Property GamesAuthor(s): Neal GoldenSource: The Mathematics Teacher, Vol. 72, No. 7 (OCTOBER 1979), pp. 510-512Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27961767 .

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sharing teaching ideas

Property Games

The purpose of these games is to give students of first- and second-year algebra the chance to practice using the real num ber (field) properties to manipulate ex

pressions. One game, the Goal Game, can also be used to introduce the idea of proof. Many variations of these games can be made.

Meander Game

The name of the game indicates that the students are not aiming toward any partic ular final expression; they are meandering.

1. To begin play, divide the class into two teams.

2. The teacher writes an expression on the board, for example,

7 + (8 + 3). 3. The teacher asks a player from one

team to go to the board.

4. The teacher calls out a real number

property, for example, associative of addi tion.

5. The player must apply that property, and only that property, in rewriting the ex

pression. For example, if we use associative of addition,

7 + (8 + 3) becomes (7 + 8) + 3.

6. The teacher calls on a player on the

opposing team to judge the correctness of

what the first player did. If it is correct, and the opponent agrees it is correct, then the first team gains a point. The player from the second team is sent to the board, and steps 4 and 5 are repeated. In this way a sequence of expressions is created. See table 1. Sometimes there is more than one correct expression for each step.

TABLE 1 Starting Expression: 7 + (8 + 3)

Teacher Calls Correct Result

Associativity of addition (7 + 8) + 3 Commutativity of addition (8 + 7) + 3 Associativity of addition 8 + (7 + 3)

7. If a player applies a property incor

rectly and the opponent spots the mistake, then the team of the first player does not earn a point. Instead the opponent who

spotted the error goes to the board and erases the last expression. If the opponent then writes a correct expression applying the property, the opponent's team earns a

point. 8. If the opponent incorrectly claims

that the first player rewrote the expression incorrectly, then the opponent's team loses its turn. (In all disputes the teacher is the fi nal judge.)

9. The players massage a given ex

pression for as many turns as the teacher

wishes; then a new expression is written

Sharing Teaching Ideas offers practical tips on the teaching of topics related to the secondary

school curriculum. We hope to include classroom-tested approaches that offer new slants on fa

miliar subjects for the beginning and the experienced teacher. Please send an original and four

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510 Mathematics Teacher

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and play continues. Another example is seen in table 2.

TABLE 2 Starting Expression: (c + d)k + ek

Teacher Calls Correct Result

Commutativity of addition ek + (c + d)k Commutativity of multiplication ek + k(c + d) Distributivity ek + (kc + kd) Associativity of addition (ek + kc) + kd Commutativity of multiplication (ke + kc) + kd Distributivity k{e -\-c) + kd

After students have played the game sev eral times, some variations can be tried. One change is to have an opponent, and not the teacher, call out the property to be

applied. The option not possible can be used as a correct answer to cover a case like

commutativity of multiplication being called for the expression {a + b) + c.

In another variation of the game, a

player rewrites the expression, and then an

opponent must name the property that was

applied. Or each player can be told to re

write the expression and state the property used.

Goal Game

After players have mastered the Mean der Game, a second version can be in troduced that prepares students more di

rectly for the concept of proof. In this

variation, the teacher gives a starting ex

pression and also a final expression or goal. Players take turns rewriting the given ex

pression until it is the same as the target ex

pression. Each time an expression is rewrit

ten, the player must state what property was applied. An opponent judges not only the correct application of the property but also whether the result brings the ex

pression closer to the goal. Here is a sample game of this form. The parenthetical notes beneath each expression indicate the rea sons. Of course, other steps are possible.

?Pr?pAND NASHVILLE MEETING

The city of Nashville successfully blends culture, a sense of history, and a

contemporary nightlife. It offers cuisine to satisfy every palate and features entertainment that ranges from the country strains of Grand Ole Opry to discos and exciting floor shows at the Opryland Hotel, and the world famous Printer's

Alley is located here. The Tennessee Mathematics Teachers Association is pleased to host this

meeting at the fabulous Opryland Hotel?showcase of the Midsouth. The meeting will feature a number of excellent sessions and workshops; will

include a Strand on Competency Testing and Math Anxieties; and will include sessions and workshops concerned with coping with learning disabilities, gifted students, and handicapped students. A delicious Friday night buffet dinner is

planned, with drawings for door prizes. Featured guests and speakers include Roy Acuff and the Smokey Mountain

Boys, NCTM President Shirley Hill, Lola May, Chuck Allen, David Wells, Stephen Krulik, and many others.

Come and enjoy a great week at the Nashville Meeting.

15-17 November 1979

October 1979 511

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Given: a + (y + ) Goal: {xa + ) + y

a + {y + ) (given)

a + ( + .y)

(commutativity of addition)

(a + ) + y (associativity of addition)

(xa + z) + y (commutativity of multiplication)

Players can be challenged to take a differ

ent route to the same goal, as follows.

ax + (y + z) (given)

xa + (y + z) (commutativity of multiplication)

a + (z + .v)

(commutativity of addition)*

( a + ) + >>

(associativity of addition)

However, a player starting the chain as fol

lows should be caught by an opponent and

penalized.

ax + (y + z) (given)

(a + y) + (associativity of addition)

This player should not earn a point be

cause his or her step will make it harder, that is, require more steps, to reach the

goal. Neal Golden Brother Martin High School

New Orleans, LA 70122

Pythagorean Theorem and Transformation Geometry

Although there are many elegant proofs of the Pythagorean theorem (Loomis 1968), the one presented below is particularly ap

pealing because it is based on only two

transformations. Its very simplicity makes

it easy to follow. Let's begin by assuming we have a plane figure (fig. 1) with AEFG, m?F = 90, EF= b, FG = a, GE = c, and

squares on sides EFand FG.

To demonstrate the proof, we will de

scribe a series of motions with figures. Each

figure represents the preceding figure after

a certain transformation (the physical term

motion is used rather than the mop formal

term transformation because of its teaching

appeal). Translations and rotations are the

This article is dedicated for children around the

world on the occasion of the International Year of the

Child, 1979.

only transformations needed for this proof. For the proof it is assumed that these two

Fig.

512 Mathematics Teacher

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