by the Math Construction Company

Hsin-Hwa Chien
Andrea Joyce
George Muñoz
Jean O'Grady
Carol Tohsaku

Instructional Objective The students will combine the calculation of products and the strategic building of proportional squares on a playing grid.

1. Given a randomly drawn multiplication problem card, the student will identify the correct product by placing a marker on an appropriate square on the playing board.

2. Given the square shape as the geometric goal, the student will attempt to place markers on the playing board to form that geometric shape. They need to use four markers and place one on each vertex of the square.

3. Given the existing pattern of played markers on the playing board, the student will use analytical thinking to determine which correct placement of his/her marker to chose, including bumping a covered square in order to complete the shape in his/her color.

Learners/Context The learners are third and fourth graders.

The game is meant to be used in the classroom, initially with the supervision of a teacher. An aide or other volunteer may supervise while the students are still learning the game. Eventually the students play by themselves, referring to the reference card to confirm the accuracy of answers, if necessary.

The students are introduced to the game after they have been taught multiplication concepts and facts up to 9 x 9. At this point in their learning curve, they need practice to reinforce their new knowledge and skills. The teacher may introduce the game to a small group of 3 to 4 students who are playing together. Or, by placing the clear acetate playing surface on an overhead projector, the teacher may introduce it to the entire class at once.

Once they understand how the game works, the students may play in either of two modes:

* Several small groups, each having its own set of playing materials, may play simultaneously.

* The whole class may be divided into four cooperative groups and may play with the teacher placing pieces for them on the overhead playing set. The coordinates that are printed on the transparency will help the students communicate which of the squares their marker is to be placed on. Each cooperative group is considered one player.

Rationale Multiplication facts are usually presented via traditional, teacher-directed, drill and practice methods. Since they form a foundation for many other skills in mathematics, they need to be practiced until automaticity is achieved.

The use of a board game provides an informal forum for practice in contrast to the classroom instruction. In addition, this board game, which is formatted as a grid, provides an element of strategy that will require a deeper level of mental involvement from students. Moreover, students must take the responsibility for verifying the accuracy of the answers given by their peers by monitoring play, challenging questionable responses, and checking answers on the reference card.

Goal of the Game The goal of the game is for a player to place four of his/her tokens on the game board so that the pieces form the corners of a square of any size (See Diagram).

Rules of the Game The game is played in the following manner:

1. The board is placed so that all of the players can see the numbers without moving it. One player shuffles the deck of cards and places it on the board in the marked square. Each player chooses a set of colored markers. To choose the starting player, each player draws a card, the products are compared, and the player with the largest product starts the game.

2. The starting player draws a card and turns it over onto the discard pile. The player reads the multiplication problem aloud, states the answer, and finds the squares on the board containing the correct answer. The player chooses a square and places a marker onto that square. Play continues to the player on the left.

3. If there is a marker already on a square, the player may remove it and replace it with one of his/her markers. The bumped marker is returned to its owner.

4. If a player places a marker on an incorrect answer, another player may challenge the play. There is a multiplication chart provided with the game to verify correct answers. If the challenge is verified, the player must take back the marker and loses that turn.

5. The play continues until one of the players completes a square (see Goal section above). If a player completes a square but does not notice, then the player must wait until his/her next turn to proclaim victory. In the meantime, another player may remove a piece of the winning square or may win by completing his/her own pattern.

6. If the cards are used up during the game, they can be reshuffled and used again.

Extensions Extensions use most of the same game pieces in a different way. The teacher and students will undoubtedly think of many other extensions as well.

* On subsequent rounds, players may freeze a square from the previous round. It is marked with double markers. The markers forming that shape may not be used by any of the players.

* On subsequent rounds of the game, vary the strategy shape. The shapes could include proportional right triangles (with all sides unequal), or isosceles right triangles (with the two sides that make up the right angle equal). Examples of these are shown below.

* Allow the students to win with a proportional rectangle, using the same guidelines as are used with the square. In other words, there should not be rectangles that run diagonally across lines, and the rectangles are defined by the vertices.

* Change the strategy by connecting four markers in a row.

* Change the strategy by making as many of one kind of shape as possible within a time limit set by the teacher.

* Play a solitaire game where the student keeps track of how many moves it takes to form a shape.

Variations Variations use the same rules with modified game pieces. The teacher and students are encouraged to think of other possibilities here too.

* Replace the equation cards with two eight-sided dice on which the 1's have been turned into 9's. This removes the safety of the one-of-a-kind products.

* Equation cards, playing board, and reference card may be created for other tables (subtraction, addition, division).

* Equation card, playing board, and reference card may be created for equivalent ratios.

* Equation card, playing board, and reference card may be created for equivalent fractions, decimals, and percents.

* Equation card, playing board, and reference card may be created for powers (squares and cubes) for middle school students.

Board Design The board was designed as a 16 x 16 inch square with the grid shown below located in the middle. The board would also include a frequency table to the right of the playing grid, several examples of winning combinations to the left of the grid, and two boxes along the bottom edge; one for the playing cards and one for the discard stack.

Design of Other Components Deck of 64 equation cards of standard flash card size:

Face up, the cards show the multiplication sentence in large print in the center; face down the cards show the name of the game.

