Objectives To review equivalency concepts for whole numbers; and to introduce factor strings and prime factorization. 1 materials Teaching the Lesson Key Activities Students use name-collection boxes to review the idea that numbers can be represented in many different ways. They represent composite numbers as factor strings; identify the prime factorization of a number; and use factor trees to find the prime factorization of numbers. Math Journal 1, pp. 25 and 26 Study Link 1 8 Teaching Master (Math Masters, p. 25) quarter sheet of paper Class Data Pad calculator Key Concepts and Skills • Find factor strings for numbers. [Number and Numeration Goal 3] See Advance Preparation • Write the prime factorization for numbers. [Number and Numeration Goal 3] • Rename numbers as factor strings or products of exponents. [Number and Numeration Goal 4] • Use exponents to rename numbers. [Number and Numeration Goal 4] Key Vocabulary name-collection box • factor string • length of factor string • prime factorization Ongoing Assessment: Informing Instruction See page 58. Ongoing Assessment: Recognizing Student Achievement Use journal page 25. [Number and Numeration Goal 3] 2 Ongoing Learning & Practice materials Students practice arithmetic operations and basic facts by playing Name That Number. Math Journal 1, p. 27 Student Reference Book, p. 325 Students practice and maintain skills through Math Boxes and Study Link activities. Study Link Master (Math Masters, p. 24) 4 each of the number cards 0–9; calculator 3 materials Differentiation Options READINESS Students identify the prime numbers between 1 and 30. ENRICHMENT Students explore square numbers that are also palindromes. Additional Information Advance Preparation Prepare a name-collection box for 16 on the Class Data Pad to use in the Math Message Follow-up. Teaching Masters (Math Masters, pp. 26–28) Technology Assessment Management System Journal page 25 See the iTLG. Lesson 1 9 57 Getting Started Mental Math and Reflexes Math Message Dictate the problems, repeating each problem once. 224 5 6 30 6 9 54 236 7 3 21 8 9 72 3 4 12 7 6 42 9 7 63 4 5 20 8 5 40 8 6 48 This concludes the multiplication facts routine. Plan to give a quiz once or twice weekly as needed. 8 8 and 4 4 are two names for the number 16. On a quarter-sheet of paper, write at least five other names for 16. Study Link 1 8 Follow-Up Have partners compare answers and resolve any differences. 1 Teaching the Lesson NOTE Some students may benefit from doing the Readiness activity before you begin Part 1 of each lesson. See the Readiness activity in Part 3 for details. 16 Math Message Follow-Up WHOLE-CLASS ACTIVITY Representing a number in many equivalent ways is an important concept in Everyday Mathematics. Display a name-collection box for 16 on the Class Data Pad. (See margin.) Ask students to share equivalent names for 16, comparing each other’s choices (and possibly thinking of new ones). Record equivalent names in the name-collection box. 42 兹256 苶 (4 6) º 6 4 º 11 XVI Ongoing Assessment: Informing Instruction Watch for students who write only addition or subtraction expressions in name-collection boxes. Encourage them to use more than one operation in their expressions, and to use other types of numbers, such as square roots and exponents. A typical name-collection box for 16. There are infinite possibilities. Introducing Factor Strings WHOLE-CLASS ACTIVITY Explain that another form of an equivalent name for a number is called a factor string. A factor string is a multiplication expression that has at least two factors that are greater than 1. In a factor string, the number 1 may not be used as a factor. For example, a factor string for the number 24 is 2 3 4. To support English language learners, write factor string and its definition on the Class Data Pad. Include the example. We compare factor strings by comparing their lengths. In the example, the factor string has three factors, so the length of the factor string is 3. Note that the turn-around rule for multiplication applies to factor strings: 2 3 4 and 3 4 2 are considered to be the same factor string. 