Pre-Calculus

Thinking!

I love it when my students think for themselves.

When learning the law of sines and law of cosines — and when it’s appropriate to use one or the other — our textbook gives pretty prescriptive directions. For example, when you are given SSA (a side, a side, and an angle opposite one of those given sides), you are supposed to use the law of sines. And depending on these values, you can actually get two possible solutions!

Let’s work this out.

If in triangle ABC you’re given that side a=6, side b=8, and angle A is 40 degrees, then let’s solve the triangle.

Using the law of sines, \frac{6}{\sin (40)}=\frac{8}{\sin B}. Rearranging, we get \sin B=0.8571. But we know that sine is positive in quadrants I and II, so we get for B=58.99 or B=121.01. And hence, we have two possible angle values for B, which leads to two different triangles. [For a more detailed explanation, see here.] Another quick application of the law of sines yields that c=9.22 or c=3.04

And looking at the picture below (cribbed from the site above), you can see that both triangles are possible!

Here’s where student thinking is awesome. The book, as I said, says that everytime you have SSA you should use the law of sines. And I agree, it is easier. But it is definitely possible to use the law of cosines too, as one of my students pointed out to me.

Let’s do it:

6^2=8^2+c^2-2(8)(c)\cos(40)

This simplifies, with some rearranging, to the quadratic: c^2-12.26c+28=0. This can be solved to get the two values for c, which are c=9.22 or c=3.04.

I love that it works, and that the student insisted that we could do it. It might be slightly more work, but not that much more, and the exploration aspect is awesome. [1]

[1] An extension for a project for next year might be: how can we use the fact that we can generate a quadratic help us in determining when we have two possible triangles, one possible triangle, or no possible triangles.

Reorganizing Trigonometry

In my trigonometry classes, I decided to deviate from the textbook ordering of concepts. The other teacher, thankfully, was on board. (And next year, I want to tweak things even more.)

Our textbook presentation of trig starts out with reviewing right triangles, SOH CAH TOA, and in the same section, introduces the reciprocal trig functions (csc, sec, cot). It then goes into application problems, involving angle of elevation and depression. Finally, it throws all the hard but juicy stuff into the next section — a section that took me 4+ days to cover. It included introducing the concept of using trig for angles greater than 90 degrees (a VERY hard concept for kids to grasp), reference angles, quadrant analysis, and a variety of different types of problems that students are expected to do.

But then the book starts veering into radians (which I covered already this year, and next year, which I’ll postpone), the graphing of trig functions, and finally goes into the translation and stretching of these graphs.

I skipped this graphing work and an entire other chapter to get to the law of sines and the law of cosines. Um, yeah, hello? Let’s think about this:

Students start by learning that trig helps with right triangles and they do application problems. Then they learn how we can extend this trig work to angles greater than 90 degrees. Me thinks that it would be natural to show them how trig can help them with all triangles at this point – including obtuse triangles. (Importantly, the law of sines also tests a student’s understanding of reference angles.)

My students seem to find doing problems with the law of sines and cosines very tedious, yes, but they also love the grounding and concreteness of it. They told me that. Which makes me think it’s better to put that sort of thing at the beginning of our exploration of trig. And leave the more abstract discussions to later, when the basics are fixed in their minds.

Rethinking the textbook makes me feel good, because it means I’m paying attention to the flow of the subject, to how I’m presenting the topics, and what students are thinking as they learn trig. And it means I’ve started fulfilling one of the goals I set before school started, one that I decried just couldn’t happen my first year.

[1] For giggles, I want to share what the section after the law of sines and cosines is in the textbook: Graphing complex numbers on a complex plane. Can we all say “jarring transition”?

Spinning on a Globe

This slideshow is the Smartboard I used in this class that I describe below:

The day my students got back from spring break, this week, I started things off with a bang! At least, that was the intention. Instead of doing the whole “Paris was great, where did you go for your vacation?” time waster, I decided to dive right in. They left for break having learned some trigonometry basics, and I had blown their minds with the idea that you can measure angles with something other than degrees (radians). Monday’s lesson was to be on central angles, arcs, and angular velocity.

On Sunday night, I downloaded GoogleEarth onto my school laptop. In an ideal world, I wanted to say:

So kiddos, I flew from Newark to Paris for spring break. Let’s all get out our laptops and find out their longitude and latitude, and the distance between them. [I do it in front of them on GoogleEarth]. Now let’s use this info to the find the radius of the earth.

