Factor Trees Objective To provide experiences with finding the greatest common factor and the least common multiple of two numbers. www.everydaymathonline.com epresentations etoolkit Algorithms Practice EM Facts Workshop Game Family Letters Assessment Management Common Core State Standards Curriculum Focal Points Interactive Teacher s Lesson Guide Teaching the Lesson Ongoing Learning & Practice Differentiation Options Key Concepts and Skills Identify the prime factorization for a number. [Number and Numeration Goal ] Use greatest common factors and least common multiples to rename fractions. [Number and Numeration Goal ] Use multiplication facts to find factor strings. [Operations and Computation Goal ] Find greatest common factors and least common multiples using factor strings. [Number and Numeration Goal ] Key Activities Students use factor trees to find all the prime factors of a number and write the prime factorization. They use prime factorizations to simplify fractions. Ongoing Assessment: Recognizing Student Achievement Use journal page 9. [Number and Numeration Goal ] Key Vocabulary prime factorization factor tree common factor greatest common factor least common multiple Playing Factor Captor Student Reference Book, p. 06 Math Masters, p. calculator paper and pencil per partnership: 0 counters Students practice finding factors of larger numbers and recognizing prime factors. Math Boxes Math Journal, p. 9 Students practice and maintain skills through Math Box problems. Study Link Math Masters, p. 8 Students practice and maintain skills through Study Link activities. READINESS Making Factor Rainbows Students factor numbers by making factor rainbows. ENRICHMENT Using a Division Method for Prime Factorizations Math Masters, p. 9 Students use division to find prime factorizations. EXTRA PRACTICE Using Factor Trees to Find Common Denominators Math Masters, p. 0 Students identify common denominators by using factor trees. ELL SUPPORT Making a Factor Tree Poster per group: chart paper, markers Students make a display to reference concepts related to factor trees. Materials Math Journal, pp. 9 96 Student Reference Book Glossary Class Data Pad Advance Preparation Teacher s Reference Manual, Grades 6 pp. 9 8 9 Unit Probability, Ratios, and Rates
Getting Started Mathematical Practices SMP, SMP, SMP, SMP, SMP6, SMP, SMP8 Content Standards.NF.a,.NF.b,.NF.c Mental Math and Reflexes Have students compute each quotient and explain their answer using the relationship between multiplication and division. Math Message Solve Problems and on journal page 9. ( _ ) = 0 because 0 ( _ ) =. ( _ ) = _ 8 because ( _ 8 ) = _. ( _ ) = _ 8 because ( _ 8 ) = _. 6 ( _ ) = because ( _ ) = 6. ( _ ) = because ( _ ) =. ( _ ) = _ because ( _ ) = _. Teaching the Lesson Math Message Follow-Up (Math Journal, p. 9; Student Reference Book Glossary) WHOLE-CLASS DISCUSSION Interactive whiteboard-ready epresentations are available at www.everydaymathonline.com to help you teach the lesson. Have students read the definitions for factor, factor pair, composite number, prime number, and factor string in the Student Reference Book Glossary. Read and discuss how the terms relate. Discussion should include the following points: A factor of a number can be any type of number, but the other terms only refer to whole numbers. Numbers can be defined by their factors. For example, a prime number has only two whole-number factors, and itself. A composite number has more than two whole-number factors. A square number has an odd number of factors. Each factor in the longest factor string of a number is prime. The longest factor string is called its prime factorization. Ask: What device can be used to find all the factor pairs for a number? Factor rainbow Have students make factor rainbows for 8 and share their solutions using the Class Data Pad. A factor tree is a device that can be used to find the prime factorization of a number. Draw one for 8 on the Class Data Pad. Ask: What is the prime factorization for 8? Have students refer to the factor tree examples for on Math Journal, page 9. Note that each factor tree begins with two different factors for but the prime factorizations are the same. Ask students to make factor trees for 6. Ask: What do