Instructor:
Farhad Saba , PhD
Office: NE-288 | fsaba@mail.sdsu.edu
inconjunction with EdTec Graduate
Karen Anderson
fxhorse@gmail.com

<title>Integer Matrix</title>
<internetlink>href="http://edweb.sdsu.edu/courses/edtec670/Cardboard/Board/I/integermatrix/integermatrix.htm"</internetlink>
<instructionalobjective><![CDATA[Integer Matrix was created to have the learner practice adding integers.]]></instructionalobjective>
<learnersandcontext>
..... <numberofplayers>2 or 4</numberofplayers>
.....<age>10</age><!--enter the youngest age that the game is appropriate for-->
.....<grade>6</grade><!--only one number is accepted here-use your best judgement-->
.....<subjectmatter>math</subjectmatter><!--The following subjects are to be utilized: math=1, english, socialstudies/history,
.....technology, science, geography, art, computing, foreign language, health, job skills, life skills, music, PE, and other-->
.....<statestandard>
.....<![CDATA[The game meets the following California State Math Standards:

5th Grade Standard
Number Sense
2.1 Add with negative integers and positive integers.

6th Grade Standard
Number Sense
2.3 Solve addition problems that use positive and negative integers
and combinations of these operations.

7th Grade Standard
Number sense
1.2 Add rational numbers (which includes integers)]]>
</statestandard><!--California State Standards-->
<nationalstandard><![CDATA[The game meets the following National Math Standards:

Grades 6-8
Understand numbers, ways of representing numbers,
relationships among numbers, and number systems.
Develop meaning for integers and represent and compare
quantities with them. ]]>
</nationalstandard>
<specialreauirements>The players needs to have basic prior knowledge on how to add integers at a very introductory level. Integer Matrix will provide the practice to perfect the players ability level.</specialreauirements>
</learnersandcontext>

<objectofgame>The object of the game is to create a net positive or negative score that is farthest away from zero through the addition of integer matrices.</objectofgame>
<gamematerials>
.....<board>Game board-black and white checkered like othello board game</board><!--please put none or detail out the board .....description based on the information give, make it short and precise.-->
.....<card>none</card><!--please put none or detail out the card description-->
.....<pieces>49 game discs (as detailed)

1,0
2,0
3,0
4,0
-1,0
-2,0
-3,0
1,-1
2,-1
3,-1
4,-1
-1,1
-2,1
-3,1
1,-2
2,-2
3,-2
4,-2
-1,2
-2,2
-3,2
1,-3
2,-3
3,-3
4,-3
-1,3
-2,3
-3,3
1,-4
2,-4
3,-4
4,-4
-1,4
-2,4
-3,4
0,0
0,2
0,4
0,-2
-4,0
-4,1
-4,3
0,1
0,3
0,-1
0,-3
0,-4
-4,2
-4,4

.....Game disc storage bagNet score sheets</pieces>
</gamematerials>
<timerequired>
.....<setup>2</setup><!--put in minutes based on the given time or your best estimate based on the information. Only provide a number!-->
.....<play>5 to 10</play></timerequired><!--put in minutes based on the given time or your best estimate based on the .....information. Only provide a number! If there needs to be a time span put "x to y" with numbers-->
<rules><![CDATA[Rules
(1) Setting The Players
Decide who is to be positive player and who is to be negative player in the game. The goal for the positive player is to accumulate a net score
that is as far away from zero in the positive direction. The goal for the negative player is to accumulate a net score that is as far away from
zero in the negative direction.

(2) Setting Up The Board
The positive player reaches into the bag and removes a disc. Place it black side up on square 1 on the center of the game board. The negative
player removes another disc from the bag and places it white side up on square 2 on the game board. Alternating positive player and negative
player, continue placing pieces on the board until the 9 center squares are filled. (see Figure 1)
(3) Players Turn
Negative player plays first.
Reach into the bag and remove a disc. Three things must then occur, the player:
(A) creates a matrix on the board;
(B) sums the matrix score; and
(C) adds the matrix score to his or her net score card.
(A) Creating a Matrix
Figure 2: Creating a Matrix
The player decides whether to place the disc white side up or black side up. The disc must be placed such that a row or column of discs is bordered at each end by the same color. Figure 2 shows an example where white disc A is already placed on the board. The placement of white disc B flanks
the row of three black discs, and the black discs are turned over so they are white.

Figure 3 shows an example where flanking occurs on multiple lines, one vertical and one diagonal.

A disc must be placed such that a matrix is formed and at least one disk is turned over (black to white, or white to black).

Figure 3: Creating Multiple Matrices

Game Note: A player may not create a matrix that will cause the board to turn all one color. For example, if there are 8 white discs on the board
and 1 black disc, the player may not place a disc that will cause the board to contain 10 white discs and 0 black discs. It must be played as a
black disc, regardless of the scoring outcome.

