Hello, reader! If you intend to post a link to this blog on Twitter, be aware that for utterly mysterious reasons, Twitter thinks this blog is spam, and will prevent you from linking to it. Here's a workaround: change the .com in the address to .ca. I call it the "Maple Leaf Loophole." And thanks for sharing!

Saturday, December 21, 2013

My Geogebra Fancy Pants

The #1 thing you should learn in Geogebra if you want to make cool-as-in-cool things is the Sequence command. If you're already familiar with the sequence command on TI's it works the same: Sequence[expression, variable, start, end, interval].

As a simple example, here is a tool I made to subdivide a segment with evenly-spaced points.

But you don't have to just make sequences of points, you can make sequences of anything! Segments, so you can make a grid with a variable number of gridlines. Or an ice cube chopped up into smaller ice cubes:

Sectors, so you can make a customizable circle graph.

And this thing for Pandemic that is just too fun:

Sunday, December 1, 2013

Catching Fire/Hell

Warning: cranky old lady rant coming. Avert your eyes if you don't like this sort of thing.

I saw Catching Fire last night, on its second weekend, at an 8 PM show on a Saturday. Normally, I go to matinees, because I am cheap. But, for complicated reasons, I was there. This is all to say, I haven't been to a crowded showing where the audience skews young in quite a while.

During the previews, I could see lots of phone screens. Maybe a dozen. They were the brightest things in the theater. Far brighter than the projected image on the screen. I thought, surely, everyone would put away his phone when the movie started.


The young man (age hard to tell...I put him at 16-20) sitting right next to me, in fact, was looking at his phone more than he was watching the movie. He was reading his Facebook feed, composing status updates, and tagging lots of people. I know because it kept distracting me from the film, so I read over his shoulder.


I noticed, though, that he kept logging out of Facebook, so when he went back to check it again, he had to log back in. I reasoned that he was trying to deter himself from checking his phone. Each time he finished, he thought, "I know, I'll log out. That way, it will be a pain to get back on, which will make me less likely to check it again." (I often use the same logic when polishing off a pint of ice cream.)

Before the film had started, he had to leave and come back twice, awkwardly stepping over my companion and me. Each time, he apologized for inconveniencing us and said thank you, and we were, of course, very polite and accommodating. This was not a rude kid.

So about twenty minutes into the film, when I couldn't take it anymore, I leaned to him and loud-whispered, "YOU KNOW. THAT SCREEN IS REALLY BRIGHT." He apologized and put his phone away. I thanked him. I was afraid that might not last very long, but no. He didn't turn it back on for the rest of the movie. There were a couple others in rows further down, and they were annoying, but they were too far away for me to yell at. And the one right next to me was the one really ruining the movie. Which cost eleven dollars.

The kids in the theater didn't make me mad. They're kids, and they need to be told. What made me mad, after the fact (I was not ruminating on this during the movie, mind you. It was great. You should go.) was remembering every dummy on the Internet, who inevitably is not someone who spends much time in a classroom, who suggests that if teachers' lessons were interesting enough, kids wouldn't be tempted to distract themselves with their phones. Therefore we shouldn't have to require kids to put away their phones sometimes during math class.

Um, Francis Lawrence can't keep kids from being distracted by their phones. With a crazy-good story about reluctant teen revolutionaries. And a $78 million budget.

Learning takes focus. Focus takes practice. Kids might never know how jaw-droppingly cool are the things we are trying to teach them, if their focus is interrupted by Snapchat every two minutes. There are some things they need to be told.

Saturday, November 9, 2013

Tell Me Why You Blog

So, as much to my surprise as anyone's, I'm not only talking at NCTM in April but they made me a featured speaker? Only freaking out a little. I applied in response to a few people who shall not be named (unless they want to out themselves) proposing "a possible blogging strand with maybe a panel or something." So as you can imagine, the idea for what I'm talking about is super well thought-out and fully baked right now. (That was sarcasm, if that wasn't clear.)

The benefits to written, public reflection are, to me, by this point, so internalized that I find them hard to articulate. And, "reflective practice" as The Thing to Do seems to have gone out of fashion. Now it's all about data. Data is the new Reflection.

Ahem. If you would, comment on this post and share with me some things:
1. What hooked you on reading the blogs? Was it a particular post or person? Was it an initiative by the nice MTBoS folks? A colleague in your building got you into it? Desperation?
2. What keeps you coming back? What's the biggest thing you get out of reading and/or commenting?
3. If you write, why do you write? What's the biggest thing you get out of it?
4. If you chose to enter a room where I was going to talk about blogging for an hour (or however long you could stand it), what would you hope to be hearing from me? MTBoS cheerleading and/or tourism? How-to's? Stories?

And, please, link your reply back to your blog, if you have one. I'll make every effort to cite appropriately. Feeling a little weird about crowdsourcing this but I should get over that already. This community has already helped me crowdsource lessons, units, math research, and recommendation letters. Lots of us like to say we got involved and stay involved in this so we can suck a little less. I only get one chance to not suck in New Orleans, and I'd love your help. Let's hear you.

Thursday, November 7, 2013

Today's Stats

Emails sent: 7
Tweets sent: 3
Times scolded by strangers on Twitter: 1
Nice @mentions by friends on Twitter: 6
Scantron forms ordered: -23,500
Phone calls made: 13
New Desmos features requested: 3
Spreadsheets dominated: 4
Cupcakes eaten: 1
Steps! 8,235
Google Hangouts: 3
Dead leaves on front porch: all of them
Earbuds destroyed by cat: 2
Pots of chile made: 1
Jamie Foxx/Channing Tatum movies watched: 3/4
Kitchens cleaned: 0

