## Alert!

**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!**## Tuesday, December 22, 2009

### I Finally Used the Cell Phones for Something

Sixty students! Two days! Two buildings! Twelve clues! One map! I went a step beyond the classroom locus treasure hunt this year and made a school-wide locus treasure hunt.

In class the students worked with large (11"x17") maps of the school, compasses, rulers, and locus-type pirate map style clues (100 feet from Miss Nowak's room, and also equidistant from the math hallway and the resource hallway) to narrow down the location of 12 stars I stuck around the school. Then they had the whole next day to look for the stars and send a picture of them standing next to it to a picasaweb drop box for a bonus point.

I'm not going to bother posting documents because obviously they would only work for my school. But, it was fun, and I was proud of myself for thinking to use the phone cameras like that.

And, it seemed much, much easier for the kids to get a hold of the locus concepts when it was tied to a concrete map of the school. For some reason saying "equidistant from room 2122 and 2126" has magical super powers that "equidistant from two points" does not. Weird.

Update: Here are the files, for whatever they are worth.

## Friday, December 18, 2009

### Introducing Logs

This is nothing earth-shattering, but I feel my Algebra 2 students are less freaked out by logarithms this year, and I think it has to do with how I first introduced them. I used to start by declaring A LOGARITHM IS AN EXPONENT, like saying it loudly and slowly would help it sink in better. I should really know better by now. Well ok maybe, I do know better by now, because this year I started by inviting them to play a fun puzzle:

(Adapted from James Tanton's monthly St. Mark's Math Institute newsletter.)

And then let them in on the dirty little trick that in math we insist on calling it a logarithm instead of a power when we write it like that.

To answer the inevitable "

Then we mention that it's an inverse of an exponential equation, play a little more with shifting between exponential equations and log equations, and we are done for the day.

Here are my filled in notes:

Here is the smart notebook file. Here is a google doc with text of a task for easy copy and pasting.

(Adapted from James Tanton's monthly St. Mark's Math Institute newsletter.)

And then let them in on the dirty little trick that in math we insist on calling it a logarithm instead of a power when we write it like that.

To answer the inevitable "

*What the hell?*", I go into a little history of how John Napier invented logarithms to make multi-digit multiplication easier for Renaissance astronomers.Then we mention that it's an inverse of an exponential equation, play a little more with shifting between exponential equations and log equations, and we are done for the day.

Here are my filled in notes:

Here is the smart notebook file. Here is a google doc with text of a task for easy copy and pasting.

## Wednesday, December 16, 2009

### Row Games Galore

I love a good Row Game but they are sure a pain to make. I just made a new one for reviewing surface area and volume, and in a minute I'm going to go lay down with a hot compress and a glass of wine.

I know some others are out there making them too, and I'd love it if we could all put them in one place, and save each other some work. So here is that place! It's a global folder on box.net and I think that means anyone with the link can bookmark it, download, upload, and we can share our little heads off. I put in four for now (but I'm pretty sure I have more floating around, I'll add as I find them.) I don't know what I'm going to do if spammers start putting garbage in there, obnoxious little soulless weasels that they are, but we'll try it for now. I used the naming convention "RG - Topic" but I don't really care what the files are named as long as they describe the topic.

I know some others are out there making them too, and I'd love it if we could all put them in one place, and save each other some work. So here is that place! It's a global folder on box.net and I think that means anyone with the link can bookmark it, download, upload, and we can share our little heads off. I put in four for now (but I'm pretty sure I have more floating around, I'll add as I find them.) I don't know what I'm going to do if spammers start putting garbage in there, obnoxious little soulless weasels that they are, but we'll try it for now. I used the naming convention "RG - Topic" but I don't really care what the files are named as long as they describe the topic.

## Friday, December 11, 2009

### If You are a Fan

of f(t), you can express your enthusiasm by voting! Here (Best Teacher Blog) or here (Best Resource Sharing Blog).

I'm not sure what the acceptable social norms are for promoting oneself (this is new to me), but I wanted to let readers know the nominations were up, and voting is open until Wednesday the 16th. I'm stunned and humbled just being nominated.

I'm not sure what the acceptable social norms are for promoting oneself (this is new to me), but I wanted to let readers know the nominations were up, and voting is open until Wednesday the 16th. I'm stunned and humbled just being nominated.

