb went off.

They weren’t telling me why the function was useful, but why the existence of the function was useful. And when I looked at my questions again, I realized why. The previous two questions had been about the existence of the function. That’s what I’d put on their minds. Of course they were going to tell me about the existence of the function!

I’m not sure how to change this problem to get what I want. On the other hand, I’m not sure that I should change the problem to get what I had in mind. Their interpretation of the question gives a question worth thinking about. The goal was to make inverse functions meaningful. If this makes it meaningful…ok. Maybe a good follow-up would be, “Is the inverse function the easiest way to figure that out?”

*One class realized that I misspelled Chewbacca on the key. This lead to a student asking me how you spell his nickname: Chewie or Chewee.

Me: “It’s Chewie.”

Him: “Are you sure?”

Me: “I spent my entire childhood reading Star Wars novels. I’ll stake my life on this.”

Different student: “I think Mrs. L might be a Star Wars fan…”

Yes, dear child, I am.

# Abandoning Code Words

At some point in the past, I decided that it was important that students know that the question “Where does this function have a maximum?” meant “At what x-value does this function have a maximum?” and “What is the value of the maximum?” meant “What is the y-value of the maximum?” I told them what those code-words meant repeatedly. It would show up on homework, on warmups, and on quizzes. And kids would still miss it on the test. They knew how to find the maximum, but they couldn’t remember how to phrase the answer, or they forgot whether I wanted the x or the y, or they gave me both. So I could tell that they knew the math, but they forgot the code word, so they didn’t get full credit.

I’m not sure where this came from. My textbook? But I’ve largely abandoned my textbook, especially in precal, so I don’t really care about how it phrases things. A glance at what the AP Calc test requires? But the AP test typically specifies when they want x-values (I think. Someone tell me if I’m wrong.) And why am I beating pre-cal kids who will never take AP Calc over the head with AP Calc code words that are keeping them from showing me how much math they know? Sure, in Calculus, we should talk about the College Board’s code words and how to phrase their answers. That’s an important thing for them to know. But why am I making everyone else learn it to?

I’ve decided that this is a stupid teaching choice. I don’t actually care about the code words. If I want to know that they can find the x-value where the maximum occurs, I should just ask that and make it clear that I mean the x-value. And then I can tell how many kids don’t know how to find the x-value, instead of how many kids forgot about a code word.

Will I continue to talk about it? Probably. It can foster a discussion of how we think about functions, about how the value of a function means the output, etc.

I wonder how many other things I require kids to know that I don’t actually think are important. What else am I imposing on them that’s getting in the way of the math? What are other areas where I’m on teacher auto-pilot and not thoughtfully evaluating the curriculum?

Oh, and if you have your students learn this and have a really good reason for it, please tell me. Maybe I just forgot why it matters.

# Permutations, Combinations, and Why?

We’ve spent a couple of days in precal talking about permutations and combinations.

The first day, we did a fairly straight-forward permutations/combinations lesson. How many ways can you arrange 5 cars in a driveway? How many distinguishable arrangements of the letters in the your full name are there? (Most people have at least one repeated letter.) How many 3-letter “words” can you form if you never repeat a letter? If you only use consonants? If you only use vowels? If I have ten books and want to take 3 on a trip, how many different sets of 3 do I have to choose from?

It was a fairly standard lesson. We’d arrange 2 cars, then 3, then they’d arrange 4 cars, then we’d find the pattern and make predictions. What was slightly different (and what I want to do more of) is this: when we found the pattern, I’d make them figure out why. If they couldn’t, I’d help, but they had to try on their own first.

Why is the number of ways you can arrange 4 cars equal to 4*(the number of ways you can arrange 3 cars)? Why would we divide by the factorial of the number of times a letter is repeated in your name? Repeat with each different topic. When we hit permutations and combinations, I introduced the vocabulary, as well as the notation ${n \choose k}$ and the phrase “binomial coefficient.”

Today, we did Pascal’s Triangle. We started by expanding binomials: $(x+y)^0, (x+y)^1, (x+y)^2, (x+y)^3,$ … . I wrote the coefficients in a triangle, they saw the pattern, and we expanded more binomials. Then I had them compute binomial coefficients, and we arranged them in a triangle. They caught on pretty quickly to the fact that the binomial coefficients we’d computed on Wednesday were, in fact, the coefficients of expanded binomials.

Why would it be true that the coefficient of $x^3y$ in the expansion of $(x+y)^4$ would be the same as the number of ways to choose 3 items out of 4?

I got a lot of blank stares, so I told them to talk about it with their neighbors for five minutes. I wandered the classroom and listened to their discussions (they’re very used to this), and I realized that almost all of them were stuck. After about four minutes, I called it, and we went back to a large group.

One pair of students came up with a pretty good explanation. They had trouble phrasing it, so I helped a bit. Most of them understood it when I explained it, although they said they couldn’t explain it yet themselves. That’s not a total victory, but I had to leave it there for the day.

