# At the end…

This is an archive of my day yesterday.

We’re nearing the close of classes with online schooling. It isn’t the end for me, with narrative comments and college letters of recommendations looming. But in terms of meeting with kids, there’s not much left, and I had my plans in order — at least well outlined in my head for some, ready to execute for others. But I felt anxious. On Thursday afternoon, I finished my last meeting, had finished my prep work for Friday, and was in a good place. But my heart was beating, racing, and I didn’t know what it was. That anxiety usually happens to me when I have something impending looming (like a stack of tests I need to grade, or a challenging email I need to write or conversation I need to have) and I am stressed about it. But I had none of that. But my heart was racing. I was a little short of breath.

I tried to relax, but I couldn’t focus on reading. My mind quickly wandered. TV worked.

A few different things had been thrown at me in the past few days. I wrote to teaching friends, as I processed and as a way to vent: “The problem about keeping boundaries in teaching is that boundaries I draw to protect myself often involves saying no to a kid who I can prop up by saying yes. I recently wanted to say no to four requests — some entitled, some respectful. I don’t have much left of me to give. I couldn’t. When it comes to me or them, I pick them. Again and again. As I slowly burn out. Teaching is hard. It will never not be hard. Because we give so much of our selves, emotionally, mentally. I don’t always want to do that. But I (honestly) don’t know how not to.” The same thing happened two weeks ago with college recommendations. I teach two junior classes. All but three juniors asked me for a recommendation. I felt utterly deflated. And honored. If someone could be in my head as I try to craft these, you’d understand the deflation. They not only take me forever, but they are emotionally draining because I want to capture my kids with integrity. I told my classes I had to limit them and I hoped they understood. I tried to be transparent and vulnerable with them about what doing this meant for me, and why I had to limit them, because they deserved that. But when I read their reflections, I couldn’t say no. I thought about the Giving Tree. I’m not that self-destructive. But for a moment I martyred myself.

Yesterday evening all advisors received an email saying that instead of meeting in our advisories, we would be meeting as an entire high school to give some time to discuss the recent tragedies involving Ahmaud Arbery  Christian Cooper  George Floyd  Nina Pop. I hadn’t heard of Nina Pop. Her name was misspelled in the email we received. Both of those facts together speaks to something about me, and something bigger than me. I entered the zoom meeting and scroll through 16 pages of faces. More, maybe. Tired, yawning. And then the meeting begins. It was planned to be a 20 minute meeting. It starts with Feel. Reflect. Act. Grow. Repeat.

Our high school principal ended the meeting before our third period class. It was two hours from when we started. I open my zoom meeting for our Algebra 2 class. Obviously we aren’t doing math today. As the large meeting was winding down, I brainstormed what would make sense and gave my kids some options. They chose to continue the conversation. Unlike in the first breakout room I was in, this conversation was mainly had by kids, where I tried to give them space to speak and I listened. I took notes. They were in the same grade and class and knew each other. They didn’t turn to me for answers. I don’t have answers. I interjected sometimes, and then stepped away. When class was officially over, they were still talking, so I told them they could leave if they wanted or continue talking. Most wanted to stay and so we held court for 20 more minutes.

I entered my last class. At this point I had very little more to give. I had very little left I could discuss. I told kids I had been on zoom all day, and I had been engaging all day, and I needed a 10 minute break to do math and recenter myself. So I did math with them for 10 minutes. And then instead of giving my kids a choice like I did for an earlier class, I made the decision that we’d read something together. I pulled up Francis Su’s keynote address to the Mathematical Association of America — his farewell address as president. A section of this address is on “Justice.” It is centered around this quotation by Simone Weil:

“Justice. To be ever ready to admit that another person is something quite different from what we read when he is there (or when we think about him). Or rather, to read in him that he is certainly something different, perhaps something completely different from what we read in him.  Every being cries out silently to be read differently.”

I posted the nightly work for my classes from today. I shared with them some articles, including  this piece on Edray Goins, which I remember having my classes read last year and gave us a lot to talk about. And this keynote speech by Marian Dingle about “centering.” I apologized in my google classroom post for not bringing in more conversations about race, gender, and other -isms into class, and said it wasn’t just the jobs of the history and English teachers (something kids had brought up in the conversations).

