Thursday, April 30, 2009

Fixed: Introducing Right Triangle Trig

I took a bunch of comments I got on my previous post and used them as a hatchet on my trig lesson. Pared it way down - less angles, took out tangent, made the problems easier, didn't give away the store. I suspect it's going to work much better next year. I wish I could teach it again, now. But unfortunately we have to fix the plane while we are flying it. (Maybe I can talk one of my colleagues into letting me take over their class for a

Here is the important part (links to download complete docs at the end):

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Word Doc
pdf (with MathType a little smushed)

Thanks again, all my collaborators. You're the best around.

NYSED's Secret Shame

Skitch is my new favorite toy.

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Wednesday, April 29, 2009

Introducing Right Triangle Trig

I am conflicted every year about how to introduce right triangle trig to my Algebra 1 classes. I am not thrilled that we have to worry about it in this course, and come at it from the perspective of similar right triangles (instead of unit circle/wrapping function), but that decision is above my paygrade. This is a topic that I find so difficult to promote a conceptual understanding. We have these three weird word abbreviations, sin, cos, and tan (that students tend to pronounce phoenetically) and they mean what exactly now? Ratios of sides? What's a ratio, again? I suspect that most Algebra 1 teachers don't even try too hard. I suspect that this topic is largely approached as a procedural exercise, with lots of practice. And I admit that every year, after giving it my best shot at illustrating it conceptually, I also revert to teaching procedure, with alot of practice.

Here is what I do, or try to do, and please, readers, I am asking you to criticize the hell out of it. I really want to do it better. (Due credit: this idea and the original document came from Dave Cox, who used to be a professor at Cornell, but I think he's retired.)

I give every student a protractor, ruler, and a 4-page packet. I usually have my desks arranged in pairs, and each pair of students is assigned an angle. Using the protractor, they draw a series of parallel lines that make that angle with a horizontal line on a given page with some axes provided. This creates several overlapping right triangles.

Then they use the ruler to measure all three sides of each right triangle. They record it in a table I provide. They use their calculator to compute the three relevant ratios, and enter them in the table. Here is the data sheet:
They notice that the ratios for the two corresponding sides all come out to be the same, and we have a discussion about how similar figures have sides that are in proportion, and that's what it means for things to be in proportion. When you divide them, the quotients are equal. They average the five ratios to determine the "real" ratio.

Then they are supposed to use the ratios to solve this problem for "their" angle:
Doing that successfully is probably the key part of the lesson, and it never goes well. They don't see right away exactly what to do, and give up. I end up doing a bunch on the board for them, using their angles and the ratios they calculated.

Next we are supposed to collect every pair's calculated ratios in a table, like this:
And they are supposed to copy them and use it to complete some problems for homework, that look like this:

I keep trying this lesson every year, because I really want it to work. I really think it should work. Part of the problem is that we run out of time. I can't get this whole thing done in 43 minutes, and there isn't really a good point to stop and pick up the next day. The next day, I just tell them "SOHCAHTOA" (*huge resigned sigh*) (at least I don't claim Sohcahtoa was a Native American princess) and start teaching procedure. And feel like I am committing malpractice, and stealing my paycheck.
Here is another link to the document.

Update: I revised this activity based on the comments and discussion on this post. Go here.

Tuesday, April 21, 2009

The Pattern Does Not Hold

Here's a little illustration if you need to show that a few examples do not make a proof.

(Context: Introducing a quick 4 days on proof by induction, because we use it in the next unit on Series & Sequences.)

First I asked them to write an expression for the nth term of the sum of the first n positive odd integers. Like this:

1 = 1
1 + 3 = 4
1 + 3 + 5 = 9
1 + 3 + 5 + 7 = 16

They did up to like n = 5, saw the pattern, and settled on n squared. They were utterly convinced. Their intuition was serving them well. For now.

Then, I drew a circle on the Smartboard, and asked them, if we draw n unique points on the circle, and connect them with as many segments as possible, in how many maximum regions is the circle divided?

2 points: 2 regions

3 points: 4 regions

(At this point, they had a few minutes to work independently or with a partner to come up with a conjecture for number of regions in terms of n points on the circle.)

4 points: 8 regions
They think they see it: the number of regions doubles. They check it with

5 points: 16 regions
And nearly everyone was satisfied with that, and conjectured that n points yield regions.

A few intrepid souls, and eventually all with some cajoling, drew 6 points, yielding 31 regions, not the expected 32.

Coming up with the actual expression is non-trivial (it ends up involving nCr, of all things), and we didn't get into it, but I found this an effective motivator for introducing proof by induction.

