6 Reasons This is My Favorite Lesson

I want to share what might be the best lesson I’ve created and a few reasons why.

I actually wrote about this a couple years ago but since we’re doing it right now I thought it useful to reflect on and share it again.

This lesson came from the following problem I was struggling with:

I had spent a lot of time thinking about how to help students understand the connection between trig ratios on the unit circle and the graphs of trig functions on the Cartesian plane. Despite a couple activities and practice I was convinced, mainly through questioning, that they didn’t fully understand it.

My solution to this was to make a giant unit circle and cartesian plane and have students use them to work out problems. This would allow us to literally walk to specific angles and equivalent places on the cartesian plane. The hope was that this would help students solidify the connection between the two.

The details of how the activity works are in the original post and the materials are linked at the end of this post, so I want to emphasize the aspects of the lesson that really make it effective, in a convenient list.


The activity is broken into two parts. The practice portion and the assessment portion. The assessment requires students, working in pairs, to come into the hallway and work through five problems (like these). This portion is vital for the following reasons.

Like any assessment, it helps me know what they know.

It makes students take the practice seriously. The assessment mirrors what the practice rounds were like. They take it seriously and practice until they’re confident.

Students work in pairs, sometimes disagree, and then must convince each other of their reasoning. Tremendous mathematical conversations come from this time.

It puts students in a position in which the teacher is there, but can’t help. This is true of assessments in general, but the format of this one means students must convince themselves and each other that their answer is “their final answer”.

No calculator. No notes.

On the assessment, and likewise on the practice, students cannot use a calculator, notes, their unit circle, or anything besides their brains and a whiteboard. This means students don’t have any crutches with which to rely on. These problems are not algorithmic. Each one is slightly different from the other ones. This means that the only way to be successful is to truly understand what is going on in the math.


This is my third year doing this activity and every year there’s nearly full engagement. Now, this is precalc and while I wouldn’t say that all of these students want to be there, it is an elective. But it’s difficult for me to get this level of engagement from them.

This is, in part, because they know there’s a test coming after they’ve practiced. But I think it’s also because each problem sparks at least a little bit of curiosity. “How do we figure this out?” Initially many students don’t have a clue about how to approach something like sec(2pi/3) with only their brains and a whiteboard. But with a good understanding of trig they can figure it out.

And figuring it out is satisfying. Students are proud of themselves when they solve one of these problems correctly. I love seeing high fives in my classes, and this is one of those activities where they happen.

Embodied Cognition

I’ve written about embodied cognition before so I won’t go into too much detail, except to say that it’s incredibly valuable if you can incorporate it effectively. There is something fundamentally different from paper and pencil when you can stand there with a student inside of a unit circle and discuss these problems. It’s something that is hard to describe, but once you’ve tried it you clearly see the value.

Purposeful practice without a book assignment

A few weeks ago students initially learned how to do these problems via a lesson and practice problems. If that was effective, then I wouldn’t have needed to do this activity. What ends up happening in this activity is that students end up doing a bunch of practice problems, that I never assigned! I just tell them they can do as many practice rounds as they feel they need. Then they work until they have convinced themselves they’ve mastered it.


The test and practice require students to work in pairs. This is incredibly valuable as students are constantly conversing and helping each other understand. Once again, the knowledge that there’s an assessment plays into this, but who cares? From my observations students are rarely begrudgingly woking through these problems. They seem to enjoy them.

I probably see more learning and teaching happening between the students in this activity than any other lesson I do, for any class.

I understand that without seeing it happen it might be difficult for you to implement this. I’ve included some images below to give you an idea of the set up. Feel free to contact me with any questions you have. I’d encourage you to look for opportunities to use embodied cognition in your classes as I think it can be an incredibly useful teaching tool.

Here are the resources for doing the activity

Description Sheet

Possible Problem Bank

Practice Cards

Assessment Cards (Yeah, I’m not posting these on the web. I, shockingly, sometimes have students read my blog. But if you reach out to me I would be happy to email them to you and save you the time of making them.)