Wednesday, November 6, 2013

Tools Smart People Made That Make My Life Easier

Boomerang. It's a Gmail plugin. If you use Gmail, you should get it. It does two really great things.
  • Write an email now, schedule it to be sent later. Say you want to get an email written so you can stop thinking about it, but you don't want the recipient to know you are a freak who writes work emails at 2:30 AM on Saturday morning. No problem. Write the email, and tell Boomerang to send it Monday at 8. Boom. Erang.
  • You email someone a question, but this person maybe doesn't always, you know, answer. Without help, it's easy for a month to go by, and then you're like, oh, snap, I really needed an answer to that. You can use Boomerang to Boomerang the email back to your inbox in a few hours, or a few days, or a week, if there is never a reply. I'm pretty sure that's why it's called Boomerang. (This feature also makes people think you have super memory powers, or at least that you are somehow On Top of Things in a way that they are decidedly Not.)
Evernote. Replaces my URL bookmarking, notebooks, filing cabinet, and to-do list. There are a zillion wonderful articles that go into great detail about why it's so great, so I won't reinvent the enthusiasm wheel, here. Here are a few specific things that I am loving it for, lately:
  • I find myself writing what are essentially the same emails over and over again. For example, instructions for teachers in our online PD project to join an Edmodo group. I have the instructions saved in a note in Evernote in large, friendly letters and upbeat prose that I wrote after a glass and a half of wine. Instead of spending ten minutes to write that email from scratch several times a week, I just copy it from Evernote, tweak a few details, and boom. (I realize that the text is agnostic about where it is stored, and you don't need Evernote for this. But I like having all the things I need to keep and remember in one place. It's just one more thing I use this app for.)
  • The Secret Weapon. Seriously. "Weapon" is not an overstatement. If you tend to forget to do things because the tasking gets buried in your email inbox, or you keep writing the same thing on hand-written to-do lists over and over again, this is worth a read. Check this out:

Tasks and to-dos are entered as notes. They're tagged with what project they are for, when I need to do them, where, and who is involved. By clicking on any of those tags in the left column, I can see what I need to get done today, or everything for one project, or everything I need to talk to Karim about. When done, I move the note from the Action Pending notebook to the Completed notebook, so I have a record of everything. EVERYTHING. It's pretty magical.

Doodle. The next time you have to schedule a meeting, avoid an interminable chain of reply-all emails that say things like "I can come Wednesday between 9 and 11, Friday after 3, or any time on Sunday." Send those people a link to a Doodle instead. They can all click on the times they're available, you can easily see a time that works for everyone, and they will think you are a genius.

Certain OSX shortcuts
  • Spotlight search. I've loved keyboard launchers for a long time, and I used Quicksilver for a long time. That functionality is in OSX now, and it's super fast. Instead of having a million icons in the dock, or opening Finder every time I have to open something, I type ⌘-Space and then the first few characters of the thing I want to open. It's so fast.
  • Insert a link with ⌘-K. Instead of sharing a gross, ugly URL with someone (Google Drive shared documents are particularly heinous), highlight text and type ⌘-K then paste in the link. Voila, pretty hyperlink.
  • This is a Chrome thing, but you can right click on a tab, and close all the other tabs. Take that, other tabs. Aaahhhhhh.

Tuesday, November 5, 2013

Treading Lessons

Mildly frustrating couple of days at the Lish. Yesterday we tried to write a lesson about Scrabble. Is the letter distribution of the tiles correct? Do the assigned points values make sense? How do these things compare to Words with Friends? Lots of data. Lots of good math questions to ask. But we couldn't find a compelling overarching structure. Frustrating, because I heart Scrabble.

Today we tried to write a lesson about Black Friday. When should stores open? Do stores pay a price for opening the earliest, because they get negative publicity? Is there a race to the bottom? Or rather, to the early? This was one of those lessons where my Economics-major colleagues start invoking concepts I don't understand, and I can't help but kind of zone out and work on something else. (I prefer things we can measure. Sorry.) Which is actually fine, because once some questions are written, I will be able to come at it fresh and help evaluate if the story in the lesson makes sense.

Yesterday we were visited by the great and powerful Max Ray and his sensei Lois Burke, and I am so glad. I'm embroiled in this online teacher professional development research study this year (I would like to write about it here, but oy. Its tentacles are many), and we're having trouble with the online part. Max has lots of experience in what works with the online part. He very patiently let me describe our setup and offered some suggestions. What I appreciate about Max is, he's very calm. Being around Max is to feel like everything is going to be okay.

For Elizabeth: my cooking lately has leaned vegetarian. I made an awesome yam coconut curry soup last weekend, and have made two of these crazy rice bowls. Yum. Right now I have a CSA half-share in my kitchen that is mostly squash, so I think there is some roasting in my near future.

Sunday, November 3, 2013

I Feel Guilty about My Blogging

Hey, Internet. What's up? I miss you. 

I only write anymore when I think I have something unique and important to say. Scratch that, I think that's been my threshold for a while. But there's been lots to learn at new job. I've been less trusting of my "what's unique and interesting" radar. 

But, here's a thought. A hubristic one, but still. There aren't that many people writing real-world lessons for middle and high school and selling them on the Internet. And coordinating a research study about the effectiveness of online teacher professional development. And helping to write unit blueprints that teachers can use to give some structure to high school CCSS courses. Things that happen around me are probably unique and interesting, and maybe occasionally important. Last week I spent 2 days on the other side of a booth on a conference exhibitor floor. I taught, over email, grown-ass adults how to use the Internet. Chris taught me how airlines make decisions about overbooking flights. I still learn alot. I believe in making learning public.

So this is what's happening: a post every day this week. I've never put myself on a blogging schedule before. I need a low threshold for commitment. Five days, starting tomorrow. Posts will be up by 10 PM, and feel free to yell at me. Let's see how this goes.

Saturday, October 12, 2013

Evens and Odds

update 10/16: Look at this coolness. Thanks for sharing, Øistein.


This is great. I shared this on Facebook, and it piqued the interest of all the math teachers, of course.

But also some elementary teachers. Who recognize that there's something important here, that kids could be doing. But who don't have much experience with proof themselves, and aren't sure what it would look like in their classes.

This has got me a little obsessed. What kinds of proofs would be appropriate for little kids to explore? If we're talking, say, third graders, lots of them don't really get multiplication yet.

I see alot of inspiring mathematical...stuff...that teachers have no idea what to do with. So I tried to write something out. Here's what I imagine this could look like. Sharing just in case anybody finds it useful. If this is all old news to you, I'm not trying to insult anyone's intelligence. Also, the standard disclaimer: I don't live in a vacuum; there's plenty going on in here that was inspired by other people.

If anyone has little kids they can try it on, I'm curious to hear how it goes.  My anticipating-response muscles are very rusty.

Suggestions for things to say out loud.
Suggestions for things to write on the board.
Other notes about what is happening.

Tools: As many of these as we can muster
  • Dot Paper
  • Counting Chips
  • Colored Pencils or Crayons
  • Blank Paper
  • Mini Whiteboards
  • Whiteboard Markers

What do you notice?
- 2 minute think time
- write them all on the board under Noticings

Some time after someone notices that they are all odd numbers

What if we add together one number from the green sack, and one from the red sack?

Students share results. What do you notice? Keep writing noticings.

Some time after someone notices that the sum is always even:

Whoa, really? Always? Did anyone get a sum that was not even?

