Building Equivalent Fractions, the Least Common Denominator, and Ordering Fractions

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$PRE-ACTIVITY PREPARATION Building Equivalent Fractions, the Least Common Denominator, and Ordering Fractions You have learned that a fraction might be written in an$ $In this section, you will explore the techniques of how to write an equivalent form of the fraction in another manner to build up the fraction with a larger$ $The skill of rewriting fractions in higher terms is valuable when comparing, ordering, adding, and subtracting fractions.$ $LEARNING OBJECTIVES Use a methodology to determine the Least Common Multiple (LCM) of a set of numbers and the Least Common Denominator (LCD) of a set of fractions.$

1 Section. PRE-ACTIVITY PREPARATION Building Equivalent Fractions, the Least Common Denominator, and Ordering Fractions You have learned that a fraction might be written in an equivalent form by reducing it to its lowest terms. In this section, you will explore the techniques of how to write an equivalent form of the fraction in another manner to build up the fraction with a larger numerator and a larger denominator, yet still retain its value. The skill of rewriting fractions in higher terms is valuable when comparing, ordering, adding, and subtracting fractions. Additionally, looking at different configurations for the same whole unit or group will expand your ability to look for patterns within sets of numbers. LEARNING OBJECTIVES Use a methodology to determine the Least Common Multiple (LCM) of a set of numbers and the Least Common Denominator (LCD) of a set of fractions. Build up equivalent fractions. Use the LCD to put a set of fractions in order from smallest to largest or largest to smallest. TERMINOLOGY PREVIOUSLY USED cross-multiply cross-product denominator equivalent fraction factors multiple multiplier numerator prime factorization prime factors primes NEW TERMS TO LEARN building up common denominator common multiple higher terms Least Common Denominator (LCD) Least Common Multiple (LCM) 9

$96 Chapter Fractions BUILDING MATHEMATICAL LANGUAGE Building Equivalent Fractions Recall that when you reduce a fraction, you divide out common factors from the numerator and denominator, the result$ $To build up a fraction, you multiply both numerator and denominator by the same number, resulting in a higher number for both.$

2 96 Chapter Fractions BUILDING MATHEMATICAL LANGUAGE Building Equivalent Fractions Recall that when you reduce a fraction, you divide out common factors from the numerator and denominator, the result being an equivalent fraction in lower terms. Building up an equivalent fraction to higher terms is the opposite process. To build up a fraction, you multiply both numerator and denominator by the same number, resulting in a higher number for both. By the Identity Property of Multiplication (multiplying a number by does not change its value) and the any number fact that the number can take the form of the same number, you can write an infinite number of fractions equivalent to a given fraction. Example :, a fraction whose value is equiv 6 alent to. shaded 6 shaded VISUALIZE The whole rectangle is now broken up into parts, and it takes 6 of them to equal the original out of parts. In fact, you can choose any fraction form of the number to build up an equivalent fraction For the same example,,,, and so on Now suppose that you want to take a fraction and build an equivalent fraction with a specific denominator. As long as the new denominator is a multiple of the original denominator, you can use the following technique.

3 Section. Building Equivalent Fractions, the Least Common Denominator, and Ordering Fractions 9 TECHNIQUE Building Up an Equivalent Fraction with a Specified Denominator Technique Step : Step : Divide the larger denominator by the denominator of the given fraction to determine the multiplier. Multiply the numerator of the given fraction by that same number. MODELS A 4 B 9 Step 4 6 Step 6 0 Step Step 9 9 You can validate that your answer and the original fraction are equivalent by applying the Equality Test for Fractions (comparing the cross-products which should be equal) C 4 8 Before Step, write the whole number as a fraction: 4 Step 8 8 Step In this case, to validate you know that, an improper fraction, is equal to 4. 8

