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 [there's an earlier story in this saga, "An Introduction to Fractions", which should logically be read first (click here)]

 ADDITION OF FRACTIONS ...

 Bart sat down at the desk with Miss Walker. "Fractions seem pretty easy so far," he said.

 "They're not bad," said Miss Walker, "so long as you understand the logic behind what you're doing."

 She reached into her bag and pulled out a medium-sized apple, a small cardboard box full of elastic bands, and a large plate, all of which she placed on the desk in front of Bart. Next came a very sharp knife and a chopping board, which she placed in front of herself.

 Bart eyed her with some degree of surprise. "I thought we were doing fractions today?" he queried.

 Miss Walker smiled. "Fractions it is," she agreed, "bear with me, you'll understand more as we go along."

 She looked at him keenly. "Still remember what we learned last time, in 'An Introduction to Fractions'?"

 Bart nodded.

 "Okay then, simplify four eighths for me."

 Bart replied swiftly, "Four eighths equals two quarters equals one half."

 4-8 = 2-4 = 1-2

 "Excellent," said Miss Walker, writing it down. "Well remembered. What about four twelfths?"

 "Four twelfths equals two sixths equals one third," said Bart.

 4-12 = 2-6 = 1-3

 "Outstanding," said Miss Walker. "Well done, Bart."

 After she had written the figures down she turned to Bart. "Remember we started with eight apples last time, when we started learning about fractions?"

 "Yes," said Bart, not quite seeing where this was going, being that there was only one apple on the desk.

 "Well, said Miss Walker, "this time we're going to look at fractions by dividing this one apple into pieces."

 She picked up the sharp knife and placed the solitary apple in front of her. "I need to cut this apple into 24 pieces," she said.

 Bart looked at her doubtfully. "It's a fairly small apple," he said. "Are you sure it can be cut that many times?"

 "I hope so," said Miss Walker. She sliced the apple in half then put the flat parts of the halves on the board, so she could slice through each of them. Soon there were four equal-sized pieces on the board.

 "Four quarters," said Bart, anticipating her question.

 "Excellent," said Miss Walker. "And if we fit them all together, we get a whole apple."

 She wrote it down.

 1-4 + 1-4 + 1-4 + 1-4 = 1

 "No problem," said Bart.

 "Okay," said Miss Walker, "I'm now going to divide each quarter into two pieces." She took a quarter and sliced it in half through the length of its shiny surface, and did the same with all the other quarters.

 "Eight eighths," said Bart. "All fitted together makes one whole apple."

 "First class again," said Miss Walker. She wrote it down.

 1-8 + 1-8 + 1-8 + 1-8 + 1-8 + 1-8 + 1-8 + 1-8 = 1

 Bart looked at the thick slices of apple. "Sure you can divide each of those eighths into three equal parts?"

 "Probably," said Miss Walker. "This is a very sharp knife and if I slice them quite thinly we should just manage it."

 She took the first eighth and sliced it carefully, longways, into three thin slices, then did the same thing with all the other eighths.

 Bart counted them. "Twenty-four," he said. "Amazing."

 He arranged the 24 thin slices around the edge of the large plate. "Pretty good for one small apple," he said.

 He looked quizzically at Miss Walker. "Where do the elastic bands come in?"

 "You'll see," laughed Miss Walker. "Right, Bart, I want you to re-make a quarter of the apple."

 Bart raised his eyebrows. Miss Walker was usually quite sensible. "You want me to use elastic bands to join together bits of apple you just sliced up," he said eventually.

 "Yes," said Miss Walker.

 "Okay, fine," thought Bart, but he didn't say anything out loud.

 He picked up six of the 24 pieces. "Six twenty-fourths equals one quarter, right?" he asked.

 "Agreed," said Miss Walker, and she wrote it down:

 6-24 = 1-4

 Bart bunched together six of the pieces in a quarter-apple shape and Miss Walker put the band round twice to hold it in place, after Bart had difficulty doing this one-handed.

 "One quarter re-made," said Bart, still wondering if his teacher was slightly mad.

