6th Grade. Factors and Multiple.

Size: px

Start display at page:

Download "6th Grade. Factors and Multiple."

Estella Davidson
5 years ago
Views:

1 1

2 6th Grade Factors and Multiple

3 Factors and Multiples Click on the topic to go to that section Even and Odd Numbers Divisibility Rules for 3 & 9 Greatest Common Factor Least Common Multiple GCF and LCM Word Problems Glossary & Standards Teacher Notes 3

4 Even and Odd Numbers Return to Table of Contents 4

5 Warm Up Exercise Think about the following questions and write your answers in your notes. 1) What is an even number? 2) List some examples of even numbers. 3) What is an odd number? 4) List some examples of odd numbers. Derived from 5

6 What do you think? What happens when we add two even numbers? Will we always get an even number? 6

7 Adding Even Numbers Drag the paw prints into the box to model Circle pairs of paw prints to determine if any of the paw prints are left over. Will the sum be even or odd every time two even numbers are added together? Why or why not? 7

8 Adding Odd Numbers Drag the paw prints into the box to model Circle pairs of paw prints to determine if any of the paw prints are left over. Will the sum be even or odd every time two odd numbers are added together? Why or why not? 8

9 Adding Odd and Even Numbers Drag the paw prints into the box to model Circle pairs of paw prints to determine if any of the paw prints are left over. Will the sum be even or odd every time an odd and even number are added together? Why or why not? If the first addend was even and the second was odd, then would your answer change? Why or why not? 9

10 1 The product of two even numbers is even. True False 10

11 2 The product of two odd numbers is A B odd even Explain your answer. Multiplication is repeated addition. If you add an odd number over and over, then the sum will switch between even and Click to Reveal odd. Since you are adding the number an odd number of times, your product will be odd. 11

12 3 The product of 13 x 8 is A B odd even Explain your answer. 13 x 8 is equivalent to saying Since you are adding it an even Click number to Reveal of times, the product will be even. 12

13 4 The sum of 32, ,521 is A B odd even Explain your answer. If you model the numbers using dots and circle all the pairs, the single dots leftover from each Click number to Reveal will create a pair and none will be leftover making the sum an even number. 13

14 5 The product of 12 x 9 is A B odd even Explain your answer. 12 x 9 is equivalent to No matter how many times Click you add to Reveal 12, since it is even the sum will always be even. 14

15 6 The sum of 8, ,999 is A B odd even Explain your answer. If you model the problem using Click to dots Reveal and circle all the pairs, then there will be one dot leftover since one of the addends is odd. 15

16 7 The sum of is A B odd even Explain your answer. The first two addends will result in an odd number. By adding another odd number, the sum Click is to even. Reveal Adding an even number will result in an even number. Since the last addend is odd, the final answer will be odd. 16

17 8 The product of 250 x 19 is A B odd even Explain your answer. The product of an odd and even number will always result in an even number. Click to Reveal 17

18 9 The product of 15 x 0 is A B odd even Explain your answer. 0 is an even number and the product of any even number and odd number is always even. Click to Reveal 18

19 Divisibility Rules for 3 and 9 Return to Table of Contents 19

20 Let's review! Below is a list of numbers. Drag each number in the circle(s) that is a factor of the number. You may place some numbers in more than one circle ,678 29, , Derived from 20

21 Divisibility Rules 2: If and only if its last digit is 0, 2, 4, 6, or 8. 4: If and only if its last two digits are a number divisible by 4. 5: If and only if its last digit is 0 or 5. 8: If and only if its last three digits are a number divisible by 8. 10: If and only if its last digit is 0. 21

22 Divisibility Rule for 3 What factor do the numbers 12, 15, 27, and 66 have in common? They are all Click divisible by 3. Now, take each of those numbers and calculate the sum of its digits = What do all these sums have in common? They are all divisible by 3! Click 66 A number is divisible by 3 if the sum of the number's Click digits is divisible by 3. 22

23 Divisibility Rule for 9 What factor do the numbers 18, 27, 45, and 99 have in common? They are all Click divisible by 9. Now, take each of those numbers and calculate the sum of its digits = What do all these sums have in common? They are all divisible by 9! Click 99 A number is divisible by 9 Click if the sum of the number's digits is divisible by 9. 23

