Tag Archives: Lesson Ideas

[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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[3 Acts] The Coefficient of Friction of My Son

Last year in Physics, I used the PhET simulation Ramp: Forces and Motion to teach about calculating the coefficient of friction in the situation of an object on a ramp (in this case, a box).

This has worked well enough for me in the past. However, as I was travelling home with my son, I had an idea to create a 3 act lesson using my experiences.

Perhaps I talked too much in the video, so I might just show them the final part where I’m sitting up with Benji.

The idea really did just hit me as I was sitting there in the train, thankful that I didn’t have to keep my arm under him, propping him up. So often, ideas will come to me, but I’m just not in the right location to jot them down to save for later.

Another thing I thought of pertaining to this lesson is how it relates better to the girls in my class than most of the lessons. Sure, some girls like talking about drag racing or shooting a basketball, but this connects better to the majority of girls than those other lessons[1].

One downside to this activity is that on the train & plane, I’m not concerned about the coefficient of friction–I’m concerned about what angle I can sit up with him without him falling down. And that’s what we’re measuring at the start, so finding the coefficient of friction has become a superfluous academic exercise despite the “real-world-ness” of the problem.

And then there are the other things that I hope students will consider: things like my chest isn’t actually flat and we both have shirts on, so I guess we’re actually finding the coefficient of friction between his shirt and my shirt. But those are great things for students to consider on their own, so I don’t want to spoil it by pointing out all the inconsistencies in the video from the start.

[1] If you think I’m being sexist here, you should try walking down an airport terminal with a 3-week old. At least 10 times more women than men will stop you and say something.

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Speed Dating: Chemistry Style

Others have posted about various “speed dating” review games, but I just wanted to share one I did recently in Chemistry which went better than I could have expected!

We’ve been working on Chemical equations, and I want them to (1) write the formulas from the Chemical names, (2) predict the products, (3) predict whether the reaction would take place, and (4) balance the equation. For many of my students, that process is overwhelming because there is a lot more to do on each of those steps, depending on the reaction.

We had a block period this day (1 hr 30 min class), so we started with the students making their own chemical equations with their lab partner and doing all 4 steps above to them. This took about 30-50 minutes depending on the class, especially as I wanted to make sure that (a) they got all the steps correct and (b) they understood how to get the right answer for their equation.

After all groups had equations ready, I had them go to the lab (so they were standing up) and pair up with another lab group. There were whiteboards at the lab tables and I had them divide the boards in half. They wrote their equation along the top of their half of the board, and then flipped the board around so that they were working on each other’s equations. See the pictures below.

photo 1

photo 2

photo 3


What’s great about this setup is that the students immediately become tutors of “their” equations. I give them a set amount of time to work, and when the time runs out, even if they’re not done, they’ve got some work and the people across from them can check their work right away.  If they get stuck, they have immediate help. Students rotate so that they are working on a different equation, but “their” equation is still across from them and they’re still available to help.

We were only able to work on this for a relatively short while (about 40-50 minutes depending on the class) because it took students so long to come up with their own equation, but it was essential for them to come to the table with the right equation and it gave them a confidence booster to help others on “their” equation. They also got faster each time they were doing different equations, which was another goal of this activity.

It also gave them ownership over their work to have an equation that they created and that they know the in’s and out’s of. They actually surprised me with how into this activity they were. It was probably partly that they were working on whiteboards (where mistakes are okay), and partly a combination of the things I mentioned above.

I could have handed out worksheets[1], but this was so much more fun and exciting for them.


[1] Not all worksheets are created equal. Some would be highly engaging, but the ones I had in mind were boring as anything.


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[Explore the MTBoS] Mission #1: Stop-Motion Video and Participation Points

Note: I just finished typing this blog post and re-read the prompt carefully and realized that Sam asked for one of these two prompts.  Oops.  Sorry if you’re reading this because my comment is above yours: don’t feel like you need to read both sections if you don’t want to. 🙂

Favorite Open-Ended/Rich Problems: The Stop-Motion Parametric Video

One of the topics in Precalculus that we cover is Parametric equations. One day I was watching a stop-motion video, and I had the idea to get my students to do stop-motion videos showing a parametric equation.  Here are some example videos from last year:

(The third one is my favorite…)

The students were asked to create a Parametric equation and then use stop-motion to illustrate it. I didn’t care which came first: the video or the equation, as long as they matched each other.  We used iPads and a stop-motion app, which takes pictures and puts them together to make a video for you.  However, you don’t need access to iPads because there are plenty of websites that can take your pictures and create a stop-motion video out of them, so all you’d need is one camera per group and access to a computer or computer lab for each group.