Assessment Rubric

X-axis “Tick Marks”


Real Questions

Every year in precalc, two things happen. First, the exponential and logarithm unit gets squished (education jargon, I know). Second, we do the “student loan” blog post, which Steve came up with when we were first working on precalculus together. The former drives me crazy because I think that unit is more useful than some of the other things we spend time on. The latter I look forward to because it’s such a great learning experience for students.

The problem goes something like this:

Suppose you take out a $5000 student loan every year for four years. How long would it take you to pay back the money? Please make sure to include research on government subsidized and unsubsidized loans and private bank loans.

Yeah, it’s vague. There’s no rubric, besides the standard blog rubric. The openness of the problem drives students a little crazy, which I’ve decided isn’t a bad thing. Let me explain.

The students in this class are, most likely, college bound. I don’t teach in a particularly affluent part of the state so most of my students are going to have to take out loans to pay for college. This means that the question, how long will it take to pay back loans, is painfully relevant.

Here are my main objectives for this assignment:

  • I want students to understand the different types of loans and some of the verbiage they’ll encounter, if only at a surface level.
  • I want students to understand the connection between exponential functions loans.
  • I want students to understand how important interest rates are in total cost of a loan over time.
  • I want students to realize that in this fairly common scenario, it’s not unlikely that they’ll be paying back loans for 20ish years.

I should also note that I don’t make students run these calculations by hand. I encourage them to use loan repayment calculators. I want them to understand that there is mathematics at play here, but I don’t want them to get too hung up in computations.

For many students this is an eye opening project. I’ve had students say they plan on paying it back in five years and they want to be a teacher. I had to gently explain to them that it was unlikely they’d be able to afford that. Steve has had students in tears doing this project. It has many great opportunities for students to learn about life, and also mathematics.


I wrote the beginning of this post when students were working on the project. I’m writing this part after grading them. Here’s a couple things I’m going to connect year to make it go better.

  • Clear up the guidelines. There’s explanations in the video and a synopsis on the website. They’re both slightly different. I also plan on making them clearer.
  • Listing things for students that are wrong and that students have done in the past. Things like only running one scenario, only computing a loan for $5000, and not providing explanations for their numbers.
  • I will include a question asking them to explain what they learned or gained from doing the projects.
  • I will look up some useful repayment calculators as suggestions. I noticed that this can be a bit help or hinder ace depending on where google leads them.

Other than that, this activity will be used next year.

I’ve underestimated the importance of vocabulary

"Words" by Shelly on Flickr

I thought for a long time that I could get by teaching math while deemphasizing vocabulary. Obviously we would discuss the meaning of words, especially the ones that come up frequently. But I thought that if I was able to help students get a feel for the math, and show kids how to do math, without getting too caught up in what the new vocabulary meant, that would be success.

Part of this was time. Or rather to save time. Spending time helping students really understand vocabulary takes more time, especially if it’s something that is more easily shown/practiced. For example, I feel like one of my struggles with helping students understand domain and range is that I don’t do a good job at really helping them understand the words. In algebra II, if I present a new type of function to them and ask them to find the domain and range, they often struggle until they see a few examples. It’s as if they’re simply replicating the process for each type of function.

At risk of this turning into a domain and range post, let me explain a bit further. When we study quadratic functions I tell students the domain is always “all real numbers”. The student thinks, “Sweet. Whenever I see a question over domain on the quiz, I’ll just write ‘all real numbers’.” When we learn a new family of functions they have no understanding of how to find the domain, beyond “that’s something with the x values, right?”.

It’s not just that topic. In fact, the concept that propelled me to write on this topic was grading a quiz over factoring polynomials and finding zeros in polynomials. Way too many of my students don’t know the difference between factors and zeros and constantly get them confused. My most significant observation was that I find students are trying to get by with the least amount of vocabulary understanding, and I don’t think I’m helping things by demphasizing it.

Since I’m having this realization at this point in the school year, I think the fix going forward will be trying to find and develop small activities to help reinforce vocabulary. Simply emphasizing it more is a start. I’ve also done some activities, like concept maps and “functions back-to-back” which help with vocabulary understanding. Next school year I’d like to take a more systematic approach and deliberately build in vocabulary activities into each unit.

Drop your favorite vocabulary activities in the comments below or send them my way on Twitter. Thanks!