## Thursday, December 10, 2009

### A Very Special Episode of f(t)

First, a student I'm close to (she was with me for three years - my only hat trick, so far) came to tell me she was accepted to the college she really wanted. She's a lovely young woman who works hard and treats people well, and it was gratifying to see the payoff. There was even a hug. Then, I had carrot cake for breakfast. Which is fine, because it's basically a muffin, you know.

And then, at the beginning of one particular class, I begged whoever swiped one of my loaner graphing calculators, that was missing for over a week, to please return it. I told them I provided them so people who forgot theirs would be able to borrow one. Because it seemed like a considerate thing to do, to make math class less stressful. I knew if someone had it, it didn't mean they were bad, but that I really needed it back, and it was within their power to make a more honorable choice. I didn't care who it was, they could put it in my mailbox in the office so I didn't have to know. I wouldn't be able to loan out calculators anymore if I didn't get it back. And it was a shame that the 120 people who came through my room each day would have to suffer for one person's actions. That people without a calculator would have to make arrangements before Monday's quiz, because they won't be able to borrow one from me.

So we had a half-period lesson, then a half-period practice / peer tutoring kind of activity. I was all over the place, the kids were up and around, everyone was working hard.

And then after the bell rang, I walked to the back of the room, and OH SNAP, SITTING THERE, IN THE MIDDLE OF MY DESK, THE MISSING CALCULATOR! It was a CHRISTMAS MIRACLE! I couldn't believe it. I still can't. Little speeches like that never work out for me. I felt like a movie teacher. I wish I knew who it was. I would hug the little bastard.

And then, at the beginning of one particular class, I begged whoever swiped one of my loaner graphing calculators, that was missing for over a week, to please return it. I told them I provided them so people who forgot theirs would be able to borrow one. Because it seemed like a considerate thing to do, to make math class less stressful. I knew if someone had it, it didn't mean they were bad, but that I really needed it back, and it was within their power to make a more honorable choice. I didn't care who it was, they could put it in my mailbox in the office so I didn't have to know. I wouldn't be able to loan out calculators anymore if I didn't get it back. And it was a shame that the 120 people who came through my room each day would have to suffer for one person's actions. That people without a calculator would have to make arrangements before Monday's quiz, because they won't be able to borrow one from me.

So we had a half-period lesson, then a half-period practice / peer tutoring kind of activity. I was all over the place, the kids were up and around, everyone was working hard.

And then after the bell rang, I walked to the back of the room, and OH SNAP, SITTING THERE, IN THE MIDDLE OF MY DESK, THE MISSING CALCULATOR! It was a CHRISTMAS MIRACLE! I couldn't believe it. I still can't. Little speeches like that never work out for me. I felt like a movie teacher. I wish I knew who it was. I would hug the little bastard.

## Tuesday, December 8, 2009

### Edublog Award Nominations 09

My reader is full of worthwhile material, so it's hard to pick, and also I'm a chickenshit and I don't want anyone to get mad at me. But then, it would be kind of chickenshit not to. I know! The ennui! It consumes! I went with anything that qualifies as an insta-click when it pops up in my reader. Here goes:

Best individual blog : dy/dan. Even though Dan went half corporate this year, he keeps feeding me lessons that are made of adolescent catnip. And if you hadn't noticed, is constructing a new framework for math ed.

Best individual tweeter : @msgregson I don't know when this girl sleeps, but she's the most reliably helpful tweeter around. She's going to be a technology coordinator to be reckoned with.

Best group blog : 360. I spend a little time trying to hack my way through some intimidating math blogs, so I enjoy the way these college professors can make sophisticated concepts relatable.

Best new blog : Questions? Dave Cox came out swinging this year, asking profound and demanding questions of his not that new practice. Take a moment to appreciate how rare that is. And he can carry a tune, too.

Best elearning / corporate education blog: Not sure what category works best for Colleen K, but http://www.mathapprentice.com/ and http://www.mathplayground.com/ both blow me away. I know I didn't do anything with my computer degree, but if I had, I hope it would look something like that.

Best resource sharing blog The Exponential Curve Blog posts are sporadic, but Dan Greene figured out how to share all of his stuff, all of the time, and keeps adding to it.