At 3pm, we had a department meeting. It felt incongruous to the day because we had no mention of the happenings of the day. We toasted to colleagues (friends) who were moving on — one to another school, one to blissful retirement. And we did some work together. When the meeting ended at 3:45pm, I moved to my couch and laid down.

A little over an hour later, I hopped onto a zoom call with three colleagues and friends. We have this tradition every Friday. It’s nice to have this normalcy and camaraderie. These are people I need to vent to, but also laugh with, because otherwise I would feel very alone as a teacher in isolation. We talked about the day, processed, a few tears were shed early on, and laughed. We lasted two hours. And we imagined ourselves on a beach adventure together, in person, and committed to making that happen when we could. It dawns on me that this trip is literally the only actual thing I have to look forward to in my life right now. A hypothetical trip to the beach.

This is an archive of my day yesterday.

A friend asked me what sorts of games might be playable over zoom.

I was going to write an email with some games I’ve played with my advisory, but I figured I might as well just blog about it in case it’s useful to anyone else.

Some important context:  I have an awesome advisory group of 8 kids. They’re juniors now but I’ve been with them since they were ninth graders. So we’re comfortable with each other and they know a bit about each other. In quarantine, we always start our distance learning school day with a 15 minute zoom advisory. Honestly, it almost always is one of my favorite parts of the day because … well, if you met my kiddos you’d know why. Sometimes I lead advisory with an activity, sometimes I assign them to come to advisory with a plan. Here are some things we’ve done and things we may do in the future. Some of them are games. The ones with * are things they’ve suggested but we haven’t done or I’ve been thinking about doing it but we haven’t done.

