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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.

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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.

8 comments:

  1. I like it. One thing I think that you don't necessarily need to include in a lesson, but should be prepared for if a student brings it up is the thought that maybe there is no square root of 2 (or 10). If you can't write a number down, is it really a number. I had a high school student last week who argued that pi and the square root of 2 were like infinity in that they weren't really numbers, just ideas. My best comeback was to draw a square with vertices at (0,0) (1,1) (0,2) and (-1,1). We could see (with Pythagoras) that the length of a side of this square should be a point on the number line (drawn diagonally) and when squared it should be 2.

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  2. I remember it taking me a while to get what was going on with irrational numbers. I understood them as decimals that went on forever with no purpose or pattern, and I didn't really get why it was interesting that they couldn't be expressed as a fraction.

    I think my problem was that I didn't properly understand how any infinite decimals could be expressed as fractions. I'd like to take a stab at adding some thoughts to your lesson along these lines. Sorry for the long-windedness.

    I'm wondering if something like that might help here. I'm imagining that we start by talking about writing 1.2 as a fraction. We bump up the number of digits, and that culminates in a procedure for converting finite decimals to fractions.

    Then we ask: is it only finite decimals? Clearly not, because 1/3 shows up as a series of repeating threes. But what sort of infinite decimals can be written as fractions? Let's start by dealing with all the infinite decimals that are like 1/3, i.e. .5555555... and .7777...... Is there a guaranteed way to always do this?

    And are there other infinite decimals that we can find a trick for? What about infinite decimals like .121212121212..? Write a decimal that goes on forever with a pattern. Can you find a fraction for it?

    OK, so what if we had an infinite decimal that didn't have a pattern? True, none of our existing tricks will work. But maybe there's another trick out there that we don't know about.

    And now I think we begin to merge into your lesson, because we need to. How do we know whether we should bother looking for another trick or not? What would serve as evidence that there is, or isn't, a trick for every decimal? True: a counter-example would do the trick.

    What numbers might serve as good candidates for counter-examples? Well, decimals that don't have a pattern would be a good place to start, and the square root of two don't have no pattern. Let's investigate it.

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  3. Good point, Michael. I think I was vaguely thinking that learning about how to turn a decimal into a fraction had to come before this. But I hadn't really thought it through. Thank you.

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  4. Here's an idea for approximating square roots of non-perfect squares. I think it may provide an interesting technique for students to investigate.

    Start with √10. We could observe that it must be larger than 3 (draw a 3x3 tiling for students) and smaller than 4 (draw a 4x4 tiling for students). Note that it would take 7 more tiles to go from a 3x3 tiling to a 4x4 tiling.

    But we want a square with area equal to 10. Let's take one tile and divide it into sevenths, each slice having dimensions 1 by 1/7. (Except for the top right corner, which is a 1/7 by 1/7 square.) Lay them along the top and right side of our 3x3 tiling. This suggests that √10 is approximately 3 and 1/7.

    Questions to ponder
    - Based on the picture, will our estimate be too large or too small?
    - Will this method be more precise with larger or smaller square roots?
    - Can we come up with a way to determine our maximum error (i.e., how far off we are from the actual square root?)

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  5. This might not be easy for 8th graders to understand, but I did some work with continued fractions and I was currently working on proving the sqrt of 2. Here is a link that somewhat explains it: http://www.cut-the-knot.org/proofs/SqContinuedFraction.shtml

    Good work! I enjoy reading your posts.

    Amanda

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  6. @complezanalytic. I used the Wheel of Theodorus to demonstrate to my students that irrational numbers have very real values (I mean, we can draw them, and see them.. which is about enough proof for my kids). I think it solidified pretty well the idea that although these numbers aren't like what we're used to, they very much exist. We we well into the Pythagorean Theorem at the point I introduced the wheel, so depending on your sequence it might not work perfectly.

    This line of questioning is great. I've bookmarked it for next fall!

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  7. What I love about this approach is how clear your irrationality proof is because of the way it avoids algebra.

    My usual story about this proof is that in class I show the students a proof that square root of 2 is irrational, for homework I ask them to follow the same method to show that square root of 3 is irrational, and then on the quiz I ask them to follow the same method to show that square root of 4 is irrational. I think with your proof they'd be a lot more likely to spot the error in the proof that comes from replacing 2 with 4.

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  8. I do what Joshua does. But I throw in little arithmetic bits. Hats off to you, this is amazingly more engaging, without giving up an inch of the mathematics.

    Jonathan

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