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The Properties of Number Systems Objective To summarize the properties of number systems and operations. a www.everydaymathonline.com ePresentations eToolkit Algorithms Practice EM Facts Workshop Game™ Teaching the Lesson Key Concepts and Skills • Identify terminating and repeating decimals. [Number and Numeration Goal 1] • Identify the absolute value of rational numbers. [Number and Numeration Goal 1] • Use the order of operations to solve problems. Family Letters Assessment Management Common Core State Standards Ongoing Learning & Practice Reviewing a Problem-Solving Diagram and Solving Number Stories Math Journal 2, p. 220 Student Reference Book, pp. 258 and 259 Students review a problem-solving diagram and solve multistep number stories. [Patterns, Functions, and Algebra Goal 3] • Apply the Identity, Commutative, Associative, and Distributive Properties of addition and multiplication. [Patterns, Functions, and Algebra Goal 4] • Apply the Multiplication Property of –1. [Patterns, Functions, and Algebra Goal 4] Key Activities Students explore the real number system and properties of various sets of numbers within it. Math Boxes 6 5 Math Journal 2, p. 221 Students practice and maintain skills through Math Box problems. Study Link 6 5 Math Masters, p. 193 Students practice and maintain skills through Study Link activities. Ongoing Assessment: Recognizing Student Achievement Use Mental Math and Reflexes. Curriculum Focal Points Interactive Teacher’s Lesson Guide Differentiation Options ENRICHMENT Investigating Properties of Rational Numbers Student Reference Book, pp. 104 –106 Math Masters, p. 194 Students explore properties of rational numbers by determining whether general statements are true or false. ENRICHMENT Renaming Repeating Decimals as Fractions Math Masters, p. 195 Students use a power-of-10 strategy to write equivalent equations and rename repeating decimals as fractions. EXTRA PRACTICE Plotting Numbers on a Number Line Class Number Line Students generate different types of numbers in real-life contexts. They plot them on a number line and compare them. [Operations and Computation Goal 4] ELL SUPPORT Key Vocabulary Naming and Comparing Numbers counting numbers whole numbers integers rational numbers terminating decimals repeating decimals irrational numbers real numbers Students write examples of various types of numbers and compare them. Materials Math Journal 2, pp. 218–220 Student Reference Book, pp. 2–24 and 94–106 Study Link 64 calculator Real Number Line Poster Additional Information When you use the ready-made Everyday Mathematics posters to support English language learners, you should display the English version only or both the English and Spanish versions simultaneously; do not display the Spanish version only. Teacher’s Reference Manual, Grades 4–6 pp. 59–77 Lesson 6 5 553 Mathematical Practices SMP2, SMP3, SMP4, SMP6, SMP7, SMP8 Content Standards Getting Started 6.NS.1, 6.NS.6, 6.NS.6a, 6.NS.6c, 6.NS.7, 6.NS.7a, 6.NS.7b, 6.NS.7c, 6.NS.7d, 6.EE.2 Mental Math and Reflexes Math Message Remind students that they have used visual models, the commondenominator method, and the invert-and-multiply method to solve fraction division problems. Pose division problems. Students write a number model and solve. Suggestions: 1 1 _ Geraldine is making curtains. Each curtain uses _ 4 yard of fabric. She has 4 yard 1 1 ÷_ = 1; 1 curtain of fabric. How many curtains can she make? _ 4 4 • How many positive numbers are there? • How many negative numbers are there? • How many numbers are neither positive nor negative? • Name one positive and one negative number, each of which is very close to 0. 7 1 Charlie made _ gallon of soup. He has containers that can store _ gallon. 8 3 5 7 1 21 _ _ _ How many containers does Charlie need to store the soup? _ 8 ÷ 3 = 8 , or 2 8 ; 3 containers 2 1 _ Larry has 2 _ 3 pints of ice cream in his freezer. One serving of ice cream is 6 pint. 2 1 _ _ How many servings of ice cream does Larry have? 2 ÷ = 16; 16 servings 3 Mental Math Ongoing and Reflexes Assessment: Recognizing Student Achievement Study Link 6 4a Follow-Up Briefly go over the answers. 