How I Did Trig Review in Adv. Precalculus this year

The Problem: Wasting Class Time Reviewing Trigonometry

Last year the Advanced Algebra II kids did a boatload of trigonometry, and this year I had to make sure my kids had a strong grasp of the basics (it’s been ages since they’d seen it) before we delved into trigonometry this year in Advanced Precalculus. In previous years, I always did a “trig review unit” which I always felt like wasted time. I like to use classtime to give kids things where they have to rely on each other — but during the review unit, kids didn’t need each other much. Different kids needed to review different things. I found ways around it, but overall, it felt like wasted time.

The Solution: Mix Review A Little Each Night During the Prior Unit (Which Posed Another Problem…)

So the other teacher and I decided that while we were working on sequences and series, we would also give kids some basic trig questions each night, maybe 10-15 minutes worth. Although I can’t see myself using this curriculum in my classroom the teach the material for the first time, I really love eMathInstruction’s packets. They are well-thought-out, and their problems highlight drawing connections among tables, graphs, and equations, and they often give forwards and backwards problems.

So each night I gave a selection of problems from one or two of these lessons — all topics they had worked on last year — and had kids do them. I chose the problems and lessons based on the specific things I needed kids to remember for what we were doing this year. And then the next day, I gave the answers and let kids resolve any difficulties.

But I wanted to be thoughtful about this. It was review, but I needed to make sure that kids really had these basics down before we jumped with both feet into the depths of trigonometry. And remember, all of this was happening during a unit on sequences and series. And I was afraid without some sort of feedback mechanism, I was going to finish this review and find that kids didn’t interalize any of it, or regain the fluency with trig basics that they had last year. So I worked with another teacher (who has been acting as my “teacher coach” this year) to circumvent this problem.

The Solution To The Problem My Solution Generated: Short Daily Feedback Quizzes

This is how it worked…

Let’s say on Friday students were asked to complete review problems from Lessons #1 and #2 from the review packet. Then on Monday, I give each group the answers, have them check their own work and talk with their group to resolve any difficulties (and if that doesn’t work, ask me!), and then the rest of the lesson is continuing on with sequences and series.

Then on Tuesday, we’d start class with a short 3-5 minute check-in super basic quiz on the trig review that was due on Monday and we had already gone over. It might look like this:

The back of the quiz would look like this (kids flip to the back when they’re done):

Then we have the rest of the class on Tuesday, consisting of going over the new trig review answers for a few minutes, and then working on sequences and series.

Tuesday night, I’ll mark up the quizzes. They are worth a whopping total of 1 assessment point (most of my assessments are 30-40 points). But here’s the catch: the score is either a 0 or a 1. To get the point, you need to get all parts correct. I’m okay with that because this is an advanced class and these questions are super basic. This is feedback for the student: do they really know the basic material, or do they merely think they know the basic material? [1]

On Wednesday, we’d start class with a short basic quiz on the review trig material we went over on Tuesday, kids would get their quizzes from Monday, and we’d go over the review trig material due today (before continuing on with sequences and series).

A note about timing… Most of our classes are 50 minutes. So about 4 minutes were spent taking the brief quiz, about 5-8 minutes were spent going over the trig review work and resolving any difficulties, and the remaining time was spent on the current unit of sequences and series.

At the very end of all the trig review, I had a mini-assessment on all the trig review material.

Framing The Quizzes

When presenting the daily quizzes to students, I expected a lot of groans. Thankfully I didn’t get any audible ones, which I attitribute to taking my time framing the plan for them. I wanted them to understand the thinking and impetus behind this approach to the review material. I wanted to be transparent.

First, I acknowledged that it was a long time ago (last year!) that they had worked on trig. So it would be unfair of me to expect them to know things like $\sin(225^o)$ or how to graph $y=-2\sin(0.5x)+3$ immediately. I wanted us to build up to it, slowly, so they had time to practice and get feedback. It was my job to make sure that before we resumed trig that they had refreshed themselves with the basics.

Second, I talked them through the idea behind the daily quizzes. I made sure that kids understood they would be short and only on material they had reviewed and had time to practice first. I highlighted that the quizzes served three purposes.