The most interesting part of class for me, though, was when one girl said, “Ugh! This is math class! I’m not supposed to have to answer why!”

I ask students to explain and justify their answers, although I probably ask that more in Calculus than in Precal. That’s something that I’ll need to work on for next year. But the beautiful thing about math is that there is a why, that things can be proven. Lots of my students want to tell me, “Because that’s just how it is!” or “Because God made it that way!” when I ask them to explain why. That’s true, but that’s not a reason. Math, more than any other subject, always has a reason. I’m not sure how to address this misconception, but it’s something I want to change.

# Local Linearity: The Big Reveal

Precal is currently doing a quick, basic intro to Calculus. We talked a little bit about limits (we can do them with graphs and with charts), we talked about continuity (three part definition, removable vs nonremovable), and today we talked about local linearity.

Disclaimer: I took this wholesale from an AP Summer Institute instructor.

I started by throwing this up on the projector:

I ask them to tell me everything they can about this function. I typically give them two or three minutes to write everything down. (If I’d been paying attention to the fundamental five, I’d probably have turned this into a small group discussion. “Make a list with your neighbor…” I didn’t, though, and since this was the very start of the lesson, I’m ok with it.)

Sometimes they need a little prompting with their ideas, but they come up with things you’d expect: it’s a line, it’s red, it has a slope of 1, it’s y-intercept is 0, it’s y=x, it’s increasing, it’s continuous, it’s odd. Someone asked me if each square was one. I shrugged – they’ve gotten used to that by this point in the year.

I ask them how much we can see. “Not much.”

I click on a point so that they can see the scale.

I ask again, how much can we see. “Barely anything! Oh, there are probably all of these curves, aren’t there?”

I zoom out, so they can see this:

They’re reactions were perfect. “Oh, wow!” “I knew it!” “You tricked us!” One girl was out of the classroom while we did this. When she walked back in, her friends told her, “You missed it! That was the graph! She tricked us!” I mention the phrase local linearity, and we’re off.

We start talking about slope. We don’t know how to find the slope of a function, so what can we do? One class wanted to find the average slope, so I had to nudge them towards finding the slope at a point. Can we do that? “Yes.” Cool, how? “Uhm….we could zoom in so that it looks like a line!” Yes, yes we can.

So we do. We find the slope at 0, at $\frac{\pi}{2}$, at $\pi$. I point out that it would be nice if we could just have a function whose output was the slope, so that we didn’t have to do all of these approximations over and over again. They agree – yes, that would be nice. (I do love it when they play along…which usually happens when I’ve set up the need before giving them something…but it also requires them to take the bait.)

We switch to $y = x^2$. I give each of them an x-value or two, and they have to find the slope by zooming in, finding two points near their x-value, and then finding the slope between them. Maybe it would be better to stick with $\sin x$, but you have to do a lot of points to see the shape, and I have really small classes.

After they all have their slopes, we put them in a table on Desmos. In one class, they did a great job finding their slopes, and it was a line. In the other, it didn’t work out so well, so we had to redo some points. Eventually, they got it, though. That would have been a spot where I could have gotten some really good estimation conversations going (why do you think that point is wrong? positive slope, decreasing function, etc.), but I didn’t. Maybe next year. They were able to figure out which points were wrong, though, and fix them without me, so I’ll take it.

If we have a bunch of points in a line, we can find the line of best fit. Desmos is really great for this – we typed in $y = mx + b$, created m and b sliders, and went to town. They got $y = 2x$ pretty quickly.

At this point, I mention the word derivative. In the advanced class, I throw up the limit definition of the derivative and talk about it. We don’t use it in precal; I just want it to simmer in their backbrain for a few months until we need it in August.

For homework, they repeat the process on two more polynomials. They get to choose which two from a list I give them. Then they have to write about what they notice, what they wonder, and how they could check those wonderings. I’m hoping some of them will notice the  Power Rule; the list they can choose from leads them their nicely. If they don’t get it, well, we repeat this with a few tweaks in Calculus, so they’ll get it then. My goal for the overview of Calculus is primarily to get them thinking about a few big ideas. We have all of next year to do stuff with it. We’ll probably also do a class discussion next time about what they noticed and wondered, and I might get them to the Power Rule that way.

Their exit ticket was to tell me why we could find the slope by zooming in. Most of them, with a bit of a nudge, could tell me that we zoomed in, saw a line, and from there we could find the slope. No one used the phrase locally linear, but I’m ok with that for this level.

I really, really liked the way this worked today. Their reactions when we zoomed out made my day. Also, in the past I’ve done this with the graphing calculator instead of Desmos (this is the first year we’re 1-1), and it was a pain. Desmos was perfect for this lesson – it’s easy to use, they’re familiar with it, and the sliders are a huge help.

I also liked that this only took about 25-30 minutes; we fit this in after a warmup and a quiz. With a whole hour, we could have done more with the theory, or I could have given them more time to practice. Things to think about….