6 1 Teaching the Lesson ▶ Math Message Follow-Up Use the Mental Math and Reflexes problems to assess students’ ability to divide fractions and mixed numbers. Students are making adequate progress if they are able to solve the suggested problems. WHOLE-CLASS DISCUSSION Most students probably know that there is an infinite amount of positive and negative numbers. Ask students to express this fact in a variety of ways: Positive (negative) numbers go on without end. It does not matter how large (small) a positive (negative) number I think of, I can always think of one that is larger (smaller). Zero is the only number that is neither positive nor negative. Have students share examples of positive and negative numbers that are close to zero. Record them on the board and have students put them in ascending order. [Operations and Computation Goal 4] Student Page Fractions Different Types of Numbers ▶ Presenting an Overview Counting is almost as old as the human race and has been used in some form by every human society. Long ago, people found that the counting numbers (1, 2, 3, and so on) did not meet all their needs. WHOLE-CLASS DISCUSSION of Our Number System ♦ Counting numbers cannot be used to express measures 1 between two consecutive whole numbers, such as 22 inches and 1.6 kilometers. (Student Reference Book, pp. 99 –102) ♦ With the counting numbers, division problems such as 8 / 5 and 3 / 7 do not have an answer. Fractions were invented to meet these needs. Fractions can also be renamed as decimals and percents. And most of the 1 1 numbers you have seen, such as 2, 56, 1.23, and 25%, are either fractions or can be renamed as fractions. With the invention of fractions, it became possible to express rates and ratios, to name many more points on the number line, and to solve any division problem involving whole numbers (except division by 0). However, even fractions did not meet every need. For example, 3 1 problems such as 5 7 and 24 54 have answers that are less than 0 and cannot be named as fractions. (Fractions, by the way they are defined, can never be less than 0). This led to the invention of negative numbers. Negative numbers are 1 numbers that are less than 0. The numbers 4, 3.25, and 100 are negative numbers. The number 3 is read as “negative 3.” Negative numbers serve several purposes: ♦ to name locations such as temperatures below zero on a thermometer and depths below sea level Every whole number can be renamed as a fraction. For example, 0 can be 0 written as 1 and 8 can 8 be written as 1. Since every whole number can be renamed as a fraction, every negative whole number can be renamed as a negative fraction. For 7 example, 7 1. ♦ to show changes such as yards lost in a football game and decreases in weight ♦ to extend the number line to the left of zero ♦ to calculate answers to many subtraction problems Student Reference Book, p. 99 554 Unit 6 Number Systems and Algebra Concepts Links to the Future This is an introduction to sophisticated ideas that students will revisit in seventhor eighth-grade algebra courses. Identifying the properties of real-number subsets is not a Grade-Level Goal for sixth grade. 1. Identify counting numbers. Draw a number line on the board with 11 evenly spaced marks on it as shown. Write the numbers 1 through 5 on the number line. As you discuss the following ideas, write them on the board along with examples. 1 2 3 4 5... These are called counting numbers. As the name implies, they are used to count things. There are infinitely many counting numbers. Ask students to name some counting numbers not shown on the number line. 2. Identify whole numbers. Add the number 0 to the number line. All of the counting numbers and 0 make up the set of whole numbers. 0 1 2 3 4 5... 3. Identify integers. Now write the numbers -1 through -5 to the left of 0. ...5 4 3 2 1 0 1 2 3 4 5... All of the whole numbers and their opposites make up the set of integers. Remind students that numbers and their opposites are located on opposite sides of 0 on the number line. 0 is its own opposite. Ask students to name some integers not shown on the number line. 4. Identify rational numbers. Ask students to name some numbers that are between consecutive numbers on the number line. Students should 1 , 2.5, -4_ 1 , and _ 1 . Write name fractions and decimals, such as _ 2 4 3 a few of these numbers in the appropriate places on the number line. These are called rational numbers. A rational number is any number that can be written as a a , where a and b are integers and b ≠ 0. simple fraction _ b Rational numbers can be