Do we think this is always true?

Write conjecture on board.

We think this is always true:
“When we add a number from the green sack and a number from the red sack, we always get an even number.”

Did we check them all? Good.

Can we make this statement any more interesting?

“When we add together odd numbers, we always get an even number.”

Can anyone find an example where this is not true? Where you add odd numbers, and get an odd number?

Possible: 3+5+7 = 15

“When we add together two odd numbers, we always get an even number.”

What would prove that this was not true? (Someone would have to find an example of odd+odd=odd.)

Have we checked all the odd numbers?

How many odd numbers are there?

Maybe we just haven’t stumbled on some odds that add up to odd. Maybe they’re out there...

(possible: give time to look for a pair of odds that add up to an odd. not sure if this is necessary.)

Have we checked all the odd numbers anywhere in the universe?

(This is part of what math is. Noticing that something seems to be always true, and convincing ourselves it's always true, even when we can’t check all the examples. We need a way to explain why odd + odd is always even, no matter what the numbers are.)

What makes an odd number odd?

What makes an even number even?

Time to play with math toys. Dot paper, counting chips, etc.


This is one example of a proof. There are others.

This first part is the important part. Kids need to stumble over it themselves. Don’t show them. If this doesn’t get proved in one session, it’s okay. Leave the loose end and let it linger. Some of them will keep thinking about it.

Even numbers can be arranged in two equal rows. Odd numbers can’t. If you arrange them in two rows, one row has an extra bit hanging off:

But when you bring two odd numbers together, the extra bits come together to make their own pair.

Can be proved with algebra, depending on grade level.

Even numbers can all be written 2 * something. Odd numbers can all be written 2 * something + 1.

Take an odd number. Can be written as 2n + 1, where n is a natural number.

Take another odd number. Can be written as 2p + 1, where p is a natural number.

Add them: 2n + 1 + 2p + 1

Gather terms: 2n + 2p + 2

Factor out a 2: 2(n + p + 1)

This is 2 * something, therefore it has to be even.

Tuesday, September 3, 2013

Building Functions, Clarified

So, I really appreciate all the thoughtful input on this previous post. I started commenting, but the comment got real big.

To give some background, I'm writing a unit flow for an introduction to quadratics unit. The big things I'd like students to remember from this unit for a long time are:
  • some situations that can be modeled with quadratics, i.e. falling objects, triangular numbers, area
  • compare/contrast with linear and exponential (which come before this). how can we tell if given information (situation, table, graph, 3 points) can be modeled with linear, exponential, or quadratic?
  • what information can easily be obtained from each of the forms (vertex, standard, factored), given an equation
  • what form (vertex, standard, factored) is it easiest to write an equation in, given different information
  • why the graphs look like that (why symmetrical? why a U-shape? why does it have a max or min?)
What I DON'T want to do in this unit is algebraically convert between equivalent forms, or solve quadratics with various techniques.

The big question is still, how do you introduce things that are more complicated than ax2?

Option 1: Okay, kids, come up with a rule that models something like {(0,3), (1, 4), (2,7) (3,12)}


Option 2: y=a(x-h)2+k with some sliders, mess around on Desmos and see how a, h, and k make it different from y=x2. Now we'll also mess around with y=a(x-p)(x-q) and y=ax2+bx+c.

The room seems to be divided on this, but leaning toward option 2.

Sunday, September 1, 2013

Pretty Big Ideas for Intermediate (Highschoolish) Mathematics

I am just jumping into this for fun. I think they all capture the idea of a big idea or pretty big idea, but feel free to argue.

variable - we can work with quantities, and learn things about them and draw conclusions from them, even when we don't know what they are.
inverse operations - operations can be undone (and sometimes they can't, at least not uniquely) and this is useful for solving all kinds of problems.
functions - Different kinds of rules that map a set of numbers to another set of numbers follow certain patterns.
transformations - rules can be changed in systematic and useful ways.
equivalence - How do we know when things are the same? How do we know when they are not the same?
proof  - Usually a big feature of a geometry class, but I'd argue at least as important in algebra. How do we know for sure that something is true?

Monday, August 26, 2013

Building Functions

How do you motivate vertex form for a quadratic? Do you just drop it on them? That's pretty much what I used to do in Algebra 1. Hey, kids, you want to model this u-shaped path, like you get when you toss a basketball. I'm just going to tell you to start with y = a(x - h)2 + k. Start messing with a, h, and k, and see what happens. Let's see what we can say about how they each affect the graph.

There's an awful lot of "building functions" in the common core, and an awful lot of modeling, and I think it's great. The whole F-BF header should be a playground. I'm just not clear on how you take a class there.

You can look at sequences of patterns easily enough that result in y = ax2, and I suppose patterns that result in y = (x - h)2 and y = x2 + k. Do those arise naturally anywhere? Or do you choose carefully something from Visual Patterns?

What else? "Fit a quadratic function to a photograph" seems to be a favorite of presenters at conferences who want to browbeat teachers for not making class real-world enough. But how do students develop those functions in the first place, in an authentic way? I feel a little awkward for asking, because I feel like I should already know this. But I also suspect that not that many people have a great answer.

Sunday, August 25, 2013

The Fault in Our Stars has some wrong math (spoilers)

"There are infinite numbers between 0 and 1. There's .1 and .12 and .112 and an infinite collection of others. Of course, there is a bigger infinite set of numbers between 0 and 2, or between 0 and a million. Some infinities are bigger than other infinities. A writer we used to like taught us that. There are days, many of them, when I resent the size of my unbounded set. I want more numbers than I'm likely to get, and God, I want more numbers for Augustus Waters than he got. But, Gus, my love, I cannot tell you how thankful I am for our little infinity. I wouldn't trade it for the world. You gave me a forever within the numbered days, and I'm grateful."

I love that this character used mathematics to express her love to her dying boyfriend. It still bugs me that the math is wrong. Gus gave her the same size infinity between 0 and 1 as she would have gotten between 0 and 2, or 0 and a million. I'm afraid this makes me a heartless asshole.

(The author acknowledges that the math is wrong, and claims it was intentional. I believe him. I don't know what to do with it.)

Saturday, August 24, 2013

A little bit of what I intend as thoughtful pushback

I went to Steve Leinwand's global math presentation this week. It was thought-provoking and worth the time. You should go watch it. I asked a question and got an answer that helped me grow.

So, Steve's all like, hey teachers, you need math smarts. Check! You need good tasks. Check! You need effective instructional practices. You suck at these!

