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PowerTeaching Math ® 3rd Edition Level 8 | Unit 2 Rational and Irrational Numbers unit guide With Student and Assessment Pages PowerTeaching Math 3rd Edition Unit Guide: Level 8 © 2015 Success for All Foundation. All rights reserved Produced by the PowerTeaching Math 3rd Edition Team Angela Watson Kate Conway Nancy Madden Angie Hale Kathleen Collins Nick Leonhardt Cathy Pascone Kathy Brune Patricia Johnson Debra Branner Kenly Novotny Paul Miller Devon Bouldin Kimberly Sargeant Peg Weigel Erin Toomey Kris Misage Rebecca Prell Irene Baranyk Laura Alexander Russell Jozwiak Irina Mukhutdinova Laurie Warner Sarah Eitel James Bravo Luke Wiedeman Sharon Clark Jane Strausbaugh Mark Kamberger Susan Perkins Janet Wisner Marti Gastineau Terri Morrison Jeffrey Goddard Meghan Fay Tina Widzbor Jennifer Austin Michael Hummel Tonia Hawkins Joseph Wilson Michelle Hartz Wanda Jackson Karen Poe Michelle Zahler Wendy Fitchett We wish to acknowledge the coaches, teachers, and students who piloted the program and provided valuable feedback. The Success for All Foundation grants permission to reproduce the blackline masters of the PowerTeaching Math unit guides on an as-needed basis for classroom use. A Nonprofit Education Reform Organization 300 E. Joppa Road, Suite 500, Baltimore, MD 21286 PHONE: (800) 548-4998; FAX: (410) 324-4444 E-MAIL: [email protected] WEBSITE: www.successforall.org table of contents Unit Overview.. .................................................................. 1 Cycle 1 Rational and Irrational Numbers. . .......................................... 3 Student Pages Teamwork, Quick Check, Homework, and Assessments. . ...... 33 Cycle 1..................................................................... 35 This project was developed at the Success for All Foundation under the direction of Robert E. Slavin and Nancy A. Madden to utilize the power of cooperative learning, frequent assessment and feedback, and schoolwide collaboration proven in decades of research to increase student learning. © 2015 Success for All Foundation PowerTeaching Math 3rd Edition | Unit Guide iii Level 8 | Unit 2: Rational and Irrational Numbers Level 8 | Unit 2: Rational and Irrational Numbers Unit Overview Vocabulary introduced in this unit: rational number radical sign square root perfect square irrational number Think Like a Mathematician practice(s) used in this unit: Make sense of it. Translate into math. Defend and review. Use your math toolkit. Be precise. Find the patterns and structure. In unit 2 of grade 8, your students will add to their knowledge of the types of numbers. In grades 6 and 7, your students added negative numbers to their knowledge and formed a full understanding of rational numbers. In grade 8, your students will learn about irrational numbers to develop a full view of what real numbers are. They will define, classify, and approximate the values of various types of numbers. Unit 2 consists of one cycle: cycle 1—Rational and Irrational Numbers. Cycle 1—Rational and Irrational Numbers Lesson 1: Defining Irrational Numbers Define and explore rational and irrational numbers. (CC 8.NS.A.1 and 2; TEKS 8.b.2.A; VA SOL 7.1d, 8.2, 8.5a) Lesson 2: Classifying Numbers Classify rational and irrational numbers. (CC 8.NS.A.1 and 2; TEKS 8.b.2.A; VA SOL 8.2) Lesson 3: Converting a Decimal Expansion Convert a decimal expansion that repeats eventually into a rational number. (CC 8.NS.A.1; TEKS 8.b.2.A; VA SOL 8.2) Lesson 4: Ordering Rational and Irrational Numbers Use knowledge of perfect squares and the number line to order rational and irrational numbers. (CC 8.NS.A.2; TEKS 8.b.2.B and D; VA SOL 8.2, 8.5a and b) Lesson 5: Comparing Irrational Expressions Use approximations of the value of irrational numbers to estimate and compare expressions containing irrational numbers. (CC 8.NS.A.2; TEKS 8.b.2.B and D; VA SOL 8.5b and A.3) © 2015 Success for All Foundation PowerTeaching Math 3rd Edition | Unit Guide 1 Access Code: zwdffn Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 1 Lesson 1: Defining Irrational Numbers Vocabulary: rational number radical sign square root perfect square irrational number Materials: calculators Lesson Objective: Define and explore rational and irrational numbers. By the end of this lesson, students will: • define irrational numbers; • explore the concept of and values of irrational numbers; and • determine whether a given number is irrational. This lesson involves an introduction to square roots with emphasis on the square root of 2. opening (3 minutes) get the goof • Ask teams to begin Get the Goof. TEACHER’S NOTE: Students learned about rational numbers in grade 7. Here they are identifying a classification error. If students get stuck, refer to grade 7, unit 2, cycle 1, lesson 1. Maya said that – 2 is a whole number. What’s wrong with her thinking? Random Reporter Rubric | Possible Answer Answer: Maya identified the number incorrectly. Explanation: – 2 is not positive or 0, so it cannot be a whole number . – 2 is an integer . Math Practice: I know that whole numbers are positive numbers that aren’t fractions or decimals and include 0. Using the definition of a whole number helped me figure out what was wrong with Maya’s thinking (TLM #6). • Use Random Reporter to have teams share. Use the Random Reporter rubric to evaluate responses and give feedback. • Record individual scores on the teacher cycle record form. • Award team celebration points. © 2015 Success for All Foundation PowerTeaching Math 3rd Edition | Unit Guide 3 Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 1 Access Code: zwdffn active instruction (10–15 minutes) set the stage • Distribute team score sheets. Have students review their scores and set new team goals in lesson 1. • Post and present the lesson objective: Today you will define and explore rational and irrational numbers. • Ask students to write this cycle’s vocabulary words in their notebooks: rational number, radical sign, square root, perfect square, irrational number. • Remind students how to earn team celebration points. interactive instruction and guided practice • Use a Think Aloud to model explaining and exploring the differences between rational and irrational numbers. Define irrational numbers. 6 layers Find the value of each. Is each value a rational or an irrational number? a) the square roots of 25 b) 2 I know that I cannot claim whether a and/or b are rational or irrational numbers until I find the value of each. Using my Think Like a Mathematician sheet, I know that definitions can help me. Being precise and using the correct definition of terms is very important in math. That’s TLM practice #6. Let’s start with a: the square roots of 25. What do I know about square roots? Show layer 1. A square root of a number is a value that can be multiplied by itself to give the original number. What are the square roots of 25? Show layer 2. Is this a perfect square? Yes. 25 is a perfect square because 5 times 5 is equal to 25. I also know that – 5 times – 5 is also equal to 25. So the number 25 has two square roots. Both 5 and – 5 are the square roots of 25. Both 5 and – 5 are rational numbers. Remember, rational numbers are numbers that can be written in fractional form. Show layer 3. When the radical sign is used, the only answer is the positive square root. Now let’s examine b: the positive square root of 2. What do I know about the number 2? Is it a perfect square? Show layer 4. So 2 is not a perfect square. How can I tell whether this is a rational or an irrational number? Show layer 5. If I use the calculator to find the square root of 2, the decimal does not repeat or terminate. So the square root of 2 is an irrational number. Can you think of another irrational number? Show layer 6. Pi is another commonly used irrational number. We use the symbol p to represent the number. Most of the time, we use the estimate 3.14 to calculate the circumference or the area of a circle. • Use Think‑Pair‑Share to ask students the following question: Can a length be an irrational number? How would you measure it? • Randomly select a few students to share. Possible answer: Yes. A length can be an irrational number. It is measured as its rational approximate. 4 PowerTeaching Math 3rd Edition | Unit Guide © 2015 Success for All Foundation Access Code: zwdffn Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 1 • Use Team Huddle to have teams practice defend‑and‑review statements about rational and irrational numbers. 1) Provide two examples that show that the statement is false. Lydia said that all square roots are irrational numbers. Random Reporter Rubric | Possible Answer 4 5 2 and 100 5 10. Answer: The statement is false because Explanation: I know the statement is false because the square roots of perfect squares are always rational numbers . Math Practice: I examined the statement that Lydia made to determine whether her argument made sense. I used what I know about the definitions of rational and irrational numbers to prove her statement as false. I know that any example where I would find the square root of a perfect square will show that Lydia’s statement is false (TLM #6). • Use Random Reporter to have teams share. Use the Random Reporter rubric to evaluate responses and give specific feedback. • Record individual scores on the teacher cycle record form. • Award team celebration points. team mastery (10–15 minutes) • Ask students to follow the Team Mastery student routine. • Circulate and use the following questions to prompt discussions. –– What is an irrational number? –– What’s the difference between a rational and an irrational number? –– How did you know this was an irrational number? –– How can a number have two square roots? –– What is a perfect square? –– How does the square root of a number relate to its classification? • When there are 5 minutes left in Team Mastery, prompt teams to prepare for the Lightning Round. Have teams discuss one Team Mastery problem that the whole team has completed. • Award team celebration points for good team discussions that demonstrate 100‑point responses. © 2015 Success for All Foundation PowerTeaching Math 3rd Edition | Unit Guide 5 Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 1 Access Code: zwdffn lightning round (10 minutes) • Tell students the Team Mastery problem that you will use for the Lightning Round. Classify the number as rational or irrational. 