Expressing Rational Numbers with Decimal Expansions LAUNCH (7 MIN) Before How can you use the data in the table to figure out which nights would cause an argument? During Why can t you evenly split the last dollar on Wednesday? After How can the manager or owner of the lasagna shack avoid these arguments? KEY CONCEPT (6 MIN) Emphasize that all integers are rational numbers (and terminating decimals) because they can be written with a denominator of 1 or as a decimal that terminates in 0s. Clarify, too, that every fraction is a rational number. Lisa Says Use the Lisa Says button to explain to students the connection between the terms ratio and rational. Distinguish between the everyday meaning of rational and its mathematical definition, which is derived from the term ratio. Why is using a bar to show repeated digits a good idea? [Sample answer: It saves space and makes the digits that repeat easy to spot quickly.] PART 1 (8 MIN) Why can you write a fraction as an equivalent decimal? How do you find that decimal? How do you keep dividing when the dividend runs out of digits? PART 2 (7 MIN) Lisa Says (Screen 1) Use the Lisa Says button to explain that terminating and repeating decimals are mutually exclusive. Emphasize that a terminating decimal has a finite number of nonzero digits to the right of the decimal point: at some point the remainder is 0. A repeating decimal, on the other hand, never terminates it has digits that repeat and a remainder that is never zero. Does 1 written as a decimal terminate or repeat? Explain. 18 How can you more precisely describe the relationship between each unequal pair? PART 3 (7 MIN) What is a span, and why might someone wish to know which span is longer? When you divide 10 by 33, the second decimal place is a 0. Is this a terminating decimal? Does your solution mean the Williamsburg Bridge is more impressive than the Brooklyn Bridge? Explain your answer. CLOSE AND CHECK (5 MIN) Why would you want to have multiple ways to write the same rational number? Can you write pi as a terminating or repeating decimal? Explain your answer.
Expressing Rational Numbers with Decimal Expansions LESSON OBJECTIVE Express rational numbers using decimal expansions that terminate in 0s or eventually repeat. FOCUS QUESTION What does being able to express numbers in equivalent forms allow you to do? MATH BACKGROUND In past courses, students worked with natural numbers, which are also called the counting numbers (1, 2, 3, ). They learned that the natural numbers were part of a larger set of numbers called integers (, 3, 2, 1, 0, 1, 2, 3, ). Students also learned to compare, order, and operate with fractions and decimals, which led to the use of an even larger set of numbers that includes integers, fractions, and many decimals called the rational numbers. Rational numbers can be compared, ordered, and operated on using the same algorithms as those for integers. In this lesson, students will learn that rational numbers can always be written as a fraction (a ratio) but may be either terminating and repeating decimals. The basis for rewriting numbers (such as using long division to convert a fraction to a decimal) is that the same number can be represented in multiple equivalent forms, which allows you to write several different rational numbers in the same form in order to compare and/or order them. In the future, students will explore irrational numbers and approximate them to various degrees. They will compare and order irrational and rational numbers and locate them on a number line. In future courses of study, students will use algebra and arithmetic to operate on irrational numbers and solve equations involving irrational numbers. They will also learn when it is appropriate to leave an answer as an irrational number. LAUNCH (7 MIN) Objective: Describe decimals that do and do not terminate. This problem uses a real-world situation how to evenly divide tips among workers in a restaurant to help students recognize that division can result in either decimals that terminate or decimals that repeat. Questions for Understanding Before How can you use the data in the table to figure out which nights would cause an argument? [Sample answer: Divide the last dollar by the number of workers sharing it to see how much each gets. There will be no argument if the money can be divided evenly, with each worker getting the same amount.] During Why can t you evenly split the last dollar on Wednesday? [Sample answer: There are 3 workers, and they cannot split the last penny.] After How can the manager or owner of the lasagna shack avoid these arguments? [Sample answers: The manager can chip in a few extra dollars so that everyone gets the same amount of money. The manager can make sure there are always 2, 4, or 5 workers each night.]