These are his major recommendations, as I understand them. And they are very good. Everybody, listen and heed.

1. Students do more justification as a regular part of math class. Note: this doesn't mean "proof" as I clumsily tried to ask about during the session. Proof, of course, has a precise meaning in mathematics with a level of ironcladness that we're not talking about here. Rather, this is more informal, could be written but often verbal, justification. Which is just as well, as far as I'm concerned. The word "proof" scares K-12 teachers more than having class outside, chaperoning dances, and unannounced classroom observations combined.

Practically, this means students solving fewer problems, but being required to write about their process and reasoning.  Additionally, student responses ought to be normally followed up by a request for justification.

2. Ask students to estimate more often. Before solving or mindlessly applying a formula, students are explicitly asked to estimate a reasonable answer. Fine. Awesome. Takes zero time. If you're not already doing this, get with it and start.

3. Collaboration centered around classroom video. The recommendation is, everyone tapes him/herself regularly, and clips are randomly selected at staff or department meetings to watch and critique.

Review: I'd be fine with this. I'd be all, look at how amazing I am, bitches. Let me hear your critiques; I am eager to learn from them. I wish Steve Leinwand would come observe me, so he could see one teacher asking kids to explain the reasoning behind their answer (which is evidently rare in the classes he observes) and approve of me and everything I do, and maybe adopt me. 

95% of my colleagues would not have been fine with this.  And I'm pretty sure that where there are strong unions, you can't make them.  And where there aren't strong unions, life sucks anyway.

Case in point: central Virginia. I worked with some great teachers this week. Whip smart. Love their kids. Doing their wholehearted best to do the best thing. In a nutshell, good people. This year, the high school teachers will be teaching 6 classes a day instead of 5, with no change in salary. Computing...20% increase in workload, 0% increase in pay. 

Talking to more people locally, evidently this is a trend in Virginia. Where unions and contracts are not commonplace features of school employment as they were in New York. School boards can basically demand whatever fuckery, and rely on compliance, because, teachers are part of their communities and are committed to their work and don't want to do something else or move.  So they're taken advantage of. 

But then, to suggest they collaborate with nonexistent time in their work day is a bit laughable.

Don't kid yourself, it's a 20% increase, because it's 20% more children, when you teach children as opposed to teaching a subject. There are 8 periods in a day. So, 3/4 of their work day, they are executing carefully-as-possible planned interactions with kids. They have two other periods to spare. One is lunch. Humans require a meal in the middle of their day. It's civilized. The other is given over to a duty, like study hall or checking hall passes or whatever. They probably have one hour after classes end when they are expected to be available to help kids. Then, they need to leave and pick up their own kids from daycare, because they are humans who have lives. Our active-on-twitter friends take a ton of time to plan instruction outside their normal workday because they are math ed nerds. And they're awesome. But can we expect more than a small fraction of our colleagues to travel the same course? Not really.

When are these folks supposed to collaborate? When are they supposed to do the cognitively demanding work of figuring out how to get another person to understand something? When exactly? Leinwand, from what I've seen, brushes these concerns off. He says, teachers typically teach 5 out of 8 periods a day, and can find time to collaborate if they really want to. But at least where I live, these are real questions and not excuses. Act like teachers are kicking back and drinking coffee during luxurious planning periods, and you'll lose them.

Please note I'm not arguing with the premise. But we need to acknowledge the dearth of and fight for collaboration time. Which I don't believe will happen without funding that gives professional educators more time in their day not in front of the children. In places that do this right like Japan and Finland, teachers spend half as much time in front of students as American teachers - and they spend all that extra time: planning, collaborating, developing lessons, and generally kicking our asses. Policy matters. Funding matters. Yelling louder at classroom teachers isn't going to solve this problem.

Sunday, June 30, 2013

Making a Gift More Valuable

Spiegel Online: Forensic Anthropologists at Work
I'm starting to feel a little like an anthropologist, but I'm finding the implications of and discussions stemming from the last post framing the MTBoS as a gift culture, to be fascinating. Logical questions are: "What is a gift? What kinds of contributions earn a person status in our culture? If you're going to participate by gift-giving, anyway, are there steps you can take to make your gift more valuable?"

I think we'd all agree that status itself, here, is not the goal. That would be silly. But it can be a motivation, and that's okay. Importantly, the gifts make us, all who are participating in many different capacities, better teachers. That's worth paying attention to.

There are different kinds of gifts this community finds valuable: curation, commentary, cheerleading. But a discussion on Twitter today made me want to write down some guidelines for what features make a gift more valuable. Several people expressed incredulity, arguing that an artifact's value is too dependent on the needs of the receiver to make this exercise meaningful. But I disagree. While you might find one gift more valuable than I do, gifts can have general features that make them objectively more valuable to the community.

I am not posting this to make anyone feel like they should do something. Let's please keep the MTBoS easy fun free. You're free to do some, none, or all of these. You're free to quit this tab right now and order a pizza. But my feeling is, the more your gift displays these features, the more useful and valuable it will be. The ever-incisive Justin Lanier stated the query thusly:


Pershan's Desk

This one is about community. It's also about leveraging MTBoS so everyone becomes better teachers much more rapidly than they would without it.
  • Allows comments; responds to direct questions, arguments, and suggestions.
  • Citing/linking others’ work as inspiration. Beyond the blog roll, can I backtrack the evolution of your idea? (see Brian's adaptation of Fawn's post about a Taboo game)
  • Is on Twitter 
  • Responds to @ questions on Twitter
(A good example that it's possible to be influential without Twitter is Shireen. Her Math Teacher Mambo blog is amazing, but Twitter doesn't seem to be her cup of tea, but that's okay.)


from Infinite Sums
  • The math content is wrapped in well-matched pedagogical moves. Instead of just some cool math problem, we can see how the learning happened (see Matt's The Mullet Ratio. See Liisa and Jessica's use of dialog.) 
  • A lesson comes from some sort of curricular or philosophical organizing structure, instead of a one-off. (There are comprehensive examples like 3Acts, but see how Bowman shares a problem to motivate Riemann Sums, but frames it as a unit anchor problem.)
  • Descriptions are illustrated with classroom photos, snapshots of whiteboards or IWBs, scans or snaps of student work. (see Fawn on any given day, Jonathan's blockheaded students, Frank doing his thing.)
  • Providing docs is more valuable than not. People rarely print out and use docs wholesale, but they value not having to start from scratch.
  • When docs are provided, editable is higher-valued than pdfs.
  • When docs are provided, being able to download them immediately from Dropbox or another server is more valuable than having to request them by email like it's 1997. 
On protecting your work: when we share something, we want and expect it to be used, adapted, and re-shared by teachers and maybe professors in teacher ed programs. We don't expect anyone to take our stuff, adapted or not, and sell it on Teachers Pay Teachers or its ilk. We certainly don't expect it to show up in a book or website of a large publisher. You can't do anything technologically to prevent this (even pdf's can be recreated by an enterprising soul). But, you can give yourself some recourse down the road, should someone seriously cash in on your work. Go here and get you one of these.