1. That they were low-stakes feedback for you on what you truly know and what you don’t know.
2. They will provide specific places for additional help if you find you don’t know something.
3. They were feedback for me on what y’all are good with and what you need work with — so I know to talk publicly about anything I’m noticing the whole class needs help with.

I did mention the grading, but I didn’t put much emphasis on that. The score wasn’t super important, except as feedback.

You might have noticed that the back of the quizzes gave specific places for students to get help with the concepts/ideas on the quiz. This was an idea my teacher coach and I generated together. The conversation we had was about feedback in general. We teachers can be good about giving feedback, but we never teach students explicitly how to use that feedback. What do they do with it? By providing specific resources/places for kids to go to get additional help (along with their teacher and classmates, of course), we thought this might highlight that we really do want these quizzes to be part of a feedback loop.

The Feedback

I took data on the quizzes. You’ll note that before each 1 or 0 are a few columns. Those are the concepts being asked. An “x” means that the student got that part incorrect. That data helped me look for trends, and what was more challenging for students, so I knew if I had to explicitly talk about any concept/idea in class.

I was planning on also using this feedback later. I was going to look at the assessment kids took at the end of the review, and see if there was any relationship between kids’ performance on the assessment and these feedback quizzes. I didn’t get a chance to do this, and truth be told, the average was so high for the review assessment (89%) I suspect it would have been a waste of time.

I also wanted to know how students felt about this process. This was an experiment for me, but to know if it succeeded, I needed feedback from students. I wanted to know (a) if they found the feedback quizzes were helpful, (b) if the feedback quizzes changed their practice in any way, and (c) if they used the feedback from the quizzes in any way. So my teacher coach and I wrote this short and pointed set of questions for them:

The results were interesting.

When asked if their preparation changed or not, it was interesting. Most kids said their preparation did change a bit, but even kids who said that it didn’t would then go on to say something that indicated that their preparation did change (highlighted in red)!

This is what kids did with the feedback (see list above from survey to see what each bar corresponds with):

And finally, here’s what kids wrote in the “anything else” box:

Overall, a success! Not only because kids found them useful on the whole, and because their practice changed because of them (for the better), but also because they did quite well on the trig review assessment (as I noted above, earning an 89% average).

More than figuring out how to deal with the annoying “how to review trig from the previous year before starting on trig in the following year” problem, this whole enterprise was an interesting excursion into feedback for me. I was hoping to find a way to create a feedback loop that was doable (and this was! it only took 10 minutes each day to mark up the simple quizzes) and created a change in student practice (which it did, because knowing there was something small students were accountable for each day changed how most kids prepared just a little bit).

To me this post and this experiment isn’t really about trig, but about now having another tool (daily feedback quizzes) in my teacher toolbelt to pull out at appropriate times.

[1] I debated whether I wanted to put a grade on these at all, or just let them be feedback with no score attached. I went back and forth about this for a long while. But ultimately, I knew that attaching a score, no matter how minimal, to the quizzes would effect more change than if I didn’t. But after introducing it, I didn’t mention the grade/score once when talking about them. I would mention common mistakes I noted and talked about ways to get extra practice with something or another. I kept my focus on the notion of feedback, and doing something with that feedback.

Bridge to Enter Advanced Mathematics (BEAM)

“Fundamentally, this is a question about power in society,” said Daniel Zaharopol, BEAM’s director. “Not just financial power, but who is respected, whose views are listened to, who is assumed to be what kind of person.”

***

My friend Dan said this to a New York Times reporter, in the context of an organization that he started called BEAM. It is a pathway for underserved middle school students to gain exposure, interest, and opportunities to see how amazing the world of mathematics can be. This is important. Why?

Guess how many math and statistics Ph.D.’s were awarded in 2015 to black students?

20.

***

So how do we change that? Dan and I were both at a lecture at Teachers College at Columbia yesterday given by Erica Walker. Her thesis? That you need mathematical socialization, spaces, and sponsoring (mentoring) to build positive, strong math communities.

Dan’s organization is doing that.

I highly recommend checking out the BEAM website to read about the ways it is trying to change the status quo.

But more than that, I recommend reading the article the New York Times reporter wrote about BEAM. She took an intimate, in-depth look at the program through the eyes of its participants — from riding the subway together to their discussion of the Black Lives Matter movement to their families. You may get welled up, as I did.