3) 48 Random Reporter Rubric | Possible Answer Answer: 48 is an irrational number . Explanation: First, I asked myself if 48 was a perfect square . 6 ? 6 5 36 and 7 ? 7 5 49, so 48 is not a perfect square. The square root of 48 is an irrational number. Math Practice: I used the calculator to find the square root of 48 and saw that the decimal did not terminate or repeat. The calculator helped to confirm my answer (TLM #5). • Use Random Reporter to have teams share. Use the Random Reporter rubric to evaluate responses and give specific feedback. • Record individual scores on the teacher cycle record form. • Award team celebration points. celebration (2 minutes) • Record team celebration points on the poster. • Have the top team choose a cheer. • Assign homework, and remind students about the Vocabulary Vault. • Ask students to follow the Quick Check student routine. (optional) Find the square roots for 64. Possible answer: 8 and – 8 6 PowerTeaching Math 3rd Edition | Unit Guide © 2015 Success for All Foundation Access Code: zwdffn Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 2 Lesson 2: Classifying Numbers Vocabulary: none Materials: none Lesson Objective: Classify rational and irrational numbers. By the end of this lesson, students will: • classify rational and irrational numbers; and • categorize numbers as members of one or more of the following sets: natural numbers, whole numbers, integers, rational numbers, or irrational numbers. opening (3 minutes) get the goof • Ask teams to begin Get the Goof. TEACHER’S NOTE: Students may assume that the square root of any number is rational. Remind them that the square roots of perfect squares are natural numbers and, therefore, rational. The square root of a number other than a perfect square is irrational. Stefan said that 110 is rational. What’s wrong with his thinking? Random Reporter Rubric | Possible Answer 110 incorrectly. Answer: Stefan classified Explanation: 110 is an irrational number . I asked myself if 110 is a perfect square . 10 ? 10 5 100 and 11 ? 11 5 121, so 110 is not a perfect square. Math Practice: I checked my thinking with the calculator to find the square root of 110 and saw that the decimal did not terminate or repeat (TLM #5). • Use Random Reporter to have teams share. Use the Random Reporter rubric to evaluate responses and give feedback. • Record individual scores on the teacher cycle record form. • Award team celebration points. homework check • Ask teams to do a homework check. • Confirm the number of students who completed the homework on each team. • Poll students on their team’s understanding of the homework. • Award team celebration points. • Collect and grade homework once per cycle. Record individual scores on the teacher cycle record form. © 2015 Success for All Foundation PowerTeaching Math 3rd Edition | Unit Guide 7 Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 2 Access Code: zwdffn active instruction (10–15 minutes) set the stage • Post and present the lesson objective: Today you will classify rational and irrational numbers. • Remind students how to earn team celebration points. interactive instruction and guided practice • Use a Think Aloud to review definitions of rational and irrational numbers to classify a number. Classify rational and irrational numbers. 3 layers Classify 121 on the Venn diagram. What do I know about rational and irrational numbers? Natural numbers are counting numbers like 1, 2, 3, 4…. Whole numbers are natural numbers and 0. Integers are whole numbers and their opposites. Rational numbers include natural numbers, whole numbers, integers, fractions, and terminating and repeating decimals. An irrational number cannot be written in fractional form. Show layer 1. Now I’ll examine the examples I’ve been given. Rational Numbers 0.5 Integers 5 Irrational Numbers 3 _ 10 10 0 Whole Numbers Natural Numbers 1 5 10 I don’t see any examples of irrational numbers, but by definition, I know that any number that cannot be written in fractional form (such as p and 2 ) is an irrational number. Show layer 2. The problem asks me to classify the square root of 121. Since this number is under the radical and, I only need the positive square root. Is 121 a perfect square? Yes, 121 is the product of 11 times 11. Show layer 3. So the square root of 121 is classified as a natural number, a whole number, an integer, and a rational number. I used the classifications precisely to answer this question (TLM #6). • Use Think‑Pair‑Share to have the students name one number for each section of the diagram. 8 PowerTeaching Math 3rd Edition | Unit Guide © 2015 Success for All Foundation Access Code: zwdffn Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 2 • Randomly select a few students to share. Possible answers: Accept reasonable responses. Natural numbers: 1, 2, 3, 4…. Whole numbers: natural numbers and 0. Integers: …– 4, – 3, – 2, – 1, 0, 1, 2, 3, 4. Rational numbers: natural numbers, whole numbers, integers, fractions, and terminating and repeating decimals. Irrational numbers: p, 2 , a number that cannot be written in fractional form. • Use Team Huddle to have teams practice classifying rational and irrational numbers. Classify the numbers by writing them in the appropriate section of the Venn diagram. 1) 10 – _ , 2 36 , _ 0 , 7, 140 ,_ 4 , 4 , – 8, 8 , – 2.89 8 9 Random Reporter Rubric | Possible Answer Answer: Rational Numbers 4 _ Irrational Numbers 2.89 9 10 _ 2 0 _ 8 Integers 140 8 8 Whole Numbers Natural Numbers 36 7 4 Explanation: I started by examining the fractions to see if I could simplify any of them. I found that one fraction was equivalent to an integer while another was equivalent to 0. I placed the third fraction in rational numbers . Next, I determined that the square roots of perfect squares are natural numbers . The square roots of numbers that are not perfect squares are irrational numbers . I placed the remaining rational numbers in the appropriate sections. Math Practice: My classification makes sense because I recalled the definition of each section before determining which number fit into that section (TLM #1). • Use Random Reporter to have teams share. Use the Random Reporter rubric to evaluate responses and give specific feedback. • Record individual scores on the teacher cycle record form. • Award team celebration points. © 2015 Success for All Foundation PowerTeaching Math 3rd Edition | Unit Guide 9 Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 2 Access Code: zwdffn team mastery (10–15 minutes) • Ask students to follow the Team Mastery student routine. • Circulate and use the following questions to prompt discussions. –– How would you classify this number? –– How many different ways can you categorize this number? –– What is the difference between a rational and an irrational number? –– What is the difference between a terminating and a repeating decimal? –– How can you distinguish between all the subsets of rational numbers? • When there are 5 minutes left in Team Mastery, prompt teams to prepare for the Lightning Round. Have teams discuss one Team Mastery problem that the whole team has completed. • Award team celebration points for good team discussions that demonstrate 100‑point responses. lightning round (10 minutes) • Tell students the Team Mastery problem that you will use for the Lightning Round. 4) Classify the numbers by writing them in the appropriate section of the Venn diagram. 44, _ 7 , 144 , – 88, _ 12 , 5.4, 24 3 2 Random Reporter Rubric | Possible Answer Answer: Rational Numbers 7 _ 3 Irrational Numbers 5.4 24 Integers 88 Whole Numbers Natural Numbers 44 144 12 _ 2 10 PowerTeaching Math 3rd Edition | Unit Guide © 2015 Success for All Foundation Access Code: zwdffn Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 2 Explanation: I started by examining the fractions to see if I could simplify any of them. I found that one fraction was equivalent to a natural number while another was a mixed number . Next, I determined that the square roots of perfect squares are natural numbers. The square roots of numbers that are not perfect squares are irrational numbers . I placed the remaining rational numbers in the appropriate sections. I used the Venn diagram to classify the numbers into their appropriate subsets. Math Practice: My classification makes sense because I recalled the definition of each section before determining which numbers fit into that section (TLM #1). • Use Random Reporter to have teams share. Use the Random Reporter rubric to evaluate responses and give specific feedback. • Record individual scores on the teacher cycle record form. • Award team celebration points. celebration (2 minutes) • Record team celebration points on the poster. • Have the top team choose a cheer. • Assign homework, and remind students about the Vocabulary Vault. • Ask students to follow the Quick Check student routine. (optional) Classify the numbers by writing them in the appropriate section of the Venn diagram. 9.3, _ 18 , 36 , – 13, _ 55 , 1.7, 41 4 9 Possible answer: Rational Numbers Irrational Numbers 55 _ 9.3 Integers 4 41 1.7 13 Whole Numbers Natural Numbers 18 _ 9 © 2015 Success for All Foundation 36 PowerTeaching Math 3rd Edition | Unit Guide 11 Access Code: zwdffn Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 3 Lesson 3: Converting a Decimal Expansion Vocabulary: none Lesson Objective: Convert a decimal expansion that repeats eventually into a Materials: none By the end of this lesson, students will: rational number. • convert repeating decimals into fractions; and • confirm that repeating decimals are rational numbers. opening (3 minutes) get the goof • Ask teams to begin Get the Goof. TEACHER’S NOTE: Students may assume that the negative decimal is an integer. Remind them that integers are whole numbers and their opposites. This does not include negative decimals or negative fractions. Omar classified – 8.4 as an integer. What’s wrong with his thinking? Random Reporter Rubric | Possible Answer Answer: Omar did not classify the number correctly. Explanation: Integers are a set of whole numbers and their opposites . Since – 8.4 is not a whole number, it cannot be classified as an integer. The correct classification of – 8.4 is a rational number . Math Practice: I used the definitions of an integer and a rational number to classify the number correctly (TLM #6). • Use Random Reporter to have teams share. Use the Random Reporter rubric to evaluate responses and give feedback. • Record individual scores on the teacher cycle record form. • Award team celebration points. homework check • Ask teams to do a homework check. • Confirm the number of students who completed the homework on each team. • Poll students on their team’s understanding of the homework. • Award team celebration points. • Collect and grade homework once per cycle. Record individual scores on the teacher cycle record form. © 2015 Success for All Foundation PowerTeaching Math 3rd Edition | Unit Guide 13 Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 3 active instruction Access Code: zwdffn (10–15 minutes) set the stage • Post and present the lesson objective: Today you will convert a decimal expansion that repeats eventually into a rational number. • Remind students how to earn team celebration points. interactive instruction and guided practice • Use a Think Aloud to introduce the process of converting a repeating decimal into a fraction by using the students’ prior knowledge. Convert a decimal expansion. 