you notice about the longest factor string you recorded for Problem and the result of your factor tree for 6? They are the same:. Finding Greatest Common Factors (Math Journal, pp. 9 9) SOLVING WHOLE-CLASS Write the numbers 8 and 0 on the board. Ask students to name all the factors of each number while you list them. Date Math Message Factors. Write all the pairs of factors whose product is 8. One pair has been done for you. 8 6 8, 8; ; 6;. One way to write 6 as a product of factors is 8. Another way is 9. Write 6 as the product of the longest possible string of factors. Do not include as a factor. Factor Trees and Greatest Common Factors One way to find all the prime factors of a number is to make a factor tree. First write the number. Underneath the number 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. Below are two factor trees for. The greatest common factor of two whole numbers is the largest number that is a factor of both numbers. Example: Find the greatest common factor of and 60. Step List all the factors of :,,,, 6, 8,, and. Step List all the factors of 60:,,,,, 6, 0,,, 0, 0, and 60. Step,,,, 6, and are on both lists. They are common factors. is the largest number, so it is the greatest common factor of and 60.. Find the greatest common factor of 8 and. Factors of 8:,,, 6, 9, 8 Factors of :,, 9, Greatest common factor: 9 Math Journal, p. 9 9 Lesson 9
Date Factor Trees and Greatest Common Factors Another way to find the greatest common factor of two numbers is to use prime factorization. Example: Find the greatest common factor of and 60. Step Make factor trees and write the prime factorization of each number. 6 60 0 Step Circle pairs of common factors. 60 60 Step Multiply one factor in each pair of circled factors. The greatest common factor of and 60 is, or.. Make a factor tree for each number below, and write the prime factorization. a. 0 b. c. 90 0 0 0 90 Math Journal, p. 9 Factors of 8:,,, 6, 9, and 8 Factors of 0:,,,, 6, 0,, and 0 Circle,,, and 6 on both lists. Explain that because,,, and 6 are factors of both 8 and 0, they are called common factors. The largest of these common factors is called the greatest common factor. The greatest common factor of 8 and 0 is 6. There are two ways to find the greatest common factor of two numbers: List all the factors of each number and identify the largest common factor. Use factor trees to write the prime factorization of each number. Identify the prime factors that these two numbers have in common, and multiply them. Ask students to make factor trees for 8 and 0, and have volunteers draw these two factor trees on the board. 6 8 0 0 Write the prime factorizations for 8 and 0 as shown below (one above the other) on the board or a transparency. Ask: What prime factors do these two numbers have in common? and Circle the pairs of common factors in the prime factorizations for 8 and 0. Explain that since = 6, the greatest common factor is 6. It is the greatest number by which both 8 and 0 are divisible. Date Factor Trees and Greatest Common Factors cont.. a. Which prime factor(s) do 0 and have in common? b. What is the greatest common factor of 0 and? 6. a. Which prime factor(s) do and 90 have in common? b. What is the greatest common factor of and 90?. a. Which prime factor(s) do 0 and 90 have in common? b. What is the greatest common factor of 0 and 90? 8. Use the factor trees in Problem to help you write each fraction below in simplest form. Divide the numerator and denominator by their greatest common factor. a. 0 0 b. 90 90 6 c. 0 90 0 90 0 0 9 9. What is the greatest common factor of 0 and? Write the fraction 0 0 in simplest form. 0. Use the space below to draw factor trees. What is the greatest common factor of,60 and,0?,60 60 0 Math Journal, p. 9 and and 0 90,0 6 8 = 0 = GCF = Greatest Common Factor = 6 Have students make factor trees for and 0, and have volunteers add these two trees to the display on the board. Ask: What is the greatest common factor of and 0? How do the factor trees confirm that is the greatest common factor of and 0? They show that is the only common prime factor. Have students refer back to the factor trees for and 6. Ask: What is the greatest common factor of and 6?, or Write the prime factorizations for and 6, circling the pairs of common factors. Point out that,, and are the common prime factors. Therefore, the greatest common factor is, or. Tell students that the greatest common factor can be used to simplify fractions. Ask students how they might use the greatest common factor and the division rule to simplify the fraction _ 6. The factor trees for and 6 show their greatest common factor to be. Divide the numerator and the denominator by the greatest common factor, _ 6 = _. The simplified fraction is _. Have students complete journal pages 9 9. 96 Unit Probability, Ratios, and Rates
