Searching for Common Ground

Transcription

1 Searching for Common Ground Common Factors and Common Multiples 4 WARM UP List all factor pairs for each number LEARNING GOALS Identify the factors of numbers and the common factors of two whole numbers. Identify the multiples of numbers and the common multiples of two whole numbers. Write and evaluate numeric expressions using the Distributive Property to model composing and decomposing the areas of rectangles. Rewrite the sum of two whole numbers with a common factor as a product using the Distributive Property. KEY TERMS common factor relatively prime greatest common factor (GCF) multiple Commutative Property of Multiplication least common multiple (LCM) Just as you can compose and decompose shapes, you can compose and decompose numbers using factors and multiples. How can you use shapes to see relationships between numbers? LESSON 4: Searching for Common Ground M1-39

2 Getting Started How Many Rectangles Can You Build? Understanding the area of rectangles is helpful when learning about factors. A rectangular area model is one way to represent multiplication. Your class is going to create area models for each number: 12, 15, 16, and 20. For the number assigned to you by your teacher, use the grid paper at the end of the lesson to create and cut out as many unique rectangles as possible with the area of your assigned number. Label each rectangle with its dimensions. Number assigned to me: 1. List the dimensions of all of the rectangles that you created for your assigned number. 2. How do you know if you have created all of the possible rectangles with the given area? 3. How are factors represented in your rectangles? 4. List all of the factors of the number that you were assigned. M1-40 TOPIC 1: Factors and Area

3 ACTIVITY 4.1 Using Rectangles to Determine Common Factors For this investigation, select a partner who has created area models for a number different from the number assigned to you. Together with your partner, combine one of your rectangles and one of your partner s rectangles to make a bigger rectangle. If possible, use this method to create additional rectangles. 1. Complete the table with the information from each larger rectangle created by you and your partner. Number assigned to me Number assigned to my partner Dimensions of Smaller Rectangle 1 Dimensions of Smaller Rectangle 2 Dimensions of the Larger Rectangle Area of the Larger Rectangle as a Sum of the Smaller Rectangles Total Area of Larger Rectangle l 3 w 1 l 3 w 2 l(w 1 1 w 2 ) A 1 1 A 2 2. How are the dimensions of the larger rectangle related to its total area? 3. For each larger rectangle you and your partner created, write a numeric expression that relates the dimensions of the larger rectangle to the sum of the areas of the smaller rectangles. LESSON 4: Searching for Common Ground M1-41

4 Consider any factors that are shared between your number and your partner s number. These are called common factors. 4. How are the common factors represented in the larger rectangles that you and your partner created? 5. How are the common factors represented in the numeric expressions that you and your partner wrote? 6. List the common factors of the two numbers. ACTIVITY 4.2 Prime Factorization Suppose you are looking for the common factors of 56 and 42, but you do not have grid paper or scissors to create rectangles. Is there another way? A factor tree is a way to organize the prime factorization of a number. Choose any factor pair to get started ? 3? 3 WORKED EXAMPLE One way to determine common factors is to use prime factorization. Start by writing each number as a product of its prime factors. Organize the prime factors into a table, where only shared factors are listed in the same column. Number Prime Factors The common factors of the two numbers are the numbers that are in both rows and the product of the numbers that are in both rows. The common factors of 56 and 42 are 2, 7, and ? 2? 2? ? 3? 7 M1-42 TOPIC 1: Factors and Area

5 1. How do you know that 14 is a common factor of 56 and 42? 2. Why is there a space between 2 and 7 in the top row of the table? 3. Create a table to identify common factors. a. Identify all of the common factors of 54 and 84. b. Of the common factors, which factor is the largest? The greatest common factor (GCF) is the largest factor two or more numbers have in common. 4. Rewrite each numeric expression using the Distributive Property and the GCF. Two numbers that do not have any common factors other than 1 are called relatively prime. a b LESSON 4: Searching for Common Ground M1-43

