Pre-Calculus

Trigonometric Pythagorean Identities

Last year in Precalculus, I had students start with:

\sin^2(\theta)+\cos^2(\theta)=1

and derive

\tan^2(\theta)+1=\sec^2(\theta) and 1+\cot^2(\theta)=\csc^2(\theta)

simply by dividing both sides of the original equation by \sin^2(\theta) or \cos^2(\theta).

I did this same this year.

Except later on, a few weeks ago, I saw a post on twitter talking about introducing trigonometric identities through graphs on the unit circle — and having kids come up with their own identities. I loved this idea and planned to make a whole thing about of it. So far I’ve given students one thing I’ve made as a result of this idea (and that worked out super well).

[2018-01-29 Beautiful Triangles .docx version]

From this, students were able to come up with the three Pythagorean Trig Identities we saw above, but also a fourth one that was totally unexpected.

strange

I had them all pick a different angle and substitute it into the left hand side and the right hand side of the last equation. KABAM! Whoa! SAMESIES! (Note to self: Next year make a dynamic visualization of this triangle on Geogebra, like this but better/cleaner.)

Instead of doing a whole unit on Trigonometric Identities, the other teacher and I are slowly giving students a problem here and a problem there to practice with and find new strategies, over a couple weeks. I hope that works! And maybe if I have time, I’ll make a follow up activity. Maybe giving the drawing below but without anything labeled but the radius of the circle, and having kids fill in each of the lengths and find various identities? They can use the ratios of similar sides… but also if triangles are similar, they can also use the ratios of the perimeters! Or knowing that the ratio of the areas of two similar figures is simply the square of the ratio of two corresponding sides? Also, maybe just maybe kids could generate inequalities — like the area of this one triangle will always be less than the area of this other triangle?

circlestuff

I don’t quite know as things aren’t fully formed in my head yet. If anyone has any ideas, or existing resources, pass ’em along!

Radians

A couple years ago, Kate Nowak asked us to ask our kids:

What is 1 Radian?” Try it. Dare ya. They’ll do a little better with: “What is 1 Degree?”

I really loved the question, and I did it last year with my precalculus kids, and then again this year. In fact, today I had a mini-assessment in precalculus which had the question:

What, conceptually, is 3 radians? Don’t convert to degrees — rather, I want you to explain radians on their own terms as if you don’t know about degrees. You may (and are encouraged to) draw pictures to help your explanation.

My kids did pretty well. They still were struggling with a bit of the writing aspect, but for the most part, they had the concept down. Why? It’s because my colleague and geogebra-amaze-face math teacher friend made this applet which I used in my class. Since this blog can’t embed geogebra fiels, I entreat you to go to the geogebratube page to check it out.

Although very simple, I dare anyone to leave the applet not understanding: “a radian is the angle subtended by the bit of a circumference of the circle that has 1 radius a circle that has a length of a single radius.” What makes it so powerful is that it shows radii being pulled out of the center of the circle, like a clown pulls colorful a neverending set of handkerchiefs out of his pocket.

If you want to see the applet work but are too lazy to go to the page, I have made a short video showing it work.

PS. Again, I did not make this applet. My awesome colleague did. And although there are other radian applets out there, there is something that is just perfect about this one.

Trig War

This is going to be a quick post.

Kate Nowak played “log war” with her classes. I stole it and LOVED it. Her post is here. It really gets them thinking in the best kind of way. Last year I wanted to do “inverse trig war” with my precalculus class because Jonathan C. had the idea. His post is here. I didn’t end up having time so I couldn’t play it with my kids, sadly.

This year, I am teaching precalculus, and I’m having kids figure out trig on the unit circle (in both radians and degrees). So what do I make? The obvious: “trig war.”

The way it works…

I have a bunch of cards with trig expressions (just sine, cosine, and tangent for now) and special values on the unit circle — in both radians and degrees.

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You can see all the cards below, and can download the document here (doc).

They played it like a regular game of war:

I let kids use their unit circle for the first 7 minutes, and then they had to put it away for the next 10 minutes.

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And that was it!

Infinite Geometric Series

I did a bad job (in my opinion) of teaching infinite geometric series in precalculus in my previous class. I told them I did a bad job. I was rushing. They were confused. (One of them said: “you did a fine job, Mr. Shah” which made me feel better, but I still felt like they were super confused.)