Last year, I used the example of Fermat's conjecture that $2^{2^n}+1$ yields only primes - a conjecture which went unproven either way until Leonhard Euler found a factorization of $2^{2^5}+1=4292967297=(641)(6700417)$. But, I thought this was much more effective, because they could confront their erroneous conjecture with the counterexample directly.

Friday, April 10, 2009

A Classic Activity and My First Videos

Yesterday in addition to letting my freshmen play with Geogebra to see if the ball landed in the can, I had my sophomores/juniors do the old Sine Spaghetti activity. Just for fun I walked around with my new Flip camera to document the festivities.

Here is one version of instructions for the activity (from Mrs H, I think?). (Not the version I used, but that one is trapped on a CD in my classroom right now.)

I don't know how instructive these will be, but it gave me a chance to use my new toy. I was careful to edit out faces - there are voice and backs of heads. Is that ok? Anyway, enjoy...

Constructing a Circle from Kate Nowak on Vimeo.

Marking Angles from Kate Nowak on Vimeo.

Marking the String from Kate Nowak on Vimeo.

Transferring Sines from Kate Nowak on Vimeo.

Finished Product from Kate Nowak on Vimeo.

Tuesday, April 7, 2009

Real World Once Again Inconvenient

Here are three attempts to model "Will the Ball Hit the Can?" in Sketchpad:

Attempt 1: I try to make the grid scale match the given scale. (1 unit = 1 yard? Or close to it?) This requires the coefficient of the squared to term to be waaaay to small for clumsy freshman fingers.

Attempt #2: Make the scale very large. Now the leading coefficient is a little more friendly, but by comparison I have to make the constant coefficient slider way too short.

Attempt #3: Using a rectangular grid, so that the y axis is a relatively smaller scale than the x axis. Now all three coefficients are lengths I think the freshmen can handle. The catch? We can't even pretend this function means anything any more.
I'm going with #3, but it will bother me; I feel like I am perpetrating a ruse. On the other hand, I have learned way more about the behavior of the standard form of a parabola than I would have had I not bothered to try.
Feel free to email me if you want the sketchpad file to play with. It's 32MB.

Build Your Own Worksheet

A simple idea: provide a structure for a skill the kids need to practice. Let them fill in the numbers to use through some randomization process (dice, playing cards, calculator random number generator...).

I find the idea appealing -
  • it's a tad more engaging than a preprinted worksheet
  • it saves paper
  • sometimes it conveniently illustrates a larger concept, such as $\sqrt{ab}=\sqrt{a}\sqrt{b}$, and the commutative property of multiplication.
But using BYOW has consequences, and what I'd like to discuss is minimizing the undesirable consequences.

For example, these are my directions for BYOW: multiplying radical expressions.

I had them remove the face cards (which got me jokingly accused of being "facist" - ha ha), but leave the aces. They need a little experience with the fact that $\sqrt{1}=1$. You wouldn't think so, but it comes up in places like evaluating the quadratic formula.

I'd overall call this a "success", because the kids worked at it for a good 15 minutes without losing focus. Quick assessment at the end indicated that they knew meant "multiply", which was one of my goals.

The biggest issue is too many too-easy problems. Like, $\sqrt{1}\sqrt{9}(7)$. Or radicands that are already simplified. Or all black cards. Yes I want them to recognize when a radicand cannot be simplified, but I don't want 7 of these easy problems, and only 3 where they have to simplify the resulting radicand. You see?

So far the only idea I have is to keep the Queens in the deck, and say they are worth 12. Also, I could have them separate red and black, and always choose 2 of each.

Any other ideas?

Thursday, April 2, 2009

They Have Me Totally Figured Out. Must Be Spring.

I, apparently, need a new schtick. Today I was doing the old, "What's the diagonal of a 1x1 square?" Students diligently think about pythagoras for 10 seconds, with a few misguided shouts of "1! 1 and a half! 2!" and settle on the square root of 2. This is right after we talk about what square root means, how it's the opposite of square, how "radical something" is "what times itself is the something".

So I say, "ok so what's the square root of 2? what times itself is 2?" Blank stares. Lunges for calculators. "I SAID NO CALCULATORS," I say.

A few speak before they think and say, "one", "one half", "four halves", but realize those aren't going to work before they get all the words out.

"One and a half?"

"OK", I say, and I diligently write on the board 3/2 times 3/2, which gives us 9/4, which we all see is more than 2.

"Hmmm. Must be less than one and a half."

"One and a quarter!" We figure out that one and a quarter is 5/4, and I write 5/4 times 5/4, which gives us 25/16, which is obviously much less than 2.

"one point four?"

"OK", I say, and start writing, to which the kid says,

"Well obviously that's not right because you wouldn't write it down so quick. When I'm right you always say 'Are you sure?'"

That little snot.