# 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.

« »

## 0 thoughts on “A Problem with the Space Inside it to Learn”

1. Deb Nichols says:

Love it! Your girls don’t know how fortunate they are! The best things happen when we are allowed to play and wonder!

1. Deb, you have no idea how happy it makes me to see you here. Thanks for commenting! Can’t wait to get back to RGS and play and wonder with you. xo Tracy

2. toby gordon says:

Not only do I love the post, I love that Sam’s now entered the scene… Math kudos for the entire Zager fam.

1. It was just a matter of time before he got looped in…

3. This is really great! This time I like reading about the adult reaction (Sam’s) even more than the kids’. I don’t really expect all my students to remember offhand how to calculate a point of intersection in a few decades, but it would be super cool if they remained curious about math and could think out ways to answer their own questions about it.

If you’re looking at this from a HS or upper MS point of view, there might be some way for a cleverer person than I to tie in the Pythagorean Theorem. I notice the point of intersection for y = x/a – a and y = x/b + b will be at ( ab(a+b)/(b-a), (a^2 + b^2)/(b-a) ). If b = a + 1, as it does here, this is ( ab(a+b), a^2 + b^2), so a, b, and sqrt(y) would make the sides of a right triangle (e.g. for a=4 and b=5, as in the picture, they’ll intersect at (20*9, 4^2 + 5^2)) or (180, 41)).

But around this point I lose track of whether this could go anywhere really interesting or is just bogging down a cool problem that gets people to develop their own thoughts about (non)parallel lines and intersections and slopes and stuff.

Also, I love hearing your perspective about Sam. He digs a good problem just as much as I do, but doesn’t have as many opportunities to play with them. I have to ask him more about his pic in the newspaper!

Thanks!
Tracy

5. Love the problem! As you illustrated, It is accessible to people with very different levels of mathematical expertise. The question “What do you think?” turns it into a sort of 3-Act problem. What I thought was “Are they parallel or not?” I simply counted the squares to notice that the top line had a slope of 1/5 and the bottom line had a slope of 1/4, so no, they weren’t parallel.

I was done…or so I thought until I started reading the comments. Someone else may care about where they will intersect. So I guess I should care, too. Julie went about it in a way that I never would have thought by generalizing the two equations. I would have not been as clever as Julie and just used the equations y = 1/5 x + 5 and y = 1/4 x – 4. Setting 1/5 x + 5 = 1/4 x – 4 and solving for x, and then y, I would get the point of intersection at (180,41), the same answer as Julie.

What if I don’t know algebra yet and so can’t find the point of intersection by setting the y=values of the two equations and solving for x? I graphed it and noticed that at x = 0, the lines were 9 vertical units apart. At x = 20, they were 8 vertical units apart. Hmmm….could the 20 come from 4 x 5? I continued the graph to x = 40 and they were 7 vertical units apart! Practice Standard # 8 tells me to “look for and express regularity in repeated reasoning” and # 7 tells me to “look for and makes sense of structure.” I’m pretty convinced that when x = 60 that the lines will be 6 vertical units apart. If I want to graph this, I can check out my hypotheses. At some point I will be convinced that for every increase of 20 units for x, the lines are 1 unit closer together. I fill out the following table. For the y-value of the upper line, I notice that for every increase of 20 units for x, there in an increase of 4. For the y-value of the lower line, I notice that for every increase of 20 units for x, there is an increase of 5! Wow! Another pattern! For every increase of 20 units, the y-value of the line with a slope of 1/4 increases by 5 and the y-value of the line with slope 1/5 increases by 4!

x-value of both lines distance apart y-value of upper line y-value of lower line
0 9 5 -4
20 8 9 1
40 7 13 6
60 6 17 11
80 5 21 16
100 4 25 21
120 3 29 26
140 2 33 31
160 1 37 36
180 0 41 41

So, the two lines intersect at (180, 41).

In the CCSSM, the coordinate plane is introduced in Grade 5. Standard 5.G.2 states “Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.”

I love problems that are accessible to a wide range of problem solvers. Thanks for sharing! I’m going to re-post this on my own blog http://www.watsonmath.com.

1. Elaine, Wow! That’s fantastic! Thank you so much for sharing your musings too. Tracy