Best teacher blog Continuous Everywhere but Differentiable Nowhere There should be a preservice course about how to improve your practice through written reflection, and a large part of it should be reading Sam's blog.

Best individual blog : dy/dan. Even though Dan went half corporate this year, he keeps feeding me lessons that are made of adolescent catnip. And if you hadn't noticed, is constructing a new framework for math ed.

Best individual tweeter : @msgregson I don't know when this girl sleeps, but she's the most reliably helpful tweeter around. She's going to be a technology coordinator to be reckoned with.

Best group blog : 360. I spend a little time trying to hack my way through some intimidating math blogs, so I enjoy the way these college professors can make sophisticated concepts relatable.

Best new blog : Questions? Dave Cox came out swinging this year, asking profound and demanding questions of his not that new practice. Take a moment to appreciate how rare that is. And he can carry a tune, too.

Best elearning / corporate education blog: Not sure what category works best for Colleen K, but http://www.mathapprentice.com/ and http://www.mathplayground.com/ both blow me away. I know I didn't do anything with my computer degree, but if I had, I hope it would look something like that.

Best resource sharing blog The Exponential Curve Blog posts are sporadic, but Dan Greene figured out how to share all of his stuff, all of the time, and keeps adding to it.

Best teacher blog Continuous Everywhere but Differentiable Nowhere There should be a preservice course about how to improve your practice through written reflection, and a large part of it should be reading Sam's blog.

## Monday, December 7, 2009

### School and Other Miscellany

1. Today in Algebra 2 we reviewed negative exponents and the children acted like they had never seen it before. I told them about the phrase "move it, lose it" for dealing with a negative exponent, as in, move the term to the other side of the fraction, and lose the negative sign. A student who moved here from another state (where, you know, they get to spend enough time on things to actually learn them) told us about the phrase she learned "cross the line, change the sign." Which the kids liked better. "You know, because it actually rhymes, Miss Nowak. Unlike yours." Um, last I checked "it" rhymes with "it." I'm not an English teacher! You can tell because I'm not wearing cool shoes and I don't give hugs.

2. I was asked to appear with a crowd of other veterans in the last scene of the school's production of White Christmas. Even though it's a neat idea, I don't really want to, because I don't like people looking at me. Well, more than 30 people. Who are over the age of 16. And they want us to not wear a whole uniform but just like a cover (that's a hat, civilians). Which I was taught is Wrong. I haven't decided yet.

3. I've seen some Edublog award nominations, and they are very cool and flattering. Thank you. I don't expect f(t) to be competitive; I think it's a little too niche. I might have a distant shot if they made a category best math teacher blog that is not dy/dan.

4. I need some more music for running. I'm in week 3 of couch to 5k and my playlist is played out. I need help. Here is my current playlist. You should post yours so I can pick through it. Also, you only get to judge mine if I get to see yours. No deleting the embarrassing tracks first.

2. I was asked to appear with a crowd of other veterans in the last scene of the school's production of White Christmas. Even though it's a neat idea, I don't really want to, because I don't like people looking at me. Well, more than 30 people. Who are over the age of 16. And they want us to not wear a whole uniform but just like a cover (that's a hat, civilians). Which I was taught is Wrong. I haven't decided yet.

3. I've seen some Edublog award nominations, and they are very cool and flattering. Thank you. I don't expect f(t) to be competitive; I think it's a little too niche. I might have a distant shot if they made a category best math teacher blog that is not dy/dan.

4. I need some more music for running. I'm in week 3 of couch to 5k and my playlist is played out. I need help. Here is my current playlist. You should post yours so I can pick through it. Also, you only get to judge mine if I get to see yours. No deleting the embarrassing tracks first.

### New Blogger: Launch the Alert-5 Subscribe Buttons

Riley Lark is doing some excellent work. If you like f(t) I can almost guarantee you will like Point of Inflection. His ideas for quick lessons with index cards alone are worth the price of admission. He deserves a bigger audience. Get thee over there and subscribe forthwith. Then show him some comment love so he can see why this little hobby is so rewarding.

## Sunday, December 6, 2009

### My Favorite Theorem

After it showed up as a Twitter thread, Nick suggested we publish a post about our favorite theorem. I think this is the season for favorite things, right? Or is that only in Vienna? I don't know. Anyway.

