Factors and Multiples. Chapter NUMBER. Big Idea. Learning Goals. Essential Question. Important Words

Transcription

1 NUMBER Factors and Multiples Chapter 4 Big Idea Understanding multiples and factors helps me describe and solve realworld problems. Learning Goals I can determine factors and multiples of numbers less than 100. I can identify prime and composite numbers. I can solve problems involving multiples. Essential Question How can understanding multiples and factors help me understand the world around me? Important Words composite number factor multiple prime number

2 Making Multiples Use the strategies of your choice to find multiples of given numbers, find multiples common to two or more numbers, and use multiples to describe numbers. Example: Explain which number does not belong: 4, 8, 12, 15, 20, 24 Bennett s explanation: All the numbers are even, except does not belong. Carlos explanation: All the numbers are multiples of 4, except does not belong. Dallas explanation: All the numbers are evenly divisible by 4, except does not belong. Edward s explanation: I skip counted by four to get 4, 8, 12, 16, 20, 24. This is the same as the list, except for does not belong. 11. Continue each pattern for three more elements. a. 10, 20, 30, b. 6, 12, 18, c. 9, 18, 27, 12. Explain which number in each list does not belong. a. 5, 10, 15, 20, 21, 30 b. 6, 12, 16, 24, 30, 36 c. 7, 14, 21, 28, 35, 40 d. 12, 20, 30, 40, 50, CHAPTER 4: Factors and Multiples

3 Making Multiples (continued) 13. A stool has three legs. Copy and complete the table to show the number of legs needed to build different numbers of stools. Number of stools Number of legs Give the total value, in cents, of each of the following collections of coins. a. two nickels b. two dimes c. five nickels d. five dimes e. twelve nickels f. twelve dimes 15. Classify each of the following statements as true or false. a. 10 is a multiple of 5. b. 10 is a multiple of 2. c. 10 is a multiple of 4. d. 15 is a multiple of 3. e. 15 is a multiple of 6. f. 15 is a multiple of Complete the following steps to find multiples that are common to the numbers 4 and 5. a. List the first ten multiples of 4. b. List the first ten multiples of 5. c. List the multiples that are common to both lists. A multiple is the answer when two whole numbers are multiplied. d. Explain how you could find other factors common to the numbers 4 and 5. CHAPTER 4: Factors and Multiples 83

4 Making Multiples (continued) 17. Find multiples that are common to the numbers 3 and 6. a. What strategies did you use? b. What sorts of numbers did you find that were multiples of 3 and 6? 18. Find the lowest multiple that is common to each set of numbers. a. 2 and 8 b. 2, 3, and 4 c. 8 and 10 d. 5, 6, and Describe each number in many ways, using the word multiple. a. 8 b. 15 c. 37 d is a multiple of 4, and a multiple of 3, and a multiple of 6! 10. The words multiple and multiply sound the same. How are the ideas similar? 11. What strategies do you like best for finding multiples? 12. Ten is the lowest multiple for the numbers 2 and 5. Is there a highest multiple for these numbers? I can determine factors and multiples of numbers less than 100. I can solve problems involving multiples. 84 CHAPTER 4: Factors and Multiples

5 Multiplicity Use your skills with multiples to solve problems from real-life contexts, such as work schedules, music, and travel. Example: Finn and Graydon each have the same amount of money. Finn only has dimes and Graydon only has quarters. How much money could they have? How many coins could each boy have? Finn coul d have 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120, 130, 140, 150 and so on. Graydon coul d have 25, 50, 75, 100, 125, 150, 175, 200, 225 and so on. They coul d both have 50. Graydon: Finn: They coul d both have 100. Graydon: Finn: They coul d both have 150 if Graydon has six quarters and Finn has fifteen dimes. This pattern can continue as long as Finn has five dimes for every two quarters Graydon has. CHAPTER 4: Factors and Multiples 85

