Monthly Archives: February 2016

Introduction to Trig

I teach Precalculus, which means that my students have seen sin, cos, and tan twice: once each in geometry and algebra II. So I decided to see how many of the students would recognize the functions from the input and output. Spoiler: very, very few.

Intro Activity

I showed students that when you drag a corner of a right triangle around, the angle and the ratio seem to change. So I created two triangles[1], showed them how I measure the angle, write it down, and measure the two sides next to that angle (I try to avoid using the words “adjacent” and “hypotenuse” so they don’t get tipped off to the trig functions that way), and divide them to find the ratio. I then typed these into a spreadsheet, which I shared with them all through Google Classroom.

Screenshot from 2016-02-19 22-07-59.png

Showing them how to enter the data.


I showed them how to type “=50/71.2” so the spreadsheet would calculate the ratio for them[2]. I then instructed them to run along and “find” (mostly create) right triangles of their own with wacky angles. They could work in groups, but they needed 2 triangles (and therefore 4 rows) per person in the group. They typed these into the spreadsheet themselves. I also showed them how the units divided out, so they could use whatever units they wanted to for this activity, as long as they were consistent within the ratio.

Their Work

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Their finished work. Ignore the names 🙂

I then typed “=radians(A2)” for them and showed them how I could drag the formula down.

Screenshot from 2016-02-19 22-04-46.png

Getting ready to do something awesome in Desmos.

Unfortunately this took the entire day. I thought that drawing 2 right triangles, measuring 4 angles, and measuring 6 sides would take roughly 15 minutes. Between my introduction (10 minutes), them making + measuring the triangles, and showing off what spreadsheets can do, it took the entire class period. I was a little bummed, but looking back I think it was worth it.

Day 2

I started off by showing them how awesome Desmos is. I copied the radians and ratio columns (see image above) and pasted it into Desmos[3]. I told them to do the same[4] and find a function to fit the data. I think I reminded them to label their axes.

Screenshot from 2016-02-19 22-23-08.png

It’s Desmos awesome?

My instructions were “find a function that fits this data”.

What (shouldn’t have) Surprised Me

Student comments: “Oh, you mean like a line?”

“Can we use the squiggly line thing you showed us in Desmos?” “Yes, but you need to pick the parent function for that.” “Oh…”

“This looks like a parabola!”

Many of the groups found complicated formulas, such as f(x)=1-.4x^2 (see image below).

Screenshot from 2016-02-19 22-25-58.png

One group’s guess at fitting the data.

In the end only 3 of the 7 groups found that it was cos(x), but I have a hunch that 2 of the 3 groups overheard the first one talking about it, and in that pair, I think one of the two students had the idea of trying cos(x) and the second though “yeah, I guess that might work”.

Pause for Teacher Reflection

On the one hand I’m really excited about the parabola above (there were 3 other parabolas that fit the data just as well, if not better) because it means they’re (1) recognizing shapes of functions and (2) able to manipulate the functions to fit any data. On the other hand, this means they really had no idea what the trig functions meant the first time they learned them.

I think that students are too often handed these functions, perhaps told what the input and output are, but mostly just memorize steps of how to use the buttons on their calculators and don’t have any deeper understanding of why we even have sin and cos. If they actually understood this function, they should have started from the very start of the introduction and said “Why are we plotting these, isn’t this just cos?” The fact that cos(x) surprised these students with how well it fit just goes to show that they don’t understand the functions. For that reason I think that every introduction of the trig functions should start with an activity like this and build up the need for the functions instead of spoon-feeding students the method solving for x on a right triangle when you have a side and an angle.

Now that my rant is over, here’s the beautiful graph with cos(x).

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Ahhhh, good ole’ cosine.

Extension with Students

Since it was a block day, we had lots of things we could do. We talked about how the size of the triangle doesn’t matter, since it’s a ratio. I foolishly made the off-handed comment of “we could make this triangle fill the basketball gym” at which point they said “yeah, let’s go do that!!” How could I say no?


Three long tape measures and three protractors is all we needed!

The triangle ended up being incredibly accurate (to within 5 hundredths of an inch when checked it with the Pythagorean Theorem!) and fell right onto our function.

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You get two points: one for each angle.

I showed them some more excel magic, moving the columns around, finding the complementary angles with “=90-A2”, and getting data to graph sin(x) instead.

The last 20 minutes, students spend practicing finding the sides, given a side and an angle. Hopefully by this point they’re thinking of it in terms of “this trig function of this angle gives this ratio” rather than “follow these steps”. I need to come up with some quiz questions that can distinguish between the two.

Summary & Thanks

This lesson was the first one this semester that I hadn’t flipped. I don’t know how to do this kind of intro over a video, and really wanted students to struggle with “what is the best fit for this function?” so they see that sin is a function created for a specific purpose. I was very glad that I didn’t flip this activity and I hope that I can identify future lessons which would benefit from not being flipped.