I immediately chose Cantor's proof of the nondenumerability of the reals, for its counterintuitive-ness and yet easiness to understand if you can hang with a pretty simple argument. It's the best way to show the uninitiated that math is beautiful that I've ever found. And not because you drew a pretty picture of a golden rectangle, but because the ABSTRACT ARGUMENT all by itself is BEAUTIFUL and once you get it makes you go ooh and aah. I've explained this to middle school kids, roommates, people in elevators, my mom, and one time (disastrously) a boy I had a crush on.

The big, impressive, revolutionary idea embedded in here is that there is more than one size of infinity. It takes most people a while to stretch their brains around that. It drove Georg Cantor to an asylum, so no need for anyone to feel bad if it takes them a while.

Children's first encounter with infinity is (usually, I think) when they realize that the counting numbers never stop. There's no last one. You can keep counting incrementing by one forever and you will never...reach...the end.

What if you count just the even numbers? Nope. By fives? Nope. Tens? Nope. Hundreds? Millions? Brazillions? You get the idea.

What if we want to compare how many counting numbers there are to how many even numbers? We obviously can't count them and see which set has more. It makes intuitive sense to conjecture that there are more counting numbers than even numbers. Seems like there should be twice as many.

Cantor's genius move was to invent a way to compare the 'sizes' of these infinite sets, which he called 'cardinality.' Instead of trying to count the number of elements in each set, which is impossible, he said let's look at it another way. For example, when I take attendance in my class, I don't count the number of students, and count the number of chairs, and compare the two counts. I just see if every chair is matched with a butt. If there are no chairs left over, I know the sets are the same size, and I mark All Present.

Cantor applied the same logic to infinite sets. He said if we can match every element in one set to an element in the other, then they are the same size. Since both sets go forever, we don't have to worry about leftovers. They both keep going forever, so every element will match something in the other set, and they are the same size. Sets that can be matched this way are said to be in a one to one correspondence.

Looked at this way, there are the same number of elements in the counting numbers as there are in the even numbers. We can match 1-2, 2-4, 3-6, 4-8, 5-10, and so on, forever. The elements of the sets can be placed in a one to one correspondence, so they are the same size.

You can make a matching to show that the cardinality of the counting numbers = the integers. The counting numbers = the multiples of ten. And even, with a little clever organization, the counting numbers = the rationals.

Intuition resists this conclusion. Sometimes with students, I have to stop here. They just aren't ready to concede the point, and I'm not about to make them push the "I believe" button on something we're doing for fun. I think it's healthier to let them be bothered by it. And if you don't believe that part, you definitely won't come along with me for the rest of the ride.

So you believe that all these infinite sets that can be matched with the counting numbers are the same size. Great! Then what? There's just that one size of infinity?

Nope.

There's an infinity bigger than that. Infinitely bigger than that. And that infinity is located, for an example, on a number line between zero and one.

It takes a little meditation to absorb that statement, too. There are infinitely more numbers between zero and one than there are counting numbers.

Here's where my favorite proof starts. :-) Quite the setup, I know.

Let's say that that conjecture is NOT true. Let's say that there are the same number of elements in the set between zero and one as there are counting numbers.

If that's true, then I can make a list of all of them. It will be infinitely long, but a list can be made. Much like you can list the counting numbers.

So here's the beginning of my list:

0.124598560

0.7463728642893

0.00033033845788

0.1111111111111....

0.7465564748383

0.123123123123123...

0.141592653589...

0.799999999999...

.

.

You get the idea. The decimal expression of every rational and irrational number between zero and one will be in my list.

Now, I'm going to write a new number. But I'm going to write it in a very particular way. The first digit will not be a 1. The second digit will not be a 4. The third digit will not be a 0. The fourth digit will not be a 1. The fifth digit will not be a 5. etc*.

By doing this, my new number must be different from any number in my list, because for the

But wait! We said we were going to list ALL the numbers between 0 and 1! What gives?!

What gives, is that the only thing that could be wrong here is our original supposition. When we said, "

Which means there must be MORE elements in the set between zero and one, than there are in the counting numbers.

Another size of infinity. Cool, right? That's why it's my fave.

*A little caveat needs to go here about not choosing 0's or 9's for your new number, since for example 0.50000.... = 0.499999....

