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Lesson 3: Existence and Uniqueness of Square Roots and Cube Roots NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 3 8•7 Lesson 3: Existence and Uniqueness of Square Roots and Cube Roots Student Outcomes . Students know that the positive square root and the cube root exist for all positive numbers and both a square root of a number and a cube root of a number are unique. Students solve simple equations that require them to find the square root or cube root of a number. Lesson Notes This lesson has two options for showing the existence and uniqueness of positive square roots and cube roots. Each option has an Opening Exercise and a Discussion that follows. The first option has students explore facts about numbers on a number line, leading to an understanding of the trichotomy law, followed by a discussion of how the law applies to squares of numbers, which should give students a better understanding of what square roots and cube roots are and how they are unique. The second option explores numbers and their squares via a Find the Rule exercise, followed by a discussion that explores how square roots and cube roots are unique. The first option includes a discussion of the basic inequality property, a property referred to in subsequent lessons. The basic inequality property states that if 푥, 푦, 푤, and 푧 are positive numbers so that 푥 < 푦 and 푤 < 푧, then 푥푤 < 푦푧. Further, if 푥 = 푤 and 푦 = 푧, when 푥 < 푦, then 푥2 < 푦2. Once the first or second option is completed, the lesson continues with a discussion of how to solve equations using square roots. Throughout this and subsequent lessons, we ask students to find only the positive values of 푥 that satisfy an equation containing squared values of 푥. The reason is that students will not learn how to solve these problems completely until Algebra I. Consider the following equation and solution below: 4푥2 + 12푥 + 9 = 49 (2푥 + 3)2 = 49 √(2푥 + 3)2 = √49 2푥 + 3 = ±7 2푥 + 3 = 7 or 2푥 + 3 = −7 푥 = 2 푥 = −5 At this point, students have not learned how to factor quadratics and will solve all equations using the square root symbol, which means students are only responsible for finding the positive solution(s) to an equation. Classwork Opening (4 minutes): Option 1 Ask students the following to prepare for the discussion that follows. Considering only the positive integers, if 푥2 = 4, what must 푥 be to make the statement a true number sentence? Could 푥 be any other number? 푥 is the number 2 and cannot be any other number. Lesson 3: Existence and Uniqueness of Square Roots and Cube Roots 36 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 3 8•7 . Can you find a pair of positive numbers 푐 and 푑 so that 푐 < 푑 and 푐2 < 푑2? Yes! Answers will vary. Can you find a pair of positive numbers 푐 and 푑 so that 푐 < 푑 and 푐2 = 푑2? No . Can you find a pair of positive numbers 푐 and 푑 so that 푐 < 푑 and 푐2 > 푑2? No Discussion (9 minutes): Option 1 (An alternative discussion is provided below.) Once this discussion is complete, continue with the discussion that begins on page 40. We will soon be solving equations that include roots. For this reason, we want to be sure that the answer we get when we simplify is correct. Specifically, we Scaffolding: want to be sure that we can get an answer, that it exists, that the answer we get The number line can help is correct, and that it is unique to the given situation. students make sense of the . To this end, existence requires us to show that given a positive number 푏 and a trichotomy law. Give students positive integer 푛, there is one and only one positive number 푐, so that when two numbers, and ask them 푐푛 = 푏, we say “푐 is the positive 푛th root of 푏.” When 푛 = 2, we say “푐 is the which of the three possibilities positive square root of 푏” and when 푛 = 3, we say “푐 is the positive cube root of for those numbers is