Repeat or Not? That Is the Question!

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Repeat or Not? That Is the Question! Exact Decimal Representations of Fractions Learning Goals In this lesson, you will: Use decimals and fractions to evaluate arithmetic expressions. Convert fractions to decimals. Represent fractions as repeating decimals. Key Terms decimals non- decimals repeating decimals non-repeating decimals bar notation Essential Ideas Decimals are classified as, non-, repeating, and non-repeating. Bar notation is used when writing repeating decimals. Common Core State Standards for Mathematics Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. Apply properties of operations as strategies to add and subtract rational numbers. Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats. 5.5 Exact Decimal Representations of Fractions 28A

Repeat or Not? That Is the Question! Exact Decimal Representations of Fractions Learning Goals In this lesson, you will: Use decimals and fractions to evaluate arithmetic expressions. Convert fractions to decimals. Represent fractions as repeating decimals. Key Terms decimals non- decimals repeating decimals non-repeating decimals bar notation Sometimes calculating an exact answer is very important. For example, making sure that all the parts of an airplane fit exactly is very important to keep the plane in the air. Can you think of other examples where very exact answers are necessary? 5.5 Exact Decimal Representations of Fractions 28

Problem 1 Students compare the answers to a problem using mixed numbers and using the decimal equivalents of the mixed numbers. They will conclude the answers are the same when the decimal terminates. Grouping Ask a student to read the introduction aloud. Discuss the context as a class. Have students complete Questions 1 through 5 with a partner. Then share the responses as a class. Share Phase, Questions 1 through What least common denominator should Jayme use? Does using Jayme s method result in an exact answer? Explain. How do you determine the decimal equivalents for each fraction? Problem 1 Not More Homework! Jayme was complaining to her brother about having to do homework problems with fractions like this: 2 1 2 1 ( 2 4 ) 1 5 2 5 5? Jayme said, I have to find the least common denominator, convert the fractions to equivalent fractions with the least common denominator, and then calculate the answer! Her brother said, Whoa! Why don t you just use decimals? 1. Calculate the answer using Jayme s method. 2 1 2 1 ( 2 4 ) 1 5 2 5 5 2 _ 10 20 1 ( 2 _ 15 20 ) 1 5 _ 8 20 5 4 _ 20 2. Convert each mixed number to a decimal and calculate the sum. 2 1 2 1 ( 2 4 ) 1 5 2 5 5 2.5 1 ( 2.5 ) 1 5.4 5 4.15 5 4 15 100 5 4 _ 20. In this case, which method do you think works best? Answers will vary, but most students will like using the decimals. Jayme said: That s okay for that problem, but what about this next one? 5 1 1 ( 24 1 6 ) 1 ( 22 1 2 ) 5 284 Chapter 5 Multiplication and Division with Rational Numbers

Share Phase, Questions 4 and 5 Does using Jayme s brother s method result in an exact answer? Explain. What is the decimal equivalent of 21 1? Is this an exact answer? Why or why not? What do you consider to be an exact answer? What do you consider to be an answer that is not exact? 4. Calculate the answer using Jayme s method. 5 1 1 ( 24 1 6 ) 1 ( 22 1 2 ) 5 5 2 6 1 ( 24 1 6 ) 1 ( 22 6 ) 5 5 2 6 1 ( 26 4 6 ) 5 21 2 6 5 21 1 5. Will Jayme s brother s method work for the second problem? Why or why not? Answers will vary. Most students should understand that for thirds and sixths the decimal representations are decimals that do not end. Problem 2 Students convert fractions to decimals. All of the decimals are classified as or non- decimals and repeating or non-repeating decimals. Bar notation is introduced to express repeating decimals. Fractions and decimals that either terminate or repeat are referred to as rational numbers. Problem 2 Analyzing Decimals 1. Convert each fraction to a decimal. a. _ 11 25 b. 1 6 The decimal equivalent is 0.44. The decimal equivalent is 0.1666... repeating: 0.1 6 c. _ 2 50 d. _ 15 64 The decimal equivalent is 0.54. The decimal equivalent is 0.245. Grouping Have students complete Question 1 with a partner. Then share the responses as a class. 5.5 Exact Decimal Representations of Fractions 285

Note Reinforce the importance of not dividing by zero. Share Phase, Question 1 How do you convert a fraction to a decimal? What operation is used to convert a fraction to a decimal? Does the decimal end or does it seem to go on forever? Can you tell before you divide if the decimal will end or go on forever? Explain. e. 9 g. f. _ 5 11 The decimal equivalent is 0. The decimal equivalent is 0.4545 repeating: 0. repeating: 0. 45 22 h. 5 8 The decimal equivalent is The decimal equivalent is 0.625. 0.181818 repeating: 0. _ 18 i. j. _ 9 60 The decimal equivalent is The decimal equivalent is 0.65. 0.4285142851 repeating: 0. _ 42851 Grouping Ask a student to read the information aloud. Discuss the classifications of decimals, and bar notations used with repeating decimals as a class. Discuss Phase, Decimal Classification Can a decimal be both and repeating? Can a decimal be both non- and non-repeating? How many decimal places are needed when expressing a repeating decimal? Can bar notation be used to express all repeating decimals? Decimals can be classified in four different ways:, non-, repeating, or non-repeating. A decimal has a finite number of digits, meaning that the decimal will end, or terminate. A non- decimal is a decimal that continues without end. A repeating decimal is a decimal in which a digit, or a group of digits, repeat(s) without end. A non-repeating decimal neither terminates nor repeats. Bar notation is used for repeating decimals. Consider the example shown. The sequence 14285 repeats. The numbers that lie underneath the bar are those numbers that repeat. _ 1 5 0.1428514285... 5 0. 14285 The bar is called a vinculum. How do you know where to place the bar when expressing a repeating decimal? Is 0. a repeating decimal? Why or why not? 286 Chapter 5 Multiplication and Division with Rational Numbers

Grouping Have students complete Questions 2 through 4 with a partner. Then share the responses as a class. 2. Classify each decimal in Question 1, parts (a) through ( j) as, non-, repeating, or non-repeating. If the decimal repeats, rewrite it using bar notation.. Can all fractions be represented as either or repeating decimals? Write some examples to explain your answer. No. Some decimals do not repeat and do not terminate. Examples may vary. 4. Complete the graphic organizer. Describe each decimal in words. Show examples. Be prepared to share your solutions and methods. 5.5 Exact Decimal Representations of Fractions 28

Non Terminating Terminating 1 = 0.5 1 = 0. 1 = 0.25 p =.141592654... = 0. = 2.64551... 2 4 10 _ 5 = 0.15625 2 9 = 0. 81 11 Decimals 1 = 0. 2 = 0. 6 _ 8 = 0. 2 11 6 = 0.85142 p is a well known non repeating decimal. ou will learn more when you study circles later in this course. 4 = 2.64551... = 0.85 8 Repeating 288 288 Chapter 5 B Chapter 5 Multiplication and Division with Rational Numbers Multiplication and Division with Rational Numbers Non Repeating 1 = 0.25 p =.141592654...

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