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Saturday, October 12, 2013

Evens and Odds

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


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.

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

“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.


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.


  1. Love the video, and love your lesson thoughts. I don't work with young kids enough to know whether the tone and pacing are just right, so I can't comment on that.

    I noticed a mistake in the video, at 1:25.If folks use this video with kids, it could provoke a great discussion about him saying "computers never make mistakes" and then showing a mistake emanating from a computer program. (I commented at youtube, so maybe he'll fix the mistake.)

  2. Do you know the Deborah Ball video Shea's Numbers? (I think that's what it's called.) It's something of a math ed classic, and is about elementary school kids discussing & proving things about even and odd numbers.

    Also check out Maggie Lampert's book Teaching Problems & The Problems of Teaching. It's all about teaching elementary school math through a proof perspective.

    Ball & Lampert are basically my math ed heros. They study math teaching and learning from the perspective of people who have actually taught kids. They get that, in education, nothing is as simple as it seems. :)

  3. Nope! Elem. ed isn't something I've paid much attention to. Thanks for the recs.

  4. I love ideas like this for student lessons. One thing I'd like to mention is that we have to be careful attaching grade levels (elementary) to lessons/tasks like this one. I have a number of seventh and eighth grade students that would struggle with the conversations here because of a lack of skills and/or a weak number sense. I'd love to see a progression plan laid out for tasks like these that flow with students' mathematical development, regardless of grade or course.

  5. I hear you. We run into that problem at Mathalicious. For example, Domino Effect is tagged with 8th grade CCSS standards, but lots of classes need to spend a little time on writing an equation of a line given two points.

    And if kids aren't used to thinking about proof, which, I can't imagine that many are until they maybe see a proof-ish like substance in a Geometry course, then yeah, use odd+odd=even at any level. I'm curious what modifications you'd make to what's in this post, so that it "flows with students' mathematical development, regardless of grade or course"?

  6. Just sent out your link to some colleagues who work with fifth graders. I think you task is a great introduction to kiddos who have had limited experience with the idea of mathematical discourse. Am super curious about how the kids will work their way through the ideas in the lesson.

    Agree with Chris: a progression would be fabulous-as we are starting to move into having our kids explore math in this fashion.

  7. Woo hoo! Psyched to see that my twitter suggestion made it into a post. Love how you fleshed out what tools kids might use to explore and how teachers might guide the discussion.
    There's an Everyday Math lesson where kids sort a set of pair collects all the dominoes with both sides even, another all with both odd, another, one side odd and one side even. Then they record the sums. That could be another way to begin. Some kids might notice that all the odd domino configurations have a dot in the center, while evens do not...another way to talk about odd numbers as pairs or multiples of two plus a leftover.
    Agree that this lesson is useful for any grade level if the students aren't comfortable yet with proof. My college math ed professor used it with us elementary ed undergrads!

  8. Kate, I especially like the visual with the blue and red dots. That added something for me, since I'm used to doing the algebraic stuff, but the dots really add powerfully. In fact, here is a small improvement: how can we bridge the gap between 1) I'm playing with dots and 2) I'm playing with alg. expressions? Here is a thought:

    There are 7 dots in group 1. To check that this is an odd number, we write 7 = 2(3) + 1. That fits our notion of "odd number," and the fact that that one extra dot is "hanging off" fits our visual notion.

    There are 9 dots in group 2. We do a similar check, noticing that 9 = 2(4) + 1. Odd!

    When we put these together, we get 3 pairs from group 1, plus the 4 pairs from group 2, which is like 7 pairs. But then there are those two extra dots, so we have 7 + 9 = 7(2) + 2. That's like one extra pair, which is 8 pairs altogether. So we have 7 + 9 = 8(2).

    I just think this could be a useful exercise, b/c if you model one like that, and then the students do one or two themselves, the transition to "symbols only" would, I expect, be nice and smooth. :-)

  9. Stacy, that was an inspired suggestion, and I'm honored that it was one of your four tweets. :-)

    James, that's a great suggestion for bridging the two representations. Thank you for sharing it.

    Judy! If anyone tries it, I hope someone will let me know how it goes.

  10. Would be great to have a repository of little interesting patterns students could prove. Know of any?


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