6 layers Write 0.4141…as a fraction. I need to write this decimal as a fraction. What do I know about fractions? Fractions are rational numbers. I know from my Think Like a Mathematician guidelines that I can translate numbers, so I can represent fractions as division expressions. That’s TLM practice #2. In every fraction, the numerator is the dividend, and the denominator is the divisor. Therefore, the quotient will either terminate or repeat. Let’s see if that’s true. I’ll look at one‑half and one‑third. Show layer 1. 0.3 _ 1.0 1 5 1 4 3 or 3 0.5 _ 1.0 1 5 1 4 2 or 2 2 3 In the expression “one‑half,” 1 is divided by 2 in the form a/b. These are integers written in fractional form where b is not equal to 0. What if I rewrote the expression as an equation to solve for x? Show layer 2. _ 1 1 5 2x 2 1 _ 5 x 2 To solve this equation, I would isolate the variable by dividing both sides by 2. So x is equal to one‑half. I have not changed the value of the number. Let’s use the equation with the decimal form of one‑half to rewrite the decimal as a fraction. Show layer 3. Example 1 Example 2 x 5 0.5 x 5 0.55 (10)x 5 0.5(10) (100)x 5 0.55(100) 10x 5 5 100x 5 55 x5_ 5 55 x 5 _ 1 x 5 _ x5_ 11 10 2 14 PowerTeaching Math 3rd Edition | Unit Guide 100 20 © 2015 Success for All Foundation Access Code: zwdffn Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 3 I have two examples showing equations with terminating decimals. I want to write the decimals as fractions. To do this, I need to multiply both sides by a power of 10. (The power of 10 corresponds to the number of places behind the decimal point.) My problem is not a terminating decimal. What kind of decimal is this? It’s a repeating decimal. Can I use the same method to write this decimal as a fraction? Let’s find out. Show layer 4. x 5 0.4141… 100x 5 0.4141…(100) 100x 5 41.4141… I multiplied both sides by 100 because two numbers repeat in the decimal. Since I still have a decimal, I cannot treat this the same way I treated the terminating decimal. I need a different way to isolate the variable. Before, I divided both sides by the same number. Division is repeated subtraction, so let’s see what happens when I subtract. Show layer 5. 100x 5 41.4141… 2 x 5 0.4141… 99x 5 41 Now I can isolate the variable because I’m no longer working with a decimal. Show layer 6. _ 99x 5 _ 41 99 99 x 5 _ 41 99 41 • Use Think‑Pair‑Share to have students check to see whether _ 5 0.4141…. 41 • Randomly select a few students to share. Yes. _ 5 0.4141…. 99 99 • Use Team Huddle to have teams practice converting a repeating decimal into a fraction. 1) Find the fractional equivalent of 0.234 . Show your work. Random Reporter Rubric | Possible Answer or _ 78 or _ 26 Answer: _ 234 999 333 111 Explanation: x 5 0.234 1,000x 5 234.234… 1,000x 5 234.234… 2 x 5 0.234… 999x 5 234 234 x 5 _ 999 I started by expressing the repeated decimal as an equation . Then, I multiplied each side of the equation by 1,000 because three digits repeat. This moved the set of repeating digits to the left side of the decimal point. Next, I subtracted x from 1,000x to get rid of the decimal. Finally, I isolated the variable and solved for x. Math Practice: I used what I know about converting a terminating decimal into an equation to convert a repeating decimal into an equation (TLM #3). © 2015 Success for All Foundation PowerTeaching Math 3rd Edition | Unit Guide 15 Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 3 Access Code: zwdffn • Use Random Reporter to have teams share. Use the Random Reporter rubric to evaluate responses and give specific feedback. • Record individual scores on the teacher cycle record form. • Award team celebration points. team mastery (10–15 minutes) • Ask students to follow the Team Mastery student routine. • Circulate and use the following questions to prompt discussions. –– How did you write this repeating decimal as a fraction? –– How do you know that repeating decimals are rational numbers? –– Why do you multiply each side by a multiple of 10? • When there are 5 minutes left in Team Mastery, prompt teams to prepare for the Lightning Round. Have teams discuss one Team Mastery problem that the whole team has completed. • Award team celebration points for good team discussions that demonstrate 100‑point responses. lightning round (10 minutes) • Tell students the Team Mastery problem that you will use for the Lightning Round. 5) Find the fractional equivalent of 0.21 . Show your work. Random Reporter Rubric | Possible Answer 7 Answer: _ 21 or _ 99 Explanation: 33 x 5 0.21 100x 5 21.21… 100x 5 21.21… 2 x 5 0.21… 99x 5 21 21 x 5 _ 99 I started by expressing the repeated decimal as an equation . Then, I multiplied each side of the equation by 100 because two digits repeat. This moved the set of repeating digits to the left side of the decimal point. Next, I subtracted x from 100x to get rid of the decimal. Finally, I isolated the variable and solved for x. Math Practice: I used what I knew about converting a terminating decimal into an equation to convert a repeating decimal into an equation (TLM #3). • Use Random Reporter to have teams share. Use the Random Reporter rubric to evaluate responses and give feedback. • Record individual scores on the teacher cycle record form. • Award team celebration points. 16 PowerTeaching Math 3rd Edition | Unit Guide © 2015 Success for All Foundation Access Code: zwdffn Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 3 celebration (2 minutes) • Record team celebration points on the poster. • Have the top team choose a cheer. • Assign homework, and remind students about the Vocabulary Vault. • Ask students to follow the Quick Check student routine. (optional) Find the fractional equivalent of 0.426 . Show your work. Possible answer: _ 426 or _ 142 999 © 2015 Success for All Foundation 333 PowerTeaching Math 3rd Edition | Unit Guide 17 Access Code: zwdffn Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 4 Lesson 4: Ordering Rational and Irrational Numbers Vocabulary: none Lesson Objective: Use knowledge of perfect squares and the number line to order rational Materials: NO calculators By the end of this lesson, students will: and irrational numbers. • compare rational and irrational numbers, and • use their knowledge of perfect squares and the number line to order rational and irrational numbers. opening (3 minutes) get the goof • Ask teams to begin Get the Goof. TEACHER’S NOTE: In the previous lesson, students learned to multiply each side of the equation by a power of 10 that corresponds to the number of decimal numbers that repeat. Sofia has to write a repeating decimal as a fraction. She multiplied each side of the equation: x 5 0.636363… by 10. What’s wrong with her thinking? Random Reporter Rubric | Possible Answer Answer: Sofia did not multiply each side of the equation by the correct power of 10 . Explanation: Sofia should have multiplied by the power of 10 with two zeroes, or 100. Math Practice: My answer makes sense because 0.63 has two numbers that repeat behind the decimal point (TLM #1). • Use Random Reporter to have teams share. Use the Random Reporter rubric to evaluate responses and give feedback. • Record individual scores on the teacher cycle record form. • Award team celebration points. homework check • Ask teams to do a homework check. • Confirm the number of students who completed the homework on each team. • Poll students on their team’s understanding of the homework. • Award team celebration points. • Collect and grade homework once per cycle. Record individual scores on the teacher cycle record form. © 2015 Success for All Foundation PowerTeaching Math 3rd Edition | Unit Guide 19 Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 4 Access Code: zwdffn active instruction (10–15 minutes) set the stage • Post and present the lesson objective: Today you will use knowledge of perfect squares and the number line to order rational and irrational numbers. • Remind students how to earn team celebration points. interactive instruction and guided practice • Use a Think Aloud to explore the process of organizing and arranging rational and irrational numbers. Use knowledge of perfect squares and the number line to order rational and irrational numbers. 4 layers Order the numbers from least to greatest. 32 15 4 62 , 9.3 10 , _ , – 76 , – _ I need to make sense of this problem. The problem asks me to order the numbers from least to greatest. Show layer 1. To convert a fraction to a decimal, I need to divide the numerator by the denominator. Show layer 2. Next I’ll take a look at the irrational numbers. I do not need a calculator to figure this out. I can use what I know about these numbers to estimate their position. 62 falls between the perfect squares 49 and 64, while the negative square root of 10 falls between – 3 and – 4. Show layer 3. Now I can put this all together and arrange these numbers from least to greatest. 