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Name ________________________________________ Date __________________ Class__________________ SECTION 4A Number Theory and Fractions SECTION A Family Letter: Number Theory Dear Family, Vocabulary The student is working with prime factors, and determining the greatest common factor of whole numbers. These skills are precursors to the more advanced applications involving fractions. These are the math words we are learning: Knowledge of equivalent expressions rules will allow the student to use mental math and number patterns to determine if a number can be evenly divided into another number. Knowing these rules will also allow him or her to see the relationship and patterns in numbers. Have the student use the guidelines and the examples in the table to explain how one determines the equivalent expressions of a particular number. Equivalent Expressions Rules A number is divisible by … 2 if the last digit is an even number. Divisible 7,992 Not Divisible 4,221 219 451 5,516 431 5 if the last digit is 0 or 5. 795 977 6 if the number is divisible by both 2 and 3. 132 561 9 if the sum of the digits is divisible by 9. 801 1,712 27,000 1,895 3 if the sum of the digits is divisible by 3. © Houghton Mifflin Harcourt Publishing Company 4 if the last two digits form a number divisible by 4. 10 if the last digit is 0. coefficient a number that is multiplied by a variable in an algebraic expression. equivalent expressions expressions that have the same value for all values of the variables. factor whole numbers that are multiplied to find a product greatest common factor the largest of the common factors shared by two or more whole numbers prime factorization the process of writing the number as the product of its prime factors term the part of an expression that is added or subtracted. Tell whether the number 846 is divisible by 2, 3, 4, 5, and 6. 2 The last digit is an even number. Divisible 3 The sum of the digits is 18, which is divisible by 3. Divisible 4 The last two digits form the number 46. 46 is not divisible by 4. Not divisible 5 The last digit is not a 0 or a 5. Not divisible 6 The number is divisible by both 2 and 3. Divisible 846 is divisible by 2, 3, and 6. 4-1 Holt McDougal Mathematics Name ________________________________________ Date __________________ Class__________________ SECTION 4A Number Theory and Fractions SECTION A Family Letter: Number Theory continued A number that is only divisible by the number 1 and itself is called a prime number. Some numbers are divisible by many factors. These are called composite numbers. All numbers can be written as a product of its prime factors. This is called prime factorization. Write the prime factorization of the number 84. Method 1: Use a tree diagram. Choose any two factors of 84. Keep finding factors until each branch ends with a prime factor. The prime factorization of 84 is 2 • 2 • 3 • 7 or 22 • 3 • 7. Method 2: Use a ladder diagram. 2 84 Choose a prime factor of 84 to begin. 2 42 Keep dividing by prime factors until the 3 21 quotient is 1. 77 1 Again, the prime factorization of 84 is 2 • 2 • 3 • 7 or 22 • 3 • 7. Notice how both methods result in the same answer. © Houghton Mifflin Harcourt Publishing Company The student will use prime factorization as one way to help find the greatest common factor, GCF, of a set of numbers. He or she will list the prime factors of each number in the set, circle the matching factors, and then multiply those common prime factors. Knowing how to find the GCF is essential when working with fractions. These are important skills that will allow the student to comprehend the concepts and applications that will be studied throughout the rest of this chapter. Review these skills often to help the student become more confident in this topic. Sincerely, 4-2 Holt McDougal Mathematics Name ________________________________________ Date __________________ Class__________________ SECTION 4A Number Theory and Fractions SECTION A At-Home Practice: Number Theory List all the factors of each number. 1. 42 ________________ 2. 88 3. 24 ________________ 4. 50 _______________ ________________ Write the prime factorization of each number. 