More Math Games & Activities from Around the Worldby Claudia Zaslavsky
Math, history, art, and world cultures come together in this delightful book for kids, even for those who find traditional math lessons boring. More than 70 games, puzzles, and projects encourage kids to hone their math skills as they calculate, measure, and solve problems. The games span the globe, and many have been played for thousands of years, such as
Math, history, art, and world cultures come together in this delightful book for kids, even for those who find traditional math lessons boring. More than 70 games, puzzles, and projects encourage kids to hone their math skills as they calculate, measure, and solve problems. The games span the globe, and many have been played for thousands of years, such as three-in-a-row games like Achi from Ghana or the forbidden game of Jirig from Mongolia. Also included are imaginative board games like Lambs and Tigers from India and the Little Goat Game from Sudan, or bead and string puzzles from China, and Möbius strip puzzles from Germany. Through compelling math play, children will gain confidence and have fun as they learn about the different ways people around the world measure, count, and use patterns and symmetry in their everyday lives.
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More Math Games and Activities from Around the World
By Claudia Zaslavsky
Chicago Review Press IncorporatedCopyright © 2003 Claudia Zaslavsky
All rights reserved.
In most parts of the world people play some form of three-in-a-row games. The object of the game is to place your three markers in a row on the game board. You have probably played Tic-Tac-Toe. Many of these games are more complicated than Tic-Tac-Toe.
All the games are for two players, sometimes for two teams. You can also play them by yourself. Pretend that you are two people, and play on both sides of the board. This is a good way to learn a new game, or to work out the fine points of strategy, as though you were solving a puzzle.
The games in this chapter gradually become more complicated. They begin with a simple game called Nine Holes from England, with three counters for each player. As you go through the chapter, the games will require more counters. Finally you come to Murabaraba from southern Africa, in which each player starts with 12 counters. Of course, the game board also becomes more complex as you go from one game to the next. In the last activity you will analyze the connection between the type of game board and the number of counters.
These games call for two kinds of counters or markers. Kings and princes used to play with beautiful pieces made of gold and ivory. Ordinary people used stones or seeds, or peeled and unpeeled twigs. You can also use red and black checkers, or two kinds of coins, or make your own special counters.
Game boards for three-in-a-row games have been found scratched in the stones of the rooftop of an ancient Egyptian temple built 3,300 years ago, and in several other ancient sites. The games in this book come from many parts of the world. Look up these places on a map or globe.
Wherever you might travel, you will probably find that people play some version of these games. Although you may not be able to speak their languages, you can make friends all over the world with three-in-a-row games.
TIPS TO THREE-IN-A-ROW FUN
Three-in-a-row games require several types of game boards. You will probably want boards that will last for a while. Draw the lines neatly with a ruler on paper, and glue the paper to cardboard or mat board. It's a good idea to make a pattern on a sheet of scratch paper first.
Some people play games just to win and they get upset when they lose. Playing a game should be fun. When one player always wins, the other player must always lose, and may give up after a while. Helping an opponent to improve his or her skills makes the game more interesting for both players.
Each player should have an equal chance of winning. In some games the first player to move is more likely to win. Players should take turns going first in this type of game.
You may want to vary the games. A slight change in the rules, or in the shape of the game board, or in the number of counters may call for an entirely different strategy. Just be sure that both players agree on the new rules before the game starts.
Nine Holes from England
Nine Holes is one of the simplest of all the three-in-a-row games. Very young children can learn to play it. I read in a book on African games that four-year-olds in some parts of Africa like to play games similar to Nine Holes. They learn to take turns and to think about their moves. They also learn that there is no point in getting upset when they lose a game. Nine Holes is a good introduction to Tic-Tac-Toe and other three-in-a-row games.
Long before anyone had heard of Tic-Tac-Toe, people were playing Nine Holes. It was a favorite among the boys who herded sheep and cattle. While the animals were feeding in the pastures, two boys would agree to draw the game board on the ground. They would dig three rows of holes, three holes in each row. They would gather three stones of one kind and three of another kind, and be ready to play.
The seventeenth-century English poet Michael Drayton described the scene:
The unhappy wags, which let their cattle stray, fit Nine Holes on the heath whilst they together play.
Some of these "unhappy wags" invented strange rules for the game. On the Salisbury Plain, in southern England, the counters were not stones, but wooden pegs stuck into the earth. The players had to get down on the ground and pull out the pegs with their teeth!
In many old English churches one can find sets of holes or lines for three-in-a-row games. Centuries ago the few boys who were lucky enough to go to school in England usually attended church schools. The boring lessons seemed to go on forever, and the boys were often tempted to sneak in a quick game of Nine Holes.
Even grown-ups were guilty of playing games when they should have been attending to the Sunday sermon. An English court record for the year 1699 tells of two men who were punished for playing Nine Holes during church services.
