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Lesson 8 NYS COMMON CORE MATHEMATICS CURRICULUM 6β’3 Lesson 8: Ordering Integers and Other Rational Numbers Student Outcomes ο§ Students write, interpret, and explain statements of order for rational numbers in the real world. ο§ Students recognize that if π < π, then βπ > βπ because a number and its opposite are equal distances from zero, and moving along the horizontal number line to the right means the numbers are increasing. Lesson Notes As a continuation of Lesson 7, students order rational numbers from least to greatest and from greatest to least. They relate the orderings to numbersβ locations on the number line. Classwork Opening Exercise (6 minutes) For this warm-up exercise, students work in groups of three or four to order the following rational numbers from least to greatest. Each group of students may be provided with cards to put in order, or the numbers may be displayed on the board where students work at their seats, recording them in the correct order. As an alternative, the numbers may be displayed on an interactive board along with a number line, and students or teams come up to the board and slide the numbers onto the number line into the correct order. Allow time for the class to come to a consensus on the correct order and for students to share with the class their strategies and thought processes. The following are examples of rational numbers to sort and order: 1 1 2 3 5 0, β4, , β , 1, β3 , 2, β4.1, β0.6, 4 Scaffolding: 23 , 6, β1, 4.5, β5, 2.1 5 Adjust the number of cards given to students depending on their ability level. The types of rational numbers given to each group of students may also be differentiated. Solution: 3 5 1 2 1 β5, β4.1, β4, β3 , β1, β0.6, β , 0, , 1, 2, 2.1, 4.5, 4 23 ,6 5 The following line of questioning can be used to elicit student responses: ο§ How did you begin to sort and order the numbers? What was your first step? οΊ ο§ Our group began by separating the numbers into two groups: negative numbers and positive numbers. Zero was not in either group, but we knew it fell in between the negative numbers and positive numbers. What was your next step? What did you do with the two groups of numbers? οΊ We ordered the positive whole numbers and then took the remaining positive numbers and determined which two whole numbers they fell in between. Lesson 8: Ordering Integers and Other Rational Numbers This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from G6-M3-TE-1.3.0-08.2015 71 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 8 NYS COMMON CORE MATHEMATICS CURRICULUM ο§ How did you know where to place οΊ Since 1 4 1 4 and 23 5 6β’3 ? is less than a whole (1) but greater than zero, we knew the rational number was located between 0 and 1. οΊ We know that 23 5 3 5 is the same as 4 , which is more than 4 but less than 5, so we knew the rational number was located between 4 and 5. ο§ How did you order the negative numbers? οΊ ο§ First, we started with the negative integers: β5, β4, and β1. β5 is the least because it is farthest left at 5 units to the left of zero. Then came β4, and then came β1, which is only 1 unit to the left of zero. How did you order the negative non-integers? οΊ 1 2 5 6 , which is to the right of β0.6 (or β ) since β0.5 is closer to zero 10 10 3 than β0.6. Then, we ordered β4.1 and β3 . Both numbers are close to β4, but β4.1 is to the left of 5 3 β4, and β3 is to the right of β4 and to the left of β3. Lastly, we put our ordered group of negative 5 We know β is equivalent to β numbers to the left of zero and our ordered group of positive numbers to the right of zero and ended up with 3 5 1 2 1 β5, β4.1, β4, β3 , β1, β0.6, β , 0, , 1, 2, 2.1, 4.5, 4 23 , 6. 5 Exercise 1 (8 minutes) 1. Students are each given four index cards or small slips of paper. Each student must independently choose four noninteger rational numbers and write each one on a slip of paper. At least two of the numbers must be negative. 2. Students order their rational numbers from least to greatest by sliding their slips of paper into the correct order. The teacher walks around the room to check for understanding and to provide individual assistance. Students may use the number line in their student materials to help determine the order. 3. Once all students have arranged their numbers into the correct order, they shuffle them and then switch with another student. 4. Students arrange the new set of cards they receive into the correct order from least to greatest. 