Every time I teach the Real Number system, I teach it using a sort-of flow chart/family tree kind-of thing. At the top of the page we have Real Numbers and Imaginary Numbers, and coming down from Real Numbers we have Rational and Irrational, and then from Rational we have Integers, Whole ... you get the idea.
The kids, I have learned, either get it or don't. For the kids who don't, they don't understand the nesting concept or even (for some of them) how to read a flowchart.
So then I supplement this with a Venn Diagram. But kids don't always get that, either.
This year I had an epiphany.
I present to you my "matryoshka-doll-style Real Number system."
|The pink can actually does fit in the green one. It's the same height, but narrower.|
In each can, I put little cardboard cutouts of numbers, so that in the natural numbers I had 2, 15, 45846876524, etc. Then for whole numbers I added zero and threw the little can inside it. Then for integers, I added negatives and put the other two cans in in. Then I added fractions and decimals. Then for the two same-size cans (rational and irrational) you get the idea ... I spent some time talking about repeating and non-terminating compared to non-repeating, non-terminiating decimals and let the kids tell me what went where. We went through this starting with the smallest cans and putting them into their next biggest cans, and then taking the cans out and talking about the groupings and what we were losing/being more specific about/more general about each time.
I am so excited to say that for the first time in my teaching career, the Real Number system was a little less boring and a little easier to understand. "The Cans" system, as one class called it, just clicked for them.
The kids loved it. They got it! It was AWESOME!
p.s. So if you're wondering, the rational/irrational numbers are from cans of Cento tomatoes. Then it's a full-size can of Progresso soup. Then a regular-sized can of mandarin oranges, then a tiny can of chipotle peppers. I have a GIANT can of tomatoes that was just slightly too small to hold both rational and irrational, so I put them in a box that I labeled Real Numbers -- and when we get to imaginary numbers in Algebra 2 later this year, I'll break these back out and add an identical box labeled "Imaginary Numbers."