Unfortunately, after a few minutes of playing around with this problem, it becomes very clear to me that it is not as simplistic and trivial as I had hoped, because the latitude and longitude are different. Picture the following. Draw a line from the center of the earth to Newark, from the center of the earth to Paris, and a curved line (along the globe) from Newark to Paris. You have a sector of a circle. You know the arc length. But what’s the central angle (the angle between the radii)? It’s not a trivial problem for students to calculate that angle.

So instead, I copped out and found two random spots on the equator (one in Brazil, one in the Democratic Republic of Congo), and the distance between them. And then I had students come up with what to do next. It took very little time before they stumbled upon the idea of setting up a proportion: angle/full circle angles = arc/full circle perimeter.

\frac{87.74}{360}=\frac{9610.10 km}{Circumference}

From that, we found the circumference of the globe (about 40,046km). And then we compared it to what Wikipedia said (40,075km), marveled at our own acumen, and talked about the sources of error. And because I truly am a big nerd, I talked about the origin of the meter (see this book).

This warmup took about 10 minutes, and was a perfect segway into a discussion of arc length and central angles.

We set up the same proportion as we did above, but for a general circle of radius r, central angle \theta, and arc length s, to discover the wonderful formula s=r\theta. Then I asked them if it made sense… (The larger the circle’s radius, the larger the arc length… check! The smaller the angle, the smaller the arc length… check!) And when I asked them when they had seen this before, most of my students noticed: “duh Mr. Shah, the Earth problem…”[1]

Finally, I got to the final topic, angular speed. I remember when I was first taught this, I was taught it as a formula. There was no conceptual understanding behind it. And students tend to love to rely on formulas without understanding them. So I introduced the ideas of “linear speed” and “angular speed” — and then asked them to calculate (without showing them any problems) the linear speed and angular speed of someone in Brazil or the Democratic Republic of Congo as the earth spun on it’s axis. And someone’s hand shot in the air, then another, and soon the problem was solved.

In addition, one student noticed that the angular velocity remained the same no matter where on the globe you stood, but your linear velocity decreased the further away you got from the equator.

Which led to the final equation of the day: v=r\omega.

Some more practice problems were done, and that was that. With that concluded one of my better, more cohesive lessons. It all tied together with the globe thing, and they really left with a sense of the concepts, more than the simply memorizing the formulas.

[1] One class noticed we could calculate the radius of the earth with this formula, since in our earlier problem, we knew the distance between the places in Brazil and the Democratic Republic of Congo, and the angle between them… So we did that, and got another wonderfully accurate answer too!

We love radians

Introducing trigonometry has become even more of a challenge than I thought. I think about each part of the lesson really hard; I want to give it a flow and focus on teaching the concepts. What I don’t want trigonometry to be is a huge mess of ad hoc rules.

Today was my introduction to radians. Looking back, my presentation was a bit more complicated than it needed to be to get the idea across. And what wasn’t clear (although that was one of my objectives) was why we use radians instead of degrees. So I’m going to start off class tomorrow with a little silent slideshow, replete with my own histrionics to make extraordinarily clear why we! love! radians!

Without further ado: why radians? (PDF file) [Unfortunately, SlideShare is only showing 18 of 23 pages for some reason. The PDF is complete.]

I’ve also come to realize that more is going on with kids than this whole forest for the trees crap that I wrote about before.

There’s a second reason things are getting mucked up, and that doesn’t have anything to do with my concept behind each lesson, or the flow, or anything like that. I realized today that a lot of the things that kids get tripped up on are (surprise surprise) basic facts about numbers. Is “1/2 times pi” the same thing as “pi over 2”? Yes. Do they know that? Possibly.

Or, for example, there’s the issue of manipulating visual and fractional information in their heads. We’re learning about radians, and we learn that there are “2 pi” radians in a circle. Then I ask them to draw an angle of “3 pi over 2” radians. It was as if I asked them to dance around like a chicken while singing Ave Maria. And since the “pi” was there, they thought that using the calculator wasn’t really going to help them.

