Learning and Fun with Geometry Games

Learning and Fun with Geometry GamesAuthor(s): George W. Bright and John G. HarveySource: The Arithmetic Teacher, Vol. 35, No. 8 (April 1988), pp. 22-26Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41193383 .

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Learning and Fun with

Geometry Games By George W. Bright and John G. Harvey

Much of the geometry taught in elementary schools deals with mea- surement and with the properties of polygons and solids. Most of this content can be included in instructional games similar to those illustrated in this article.

In general, games can help to improve mathematics achievement (cf., Bright, Harvey, and Wheeler 1985), although any particular game cannot guarantee achievement. Geometry games to be used by elementary and middle school students must be care- fully chosen since (a) the geometry backgrounds of these students are often uneven and (b) it is important that the level of the game be consistent with what students know and can do with their knowledge.

Geometry games can accomplish a number of desirable instructional goals. For instance, they can furnish settings in which students visualize or construct geometric figures, see specific examples of general results or theorems, and apply logical reasoning skills in informal situations. Games also allow teachers to assess informal- ly how well students are achieving what is taught by the games and to determine how well students are applying this knowledge to new situations. More formal assessments can

George Bright is director of the Education Microcomputer Center at the University of Houston, Houston, TX 77004. He maintains active involvement with mathematics teachers in grades K-8. John Harvey has a continuing interest in the use of mathematics games. He is presently directing a project to develop college- level, calculator-based placement tests. He is on the faculty of the University of Wisconsin - Madison, Madison, WI 53706.

be made after a game has been played several times. Later in this article we give some examples of the kinds of test questions we have used to do just that.

One of the strongest features of game playing is that a game presents students with many very similar problems that are therefore solved using the same problem-solving techniques. In most instances the problems differ only in noncritical features; for example, the specific numbers change. Ex- posure to many similar problems might reasonably be expected to develop problem-solving skills that will transfer to new situations.

Geometry Games We present three games in this article: Polyhedron Rummy, "polygon rummy," and That's Stretching It. For each one we will discuss the game itself and its instructional objectives.

The instructional objective of Poly- hedron Rummy (Peterson 1971) is to choose, from among different kinds and numbers of faces, those that com- prise a given solid (fig. 1). To be successful, students need to compre- hend the relationships between a whole (i.e., the solid) and its various parts (i.e., the faces) and to reorder or rearrange those faces mentally so as to secure some total view of the solid. We used this game in grades 7-8 with 109 students who had not yet been given instruction designed to produce mastery on this content. The students played the games for twenty minutes a day on eight different days over a two- week period. They remained enthusi- astic about playing throughout this

period. The students were not paired for playing in any systematic way. Because of the somewhat unusual na- ture of the content, special test items were written to assess learning. Sam- ples are given in figure 2; other kinds of items could certainly be used. The average scores on the forty-two-item test improved from 20.5 to 26.2, a result that is both statistically and educationally significant.

The test items encouraged students to think about three-dimensional figures in several ways. First, three- dimensional figures had to be decom- posed into their faces. Second, figures had to be constructed from the faces. Third, extra faces had to be ignored. Fourth, figures were compared ac- cording to the number of faces needed to complete a construction.

Polyhedron Rummy might be incor- porated in instruction in a variety of ways. We used it as the only instruction on the identified instructional objective, but it might also be used along with other regular instruction on geometry. You might introduce two-dimensional figures for the purpose of having students classify them or iden- tify their properties. Students could extend those analytical skills by playing Polyhedron Rummy.

Polyhedron Rummy might also be varied for plane geometry by making a card deck of lines and angles and replacing the polyhedrons card with a polygons card. This game, "polygon rummy," would play like Polyhedron Rummy except that to make a polygon, the proper side and angle cards all need to be played. Students would then play cards to make the plane figures (fig. 3), just as they did to

22 Arithmetic Teacher



Fig. 1 Polyhedron Rummy rules and cards

i g»n - i mh Polyhedron Rummy HTv^ <^> ' I W Rules I 1<J^ ^^J '^__ 3-^_ ^ fl 1 . Decide who will be the dealer. The dealer gives each player a

polyhedrons card. This polyhedrons card should help players visualize the polyhedrons that can be built.

