SAMPLE SAMPLE SAMPLE. Advanced Fractions. I d rather have a fraction Multiplying, Cancelling, Dividing, Inverting
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1 Advanced Fractions I d rather have a fraction Multiplying, Cancelling, Dividing, Inverting Copyright 2010 John McCormick All Rights Reserved
2 Advanced Fractions I d Rather Have a Fraction Multiplying, Cancelling, Dividing, Inverting Word Corner Publishing Burwood Drive Lake Mathews, CA Advanced Fractions: Carlynn McCormick Cheat Sheet: John McCormick Illustrations: Microsoft Clipart Copyright 2010 All Rights Reserved. No part of this work may be copied or duplicated in any form without the express permission of the publisher.
3 Table of Contents To the Student Multiplying, Cancelling Dividing Inverting A Note About Retaining the Ability to do Fractions. 15 Fraction Cheat Sheet
4 To the Student Purpose: This study guide has the purpose of helping student review advanced fractions: how to cancel before multiplying fractions and how to invert when dividing fractions to turn them into a multiplication problem. This guide has attempted to make the review of advanced fractions as easy as eating fruit tarts. Pre-requisite: You will want to complete the beginning fractions book: I d rather have a fraction, thank you before starting the advanced fraction book. How to do the Study Guide: Do the steps in order and track your progress by putting your initials and the date next to each step after completing it. Some steps are done with your teacher or study partner. If you do not have a teacher or study partner, you may do the study guide by yourself. Important Note: To be competent in any subject you must know the vocabulary of that subject. Be sure you know the meaning of all the words used in this study guide.
5 Cancelling Before Multiplying Fractions How to Take a Shortcut Personally, I d rather have shortcake than have a shortcut READ: When it comes to multiplying fractions there is a shortcut that should be taken. Before we learn the steps of the shortcut, let s review some definitions: Lowest Terms: When a fraction has been put in its simplest form possible. Example: 4/6 is put in lowest terms as 2/3 (also called simplify or reduce). Factor: A number that is being multiplied in a multiplication problem. Example: In the problem 3 x 4 = 12, one factor is 3 and another factor is 4. Common Factor: A number that is common to two different numbers when those numbers are found by multiplying two numbers one of which is the
6 common factor. Example: 2 is a common factor of the numbers 4 and 6 since 4 has the factors 2 x 2 and 6 has the factors 2 x 3. Numerator: A fraction is made up of two numbers, one number on top and another number on the bottom. The top number is the numerator. The numerator tells how many of the parts are being counted. Denominator: The bottom number is the denominator. The denominator tells how many equal parts are in the whole. Example: 1/5 would require 5 parts to equal a whole. Cancel: When you are multiplying fractions you should see if you can divide the numerator of one fraction and the denominator of the other fraction by a common number (one of the factors) which will give an answer that is already in its simplest terms. If you don t cancel before you multiply, then your answer will need to be reduced to its simplest term. EXERCISE: Demonstrate the following to a study partner or your teacher: lowest terms, factor, common factor, numerator, denominator and cancel. OPTIONAL: Make the definitions lowest terms, factor, common factor, numerator, denominator and cancel in clay.
7 READ: Now let s look at the steps we take to get fractions in their lowest terms. This is a shortcut that will make multiplying fractions quick and easy. We will use 3/8 x 5/12 for the demonstration: Step 1: Find a common factor of a numerator and the opposite denominator. 3 x 1 = 3 (numerator factors) 3 x 4 = 12 (denominator factors) 3 is a common factor Step 2: Divide the numerator by that common factor. This is now the new numerator of that fraction. 3 3 = 1 Step 3: Divide the denominator by that common factor. This is now the new denominator of that fraction = 4 Step 4: If able, do steps 1-3 with the other numerator and opposite denominator. You can now do the multiplication problem: 1/8 x 5/4 (newly canceled fractions) Step 5: Multiply the 2 numerators to get the product s numerator. 1 x 5 = 5 (numerator) Step 6: Multiply the 2 denominators to get the product s denominator. 8 x 4 = 32 (denominator) This gives you the answer: 5/32 Note: If you don t cancel the fractions before multiplying, the product of 3/8 x 5/12 would be 15/96. You would then need to reduce it to its lowest term of 5/32.
8 EXERCISE: Work out how to cancel the fractions 5/12 x 4/15 using the six steps above. You might see instantly that 5 and 15 have a common factor of 5 and that 4 and 12 have a common factor of 4. But go ahead and do the six steps, just to see how they work. READ: Cancelling is like working a puzzle. It is actually fun to do. You may not need to do all six steps when problems are very easy. In the problem 6/8 x 2/9 you might instantly know that that 6 and 9 have a common factor of 3; and that 2 and 8 have a common factor of 2. You would have the newly calculated fraction of 2/4 x 1/3 = 2/12 which can be reduced to 1/6. But even before you multiply the newly calculated fraction, you might see that 2/4 can be reduced to 1/2 so you would multiply 1/2 x 1/3 to get 1/6.
