Part I of Machines: A Useful Algebra 2 Representation

So last year, I started teaching Algebra 2 again after years of not teaching it. I worked with a colleague on the curriculum, and one thing we really wanted to make sure kids were continually exposed to were various representations of functions and relations. Of course this includes equations, graphs, tables, and words. 

But in addition to these representations, I was inspired to include a fifth representation. It has a few drawbacks, but I can’t even express to you how many positive aspects it has going for it. It is the “machine.” I remember seeing images of these machines in middle school textbooks, and they really emphasize the idea of an input, output, and rule. Here’s one I randomly found online:


In this blogpost, I’m going to share how I introduced this representation, and how I subsumed the others in it. In future blogposts, I’ll share all the ways I’ve exploited this representation. It’s pretty magical, I have to say. So stay with me…

At the very start of the course, I introduced this machine representation also. Just not as fancy and cartooney.


I thought a lot about whether I wanted the machine to allow multiple inputs or allow multiple outputs. In my first iteration of drafting these materials, I did that, but then I backtracked. Things started to get pretty complex with an expanded definition for a machine, and I wanted to start the course simply. And, of course, I really wanted to emphasize the idea of a function and a non-function. So I started with the definition above. And started with things like this…


Notice these are “non-mathy” examples of machines. They eased kids into the idea, without throwing them into the deep end.

What was nice is that we got to understand and interrogate the idea of domain and range from this… where I described the domain as the “the bucket of all possible items that can be put into the machine and give you an output” and the range as “the bucket of all possible items that comes out of the machine.”

image4.pngSo for the sandwich one, we know the range is {yes, no}. And the domain might be {all foods} or {every physical thing in the universe}. We talked about the ambiguity and how for these non-math ones, there might be multiple sets of domains that make sense. But then for the math-y ones, we saw there was only one possible domain and range.

In fact, to really drive home the idea of inputs, outputs, domain, and range, I created an activity. I paired up the kids and one kid was the machine, and one kid was the guesser.

The machine got a card like this, with the rule:


The guesser got a card with the domain and range:


And the guesser would give words to the machine, and get a result. And their goal: figure out the rule. Then I would switch the machine and guesser, and give a new set of cards. It was crazy fun! I did it a long time ago, but I distinctly remember kids wanting to play longer than the time I had allotted. (If you want the cards I made, here’s a PDF I created Domain and Range Game.)

Next I showed how the “rule” in the machine could take a number of different forms — tables, words, equations, and graphs — and this is how I introduced the various representations. Kids were given these and asked to fill in the missing information…


… and then they were asked to find the domain and range for these same rules…


To drive home the various representations, I gave kids questions like these, where kids were given one representation and were asked to come up with the equivalent other representations.


So this was the gentle introduction my kids had to machines. I’ll explain where we went from there in future posts… and I promise you it’s going to be good… I’m really proud of it!

A short whiteboard activity to check understanding

So I’m not great at coming up with activities. Not in the way people talk about. But I recently was in a moment in my Algebra 2 classes where we had discussed function notation and how we use it, and also we had introduced interval notation to discuss things like domain and range. I wanted to challenge students to test their understanding. So I came up with this activity!

Kids have to come in knowing: 

(a) what a function is, what interval notation is
(b) what domain and range are conceptually, and how to write them in interval notation
(b) how to read/understand function notation

Here’s what I did.

Kids in each group got a giant whiteboard. In one color, they were asked to draw x- and y-axes and put tick marks so each axis went from -5 to 5.

Then they were asked to draw a function. The requirements: it had to be complicated and interesting. I made it into a small competition, with my subjective interpretation deciding which group won. They also had to be able to determine pretty clearly what the domain and range for their function were. They were told that their graphs would be given to other groups to stump them. So make ’em good!

Kids rose to the challenge. Here are three examples:

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Cool, right? I had each group write what the domain and range was for their functions on a post-it note on the back of the whiteboard.

Then I assigned each group a different group’s graph. Everyone in class took a crisp photograph of the graph they were assigned. And then class was over.

That night, kids did problems #1-#4 in this sheet I created. I’m pretty proud of this sheet! (Here it is in .docx form to download/edit.)

The next day, kids in groups compared their answers to #1-#4 with each other. They made revisions. They checked to see if their domain/range matched the post it on the back (the post-it the original group made when they created the graph). Then they worked collaboratively on #5 and #6.

When they were all done, I went around and checked their answers. (I had filled out an answer sheet for all the graphs so this part could be smooth.) I had discussions with groups about misconceptions they had. These conversations helped me see precisely where kids were getting tied up.

