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.
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.