# Scheduling Conics and Parametric Equations

Last year, I taught conic sections right before spring break. We took a week off, came back and spent two weeks on polar coordinates, and then dove into parametric equations. At the end of parametrics, we wrote parametric equations for conics.

This did not go well.

In hindsight, I can see that it was destined to not go well. They hadn’t seen a conic section in four weeks, one of which was spring break. They were struggling with an entirely new idea, and then I asked them to remember details of things from a while ago. In an ideal world, would students be able to do this? Yes. But spring break makes people forget things. It’s true. I’m sure there’s a study on it somewhere, but I probably read it before spring break, so now I’ve forgotten about it.

I made a note to myself somewhere: “Next year, don’t break up conic/polar/parametric so much.” Our textbook actually puts all of these ideas in one chapter. While that may be overkill – there are some major differences between these ideas – I didn’t tie them together very well, then suddenly expected kids to see connections at the end without weaving them all the way through.

This year, things were different. We pushed sequences and series to before spring break (still not sure if I like that plan), which freed up a large chunk of continuous time after spring break for polar, conic, and parametric. So we did things in that order: a few weeks on polar, a week on conics, and now parametrics.

It’s going so much better! They just learned about hyperbolas last week, so when I asked them yesterday to write parametric equations for a hyperbola, they were so on top of it. And because conics and parametrics are on the same test, they don’t feel like they should have been able to forget about it yet and see this as review.

(It’d be great if students saw the connections between everything, and didn’t see each unit as stand-alone or something to forget about until the final. It’s a long-term goal of mine to make progress on that front, and we’re getting places. But for now, I’ll work with where they are.)

You’d think that teaching related concepts right next to each other would be obvious, wouldn’t you? But I probably didn’t make the decision to do parametric equations of conics until I’d already taught conic sections, and it was a little late by then. But that’s the wonderful thing about re-teaching a class: I’ve found some of the major problems and can fix them. And next year, I can work on more, and maybe even some of the medium sized problems, too.

For instance, now that conics and parametrics are tied together, I’d like to loop polar in to this. I think Sam Shah starts polar by looking at polar equations of conics, and that’s a definite possibility. I’d probably have to move polar to after conics to do that, but it’s feasible. Things to think about it…

# Talking Less and Planning More

Wednesday featured an object lesson on why I should plan more but talk less.

In precal, we started parametric equations. My typical introduction for parametric equations is to throw a line on the board labeled with time, like this:

I tell the kids to imagine an ant crawling across a picnic blanket. At t = 0, he’s at (0,0). At t = 1, he’s at (2,3). At t = 2, he’s at (4, 6). And so on. What are equations that describe his x and y movement?

Over the last couple of days, though, I’ve been thinking about how we might get students moving or touching things or doing something different with this. I thought maybe the tile floors in the atrium might be a grid, and we could go out and turn it into a coordinate plane and move around on it. But the tiles didn’t align. Or if I’d planned ahead, we could have laid out tape on the floor and made a grid. But yesterday was busy and I didn’t get it done.

So, while the students are doing their warmup, I walking around the room trying to figure out where I can find a coordinate plane. I stare out the window into the parking lot and see the grid in the parking lot from the parking lines. It wasn’t perfect, but it worked.

First period didn’t have time, but in third period, I told them to leave everything and follow me for a field trip. I stood on a point and declared myself the origin. I gave them a set of parametric equations and told them to go figure out where they’d be when t was 0. Then we did t = 1, 2, 3, 4. Then I gave them another set of equations that would produce the same path, but a different orientation and a different starting point. Again, t = 1, 2, 3, 4. I started asking them, “Have you been here before?” They eventually realized that it was the same path, but they were going the opposite direction. Then we trooped back inside and had the normal lesson.

Over all, it went well. When we were talking about orientation and parametrizing equations, I could talk about how those two sets of equations produced the same path, but different orientation and different starting points. I got a lot of “Oh, that makes sense,” looks – a lot more than I did last year when I just drew colored arrows on the board. Orientation is where they usually got lost, so I’m hoping it’ll pay dividends.

On the other hand, planning more would have been good. For one, standing on a random point for the origin was confusion. I should have declared a light pole the origin because it was an easier reference. Also, they kept forgetting the equations. I should have thought them out ahead of time and had them write them down. Little things, and they were willing to bear with me, but it could have been smoother.

As for talking less…we worked on eliminating the parameter today. I split that up into two methods: substitution for algebraic equations and matching them with trig identities for equations with trig functions in them. In the past, I’ve done a lot of prompting, or assumed that they wouldn’t get it on their own.

This year, I kept quiet. I threw a problem up on the board and asked for ideas.

They came up with it.

In each class, a student said, “Well, can we use $sin^2 \theta + \cos^2 \theta = 1$?” I said, “Sure, what do you want to do with it?” “Well, we could solve for the trig part and plug in…” I was so proud of them! And it wasn’t from the students I would expect it from, either. I need to have more faith in them, and give them more opportunities to show me that they can think.