Date Ongoing Assessment: Recognizing Student Achievement Journal Page 9 Problem Use journal page 9, Problem to assess students ability to factor numbers and identify the prime factorizations. Students are making adequate progress if they correctly form the factor pairs in each factor tree and write the correct prime factorizations. [Number and Numeration Goal ] Finding Least Common Multiples (Math Journal, p. 96; Student Reference Book Glossary) PARTNER Have students locate the term least common multiple in the Student Reference Book Glossary and read the entry. Ask: What is a common multiple? A number that is a multiple of two or more given numbers The quick common denominator is an example of a common multiple. The smallest number that is a multiple of two or more numbers is the least common multiple. Have students list the first seven multiples of 6 and 9. Ask: What numbers are in both lists? The common multiples are 8 and 6. What is the least common multiple of 6 and 9? 8 Tell students that factor trees also can be used to find the least common multiple of two numbers. This procedure is similar to finding the greatest common factor. Step : Refer to the prime factorizations for 8 and 0 from the previous activity, circling the pairs of common factors. Step : Draw a line through one factor in each of the pairs of common factors. 8 = 0 = Least common multiple: = 90 Step : Write the remaining factors in a multiplication expression, = 90. The least common multiple of 8 and 0 is 90. Have partners complete journal page 96. Circulate and assist. Ongoing Learning & Practice Playing Factor Captor (Student Reference Book, p. 06; Math Masters, p. ) Students practice finding factors of larger numbers and recognizing prime factors by playing Factor Captor on the 0 Grid. PROBLEM SOLVING PARTNER Factor Trees and Least Common Multiples The least common multiple of two numbers is the smallest number that is a multiple of both numbers. Example: Find the least common multiple of 8 and. Step List the multiples of 8: 8, 6,,, 0, 8, 6, and so on. Step List the multiples of :,, 6, 8, 60, and so on. Step and 8 are in both lists. They are common multiples. is the smallest number. It is the least common multiple for 8 and. is also the smallest number that can be divided by both 8 and. Another way to find the least common multiple for two numbers is to use prime factorization. Example: Find the least common multiple of 8 and. Step Write the prime factorization of each number: 8 Step Circle pairs of common factors. Then cross out one factor in each pair as shown below. 8 Step Multiply the factors that are not crossed out. The least common multiple of 8 and is, or.. Make factor trees and write the prime factorizations for each number. a. b. 9 c. 0 9 0. What is the least common multiple of 0 a. 9 and? b. and 0? c. 9 and 0? 90 Math Journal, p. 96 Name Date Factor Lesson Captor Title 0 Grid 6 8 9 0 6 8 9 0 6 8 9 0 6 8 9 0 6 8 9 0 6 8 9 60 6 6 6 6 6 66 6 68 69 0 6 8 9 80 8 8 8 8 8 86 8 88 89 90 9 9 9 9 9 96 9 98 99 00 0 0 0 0 0 06 0 08 09 0 Math Masters, p. Game Master Lesson 9
