Tag Archives: Precalculus

[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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[2016 Blogging Initiative] Week Three: Understanding Questions


After making the transition to Standards Based Grading a few years ago (which is awesome!), over time I realized that my class had become too skill-oriented. To fix this I first tried to create standards that were “understanding standards”, but this overwhelmed my students with too many grades.

It took me an entire semester, but I realized that what I should be doing is asking “understanding questions” on assessments instead of only skill-oriented questions.

I try to limit my assessments to three questions (sometimes a question might have multiple parts), but I now always try to include an “understanding question” as one of those three questions. I never grew up answering these on math assessments, and they’re harder to grade because there’s usually not “one right answer”, but it has helped me get a better grasp of what my students understand (or don’t).

What do these look like? Here are some examples.


Exponents: Explain why b^x \cdot b^y = b^{x+y} is true.[1]

Polynomials: What does multiplying polynomials have to do with the distributive property?

Polynomials: Why can you combine some terms of a polynomial but not others? (3x^2 + 4x^2 can be added but 3x^2 + 4x^3 cannot)

Rational Expressions: Before factoring was the opposite of simplifying. What has changed and why do we factor first to simplify rational expressions?

Functions: Give an example of a function and a non-function outside of math class.

Transformations: Why does (x+3) move a graph left and (x-3) move a graph right? Isn’t that that the opposite of what you would expect?

Logarithms: Explain why \log_b{M} + \log_b{N} = \log_b{(M \cdot N)} is true.[1]

Reflection/My Own Questions

(1) Are these “understanding questions” enough to check for understanding? Probably not by themselves. So I need to get better at assessing repeatedly over time to check for retention of understanding.

(2) Should I give students the questions beforehand? Right now I do because if they want to figure out the answers on their own, great! As long as I have enough possible questions so they’re not simply memorizing and spitting back what I say, but really understanding it. (Or should they be able to get these questions even without me providing them ahead of time?)

(3) Is there a place to get these types of questions? I primarily look in the textbook or come up with my own questions, but surely there’s a bank of these somewhere online that I haven’t found yet.


These questions are ones that get at understanding, though harder to grade (at least they take longer), are worth it. When I started these questions, I was sorely disappointed how little my Precalculus students understood (even though they have seen some things, like exponents, in Algebra II and probably even in Algebra I!). I’m really curious what people think about the three reflection questions above.

Thanks for reading!


[1] I’ve flip-flopped between using the vocabulary “prove” and “explain”. The former suggests there’s one right answer to students whereas the latter allows for various explanations. “Explain” also is harder to grade, but I’m very excited if I see students start to write out examples in their explanations. No, it’s not as rigorous as professional mathematicians, but it shows me that they’re starting to understand.


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[2016 Blogging Initiative] Week Two: My Favorite = Desmos


Desmos. If you’re a math teacher and you don’t know about it, stop reading this and go to their website now.

It took me a long time to even think of Desmos as one of “my favorites” because it’s so integral and every-day in my math classes that I guess I’ve started taking it for granted. I forget the days when TI button mashing ruled the day.

Here are a few things I appreciate about Desmos.

Browser Based & Free

The fact that students can access these from any computer with internet, or install the app and have them without internet access is incredible.

I have a class set of iPads, but increasingly students are asking to use their own phones. I point out to them that their eyes are going to go bad, but many of them have become proficient at using the tiny screens.

Sharing Graphs

I am increasingly using the “share this graph” feature. A student will create an awesome graph and instead of just presenting with the projector, all the students in the class are interacting with their own version of the graph on their device. Or I’ll create a graph (usually a table that I don’t want to waste class time having students type in the data points) and share it quickly with the whole class. It’s great!


Nothing beats building intuition with function transformations like having students move the sliders to manipulate the values. Before Desmos, I used Geogebra, but I spent a lot of class time showing students how to make sliders–it was not nearly as streamlined and intuitive as in Desmos.

Activities & Activity Builder

I was a huge fan of Function Carnival, Penny Circle, and Central Park when they first came out. I even had my students do these, even though we weren’t exactly on those topics when each activity came out (it was review, okay?). Now Desmos has Polygraph[1] and, the latest that I’ve yet to try with my class, Marbleslides, each excellent activities.

