Pre-Calculus

Counting Without Counting: An Introduction

On the second day of Precalculus class, before embarking on our starting unit on Combinations and Permutations, I put students in groups and had them work on this packet. I’m including it below with my teacher notes in the margin, and also without my teacher notes in the margin:

(Also the .docx file so you can modify, cut, paste, hack apart! To see with teacher notes, go to the “Review” tab and click on “Final: Show Markup” and to see it without teacher notes, click on “Final.”)

This was a two day activity, with the groups working together, and every so often, I would stop and we would go over some of the problems. Some problems were assigned for nightly homework, and because of time issues, I had to cut out the Applebee’s problem completely. I’m quite proud of some of the problems… namely the Applebee’s problems, the bit.ly / QR code problems, and the Mozart’s Minuet problem (largely taken from here, and modified with some extensions).

The goal of the packet was fourfold:

1) I wanted problems which promoted thinking, conversation, etc., before students were introduced to formulas, notation, etc.

2) I wanted students to understand that “counting without counting” means that instead of listing all possibilities and counting, there are often other faster ways to get answers. This is the essential understanding that I would hope for in any unit on combinations and permutations. In order to do this, students will need to organize your information in some special, logical way. Usually this requires students to multiply numbers. But students need to really understand why multiplication (rather than another operation, like addition).

3) I wanted students to work in groups, so the problems were designed to be conducive for groupwork.

4) I wanted students to get some sense of what huge numbers mean.

How did it work? Overall, I think it worked pretty well. I gave groups “hint tokens” and most didn’t rely on them for a hint. Most students were able to see that you have to multiply for most of the problems, but most had trouble explaining why. Finally, most had never seen a tree diagram. In the future, I honestly think having students draw a complete tree diagram and explain what each leaf means would be useful. Adding two questions like the following would help:

1. If you have the letters A, B, C, D, E, and F and you want to write a three letter code and you are allowed repetition, what would the tree diagram look like. Make all the branches. Then pick a single “leaf” of the tree and explain what that leaf means.

2. If you have the letters A, B, C, D, E, and F and you want to write a three letter code and you are not allowed repetition, what would the tree diagram look like. Make all the branches. Then pick a single “leaf” of the tree and explain what that leaf means.

The reason I say this is that I’ve been collecting homework problems, and the tree diagrams some of the kids are constructing are just nonsense.

I probably should write more about this, but I’m exhausted and all I want to do is sleep. (I’ve been sick since Monday.) I suppose I should end adding that I’m teaching a Precalculus Advanced class. I don’t think I would have these problems for a standard Precalculus class… I would use fewer of them, and I would scaffold them more, and build in more “listing” of things, rather than go straight into “how many different ways…?”

Sequences and Series: An Exploratory Unit

I had great ambitions to do a lot of schoolwork this summer. Instead I started, abandoned, and restarted a unit for a course that I’m teaching next year. That’s about it. It’s a new course for me: Advanced Precalculus. The other teacher and I have decided to totally mess around with the ordering of topics, and we put sequences and series as the second unit. Our department is also trying to integrate more problem solving in the curriculum, and so I tried to make this unit involve as much problem solving as possible [1]. I like that we’ll be doing it early in the year, because I want them to see immediately that we are not going to be focusing on plug-and-chug but real thinking.

Those of you who know me know that I am a pretty traditional teacher, and I have gotten in the habit of creating guided worksheets as a structural backbone for a lot of my classes. This is the first time I’m creating an entire guided unit. It isn’t flashy or have a good hook. It’s simply a scaffolded way to help kids think in an increasingly abstract way. It also gets at almost all the standard things in a sequences and series unit (except for recursive equations, which I threw out). To put it out there: I would never say that what I do is inherently engaging for my kids. But it does get kids talking. I guess what I mean to say is: these packets/worksheets that I tend to create don’t make kids like/love math, but it does get them to think about math. I’m not great at the former, but I’m definitely getting better at the latter.

The last thing I have to say is that although it may look pretty traditional (the questions), try to think about the packet if you were a student and you were in a class going through it. It builds up elegantly, in my opinion. The motivation for sequences comes out of a series of IQ-test-ish puzzles, and the motivation for series comes out of a lottery problem. No formula is given to students. There are connections drawn to graphs, and to a few geometric visualizations of sequences and series. Students are asked to conjecture and defend their conjecture at various times.

