# Category Archives: Elementary

## 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?!

We followed it up with a “Coin Problem,” which involves similar reasoning to solve. This provides a natural opportunity for students to utilize Mathematical Practice 8 “Look for and Express Regularity in Repeated Reasoning.” This generalized problem can get quite complicated, but the basic example is accessible to early elementary. Here is what I started them with.

“If you had an unlimited number of 3 cent and 5 cent coins, which values could you make? Which values would be impossible?”

Here is the graphic organizer I gave them: Coin Problem

Try this one out for yourself, and see if you can figure out where students could employ a similar kind of reasoning as in the squareable number puzzle.

Filed under Elementary, MP.1, MP.2, MP.3, MP.7, MP.8, Proof

## Revisiting Multiplication Tic-Tac-Toe: Common Factors and Multiples

After reading about Ultimate Tic Tac Toe, I was inspired to post a bit more about Multiplication Tic-Tac-Toe. Multiplication TTT offers similar constraints to Ultimate tic-tac-toe, but the constraints are tied to the common factors. Check out my earlier post for rules and notes about classroom use.

One main difference though, is that you can claim more than one square at a time. Below I have shared a filled out board, color coded by the amount of numbers you claim with that move. For example, if you place a token on 6 and 4 you in fact get 4 positions on the Tic-Tac-Toe board as 24 shows up 4 times. In the image below, blue represents “quadruple plays”, yellow’s represent “triple plays”, oranges represent “double plays”, and white represents “single plays.”

Here is a google doc of the Color Coded Multiplication Tic-Tac-Toe Board.

Recently I introduced the game to my 3rd and 4th graders. I mentioned you could analyze the game based on how many positions each number occupies. Then, the other day I was walking down the hallway and saw a multiplication tic-tac-toe board on the floor next to their cubbies. The cool thing was, the student had begun to color code the board herself (I have a feeling I know which student it was). This kind of investigation could naturally lead students towards questions related to common factors and multiples.

One of these days, when I have the time, I’d like to figure out the optimal strategy for the game. If anyone out there discovers it, please let me know.

## Fun with Collatz Conjecture

You may have heard of Collatz Conjecture, it’s simple enough to explain to a 2nd grader, yet has stumped mathematicians for the last 80 years. Paul Erdos famously referred to it when he said “Mathematics is not yet ready for such problems.” I like exposing students to unsolved problems in mathematics, because it gives them a real sense for what mathematicians do (also their’s no pressure to solve it).

I have explored it with a variety of ages, last year I had a fruitful experience with my 5th-6th grade class and thought others would enjoy the investigation. Below you will find two ways to introduce the conjecture, and a method for reversing the recursive formula to “grow the Collatz tree.” This investigation also hits a few of the Expressions and Equations standards, such as 6.EE.A.1-2, and 6.EE.B.5.

Collatz Conjecture:

1. Take any natural number n.

2. If its even divide by two, if its odd multiply by three and add one.

Repeat step 2 indefinitely. The conjecture states that you will always reach 1 eventually.

Introduction:

The Game Intro: One way I’ve introduced it in the past is to have the students play a game where they roll a 10 sided die for a starting number. Then the person who gets to one in the most number of steps wins. Students quickly realize they can create a diagram, or tree network, which tells them right away who will win. Sometimes we extend it, by playing the game with 20-sided die and trying to solve that version as well. Here’s what the tree network might look like for the game with a ten sided die.

Collatz Tree for a game with ten-sided dice

The Human Tree Intro: Another way to introduce it that is more kinesthetic, is to give each student a number on a sticky note, name tag, or index card. Then say if your number is even, look for the person that is half your number and if your number is odd find the person with one more than triple your number. Once they find their person they can connect with them by holding their hand or putting their hand on their shoulder. I usually make a stack starting with one, going up the “Collatz Tree” until I have enough numbers for the amount of students and teachers in the group. I also try to make it so there are several branchings. When everyone is “connected,” the group should work to untangle themselves so that the overall structure can be seen. I wish I had a picture of the process, and outcome, but alas I couldn’t find one.

Growing the Collatz Tree (doubling)

Once students get familiar with the structure of the Collatz tree we talk a bit about what it would mean for the conjecture to be true. Can we prove that a certain set of numbers will always go to one. Quickly students discover that as soon as you hit a power of 2, you are dividing by two all the way down. I like calling this the “tower of powers.”

We use the powers of 2 as the “trunk” of our Collatz tree, and the first example of a way to grow the Collatz Tree. You can reverse the recursive formula and double any number to “grow a branch” of the tree. For example, 5 could have come from 10, which could have come from 20, which could have come from 40, etc.

Branching Rule (n-1)/3

Then I ask the students to notice when there are “branchings” in the tree, in other words when are there two numbers that will lead to the same number such as 5 and 32, both lead to 16. Here students have to think about when (x-1)/3 will have an integer solution. This happens when a number is one more than a multiple of three. So we reviewed the divisibility rule for 3, and tried it out on a few examples.

Students worked in groups to apply these two methods to grow the Collatz tree. There are lots of opportunities to differentiate the process, as students noticed patterns in branchings and some wrote algebraic expressions to describe the branches. For example, the 3-6-12… branch above can be described as 3*2^n.

Here is an example of what one group made:

Extensions and Connections

1. If you extend the recursive formula into the complex plane you get the fractal below. For a thorough explanation check out the blog post by Nathaniel Johnston.

2. Dan Finkel over at mathforlove.com has a variation called “The Dr Squares Puzzle” where they come up with a similar recursive process, with a few loops.

3. Here’s another recursive process involving numbers and their written form (It also has a tree structure):

Step one: choose a natural number N

Step two: write the number in words, count the number of letters in the word and write that number.

