Teaching Mathematics with Technology: Introducing Factors with aTiling SimulationAuthor(s): Robert J. JensenSource: The Arithmetic Teacher, Vol. 35, No. 5 (January 1988), pp. 36-38Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41194314 .

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Teaching mathematica coith Technology

Fig. 1

Introducing Factors with a Tiling Simulation

The concept of factor is often confusing for students when it is first introduced in the elementary curriculum. Motivating the development of new ideas through problem-solving episodes can be a fruitful approach before formal instruction begins. A tiling problem is posed here for your students to investigate using the microcomputer as a tool. Since factors, common fac- tors, and the greatest common factor play a crucial role in the solution to problems of this type, this ad- vance activity will give students a specific context upon which to build meaning for these concepts when they are formally introduced.

A Tiling Problem

Begin by telling the following background story:

You own a company that specializes in laying square floor tiles to cover rooms with rectangular-shaped floors. Your available stock of floor tiles consists of squares of any whole-number dimension measured in feet (i.e., 1 x 1, 2 x 2, 3 x 3, and so on). To save money, you wish to make the task of laying these tiles as simple as possible. Therefore, let's agree to follow two rules in selecting the size of tile to use. First, you do not wish to cut any tiles, and second, you wish to cover the room with as few identical square tiles as possible.

After this scenario is explained, pose the problem of selecting the size of square tile that would be most efficient for tiling a 6 x 9 storage room. Experimenta- tion should lead to the conclusion that using either 1 x 1 or 3 x 3 tiles will allow you to tile the floor area without cutting tiles. Then, since the larger the tile

Prepared by Robert J. Jensen, Emory University, Atlanta, G A 30322

selected, the fewer tiles needed, the 3 x 3 square tiles would represent the optimum size for this job (see fig. 1).

i i i i i i i i I I

6x9 room

A 6 x 9 room tiled with 1 x 1 square tiles

I I I I I I I I I I

A 6 x 9 room tiled with 3x3 square tiles

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Fig. 2 Ь '

I After your students understand the nature of the

problem, assign small groups to computers and give them a disk containing the Apple Logo II procedures that appear in the Appendix. Since the programming logic is not important for this lesson, you should save these procedures in a file named "TILES" before class rather than have students type in these procedures themselves. Then, after they load this file, instruct your students to begin their investigation by typing the master procedure, ROOM :L :W. Explain that ROOM is a command that requires two inputs that must be whole numbers representing the length and width in feet of the floor to be tiled.

Typing this command will produce scaled drawings of the floor for rooms with the dimensions specified. The maximum room size is limited only by the resolu- tion of your monitor - typically around 60 x 60. After the room is drawn, the student is asked by the com- puter to suggest the size of the largest square that will evenly tile the room. Figure 2 shows a sample screen after typing ROOM 24 18.

The computer then simulates the tiling of the room with whatever size tile is typed. Four different out- comes are possible. First, the tile does not fit evenly into either dimension; second, the tile fits evenly in one dimension but not the other; third, the tile fits evenly in both dimensions but is not the largest tile

TYPE THE LENGTH OF THE SIDE OF THE LARGEST SQUARE THAT WILL EVENLY TILE A ROOM WITH DIMENSIONS 24 BY 18. ?_

possible; or fourth, the tile fits in both dimensions and is the largest tile possible. Each of these possibilities generates a different message to accompany the graphic display. Figure 3 displays four possible tiling outcomes for a room with dimensions 24 x 18.

Fig. 3 Outcome 1 M ■ ■ ■ ■ • I ' • « ' • ' I ■ • ' I Outcome 2 I i ■ ■ ■ ■ ' ' I ' ' ' ' ' ' ' I ' I

7 BY 7 SQUARES 8 BY 8 SQUARES CAN - CANNOT TILE EITHER "

TILE ONE DIMENSION "

DIMENSION EVENLY. I EVENLY BUT NOT THE I TRY A 24 BY 18 - OTHER. TRY A 24 BY 18 -

ROOM AGAIN. - ROOM AGAIN. "

? Q | | | ? Q | | |

Outcome 3 I ■ ' I ' ' I ' ■ I ■ ' I ' ' I ' ' I Outcome 4 I ■ ■■■■■■■■■■■■■■■*■

3 BY 3 SQUARES TILE I I BOTH DIMENSIONS - YES, 6 BY 6 SQUARES -

EVENLY BUT ARE NOT - CAN TILE BOTH -

THE LARGEST DIMENSIONS EVENLY

POSSIBLE SIZE FOR A I AND ARE THE LARGEST I 24 BY 1 8 ROOM. TRY POSSIBLE SIZE FOR A -

AGAIN. - 24 BY 18 ROOM. -

? Г I I I I I I ? L I I I

January 1988 37

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qaadary Dktfogua

(Continued from page 3)

Correction

In the "One Point of View" department in the Arithmetic Teacher (October 1987), Robert Madell was kind enough to cite my book Chil- dren's Arithmetic. Unfortunately the publisher was listed incorrectly. The correct publisher is Pro-Ed, 5341 Industrial Oaks Boulevard, Aus- tin, TX 78735.