But I think the best thing that Desmos has done in this area is the Activity Builder. I haven’t had enough time to dig in and create activities, but the possibilities are endless. And no need to reinvent the wheel–activities that other teachers have created are available for you to see, too! Holy smokes!

To access all these cool things, go to teacher.desmos.com. I haven’t even started talking about the awesome teacher-view for when all your students are working on these activities.

And So Much More

It would take me way too long to mention all the incredible things that Desmos can do and is doing[2]. And they’re constantly improving things. They respond quickly to feedback, both in communication and by doing the thing you asked for within Desmos.


Thank you, Desmos!


[1] There’s a huge bank of polygraphs since people can make their own!

[2] To list a few: recognizing function notation, derivatives, inserting images,intuitive click-on-the-point to find the intersection or x & y-intercept, domain & range restrictions, lists, click & drag points, regressions for any equation, implicit function, colors!, labeling axes, easy animation, and converting equations to tables. I’m sure I forgot ~90% of the features that I like.


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Flip the Class?

I’ve heard pro’s and con’s of “flipping” a classroom (basic idea: having students watch videos of lectures at home and doing “homework” or practice problems at school instead). Here are a few that come to mind:


  • Low student HW completion rate, but students are more likely to watch a video than do actual work.
  • Students need help when they’re doing the work, but if it’s homework that’s when you (the expert) are not with them.
  • More time in class to do practice or other things, rather than students sitting there listening to you lecture.


  • For teachers that do Project Based Learning or some other sort of exploratory work where students are working with peers, not much lecturing is happening anyways.
  • Lectures with feedback (Plickers, anyone?) is the best way to lecture, but students don’t get that sort of feedback on videos.
  • Students don’t learn much (or worse!) with lectures OR math videos (great explanation here!) so it’s bad pedagogy at school or at home.
  • Roughly 20% of my students don’t have internet at home, and that’s a large enough chunk to cause headaches when requiring students to watch videos online.

After hearing another good speaker at a conference on flipping the classroom, I decided to give it another shot, but with some significant differences.

Students watch the video while filling out “Take Home Notes”

Here’s my first attempt at this. The idea is that if students are filling out notes, they’re more likely to pay significant attention to the video.

I periodically ask students to pause the video and attempt problems on their own.

Similar to the previous point, if students are doing the problems on their own, they can then check their answers against what I have in the video. I then ask them to follow that up and complete a few more problems before coming to class. This makes them pay attention to the video in a way that they otherwise probably wouldn’t, and provides good feedback during the video.

Students begin class by checking problems from the “Take Home Notes” with others in their group.

I’m still undecided whether to split students into “filled out the notes” and “didn’t fill out the notes” groups when they get to class. Part of me wants to do random grouping, but part of me wants to reward students for doing work at home. That time at the start of class (sharing notes) could become “the few who do the work always filling in the majority who didn’t”, which would be bad for many reasons.

This year we have mandatory Study Hall, so any student has, twice a week, built-in time to watch videos.

This takes care of the “I don’t have internet at home” problem. As I’m the only flipped class in the school, if they don’t have internet, they’ll have to prioritize watching my video over whatever else they wanted to work on. If they don’t, their parents will hear about it.

We still do 3 Act Lessons and Projects in class.

The things I find important–problem solving, communication of math ideas, applying math to real-world situations, building ideas and math concepts before introducing them via lecture–can all still happen. I just need to find a balance between “practice days” and “3 act lesson days”.

Now the only “con” is making the guided notes and the videos. When will I find time for that?!?

Here’s my first video. Please tell me what you think!

Some other Notes

I was worried about how long the video would be when I finished–I didn’t want long videos (longer than 15 minutes) to deter students from watching. When I finished my first video above, as you can see, it was 8 minutes and 45 seconds. Condensing all I do in a 45 minute period [1] into 8 minutes is ridiculous. Of course, students should be pausing the video to work out practice problems, so the actual time to watch the video should be between 15 and 30 minutes. But it made me realize how slow I lecture because I want to make sure that a majority of students follow what I’m saying. The ability for students pause and replay parts of the video help tremendously. The other incredible thing is ideally [2] that I now have a full class to devote to the questions that slowed down the lectures before. But since students will be working in small groups, the entire class doesn’t have to wait for the questions of the few.