1. We’ve done a pet share. Kids brought their pets (or some sort of stuffed animal or other object) and introduced their pet to the advisory.
2. We’ve shown our workspaces during digital learning to each other.
3. We shared things that were difficult about online learning and problem solved together.
4. I posted a list of 8 random items (a balloon, a jar of peanut butter, the playing card the 7 of hearts, etc.). I assigned point values to the items based on their obscurity.  Students took a photo of the list and had 10 minutes to gather everything and come back at a certain time. Then they tallied up their points, and we declared a winner. (I hope a student does this when leading their advisory… I wanted to play!)
5. Played Kahoot (with pre-made Kahoots)
6. Each of us bring baby pictures of ourselves and share them with each other
7. We break up into two teams on zoom. We pick a word (e.g. “love” or “red”). Each team has 30 seconds to come up with a song lyric or title with that word in it. We go back and forth until a team can’t come up with another one.
8. The same thing as the previous one, but we changed it from song lyrics to song titles, movie titles, book titles, magazine titles, etc. (media titles).
9. Played Mad Gab. I projected a card and kids tried to figure out what the card was saying. We played this fine with kids just sounding things out on their own, but I was thinking they could mute themselves and then they could unmute themselves when they think they got it.
10. Played drawphone. This is like telephone, but with words/phrases and drawings. Initially we started out with pre-written words from telestrations/pictionary, but recently we’ve done it where we ourselves write the initial words/phrases. This is one of my favorites! Caution: there is a cards against humanity version that you can play on this site — so you should always initiate the game, and ask kids the join (and not let kids create the game).
11. Played pictionary/skribbl.io. Initially, we did this using the pictionary words built-in, but then one of the students (when they were leading the advisory) came with words associated with our school (people, places, terms, etc.). That was so sweet.
12. Picked a Sporcle quiz, I projected it, and we collectively tried to see how good we could do working together. I typed what they told me to type. We did this today and we did a Pixar quiz (3 minutes) and a Harry Potter quiz (8 minutes). Also, one of my favorites!
13. Projected some “would you rather” questions I found online and had kids discuss them.
14. Since kids are doing a lot of cooking, I told them to take photos of their cooking process and their final foodstuffs… and then we shared those photos.
15. *One person gets put in a breakout room alone. The rest of the people get assigned some weird feature (e.g. they can only speak in questions, all their sentences have to start with the letter T, they have to cough each time they speak, they have to mention a color each time they speak, they have to address each person they speak to by their name, they have to do a strange hand movement at the camera each time they speak, etc.). Then the person has to come out of the breakout room and talk with everyone and try to determine what the person’s quirk is.
16. *Play Spyfall.
17. *Play Evil Hangman.
18. Update: We did this! (But I just found it online too!) We can do it with the annotation/whiteboard feature on zoom. Me (not playing) secretly emails everyone but one person the same object (e.g. A teddy bear) and one person gets an email saying they don’t get told what the object is. We go around and each person draws one line/shape on the board. A line is considered anything you draw without picking your pencil/pen up. The group has to decide — after two rounds of drawing — who wasn’t told the object being drawn. For example, let’s say everyone but one person was told the object was “teddy bear.” So the person who wasn’t told “teddy bear” has to be careful about what they draw… they’re always taking a risk when drawing… but the rest of the people can’t be too obvious either, because if it’s clear early on it’s a creature of some sort, the person who wasn’t told could draw an eyeball or something. If the group correctly guesses what the item is (watch the video to determine how the group guesses) then the fake person can still win if they can identify what the object was supposed to be.
(This is like Spyfall, but with drawings!)
Update: We did this today! It’s based on a game called A Fake Artist Goes to New York [video tutorial]
19. Update: My advisory played scattegories online today! I admit to not really understanding how it worked at the beginning, but by the end I think I had the hang of it.
20. Update: Today one of my advisees created a playlist on spotify. She shared her desktop with audio but had a random chrome page shared so we couldn’t see her spotify. She then controlled her spotify from her phone and played the music. The first person to identify the song won a point. I kinda loved this even though I definitely didn’t know many songs… and then someone else suggested next time we all rotate who shares a song so we can get lots of different genres and get to know other peoples’ musical tastes also. Without screensharing with audio, I tried just playing a song from my computer to see if kids could hear it, and they said they could hear it fine… so it seems like that would work pretty seamlessly.
21. Update from an advisor colleague at school (Carla K): Go visit a museum tour virtually together!
22. Update: We played “Charades” today! We started by each person just acting out whatever they wanted to and everyone else trying to guess. And then when a student had trouble coming up with an idea, I private messaged them on zoom to give them 3 choices to choose from. And then we played that way! No teams. Nothing serious. Super fun!

If anyone has anything they’ve done that’s fun and low-key like this with their kiddos in advisory or class, please throw down ideas in the comments! I’d love to add to our list!

Update: John Golden shared, in a recent culling of neat math/teaching things, this google doc which not only has games but also digital escape rooms!

Update: My friend Mattie Baker sent me this image which has lots of boardgames online! It includes some that I’ve listed above, and a lot more!

Update: A friend sent me this spreadsheet with virtual games to play.

# Distance Learning: Sorting Through It All

Technically I’m still on Spring Break, but this all ends next week, when we go back to school remotely. I’m one of the lucky few who didn’t have to get thrown into the fire immediately, so I’m using this blogpost as a way for me to sort through what I’ve done and what my take-aways are. I’ll be updating this as new things come my way, so I can keep track of everything useful in one spot.

## My dear friend @rdkpickle

So my friend @rdkpickle had to start distance teaching already. She’s kinda amazing in all the ways, and so on twitter she shared out how she was doing her lessons — and noted that they’ve been going well. They are low-tech in that they use Zoom and Google Docs, and use a Google Doc as an anchor for the lesson. I love that the doc allows kids who have to miss the lesson for whatever reason (emotional/anxiety issues, having to take care of a sibling, etc.) have a way to keep up.

Before sharing it, I want to say: seeing what she did was the very first thing that I saw that made me feel like: “okay, I can do this. It’s doable.” BLESS. When talking briefly with her online, she was saying right now she couldn’t be all investigatory in the same way she was in class, almost like she was ashamed. BUT very little of we’re doing is going to be like what we do in class. The ballgame has changed (from basketball to some other sportsball!). Right now, for me, the question is can I give space and structure and community to kids where they feel they can learn a few things. And @rdkpickle’s low-tech approach allows for that!