 "Okay," said Miss Walker. "Now I'd like you to leave that piece where it is, and make me up a piece that is one sixth of the original apple."

 Bart looked at her steadily, a glimmer of understanding visible in his eyes. "Four pieces, right?" he asked. "Four twenty-fourths is one sixth, yes?"

 "It is indeed," said Miss Walker, and she wrote it down.

 4-24 = 1-6

 Bart made up the one-sixth piece with four thin slices of apple, and again Miss Walker put the elastic band around.

 "I'd like you to make me up an eighth now," said Miss Walker.

 Bart thought rapidly. Twenty-four pieces divided by eight was three pieces. "Three pieces," he said, fairly confidently.

 "Sure is," said Miss Walker, and together they made up the eighth, and then she wrote the figures down again.

 3-24 = 1-8

 She looked at Bart. "How many pieces have we used so far?"

 Bart added them up. "Six plus four plus three - thirteen altogether," he said. "Thirteen twenty-fourths."

 Miss Walker wrote it down:

 6-24 + 4-24 + 3-24 = 13-24

 "And how many do we have left?" she asked.

 Bart worked it out in his head, without counting them. "24 take away 13 leaves 11," he said. "Eleven pieces - eleven twenty-fourths."

 "Outstanding," said Miss Walker, and wrote it down:

 24-24 - 13-24 = 11-24

 She looked carefully at him. "We could write that another way as well," she said, showing him.

 1 - 13-24 = 11-24

 "24 over 24 can be written simply as '1', because when you put all the slices of the apple together you get the whole apple, with no slices missing - and you already know how to write whole numbers."

 "I do?" said Bart doubtfully.

 "Yes," said Miss Walker. "You know you learned to count one, two three, four, five .... ?"

 "Yes," said Bart.

 "Well, they're whole numbers," said Miss Walker.

 "Hold on," said Bart. "You mean these numbers I've known since I was a little kid, 1, 2, 3, 4, 5, actually mean more than I thought they did?"

 "In a way," said Miss Walker.

 Bart thought for a moment. "Each slice is a twenty-fourth and there are 24 of them and when I put them all together I get a whole apple which we write as '1'," he said.

 "Yes," said Miss Walker. She wrote it down on a separate piece of paper:

 24-24 = 1

 "You could work that out in the usual way," she said. "24 over 24 - both top and bottom can be divided by 24 which gives you one over one (one divided by one) which is one whole."

 She wrote it down:

 24-24 = 1-1 = 1

 "So if we had another apple," said Bart, "and we divided the second apple into 24 pieces as well, we'd have 48 pieces altogether."

 "Mmm," said Miss Walker. "I don't want this to get too complex, Bart - where are you going with this one?"

 Bart looked at her. "All I meant was that if you have two apples each in 24 pieces you'd have 48 pieces altogether. 48 slices, that's 48 twenty-fourths. But still only two whole apples if you fit the slices back together."

 Miss Walker wrote it down:

 24-24 + 24-24 = 48-24 = 2-1 = 2

 She smiled at Bart. "That's what you meant?" she asked.

 He grinned. "Yes," he said. "But I wouldn't have written in the 2 over 1, why do we need that?"

 "It's just a logical step in the working out," said Miss Walker. "Two over one means two-divided-by-one, which is indeed equal to the whole number '2'. A whole number divided by one always gives the answer of the whole number you started with."

 She wrote down a few examples:

 1-1 = 1
 2-1 = 2
 3-1 = 3
 4-1 = 4

 "You got that okay?" she asked.

 "No problem," said Bart.

 She wrote some more down:

 1-1 = 1
 2-2 = 1
 3-3 = 1
 4-4 = 1

 Bart read them. "The number underneath tells you how many pieces the item is cut into," he said eventually. One whole equals one, two halves equal one whole, three thirds equal one whole, four quarters equal one whole."

 "Excellent," said Miss Walker.

 "But," said Bart, following through, "if you only have the number '1' underneath the line, you're actually dealing with whole numbers."