2. After shuffling the cards, the dealer deals each player seven cards. The next four cards are turned faceup on the table. If a wild card should be one of the four cards placed faceup, it is placed in the middle of the deck and replaced with a card from the top of the deck.

3. The play moves in a clockwise direction. Begin with the player to the left of the dealer.

4. A player takes a turn by first drawing a card from the top of the deck and then (a) playing one card needed to build a polyhedron; or (b) playing one or more cards to complete a polyhedron; or (c) passing, if nothing can be played on any of the four

faceup cards.

5. A wild card can be used in place of any card. A player who uses a wild card must tell which card it replaces.

6. When a polyhedron is completed, the score is computed and the cards used to make the polyhedron are placed in the discard pile. A new card to be placed faceup is drawn from the top of the deck.

7. A player can play a card or cards during only her or his turn.

8. A player can play more than one card per turn only when completing a polyhedron.

9. A player can play on only one polyhedron per turn.

10. When the last card is drawn from the deck, the discards are shuffled and used again.

1 1 . The game is over when a player has played all the cards in her or his hand.

12. The winner is the player with the greatest number of points at the end of a game or when time is called.

Scoring

1 . A player gets one point for each card in a polyhedron that he or she has completed.

2. The player who is first to play all of her or his cards gets one point for each card left in the opponents' hands.

p Scott, Foresman and Company. Used by permission.

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Fig. 2 Sample items . ,

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Ly Figure I /' Figure II Vîy Figure III

3. Three surfaces are pictured above. 10. Six faces are shown above.

Which of these surfaces might include the rectangle shown Which surface(s) shown below can be built using only some or below as three or more of its faces? all of these six faces?

(a) Only figures I and II (b) Only figures I and III | 1 r p^_ r^ ^ .^^ r^^_ (c) Only figures II and III

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21. Select an answer that is not true about the surface pictured.

(a) Has four congruent triangular faces .a (a) Only figure I (b) Has five polygonal faces /i ' (b) Only figure II (c) Has a square base /-''r' (c) Only fjgure m (of) Has six congruent square faces ^JL<^ ^ Only fj9ures • and " (e) Has eight edges (e) Only figures II and III

April 1988 23



Fig. 3 Polygon rummy cards

0 H Number of cards: 18 8

/6o° 'fl 'î°°

Number of cards: 10 8 4

Wild cards

Number of cards: 4 2

. 1 . I I 2

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2 2 i| |i ^ Shapes 1 2 1

to build 2 1

2

make the polyhedra in Polyhedron Rummy. If challenged, students would have to draw the polygon. This modified game could be used in elementary or junior high school.

Another example of a geometry game is That's Stretching It (Rom- berg, Harvey, Moser, and Montgom- ery 1976). This game requires special equipment (a triangle geoboard in- stead of the more common square geoboard), so teachers must do some planning in order to play it. Very able children might be able to play this

game with triangular dot paper and colored pencils, but drawing many figures on one sheet of paper is potentially quite confusing. Adapting it to a rectangular geoboard would require changing some of the properties that are at the bottom of the rule sheet.

The primary objective of this game is creating a polygon to satisfy a given property. This objective is at a higher level than that of Polyhedron Rummy, since students must predict the proba- ble effects of changes in the figure currently on the geoboard. Too, since

the scoring rule rewards minimal changes in the existing figure, it is expected that students will search through several alterations of that figure during a move. The game could be played either as a part of regular instruction or, after that instruction, as reinforcement for identifying properties of plane figures.

We've not gathered achievement data to assess the effectiveness of this game, but we believe that it is an effective instructional tool. It was thoroughly field-tested as part of the




Fig. 4 That's Stretching It!

That's Stretching It! (Two or three players)

You will need • geoboard and rubber band • numbered die • paper and pencil for keeping score

Game rules

1. Place the rubber band on the geoboard to make this parallelogram near the center of the board.

2. Play in turn. On your turn, roll the die twice and get an attribute from the grid below. Change the existing figure on

the board to a new quadrilateral that has the attribute you rolled. Even if the existing figure has the attribute you rolled, you must change. Try to change as few corners as possible.

