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”


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

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.

Getting A “Feel” for Trig Functions

My good friend Steve Kelly always talks about getting creative ideas in the shower and to be honest I’ve always been jealous. It’s not frequent that I get a good idea in the shower. However, that changed a couple weeks ago. (Having been entrenched in learning about creativity for the last several months I realized that maybe the problem was I was trying to find ideas in the shower as opposed to letting my subconscious do the work. So, I stopped doing that, not only during showers but also workouts, driving, etc. and the ideas have come more frequently.) Prior to the idea 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. The idea I had was to somehow make a massive unit circle and Cartesian plane and show students the connections in a life size context. Below is the outline of the activity and below that, my reflections. Continue reading

Dimensional Thinking – Functions

I’ve encountered assignments in this course that I’ve struggled to apply to my topic area (functions) but this assignment seems built for it! Dimensional thinking is thinking about a topic or concept in various dimensions and possibly coming up with models to represent that concept. One of the great things about modern graphing technology is that it allows us to push, pull, and explore functions. I want to explore functions both at the minuscule scale, the large scale, in three dimensions, and even explore higher dimensions.

We often look at functions in a standard viewing window. We generally like integers, especially numbers -10 to 10, when we first look at functions. Consider the image below that I’m sure most people remember from high school algebra, y=x^2 in the standard viewing window.

Dimensional Thinking Image 1

y=x^2, Standard Viewing Window

There is nothing special about this viewing window. In fact, as many math students find out, lot’s of other viewing windows are more useful. For instance, suppose this graph was modeling position (in meters) versus time (in seconds) and we wanted to know the velocity at exactly 2 seconds. This problem becomes difficult without calculus unless we change our window to make finding the exact velocity easier. See the image below.  Continue reading

Abstracting the Abstract – More Math GIFs and Function Talk

Creative individuals often take something that is concrete or complex and abstract it in some way that makes it more meaningful or provides a more useful perspective. When charged with the task of creating an abstract representation of my concept (functions) I was first at a loss. How do I create an abstract representation of something that is fundamentally abstract? The descriptions and media below describe the two methods I chose. For a summary of how these abstractions impact my teaching and understanding of functions scroll to the end of the media.


Method 1

As I thought about this, I realized that when we write a function, we often only talk about a specific function. Yes, occasionally we will discuss a “family” of functions. But what does that mean and students really understand it? When we discuss a family of functions we are really talking about an infinite number of functions. What does that mean? How can we visualize such an abstract concept? Yes, sliders can help, but it can be visualized more effectively. Below are my ideas for abstracting various families of functions.

This also made me think…Could an abstraction be (at the surface level) more complex than what you’re abstracting? The GIFs show a more complex picture than just a curve with a slider (y=ax^2 with “a” as a slider). A family of functions is an infinite number of functions, right? How do we abstract an idea that is so complex? The GIFs attempt to abstract infinity (or an infinite number of curves) by suggesting what the escalation to an infinite number of curves would look like. (Click a GIF to look at a slideshow of each one individually.)










Method 2

The task set before me was to abstract my concept in two different ways. The second method I chose for abstracting the concept of functions was to pare downs functions down to their simplest…well…abstraction. Then, I sketched these abstractions and below are the images of my sketches. I decided that at their fundamental level functions have three things: an input, an operation (something that modifies the input), and an output. The images all contain these fundamental pieces.

Photo Oct 22, 2 22 05 PM

Photo Oct 22, 2 22 17 PM

Photo Oct 22, 2 22 57 PM

Photo Oct 22, 2 22 41 PM

Photo Oct 22, 2 23 10 PM

I think both abstractions can impact my teaching. The GIFs would be something that I could either show my students to give them a visual for the concept of a family of functions. A good project would be to have students create similar GIFs for different functions and their transformations. This solidifies the concept in a visual way and gives us something we can anchor back to throughout the rest of each unit.

The second abstraction might be useful in a different context. I like that the images provide visuals for the simplicity of functions. Often students don’t realize the vastness of functions and their stretch throughout the world. Most things with an input, operation, and output is a function. When you ask a student to give you a function they are likely to say something like y=2x+3 or f(x)=sin(x). They are unlikely to say “The number of cars in the school parking lot is a function of the time of day.” If students have the abstract concept of a function as their fundamental understanding then they might start to see function behavior in their worlds more frequently.

Observe and Re-imagine

My current project in CEP 818 – “Creativity in Teaching and Learning” was to observe an object tied to my topic area (functions in mathematics) and to re-imagine it, appealing to a different sense than originally intended.

First, a note on the importance of observation:  Robert and Michele Bernstein, authors of Sparks of Genius, point out that “All knowledge begins in observation. We must be able perceive our world accurately to be able to discern patterns of action, abstract their principles, make analogies between properties of things, create models of behavior, and innovate fruitfully .”

And now, the re-imagining of the graphing calculator.

Any math student that graduated from high school in the last two decades probably used one and I can guess (although I might be wrong in some cases) that it was used to get answers quickly and easily. In fact, as I was observing this device from every possibly angle and sense (okay, I didn’t lick it…) I noticed that it screamed simplicity. It is a no frills, unexciting, almost heartless object. Take 30 seconds and look at the image below. There is not much about that calculator that screams “Hey you, let’s do math and it’ll be awesome!” To be clear, it is incredibly powerful, but it is often underutilized as an exploration device and over used as a “shortcut machine”.

CalculatorThis may seem irrelevant to the learning of mathematics but it doesn’t help the plight of the math teacher that is trying to motivate students. I hadn’t considered this before, but I started thinking “Is this a device that I want to use?” I love mathematics but this device, although it is simple, is not elegant and not a device that I look forward to using. It is fraught with ugliness. From the black and green display to keys that seem randomly colored, there is much to be desired in terms of design. I believe that if we redesigned this device to be aesthetically pleasing and still powerful, functions and mathematics in general would be more accessible.

If you haven’t bought into this yet, consider your favorite technological device. My guess is it’s your phone (and if it isn’t, humor me for a minute). First, I bet the ratio of the sides is close to the Golden Ratio, which creates the most visually appealing rectangle. I would also guess that it fits nicely in your hand, that the interface is intuitive yet attractive, and that it balances power with beauty. The feel and design of this device make you want to use it. What if a calculator had that same appeal for math students? (The GIF below represents the potential of moving mathematics to the devices we use most often.)

In addition, this is not how I see mathematics.  I see math as a beautiful framework within which we can describe the natural world (and even if we couldn’t, it is still beautiful). Functions are like an engine that can be constantly modified to describe an infinite number of situations. If we can find ways to relay the beauty in mathematics and functions to students then I think (and admittedly I’m aiming high) we could ultimately shift the culture to one that views math as elegant and powerful, not just an obnoxious requirement to get a diploma.

Desmos gif


A note on the above GIF: Each one of these graphs (with the exception of the smiley face) is a function or two functions graphed together. I used desmos.com to create the graphs. I think that tools like Desmos, which runs on all smartphones (the devices we actually like to use) can help students to visualize functions as well as other mathematics. (Here are some of the functions I used.)


Bernstein, R., & Bernstein, M. (1999). Sparks of genius: The thirteen thinking tools of the world’s most creative people (p. 30). Boston, Mass.: Houghton Mifflin.