Long/fraction-FM 11/28/00 7:54 AM Page ii Also in the Magical Math series Dazzling Division Marvelous Multiplication Measurement Mania Long/fraction-FM 11/28/00 7:54 AM Page iii Magical Math FA B U L O U S FRACTIONS Games and Activities That Make Math Easy and Fun Lynette Long John Wiley & Sons, Inc New York • Chichester • Weinheim • Brisbane • Singapore • Toronto fcopyebk.qxd 1/25/01 11:04 AM Page iv Copyright © 2001 by Lynette Long All rights reserved Illustrations copyright © 2001 by Tina Cash-Walsh Published by John Wiley & Sons, Inc Design and production by Navta Associates, Inc No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4744 Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 605 Third Avenue, New York, NY, 10158-0012, (212) 850-6011, fax (212) 850-6008, email: PERMREQ@WILEY.COM The Publisher and the author have made every reasonable effort to ensure that the experiments and activities in this book are safe when conducted as instructed but assume no responsibility for any damage caused or sustained while performing the experiments or activities in the book Parents, guardians, and/or teachers should supervise young readers who undertake the experiments and activities in this book This title is also available in print as ISBN 0-471-36981-0 For more information about Wiley products, visit our Web site at www.Wiley.com Long/fraction-FM 11/28/00 7:54 AM Page v Contents I The Magic of Fractions II The Facts on Fractions 5 10 11 12 13 Half Happenings Sandwich Quarters Fraction Stars Fraction Strips Fraction Match Lowest-Terms Picture Reduce It! The Lineup Fraction War! Domino ESP Flip It! Cut Up! Zero III Adding and Subtracting Fractions 14 15 16 17 18 19 20 21 22 23 12 15 18 20 22 24 27 29 31 34 36 39 Combination Pizza Addition Chip Trading 41 44 Pattern Pictures Circles inside Circles Subtraction Chip Trading Least Common Multiple Tic-Tac-Toe Quick Change Fraction Roll Up-and-Down Dominoes Missing Numbers 47 50 52 54 57 60 63 65 v Long/fraction-FM 11/28/00 7:54 AM Page vi I V M u l t i p l y i n g a n d Dividing Fractions 24 25 26 27 28 29 30 31 32 33 Graph Paper Multiplication Splitting the Deck Fraction Sale! Mixed-Number Products 69 72 74 76 Add Them Up Domino Match Magic Fraction Squares Number Line Division Shoe Box Fractions Three-in-a-Row Bingo 79 81 84 88 91 95 V F r a c t i o n s , D e c i m a l s , a n d M o r e 34 35 36 37 38 39 40 vi 67 99 Order Dancing Decimals Fraction Jeopardy! Problem Generator Crossword Puzzle 101 104 107 110 111 Challenging Dot-to-Dot Fraction Words 113 115 Fractions Master Certificate 116 Index 117 CONTENTS PartI 11/28/00 7:55 AM Page I THE MAGIC OF FRACTIONS T he world is full of fractions If there weren’t any fractions, you couldn’t have half of your friend’s cookie, or leave one-third of your spinach on your plate, or eat one-eighth of a pizza What if your father sent you to the market to pick up a few groceries? On the list he might write: PartI 11/28/00 7:55 AM Page • 1⁄4 pound of American cheese • 3⁄4 pound of ham • 1⁄2 gallon of milk • half a dozen eggs • one loaf of bread If there weren’t any fractions, you’d have to get either a whole of everything or nothing Neither of these options would make your dad very happy What are fractions? “Fractions” are just another way to write numbers Fractions show that a number is part of a set For example, if you ate one slice of pizza out of an eight-slice pie, you can show that in a fraction by saying that you ate one-eighth (1⁄8) of the pizza The word fraction means “part of a whole,” from the Latin fractio, meaning “to break into pieces.” A fraction has two parts, a “numerator” and a “denominator.” The numerator and the denominator are separated by a bar, as in , or a slant line, as in 1⁄2 or 1/2 The numerator is written on top of the bar or to the left of the slant line, and the denominator is written below the bar or to the right of the slant line However the fraction is written, one-half means one of two equal parts of a whole Look at the parts of these fractions: • In the fraction , is the numerator and is the denominator • In the fraction 2⁄3, is the numerator and is