58 Unit 1 Number Theory Student Page Ask students to find other factor strings for 24. Remind students to exclude 1 as a factor. Record all possible factor strings on the board. (See margin.) Ask students to find the factor string for 7. Most students will automatically answer 1 * 7. Ask: What type of number is 7? Prime number The number 1 may not be used in a factor string, so there are no factor strings for prime numbers. Have students find all possible factor strings for other numbers, such as 30, 50, 54, and 72. Record them in tables on the board. Date Is 2 a factor of 36? Yes; so 36 2 18 ● Is 2 a factor of 18? Yes; so 36 2 2 9 ● Is 2 a factor of 9? No ● Is 3 a factor of 9? Yes; so 36 2 2 3 3 Factor Strings 19 A factor string is a name for a number written as a product of two or more factors. In a factor string, 1 may not be used as a factor. Example: K I N E S T H E T I C Factor Strings Length 2 º 10 2 4º5 2 2º2º5 3 The order of the factors is not important. For example, 2 10 and 10 2 are the same factor string. The longest factor string for 20 is 2 2 5. So the prime factorization of 20 is 2 2 5. 1. Find all the factor strings for each number below. a. b. Number Factor Strings Length Number Factor Strings Length 12 3º4 2º6 3º2º2 2 2 3 16 4º4 4º2º2 8º2 2º2º2º2 2 3 2 4 Number Factor Strings Length Number Factor Strings Length 16 18 º 42 49 º 6º 32 4º 2º 38 ºº 32 º2 2º2º2º2 22 32 23 4 28 16 7º4 4 4 14 º 2ºº2 2 8º 22 7º 2º 2º2º2º2 2 2 3 3 2 4 c. d. 25 Math Journal 1, p. 25 Number Factor String Length 24 234 3 46 2 2 12 2 38 2 226 3 2223 4 Remind students that divisibility rules can be used to find the prime factorization of a number. Model using the divisibility rules to find the prime factorization of 126. 2 3 3 7 Number 20 Adjusting the Activity A U D I T O R Y The length of a factor string is equal to the number of factors in the string. The longest factor string for a number is made up of prime numbers. The longest factor string for a number is called the prime factorization of that number. Share students’ strategies for finding the longest factor strings. For example, for the number 36, you might start with 4 9 and then rename 4 as 2 2 and 9 as 3 3. Another strategy is to try the prime numbers in order: ● Time LESSON T A C T I L E V I S U A L Ask: What kind of numbers are the factors in the longest possible factor string for any number? Prime numbers The longest factor string for a number is called the prime factorization of the number. For example, the prime factorization of 24 is 2 2 2 3. To support English language learners, write prime factorization and its definition on the Class Data Pad. Include the example. Student Page Date Time LESSON Factor Strings 19 continued 2. Write the prime factorization (the longest factor string) for each number. Finding Factor Strings PARTNER ACTIVITY and Prime Factorization (Math Journal 1, pp. 25 and 26) a. 27 333 b. 40 c. 36 2233 d. 42 237 e. 48 22223 f. 60 2235 2255 g. 100 An exponent is a raised number that shows how many times the number to its left is used as a factor. Examples: Have partners complete both journal pages. Circulate and assist. Problems 3 and 4 offer additional practice in writing and decoding exponential notation. To support English language learners, clarify the meaning of exponent before students complete the journal page. 2225 52 ∑ exponent 52 means 5 º 5, which is 25. 52 is read as “5 squared” or as “5 to the second power.” 103 ∑ exponent 103 means 10 º 10 º 10, which is 1,000. 103 is read as “10 cubed” or as “10 to the third power.” 24 ∑ exponent 24 means 2 º 2 º 2 º 2, which is 16. 24 is read as “2 to the fourth power.” 3. Rewrite each number written in exponential notation as a product of factors. Then find the answer. 2 Examples: 23 22 º 9 a. 10 4 2 º 2 º 2 º 10 10 10 10 8 9 36 10,000 335 2 b. 3 º 5 4 2 c. 2 º 10 º 2 45 2 2 2 2 10 10 1,600 4. Rewrite each product using exponents. 