I’m including two copies below. The packet with my teacher notes, and the packet without my teacher notes.

With Teacher Notes

Without Teacher Notes (Blank)

[Word version of this to download: .docx… to see my teacher notes, go to “Review” and go to “Final Showing Markup”]

Huge thanks in the creation of this goes to @JackieB, who went through a lot of it page by page and gave excellent suggestions! Precalculus guru! Also I included a few blogposts at the end of the document which I stole wholesale from or adapted in my own way.

Lastly, yes, I know this is a long packet. Usually I think classes do this whole unit in single week, and there’s no way we’ll be done with it in that time. It’s an experiment. From what I’ve heard from teachers everywhere, sequences and series always get short shrift in precalculus classes because they come at the end of the year. But I think there is so much depth and abstract thinking that can be brought out of a unit properly done. I’m super nervous, but we’ll see if this is an experiment that fails or not.

[1] I’m liberally defining problem solving as having students deal with situations they have never dealt with before, and generalizing from those situations. But I understand I am giving them A LOT of scaffolding with which to do it.

The Calculus of Saying I LOVE YOU!

I found — when searching for something else — this page on the calculus of love. It’s actually really cute, and totally accurate mathematically. Both big plusses in my book.

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The article analyzing these graphs is here. Definitely something to check out if you’re teaching precalculus or calculus.

Okay, it is 6:54 and I have to run to school for a 7:30 meeting! Yikes!

Lates.

Algebraic Manipulation Is Overrated

An intuition question.

Look at the function below. It may surprise you that it is a constant! For any value of x, the function g will have the same value. I’m wondering, now that you know this, if you can get a sense of why it would be a constant, without (a) using your graphing calculator, or (b) taking the derivative to show that it is 0 [that is what I did, and as a side note, I have to use this on a test or homework next year].

g(x)=\frac{\sin(x)+\sin(x+a)}{\cos(x)-\cos(x+a)}

Can you find some geometric way to see that?

It took me somewhere between a half hour and an hour of playing around to get it. I can post my solution in a couple days, but right now I don’t have the energy to find a program to draw my solution [1]. But let me just tell you: it’s beautiful. You’ll be stunned when you first do it. Yeah, the calculus way tells you it is a constant, but seeing the “why” is still a mystery. The geometric way takes a bit, but whoa nellie, you won’t regret spending the time!

[1] Or maybe I should claim there is no room in the margin! (JK)

Update: I did finally write up my solution. I quickly did something I never have done before: do my work in powerpoint. It worked fine.

Update: Mr. K solved the problem in 3 minutes and found a way to show the geometric solution. Head over to his very excellent blog to see it in all it’s glory.

Update: Besides mine and Mr. Ks, a third and perhaps more elegant solution is up at 11011110.

Of the three, I think I like Mr. K’s visualization best, even though it might not be a proof in the formal sense.

Generating Fibonacci: Part II

We left off in our quest for an explicit formula for the nth Fibonacci number having created this amazing generating function: g(x)=\frac{1}{1-x-x^2}.

To do what we’re about to do, I need to remind you of two precalculus things:

First, that \frac{1}{1-pq}=1+(pq)+(pq)^2+(pq)^3.... If you don’t know why, I suggest doing long division!

Second, partial fractions.

I’m going to go through this explanation assuming you know these two things.

So let’s look at the denominator of our g(x) and factor it. Okay, okay, you got me. There isn’t a nice factoring with integers. But it can still be factored, of course.

g(x)=\frac{1}{(1-x\phi)(1-x\overline{\phi})}, where \phi=\frac{1+\sqrt{5}}{2} and \overline{\phi}=\frac{1-\sqrt{5}}{2}. Using partial fractions, we get:

g(x)=\frac{a}{1-x\phi}+\frac{b}{1-x\overline{\phi}}=a(\frac{1}{1-x\phi})+b(\frac{1}{1-x\overline{\phi}})

We’ve made good headway, but what we don’t know are a and b! However, noticing that we can use the first precalculus topic above, that

g(x)=a(1+x\phi+x^2\phi^2+...)+b(1+x\overline{\phi}+x^2\overline{\phi}^2+...).

