Addition and Multiplication Squares

Do you remember this old game that we used to play as kids: We needed only a squared paper and different color pens or pencils for each player. Then, each player, on his turn, draws one side of a square. The player who draws the last side of a square, colors that square in his color and draws another line. When all squares are colored, the player with the most colored squares is the winner. Using the same strategy, adding some numbers, uncertainty and math, the Squares Game can become more interesting, playful, and educational at the same time. Estimated level: 1st Grade Math for Addition Squares and 3th Grade Math for Multiplication Squares.


First, I would like to give credits to Games 4 Gains for "mathification" of the Squares Game. On their site they have different Squares Games for practicing math skills, such as squares games for addition, subtraction, multiplication, division, factors, multiples, and prime and composite numbers. The idea and the basic game rules are the same with the old Squares Game, and the main difference is the way how numbers are obtained, which depends on which math skill is being practiced. I chose two "mathified" variants of the game to present - Addition and Multiplication Squares. Here, you can find game boards for these games that differ from the Games 4 Gains' game boards in size, frequency and distribution of numbers, which I will explain later.

Addition Squares Game


Let me explain the game rules for the Addition Squares Game. The game is for 2 or more players, age 7 and up. You will need a game board (squares with numbers which you can download from the links below), 2 dice and a different colored pencil for each player. Here are the rules of the game:
  1. First, by rolling a dice, decide who will start the game. Player with the highest number starts first.
  2. On each turn, the player rolls the both of the dice and adds the numbers. 
  3. Then the player looks for a square on the board that contains the sum of the numbers and draws one side of that square.
  4. If the player has drawn the last side of the square, then he colors that square in his color and rolls again the both dice, otherwise it's the next player's turn to roll the dice.
  5. If there is no available square on the board that contains the sum of the numbers, the player’s turn is over, and the next player rolls the both dice.
  6. The game is over when all squares on the board are colored, and the player with the most colored squares is the winner.

Why is this game attractive for kids and their teachers? It is a game and the game can be won. So, a strategy is needed every time when decision for coloring a side of a square is made. And the most important - even the kids with lower math skills can win the game! There is a uncertainty in rolling the dice, and favorable numbers can be rolled in spite of the math skills. This feature makes the game more enjoyable for these kids. They practice the math skill without focusing on practicing, which gives teachers another efficient educational tool for their math class.

I would like to note that the Addition Squares Game can be time consuming, if the large game board 10×10 is used. So, I made two smaller game boards, a medium game board 7×7 and a small game board 5×5. I recommend to start playing the game on smaller boards. You can download the game boards and the game instructions by clicking on the download link next to each Addition Squares Game:

Multiplication Squares Game


As you suppose, with Multiplication Squares, instead of adding the numbers rolled on the both dice, we multiply them. Again, the game is for 2 or more players, but now age 8 and up. You will need a game board for Multiplication Squares (you can download from the links below), 2 dice and a different colored pencil for each player. These are the rules for this game:
  1. First, by rolling a dice, decide who will start the game. Player with the highest number starts first.
  2. On each turn, the player rolls the both of the dice and multiply the numbers. 
  3. Then the player looks for a square on the board that contains the product of the numbers and draws one side of that square.
  4. If the player has drawn the last side of the square, then he colors that square in his color and rolls again the both dice, otherwise it's the next player's turn to roll the dice.
  5. If there is no available square on the board that contains the product of the numbers, the player’s turn is over, and the next player rolls the both dice.
  6. The game is over when all squares on the board are colored, and the player with the most colored squares is the winner.

Again, it is recommended to start with the smaller game boards. You can download the game boards and the game instructions by clicking on the download link next to each Multiplication Squares Game:

How did I make the game boards?


I will explain here the construction of the game board for Addition Squares. Since we play with two dice it means that the sum of the numbers on dice is a number between 2 and 12 inclusive. But, not every sum is equally likely to appear. For example, the sum that equals to 2 can appear only if on the both dice we have 1, and the sum that equals to 4 can appear if we have 1 and 3, 2 and 2, or 3 and 1. And since each number from 1 to 6 is equally likely to appear on a dice, it means that the sum that equals to 4 is three times more likely to appear then the sum that equals to 1. Consequently, if we want to have more realistic game board, we have to put three times more 4s then 2s on it.

The following table shows the number of different ways to obtain each of the sum between 2 and 12 by rolling two dice and the chance (in percentage) that the respective sum can appear.


 Now, on the 10×10 board there are 100 squares, and 2.78% of them should be 2s, 5.56% of them should be 3s, 8.33% of them should be 4s and so on, or 2.78 squares should be with 2s, 5.56 squares should be with 3s, 8.33 squares should be with 4s and so on. But, the number of squares should also be a whole number, so I rounded them to the nearest whole number, and then I made some corrections to retain their sum equals to 100. In that way I got the following approximation of the number of squares for each sum that should be put on the 10×10 board:


While I was distributing the numbers on the board, I put the numbers with lower chance to appear in the interior of the board. On the edges and the corners of the board, I put the numbers that have greater chance to appear, since these squares have less common sides with other squares on the board. In that way I got the following large board for Addition Squares:


I did the same reasoning for 7×7 and 5×5 boards that have 49 and 25 squares respectively. You can find these boards on the above download links.

In Multiplication Squares, the lower product is 1, and the greater product is 36, but not every number between these numbers can appear as a product of two numbers rolled on the dice, and we have to be careful not to put those numbers on the board. The following table shows the number of different ways to obtain each of the possible products by rolling two dice and the chance (in percentage) that the respective product can appear.


After finding the chance for appearance of all possible products, I made similar corrections in the number of squares with the same product, as I did with Addition Squares, and for the 10×10 board I got the following approximation of the number of squares for each possible product that should be put on the board:


And again I distributed more likely products on the edges and the corners of the board, and the less likely products in the interior of the board. In that way I got the following large board for Multiplication Squares:


You can find the other two smaller boards, 7×7 and 5×5, on the above download links. You can also make your own Addition and Multiplication Squares, even with the different board sizes.

I hope you and your kids will enjoy these math games.

Comments

Post a Comment