Sample Lesson

From the book: World of Numbers

Introduction to Multiplication

An Introduction to Multiplication for the First Year

Mul de Plier
The Girl from the Land of Plenty
by B.B.

Mul lived in the little town of Plier in the breezy highlands at the foot of the Hohen mountains. She was a happy child and spent her days in play without care or worry. She was brought to live in Plier when she was little more than a year old. Her step parents loved her as their own child. Some say that her father and mother were a great King and Queen from a land beyond the Hohen mountains, who had to hide her from a threat of evil that had infected the kingdom.

The weather in Plier was fair and sunny much of the year and breezes danced about over the rolling hills blowing in from the distant sea. Often while Mul was eating her breakfast, she could count the many sails in the distance as the fishermen set out for the day. During the rest of the day she played like a little breeze herself, moving from one thing to another and not lingering anywhere for very long. Toys were strewn everywhere –on the floor of her room, throughout the house, and all around the yard outdoors. When she became seven, her stepfather began to teach her to read. She learned quickly, but could never finish a single book. There is a bookcase in her room that contains 127 volumes of the best children's books in the world. By the age of 12 Mul had not read a single volume, but she had visited every book on the shelves, and had even read the first two or three chapters on many of them. Mul was that way in school as well. By the time she was 16, she knew a little bit about many things, but really didn't know anything in much depth. But this didn't bother her, for a care-free life cannot be burdened with knowing the world too deeply.

Mul loved to explore the forests and meadow in the surrounding hills of Plier. One day while she was gathering wildflowers along the edge of a meadow, she noticed some unusually shaped mushrooms and while intent on discovering more, she ventured into the woods far from her accustomed pathway. After some time she realized that she did not know the way back to the path and without meaning to do so, strayed farther into the deep woods. After some time she grew tired and sat down upon a fallen tree. A trickle of tears began to run along her cheeks for now she realized that she was lost. Then she heard a soft gentle voice. An old woman was calling to her from behind a thick clump of trees. "Come child, I will show you something the likes of which you have never seen before" said the old woman. "But can you help me find my way back to Plier?" said Mul. "Yes, it's not far from here, but first come see what I want to show you." Mul was curious to see what it was the woman wanted her to show and so followed her into the thicket of wood. Eventually the trees gave way to a view from a precipice overlooking a great valley. A broad river meandered through the farmlands and in the distance beyond the villages, one could see the towers of a mighty castle. "I've played in the nearby meadows and forests many times, but I never knew this was here," said Mul with surprise. "What is that place?" she asked.

"Close your eyes for a moment, and I will show you," said the old woman. Mul closed her eyes and suddenly was as light as the wind. She opened her eyes and was amazed by what she saw, for she could look closely at the people in the villages below, even though they were far away. "What do you see?" said the old woman. "I see villages full of people," said Mul. "Look again –this time more carefully," said the old woman. Mul looked closely, then suddenly caught her breath. " I see the children with tears in their eyes, and the mothers and fathers with long lines upon their faces," she said. "Why do the children have tears in their eyes and the fathers and mothers have long lines upon their faces?" asked the old woman. Mul looked closely again. " The plates are empty. The pots are filled with watery soup. And the fields are barren." And for the first time in her life, care began to well up in her heart. " What can I do to help them?" she asked.

"That is the land where you were born," said the old woman. "If you want to help them, then you must leave Plier. I will show you a path along the cliff that leads into the caves of the earth. A dwarf named Gnomis lives there. If you still carry the feeling of care for others in your heart, he will lead you out of the cave and into a hidden valley that no one knows, where grows the tree of life. Perhaps, there you will discover what to do."

Mul followed the path pointed out by the old woman. By sundown she had reached the bottom of the mountain, and there just like the woman described, was the cave. An old dwarf with long ears, a humped back and gangly arms that hung all the way down to his knees greeted her at the mouth of the cave. "What's a pretty thing like you want?" he asked showing his black teeth with a big grin. "Are you Gnomis?," she asked. " Who wishes to know?" he replied. "I am Mul from the village of Plier," she said politely, suddenly remembering her manners. "The old woman of the forest said that you could show me the way to the tree of life." The old dwarf shook his head "No mortals who dwell on the earth know of the tree of life," he said and turned to walk away. " Wait !," said Mul "it's not for me, it is for the hungry people of the village." The dwarf's pointed ears suddenly perked upright. " Well then Mul of Plier -I shall call you Mul de Plier from now on–I shall show you the way. But you must stay close to me and not turn around no matter what happens."

Together they passed through passages that were so dark that she could not see the dwarf in front of her as he picked his way along a narrow ledge with his large feet. She clung with both hands to his rumpled coat. "Couldn't we light a torch or lantern so that we might see?" she asked. " You don't want to see what's on either side of us this very moment," he whispered. And suddenly a cold chill passed along the girl's spine for she could hear a low rumbling on either side of the narrow path, but could not see what was there. There seemed to be something behind her and though she yearned to turn around to see what it was, yet remembering the dwarf's words, for the first time in her life she resisted the urge to follow her curiosity though she continued to wonder. They began to ascend up the path to where a small beam of light shone in the dark like a tiny star. " You must go the rest of the way on your own, Mul de Plier," said the dwarf.