5. 64 ________________ 6. 18 7. 55 ________________ 8. 48 _______________ ________________ Find the GCF of each set of numbers. 9. 12 and 50 ________________________ 10. 15 and 75 _______________________ 11. 18, 24, and 30 ________________________ 12. Brenda is making toy baskets to donate to charity. She has 24 toy cars, 30 jump ropes, and 42 teddy bears. What is the greatest number of baskets she can make if each type of toy is equally distributed among the baskets? _________________________________________________________________________________________ Write four equivalent expressions for each given expression. © Houghton Mifflin Harcourt Publishing Company 13. 15x + 25x 14. 9 + 36 15. 2(1 + 16y) ________________________ _______________________ ________________________ ________________________ _______________________ ________________________ ________________________ _______________________ ________________________ ________________________ _______________________ ________________________ Answers: 1. 1, 2, 3, 6, 7, 14, 21, 42 2. 1, 2, 4, 8, 11, 22, 44, 88 3. 1, 2, 3, 4, 6, 8, 12, 24 4. 1, 2, 5, 10, 25, 50 5. 2 • 2 • 2 • 2 • 2 • 2 or 26 6. 2 • 3 • 3 or 2 • 32 7. 11 • 5 8. 2 • 2 • 2 • 2 • 3 or 24 • 3 9. 2 10. 15 11. 6 12. 6 baskets 13. 3.5x + 5.5x, (3 + 5)5y, 15.5y, 75y 14. 9.1 + 9.4, 9(1 + 4), 9(4 + 1), 32(1 + 4) 15. 2.1 + 2.16y, 2 + 32y, 32y + 2, 25y + 2 4-3 Holt McDougal Mathematics Name ________________________________________ Date __________________ Class__________________ SECTION 4A Number Theory and Fractions Family Fun: Divisibility Squares Directions • If a number is divisible by 5, color it blue. • If a number is divisible by 6, keep it white. • If a number is divisible by 2 but not by 5 or 6, color it red. 154 32 44 254 392 76 322 912 36 822 762 712 442 18 55 975 108 916 184 444 305 415 234 886 62 12 336 888 126 16 746 98 202 694 242 512 © Houghton Mifflin Harcourt Publishing Company Create your own divisibility square design and pass it on to a friend. Answers: Blue: 55, 975, 305, 415; White: 912, 36, 822, 762, 18, 108, 444, 234, 12, 336, 888, 126; Red: 154, 32, 44, 254, 392, 76, 322, 712, 442, 916, 184, 886, 62, 16, 746, 98, 202, 694, 242, 512 4-4 Holt McDougal Mathematics Name ________________________________________ Date __________________ Class__________________ SECTION 4B Number Theory and Fractions SECTION B Family Letter: Understanding Fractions Dear Family, Vocabulary The student is beginning to study fractions. Some of the basic skills he or she will learn in this section are converting between decimals and fractions and writing equivalent fractions. These are the math words we are learning: Decimals are another way to write fractions and mixed numbers. Mixed numbers are special fractions that consist of a whole number and a fraction. The student will use place value concepts to convert between decimals and fractions. Write each decimal as a fraction or mixed number. A. 0.17 B. 2.3 0.17 Identify the place 2.3 Identify the place value of the digit value of the digit farthest to the right. farthest to the right. © Houghton Mifflin Harcourt Publishing Company 17 100 The 7 is in the hundredths place, so use 100 as the denominator. 2 3 Write the whole 10 number, 2. The 3 is in the tenths place, so use 10 as the denominator. When converting from a fraction to a decimal, you may discover that the decimal part will repeat itself. This type of decimal is called a repeating decimal. A decimal that has a finite number of decimal places is called a terminating decimal. repeating decimal 0.35 35 terminating decimal 0.64 To convert from a fraction to a decimal, divide the numerator by the denominator. The quotient will be the decimal. Write 3 as a decimal. 5 0.6 3 5 3.0 5 3 0 0 Divide the numerator by the denominator. 3 0.6 5 The quotient is the decimal. 