5. The pairs of students who exchanged cards discuss their solutions and come to a consensus. Example 1 (3 minutes): Ordering Rational Numbers from Least to Greatest Example 1: Ordering Rational Numbers from Least to Greatest Sam has $ππ. ππ in the bank. He owes his friend Hank $π. ππ. He owes his sister $π. ππ. Consider the three rational numbers related to this story of Samβs money. Write and order them from least to greatest. βπ. ππ, βπ. ππ, ππ. ππ Lesson 8: Scaffolding: Provide a number line diagram for visual learners to help them determine the numbersβ order. Ordering Integers and Other Rational Numbers This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from G6-M3-TE-1.3.0-08.2015 72 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 8 NYS COMMON CORE MATHEMATICS CURRICULUM ο§ Explain the process you used to determine the order of the numbers. οΊ ο§ 6β’3 There is only one positive number, 10.00, so I know that 10.00 is the greatest. I know 2.25 is farther to the right on the number line than 1.75; therefore, its opposite, β2.25, will be farther to the left than the opposite of 1.75. This means β2.25 is the least, and β1.75 is between β2.25 and 10.00. How would the order change if you were asked to write the numbers from greatest to least? οΊ The order would be reversed. I would list the numbers so that the number that comes first is the one farthest to the right on the number line, and the number that comes last is the one farthest to the left on the number line. The order would be 10.00 (the greatest), followed by β1.75, and then followed by β2.25 (the least). Exercises 2β4 (10 minutes) Allow time for students to share their answers with the class and explain their reasoning. Exercises 2β4 For each problem, list the rational numbers that relate to each situation. Then, order them from least to greatest, and explain how you made your determination. 2. During their most recent visit to the optometrist (eye doctor), Kadijsha and her sister, Beth, had their vision tested. Kadijshaβs vision in her left eye was βπ. ππ, and her vision in her right eye was the opposite number. Bethβs vision was βπ. ππ in her left eye and +π. ππ in her right eye. βπ. ππ, βπ. ππ, π. ππ, π. ππ The opposite of βπ. ππ is π. ππ, and π. ππ is farthest right on the number line, so it is the greatest. βπ. ππ is the same distance from zero but on the other side, so it is the least number. βπ. ππ is to the right of βπ. ππ, so it is greater than βπ. ππ, and π. ππ is to the right of βπ. ππ, so it is greater than βπ. ππ. Finally, π. ππ is the greatest. 3. There are three pieces of mail in Ms. Thomasβs mailbox: a bill from the phone company for $ππ. ππ, a bill from the electric company for $ππ. ππ, and a tax refund check for $ππ. ππ. (A bill is money that you owe, and a tax refund check is money that you receive.) βππ. ππ, βππ. ππ, ππ. ππ The change in Ms. Thomasβs money is represented by βππ. ππ due to the phone bill, and βππ. ππ represents the change in her money due to the electric bill. Since βππ. ππ is farthest to the left on the number line, it is the least. Since βππ. ππ is to the right of βππ. ππ, it comes next. The check she has to deposit for $ππ. ππ can be represented by ππ. ππ, which is to the right of βππ. ππ, and so it is the greatest number. MP.2 4. Monica, Jack, and Destiny measured their arm lengths for an experiment in science class. They compared their arm lengths to a standard length of ππ inches. The listing below shows, in inches, how each studentβs arm length compares to ππ inches. Monica: β Jack: π π π π π Destiny: β π π π π π β , β ,π π π π π π π π I ordered the numbers on a number line, and β was farthest to the left. To the right of that was β . Lastly, π π π π π π π is to the right of β , so π is the greatest. Lesson 8: Ordering Integers and Other Rational Numbers This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from G6-M3-TE-1.3.0-08.2015 73 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 8 NYS COMMON CORE MATHEMATICS CURRICULUM 6β’3 Example 2 (3 minutes): Ordering Rational Numbers from Greatest to Least Example 2: Ordering Rational Numbers from Greatest to Least Jason is entering college and has opened a checking account, which he will use for college expenses. His parents gave him $πππ. ππ to deposit into the account. Jason wrote a check for $ππ. ππ to pay for his calculus book and a check for $ππ. ππ to pay for miscellaneous school supplies. Write the three rational numbers related to the balance in Jasonβs checking account in order from greatest to least. πππ. ππ, βππ. ππ, βππ. ππ ο§ Explain the process you used to determine the order of the numbers. οΊ There was only one positive number, 200.00, so I know that 200.00 is the greatest. I know 85.00 is farther to the right on the number line than 25.34, so its opposite, β85.00, will be farther to the left than the opposite of 25.34. This means β85.00 is the least, and β25.34 would be between β85.00 and 200.00. Exercises 5β6 (6 minutes) Allow time for students to share their answers with the class and explain their reasoning. Exercises 5β6 For each problem, list the rational numbers that relate to each situation in order from greatest to least. Explain how you arrived at the order. 5. The following are the current monthly bills that Mr. McGraw must pay: $πππ. ππ Cable and Internet $ππ. ππ Gas and Electric $ππ. ππ Cell Phone βππ. ππ, βππ. ππ, βπππ. ππ Because Mr. McGraw owes the money, I represented the amount of each bill as a negative number. Ordering them from greatest to least means I have to move from right to left on a number line. Since βππ. ππ is farthest right, it is the greatest. To the left of that is βππ. ππ, and to the left of that is βπππ. ππ, which means βπππ. ππ is the least. 