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## Saturday, October 12, 2013

### Evens and Odds

update 10/16: Look at this coolness. Thanks for sharing, Øistein.

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This is great. I shared this on Facebook, and it piqued the interest of all the math teachers, of course.

But also some elementary teachers. Who recognize that there's something important here, that kids could be doing. But who don't have much experience with proof themselves, and aren't sure what it would look like in their classes.

This has got me a little obsessed. What kinds of proofs would be appropriate for little kids to explore? If we're talking, say, third graders, lots of them don't really get multiplication yet.

I see alot of inspiring mathematical...stuff...that teachers have no idea what to do with. So I tried to write something out. Here's what I imagine this could look like. Sharing just in case anybody finds it useful. If this is all old news to you, I'm not trying to insult anyone's intelligence. Also, the standard disclaimer: I don't live in a vacuum; there's plenty going on in here that was inspired by other people.

If anyone has little kids they can try it on, I'm curious to hear how it goes.  My anticipating-response muscles are very rusty.

Fonts:
Suggestions for things to say out loud.
Suggestions for things to write on the board.
Other notes about what is happening.

Tools: As many of these as we can muster
• Dot Paper
• Counting Chips
• Colored Pencils or Crayons
• Blank Paper
• Mini Whiteboards
• Whiteboard Markers

What do you notice?
- 2 minute think time
- write them all on the board under Noticings

Some time after someone notices that they are all odd numbers

What if we add together one number from the green sack, and one from the red sack?

Students share results. What do you notice? Keep writing noticings.

Some time after someone notices that the sum is always even:

Whoa, really? Always? Did anyone get a sum that was not even?

Do we think this is always true?

Write conjecture on board.

We think this is always true:
“When we add a number from the green sack and a number from the red sack, we always get an even number.”

Did we check them all? Good.

Can we make this statement any more interesting?

“When we add together odd numbers, we always get an even number.”

Can anyone find an example where this is not true? Where you add odd numbers, and get an odd number?

Possible: 3+5+7 = 15

Refinement:
“When we add together two odd numbers, we always get an even number.”

What would prove that this was not true? (Someone would have to find an example of odd+odd=odd.)

Have we checked all the odd numbers?

How many odd numbers are there?

Maybe we just haven’t stumbled on some odds that add up to odd. Maybe they’re out there...

(possible: give time to look for a pair of odds that add up to an odd. not sure if this is necessary.)

Have we checked all the odd numbers anywhere in the universe?

(This is part of what math is. Noticing that something seems to be always true, and convincing ourselves it's always true, even when we can’t check all the examples. We need a way to explain why odd + odd is always even, no matter what the numbers are.)

What makes an odd number odd?

What makes an even number even?

Time to play with math toys. Dot paper, counting chips, etc.

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This is one example of a proof. There are others.

This first part is the important part. Kids need to stumble over it themselves. Don’t show them. If this doesn’t get proved in one session, it’s okay. Leave the loose end and let it linger. Some of them will keep thinking about it.

Even numbers can be arranged in two equal rows. Odd numbers can’t. If you arrange them in two rows, one row has an extra bit hanging off:

But when you bring two odd numbers together, the extra bits come together to make their own pair.

Can be proved with algebra, depending on grade level.

Even numbers can all be written 2 * something. Odd numbers can all be written 2 * something + 1.

Take an odd number. Can be written as 2n + 1, where n is a natural number.

Take another odd number. Can be written as 2p + 1, where p is a natural number.

Add them: 2n + 1 + 2p + 1

Gather terms: 2n + 2p + 2

Factor out a 2: 2(n + p + 1)

This is 2 * something, therefore it has to be even.