24 Try these! Check if the numbers in the chart are divisible by 3 or 9. Put a check mark in the box in the correct column ,476 Divisible by 3 Divisible by 9 24

25 is divisible by: (choose all that apply) A 2 B 3 C 4 D 5 E 8 F 9 G 10 25

26 11 Is any number divisible by 9 also divisible by 3? Explain. Yes No 26

27 12 Is 135 divisible by 3? Yes No 27

28 13 Any number divisible by 3 is also divisible by 9. True False 28

29 14 The number 129 is divisible by 9. True False 29

30 15 Is 24,981 divisible by 3? If it is, type the quotient. If it is not, type

31 1. Make a table listing all the possible first moves, proper factors, your score and your partner's score. Here's an example: First Move Discussion Questions Proper Factors My Score 2. What number is the best first move? Why? Partner's Score 1 None Lose a Turn , Choosing what number as your first move would make you lose your next turn? Why? 4. What is the worst first move other than the number you chose in Question 3? more questions 31

32 Discussion Questions Continued 5. On your table, circle all the first moves that only allow your partner to score one point. These numbers have a special name. What are these numbers called? Are all these numbers good first moves? Explain. 6. On your table, draw a triangle around all the first moves that allow your partner to score more than one point. These numbers also have a special name. What are these numbers called? Are these numbers good first moves? Explain. 32

33 Activity Party Favors! You are planning a party and want to give your guests party favors. You have 24 chocolate bars and 36 lollipops. Discussion Questions What is the greatest number of party favors you can make if each bag must have exactly the same number of chocolate bars and exactly the same number of lollipops? You do not want any candy left over. Explain. Could you make a different number of party favors so that the candy is shared equally? If so, describe each possibility. Teacher Notes Which possibility allows you to invite the greatest number of guests? Why? Uh oh! Your little brother ate 6 of your lollipops. Now what is the greatest number of party favors you can make so that the candy is shared equally? 33

34 Greatest Common Factor We can use prime factorization to find the greatest common factor (GCF). 1. Factor the given numbers into primes. 2. Circle the factors that are common. 3. Multiply the common factors together to find the greatest common factor. 34

35 16 Is 54 divisible by 3 and 9? Yes No 35

36 17 Is 15,516 divisible by 9? If it is, type the quotient. If it is not, type

37 18 Which of the following numbers is divisible by 3, 4 and 5? A 45 B 54 C 60 D 80 37

38 19 The number 126 is divisible by: (choose all that apply) A 2 B 3 C 4 D 5 E 8 F 9 G 10 38

39 20 The number 120 is divisible by: (choose all that apply) A 2 B 3 C 4 D 5 E 8 F 9 G 10 39

40 Greatest Common Factor Math Practice Return to Table of Contents 40

41 Prime Factorization Use prime factorization to find the greatest common factor of 12 and = 2 x 2 x 3 16 = 2 x 2 x 2 x 2 Teacher Notes The Greatest Common Factor is 2 x 2 = 4 41

42 Teacher Notes Another way to find Prime Factorization... Use prime factorization to find the greatest common factor of 12 and = 2 x 2 x 3 16 = 2 x 2 x 2 x 2 The Greatest Common Factor is 2 x 2 = 4 42

43 Example Use prime factorization to find the greatest common factor of 60 and Teacher Notes 60 = 2 x 2 x 3 x 5 72 = 2 x 2 x 2 x 3 x 3 GCF is 2 x 2 x 3 = 12 43

44 Example Use prime factorization to find the greatest common factor of 60 and = 2 x 2 x 3 x = 2 x 2 x 2 x 3 x 3 GCF is 2 x 2 x 3 =

45 Example Use prime factorization to find the greatest common factor of 36 and = 2 x 2 x 3 x 3 90 = 2 x 3 x 3 x 5 Teacher Notes GCF is 2 x 3 x 3 = 18 45

46 Teacher Notes Example Use prime factorization to find the greatest common factor of 36 and = 2 x 2 x 3 x 3 90 = 2 x 3 x 3 x 5 GCF is 2 x 3 x 3 = 18 46

47 21 Find the GCF of 18 and

48 22 Find the GCF of 28 and

49 23 Find the GCF of 55 and

50 24 Find the GCF of 52 and

51 25 Find the GCF of 72 and

52 26 What is the greatest common factor of 16 and 48. Enter your answer in the box. From PARCC EOY sample test non calculator #13 52

Interactive Website Review of factors, prime numbers and composite numbers. Play the Factor Game a few times with a partner. Be sure to take turns going first.

