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Wednesday, February 13, 2013

On Not Being Irrational

From your friendly neighborhood Common Core eighth grade standards:

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.


  1. There are a few places where I think the Common Core standards break down.

    Here's another example:

    Complex numbers are apparently at the top end of the standards and are a required topic.

    Vectors (and matrices) are an optional topic. It seems to me that you cannot understand a complex number without understanding what a vector is.

    I'm with you, I'm not clear that 8th graders have enough experience with decimals & fractions to really come to a clear understanding of what an irrational number is. That particular standard is in 10th grade here, and students mostly don't get it either.

  2. This task off Illustrative Mathematics gives a pretty good idea of sqrt(2):

    I especially like "Explain, in terms of the structure of the expression (1.4142136)^2, why it can not be equal to sqrt(2)." The problem gives a good sense of why we care about this rational / irrational stuff in the first place.

    There are two other illustrations on the standard but not all the nuances are covered yet.

    Incidentally, I wrote this one for the high school level:

  3. Just a few comments. First of all, I think it helps to actually draw a square of area 2 on the grid (or better yet, see if the students can find one--hint your squares do not need to have horizontal or vertical sides). Second, if the students find a decimal approximation with a calculator, as a class start to square that decimal by hand. You don't have to go far to see that the answer can't possibly be 2. Discuss why calculators often give only approximations. (Taking 1 divided by 3 is an example of how a calculator does this even for nice rational numbers). Students should be aware that this is one drawback to calculators. Still that doesn't show that the number is irrational (see 1 divided by 3). I think 8th graders should have a definition of a rational number (a point on the number line that can be written as a fraction with one integer over another). So one-third is clearly a rational number. Then it would be nice for even 8th graders to see a proof that the square root of 2 is irrational. To understand the idea of the standard proof requires only the concepts of equivalent fractions and even and odd numbers. You can tell the students that it took mathematicians thousands of years to show that pi was irrational.

  4. This isn't so much about the Common Core but my daughter (an advanced 7th grader) just saw the proof that the square root of 2 was irrational. She thought it was cool. It is a little hard for me to see how to use it for the general middle school-er but perhaps it's worth a shot.

  5. "(But someone is heard about it, or will find it, and spill the beans. (Intentional nod to the Pythagoreans.))"

    This is why I will love you forever.

    To be fair to our students (or maybe to embarrassingly admit how much math I'm still learning every day) I'm not sure I FULLY grasped how weird irrational numbers are until after high school. The idea that you can keep subdividing a 1 unit measuring stick in an attempt to "measure" sqrt(2), and NEVER be done measuring it. (incommensurate...)

    When I finally started to get a mental handle on that, it was actually kind of mind-blowing for me. (I think the first time I saw the proof for square root of 2 being irrational was in college?)

  6. I prove that the square root of 2 is irrational with my 8th graders (who are advanced). Some of them get it, some of them don't, and I'm okay with that. To make sure everyone gets something though, I spend a long time having the class try to come up with fractions that approximate the square root of two by constantly subdividing the number line. (Oh no, 3/2 was too much? Let's try 5/4, then.) Then we talk about what it means to say that this process will never end.

  7. Here's an exercise I have tried with teachers, which might also be good for high school students.

    1) Make a list of square numbers, 1, 4, 9, 16, 25, 49, 64, 81, 100, ...

    2) Point out that 49 is almost exactly twice 25, 100 is almost exactly twice 49, and ask if they can find any other instances where a square is almost exactly 2 times another square. Playing with their calculators they might find that 29^2=841 and 41^2 = 1681.

    3) Ask them if it's possible to find a square that is exactly two times another square. At this point some of them might be convinced that it is impossible (you can invite the ones who think it is possible to keep looking).

    4) Point out that this is the same as asking if there is a rational number whose square is 2. If (a/b)^2 = 2, then a^2=2b^2, and vice versa.

    The aim of this exercise is not to prove the sqrt(2) is irrational, but to help students see what that means and convince them that it is true. If you want a proof, you can examine the claim in (3) more carefully, and use unique factorization into prime numbers to prove that it is impossible. That would be a good extension module or a topic for a talk at a math club. (But note that unique factorization into prime numbers is not in the standards, strange when you consider that one of the lead writers is a number theorist.)

  8. I don't know if it helps, but the first non-square-root type number to be proved irrational wasn't pi, it was .101001000100001... It's a fairly straightforward way to show that non-repeating decimals exist.

  9. Proving that sqrt(2) is not rational is hard for 8th graders to do, but not too hard for them to follow if they can be made to care about it.

    Calculators are definitely the wrong way to go about discussing irrational numbers, as they inherently do approximate arithmetic on rational numbers.

  10. It should be possible to explain the irrationality of the square root of 2 at that level. The key is to lead them to discover that when you square a reduced fraction, the result is also reduced. For example, if 7/5 is reduced, then 49/25 is also reduced. But 2 cannot be written as a reduced fraction, except as 2/1. Therefore the square root of two cannot written as a reduced fraction.

  11. What your intuition is telling you is that 8.NS.1 is an awful standard, and you are right. We'll eventually get around to our take on it, too.