6. π π π π β , π, β , π π π π π , π, β , β π π π I graphed them on the number line. Since I needed to order them from greatest to least, I moved from right to left to π record the order. Farthest to the right is , so that is the greatest value. To the left of that number is π. To the left π π π π π of π is β , and the farthest left is β , so that is the least. Lesson 8: Ordering Integers and Other Rational Numbers This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from G6-M3-TE-1.3.0-08.2015 74 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 8 6β’3 Closing (3 minutes) ο§ ο§ If three numbers are ordered from least to greatest and the order is π, π, π, what would the order be if the same three numbers were arranged in order from greatest to least? How did you determine the new order? οΊ π, π, π οΊ This is the correct order because it has to be exactly the opposite order since we are now moving right to left on the number line, when originally we moved left to right. How does graphing numbers on a number line help us determine the order when arranging the numbers from greatest to least or least to greatest? οΊ Using a number line helps us order numbers because when numbers are placed on a number line, they are placed in order. Lesson Summary When we order rational numbers, their opposites are in the opposite order. For example, if π is greater than π, βπ is less than βπ. Exit Ticket (6 minutes) Lesson 8: Ordering Integers and Other Rational Numbers This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from G6-M3-TE-1.3.0-08.2015 75 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 8 NYS COMMON CORE MATHEMATICS CURRICULUM Name 6β’3 Date Lesson 8: Ordering Integers and Other Rational Numbers Exit Ticket Order the following set of rational numbers from least to greatest, and explain how you determined the order. 1 2 1 3 β3, 0, β , 1, β3 , 6, 5, β1, Lesson 8: 21 Ordering Integers and Other Rational Numbers This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from G6-M3-TE-1.3.0-08.2015 5 ,4 76 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 8 NYS COMMON CORE MATHEMATICS CURRICULUM 6β’3 Exit Ticket Sample Solutions Order the following set of rational numbers from least to greatest, and explain how you determined the order. π π βπ, π, β π, π, βπ π, π, π, βπ, βππ βπ βπ βπ βπ βπ βπ βπ βπ βπ π π π π π π ππ π π βπ , βπ, βπ, β , π, π, π, ,π π π π π π π π ππ ππ π , π, π I drew a number line and started at zero. I located the positive numbers to the right and their opposites (the negative numbers) to the left of zero. The positive integers listed in order from left to right are π, π, π, π. And since π π π , I know that it is π π more than π but less than π. Therefore, I arrived at π, π, π, ππ π is equal to ππ , π, π. Next, I ordered the negative π numbers. Since βπ and βπ are the opposites of π and π, they are π unit and π units from zero but to the left of zero. And π π π π βπ is even farther left, since it is π units to the left of zero. The smallest number is farthest to the left, so I arrived at π π π π the following order: βπ , βπ, βπ, β , π, π, π, ππ , π, π. π Problem Set Sample Solutions 1. a. In the table below, list each set of rational numbers from greatest to least. Then, in the appropriate column, state which number was farthest right and which number was farthest left on the number line. Column 1 Column 2 Column 3 Column 4 Rational Numbers Ordered from Greatest to Least Farthest Right on the Number Line Farthest Left on the Number Line βπ. ππ, βπ. ππ βπ. ππ, βπ. ππ βπ. ππ βπ. ππ βπ. π, βπ βπ, βπ. π βπ βπ. π π , π π π , π π π π π βππ, β ππ π π βππ, β ππ π π βππ βππ π π βππ, βπ βπ, βππ βπ βππ π , βπ π π , βπ π π π βπ βππ, βπππ, βππ. π βππ, βππ. π, βπππ βππ βπππ π. ππ, π. π π. π, π. ππ π. π π. ππ π π π, β , β π π π π π, β , β π π π βπ. ππ, βπ. ππ βπ. ππ, βπ. ππ βπ. ππ Lesson 8: Ordering Integers and Other Rational Numbers This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from G6-M3-TE-1.3.0-08.2015 β π π βπ. ππ 77 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 8 NYS COMMON CORE MATHEMATICS CURRICULUM b. 6β’3 For each row, describe the relationship between the number in Column 3 and its order in Column 2. Why is this? The number in Column 3 is the first number listed in Column 2. Since it is farthest right on the number line, it will be the greatest; therefore, it comes first when ordering the numbers from greatest to least. c. For each row, describe the relationship between the number in Column 4 and its order in Column 2. Why is this? The number in Column 4 is the last number listed in Column 2. Since it is farthest left on the number line, it will be the smallest; therefore, it comes last when ordering the numbers from greatest to least. 2. If two rational numbers, π and π, are ordered such that π is less than π, then what must be true about the order for their opposites: βπ and βπ? The order will be reversed for the opposites, which means βπ is greater than βπ. 3. Read each statement, and then write a statement relating the opposites of each of the given numbers: a. π is greater than π. βπ is less than βπ. b. ππ. π is greater than ππ. βππ. π is less than βππ. c. β π π 4. π π is less than . π π π π is greater than β . Order the following from least to greatest: βπ, βππ, π, π π , . π π π π βππ, β π, π, , π π 5. π π Order the following from greatest to least: βππ, ππ, βππ, π , π. π ππ, π, π , β ππ, β ππ π Lesson 8: Ordering Integers and Other Rational Numbers This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from G6-M3-TE-1.3.0-08.2015 78 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.