53 Examples: 5 º 5 º 5 a. 3 º 3 º 3 º 3 c. 2 º 5 º 5 º 7 4 3 2 52 7 5º5º3º3 b. 4 º 7 º 7 52 º 32 4 72 d. 2 º 2 º 2 º 5 º 5 23 52 26 Math Journal 1, p. 26 Lesson 1 9 59 Teaching Master Name Date Time Using Factor Trees LESSON 19 Ongoing Assessment: Recognizing Student Achievement Factor Trees One way to find all the prime factors of a number is to make a factor tree. First write the number. Then, underneath, write any two factors whose product is that number. Then write factors of each of these factors. Continue until all the factors are prime numbers. 6 36 º 6 36 3 º 12 9 3 º 3 º 4 3 º 2 º 3 º 2 º Use journal page 25 to assess students’ facility with finding factors of a number. Students are making adequate progress if they have correctly written more than one factor string for each number. Below are three factor trees for 36. 36 Journal Page 25 4 [Number and Numeration Goal 3] 3 º 3 º 2 º 2 3 º 3 º 2 º 2 It does not matter which two factors you begin with. You always end with the same prime factors—for 36, they are 2, 2, 3, and 3. The prime factorization of 36 is 2 º 2 º 3 º 3. Make a factor tree for each number. Then write the prime factorization for each number. Sample answers: 24 2 º12 2 º 25 2º 4º3 2 º 5º 5 2º 2 º 2 º3 2º2º2º3 24 WHOLE-CLASS ACTIVITY Using Factor Trees (Math Masters, p. 25) 2º5º5 50 48 100 4 º 12 10 º 10 2 º 2º 3 º 4 2º5º2º 5 2 º 2º 3 º 2 º 2 2º2º2º2º3 48 Finding the Prime Factorization 50 Explain that sometimes it is helpful to organize factors when we are looking for the longest factor string (the prime factorization). One way to do this is to make a factor tree. As a class, read the directions on the teaching master. Students then work together to find the prime factorization for the numbers 24 and 50. Circulate and assist. 2º2º5º5 100 Math Masters, p. 25 2 Ongoing Learning & Practice Playing Name That Number PARTNER ACTIVITY (Student Reference Book, p. 325) Students practice applying number properties, equivalent names, arithmetic operations, and basic facts by playing Name That Number. Student Page Date Time LESSON 19 1. Circle the numbers that are divisible by 9. 360 252 819 426 651 2. Round 385.27 to the nearest … a. hundred. b. whole number. c. tenth. 4 28–30 249 4. a. Write a 6-digit numeral with 3. Complete the table. Fraction Decimal Percent 1 2 3 8 0.375 37 % 60% 0.6 0.4 40% 0.55 55% 0.08 8% 6 10 2 5 5 5 100 8 100 7 in the thousands place, 5 in the hundredths place, 4 in the tenths place, 3 in the tens place, and 9s in all other places. b. c. d. e. Study Link 1 9 Seven thousand, nine hundred thirty-nine and forty-five hundredths 4 30 31 Home Connection Students write numbers with exponents as factor strings and vice versa. Then they find the prime factorization of numbers and express the prime factorization using exponents. if each bag contains 8 marbles? 56 marbles (unit) b. How many marbles are in 700 bags if each bag contains 8 marbles? 5,600 marbles (unit) 18 18–20 27 Math Journal 1, p. 27 60 INDEPENDENT ACTIVITY (Math Masters, p. 24) 6. a. How many marbles are in 7 bags 300 40 12,000 16,000 80 200 540,000 900 600 80 80 6,400 60 36,000 600 Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lessons 1-5 and 1-7. The skills in Problems 5 and 6 preview Unit 2 content. 7, 9 3 9 . 4 5 b. Write this numeral in words. 80 90 5. Complete. INDEPENDENT ACTIVITY (Math Journal 1, p. 27) 400 385 385.3 11 a. Math Boxes 1 9 Math Boxes Unit 1 Number Theory Study Link Master Name 3 Differentiation Options Date STUDY LINK 19 Using the Sieve of Eratosthenes Exponents An exponent is a raised number that shows how many times the number to its left is used as a factor. Examples: READINESS Time INDEPENDENT ACTIVITY 1. 52 103 24 Ò exponent Ò exponent Ò exponent Write each of the following as a factor string. Then find the product. Example: 23 15–30 Min (Math Masters, pp. 26 and 27) 3. a. 104 10º10º10º10 c. 203 20º20º20 3 9 9º9º9 c. 10,000 8,000 112 50 º 50 º 50 º 50 504 Write each of the following as a factor string that does not have any exponents. Then use your calculator to find the product. Example: 23 º 3 4. 