76 _ 15 10 , – _ , , 62 , 9.3 Show layer 4. My answer is – . To solve this I converted to 32 4 equivalent forms so my ordering was accurate (TLM #6). • Use Think‑Pair‑Share to have students answer the following question: Would the answer be different if we converted to equivalent fractions instead of decimals? • Randomly select a few students to share. Possible answer: No. The type of conversion, whether to decimal or to fraction, would not change the value of the number. Therefore, the answer would not be different. • Use a Think Aloud to compare two numbers. Compare rational and irrational numbers. Use ,, ., or 5 to compare the following numbers. 6.92820323… ____ 42 Again, before I solve the problem, I have to make sense of the problem. What do I know about these numbers? I can see that one number is a decimal, but I cannot tell if the decimal repeats. The decimal does not appear to terminate. I also know that the square root of 42 is an irrational number. I’m going to use a number line to help me visualize the position of each number. That way I can determine whether one number is greater than, less than, or equal to the other. 20 PowerTeaching Math 3rd Edition | Unit Guide © 2015 Success for All Foundation Access Code: zwdffn Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 4 Show layer 1. I’ve got a good idea where the decimal is located. I know that the square root of 42 is between 6 and 7 because 42 falls between 36 and 49. Show layer 2. So the square root of 42 is about half the distance between 36 and 49. I can put it around the half mark between 6 and 7. Show layer 3. Now I can make a reasonable conclusion and answer the original question. 42 . I used a number line as a tool to help me Show layer 4. 6.92820323… . compare the numbers (TLM #5). • Use Think‑Pair‑Share to have students answer the following question: Which is greater: p or 3.14? Explain. • Randomly select a few students to share. p is greater than 3.14. Possible answer: The number pi continues on past 3.14 as a never‑ending, nonrepeating decimal. This is why it is greater than 3.14. • Use Team Huddle to have teams practice arranging a group of rational numbers on a number line. 1) Graph each number on the number line below. 2.55, _ 9 , 24 , p, 9 5 –5 –4 –3 –2 –1 0 1 2 3 4 5 Random Reporter Rubric | Possible Answer Answer: 9 5 –5 –4 –3 –2 –1 0 1 9 2 3 24 4 5 2.55 Explanation: I started by plotting the rational numbers on the number line. I converted _ 9 5 to 1.8 and 9 to 3. Then, I plotted both of them. Next, I approximated 24 as close to 5 and p as 3.14 and plotted them also. Math Practice: My answer makes sense because I used the decimal estimates of each number to arrange the numbers on the number line (TLM #1). • Use Random Reporter to have teams share. Use the Random Reporter rubric to evaluate responses and give feedback. • Record individual scores on the teacher cycle record form. • Award team celebration points. © 2015 Success for All Foundation PowerTeaching Math 3rd Edition | Unit Guide 21 Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 4 Access Code: zwdffn team mastery (10–15 minutes) • Ask students to follow the Team Mastery student routine. • Circulate and use the following questions to prompt discussions. –– Is this number rational or irrational? How do you know? –– Between which numbers does this square root fall? –– How did you determine which number is greater/lesser? –– How can you compare decimals to fractions? –– How did you determine the order of this set of numbers? • When there are 5 minutes left in Team Mastery, prompt teams to prepare for Lightning Round. Have teams discuss one Team Mastery problem that the whole team has completed. • Award team celebration points for good team discussions that demonstrate 100‑point responses. lightning round (10 minutes) • Tell students the Team Mastery problem that you will use for the Lightning Round. 4) Write the following numbers in order from greatest to least. – 2.3, 1 0, _ 23 , p, – 7 _ 7 5 Random Reporter Rubric | Possible Answer 1 Answer: _ 23 , p, 0, – 2.3, – 7 _ 7 5 Explanation: I started by converting the improper fraction and mixed number into their decimal forms. I also wrote the decimal approximate for p. Since we are ordering the numbers from greatest to least , I used a number line to approximate each number’s location. Then, I wrote my answer in the reverse order, starting with the largest positive number and ending with the negative number at the far left of the number line. Math Practice: My answer makes sense because on a number line, moving to the right means the numbers increase in value and moving to the left means the numbers decrease in value (TLM #1). • Use Random Reporter to have teams share. Use the Random Reporter rubric to evaluate responses and give feedback. • Record individual scores on the teacher cycle record form. • Award team celebration points. • Tell students that it’s time to power up Random Reporter. Use the layers on the page to guide discussion. 22 PowerTeaching Math 3rd Edition | Unit Guide © 2015 Success for All Foundation Access Code: zwdffn Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 4 celebration (2 minutes) • Record team celebration points on the poster. • Have the top team choose a cheer. • Assign homework, and remind students about the Vocabulary Vault. • Ask students to follow the Quick Check student routine. (optional) Use ,, ., or 5 to compare the following numbers. 6.1678203027… _____ 32 Possible answer: 6.1678203027… . 32 © 2015 Success for All Foundation PowerTeaching Math 3rd Edition | Unit Guide 23 Access Code: zwdffn Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 5 Lesson 5: Comparing Irrational Expressions Vocabulary: none Lesson Objective: Use approximations of the values of irrational numbers to estimate and Materials: none By the end of this lesson, students will: compare expressions containing irrational numbers. • approximate the value of irrational numbers with rational numbers; and • use number sense and what they know about the number line to estimate and compare expressions that contain irrational numbers. opening (3 minutes) get the goof • Ask teams to begin Get the Goof. Luigi said that the value of 93 is between 81 and 100. What’s wrong with his thinking? Random Reporter Rubric | Possible Answer Answer: Luigi did not approximate the value correctly; he didn’t account for the square root sign. Explanation: The square root of 93 is a number that can be multiplied by itself to make 93. This means the value is actually between 9 and 10. Math Practice: I used what I know about the definition of square root to make sense of this problem (TLM #3). Because 9 2 5 81 and 1 0 2 5 100, 93 is between 9 and 10, not between 81 and 100. • Use Random Reporter to have teams share. Use the Random Reporter rubric to evaluate responses and give feedback. • Record individual scores on the teacher cycle record form. • Award team celebration points. homework check • Ask teams to do a homework check. • Confirm the number of students who completed the homework on each team. • Poll students on their team’s understanding of the homework. • Award team celebration points. • Collect and grade homework once per cycle. Record individual scores on the teacher cycle record form. © 2015 Success for All Foundation PowerTeaching Math 3rd Edition | Unit Guide 25 Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 5 Access Code: zwdffn active instruction (10–15 minutes) set the stage • Post and present the lesson objective: Today you will compare numbers that contain irrational values. • Remind students how to earn team celebration points. interactive instruction and guided practice • Show the “Think Like a Mathematician” video clip for practice #7, find the patterns and structure. • Use Think‑Pair‑Share to have students discuss situations in which it would be helpful to use this practice and, if possible, to give their own personal usage examples. Randomly select students to share. • Use a Think Aloud to model comparing expressions that contain irrational numbers. Estimate and compare the values of expressions that contain irrational numbers. 6 layers In each pair, decide which number is greater without using a calculator. What’s going on in this problem? I have to compare these numbers to see which one is greater, so this is just basic comparison. But I see that the first pair has pi squared in it, and the second pair has the square root of 30 in it, so I have to do some approximation here because these are irrational numbers. Show layer 1. To make sense of the first problem, I have to make sense of the opposite of pi squared. I know that a good approximation of pi is 3.14, so I’ll start there. Show layer 2. Of course, I’m not supposed to use a calculator to solve this, but even if I did, my answer would be an estimate. Even a calculator can’t accurately represent an irrational number. So I’ll indicate that I’m making an estimate. Let me think about the value of pi squared. Squaring any number means using that number as a factor twice, so I have to estimate 3.14 ? 3.14. Just by thinking about this, I know that pi squared will be greater than 9 because 3 2 5 9, and 3.14 . 3. 26 PowerTeaching Math 3rd Edition | Unit Guide © 2015 Success for All Foundation Access Code: zwdffn Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 5 Show layer 3. Now I have to compare the opposite of pi squared with – 6. I’ll use a number line because sometimes I get confused when comparing negative numbers. This makes sense. Negative 6 is closer to zero than a value to the left of negative 9, so I know that – 6 is greater in this first pair. Show layer 4. Now on to the next pair. I have to compare two square roots of 30 and 10. Once again, I have to make sense of an irrational number. Show layer 5. To take this one step at a time, first I’ll focus on the square root of 30. I know that value is between the square root of 25 and the square root of 36. 30 is almost exactly halfway between 25 and 36, so I know a good approximation is 5.5. Now, two square roots of 30 is just 2 times that value. 2 ? 