Download Polygon Capture overview

Polygon
Capture
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.
Learning Objectives
Students will:
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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
Materials
Polygon Capture Game Rules
Polygon Capture Game Cards, photocopied onto cardstock
Polygon Capture Game Polygons, photocopied onto cardstock
Instructional Plan
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 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
nonrectangular 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 angleby 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.
Polygon Capture Game Rules
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.
(c) Teacher selects cards.
Angle card: There is at
least one right angle.
Side card: No sides are
parallel.
(b) Students select cards.
Angle card: At least two
angles are acute.
Side card: It is a
quadrilateral.
(d) Students capture piece
that teacher missed.
Sample Steps in a Game
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.
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 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 nonrectangles), 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.
3. Several other extensions of the game are possible. More polygons can be added, either
by the teacher or by the students, including some that are more difficult to capture,
such as a kite or a nonconvex hexagon. Nonpolygons, such a figures with curves, can
be added for sorting activities. Additional property cards can also be added to the basic
deck. For example, as students learn more about polygons, you may wish to add angle
cards, such as Opposite angles have equal measure or The number of vertices is a
prime number. Similarly, questions about diagonals can be added to the side cards,
such as All diagonals have the same length. If you have a set of geometric solids
available, you can adapt this game to to three-dimensional geometry. Instead of side
and angle cards, make one set of surface and face cards ("I have one curved surface")
and edge and vertex cards ("I have an even number of vertices"). If three-dimensional
solids are not available, make a third set of picture cards. Instead of polygon cards,
students choose the geometric solids.
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.
NCTM Standards and Expectations
Geometry 6-8
1. Create and critique inductive and deductive arguments concerning geometric ideas and
relationships, such as congruence, similarity, and the Pythagorean relationship.
2. Understand relationships among the angles, side lengths, perimeters, areas, and
volumes of similar objects
3. Precisely describe, classify, and understand relationships among types of two- and
three-dimensional objects using their defining properties.
References
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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.
This lesson prepared by William Carroll.