Here’s a PDF her googledoc, which she said I could share. (And here’s the google doc.)

## Zoom

Mike Flynn (helped by Sarah Bent) gave two wonderful webinars on distance learning that he has put online — March 11th and March 17th. (If you only have time for one, I’d watch the second one, but both are great.) They were some of the first things that made me realize distance learning was possible, by showing how to do it through his webinar. (Unlike, say, dry powerpoint lectures on teaching active learning strategies. Ahem. We’ve all been there. I just think over and over, “Physician! Heal thyself!”) My takeaways were both about distance learning and about zoom, so I’ll list them here. Fundamentally, though, the best way to learn zoom is to actually just get a few friends and all try it out together (each of y’all practicing being the leaders/hosts of the meeting).

• If you can, start the zoom meeting 10-15 minutes early and let kids know you’ll be there. You can just have informal chats like you do before a normal class, and you can use that as an easy way to start building community.
• You can record your sessions, but if you do that, don’t start recording during that informal chat time. (Right now, since Zoom is overloaded, it’s taking them a long time to get the recorded sessions on their website, FYI. But you can have zoom do a “local recording” on your laptop… so I was thinking if the file were small enough, I could just upload that to a google drive folder my kids could access.) Note that the chat box doesn’t show up on the recording.
• Talk with kids explicitly about the weirdness of talking on Zoom. There are going to be awkward pauses because we can’t use facial cues and body movement to figure out if we’re going to talk or not (we’re all sort of trained to sort of check before we talk so we don’t start at the same time as someone else). So name that, and say that awkwardness is normal in zoom. You should also mention (and give) lots of wait time — just like we should be doing in our regular teaching.
• It’s okay if you’re having kids use chat to stop every so often and take a few minutes (in silence) to read over the chat so you can respond to what you’re seeing.
• The chat can be the “lightest lift” for interactivity, but it’s effective! One tip I got on twitter is that you can ask everyone to write a response to a question, but not press enter until you give the command. Then you’ll get a quick flood of responses that you can go through, and students can also read.
• You can also set the zoom meeting to have the chat be private – so students are talking to you but not each other… then as you see the responses, you can say “Nice thinking, Jake!” or “If you’re thinking about a parabola instead of an exponential function, you’re going in the wrong direction!” This came directly from Michael Pershan’s experience teaching online this past week:
• If you have pre-determined questions you want to ask at a particular time during the lesson, have them written in a google doc/notepad, so when you want to ask it, you can just copy/paste them in.
• Have everyone use their own regular names in zoom (and not emails or userhandles) to make life easier for you.
• There is a way to include “polls” in your zoom meetings, but I couldn’t figure that feature out when trying it out!
• You can divide your class into groups (either randomly or pre-determined) and send them to breakout rooms. You can visit any of those rooms and join in the conversations. Each breakout room is given a number when students join. You can have one person in each group (e.g. the person whose last name comes first alphabetically) to create a Google Doc in a Google Drive Folder you share with them in the chat window… And title it “Group 5, March 25, 2020.” Then all participants can write answers in their google doc and you have access to all of them in an organized way.
• When students are first put into a breakout room, if they’re new to working with each other, start with a non-mathy but quick ice-breaker to get everyone talking (e.g. what’s your favorite pizza topping?) and build a tiiiiny bit of community before diving in.
• SUPER COOL DISCOVERY: When I did this in Mike Flynn’s webinar, one person in my breakout room showed me a ridiculously cool feature. In any google doc, you can go to INSERT > IMAGE > CAMERA

And then you just take a picture of your work using the webcam, and it automatically inserts the picture in the google doc!

Bam!
• Don’t go crazy with the new technology. There are so many apps and websites. Limit yourself to just a few, like two, for your own sanity and your students’ sanity. Keep it simple and easy — don’t go down the rabbit hole of looking for “the perfect way to do x, y, or z.” Be okay with the tradeoff of having “good enough.”
• When designing online learning, start with the question “how do we want our students to learn?” Then choose your technology based on that.
• Screensharing is awesome (so you can set up a google slideshow, and in zoom you can screenshare that slideshow to the kiddos… And you can show kids how to annotate so individuals or the whole class and write/type/draw on a screen you’re sharing (and you can save that).

