# Category Archives: MP.3

## Number Bracelets

“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.

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 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.

## The Thrill of Proof in the Early Grades

What kinds of problems allow young students an opportunity to experience the, dare I say it, exhilaration of proof? Certainly there are several classics, which can give young students a sense of the beauty, elegance, and thrill of discovering a proof. The Seven Bridges of Konigsberg is a classic problem many elementary school students can figure out.

Last year, while working with a group of gifted 2nd graders, I presented an interesting problem that appeared in a New York Times NumberPlay article last April.

The puzzle asks which numbers can you draw a square made up of that many squares. The squares need not be the same size. See the article for a few examples, or read on to see the results from my students.

This group met each Friday for 30 minutes. At the end of the first session, most agreed that 2, 3 and maybe 5 were not possible. So I asked them to try to find other “squareable” numbers at home, and bring the results in for the next week. Students spent the following two Fridays investigating which numbers were squareable.

Finally by the fourth Friday we had a strong feeling that only 2, 3, and 5 were impossible. I asked them to work together to demonstrate which of the first 25 numbers were squareable. By the end of that session we had the following chart, each student participated in the creation of one or more “squareable” numbers.

In the final session, I showed them the chart and asked them “so what about the next five numbers, are they ‘squareable’?”

Students quickly started raising their hands to explain how you could make 26, 27, 28, 29 and 30 using a pair of “operations”.

Students proceeded to explain two ways of adding three squares, they dubbed these ways the “banana split” operation and the “gigantor” operation.

One student offered her “banana split” operation which cuts an existing square into four quarters (each a square).

Banana Split operation on 4, to make 7.

One student directed our attention to 23, which is basically a 20 with three squares added to make a square four times the area. We called this the “gigantor” operation to formalize it, and decided we could do this with any square we’ve made to show the “squareability” of that number plus three.

“Gigantor Operation” on 20, to make 23

Whole Class Discussion:

What follows is roughly how the conversation went:

me: “Great, now what about 30 – 35, and the next 5, etc?”

students: Yes, they are all possible!

me: “At what point can we tell that the rest of the numbers will be squareable?”

students: “What do you mean?”

me: “When is the earliest number you could stop, and build the rest using your ‘banana split’ or ‘gigantor’ operations?”

students: “Eight!”

me: “why eight?”

students: “because using the banana split operation you could build 9 from 6, 10 from 7, and 11 from 8, and so on”

me: “Awesome! lets record your proofs on paper!”

students: “huh?”

So once I felt confident that they all understood this reasoning, I asked them to write their explanations down on paper. Most struggled to put it into words, many just made diagrams with numbers. Perhaps it was the limited amount of time, or that they were just not ready or practiced in this kind of written task. Unfortunately our group didn’t meet for a few weeks after that, and the kids were ready to move on to another problem. Next time I will make sure to set aside more time to support their emerging mathematical writing skills.

While the proof that 2, 3, and 5 are impossible requires quite a bit more experience. These young students were able to come up with an explanation for why there are an infinite set of “squareable” numbers, starting at most 6. That was thrilling for me to watch, I could see the excitement in their eyes as they had the “AHA!” moment. What’s more thrilling (in mathematics) than coming up with a simple and elegant argument that applies to a infinite set of number?!