Getting familiar with the Unit Circle

In our standard precalculus class, we’ve spent 4 days “getting ready” for trigonometry. Which sounds crazy, until you see what awesome thing we’ve done. But I’ll blog about that later. Right now I want to share what I created to help kids start learning the unit circle.

Here were the hurdles:

1. We are introducing radians for the first time this year. So they’re super unfamiliar.
2. The unit circle feels overwhelming.
3. Although I am familiar with the special angles in degrees and radians, kids aren’t. So I know when I hear 210 degrees that’s “special” but kids don’t know that yet.

Here is what we have done:

1. Filled in a “blank unit circle” using knowledge of 30-60-90 degree triangles, 45-45-90 degree triangles, and reflections of 1st quadrant points to get the points in the other quadrants.

In this post, I’m going to do here is to share what I’m going to be doing to help kids learn the unit circle.

Phase I: Get confident with angles

I am going to talk about these like pizza. And to start, focus on radians.

I’m going to remind kids that $\pi$ radians is a half rotation about the circle. Then we can see that each pizza pie slice is $\frac{\pi}{2}$,$\frac{\pi}{4}$, $\frac{\pi}{3}$, and$\frac{\pi}{6}$ radians. [The “top” half of the pizza is divided into two, four, three, or six pieces! And the top half is $\pi$ radians!]

Then I’m going to work on the “easy-ish” angles by pointing at various places on the unit circle and have kids figure out the angle. I am going to have kids not only state the angle in radians, but also explain how they found it. For each angle, I will ask for a few different ways one could determine the angle measure. Then I’m going to repeat the same thing with the “easy-ish” angles, except I am going to do it in degrees.

And then… you guessed it… I’m going to do the same exercise but with the “harder-ish” angles. Start with radians. Then again with degrees. Always justifying/explaining their thinking.

Finally, I am going to let them practice for 5-8 minutes using this Geogebra applet I made. The goal here? To focus on getting kids familiar with the important special angles. Not only what the values are of these angles, but also to get them to start finding good ways to “see” where these angles are.

Phase II: Start Visualizing Side Lengths — utilizing short/long

Next comes getting kids to quickly figure out the coordinates of these special angles.

We’ve already been working on special right triangles, so I think this should be fine. And then…

Kids are asked to visualize the side lengths/coordinates based on the drawing. So, for example, for the first problem, kids will see that the angle is $\frac{4}{3}\pi$. They hopefully would have mastered that from the previous exercise. They also will see that if they would draw the reference triangle, the x-leg is shorter than the y-leg, so they know the x-coordinate must be $\frac{1}{2}$ (but negative), and the y-coordinate must be $\frac{\sqrt{3}}{2}$ (but negative).

After practicing with this for these four problems, kids are going to practice some more using this second geogebra applet I created.

Phase III: Putting It All Together

It’s now time to take the training wheels off. No longer do I give the picture to help visualize things. Now, I give the angle. This is more like what kids are going to be seeing. They need to know $\sin(315^o)$ and $\cos(3\pi/4)$. No one is going to be giving them nice pictures!

So this is what they’re tasked with:

I have a strong feeling that breaking down the unit circle in this way is going to make all the difference in the world. Fingers crossed!

If you want the file I created for my kids, here you go (.docx2017-02-xx Basic Trigonometry #2.docx:  , PDF: 2017-02-08-basic-trigonometry-2)!

Profound Impact

Here is my belated (and going to be short) Round 3 blogging initiative post — where we share our favorite post(s) from other bloggers.

To be truthful, this year I’ve been totally negligent on the #MTBoS front. I stopped checking in on twitter. I stopped checking in on Feedly. I stopped blogging regularly. And that’s okay because the #MTBoS is there for us when we need it, and it is there when we don’t. I make it a point to engage as long as it comes naturally, as I am compelled, and as I have time. This year I haven’t had much time. But I’m slowly realizing how much I need it — for personal reasons to stay invigorated — and so I’m making time. All of this prelude is to say that I haven’t been reading blogposts lately.

So I started thinking: what blogposts have had the biggest impact on my teaching?

I’ve had three major moments in my teaching that have pushed me forward as a teacher by leaps and bounds. The “small steps” mantra works, but you only get incremental improvement. There are three things I’ve done that have changed my teaching radically, for the better.