## Desmos Activity Builder

Julie Reulbach led a webinar on using desmos for assessments, but basically she outlined all the ways we could create activity builders to actually teach content also, and bring students along with us as they navigate the pages, and we talk through what they’re doing. Her resource page is clipped below so you can see what’s there…

But importantly, her page includes links to various activity builders where you can simply copy and paste! Here’s how you copy and paste screens from better activities that your own into your own! They can even have computational layer in them!

Some key tips for creating Activity Builders (but not necessarily for assessments in particular):

• Steal steal steal screens from other activity builder assessments if you’re doing anything fancy (e.g. self checking, anything with computation layer), because there’s no need to reinvent the wheel right now. Julie has curated a whole list of activities that she takes screens from! And desmos has curated a bunch of starter screens(e.g. “graph how you’re feeling today?”) that you can take!
• DESMOS NOW ALLOWS FEEDBACK – so you can write a note to individual students.
• Importantly, let’s say kids are doing a desmos activity or assessment, and you want them to show their work that they’ve done on paper. All you need to do is create a screen which has a blank graph, and instruct kids to insert an image (see sidebar instructions in the image below) and they can actually INSERT a picture of their work! The workflow is a little clunky because they have to take a picture on their phones and then email/airdrop it to their laptops, and then select that file. But it took me only like 20 seconds after I had done it once.

Some key tips for using Desmos for assessments:

• Have a fun introductory “hi there!” screen
• Have students fill out an honor code/statement screen first if doing a formative assessment
• After you finish the activity, have two screens at the end. First, a feedback screen so you can find out how they felt it went. Second, a screen asking them if they have any questions or anything they want you to know.
• If a formal assessment, you should PAUSE the activity at the end — so kids can’t go back and change their answers or share the class code with other kids

## Michael Pershan’s blogpost

My friend Michael Pershan has been in the thick of online teaching. He wrote a detailed blogpost about what he’s discovered thus far. I highly recommend reading it! Big takeaways:

• His school is using Google Classroom (like ours does), so he’s using that to create a system of organization for the kids, with instructions given day-by-day (within a unit):
He noted: “The most important thing, though, is that each learning activity becomes its own “assignment.” During week 1 I was creating large documents that students were working on over multiple days. This was good in one sense, because I had to post only one thing. But it became very difficult to monitor the progress of kids through the assignment at all. And then it became tricky to modify the plan in the middle of the week by adding on other bits of classwork.”
• He’s using google classroom to teach kids how to upload their written work. (Note: my kids always submit PDFs of their work on google classroom, so they’re very familiar with this!)
• To give feedback on google classroom: “Google lets you comment on the work itself via highlighting and commenting, but I’ve found it more useful to give a quick written comment that appears under the assignment itself.”

Lots of great things being shared on twitter. It was so overwhelming that I stopped looking at twitter for a while, but I did save a few things:

## What Wasn’t That Useful For Me – But Here are the Nuggets I’ve Taken Away From These

What I have below doesn’t mean these aren’t good for others. It just means that for me, I like to jump in and these things didn’t quite pan out fully.

Alice Keeler had a webinar (“Oh Crap, I’m Teaching Math Online Now“) that wasn’t crazy useful for me because it was a brief overview of many things I already knew about. It was just super tech happy (look at Pear Deck! Look at Geogebra! Look at Desmos! Look at …) and didn’t give me the focus or vision I’m searching for.