Humanity and Hilarity
  • Just like your kids don't want you to be a teacherbot, no one wants to read a bloggerbot. People feel more connected to a personality. Let your voice come through. If you don't feel like you have a voice yet, the answer is to write more. (see: Mimi, Sophie)
  • Earestness and seriousness beats work-a-day, but earnestness + a sense of humor is killer (too many examples, but I'm thinking the Platonic ones are Shawn and Fawn.) 
So, let me know what you think! Did I miss anything? Do you have any better examples than the ones I cited? Am I way off base even trying to write these down? 

Many thanks to Justin Lanier who basically deserves a byline on this post, and to @algebrainiac1 @vtdeacon @JJJsally @jybuell and @samjshah whose help on Twitter planted some of these seeds.

Thursday, June 27, 2013

Rock Stars and the Gift Culture

I just re-read Eric S. Raymond's Homesteading the Noosphere, a work I first read in approximately 1999 right around the time I was installing my first Linux distro. I was never anything more than a baby hacker, but lots of stuff from that work stuck with me. And the parallels to my experiences blogging about teaching are kind of amazing.

Through this week, I've come across multiple occurrences of a particular refrain: "There are people in this community regarded as rock stars.  This is bullshit.  They are just teachers, we are all teachers, no one's writings or opinion should carry more weight than another's." On one hand, this makes rational sense, and I don't know anyone who's had the term rockstar (or a synonym) lobbed at them and, like, welcomed it. They're all, "Really? Me? That's weird. Okay. Please stop that now." 

The MTBoS is not a school. It's not a marketplace, not academia, not a hierarchy. It's not like any other thing most people are familiar with, and so resists explanation. But it is...something. It's a culture, with norms and taboos and status. I think, and humbly propose, that it's very much like open-source hacker culture. Understanding how might go some distance toward dispelling the bewilderment over the rockstardom phenomenon, and why we're all so nice to each other, and what motivates us to give so much time and energy for no tangible reward. 

I'm about to quote liberally from this page.
Most ways humans have of organizing are adaptations to scarcity and want. Each way carries with it different ways of gaining social status. 
The simplest way is the command hierarchy. In command hierarchies, scarce goods are allocated by one central authority and backed up by force. Command hierarchies scale very poorly; they become increasingly brutal and inefficient as they get larger. For this reason, command hierarchies above the size of an extended family are almost always parasites on a larger economy of a different type. In command hierarchies, social status is primarily determined by access to coercive power. 
Our society is predominantly an exchange economy. This is a sophisticated adaptation to scarcity that, unlike the command model, scales quite well. Allocation of scarce goods is done in a decentralized way through trade and voluntary cooperation (and in fact, the dominating effect of competitive desire is to produce cooperative behavior). In an exchange economy, social status is primarily determined by having control of things (not necessarily material things) to use or trade. 
Most people have implicit mental models for both of the above, and how they interact with each other. Government, the military, and organized crime (for example) are command hierarchies parasitic on the broader exchange economy we call `the free market'. There's a third model, however, that is radically different from either and not generally recognized except by anthropologists; the gift culture. 
Gift cultures are adaptations not to scarcity but to abundance. They arise in populations that do not have significant material-scarcity problems with survival goods. We can observe gift cultures in action among aboriginal cultures living in ecozones with mild climates and abundant food. We can also observe them in certain strata of our own society, especially in show business and among the very wealthy. 
Abundance makes command relationships difficult to sustain and exchange relationships an almost pointless game. In gift cultures, social status is determined not by what you control but by what you give away.
The following section is a direct quote, but I substituted teacher stuff for hacking stuff: 
it is quite clear that viewed this way, the society of blogging math teachers is in fact a gift culture. Within it, "survival necessities" are abundant -- collaboration tools (blogging platforms), content creation tools (too numerous to elaborate), lessons carefully crafted and freely shared.  This abundance creates a situation in which the only available measure of competitive success is reputation among one's peers. 
(Onto the next chapter, The Joy of Hacking:)
In making this `reputation game' analysis, I do not mean to devalue or ignore the pure satisfaction of learning and teaching well in the privacy of one's own classroom. Teachers all experience this kind of satisfaction and thrive on it. People for whom it is not a significant motivation never become teachers in the first place, just as people who don't love music never become composers. 
Imagine your beautiful lesson locked up in your room and used only by you. Now imagine it being used effectively and with pleasure by many people. Which dream gives you satisfaction?
Back to verbatim-quoting the next chapter, The Many Faces of Reputation:
There are reasons general to every gift culture why peer repute (prestige) is worth playing for: 
First and most obviously, good reputation among one's peers is a primary reward. We're wired to experience it that way for evolutionary reasons touched on earlier. 
Secondly, prestige is a good way (and in a pure gift economy, the only way) to attract attention and cooperation from others. If one is well known for generosity, intelligence, fair dealing, leadership ability, or other good qualities, it becomes much easier to persuade other people that they will gain by association with you. 
Thirdly, if your gift economy is in contact with or intertwined with an exchange economy or a command hierarchy, your reputation may spill over and earn you higher status there.
I swear I'm almost done, here, The Problem of Ego:
I have observed another interesting example of this phenomenon when discussing the reputation-game analysis with hackers. This is that many hackers resisted the analysis and showed a strong reluctance to admit that their behavior was motivated by a desire for peer repute or, as I incautiously labeled it at the time, `ego satisfaction'. 
This illustrates an interesting point about the hacker culture. It consciously distrusts and despises egotism and ego-based motivations; self-promotion tends to be mercilessly criticized, even when the community might appear to have something to gain from it. So much so, in fact, that the culture's `big men' and tribal elders are required to talk softly and humorously deprecate themselves at every turn in order to maintain their status. How this attitude meshes with an incentive structure that apparently runs almost entirely on ego cries out for explanation. 
Okay, not done. The Value of Humility:
Having established that prestige is central to the hacker culture's reward mechanisms, we now need to understand why it has seemed so important that this fact remain semi-covert and largely unadmitted. 
The contrast with the pirate culture is instructive. In that culture, status-seeking behavior is overt and even blatant. These crackers seek acclaim for releasing ``zero-day warez'' (cracked software redistributed on the day of the original uncracked version's release) but are closemouthed about how they do it. These magicians don't like to give away their tricks. And, as a result, the knowledge base of the cracker culture as a whole increases only slowly. 
In the hacker community, by contrast, one's work is one's statement. There's a very strict meritocracy (the best craftsmanship wins) and there's a strong ethos that quality should (indeed must) be left to speak for itself. The best brag is code that ``just works'', and that any competent programmer can see is good stuff. Thus, the hacker culture's knowledge base increases rapidly. 
For very similar reasons, attacking the author rather than the code is not done. There is an interesting subtlety here that reinforces the point; hackers feel very free to flame each other over ideological and personal differences, but it is unheard of for any hacker to publicly attack another's competence at technical work (even private criticism is unusual and tends to be muted in tone). Bug-hunting and criticism are always project-labeled, not person-labeled.