Image Credit: “Words” by Shelly on Flickr

How to Factor Quadratics

After years of teaching how to factor quadratics and then getting in my car and banging my head on the steering wheel, I decided that enough was enough. I was going to spend some time finding a better method. I took to my favorite community of math educators, the #MTBoS.

Several different ideas were thrown my way, but the one that was most attractive was Mary Bourasa’s method, sent to me by Helene Matte.

Last year I had tried the “diamond” method, which worked a bit better than simply guessing and checking, which I’d done in previous years. The first problem I ran into was that I had trouble remembering what went in the top of the x and what went in the bottom. In videos I watch online teachers did it different ways. I think this was because of the second problem I ran into, which is where the hell did this giant “x” thing come from anyway? It’s not a “trick” really, but it does seem to have no connection to other things we do in math.

I might as well have said, “Today we are going to factor quadratics. Draw a random shape, fill it with numbers in the recipe I give you, then get your answer.” And kids weren’t that good at it.

Enter Box Method (or area method, or whatever)

A few years ago I learned about the “box method” for multiply polynomials, binomials included. Put the first polynomial along the top, the second along the left side, multiple rows by columns. Very similar to multiplying actual numbers with this method.

Example with Numbers

Example with Binomials

The approach to factoring using this method is attractive because it feels like working the box method, in reverse. If students are familiar with the box method for multiplying binomials, it’s a natural extension to use this method to factor them (as I often talk about factoring as the reverse of distributing).

Step one

The first step in this process is writing down a*c (M), the coefficient on the middle term (A), and then finding two Numbers that multiply to give you M and add to give you A.

Step Two

Once you have your numbers, fill the box. The upper left corner and lower right corner have to contain the squared term and constant term respectively. Fill the upper right and lower left with the two numbers you found in step one.

Step Three

The last step is to factor out the GCF of each row and column. Then you’re done. You have the factors that multiply to give you the quadratic.

A couple notes

This method fails miserably if you don’t factor out any common factors at the beginning. For instance, if you have a 2 in each term that can be factored out, you have to do that first before using this method.

I still have to grade the quizzes that cover this section but the kids seemed to respond a lot better than they have in previous years. I’ll update this post once I know more.

Thoughts on “GPSing our Students”

In June Dan Meyer posted Your GPS is Making You Dumber and What that Means for Teaching. In it he makes the argument that providing step by step instructions for math concepts results in students being able to get from point A to B, while not understanding much about the concepts they’re supposed to be learning. His argument can be summed up with this paragraph, and is somewhat inspired by what Ann Shannon wrote in what teachers should Look for in the CCSS Mathematics Classroom.

Similarly, our step-by-step instructions do an excellent job transporting students efficiently from a question to its answer, but a poor job helping them acquire the domain knowledge to understand the deep structure in a problem set and adapt old methods to new questions.

I would tend to agree. I do give students steps occasionally but it’s often in order to simplify concepts and, if I’m being honest, to some degree avoid students truly struggling and grappling with the concepts.

I’m curious as to what others think about his post and the notion that GPSing students leads to less learning.

Panera Bread and Learning

In the last month I’ve become a Panera regular. We’ve been doing a lot of traveling and Panera was as close to fast food as we were willing to go. Until the last month however I’d only been there a few times in several years, without being particularly impressed.


image credit: nbcnews.to/29POCSz

Take a look at the menu above. This is what you encounter when you walk into Panera Bread for the first time. Seven sections of menu packed with different dining options. For the less food savvy among us, not only is the menu packed with stuff, but it’s packed with a lot of stuff that is unfamiliar. In my first couple of visits I stood there bewildered for a while, let my friend order, then picked something with turkey in it.

Because I was pretty sure I knew what turkey was.

Well, I was wrong. I mean, I think the thing had turkey in it. But everything else did not complement the turkey in a positive way.

The next time I went there with a few colleagues I ordered a caesar salad. Why? Because every time I’ve ever ordered a caesar salad in a restaurant I haven’t been surprised by what came out. The same was true for this visit.

Sweet. I found something on the menu without holding up the four people behind me.But after that I avoided Panera for a while. A long while. I could get a caesar salad from a lot of places. The notion of standing there in line trying to find something different felt as though it would only bring frustration.