6 Multiplicity (continued) 11. Dana s bean plant grows 3 cm each week. Matthew s bean plant grows 4 cm each week. a. How much will each plant grow in two weeks? b. How tall will Dana s plant be after five weeks? c. How tall will Matthew s plant be after four weeks? d. At what height will the two plants first be equal heights? e. When could each student plant their bean plants to have them reach the same height at the same time? 12. Mr. Yee works as an air traffic controller. He noticed that, beginning at 6 p.m., West Air flights arrive every 15 minutes and Air Saskatchewan flights arrive every 12 minutes. At which times will Mr. Yee be most occupied? 13. In her band, Johanna has to play the drum on every sixth beat and the cymbal on every tenth beat. On which beats does she play both? 14. The Tipaskan Tigers have to transport 96 basketball players to a tournament. They can use cars that carry four passengers, vans that carry 12 passengers, or buses that carry 42 passengers. a. If they only used cars, how many cars would be needed? b. If they only used vans, how many vans would be needed? c. If they only used buses, how many buses would be needed? d. What is the best way to transport the players? 15. Adrian is planning a class picnic. He wants to buy an equal number of hot dogs and hot dog buns. Hot dogs come in packages of 12. Hot dog buns come in packages of eight. a. What is the fewest number of packages he could buy to have an equal number of each? b. Adrian is expecting 60 people at the picnic. How many packages should he buy to have an equal number of each, and have enough for all the people at the picnic? 16. Trevor visits his Opa at the Senior s Lodge every two months. He visits his Tante every three months. How may times in a year does Trevor visit both his Tante and his Opa in the same month? 86 CHAPTER 4: Factors and Multiples

7 Multiplicity (continued) 17. Lorna, Mary, Nancy, Opal, and Penny all work together on the school safety patrol. Lorna works every day, Mary works every second day, Nancy works every third day, Opal works every fourth day, and Penny works every fifth day. Today (Day 0) they all worked together. How many days will it be until they all work together again? 18. Hayden and Ibrahim are running laps to practise for the upcoming track meet. Hayden runs seven laps in five minutes and Ibrahim runs four laps in five minutes. They both run for 25 minutes. a. How far has each boy run? b. How much farther has Hayden run than Ibrahim? c. If they each want to run 20 laps, how long should they each run? 19. Jace is counting the change in his money bank. To count faster he is making piles of the same type of coin that total $1. How many of each coin would be in each pile? 10. Two families have the same number of children. Karim s family has only twins. Landon s family has only triplets. How many children could be in each boy s family? 11. How many packages do you need to have a class set of each of the following? a. cartons of one dozen eggs b. six-packs of cola c. 30-packs of glitter pens d. bags of three soccer balls e. sets of eight puppets f. packages of ten mini-whiteboards 12. Explain the strategies you used to solve these problems. 13. Create a word problem that uses multiples in the solution. Exchange problems with a classmate and solve each other s problem. I can determine factors and multiples of numbers less than 100. I can solve problems involving multiples. CHAPTER 4: Factors and Multiples 87

8 Finding Factors Use the strategy of your choice to find factors of a given number, to find factors common to two or more numbers, and to write numbers as multiplication expressions. Example: Find all the factors of 72. Eleanor s strategy: I drew rectangles with an area of 72 to find the factors. I know 1 and 72 are factors, so I don t have to draw that rectangle. 2 x 36 3 x 24 4 x 18 6 x 12 8 x 9 Faith s strategy: I started with a fact I knew, 8x 9, and then changed it to find other factors. I halved 8 and doubled 9 to get 4 x 18. I halved 4 and doubled 18 to get 2 x 36. I halved 2 and doubled 36 to get 1 x 72. I took a third of 9 and tripled 8 to get 3 x 24. I doubled 3 and halved 24 to get 6 x CHAPTER 4: Factors and Multiples

9 Finding Factors (continued) Gemma s strategy: I made a factor rain bow. Hadassah s strategy: I used division to find the factors = = = = = = 9 Because nine is the answer to the last division, I don t need to go any further - I already have all the factors: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72. Ireland s strategy: I used 72 counters and made groups with them. 2 groups of groups of groups of groups of 12 8 groups of = a number with a remainder. 5 is not a factor of = a number with a remainder. 7 is not a factor of CHAPTER 4: Factors and Multiples 89

10 Finding Factors (continued) 11. Write the missing factor in each equation. a. 1? = 6 b. 2? = 20 2? = 6 4? = 20 3? = 6 5? = 20 6? = 6 10? = 20 A factor of a number divides evenly into that number. 12. Explain which of the following number(s) will have a remainder when divided by five. a b. 345 c. 650 d Write each number as a product of two different factors. a. 32 b. 17 c. 45 d Explain which of the following number(s) will have a remainder when divided by two. a. 44 b. 393 c. 985 d Classify each of the following numbers as even or odd. a b c d Draw all the possible rectangles that have the following areas. a. 10 b. 21 c. 16 d Use the strategy of your choice to find all the factors of each number. a. 8 b. 33 c. 13 d CHAPTER 4: Factors and Multiples