Solution Notes Students might suggest alternative solutions, such as flipping a coin for the extra few cents or taking turns getting the leftover money. Students may write fractions in the Even Split column. Encourage them to use decimals, particularly a format in cents. Connect Your Learning Move to the Connect Your Learning screen. In the Launch, students learned that some integers do not divide evenly. Discuss whether it is better to express these situations as fractions or decimals, especially in the context of the Launch. Use the Focus Question to show that you need a way to express all kinds of decimals, including ones that repeat, to use a form that is most appropriate for the situation. KEY CONCEPT (6 MIN) Teaching Tips for the Key Concept Point out to students that this definition of a rational number is a more sophisticated version of the definition they may have been given in earlier grades. You can have students click on the radio buttons to reveal each type of decimal. Emphasize that all integers are rational numbers (and terminating decimals) because they can be written with a denominator of 1 or as a decimal that terminates in 0s. Clarify, too, that every fraction is a rational number. When you help students distinguish between terminating and repeating decimals, you can start a debate about whether a terminating decimal can be considered a repeating decimal, for which 0 is the repeating digit. Emphasize the examples of repeating decimals that show some have more than one digit that repeats and some do not have a bar over every decimal place (because not every decimal place repeats). Lisa Says Use the Lisa Says button to explain to students the connection between the terms ratio and rational. Distinguish between the everyday meaning of rational and its mathematical definition, which is derived from the term ratio. Why is using a bar to show repeated digits a good idea? [Sample answer: It saves space and makes the digits that repeat easy to spot quickly.] PART 1 (8 MIN) Objective: Express rational numbers using decimal expansions that terminate in 0s. Students are asked to convert a fraction to a decimal. They use long division until the decimal terminates, instead of rounding or truncating, to prepare students for distinguishing terminating and repeating decimals. Questions for Understanding Why can you write a fraction as an equivalent decimal? How do you find that decimal? [Sample answer: Every rational number can be expressed as either a fraction or a decimal. You divide the numerator by the denominator.] How do you keep dividing when the dividend runs out of digits? [Sample answer: Add as many zeros after the decimal point as you need until the remainder is zero. In this case, you need to add 4.]
Solution Notes Some students may use number sense to determine the answer using mental math. One particular way to find the decimal without long division is to start with 1 8 0.125. So 1 0.0625, and 3 is three times as much. Show students that memorizing the 16 16 decimal forms of some basic fractions can help you solve problems more quickly. Differentiated Instruction For struggling students: As needed, help students distinguish between the numerator and denominator of the fraction in order to correctly set up and use the long division algorithm. For advanced students: Have students explain, for example, why they need to insert 4 zeros after the decimal point of the dividend. Error Prevention Students may set up the division incorrectly (i.e., dividing the denominator by the numerator), stop dividing before the remainder equals zero, or place the decimal point in the quotient incorrectly. Got It Notes You might want to have students perform long division before revealing the answer choices because choices A and B are so similar. Alternatively, you can show the answer choices to help students practice estimation in a test-taking situation. If students understand that 5 32 is less than 1, they can eliminate C and D. Students who answer incorrectly may have made the following errors: If students choose A, they are not dividing fully they should have written additional zeros in the dividend. Students who select C may be simply replacing the fraction bar with a decimal point. Students who choose D are dividing the denominator by the numerator because they are used to dividing the greater number by the lesser one. PART 2 (7 MIN) Objective: Express rational numbers using decimal expansions that eventually repeat. Students continue to write fractions as decimals in order to compare rational numbers. They apply the bar notation for repeating decimals and use equality and inequality symbols to describe the relationship between each pair of rational numbers. This problem prepares students to work with different kinds of rational numbers in real-world situations. Instructional Design You can call on students to write each fraction as a decimal using long division. Have them drag a tile to show the either equal or unequal relationship between the two rational numbers. When all parts are complete, click the Check button. Any incorrect answers will snap back to the tile banks. Give students an opportunity to find their mistakes and place the correct tiles.