 "Precisely," said Miss Walker. "Well described, Bart."

 She looked at him. "I'm glad you've understood about whole numbers - we've gone a little further down that track than I intended," she said with a smile.

 "What we need to do now is to go back to our original apple, the one with the 24 pieces going brown .... "

 She picked up the original piece of paper, the one where she'd written down the fractions of the sliced apple. The page was nearly full up.

 "We'd better write this on a new sheet," she said, copying down the last line of figures onto a new sheet of paper.

 6-24 + 4-24 + 3-24 = 13-24

 Miss Walker studied the figures for a moment, and then she looked up at Bart. "Have you any idea what one twelfth of the whole apple would look like?"

 Bart thought for a second or two. "One twelfth of 24 pieces is 2 pieces," he said, picking up two of the remaining slices and holding them together for Miss Walker to put an elastic band around.

 "Well done, Bart," said Miss Walker. "Now, how many pieces of the apple have we used altogether?"

 Bart counted them. "Six plus four plus three plus two - fifteen altogether," he said. "Fifteen twenty-fourths."

 Miss Walker wrote it down.

 6-24 + 4-24 + 3-24 + 2-24 = 15-24

 "Right," she said. "Fifteen slices used. How many slices of the original 24 do we have left?"

 "Nine," Bart replied promptly. "Nine twenty-fourths."

 Miss Walker wrote it down.

 24-24 - 15-24 = 9-24

 "Let's parcel those up," she suggested.

 Bart fitted the nine slices together and Miss Walker put the elastic band around them twice.

 "Nine over twenty-four," said Miss Walker. "Can we reduce that down to any smaller numbers?"

 "Divide top and bottom by three!" said Bart instantly.

 "Exactly," said Miss Walker. "What do you get then?"

 Bart worked it out. "Nine over twenty-four becomes three over eight - that's three eighths," he said confidently.

 "Excellent," said Miss Walker. She wrote it down.

 9-24 = 3-8

 "Hold on," said Bart. "That means that if there's three eighths left, all the other bits we took out first must add up to five eighths, because there are eight eighths altogether."

 "First class," said Miss Walker. "Let's prove that."

 She wrote down the long line of fractions again:

 6-24 + 4-24 + 3-24 + 2-24 = 15-24

 "Simplify fifteen over twenty-four," she said to Bart.

 Bart thought for a moment. "Divide top and bottom by three - fifteen over twenty-four becomes 5 over 8 - five eighths," he said. "So I was right!"

 "You were indeed," smiled Miss Walker. She wrote it down.

 15-24 = 5-8

 "Right then," she said. "We know five eighths plus three eighths equals one whole."

 She wrote it down:

 5-8 + 3-8 = 8-8 = 1

 "Let's write all the individual parts of this sum down," she suggested.

 6-24 + 4-24 + 3-24 + 2-24 + 9-24 = 24-24 = 1

 "Or we could say that another way," she said, writing some more:

 sixpieces + fourpieces + threepieces + twopieces + ninepieces = 24pieces = onewholeapple

 She looked at Bart. "With me so far?" she asked.

 He nodded.

 "Okay then," she said, "let's modify it a little by leaving out the 24 twenty-fourths, now that you know about whole numbers."

 She wrote it down.

 6-24 + 4-24 + 3-24 + 2-24 + 9-24 = 1

 Bart looked at her. "It's a bit confusing like that - somehow it's easier if you can see how you got to the final number," he commented.

 Miss Walker smiled. "I would agree with you there," she said, "and to be truthful it's a lot easier for teachers to be able to see the steps you used to reach your final answer, but bear with me a little longer so I can show you the sum we've actually done, without all the working-out steps we used."

 She wrote down one final line of figures.

 1-4 + 1-6 + 1-8 + 1-12 + 3-8 = 1

 "Awesome," said Bart. "We've come a long way, haven't we!"

 "We have indeed," said Miss Walker.

 © Penny Midas Rollo 16th October 2001

 [to read the next story in this saga, "Fractions and Percentages", please click here], or return to Fractions index

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