## Polygon Capture

• Lesson
6-8
2

In this lesson, students classify polygons according to more than one property at a time. In the context of a game, students move from a simple description of shapes to an analysis of how properties are related. This lesson was adapted from an article which appeared in the October, 1998 edition of Mathematics Teaching in the Middle School.

The purpose of this game is to motivate students to examine relationships among geometric properties. From the perspective of the Van Hiele model of geometry, the students move from recognition or description to analysis (Fuys 1988). Often, when asked to describe geometric figures, middle school students mention the sides ("The opposite sides are equal") or the angles ("It has four right angles"), but they rarely use more than one property or describe how two properties are related. For example, is it possible to have a four-sided figure with opposite sides not equal and four right angles? Or a triangle with three right angles? What geometric relationships make such figures possible or impossible? By having to choose figures according to a pair of properties, players go beyond simple recognition to an analysis of the properties and how they interrelate.

Choosing all figures in the Polygon Capture Game Polygons Activity Sheet that have parallel opposite sides is relatively easy. Choosing all figures with parallel opposite sides and at least one obtuse angle requires reasoning, and a good analysis of such figures leads to the inference that all non-rectangular parallelograms have these two properties, as does the regular hexagon.

Another purpose of the game is to give students a format for using important geometric vocabulary-parallel, perpendicular, quadrilateral, acute, obtuse, and right angle-in a playful situation. The basic game is described below and is followed by warm-ups and extensions.

To get ready for the game, distribute copies of Polygon Capture Game Rules, Polygon Capture Game Cards, and Polygon Capture Game Polygons. You will need only one copy of each master for every two students. Before introducing the game, have the students cut out the polygons and the cards. They should also mark each card on the back to designate it as an "angle" or "side" card. The eight cards from the top of  Polygon Capture Game Cards sheet should be marked with an "A" for angle property; the eight cards from the bottom should be marked with an "S" for side property.

Before the game, assess the students' familiarity with the vocabulary used in this game, such as parallel, perpendicular, polygon, and acute angle by engaging students in a class discussion in which they define, illustrate, or find examples of the geometry terms.

Basic Rules of the Game

Have the students read the rules on Polygon Capture Game Rules sheet.

Teachers have found it helpful to begin by playing the game together, the teacher against the class. You may want to do so a few times until the class is confident about the rules. For the first game, remove the Steal Card to simplify the game.

To introduce the game as a whole-class activity, lay all twenty polygons in the center of the overhead projector. Students may lay out their shapes and follow along. An introductory game observed in one of the classrooms (as shown in step 4, below) proceeded as follows.

1. The teacher draws the cards All angles have the same measure and All sides have the same measure. She takes figures D, G, Q, and S, placing them in her pile and out of play.
2. Students then pick the cards At least two angles are acute and It is a quadrilateral. They choose figures I, J, K, M, N, O, and R.
3. On her second turn, the teacher picks the cards There is at least one right angle and No sides are parallel. She chooses figures A and C and then asks students to find a figure that she could have taken but forgot. One student points out that figure H has a right angle and no parallel sides. Other students are not sure that this polygon has a right angle, which leads to a discussion of how they might check.
4. The students then proceed to take two new cards
 (a) Teacher selects cards. Angle card: All angles have the same measure. Side card: All sides have the same measure. (b) Students select cards. Angle card: At least two angles are acute. Side card: It is a quadrilateral. (c) Teacher selects cards. Angle card: There is at least one right angle. Side card: No sides are parallel. (d) Students capture piece that teacher missed.

When no polygons remain in play that match the two cards chosen, the player may turn over one additional card-either an angle or a side card. This move calls for some planning and analysis to determine whether an angle card or a side card is most likely to be useful in capturing the most polygons. If the player still cannot capture any polygons, play moves to the opponent. When all cards in a deck are used up before the end of the game, they are reshuffled. Play continues until two or fewer polygons remain. The player with the most polygons is the winner.

When the "Wild Card" is selected, the player may name whatever side property he or she wishes; it need not be one of the properties listed on the cards. Again, a good strategy to capture the largest number of polygons requires an analysis of the figures that are still in play.

Steal Card

When the "Steal Card" comes up, a card from the deck is not drawn. Instead, the player has the opportunity to capture some of the opponent's polygons. The person who has chosen the Steal Card names two properties (one side and one angle) and "steals" the polygons with those properties from the opponent. The students may select their own properties, not necessarily those on the game cards. If the opponent has no polygons yet, the Steal Card is put back in the deck and a new card chosen.

Teacher Notes

One interesting aspect of the game is the various strategies that students use. Some students go through the figures one at a time, using a trial-and-error method to match them to properties on the cards. Some students perform two sorts; they find the polygons that match the first card and, of this group, those that also match the second card. Others seem to analyze the properties and mentally visualize the polygons that are possible. In analyzing properties ("Is this angle acute?"), students quickly learn to use angles and sides in other figures as benchmarks, for example, using the right angle in a rectangle to check whether a triangle has a right angle. Generally classes play with no time limits, although students could choose a limit as an option.

### In the Classroom

"I first played the game against the students and then had them play against each other. They were arguing about whether they could take certain polygons—it allowed them to solidify their understanding by explaining their reasoning to someone else." -Ava T. Parnes, MS Math/Science Teacher (New York, NY)

### References

• Carroll, William. "Polygon Capture: A Geometry Game." Mathematics Teaching in the Middle School, Volume 4 (Ocober 1998), pp. 90‑94.
• Fuys, David, Dorothy Geddes, and Rosamond Tischler. The Van Hiele Model of Thinking in Geometry Among Adolescents. Journal for Research in Mathematics Education Monograph Series, no. 3. Reston, Va.: National Council of Teachers of Mathematics, 1988.

Extensions

1. Some teachers have found that coordinating two properties is initially too difficult for their students and have simplified the game by placing all cards into a single pile. For this simpler version only one card is turned over, and students choose all polygons with that property. In this adaptation, it is probably best to remove the Wild Card and the Steal Card. The other rules are the same as described previously. Because only one property is being analyzed at a time, this game will go more rapidly.
2. The polygons on the Polygon Capture Game Polygons Activity Sheet can also be used for various sorting games and activities. For example, students may work in pairs, with one student separating the shapes into groups based on some rule or set of rules, and the other student trying to deduce the rules. Whereas some students may begin with simple classifications (rectangles and non-rectangles), others may use more complex relationships (regular polygons, polygons with equal sides but not equal angles, and other figures). With a little experience, many students will find interesting ways to sort the polygons. You may also use the figures to review geometry vocabulary before the game: "Find all of the figures that have a pair of perpendicular sides." "Pick all regular polygons." These activities provide a nice warm-up to the game and other geometry activities.
4. The Polygon Capture game cards can also be used to generate figures. As in the game, students turn over two cards. Instead of capturing polygons, they use a geoboard or dot paper to make a figure that has the two properties. Rather than a game, this is simply an activity to help students learn to coordinate the features of a polygon.

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### Learning Objectives

Students will:

• Precisely describe, classify, and understand relationships among types of two- and three-dimensional objects using their defining properties.
• Create and critique inductive arguments concerning geometric ideas and relationships.
• Progress from description to analysis of geometric shapes and their properties.

### NCTM Standards and Expectations

• Precisely describe, classify, and understand relationships among types of two- and three-dimensional objects using their defining properties.
• Understand relationships among the angles, side lengths, perimeters, areas, and volumes of similar objects.
• Create and critique inductive and deductive arguments concerning geometric ideas and relationships, such as congruence, similarity, and the Pythagorean relationship.