positive, negative, or 0. They can be whole numbers, because any whole number can be expressed 3 ). They can also be mixed as a fraction (for example, 3 = _ 1 numbers or percents, because any mixed number or percent can be renamed as a fraction. Students have learned that any fraction can be renamed as a 1 = 0.5). Dividing 1 by 2 to rename _ 1 decimal (for example, _ 2 2 results in a remainder of 0. In other words, the division comes to an end. These decimals are called terminating decimals. Student Page Date Time LESSON Scavenger Hunt 65 Use Student Reference Book, pages 2–24 and 94–106 to find answers to as many of these questions as you can. Try to get as high a score as possible. An infinite number Sample answers: 1. How many rational numbers are there? (10 points) 2. Give an example of each of the following. (5 points each) 3. 5 a. A counting number b. A negative rational number c. A positive rational number d. A real number e. An integer f. An irrational number -_12 3 _ 4 _ √5 0 π Name two examples of uses of negative rational numbers. (5 points each) Sample answer: To express measures below sea level; and to express temperatures below the freezing point. 4. 3 Explain why numbers such as 4, _ 5 , and 3.5 are rational numbers. (10 points) Sample answer: They can be expressed as simple fractions. 4 can be expressed as _41 , _35 is already a fraction, and 3.5 can be expressed as _72 . 5. Explain why numbers such as π and √2 are irrational numbers. (10 points)Sample answer: They are nonterminating and nonrepeating decimals. 0 0 6. n+n=n What is n? 7. k = OPP(k) What is k? 8. j∗j=j Which two numbers could j be? 9. a + (-a) = 0 (15 points) (15 points) (15 points) 1 or 0 10. 1 = b∗_ b (15 points each) 1 (15 points) Math Journal 2, p. 218 205_246_EMCS_S_G6_MJ2_U06_576442.indd 218 8/29/11 10:43 AM Lesson 6 5 555 Student Page Date 1 does not result in a zero Dividing 1 by 3 to rename _ 3 remainder despite how many numbers follow the decimal point 1 = 0.333333...). Not only do the digits continue, but they do (_ 3 so in a repeating pattern—in this case, 3 is repeated over and over. Such decimals are called repeating decimals. Time LESSON Scavenger Hunt 6 5 䉬 continued Match each sentence in Column 1 with the property in Column 2 that it illustrates. (5 points each) 11. Column 1 Column 2 A. a (b c) (a b) c B. abba C. a ⴱ (b c) (a ⴱ b) (a ⴱ c) D. a ⴱ (b c) (a ⴱ b) (a ⴱ c) E. aⴱbbⴱa F. a ⴱ (b ⴱ c) (a ⴱ b) ⴱ c D B C F E A Distributive Property of Multiplication over Subtraction Commutative Property of Addition Distributive Property of Multiplication over Addition 5. Identify irrational numbers. Nonterminating decimals whose digits do not follow a repeating pattern are called irrational numbers. The number π is an example of an irrational number. Pi (3.14159265389…) has long fascinated mathematicians. Teams of researchers have calculated the digits of pi to 1.24 trillion places and found no pattern in the digits to the right of its decimal point. Irrational a , where a and b are numbers cannot be written as a fraction _ b integers (b ≠ 0). Thus, the set of rational numbers and the set of irrational numbers have no numbers in common. Associative Property of Multiplication Commutative Property of Multiplication Associative Property of Addition a 0. How can that be? (15 points) 12. If a is a negative number, then its opposite will always be greater than 0. For example, (8) 0, because (8) 8. Complete. (2 points each, except the last problem, which is worth 25 points) 13. OPP(1) 1 1 OPP(OPP(1)) OPP(OPP(OPP(1))) 1 1 OPP(OPP(OPP(OPP(1)))) OPP(OPP(OPP(OPP(OPP(OPP … (OPP(OPP(1))))))))) 1 100 OPPs Explain how you found the answer to the last problem. Sample answer: Taking the opposite an even number of times gives a positive number, 6. Identify real numbers. The set of numbers consisting of all rational numbers and all irrational numbers is called the set of real numbers. Examples of these numbers are displayed on the Real Number Line Poster. Students have worked with all the notations shown on the number line except the tangent of 30° (tan 30°), the number e, and the irrational square roots, all of which are represented by nonterminating, nonrepeating decimals. because every pair of OPPs gives a positive number. Math Journal 2, p. 219 Adjusting the Activity ELL Summarize this section of the lesson by drawing a diagram that shows the relationships among the six sets of numbers. Include examples in the diagram. Ask students to identify the sets to which a number belongs. Have them explain why a particular number belongs to some sets but not to others. (See below.) AUDITORY KINESTHETIC TACTILE ▶ Reviewing Absolute Value Ask students to find the absolute values of some of the numbers you plotted on the number line. Then ask questions such as the following. Tell students to use absolute value to justify their answers. 