# Cultivating Teamwork in Math Class

Establishing community in the classroom can be a challenge. Here's an activity my students participated in on the first day of school. I learned about this activity while participating in the Advanced Educator International Space Camp in Huntsville, Alabama. The objective is for two crews of astronauts to exchange positions in cramped quarters when a new crew shows up to relieve the old crew at the International Space Station.

## How the Game is Played

Players must follow these rules:

1. Only one person can move at a time.
2. Only movement forward (in the direction a person faces) is allowed. In the above diagram, orange players can only move right, while blue players can only move left.
3. A person can move to an empty space in front of them.
4. A person can jump an opposing team member in front of them.

## What Does It Mean to Win the Game?

The teams win the challenge when they have exchanged their original positions. See the ending position diagram for an example of what this looks like.

## What If...?

A teacher could use an agility (speed) ladder for this activity. Or if a ladder isn't handy, use tape. This is a shot of my classroom the first day. It's pretty unlikely a teacher would have only eight students. I put down three tape ladders in my classroom on the floor. If the number of students is not a multiple of eight (like my class was), the teacher could place the extra students on the side as coaches. To up the responsibility of the coach, add the rule that no one inside the ladder can talk to anyone else.

## Teacher Moves

While this activity was going on, I floated between the groups and listened very carefully. I wanted to learn about which of my students would step up and take initiative; which would be a leader; which would be concerned about the frustration of others and take action to minimize other students' discomfort/anxiety. This activity helped me better understand how to assign groups for course work in a meaningful way.

## Examples of Student Moves

Below is an example of what some students might do.

If the third blue player from the right jumps the lone orange player, the blue team has a problem. With two blue players in adjacent cells, the game is gridlocked and ends.

## Computational Thinking

Once the students came up with the solution, I gave them the sequence "1-2-3-4-4-3-2-1" and asked them how it relates to this situation. Think of this sequence as the answer key.

1: Orange moves first
2: Blue moves next - twice
3: Orange moves three times
4: Blue moves four times
4: Orange moves four times
3: Blue moves three times
2: Orange moves two times
1: Blue moves one time

## Low Entry, High Ceiling (Extending the Task)

• Ask the students to come up with some pseudo-code to describe how they would build this game on a computer using programming applications.
• Ask the students whether the strategy remains the same if there are teams of five? Or if there are two empty middle squares? Three empty middle squares?
• Ask the students to write a program that allows the user to watch the game. Then ask the students to write a program that allows the user to play the game.
• Example from my classroom I had two students come up with different lines of thinking for coding this game on a computer. One student thought of a number line to label each cell, using the values -4, -3, -2, -1, 0, 1, 2, 3, 4. Another student thought of simply have numbers represent each student. The starting configuration would be
1 2 3 4 _ 5 6 7 8. Then, each move would be a shuffling of the sequence. The second row would be 1 2 3 _ 4 5 6 7 8. The third row would be 1 2 3 5 4 _ 6 7 8. The fourth row would be 1 2 3 5 _ 4 6 7 8. We had a really spirited discussion of the issues that could arise from each organizational coding strategy.

# 7 Games for the High School Math Classroom

## Euclid: The Game

Euclid: The Game, designed by Kasper Peulen, takes the basic compass and straightedge constructions featured in geometry class and makes a game of them. There are 20 levels to complete. A teacher can directly access a particular level by modifying the URL. http://euclidthegame.org/Level1.html (for example, change the "1" to a "7") I would have loved the opportunity to learn geometric constructions in this format when I was in high school.

## 2048

The premise of 2048 is very simple. Use the arrow keys to slide tiles about the game board. When two tiles with the same number touch, they merge into a new tile. For example, when two tiles with the number 64 on them touch, they merge into a 128 tile. The objective is to make a "2048" tile, which would require two 1024 tiles, and each of those require two 512 tiles, and each of those requires two 256 tiles, and each of those requires two 128 tiles, and each of those requires two 64 tiles, and each of those requires two 32 tiles, and each of those requires two 16 tiles, and each of those requires two 8 tiles, and each of those requires two 4 tiles, and each of those requires two 2 tiles. Each move in the game yields a new number tile on a random space. A user may continue with the game even beyond the 2048 tile. I have personally witnessed one of my students with a 65,536 tile. This game has some uses in the math classroom. The obvious is knowing the positive integer powers of 2 and its parallels to data storage capacity in computer hard drives. We can also use this game to teach an introduction to game theory and optimal strategies. After playing this game for a while, a user tends to see patterns and trends, cycles of values and positions that appear frequently. Students can intuit optimal strategies through trial and error, but this could also allow students to use formal mathematics to move towards establishing optimal strategies systematically.

## SET (card game)

While the card game isn't free, the daily set puzzle is free. My students avoid trying to explain SET to a novice player. It is much easier to play and learn through trial and error than to learn all the rules before doing anything... (sounds a bit like mathematics, in my opinion). Each card has four attributes - shape, color, shading, and number of shapes. A set is a collection of three cards for which each individual attribute is all the same or all different. Here are two examples of possible sets.

There are some great combinatorial features associated with this game. A great strategy lends itself incredibly well to mathematical statement: For any two selected cards, there exists a unique third card in the deck which completes the set.

## ALICE

Alice isn't a game, but rather a software package which teaches computer programming as if it were a game. Students can use previously constructed environments and characters to create animated videos. Students can use the interface to make their own games. I have used this software in my class for five years. I have had students make a soccer game that plays a victory sequence for the first player to three goals, a tank rolling through a world that enables collision detection (the default objects can pass through walls; writing code to detect collisions requires some great geometric reasoning), and a first person zombie shooter game *complete* with a zoom-in rifle scope. All my work with students has been with Alice 2.0. I plan to use Alice 3.0 this upcoming school year!

## SCRATCH

Scratch is like Alice in that it's a programming interface. I have not used it personally with students, but it's another resource I do plan to explore with some of my students this upcoming school year.

## KenKen

Many are familiar with Sudoku. KenKen is a similar reasoning number puzzle game involving operations in addition to populating the digits.

## Alcumus

I've mentioned Alcumus previously on my blog, but I can't say enough about the role Alcumus plays in addressing learners' needs. Teachers can now register for a teacher account to monitor students' progress through the self-paced, differentiated curriculum. Students can sign up for free, only an email account is needed. Students can take a pre-assessment to determine areas of strength and deficiency. Quests and experience points provide instant feedback, instant gratification, and a mechanism to keep students engaged and to avoid tedium.