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 AN INTRODUCTION TO FRACTIONS ...

 "Hello, Bart," said Miss Walker. How much do you know about fractions?"

 "Not much," said Bart.

 "Okay," said Miss Walker. "Let's suppose you have eight apples. If I take one apple, do you know what fraction that represents? A quarter, a half, an eighth, a third?"

 Bart thought for a moment. "An eighth," he said confidently.

 "Excellent," said Miss Walker. "Now, do you know how that is written? I'll show you. The figure one with a line underneath, and the figure eight, underneath that, like this:"

 She wrote at the top of a large piece of paper:

 1-8

 "That's the way we write one eighth," she said.

 "Fair enough," said Bart.

 "Now," said Miss Walker, "what happens if I take two apples out of the eight, instead of one? How many eighths would I have then?"

 Bart thought for a moment. "Two eighths," he said confidently.

 "Excellent," said Miss Walker, and she wrote some more figures a little way down the large piece of paper:

 2-8

 "Now," said Miss Walker, "that proportion of the apples can also have a different name. If you take two apples out of eight apples, have you taken away half the apples or a quarter of the apples?"

 Bart thought for a moment. "A quarter of the apples."

 "Excellent," said Miss Walker. "We write a quarter like this."

 She wrote further down the piece of paper:

 1-4

 "Fair enough," said Bart.

 "Now," said Miss Walker, "two apples out of eight was two eighths of the apples, as you'll remember."

 "Yes," said Bart confidently.

 "And it was also one quarter of the apples," said Miss Walker.

 "Yes," agreed Bart.

 "So two eighths is the same as one quarter?" suggested Miss Walker.

 Bart thought for a moment. "Yes, it has to be," he said.

 Miss Walker wrote some more figures on the large piece of paper.

 2-8 = 1-4

 "Two eighths is the same as one quarter - two eighths *equals* one quarter. That's how we write this."

 Bart thought for a moment. "Yes," he agreed.

 Miss Walker looked at him. "Fractions aren't that hard, are they?"

 Bart grinned. "No," he said.

 "Shall we take this a little further?" said Miss Walker.

 "Why not," said Bart.

 "Okay then, said Miss Walker, "if I gave you the figure of two eighths, that we started off with, two over eight, can you think of any way of getting it to one over four - the actual "method" you would use to do this?"

 Bart thought for a moment. "You could divide the figure at the top and the figure at the bottom by the same number, the number two."

 "If you divide the top number 2, by two, you get one, and if you divide the bottom number, 8, by two you get four. So you end up with one over four."

 "Excellent," said Miss Walker.

 She paused, and then said, "If you took four of the apples, how many eighths do you think you would have?"

 Bart didn't hesitate. "Four," he said.

 "Well done," said Miss Walker. "And if one quarter of the apples is two apples, and you have four apples (twice as many as the original two), how many quarters would you have?"

 Bart thought for a moment. "Two quarters," he said.

 "Pretty good," said Miss Walker.

 "Can I ask something?" said Bart.

 "If four apples is four eighths *and* it's also two quarters, that means four eighths equals two quarters, right?" said Bart.

 "Exactly right," said Miss Walker. She wrote down some more figures, further down the large sheet of paper:

 4-8 = 2-4

 "Right on," said Bart.

 Miss Walker looked at him. "What would you say the 'method' was, for getting from the first fraction to the second one?"

 Bart looked back at her. "Same as the first time, you divide the top figure and the bottom figure by the same number, the number two. Four divided by two is two, eight divided by two is four."

 He looked at her. "You could do that again," he said. You could divide the top and bottom numbers on two quarters, two over four, by two - so you would get the figure one at the top, and the figure two underneath."

 "Good thinking, Bart," said Miss Walker. "What fraction would that give you?"

 "One over two," said Bart.

 "Excellent," said Miss Walker. "Do you know what one over two means?"

 "One half," said Bart.

 "First class," said Miss Walker. She wrote some more figures near the bottom of the piece of paper, underneath all the others.

 4-8 = 2-4 = 1-2

 Bart leaned over her shoulder as she was writing them. "Four eighths equals two quarters equals one half," he said.

 "Outstanding," said Miss Walker. "Now, Bart, one last question - can you think of a "quick way" to get from four eighths to one half?"

 "Dead easy," said Bart. "You divide the top number by four and the bottom number by four as well."

 "You've got it," said Miss Walker. "We always use the *same* number for dividing into each of the top and bottom figures, to simplify a fraction, to reach the point where the lowest possible number is on the top of it."

 She wrote down the figures:

 4-8 = 1-2

 She looked at him. "One last challenge," she said. "How do you think we could simplify four over twelve?"

 Bart thought for a moment. "Divide both the numbers by four - so you get one over three - that's one third!" he said.

 Miss Walker wrote it down.

 4-12 = 1-3

 "Exactly right," she said. "Bart, you are a class A student!"

 Bart grinned widely. "I like fractions," he said. "Can we do some more?"

 "That's enough for one day," said Miss Walker, as the bell sounded for the end of class. "We'll do some more tomorrow."

 © Penny Midas Rollo 30th September 2001