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## 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 day...hm.)

Word Doc
pdf (with MathType a little smushed)

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

1. Hi Kate,
I am new to your blog. I love the way to put out this lesson for comments. The process is fantastic.
I have one question. How are you introducing the lesson? Do your students know from the very beginning that they are trying to determine the length something which is not practical to measure directly? I think it is important for them to know this, so they can understand why they are doing every step as they are doing it (steps 1-3), and they can evaluate as they go along how each step is useful. They might speculate about how one piece might be useful and get it wrong, but they can correct it on their own as proceed and see how it fits together.

Burt

2. Meh. No. The day before this, we are working with finding missing sides in similar figures, so they're kind of primed that way anyway.

I'd love to make this more problem-solve-y, but I haven't been able to think of a sufficiently hooky, relevant, accessible problem for it.

Historically, it was all motivated by wanting to find the length of a chord for a given central angle and radius. Maybe I should be looking there.

Do you have any ideas?

3. Kate, I haven’t worked with this age group in a long time, so maybe I am way off, but I don’t think it needs to be “just right.” How about Romeo and Juliet, or just a triangle? I think what I am saying ties in with Alex’s comment about their realization that the ratios are fixed and useful. That could start happening when they calculate the ratios, not when you present the problem in #4, at which time their attention is taken away from the ratios. From their point of view, while they are calculating the ratios in #1 to #3, what is the purpose? Are they just doing a worksheet, or are they searching for some other meaning, some way that the ratios are useful in finding the length of a side of a triangle?

4. Hi Kate,

Your lesson had time to incubate and a couple of thoughts popped in my head. Here’s a simple introduction that can be placed at the top of your worksheet:
“Today we’ll solve problems like the following, where we know one angle of a right triangle and the length on one side.

Romeo and Juliet
plain triangle like one from #4
(problems side by side if possible)

We’ll start by exploring the properties of right triangles.

Comments: I’d use both a word problem and a plain triangle so that I have the more difficult problem but help them see the simple triangle commonality.
I’ll speculate on their thought process.

WITHOUT THE INTRODUCTION
on #2 after computing the first ratio: “I wonder if the ratios are the same.”
After computing the second ratio: “Maybe the ratios are the same.”
On 3# after computing the first ratio: “I think they are the same.”
After computing the second ratio: “Yes the ratios are equal.” #3 confirms and gives closure. Done with this problem, on to the next.
#4: “This is hard. Not enough data. I don’t know”

WITH THE INTRODUCTION
on #2 after computing the first ratio: “I wonder if the ratios are the same.”
After computing the second ratio: “Maybe the ratios are the same. What does that mean?” Incubation time, especially when measuring the third triangle.
On 3# after computing the first ratio: “I think they are the same. I can use this to calculate the other side. I got to think about this”
After computing the second ratio: “Yes the ratios are equal.” Maybe they can see how to do the calculation right away, maybe they have to think about it more. But the moment they realize that the ratio is fixed is the most likely moment that they will see the usefulness of that fact. Without a problem solving context, that moment passes with much less chance of seeing the implications of a constant ratio in similar triangles.

5. I follow your logic, I'm just not convinced it would help much. These are freshmen...they tend to skim/skip over "the blah blah" at the top of a worksheet and look for what they are supposed to "do".

BUT, I have an idea. At the start of class you could project a problem like the Romeo/Juliet on the screen, and ask them to try to solve it. They might mention pythagoras and realize it isn't going to work, they might notice they can find the measure of the other acute angle... and then say "by the end of class we will have the tools to figure this out", and then pass out the worksheet.

6. Ok I plan to use this lesson but when I worked it out myself, I got different rations every time. For vertical over hypotenuse, I got .625, then .6869 and then .655. If I'm not mistaken, I'm supposed to get the same ratio every time, correct? And then they find the missing length in the homework problems by cross multiplying and solving the ratio right? So we are introducing the solving and the ratios but not actually using sin, cos, or tan, correct again? I just wanted to make sure before I use it in the trenches.

7. If you draw the triangles by making the third sides perfectly parallel intersecting at the exact angle, and you measure perfectly, your ratios must be exactly the same. The triangles are similar! Sides are in proportion! Obviously perfection is really hard to do for mortals.

I'd suggest trying again, and being painstaking about your measurements. Also when you conduct the lesson, now you will know the parts where your kids will have trouble. I find the protractor use the most error-prone part of this whole thing. And also, sometimes they will measure with the inches side of the ruler, but try to use tenths for the 1/16" division markings.

To avoid that, you could put the third sides on before you give them the copy to measure. I don't think it has the same punch, but it would avoid some error.

You could also know what the ratios should be, secretly, from sine and cosine, and tell them after they calculate and average "If you did this perfectly, you would have gotten vertical/hypotenuse for 45 degrees to be 0.7071" etc etc.

And yes, on the first day, you should be using the ratios gleaned from the measurements. You don't even have to mention sine or cosine.

Good luck - let me know how it goes.

8. Ok I tried it and they were super confused on just drawing the triangles. I used the nested triangle approach and literally had them draw on a giant protractor so that the angles would be correct. We didn't get past finding the rations but I noticed that no one had the same ratios. Quite a few students had similar ones but now I don't know where to go from that point since no one can notice the ratios are constant.

9. Maybe next time, you put the third sides on for them, and then all they have to do is measure the angles and sides? Chaos always ensues whenever I ask kids to do anything where precision is required, too. It's not just you.

For now, in your shoes I'd just be like "OK if you measured perfectly, your ratio would have been this" and go from there.

10. This is a bit past comment expiration date, but I just want to add:

I do something very similar, but avoid having them draw triangles themselves (partly because our school does not have protractors).
Rather I give the students an investigation worksheet which has three pairs of right triangles - 30, 45, 60 degrees - designed by me and have students in pairs measure and calculate the ratios. This saves a lot of time and so far all students have "gotten the point" rather easily.

11. This. Was. AWESOME! We just started our Trig unit today and I used this as the jumping off point. We have a modified block schedule and today's my 80 min. class, so between some Warm Up problems, going over our homework from the night before, and then hitting this lesson, it was about perfect timing-wise. Most were able to get into the first part of the homework problems. Anyway, after making that connection between the common ratios for each angle measure and solving the given triangles with hypotenuse equal to 100, they were good-to-go for the homework portion and seemed quite confident.

Also, I led with a picture of myself standing next to a light pole in a local park with the question, "How high is the pole?" After estimating using Dan's method (too high, too low, just right) and getting my height put in there, I then added a little more info with a triangle that had the angle of elevation and bottom side labeled. Of course, we came to the conclusion that we still couldn't figure out the height... However, this gives me a perfect spot to start tomorrow with, as I can flash that photo again and ask, "Ok, what would we need to know in order to solve this?" Hopefully, I'll get some answers along the lines of "Well, there must be a common ratio for 25º too" and roll from there with the formal trig functions... We'll see! Anyway, thanks again for being such a great, transparent blogger.

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