6 ACTIVITY 4.3 Common Multiples Rectangular arrays can also be used to determine multiples and common multiples. A multiple is the product of a given whole number and another whole number. Consider the area model for 6? The Commutative Property of Multiplication states that for any numbers a and b, the product a? b is equal to the product b? a. One way to think about the area model is to analyze the collection of columns. As you look at how the area model builds from left to right, the addition of each new column creates a multiple of 6. So, column 1 alone is a rectangle, which represents the first multiple of 6, which is 6. By adding column 2, the rectangle is now 6 3 2, which represents the second multiple of 6, which is 12. The whole rectangle represents 6 3 8, or List the first eight multiples of 6 by labeling each column of the area model. Next, think about the area model as a collection of 6 rows. The first row alone creates an rectangle, which represents the first multiple of 8, which is 8. Including all rows of the rectangle represents the sixth multiple of 8, which is List the first six multiples of 8 by labeling each row of the area model. M1-44 TOPIC 1: Factors and Area

7 While 48 is a multiple shared by both 6 and 8, it is not the least common multiple (LCM). The LCM is the smallest multiple (other than zero) that two or more numbers have in common. NOTES 3. Analyze the multiples of 6 and 8 that you labeled on the area model. Identify the least common multiple of 6 and 8. As demonstrated by the rectangular array, for any two whole numbers a and b, a common multiple is a? b. However, this number may not be the least common multiple of a and b. 4. Determine the least common multiple of 6 and 9. a. List the first 9 multiples of 6. b. List the first 6 multiples of 9. c. What is the least common multiple of 6 and 9? 5. Determine the least common multiple of 7 and Using prime factorization, how can you determine whether the least common multiple of two numbers is the product of the two numbers, or is less than the product of the two numbers? LESSON 4: Searching for Common Ground M1-45

8 NOTES TALK the TALK Bringing It Back Around Answer each question to show how to use the Distributive Property to decompose numbers. 1. Consider the sum a. Express the sum as many ways as possible as the product a(b 1 c). b. How can you use factors to determine if you have listed all possible products a(b 1 c) that are equivalent to ? 2. Suppose you have a composite figure composed of a rectangle and another parallelogram with a shared side. The area of the rectangle is 72 square centimeters and the area of the parallelogram is 84 square centimeters. Explain how to use factors and multiples to determine all possible dimensions a, b, and h for the figure. h a b M1-46 TOPIC 1: Factors and Area

9 Grid Paper LESSON 4: Searching for Common Ground M1-47

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11 Assignment Write 1. Match each definition to its corresponding term. a. a rectangular arrangement that has an equal number of objects in each row and an equal number of objects in each column b. the product of a given whole number and another whole number c. two natural numbers other than zero that are multiplied together to produce another number d. one of the two numbers being multiplied together in a factor pair e. changing the order of two or more factors in a multiplication problem does not change the product i. factor pair ii. array iii. Commutative Property of Multiplication iv. factor v. multiple 2. Select the word that makes the following statement true. Then, use complete sentences to explain your choice: The LCM of two numbers is (always, sometimes, never) the product of the two numbers. Remember Numbers can be decomposed into a product of their prime factors. Numbers can be composed into multiples. Numbers can be compared by their greatest common factor and their least common multiple. Practice 1. Consider the numbers 18 and 30. a. List all of the factors of 18. b. List all of the factors of 30. c. What factors do 18 and 30 have in common? d. What is the greatest common factor of 18 and 30? 2. Consider the numbers 54 and 72. a. Complete a prime factorization of 54 and write it as a product of primes. b. Complete a prime factorization of 72 and write it as a product of primes. c. Put the prime factors of 54 and 72 into a table. d. What are the common factors of 54 and 72? e. What is the greatest common factor of 54 and 72? 3. For each pair of numbers, determine the least common multiple and at least one other common multiple. a. 3 and 5 b. 4 and 6 c. 8 and 12 LESSON 4: Searching for Common Ground M1-49

12 Stretch 1. Determine the LCM for each group of numbers. a. 4, 8, 14 b. 9, 15, Determine the GCF for each group of numbers. a. 8, 27, 35 b. 20, 90, 50 Review Determine the area of each figure m 4 m 2. In the given kite, SZ 5 WZ 5 10 yards, TZ 5 12 yards, and RZ 5 32 yards. S 16 m R Z T 6 m W 3. The polygon is a rhombus yd 5 yards 2 yards 8 yards 11 yd 5. 7 ft 7 ft M1-50 TOPIC 1: Factors and Area