Last year that’s all I had them do: make a video and tell me what the associate equation is. This year what I want them to think about it a little more, so I’m going to have them create a Google power-point which includes various thing, such as a graph, table, equation of the situation, explanation of the variables, and a few challenging questions for their classmates.  Here’s my example Google Presentation that you can check out (yes, I used one of my student’s videos from last year as my example, but don’t worry: I’ll give them credit!).


One Thing That Makes My Classroom Distinctly Mine: Participation Points

Something I call Participation Points.  I am the only one at my school who uses this system, and I’ve yet to convince any other teachers at my school to use them, but I may have some converts in the MTBoS–we’ll see.

PPs (Participation Points) are something that students must keep track of and they must earn 100 each week. My premise:

  • Students should often be able to choose what level and type of work they complete to “earn” their grade.
  • When allowed to choose, students will typically choose work that challenges them “just enough”.
  • Homework should be option and seen as “practice if needed” (and usually it should be needed).
  • I should have the flexibility to create opportunities to learn on the spot, and have it count toward some part of their grade even if I do not have a standard for exactly what they want to do (I use SBG).

PPs are 30% of their grade, while their Standards make up the other 70%. Students hold onto a grid which documents how many points they earn each week for 9 weeks.  At the end of each class, students come up and tell me what kind of points they’ve earned.  On Friday I collect their sheets and put their grade out of 100 points into the computer, and return the sheet to them on Monday.  Here’s what their sheets look like:

Some of the things students can earn points for include:

  • Speaking up in class (making a positive contribution).
  • Reflecting about class through their blog.
  • Working on exercises on Khan Academy.
  • Working on the Warm-up on time.
  • Putting your answers to the HW on the board for others to check.
  • Answering a handful of thought-provoking questions I have on my website.
  • Coming to Tutoring (at lunch on Tuesdays and Thursdays).
  • Signing up to be a Tutor at those times.

There are many others: I have an incomplete (but more comprehensive) list on my website here.

The tutoring, especially, has helped students grow in the areas they need to grow.  At my school I am currently the only teacher to have students come regularly to tutoring, and there will be 30+ students in my classroom at lunch on those days working because they know it’s good for their grade in multiple ways.

I’ve blogged before about Participation Points, but they are something that I’ll probably do for as long as I teach because of all the little ways it helps my classes. I don’t do PPs in all my classes, but there is a big difference in the attitude of the students toward learning and work in those classes that I do use PPs. A small example of this is in the warm-up: I’ve got a facebook-style Like-Stamp that I use, which gives students 5 PPs, and I’ll stamp students’ papers who are working on the Bell-Ringer (warm-up) before the bell rings.  Students know this so they come quickly into the room and get to work right away, begging for me to come look at what they’ve done so they can earn that stamp.

Please ask if you have more questions about PPs, as I’m eager for another teacher to try them out and give feedback on them! They’re up there with SBG (Standards Based Grading) for “Most-Impact-On-My-Classroom” ideas.


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Entire Lesson Videotaped: Intro to Parametrics

On a whim, I decided to videotape an entire 45 minute lesson in one of my classes. The only thing I had on hand was an iPad (2nd Gen), so the video quality is low, but I’m also kinda banking on that so I don’t have to blur out students’ faces.

Other information possibly of importance:

  1. There are 15 students in this Precalculus class.
  2. This class is the last period of the day. [1]
  3. Yes, that is my principal who strolls in at [29:43], and yes he picks up a student’s guitar and starts playing in the last 10 minutes of class.   (Oh, and it was his birthday.) Don’t you wish you had a principal as cool as mine?

Outline of the Video

[0:00] to [3:46] Waiting for class to start. (I should have edited it out, but this is why the video starts at [3:46].

[3:46] to [5:40] Waiting for student let out of choir to get to my room (they’re late most of the time because our choir teachers let them out late).  Because I have all these students for Chemistry, we discuss a little Chemistry while waiting for everyone else to get here.

[5:40] to [8:36] I show students how they can find all the standards from the class on my website.  We also discuss Inverses of functions and they convince me to give the quiz on Tuesday instead of Monday of next week.

[8:36] to [12:30] 1st Act: I develop a reason for Parametric Equations and we do a really rough experiment of a student walking into the room.

[12:30] to [31:47] 2nd Act: Through questions I build an intuition for Parametric Equations through graphs and the measurement of our “experiment”.

[31:47] to [40:04] 3rd Act: We answer the questions I provided for them at the start of the lesson (although I didn’t vocalize them, and I didn’t have a “hook” for them like your typical good 3 Acts lesson. We look at two ways to use technology to graph this equation and I give them two of these types problems for homework (see the sheet below). I use the TI Calculator because they’ll need to know how to use that for most standardized tests. I use Desmos because it is awesome and easy to use (and way cooler than the TI Calculators).