That’s where we are right now. A great finishing activity to function notation, domain, and range. I was so so so happy with the strong work kids were doing with such tricky functions! It was incredible! I even found a few mistakes in my own answer key!

At the start of our next class, I’m going to project a few questions like these to draw together our understandings and talk through some larger things that I realized I needed to highlight from my smaller conversations with groups:


Overall, this was relatively simple to execute. It broke up the monotony of class. And I love what I got out of it in terms of student thinking/analyzing.


Some notes from doing this:

  • I loved kids working on the whiteboards to create their functions, with the easy ability to erase and recreate parts of their graphs. And I’m glad the whiteboards are large. I only wish that the whiteboards had gridlines on them to make the graphs extra neat and easier to read.
  • I wondered if, after a group themselves finishes drawing their graphs, they should be given the worksheet to fill out on their own work. (In addition to a new group.) Then the worksheets could be compared and discussions/debates could happen.
  • That being said, I liked that the worksheets/questions were hidden from kids, so they felt like extensions beyond the domain/range.
  • I thought a lot about how Desmos activity builder could probably be harnessed to make this happen… where kids create their own graphs to challenge classmates with… But even if kids don’t come up with their own graphs, a Desmos activity with well-created graphs could also be neat to have at my fingertips.
  • It took kids about 20 minutes to draw the axes and come up with their graphs. And some took a little longer if they had a tough time identifying the domain/range for their post-it.

Ally Week

I was getting the math space ready with some climate change infographics when a health teacher walked by and mentioned that the following week (now this week) is Ally Week. I wasn’t aware. So thanks to twitter (@benjamindickman and @annie_p and @LauraVHawkins) I put up some information on some gay and trans mathematicians in the math space.

But I also had my classes engage in such a simple way. I just added this to their nightly work:


Those links, so you can click on them, are:




The way I facilitated this in my classes was very similar, and really informal. In one, I had people “popcorn” their thoughts about the article they read, share something that struck them, ask questions they had, make note of something they never considered until reading the article. In another, I had groups talk about what they read, and then we talked as a whole class, people sharing out what their groups said. (I think if I did this in the future, I might have everyone pull up the article on their laptops so they could refer to them for quotations, or to de-stress kids who don’t want to mischaracterize something they read.) I did almost no talking. I just kept silent, and let kids share in the whole class. Pro tip: sometimes I find it’s effective to a little silence go for 20 or 30 seconds, and then someone else will say something. And if I think things are coming to an end with a tremendous silence, I say “okay, we have time for one more thing.”

Then after they shared, I brought us together as a class. To talk a bit about allyship specifically, I read this paragraph from Anthony Bonato’s piece on being a gay mathematician aloud (this was something most students mentioned when talking about the pieces… it stood out to them):

I didn’t experience explicit discrimination until I was working on my doctorate. One of the professors in my Department told me to be careful about being open about my sexuality, as it would make professors and students uncomfortable.  He thought he was doing me a favor, I think. I nodded politely and buried the incident away in my memory. Being gay often involves so many of these small defeats, these small let downs, that it becomes part of our everyday experience.

In two classes, I asked students “If you wanted to be a strong ally and they were in the professor’s shoes, what are actions you could take to support Anthony?”

In one class, we talked more about Autumn Kent (who is a mathematician who is trans). And I asked where in the piece she mentions how to be an ally, and we reviewed that together

Sometimes we need a shoulder, or an ear. Or just some normalcy.

The thing I think most people don’t see is the constant underlying dread, anxiety, stress, and anguish that a lot of us are carrying around. A lot of the time I am walking to and from my daily tasks, my inner voice hoarse from screaming. After the election I would be out and hear people making small talk about the sunshine and I’d want to tear out my hair. When I am doing bureaucratic tasks at work, I am carrying all of my anguish. When I am teaching and getting a laugh from my class I am carrying my anguish. When I am writing that email. When I am in the elevator or at the water fountain. When you ask how it’s going I am frozen. I am saturated with grief.

Listen to us.

In all my classes, I had to come out during this discussion. And I did it by explaining why I had them do this assignment… I talked about when I was in high school, there weren’t really any out kids, and definitely no out teachers (that I knew about), and no real representation of the queer community. And how I’m so happy times have changed, and hearing stories is such a big part of that. And so I wanted to share these stories, to show that being a mathematician does have something to do with sexuality and gender identity… because being a mathematician means being in a community with other people.  I ended by sharing the quotation that stuck out the most to me, because I felt it a lot in my first years of teaching. It was from Anthony Bonato’s piece:

We edit ourselves by asking internally a series of questions.  While lecturing does the audience think less of me if they know I am gay? When colleagues talk about their family over dinner is it OK for me to join in and talk about [my] husband too? Am I acting too queer in front of my students?