  12. Great issue to raise!

    I don't see how using the calculator or not using the calculator makes any difference. After all, the square root of 2 does have a decimal expansion. (The Illustrative Mathematics task could be used to explain why its decimal expansion doesn't terminate -- but in theory it could still have a repeating expansion.)

    Similar to Bill McCallum's, you can prove it's irrational by supposing that the square root of 2 could be written as the fraction A/B (where A and B are whole numbers). Then you'd get that A^2 = 2B^2. Now think about factoring A and B into products of prime numbers. When you square them, each prime factor now appears twice as often. But there is the extra prime factor 2 on the right side. So the two sides of the equation can't possibly be equal because you have an even number of twos on the left side and an odd number of twos on the right. Therefore you couldn't have written the square root of 2 as A/B in the first place, and so the square root of 2 is not rational.

    Note that this uses the fundamental theorem of arithmetic (uniqueness of factorization into products of primes).

    Despite all the above, I actually think it's ok at this grade level for students to learn that it turns out to be a surprising and amazing fact that there are some numbers that actually can't be expressed as fractions (or their negatives), such as the square root of 2, without being responsible for knowing why that is so. If they are interested, great, go for it. If not, let them know that it's something they can learn later.

  13. For my 8th grade we mostly focus on approximating, and why we approximate instead of giving an exact answer.

    I like numberwarrior's link, thanks!

  14. When I think of classifying something in mathematics, whether it be quadrilaterals or types of numbers, I think it's worthwhile to discuss with students both examples and non-examples. That said, it's hard to grasp the significance of irrational numbers without a solid understanding of rational numbers.

    Some concepts I would expect an 8th grader to understand about rational numbers:

    1) All rational numbers have a repeating decimal expansion (they just look like they terminate because we don't write the repeating zeros. Changing bases illustrates this point nicely.

    2) We can convert any fraction into its decimal expansion via long division.

    3) We can convert any repeating decimal expansion into a fraction via some simple algebra. Let x = 0.111... and then 10x = 1.111... so 9x = 1, and therefore x = 1/9 is within the grasp of an 8th grader.

    4) Given this flexibility moving between any rational fractions and decimals, students are prepared to consider what a non-example of a rational number might look like. I would challenge students to come up with their own non-example(s) for homework and then discuss a few days later. It's at this point that I would introduce some of the ideas other readers have suggested.

    An engaging formative assessment I've given students in the past is a RAFT writing assignment:

    Role: a rational number
    Audience: an irrational number
    Format: breakup e-mail, letter, note, etc.
    Topic: our differences and why things won't work out between us

  15. Not interesting to me:

    • The square root of 2 is not quite equal to the number on your calculator screen.

    • Proof of the above without giving a sense of the mystery of the problem.

    But, I do think the idea of infinity and the unkown that is tied up in root 2 is very interesting.

    • If you look at the first 20 digits of root 2 is there any structure? How about the first 100 digits? Any repeating patters?

    • If the digits go on forever, won’t the pattern have to repeat itself eventually?

    • Can you come up with a way to create your own decimal number that never repeats?

    • How many digits have “they” checked?

    • How do you check to see if it repeats if it goes on forever? How will you ever finish checking?

    • Are any digits more or less popular than others in the expansion?

    • Does the expansion not like repeats like two fours in a row?

    • What would a stem & leaf plot of the digits look like? How about a stem a leaf plot taking two digits at a time?

    I don’t think you need to answer all of these questions. But, it seems like it is rare that we are exposed to these really mysterious sorts of things that are actually common in math.

  16. I dig numberwarrior's solution. It adds a new dimension to my thinking about things.

    But it doesn't hold up to rigor, does it?

    1/6 is rational, and its decimal expansion is 0.1666... It is sixes all the way out. Truncate at any point, multiply by 6 and you get a 6 as the last digit, when we KNOW it needs to be a 0. That is, the behavior looks exactly like the behavior of sqrt 2.

    To state explicitly, the behavior that the Illustrative Mathematics task points to is this: Finite decimal expansions do not have the same characteristics as the infinite decimal expansions they approximate. But nothing in that observation demonstrates that there are two types of infinite decimal expansions.

    We need to be cautious, I think, of presenting arguments that are misleading.

  17. I like the approach that involves repeating decimals and moving between those and fractions.

    Maybe at some point you can move to many-digit calculators, or lists of digits like this one:

    The question that comes to my mind is, how did they get those digits?

  18. Re: "But it doesn't hold up to rigor, does it?"

    One can make the solution by numberwarrior rigorous by noticing that the argument is not restricted to base 10. Same reasoning shows that the expansion of the square root of 2 does not terminate in any base. (Trying bases 2, 8 and maybe 11 will make this plausible enough).

    On the other hand, every rational number has finite expansion in some base, e.g., in the base equal to its denominator.

  19. As someone who just spent the first two of eight days working on units for the CCSS 8th grade standards, I'm particularly interested in where 8NS1 fits with the rest of the 8th grade standards. I mean, I suppose approximating irrationals can come up naturally when dealing with the Pythagorean Theorem, but it still seems like an in-depth discussion of rational vs. irrational will be forced.


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