5 38 Whole 6.75 –5 1 3 –29 0 Counting 1 If Brian’s account balance is -$20, what is the size of his debt? The size of his debt is $20 because |-20| = 20. ● The floor of Lisa’s basement is 12 feet underground. The floor of Jeremy’s basement is 15 feet underground. Whose basement is deeper? Jeremy’s basement because |-15| > |-12|. As students share their answers, be sure that the following points are discussed: Irrational Integers 16 8 ● VISUAL Real Numbers Rational WHOLE-CLASS DISCUSSION The absolute value of a number is the distance of that number from zero on the number line. 2 π Absolute value can be used to indicate the size, or magnitude, of numbers representing certain real-world situations. º 5 In some contexts, it is useful to compare the absolute value of two numbers instead of the numbers themselves. 3 ▶ Taking Part in a Mathematical PARTNER ACTIVITY Scavenger Hunt (Math Journal 2, pp. 218–220; Student Reference Book, pp. 2–24 and 94–106) Explain that students will work in pairs to complete the scavenger hunt on journal pages 218–220. 556 Unit 6 Number Systems and Algebra Concepts Student Page Encourage students to read all the questions first and then to search the Student Reference Book for answers. Most answers require thinking beyond the information provided. Let students know that some questions are very challenging. When most partnerships have finished, review the answers. You may also want to discuss some of the following ideas: _ _ Numbers such as √12 and √5 are irrational numbers, but not all square roots are irrational; for example, the square root of 16 is 4, which is a counting number. Ask students to identify other square _ _ roots_that are counting numbers. Sample answers: √9 = 3, √4 = 2, √1 = 1 All square _roots have a positive and a negative value. For example: √9 = +3 or -3, because 3 ∗ 3 = 9 and -3 ∗ -3 = 9. _ Note that -√1 _ is shown at the -1 point on the number line, because OPP(√1 ) = OPP(1) = -1. 1. 10–1 is a positive number because it is equivalent to _ Date Time LESSON Scavenger Hunt 6 5 䉬 continued Is 52 a positive or negative number? Explain. (15 points) 14. Sample answer: It is positive. By the Powers of a 2 Number Property, 5 is 1 divided by 5 used as a 1 1 factor 2 times, which is 5 º 5 , or 25 . 15. Two numbers are their own reciprocals. What are they? 16. What number has no reciprocal? 0 1 and 1 (15 points each) (15 points) Number Stories Diana wants to make a 15 ft by 20 ft section of her yard into a garden. She will plant 2 flowers in 3 of the garden and vegetables in the rest of the garden. How many square feet of vegetable garden will she have? 2 1. 100 ft Sample answer: I multiplied 15 by 20 to get the total square footage of 300 ft2; then I 2 multiplied 300 by 3 to find the area for the flower garden: 200 ft2. Finally, I subtracted 200 from 300 to find the area used for vegetables. Explain how you got your answer. 2. Leo is in charge of buying hot dogs for his school’s family math night. Out of 300 parents and 3 children, he expects about 5 of them to attend. Hot dogs are sold 8 in a package, and Leo figures he will need to buy 22 packages so that each person can have 1 hot dog. a. How do you think he calculated to get 22 packages? 3 Sample answer: He first multiplied 5 by 300 to find out that about 180 people will come. Then he divided 180 by 8, which is 22.5. Will Leo have enough hot dogs? No. He did not interpret the remainder properly. He needs to buy 23 packages. b. 10 Math Journal 2, p. 220 2 Ongoing Learning & Practice ▶ Reviewing a Problem-Solving INDEPENDENT ACTIVITY Diagram and Solving Number Stories (Math Journal 2, p. 220; Student Reference Book, pp. 258 and 259) With the class, review the general approach to problem solving described on the Student Reference Book pages. Use the strategies and diagram to discuss how students might solve the Check Your Understanding problem at the bottom of page 259. Over the next few days, students can estimate an answer and write an explanation of how they arrived at their estimate. Student Page Date Time LESSON Math Boxes 6 5 䉬 Have students solve the multistep number stories on the bottom half of journal page 220 and explain their thinking. ▶ Math Boxes 6 5 1. Simplify. of 80 9 b. 8 of 2 2 c. 3 of 32 3 d. 8 INDEPENDENT ACTIVITY 2. 