9 EXERCISE: For the following nine problems, cancel the fractions wherever possible before multiplying them. Check your answers 1 in the footnote below. (1) 3/4 x 2/9 = (2) 5/8 x 3/20 = (3) 1/2 x 8/5 = (4) 2/5 x 5/4 = (5) 9/2 x 8/21 = (6) 6/2 x 8/12 = (7) 18/3 x 15/27 = (8) 2/5 x 5/2 = (9) 2/6 x 6/12 = I do not wish to reduce bonbons into fractions 1 Answers: (1) 1/6 (2) 3/32 (3) 4/5 (4) 1/2 (5) 1 5/7 (6) 4/2 = 2 (7) 10/3 = 3 1/3 (8) 1/1 = 1 (9) 1/6
10 EXERCISE: If you find cancelling and multiplying fun, then you will enjoy doing the following exercise as well. Solve these problems by cancelling the fractions wherever possible before multiplying them. Check your answers 2 in the footnote below. (1) 11/25 x 5/7 = (2) 23/52 x 43/ 46 = (3) 20/23 x 7/2 = (4) 11/3 x 12/10 = (5) 3/4 x 2/3 = (6) 150/144 x 12/15 = When it comes to bonbons, I prefer improper fractions 2 Answers: (1) 11/35 (2) 43/104 (3) 70/23 = 3 1/23 (4) 44/10 = 4 2/5 (5) 1/2 (6) 10/12 = 5/6
11 Inverting Before Dividing Fractions READ: When it comes to dividing fractions there are some tricks to learn. Before we start, let s review a definition and learn one: Divide: To separate something into equal-sized parts. Example: You have a pizza and six hungry people you want to divide the pizza into sixths so each person gets an equal share of the pizza. Invert: To invert means to reverse the denominator and numerator of a fraction. Example: If you invert 2/6, you get 6/2. If you invert 3/4, you get 4/3. EXERCISE: Show your teacher or study partner how easy it is to invert the fraction 1/2. Now have your teacher or study partner invert 5/8 and show you. READ: Here is another definition to learn: Reciprocal: When two fractions are related to each other, such as 2/3 and 3/2, they are said to be reciprocal. When two reciprocal fractions are multiplied together the answer is always 1. Example: 2/3 times 3/2 equals 6/6 or 1. EXERCISE: Draw out, make in clay or demonstrate the following to your teacher or study partner: If you multiply 1/2 by its reciprocal 2/1 you get 2/2, which equals 1.
12 EXERCISE: Explain to your teacher or study partner why multiplying the fraction 5/8 by its reciprocal will give you 1. READ: Here are the steps that tell you how to divide fractions: 3/7 1/3 = Step 1: Invert the fraction divisor (the divisor is the number you are dividing by) to make a multiplication problem. 1/3 = 3/1 (invert divisor) 3/7 x 3/1 (division problem converted to multiplication problem) Step 2: Multiply the numerators (top numbers): 3 x 3 = 9 Step 3: Multiply the denominators (bottom numbers): 7 x 1 = 7 3/7 x 3/1 = 9/7 Step 4: As needed, reduce the fraction to the lowest terms. If it is an improper fraction change it to a mixed number: 9/7 = 1 2/7 EXERCISE: Draw out, make in clay or demonstrate the four steps to your teacher or study partner using the problem: 5/8 2/3 = (by inverting the divisor and multiplying, you will get the answer 15/16).
13 READ: How do you divide if you have a mixed number such as 1 5/8 2/3? You simply change the mixed number into and improper fraction (8/8 + 5/8 = 13/8) and divide: 13/8 2/3; you get your answer by using the four steps above (which include inverting and then multiplying). EXERCISE: Dividing fractions is similar to working a crossword puzzle when you know the guidelines and can figure out the answers it is fun to do. On the next page there are eight problems to use the four steps on. Remember if you start out with a mixed number such as 4 9/18, you must change it to an improper fraction: 81/18. (Here is the solution for getting the numerator: 4 x 18 = = 81).
14 You can also reduce the fraction 81/18 see problem 6 below. Check your answers 3 in the footnote below. If you get an improper fraction for an answer, change it to a mixed number. (1) 1/8 1/3 = (2) 3/4 1/3 = (3) 3/4 5/7 = (4) 3/7 1/2 = (5) 7/4 21/16 = (6) 4 9/18 9/8 = (Hint: 4 9/18 = 4 ½ = 9/2) (7) 3 4/5 2 29/31 = (Hint: 3 4/5 = 19/5 and 2 29/31 = 91/31) (8) 15/7 2/4 = 3 Answers: (1) 3/8 (2) 9/4 = 2 ¼ (3) 21/20 = 1 1/20 (4) 6/7 (5) 4/3 = 1 1/3 (6) 4/1 = 4 (7) 589/455 = 1 134/455 (8) 60/40 = 4 4/14 = 4 2/7
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