Mul continued upward. As she got closer, the little star began to grow until it seemed like a glowing moon. Then the moon became like a bright sun, and as she approached it she realized that the bright light was a doorway leading out of the cave and into a sunlit valley. All was still and quiet. In the midst of the valley was a tiny spring and from it trickled four rivulets in four directions that filled four shimmering pools. Above the spring was a fantastic tree that seem to grow before her very eyes. It grew so high that she could not see the top of it. She waded over one of the little streams that seemed to giggle as it bubbled around her feet, and she would have loved to gather handfuls of the sand that sparkled as though it were made from gold and precious gems. On either side of the stream there were brightly colored wildflowers as bright as the sunrise that she would have liked to gather, but instead, remembering her aim, she continued to the trunk of the fantastic tree. When she touched it, the bark felt warm and seemed to ripple like living skin, as though it could free itself from the earth and walk over the ground. A branch swept down beside her face and suddenly the leaves began to flutter as though a hundred tiny breezes twirled through the branches. Though she did not hear any words –like the words you and I use when we talk to one another–she knew that the tree was talking with her, and she understood what it was saying. She reached up and broke off two branches from the limb that had swept down beside her face.

All at once the ground gave way beneath her, and she was as light as the wind. The tree branches began to wave violently swirling above her faster and faster until they became a great swirling mass like a tremendous dark cloud that covered over everything. She began to slip away into the dark cloud, drawn up into it until everything became dark and she fell into a deep sleep.

When she awakened, she found herself lying in the meadow in the woods outside of her native village of Plier. She looked around and saw that she was alone. Beside her lay the wild flowers she had been picking at the edge of the meadow. In her hand she grasped two little branches broken off from the tree of life.

She held the branches out before her like two little drum sticks. Then she began to wave them about like two conductor's batons. While waving the branches, they inadvertently crossed and made a little clicking sound that reminded her of the laughing streams that poured from the spring at the foot of the tree of life. Merrily she clicked the sticks together again, this time however, she clicked them twice. Much to her surprise, the bunch of wild flowers that she had gathered suddenly became two bunches of wild flowers. When she looked carefully at the flowers, she saw that they were both exactly alike with the same number of flowers in each bunch, and all the same colors and sizes as the other. How great was her wonder when she clicked the little branches twice together again, and the two bunches of flowers became four bunches. She pulled three shiny stones from her pockets that she had gathered on the path to the meadow, and setting aside the four bunches of flowers, she placed the three stones on the ground before her and clicked the branches two times together. How many stones do you think appeared before her? There were now six shiny stones, for three doubled or multiplied by two makes six.

Mul de Plier spent the rest of the afternoon playing in the meadow, multiplying by two, little saplings growing at the edge of the meadow, little groups of mushrooms, piles of acorns for the squirrels, and of course the wildflowers that she loved so well. When it was time for her to leave, she noticed a shadow among the trees at the edge of the meadow, which disappeared when she tried to walk closer. Then she remembered the old woman and the path to the dwarf Gnomis, and the frightening cave. And the tree of life. Then she remembered the tears in the eyes of the children and the long lines on the faces of the mothers and fathers. And the empty plates, and the pots with watery soup, and the barren fields. And for the second time in her life, care welled up in her heart. "Now I know why I have these two branches from the tree of life," she said.

And thus began the adventures of Mul de Plier and how she helped to transform the barren land into the land of plenty.

After telling this introductory story to the students, the teacher held aloft two smooth dowels (about 15 inches long; these could simply be two small branches) that she had prepared in advance. Then she told the students that whenever Mul de Plier traveled to a village and helped to multiply the seeds to be planted, she would hold the two branches from the tree of life over her head like this. Then she crossed the two dowels over her head forming an "X". The teacher continued to tell the students that after Mul de Plier had visited a village the people were so happy that they now had many seeds to plant and soon the land would no longer be barren. Whenever the people spoke of her, they would make her sign –the sign of the crossed branches. The teacher then moved to the blackboard and made a large "X" on the board. Then she continued to tell the students how this sign "X" of Mul de Plier, later was used to show that something is multiplied. "Here's how they would write it," said the teacher, and then she wrote the expression 3 X 2 is ____. "Remember when Mul was in the meadow and she placed three stones on the ground, then she clicked her stick two times. So this number sentence says "three stones multiplied by two is how many?" How many stones appeared? Yes, there were six, so we can finish writing this sentence by putting the number six in the blank spot.

"Now let's play a little game" she continued. I will give you the sticks and you can pretend that you are Mul de Plier. I will place some seeds on the table, and you tell me how many are there." The teacher placed some seeds on the table and the student noted that there were six of them. "Now," said the teacher, "strike your sticks two times Mul de Plier!" After two clicks, the teacher placed six more seeds on the table. "Now how many seeds are there?" she asked. "Twelve," answered the student. "So we can say that six seeds multiplied by two are twelve seeds. Let's see if you can write that on the board with numbers. Don't forget to use the sign of Mul de Plier." Then with some guidance, the student wrote the expression 6 X 2 is 12. This concluded the first day lesson on multiplication.

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