5.5 is 11. Even though I don’t know what 30 is exactly, I can just use what I know about the structure of the expression 2 30 and multiplication to estimate the value as about 11. That’s TLM practice #7, finding the patterns and structure. Show layer 6. Now I can complete my estimate. In this pair, 11 is greater than 10, so two square roots of 30 is greater. I made sense of the comparison by approximating the irrational numbers with rational numbers and then using what I know about multiplication to estimate their values. Once I figured out approximations for the irrational numbers, I could find products with those approximations to complete the comparison. That’s TLM practice #7. • Use Think‑Pair‑Share to have students answer the following question: If 45 is 45 3 about 6.75, what is _ ? • Randomly select a few students to share. Possible answer: It’s about 2.25. • Use Team Huddle to have teams practice estimating expressions that contain irrational numbers. Decide which number is greater without using a calculator. 1)4 20 or 2 20 Explain your thinking. Random Reporter Rubric | Possible Answer 20 . 2 20 Answer: 4 Explanation: Estimate: 4(4.5)or 2(6.4) 18 . 12.8 Math Practice: I made sense of the comparison by approximating the irrational numbers with rational numbers and then using what I know about multiplication to estimate their values. Once I figured out that 20 is about halfway between 4 and 5, I could find 4 times that value (TLM #7). © 2015 Success for All Foundation PowerTeaching Math 3rd Edition | Unit Guide 27 Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 5 Access Code: zwdffn • Use Random Reporter to have teams share. Use the Random Reporter rubric to evaluate responses and give feedback. • Record individual scores on the teacher cycle record form. • Award team celebration points. team mastery (10–15 minutes) • Ask students to follow the Team Mastery student routine. • Circulate and use the following questions to prompt discussions. –– What information do you need to solve this problem? –– How did you approximate the square root of this number? –– Which perfect squares does the square fall directly between? –– How did you figure out which expression was greater? • When there are 5 minutes left in Team Mastery, prompt teams to prepare for the Lightning Round. Have teams discuss one Team Mastery problem that the whole team has completed. • Award team celebration points for good team discussions that demonstrate 100‑point responses. lightning round (10 minutes) • Tell students the Team Mastery problem that you will use for the Lightning Round. Decide which number is greater without using a calculator. 4) – 5 5 or – 25 Explain your thinking. Random Reporter Rubric | Possible Answer 5 is greater Answer: – 5 Explanation: Estimate: – 5(2.25) or – 25 – 11.25 or – 25 – 11.25 . – 25 Math Practice: I made sense of the comparison by approximating the irrational number with a rational number and then using what I know about multiplication to estimate the value. Once I figured out that 5 is about a quarter of the way from 2 to 3, I could find – 5 times that value (TLM #7). • Use Random Reporter to have teams share. Use the Random Reporter rubric to evaluate responses and give feedback. • Record individual scores on the teacher cycle record form. • Award team celebration points. 28 PowerTeaching Math 3rd Edition | Unit Guide © 2015 Success for All Foundation Access Code: zwdffn Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 5 celebration (2 minutes) • Record team celebration points on the poster. • Have the top team choose a cheer. • Assign homework, and remind students about the Vocabulary Vault. • Ask students to follow the Quick Check student routine. (optional) Decide which number is greater without using a calculator. 3 12 or 3 11 Possible answer: 3 12 is greater. © 2015 Success for All Foundation PowerTeaching Math 3rd Edition | Unit Guide 29 Access Code: brdpsg Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Assessment Day Assessment Day: Unit Check on Rational and Irrational Numbers Materials: extra blank copies of the assessment Lesson Objective: Demonstrate mastery of unit content. assessment (20–30 minutes) • Confirm the number of students who completed the homework on each team. Award team celebration points. • Remind the students that the test is independent work. • Distribute the tests so students can preview the questions. • Tell students how much time they have for the test and that they may begin. Give students a 5‑minute warning before the end of the test. • Collect the tests. team reflection (5 minutes) • Display or hand out blank copies of the test. • Explain or review, if necessary, the student routine for team discussions after the test. • Award team celebration points. prep points (5–10 minutes) • Assign prep points for each team for the five questions indicated (#s 2, 6, 7, 11, 14). • Score individual tests when convenient. vocabulary vault (2 minutes) • Randomly select Vocabulary Vouchers, and award team celebration points. • Ask students to record the words that they explain on their team score sheets. team scoring (5 minutes) • Lead the class in completing the team scoring on their team score sheets. © 2015 Success for All Foundation PowerTeaching Math 3rd Edition | Unit Guide 31 Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Assessment Day celebration Access Code: brdpsg (2 minutes) • Announce team statuses, and celebrate. • Poll teams about how many times they have been super teams. Celebrate those teams, and encourage all teams to work toward super team status during the next cycle. • Play the video “Practice Active Listening.” • Use Think-Pair-Share to have students discuss how this goal can help them reach super team status. Randomly select a few students to share. 32 PowerTeaching Math 3rd Edition | Unit Guide © 2015 Success for All Foundation student pages Level 8 8 || Unit 2: Rational and Irrational Numbers Numbers || Cycle 1 1 Lesson Lesson 1 1 teamwork 1) Provide two examples that show that the statement is false. Lydia said that all square roots are irrational numbers. Directions for questions 2 and 3: Classify the numbers as rational or irrational. 2) 77 3) 48 Directions for questions 4 and 5: Find the square roots for each number. 4) 121 5) 81 Directions for questions 6–8: Classify the numbers as rational or irrational. 6) – 8.875 7) 16 8) 2.67034165508… 9) Provide two examples that show that the statement is false. Sebastian said that if a number is a perfect square, then the number is even. Directions for questions 10–12: Find the square roots for each number. 10) 25 11) 49 12) 144 Challenge 13) Is the product of a rational and an irrational number rational or irrational? Give an example to support your answer. 14) Do the expressions 100 2 64 and 100 2 64 have the same value? Explain your thinking. © 2015 Success for All Foundation Foundation Math 3rd Edition PowerTeaching Math PowerTeaching 3rd Edition | Unit Guide 35 Level 8 8 || Unit 2: Rational and Irrational Numbers Numbers || Cycle 1 1 Lesson Lesson 1 1 teamwork answers 1) Possible answers: Example 1: 4 Example 2: 100 2) irrational 3) irrational 4) 11 and – 11 5) 9 and – 9 6) rational 7) rational 8) irrational 9) Possible answers: Example 1: 49 Example 2: 121 10) 5 and – 5 11) 7 and – 7 12) 12 and – 12 13) The product of a rational and an irrational number is irrational. Accept reasonable explanations. 14) No, these expressions do have not the same value. Possible explanation: To solve the first expression, I subtracted 64 from 100 and got 36. The square root of 36 is equal to 6. For the second expression, I found the square roots of 100 and 64 and got 10 and 8. Then, I subtracted 8 from 10 and got 2. Therefore, the values of the expressions are not the same. PowerTeaching Math 3rdMath Edition 36 PowerTeaching 3rd Edition | Unit Guide © 2015 Success for All Foundation quick check Level 8 8 || Unit 2: Rational and Irrational Numbers Numbers || Cycle 1 1 Lesson Lesson 1 1 Name Find the square roots for 64. © 2015 Success for All Foundation quick check PowerTeaching Math 3rd Edition Name Find the square roots for 64. © 2015 Success for All Foundation Foundation Math 3rd Edition PowerTeaching Math PowerTeaching 3rd Edition | Unit Guide 37 Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson Level 81 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 1 homework Quick Look Vocabulary words introduced in this cycle: rational number, radical sign, square root, perfect square, irrational number Today we defined and explored irrational numbers. An irrational number is a number that cannot be written in fractional form. We know a number is irrational if it is a decimal number that is infinitely long and has no repeating pattern. We also learned the difference between rational and irrational numbers. For example: Number Type and Explanation 2 Irrational; 2 is not a perfect square. 9 Rational; 9 is a perfect square, 9 5 3. 0.0101010101… 0.01001000100001… Rational; repeating decimal, it has a pattern Irrational; nonrepeating, nonterminating decimal We learned that the square root of a number is a number that when multiplied by itself, equals the original number. Square roots include both positive and negative numbers. For example, the square root of 2 2 25 is 5 because (5) 5 25, and (– 5) 5 25. However, if we write it as 25 , then we are only talking about the positive root, 5. 1) Provide two examples that show the statement is false. Explain your thinking. Zoe said that an irrational number can be expressed as a terminating decimal. Directions for questions 2–6: Classify the numbers as rational or irrational. 2) p 3) 110 4) 81 5) 14 6) 2 –_ 3 Directions for questions 7 and 8: Find the square roots for each number. 7) 100 8) 49 © for All Foundation 382015 Success PowerTeaching Math 3rd Edition | Unit Guide Math 3rd Edition ©PowerTeaching 2015 Success for All Foundation Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson Level 8 1 | Unit 2: Rational and Irrational Numbers | Cycle 1 Homework Lesson 1 Mixed Practice 9) Evaluate the expression. 229?123?9 10) Solve for x. 6x 2 5 5 59 11) You have a number cube labeled 1–6. What is the probability of rolling an even number? 12) What is the measure of the radius of the circle whose circumference is 21.98 inches? Use 3.14 for p. Word Problem 13) Tell what an irrational number is in your own words. Give an example of an irrational number and a rational number. PowerTeaching Math Edition © 2015 Success for All3rd Foundation 2015 Success for Guide All Foundation PowerTeaching Math©3rd Edition | Unit 39 Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson Homework Level 81 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 1 For the Guide on the Side Today your student defined irrational numbers. An irrational number is a number that cannot be written in fractional form. Any infinite nonrepeating decimal is an irrational number. Your student also discussed square roots by learning to identify perfect squares and nonperfect squares and to tell when a nonperfect square is irrational. Your student learned that the square root of a perfect square is a rational number. Furthermore, the square root of a number that is not a perfect square is an irrational number. Your student should be able to answer the following questions about irrational numbers: 1) What is an irrational number? 