This makes an interesting contrast with many parts of academia, in which trashing putatively defective work by others is an important mode of gaining reputation. In the hacker culture, such behavior is rather heavily tabooed. 
The hacker culture's medium of gifting is intangible, its communications channels are poor at expressing emotional nuance, and face-to-face contact among its members is the exception rather than the rule. This gives it a lower tolerance of noise than most other gift cultures, and goes a long way to explain both the taboo against posturing and the taboo against attacks on competence. 
Um, sound like any cultures you know?

So, look. I'm seeking ways to make newcomers feel welcome, indeed we all should. But if we agree that some parallels can be drawn between hacker culture and MTBoS, the fact that those who have been around awhile become well-known, and that status is correlated with contribution, doesn't come as a surprise.  The very cool thing is, there's so much going on in our global, stateless army of awesome! So there are so many ways to contribute, make some friends, and learn something. And the best part is, you can't help but grow as a teacher, so the kids directly benefit. Which is what got us all here in the first place.

Update: thanks to Megan Hayes-Golding for writing some missing pieces of this post.
On reputation: So if the MTBoS is a gift culture, who I was previously calling the rockstars are really just the most generous members of our community. They've earned high reputations because of their prior contributions. Most hackers who contribute to open source software do it not for reputation but do it to make the code work for them. Most teachers who contribute to the MTBoS do it because they need something for their classroom. We share it because, why not? But after we share, and someone else picks it up for their classroom, we gain in reputation. Reputation wasn't the original goal, but is a natural byproduct.

On social capital: Have you or anyone else here read Down and Out in the Magic Kingdom by Cory Doctorow? In it, he describes a utopian currency called whuffie. It's a reputation based system where the highly-respected have high whuffies and the universally hated have low. When it comes to giving reputation points, "A person with a score of 0 is just as capable of giving and revoking Whuffie as someone with a score of 1,000,000." Reddit gold and Slashdot karma are popular online social capital-measuring sticks. gasstationwithoutpumps also mentioned stackoverflow as a third.
On secrecy: Rebecca commented about the secretiveness of her local colleagues. Yes, to that point! Folks who know little about open source software often comment that they don't understand why people would volunteer to write software when they could get paid for doing the same work under a different banner. Why would you want your work out in the open? Because gifting makes us better teachers.

Tuesday, June 25, 2013

Hello Out There (HELLO Hello helloooo....)

There was a great discussion in Global Math Department today, lead by the intrepid Chris Robinson, about how we facilitate, and how we sometimes fail, new people participating in the little online math teacher thing we have going on here.  At one point I tried to take the mic to add a few thoughts, but the technology failed me.  (I really wish I knew what I did to anger the technology gods.  What's an appropriate sacrifice for that?  Do I have to throw my iPhone into a volcano?  Would they settle for a nano?  Will they settle for a hibachi?)

So, here they are, the thoughts.  Embellished, just because I'm way better at writing than speaking.  Broadly categorized as assumption-disspelling:

  • I'm the worst at keeping up with blog posts.  The worst.  I'll let Reader go unchecked for weeks on end.  And now Reader is going away, and everything else sucks.  (Yes even Feedly.  Feedly is not good.  There, I said it.  I want to be able to mark either a post or a whole blog as *&^%ing "read" and I want it to go away forever.  I want to see all the subscriptions all the time, and I want to know which ones have unread posts.  I don't want you making decisions about what to show me.  This is basic reader stuff, Feedly.  You should have learned this in reader Kindergarten.  Get it together.)
  • I blame Twitter.  I just assume if a post is noteworthy enough, someone will mention it there and it will catch my attention that way.  This is all to say, if I didn't read your post, I'm sorry.  And if I've never interacted with your blog, I'm also sorry.  It's not personal.  I'm just disorganized and lazy and over-reliant on Twitter to act as my newspaper.  And also, because of new job, I'm not desperately Googling "how to teach factoring" at 6AM anymore.
  • I don't think there's a single thing on this blog that I thought up all by myself.  Look around if you don't believe me.  It's all adaptations of the work of other people, or there's a vague statement apologizing for not remembering where I first saw something, or it's a dumb story about how a kid accidentally grabbed my boob.  That is to say, you don't have to wait until you have the next great inspired original lesson idea to write a blog post.  Write about a thing you tried that bombed.  Write about a thing you stole from someone else and adapted for your unique situation.  Write about how the light looks outside your classroom at 5:30 when you are all alone and cutting out laminated blah blah.  Write about how that kid in third period asked you for crush advice and you felt unqualified to help because you have been divorced forever, so you just told him to smile and try not to act like a spaz.  People love reading about humans.  It hardly matters what they are even doing.  
  • I should take my own advice.  The last post on here was like a million weeks ago.
  • People aren't fans of f(t) because there's awesome stuff on it.  People are fans because it's honest, and I try to pay at least a little attention to telling a story.  Plus it's been around forever because I'm a super nerd who knew what a blog was in 2005.  That's all.  That's really all.
  • It is always a good idea to add some pictures.
  • Everyone should comment more.  Everyone.  Me, you, YOU.  WITH THE FACE.  Deal?  Deal. 

Update: Featured Comments (in which I steal a(nother) good idea from dy/dan)
[Lots of good Reader alternative / Feedly customization advice.]
GregT:  I think some of it is just EXPERIENCE, which is something you can't really teach or help with. But at the same time there's a perception that we're LOSING promising people to misconceptions, which is not good either (is it even really the case?). 
Christopher Danielson:  If you're seeking engagement with a larger audience, do more of what they have responded to. It turns out not always to be what you might rather be writing about, nor what you expect will resonate. 
Josh Giesbrecht: "F you NSA"
Everything Elizabeth said. 