A reference card:

Face up, it shows a multiplication reference table for the facts 2 through 9; face down it is plain.

Transparent, colored markers:

There are four colors, one for each player or cooperative group. Each player or group needs 20.

Design Process The Math Construction Company's designers experimented with a variety of design issues regarding the board, its contents, and its use. The fact that our target audience consisted of beginners with multiplication influenced our final decisions. Our goal was to keep success with multiplication within reach while injecting the game with the challenge of strategy, whereby a player bumps a marker that's on a number he or she needs or to block an opponent. Our simplest ideas became the basic game. Our other ideas became extensions and variations.

Grid format was used for the board for a number of reasons. First, it fit the format of the knowledge contained in multiplication tables. Secondly, it lent itself to plotting the vertices of angled geometric shapes. In addition, grid format has always been basic to strategy-type games. Finally, the square rather than the hexagon was selected because the game already had enough age-appropriate challenges.

We questioned how many cells should be in the grid. Our original 144-square playing board required too much time to search for numbers and caused a deceleration in the pace of the game. We also discussed and rejected the concept of a generic 144-square board where various sets of 8 by 8 playing areas could be defined. This idea was too complex. Finally, we settled on a single 8 by 8 board as the best size.

Another board design issue concerned permanent vs. student-written numbers. At the core of this issue was the idea of loose pieces that could easily be bumped vs. written ones that could get messy. We settled on permanent numbers on which students would place transparent colored markers.

A third board design issue involved what to write in the grid; products or equations. Since our target group consists of beginners, we decided that products would be the better choice.

Next, we had to finalize the set of multiplication facts. Since 1's and 10's establish simple patterns and are easy to learn, we decided to leave them out. We also left out the 11's and 12's because they are used less than the facts between 2 and 9. The products for these factors were placed randomly on the grid. The bulk of the randomizing effort was left to the shuffling of the equation cards. There is one box on the grid per equation card.

A friendly feature that we added to one side of the grid was a frequency table itemizing the number of times a product appears. Students may use this table to keep track of how many possibilities exist on the board, how many have been used, and how many are left. Indirectly, they may discover a way to use this information to their strategic advantage.

Another friendly feature was incorporated into the transparency for the introduction of the game. To ease in the location of specific squares, a coordinate axis was included.

Regarding the geometry aspect of the game, we experimented with squares, rectangles, and triangles. Eventually we decided that proportional squares presented a sufficient strategy challenge for the basic game.

ARCS Discussion Attention:

The teacher can get the attention of the learners by playing a class game with the transparency provided, and by soliciting responses to questions about the patterns involved in the facts they will be practicing. For instance, they could be asked why there is only one square for numbers like 64 and 81. This should arouse their curiosity and help keep them interested.


Most students in the third and fourth grades will be able to see the relevance in the subject matter of this game. Much of what they do in the classroom, both in their daily lessons and when they take tests, depends on a firm knowledge of the multiplication facts. Hearing the facts as well as using manipulatives in a game situation should help the students become quicker and more accurate with these facts.


The students should be confident that they have a chance to win since the game board and the cards (or dice) are also randomized. In the beginning, students could be allowed to ask for help or to look on their individual multiplication charts if they're stumped. Then all they have to worry about is the shape they are trying to construct. There will be examples of the shapes available for reference.

A variation was suggested where adjacent squares revert to the color of the last marker played. This variation should be used sparingly, if at all, because it could undermine the confidence that beginning players have in determining the outcome on the game.


The game has feedback built into it by the types of markers being used, among other things. The students can easily see how well they are doing in relation to the other players, and in relation to fulfilling the goal of building the shape.

Flow Discussion As the players improve their multiplication skills, they can concentrate even more energy on the strategy that goes into winning the game. As Csikszentmihalyi pointed out, we can't enjoy the same activity at the same level for too long. Once the students know their facts, they can concentrate on what strategies are the most effective for each game. They can also try out their skills against one or more opponents.

This game was initially tested by a group of 5th and 6th grade students. These students were slightly older than our target audience, and could have been bored with the game since their multiplication skills were quite strong. They weren't bored, however, because of the strategy involved in the game. They were interested enough that they wanted to continue playing to see who could create the next winning shape after the game should have ended. When tested with a group of 4th and 5th graders, they also remained interested and wanted to play multiple games.

The students also offered many suggestions about extensions and variations that might make interesting games, and some of those suggestions were incorporated into our project. Allowing the students to regularly suggest and create variations could add to the interest in the game by adding new challenges once previous games had been mastered.

There were several variations that were interesting. One consisted of allowing the students to cover all of the available squares with their markers when they solve an equation. Another one involved letting the players "freeze" a strategic piece by placing a second marker on top of the first one if that player gets the same product a second time. The last one involves allowing a player to "capture" all of an opponent's markers that share an edge with a piece that has just been played, and then replacing the captured pieces with his/her markers. Pieces on the diagonals were safe from capture, however. This variation was inspired by a video game one of the students had played called Ataxx.

Many of the variations were tried by the students, and most seemed to add to the flow experience. The only one that might add a little too much anxiety for some students was the one based on Ataxx. It seemed to make some of the students anxious about the fact that many of their pieces were being captured.