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Lesson 8: the Long Division Algorithm NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 8 8•7 Lesson 8: The Long Division Algorithm Student Outcomes . Students explore a variation of the long division algorithm. Students discover that every rational number has a repeating decimal expansion. Lesson Notes In this lesson, students move toward being able to define an irrational number by first noting the decimal structure of rational numbers. Classwork Example 1 (5 minutes) Scaffolding: There is no single long division Example 1 algorithm. The algorithm commonly taught and used in ퟐퟔ Show that the decimal expansion of is ퟔ. ퟓ. the U.S. is rarely used ퟒ elsewhere. Students may come with earlier experiences Use the example with students so they have a model to complete Exercises 1–5. with other division algorithms that make more sense to them. 26 . Show that the decimal expansion of is 6.5. Consider using formative 4 assessment to determine how Students might use the long division algorithm, or they might simply different students approach 26 13 observe = = 6.5. long division. 4 2 . Here is another way to see this: What is the greatest number of groups of 4 that are in 26? MP.3 There are 6 groups of 4 in 26. Is there a remainder? Yes, there are 2 left over. This means we can write 26 as 26 = 6 × 4 + 2. Lesson 8: The Long Division Algorithm 104 This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This work is licensed under a This file derived from G8-M7-TE-1.3.0-10.2015 Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 8 8•7 26 . This means we could also compute as follows: 4 26 6 × 4 + 2 = 4 4 26 6 × 4 2 = + 4 4 4 26 2 MP.3 = 6 + 4 4 26 2 1 = 6 = 6 . 4 4 2 (Some students might note we are simply rewriting the fraction as a mixed number.) 26 . The fraction is equal to the finite decimal 6.5. When the fraction is not equal to a finite decimal, then we 4 need to use the long division algorithm to determine the decimal expansion of the number. Exploratory Challenge/Exercises 1–5 (15 minutes) Students complete Exercises 1–5 independently or in pairs. The discussion that follows is related to the concepts in the exercises. Exploratory Challenge/Exercises 1–5 1. ퟏퟒퟐ a. Use long division to determine the decimal expansion of . ퟐ 71.0 2 142.0 ퟏퟒퟐ b. Fill in the blanks to show another way to determine the decimal expansion of . ퟐ ퟏퟒퟐ = ퟕퟏ × ퟐ + ퟎ ퟏퟒퟐ ퟕퟏ × ퟐ + ퟎ = ퟐ ퟐ ퟏퟒퟐ ퟕퟏ × ퟐ ퟎ = + ퟐ ퟐ ퟐ ퟏퟒퟐ ퟎ = ퟕퟏ + ퟐ ퟐ ퟏퟒퟐ = ퟕퟏ. ퟎ ퟐ ퟏퟒퟐ c. Does the number have a finite or an infinite decimal expansion? ퟐ ퟏퟒퟐ The decimal expansion of is ퟕퟏ. ퟎ and is finite. ퟐ Lesson 8: The Long Division Algorithm 105 This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This work is licensed under a This file derived from G8-M7-TE-1.3.0-10.2015 Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 8 8•7 2. ퟏퟒퟐ a. Use long division to determine the decimal expansion of . ퟒ 35.5 4 142.0 ퟏퟒퟐ b. Fill in the blanks to show another way to determine the decimal expansion of . ퟒ ퟏퟒퟐ = ퟑퟓ × ퟒ + ퟐ ퟏퟒퟐ ퟑퟓ × ퟒ + ퟐ = ퟒ ퟒ ퟏퟒퟐ ퟑퟓ × ퟒ ퟐ = + ퟒ ퟒ ퟒ ퟏퟒퟐ ퟐ = ퟑퟓ + ퟒ ퟒ ퟏퟒퟐ ퟐ = ퟑퟓ = ퟑퟓ. ퟓ ퟒ ퟒ ퟏퟒퟐ c. Does the number have a finite or an infinite decimal expansion? ퟒ ퟏퟒퟐ The decimal expansion of is ퟑퟓ. ퟓ and is finite. ퟒ 3. ퟏퟒퟐ a. Use long division to determine the decimal expansion of . ퟔ 23.666 6 142.000 12 22 18 40 36 40 36 40 36 4 ퟏퟒퟐ b. Fill in the blanks to show another way to determine the decimal expansion of . ퟔ ퟏퟒퟐ = ퟐퟑ × ퟔ + ퟒ ퟏퟒퟐ ퟐퟑ × ퟔ + ퟒ = ퟔ ퟔ ퟏퟒퟐ ퟐퟑ × ퟔ ퟒ = + ퟔ ퟔ ퟔ ퟏퟒퟐ ퟒ = ퟐퟑ + ퟔ ퟔ ퟏퟒퟐ ퟒ = ퟐퟑ = ퟐퟑ. ퟔퟔퟔ… ퟔ ퟔ Lesson 8: The Long Division Algorithm 106 This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This work is licensed under a This file derived from G8-M7-TE-1.3.0-10.2015 Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 8 8•7 ퟏퟒퟐ c. Does the number have a finite or an infinite decimal expansion? ퟔ ퟏퟒퟐ The decimal expansion of is ퟐퟑ. ퟔퟔퟔ… and is infinite. ퟔ 4. ퟏퟒퟐ a. Use long division to determine the decimal expansion of . ퟏퟏ 12.90909 11 142.00000 11 32 22 100 