At the start of the lesson, I gave each group one colored piece of paper. (I got this idea last year from my friend Bowen Kerins on Facebook! He is not only a math genius but he’s also a 5 time world pinball player champion. Seriously.) I don’t know why but it was nice to give each group a different color piece of paper. Then I had them designate one person to be the “paper master” and two people to be the friends of the paper master. Any group with a fourth person simply had to have the fourth person be the observer.

I did not document this, so I have made photographs to illustrate ex post facto.

IMAG4465

I started, “Paper master, you have a whole sheet of paper! One whole sheet of paper! And you have two friends. You feel like being kind, sharing is caring, so why don’t you give them each a third of your paper.”

The paper master divided the paper in thirds, tore it, and shared their paper.

IMAG4466

Then I said: “Your friends loveeeed their paper gift. They want just a little bit more. Why don’t you give them each some more… Maybe divide what you have left into thirds so you can keep some too.”

And the paper master took what they had, divided it into thirds, and shared it.

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To the friends, I said: “Hey, friends, how many of you LOOOOOVE all these presents you’re getting? WHO WANTS MORE?” and the friends replied “MEEEEEEEEEEEEEEE!”

“Paper master, your friends are getting greedy. And they demand more paper. They said you must give them more or they won’t be your friends. And you are peer pressured into giving them more. So divide what little you have left and hand it to them.”

They do.

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“Now do it again. Because your greedy friends are greedy and evil, but they’re still your friends.”

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“Again.”

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“Again.”

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Here we stop. The friends have a lot of slips of paper of varying sizes. The paper master has a tiny speck.

I ask the class: “If we continue this, how much paper is the paper master going to eventually end up with?”

(Discussion ensues about whether the answer is 0 or super duper super close to 0.)

I ask the class: “If we continue this, how much paper are each of the friends going to have?”

(A more lively short discussion ensues… Eventually they agree… each friend will have about 1/2 the paper, since there was a whole piece of paper to start, each friend gets the same amount, and the paper master has essentially no paper left.)

I then go to the board.

I write \frac{1}{2}=

and then I say: “How much paper did you get in your initial gift, friends?”

I write \frac{1}{2}=\frac{1}{3}+

and then we continue, until I have:

\frac{1}{2}=\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+\frac{1}{81}+...

Ooohs and aahs.

Next year I am going to task each student to do this with two friends or people from their family, and have them write down their friends/family member’s reactions…

I love this.

Inverse Trig Functions

At TMC13, I was in a group of people talking about precalculus. One of the exercises we did was make a list of some of the topics we found challenging to teach as teachers — and we broke out in groups to try to come up with ways to tackle those topics.

My group’s topic was inverse trig functions. (This was with April, Dan, Greg, and Andrew.)

Our initial task was to find the deep mathematical idea behind the topic… why we teach it, what we think we can get out of it conceptually… and what we sort-of converged on is that the topic really illuminates the idea of inverses and restricted domains. And that’s about it. And when push came to shove, we decided we didn’t find that restricted domain is something we really care about. We decided we didn’t really care about the inverse trig graphs, and the work we put into that side of things wasn’t really worth what we little we were able to squeeze out of it. It’s not that it is horrible, but we just didn’t couldn’t justify it.

So, honestly, we decided to just focus on inverses, and the idea of them as “backwards problems.”

Thus, we came up with two things:

1. A packet that has students secretly engage with inverse problem work before they even know what they’re doing. So the first packet is meant to be used before any unit circle trig is introduced. (A few of us, especially April, did something similar in her classes, and randomly, Greg Taylor did a my favorites on the same essential idea!)

In fact, if I were to use this in the classroom, I would not even mention the words “trigonometry.” I would focus on the idea of coordinate planes and circles, and simply leave it there.

2. A packet that students work on after they learn unit circle trig — and that more formally introduced the idea of the inverse trig functions. It tries to draw connections between the unit circle, the sine/cosine graphs, and their calculators.

There are concept-y questions for both packets. I’m including both packets below in one document. I’m posting one with a few teacher notes, and one with the teacher notes hidden. (The .docx is here if you want to edit!)

Packet with teacher notes 

Packet without teacher notes

We did all this planning in pretty much an hour and a bit — from start to finish. And then I pulled together the ideas to make this document. I’m not sure I was able to capture everything we talked about, but I think I got most of the big things. Apologies to my collaborators if I totally botched the translation of our vision to reality!

