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:
- 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.
- 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.
- 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.
- 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.
- 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.))
- 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.
- 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:
- 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.
- I tell you a square has an area of 81/4, and you can easily find the root.
- 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.
- 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.
- Just trust me, kid.
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 Number||Potential Misconception|
|non-repeating decimal displayed by calculator||1/19 is irrational|
|anything with a √ in it||√2.25 is irrational|
|weird looking numbers like π and √2||π 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.