53 Interactive Website Review of factors, prime numbers and composite numbers. Play the Factor Game a few times with a partner. Be sure to take turns going first. Find moves that will help you score more points than your partner. Be sure to write down strategies or patterns you use or find. the Discussion Questions. 53

54 Game (Rows and Columns can be adjusted prior to starting the game) Player 1 chose 24 to earn 24 points. Player 2 finds 1, 2, 3, 4, 6, 8, 12 and earns 36 points. Player 2 chose 28 to earn 28 points. Player 1 finds 7 and 14 are the only available factors and earns 21 points. 54

55 Relatively Prime Two or more numbers are relatively prime if their greatest common factor is 1. Example: 15 and 32 are relatively prime because their GCF is 1. Name two numbers that are relatively prime. 55

56 27 Seven and 35 are not relatively prime. True False 56

57 28 Identify at least two numbers that are relatively prime to 9. A 16 B 15 C 28 D 36 57

58 29 Name a number that is relatively prime to

59 30 Name a number that is relatively prime to 5 and

60 31 Choose two numbers that are relatively prime. A 7 B 14 C 15 D 49 60

61 Least Common Multiple Math Practice Return to Table of Contents 61

62 Text to World Connection (Click for Link to Video Clip) 1. Use what you know about factor pairs to evaluate George Banks' mathematical thinking. Is his thinking accurate? What mathematical relationship is he missing? 2. How many hot dogs came in a pack? Buns? 3. How many "superfluous" buns did George Banks remove from each package? How many packages did he do this to? 4. How many buns did he want to buy? Was his thinking correct? Did he end up with 24 hot dog buns? Teacher Notes 5. Was there a more logical way for him to do this? What was he missing? 6. What is the significance of the number 24? 62

63 Least Common Multiple A multiple of a whole number is the product of the number and any nonzero whole number. A multiple that is shared by two or more numbers is a common multiple. Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48,... Multiples of 14: 14, 28, 42, 56, 70, 84,... The least of the common multiples of two or more numbers is the least common multiple (LCM). The LCM of 6 and 14 is

64 Least Common Multiple There are 2 ways to find the LCM: 1. List the multiples of each number until you find the first one they have in common. 2. Write the prime factorization of each number. Multiply all factors together. Use common factors only once (in other words, use the highest exponent for a repeated factor). 64

65 EXAMPLE: 6 and 8 Multiples of 6: 6, 12, 18, 24, 30 Multiples of 8: 8, 16, 24 LCM = 24 Least Common Multiple Prime Factorization: LCM: = 8 3 = 24 65

66 Find the least common multiple of 18 and 24. Multiples of 18: 18, 36, 54, 72,... Multiples of 24: 24, 48, 72,... LCM: 72 Prime Factorization: Example LCM: = 8 9 = 72 66

67 32 Find the least common multiple of 10 and 14. A 2 B 20 C 70 D

68 33 Find the least common multiple of 6 and 14. A 10 B 30 C 42 D

69 34 Find the least common multiple of 9 and 15. A 3 B 45 C 60 D

70 35 Find the least common multiple of 6 and 9. A 3 B 12 C 18 D 36 70

71 36 Find the least common multiple of 16 and 20. A 80 B 100 C 240 D

72 37 Find the LCM of 12 and

73 38 Find the LCM of 24 and

74 39 Find the LCM of 15 and

75 40 Find the LCM of 24 and

76 41 Find the LCM of 15 and

77 42 Find the LCM of 20 and

78 Interactive Website Uses a venn diagram to find the GCF and LCM for extra practice. 78

79 GCF and LCM Word Problems Return to Table of Contents 79

80 Question How can you tell is a word problem requires you to use Greatest Common Factor or Least Common Multiple to solve? Math Practice 80

81 GCF Problems Do we have to split things into smaller sections? Are we trying to figure out how many people we can invite? Are we trying to arrange something into rows or groups? 81

82 LCM Problems Do we have an event that is or will be repeating over and over? Will we have to purchase or get multiple items in order to have enough? Are we trying to figure out when something will happen again at the same time? 82

