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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.


  1. This is a tough one. I started this year by introducing the unit circle and having kids play with the values. I created this GeoGebra interactive worksheet ( I think it still needs some work.

    Thanks for the credit on the worksheet, but I am not retired. ;)

  2. On the one hand, I'm tempted to say "they're too young" for this. On the other hand, I know hundreds of years ago, young men their age were navigating boats using this very concept with only an astrolabe and the sun.

    I have not taught this stuff, before, but maybe give them a "standard" for some of the angles? Then they fill out a small chart like in the opening activity with the standard and then underneath fill in the known values and try to find the others? It will connect the activities together, maybe.

  3. In algebra 1?! I thought trig always went with algebra II. It doesn't seem right to do it with algebra 1. Am I missing something?

    I love what you're doing and may steal it for one of the first day exercises for my trig class at the college.

  4. Well, the difficulty is that students have trouble with "word problems", and always have done. I've never understood why it's such a leap to turn the problem into the correct mathematical formula, but perhaps that's why I've been a computer scientist/programmer for more than 30 years. I never had trouble with it. My classmates certainly did.

    You know "The Far Side" cartoon depicting "Hell's library", right?: the shelves are stocked full of books with titles like "Word Problems", "More Word Problems", and "Word Problems Volume LXVIII".

    And, see, then you throw them off by flipping the flippin' triangle around: after giving them a bunch of problems where "the angle" is in the lower left, you throw the ladder one at them, and the angle in question is now at the top.

    It really seems to be an analytical skill that most students lack. We who are good at it don't understand.

  5. Afraid I can't help here either. My son pretty much picked up this level of trig on his own or in discussions we had walking to school when he was in 5th or 6th grade. Like Barry Leiba, I have trouble visualizing the difficulty, since it seems so easy to me. Luckily, my son seems to think a lot like me, so I rarely have difficulty teaching him math or science.

    I don't think I'd do well teaching a non-honors Alg 1 class.

  6. Here's my suggestion for the problems you've encountered with the "Romeo and Juliet" part of the lesson. Take with a pinch of salt, of course. The main idea is to focus on the moments when they realise that these ratios are (1) fixed and (2) useful. The bits where they take averages and collect other people's results are, by comparison, distractions.


    Remove all the text between the table and the problem (including the bit that says "estimated true value...". Now nobody else has told them that the ratios will always be the same.

    Replace that text with a question 4 like "here is a triangle with a base of 40cm. You may not draw that triangle. Estimate its altitude."

    The deductions aren't hard. I'm pretty sure your students will be able to notice the ratios are the same by themselves (aha!). The brighter ones - or all of them with the correct prodding - should be able to use that to get the altitude... helped by the fact that it isn't a word problem.

    If they do that then at this point the students should feel more involved because they've discovered the key points themselves (ratio stays the same, can be applied to problems). It should also be more memorable. As for the length of the lesson? Leave the rest, you've met your top priorities. Have a discussion about what they found out and how they could apply that to problems, then set them just one or two for homework. Practice can (and will) come the following day.


    So... do you think that might help?

  7. Sue - NY has included sohcahtoa to solve right triangle problems in Algebra 1 forever.

    Dave - Must have been a different Dave Cox :) (We also have a sub named Dave Cox. Do you run into this often?). Unit circle is a good way to go, but I don't know if it's appropriate to the learning goals in this case.

    Calculus Dave - I'm not sure I follow. You mean, don't have them draw the angle? Just give them some triangles?

    Barry - I think non word problems aren't really problems. They are "exercises". Kids aren't good at it because they aren't expected to do it very much, and often aren't taught very well in primary grades. You are right, moving the angle on them is too high a level of difficulty.

    Kevin - Yeah, this needs to work for kids for whom insight doesn't come very naturally. And/or whose parents don't or aren't equipped to discuss it with them in 5th grade.

    Alex - I like that idea. I'm going to incorporate it. It will make it more gradual and more obvious. They do, by the way, notice they are the same by themselves. I'm not sure I want to use the word "estimate". They think it means "take a wild guess and just write something, 'they' can't mark it wrong." If I truncate the lesson, just to clarify, are you suggesting to assign a few problems just using their angle? Otherwise other students' data would be necessary.

  8. Kate
    If the learning goals is for students to recognize that the trig ratios are just the comparison of two sides, the unit circle is the perfect place to start. That is where it all comes from. Once you deal with the unit circle, you can change the radius but the ratios don't change. Once kids understand that the ratios have a meaning, they may be more apt to apply them and generalize (ie. sohcahtoa).

    If I am misunderstanding your goal, my apologies.

  9. I guess I just think that getting into the unit circle will introduce a bunch of other stuff they would have to understand, or I would have to teach them (and don't have the time to). For example, they're not strong on the cartesian plane. But maybe I should try it and see what happens.

  10. From the perspective of a calculus teacher, the unit circle is the way, bar none, to understand trigonometric functions.

