Thinking Rationally. Identifying and Ordering Rational Numbers
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1 Thinking Rationally Identifying and Ordering Rational Numbers 1 WARM UP Determine the fraction represented by the shaded part of each grid. If necessary, rewrite in lowest terms LEARNING GOALS Understand that counting numbers, fractions, and decimals are rational numbers. Identify properties of rational numbers. Identify models for rational numbers. Fluently compare and order rational numbers. KEY TERMS positive rational number benchmark fraction You have learned about whole numbers, fractions, and decimals. How can you compare these types of numbers? LESSON 1: Thinking Rationally M1-71
2 Getting Started How Many Can You Name? You have learned about many different types of numbers. List as many types of numbers as you can. Give an example of each number type. ACTIVITY 1.1 Identifying Positive Rational Numbers You can group numbers in many different ways. Keep these cards when you're finished. You'll need them in a future lesson. 1. Cut out the cards at the end of this lesson. Sort the cards into different groups. You may sort the cards in any way you think is appropriate, but you must sort them into more than 1 group. Give each group of cards a title. Explain how you sorted the numbers and diagrams on the cards, including why you gave each group its title. 2. Compare your groupings with your classmates groupings. Create a list of some of the different ways to group the numbers. M1-72 TOPIC 2: Positive Rational Numbers
3 3. Vivianne grouped these cards together. What reason could she give for why she put these cards into the same group? NOTES, 1, Danika and Josh explained how they sorted the numbers. Danika I grouped these numbers together because they all represent whole numbers. 8_ 8, _, 10, 1, 0 0 _ 1 Show why Danika s reasoning is correct. Identify other numbers or diagrams that belong in Danika s group. Josh I grouped these numbers together because they are all equal. 3_, _ 3 4, _ 3 8 c. Explain why Josh s reasoning is not correct. d. Identify pairs of cards which show equal values. How many pairs can you find? LESSON 1: Thinking Rationally M1-73
4 ACTIVITY 1.2 Writing Positive Rational Numbers A positive rational number is a number that can be written in the form a, where a and b are both whole numbers greater than 0. b WORKED EXAMPLE Is 0.7 a rational number? To write a decimal like 0.7 in the form a, where a and b are both b whole numbers and b is not equal to 0: Read the decimal using place value. 0.7 seventy-five hundredths Write the decimal as a fraction The fraction is written in the form a, where a is equal to b 7 and b is equal to 100. The numbers 7 and 100 are both whole numbers greater than 0. So, 0.7 is a rational number. Any decimal greater than 0 that has a limited number of nonzero digits after the decimal point (like 0.) or whose digits repeat in a pattern (like ) is a positive rational number. 1. Show that the decimals 0.6, 0.1, 0.2, and 0.32 are positive rational numbers. 2. Which numbers, if any, that you sorted are not positive rational numbers? Explain your answer. M1-74 TOPIC 2: Positive Rational Numbers
5 ACTIVITY 1.3 Benchmark Fractions Benchmark fractions are common fractions you can use to estimate the value of fractions. Three common benchmark fractions are 0 1, 1 2, and A fraction is close to 0 when the numerator is very small compared to the denominator. A fraction is close to 1 when the 2 numerator is about half the size of the denominator. A fraction is close to 1 when the numerator is very close in size to the denominator. 1. Name the closest benchmark fraction for each fraction given c. 6 9 d. 67 e. 7 1 f. 7 g. 6 h i. 13 j k. 11 l Write the unknown numerator or denominator so that each fraction is close to but greater than 0. ( ) ( ) 27 c. 8 ( ) d. 7 ( ) LESSON 1: Thinking Rationally M1-7
6 3. Write the unknown numerator or denominator so that each fraction is close to but less than 1 2. ( ) ( ) 27 c. 8 ( ) d. 7 ( ) 4. Write the unknown numerator or denominator so that each fraction is close to but less than 1. ( ) ( ) 27 c. 8 ( ) d. 7 ( ). Describe the relationship between a and b when the fraction a b is: close to 0. close to 1 2. c. close to 1. An inequality is a statement that one number is less than or greater than another number. 6. Compare each pair of fractions using benchmark fractions. Insert a. or, symbol to make the inequality true. Explain your reasoning c d Compare the fractions in each pair. Think about how close the fractions are to 0, 1 2, or 1. 8 and and 7 8 c. 1 9 and 1 23 M1-76 TOPIC 2: Positive Rational Numbers
7 ACTIVITY 1.4 Ordering Rational Numbers Felipe and Corinne ordered the rational numbers 0.8, 0.06, and 3 from least to greatest using different strategies. Felipe used benchmark numbers, and Corinne used equivalent fractions. 1. Use Felipe s strategy of benchmark numbers to order the rational numbers from least to greatest. 2. Use Corinne s strategy of equivalent fractions to order the rational numbers from least to greatest. 3. Use any strategy to order the rational numbers 0.6, 3 4, and 8 from least to greatest. 4. List the fractions in each set in ascending order. 1 8, 1 11, 1 9, 1 4, 1 7, 1 4, 4 10, 4, 4 7 c. 3 8, 3 11, 3 9, 3 4, 3 7, 3. What do the fractions in each part of Question 4 have in common? Explain how you determined the order of the fractions in each. LESSON 1: Thinking Rationally M1-77
8 NOTES TALK the TALK Close to Half Consider the fractions shown. 9, 7 13, 2 7, Write the fractions in ascending order. Use what you know about benchmark fractions to determine the order. Explain your reasoning. M1-78 TOPIC 2: Positive Rational Numbers
9 LESSON 1: Thinking Rationally M1-79
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11 Assignment Write Describe a way to compare two positive rational numbers that are not written in the same form. Remember A positive rational number is a number that can be written in the form a, where a and b are both whole numbers, and b is not equal b to 0. An inequality is a statement that one number is less than or greater than another number. Practice Order the rational numbers in each group from least to greatest , 0.1, , 8, , 3, , 3.10, 4 1 8, 3.01, 2.3, 2 4, , 8 7, 6.34, 6 1 4, , , 0.23, 0, 1.89, 1 3, 1.02, , 2.4, , 3, 9.90, , 3.78, 3.9, , 0, 6.98, 2 1, 2.2, 6.89, Stretch Use reasoning to compare the fractions. Do not use common denominators. Explain your reasoning LESSON 1: Thinking Rationally M1-81
12 Review 1. In a video game, a character needs to shine a light through two spinning wheels that have holes in them. The first wheel makes a complete rotation in 7 seconds. The second wheel makes a complete rotation in 9 seconds. The holes are lined up at 0 seconds. How many seconds will pass before they are lined up again? 2. Your aunt s club is planning to sell small bags of different types of beads to people who want to make their own bead jewelry. The table below lists the different types of beads and how many they have. Type of Bead Quantity Oval bead 24 Metal bead 18 The club wants to divide these beads into bags so that each bag has exactly the same number of oval beads and metal beads. What is the greatest number of bags that they can make so that all of the beads are used and there is the same number of each bead in each bag? 3. Determine each sum or difference M1-82 TOPIC 2: Positive Rational Numbers
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