My Introduction to Rational Functions

Going into Rational Functions

My impression is that most people introduce rational functions by showing something like…

y=\frac{(x+3)(x+4)(x-3)}{(x-1)(x-3)}

… and then spend the rest of the time asking kids some questions, like “what’s the x-intercept(s)?” “what’s the y-intercepts?” “what’s the vertical asymptotes?” And so from one big equation, you pull out all this individual stuff…

But from what I’ve seen when most kids approach rational equations, it is all very procedural. And every time I dug a little deeper to see what they truly understood about these equations, it became clear that a procedure to “solve” these questions was taking the place of understanding what was going on. So pay attention. You might recognize students don’t know what a hole truly is and why it appears in a graph… or they might not understand why vertical asymptotes appear… at least not on a deep level. The answers I have heard from kids are procedural, and rarely have any deep stuff underneath.

To counter this, I made two major changes to how I approach/introduce rational functions this year.

First, initially, I focus heavily on the graphical side of things. To the point where on the first day, students do not see a single equation, and are asked (entreated!) not to write a single equation down.

Second, I want kids to build up rational functions, instead of breaking them down. I want them to see how they are constructed term by term by term.

So  for example, when we see the rational function listed above, we find it easier to view it as:

y=\frac{x-3}{x-3}\cdot\frac{1}{x-1}\cdot\frac{x+4}{1}\cdot\frac{x+3}{1}

Kids need to understand what the first term is doing — not just “as a rule” but conceptually/graphically. I expect them to say that for any x value other than 3, the fraction will evaluate to be 1 (thus it will not affect the rest of the multiplication), but when x is 3, we clearly get something undefined.

Kids need to understand that the second term is creating the function to blow up in a certain way at x=1. Not just because we’re dividing by zero so things go crazy and explode, but being able to articulate precisely why the function blows up. (The explanation I’m looking for says that at x-values closer and closer to 1, the denominator is getting smaller and smaller, but the numerator is staying at 1. Thus the output is getting bigger and bigger and bigger.)

And of course kids need to understand how the third and fourth terms are (graphically) creating x-intercepts in the final graph.

Of course once this is done, you can throw in the other stuff…

Here are my files in .doc form [Rational Fxns 1, Rational Fxns 2, Rational Fxns 3, Rational Fxns 4]

My Awesome Introduction

Although there are definitely ways I can improve this, here is how I started off rational functions. My goal — gentle reader, to remind you — is to do very little explaining and have the kids figure as much out on their own as they can. I felt wildly successful with this when it came to the introductory materials for rational functions.

It took my kids about a class period to do this first packet (they finished the rest up at home). I started with the admonition that no equations should be used and everything needed to be thought of graphically if it was going to be an effective exercise.

Out of this came nice discussions of holes and vertical asymptotes.

For their nightly work on the first day, I had kids finish this packet and then write down all the equations for each of the graphs.

The next day, we went through our answers, and started working on this, which they were crazy adept at doing:

Taking Things Further

I link to a couple more sheets I created above if you want to see what came after… how I introduced end behavior and horizontal asymptotes, and how I introduced graphing.

It wasn’t anything innovative, and could use a lot of work to refine it, but maybe you’ll find something you can work with?

The two things I did like that happened when going over this less basic stuff is:

(1) When kids make sign analyses, they don’t always understand why they are plotting the points they are plotting on the sign analysis. Why do they plot x-intercepts, holes, and vertical asymptotes? I like having my kids discuss why those particular graphical features, and then draw pictures of various graphs where the function does switch from positive to negative (or vice versa) at these points… and have kids draw pictures of various graphs where the function does not switch from positive to negative at these points. Kids, from this, start to understand that if we wanted a sign analysis, these special places (vertical asymptotes, holes, x-intercepts) are places we want to look… and they also start to understand that a function doesn’t necessarily have to change signs at these special places.

(2) I have found in the past that students find it challenging to go from a rational equation to a rational graph, without any scaffolding. So before throwing them in the deep end I like to give them sign analyses and end behaviors, and ask them to sketch a graph that matches the information we know. They start to think of it like a puzzle. Once they have practiced that a few times, they can start doing everything from an equation.

I wish I were less exhausted and could explain more. I literally passed out for a few seconds while typing the end of this post. However, check these lessons out. See if you might want to join me in switching up how we think about rational functions.