The religious beliefs of the New England colonists did not permit gambling or games played with dice or cards. Children who spent time in play of any kind were warned: "Satan finds some mischief still for idle hands to do." Still, both grownups and children in the New England colonies did play Nine Holes, as well as more complicated three-in-a-row games like Nine Men's Morris and Twelve Men's Morris, in which each player had either nine or twelve counters.
Children in other countries play games that have the same rules as Nine Holes, but different names. Here are some of them:
Nulochen and San-nokunarabe
Sheet of unlined paper, at least 8 inches (20 cm) square
Colored markers or crayons
Piece of cardboard, at least 9 inches (22.5 cm) square
Three counters for each player, of two different kinds (beans, buttons, checkers, or coins)
DRAWING THE GAME BOARD
1. On the sheet of paper, draw a square that measures 6 inches (15 cm) on each side.
2. Draw lines that connect the mid-points, or middles, of the opposite sides.
3. Use a marker or crayon to mark the nine points where the lines meet, as shown in the diagram. Figure 1
4. Glue the paper to the cardboard and decorate your game board. You will want to use it over and over again.
PLAYING THE GAME
The game is played on the nine points where the lines intersect. The two players take turns going first. We will call the counters white and black, although they may be other colors. Player One places a white counter on any point. Then Player Two places a black counter on any point. They take turns, until each player has placed all three counters on the game board. After that, the players take turns moving their counters around on the game board. On each turn, a player moves one of his or her own counters to any empty point on the board.
Each player tries to make a row of three counters of one color, and to block the other player from making a row of three. There are six different ways to make a row: three across, and three up and down. Figure 2
The winner is the first player to make an unbroken row of three across or up and down. If neither player makes a row, the game is a draw. The players can decide to call the game a draw at any time. If both players are careful, a game can go on for a long time.
The players try to move their counters so that they have two different ways to make a row on their next move. This is called a trap. Figure 3
a. White has set a trap
b. Black can block only one row
c. White makes a row and wins the game
THINGS TO THINK ABOUT
Can you find a different way to set a trap?
Where should the first player start? Is it easier for Player One to win the game than for Player Two?
HOW NINE HOLES BECAME TIC-TAC-TOE
Imagine how the game of Nine Holes might have become Tic-Tac-Toe. It could have happened in an English schoolroom. The schoolmaster had given the students several long addition exercises to work out on their slates. While the master was calling on the children on the front bench, a boy in the back nudged the student next to him.
"Got a knife to cut lines in the bench for Nine Holes?"
"No. But we can draw the lines on my slate."
They soon had a lively game going, with pebbles and pieces of chalk as playing pieces.
Suddenly they heard the school-master call out their names. Startled, they let the buttons and chalk clatter to the floor. The master gave them 10 more sums as punishment for playing games in class.
The next morning one of the boys had a wonderful idea. They could make marks on the slate instead of moving buttons and pebbles. One player could mark O's (or noughts, as they say in England) and the other could mark X's (or crosses). And the marks would go in the spaces, not on the points of the game board. As soon as the school-master had turned his back, they tried out the new game on a board with nine spaces, like this. Figure 4 Before the end of the day, the whole class was playing Noughts and Crosses. That's the name they gave Tic-Tac-Toe in England. Soon the game had spread all over England.
Tic-Tac-Toe, also called Noughts and Crosses in England, was a favorite game of English school-boys. They would draw tiny game boards in the corners of their slates. When a boy was called upon to show his work, all he had to do was wet his finger and wipe away all the evidence. But it was not long before sharp-eared schoolmasters learned to recognize the click-clack of Tic-Tac-Toe games going on behind their backs!
In their free time, two children would agree at the start to play a certain number of games, usually 20. At the end of each game, they marked the score at the top of the slate — one point for the winner of each game. Then they erased the game board and drew a new one for the next game. A tied game was scored in the center space at the top of the slate. Children called it "one for Old Nick." Figure 5
Today Tic-Tac-Toe is the most popular three-in-a-row game in the world. Figure 6
Some English children, when they win, call out:
Here I go,
Three jolly butcher boys
All in a row!"
When the game ends in a draw, children in the United States may say, "It's a tie, cat's eye!"
Sheet of paper
DRAWING THE GAME BOARD
Draw this diagram on a sheet of paper. Figure 7
PLAYING THE GAME
1. Each player has a pencil. Toss a coin, or decide in some other way who will make the first move. The players should take turns going first, because the first to go has a much better chance of winning. Player One uses the mark X. Player Two uses the mark O.
2. The game is played in the nine spaces set off by the lines. Player One writes X in any of the nine spaces. Then Player Two marks O in an empty space. The players take turns placing their marks in the spaces.
3. Each player tries to get three of his or her marks in a line — across, or up and down, or along a diagonal. This line of three marks is called a row. There are eight different ways to make a row: Figure 8
a. Across: three ways
b. Up and down: three ways
c. Diagonally: two ways
4. The winner is the first player to get three in a row. If neither player can make a row, the game is called a draw, or a tied game.
