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Math 6 Unit 1 Lesson 4 Factors of Whole Numbers Comparing Ages Alyssa is twelve years old. She has three cousins. They are Daron, age 2, Rachel age 3, and Matt age 8. Alyssa wrote the following mathematical sentences to show her age: Alyssa Age 12 Math 6 Six times as old as Daron: 12 = 6 x 2 Four times as old as Rachel: 12 = 4 x 3 % ∆ Daron Age 2 Rachel Age 3 One and one half times as old as Matt: 12 = 11 2 x 8 Matt Age 8 1-33 Math 6 Unit 1 Lesson 4: Factors of Whole Numbers Reflection How are the first two examples different from the last? Daksha is also 12 years old. He has four cousins. They are Anoop, age 2, Bishnu age 4, Dayita, age 5 and Rani, age 6. Can you write mathematical sentences for Daksha’s age like those that Alyssa wrote? The easy ones should be Anoop, Bishnu and Rani. Daksha is six times as old as Anoop: 12 = 6 x 2 Daksha is three times as old as Bishnu: 12 = 3 x 4 Daksha is two times as old as Rani: 12 = 2 x 6 These are easy because the numbers 2, 4, and 6 are factors of the number 12. Since 5 is not a factor of 12 it is harder to write a description. You cannot find another whole number to multiply by 5 to get a product of 12. Daksha Age 12 Rani Age 6 Dayita Age 5 Bishnu Age 4 Anoop Age 2 ∏ 1-34 Math 6 Unit 1 Lesson 4: Factors of Whole Numbers Reflection Are there any other factors of 12? Objectives for this Lesson In this lesson you will explore the following concepts: • • • • • Use arrays to find all the factors of a number Explain strategies to find factors of a number Use concrete and visual representations to find factors of a number Use repeated division to find factors of a number Solve problems involving factors or multiples Factors A whole number that divides evenly into a number is called a factor of the number. 3 is a factor of 12 because when 3 is multiplied by 4 the product is 12. 12 = 3x4 Product Factors 4 is also a factor of 12 because when 4 is multiplied by 3 the product is 12. Math 6 1-35 Math 6 Unit 1 Lesson 4: Factors of Whole Numbers Example 1 Identify the factors in the following number sentences: a. 12 x 7 = 84 b. 15 x 3 = 45 c. 76 x 4 = 304 a. The factors of 84 in this number sentence are 12 and 7. b. The factors of 45 in this number sentence are 15 and 3. c. The factors of 304 in this number sentence are 76 and 4. In some cases you can use your knowledge of multiplication facts to 12 x 12 to find the factors of a number. You may also need to use some methods when the numbers are a little harder. Example 2 Identify the factors of 24. Start with 1 and list multiplication sentences that have a product of 24: 1 x 24 = 24 2 x 12 = 24 3 x 8 = 24 4 x 6 = 24 At some point the factors will reverse and you will be listing the same factors all over again: 6 x 4 = 24 8 x 3 = 24 12 x 2 = 24 1 x 24 = 24 The factors of 24 are: 1, 2, 3, 4, 6, 8, 12, 24 You may also notice a pattern to listing the factors of a number: 1-36 Factors of 18: 1, 2, 3, 6, 3 x 6 = 18 2 x 9 = 18 1 x 18 = 18 9, 18 Sometimes, this type of pattern can help you list your factors. Math 6 Unit 1 Lesson 4: Factors of Whole Numbers Using Arrays to Find Factors Just as you used arrays to find the multiples of a number you can also use them to find factors of a number. Example 3 Use the grid paper to find all of the arrays that are made using a total of 18 squares. List the factors of 18. You need to make as many unique rectangles as possible that have a total of 18 squares: 18 1 9 2 6 3 When you cannot find any more unique rectangles for the number of squares needed, you have found all the factors of the number. The factors of 18 are: 1, 2, 3, 6, 9, 18 Some numbers have more factors and some have fewer factors than 18. Math 6 1-37 Math 6 Unit 1 Lesson 4: Factors of Whole Numbers Reflection Do you think the size of the number determines the number of factors? Why or why not? Let’s Explore Exploration 1: Arrays to Factors Materials: Unit 1, Lesson 4, Exploration 1 page in your Workbook, Grid Paper from the back of this Unit in your Workbook (you may need two pieces), 5 Pencil Crayons, Pencil For 1 – 5: Use a different coloured pencil crayon for each. 1.Make all possible arrays of 36 squares on your grid paper. List the factors of 36. 