# Equations Squared

• Click or touch a symbol to select it, then click or touch the board to place it.

• Play a straight line of symbols to create balanced algebraic equations.

Example
• new Board(5).horizontal(0, 2,'1+1=2').caption('Horizontal')
• new Board(5).vertical(2, 0,'3=4-1').caption('Vertical')
• new Board(5).horizontal(0, 2,'1+1').vertical(2,3,'=2').highlightArray( [0,2,1,2,2,2,2,3,2,4] ).caption('Illegal! Tiles must be placed on one axis')
• Every turn after your first, your new equation must touch an existing one.

Example
• new Board(5).horizontal(0, 2,'1+1=2').vertical(0,0,'4*1=1').highlightArray([0,0,0,1,0,3,0,4]). caption('Added vertical equation')
• new Board(5).vertical(2, 0,'3=4-1').horizontal(0,4,'8/1=8').highlightArray([0,4,1,4,3,4,4,4]). caption('Added horizontal equation')
• new Board(5).horizontal(2,1,'5=5').vertical(1,0,'10=10').caption('You can alter existing equations as long as they are still true')
• Symbols must always form balanced equations in a line with their neighbors; you cannot have two symbols next to each other that are not part of an equation.

Example
• new Board(5).horizontal(0, 1,'5=5').horizontal(2,2,'0=0').caption('Illegal! \"50\" is not an equation.')
• If you play a variable, you will have to give it a value that makes the equation true.

Example
• new Board(5).horizontal(0, 2,'A+1=2').caption('You will have to give a value to A')
• new Board(5).horizontal(0, 2,'A+1=2').vertical(0,0,'B-A=8').caption('You will have to give a value to B, but A was already set')
• new Board(5).horizontal(0, 2,'3A=15').caption('You can also multiply by juxtaposition')
• new Board(5).horizontal(0, 2,'2A=24').caption('Variables are not digits! Here, A must be 12, not 4')
• You get points for each operation, and combining them into complex expressions earns a big bonus!

Example
• new Board(7).horizontal(0, 3,'3+2*2=7').caption('Bonus points for combining operations...')
• new Board(7).horizontal(0, 3,'3*2+2=8').caption('but be careful with order of operations!')
• Addition, subtraction, multiplication, and division: each is worth more points than the previous one! When you combine them in the same expression, the effect is multiplicative.

• Earn badges by playing interesting and complicated equations, but avoid demerits for making mathematical errors.

• The game is over when there are not enough digits and variables left to fill up your tray. The number remaining is shown below your score.

This game was designed to be usable as a math learning assessment instrument as part of the ETS Assessment Games Challenge. It is structured around the Equality and Variable: Equations and Expressions Model learning progression. There are two ways to use game results as part of a learning assessment: through game summary and through the score.

### Assessment by Game Summary

Many of the badges and demerits earned during the game are designed to correspond to the levels of the Equations and Expression learning progression. The game summary screen indicates how many times each badge and demerit was earned, and this should be taken into account when using the game to assess a player's place in the learning progression.

Demerit: Not an Equation

This demerit is earned when a player commits a sequence that does not form an equation at all. The sequence would otherwise be a legal move, so this demerit does not represent a misunderstanding of the rules of the game, but rather the rules of mathematics. As such, it corresponds to Level 1 in the learning progression.

Demerit: Unbalanced Equation and Unbalanced Variable Equation

These demerits are earned by playing well-formed equations that are false, such as 5=1+1 or A=10 with A bound to anything besides 10. The difference between the two demerits is simply whether the unbalanced equation used variables or not. A player who earns this demerit a few times may simply be making computational errors, but a player who earns it consistently demonstrates Level 1 of the learning progression: a fundamental misunderstanding of the relationship between expressions represented by equality and, for the latter demerit, variables.

This badge is earned by creating an equation that uses a variable that the player had previously bound to a value. Earning this badge reflects a minimum of Level 2 of the learning progression; the player understands that the variable can have a specific value, and that value "sticks."

This badge is earned by playing at least two non-trivial equations in which a single integer value is on the left and the right. For example, a student could earn this by playing 1+1=2 and 4=2+2. Earning this badge corresponds to Level 3 of the learning progression: it demonstrates an understanding that the two sides of an equal sign can be exchanged— that there is not a dependency on having operations on one side or the other.

This badge is earned by playing an equation in which operators are used on both sides of an equal sign, such as 1+1=2-0. It corresponds to Level 4 of the learning progression, a richer understanding of equality and the equivalence of the two sides of an equation, regardless of their structure.

This badge is earned by playing an equation with more than one equal sign, such as 5=3+2=1+4. It also corresponds to Level 4 of the learning progression, where a player is using a rich understanding of equality as a relationship among expressions.

Other demerits included in the game may provide an educator with insight into a player's weak points. For example, a player who earns Divide by Zero more than once likely does not have a good understanding of why this cannot be done,

### Assessment by Score

Players' scores also provide a rough approximation of their understanding of equations and expressions. Simple identity equations such as 5=5 are worth zero points, and while punting does not incur a penalty, it does have an opportunity cost. Hence, a low score probably indicates that the player is not savvy with manipulating mathematical symbols into valid equations. Similarly, players with high scores are likely to have demonstrated significant ability with mathematical manipulations. More detailed assessment requires looking at the game summaries, as described above.

Why isn't the board showing up?

The game is written to a draft specification of HTML5, and since there is no definitive standard, different browsers may behave differently. We've tested the game primarily using the Google Chrome browser, which is free and recommended for playing the game. The game should be playable on Mozilla Firefox, but may not run properly on Microsoft Internet Explorer

Is the set of symbols used the same each game?

Each game, you have the same number of digits and variables. However, the set of operators is different each time. The most common operation is addition, with decreasing probabilities of drawing subtraction, multiplication, and division.

Why are division equations such as 3÷2=1 accepted?

We are doing integer arithmetic, so division gives you the answer in whole numbers, not fractions or decimals. How many 2s go in 3? Only one. There is a remainder of one, but that's the result of a different operation—modulo—which is not included in the game.

This game was designed and developed in Summer 2012 by Paul Gestwicki as an entry into the ETS Math Assessment Game Challenge, specifically to align with the Equality and Variable: Equations and Expressions Learning Progression.

Testers and friends have recommended a host of features, including a persistent high score table, additional mathematical operations, and multiplayer and campaign modes. These are great ideas, especially those that would increase both the fun and utility of the game with respect to skills assessment. I have also been looking at native support for mobile devices, which requires a bit more platform testing than I can execute at this moment. I would like to add some of these features and platform support, if there is sufficient community interest; however, as of this writing, the judging for the competition is taking place, and so I want to leave the current build alone for a few weeks.

This game was created using PlayN and Tripleplay. Thanks to the good folks at Google and Three Rings as well as all the other contributors.

Thanks to all my alpha and beta testers!