(B) Adding The Matrix Score
Each disc has two numbers, one side positive (or zero) and the other side negative (or zero). When the discs have all been flipped in a turn, the
matrix of zeros, positive numbers, and negative numbers is added and the player receives that score as a matrix score.
Figure 4 shows the sum of a white matrix. Notice that the black disc (-4) is not included in the matrix score.
Figure 4: Adding a matrix
Figure 5 shows the addition of multiple matrices (assuming the player placed the black –3 disc and created 2 matrices. Notice that the white discs
are not included in the total matrix score.
Figure 5: Adding multiple matrices

(C) Net Score Card
Figure 6: Using the Net Score Card
At the end of a players turn, the matrix score is added to the net score for the player. For example, if the negative player received a matrix
score of –2 on his first turn, and scores of –3, +1, -11, and 0 on his subsequent turns, his Net Score Card would look like figure 6.

Thus, after 5 turns, the negative player would have a score of –15.

Play continues back and forth, negative player to positive player, until the game board is filled with 49 discs, or until it is no longer possible
to place a disc such that another disc can be turned, creating a new matrix.

Net Score
-2

-3

-5

+1

-4

-11

-15

0

-15


Winning The Game
At the completion of the game, the player with the score farthest from zero (greatest absolute value) is declared the winner. Refer to the example
below.

Game A
Positive Player Score: +64

Positive Player Score: +44
Winner: Positive Player

Game B
Negative Player Score: -50

Negative Player Score: -78

Winner: Negative Player]]></rules>
<designprocess><![CDATA[Brain Storm for Content Analysis
All ideas and thoughts were compiled onto a paper.
The focus of the session created a focus on integers as a topic.
Several game ideas came to mind.

Incubation
Ideas were left to simmer in the gray matter.

Chunking
Pieces: chips
Patterns: Linear lines such as seen in mathematical equations
Paths: relate to mathematical equations that contain multiple numbers
Probabilities: when a player chooses at random the game chips from a bag
Prizes: the player with the largest score is the winner
Principles: In theory if the player understands the concept of adding integers she/he
should be able to play the game at a basic level. Then the player can work on mastering the subtleties of the board game.

Aligning
The general theme of a Othello game was used as a template to start with. The main pattern within the game was adding integers in a linear manner.
This transferred to the game board by selecting a linear pattern for the game board by considering plays both horizontally, vertically, and
diagonally (see rules). This gave the player various options of where to place their chip and determine their score. Initially when there are few
chips on the board a player can remember what number is on the back of the chips, but after a few rounds the player only knows that if the chip is
positive on the front the back must be negative or zero and visa versa. The game offers changing conditions with each play. As the chips are turned over the numbers change and so does the possibility of the score for that round. This also affords a bit of random unpredictability to the game
based on the fact that you can not predict how the board will look for your next turn. The player must always be thinking and planning there is no
down time in this game.

Drafting
Initially we used some foam pieces and wrote numbers on them. A eight by eight game board similar to checkers or Othello was used. The game started out with four pieces two black and two white placed in the middle four squares just like one would do when playing Othello After about four rounds
it was determined that four chips to start was too few. We also found that the game board ended up turning one color or the other quite quickly
thus ending the game.

Incubating
At this point we walked away from the game. Further research was conducted and a Othello game was purchased, played, and the rules analyzed.

Prototype testing
After further research a board that was 7 by 7 squares was selected. Nine squares in the middle were to be predetermined by random draws and
placement based on the players color (see rules). The rule that the game board could not be turned completely one color alleviated the issue of the game ending to quickly. The game was then tested by the creators extensively.

Beta Testing
Several individuals were selected to test the game. The individuals who tested the game were asked how they liked the game afterwards. They stated
that it was engaging and after an initial learning curve they were intrigued by the strategy of the game beyond the simple adding of integers.
Where you placed your chip had to be well thought out so as to attain the highest possible score and thwart your opponents next move. Further Beta
testing is to take place in 6th and 7th grade math classrooms. Currently game production is taking place so as to provide a classroom with an ample supply of games to play in teams of four.

Further Thoughts
After completing the game we thought we had made too simple of a game, but that is actually the beauty of it. It is clear, concise, and easy to
learn. In looking toward making another game the challenge would be to maintain the simplicity in the game yet provide a game that is engaging and
challenging. Aligning the game with the what is to be learned is the most important challenge of a game designer. It is easy to wander from the
main objective if you are not careful.]]></designprocess>
<references>
<booksandjournals>Rouse, Richard. Game Design: theory and practice, Worldware publishing, Inc. Plano Texas. 2001.</booksandjournals>
<electronic>Othello Game Site
http://www.maths.nott.ac.uk/othello/beginners.htm
Official Mattel Othello Game Site
http://www.mattelothello.com</electronic>
</references>