Then my wife and I were looking for something quick and healthy to eat, so we decided, to my slight dismay, to give it another try.

This time however, on the way there my wife looked up the menu on her phone and read off several items, describing them from their website. When we walked in I knew several of the items on the menu, how it worked (that “pick 2” thing for example) and was basically ready to order when I walked up. We’ve been there a couple of times since and now it’s one of my favorite restaurants. Since I’m confident in my understanding of the menu, I try new items, ask questions about different types of food, etc.

How does this relate to learning?

Think about the first time you encountered an unfamiliar topic and tried to learn it. This was difficult at first but I can certainly think back to math lectures (here’s lookin’ at you, Linear Algebra) in which there was new notation, vocabulary, and concepts and it all felt unfamiliar.

However, once I got with my peers and we began working through the problems (encountering the “menu” multiple times) the concepts began to feel more familiar. I realized that I knew more about them than what I thought (romaine and kale! Hey, I know what those are…). The more I worked individually and with my peers the clearer the concepts became.

If I avoided doing the exercises or only did it individually, that feeling of everything being foreign never really went away. Usually parts of the notation would be confusing. Or the instructions around a problem wouldn’t make sense. Or I could start a problem, but get lost in trying to solve it, etc.

A couple points can be pulled from this. First, a student’s first exposure to a topic is incredibly important. If you drop seven new vocab words and a gaggle of new notation on students at 8:00am on Monday morning you’re bound have a large group of students not wanting to come back to the menu you just presented them.

Even if they should know most of the food on there.

So we have to think carefully about how students first engage with content. The second point is helping students understand that multiple engagements with a concept will (usually) alleviate this feeling. I think many students never go back to the restaurant because they don’t want to be embarrassed for not knowing what their peers may already know. We have to help students be comfortable with this phase of unfamiliarity (my study group in college was a place I felt comfortable being wrong), and develop tasks that help them engage with the concept in safe, productive ways (investigating the menu on the way).

For a great post on how students see unfamiliar mathematics check out Ben Orlin’s recent post entitled What Students See When They Look at Algebra.

Alternative Titles

“What’s on that sandwich? Oh… will that be on the test?”

“How Avocado Ruins Education”

“I don’t even like lettuce so why should I be learning about salads?”

My Precalculus Problem

This is Lucas’s first year in my school. He’s a senior in my precalculus class. After a few assessments his grade begins to tumble. I know little about his previous math education. I meet with him during lunch a few times to help on the content and see that he’s missing several foundational math and reasoning skills. He ends up with a D in my class.

Jennifer is bored in my class. She’s easily getting an A. I can tell that she knows most of the content because she remembers it from algebra II. At some point in the first trimester she asks me if the second trimester will be review as well. I said that a lot of this is not review and that there will less familiar content in the next trimester (which is mostly true). She ends up with an A-.

In the high school in which I teach there is honors precalculus and “regular” precalculus. I teach the regular level. This means that I get kids like Jennifer, kids like Lucas, and everyone in between. I have college bound students and students that will go into a trade. I have students that love math and some that are only there because their parents made them. I get a lot of students that have scooted by with A’s and B’s without trying and would prefer to continue not trying.

So, basically a typical high school class.

The last couple years I’ve struggled to differentiate for the diversity in this class. I’ve failed many of these students because the content either goes too deep or not deep enough. This summer I’m working on solving this problem, or at least minimizing it. The flowchart below is what I’m currently thinking, although I’m sure this will change as I continue to work on it.

Screenshot 2016-06-27 06.36.30

My idea is to pre-assess over algebra II skills that are needed for the unit. If students have mastered most of those skills then they take a different track then those that haven’t. I haven’t worked out a full module yet but I’m thinking I have most of the track 1 materials made and need to make most of the track 2 materials.

Most of my direct instruction is on video which means not every student has to be at the same place at the same time. I just need them to be ready for the summative assessment on a certain day.

With this model I can patch the conceptual holes for the kids that need it, and push the kids that don’t.

What problems do you see cropping up with this idea? For example, I’m worried some less motivated kids might intentionally do poorly on the pre-assessment so they have the “easier” track.

Any feedback is welcome and appreciated. Thanks for reading.