Herbert P. Ginsburg Teachers College, Columbia

University New York, 'NY 10027

38

Book review update

We enjoyed reading your review of the Mid- dle Grades Mathematics Project in the Septem- ber 1987 issue of the Arithmetic Teacher. Thank you for your interest in this project and your clear summation of each component of the project.

We want to bring to your attention that the review indicated that the books are out of print. The books were temporarily out of stock early this summer; a reprint has been completed, and all are currently available.

Tina A s ht on Addison-Wesley Publishing

Company Menlo Park, С А 94025

In NCTM journals

Readers of the Arithmetic Teacher might en- joy the following articles in this month's Mathematics Teacher:

• "Using Percent Problems to Promote Criti- cal Thinking," Beverly A. Rossini Osiecki

• Cryptarithms: Math Made Me Daft, Mom- ma," William A. Ewbank

And in the Journal for Research in Mathe- matics Education:

• "Race, Sex, Socioeconomic Status, and Mathematics," Laurie Hart Reyes and George M. A. Stanic

Arithmetic Teacher

Relating Outcomes to Mathematical op 195 / w Terminology

END TO MAKE.GUESS

After your students have worked through several prob- < pr<^IJ!Sa^ ЬАЯ? est lems of this type, introduce the concepts and vocabu- ^^ж-^fí^u lary of factors, common factors, and the greatest com- make • «guess first reàdlist mon factor in the context of these tiling problems. P-gSesI = gcf l w [response.i stopí Explain that the size of any square tile fitting evenly if ( remainder gcf 1 :W :GUESS ) = 0 [responses stopí across the room can be described as a factor of the if or ( REMMnder :l ¿guess ) = 0 ( remainder :w iGuesS ) width and, similarly, that the size of any tile fitting response^ evenly across the length of the room can be described end as a factor of the length. In the case of a square that JQ T|LE evenly tiles in both directions, its size is said to be a

JQ repeat

T|LE :l / :Guess [fd scale * :Guess rt 90 fd :W ♦ I

common factor for the two dimensions of the room. scale bk :W * scale lt 90] The size of the biggest square that will tile the room is repIatw / ̂guess fiscale9* guess lt 90 fd l * therefore logically described as the greatest common scale bk :l * scale rt 90] factor. Stress that the word greatest is an adjective END that is specifying the one common factor, from among T0 GCF :L :W what could be many other common factors, that hap- if :W = ò [OP :li pens to be the

many largest. opgcf :W ( remainder :l :w )

With this background, these terms will have more I meaning for your students. They should now be moti- то responses I vated vaiea to ю learn learn emcient efficient metnoas methods ror for f nnaing indina me the Greatest greatest < PR l^J :GUESS l?Y) :GUESS [SQUARES CAN TILE BOTH vated vaiea to ю learn learn emcient efficient metnoas methods for ror f nnaing indina the me greatest Greatest dimensions evenly l?Y) and are the largest possible common factor, since they will then be able to apply size for a] :l [by] :W [room.] ) I these new techniques in solving more complicated END I tiling problems. T0 response.2 I

( PR :GUESS [BY] :GUESS [SQUARES TILE BOTH DIMENSIONS EVENLY BUT ARE NOT THE LARGEST POSSIBLE SIZE FOR

Appendix A] :L [BY] :W [ROOM. TRY AGAIN.]) END I

TO ROOM :L :W I CS SETBG 6 HT PU SETPOS Г - 100 - 75] PD ТО RESPONSES REPEAT :L [FD SCALE RT 90 FD 3 BK 3 LT 90] RT 90 ( PR ¡GUESS [BY] :GUESS [SQUARES CAN TILE ONE I REPEAT :W [FD SCALE RT 90 FD 3 BK 3 LT 90] RT 90 DIMENSION EVENLY BUT NOT THE OTHER. TRY A] :L [BY] FD :L * SCALE RT 90 :W [ROOM AGAIN.] ) FD :W * SCALE RT 90 END MAKE.GUESS END TO RESPONSES

( PR :GUESS [BY] ¡GUESS [SQUARES CANNOT TILE EITHER TO SCALE DIMENSION EVENLY. TRY A] :L [BY] :W [ROOM AGAIN.] ) IF:L>:W[OP195/:L] END I

m I

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