11 Finding Factors (continued) 18. Copy and complete the following table. Number Is two a factor? Is three a factor? Is six a factor? a. b. c. d What do you notice about the numbers in question 8 that have 6 as a factor? 10. Complete each sentence with the word multiple or factor. a. 6 is a? of 12 b. 15 is a? of 5 c. 12 is a? of 6 d. 5 is a? of Use the numbers 18 and 24 to complete the following steps. a. Find the factors of 18. b. Find the factors of 24. c. List any numbers that are factors of both numbers. d. Explain how these two numbers can have some of the same factors. 12. Use the numbers 14 and 28 to complete the following steps. a. Find the factors of 14. b. Find the factors of 28. c. List any numbers that are factors of both numbers. d. Explain how these two numbers can have some of the same factors. 13. Find two numbers that have 7 as a factor. Explain the strategy you used. 14. Find two numbers that have exactly three factors. What is special about these numbers? 15. Find each product. a b c d CHAPTER 4: Factors and Multiples 91

12 Finding Factors (continued) 16. Build all the possible prisms that have the following volumes. a. 30 b. 72 c. 56 d Write each number as a product of three different numbers. a. 36 b. 75 c. 52 d Create a factor tree for each number. a. 27 b. 60 c. 35 d Compare your factor trees from question 18 with the factor tree of a classmate. a. How were they the same? b. How were they different? c. Did you write the same multiplication expression for each number? Explain. 20. Explain which number is a factor of every number. 21. Can two numbers ever have all the same factors? Why or why not? I can determine factors and multiples of numbers less than 100. I can solve problems involving multiples. 92 CHAPTER 4: Factors and Multiples

13 Factor Fun Use your skills with factors to solve problems from real-life contexts, such as packaging, music, and travel. Example: Pete s Produce has 36 oranges. The owner wants to package the oranges in bags with the same number of oranges in each bag. What are all the ways he could package the oranges? He could make 1 bag of 36 oranges. He could make 2 bags of 18 oranges. He could make 3 bags of 12 oranges. He could make 4 bags of 9 oranges. He could make 6 bags of 6 oranges. He could make 9 bags of 4 oranges. He could make 12 bags of 3 oranges. He could make 18 bags of 2 oranges. He could make 36 bags of 1 orange. 11. Cain is organizing the spring banquet. He expects 40 guests. Cain would like to seat the guests with the same number of people at each table. a. What are all the possible seating arrangements? b. Which seating arrangement do you think Cain should pick? Why? 12. Find the number less than 100 has the greatest number of factors. 13. There are 48 members in the senior choir. The conductor wants to arrange them in equal rows. How could the choir members be arranged? 14. Kamryn has $300 and Abel has $260. They each have only one kind of bill. a. What kinds of bills could Kamryn and Abel have? b. What is the largest bill Kamryn and Abel could both have? CHAPTER 4: Factors and Multiples 93

14 Factor Fun (continued) 15. A perfect number is equal to the sum of its factors (other than itself). Four is not a perfect number. Its factors are 1, 2, and 4. The sum of = 3. Find the two perfect numbers less than Danica is making a craft and needs to cut squares out of the piece of fabric shown. 36 cm 24 cm a. What different-sized squares could Danica make without wasting any fabric? b. What is the largest square Danica could make without wasting any fabric? 17. Jacinta says that the only factor greater than half of a number is the number itself. a. Find some numbers for which this is true. b. Explain whether you think Jacinta is always correct. 18. Mr. Abner and Mrs. Pawluk are neighbours. They each want to build a fence around their garden and want to share one side of the fence to reduce their costs. Mr. Abner wants his garden to be 36 m 2 and Mrs. Pawluk wants her garden to be 48 m 2. a. What possible dimensions could Mr. Abner s garden be? b. What possible dimensions could Mrs. Pawluk s garden be? c. What length could the neighbours make the shared side of the fence? 19. What is the smallest number that has a remainder of one when divided by 2, 3, 4, 5, and 6? 10. Ms. Snider has 30 desks in her classroom. She would like to arrange the desks in rows of equal length. a. Find all the possible arrangements of Ms. Snider s desks. b. Explain which arrangements you think are the most reasonable. c. Explain which arrangements you think are the least reasonable. 11. Draw rectangles to show the factors of the numbers 1 through 10. a. Which numbers have an odd number of factors? b. Which numbers can be drawn as squares? c. What other numbers have an odd number of factors? 94 CHAPTER 4: Factors and Multiples