83 Example Samantha has two pieces of cloth. One piece is 72 inches wide and the other piece is 90 inches wide. She wants to cut both pieces into strips of equal width that are as wide as possible. How wide should she cut the strips? What is the question: How wide should she cut the strips? Important information: One cloth is 72 inches wide. The other is 90 inches wide. Math Practice Is this a GCF or LCM problem? Does she need smaller or larger pieces? This click is a GCF problem because we are cutting or "dividing" the pieces of cloth into smaller pieces (factor) of 72 and

84 Bar Modeling Use the greatest common factor to determine the greatest width possible. The greatest common factor represents the greatest width possible not the number of pieces, because all the pieces need to be of equal length. This is called making a Bar Model. 72 inches 90 inches Teacher Notes click 18 inches 84

85 Example Ben exercises every 12 days and Isabel every 8 days. Ben and Isabel both exercised today. How many days will it be until they exercise together again? What is the question: How many days until they exercise together again? Important information: Ben exercises every 12 days Isabel exercises every 8 days Is this a GCF or LCM problem? Math Practice Are they repeating the event over and over or splitting up the days? click This is a LCM problem because they are repeating the event to find out when they will exercise together again. 85

86 Bar Modeling Use the least common multiple to determine the least amount of days possible. The least common multiple represents the number of days not how many times they will exercise. Ben exercises in: 12 Days Isabel exercises in: Teacher Notes 8 Days 86

87 43 Mrs. Evans has 90 crayons and 15 pieces of paper to give to her students. What is the largest number of students she can have in her class so that each student gets an equal number of crayons and an equal number of paper? A B GCF Problem LCM Problem 87

88 44 Mrs. Evans has 90 crayons and 15 pieces of paper to give to her students. What is the largest number of students she can have in her class so that each student gets an equal number of crayons and an equal number of paper? A 3 B 5 C 15 D 90 88

89 45 How many crayons and pieces of paper does each student receive if there are 15 students in the class? A B C D 30 crayons and 10 pieces of paper 12 crayons and pieces of paper 18 crayons and 6 pieces of paper 6 crayons and 1 piece of paper 89

90 46 Rosa is making a game board that is 16 inches by 24 inches. She wants to use square tiles. What is the largest tile she can use? A B GCF Problem LCM Problem 90

91 47 Rosa is making a game board that is 16 inches by 24 inches. She wants to use square tiles. What is the largest tile she can use? 91

92 48 How many tiles will she need? 92

93 49 Y100 gave away a $100 bill for every 12th caller. Every 9th caller received free concert tickets. How many callers must get through before one of them receives both a $100 bill and a concert ticket? A B GCF Problem LCM Problem 93

94 50 Y100 gave away a $100 bill for every 12th caller. Every 9th caller received free concert tickets. How many callers must get through before one of them receives both a $100 bill and a concert ticket? A 36 B 3 C 108 D 6 94

95 51 There are two ferris wheels at the state fair. The children's ferris wheel takes 8 minutes to rotate fully. The bigger ferris wheel takes 12 minutes to rotate fully. Marcia went on the large ferris wheel and her brother Joey went on the children's ferris wheel. If they both start at the bottom, how many minutes will it take for both of them to meet at the bottom at the same time? A B GCF Problem LCM Problem 95

96 52 There are two ferris wheels at the state fair. The children's ferris wheel takes 8 minutes to rotate fully. The bigger ferris wheel takes 12 minutes to rotate fully. Marcia went on the large ferris wheel and her brother Joey went on the children's ferris wheel. If they both start at the bottom, how many minutes will it take for both of them to meet at the bottom at the same time? A 2 B 4 C 24 D 96 96

97 53 How many rotations will each ferris wheel complete before they meet at the bottom at the same time? (Input the answer for the small ferris wheel.) 97

98 54 Sean has 8 inch pieces of toy train track and Ruth has 18 inch pieces of train track. How many of each piece would each child need to build tracks that are equal in length? A B GCF Problem LCM Problem 98

99 55 What is the length of the track that each child will build? 99

100 56 I am planting 50 apple trees and 30 peach trees. I want the same number and type of trees per row. What is the maximum number of trees I can plant per row? A B GCF Problem LCM Problem 100