true. In 푏.” Uniqueness requires us to show that given two positive numbers 푐 and 푑 each case, only one of three is and 푛 = 2: If 푐2 = 푑2, then 푐 = 푑. This statement implies uniqueness because true for any pair of numbers. both 푐 and 푑 are the positive square root of 푏, that is, 푐2 = 푏 and 푑2 = 푏; since For example, given the 푐2 = 푑2, then 푐 = 푑. Similarly, when 푛 = 3, if 푐3 = 푑3, then 푐 = 푑. The numbers 푐 = 2 and 푑 = 3, reasoning is the same since both 푐 and 푑 are the positive cube root of 푏, that is, which of the following is true: 푐3 = 푏 and 푑3 = 푏; since 푐3 = 푑3, then 푐 = 푑. Showing uniqueness also shows 푐 = 푑, 푐 < 푑, or 푐 > 푑? existence, so we focus on proving the uniqueness of square and cube roots. MP.3 . To show that 푐 = 푑, we use the trichotomy law. The trichotomy law states that given two numbers 푐 and 푑, one and only one of the following three possibilities is true. (i) 푐 = 푑 (ii) 푐 < 푑 (iii) 푐 > 푑 We show 푐 = 푑 by showing that 푐 < 푑 and 푐 > 푑 cannot be true. If 푥, 푦, 푤, and 푧 are positive numbers so that 푥 < 푦 and 푤 < 푧, is it true that 푥푤 < 푦푧? Explain. Yes, it is true that 푥푤 < 푦푧. Since all of the numbers are positive and both 푥 and 푤 are less than 푦 and 푧, respectively, then their product must also be less. For example, since 3 < 4 and 5 < 6, then 3 × 5 < 4 × 6. This basic inequality property can also be explained in terms of areas of a rectangle. The picture on the right clearly shows that when 푥 < 푦 and 푤 < 푧, then 푥푤 < 푦푧. Lesson 3: Existence and Uniqueness of Square Roots and Cube Roots 37 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 3 8•7 . We use this fact to show that 푐 < 푑 and 푐 > 푑 cannot be true. We begin with 푐푛 < 푑푛 when 푛 = 2. By the basic inequality, 푐2 < 푑2. Now, we look at the case where 푛 = 3. We use the fact that 푐2 < 푑2 to show 푐3 < 푑3. What can we do to show 푐3 < 푑3? We can multiply 푐2 by 푐 and 푑2 by 푑. The basic inequality guarantees that since 푐 < 푑 and 푐2 < 푑2, that 푐2 × 푐 < 푑2 × 푑, which is the same as 푐3 < 푑3. Using 푐3 < 푑3, how can we show 푐4 < 푑4? 3 3 3 3 MP.3 We can multiply 푐 by 푐 and 푑 by 푑. The basic inequality guarantees that since 푐 < 푑 and 푐 < 푑 , 푐3 × 푐 < 푑3 × 푑, which is the same as 푐4 < 푑4. We can use the same reasoning for any positive integer 푛. We can use similar reasoning to show that if 푐 > 푑, then 푐푛 > 푑푛 for any positive integer 푛. Recall that we are trying to show that if 푐푛 = 푑푛, then 푐 = 푑 for 푛 = 2 or 푛 = 3. If we assume that 푐 < 푑, then we know that 푐푛 < 푑푛, which contradicts our hypothesis of 푐푛 = 푑푛. By the same reasoning, if 푐 > 푑, then 푐푛 > 푑푛, which is also a contradiction of the hypothesis. By the trichotomy law, the only possibility left is that 푐 = 푑. Therefore, we have shown that the square root or cube root of a number is unique and also exists. Opening (6 minutes): Option 2 Begin by having students find the rule given numbers in two columns. The goal is for students to see the relationship between the square of a number and its square root and the cube of a number and its cube root. Students have to figure out the rule, find the missing values in the columns, and then explain their reasoning. Provide time for students to do this independently. If necessary, allow students to work in pairs. The numbers in each column are related. Your goal is to determine how they are related, determine which numbers belong in the blank parts of the columns, and write an explanation for how you know the numbers belong there. Opening Find the Rule Part 1 ퟏ ퟏ ퟐ ퟒ ퟑ ퟗ MP.8 ퟗ ퟖퟏ ퟏퟏ ퟏퟐퟏ ퟏퟓ ퟐퟐퟓ ퟕ ퟒퟗ ퟏퟎ ퟏퟎퟎ ퟏퟐ ퟏퟒퟒ ퟏퟑ ퟏퟔퟗ 풎 풎ퟐ √풏 풏 Lesson 3: Existence and Uniqueness of Square Roots and Cube Roots 38 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 3 8•7 Note: Students will not know how to write the cube root of a number using the proper notation, but this activity will be a good way to launch into the discussion below.
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