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Explore Rational Numbers Name: _________________________________ Date: __________________________________ California Common Core Standard The Number System 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. A rational number is a number that can be written as a fraction , where n and d are integers and d ≠ 0. You can use long division to write the decimal expansion of a rational number. For example, to write the decimal expansion of 7 , divide 7 by 11 as shown at right. 11 Notice that the division process could be continued forever. 7 You write = 0.63636363… to show that the decimal 11 continues, repeating 63 over and over. ACTIVITY 1 1. In table groups, use long division to find the decimal expansion of each rational number listed below. NO CALCULATORS ALLOWED! 3 1 1 2 3 5 7 3 5 6 7 7 8 12 16 20 2. Write each rational number and its decimal expansion on an index card as shown. 3. Sort the index cards into three stacks: Decimals that terminate, or end (for example, ½ = 0.5, which is a terminating decimal) Decimals that repeat (for example, 7/11 = 0.6363636363…, which is a repeating decimal) Decimals that neither terminate nor repeat 4. Which rational numbers are in each stack? List the rational number in the appropriate column below. Terminating Repeating Neither THINK AND DISCUSS 1. When you use long division to write a decimal expansion, how can you tell that the decimal terminates? How can you tell that the decimal repeats? 2. What do you notice about your stacks of index cards? Are there any cards in the stack for decimals that neither terminate nor repeat? 3. What can you say about the decimal expansion of any rational number? TRY THIS Determine whether the decimal expansion of each rational number terminates or repeats. 4 7 ______________ 7 8 _______________ 2 9 _______________ 4 5 _______________ ACTIVITY 2 1. The period of a repeating decimal is the shortest set of digits that repeats over and over. For example, in the decimal 0.636363…, the shortest set of digits that repeats over and over is 63, so the period is 63. 2. On each of your index cards that contain a repeating decimal, circle the period as shown at right. 3. Write the length of the period on the card as shown at right. THINK AND DISCUSS 1. What is the longest period that you found? Which of the rational numbers have the longest period? 2. For a rational number , do you think that the period can ever have more than d digits? Explain. 3 3. A student writes the decimal expansion of and claims that the period has 12 7 digits. Explain why the student is incorrect. What mistake do you think the student made? TRY THIS Find the period of the decimal expansion of each rational number. 5 9 9 11 4 7 1 13 Find the length of the period of the decimal expansion of each rational number. 8 11 7 9 6 7 1 99