I immediately chose Cantor's proof of the nondenumerability of the reals, for its counterintuitive-ness and yet easiness to understand if you can hang with a pretty simple argument. It's the best way to show the uninitiated that math is beautiful that I've ever found. And not because you drew a pretty picture of a golden rectangle, but because the ABSTRACT ARGUMENT all by itself is BEAUTIFUL and once you get it makes you go ooh and aah. I've explained this to middle school kids, roommates, people in elevators, my mom, and one time (disastrously) a boy I had a crush on.

The big, impressive, revolutionary idea embedded in here is that there is more than one size of infinity. It takes most people a while to stretch their brains around that. It drove Georg Cantor to an asylum, so no need for anyone to feel bad if it takes them a while.

Children's first encounter with infinity is (usually, I think) when they realize that the counting numbers never stop. There's no last one. You can keep counting incrementing by one forever and you will never...reach...the end.

What if you count just the even numbers? Nope. By fives? Nope. Tens? Nope. Hundreds? Millions? Brazillions? You get the idea.

What if we want to compare how many counting numbers there are to how many even numbers? We obviously can't count them and see which set has more. It makes intuitive sense to conjecture that there are more counting numbers than even numbers. Seems like there should be twice as many.

Cantor's genius move was to invent a way to compare the 'sizes' of these infinite sets, which he called 'cardinality.' Instead of trying to count the number of elements in each set, which is impossible, he said let's look at it another way. For example, when I take attendance in my class, I don't count the number of students, and count the number of chairs, and compare the two counts. I just see if every chair is matched with a butt. If there are no chairs left over, I know the sets are the same size, and I mark All Present.

Cantor applied the same logic to infinite sets. He said if we can match every element in one set to an element in the other, then they are the same size. Since both sets go forever, we don't have to worry about leftovers. They both keep going forever, so every element will match something in the other set, and they are the same size. Sets that can be matched this way are said to be in a one to one correspondence.

Looked at this way, there are the same number of elements in the counting numbers as there are in the even numbers. We can match 1-2, 2-4, 3-6, 4-8, 5-10, and so on, forever. The elements of the sets can be placed in a one to one correspondence, so they are the same size.

You can make a matching to show that the cardinality of the counting numbers = the integers. The counting numbers = the multiples of ten. And even, with a little clever organization, the counting numbers = the rationals.

Intuition resists this conclusion. Sometimes with students, I have to stop here. They just aren't ready to concede the point, and I'm not about to make them push the "I believe" button on something we're doing for fun. I think it's healthier to let them be bothered by it. And if you don't believe that part, you definitely won't come along with me for the rest of the ride.

So you believe that all these infinite sets that can be matched with the counting numbers are the same size. Great! Then what? There's just that one size of infinity?

Nope.

There's an infinity bigger than that. Infinitely bigger than that. And that infinity is located, for an example, on a number line between zero and one.

It takes a little meditation to absorb that statement, too. There are infinitely more numbers between zero and one than there are counting numbers.

Here's where my favorite proof starts. :-) Quite the setup, I know.

Let's say that that conjecture is NOT true. Let's say that there are the same number of elements in the set between zero and one as there are counting numbers.

If that's true, then I can make a list of all of them. It will be infinitely long, but a list can be made. Much like you can list the counting numbers.

So here's the beginning of my list:

0.124598560

0.7463728642893

0.00033033845788

0.1111111111111....

0.7465564748383

0.123123123123123...

0.141592653589...

0.799999999999...

.

.

You get the idea. The decimal expression of every rational and irrational number between zero and one will be in my list.

Now, I'm going to write a new number. But I'm going to write it in a very particular way. The first digit will not be a 1. The second digit will not be a 4. The third digit will not be a 0. The fourth digit will not be a 1. The fifth digit will not be a 5. etc*.

By doing this, my new number must be different from any number in my list, because for the

*n*th list item, it differs in the*n*th digit.But wait! We said we were going to list ALL the numbers between 0 and 1! What gives?!

What gives, is that the only thing that could be wrong here is our original supposition. When we said, "

*Let's say that there are the same number of elements in the set between zero and one as there are counting numbers.*" that must have been wrong.Which means there must be MORE elements in the set between zero and one, than there are in the counting numbers.

Another size of infinity. Cool, right? That's why it's my fave.