2.Make all possible arrays of 21 squares on your grid paper. List the factors of 21. 3.Make all possible arrays of 32 squares on your grid paper. List the factors of 32. 4.Make all possible arrays of 17 squares on your grid paper. List the factors of 17. 5.Make all possible arrays of 29 squares on your grid paper. List the factors of 29. 1-38 Math 6 Unit 1 Lesson 4: Factors of Whole Numbers The proper factors of a number are all of the factors less than the number but greater than 1. The proper factors of 18 are: 2, 3, 6, 9 6. How many proper factors does the number 29 have? 7.Reflect: What is different about the factors of 29 compared to the factors of 32? 8. Reflect: How many numbers from 1 to 20 have no proper factors? Using Repeated Division to Find Factors You can also use division to identify factors of a number. The factors of 18 using division: 18 ÷ 1 = 18 18 ÷ 2 = 9 18 ÷ 3 = 6 18 ÷ 6 = 3 18 ÷ 9 = 2 To start finding factors of larger numbers you may want to use division. A number is always divisible by 1 and itself. To find the factors you may want to start with a number larger than 1. Example 4 Zach has 88 baseball cards. He wants to arrange them in rows with an equal number of cards in each row on his wall. What are the different arrangements can he make for the cards? You need to know all of the multiplication sentences for a product of 88. You can use upside down repeated division to find the factors of 88. Only divide by numbers that are divisible by 1 and itself. Math 6 1-39 Math 6 Unit 1 1. D ivide by a number and write the division upside down like this: 2 88 Lesson 4: Factors of Whole Numbers 2. N ow divide the 2 into 88 and write the answer beneath the 88: 2 88 44 3. R epeat the process going down the page until you cannot divide anymore: 2 88 44 22 2 88 44 22 11 The only whole numbers that divide 11 evenly are 1 and itself, so you have finished dividing. 4. List the factors that you found: 1, 2, 11, 22, 44, 88 5.List all number sentences: You have the following number sentences that are complete: 1-40 1 x 88 = 88 2 x 44 = 88 And you have the following that are incomplete: __ x 11 = 88 __ x 22 = 88 Complete these sentences: 8 x 11 = 88 4 x 22 = 88 Math 6 Unit 1 Lesson 4: Factors of Whole Numbers 6. Answer the question: Zach can make the following arrangements: 1 row of 88 cards 2 rows of 44 cards 4 rows of 22 cards 8 rows of 11 cards 88 rows of 1 card 44 rows of 2 cards 22 rows of 4 cards 11 rows of 8 cards You can also use factor trees to track the numbers that you have divided into your number. One Factor tree of 88: Another Factor tree of 88: 88 88 2 8 44 2 2 22 2 4 2 11 11 2 Both methods allow you to find several factors of 88. You can then use them to identify the rest: 1, 2, ____, ____, 11, 22, 44, 88 1, 2, 4, 8, 11, ____, ____, 88 ? X 11 = 88 4 x ? = 88 ? x 22 = 88 2 x ? = 88 1, 2, 4, 8, 11, 22, 44, 88 Math 6 1, 2, 4, 8, 11, 22, 44, 88 1-41 Math 6 Unit 1 Lesson 4: Factors of Whole Numbers Comparing Factors You can use Venn diagrams to show the relationship between factors of two numbers. Example 5 Complete the Venn diagram with all possible factors of each number. Factors of 24 Factors of 36 List the factors of each number. Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36 The factors they have in common go in the intersection of the circles. Factors of 24 Factors of 36 1 8 24 1-42 2 3 4 6 12 9 18 36 Math 6 Unit 1 Lesson 4: Factors of Whole Numbers You can use your Venn diagram to answer questions like: • • • • Do the numbers share any factors? What is the largest factor these two numbers share? How many factors do the numbers share? Which number has more factors? Let’s Practice • G o online to watch the Notepad Tutor Lesson: Comparing Factors Using Graphic Organizers. • Turn in your Workbook to Unit 1, Lesson 4 and complete 1 to 21. Math 6 1-43 Math 6 Unit 1 1-44 Lesson 4: Factors of Whole Numbers