the denominator • In the fraction 1/10, is the numerator and 10 is the denominator • In the fraction 4⁄3, is the numerator and is the denominator The denominator tells you how many equal parts make up the whole If the denominator is 4, the whole is divided into four equal parts If the denominator is 3, the whole is divided into three equal parts The numerator represents how many parts are being used If the numerator is 2, two parts are being used If the numerator is 1, one part is being used THE MAGIC OF FRACTIONS PartI 11/28/00 7:55 AM Page Fractions can be FRACTION WORDS expressed in words, Number Forms Word Forms such as “I’d like half a sandwich” for 1⁄2 or “I’ll 1⁄ one-half or half be back in a quarter of 1⁄4 one-fourth or one quarter an hour” for 1⁄4 For most fractions, the 11⁄ one and a half word is just a combina13⁄ 16 thirteen-sixteenths tion of the numerator and the denominator, so 1⁄6 is said as “one-sixth.” If you want to eat one-fourth of a sandwich, first divide the sandwich into four equal parts, and then eat one of them: 1⁄4 If you want to eat three-fourths of an apple, first divide the apple into four equal parts, and then eat three of them: 3⁄4 If you want to drink half a glass of milk, first mentally divide the milk in the glass into two equal parts, and then drink one of them: 1⁄2 Fractions are an essential mathematical skill You will use fractions every day of your life, so start practicing with the 40 fun fraction activities in this book After the first one, you will have done one-fortieth of the activities After the last one, you’ll be a fractions master Then you can proudly display the fractions master certificate at the back of this book THE MAGIC OF FRACTIONS PartI 11/28/00 7:55 AM Page PartV 12/6/00 9:23 AM Page 108 For example, in the Multiplication of Fractions set, one card should have a 10, one a 20, one a 30, one a 40, and one a 50 Set up each set this way A sample card is shown here Game Rules Shuffle the cards and deal them out to the players There are 25 cards, so one player gets 12 cards and the other gets 13 Players create problems and write one problem on the front of each dealt card The category labeled on the back of the card determines the type of problem For example, a “Lowest Terms” card should have a fraction to reduce Vary the difficulty of the problems based on the number on the back of the card, which is the number of points earned for correctly solving the problem Cards with higher numbers should have more difficult problems, while cards with lower numbers should have easier problems A level 20 problem on an Addition of Fractions card is shown 108 FRACTIONS, DECIMALS, AND MORE PartV 12/6/00 9:23 AM Page 109 Place all the cards on the table so that the problems are facedown Arrange the cards in five columns and five rows according to category and point value Put the Lowest Terms cards in column 1, Addition of Fractions cards in column 2, Subtraction of Fractions cards in column 3, Multiplication of Fractions cards in column 4, and Division of Fractions cards in column Put the 10-point cards in row 1, 20-point cards in row 2, 30point cards in row 3, 40-point cards in row 4, and 50-point cards in row as shown Player turns over any card, and both players try to solve the problem The first player to correctly solve the problem keeps the card and turns over another card Both players try to solve this new problem If a player solves a problem incorrectly, that player must give one of the cards he or she has won, of equal or greater value, to the other player The game continues until all cards have been played At the end of the game, players add up the points on the back of their cards The player with the most points wins SUPER FRACTION JEOPARDY! Make 25 new cards with harder problems that award point values of 20, 40, 60, 80, and 100 FRACTION JEOPARDY! 