CALCULATOR TIC-TAC-TOE: A GAME OF ESTIMATION

CALCULATOR TIC-TAC-TOE: A GAME OF ESTIMATIONAuthor(s): William A. MillerSource: The Mathematics Teacher, Vol. 74, No. 9 (December 1981), pp. 713-716, 724Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27962694 .

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Edited by Evan M. Maletsky, Montclair State College, Upper Montclair, NJ 07043 Christian Hirsch, Western Michigan University, Kalamazoo, MI 49008 Daniel Yates, Mathematics and Science Center, Richmond, VA 23223

CALCULATOR TIC-TAC-TOE: A GAME OF ESTIMATION

By William A. Miller, Central Michigan University, Mt. Pleasant, MI 48859

Teacher's Guide

Grade level: 7-12. Materials: One set of activity sheets

and a hand-held calculator for each pair of students.

Objective: To provide experience in es

timating products, quotients, and powers. Problem-solving abilities are developed as

the individual focuses on finding a game

winning strategy. Experience is also

gained in working with calculators. Directions: Distribute the activity

sheets, one at a time, to pairs of students. Have the students play each game until a

winner emerges. Let them complete sheet 1 before working with sheets 2 or 3. Sheet 1: The initial activity on sheet 1

familiarizes the students with some three in-a-row alignment games that are similar to traditional tic-tac-toe. Students quickly discover that they must estimate a prod uct if they want to mark a desired cell. The activity also provides practice in us

ing a calculator and in developing block

ing and winning strategies.

Sheet 2: This activity provides practice in estimating quotients, using the calcula tor, and developing game-winning strate

gies. Sheet 3: Exponents of 2, 3, 4, and 5 are

used. They can be handled as repeated multiplication on all calculators. How ever, you may wish to develop other activities using different exponents with classes that have an automatic constant

capability or a y* key on their calculators. A nice extension of these activities is to

let your students select th?ir own opera tion and numbers and fill in their own gameboards. Constructing a game moti vates students to estimate and think about what makes a good problem.

In game theory terminology, these are

two-person games that are finite (come to a definite end), minimize the element of chance, and are played with "perfect in formation" (each player knows the oth er's moves). If both players are familiar

(Continued on page 724)

This section is designed to provide mathematical activities suitable for reproduction in worksheet and transparency form for classroom use. This material may be photoreproduced by classroom teachers for use in their own classes without requesting permission from the National Council of Teachers of

Mathematics. Laboratory experiences, discovery activities, and model constructions drawn from the

topics of seventh, eighth, and ninth grades are most welcomed for review.

December 1981 713



SHEET 1 PRODUCT TAC-TOE

Two players take turas choosing two numbers, one in a triangle and one in a

rectangle. They find the product on a calculator and place their marks (X or O) on the number closest in value to the actual product. If the number is already marked,

they mark the unmarked number closest in value. The first player with three marks in a row wins. The loser plays first in the next game. If neither player gets three in a

row, the game is a draw and should be repeated.

Tic-Tac-Toe Star-Tac-Toe

691 77 87 W] 53 67

Tri-Tac-Toe Tri-Row-Toe



SHEET 2 QUOTIENT TAC-TOE

Two players take turns choosing two numbers, a divisor from a triangle and a dividend from a rectangle. They find the quotient on a calculator and place their marks (X or O) on the unmarked number that is closest in value to the quotient. The first player with three marks in a row wins. The loser plays first in the next game. If neither player gets three in a row, the game is a draw and should be repeated.


60 84 420 385 5009 2310

Tri-Tac-Toe Tri-Row-Toe

48? 1125 150CS ?A?



SHEET 3 POWER TAC-TOE

Two players take turns choosing two numbers, an exponent from a triangle and a base from a rectangle. They find the power on a calculator and place their marks (X or O) on the unmarked number that is closest in value to the power. The first player with three marks in a row wins. The loser plays first in the next game. If neither

player gets three in a row, the game is a draw and should be repeated.




chart, you may use the formula

252D '

where/is the coefficient of friction, S the speed of the car (km/h), and D the braking or skidding distance in meters.

For example, for a stop in 12 m at a

speed of 50 km/h,

252D

SO2 252 X 12

_ 2500 ~" 3024

= 0.83.

A nomogram provides a simple way of

determining the speed of an automobile

using the skidding distance. For example, if a car skidded 21 m on dry concrete, we

draw a line from 0.8 on the left bar in the chart to 21 on the right bar. The line would cross the center bar at 65, indicating the

speed of the car at the time the tires started to skid. Of course the car may have been traveling at 80 km/h and slowed down to 65 km/h before the tires started to skid. For this reason a speed determined from the chart is really a minimum speed.

The slope of a road is called its grade:

rise GRADE =

run

It is easier to stop a car when driving uphill than it is when driving downhill. An uphill grade increases the coefficient of friction (makes it easier to stop) by the value of the grade (slope). A downhill grade decreases the coefficient of friction

by the value of the grade (slope).

Example

A car is traveling uphill. The grade (slope) of the hill is 0.2. The road surface is wet concrete. The coefficient of friction is then 0.6 4- 0.2 = 0.8. If the same car was

traveling downhill on the same road, the

coefficient of friction would be 0.6 - 0.2 =

0.4.

Problems

1. A car skidded 24 m on a level gravel road. What was the speed of the car?

2. A car traveling uphill on a dry con crete road with a grade of 0.1 skidded 27 m. Find the speed of the car.

3. A car traveling downhill on a wet

asphalt road with a grade of 0.2 skidded 30 m. Find the speed of the car.

4. If road conditions remain the same, what effect on the skidding distance does doubling the speed have?

Answers: 1. 55 km/h, 2. 78 km/h, 3. 55 km/h, 4. increases by a factor of 4.

(Continued from page 713)

with the common tic-tac-toe game, they Joiow it must end in a draw. Likewise, the first game here, called Tic-Tac-Toe, also ends in a draw when properly played. However, in the other three games, there is a winning strategy for the first player. These strategies, including the tie strategy for Tic-Tac-Toe, are too extensive to in clude in this article. An analysis of Tic Tac-Toe appears in Gardner's book (1959) and the other three are treated in O'Beirne's article (1962).

BIBLIOGRAPHY

Gardner, Martin. The "Scientific American" Book

of Mathematical Puzzles and Diversions, pp. 37 46. New York: Simon & Schuster, 1959.

Krulik, Stephen. "Problems, Problem Solving, and

Strategy Games." Mathematics Teacher 70 (No vember 1977):649-52.

O'Beirne, T. H. "Puzzles and Paradoxes." New Scientist, no. 269, 11 January 1962, pp. 98-99.

724 Mathematics Teacher



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CALCULATOR TIC-TAC-TOE: A GAME OF ESTIMATION