Reflecting over a circle

If you’re reading this blog, you probably know something about reflections in geometry. You have a point (purple in diagram below) and a line (solid). If you reflect the point over the line, you get a new point (orange) on the other side of the line, the same distance as the original point. The segment connecting the two points (dashed) is perpendicular to the line of reflection. (Link to live graph.)

What if you reflect over a circle? I played today with the circle I understand best—the circle of radius 1, centered at the origin.

You still want the line segment to be perpendicular, which is tricky enough. But distance gets messy. Reflecting over a line means taking everything on one side of the line and matching it to something on the other side. Easy to do with two half-planes.

But with a circle? You need everything outside the circle to match up with everything inside the circle. The basic idea is a point outside the circle will match up with a point inside the circle, and that their distances will be reciprocals of each other. If the original point is 2 units from the origin, its reflection will be 1/2 unit from the origin. 3/4 matches with 4/3, 5 with 1/5 and so on.

Whether infinity and zero match up is open to interpretation and not important right now.

Here’s what this looks like.

Go play with the graph. Move the orange point around and start to get a feel for the relationship between the original and its reflection.

Now we’ll do two points and connect them with a straight dotted line segment. Each endpoint is reflected in the circle. (Play with it here.)

What does the reflection of the in-between points look like?

Imagine it. Sketch it. Then go see.

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My favorite quadrilateral

I am a known fan of triangles and hexagons. I have also been having quite a bit of fun with laser cutters at a local maker space.

A while back I wondered what it would be like to decompose a triangle and play with its parts. So I cut up a triangle and got busy.

My play started simply.

rhombus

trapezoid

hexagon

square

Things quickly got more complicated, with symmetry, patterns, and tilings.

I saw happening in myself what I kept telling people I saw in children at Math On-A-Stick last summer. The longer I persisted, the richer the ideas I had. These are in a bowl on the dining room table (along with my favorite pentagon and some materials for Tabitha’s decimal study), available for play whenever we like.

You may not have a laser cutter, but you certainly have access to a compass, cardstock, and scissors. I recommend getting down to business so you can play with my favorite quadrilateral.

Project Pentagon

Pentagons are taking over my life.

You may have heard the announcement this summer that mathematicians found a new tiling pentagon. Previously, there were 14 known classes of convex pentagons that tile the plane. Now there are 15. Maybe that’s all there is; maybe there is another class, or even infinitely many classes, remaining. No one knows.

My Normandale colleague Kevin Lee brought some samples of this new pentagon to Math On-A-Stick this summer, mere days after the announcement. This led to discussing the nature of sameness of the pentagons with my father, which led to further reading, and so on…

I am now drawing an example of each of pentagon type using Geometer’s Sketchpad and Adobe Illustrator, cutting them out of wood on a laser cutter, and then figuring out how they go together. No phase of this project is simple.

I consider a pentagon “solved” if I have at least once figured out how it tiles.

I have successfully drawn and cut pentagons 1 through 11. I have solved all of these but number 9.

The project is making me think a lot about learning.

For example, tonight I was working on pentagon number 8. I solved it.

These sets of four can continue to go together in a way I see and can describe.

But that’s not the only way to view the solution. Maybe someone else solves it using sets of three.

This is the exact same arrangement—the same solution—organized differently. The threes are meaningful here, whereas the fours were meaningful in the first solution. Which is better? Which is right?

Another solution uses sixes.

With that set of six pentagons, you can tessellate by translation only. The three pentagons at lower right are the beginning of the next set of six. Each of these has the same orientation as its corresponding pentagon above it. Does that make it a better solution?

I’m thinking a lot these days about the kinds of questions I’ve posed here. I’m trying to sort out my answers to a larger question:

What is (or should be) the relationship between informal outside-of-school math, and school math?

I have given a couple versions of a talk that asks four basic questions about people’s mathematical activity that occurs outside of school:

• Is this math?
• Is it school math?
• Do we value it?
• Why or why not?

I invite you to join me on this journey.

I’ll keep you posted on the pentagon project.

The tale of Tabitha and the two division problems

Consider these two division problems:

Problem A: 22 cookies. Each kid gets 10 cookies. How many kids can get a full share? How many are left over?

Problem B: 22 cookies. There are 10 kids. How many cookies does each kid get? How many are left over?

These are not copied verbatim from Tabitha’s third-grade homework this week, but the numbers and context are the same. (Forgive me; I didn’t think about the potential for large-group discussion until the homework went back to school.)

The point is this: One of these problems was very easy for Tabitha, and the other was very challenging.

Do you know which is which?

We talked about this on Twitter today. (Click through for some really outstanding discussion….seriously.)

I have written about the two major types of division problems before, and they are relevant here.

Problem A was a snap for Tabitha. She skip counts well, and she is a whiz with place value. How many 10s in 268? Why 26 of course! This is the sort of thing I’m talking about.

So Problem A above is a piece of cake for her. This problem—for Tabitha—is very clearly asking How many tens are in 22? For her, this isn’t really even a question worth asking. Each kid gets one ten. There are two tens. QED.

Problem B doesn’t submit to this strategy in an obvious way. It requires her to keep track of 22 things as they get shared among 10 kids. One for you, one for you, one for you, etc. That’s taxing work, and so it’s a much harder problem for her.

When we discussed this problem together the other night, I made the argument that you use up 10 cookies each time you give everybody one cookie. I wanted to help her see how her strategy from Problem A would be useful in Problem B, while respecting that—for her—the sameness of these two problems is not at all obvious.

What’s the moral of the story? Let me know your thoughts in the comments.

Fruit snacks

Kellogg’s has issued Froot Loops fruit snacks in the shape of digits. (Side note: Cheez-Its need to get on board with this! There have been Scrabble tile Cheez-Its for years. We want numbers, operations and relational symbols!)

Naturally I bought some.

Tabitha (8 years old) asked—as she does in these scenarios which occur with great frequency—Are you just buying that because it’s mathy?

Yes, sweetie. Yes I am.

But how to put them to use?

After many rejected ideas, here’s my favorite.

The task

Here are the contents of one pack.

That’s 5, 2, 9, 1, 3, 2, 4, 3, 9. Their sum is 38.

I’m setting the over/under on the sum of the next pack at 41. Do you want the over or the under? Why?

Play along with your questions and answers in the comments.

I’ll open the pack on Wednesday, May 27.