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!

## 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.

## 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.

## Wednesday, February 13, 2013

### On Not Being Irrational

I am particularly intrigued by what students in eighth grade are meant to understand about what it means for a number to be irrational.

Okay hypothetical eighth grader, come with me down this road. As you work through some classroom tasks, this is what you will discover:
1. If you build a square with 3 things on a side, the square will have 9 things in it. 4 to a side, 16 things in it. A shortcut to how many things in the square is the side times itself. Notation for something times itself is something2.
2. If you try to arrange a certain number of things into a square, you can't do it with any old number of things. only numbers like 9 and 16 and 25 will work. We call these numbers of things "perfect squares". To decide if a number is a perfect square, see if you can find something times itself that equals it. We call this function square root and use a funky symbol √ which is really a stylized r because it's a root.
3. There's no reason to restrict our side lengths to discrete values. If I can transition you to thinking about area, you can see that if I build a square on a grid with a side that's 2.5, there is an area of 6.25 square units inside the square. The 2.52 shortcut still works.
4. Likewise, if I tell you a square has an area of say 20.25, you can find the length of a side of that square. The square root thing again. Keep trying to square numbers until you hit on the one that gives you 20.25.
5. Now you will look for the square root of two. Sure you can use your calculator. Only use the multiplication function, please. I know there's the funky symbol. Just ignore it for now please. (Or maybe I didn't tell you about the funky symbol. But someone is heard about it, or will find it, and spill the beans. (Intentional nod to the Pythagoreans.))
6. No matter what, the class will quickly discover that they can ask their calculator for the square root of two. The calculator will give them a nine- or ten-digit number. If they think to square that number, the calculator will say 2. They will think they have found it.
7. Nothing I do will convince you that irrational numbers are a really different kind of number.

So I try to get around this, the most extreme version of that goes like this, picking up at 3:
1. No calculators. We build a square on a grid with a side that's 2 and 1/2, which I will try to give as 5/2. There is an area of 25/4 square units inside the square. You will probably write this as 6 and 1/4. Maybe you will see that (5/2)2 still works, if I can convince you to just work with improper fractions.
2. I tell you a square has an area of 81/4, and you can easily find the root.
3. Now you will look for a square root of two. Still no calculators. We guess 3/2, but (3/2)2 is 9/4, and that's too big. Maybe you reason that 3/2 is the same as 6/4, so 5/4 is a little bit smaller. but (5/4)2 is 25/16, and that's too small. Okay let's try (11/8)2. Still too small.
4. You give up after a while. I tell you that, surprise, there is no fraction whose square root is two. The square root of two can not be expressed as a ratio. We call numbers like that irrational. You know how when we divided out fractions to express them as a decimal, and the decimals always ended up ending or repeating a pattern? Irrational numbers don't do that.
5. Just trust me, kid.
There are in-between methods, like working with decimals but not calculators. It seems to me that no matter what, we are going to run into the same problem. We'll be looking for something that is not there, and I'll have to just tell you it doesn't exist. CCSS doesn't expect us to prove it, and that seems too hard for eighth grade.

8.NS.1 says "Know that numbers that are not rational..." hold it right there. Is it even possible for an eighth grader to grok that there are numbers that are not rational? For that to mean anything and not just be a memorized definition? What definition would they be able to hold onto?

Potential Definition of Irrational NumberPotential Misconception
non-repeating decimal displayed by calculator1/19 is irrational
anything with a √ in it√2.25 is irrational
weird looking numbers like π and √2π and √2 are the only example of irrational numbers I know

This is something that has been breaking my brain for a while, it's just freshly breaking it this week. I know lots of really smart people, and there doesn't seem to be a right answer. But, you know, it's okay. Questions are cool, too.