2) What’s the difference between a rational and an irrational number? 3) How did you know this was an irrational number? 4) How can a number have two square roots? 5) What is a perfect square? 6) How does the square root of a number relate to its classification? Here are some ideas to use to practice working with irrational numbers: 1) Understand and apply the definition of irrational numbers: http://learnzillion.com/lessons/220‑understand‑and‑apply‑the‑definition‑of‑irrational‑numbers 2) What’s an irrational number?: http://virtualnerd.com/pre‑algebra/real‑numbers‑right‑triangles/real‑and‑irrational/define‑real‑numbers/ irrational‑number‑definition 3) Understanding Square Roots: www.khanacademy.org/math/arithmetic/exponents‑radicals/radical‑radicals/v/understanding‑square‑roots 4) Think of two integers on a number line. What rational numbers can you think of that fall between the two integers? What irrational numbers fall between them? © for All Foundation 402015 Success PowerTeaching Math 3rd Edition | Unit Guide Math 3rd Edition ©PowerTeaching 2015 Success for All Foundation Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson Level 8 1 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 1 homework answers 1) Possible answers: Example 1: p Example 2: 2 Possible explanation: I know the statement is false because an irrational number cannot be expressed in fractional form. A terminating decimal is a number that can be written over a power of 10 in fractional form. I examined the statement Zoe made to determine whether her argument made sense. I used what I know about the definitions of irrational numbers and terminating decimals to prove her statement false (TLM #3). I know that any example of a terminating decimal cannot be an irrational number, and an irrational number (since its decimal does not repeat or terminate) cannot be expressed as a terminating decimal. 2) irrational 3) irrational 4) rational 5) irrational 6) rational 7) 10 and – 10 8) 7 and – 7 Mixed Practice 9) – 34 2 10) 10 _ 3 1 11) 0.5 or _ 2 12) r 5 3.5 in. Word Problem 13) Possible answer: An irrational number is a number that cannot be written in fractional form. It includes the square roots of nonperfect squares and decimals that keep going without any pattern. Accept appropriate examples. PowerTeaching Math Edition © 2015 Success for All3rd Foundation 2015 Success for Guide All Foundation PowerTeaching Math©3rd Edition | Unit 41 Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson Level 82 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 2 teamwork Classify the numbers by writing them in the appropriate section of the Venn diagram. 1) 10 , –_ 2 0 8 4 9 36 , _, 7, 140 , _, 4 , – 8, 8 , – 2.89 2) – 25, 72 21 _ , 14 , 6.5, _ , 81 9 6 6 3 12 , 3) 53 , – _ 19 , 8, _ , p, 0, _ 7 12 4) 44, _ , 144 , – 88, _ , 5.4, 24 80 45 0 5) _ , 72 , _ , 50 , _ , 49 36 2 6) 4 , – 7, _ , 97 , _ , 3, 30 , p 45 6 1 6 2 94 © for All Foundation 422015 Success PowerTeaching Math 3rd Edition | Unit Guide 3 2 9 9 Math 3rd Edition ©PowerTeaching 2015 Success for All Foundation Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson Level 8 2 | Unit 2: Rational and Irrational Numbers | Cycle 1 Teamwork Lesson 2 3 7 7) 1, _ , 37 , _ , 23 , 16 8) 25 2 9) 83, – 9.3, 12, _ , 90 , 9.8, _ , 63 0 7 1 10) _ , 32, 6.4, _ , 87 , – 30, _ 45 19 26 2 9 , – 88, –_ 1 6 2 9.2, 26, 71 , – 67, _ , 1 8 41 3 Challenge Find the square roots. 11) 0.25 1 12) _ 4 PowerTeaching Math Edition © 2015 Success for All3rd Foundation 2015 Success for Guide All Foundation PowerTeaching Math©3rd Edition | Unit 43 Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson Level 82 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 2 teamwork answers 1) 2) 3) 4) 3) 6) 7) 8) © for All Foundation 442015 Success PowerTeaching Math 3rd Edition | Unit Guide Math 3rd Edition ©PowerTeaching 2015 Success for All Foundation Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson Level 8 2 | Unit 2: Rational and Irrational Numbers Teamwork | Cycle 1 Lesson Answers 2 9) 10) 11) 0.5 and – 0.5 1 1 12) _ and – _ 2 2 PowerTeaching Math Edition © 2015 Success for All3rd Foundation 2015 Success for Guide All Foundation PowerTeaching Math©3rd Edition | Unit 45 Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson Level 82 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 2 quick check Name Classify the numbers by writing them in the appropriate section of the Venn Diagram. 18 55 9.3, _ , 36 , – 13, _ , 1.7, 41 9 4 © 2015 Success for All Foundation PowerTeaching Math 3rd Edition quick check Name Classify the numbers by writing them in the appropriate section of the Venn Diagram. 18 55 9.3, _ , 36 , – 13, _ , 1.7, 41 9 4 © for All Foundation 462015 Success PowerTeaching Math 3rd Edition | Unit Guide Math 3rd Edition ©PowerTeaching 2015 Success for All Foundation Level 8 8 || Unit 2: Rational and Irrational Numbers Numbers || Cycle 1 1 Lesson Lesson 2 2 homework Quick Look Vocabulary words introduced in this cycle: rational number, radical sign, square root, perfect square, irrational number Today we delved further into understanding rational versus irrational numbers. We also used a Venn diagram to help us classify rational and irrational numbers and see the relationships between classifications. For example, the Venn diagram shows how we classify numbers. Directions for questions 1–5: Classify the numbers by writing them in the appropriate section of the Venn diagram. 8 9 , 47 , – 7, _ , 81 , 10.6, 2 1) _ 3 3 © 2015 Success for All Foundation Foundation 14 – 4 2) _ , 63, 2.8, 0, 25 , – 3.9, _ , 75 , 24 27 9 Math 3rd Edition PowerTeaching Math PowerTeaching 3rd Edition | Unit Guide 47 Level 8 8 || Unit 2: Rational and Irrational Numbers Numbers || Cycle 1 1 Lesson Lesson 2 2 Homework 8 – 0 24 2 3) _ , 80, 5 , _ , 55, _ , 67 , – 48, _ , 91 7 16 6 6 Explain your thinking. 5 12 – , 96 , – 87, _ , 8.4, 49 4) 14, _ 6 2 4 – 4 5) _ , 2.6, 11, 2 , – 82, _ , 32 4 9 Mixed Practice 6) Write 0.35 as a fraction in simplest form. 7) Anita purchased 3 pairs of shoes for a total of $89.13. Calculate the average cost that Anita spent for 1 pair of shoes. 8) Is 26,877,240 divisible by 2, 3, 4, 5, 6, 7, 8, 9, and 10? 6 2 9) _ 4_ 7 3 Word Problem 10) Siddiquah said that zero is not a rational number because you cannot write zero in fractional form. Is Siddiquah correct? Explain your thinking. PowerTeaching Math 3rdMath Edition 48 PowerTeaching 3rd Edition | Unit Guide © 2015 Success for All Foundation Homework Level 8 8 || Unit 2: Rational and Irrational Numbers Numbers || Cycle 1 1 Lesson Lesson 2 2 For the Guide on the Side Today your student classified numbers as rational, integers, whole, natural, or irrational. In a previous lesson, your student learned that irrational numbers are numbers that cannot be written in fractional form and that an infinite nonrepeating decimal is an irrational number. He or she recalled the following: • Natural numbers 5 counting numbers: 1, 2, 3, … • Whole numbers 5 natural numbers and zero: 0, 1, 2, 3, … • Integers 5 positive and negative whole numbers: – 3, – 2, – 1, 0, 1, 2, 3, … 72 • Rational numbers 5 numbers that can be written in fractional form: – 1.2, _ ,4 5 In the next lesson, your student will confirm that repeating decimals are rational numbers. Your student should be able to answer the following questions about classifying numbers: 1) How would you classify this number? 2) How many different ways can you categorize this number? 3) What is the difference between a rational and an irrational number? 4) What is the difference between a terminating and a repeating decimal? 5) How can you distinguish between all the subsets of rational numbers? Here are some ideas to practice classifying numbers: 1) Learn Zillion: Distinguish Between Rational and Irrational Numbers: http://learnzillion.com/lessons/221‑distinguish‑between‑rational‑and‑irrational‑numbers 2) Virtual Nerd: How do different categories of numbers compare to each other? http://virtualnerd.com/algebra‑1/algebra‑foundations/number‑category‑comparison.php © 2015 Success for All Foundation Foundation Math 3rd Edition PowerTeaching Math PowerTeaching 3rd Edition | Unit Guide 49 Level 8 8 || Unit 2: Rational and Irrational Numbers Numbers || Cycle 1 1 Lesson Lesson 2 2 homework answers 1) 2) 3) Possible explanation: I started by examining the fractions to see if I could simplify any of them. I found that one fraction was equivalent to a natural number while another was equivalent to 0. I placed the other fractions in the section for rational numbers. Next, I determined that the square roots were not for perfect squares and, therefore, were irrational numbers. I placed the remaining rational numbers in the appropriate sections based on their definitions. My classification makes sense because I recalled the definition of each section before determining which number fit into that section (TLM #1). 4) 5) PowerTeaching Math 3rdMath Edition 50 PowerTeaching 3rd Edition | Unit Guide © 2015 Success for All Foundation Homework Answers Level 8 8 || Unit 2: Rational and Irrational Numbers Numbers || Cycle 1 1 Lesson Lesson 2 2 Mixed Practice 7 6) _ 20 7) Anita spent $29.71 on 1 pair of shoes. 8) 26,867,240 is divisible by 2, 3, 4, 5, 6, 8, 9, and 10. 9 2 9) _ or 1_ 7 7 Word Problem 10) No. Siddiquah is not correct. 