49 Write each factor string using an exponent. 64 Example: 6 º 6 º 6 º 6 a. 11 º 11 b. Social Studies Link To explore the concept of prime numbers, have students use the Sieve of Eratosthenes to find the prime numbers. Math Masters, page 27 provides a grid that will allow students to identify all of the prime numbers from 1 to 30. Have students describe patterns they see in the marked-up grid. Encourage them to use vocabulary words such as row, column, and diagonal. 2º2º2 8 7º7 72 b. 2. 16 52 means 5 º 5, which is 25. 103 means 10 º 10 º 10, which is 1,000. 24 means 2 º 2 º 2 º 2, which is 16. a. 2 º 33 º 52 b. 24 º 42 2º2º2º3 2º2º2º2º4º4 24 2º3º3º3º5º5 1,350 256 Write the prime factorization of each number. Then write it using exponents. 2 º 32 2 º3º3 Example: 18 3 2 º 2 º 2 º 5 2 º 5 a. 40 2 º 32 º 5 b. 90 2 º 3 º 3 º 5 Practice ENRICHMENT Exploring Palindromic Squares INDEPENDENT ACTIVITY 15–30 Min 5,041 50 R4 12 9. 84 7 720 99,140 47,668 701 68 5. 6,383 1,342 6. 48 15 7. 7冄3 苶5 苶4 苶∑ 8. 50,314 48,826 10. Math Masters, p. 24 (Math Masters, p. 28) To explore the relationship between numbers and their squares, have students find palindrome numbers by looking at the arrangement of their digits. Students list 3- and 4-digit palindrome numbers. Then they square these to find palindromic squares—the square of a palindrome number that is also a palindrome number. Teaching Master Name LESSON 19 Date Teaching Master Time The Sieve of Eratosthenes The mathematician Eratosthenes, born in 276 B.C., developed this method for finding prime numbers. Follow the directions below for Math Masters, page 27. When you have finished, you will have crossed out every number from 1 to 30 in the grid that is not a prime number. 1. Since 1 is not a prime number, cross it out. 2. Circle 2 with a colored marker or crayon. Then count by 2, crossing out all multiples of 2—that is, 4, 6, 8, 10, and so on. 3. Name LESSON 19 Skip 4 on the grid because it is already crossed out, and go on to 5. Use a new color to circle 5 and cross out the multiples of 5. 5. Continue. Start each time by circling the next number that is not crossed out. Cross out all multiples of that number. If a number is already crossed out, make a mark in a corner of the box. If there are no multiples for a number, start again. Use a different color for each new set of multiples. 6. Stop when there are no more numbers to be circled or crossed out. The circled numbers are the prime numbers from 1 to 30. 7. List the prime numbers from 1 to 30. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 Time Palindrome numbers are numbers that read the same forward or backward. A single-digit number is also a palindrome. The two-digit palindrome numbers are 11, 22, 33, 44, 55, 66, 77, 88, and 99. The table below lists samples of 3-digit and 4-digit palindromes. 1. Find 3-digit and 4-digit numbers to add to the table. Sample answers: Palindrome Numbers Circle 3 with a color different from Step 2. Cross out every third number after 3 (6, 9, 12, and so on). If a number is already crossed out, make a mark in a corner of the box. The numbers you have crossed out or marked are multiples of 3. 4. Date Palindromic Squares 3-digit 4-digit 101, 111 1,001; 1,111 202, 222 2,002; 2,222 303, 333 3,003; 3,333 404, 414, 424, 434, 444 454, 464, 474, 484, 494 4,004; 4,114; 4,224; 4,334; 4,444; 4,554; 4,664; 4,774; 4,884; 4,994 Sometimes finding the square of a palindrome number results in a square number that is also a palindrome number—a palindromic square. For example, 1112 12,321. 1, 2, and 3 2. Which 3 single-digit numbers have palindromic squares? 3. Which 2-digit numbers have palindromic squares? 4. Find the numbers from the table that have a palindromic square and write the number model. 11 and 22 Example: 1012 10,201 Sample answers: 1112 = 12,321; 2022 = 40,804; 1,0012 = 1,002,001; 1,1112 = 1,234,321 Math Masters, p. 26 Math Masters, p. 28 Lesson 1 9 61

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