Tuesday, April 9, 2013


Educators seem to be a mixed bag of afraid of vs. jazzed about the CCSSM Standards for Mathematical Practice. At first read-through, they seem very sensible, and like things math students should be doing as a matter of course.

But if you think you really get them, try a little experiment: ask a small group of math teachers what they think "Attend to Precision" means. What does it look like if a classroom task requires it? What does it look like when a teacher is facilitating it? What does it look like when students are doing it? Here are some responses you might hear:

  1. Rounding correctly according to the directions
  2. Rounding sensibly based on the problem's context
  3. Being careful when plotting points
  4. Labeling axes and diagrams correctly
  5. Drawing sketches and diagrams to scale
  6. Using an appropriate number of sig figs based on the precision of the measuring device
  7. Using precise mathematical terms in written and verbal communication
  8. Defining variables and symbols
I've spoken to teachers who express their understanding with numbers 2, 6, and 7, but I've talked to teachers whose understanding hews closest to numbers 1 and 3. Which is not to pass judgment, but is to say: it might be wise to be aware that you and your colleagues could have different, and potentially incorrect, assumptions about the SMPs. 

And "Attend to Precision" seems like one of the more concrete ones. See what your colleagues have to say about "Look for and express regularity in repeated reasoning," and I bet the answers will be even more all over the place.

Another observation: it can be really hard to evaluate which SMPs are highlighted or emphasized in a classroom task. When I try, I tend to go "uummmm...all of them...?"

So what kind of task lends itself to "Modeling with Mathematics"? What does it look and sound like when teachers and kids "Look for and Make Use of Structure"? 

I'd like to point you to a recently published resource: A Rubric for Implementing Standards for Mathematical Practice. It was written in July of 2011 by Danielle Maletta, Mimi Yang, and Mariam Youssef as part of the Visualizing Functions working group at PCMI. It gives an observer specific items to look for in a task, as well as specific teacher behaviors, to help evaluate how faithfully a standard is being met in a particular lesson. The accompanying Resources document will also give you a deeper understanding of each standard.

Also, heads up that Illustrative Mathematics, in addition to the Herculean undertaking of trying to illustrate every K-12 content standard, has put a significant amount of effort into illustrating the Standards for Mathematical Practice using both sample videos and classroom tasks. 

Check them out. Share widely.

Monday, April 8, 2013

Pop Quiz

Was this published in 2001, or this morning?
New York State United Teachers, the state's largest teachers union, is urging members and parents to call on the state Education Department to stop implementation of this year's tests, which will be more challenging, because schools have not received all of the necessary curriculum. 
"If we want our children to be ready for college and meaningful careers, we need higher standards — and a way to measure whether those standards are being met — and we need them now," Education Department spokesman Dennis Tompkins said.
Give up?

Saturday, April 6, 2013

The Tests Matter

Here is what is going on right now, in the time before the Common Core Standards have really hit high schools, and before a common assessment has been inflicted on any live children. The non-teachers in education are going: "Just start teaching the right way. Pay no attention to the tests. If you teach right, you don't have to worry about the tests. The tests will take care of themselves." The teachers are saying: "The way I teach is basically fine, anyway, so I'll make whatever adjustments I need to make once I see what they want kids to do on these new tests." I know there are probably some teachers changing their practice, and some non-teachers with half an eye on assessment. I'm painting with a broad brush. Go with it.

This is what I am afraid of: the thing that happened in New York State, starting in 1999. That's when NY changed from Course1/2/3: a decontextualized, integrated curriculum with very predictable though rigorous exams that were none of them a graduation requirement... to Math A/B, standards with more focus on applications and much less predictable tests -- also, kids had to pass the Math A exam to graduate. (This was a huge deal. Regents exams had traditionally been taken by your college-bound academically-oriented students, and suddenly everybody had to take one of them.) The new requirements were supposed to make things tougher, with all the rhetoric that comes with such changes. June 1998:
Yesterday, officials at New York City public schools welcomed the tougher tests, while some education advocates worried about the lack of resources to train teachers to teach for the higher standards.
If it sounds familiar, that's because it's straight from whatever school-reform-article-generating-machine the news has been using for thirty years. Moving on.

Some shit started hitting some fans. October 2000:

Mr. Mills said middle schools ''need to rethink what they are doing'' and quickly figure out how to teach students the skills they need to meet the new standards. He said he had no intention of backing down on the standards, which as of last June required every high school student to pass an English Regents exam to graduate, and by next June will require every high school student to pass a Regents math exam as well.
People started freaking out when they realized that requiring a passing score on an algebra test was going to be a graduation-rate debacle:
Students in the next class, which entered in fall 1997, will have to pass both the English and Math Regents to get their high school diplomas. If the results hold steady, about a quarter of this year's seniors will not be allowed to graduate.
There were protests (May 2001). There were districts trying to opt out (Nov 2001). 

I don't know what happened to all the kids in the early 2000's who were denied a diploma because they couldn't pass the Math A Exam. A bunch of heartbreaking shit, I'm sure. 

In June 2003, there was TESTMAGEDDON. The Math A Regents exam was the straw that broke New York's resolve
Though many districts have not finished tabulating their scores, superintendents, principals and math department heads are reporting preliminary results that some described yesterday as ''abysmal,'' ''disastrous'' and ''outrageous.''
It was not a good test. Post-Course 1/2/3 exams were not good tests, generally: problems that didn't make sense, weird, contrived contexts, a fetishization of goofy vocabulary and notation. Too much content was a huge problem. A test that didn't know whether it was an algebra or geometry test was a huge problem. A test that didn't know what it was measuring -- readiness for higher mathematics courses? Basic skills that should be expected of every graduate? -- was a huge problem. In the end, the test measured nothing but whether or not a kid had passed that test. The accountability movement compelled schools with lower scores to make their math courses all about passing the test. Math A became a de facto curriculum, and a horrible one. 

NY tried to raise the bar. Then, a whole mess of kids ran head-first into the bar and fell on their asses. Then, instead of re-evaluating any of their faulty premises, NY responded by lowering the bar.

On the June 2003 exam, they relented and lowered the cut score

Then, they eased up on subsequent tests

New York State's education commissioner, Richard P. Mills, said Wednesday that the state would loosen the demanding testing requirements it has imposed for high school graduation in recent years, including the standards used to judge math proficiency.
They made the tests easier. Lots easier. Also, the thing happened that took all the respectability out of the historically respected  regents exams: for the tests required for graduation, the score you needed to pass got dramatically lower. They said it was a 65, but after June 2003, you only needed a raw score of around 42% to pass the Math A with a scaled score of 65. (The raw scores in the linked table are not percentages -- they are out of 84 points.)