*A little caveat needs to go here about not choosing 0's or 9's for your new number, since for example 0.50000.... = 0.499999....

## Wednesday, December 2, 2009

### Reporting from the EduTech Front

I take Will's point, and agree that most districts are not planning with the intention to exploit available technology. But if the impulse is "Every kid has a cell phone! Full speed ahead!" can I just urge some circumspection before we throttle up.

I tried a little polleverywhere experiment earlier this year. I am in love with the idea of this technology. My school has a few sets of clickers, and they are a total pain. All hail clicker functionality using the tiny computer the kids already have in their pockets! The kids were amped, too. When I started talking about how I wanted us to try out polleverywhere, and giving them instructions like "find the slope, and text your answer to this number!" there was a palpable "this is so cool" energy running through the room.

Except! 1. Not every kid has a cell phone. My students are predominantly middle class, but we are a large public high school and serve

And 2. Turns out my classroom is a Verizon dead zone. AT&T and T-Mobile work fine, but Verizon is the dominant carrier around here. More than half the kids were not getting enough bars to send a text.

So much for my grand polleverywhere plans.

My other jaw dropping technology moments this year have come from the class blogs. Or rather, from panicked and frustrated kids the morning after they tried to access the class blogs. (This goes double if Geogebra was embedded - I've basically given up on that.) Here are some choice quotes: "I am not good at logging into things." "Our computer at home runs Windows 98." "It wouldn't load the page. Something about cookies." And my personal favorite, "Whenever something is on the Internet, I can't take it seriously." By this point, I've been able to plan with these kids a way for them to use a school computer during the school day. But this was not nearly as easy as the "They're All Digital Natives! They Will Teach US!" propaganda would have us believe.

Maybe it's just a matter of time before every single student, indeed, carries a cell phone. Maybe Verizon will install another cell tower near my school. But for what it's worth, these are my realities on the ground.

I tried a little polleverywhere experiment earlier this year. I am in love with the idea of this technology. My school has a few sets of clickers, and they are a total pain. All hail clicker functionality using the tiny computer the kids already have in their pockets! The kids were amped, too. When I started talking about how I wanted us to try out polleverywhere, and giving them instructions like "find the slope, and text your answer to this number!" there was a palpable "this is so cool" energy running through the room.

Except! 1. Not every kid has a cell phone. My students are predominantly middle class, but we are a large public high school and serve

*plenty*of families who consider cell phones for their kids a luxury. Also, ironically I suspect some of these non-cell-phone-having kids are from families trying to adhere to what the school is telling them, that cell phones are technologia non grata up in here. Even if you bring them, you are expected to keep them turned off and out of sight during the school day (unless a teacher gives you explicit permission to use them for a classroom activity.)*Not every kid has a cell phone.*Not every kid with a cell phone has texting enabled. How to employ them as a learning tool when not every kid has one is a major teacher training challenge that needs to be addressed. It basically limits us to opt-in kind of use, like a project where students can choose from several options, and one of them involves a cell phone. Or small group work, where only one group member needs access to unlimited texting. I can not think of a good solution for kids without a phone if you are trying to implement it as a frequent whole-group feature.And 2. Turns out my classroom is a Verizon dead zone. AT&T and T-Mobile work fine, but Verizon is the dominant carrier around here. More than half the kids were not getting enough bars to send a text.

So much for my grand polleverywhere plans.

My other jaw dropping technology moments this year have come from the class blogs. Or rather, from panicked and frustrated kids the morning after they tried to access the class blogs. (This goes double if Geogebra was embedded - I've basically given up on that.) Here are some choice quotes: "I am not good at logging into things." "Our computer at home runs Windows 98." "It wouldn't load the page. Something about cookies." And my personal favorite, "Whenever something is on the Internet, I can't take it seriously." By this point, I've been able to plan with these kids a way for them to use a school computer during the school day. But this was not nearly as easy as the "They're All Digital Natives! They Will Teach US!" propaganda would have us believe.

Maybe it's just a matter of time before every single student, indeed, carries a cell phone. Maybe Verizon will install another cell tower near my school. But for what it's worth, these are my realities on the ground.

### Tales from the Google Forms

I think this means they couldn't find the sides of the square with the integral diagonal. We shall find out tomorrow.

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