101 Glossary & Standards Teacher Notes Return to Table of Contents 101

102 Standards for Mathematical Practice MP1: Making sense of problems & persevere in solving them. MP2: Reason abstractly & quantitatively. MP3: Construct viable arguments and critique the reasoning of others. MP4: Model with mathematics. MP5: Use appropriate tools strategically. MP6: Attend to precision. MP7: Look for & make use of structure. MP8: Look for & express regularity in repeated reasoning. Math Practice Additional questions are included on the slides using the "Math Practice" Pull tabs (e.g. a blank one is shown to the right on this slide) with a reference to the standards used. If questions already exist on a slide, then the specific MPs that the questions address are listed in the Pull tab. 102

103 Bar Model A diagram that uses bars to show the relationship between two or more numbers. Part Whole Part Part + Part = Whole Whole Part = Part Larger Amount Smaller Amount Difference Large Small = Difference Large Difference = Small Whole Part One part x # of parts Whole Back to Instruction 103

104 Composite Number A number that has more than two factors x 12 2 x 6 3 x 4 6 factors 3 x 5 = 15 Any number with factors other than one and itself is composite x 13 Only 2 factors. Back to Instruction 104

105 Exponent A small, raised number that shows how many times the base is used as a factor. Exponent 3 2 Base "3 to the second power" 3 2 = 3 x = 3x 3x x x 3 3 Back to Instruction 105

106 Factor A whole number that can divide into another number with no remainder. A whole number that multiplies with another number to make a third number x 5 = R is a factor of 15 3 and 5 are factors of 15 3 is not a factor of 16 Back to Instruction 106

107 Greatest Common Factor (GCF) The largest number that will divide two or more numbers without a remainder. 12: 1, 2, 3, 4, 6, 12 16: 1, 2, 4, 8, 16 Common Factors are 1, 2, 4 GCF is 4 Using Prime Factorization 12 = 2 x 2 x 3 16 = 2 x 2 x 2 x 2 GCF = 2 x 2 GCF is 4 1 and 2 are common factors, but not the greatest common factor. Back to Instruction 107

108 Least Common Multiple (LCM) The smallest number that two or more numbers share as a multiple. 9: 9, 18, 27, 36, 45 15: 15, 30, 45 LCM is 45 Using Prime Factorization 9 = 3 x 3 15 = 3 x 5 LCM = 3 x 3 x 5 LCM is 45 2: 2, 4, 6, 8 4: 4, 8 4 is the LCM, not 8 Back to Instruction 108

109 Multiple The product of two whole numbers is a multiple of each of those numbers. 3 x 5 = is a multiple of 3. 2 x 6 = 12 Factors Product / Multiple 4 x 5 = 20 5 and 4 are factors of 20, not multiples. Back to Instruction 109

110 Prime Factorization A number written as the product of all its prime factors. 18 = 2 x 3 x 3 or 18 = 2 x 3 2 There is only one for any number. 18 = 1 x 2 x 3 x 3 Only prime numbers are included in prime factorizations. Back to Instruction 110

111 Prime Number A positive integer that is greater than 1 and has exactly two factors, one and itself. Prime #s to 30 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 2 Two is the only even prime number. 1 One is not a prime number, because it has only one factor. Back to Instruction 111

112 Proper Factor All of the factors of a number other than one and itself. 6: 1, 2, 3, 6 Proper Factors: 2 and 3 9: 1, 3, 9 Proper Factor: 3 7: 1, 7 The number 7 does not have any proper factors. Back to Instruction 112

113 Relatively Prime Two numbers who only have 1 as a common factor. 8: 1, 2, 4, 8 15: 1, 3, 5 Only Common Factor is 1 All prime numbers are relatively prime to every other number. 9: 1, 3, 9 15: 1, 3, 5, 15 Common Factors: 1 and 3 Back to Instruction 113

5th Grade. Divisibility Rules. Slide 1 / 239 Slide 2 / 239. Slide 3 / 239. Slide 4 / 239. Slide 6 / 239. Slide 5 / 239. Division. Division Unit Topics

5th Grade. Divisibility Rules. Slide 1 / 239 Slide 2 / 239. Slide 3 / 239. Slide 4 / 239. Slide 6 / 239. Slide 5 / 239. Division. Division Unit Topics Slide 1 / 239 Slide 2 / 239 5th Grade Division 2015-11-25 www.njctl.org Slide 3 / 239 Slide 4 / 239 Division Unit Topics Click on the topic to go to that section Divisibility Rules Patterns in Multiplication