4 98 Chapter Fractions Common Multiples and the Least Common Multiple A common multiple of two or more numbers is a multiple of each of them. That is, each of the numbers will divide evenly into their common multiple. For example, 90 is a common multiple of the numbers, 6, and 9 because it is a multiple of (8 90), of 6 ( 6 90), and of 9 (0 9 90). Another common multiple of, 6, and 9 is 80, because, 6, and 9 each divide evenly into 80. In fact, there are infinitely more common multiples of, 6, and 9, among them 0, 60, 40, and so on. The smallest multiple that two or more numbers have in common is called their Least Common Multiple (LCM). For the previous example, the Least Common Multiple (LCM) of the numbers, 6, and 9 is 90, the smallest number that all three numbers can divide into evenly. Determining the Least Common Multiple For two or more given numbers, how can you determine their LCM Example: Find the LCM of 9,, and. You could list the multiples of each number and pick out the smallest they have in common The multiples of 9,, and (which you would have to compute) are: 9, 8,, 6, 4, 4, 6,, 8, 90, 99, 08,, 6,, 44,, 6,, 80,, 4, 6, 48, 60,, 84, 96, 08, 0,, 44, 6, 68, 80,, 0, 4, 60,, 90, 0, 0,, 0, 6, 80, Their LCM is 80. However, as the example demonstrates, this approach to determining an LCM is inefficient and prone to computational errors when finding the multiples. There are more efficient methods for determining the LCM. When the Least Common Multiple is not readily apparent to you, use either of the following two methodologies to determine the LCM.

5 Section. Building Equivalent Fractions, the Least Common Denominator, and Ordering Fractions 99 METHODOLOGIES Determining the Least Common Multiple (LCM) of a Set of Numbers by Prime Factorization Find the LCM of each set of numbers: Example : 9,, and Example :, 4, and Try It! Steps in the Methodology Example Example Step Prime factor each number. Determine the prime factorization of each number. Special Case: Special Case: Largest number is divisible by every other number (see page 00, Model ) All are prime numbers (see page 00, Model ) Readily apparent that the Special numbers share no common Case: factors (see page 0,Model ) Step Identify primes. Identify all the primes that are factors in the prime factorizations.,, Step Choose necessary factors. Step 4 Multiply the factors. Use each prime as a factor of the LCM the greatest number of times it appears in any one of the prime factorizations. Why do you do this Multiply these prime factors. The result is the LCM of the original numbers. 9 THINK two s and one are needed. is a factor of 9 twice and of and once each need two s. LCM 80

6 00 Chapter Fractions Why do you do Step There must be the correct number of each of the prime factors in the LCM to make it divisible by each of the numbers in the set. In the worked example, two s are needed as the factors of the LCM (80). If you would choose, for example, only one as a factor of the LCM, the number (which is ) would not divide into it. At the same time, there are no extra factors in the Least Common Multiple only those necessary to make it divisible by all three numbers in the set. The following illustrates how all necessary factors are included in the LCM of Example. } } } 9 MODELS Model Special Case: Largest Number is Divisible by Every Other Number Determine the LCM of,, 6, and. If, by inspection and or/application of the Divisibility Tests, you can readily determine that the largest number is divisible by all other numbers in the set, it is the LCM (no need to do Steps -4)., the largest number, is divisible by, by, by 6, and, of course, by itself. Therefore, the Least Common Multiple (LCM) is. Model Special Case: All are Prime Numbers Determine the LCM of,, and. If the numbers are all distinct prime numbers, there are no common factors. The LCM is the product of the prime numbers.,, and are all prime. The LCM 8.

7 Section. Building Equivalent Fractions, the Least Common Denominator, and Ordering Fractions 0 Model Special Case: Readily Apparent that the Numbers Share No Common Factors Determine the LCM of, 8, and 9. If, by inspection and or/application of the Divisibility Tests, you can readily determine that the numbers share no common factors, the LCM is simply the product of the numbers. THINK is prime } 8 is only divisible by no common factors; the LCM is only divisible by Determining the Least Common Multiple (LCM) of a Set of Numbers by Pulling Out Primes This methodology presents another efficient yet more compact process for determining the LCM when it is not readily apparent. It condenses the first methodology by pulling out only the necessary prime factors of the numbers from smallest to largest. It may remind you of the factor ladder process of prime factoring. Find the LCM of each set of numbers: Example : 9,, and Example :, 4, and Try It! Steps in the Methodology Example Example Step Write the numbers. Set up the numbers in a row with enough space below for many divisions. 9 See Special Cases in Models,, (see pages 00 & 0).