Step 3: Repeat Step two.

For example: One-3-Three-5-Five-4-Four… stays at four indefinitely. You can also try other languages. 🙂

For English it has a tree like structure.

Finally, a post on Collatz Conjecture wouldn’t be complete without this from XKCD:

## Subtraction Reversal Game and Investigation

In this lesson, students start by playing a game with two digit subtraction. Patterns emerge, data is organized, and a “trick” (rule or algebraic equation) is discovered for quickly solving these special kinds of subtraction problems. I love this because it gives an experience of discovering a pattern and then describing it either with words or algebraically. I’ve mostly used this lesson with 2nd-4th graders, but it is highly differentiable. Finding a general solution involved quite a bit of elegant algebra.

Intro: Subtraction Reversal Game Instructions:

•Each player rolls a 10-sided die two times.
•Find the difference between the largest number you can make using both numbers and the smallest number you can make. For example: (3,5): 53-35= 18
•Whoever has the largest difference wins.
•Play several rounds, Recording your results on paper.
Organizing the Data:
Students form groups and share the results of their games. They will notice that certain numbers come up more than once. Some may even notice they are all multiples of 9, or that the digits add up to 9. Others may need to organize the data to see the patterns.
I ask the groups to come up with a way of combining all their data on one piece of paper. You will notice that certain ways of organizing the data better displays the patterns, this can be a point of discussion when groups share their way of organizing the data.
Below is an example of what a groups partially filled out table might look like:

0 (3,3), (4,4)
9 (4,5), (7,8)
18 (7,9), (2,4)
27
36
45
54 (9,3)
63
72
81
•What patterns do you notice?
•What other numbers would you like to try?
•Is there a way to be more systematic about which numbers you try?
Discovering and Describing “The Trick”

Students work to describe the pattern either in words or using algebraic expressions.

• Is there a rule you could write to describe the patterns you noticed?
• What’s the trick/rule for finding the difference quickly?
Research Poster Project
After students have enough time exploring the patterns, I have students create posters individually or in partners to share their discoveries. Below are a few examples of posters students have made.
Differentiation
Some students may only be ready to describe “a trick” using words. Others may be ready to write an algebraic expression to describe the trick. A deeper level would be to try to explain using place value, and/or algebra to explain why this trick always works for 2 digit subtraction when the numbers are reversed.
When students quickly come up with a rule for 2 digits, I let them work on the 3 digit version. For example 632-236. They often immediately make the conjecture that the answers will be multiples of 99, and that for 4 digits it will be 999. The 4-digit version gets a bit harder to predict because the middle two numbers interact, representing the place values algebraically helps with this.

Filed under Elementary, Mathematical Investigations

## Multiplication Tic Tac Toe

I adapted this tic-tac-toe game from one I saw in a workshop claiming to be a multiplication tic-tac-toe game. It was similar to this multiplication tic-tac-toe game. It was fun enough, but it had no resemblance to a tic-tac-toe board.

I think my version has what it takes to be considered a “tiny math game.” All you really need to play this is paper, pencil, and two tokens (which could be two pieces of paper).

Embedded in the game is practice with multiplication facts, common multiples, and some good old fashion tic-tac-toe strategy, with a twist.

Instructions:

-Make a big Tic-Tac-Toe board, then make a tic-tac-toe board in each of the 9 squares.

-Now fill in the 9 squares with the multiples of 1-9 (see example below).

-Write the numbers 1-9 underneath your board.

-First player places two tokens (pennies in this case) each on one of the 9 numbers at the bottom. Multiplies these together, and places an “X” anywhere that multiple is found on the tic-tac-toe board. (in the example below player one has chosen 6 and 4, and has placed an X on all four “24s”)

-After the first move, players take turns choosing to move only one of the pennies to select their multiple to “X” or “O.” (For example 2nd player could move the “6” to a “3” and put an “O” over every 12 on the board)

-If you win a small tic-tac-toe game, you win that square on the larger board. The goal is to get Tic-Tac-Toe on the large board, by getting three of these smaller boards in a row.

-Also, I like to say any “Cat’s game” is a wildcard spot once it is completely filled in. This allows for player one and two to win simultaneously, which I like because I’m a sucker for win-win situations. 🙂

Notes For the Classroom:

I usually start by handing out this blank Multiplication Tic-Tac-Toe board. Students then take a few minutes to fill out the multiples of one in the top left corner, then multiples of 2 in the top middle, multiples of 3 in top right corner, etc (See example above).

First player places two tokens (pennies in this case) each on one of the 9 numbers. They multiply those two numbers together, and put an “X” anywhere that multiple is found on the tic-tac-toe board. In the example above, the first player has chosen to place the two tokens on 6 and 4, allowing her to place an “X” on the four places where “24.” I call these “quadruple plays,” and tell students to try to get as many of those as possible. Sometimes I have them color code the quadruple plays, and triple plays (like 36 or 9) so they are aware of them. This introduces and/or reenforces the concept of common multiples and factors. Check out my follow up post for more information on this.

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Filed under Elementary, Games

## 100 Game Theory

This is a great game to introduce game theory at a young age while simultaneously assessing place value and addition concepts.

Race to 100

How to Play:

Two players start from 0 and alternatively add a number from 1 to 10 to the sum. The player who reaches 100 wins.

I usually give students a hundreds chart to help keep track of where they are, as well as a “Race to 100” handout with tables to record what number each person said. Also once they discover that saying “89” guarantees they will be able to say 100, I have them star that number. I ask them to think if there are any other numbers that guarantee a win.

A 3rd grader illustrates the winning strategy visually on a hundreds chart. He put an X on the numbers he wants to say, and then shaded in the numbers his opponent “him” would be able to say.

Extensions:

How would the strategy change if it were a race to 99?