0 Possible explanation: Zero can be written in fractional form as _ x , where x stands for any number other than 0 because zero divided by any number other than zero is equal to zero. When zero is the denominator (or divisor) the answer is undefined. © 2015 Success for All Foundation Foundation Math 3rd Edition PowerTeaching Math PowerTeaching 3rd Edition | Unit Guide 51 Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson Level 83 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 3 teamwork 1) Find the fractional equivalent of 0.234 . Show your work. 2) Find the fractional equivalent of 0.91 . Show your work. 3) Find the fractional equivalent of 0.4 . Show your work. 4) Find the fractional equivalent of 0.5 . Show your work. 5) Find the fractional equivalent of 0.21 . Show your work. 6) Find the fractional equivalent of 0.37 . Show your work. 7) Find the fractional equivalent of 0.719 . Show your work. 8) Find the fractional equivalent of 0.372 . Show your work. Challenge 9) Find the fractional equivalent of 1.2345 . Show your work. © for All Foundation 522015 Success PowerTeaching Math 3rd Edition | Unit Guide Math 3rd Edition ©PowerTeaching 2015 Success for All Foundation Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson Level 8 3 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 3 teamwork answers 234 78 26 1) _ or _ or _ 999 333 111 91 2) _ 99 4 3) _ 9 5 4) _ 9 21 7 5) _ or _ 99 33 37 6) _ 99 719 7) _ 999 372 124 8) _ or _ 999 333 12,222 2,322 1,161 387 129 9) __ or 1 _ or 1 _ or 1 _ or 1 _ 9,900 9,900 PowerTeaching Math Edition © 2015 Success for All3rd Foundation 4,940 1,650 550 2015 Success for Guide All Foundation PowerTeaching Math©3rd Edition | Unit 53 Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 3 quick check Level X | Unit X: UnitTitle | Cycle X Lesson X Name Find the fractional equivalent of 0.426 . Show your work. © 2015 Success for All Foundation quick check PowerTeaching Math 3rd Edition Name Find the fractional equivalent of 0.426 . Show your work. © for All Foundation 542015 Success PowerTeaching Math 3rd Edition | Unit Guide Math 3rd Edition ©PowerTeaching 2015 Success for All Foundation Level 8 8 || Unit 2: Rational and Irrational Numbers Numbers || Cycle 1 1 Lesson Lesson 3 3 homework Quick Look Vocabulary words introduced in this cycle: rational number, radical sign, square root, perfect square, irrational number Today we explored why repeating decimals are rational numbers and how to convert them from repeating decimals to their fractional equivalents. For example: To find the fractional equivalent of 0.158 , x 5 0.158 1,000x 5 158.158 1,000x 5 158.158 2x5 0 .158 Multiply by a power of 10. Subtract. 999x 5 158 158 x5_ 999 Divide to isolate the variable. 1) Find the fractional equivalent of 0.57 . Show your work. 2) Find the fractional equivalent of 0.238 . Show your work. 3) Find the fractional equivalent of 0.63 . Show your work. Explain your thinking. 4) Find the fractional equivalent of 0.7 . Show your work. 5) Find the fractional equivalent of 0.374 . Show your work. Mixed Practice 6) Simplify the expression: 3t 4t 5t. 7) Evaluate the expression. 49442528 8) Classify 120 as rational or irrational. 9) Find the surface area of a cube whose side length is 1.85 in. Word Problem 10) In your own words, describe how to convert a repeating decimal into a fraction. © 2015 Success for All Foundation Foundation Math 3rd Edition PowerTeaching Math PowerTeaching 3rd Edition | Unit Guide 55 Level 8 8 || Unit 2: Rational and Irrational Numbers Numbers || Cycle 1 1 Lesson Lesson 3 3 Homework For the Guide on the Side Today your student confirmed that repeating decimals are rational numbers because they can be written in fractional form. To develop a clear concept of the differences between rational and irrational numbers, your student will examine, compare, estimate, and order numbers and use his or her number sense. In the next lesson, your student will approximate the decimal value of irrational numbers. Your student should be able to answer the following questions about classifying numbers: 1) How did you write this repeating decimal as a fraction? 2) How do you know repeating decimals are rational numbers? 3) Why do you multiply each side by a multiple of 10? Here are some ideas to practice classifying numbers: 1) Learn Zillion: Convert repeating decimals to fractions: http://learnzillion.com/lessons/223‑convert‑repeating‑decimals‑into‑fractions 2) Virtual Nerd: How do you turn a repeating decimal into a fraction?: http://virtualnerd.com/pre‑algebra/rational‑numbers/repeating‑decimal‑to‑fraction‑conversion.php PowerTeaching Math 3rdMath Edition 56 PowerTeaching 3rd Edition | Unit Guide © 2015 Success for All Foundation Level 8 8 || Unit 2: Rational and Irrational Numbers Numbers || Cycle 1 1 Lesson Lesson 3 3 homework answers 57 19 1) _ or _ 99 33 238 2) _ 999 63 7 3) _ or _ 99 11 Possible explanation: I started by expressing the repeating decimal as an equation. Then, I multiplied each side of the equation by 100 because two digits repeat. This moved the set of repeating digits to the left side of the decimal point. Next, I subtracted x from 100x to get rid of the decimal. Finally, I 63 isolated the variable and solved for x. So 0.63 is equivalent to _. I used what I knew about converting a 99 terminating decimal into an equation to convert a repeating decimal into an equation (TLM #1). 7 4) _ 9 374 5) _ 999 Mixed Practice 6) 12t 7) – 6.75 8) irrational 9) 20.535 in. 2 Word Problem 10) Accept reasonable answers. © 2015 Success for All Foundation Foundation Math 3rd Edition PowerTeaching Math PowerTeaching 3rd Edition | Unit Guide 57 Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson Level 84 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 4 teamwork 1) Graph each number on the number line below. 9 2.55, _ , 24 , , 9 5 –5 –4 –3 –2 –1 0 1 2 3 4 5 2) Use , , or to compare the following numbers. 55 _______ 7.874007874… 3) Write the following numbers in order from least to greatest. Explain your thinking. 9, –_ 6 6 8.9, – 12 , _ , 7 , – 8.3 3 4) Write the following numbers in order from greatest to least. – 2.3, 23 1 0, _ , , – 7 _ 7 5 5) Graph each number on the number line below. 10 _ , 10 , 4.75, 16 4 6) Use , , or to compare the following numbers. 40 _______ 5.916079783… 7) Write the following numbers in order from least to greatest. 3, –_ 9 9 6.6, – 54 , _ , 75 , 5 8) Write the following numbers in order from greatest to least. 9 9 2 5 34 , – 4.2, 9.6, _, – 85 , – 7 _ 9) Graph each number on the number line below. 24 3 85 , _, 8.85, 100 10) Use , , or to compare the following numbers. 99 ________ 9.38083152… Challenge 11) Find five irrational numbers and five rational numbers between 5 and 6. Explain your thinking. © for All Foundation 582015 Success PowerTeaching Math 3rd Edition | Unit Guide Math 3rd Edition ©PowerTeaching 2015 Success for All Foundation Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson Level 8 4 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 4 teamwork answers 1) 2) 3) 55 , 7.874007874… – 8.3, – 6 6 3 9 , _, 7 , 8.9 12 , – _ 23 1 4) _ , p, 0, – 2.3, – 7 _ 5 3 5) 6) 7) 40 . 5.916079783… – 9 9 5 3 , _, p, 6.6, 75 54 , – _ 9 – 2 – 8) 9.6, 34 , _ , 4.2, – 7 _ , 85 9 5 9) 10) 99 . 9.38083152… 11) Possible irrational answers: 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 26 339 987 2 Possible rational answers: 5.1, _ , 5.313, 5 _ , 5.58, 5 _ , 5.7, 5.81, 5 _ 5 5 500 1,000 Possible explanation: 5 is the square root of 25 while 6 is the square root of 36. Therefore, I know that I’m looking for square roots between 25 and 36. None of these numbers are perfect squares, so their square roots are irrational. PowerTeaching Math Edition © 2015 Success for All3rd Foundation 2015 Success for Guide All Foundation PowerTeaching Math©3rd Edition | Unit 59 Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson Level 84 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 4 quick check Name Use ,, ., or 5 to compare the following numbers. 6.1678203027… _____ 32 © 2015 Success for All Foundation quick check PowerTeaching Math 3rd Edition Name Use ,, ., or 5 to compare the following numbers. 6.1678203027… _____ 32 © for All Foundation 602015 Success PowerTeaching Math 3rd Edition | Unit Guide Math 3rd Edition ©PowerTeaching 2015 Success for All Foundation Level 8 8 || Unit 2: Rational and Irrational Numbers Numbers || Cycle 1 1 Lesson Lesson 4 4 homework Quick Look Vocabulary words introduced in this cycle: rational number, radical sign, square root, perfect square, irrational number Today we arranged rational and irrational numbers in order from least to greatest and/or from greatest to least. We also compared numbers, determining which value was the least and which was the greatest. We used a number line to approximate an irrational number’s location. For example: 9 We can compare _ to 9 by approximating each numbers location on a number line. 5 9 So _ . 9 . 5 1) Graph each number on the number line below. 17 5.5, 40 , _ , 55 4 2) Graph each number on the number line below. 78 7.25, 45 , _ , 70 8 3) Use ,, ., or 5 to compare the following numbers. – 10 ____ – 104 4) Use ,, ., or 5 to compare the following numbers. 33 _____ 5.19615242… 5) Write the following numbers in order from least to greatest. Explain your thinking. 9 – 5 _ , 9 , _ , 24 , p 6 5 5 1, –_ © 2015 Success for All Foundation Foundation Math 3rd Edition PowerTeaching Math PowerTeaching 3rd Edition | Unit Guide 61 Level 8 8 || Unit 2: Rational and Irrational Numbers Numbers || Cycle 1 1 Lesson Lesson 4 4 Homework 6) Write the following numbers in order from greatest to least. – 6.3, 8 4 0, _ , 7.1, – 6 _ 7 5 Mixed Practice 7) Find the unit rate. 15 books in 3 days 8) – 68.86 85.18 9) What is 75% of 96? 4 2 10) Multiply. 3 _ 8_ 7 5 Word Problem 11) A local television station recorded the daily low temperature for a week in January. Sunday Monday Tuesday Wednesday Thursday Friday Saturday – 15C – 16.7C – 17.8C – 21.1C – 20.6C – 18.9C – 17.8C Which temperature is the highest? Which is the lowest? For the Guide on the Side Today your student learned how to compare and order numbers and approximate their locations on a number line. Your student worked with both rational and irrational numbers. Your student should be able to answer the following questions about comparing and ordering numbers: 1) Is this number rational or irrational? How do you know? 2) Between which numbers does this square root fall? 3) How did you determine which number is greater/lesser? 4) How can you compare decimals to fractions? 