THINGS TO THINK ABOUT AND DO
Here are three diagrams to show the first three moves in a game. Copy the diagrams on your own sheet of paper. Play with a friend, or pretend that you are two players taking turns. Finish the game so that Player One wins with a row of three X's. Figure 9
a. Player One marks an X
b. Player Two marks an O
c. Player One marks an X
Then start all over again. Copy the three diagrams, and finish the game so that it ends in a draw — nobody wins or loses.
Then start again. This time, try to finish so that Player Two is the winner with a row of three O's. It won't be easy!
HOW TO BE A GOOD PLAYER
Player One can set a trap for Player Two. Player One can plan her second move so that Player Two is forced to go in a particular space. Here is one way to do it. Figure 10
a. First move
b. Second move
c. Player One prepares to make a row
d. Player Two blocks that row
On the fifth move in the game Player One tries to set up two possible ways to make a row. This is a trap because Player Two can block only one of these rows. Player One is the winner. Figure 11
a. Player One sets a trap
b. Sixth move
c. Player One wins.
There are several other ways for Player One to set traps. See how many you can find.
If you go first, you probably won't lose. If you go second, you probably won't win. For Tic-Tac-Toe to be a fair game, the players should take turns going first.
CHANGING THE RULES
After you have figured out all the good moves, Tic-Tac-Toe can be rather boring. Here are three different ways to play the game.
1. Rule that neither player may make the first move in the center.
2. Play eight-move Tic-Tac-Toe. Each player makes exactly four moves, and gets a point for each row he or she makes.
3. Play Toe-Tac-Tic. The first player to make a row is the loser.
Magic Square Tic-Tac-Toe from Ancient China
Some people believe that a certain arrangement of numbers on a Tic-Tac-Toe diagram can bring good luck. Every line of three numbers has the same sum. This arrangement is called a magic square.
It is said that a Chinese emperor was the first person to see a magic square. Over 4,000 years ago a large turtle swam close to his ship. The pattern on the turtle's back was a wonderful arrangement of numbers. Every row, every column, and each diagonal added up to 15. No wonder the Chinese thought it was magic. In time, people all over the world were making magic squares. Figure 12
A Chinese magic square, with knots in black and white cord showing the numbers
Sheet of unlined paper, at least 7 inches (17 cm) square
Pencil and markers
2 sheets of heavy paper or cardboard, at least 8 ½ inches (22 cm) square
MAKING THE GAME BOARD AND COUNTERS
1. On the sheet of paper draw a 6-by-6-inch (15 cm X 15 cm) square.
2. Subdivide the square into nine small squares, each 2-inch (5 cm) square.
3. Glue the paper to the cardboard. Decorate your game board with markers or crayons.
4. Cut out nine small disks or squares from heavy paper and number them from one to nine. Figure 13
PLAYING THE GAME
Arrange the nine counters so that each row, each column, and each diagonal has a sum of 15. You can play this game by yourself or take turns with a partner and cooperate to make a magic square. Each player may place any one of the nine counters on the game diagram.
THINGS TO THINK ABOUT AND DO
Complete the magic squares below. Then try to arrange the counters to make other magic squares. There are eight different magic squares. Figure 14
The object of this game is to form rows, columns, or diagonals that add up to 15.
Sheet of unlined paper
Pen or marker
Nine counters, numbered from one to nine (see Magic Square Tic-Tac-Toe)
DRAWING THE GAME BOARD
Draw the diagram for Tic-Tac-Toe. Figure 15
PLAYING THE GAME
1. Player One takes the five odd-numbered counters: 1, 3, 5, 7, and 9. Player Two takes the four even-numbered counters: 2, 4, 6, and 8.
2. The players take turns placing a counter on the game board. All nine counters are used. The object is to make a row of three counters that has a sum of 15. A row may contain both odd and even numbers.
3. A player scores one point for each row he or she completes. A player can sometimes make two or even three rows in one move.
4. The player with more points is the winner. Figure 16a and Figure 16b
For the next game, the players exchange counters.
a. Player One gets ready to move
b. Player One has made two rows in one move, and earns two points
Achi from Ghana
Ghana, in West Africa, became an independent country in 1957. Before that year, the country was called the Gold Coast and was a colony of England. As you might guess, Europeans traded their wares for African gold along this coast. The name Ghana came from an ancient kingdom of that name, located farther north in Africa. It was known to the Arabs as the "land of gold." Over a thousand years ago, Ibn Hawkal, an Arab traveler to that land, described the king of Ghana as the wealthiest in the world.
Schoolchildren in Ghana play Achi. They draw the game diagram in the dirt and pick up pebbles to use as counters.
Excerpted from More Math Games and Activities from Around the World by Claudia Zaslavsky. Copyright © 2003 Claudia Zaslavsky. Excerpted by permission of Chicago Review Press Incorporated.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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Meet the Author
Claudia Zaslavsky has been a math teacher and math teacher's teacher since 1959 and was one of the first educators to emphasize multicultural perspectives in math. She is the author of 13 books, including Math Games & Activities from Around the World, Number Sense and Nonsense, Africa Counts and Multicultural Math.
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