Global Online Academy (GOA)’s 1 week course on Designing for Online Learning. Since this was designed to be “big picture” (so it can accommodate people from many schools and teachers of all stripes and many disciplines), I had trouble getting specifics that I wanted to latch onto. Here’s what I did get:

• They recommended Loom for laptop screen recording, if you were going to be making videos from your laptop. It seemed pretty seamless and easy to use, based on this short video tutorial:
• The basics of good teaching are still important — clarity and being organized is paramount. Specifically for online learning, they highly recommend:
• building routines early (just like with regular teaching!)
• share the “learning goals” at the start of each lesson explicitly
• don’t get over-excited and share too much… curate what you share and make it super easy to follow
• using a lot of whitespace and images
• don’t include anything that isn’t super important — focus on key ideas
• not using too many fonts
• everything you share with your students should be “crisp” and “clean” (not “busy”)
• Be present for students. Create or adopt an online persona. Don’t leave them hanging, but show them continual engagement so they know you’re with them on this journey.

# Part II of Machines: Helping Us Understand Inverses

Here is Part I. Read that first! Also, I’m trying to write and post this quickly, so sorry if it is incoherent, monotonous, etc.

Okay, so now kids understand machines have inputs and outputs, and they understand that the “rule” can take different forms: words, equations, tables, and graphs. Wonderful.

Machines to think about Functions vs. Non-Functions

So recall that our definition of machines was:

So I had kids try to see what might go wrong with these machines…

From our conversations over these problems, students were able to see which machines were “problematic.” At this point, I told them machines that worked were called “functions” and machines that didn’t work were called “non-functions.” Conversations we had:

• We talked through what made something a function (every allowable input had a single output) and which made something a non-function (there was one allowable input that has multiple outputs).
• I had kids look at the graphs and come up with a quick way to “see” if a graph was a function or not… so from this, they came up with the vertical line test on their own.

We did a lot of practice with this idea. Kids were asked to look at a bunch of representations and decide if they were functions or not. And if they weren’t functions, they had to provide a concrete example showing where they failed (e.g. for (g) below, an input of 2 gives two ouptut of 2 and 4.)

By the end of this, I was very confident kids understood the idea of functions and non-functions.

Combining Basic Machines

Okay. Here’s where we start to get more abstract. I start telling kids that for now, we’re going to focus on four basic machines (machines with add, subtract, multiply, and divide)… but because I’m lazy and can’t make cartoon machines all the time, I’m going to come up with a simplified notation for them…

You, dear reader, might wonder why I’m using “blah” in these machines. That’s because it is helpful when we start combining machines:

Yes! It’s like a conveyor belt. Each machine takes in an input and spits out an output… that then becomes the input of the new machine… So they could start figuring out questions like these, which made me happy!

Now #17 was really tough for kids. But I let them struggle before guiding them. From this one problem, students could start seeing how equations and machines were related. By the end of our conversation, kids knew the line was $y=2x+4$. So if they have an input of $x$, they multiply that input by 2, and then they add four to the result. Which means the machine would be:

____[blah*2]____[blah+4]_____

And so students started plugging in various values for the initial input (x value) and saw they got the final output (y value). Then we substituted x into the machine… and got 2x for the middle blank, and then 2x+4 for the final blank! Seeing that really helped kids drive home the connection.

Some kids got the second way to write these machines by trial and error. But I was hoping they’d rewrite $y=2(x+2)$. And then think if they have an input of $x$, they first add two to it, and then they multiply that result by 2. Which means the machine would be:

____[blah+2]____[blah*2]_____

Creating Machines from Equations and Vice Versa

We then became comfortable going from a machine representation to an equation representation, and vice versa.

If I gave students: $y=2x^3-4$, they would say: we cube the input, multiply it by two, subtract 4. So the machine would be _____[blah^3]____[blah*2]____[blah-4]_____.

Or if I gave them: $y=-\sqrt{-x+3}$, they would say: we take the input and multiply it by -1, we add three, we take the square root of that, and then we multiply by -1 again. So the machine would be _____[blah*-1]____[blah+3]____[sqrt(blah)]____[blah*-1]_____.

And also in reverse, students could be given a machine, and easily come up with the equation by substituting in $x$ for the input, and come up with the output. So:

___[blah-3]___[blah^2]___[blah-4]___

would look like: $x$, $x-3$, $(x-3)^2$, $(x-3)^2-4$. So the equation represented by this machine is $y=(x-3)^2-4$.

Basically, students are seeing how the basic equations are built up and broken down. What’s nice about this is order of operations starts to really get emphasized and naturalized.