Date. Solve. Math Boxes 00 a. If marbles are of the marbles in a bag, how many are in the bag? marbles b. If pennies are % of a pile of pennies, how many are in the pile? pennies c. students are absent today. This is 0% of the students enrolled at the school. How many students are enrolled at the school? 0 students d. Tyesha paid $90 for a new radio. It was on sale for of the regular price. What is the regular price of the radio? $0.00. A scooter is on sale for 0% of the list price. The sale price is $8. What is the list price? $80.00. Write or. a. 0. 8 9 b. 0. 6 c. 8 d. 9 0.9 6 e. 9 89 90 Math Boxes (Math Journal, p. 9) INDEPENDENT Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson -. Writing/Reasoning Have students write a response to the following: Explain your solution strategy for Math Boxes, Problem. Since $8.00 represents 0%, I divided 8 by to find 0% of the list price. Then I multiplied that result by 0 to find the total list price; 8 0 = $80.00.. Solve. a. 8X One orange One cube weighs as much as Xs. weighs as much as Xs. Study Link (Math Masters, p. 8) INDEPENDENT b. 6X X One triangle One paper clip weighs as much as X. weighs as much as 6 X. Math Journal, p. 9 8 9 Home Connection Students work with factor trees. They use these to write prime factorizations and to write fractions in simplest form. Differentiation Options READINESS Making Factor Rainbows SMALL-GROUP Min Name Date STUDY LINK Factor Trees Study Link Master. Make factor trees and find the prime factorization for the following numbers. Example: 0 0 0 To provide experience finding all the factors of a number, have students make factor rainbows. For example, to make a factor rainbow for the number 8, list all the factors of 8 in ascending order. Then connect factor pairs. Every factor should be paired with another factor. If there is an odd number of factors, the middle factor is paired with itself. The product of each pair of factors should be 8. 66 a. 66 b. 6 6 8 6 8 66. Write each fraction in simplest form. Use factor trees to help you. Show your work. a. 0 66 0 b. 6 6 c. 0 8. Find the prime factorization for 0.. a. Circle the number that has the most prime factors. 6 9 00 b. Which has the fewest prime factors? 9. Simplify the fraction to the right. 0 Practice 6. 6 9. 0. 60 90 8. 0 0 90 9. % of 90 Math Masters, p. 8 8 6 0 Have students make factor rainbows for given numbers. Suggestions:,, 0, 9, 80, 00 ENRICHMENT PARTNER Using a Division Method for Prime Factorizations (Math Masters, p. 9) Min To apply students understanding of factors, have them use a division method to find prime factorizations. When students have finished, discuss any questions or curiosities they encountered. 98 Unit Probability, Ratios, and Rates
EXTRA PRACTICE Using Factor Trees to Find Common Denominators (Math Masters, p. 0) INDEPENDENT Min Students practice making factor trees to find prime factorizations. They use factor trees to find common denominators, identify the least common multiple of the denominators, and use the least common multiple as a common denominator. Name Date The Division Method for Prime Factorization Use the method below to find the prime factorization of the following numbers. Example: Find the prime factorization for. Step Divide, using the smallest prime factor of the number as the divisor. Step The quotient becomes the dividend. Use the smallest prime factor as the divisor, and continue dividing until the quotient is a prime number. Step Write the divisors as a multiplication expression. = 6 This is the prime factorization of. Teaching Master Divide: 66 66 Divide: 66 8 is not a factor of 8. The next smallest prime factor is. 8 Divide: 8 6 6 6 is a prime number. The prime factorization of is 6 Use the division method to find the prime factorizations. Show your work. ELL SUPPORT SMALL-GROUP Making a Factor Tree Poster 0 Min To provide language support for factors, guide students to make a poster showing important vocabulary words related to factor trees. Display the poster during lessons in this unit..,06.,90.,98 9,06,90 8,9 6 9 9 66 Math Masters, p. 9,98,99, Factors:,,,, 6, Factor Tree: Factors 0,,,, 0, 0 0 Greatest Common Factor for and 0 is Least Common Multiple for and 0 is 60 ( = 60 ) Teaching Master Name Date Factor Trees and Adding Fractions. Make factor trees and write the prime factorization for each number below. a. b. c. 6 6 6 8. Add the following fractions. Use the factor trees above to help you find the least common multiple of the denominators. Use this least common multiple as a common denominator. a. b.. Use factor trees or some other method to find a common denominator for the fraction pairs below. If you do not use factor trees, explain how you found the least common denominators. a. and b. and 6 8 6 9 c. and 6 0 96 96 6 96 6 9 8 8 8 89 8, or 8 Math Masters, p. 0 Lesson 99