    It gives a parameterization of any circle.
    It reminds that the sum of the squares is one.
    It provides the perfect memory aid for the sine and cosine of the common angles 0, pi/6, pi/4, pi/3, and pi/2.
    It clearly shows the signs of the basic trigonometric functions.
    And much more!

    That all said, I really don't know if there's a better way to teach trigonometry starting from circles instead of from right triangles. On the one hand, trigonometry starting with triangles goes back to the Greeks, and there's something to be said for the refinement of thousands of years of tradition. On the other hand, maybe trigonometry always starts with triangles because it always has started with triangles...

  11. I did a similar exercise last year in my Geometry class. They had briefly seen right triangle trigonometry in previous classes, so I wanted a way to break into the topic without throwing the words "trigonometry", "sine", "cosine", or "tangent" out and having the kids call up their previously associated conceptions of what those words meant.

    Instead of drawing a series of parallel hypotenuses joining two perpendicular lines, I used a figure of a right triangle that had a number of segments drawn parallel to one of the legs, dividing it into a number of nested right triangles. Students had to measure segments and calculate ratios for each triangle, though I referred to all of them by names of segments rather than "opposite", "adjacent", and "hypotenuse". I left it to them to figure out that the same ratios showed up in each triangle.

    Having established that the sides gave the same ratios (and why that happened), I gave them another figure where they had to measure all sides of one triangle and the hypotenuses of the other triangles, use the ratios to predict the lengths of the remaining sides, and then measure the sides to see how accurate their predictions were. Getting from there to talking about sine, cosine and tangent involved a discussion where I asked questions like "If we keep looking at right triangles, what does the ratio of a given pair of sides depend on?" and "Would having a list of these ratios for all angles be helpful?"

    It took a lot longer than I expected it too, but it seemed more rewarding than just rattling off definitions.

  12. Kate, I have to play devil's advocate here: I understand that the standards say you have to teach this, but have you considered just teaching the absolute minimum necessary and then moving on? Even if it's tested this year, pare down your unit to whatever they'll be asked on the Regents, and that's it. You might say that's teaching to the test, and perhaps you'd be right, for this particular topic.

    Besides, if they are asking "What's a ratio?" and can't move around the coordinate plane, should you really be spending a lot of time on something this advanced?

    In other words, and I know it's pure heresy to say this, you don't always have to do everything you're told to do! You have to do what's in the best interest of your students. Sometimes districts and states tell you to teach things (or test students on things) for no good reason, and I believe it's our job to decide what's most important.

    I think this might not be helpful, as you have already started teaching it. So I will add that I think going into the unit circle at this point would just make things confusing for your students. There's worse things they could be memorizing other than sohcahtoa.

  13. Hihi,


    My thoughts:

    (1) I honestly think that no way should you use the unit circle for Alg I. It's too much, and combining circles and triangles really would be better to do in geometry. I think this work with triangles is really a lot better.

    (2) I am really irked by #4 ("While the three ratios will be different, the values in each column should be close. Why?") It gives away the prize, for free.

    (3) I don't know the use of #5 or how they are supposed to get the "actual value of each ratio." I mean, it could lead to a good discussion of error and why using the biggest triangle would be best, but it isn't keeping up with the topic.

    (4) If the word problem is getting to them, if the kids can't seem to make that final precious leap which is actually what the whole class is about, I would maybe give one additional question without Romeo and Juliet/context beforehand. Which was actually Alex's suggestion, now that I think about it..

    (5) To me, it seems like you are trying to do two distinct things in this lesson, at the same time, and that could be tricky for non-accelerated kids. They are: (1) teaching about similar triangles and (2) a baby-version of trigonometry, where students generate their own trig tables without knowing it.

    Do you think that maybe separating the two completely -- at least initially -- might help? First get them to understand the concept of similar triangles (they are the same, just one is scaled up or scaled down, the angles are the same too!).

    Then get them to talk about what happens as you raise or lower the angle... what happens to the ratio of the sides of a single right triangle.

    Finally have them draw, say, two right triangles with a small angle of elevation (like 20 degrees), and two right triangles with a large angle of elevation (like 80 degrees). Then say: I want you to tell me the sides of two HUGE triangle with your same two angles of elevations, and base length 5,000,000. (But you could first prompt them by asking which opposite side of the two huge triangles would be bigger and why? And similarly, which of the hypotenuses of the two huge triangles would be bigger and why?)

    Then it could take you to where you want to go?

    And I say if you are excited about it, and you know your students are capable of doing it, then keep on keeping on! Don't scrap it.

    My 2 cents.

  14. Oh, I also forgot to say: you might want to take out all reference to tan. Since sine and cosine haye opp/adj/hypotenuse, students will be able to solve all the problems with just sine and cosine. They don't NEED tan.

    It's also nice because they are just between 0 and 1, so students will have a better grasp of what they mean -- because they give the size of a side in proportion to the hypotenuse. (A sine of an angle yielding .56 would indicate that the opposite side is 56% the length of the hypotenuse.)


  15. The inference that all the ratios are nearly the sme doesn't transfer to a generalization that most of my students are prepared to make.