3 a. 4 1 of 60 9 1 , or 2 4 4 7 , 3 1 or 23 1 6 4 9 (Math Journal 2, p. 221) 8 a. 9 3 4 7 b. 8 6 c. 9 1 3 8 d. 24 1 2 4 24 5 127 5 28 1 1 3 2 87–89 3. Give a ballpark estimate for each quotient. 93 4. Sample estimates: Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lessons 6-4a and 6-7. The skills in Problems 5 and 6 preview Unit 7 content. Writing/Reasoning Have students write a response to 3 of 80 is less than 80, while the following: Explain why _ 4 9 of 2 is greater than 2. Sample answer: Multiplying a _ 8 number by a fraction less than 1 yields a product less than the number; multiplying a number by a fraction greater than 1 yields a product greater than the number. Divide. Simplify if possible. a. 643.27 5 728.09 b. 7 c. 432.67 82 d. 2,091.05 / 53 130 100 5 40 Complete each sentence using an algebraic expression. If Mark earns t dollars for each hour he tutors, then he earns 1 1 3 2 t, or 32 ⴱ t dollars when he tutors a. 1 2 for 3 hours. b. Madison’s dog is 3 years older than her cat. If the dog is d years old, then d 3 years old. the cat is 261 5. Write each decimal as a fraction in simplest form. a. 0.06 b. 0.52 c. 0.09 d. 0.64 240 6. 3 50 13 25 9 100 16 25 Roll two 6-sided dice, one red, one green. Give the probability of rolling the following totals. 1 6 a. 11 1 18 b. 7 c. 0 0 d. 3 or 4 e. an even number 5 36 1 2 59–60, 74 148 149 Math Journal 2, p. 221 Lesson 6 5 557 Study Link Master Name Date STUDY LINK 65 ▶ Study Link 6 5 Time Fill in the missing numbers in the tables. Look for patterns in the results. 1. x y 7 9 -2 12 -3 -9 _2 _5 3 6 2.7 -1.9 22 23 OPP(y) x+y y+x x-y -7 –9 16 16 -2 2 10 -12 1_12 0.8 12 -14 6 -_16 14 -6 -12 9 -_56 10 -12 1_12 1.9 0.8 -(23) 12 -2.7 -(22) (Math Masters, p. 193) 105 OPP(x) 2 3 -_23 INDEPENDENT ACTIVITY Turn-Around Patterns y-x Home Connection Students explore the relationships between addition and subtraction and between multiplication and division of rational numbers. 1 _ 6 -4.6 4 4.6 -4 Which patterns did you find in your completed table? Sample answer: Addition answers are the same regardless of order. Subtraction answers are opposite when the order of the numbers is reversed. 2. x _1 y 7 9 -2 12 -3 -9 _2 _5 3 6 2.7 -1.9 22 23 _1 y x _1 _1 7 9 -_12 -_13 1 _ 12 -_19 6 _ 5 1 -_ 1.9 1 _ 23 3 _ 2 1 _ 2.7 1 _ 22 xºy yºx x÷y 9 _ 7 _ 63 -24 27 -_16 3 -6 3 5 _ 5 _ 4 _ 5 _ 9 9 63 -24 27 -5.13 25 7 9 1 _ 5 -5.13 25 3 Differentiation Options y÷x -1.42 -0.70 2 1 _ 2 ▶ Investigating Properties of Which patterns did you find in your completed table? Sample answer: Multiplication answers are the same regardless of the order of the factors. Division answers are reciprocals when the order of the divisor and dividend is reversed. Math Masters, p. 193 EM3cuG6MM_U06_180-216.indd 193 3/29/10 5:17 PM (Student Reference Book, pp. 104–106; Math Masters, p. 194) To further explore properties of rational numbers, students decide whether general statements about rational numbers are true or false. They revise false statements to make them true for all special cases. Students should check their work by referring to the Properties of Numbers on Student Reference Book pages 104–106. Teaching Master Name Fractions Date 6 5 Properties of Numbers 䉬 The following properties are true for all numbers. The variables a, b, c, and d stand for any numbers (except 0 if the variable stands for a divisor). Time Properties of Numbers LESSON For each statement below, indicate whether it is always true or can be false. If the statement can be false, give an example. True or false? Properties Examples 5 7 12 Binary Operations Property 3 8 3 523 0.5 * (4) 2 235 83 3490 7 8 8 7 15 Commutative Property The sum or product of two numbers is the same, regardless of the order of the numbers. abba a*bb*a 5 * (6) 6 * (5) 30 3 4 4 4 * (5) 5 * 3 4 a 2. b ⴱ a (b c) (a b) c a * (b * c) (a * b) * c a * (b c) (a * b) (a * c) a * (b c) (a * b) (a * c) 2 * (8 3) (2 * 8) (2 * 3) 2 * 5 16 (6) 10 10 Addition Property of Zero 5.37 0 5.37 The sum of any number and 0 is equal to the original number. a00aa 0 (6) 6 The product of any number and 1 is equal to the original number. a*11*aa 2 3 *1 Unit 6 2 5 5 9 9 10 4 8 5 2 Explain why giving only one example for a true statement is not enough to prove that it is true. Sample answer: True means true in every case, not just true in one case. Try This Correct each false statement in Problems 1–4 so the statement is true for all special cases. Give one example for each statement. 