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!

I tried 1-2-3-4 ….slightly surprised.

Then I tried 1-2-3-4-5 …..ah-ha, or is it aha?

A conjecture entered my head. Ask!

What’s your conjecture? Do tell!

1-2-3-4 certainly is interesting… Stay tuned for a post about that.

Conjecture: Apart from uninteresting cases such as 1,2,1,2 the path will only return to the starting point, and generate a full pattern, if the number of steps in the repeat

and the number of turns to give a full turn are co-prime, ie with no common factor.

I tried 1,2 with a 240 degree turn angle ( exterior angle of an equilateral triangle). Result was a version of the nuclear hazard sign! I want to try the “5 turns to get round next”. This is so easy to code up in Logo/Scratch.

For howardat58’s conjecture, how about 1,2,3,4,8,7,6,5 for example?

Anna,

I just saw your book/Loop de Loop activity on Dan’s blog and immediately ordered the book. I love this so much already.

I am a teacher educator, and have been spending lots of time recently trying to rethink how we teach math. I have been incorporating computer programming more into what I do (especially with Scratch and Turtle Blocks), and I can see some extensions to these activities there.

Thanks so much.

Gerald

Hi Gerald,

Thanks so much for reading! I’m glad you like the activity. I think that there are so many ways you could use these activities with Scratch (and I’m sure Turtle Blocks would be great, too, but I don’t know anything about them– time to learn!). I hadn’t thought of that myself, but I’ve gotten this suggestion from several people. I think I’ll do a blog post about that in the next week or so– if you come up with anything and you’d like to share, please let me know!

I’ve actually left the classroom to become a teacher educator myself. I’d love to hear about how you’re rethinking math teaching and incorporating your new ideas into teacher education.

Stay in touch! Thanks,

Anna

I’ve been trying a lot of these. I’m not a math teacher, but this is fun! I really like the shapes you get when you use 6 numbers (i.e. 1,2,3,4,5,6). I’ve tried the first few primes numbers (1,2,3 then 2,3,5 then 3,5,7 and so on) and am trying to figure out a relationship between the final products of each sequence. I’ve also tried the first digits of Pi up through the first nine digits. Interesting things happen for sure!

Awesome! Six numbers is actually my favorite. 🙂 Let me know if you figure out anything about primes!! Thanks for the comment!

Got inspired to create a simple Geogebra app to explore the loops:

http://tube.geogebra.org/student/m1741239

Thank you so much for sharing these puzzles! I had heard of spirolaterals before, but it had been awhile. These worksheets hit at just the right time for us. I gave them to my (11th grade) daughter for her math lesson last Friday, and after playing for awhile, she noticed a pattern in which number series loop and which ones spiral away into infinity.

She wrestled with it for awhile and pinned it down, classifying three possible types of loop-de-loop patterns (I later found one classification she’d missed) by how many copies make up a completed loop. And she came up with a vector-based proof for why they came out that way, and made a conjecture about how longer series will behave.

I love how these patterns made her mind work! 🙂

Her proof was oral, but if I can get her to write it down, I’ll share…

OK. This is everything I’ve been teaching my grades 3-5 gifted students, but all in one place (except…if Fibonacci in here?). I feel completely validated. 😉 Bought the book! Thank you!

Like everyone else with a little programming knowledge, I felt obliged to make some loop-de-loop animations. Here are two you might find amusing (just push the blue play triangle to run):

(1) Draw a rectangular loop-de-loop with your choice of step lengths

(2) Draws a randomly generated loop-de-loop, not necessarily rectangular

I particularly find the second one fun to run, with some fascinating surprises. For example, one sees a lot of busy stars when the turn angle is 180 * 5/7. Then, suddenly the figure generated by that turn angle and steps 6,6,3,9,8,8,1 pops up and is a big surprise (to me, at least).

You asked “What different number sequences make identical Loop-de-Loops? What do these sequences have in common?” — I was super surprised when I found out that there were 3-number sequences and 5-number sequences that made identical loop-de-loops! Also there are different versions of the 5-number sequence, some that trace each segment only once and another that looks the same in the final result but with some of the segments having been covered more than once.

There’s so much to discover here!