109 PartV 12/6/00 9:23 AM Page 110 37 Problem Generator Practice working with fractions by creating fractions M AT E R I A L S 55 20 40 35 Game Preparation Write one of the following fractional expressions on each of 15 index cards: 25 12 17 12 1 34 43 10 15 50 45 25 30 pencils 15 index cards timer several pieces of paper for each player players 22 66 Game Rules Shuffle the cards and place them facedown in the center of the table Player turns over the top card, and player starts the timer Player has minutes to create as many fraction problems as he or she can in which the fraction on the card is the answer to the problem The player must create an addition problem, a subtraction problem, a multiplication problem, and a division problem in that order When the player has created four problems in this order, he or she can create four new problems in the same order Players may use only a pencil and paper to create the problems When minutes is up, the player earns point for each correct problem Player turns over the top card and creates as many problems as he or she can in minutes The player with the most points after three rounds is the winner 110 PartV 12/6/00 9:23 AM Page 111 38 Crossword Puzzle Convert fractions to decimals to solve a crossword puzzle M AT E R I A L S pencil paper calculator Procedure To convert a fraction to a decimal, divide the numerator by the denomi3 nator For example, to convert to a decimal, divide by Place a decimal point and as many 0’s as necessary after the dividend (3) until it is evenly divided 0.6 ) 3.0 – 3.0 These fractions are clues to the crossword puzzle The puzzle can be solved by converting the fractions to decimals 111 PartV 12/6/00 9:23 AM Page 112 Across 10 10 11 75 10 32 207 10 Down 1 8 10 32 Create your own crossword puzzle of fraction-to-decimal problems Use a pencil, paper, and a calculator to make up the clues and the solution =W ? ? = ? ? hat you when you divide the numerator of a fraction by the denominator and the decimal repeats? Watch what happens when you convert to a decimal: ? = ? quarter The dividend (1) will never divide evenly What you do? Write three dots after the quotient, or 0.3333 , to ?9 ?9 indicate that the number continuously repeats itself This type of decimal is called a “repeating decimal.” Another A way to write a repeating decimal is 0.33, with a bar over the last The bar means the repeats 112 7/8 FRACTIONS, DECIMALS, AND MORE 0.3333 3) 1.0000 –9 10 –9 10 –9 10 –9 ? = PartV 12/6/00 9:23 AM 39 Page 113 Challenging Dot-to-Dot Challenge your fractions skills while you solve a dot-to-dot picture puzzle Procedure M AT E R I A L S pencil paper calculator Copy the dot-to-dot puzzle on a piece of paper Convert the fractions to decimals Starting at 0, connect the dots in order from smallest decimal to largest An interesting picture is revealed 113 PartV 12/6/00 9:23 AM Page 114 ?C ? = ? ? ? reate your own fraction dot-to-dot Make a list of fractions Put them in order from smallest to largest Place a series of dots on a piece of paper that = quarte will make a picture when they are connected Label the dots with the fractions so that they can be connected in the correct order 114 FRACTIONS, DECIMALS, AND MORE ?9 ? PartV 12/6/00 9:24 AM Page 115 40 Fraction Words Find fraction words in this word search puzzle M AT E R I A L S Procedure pencil paper Fourteen mathematical terms relating to fractions and at least two fun terms are hidden in this grid The words can be arranged horizontally, vertically, or diagonally, and forward or backward Make a photocopy of the grid Find the words and circle them with your pencil 115 PartV 12/6/00 9:24 AM Page 116 F R A C T I O N S M A S T E R C E R T I F I C AT E Now that you’ve mastered all the fraction facts, problems, and games in this book, you are officially certified as a fractions master! Make a photocopy of this certificate, write your name on the copy, and hang it on the wall Fabulous Fractions 116 fract-BM 12/6/00 9:27 AM Page 117 Index addition, 41–46 converting to equivalent fractions, 57–59 different denominators, 60–62 fractions with common denominators, 41–43 least common multiple, 54–56 mixed numbers with common denominators, 44–46 proper and improper fractions, 63–64 repeating to multiply whole number by mixed number, 79–80 skills practice, 95–97 Addition Chip Trading, 44–46 Add Them Up, 79–80 bingo, 95–97 brain stretchers conversion of