99 10 00 100 99 10 00 100 99 10 ퟏퟒퟐ b. Fill in the blanks to show another way to determine the decimal expansion of . ퟏퟏ ퟏퟒퟐ = ퟏퟐ × ퟏퟏ + ퟏퟎ ퟏퟒퟐ ퟏퟐ × ퟏퟏ + ퟏퟎ = ퟏퟏ ퟏퟏ ퟏퟒퟐ ퟏퟐ × ퟏퟏ ퟏퟎ = + ퟏퟏ ퟏퟏ ퟏퟏ ퟏퟒퟐ ퟏퟎ = ퟏퟐ + ퟏퟏ ퟏퟏ ퟏퟒퟐ ퟏퟎ = ퟏퟐ = ퟏퟐ. ퟗퟎퟗퟎퟗ… ퟏퟏ ퟏퟏ ퟏퟒퟐ c. Does the number have a finite or an infinite decimal expansion? ퟏퟏ ퟏퟒퟐ The decimal expansion of is ퟏퟐ. ퟗퟎퟗퟎퟗ… and is infinite. ퟏퟏ 5. In general, which fractions produce infinite decimal expansions? We discovered in Lesson 6 that fractions equivalent to ones with denominators that are a power of ퟏퟎ are precisely the fractions with finite decimal expansions. These fractions, when written in simplified form, have denominators with factors composed of ퟐ‘s and ퟓ‘s. Thus any fraction, in simplified form, whose denominator contains a factor different from ퟐ or ퟓ must yield an infinite decimal expansion. Lesson 8: The Long Division Algorithm 107 This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This work is licensed under a This file derived from G8-M7-TE-1.3.0-10.2015 Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 8 8•7 Discussion (10 minutes) 142 . What is the decimal expansion of ? 2 If students respond 71, ask them what decimal digits they could include without changing the value of the number. 142 The fraction is equal to the decimal 71.00000…. 2 . Did you need to use the long division algorithm to determine your answer? Why or why not? No, the long division algorithm was not necessary because there was a whole number of 2’s in 142. 142 . What is the decimal expansion of ? 4 142 The fraction is equal to the decimal 35.5. 4 . What decimal digits could we include to the right of the “. 5” in 35.5 without changing the value of the number? We could write the decimal as 35.500000…. Did you need to use the long division algorithm to determine your answer? Why or why not? 142 2 2 No, the long division algorithm was not necessary because = 35 + , and is a finite decimal. We 4 4 4 2 can write as 0.5. 4 142 . What is the decimal expansion of ? 6 142 The fraction is equal to the decimal 23.66666…. 6 . Did you need to use the long division algorithm to determine your answer? Why or why not? 142 2 2 Yes, the long division algorithm was necessary because = 23 + , and is not a finite decimal. 6 3 3 2 Note: Some students may have recognized the fraction as 0.6666… and not used the long division 3 algorithm to determine the decimal expansion. How did you know when you could stop dividing? I knew to stop dividing because the remainder kept repeating. Specifically, when I used the long division algorithm, the number 40 kept appearing, and there are 6 groups of 6 in 40, leaving 4 as a remainder each time, which became 40 when I brought down another 0. 142 . We represent the decimal expansion of as 23. 6̅, where the line above the 6 is the repeating block; that is, 6 the digit 6 repeats as we saw in the long division algorithm. 142 . What is the decimal expansion of ? 11 142 The fraction is equal to the decimal 12.90909090…. 11 . Did you need to use the long division algorithm to determine your answer? Why or why not? 142 10 10 Yes, the long division algorithm was necessary because = 12 + , and is not a finite decimal. 11 11 11 Lesson 8: The Long Division Algorithm 108 This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This work is licensed under a This file derived from G8-M7-TE-1.3.0-10.2015 Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 8 8•7 . How did you know when you could stop dividing? I knew to stop dividing because the remainder kept repeating. Specifically, when I used the long division algorithm, I kept getting the number 10, which is not divisible by 11, so I had to bring down another 0, making the number 100. This kept happening, so I knew to stop once I noticed the work I was doing was the same. Which block of digits kept repeating? The block of digits that kept repeating was 90. 142 . How do we represent the decimal expansion of ? 11 142 The decimal expansion of is 12.
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