[40:04] to [50:03] I promised them from the day that I would do the Fibonacci Magic trick.  If you haven’t seen it, here’s a video of it being done, long with an explanation of how to do it (however, I don’t like his explanation of how to multiply a 3-digit number by 11).  If you prefer to read, here’s an good explanation of how to do it.

Supplemental Materials

Since the board is hard to see in the video, here are two pictures of what I put on there.  For the most part, the black is what I had up before class started and green is what I added during class.

Picture of Classroom

Four Representations of a Parametric Equation

Here’s the worksheet I gave for homework.  Notice how I set up the board so all 4 parts match the worksheet.

Please leave feedback on my lesson and on my teaching style: both constructive and destructive comments are welcome, so please let me know what you think!


[1] The students recognize their tiredness at this point and frequently complain when I ask them to do a particularly mentally challenging problem or task.  I suppose I am fortunate that they recognize this, though I wish they had a bit more motivation and didn’t use it quite so often as an excuse.  Fortunately these are all good kids who try despite their tiredness, though they have vocalized to me that they wish this class was taught earlier in the day.

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What Does THAT Have to Do With Math?

The other day, the Spanish teacher at our school showed me this video:

So, being the wise-guy that I am, I decided to throw it into an ACT-style warm-up (see question #2).

Screenshot from 2013-09-12 14:19:31

The purpose of that question was that “some questions we don’t know the answer to, we can only eliminate guesses, and (on the ACT) we should always guess”. Shamelessly teaching how to take a test, but yeah, I do that every now and then.

Anyway, by the next day, only a few students seemed to have watched the video, so we watched it together. I pointed out the interesting trend of how many people have seen it: 14 million when I saw it in the morning, 15 million that afternoon, and 16 million (roughly) that evening. I asked them  for questions, followed by high and low guesses for how many views there’d be after the video has been out 1 month (yeah, delayed gratification–not the best, I know). And thus started an impromptu 3-Act lesson.

Fortunately Youtube not only gives the number of views, but it even has a few graphs over time for the number of watches of a video. This led to a short discussion of “What should the graph of views look like over time?” and we were able to rule out quite a few.

We stopped when we realized that, even if we had the points, we haven’t learned how to fit functions yet, so we decided to go learn that and put the question on hold.  We’ll see how well this lesson runs when it’s returned to like stale bread: hopefully it won’t be too long and hopefully it doesn’t go stale as fast as French bread (it tastes so good, but it’s like a rock 3 days later!).


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Improved Lesson: Review of Function Families

In Precalculus, it’s important to learn the shapes, equations, names, and domain & range of all those graphs that teachers spend weeks upon weeks on in Algebra I and II. See below for the specific names.

Here’s how the lesson used to go. I would hand them the worksheet and they I would write on the board the name and shape of the first one.  As I started to get “creative”, I’d ask them if they remembered the shape.


So this year I had a spur-of-the-moment-night-before-idea to spice it up.  The idea probably came from the 3 Acts lesson that just went so successfully not a few days before, but it’s equally likely that it came from being immersed in the MTBoS over the past year (well, “immersed” is a strong word for someone who mostly just reads others’ blogs and steals way more ideas than he gives).

Instead, I decided I would give the students a situation and have them come up with a graph of the situation.  Working in groups of 3 on individual large (2′ x 3′) whiteboards, they would sketch a graph and decide on the domain and the range.  I’d walk around and give suggestions or ask specific questions.  Then after 5-10 min of this (depending on the difficulty of the graph), we’d come together as a class, briefly share the graphs, and I would sketch a class graph which they’d have time to copy down into their notes.

After sketching the graph specific to the situation, we’d then talk as a class and decide on the generic fuction and its respective equation, name, and domain & range (domain & range are almost always different for situations than they are for generic function).

Example of situations we’d graph and talk about are: Coffee temperature over time; Distance from the start points over time of x-country runner running at a constant speed; Runner runs out to a point and then turns around and runs back: distance from that point over time, again at a constant speed; Basketball shot: height over horizontal distance traveled.  Yes, I do need more (& better) ideas for the rest of the equations, and I would welcome better ones for the equations I’ve already done so I can show the students other situations where these equations might arise.

We only got through 3 types of graphs in about 30-40 minutes, but I think it was much better spent time and the students have a better understanding of D & R, not to mention a specific situation where each graph applies.  Now, when I ask them to create a function out of thin air, I think they’ll do better this year than the unfortunate students last year.  Here’s a picture of our class’s work (my writing) on the board.


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