decimal to fraction, 106 conversion of one-third to decimal, 112 crazy halves, division of two fractions, 87 division of whole number by fraction, 90 equivalent fractions, 17, 59 four equal squares, 11 four or five mixed numbers addition, 46 fraction representation, 13, 14 fractions between zero and one, 26 lowest–terms fractions, 21 missing-number problems, 66 multiplication with graph paper and colored pencils, 71 non-reducible fractions, 23 reciprocal of mixed number, 83 calculator, 102 Challenging Dot-to-Dot, 113–14 Circles inside Circles, 50–51 Combination Pizza, 41–43 common denominator adding fractions with, 41–43 adding mixed numbers with, 44–46 comparative value fractions, 28 creating equivalent fractions for, 57–59 defined, 28 finding least common, 55 lowest common, 28, 55 in order of fractions, 102 subtracting fractions with, 47–49 subtracting mixed numbers with, 52–53 common factor, finding, 20–21 commutative process, 87 comparative value, 27–28 tips for finding, 28 117 fract-BM 12/6/00 9:27 AM Page 118 Crossword Puzzle, 111–12 Cut Up!, 34–35 Dancing Decimals, 104–6 decimal point, 106 decimals, 102–106, 111–13 converting fraction to, 111–12, 113 converting fraction to with calculator, 102 converting one-third to, 112 converting to fraction, 104–6 fraction denominator as multiple of ten, 106 repeating, 112 denominator adding with different, 60–62 creating equivalent fraction, 59 defined, dividing numerator by, 111 in equivalent fraction, 15 finding least common multiple, 54–56 lowest common, 28, 55 as multiple of ten, 106 non-reducible, 23 reduced to lowest term, 20 switching to find reciprocal, 82 zero as unacceptable, 26, 37 zeros in decimal conversion, 106 See also common denominator diagonal halves, quarters, 10 different denominator, adding with, 60–62 dividend, 86 division, 84–94 of denominator by numerator, 111 118 INDEX factor, 20 fraction by fraction, 85, 91–92 fraction squares, 84–87 mixed number by mixed number, 93–94 as non-commutative, 87 number line used in, 88–89 quotient, 84, 90 skills practice, 95–97 divisor, 86, 90 Domino ESP, 29–30 domino games, 29–30, 63–64, 81–82, 95–97 Domino Match, 81–82 dot-to-dot puzzle picture, 113–14 equal parts halves, 7–8 quarters, 9–11 equivalent fractions, 15–17 for addition with different denominators, 62 converting to, 57–59, 62 defined, 15 matching, 18–19 reducing, 24–26 factor defined, 20 finding common, 21 fantasy fractions, 26 finding missing number, 65–66 Flip It!, 31–33 Fraction Jeopardy!, 107–9 Fraction Match, 18–19 Fraction Roll, 60–61 fract-BM 12/6/00 9:27 AM Page 119 fractions, 1–3 adding, 41–46, 54–56, 63–64, 95–97 comparative value, 27–28 converting decimal to, 104–6 converting to decimals, 102, 111–13 converting whole numbers to, 73, 82, 90 creating, 110 defined, dividing, 84–97 dot-to-dot pictures, 113–14 equivalent, 15–19, 24–26, 57–59, 62 fantasy, 26 half, 3, 7–8 improper, 29–35, 63–64, 78 least common denominator, 55 least common multiple of denominator, 54–56 lowest common denominator, 28, 55 lowest terms, 20–26 multiplying, 69–75, 79–83, 95–97 non-reducible, 23 ordering, 101–3 parts of, proper, 29–30, 63–64 quarter, 3, 9–11 reciprocals, 81–83 reducing, 20–26, 51, 62, 86, 90 representing, 12–14 skills practice, 22–23, 95–97, 107–109, 113–114 squares, 11, 84–87 subtracting, 47–59, 65–66, 95–97 undefined, 37 words for, 3, 115 writing, 2, 12–14 between zero and one, 26 zero in, 36–37 Fraction Sale!, 74–75 Fractions Master Certificate, 116 Fraction Stars, 12–14 Fraction Strips, 15–17 Fraction War!, 27–28 Fraction Words, 115 Graph Paper Multiplication, 69–71 half, ways of making, 7–8 Half Happenings, 7–8 improper fractions, 29–35 adding, 63–64 changing mixed numbers to, 34–35, 78 changing to mixed numbers, 31–33 conversion tip, 33, 35 determining, 29–30 Jeopardy!, 107–9 least common denominator, finding, 55 least common multiple, 54–56 with prime numbers, 56 Least Common Multiple Tic-TacToe, 54–56 Lineup, The, 24–26 INDEX 119 fract-BM 12/6/00 9:27 AM Page 120 lowest common denominator, 28 tip for finding, 55 lowest-terms, reducing to, 20-26 adding fractions with different denominators, 62 common