Algebra Card Game Playtest Report
Luke Zaccaro
March 23, 2016


This is a test of my Algebra card game on the target audience of 7th grade students in the early stage of learning algebra.  Students were given the game materials and instructions, and asked to play as best they can.  While I did demonstrate play for about 3 minutes prior to distributing the game materials, I gave no verbal instructions with regard to the play of the game itself, unless asked by the players.  However, while testing, I did announce one change to a game constraint when it became obvious that the current form was imbalanced.  

There were 3 simultaneous games that were played for approximately 25 minutes.  All were filmed so that I could later watch and tally the frequencies of various occurrences.  The full tally of documented game occurrences can be found at the end of this document.

Anonymity precautions:
-All faces have been blurred out in photographs

About the human subjects:
Students are 7th grade students, and most are relatively advanced for their age with regard to academic ability level, though a few are not.  Their experience level with Algebra is fairly consistent:  All have had a pre-algebra class, and a few are currently in an accelerated math class.  This particular school does not have a large number of math class offerings for 7th graders.

Game Materials

Game Materials for the Red Player

Addition: 10 Cards          Subtraction: 10 Cards    Division:  10 Cards       Parenthesis:  20 Cards

Game Materials for the Blue Player

Addition: 10 Cards          Subtraction: 10 Cards    Division:  10 Cards       Multiplication:  10 Cards

Game materials for both players:

-One variable card
-36 chips - each with a digit (1-9)

All post-test observations and comments are in red.

Notes on Game constraints

The main purpose of this playtest was to determine the validity of the formal game constraints listed below.  It is likely they will need to be tweaked.  These values are ways to give advantages to one player or another, and/or affect the distribution of operator cards in their decks.  Constraints that could still be altered are in blue.


Addition cards –
Subtraction cards –
Division cards –
Parentheses cards –

The distribution of cards in the decks seems to be fine.  I think the numbers could be reduced slightly, perhaps by about 20%.  However, I don’t want to reduce the size more than that, since that would increase the likelihood and the effect of statistical anomalies with regard to card distribution.

Beginning of turn - draw until you have 7 cards
This was fine.  Increasing by 1 could confer slightly greater advantage to Red, and it is still possible this is needed, so I will leave it highlighted for now.
Can play up to
 4 operators each turn (a pair of parentheses counts as one operator)
This was fine.  

You can trade in any number of cards to draw an equal number of cards, but you can play one less operator on your turn for each time you do this.  If you’ve already played 3 or 4 operators this turn, you cannot do this.

This was fine.  It is an effective way to rebalance Red’s hand if they have too many or too few parenthesis cards

Ways to win the round (goals)
There are 3 operators in play at the end of Blue’s turn
There are 4 or more operators on the board at the end of Blue’s turn

I changed this to 4 during the test when it became clear that Red was overpowered.


Addition cards –
Subtraction cards –
Division cards –
Multiplication cards –

The distribution of cards in the decks seems to be fine.  I think the sizes of the decks could be reduced slightly, perhaps by about 20%.  However, I don’t want to reduce the size more than that, since that would increase the likelihood and the effect of statistical anomalies with regard to card distribution.

Start of Play
Draws 5 operators

This was fine

Beginning of turn – must draw until they have 5 operators in their hand
No restriction on the number of operators you can play per turn.

1. Can trade in
2 cards to draw 1 card.
This was fine, although one other possibility could be:  “Can trade any number of cards to draw the same number minus one.  Example:  can trade in 3 cards to draw 2 cards.”  This would confer a very slight advantage to Blue
2. Can directly simplify the expression, if possible.  Example:  x+9-4+5  can be simplified to x+10
This ability is central to the game - it forces both players to look at the expression and understand it more deeply.  

Way to win the round (goal)
Isolate the variable 2 times
This was fine.  While Blue seemed slightly weaker in the test, changing this to just once would give Blue too much power - the win condition would be too easy to meet.  


Do the size and shape of the cards seem to make them easy and pleasant to handle?

They are currently 5.5cm x 8.5 cm.  The kids seemed fine with the size.  I think they could be slightly larger, though - perhaps 7cm x 10cm.

Card Layout

Do the instructions and examples adequately show/explain how to use the cards to assemble algebraic expressions?  Do the players arrange them in an algebraically literate way?

For the most part, yes.  I did not explain or display rules for how to arrange an expression properly - My game assumes that knowledge, and is intended to be used in a supervised classroom setting where an instructor can act as arbiter.  

While watching the videos, I tallied 6 instances of incorrect uses of Algebra that were not detected by the opposing player.  However, 5 of these were committed by one student.  (There were 3 instances that
were detected by the opposing player, and 2 of these were again committed by that same student.) Because the rest of the students totaled a combined 2 instances of incorrect play for the entire playtest, I believe the amount of assumed knowledge for this particular age (7th grade students) is well matched to successful play of this game.