I am using Doceri–something you should definitely check out if you have an iPad, regardless of how you teach. Before flipping, I used Doceri primarily for projecting notes to the front of the room from wherever I stood in the classroom.  In my opinion it is vastly superior to the technology that Khan uses for his videos. It took me about 1 hour to make the “Take Home Notes” (Google Drive), and almost another hour to write out all the work on Doceri, during which time I wasn’t recording, simply marking stopping points for the writing. Then I hit play and recorded my voice while pressing “play” at each stop. I thought I’d have to constantly pause and hit play again to recollect my thoughts, but I was able to dictate 8 min and 45 seconds straight through the entire lesson because of all the setup I did.

One more advantage of doing it this way is that I put way more thought into what I wanted to say and how I wanted to say it than I would in a normal lesson. I have a bad habit of “winging” lessons more than I should, mostly because I (in my own mind) want to retain the flexibility of responding to student questions. [3] However, when I have no students in front of me, it forces me to choose my words carefully ahead of time. During class I’ll often forget to do something, such as show students how they can check their work. I did a little better on the video with that, though hopefully it’ll continue to improve as I do more and more of these.

Thanks for reading all of this and let me know if you’ve tried something like this and/or have thoughts on it!


[1] That’s assuming I get through everything, which doesn’t always happen! I would probably actually spend ~2 days on this.

[2] This all works great in my head. I’ll try to remember to blog when I’ve actually done it.

[3] Yes, I know that the more prepared a teacher is, usually the more ready they are in responding to student questions, and therefore the more flexible they are. That’s why I put “in my own mind”. Ha.





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Function Transformations

I tried something new this year when teaching function transformations: I had students come up with the “rules” in groups with little help from me.

Day 1

Students created a DesMan (person using Desmos) and built up an intuition for how functions move[1]. We spent a block day (1.5 hours) doing this. Students were free to share what they discovered with one another and that helped tremendously. Here are some of their creations:

And here’s one that has moving parts:


Day 2

No students finished the DesMan on the first day, but I told them that they could get credit for “Modeling with Math” if they finish it.

We split into random groups of three and I had them move to their vertical whiteboards with these instructions:

  • Split your whiteboard into 4 parts.
  • On each part you should explain to a Freshman (these are Juniors and Seniors in Precalculus) how to transform a function in these four ways: Translate up (they suggested the word “translate”, I was going to use “shift”), translate down, translate right and translate left.
  • Please include a graph, use function notation, and an explanation using the ideas “input” and “output”.
  • You should use Desmos to confirm your explanations with some examples.

Here are some of the whiteboards they created.

IMG_20151002_150615906 IMG_20151002_150229253 IMG_20151002_150601495

As groups finished[2], I approached them and made sure everyone in the group understood each idea, especially the input/output language.  I asked them to do the same thing on a second whiteboard, but this time with stretching vertically, shrinking vertically, stretching horizontally, and shrinking horizontally. Here are some of those whiteboards:

IMG_20151002_154405792 IMG_20151002_152324286 IMG_20151002_151618290

Even though their whiteboards are great, that wasn’t the best part of the activity. The best part was the discussions that I overheard. Every student was involved because every student (a) had some experience with all of this through their work on the DesMan and (b) felt comfortable asking “why?”. I felt as though we had reached a classroom culture of safe inquiry and curiosity, where “why does that work?” questions are empowering rather than embarrassing.

It also helped tremendously having Desmos as an outlet

Some groups finished this part before the end of class, so I had them examine vertical and horizontal reflections. Here’s their work:


Or maybe I drew those pictures in red, I forget.

Day 3

We did a short game of “guess the type of function transformation from the graph”. It helps having 2 screens: one that I only I can see, which I use to edit Desmos and the other that the class sees, when the function is hidden.[3]


In three short days, I feel confident that students will be able to use these transformations the rest of the year[4]. It’s important to note that I’m not introducing this to students the first time (it is Precalculus and they should have seen this at least once in Alg II), though they often act as though they haven’t seen it before.


[1] I didn’t use this Desmos page, but it’s another way to do this: https://teacher.desmos.com/desman. I just had them graph a parabola, restrict the domain {-3<x<3}, then move it and they were off!

[2] This is where the vertical whiteboards was tremendous: I could catch groups right as they got to finishing their work.