8 0 Chapter Fractions Steps in the Methodology Example Example Step Divide by the smallest prime factor. Step Divide the next row by the smallest prime factor. Step 4 Divide until the quotients are all s. Divide by the smallest prime factor of any of the numbers. If the chosen factor does not divide into a number evenly, bring down that number into the next row, indicating this with an arrow. Look at the numbers in the second row. Divide by the same prime number if it is still a factor of any of the numbers in the row, or by the next higher prime number that is a factor of any of the numbers in the row. Bring down the numbers not divisible by the prime. Continue this process with the third row and so on until you have only s remaining. Divide by 9 THINK 9 is not divisible by. is divisible by. is not divisible by. Divide row by THINK 6 is divisible by. 9 and are not. Divide row by THINK ,, and are all divisible by. is divisible by. and are not. is divisible by. Step Multiply the factors. Collect all of the factors on the outside and multiply. The product is the LCM of the original set of numbers. LCM 80

9 Section. Building Equivalent Fractions, the Least Common Denominator, and Ordering Fractions 0 MODEL Find the LCM of 4, 6, 60, and by pulling out primes. Step THINK Step divides into 4, 6, and 60, but not 8 0 Step THINK 8 0 divides into, 8, and 0, not 6 9 divides into 6 9 divides into, 9,, and divides into divides into and divides into all prime factors found Step 4 LCM The Least Common Denominator In order to compare, add, and subtract fractions, you will most often find it necessary to build them up to equivalent fractions with the same denominator. This is because the rewrite allows you to easily compare, add, or subtract parts (the numerators) when you represent the same number of equal parts in a whole (the denominators) by the same number for each fraction. The first step, therefore, is to determine which denominator is suitable to use for your entire set of fractions. Recall that you can build up a fraction to a specified denominator only if the new denominator is a multiple of the original one. This new common denominator, therefore, must be a multiple of each of the given denominator numbers their common multiple. To avoid working with larger than necessary numbers, it is most efficient to use the smallest, or Least Common Denominator (LCD); that is, the LCM of the denominators. The following methodology presents the steps necessary to rewrite a set of fractions, using their Least Common Denominator.

10 04 Chapter Fractions METHODOLOGY Building Equivalent Fractions for a Given Set of Fractions Determine the LCD and build equivalent fractions for: Example : Example :,, and 4. 6, 9, and. Try It! Steps in the Methodology Example Example Step Find the LCD. Step Identify multipliers. Step Build up fractions with LCD. Step 4 Present the answer. Determine the LCD for the given denominators. Identify the multipliers for the numerators and denominators of each fraction by dividing the LCD by each denominator. Shortcut: Using prime factors of the LCD to determine the multiplier (see pages 0 & 06, Models,, & ) Build each fraction to have the LCD as its new denominator. Use the multipliers determined in Step and apply the Identity Property of Multiplication for the building up process. Present your answer. LCD 0 0 ) 0 0 ) ) 0 0 multiplier for multiplier for multiplier for 4 4, 0 8, 0 0

11 Section. Building Equivalent Fractions, the Least Common Denominator, and Ordering Fractions 0 Steps in the Methodology Example Example Step Validate your answer. You can validate each equivalent fraction by applying the Equality Test for Fractions (cross-multiplying) MODELS Model Shortcut: Using the Prime Factors of the LCD to Determine the Multipliers Rewrite the equivalent fractions, using the LCD, for Step 4 0 LCD 0 4, 9 0, and 4. Step for 4: 0 4 for 0: 0 0 for : ) ) ) 0 Shortcut (optional): Instead of doing the divisions, use the prime factors of the LCD to determine the multipliers. 0 and 4 so 0 4, the remaining factor 0 and 0 so 0 0, the remaining factor 0 and so 0 6, the product of the remaining factors