I think we’re slowly getting it, but I’m not sure. I’m going slowly, but I have now started identifying key skills and concepts that need to be honed before we move on with radians. For example, because of what I noticed in my lesson on radians, we’re going to be practicing working with fractions (the eternal scourge of math teachers) and pi. (See my thrown-together worksheet here.)

Imagine (for surely, gentle reader, this has never happened to you before) that you’re at a mini golf course and you’re putting at the infamous and dreaded windmill hole. By mandate from the PuttPutt gods, you are not allowed to leave until you get the ball into the windmill. There’s a mini golf coach there, trying to give you advice and show you how to hold the club and how to swing. However, after 20 tries you aren’t getting it. And then you try another 2o times. No luck. Now tell me how you think you’d feel at your mini golf coach who has been standing there trying to help you.

There will be whining, complaining, anger, and frustration — anxiety — all directed to this coach.

The analogy isn’t quite right, and I hope that my students don’t direct those feelings to me (this was an extended allegory, duh), but I can’t help but notice that the anxiety level has shot up in my room in the past two weeks, when I feel that one of my teaching talents is keeping a totally relaxed atmosphere.

UPDATE: My presentation (see above) on”Why Radians?” took 5 minutes and I think did the trick. I did it in both classes, and both seemed to get it. And the levity of it all made the classroom less tense. And with Spring Break descending upon us, we’re going to have a much needed break.

But why?

Trigonometry is one of those topics that if you get the basics, the rest of it will make a heck of a lot of sense. But if you miss it, you’re going to be trying frantically to come up with ad hoc ways to understand each new concept.[1]

I have been teaching the beginning of trig, and I’ve noticed a few things that I have to watch out for next year:

  1. Have a really good reason prepared for explaining why we care about angles greater than 360. In general, the kids don’t have a good idea of why we’re doing what we’re doing. (“Why do we want to find csc(421)?”; “Why don’t we just say we have an angle of 1 instead of 361? When would we ever need 361 degrees if a circle has only 360?”)[My explanation didn’t hold over well, but it’s true. I said that often times we use angles to measure time, like in a clock. So if we have something repeating — like a spring with a mass in physics, or a ferris wheel going around and around, or a bike wheel spinning — we will be able to model how far it’s gone or how many oscillations its made by using this angle.]
  2. In fact, have a good reason to explain why we care about angles greater than 90. I started out teaching triangles and SOH CAH TOA, and they got it. Then I started teaching how to see angles on the coordinate plane, and I lost some of them. They can — I hope — calculate the sine of 210. But they don’t get why the sine of 210 is at all related to the sine of 30. They see the 210 angle, and they say “where’s our triangle”? And I show them the 30 angle, and they understand that we can form a right triangle with it, but they don’t get why we use the triangle with 30 to deal with the 210 angle.[My explanation dealt with looking at triangles made in the first quadrant, like the sine of 45. I showed them that the opposite side of the triangle was the y-coordinate, the adjacent side with the x-coordinate, and that the hypotenuse was the radius. Then I said the problem with the whole “opposite,” “adjacent,” “hypotenuse,” method of things was that it restricted our angles to lie between 0 and 90. So to expand the domain of these trig functions, we put them on the coordinate plane, and defined sine to be y/r, cosine to be x/r, and tangent to be y/x.But then they asked: “Why? Who cares about angles greater than 90?” Which then takes us back to #1.]
  3. I haven’t given them a big picture, which is part of the problem. Right now they’re learning smaller skills, but they don’t know what the whole point of it is. So what if you can find cot(260)? Why do we care? What does cotangent mean in the real world?

Next year… I might want to motivate trigonometry on the first day, and give a hard application problem that we’ll be able to solve by the end of the unit. Then I’ll give an schematic diagram of what we’ll be doing and try to motivate each step. That will give them the big picture. And hopefully the rest will fall into place once we have the big picture.

[1] It’s like that old adage… “I must’ve been absent that day.” In this case, it’s pretty disastrous.

UPDATE: A student asked me today, “Mr. Shah, will you promise me something?” “Not without knowing what it is.” “Well, will you promise me that you’ll explain why we’re doing all of this at some point?” So I was right on the ball, in terms of students just not knowing what’s going on because they can’t see the forest for the trees. And I went looking in the book to see what applications they have, and they’re awful (kites, ferris wheels, and bicycles). I’ve got to get on this right away!