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Name _______________________________________ Date __________________ Class __________________ Number Theory and Fractions Review for Mastery: Factors and Prime Factorization Factors of a product are the numbers that are multiplied to find that product. A factor is also a whole number that divides the product with no remainder. To find all of the factors of 24, make a list of multiplication facts. 1 • 24 = 24 2 • 12 = 24 3 • 8 = 24 4 • 6 = 24 The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24. Write multiplication facts to find the factors of each number. 1. 20 2. 16 _______________________________________ ________________________________________ _______________________________________ ________________________________________ 3. 35 4. 31 _______________________________________ ________________________________________ _______________________________________ ________________________________________ A number written as the product of prime factors is called the prime factorization of the number. To write the prime factorization of 24, first write it as product of 2 numbers. Then rewrite each factor as the product of 2 numbers until all of the factors are prime numbers. 24 = 4 • 6 (Write 24 as the product of 2 numbers.) =2•2•6 (Rewrite 4 as the product of 2 prime numbers.) =2•2•2•3 (Rewrite 6 as the product of 2 prime numbers.) So, the prime factorization of 24 is 2 • 2 • 2 • 3 or 23 • 3. Find the prime factorization of each number. 5. 28 _______________ 6. 45 _______________ 7. 50 _______________ 8. 72 ________________ Holt McDougal Mathematics Name _______________________________________ Date __________________ Class __________________ Number Theory and Fractions Review for Mastery: Greatest Common Factor The greatest common factor, or GCF, is the largest number that is the factor of any set of at least two numbers. You can use prime factorization to find the GCF of two or more numbers. To find the GCF of 18 and 24, first write the prime factorization of each number. Then identify the common prime factors. 18 = 2 • 3 • 3 24 = 2 • 2 • 2 • 3 Next, find the product of the common prime factors. 2•3=6 The GCF of 18 and 24 is 6. Find the GCF of each set of numbers. 1. 32 and 48 2. 45 and 81 3. 18 and 36 32 = _________________ 45 = _________________ 18 = _________________ 48 = _________________ 81 = _________________ 36 = _________________ _______________________ _______________________ _______________________ 4. 14 and 35 5. 42 and 72 6. 56 and 64 14 = _________________ 42 = _________________ 56 = _________________ 35 = _________________ 72 = _________________ 64 = _________________ _______________________ _______________________ _______________________ 7. 28, 56, and 84 8. 30, 45, and 75 9. 36, 45, and 54 28 = _________________ 30 = _________________ 36 = _________________ 56 = _________________ 45 = _________________ 45 = _________________ 84 = _________________ 75 = _________________ 54 = _________________ _______________________ _______________________ _______________________ Holt McDougal Mathematics Name _______________________________________ Date __________________ Class __________________ Number Theory and Fractions Review for Mastery: Equivalent Expressions The terms of an expression are the parts of the expression that are added or subtracted. The expression 28 + 12x has two terms, 28 and 12x. Term Term 28 + 12x A number that multiplies a variable is a coefficient. In the term 12x, the number 12 is the coefficient. Equivalent expressions are expressions that have the same value for all values of the variables. You can write equivalent expressions by factoring a given sum of terms as a product of the GCF and a sum. Coefficient Factor 28 + 12x as a product of the GCF and a sum. 28 + 12x The terms are 28 and 12x. =2•2•7+2•2•3•x The GCF of the terms is 2 • 2, or 4. = 4 • 7 + 4 • 3x Rewrite each term as a product with the GCF. = 4(7 + 3x) Apply the Distributive Property. Complete to factor the sum of terms as a product of the GCF and a sum. 1. 20 + 8y = ____ • ____ • ____ + ____ • ____ • ____ • y = 4 • _____ + 4 • _____ = 4(__________) 2. 30m + 42 = ____ • ____ • ____ • m + ____ • ____ • ____ = 6 • _____ + 6 • _____ = 6(__________) Factor the sum of terms as a product of the GCF and a sum. 3. 6c + 10 __________________ 6. 7x + 35 __________________ 9. 36 + 28z __________________ 4. 12 + 9y __________________ 7. 42k + 21 __________________ 10. 32x + 20 __________________ 5. 8 + 6n __________________ 8. 18p + 33 __________________ 11. 