As some of you might know, I recently wrote a book!

It’s a math and art activity book, containing all of the amazing MArTH activities developed by me and the Saint Ann’s School math department over the five years that I taught there. There are more than 30 different activities in the book, ranging from favorites like Pascal’s Triangle coloring to things you may not have seen before, like the Stomachion puzzle. I had SO MUCH FUN writing it– and I hope that someone out there has as much fun using it and I did putting it together.

(If you’d like to browse the book and maybe even buy one, click here.)

I’m posting this week not just to make this announcement, but because of an exciting opportunity! Dan Meyer had the amazing idea to run a contest for teachers and their students on his blog featuring one of my favorite activities from the book– Loop-de-Loops.

Visit Dan’s blog to find out more about the contest! In a nutshell, it involves making Loop-de-Loops with your math class, sending them to Dan, and some math art friends of Dan’s choosing their favorite loop. You and your students could win a class set of *This Is Not a Math Book*s!

Now, say you’re a teacher thinking of making Loop-de-Loops with your students, hopefully to enter Dan’s contest. Say you’re also wondering, what else can I do with Loop-de-Loops other than just have my students draw some? That’s a fun, mathematically interesting activity on its own– and I’m sure your students will have a blast if you stick to that. But say you wanted more…

So, for the next few posts, I’ll be sharing some more mathematically rich things you and your students can do with Loop-de-Loops. The first is pretty simple. I call it “Detective Loop-de-Loop.”

It’s one thing to come up with your own number sequence (like 2, 3, 4) and make a Loop-de-Loop out of it. It’s another thing entirely to look at a Loop-de-Loop and figure out what number sequence was used to create it!

For instance, what number sequence made this Loop-de-Loop? (Hint: It had three digits.) This one is easy compared to…

(Hint: This one uses five digits!) Figuring out how a Loop-de-Loop was made requires looking for patterns, thinking backwards, and checking your hunches with a diagram– all math habits of mind that are important for elementary and middle school students.

“Detective Loop-de-Loop” doesn’t have to stop with deciphering and trying to recreate tricky Loop-de-Loops. Lots of interesting questions come up. Like…

These two Loop-de-Loops look pretty similar– the only difference is that they seem to be looping in different directions, one clockwise and the other counter-clockwise. How were they made? What difference between their number sequences caused them to have different “chirality,” which is the technical math and science term for the direction something rotates?

Once you start making Loop-de-Loops and playing Detective Loop-de-Loop, you’ll notice that there are many interesting questions to ask about how different loops were made– and why small changes in the number sequence behind a loop make such big changes in the way the loop looks.

Or doesn’t… Here’s a 3-2-5 Loop-de-Loop. What would a 2-5-3 Loop-de-Loop look like? How would it be different, if at all? Can we figure out how to make a Loop-de-Loop that looks exactly like the 3-2-5, but turns in the opposite direction?

I think Loop-de-Loops are rich mathematical objects– meaning that playing with them can lead to an enormous number of interesting questions and investigations that take you and your students into the worlds of geometry, number theory, patterns, functions and more. Even just making up your own Loop-de-Loops and playing Detective Loop-de-Loop presents many opportunities for students to wonder about math and use their wondering to create something beautiful.

You can print these Loop-de-Loops for you and your students to play Detective Loop-de-Loop with. But you should also make more of your own– it’s fun, so why not?

To wrap up, I’ll give you a starters’ list of interesting questions to investigate while you and your students make and play detective with Loop-de-Loops. Of course, you might not use any of them– you might have so much fun drawing, coloring, and decorating your loops for Dan’s contest that you forget all about questions. And that’s great, too! Math class needs more time for free creativity, in which kids can feel proud of what they’ve made. At least, that’s what I think.

- What different number sequences make identical Loop-de-Loops? What do these sequences have in common?
- What different number sequences make Loop-de-Loops with opposite chirality? What patterns can you find in those sequences?
- What kinds of number sequences make Loop-de-Loops with long arms, like the first one in this post, or that are tightly packed, like the loop to the right? What about the sequences causes this difference?
- Have you made a Loop-de-Loop that surprised you? What about it was surprising?

Post your loop drawings and classroom activity ideas in the comments section! Don’t forget to enter Dan’s contest, and have a lovely loopy day!