[3] I only have this setup in one of my classrooms. In the other I have to use the “freeze” button on the projector to work out the next graph.

[4] And they will use them them since I like 3-act lessons where students have to figure out what the function is. It’s important that they practice this throughout the year.  Maybe one day I’ll show them how to approximate data using the tilde (~) in Desmos.

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“What’s a Good Question?” Activity

Today I did Alex Overwijk’s “What’s a Good Question?” activity, down to using the very same image.

Here’s the Google Presentation I made to go with the activity:

(Link to the document if the above embedding doesn’t work)

One think I did to add to Alex’s activity was at the beginning, when my students (who are seniors) were having trouble writing down more than a few questions (even though I asked for any and all questions!). I decided to talk about the paper clip challenge involving divergent thinking. They were inspired when they heard that kindergartners do the best on the “test”.

After that short 5-ish min test (I had them do the challenge and then just talked about it, but next time I might show the video!), students went back to the question-writing with more vigor, and the rest of the lesson went very well.

Here are some questions:



How much glycogen does it need to survive?

How much glycogen does it need to survive?

Best question (my opinion) and the one we actually tackled:

How many ppl would it take to hug the entire trunk?

How many ppl would it take to hug the entire trunk?

Here are some of the answers to “What makes a good question?” snowball:






And then we had time for the “tree hugging” question!





There were 15 people in the class at the time (of course nearly half the class is dismissed on a field trip when I go to do this critical lesson!) so we were able to test out and see if our estimations (ranging from 13 to 16 people) was reasonable.

One Big Tree!

One Big Tree!

They agreed that it looked pretty reasonable!

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Understanding Radicals

Vertical non-permanent surfaces (aka whiteboards hanging from command strips on the wall) gives every activity a “new look”. This activity, done during the early “let’s review our algebra” period of Precalculus gave me a new appreciation for how little students understand the procedures we teach them through the years.

Here’s the website that gave me the idea:

Nrich’s Nested Surds

Here’s the Google presentation I made from it:

(Link to the document in case the above embedding doesn’t work.)

I did visible random grouping (I’ve started to use this website because how straightforward it is. Make sure you select the “new window” output format.) in groups of 3. With 21-22 students, I had 7 or 8 groups. I showed students the first slide and had them “test out” numbers. Once they did a handful of perfect squares, I let them try it out on their calculators, for non-perfect squares.

\sqrt{a} \times \sqrt{b} = \sqrt{ab}


Here’s what shocked me: these precalculus students worked on this first problem for about 15 minutes[1]. Nobody had gotten even remotely close (okay, one group came up with a rule “if a is even and b is even, it works!”). I was within a few minutes of pulling the entire class in and having them sit down while I explain it all (as if they’d learn & understand any better, but I was panicking!). Then, within a minute or so, every group started “getting it”. It helped that they could look around (since all the whiteboards are vertical), but that didn’t help as much as I thought it would. Students must still have some inhibition along the lines of “seeing other groups’ work is cheating.”

The second problem (\frac{\sqrt{a}}{\sqrt{b}}=\sqrt{frac{a}{b}} ) went a little faster as did the third (a\sqrt{b}=\sqrt{ab} ). I especially liked the third because it is a common mistake, so this activity gives us some ground for discussion when that type of mistake occurs in the future. Unfortunately that was as far as we got before we ran out of time.

It surprised me how difficult this activity was for these students, who should have seen radicals 2 or 3 times before in their math classes. The Alg II teacher commented that perhaps it was the open-endedness of the activity that tripped them up, but to me that simply reveals a weak understanding of the underlying math. Regardless of how open-ended an activity is, students should know and be able to apply how radicals work.

Using the vertical whiteboards, I appreciated (A) how quickly students got to work, (B) how long they worked, and (C) how I was able to quickly assess where groups were and get them back on task. Unfortunately this particular group seemed especially prone to getting off-task, distracting other groups, having conversations that were not related to math. I’m sure that standing contributed a little to this, if only because they aren’t used to standing during most classes. However, we had a good “chat” and I’m happy to report more on-task-ness more recently.

Thanks to John Golden for sharing about the Nrich site and thanks to Alex Overwijk for sharing about vertical whiteboards & random grouping!


[1] Actually, I have no idea how long it took–it might have been 30+ minutes. I just know it felt like a long time and it was at least 15 minutes.

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