## by Mike Johnson

Mike is a math teacher at Mt. Miguel High School,

former lifeguard, and an internationally published

underwater photographer.

Instructional Objective The learners shall be able to make associations among properties that determine unique classes of quadrilaterals and identify those shapes.

Learners/Context The learners are senior high geometry students preparing for an examination over the properties of special quadrilaterals.

Rationale Students receive instruction and guided practice in the classifications of parallelograms and trapezoids as part of a unit on quadrilaterals.

Students must exercise analysis skills in order identify similar traits and differences among classes of parallelograms and trapezoids. They must make subsequent discriminations among subclasses of these quadrilaterals. It is important to recognize when one has a significant enough body of properties to determine a unique class and then identify that class, e.g., squares.

The combinative nature of playing cards lends itself well to practicing this skill. Being able to trump another player's hand facilitates demonstration that a student understands the interrelations among specialized figures.

Rules Two to four persons play Quadmania with one person keeping score. The object is to acquire the most points by playing combinations of properties that determine unique categories of quadrilaterals. Play proceeds in this manner:

1. Deck is shuffled and placed face down in the middle of playing surface.

2. Each player draws top card at beginning of turn. Play continues clockwise, (or counterclockwise when playing in Australia).

3. When a player has a minimum combination of properties that identify a unique class of quadrilaterals, he or she may place card(s) on table face-up and verbally identify the quadrilateral. Other players must confirm play is accurate before adding the sum of the point values of the cards to the individual's score. Disputes are settled by consulting reference sheet (matrix) provided with the deck.

4. Play proceeds to the next person regardless of whether a player placed cards or not.

5. A player may add a card or cards to an existing trick and earn the point value of the new card(s). Again, the player must identify the class of quadrilateral correctly. Example: certain cards may trump a parallelogram. player B trumps player A's parallelogram with "4= angles" to determine a rectangle. By doing so, player B earns the sum of all points in that trick.

6. Players are not required to play any of the cards they hold until there are no longer cards left to draw from. This is a matter of strategy.

7. The game is over when all cards have been played. Individuals that go out ahead of others simply lose their turn.

Sample plays Two sample plays illustrate differences in relationship. The vertical placement of cards on the right indicates that the properties must apply to the same pair of opposite sides.

Card and Deck Design The design of the deck and rules of play were synthesized from a list of descriptors and a Venn diagram illustrating the relationships among quadrilaterals. Forty-one cards each contain an element of the complete matrix. The obverse of each card contains a standard playing card pattern. Because the properties are not equally represented in the deck, varying point values are assigned to the cards. Less repeated properties are worth more points. The quadrilateral matrix is printed on the back of the rules sheet which accompanies the deck

Design Process Initial analysis of content matrix raised a concern that playing unequal books of cards would make certain quadrilaterals less desirable to play. The remedy chosen was to assign higher point values to cards played in smaller books, thus leveling the desirability of each type. I would consider allowing learners less practiced in the analysis of quadrilaterals to use the reference sheet at any stage of play.

Another iteration of the design process might add kites as a third sub-family of quadrilaterals. This would require the addition of two more cards. The game could be adapted to practice discrimination between solid figures or to synthesize new shapes.