Today, you can take your coins to the grocery store to have them counted by a

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Stewart Shields
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1 What s My Cut? Equivalent Fractions Learning Goals In this lesson, you will: Determine equal portions of a whole. Determine equivalent fractions. Calculate equivalent fractions using a form of 1. Key Terms simplest form Multiplicative Identity Property Simplify fractions. Order fractions. Today, you can take your coins to the grocery store to have them counted by a machine. You can exchange your coins for a cash voucher, gift cards, or you can even donate your coins to charity. Careful, though. Unless you re donating your money, these machines usually take a cut of about 10 cents for every one of your dollars that it counts. One of the largest transactions recorded for one of these machines was in San Dimas, California. A customer turned in over $8000 worth of coins! What was the machine s cut? 3.5 Equivalent Fractions 151

2 Problem 1 Pizza Cuts 1. Coach Finley buys pizzas to share with everyone who participated in the volleyball intramural program. She buys 21 pizzas to share among the 28 students who are sitting at tables of four. a. How many pizzas should each table receive if she wants each table to receive the same amount of pizza? Explain your reasoning. b. Coach Finley suggests dividing each pizza into fourths. Draw a diagram that represents the problem. Then, explain her reasoning. c. What fractional part of a pizza will each student receive? 152 Chapter 3 Fractions

3 d. Mario suggests dividing the pizzas in half. Will Mario s method work? Why or why not? If Mario s suggestion is not possible, explain what he can do to make his suggestion work. Draw a diagram that supports your explanation. e. Sydney tries to divide his group s pizzas into thirds. Will Sydney s method work? Why or why not? If Sydney s method is not possible, explain what he can do to make his method work. Draw a diagram that supports your explanation. f. Natalie tries to divide her group s pizzas into eighths. Will Natalie s method work? Why or why not? If it is not possible, explain what she can do to make her method work. Draw a diagram that supports your explanation. 3.5 Equivalent Fractions 153

$g. Name the student(s) whose method did not require your suggestion. Then, name the fractional part of the pizzas each student at the table will receive. h.$

4 g. Name the student(s) whose method did not require your suggestion. Then, name the fractional part of the pizzas each student at the table will receive. h. Juanita claims that dividing the pizzas into any number of equal-sized pieces will work to divide the pizzas equally among the 28 students. Is she correct? Explain your reasoning and offer a solution if Juanita s method does not work. 2. Casey and Jamal are talking about pizza parties they had in each of their classes. Casey said, In my class, four people shared three pizzas. Jamal said, In my class, three people shared two pizzas. a. Which student s classmates each got more pizza? Use a drawing or diagram to explain your reasoning. Assume the pizzas are the same size. All this talk of pizza is making me hungry! 154 Chapter 3 Fractions

5 Problem 2 Equivalent Fractions Throughout this chapter, you have encountered equivalent fractions. Recall that equivalent fractions are fractions that represent the same part-to-whole relationship. When determining equivalent fractions, you must multiply the numerator and the denominator of a fraction by the same number. This process is the same as multiplying the given fraction by a To change a fraction to an equivalent fraction with the same numerator and fraction with a larger numerator denominator, such as 3 3. and denominator, you multiply Recall that any fraction whose the fraction by a form of 1. numerator and denominator are the For example, same number is equivalent to Multiplying any number by 1 does not change that number. 1. Complete each number sentence with the correct fraction to make it true. Explain your reasoning. Use your fraction strips if you need help determining the missing fraction to make equivalent fractions. a. 5 3 ) 6 ( ) 5 15 b. 3 3 ) ( ) 20 c. 5 3 ) 8 ( ) 5 15 d. 9 ( ) 3 ) 5 ) 27 ( ) Equivalent Fractions 155

6 2. You just multiplied each fraction by the same number. What number was it? The Multiplicative Identity Property states: a 1 5 a, where a is a nonzero number. 3. Complete each equation to make the fractions equivalent. Explain your reasoning. Use your fraction strips if you need help determining equivalent fractions. a. 1 5 ) 4 16 b. 2 5 ) 3 6 c. 7 5 ) d. ( ) e ( ) 4. Write the first 10 equivalent fractions of each using what you know about equivalent fractions. The first example is done for you. a. 1, 2, 3 4 8, 4 12, 5 16, 6 20, 7 24, 8 28, 9 32, b. 3 5, c. 2 3, d. 5 8, e. How did you determine the order to list your equivalent fractions? 156 Chapter 3 Fractions

7 To change a fraction to an equivalent fraction with a smaller numerator and denominator, you must divide the numerator and denominator by a form of 1. For example, Whenever you determine an equivalent fraction whose numerator and denominator are smaller than the original fraction s numerator and denominator, the new fraction is simpler. When a fraction cannot be simplified further, the fraction is in simplest form, or completely simplified. Simplest form is a way of writing a fraction so that the numerator and denominator have no common factors other than 1. You can say a fraction is in a simpler form in several ways: 1 is simpler than is a simplified form of in simplest form is Sometimes it is read: 1 is the simplified form of is in lowest, or simplest, terms Determine if each fraction is simplified completely. If the fraction is not simplified completely, write the fraction in simplest form. a b c Equivalent Fractions 157

8 6. Circle the fractions that are simplified completely. How do you know? Recall that the factors of a number are those numbers that divide into the number with no remainder. 7. How can you use factors of a number to simplify fractions? 8. Write each fraction in simplest form. a. 6 b c. 8 d e. 18 f g. Explain how you simplified each fraction. 158 Chapter 3 Fractions

9 9. Ms. Glick asked her students to simplify 24. Jose, Sara, and Clifton s methods are 36 shown. Analyze each method and solution to determine if it is correct. If one of the methods is incorrect, what would you tell the student to do to correct his or her method? Jose: I simplified by first dividing both the numerator and denominator by 2. I then continued dividing by 2 until the fraction could not be evenly divided by = 24 :_ 2 36 _ : 2 = = 12 :_ 2 18 _ : 2 = 6 9 Sara: I divided both the numerator and denominator by the GCF of 12 and got a quotient of = 24 :_ _ : 12 = 2 3 Clifton: I wrote the prime factorization of the numerator and denominator, and then I divided out the common prime factors = 2 x 2 x 2 x 3 2 x 2 x 3 x 3 = Equivalent Fractions 159

10 10. Simplify each fraction using prime factorization. a. 15 b c. 24 d e. 24 f Simplify each fraction using the GCF. State the GCF used. a. 24 b c. 33 d e. 63 f Chapter 3 Fractions

Operations, vocabulary terms and examples

Operations, vocabulary terms and examples oh my Fraction- a number that names a part of a whole numerator- the part or top number in a fraction denominator-the bottom or the total number in a fraction