I wanted to thank Alex Overwijk ( for giving me the idea while he was sharing a similar activity at TMC15 this past summer. He even gives the students a table of ratios for sin and cos and they prefer using that all year instead of typing it in a calculator!

[1] I used the 45-45-90 triangle and the 30-60-90 triangle since we had just done these in class the day before.

[2] A side goal of today’s lesson was showing them how awesome spreadsheets are and how powerful they can be. Originally I even made the table in the spreadsheet so they’d see the graph, but between days I remembered how awesome Desmos is.

[3] No, you don’t have to even create a table, Desmos is that awesome. Oh, and it will adjust the domain and range automatically if it’s larger than the default window. I didn’t know this, otherwise I might have just gone with degrees instead of changing it to radians. But radians works better since Desmos defaults to that and typing in “cos(x)” gets the perfectly fitting graph immediately.

[4] There are almost too many ways to share it. I could have shared the Desmos graph, but (a) they already had access to the spreadsheet and (b) I wanted them to experience how awesome Desmos was and what it could do.

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[2016 Blogging Initiative] Week Four: A Lesson Introducing the Unit Circle

My Relationship with Textbooks: “It’s Complicated”

My first several years of teaching I avoided the math textbook as much as possible[0]. One year I even waited to hand out textbooks to students until the second quarter. I assumed (incorrectly) that using the textbook would make me a lazy, bad teacher. However, at the start of this year I decided to embrace the textbook for the good resource that it can be: a bank of practice problems[1] not a replacement for my teaching[2].

Background: My Classroom 

One other thing I’m doing this year is flipping my classroom. The flip, however, isn’t just lecture. I’m trying to challenge my students do problem solving through the vidoes, and I hope to show how I’m trying to achieve that in this lesson. For one thing, I provide guided notes for the students to fill out as they watch the lesson. I also don’t do every problem: I ask them to pause the video and try some in the middle of the video. To that end, I’m also using EDpuzzle which pauses the video and asks them questions that I’ve created at a variety of levels.

When we get back together in class the following day, the students are randomly assigned into groups of 3 or 4. Students spend about 10 minutes going over the notes and making sure each students’ notes agree with one another and that students understand the topic. After that students work on practice problems, from the textbook, on the same topic. [3]

The Challenge

So we’re chugging along and we get to the Unit Circle. This is the first lesson that I disagree with how Blitzer (our textbook) approaches it. I’ve had success with students in the past by teaching special right triangles first because students see them in the Unit Circle. So I decided to create my own “chapter” and left the textbook, like old times.

The link below is a short (<13 minute) video so you can see what the students will do for HW prior to class. But you should watch it because that’s the interesting part of my lesson. 🙂

Here’s the handout:

As you can tell from the video, I show students the special right triangles and where the values come from. My hope is that they use the Pythagorean theorem if they ever forget the shortcuts in the future, but most students will, unfortunately, probably forget that. I’m not sure how to share that with them differently.  However I only give students a few points from the Unit Circle, and ask them to “figure out the rest”. If they can figure it out on their own before coming to class, and they understand the special right triangles, then I think that it will be more likely that the Unit Circle will stick.

Since I’ve deviated from the textbook here, I had to find practice problems online, but that wasn’t too difficult. Students will go to my website and simply click on the worksheet links (complete with answers) to practice this in class. I’ll only print out the sheets for those students who want more practice beyond class and have no internet at home.

I’ve assigned the video (only 2 students have watched it so far), but we’ll meet in class Monday to see how well they did filling out the rest of the Unit Circle.[4]

Request for Feedback

How can I improve this approach? How can I teach special right triangles in the video so that they do more of the “heavy lifting”?

How are the quality of the questions in the EDpuzzle video? Are there others you thought of that I could do?

Is there a better way to approach the Unit Circle that you’ve seen/used other than special right triangles?

If you could answer any of the questions above, I’d greatly appreciate it. Thank you for reading! (and watching??)


[0] I still avoid it in Physics–I haven’t handed out a textbook in 3 years, with the exception of one student who begged for it. It didn’t help her.

[1] It’s also a good resource for ideas for 3-act lessons.

[2] I’ve seen some teachers teach how to read a textbook, which is a valuable skill, but one that I’ve decided pass on for now. I want my students to understand the math first and foremost. I’m still not sure how I feel about not teaching students to use a textbook effectively and efficiently.

[3] Because I believe that HW is practice, earlier this year (before I flipped), I don’t grade HW. Students also didn’t do the HW (with very few exceptions). Now, I still don’t grade that they watch the video, but I’m not afraid to email or call home if students are missing it chronically. Also when students get to class, they recognize that they’re responsible for learning the material at home, and so will work harder at the start of class to understand what they didn’t watch. It’s amazing how much more “HW” (practice) they’re doing now just because it’s happening during class.

[4] And if I’m on my blogging game, I’ll blog about how it went. Unfortunately it’s tennis season, so I probably won’t find time to soon.


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