denominator, 20–21 equivalent fractions, 24–26 skills practice game, 22–23 tips and tricks, 21 Lowest-Terms Picture, 20–21 Magic Fraction Squares, 84–86 matching equivalent fractions, 18–19 Missing Numbers, 65–66 Mixed-Number Products, 76–78 mixed numbers, 31–35 adding repeatedly to multiply whole number by, 79 adding with common denominator, 44–46 changing fraction to, 31–33 changing to improper fraction, 34–35 defined, 31 dividing by mixed number, 93–94 finding reciprocal of, 83 multiplying two, 76–78 subtracting, 65–66 subtracting with common denominators, 52–53 multiples finding least common, 54–56 with prime numbers, 56 of ten, 106 multiplication, 69–75, 79–83 as commutative, 87 120 INDEX to create equivalent fraction, 59 to find reciprocal, 81–83 fraction by whole number, 72–73 with graph paper and colored pencils, 69–71 repeated adding as, 79–80 skills practice, 95–97 two mixed numbers, 76–78 of two prime numbers to find least common multiple, 56 whole numbers and fractions, 74–75 non-commutative process, 87 non-reducible fractions, 23 number prime, 56 reciprocal of, 81–83 See also fractions; mixed numbers; whole numbers Number Line Division, 88–89 numerator in creating equivalent fraction, 59 defined, dividing by denominator, 111 in equivalent fraction, 15 for finding larger fractions, 28 non-reducible, 23 one as, 26 reduced to lowest term, 20 switching to find reciprocal, 82, 83 zero as, 37 one as any number times its reciprocal, 83 fract-BM 12/6/00 9:27 AM Page 121 fractions between zero and, 26 value in proper and improper fractions, 29 one-fourth, ways of making, 9–11 one-half, ways of making, 7–8 one-sixth, one-third, converting to decimal, 112 Order, 101–3 ordering fractions, 101–3 Pattern Pictures, 47–49 prime numbers, defined, 56 Problem Generator, 110 product See multiplication proper fractions, 29–30 adding, 63–64 quarter, ways of making, 9–11 Quick Change, 57–59 quotient, 84, 90 reciprocal defined, 81 of fraction, 81–83 of fraction divisor, 86, 90 Reduce It!, 22–23 reduction, 20–26 in adding fractions with different denominators, 62 common factor, 20–21 in dividing fraction by fraction, 86 in dividing whole number by fraction, 90 of equivalent fractions, 24–26 non-reducible, 23 in subtracting fraction from whole number, 51 representing fractions, 12–14 Sandwich Quarters, 9–11 Shoe Box Fractions, 91–93 Splitting the Deck, 72–73 squares dividing fraction by fraction into, 84–86 four equal, 11 subtraction, 47–59 converting to equivalent fractions, 57–59 fraction from whole number, 50–51 fractions with common denominators, 47–49 least common multiple, 54–56 mixed numbers, 65–66 mixed numbers with common denominators, 52–53 skills practice, 95–97 Subtraction Chip Trading, 52–53 ten, multiples of, 106 three-fourths, Three-in-a-Row Bingo, 95–97 tic-tac-toe, 54–56 tips and tricks adding fractions with different denominators, 62 adding mixed numbers with common denominator, 45 calculator to convert fraction to decimal, 102 INDEX 121 fract-BM 12/6/00 9:27 AM Page 122 tips and tricks (continued) dividing fraction by fraction, 85 dividing fraction by whole number, 93 dividing mixed number by mixed number, 94 finding comparative value, 28 finding least common multiple of two numbers, 46, 55 finding least common multiple with both prime numbers, 56 finding reciprocal of fraction, 82 finding reciprocal of whole number, 82 multiplying fraction by whole number, 73 multiplying mixed number by mixed number, 78 reducing to lowest terms, 21 subtracting fraction from whole number, 51 subtracting mixed numbers with common denominators, 53 zero as numerator or denominator, 37 undefined fraction, defined, 37 Up-and-Down Dominoes, 63–64 122 INDEX whole numbers converting to fraction, 73, 82, 90 dividing by fraction, 90 dividing fraction by, 92–94 factor of, 20 finding reciprocal of, 82 multiply fraction by, 72–75 number line to divide by fraction, 88–89 repeated addition to multiply by mixed number, 79 subtracting fraction from, 50–51 zero as, 26 words, for fractions, 3, 115 x (for missing number), 65 Zero, 36–37 in decimal-to-fraction conversion, 104, 106 finding fractions between one and, 26 in numerator or denominator, 36–37 [...]