1. Deciding to put my desks into groups, instead of pairs. I just made the leap — recognizing that all the changes that I would need to make as a teacher would be things I would naturally have to do to adjust for having kids working together, instead of individually and in pairs. This was huge.
2. Writing my own curricula from scratch. Whole courses. Dropping the textbook. I did this for calculus, for advanced geometry, for advanced precalculus, and I’m doing this now for standard precalculus.
3. Doing a few years of Standards Based Grading in standard calculus.

Although I’m not teaching standard calculus anymore, so I’m not doing Standards Based Grading (SBG), I would have to credit the #MTBoS more than anyone else for taking the leap. As a result, I totally rethought the meaning of grades, and in turn, the meaning of what I do, and what I should prioritize as a teacher. It was a huge shift. And a huge amount of work.

Instead of waxing euphoric, looking back with rose tinted glasses, you can read all my posts about SBG here.

But I wanted to shoutout the blogpost which got this percolating in my brain. Dan Meyer. 2006. “How Math Must Assess.

I also wanted to link to Shawn Cornally’s blog, because it kept me thinking deeply about SBG and the larger questions around the philosophy, but it seems to have disappeared! One last shoutout goes to Matt Townsley (his old blog, with lots of posts on SBG, is here). I am grateful that there was such a buzz and an ongoing set of conversations about SBG in the #MTBoS as I was trying it out, because the devil is in the details, but so are the angels.

So there you go. Profoundly impactful blogposts that changed my teaching.

Girls and Math

Act I:

In one of our department meetings near the start of the year, we started talking about the representation of girls in our math club and our math team. In years past, there was a higher representation than this year. And although I suspect that the distribution of boys/girls in our math classes are probably relatively even — based on my own anecdotal evidence — I will readily admit that in all years past, there were fewer girls than boys in math club and on our math team.

We as a department brainstormed different possible reasons. One teacher (it may have been me? maybe not though) said we could just ask students. But we agreed that this is something we should be cognizant of. And we all agreed that by reaching out individually, we as teachers might be able to make a difference. So we all committed to doing so.

And so I did. I emailed the girls in one of my classes.

Hi [Stus],

I wanted to send y’all a personal invitation. Among y’all, I see a large amount of mathematical curiosity and intellectual firepower. I can say with complete honesty that each of you have different qualities that are important in doing mathematics, and you all have impressed me thus far. Some of these qualities include dedication and the ability to work through initial frustration, the ability to math intellectual leaps/discoveries/connections, and the ability to express abstract conceptual ideas in written form.
I strongly believe that mathematics is not and should not be seen as a “boys club.” Among the students who I look back over my teaching career and think “wow, they were original and deep thinkers,” the majority of them are girls. Which is why I was surprised to see how few girls are involved in our math club and our math team this year. I want to encourage and nurture mathematical talent in girls in your generation so that future generations can see more women role models. And for me, that starts with me reaching out to you.
I wanted to give you a special and personal push/nudge in case you were interested in joining either math club or math team to talk with me so I can tell you more about these activities. (And if you’re not interested — which is totally fine, this email comes with no pressure– I’d also love to hear why you might not be interested in joining them. That would help me understand things better so I can think more broadly about things.)
Always,
Mr. Shah
It didn’t really work. The two replies I got were from students who were interested. But their schedules seemed to preclude their participation.
Act II:
I was alerted to an essay contest run by the Association for Women in Mathematics that I thought I could entice some students in my class to participate in. I threw this opportunity in Google Classroom, but … nothing. (As I write this post, it inspired me to re-post the opportunity since the deadline has not yet passed!)

Hi all,

The Association for Women in Mathematics (AWM) has an essay contest. You can interview a woman who is a mathematician or in a mathematical sciences career, and write a 500 to 1000 word essay based on that. And if you need help, AWM will even help you find someone to interview! More information is provided at the link below (along with some winning essays from previous years). From my reading of this website, this opportunity is open to everyone — not just women. If you are at all interested in hearing what a mathematician does (or what higher level math actually is!), or how gender plays a role in a mathematical career, this could be an amazing opportunity for you to find out.