And I mentioned that lots of people in all sorts of marginalized communities do this kind of editing, because they don’t know what’s in the hearts and minds of people who are around them. Which is why being an ally can be so important — to show what’s in your heart and mind.

I remember when I first started teaching, saying I was gay was something I thought I didn’t need to do to students or other faculty members. I’m a math teacher. I was still getting my bearing and earning my stripes. Would it come in the way of me getting my stripes? Would students use it against me? Why would it ever matter in our math class? But I know it does matter. Because when I was younger, there was no one. No representation. It’s not that I didn’t think I could be a mathematician. It was worse. I thought that I was alone in the world, and I was wrong in the world. And so in this activity, it would have been disingenuous to not come out, if only to honor the transformation I’ve gone through from my youth to today.

Regardless, if you are a math teacher and are wondering if there is a way to push the needle forward in your classroom on LGBTQ+ issues, maybe try something simple like this. You don’t have to be queer or trans, it only takes 10 or 15 minutes of class time, and it sends a signal out there that you are an ally and care. About these stories, and implicitly to your students, about your students’ stories.

Update: It’s Friday evening now. It has been an impossibly long week, with late nights. And parent night on Thursday night keeping me at school until 9pm. I’ve been exhausted. But I want to archive one more moment from Friday.

It was after a lunch meeting. Our school has something called “CCEs” (continuing the conversation events) where student leaders lead discussions on important topics. Today’s CCE was around Ally Week and pronouns. I go to the room. It’s a large classroom for our school, but designed for maybe 20 students. By the time everyone got there, I’d estimate there were over 60 students/faculty in attendance. We watched two videos and then have an interesting discussion. I don’t want to share what was said, since confidentiality was one of our norms. I do want to share what happened to me. I was in the room, and I was overwhelmed by the attendance and the seriousness by which everyone was taking things. And while watching the videos about trans vocabulary and getting everyone on the same page, the video had a section on transitioning. And for some reason, with that word, I was flooded with emotion. My eyes were literally tearing, and I had to keep wiping them for the remaining 20 minutes. I was afraid if I spoke aloud to share my thoughts, I would start speaking and my voice would start warbling and then descend into sobs. So I didn’t speak. I don’t know what evoked this big emotion. In my mind were the multiple memoir books I’ve read about trans women. In my mind was the TV show Pose and documentary Paris is Burning. In my mind I was surrounded by kids who cared, and maybe a kid or two who identified or were in the process of identifying as trans. In my mind was all the hardships trans people face that we couldn’t even start to understand. In my mind was the many murders of trans people. In my mind was simultaneously hope and despair. And so I kept looking up at the ceiling, and wiping my eyes, hoping no one would see me. Because we were watching videos and everyone was sharing super interesting things.

I was happy when it was over and I could go wipe my eyes properly in the math office. I tissued my eyes dry, no one was around. Then a colleague/friend walked in and we started speaking about the CCE, and just a few words out of my mouth and what I feared would happen in the CCE happened. My voice warbled and the tears just started flowing. I don’t know why. I was exhausted. I was emotional. I couldn’t speak. I tried to explain in the 30 seconds we had, then I wiped my tears away, stuffed a few extra tissues into my pocket, and went to teach my 90 minute class. I walked into the door, took a few breaths, and said “Happy Friday Everyone,” before getting us started on combinatorics.

What I Owe My Students

This year, in my Algebra II classes, I used a chunk of classtime to set some norms and to get some vulnerability together. I honestly felt like it brought us closer together, and I’m so grateful I did it. I hope to write a longer post about that soon, but I wanted to share the very last part of what I did.

I asked my students for three things of what I owed to them. I loved what they came up with. These kids are already so awesome. And these are all things I know I can work hard on giving them.



I see a lot of commonalities in what they wrote.

Understanding that math doesn’t always come easily to my students. Patience and kindness. Making mistakes and being lost and confused something that is okay and not something to be shamed. Recognizing that math is just one part of students lives. Promising to help students. Clarity. Fairness. Engagement. Encouragement.

I am getting these made into posters which I am going to hang up in the back of my classroom, where I can see them each day… reminders of what my responsibilities are, and seeing where I might be falling short.

