Student Probe Terminating and Repeating Decimals At a Glance Write as a decimal. Answer: 0.625 Lesson Description In this lesson students use long division to convert fractions into repeating or terminating decimals. Calculator use in encouraged. Rationale All rational numbers, including mixed numbers, can be converted into repeating or terminating decimals by dividing the numerator by the denominator. This division process can be done with or without the use of a calculator. It is important for students to realize that there situations where a fractional representation is more efficient and there are times when a decimal representation is more efficient. Students need to be flexible in their thinking to decide which is more appropriate in a given situation. Additionally, some fractions and their decimal equivalents are used so often that it is efficient for students to know them without having to convert from one form to another. Some examples of these include: Wit
h repeated use students will become familiar with these conversions, so it is not necessary to require memorization. Preparation What: Converting fractions into repeating or terminating decimals Common Core Standard: CC.7.NS.2d 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. Matched Arkansas Standard: AR.8.NO.1.4 (NO.1.8.4) Rational Numbers: Understand and justify classifications of numbers in the real number system Mathematical Practices: Use appropriate tools strategically. Attend to precision. Look for and express regularity in repeated reasoning. Who: Students who cannot convert a fraction into a repeating or terminating decimal Grade Level: 8 Prerequisite Vocabulary: numerator, denominator, rational number, equivalent fractions Prerequisite Skills: division of whole numbers Delivery Format: Individual, small group Lesson Length: 30 minutes Materials, Resources, Technology: calculator Student Worksheets: none Allow students to have access to calculators. Simple, four-­‐function calculators are appropriate for this lesson. Lesson The teacher says or does… 1. What are some fractions that are equivalent to ? 2. How can we write as a Expect students to say or do… If students do not, then the teacher says or does… Answers will vary, but listen What about ? Is it for . equivalent to ? 0.5 Prompt students, if necessary. Division. 0.5 What operation does the fraction bar indicate? Monitor students. decimal? 3. So we are saying or , . 4. What does the “fraction bar” mean? That means is the same as . Put in your calculator and tell me what answer it returns. That is another way to figure out . 5. What does mean? Use your calculator to find the decimal equivalent of . . … 6. Find the decimal equivalent of . What operation is the fraction bar telling us to do? Monitor students. Monitor students. Prompt if necessary. The teacher says or does… 7. Notice that the 3’s keep going! This is called a repeating decimal. Can we write all of the 3’s? Mathematicians have a way to show that the 3’s repeat forever. It is written like this: . The bar is placed over the part of the decimal that repeats to show that it repeats those digits forever. 8. There are two kinds of rational numbers: the ones Expect students to say or do… If students do not, then the teacher says or does… No. that repeat such as , and the ones that end, or terminate, like and . 9. Find the decimal equivalent of . Is it a repeating decimal or a terminating decimal? How do you know? 10. Find the decimal equivalent of . Is it a repeating decimal or a terminating decimal? How do you know? 11. Let’s find the decimal equivalent of . We know means . Repeating, because the 3 repeats forever. means Terminating, because it ends. we have 2 wholes, so we only need to consider the . Is it terminating or repeating? Terminating . . The teacher says or does… Expect students to say or do… If students do not, then the teacher says or does… 12. Repeat the steps above with a variety of fractions and mixed numbers. Include both terminating and repeating decimals. Teacher Notes 1. This lesson relies on the interpretation of fractions as indicted division, . This can seem unusual to students who have been thinking of fractions as parts of wholes. 2. Make sure that students understand the notation for repeating decimals and use it correctly. 3. When converting mixed numbers to decimals, students should understand that only the fractional part of the number is converted. The whole number remains unchanged. 4. Notice that any fraction with a denominator of 7 will have a repeating block of 6 digits. Students will sometimes fail to notice all of the digits. Variations Students may be interested in the types of rational numbers that have repeating or terminating decimal forms. Let students create a table of the first 20 unit fractions, , and their decimal equivalents (using a calculator). Students should then investigate the patterns that emerge. (Teacher Note: denominators of fractions which terminate have prime factors of only 2 and/or 5.) Formative Assessment Write Answer: as a decimal. References
Mathematics Preparation for Algebra. (n.d.). Retrieved August 10, 2010, from Doing What Works: http://dww.ed.gov/practice/?T_ID=20&P_ID=48 Van de Walle, J. A., & Lovin, L. H. (2006). Teaching Student-­‐Centered Mathematics Grades 5-­‐8 Volume 3. Boston, MA: Pearson Education, Inc.