Update: 

Some questions that you might want to bring up in your study of rational functions…

1. Why do we plot x-intercepts, vertical asymptotes, and holes on the number line when doing a sign analysis?

2. Why do you only have to test one number in each region in a number line… how do we know all the rest of the numbers in that region (when plugged into the equation) will result in the same sign?

3. What is a hole?

4. Why is a hole created? Why does the creation of a hole not affect the “rest” of the equation when it’s graphed?

5. Why do vertical asymptotes appear?

6. Why does the end behavior look like it does?

7. Why can rational function cross a horizontal asymptote and why can it not cross a vertical asymptote?

The key to all of these is why, and if kids can give a procedural answer (e.g. “A hole is created when you see the same factor in the numerator and denominator”) then you know you need to dig more and ask the next question (“so why does having the same factor in the numerator and denominator create a hole?”).

 

Some Random Things I Have Liked

The Concept of Signed Areas

In calculus, after first introducing the concept of signed areas, I came up with the “backwards problem” which really tested what kids understood. (This was before we did any integration using calculus… I always teach integration of definite integrals first with things they draw and calculate using geometry, and then things they do using the antiderivatives.)

I made this last year, so apologies if I posted it last year too.

[.d0cx]

Some nice discussions/ideas came up. Two in particular:

(1) One student said that for the first problem, any line that goes through (-1.5,-1) would have worked. I kicking myself for not following that claim up with a good investigation.

(2) For all problems, only a couple kids did the easy way out… most didn’t even think of it… Take the total signed area and divide it over the region being integrated… That gives you the height of a horizontal line that would work. (For example, for the third problem, the line y=\frac{2\pi+4}{7} would have worked.) If I taught the average value of a function in my class, I wouldn’t need to do much work. Because they would have already discovered how to find the average value of a function. And what’s nice is that it was the “shortcut”/”lazy” way to answer these questions. So being lazy but clever has tons of perks!

Motivating that an antiderivative actually gives you a signed area

I have shown this to my class for the past couple years. It makes sense to some of them, but I lose some of them along the way. I am thinking if I have them copy the “proof” down, and then explain in their own words (a) what the area function does and (b) what is going on in each step of the “proof,” it might work better. But at least I have an elegant way to explain why the antiderivative has anything to do with the area under a curve.

Note: After showing them the area function, I shade in the region between x=3 and x=4.5 and ask them what the area of that bit is. If they understand the area function, they answer F(4.5)-F(3). If they don’t, they answer “uhhhhhh (drool).” What’s good about this is that I say, in a handwaving way, that is why when we evaluate a definite integral, we evaluate the antiderivative at the top limit of integration, and then subtract off the antiderivative at the bottom limit of integration. Because you’re taking the bigger piece and subtracting off the smaller piece. It’s handwaving, but good enough.

Polynomial Functions

In Precalculus, I’m trying to (but being less consistent) have kids investigate key questions on a topic before we formal delve into it. To let them discover some of the basic ideas on their own, being sort of guided there. This is a packet that I used before we started talking formally about polynomials. It, honestly, isn’t amazing. But it does do a few nice things.

[.docx]

Here are the benefits:

  • The first question gets kids to remember/discover end behavior changes fundamentally based on even or odd powers. It also shows them that there is a difference between x^2 and x^4… the higher the degree, the more the polynomial likes to hang around the x-axis…
  • The second question just has them list everything, whether it is significant seeming or not. What’s nice is that by the time we’re done with the unit, they will have a really deep understanding of this polynomial. But having them list what they know to start out with is fun, because we can go back and say “aww, shucks, at the beggining you were such neophytes!”
  • It teaches kids the idea of a sign analysis without explaining it to them. They sort of figure it out on their own. (Though we do come together as a class to talk through that idea, because that technique is so fundamental to so much.)
  • They discover the mean value theorem on their own. (Note: You can’t talk through the mean value theorem problem without talking about continuity and the fact that polynomials are continuous everywhere.)

The Backwards Polynomial Puzzle

As you probably know, I really like backwards questions. I did this one after we did  So I was proud that without too much help, many of my kids were really digging into finding the equations, knowing what they know about polynomials. A few years ago, I would have done this by teaching a procedure, albeit one motivated by kids. Now I’m letting them do all the heavy lifting, and I’m just nudging here and there. I know this is nothing special, but this course is new to me, so I’m just a baby at figuring out how to teach this stuff.

[.docx]