If the players do not naturally arrange the cards appropriately, is this due to the instructions and examples, or due to their inexperience with algebra?

There was little confusion as to how the cards should be arranged.  The confusion that did exist, however, was mostly with regard to knowledge of Algebra.  There are, of course, multiple correct ways to display an algebraic expression, and students seemed to display an adequate level of flexibility in doing so.


Paired players switched roles after each round, so if the game is balanced, wins and losses between the complicators and the simplifiers should be close to equal.  If the game is unbalanced, the formal game constraints will need to be tweaked.  

As noted above, I changed the win conditions for Red to be:  4 or more operators are in play at the end of Blue’s turn, instead of 3.  (A pair of parenthesis counts as 1 operator)  This made the sides more equal.  However, this was about 6 minutes before the end of the test, so I do not feel confident that the sides are close enough to equal.  My hunch is that Red is still a little stronger, but not by very much.  I will need another playtest to ascertain this with greater accuracy.

Number of rounds Red won:  3
Number of times Blue won:  2  (sort of)
In all game play, the variable was isolated 4 times.  Three of these instances were by one player, making for only 1 win.  Another player isolated the variable once, but did not do it a second time, so she did not technically win.  

Verbal assistance

I gave no verbal instructions on gameplay prior to the start of the test.  However, there were some things the subjects were not able to resolve on their own that I had not anticipated in the written instructions.  

Players did occasionally ask whether or not something was “legal.”  These questions were mostly algebraic in nature, and not necessarily related to confusion with the game itself.  I gave verbal assistance in helping students to understand rules for creating an algebraic expression, which helped them play the game more efficiently


Did the subjects appear to have fun?  

Yes.  Students were disappointed when the period was over, and clearly expressed a desire to play more.  In the post game surveys, the mean rating of fun on a 10-point scale was 8.05.

Did they get past the rules and instructions quickly enough to enjoy the game?

They did this much faster than I had anticipated - they were fully engaged in a matter of minutes.

Did the players exhibit competitiveness?

Yes.  One player threw his cards down in mock anger after not drawing the card he was hoping for.  Another delighted in making weirdly complicated expressions, and shrieked in mock anger when his opponent directly simplified them.  I have a short video of this fun outburst here.  There are some awkward edits to preserve the anonymity of the participant, but it gets the point across.

Other observations

Any other observations regarding the playtest are documented here:

Number Chips
One aspect I had given little thought to was the number chips.  However, during the test, it became clear that I need to address them.  I’ve written the student-driven concerns and my own observations about the number chips below

How many?
How many of each number should I include?  Too many, and the pile becomes burdensome, as does searching for a specific number.  Too few, and the players could run out, or be severely limited.  I think there should be
2 or 3 of each digit per game, although I would need to test that assertion.

The issue with limited numbers is that Red could use up all 2 or 3 copies of one digit, making it impossible for Blue to use it.  In this instance, Blue would need to get creative in their use of numbers:  They need to multiply by 12, but there are no longer any “2’s,” so they instead multiply by 3 and by 4.  This forces them to use another card, but it also forces them to be more mathematically flexible in their thinking - which is one of the serious goals of the game.

Limiting the number chips would change a factor of gameplay, and I would need to test again in order to determine what those limits should optimally be.  

        Number Chip Organization
I think a small box, or other organizational tool is needed to visually delineate a pile for the available number chips - the play area got visually cluttered at times.

Students asked why there was no zero in the pile.  This was a conscious choice of mine, as I didn’t want players to multiply by zero, thus negating the entire expression, and I didn’t want them to divide by zero, creating a mathematical error.  However, there was an instance where Blue was simplifying, and needed to change some numbers to “10.”  Clearly, I will need to add zero to the pile and simply add a rule that neither player can multiply or divide by zero.

Using the subtraction card as a negative sign
This had not even occurred to me.  One of the students had brought it up shortly into the game, and I said yes - try it out and see what happens.  It ended up being completely fine.

There is definitely an element of luck here - namely which cards are drawn.  However, there are methods that both players have to mitigate the effect of luck.  Luck in a card game, however, is a feature, not a bug.  The fun is in using the cards you’ve been given to beat your opponent.

Survey Results:

What math class are you in right now?

All were either “normal math,” (Pre-Algebra for 7th graders here) or “Advanced math.” (Also pre-algebra, but more advanced.)

What grade are you getting in your math class right now?

All were A’s and B’s.  Somewhat unexpectedly, grades did not seem to correlate to game success:  Students with A’s did not, on the whole, have more wins than those with B’s.  (The sample size, of course, is too small to draw any conclusions, but it is an interesting observation)

Please rate the clarity of instructions.  1=very unclear, 10=very clear

Mean rating was 8.1

How easy was it to learn how to play?   1=very hard, 10=very easy

Mean rating was 8.53

Please rate the level of fun.  1=not fun, 10= very fun

Mean rating was 8.05