Later in this year, you will be doing a set of “mini math explorations” based on your interests. They are very open-ended. If you do end up doing this essay, which would be so awesome, it would count as two of these mini math explorations!

Always,
Mr. Shah

Act III:
Over winter break, I saw the film Hidden Figures, about black women mathematicians who helped put a man into space. I had quibbles with this and that about the movie, but my final judgment: I want all my kids to see this.
And then this week, I had an idea. I posted this on our google classroom site:

In case you haven’t heard about the movie _Hidden Figures_, I wanted to make you aware of it. I saw it over winter break and wanted to recommend it to all y’all! Check the trailer out:

I was thinking about reading the book it’s based on [https://www.amazon.com/Hidden-Figures-American-Untold-Mathematicians/dp/006236359X] sometime in the 2nd semester. If there are three or more of you that want to read it at the same time and have an informal book club around it, with me, I’m totally down. Just let me know who you are and we can make a plan!

Always,
Mr. Shah

And… I got a bite.Two actually. One student emailed me saying she was interested in joining the book club. And she told me that another student in our class was also interested! I asked ’em if they knew of any others — in our class or outside of our class — who might be interested. They got back to me with some names, and they agreed to reach out to them.

So it looks like we’re going to be having a book club around a book that talks about gender, race, and mathematics! I don’t know if it will be large or small. But I’m psyched that it will happen. This story was going around twitter today, and it made me emotional. Because I saw the relevance between this post which I have been working on, and this story.

Act IV:
One student in my class received a book called Math Girls as a present from another student in my class. Yes, my heart skipped a beat. Because how deliciously geeky was this gift exchange?! I have a project called Explore Math which allows kids to investigate something they are interested in that is related to math. The student who received the book wanted to meet with me to see if reading this book could be her exploration for the project. That particular book would have been too much to bite off for the mini-exploration, but I gave her a different but similar book (by the same author) to read for the exploration, and told her that when she had the time, I would help her work through Math Girls.
Epilogue:
Also making the rounds twitter tonight was a keynote address from the president of the Mathematical Association of America. He gave this talk two days ago. The purpose of the talk: Why do mathematics? And as I read, and continued to read, I welled up with emotion. There’s a lot there to unpack. He has a call to action for college professors.

Find one student and be their advocate!  Be the one who says “I see you, and I think you have a future in math.”  Be the one who searches out opportunities for them.  Be the one who pushes them towards virtue.  Be the one who calls them up when they’ve skipped class, and asks “is everything okay? what are you going through?”

I know what I’m asking you to do is hard and takes time.

But we’re mathematicians… we know how to tackle hard problems.  We have the perseverance to see it through.  We have the humility to admit when we mistakes, and learn from them.  We have hopefulness that our labor is never in vain and that our work will bear fruit in the flourishing of our students.

Because what I am asking you to do is something you already know, at the heart of the teacher-student relationship, pushes us towards virtue.  I’m asking you to love.

But this call to action applies to us math teachers too, not just college professors. Except we don’t get to find one student. We are given many students. And being all of their advocates is harder, and takes time. And is Herculean. Perhaps Sisyphean. But people like Fawn and Rebecka and Annie and Sara remind us that gaining a deep student-teacher relationship with our kids–  having our connection go beyond math and to a position of mentor and trusted ally — is possible. Someone our kids can look up to, as a human being, not just as a font of knowledge. I have a suspicion that figuring this out separates us mere mortals from the master teachers we look up to.

Sequences and Series

A few days ago I posted a card sort I did to start my unit on Sequences. I figured I’d share my entire packet for Sequences (2016-10-31-sequences [doc] 2016-10-31-sequences [pdf]) in case it’s any help for you. This was designed for our standard precalculus classes, and I have to say it worked pretty well.

1. It allows for kids playing with math at the start (card sort, some visual patterns, a 3-act)
2. It doesn’t “tell” kids anything. They discover everything. And sometimes asks for a couple different ways to do things.
3. Kids messing up notation is always an issue with sequences. So I introduced notation way after kids started playing around with sequences — and got a handle on them. I did still see some kids confuse $n$ and $a_n$ (the term number and the value of the term), but it wasn’t a huge number. I actually was pretty proud of that.