60h + 24 __________________ Holt McDougal Mathematics Name _______________________________________ Date __________________ Class __________________ Number Theory and Fractions Review for Mastery: Decimals and Fractions You can write decimals as fractions or mixed numbers. A place value chart will help you read the decimal. Remember the decimal point is read as the word “and.” To write 0.47 as a fraction, first think about the decimal in words. 0.47 is read “forty-seven hundredths.” The place value of the decimal tells you the denominator is 100. 0.47 = 47 100 To write 8.3 as a mixed number, first think about the decimal in words. 8.3 is read “eight and three tenths.” The place value of the decimal tells you the denominator is 10. The decimal point is read as the word “and.” 8.3 = 8 3 10 Write each decimal as a fraction or mixed number. 1. 0.61 _______________ 5. 1.5 _______________ 2. 3.43 _______________ 6. 0.13 _______________ 3. 0.009 _______________ 7. 5.002 _______________ 4. 4.7 ________________ 8. 0.021 ________________ Holt McDougal Mathematics Name _______________________________________ Date __________________ Class __________________ Number Theory and Fractions Review for Mastery: Equivalent Fractions Fractions that have the same value are equivalent fractions. 3=6 4 8 3 and 6 are equivalent fractions. 4 8 You can use fraction strips to help you find equivalent fractions. To solve 2 = ? , first use fraction strips to model the first 3 12 fraction. Then use 1 fraction pieces to make a length as long as 12 2 the strip. 3 You need eight 1 pieces, so 2 = 8 . 12 3 12 Find the missing number that makes the fractions equivalent. 1. 3 = ? 4 12 _______________ 2. 8 = ? 10 5 _______________ 3. 6 = ? 9 3 _______________ 4. 5 = ? 6 12 ________________ A fraction is in simplest form when the GCF of the numerator and the denominator is 1. To write 4 in simplest form, divide the numerator and the 6 denominator by their GCF, 2. 4÷2=2 6÷2=3 So, 4 in simplest form is 2 . 6 3 Write each fraction in simplest form. 5. 3 9 _______________ 6. 12 16 _______________ 7. 14 18 _______________ 8. 8 20 ________________ Holt McDougal Mathematics Name _______________________________________ Date __________________ Class __________________ Chapter 04: Number Theory and Fractions Section 06: Mixed Numbers and Improper Fractions Notes A proper fraction is a fraction whose numerator is less than its denominator. 2 , 1 , and 2 are examples of proper fractions. 7 3 4 An improper fraction is a fraction whose numerator is greater than or equal to its denominator. 3 , 8 , and 5 are examples of improper fractions. 2 3 5 Some improper fractions can be written as mixed numbers. To write 7 as a mixed number, draw circles divided into 1 sections. 4 4 Then shade in 7 of the 1 sections. 4 There is one circle and 3 of a circle shaded. 4 So, 7 = 1 3 . 4 4 Write each improper fraction as a mixed number. 1. 14 3 _______________ 2. 11 2 _______________ 3. 15 4 _______________ 4. 19 6 ________________ Mixed numbers can be written as improper fractions. To write 2 1 as an improper fraction, draw 3 circles. Divide each 3 circle into 1 sections. Next, shade in 2 whole circles and one 1 3 3 section of the last circle. Then find the total number of 1 sections that are shaded. 3 Seven 1 sections are shaded, so 2 1 = 7 . 3 3 3 Write each mixed number as an improper fraction. 5. 3 1 4 _______________ 6. 5 2 3 _______________ 7. 4 1 2 _______________ 8. 1 5 6 ________________ Holt McDougal Mathematics Name _______________________________________ Date __________________ Class __________________ Number Theory and Fractions Review for Mastery: Comparing and Ordering Fractions 3 and 2 are unlike fractions because they have different 4 3 denominators. To compare these fractions, graph the fractions on the same number line. The fraction that is farther to the right on the number line is greater in value. 9 > 8 , so 3 > 2 . 12 12 4 3 Use the number line to compare the fractions. Write < or >. 1. 2. 1 3 ___ 5 6 5 10 3. ___ 1 4 4. 11 12 ___ 6 9 4 6 ___ 1 6 You can use number lines to order fractions from least to greatest. To order 1 , 5 , and 3 , graph the values on 2 6 4 a number line. Then list the fractions as they appear from left to right. From least to greatest the fractions are 1 , 3 , and 5 . 2 4 6 Use the number line to order the fractions from least to greatest. 5. 6. 7 , 1, 4 12 4 8 _________________ 7. 2, 7 , 2 3 12 8 _________________ 5 , 1, 5 12 3 6 _________________ 8. 1, 3, 3 2 4 9 _________________ Holt McDougal Mathematics Name _______________________________________ Date __________________ Class __________________ th 6 Grade Chapter 04 Section 06 Notes