Our new “Math Space”

‘For the last year or two, I’ve seen so many people tweet about they have tables in their math classrooms where they put math or math-adjacent things for kids to fiddle around with before/after school or during their free time. Here is a recent tweet thread:

So of course, color me insanely jealous. (I think I first heard of this idea from Sara Van Der Werf on this blogpost.) The thing is… I really want kids to see math as something that exists outside of the math classroom. And anytime I see an opportunity to do that, I go for it. So things like math club and math team, yes, I’ve led those in the past. Independent studies/work with kids, yes. But I like the idea of opening up the umbrella of what counts as math. So a few years ago I helped organize a math-art gallery (with real mathematical artists!) at my school — with an exhibition called Technically BeautifulOr organizing math-related book clubs with kids (from Flatland, to Hidden Figures, to How Not To Be Wrong, to whatever.) Or assigning my “explore math” project to some of my classes.

The appeal of the math play space was so strong that last year I decided I would make one for this year. The tricky part is that in my school, we don’t have our own classrooms. Last year, I taught in four different classrooms. But luckily outside of the math office, we used to have a long bench where only a few kids sat on when waiting for class or a meeting. So my plan: remove the bench and make a math play table/space right there.

My colleague and friend Danielle was interested in the idea, so we basically just did it. We asked maintenance to remove the bench. We set up three card tables. And we had the space ready for the first day of school. Ready to see what it looks like?!?


I’ll go through what exists in our space now.

When discussing the space, we agreed that it had to look cozy and inviting. So with our limited artistic skills, we put together this beautiful sign. We tried hard to come up with a better name, but we kept on converging on this simple one… so we went with it. We literally crumpled paper of different colors and tacked them up to write the word space. I’m actually in love with the way it looks. It was what we had around, and we got creative!


Now on the left side we have this:


This little cart was being thrown away by a third grade teacher, so we stole it! We put showerboard on it so it can act as a whiteboard, and if you look closely, we have some whiteboard markers below for students to us. On the board itself is a number game lifted totally wholesale from David Butler (his post about it is here). The idea is that with four small numbers (e.g. 1, 10, 10, 7) and two large numbers (e.g. 60, 120), students should attempt to make the target number 121.

After showing this to one of my precalculus classes, a student was obsessed with trying to get the target number using all six numbers, and came up quickly with a way to do it. He was super proud, and rightfully so!

Next we have two card tables covered with some fun cloth I found at home.


These are books that I brought in for kids to thumb through (though they just have to ask and they can take it home to read!). I have a zillion books that could go here… My criteria was nothing that could turn off a student easily. So a book of math poems, a childrens book about Sophie Germain, a math book based in funny comic strips, women in mathematics book, and a couple “math novels.” I even had a math department colleague/friend write a “recommendation note” that we stuck in The Housekeeper and the Professor, like this was a book store! (I asked our school librarians if they had the little book stands, and they were happy to give me some!)


Ikea had some $1 picture frames, so we used them to post some puzzles and jokes!

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We also put out some puzzles from Play With Your Math which we thought had a low barrier of entry but that kids might enjoy!

We also have a little estimation station (currently of jars with rice in them):


And of course we saw that Sarah Carter had provided us with a lot of math jokes that we could steal and use in our math space… So we have that up also! Because how could we not?!?!

Lastly, we have a “tinker table” where we have some tiling turtles, other tiles, and a weird set of puzzle pieces which need to get put back into a square shape.


And that… is about it!

At the start of putting things together, we realized we needed a bit of a formal vision for us to stick to… so we drafted this super quickly, but it was something we both felt was approximately right:

Vision: To create an unstructured public space where kids can relax and fiddle/tinker around with fun math things that might not be related to things in the formal curriculum. The hope is that this allows for the experiencing of math as something casual and playful. We want this space to encourage students to want to talk mathematically with each other.

Through this space, which will be curated and changed periodically, we want to widen the umbrella of what gets counted as “math” and “doing math,” and who gets to be counted as a mathematician. 

We encouraged teachers at the start of the year to share information about the math space with their classes, even writing them a blurb they could read in their classroom but also encouraging them to leave their class five minutes early to bring kids over to just look around. What we wanted teachers to emphasize? “Most importantly, we don’t want you to be scared to sit down there. We spent time making this space for you. We want to say that again — this space is for you! Pick up books and see what they’re about. Make designs with the tiles. Flip the joke page over to see what the groan-worthy punchline is. Try the number game puzzle out, or pick up the paper folding puzzle that we have there for you. Make an estimate for the estimation challenge. We want you to feel comfortable here — not treat it like a museum.”