Creeping Up To Inverses

At this point, kids are comfortable with combined machines. And so I throw them a backwards question, something they’re used to (since they’re my favorite type of question to give kids). First I start off concrete…

… where they were doing a lot of thinking about inverse operations. But then I had an activity where students were trying to create machines that would “undo” another machine. By the end of this activity, students were starting to create their own inverses. They could do problems like this:

I give you this machine which I will call machine $M$: ___[blah-5]___[blah*7]____,
You need to tell me what machines I could append to the end of machine $M$ would make any input be the same as the final output.

So kids eventually saw the appended machine would be: ____[blah*1/7]___[blah+5]_____

So the big machine would be: ___[blah-5]___[blah*7]____[blah*1/7]___[blah+5]_____

And any input would also be the output (e.g. if we put in 1, we’d get 1 –> -4 –> -28 –> -4 –> 1.)

We called the machine created the inverse machine… and we named it machine $M^{-1}$.

So the inverse machine of ___[blah-5]___[blah*7]____ was ____[blah*1/7]___[blah+5]_____. Kids saw we “read” the original machine backwards, and did the inverse operations.

From this, we started going into inverses of lines:

Eventually, we got to the point where kids would be given $y=x^3+4$. To find the inverse, they would create the machine for this: ___[blah^3]___[blah+4]___. And then they’d create the inverse machine: ____[blah-4]____[cuberoot(blah)]___. And then they’d find the equation that this machine represents by substituting in $x$: $y=\sqrt[3]{x-4}$.

So questions like these didn’t really faze them:

Inverses with more representations…

From here, I started having kids come up with inverses of tables. I reminded them that if we combine a machine with its inverse, whatever we input into the big machine should be the same output… So let’s see what happens…

And this was lovely… We’re combining two tables… and our goal is to create a large machine that when an input goes through both of machines, the output would be the same!

So look at the two tables above as rules. And we’re going to combine both tables to make a big machine. So kids saw from this that if we put an input of -3 into $M$, we’d get 5 as an output… and when we feed 5 into $M^{-1}$, and we must have -3 as an output (since the output after going through both machines have to be the same as the initial input). So the inverse table is going to look like the original table, but with inputs and outputs reversed.

Why is this so beautiful? Because from this, kids saw that for inverses, the domain and range swap. And they also saw that to create an inverse, you simply have to switch the x-values and the y-values with each other. You get all of this for free!

And then I gave them graphs, and told them (with no instructions) to come up with the inverses… But since they had done tables, and see how the tables just swapped the inputs and outputs to get the inverse, they had no trouble drawing the inverse graphs.

And it’s lovely. Because they figure this out all naturally. I didn’t have to tell them anything but kids were accurately drawing inverse graphs. Putting the graphs right after them doing inverse tables was genius! And some kids came up with the fact that inverse graphs were reflections over the line $y=x$ themselves!

Of course, sometimes inverses exist but aren’t functions… So I threw everyone some curveballs…

And they saw how they could create the inverse… they could fill in the table or graph… But they saw why the inverse was “problematic” (a.k.a. not a function).

So now kids were thinking: okay, what’s the inverse? Is the inverse a function or not?

I drove this home with lots of questioning…  We had previously looked at these questions and decided if these each were functions or not. But now kids were able to decide if their inverses were functions or not.

They immediately were looking to see if any outputs had multiple inputs associated with it. And they came up with the horizontal line test on their own. It was glorious.

Going The Very Last Step with Inverses

From all of this, kids learned so much. They saw how to graph inverses. They saw the inverse graph is a reflection over the line $y=x$. And then we drove home the idea that the inverse graph is the same as the original graph, but with every x-coordinate swapped with every y-coordinate. To polish everything off, we saw the equation for the inverse graph can easily be found by swapping the x variable with the y variable.

So the inverse of $y=x^2$ was $x=y^2$.

So finally, my kids could answer questions like:

Sorry this was so long and scattered. But stay tuned. My favorite thing is coming up… whenever I get a chance to write the next post!

# 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.”

So 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:

and

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.