“Doing mathematics should always mean finding patterns and crafting beautiful and meaningful explanations.” -Paul Lockhart,

A Mathematician’s Lament: How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form.

Recently I got a chance to revisit one of my favorite activities with a fresh group of 3rd and 4th graders. I love this activity because it includes so many of the things I value. To borrow some of Dan Meyer’s Ten Design Principles for Engaging Math Tasks; perplexity is paramount, it begins with a concise description, has a low floor for entry and a high ceiling for exit, it naturally encourages a collaborative approach, it highlights the limits of students existing skills and knowledge, and students often begin making predictions and hypothesizing without any directions to do so. In summary, this task rocks!

This was the first major mathematical investigation of this year for my 3rd/4th multigrade class. I dedicated the first two block periods (2 hours each) to this investigation. Skills-wise, at its most basic level, students are practicing their single digit addition facts, as well as breaking apart the tens-place and ones-place of a sum. This serves as a perfect activity before reviewing/solidifying multi-digit addition algorithms. There are a variety of ways to extend the investigation, for example using other bases or operations. Also, once students begin to try to figure out what’s going on with all the patterns, they utilize virtually all the Mathematical Practices in a meaningful way.

**The Game Intro**

I introduce the number bracelets with a game. Students pair up and take a 10-sided die. First player rolls twice, writing the two number down on a piece of inch grid paper. The second player does the same. Then students add the two numbers together, but discard the tens-place, only writing the ones-place in the next number. Players continue this process until they get back to the two numbers they started with (in the same order). If students get into a loop, that means they made a mistake somewhere. What’s cool about those mistakes, is that they hint at the overall structure of the number bracelets, so the mistakes are truly valuable.

Within minutes students start to notice patterns, and making predictions. I have done this several times over the years, and each time I see new patterns myself. After about 20 minutes, I bring everyone back in for a whole class discussion. I ask them “What did you notice?” and “What do you wonder?”

Here is a list of some of the responses, compiled from two groups of 3rd/4th graders:

**What do you notice?**

I had 4 sequences that were the same length.

When I kept going past the first two numbers, I got into a loop, it was a pattern.

I notice that #0## comes up a lot, like 2022.

I can’t get started w/ (0,0)

Its fibonacci, starting w/random numbers.

There are patterns vertically and horizontally.

The shortest patterns start with 0.

If you mess up one number, you can go in a loop, it affects everything else.

**What do you wonder?**

What happens if you get a zero as one of your first two numbers? How long will it be?

What happens if you get (1,2)?

I think for double numbers, 5,5 would be the shortest.

Could you make a sequence that doubled each time?

What happens when you get into an infinite loop?

What if you start with (0,1)?

What would the shortest one be, other than (0,0)? How can you be sure? How many are there to check?

Why do the zeros line up horizontally sometimes?

How is it that it repeats? How can you get out of a loop?

Why is it called number bracelets game?

What’s the longest pattern?

Are double numbers best?

Can you have a pattern that never repeats?

**Number Bracelets Investigation **

After a snack recess, students returned with the goal of investigating one of these questions. I asked them to choose a question to investigate, and work individually or in small groups (2-3 students). We brought out some markers, and colored pencils, so students could color code the patterns they noticed. When a student made a discovery, or had a prediction, I would ask them to write it down in their math notebook. For predictions, I asked them how they could verify this prediction?

In this first session, I tried not to steer their inquiry very much. I focused more on observing their style of inquiry, what they are noticing, how they communicate and record their findings, and how they work with their peers.

Here are some of the patterns they found.

Here’s an example from a student that had a question, and did a few experiments to answer the question. The classroom was abuzz with kids noticing patterns, making predictions, and describing their theories. If I hadn’t stopped them for lunch, they probably would have kept working through their recess. In fact, over the next few days many of them did stay in during their recess to explore further.

**Week 2: **

The following week we returned to this investigation to see what else we could draw out, focusing on the following question.

**How many pairs of starting numbers can there be?**

I first had students work in their notebooks on the question. Then had them share their findings with other students at their table. Finally we had a whole class discussion. Students came up with varying answers, including 20, 180, 100. We talked about the various methods students used, and they debated the merits of each method. I was pleasantly surprised with the amount of debate that ensued, with students changing each other’s minds.

As snack recess approached, I decided to introduce a couple classic problem solving strategies. The first strategy is to try to solve a simpler version of the problem. So what if we had fewer possible numbers to choose from, how many possible pairs would their be. We started by considering how many pairs would exist if you only had zero to work with. Then we looked at if you could use {0,1}, and then {0,1,2}. Enumerating the possibilities started to get complicated, so I introduced a second problem solving strategy: “Draw a visual model.”

The visual representation of this is a beautiful model from graph theory called a “complete bipartite graph.” This really seemed to solidify the pattern for the students.

One student then noticed something that she could hardly contain, with the classic “uu, uu, uu!” raising her hand. She shared “The first case was 1×1, then 2×2, then 3×3!”

“That’s right, so what would come next in that pattern?”

Using this model, students were able to come to an agreement that there are 100 possible starting pairs. To wrap up this discussion, we reviewed the process we went through to organize and model the information we had to answer the question.

**Research Project Poster**

Next, I wanted to give them a chance to explore a bit more, but with a focus on the possible lengths that a number bracelet could be. So we placed them in small groups of 2,3 students and had them focus on the following questions.

*What are the different lengths a number bracelet can be? *

*What is the longest/shortest bracelet, and how can you be sure?*

The poster needed to have examples of bracelets as well as written explanation for their reasoning. Here’s how one group began their poster:

I was hoping that students would notice that each of the different lengths contain multiple number bracelets. One group realized this for the bracelet that is 60 long (62 if you include the starting number twice).

**Reflections and Extensions**

Each time I’ve done this investigation, I play around a bit with how much coaching I give towards a more complete understanding of the puzzles. If time permitted, I would love to give kids as much time as they needed to fully explore the root of the patterns they find. Unfortunately, I have typically been limited to 2 or 3 block periods, which does not seem to be enough for my 3rd/4th graders to independently come up with complete explanations for the patterns they find. I still see the process as a meaningful one, despite students not fully understanding all the implications of the patterns. I think this reflects what happens often in real mathematical research and real world problem solving, when you explore a pattern to the extent you have time to, and then you move on.

As for extensions, I would ask the students how you could change the rules and see what they notice and wonder. For example what if instead of using addition modulo 10, you could try addition modulo 5, or some other modular arithmetic. Another way you could change the rules is by using an operation other than addition, and see if there are cycles.