12 06 Chapter Fractions Steps & Step Validate: Model Change the following fractions to equivalent fractions, using the LCD: 8,, and 4. Step OR 4 LCD 6 Step Use the shortcut to finding multipliers: for 8: ( ) for : 6 6 ( ) 6 for 4: ( ) 9 Steps & Step Validate: Model Write equivalent fractions, using the LCD of the factions:,, and. Step The denominators,, and are all prime. The LCD Step for : for : Note: shortcut to finding multipliers used for :

13 Section. Building Equivalent Fractions, the Least Common Denominator, and Ordering Fractions 0 Steps & Step Validate: Ordering Fractions The most reliable way to order a set of fractions is to determine the LCD of the fractions, build equivalent fractions using the LCD, and then easily compare the numerators, as in the following methodology. METHODOLOGY Ordering Fractions Put the following sets of fractions in order from smallest to largest: Example : Example :,, and 0,, and 8 8 Try It! Steps in the Methodology Example Example Step Identify the order. Step Find the LCD and multipliers. Identify the order requested smallest to largest, or largest to smallest. Determine the LCD of the fractions and identify the multipliers. smallest to largest LCD 40 Identify the multipliers for : 40 8 for 0: 40 0 for 8: 40 8

14 08 Chapter Fractions Steps in the Methodology Example Example Step Build up fractions. Change the equivalent fractions with the LCD as the new denominator and validate by cross-multiplying Validate: 400 and and and 800 Step 4 Order numerators. Compare the numerators of the new equivalent fractions and rank them according to the order identified in Step. Verfiy the ranking. Why can do you do this smallest to largest Rank Verify: 4 < < 6 Step 4 Present the answer. Present your answer with the original fractions in the proper order., 8, 0 Why can do you do Step 4 Fractions with different numerators and denominators can best be compared when the same common denominator is the basis for comparison. Once a common denominator has been determined as the basis for comparison, you have specified how many parts are in one whole. 4 6 For example, for,, and, the whole consists of 40 equal parts. Therefore, when you look at how many parts out of the whole to consider (the numerators), you can easily determine the smaller the numerator, the smaller the fractional part of the whole that fraction represents.

$the denominator when building up equivalent fractions Change / to fifteenths. Use the Identity Property of Multiplication by multiplying both numerator and denominator by the same factor.$

15 Section. Building Equivalent Fractions, the Least Common Denominator, and Ordering Fractions 09 ADDRESSING COMMON ERRORS Issue Incorrect Process Resolution Correct Process Not changing both the numerator and the denominator when building up equivalent fractions Change / to fifteenths. Use the Identity Property of Multiplication by multiplying both numerator and denominator by the same factor. Validate that the built-up fraction is equivalent to the original fraction. Change / to fifteenths. 9 Validate: Guessing the order of fractions without finding a common denominator Rank from smallest to Compare fractions by Rank from smallest to largest: rewriting in equivalent largest: 4 4,, forms with a common 8 8 denominator. Since < 4 < and < < 8, then Rank 4 < < LCD 990 Answer: 4,, 8 PREPARATION INVENTORY Before proceeding, you should have an understanding of each of the following: the terminology and notation associated with building equivalent fractions and ordering fractions how to apply the Identity Property of Multiplication when building fractions how to use the Methodology for Finding the LCM of a Set of Numbers to determine the LCD the reliable way to order fractions

$Fractions PERFORMANCE CRITERIA Building equivalent fractions to a given denominator$ $correctly built-up equivalent fractions, each with the LCD correct order as specified$ What is the most important difference between a common factor of a set of numbers and

What is the most important difference between a common factor of a set of numbers and