4-29 equivalent fractions fractions that represent the same value improper fraction a fraction in which the numerator is greater than or equal to the denominator mixed number a number that contains both a whole number greater than zero and a fraction proper fraction a fraction in which the numerator is less than the denominator repeating decimal a decimal with a block of one or more digits that repeat without end simplest form a fraction where the greatest common factor of the numerator and the denominator is 1 terminating decimal a decimal with a finite number of decimal places unlike fractions fractions that have different denominators Holt McDougal Mathematics Name ________________________________________ Date __________________ Class__________________ SECTION 4B Number Theory and Fractions SECTION B Family Letter: Understanding Fractions continued Equivalent fractions can help the student understand how fractions are related to each other. To find a fraction equal to another fraction, multiply or divide the numerator and the denominator of a fraction by the same number. Find the missing number that makes the fractions equivalent. 3 8 40 35 15 40 85 In the denominator, 8 is multiplied by 5. Multiply the numerator, by the same number. So 3 is equivalent to 15 . 8 40 The fractions the student has seen so far have been proper fractions, or fractions where the numerator is less than the denominator. A fraction with a numerator greater than or equal to its denominator is an improper fractions. Write 19 as a mixed number. 7 Divide the numerator by the denominator. 2 57 14 5 To form the fraction part of the quotient, use the remainder as the numerator, and the divisor as the denominator. © Houghton Mifflin Harcourt Publishing Company 7 19 19 written as a mixed number is 2 5 . 7 7 Fractions with different denominators are called unlike fractions. When asked to compare unlike fractions, the student will need to use the least common denominator (LCD) to create equivalent fractions. The student will continue to learn about fractions in the future. Sincerely, 4-30 Holt McDougal Mathematics Name ________________________________________ Date __________________ Class__________________ SECTION 4B Number Theory and Fractions SECTION B At-Home Practice: Understanding Fractions Write each decimal as a fraction or a mixed number. 1. 0.37 ________________ 2. 0.09 3. 6.11 ________________ 4. 1.2 _______________ ________________ Write each fraction or mixed number as a decimal. 5. 3 2 5 ________________ 6. 3 8 7. 4 1 9 ________________ 8. 7 12 _______________ ________________ Order the fractions and decimals from least to greatest. 9. 0.38, 1 , 3 3 10 10. 8 , 1 , 0.75 15 2 ________________________________________ ________________________________________ Find two equivalent fractions for each given fraction. 11. 12 16 ________________ 13. 5 9 12. 11 22 ________________ 14. 14 21 _______________ ________________ Find the missing number that makes the fractions equivalent. © Houghton Mifflin Harcourt Publishing Company 17. 9 36 10 ? 16. 9 3 12 ? 15. 4 ? 5 20 ________________________ _______________________ ________________________ Write each fraction in simplest form. 18. 6 18 20. 25 40 19. 12 15 ________________________ _______________________ ________________________ A Compare. Write <, >, or =. 21. 3 ______ 4 5 5 22. 11 ______ 2 15 3 23. 9 ______ 3 33 11 14. 3 ______ 7 5 9 Answers: 1. 37 2. 9 3. 6 11 4. 1 2 5. 3.4 6. 0.375 7. 4.11 8. 0.583 9. 3 , 1 , 0.38 10. 1 , 8 , 0.75 100 10 2 15 100 100 10 3 Possible Answers for 11–14 11. 3 , 24 12. 1 , 33 13. 10 , 15 14. 2 , 28 15. 16 16. 4 17. 40 18. 1 19. 4 2 66 3 42 3 5 4 32 18 27 20. 5 21. < 22. > 23. = 24. < 8 4-31 Holt McDougal Mathematics Name ________________________________________ Date __________________ Class__________________ SECTION 4B Number Theory and Fractions Family Fun: First Team Home Wins! Materials 1 number cube Game markers (coins, paper clips, etc) Directions • Roll the number cube to see which team goes first. The lowest roll goes first. Each team takes turns. • Team 1 rolls the cube and moves that many places on the game board. • Team 1 then has to name a fraction equivalent to the fraction in the space where they landed. • Team 2 checks to make sure the two fractions are equivalent. If Team 1 is correct, they may stay on that space. If Team 1 is incorrect, they must roll the number cube and move back the number of spaces it shows. © Houghton Mifflin Harcourt Publishing Company • The first team to reach the finish line wins! 4-32 Holt McDougal Mathematics