... T S O N FRACTIONS I n this section, you’ll learn everything you need to know about fractions, starting with what fractions look like and how to write them You’ll learn about halves, quarters, and more complex fractions You’ll also be able to reduce fractions, change a mixed number to an improper fraction and vice versa, and create equivalent fractions While you’re learning, you’ll be making sandwiches,... two rolls of assorted fruit-flavored candies Open one roll and count the total number of pieces in the roll Make a list of the number of pieces = 9 there are of each color quarte 1 What fraction of the total number of pieces is red? 2 What fraction of the total number of pieces is green? Yellow? A ?9 3 Open the other roll of candies and list the number of pieces of each color Does each roll of candies... picture using lowest-terms fractions and their equivalents Color all the lowest-terms frac- quart tions one color Color all their equivalent fractions a second color T o determine whether a fraction is in lowest terms, find the factors of the numerator and the denominator and see if they have any common factors other than 1 For example, take the fraction 2/5 The factors of 2 are 1 and 2, and the factors... face cards (kings, queens, and jacks) from a deck of playing cards Game Rules 1 Shuffle the cards and deal each player 20 cards 2 Each player places his or her cards facedown in a pile on the table 3 Player 1 turns over the top card on his or her pile and places it faceup in the center of the table 4 Player 2 turns over the top card in his or her pile and places it beside player 1’s card These two cards... Sandwich Quarters How many ways can you cut a sandwich into four equal parts? M AT E R I A L S 8 slices of bread sandwich fixings knife 4 plates Procedure 1 Make four sandwiches out of whatever you like Put each sandwich on a plate 2 Cut one of the sandwiches into four equal pieces by cutting it in half and in half again It should look like the one shown Each square piece is one-fourth of the whole... whole sandwich 9 PartII 12/6/00 9:25 AM Page 10 3 Can you think of another way to cut a sandwich into four equal pieces? Cut the second sandwich along both diagonals as shown Each triangular piece is one-fourth of the whole sandwich 4 Can you think of another way to cut a sandwich into four equal pieces? Cut a third sandwich into four long strips as shown Each long piece is one-fourth of the whole sandwich... equivalent, the player places the cards facedown in their original position and player 2 takes a turn 3 Play continues until all the pairs of equivalent fractions are found The player with the most pairs of equivalent fractions wins the game F R A C T I O N M AT C H 19 PartII 12/6/00 9:25 AM 6 Page 20 Lowest-Terms Picture A “factor” is any whole number that divides another number evenly For example,... or her old one and forms a new fraction 8 Play continues until one player has won all the cards SUPER REDUCE IT! Instead of shouting “Reduce it!” players reduce the fraction to lowest terms and shout the reduced fraction For example, if the 4 of hearts and the 8 of spades are played to make the fraction 4/8, the player would slap the cards and shout, “One-half!” ?T ? = 9 ? ? ? hink of some fractions. .. 12/12 Game Rules 1 Shuffle and deal the cards Each player places his or her cards facedown in a pile in front of himself or herself 2 Both players turn over their top card at the same time The player whose card has the larger fraction wins both cards Players can use a pencil and paper, and the Tips and Tricks box, to determine which fraction is larger 27 PartII 12/6/00 9:25 AM Page 28 3 If the players turn... decks contains only E cards Game Rules 1 Shuffle each deck of cards and place them all facedown on the table in two rows, one of cards with E’s and one of cards without 2 Player 1 turns over two cards, one from the E row and one from the other row If the two cards show equivalent fractions, the player picks them up, puts them faceup in a pile on the table, and takes another turn If the fractions on the