Lastly, you might have noticed that in the vision we mentioned that the math space is designed to changed periodically. That’s the goal. Of course the jokes will change each week as will the numbers for the number challenge. But everything else — books, estimation, picture frames, tiles — will be swapped out. We have a giant list we’ve brainstormed of things that we could put in this space, and we’ll make decisions as we see if and how kids are using it. Some ideas include:

  • Instructions for the game of SET, and space for kids to play the game!
  • Wooden “put these together to form this neat shape” puzzles
  • Legos
  • A variety of math poems that students can take and put in their pockets during Poetry Week at our school
  • A spirograph or two
  • Math and Climate Change coloring books with lots of colored pencils (where we hang up the pages on the bulletin board after things get filled in)
  • Towers of Hannoi
  • Origami paper and instructions
  • 3D printed mathematical objects, including cool math based optical illusions (like these!)
  • A museum of WEB Du Bois stunning and eye-opening infographics involving race in America
  • Geoboards
  • Information on women mathematicians and mathematicians of color and mathematicians that are LGBTQ+ and…
  • Fun little math problems (the size of a business card) that kids can pick up and bring with them

I actually have so many more ideas on my list, but it’s all written so informally no one would ever fully make sense of things. But these are just some. But if you have ANY other ideas that you think would make sense here, I’d love to get a nice long public list for math play spaces — so throw any ideas down in the comments.

With that, I’m out!




The Finals for the Big Internet Math Off 2019

This summer, I’ve been “competing” in the Big Internet Math Off 2019. It’s a competition where 16 math-y people share their favorite or interesting bits of mathematics, and each day people vote. Believe it or not, I made it to the final two competitors — and today is THE FINALS.

If you’d like to read my post and my competitor’s post and vote, I’d appreciate it:

The Big Internet Math-Off: The final – Sameer Shah vs Sophie Carr

It will only take a short time (no need to login or anything to vote, the only time it will really take is the reading).

My mathematical tidbit today attempts to have you look at these two squares, a 17×17 colorful square and a 127×127 greyscale square.

squares.pngBoth are… slightly uninteresting.

My goal, through the post, is to show you that both of these squares are insanely interesting. I call them the most beautiful 17×17 and 127×127 squares ever. And my conclusion: once you learn about the mathematics embedded in these squares, you’ll never look at them the same way again. You can’t.

It’s like having a huge a-ha moment when learning something. It completely transforms the way you look at something, so you can’t see it in its original form again.

I hope you enjoy!

If you want to see my five entries into the Big Internet Math Off 2019:

Entry 1: a counfounding conundrum:

Entry 2: a card trick:

Entry 3: a magical property of circles:

Entry 4: an unexpected break in a mathematical pattern:

Entry 5 (the one outlined in this post): two beautiful squares:

A Much Belated Overview of My Advanced Geometry Curriculum

I recently got an email from someone who saw some of my many posts on geometry (you can see all my posts about geometry by clicking here). I realized I never shared them formally and everything is a bit scattered. So I’m going to try to include a few resources here. But the real joy is in all the blogposts, honestly.

I taught Advanced Geometry at my school for two years (2014-2016), and I wrote the curriculum with a good friend and dear colleague. We both hadn’t taught geometry before and decided we’d do a super deep dive and come up with a sequencing that made sense to us, and that prioritized conjecturing and noticing. In fact, we were so excited by this process that we shared our thinking both about how we built up the curriculum but also how we collaborated at a conference. Below are our slides, but you can also click here and go to the slideshow and read some of our presenter notes for each slide for more detail.

We were super intentional about everything. We carefully thought through how we wanted to motivate everything, and we didn’t want to give anything away throughout the course. In other words, we wanted kids to do all the heavy lifting and to be the mathematicians that we knew they could be.

Below is a word document with all our skills/topics (you can download the .docx file here: All Topic Lists Combined). The order might seem a little strange (we end, for example, the year with triangle congruence), but it worked for us! Everything was done on purpose (in this case, congruence is just a special case of similarity… so that came beforehand, along with trig which is all about exploiting similarity!). We eschewed two-column proofs for different forms (paragraph proofs, flowchart proofs, and anything else that showed logical reasoning).

Oh wait! For some reason our work on Area and Volume didn’t have a topic list. And I just looked and my core packet for Area and Volume derivations (where kids just figure things out on their own) has handdrawn images in it, but I didn’t scan a PDF of those. Well, at some point in the future if I remember, I’ll try to write a post to share that. (We did it after kids learned trigonometry, so they had a lot of flexibility. For example, I think kids came up with like 6 different methods to find the area of a trapezoid when they were asked to create a formula and justify it!)

I hope this is helpful for anyone trying to think through geometry. As I said before, the best thing might be to just read the blogposts, but this is a bit of an overview.