Rewriting this as a simple polynomial, we get:

g(x)=(a+b)+(a\phi+b\overline{\phi})x+(a\phi^2+b\overline{\phi}^2)x^2+...

Now we’re almost done! We use the initial conditions to find out a and b.

Since we know F_0=1 and F_1=1, we can say a+b=1 and a\phi+b\overline{\phi}=1. Solving these two equations simultaneously yields a=\frac{1}{\sqrt{5}} and b=-\frac{1}{\sqrt{5}}.

So the nth Fibonacci number is: \frac{1}{\sqrt{5}}(\phi^n)-\frac{1}{\sqrt{5}}(\overline{\phi}^n)

which simplifies to: \frac{1}{\sqrt{5}}(\phi^n-\overline{\phi}^n).

Which is what we set out to show! Huzzah! What’s also nice about this (besides the fact that it’s an integer, which is surprising) is that it shows that the Fibonacci sequence grows exponentially!

Generating Fibonacci: Part I

Yes, this is a YAFPOTW (Yet Another Fibonacci Post On The Web).

People have talked about and analyzed the Fibonacci sequence to death. So I’m clearly not going to be doing something new and novel for everyone. But I remember one stunning way to look at Fibonacci is through Generating Functions. You only really need precalculus to understand it, as long as you allow for infinite degree polynomials. And the best part is: you can apply this method to any simple recurrence relation.

To refresh your memory, the Fibonacci sequence is: 1,1,2,3,5,8,13,…, where the nth Fibonacci number is the sum of the two preceeding Fibonacci numbers:

F_n=F_{n-1}+F_{n-2}.

That recursive formula, with the initial conditions (F_0=1 and F_1=1), defines the entire sequence.

Let’s create a polynomial function from these Fibonacci numbers:

g(x)=F_0+F_1x+F_2x^2+...+F_{n-2}x^{n-2}+F_{n-1}x^{n-1}+...

We call this function g(x) the generating function of the sequence F_0, F_1, F_2,....

So what if we add g(x)+xg(x)? First we find xg(x):

xg(x)=F_0x+F_1x^2+F_2x^3+...+F_{n-3}x^{n-2}+F_{n-2}x^{n-1}+...

and we add, term by term, to get:

g(x)+xg(x)=F_0+(F_0+F_1)x+(F_1+F_2)x^2+...+(F_{n-3}+F_{n-2})x^{n-2}+...

Excellent! Do you see where this is going? HUZZAH We know those coefficients! They use the recursion formula that defines the Fibonacci numbers! So we simplify:

g(x)+xg(x)=F_0+F_2x+F_3x^2+...+F_{n-1}x^{n-2}+...

And we know F_0=1=F_1, we replace that F_0 above to get:

g(x)+xg(x)=F_1+F_2x+F_3x^2+...+F_{n-1}x^{n-2}+...

Notice the right side of the equation is equal to:

\frac{1}{x}[-F_0+F_0+F_1x+F_2x^2+F_3x^3+...+F_nx^n+...]

which immediately shows us a g(x)! So we can rewrite that to as the much simpler:

g(x)+xg(x)=\frac{1}{x}[-F_0+g(x)]

And we’re almost there! Replacing F_0 with it’s value of 1, multiplying both sides by x, and rearranging, yields: (x^2+x-1)g(x)=-1. So we get:

g(x)=\frac{1}{1-x-x^2}

This is our generating function! The coefficients of the polynomial expansion will give you the Fibonacci sequence! But this method of generating functions will prove a nice and general way to find explicit solutions for any basic recursive relation. (What if we had the Lucas Numbers, for example? Or some recurrence like M_n=2M_{n-1}+M_{n-3}?)

In a later post, I’ll write about how to get an explicit formula for the nth Fibonacci number from this. In case you need a refresher or never knew:

F_n=\frac{1}{\sqrt{5}}((\frac{1+\sqrt{5}}{2})^n-(\frac{1-\sqrt{5}}{2})^n)

Where clearly we see some popping up of the golden ratio and its “opposite”! And in our generating function, we see an 1-x-x^2 which has zeros which are the golden ratio and its “opposite.”