I wasn't around when this all happened. I didn't start teaching until 2005. And I don't think we're getting exactly a repeat with the Common Core. For one, there does seem to be a coordinated, genuine effort to support teachers in changing their practice, independent of testing. For two, there's a coherence and focus in the CCSS that New York was sorely lacking. But also, there's the whole added wrinkle that tests are trying to fulfill still another purpose: teacher evaluation. The disaster story might not be "so many kids can't graduate", it might be "so many teachers are being rated poorly, even good ones that kids, parents, colleagues respect."

But I still think it serves as a cautionary tale, and I'm still curious about how this is going to play out once the new tests hit a computer lab near you. If they really measure the stated goals of the new standards, they're going to be very different. Because of that, they're going to be perceived as too hard. How the test-writing consortia, DoE, states, districts, etc react to that is going to be really interesting.

Sunday, March 10, 2013


Happy spring forward day! In Syracuse, they don't get sun for another two months or so. Or birds. And in Buenos Aires, it's the oppressive mosquito-thick shank of late summer. It was 65 degrees in Charlottesville today, the sun is out, and birds are singing. I'm 10% of the way through a 30-day Bikram challenge, and all I can move right now is my fingers. Things are good.

Maybe you all have heard of all of these things already, maybe not. Here you go:

Jason Dyer is on a tear lately.

1ucasvb makes pretty and interesting things out of code.

Daily Desmos has given me a right headache a number of times already.

Michael and Tina are taking the best kind of teacher blog post and making it into another blog.

Ben has reminded us why we do this, and what makes it fun.

We're getting a podcast!

Collaborative Mathematics is so great.

Thursday, February 28, 2013

ICYMI : Math Teachers Get Down With Their Bad Selves

This happened.

Sometimes I forget that things happen not-on-Twitter. It didn't occur to me to post here until Sam posted it. Which I only noticed because I opened my Reader for the first time all week.

Fun facts:
  • The dog's name is Hershey.
  • There are two Rubik's cubes.
  • The equation on the board is a nod to the Simpsons.
  • Greg is actually doing the Harlem Shake.
  • Timon has some seriously underrated breakdancing skills. (Until yesterday, I suppose.)
  • Julie did not have to dress up special. She happened to be wearing a cowgirl outfit that evening.
  • The Matt in the first half on the monitors is the same Matt in the second half in person. We had him on the G+ Hangout for the first half because that's normally what he looks like in our office.
  • You might have to look hard for Sam, since he is wearing a disguise.
  • At the very end, Christopher is getting ready to no-kidding launch Tabitha across the room. That part got accidentally cut as a result of the slow-mo.
  • The math twitterblogosphere is the best twitterblogosphere.

Saturday, February 16, 2013

Two Tens for a Five

Thanks for all the ideas about how to talk to eighth graders about irrational numbers. Here is my stab at a question progression.

I don't know how to credit people who shared ideas that made it in here - they are so overlapping. Also, several people didn't provide their names.

I do want to give a shout-out specifically to Justin Lanier, as I copied his even/odd irrationality of √2 proof basically verbatim.

Thoughts appreciated.


Calculators away!

Let’s try to figure out exactly where √10 is!

What two integers is √10 between? Label them on the points plotted below.

Which numbers is √10 between, rounded to the nearest tenth? Find these by hand. Place them CAREFULLY on the number line.

Which numbers is √10 between, rounded to the nearest hundredth? Find these by hand. Place them on the number line.

How much more precise can you get?

You may have learned that when you turn a fraction into a decimal, the decimal eventually either ends altogether, or ends in a chunk that repeats over and over forever. 

For example, 3/8 = 0.375 and 1/7 = 0.142857142857142857… However, the square root of ten never ends or makes a repeating pattern! You can compute its value as precisely as you want, but there is no way to write it exactly as a decimal. (If you think about it, a decimal is really a bunch of fractions: tenths and hundredths and thousandths, added all up.)

This may seem too weird to be believed. However, we can come up with possible, theoretical non-repeating decimals. 

For example, can you spot a rule suggested by the start of this number, and write more digits?   0.13113111311113__________________

Can you come up with your own rule to create a decimal that will never be just a repeating chunk of numbers?

Many calculators claim that √10 = 3.16227766 (maybe even yours!) Explain how you can tell, beyond any doubt, that this can’t POSSIBLY be true.

Recall that the algorithm for multiplying fractions is stupid-easy. For example, 2/3∙4/5=8/15 and 8/7∙8/7=64/49

Let’s try and pinpoint √2

Explain how you know for sure that 3/2 is too big.

Explain how you know for sure that 5/4 is too small.

Carefully plot 3/2 and 5/4 on the number line below. The points are plotted exactly at 0, 1 and 2.

Do any of these fractions exactly equal √2 ?   7/5, 11/8, 10/7

Plot them as precisely as possible on the number line.

Can you find any fractions that are even closer to √2 ?

As you may have guessed, there is no fraction, that when you square it, equals 2 exactly. √2 can not be expressed as a fraction – a ratio of numbers. That is why it is known as irrational.

But how do we know? Maybe we just haven’t looked hard enough for the fraction. Maybe if we could look nonstop for a week, we would find it! How can we know for sure that it doesn't exist?

Any fraction has to be one of only four kinds: odd/odd, odd/even, even/odd, and even/even. What can you say for sure about any even/even fraction?

Of course, even/even can be reduced to one of the other three kinds, so we only need to consider these. We’re going to show that none of these kinds of fractions could be √2—that is, that none of them squared is 2.

One example of a fraction that equals 2 is 18/9. Can you think of three more examples of fractions that equal 2? How can you describe them in general?

We’re just going to look at three cases of candidates. Odd/odd, odd/even, and even/odd.

Well, when you square odd/odd, what do you always get? Could one of these possibly equal 2?

When you square odd/even, what do you always get? Could any of these possibly equal 2?

So the remaining case is even/odd. When this is squared, we get even/odd—so it looks like it might be possible for the top to be the double of the bottom. But consider this: when an even number is squared, the result is a multiple of 4. (Pause a moment and convince yourself this is true.) And a multiple of 4 is never the double of an odd number.

So √2 can’t be a fraction that’s even/odd.

But then there’s no option left! So √2 is irrational.