5) How did you determine the order of this set of numbers? Here are some ideas to practice comparing and ordering numbers: 1) Learn Zillion: Comparing Irrational and Rational Numbers: http://learnzillion.com/lessons/222‑compare‑irrational‑and‑rational‑numbers 2) Virtual Nerd: How do you put real numbers in order?: http://virtualnerd.com/common‑core/grade‑8/8_NS‑number‑system/A/2/order‑real‑numbers‑example PowerTeaching Math 3rdMath Edition 62 PowerTeaching 3rd Edition | Unit Guide © 2015 Success for All Foundation Level 8 8 || Unit 2: Rational and Irrational Numbers Numbers || Cycle 1 1 Lesson Lesson 4 4 homework answers 1) 2) 3) – 10 . – 104 4) 33 . 5.19615242… 5) – 5 9 5 6 5 1 , _, _, p, 9 , – _ 24 Possible explanation: I started by converting the fractions and square roots into decimal form. I also wrote the decimal approximation for p. Since I am ordering the numbers from least to greatest, I used a number line to approximate each number’s location. My answer makes sense because as you go left, the values decrease, and as you go right, the values increase. 8 4 6) 7.1, _ , 0, – 6.3, – 6 _ 7 5 Mixed Practice 7) 5 books per day 8) 16.32 9) 72 10) 30 Word Problem 11) The highest temperature is – 15C while the lowest temperature is – 21.1C. © 2015 Success for All Foundation Foundation Math 3rd Edition PowerTeaching Math PowerTeaching 3rd Edition | Unit Guide 63 Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson Level 85 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 5 teamwork Directions: Decide which number is greater without using a calculator. 1) 4 20 or 2 40 Explain your thinking. 2) – 12 or – 8 8 3) 3 5 or 88 4) –5 5 or – 25 Explain your thinking. 5) 2 30 or 3 20 6) –7 7 or – 27 7) 6 50 or 5 60 Challenge 8) Without using the square or cube root button on your calculator, approximate the cubic root of 25 to the nearest hundredth. © for All Foundation 642015 Success PowerTeaching Math 3rd Edition | Unit Guide Math 3rd Edition ©PowerTeaching 2015 Success for All Foundation Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson Level 8 5 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 5 teamwork answers 1) 4 20 . 2 40 Possible explanation: Estimate: 4(4.5) or 2(6.4) 18 . 12.8 I made sense of the comparison by approximating the irrational numbers with rational numbers and then using what I know about multiplication to estimate their values. Once I figured out that 20 is about halfway between 4 and 5, I could find 4 times that value (TLM #7). 2) – 12 is greater. 3) 3 5 , 88 4) –5 5 is greater. Possible explanation: Estimate: – 5(2.25) or – 25 – 11.25 or – 25 – 11.25 . – 25 I made sense of the comparison by approximating the irrational number with a rational number and then using what I know about multiplication to estimate the value. Once I figured out that 5 is about a quarter of the way from 2 to 3, I could find – 5 times that value (TLM #7). 5) 2 30 , 3 20 6) –7 7 is greater. 7) 6 50 . 5 60 8) 2.92 PowerTeaching Math Edition © 2015 Success for All3rd Foundation 2015 Success for Guide All Foundation PowerTeaching Math©3rd Edition | Unit 65 Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson Level 85 | Unit 2: Rational and Irrational Numbers | Cycle 1 Lesson 5 quick check Name Decide which number is greater without using a calculator. 3 12 or 3 11 © 2015 Success for All Foundation quick check PowerTeaching Math 3rd Edition Name Decide which number is greater without using a calculator. 3 12 or 3 11 © for All Foundation 662015 Success PowerTeaching Math 3rd Edition | Unit Guide Math 3rd Edition ©PowerTeaching 2015 Success for All Foundation Level 8 8 || Unit 2: Rational and Irrational Numbers Numbers || Cycle 1 1 Lesson Lesson 5 5 homework Quick Look Vocabulary words introduced in this cycle: rational number, radical sign, square root, perfect square, irrational number Today we estimated and compared the value of expressions that contained irrational numbers. For example: In each of these pairs, we were able to determine which number was greater without using a calculator. Directions for questions 1–4: Determine which number is greater without using a calculator. 1) 7 80 or 8 70 2) 3 50 or 5 30 Explain your thinking. 3) 40 or 12 10 4) –1 30 or – 2 Mixed Practice 5) The ratio of boys to girls in grade 8 at Robinson Middle School is 5:7. If there are 600 students in grade 8, how many are girls? 6) Find both square roots for 81. 7) Find the area of a square whose side is 8.82 centimeters. 8) Classify 100 as: natural number, whole number, integer, rational number, irrational number. (Use all that apply.) Word Problem 9) A square‑shaped pool has an area of 234 square meters. Find the approximate length of each side of the pool. © 2015 Success for All Foundation Foundation Math 3rd Edition PowerTeaching Math PowerTeaching 3rd Edition | Unit Guide 67 Level 8 8 || Unit 2: Rational and Irrational Numbers Numbers || Cycle 1 1 Lesson Lesson 5 5 Homework For the Guide on the Side Today your student learned how to approximate the square root of irrational numbers. Your student used his or her number sense to estimate a good rational approximation of the square root of an irrational number. He or she started with the perfect squares that the number falls between. Next, he or she focused on the square roots of the perfect squares to narrow the range of the approximation. Even though we cannot calculate an exact rational equivalent for the square root of an irrational number (because the decimal does not terminate or repeat), your student was able to calculate a good approximation. In the next lesson, your student will compare and order rational and irrational numbers. Your student should be able to answer the following questions about approximating irrational numbers: 1) What information do you need to solve this problem? 2) How did you approximate the square root of this number? 3) Which perfect squares does the square fall directly between? 4) How did you figure out which expression was greater? Here are some ideas to practice approximating irrational numbers: 1) Khan Academy: Approximating Square Roots: www.khanacademy.org/math/arithmetic/exponents‑radicals/radical‑radicals/v/approximating‑square‑roots 2) Learn Zillion: Estimate the value of square roots: http://learnzillion.com/lessons/3136‑estimate‑the‑value‑of‑square‑roots PowerTeaching Math 3rdMath Edition 68 PowerTeaching 3rd Edition | Unit Guide © 2015 Success for All Foundation Level 8 8 || Unit 2: Rational and Irrational Numbers Numbers || Cycle 1 1 Lesson Lesson 5 5 homework answers 1) 7 80 , 8 70 2) 3 50 , 5 30 Possible explanation: Estimate: 3(7.1) or 5(5.5) 21.3 , 27.5 I made sense of the comparison by approximating the irrational numbers with rational numbers and then using what I know about multiplication to estimate their values. Once I figured out that 30 is about halfway between 5 and 6, I could find 5 times that value. That’s TLM practice #7. 3) 40 is greater. 4) –2 is greater. Mixed Practice 5) There are 350 girls in grade 8. 6) 9 and – 9 2 7) 77.7924 cm 8) natural number, whole number, integer, rational number Word Problem 9) The length of each side of the pool is approximately 15.29 meters. © 2015 Success for All Foundation Foundation Math 3rd Edition PowerTeaching Math PowerTeaching 3rd Edition | Unit Guide 69 Level 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 AssessmentLevel Day 8 | Unit 2: Rational and Irrational Numbers Cycle 1 unit check Directions for question 1 and 2: Find the square roots for each number. 1) 49 2) 81 Directions for question 3–5: Classify each number as rational or irrational. 3) 63.88 4) 35 5) 36 Directions for question 6: Classify the numbers by writing them in the appropriate section of the Venn diagram. 6) 3 0, 144 , 1.1, 2 , _ , 8 , p, 2 16 , – 64, 128 Directions for questions 7 and 8: Find the fraction equivalent of each number. Show your work. 7) 0.27 8) 0.11… Directions for questions 9–13: On the number line, approximate the locations of the numbers. 9) 53 10) 22 11) 74 12) 39 13) 89 14) Determine which number is greater without using a calculator. Explain your thinking. 3 20 or 5 10 © for All Foundation 702015 Success PowerTeaching Math 3rd Edition | Unit Guide Math 3rd Edition ©PowerTeaching 2015 Success for All Foundation Level 8 | Unit 2: Rational and Irrational Numbers Cycle Level 1 8 | Unit 2: Rational and Irrational Numbers | Cycle 1 Assessment Day unit check answers Lesson 1: Define and explore rational and irrational numbers. [20 points] 1) 7 and – 7 [4 points] 2) 9 and – 9 [4 points] 3) rational [4 points] 4) irrational [4 points] 5) rational [4 points] Lesson 2: Classify rational and irrational numbers. [20 points] 6) [2 points each] Lesson 3: Convert a decimal expansion that repeats eventually into a rational number. [20 points] 7) 3 0.27 5 _ [5 points] Possible work: [5 points] 11 8) 1 0.11… 5 _ [5 points] Possible work: [5 points] 9 x 5 0.2727… 100x 5 27.2727… x 5 0.11… 10x 5 1.11… 100x 5 27.2727… 2 x 5 0.2727… 2 99x 5 27 9x 5 1 3 27 5 _ x5_ 99 10x 5 1.11… x 5 0.11… 11 PowerTeaching Math Edition © 2015 Success for All3rd Foundation 1 x5_ 9 2015 Success for Guide All Foundation PowerTeaching Math©3rd Edition | Unit 71 Level Cycle 8 Check | Unit Answers 2: Rational and Irrational Numbers | Cycle 1 AssessmentLevel Day 8 | Unit 2: Rational and Irrational Numbers Cycle 1 Lesson 4: Use knowledge of perfect squares and the number line to order rational and irrational numbers. [20 points] 9–13) [4 points each] Lesson 5: Use approximations of the value of irrational numbers to estimate and compare expressions containing irrational numbers. [20 points] 14) 3 20 , 5 10 [10 points] Possible explanation: [10 points] Estimate: 3(4.5) or 5(3.1) 13 or 15.5 13 , 15.5 I made sense of the comparison by approximating the irrational numbers with a rational number and then using what I know about multiplication to estimate the value. That’s TLM practice #7. Prep Points Analysis Question Number Team Scores (out of 20 points) Core Objective 2 Define and explore rational and irrational numbers. 6 Classify rational and irrational numbers. 7 Convert a decimal expansion that repeats eventually into a rational number. 11 Use knowledge of perfect squares and the number line to order rational and irrational numbers. 14 Use approximations of the value of irrational numbers to estimate and compare expressions containing irrational numbers. © for All Foundation 722015 Success PowerTeaching Math 3rd Edition | Unit Guide 1 2 3 4 5 Class Results (check if 16 out of 20 points or better) Math 3rd Edition ©PowerTeaching 2015 Success for All Foundation