16 Section. ACTIVITY Building Equivalent Fractions, the Least Common Denominator, and Ordering Fractions PERFORMANCE CRITERIA Building equivalent fractions to a given denominator Ordering a set of fractions identification of the Least Common Denominator of the set correctly built-up equivalent fractions, each with the LCD correct order as specified CRITICAL THINKING QUESTIONS. What is the most important difference between a common factor of a set of numbers and a common multiple of the numbers. What are three characteristics of a Least Common Denominator. Even though you can use any common denominator for comparing fractions, what are the advantages of using the lowest common denominator 0

17 Section. Building Equivalent Fractions, the Least Common Denominator, and Ordering Fractions 4. What is the relationship between finding the LCM and finding the LCD. How do you determine what factors to multiply the numerator and denominator by in order to build up an equivalent fraction 6. Why should you use a common denominator to compare two or more fractions. Why must the numerator change when building up a fraction

$up fractions, use effective notation: original denominator multiplier Use the$ $Least Common Denominator to easily compare the size of fractions rather than$ $Use cross-products to validate equivalent fractions.$ $Supply the missing numerator for each pair of equivalent fractions.$

18 Chapter Fractions TIPS FOR SUCCESS original numerator multiplier When building up fractions, use effective notation: original denominator multiplier Use the Least Common Denominator to easily compare the size of fractions rather than trying a visualize and guess approach. Use cross-products to validate equivalent fractions. DEMONSTRATE YOUR UNDERSTANDING. Supply the missing numerator for each pair of equivalent fractions. 4 a) b) c) For each of the following fractions, write three equivalent fractions. a) b) c) a) Determine the LCD of 6 4,, and. b) Write their equivalent fractions and order them from smallest to largest.

19 Section. Building Equivalent Fractions, the Least Common Denominator, and Ordering Fractions 4. a) Determine the LCD of 8, 4 0,, and 0. b) Write their equivalent fractions and order them from largest to smallest.. Use the Methodology for Ordering Fractions to put the fractions,, and in order from smallest to largest. 8

$4 Chapter Fractions TEAM EXERCISES. In the grids below, fill in the correct numbers of rectangles to represent the following fractions. (Hint: use your knowledge of equivalent fractions.$ IDENTIFY AND CORRECT THE ERRORS Identify the error(s) in the following worked solutions. If the worked solution is correct, write Correct in the second column.

IDENTIFY AND CORRECT THE ERRORS Identify the error(s) in the following worked solutions. If the worked solution is correct, write Correct in the second column.

20 4 Chapter Fractions TEAM EXERCISES. In the grids below, fill in the correct numbers of rectangles to represent the following fractions. (Hint: use your knowledge of equivalent fractions.) a) Use a pencil. c) Use a highlighter b) Use a pen. d) Use a different color highlighter.. Find two fractions between and (greater than and less than ). IDENTIFY AND CORRECT THE ERRORS Identify the error(s) in the following worked solutions. If the worked solution is correct, write Correct in the second column. If the worked solution is incorrect, solve the problem correctly in the third column. Worked Solution What is Wrong Here Identify the Errors Correct Process ) Determine the LCD of 44 is a common 8 and. 8 denominator, but not the least (smallest) common denominator of 8 and Answer: LCD 8 9

21 Section. Building Equivalent Fractions, the Least Common Denominator, and Ordering Fractions Worked Solution What is Wrong Here ) Order these fractions from largest to smallest:,,. Identify the Errors Correct Process ) Put these fractions in order from smallest to largest:, 8, 0.

22 6 Chapter Fractions Worked Solution What is Wrong Here 4) Determine the LCD of 4 9, 8, and 4. Identify the Errors Correct Process ADDITIONAL EXERCISES. Supply the missing numerator: a) b) c) Order the fractions from smallest to largest:,, and 8. Order the fractions from largest to smallest: 4. Order the fractions from smallest to largest:,, and 4,,, and 6 4

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