Questions for Understanding During the Intro Lisa Says (Screen 1) Use the Lisa Says button to explain that terminating and repeating decimals are mutually exclusive. Emphasize that a terminating decimal has a finite number of nonzero digits to the right of the decimal point: at some point the remainder is 0. A repeating decimal, on the other hand, never terminates it has digits that repeat and a remainder that is never zero. Does 1 written as a decimal terminate or repeat? Explain. [Sample answer: It 18 repeats; the first decimal place is a 0, and every decimal place after that is a 5.] How can you more precisely describe the relationship between each unequal pair? [Sample answer: You can use either a greater than or less than symbol.] Solution Notes Another way to compare the two numbers is to first express each decimal as a fraction and then compare the two fractions. Let students try to convert a few repeating decimals, to see that this strategy is more cumbersome and therefore not sensible. Error Prevention Watch for students who divide the denominator by the numerator when writing a fraction as a decimal, or who use the incorrect inequality symbol to show the comparison. Remind students that they need to make the numerator the dividend and cannot look at which number is greater. Got It Notes Students may realize that they can simplify the fraction before dividing. If students know the decimal expansion for 1, they do not have to divide. 3 Students who answer incorrectly may have made the following errors: If students choose A, they may not have remembered to write a bar over the repeating digit. Students who round the exact answer or prematurely stop dividing may choose A or B. If students choose D, they are dividing the denominator by the numerator. PART 3 (7 MIN) Objective: Compare rational numbers by using decimal expansions that terminate in 0s or eventually repeat. Students now compare terminating and repeating decimals in a real-world context. They choose which form, fraction or decimal, is most useful to compare and write both rational numbers in that form. Questions for Understanding What is a span, and why might someone wish to know which span is longer? [Sample answer: A bridge s span is not its length, but rather the distance between supports. Because longer span lengths mean fewer supports, engineers acknowledge long span lengths as engineering achievements.]
When you divide 10 by 33, the second decimal place is a 0. Is this a terminating decimal? [Sample answer: No; you only care if the remainder is 0, not the decimal places of the quotient. It is a repeating decimal; the digits 3 and 0 repeat.] Does your solution mean the Williamsburg Bridge is more impressive than the Brooklyn Bridge? Explain your answer. [Sample answer: No; there are many other characteristics that make a bridge impressive, such as its total length or width, its architecture, or the view as you drive over it.] Solution Notes Alternatively, students can compare these two rational numbers by expressing each as a fraction with the same denominator and then comparing numerators. One way to do so is to first write 0.3 as the equivalent fraction 3 and then write both fractions using 10 the lowest common denominator of 330. Error Prevention Caution students that when looking for a terminating or repeating decimal, they should not stop dividing just because there is a zero in the decimal expansion. Emphasize that the remainder needs to be zero, not the decimal place. Got It Notes Like the Example, this problem can be solved using either decimal or fractional form. Let students choose their preferred method and then discuss how both lead to the same answer. Regardless of the method students choose, make sure they are able to express 5 6 as a repeating decimal. Students can solve problems like this one mentally and quickly if they memorize the decimal equivalents of common fractions like halves, thirds, fourths, and tenths. You might want to discuss with students why someone might prefer to park further away. They should not assume that people will automatically park at the closer lot. CLOSE AND CHECK (5 MIN) Focus Question Sample Answer Being able to express numbers in equivalent forms allows you to write numbers in the form you find easiest to use in a given situation. Being able to rewrite a fraction as a decimal can help you to identify repeating and terminating decimals. Focus Question Notes Good answers should mention that writing rational numbers in different forms is useful for making comparisons. Students may relate this skill to their previous work with ratios in multiple forms. Make sure students know that division is still the process to convert a fraction to a decimal. Unlike previous topics in which students rounded or truncated decimals, emphasize that you do not stop dividing until you get a remainder of 0 or reveal a pattern that repeats.
Essential Question Connection Students have learned more about rational numbers, particularly writing them in different forms. They are more prepared now to learn about the first part of the Essential Question: What other types of numbers [besides rational numbers] are there? Use the questions below to get students ready to learn about irrational numbers. Why would you want to have multiple ways to write the same rational number? [Sample answer: Each form of a ratio is useful depending on how you want to represent the number or what you are going to use it for.] Can you write pi as a terminating or repeating decimal? Explain your answer. [Sample answer: The digits of pi are infinite, so it is not a terminating decimal, but the digits do not repeat, so it is not a repeating decimal either.]