# A Problem with the Space Inside it to Learn

Justin Lanier shared this bit of loveliness on Twitter the other day:

I love everything about it. I printed out a couple of copies this morning and made them available at breakfast. My kids are 5 and 7. Maya, the 7-year old, has read A Fly on the Ceiling a time or two, but has otherwise not yet been introduced to coordinate grids, graphing, slope, parallel and perpendicular lines, etc. Daphne, the 5-year old, is in the same boat, except she hasn’t read A Fly on the Ceiling. I was curious what would happen.

After studying it for a bit, Maya went straight for the rulers. She spent some time measuring the axes, trying to figure out how far apart the numbers were. She was surprised they weren’t an even centimeter or inch apart. She figured out their distance, then compared the x-axis to the y-axis for a while. For those of us familiar with coordinate grids, the axes quickly recede into the background so we can see the information laid on top. For someone new to coordinate grids, they carried lots of information.

“Mommy, I see these faint lines–like graph paper–that line up with the numbers. I think that helps you see where you are. It’s like the tape lines in A Fly on the Ceiling.”

After a while, she seemed content-for-now about the axes, and moved on to looking at the two lines. She started measuring the distance between them.

I got up to make my oatmeal, and noticed someone else in the family was using a different approach.

When I asked Sam what he’d been doing, he told me he was thinking about ways to see if the lines were parallel. Maya overheard him, and said, “Oh! I don’t think they’re parallel. She started showing us with her hands how the lines were ever so slightly tipped toward each other. She said, “They’ll meet someday, way, way over here.” I asked her how she knew. Back to work she went.

Meanwhile, Daphne got in on the action with her own ruler.

And I noticed this sketch in the newspaper. Sam wasn’t done.

This problem is so lovely because it’s tantalizing, and open, and full of possibilities. It reminded me of this tweet that went by yesterday.

This problem has plenty of space inside it to learn. What do you think Daphne learned? What’s she thinking about?
Maya kept wanting to measure the distance between the lines, and we ended up playing around with that idea some. She isn’t there yet, but she’s starting to see the helpfulness of right angles. She’s really into rulers right now, and she spent quite a bit of time figuring out how to write down the lengths she found. She told me one was 3 1/4″ and the other was 3 1/8.”

I asked her which was longer.

“3 1/4.”

I said, “Wait a minute! Eight is more than 4! That can’t be right!”

She said, “The 8 tells me that the inch is cut into 8 pieces, so each piece is smaller because there are more of them. Actually, 1/8 is half the size of 1/4.”

Happy dance. Of course we spent a little more time playing with that idea. Then she said, “I want to write down what I think. It asks, ‘What do you think,’ and I want to write about that. I think I want to write about whether the lines are parallel or not.”

What a lovely change from her curriculum at school, which is nothing but bubbles, boxes, and blanks to fill in.

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