    Most of them are still stuck with even being able to identify the opposite, adjacent, and hypotenuse consistently. I actually practice just that for a while.

    One of the earlier commenters suggested not asking too much up front and I'm inclined to agree.

    I didn't really get the unit circle stuff until pre-calc in 11th grade. Even in Algebra 2/trig, the law of sines and cosines, sum of angles, etc. were derived from a triangle...

  16. For a discovery lesson, that may be overkill for Algebra I students using that many numbers. I do a discovery section of my lesson, but only on a 30-60-90 triangle -- that's enough to motivate the concept (and link to unit circles later).

    To start things out, I do a similarity / scale factor type lesson. The students get the idea of scaling very intuitively.

    Then we do the similarity ratio trick on two triangles...

    a/b = c/d

    but note that with some algebra

    a/c = b/d

    ...we can talk about ratios *within* each triangle being scaled identically.

    Also, get your students to actually do the tree thing -- that is, go outside and pick a tree and measure it.

  17. I wasn't suggesting that you throw a unit circle up on the board and have them go at it. I was suggesting using somethign like the GeoGebra file I linked to. I agree that the unit circle by itself is a bit too much for algebra 1 kids. My question to you is can my worksheet, with modifications, help? I can re do the dynamic sheet and allow for the GeoGebra tools to be seen and the questions can be changed.

  18. It's hard to get past the poor thinking that went into putting this topic into algebra (they CAN do it, so they should; just a corollary to "as early and superficially as possible")

    Honestly, your stuff looks okay.

    Something different: I give them a rt triangle with hypotenuse 1 and angle (I pick a number, maybe 22) degrees. I label the legs (with sin22 and cos22, rounded). Then I ask them to use similar triangles to find missing sides in a bunch of 22-68-90 triangles. Everyon's happy. Then I put up a 27-63-90 (or something like that) and someone complains that they are not similar, so I put up a hypotenuse 1 triangle for that...

    And after another triangle or so, I show them how to look up the values...

    Clumsy, but something nice.

    Better, of course, to push it off for a couple of years. But not my choice.


  19. I'm 3/4 braindead from exhaustion right now. Haven't read through all comments yet. Forgive me in advance?

    Trig in Algebra 1? Really? Crazy New Yorkers. I'm just getting to it for my first time as a teacher and have realized how much I internalized my Precalculus teacher's pneumoic devices. Instead of SOHCAHTOA (which seems stranger on the rez), I learned "Some old hippie caught another hippie tripping on acid."

    As I've worked through problems in the past week, I keep saying, "Tripping on..." My student just remembers "opp hyp" "adj hyp" "opp adj." She has amazing recall though.

  20. 3♣ sez:

    I didn't really get the unit circle stuff until pre-calc in 11th grade.This is exactly the problem I see in calculus. The unit circle has many more uses, and makes it easier to memorize the basic facts about trigonometric functions than the triangle definition. It's at the very least a crying shame that it isn't taught more thoroughly.

  21. Thanks everyone who has commented so far. I'm overwhelmed by your willingness to help me improve. I'll try to respond to everyone as I read through and process your comments. (And if you are just getting here and have more to add, don't let me stop you, keep them coming.)

  22. I tried to avoid SohCahToa, and I heaved ahuge sigh when I finally had to resort to it

    I'm glad I'm not the only one.

    It does mean, however, that I have no answers for you.

  23. Mr. D I'm normally as big a rebel as the next guy, but simply leaving out this whole topic is not feasible. The regents exam is unpredictable, and could conceivably include 6 or 8 trig questions. It wouldn't be responsible to hamstring the kids by not even mentioning it.

    Unfortunately (or fortunately), I am constitutionally averse to knowingly doing a bad job at something. If I'm going to do it, I'm going to do it right, or at least do everything in my power.

    Sam - this lesson is sequenced immediately after a few days about similar figures, as you suggest. I also incorporated some of yours and Alex's ideas in the new worksheet (see newer post).

    Jason and Sam - I pared it way down, removing tangent and the number of angles the class is working with.

    Thanks again, everyone, all the comments were very helpful. Take a look at the new lesson, if you're so inclined.

  24. Teach them tangent first. Do this by talking about road grades. Road grades are the slopes of roads expressed as percentages. You see these on very steep roads. They are equivalent to tangent of course. There are lots of pictures on the internet of road grade signs, and also the wiki on Lombard street or other steep roads may be helpful.

    I introduced my pre-algebra students to tangent in this fashion and they are quite comfortable with the concept.

    A good project would be to take a map with elevation curves and have students calculate the road grades of a hypothetical road on the map. They can then use a trig table to find the angle on the road. In this way they can learn how tangent connects these ratios to measurements of angles.

    Only once students have mastered tangent, should they be introduced to the other trig functions. To throw them all at them at once would be foolish.

    By the way, my school is cutting teachers due to state budget shortfalls. I am the newby, so if anyone needs a math teacher then drop me a line.




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