3. Score: Four points if you changed only one corner Three points if you changed two corners Two points if you changed three corners One point if you changed four corners (had to move the figure entirely)

4. Check each other.

5. Play five rounds.

6. The winner is the player with the most points at the end of five rounds.

« fourriaht (îy&WOKiUK*) « 1 fourriaht angleS anaïls V A JOB FOR X ^vQ angleS f ARCHITECTS. ¿ .5^=?

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^f' MffUi^ K exactly ¡ Â - 'T^-^GC?^ ' 3 3 no line two pairs 3^SSL

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exactly exactly TW - no sides two pairs no sides one pair exactly two A^LN s parallel of opposite equal of touching right angles ^

angles equal sides equal

two pairs one angle all noanaies dSdesit two pairs 6. 6 of touching greater than four sides noanaies

¿S «*"* into So of opposite 6. 6 sides touching

equal greater

180° equal «*"* equal into

îlarts sides equal

12 3 4 5 6 Taken from Romberg, Harvey. Moser, and Montgomery (1976)

April 1988 25



development of Developing Mathe- matical Processes (1976). Again, however, special items might need to be developed if it is necessary to test. One way would be to show a picture of a figure on the geoboard as in rule 1 and to state one of the properties; students could be asked to sketch alterations of the given figure so as to satisfy the given property. A second way would be to have students match figures and properties. A third way would be to ask students to sketch several figures on triangular geoboard grid paper that satisfy a given property, perhaps with one side of each of these figures being kept fixed.

One advantage to this game seems to be that students will likely do men- tal, visual manipulations as they search for a good move. The game might, therefore, support improve- ment in two-dimensional visualiza- tion. Another advantage is that students may draw sketches of possible moves. The heuristic of drawing a picture seems to be important for problem solving (e.g., Polya 1957). Flexibility in dealing with information in a plane may have considerable pay- off in later geometry instruction.

Another geometry game that teachers can use is "geogolf," which now exists in several versions. (See the article "Golfing with a Protractor" in this issue, or the article ' 'Games, Ge- ometry, and Teaching" in the April 1988 issue of the Mathematics Teach- er.) Students get practice applying their knowledge of estimating lengths and angles. The data that we gathered on one version (Bright, Harvey, and Wheeler 1985) show that this game provides effective instruction.

Conclusions Geometry games are potentially effective instructional activities for all grade levels. One reason is that geometry games may be useful in helping students in a concrete way to visualize geometric properties, to analyze those quantities, and to make informal deductions. In all of the games we have discussed, students are required to visualize and to analyze results in order to play the games effectively.

Many games can be used by them- selves as the only vehicle of instruction for some parts of geometry. Such use might be with whole classes (our personal choice) or in activity centers. Indeed, our data (Bright, Harvey, and Wheeler 1985) clearly show that games alone can be used effectively to teach whole classes. We also encour- age the use of games as a supplement to regular instruction to extend content or to apply already acquired con- cepts, skills, or procedures. However games are used, some assessment should be made of what students learn from playing. Frequently, specific test items must be designed to measure the learning that takes place during game-playing situations. Only a few items are probably necessary for any game, so this task is not particularly time-consuming.

Instructional games are fun, and we know that students often find games more fun than other instructional activities. Students will often repeatedly practice instructional objectives in games when they would find this prac-

tice dull and boring otherwise. In ad- dition, within a game the concrete situations are constantly changing, and thus, geometry games may help students learn to use their knowledge more flexibly. References Bright, George W., John G. Harvey, and Mar-

gariete M. Wheeler. Learning and Mathe- matics Games. Journal for Research in Mathematics Education Monograph No. 1. Reston, Va.: National Council of Teachers of Mathematics, 1985.

Peterson, W. H. Polyhedron Rummy. Glen- view, 111.: Scott, Foresman & Co., 1971.

Polya, George. How to Solve It. 2d ed. Garden City, N.J.: Doubleday Anchor Books, 1957.

Romberg, Thomas A., John G. Harvey, James M. Moser, and Mary E. Montgomery. Devel- oping Mathematical Processes. Chicago, 111.: Rand McNally & Co., 1976. m

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Learning and Fun with Geometry Games