2 c 3 (a º d ) (c º b) 3 (3 º 5) (2 º 4) ; 1 1. ab d 5 20 bºd 4 20 4 32 (9 º 8) (4 º 10) c (a º d ) (c º b) 9 ; 3. ab d 8 80 10 80 bºd 2 4 10 5 2 3 1 * 19 19 Student Reference Book, p. 104 558 ac bd 13 5 * (8 2) (5 * 8) (5 * 2) 5 * 10 40 10 50 50 Multiplication Property of One c d a 4. b 3 6. When a number a is multiplied by the sum or difference of two other numbers, the number a is “distributed” to each of these numbers. ac bd 212 * (2 * 3) (212 * 2) * 3 212 * 6 5 *3 15 15 Distributive Property aⴱc bⴱd c d false true false true Example 3 4 5 87 20 20 12 c d a 3. b 5. The sum or product of three or more numbers is the same, regardless of how the numbers are grouped. ac bd c d a 1. b (7 5) 8 7 (5 8) Associative Property 15–30 Min Rational Numbers Student Page When any two numbers are added, subtracted, multiplied, or divided, the result is a single number. a b, a b, a * b, and a b are equal to single numbers. INDEPENDENT ACTIVITY ENRICHMENT 4 Number Systems and Algebra Concepts Math Masters, p. 194 Teaching Master ENRICHMENT ▶ Renaming Repeating Decimals INDEPENDENT ACTIVITY 15–30 Min Name LESSON 6 5 䉬 Date Renaming Repeating Decimals You can use a power-of-10 strategy when renaming a repeating decimal as a fraction. Work through each of the examples shown below. as Fractions – Example 1: Rename 0.3 as a fraction. Let 1x 0.3333… Because one digit repeats, multiply both sides by 10 to eliminate the repeating digits to the right of the decimal point. (Math Masters, p. 195) Subtract To further explore repeating decimals, students use a power-of-10 strategy to write equivalent equations that they use to rename a repeating decimal as a fraction. Some students may notice that if the repeating decimal has (a) as the repetend, then the fraction represented by that repeating (a) decimal is _ , where R is a number with the same number of R digits as (a), but all the digits are 9s. For example: In the decimal 45 = _ 5 . Similarly, ⎯⎯, if (a) is 45, then R is 99. So, 0.45 ⎯⎯ = _ 0.45 99 11 153 = _ 17 . ⎯⎯⎯ = _ 0.153 999 111 ▶ Plotting Numbers on a SMALL-GROUP ACTIVITY 15–30 Min If 1x 0.333..., then 10x 3.33... . 10x 3.333 1x 0.333 9x 3 9x 9 Divide to solve for x. Simplify. – 1 0.3 renamed as a fraction is . 3 9 3 9 1 3 x 3 — Example 2: Rename 0.45 as a fraction. Let 1x 0.4545… Because two digits repeat, multiply both sides by 100 to eliminate the repeating digits to the right of the decimal point. Subtract. If 1x 0.454545..., then 100x 45.45... . 100x 45.4545 1x 0.4545 99x 45 99x 99 Divide to solve for x. Simplify. — 5 0.45 renamed as a fraction is . 45 99 45 99 5 11 x 11 Rename each repeating decimal as a fraction. 7 – — 9 1. 0.7 2. 0.25 3. EXTRA PRACTICE Time 25 99 Compare the denominators in the examples to the denominators of your answers for Problems 1 and 2. Use any patterns you notice to mentally – — rename 0.5 and 0.32. Check your answers with a calculator. 5 32 – — 9 99 a. 0.5 b. 0.32 Math Masters, p. 195 Number Line To provide additional practice with the number system, have students generate real-world contexts that involve different kinds of numbers. Encourage them to come up with a variety of contexts that use integers, rational numbers, and irrational numbers. Tell students to name several numbers that make sense in each of their contexts. Have them locate each number on the Class Number Line and compare the numbers using their locations on the number line. Ask students to make a general statement about the size of a number and its location on the number line. Sample answer: The larger the number, the farther to the right it appears on the number line. ELL SUPPORT ▶ Naming and SMALL-GROUP ACTIVITY 5–15 Min Comparing Numbers Have students write three examples of counting numbers, whole numbers, integers, terminating decimals, and repeating decimals within the following ranges: -5 ≤ x ≤ 10 and 54 ≤ x ≤ 56. Ask students to compare these types of numbers and to write about the differences. Lesson 6 5 559