Note: Problem #17 on page 23 is actually ill-posed. So I didn’t have my kids work on it. I have a replacement warm-up question (included at the bottom of this post) which worked wonders!

Also in the packet, there is a blank page (on page 4) which simply says “A Pixel Puzzle.” For that, I used Dan Meyer’s 3-Act.

For me, the 3-Act was a mixed bag. Mainly because I thought it would be easier than it ended up being for my kids, so I didn’t plan much about how to help groups get started. At the beginning of the 3-Act, kids asked great questions, but not the one I was hoping for (when does the pixel hit the border). So I didn’t have a smooth way to deal with that. And I anticipated it would take less time than it actually did, so the 3-Act got split up between multiple classes. And when kids were working, I don’t think I was strong at facilitating them working. I should have added in a wee bit more structure to help kids out. Even if it’s something as simple as coming up with ways to get kids to think about making a table or a graph — and what would be useful/important to include in the table/graph. And I didn’t have a good way for kids to share their findings. So I’d need to think about the close to the 3-Act better. I’d give myself a “C.” Would I do it again? Yes. I saw the potential in it. It got kids thinking about sequences visually. It had kids thinking temporally. And had them relate tables/graphs/equations. All good things!

One last thing I want to include in this post… This is related to the ill-posed Problem #17 on page 23 I made note of before. There is one type of question which tends to flummox kids when it comes to geometric sequences. It reads something like “The third term in a geometric sequence is 6 and the seventh term is 96. Come up with a formula for the $n$th term in this sequence.”

The reason this is tricky is because there are two possible sequences! (The common ratio could be 2 or -2.)

Thus, I created this warm-up (.docx) for my kids to check if they truly understood what we were doing. The conversations were incredible. Some groups were done in 7 minutes… Others had a solid 15 minutes of discussion.

2, 4, 6, 8, what do we appreciate? A Card Sort!

This year I’m teaching both Advanced Precalculus and Standard Precalculus. (Totally confusing on a daily basis? Yup.) And I’m working with two other teachers to write the Standard Precalculus curriculum from scratch. Of course this is something that is daunting, but I love to do when I have the time and like-minded colleagues.

I was in charge of spearheading our sequences and series unit. In this post, I want to briefly share how we started the unit. Instead of diving right in, or doing something intense, I wanted to gently get some good conversations percolating. So I handed out this set of cards:

and gave them this set of instructions:

I debated having kids use Desmos for the card sort, but since I have kids work in groups (mostly groups of 3), and I wanted the entire group to be working together, and I wanted them to actually physically move and shuffle cards, I decided to use physical cards. I also had all kids stand up while doing the card sort. I had a feeling that would be magical, in terms of getting kids talking, moving, and engaging with each other (even thought they were all at the same table), and it was! So I highly recommend that.

These cards end up having three different types: arithmetic, geometric, and recursive.

Most kids got the arithmetic sequences quickly, but it was interesting to watch them struggle with the geometric and recursive. There were great conversations, and because I demanded the next number for each of the sequences, kids had to really think through what the pattern was (and in geometric sequences, how to find the common ratio). I had thought that kids would finish this really quickly, but I was totally wrong. It took about 20 minutes. So plan accordingly. (A few groups needed to do a little bit more at the end of class, so I had them take a photo of their card sort and use that photo to finish things up!)

One note: Card H which has the sequence 0,0,0,0,0,0 fits all three categories. So it’s great fun to watch kids try to place it.

I wanted to share this activity because I haven’t really done many card sorts before —  and I was so pleased that this particular one generated productive conversations. So I need to keep this teaching tool in my arsenal for generating conversations about something new. (Example: I just thought of giving a bunch of graphs of rational functions to kids on cards, before we start that unit, and say “find different ways to sort these!” There are so many features, so that could lead to so many different ways to sort the cards. I suspect that Desmos would be good for that particular card sort, since there would be many different ways to sort those pictures, and I’d want to project the different ways kids did it to the entire class. I bet through that sort, we could actually recognize vertical asymptotes, horizontal asymptotes, oblique asymptotes, and holes!)

Here is the .docx (2016-10-31-card-sort-for-sequences) and .pdf (2016-10-31-card-sort-for-sequences) for the cards.

I will try to write up some more about my sequences and series unit soon!