15 Factor Fun (continued) 12. To celebrate his birthday, Michael is making treat bags for 24 of his friends. He wants to buy small boxes of candies, small bags of potato chips, and kazoos. The candies come in cartons of six boxes. The chips come in packages of eight bags, and the kazoos come in sets of two. a. How many cartons of candies does he need to buy? b. How many packages of chips does he need to buy? c. How many sets of kazoos does he need to buy? 13. An abundant number is less than the sum of its factors (other than itself). The number 8 is not an abundant number; its factors are 1, 2, 4, and 8. The sum of = 7. There are 21 abundant numbers less than 100. How many can you find? 14. Thomas and Andrew collect hockey cards. Thomas has 45 cards in his collection and Andrew has 30 cards in his collection. The cards in both boys collections come in packages of the same number of cards. a. How many hockey cards could come in each package? b. What is the greatest number of hockey cards that could come in each package? 15. Create a word problem that uses factors in the solution. Exchange problems with a classmate and solve each others problems. 16. How are factors and multiples related? I can determine factors and multiples of numbers less than 100. CHAPTER 4: Factors and Multiples 95

16 Clearly Composite Use your skills with factors and multiples to classify numbers as prime or composite and to solve problems with prime and composite numbers. Example: Write 90 as a product of factors that are prime. Malachi s strategy: I know that 2, 3, 5, and 7 are the first few prime numbers. I ll divide 90 by prime numbers I can write 90 = 2 x 3 x 5 x 3. Nathaniel s strategy: I made a factor tree. 90 / \ 9 x 10 / \ / \ 3 x 3 x 2 x 5 I can write 90 = 3 x 3 x 2 x 5. Omar s strategy: I can start with a fact I know and then rewrite it using other facts. 90 = 3 x 30 = 3 x (3 x 10) = 3 x 3 x (5 x 2) I can write 90 = 3 x 3 x 5 x CHAPTER 4: Factors and Multiples

17 Clearly Composite (continued) Paolo s strategy: I started by drawing a rectangle with an area of 90. From this I know 90 = 15 x 6. Because neither of these numbers is prime, I can draw a rectangle for each new number. 15 x 6 3 x 5 2 x 3 I can write 90 = 3 x 5 x 2 x 3. 3 Quaid s strategy: I can build a prism with 6 a volume of 90. From this I know 90 = 3 x 5 x 6 5 The 2 and the 5 are prime, but the 6 isn t. I need to write the 6 using prime numbers. 6 = 3 x 2 I can write 90 = 3 x 5 x 3 x Look at the strategy each boy used. a. How are their strategies the same? b. How are their strategies different? c. How are their answers the same? d. How are their answers different? 12. Using prime numbers, make a factor tree for each number. Then write each number as a product of prime numbers. a. 24 b. 100 c. 48 d. 144 A prime number has exactly two factors: 1, and the number. e. 56 f. 360 CHAPTER 4: Factors and Multiples 97

18 Clearly Composite (continued) 13. Is your age a prime number or a composite number? a. Will your age be prime next year? b. Was your age prime last year? A composite number has at least three factors. 14. Classify each of the following numbers as prime or composite. a. 15 b. 57 c. 96 d. 38 e. 36 f. 43 g. 27 h. 37 i If you reverse the digits of the prime number 17 you get 71, another prime number. There are four other pairs of prime numbers less than 100 that are like this. a. How many can you find? b. These reversed numbers are sometimes called emirp numbers. Why do you think that is? 16. Some numbers can be written as the sum of prime numbers. For example, 4 can be written as 4 = a. Show all the numbers between 5 and 25 that can be written as the sum of two prime numbers. b. Show all the numbers between 5 and 25 that can be written as the sum of three prime numbers. c. Show all the numbers between 5 and 25 that can be written as the sum of four prime numbers. 17. A number can be expressed as a. Draw what this factor tree might look like. b. Explain how you can tell what the number is. 18. Use the strategy of your choice to write each number as a product of factors that are prime. a. 45 b. 123 c. 60 d Twin primes are two prime numbers that are two digits apart; 3 and 5 are twin primes. a. How many twin primes are there between 1 and 100? b. Are there any groups of three twin primes? If so, what would you call them? 98 CHAPTER 4: Factors and Multiples

19 Clearly Composite (continued) 10. Use what you know about fractions, decimals, and prime numbers to complete the following activity. a. Create a table like the one below. Write the fractions to in the table Fraction Decimal Repeating? Denominator Factored denominator No Yes b. Use a calculator to divide to write each fraction as a decimal. c. Record whether each decimal repeats. d. Write each denominator as a product of prime numbers. e. Make a conclusion about the types of fractions with decimals that repeat. 11. Lacey and MacKenzie share a room. To share the chore of cleaning their room, Lacey suggests that she will clean the room on prime numbered days if MacKenzie cleans the room on composite numbered days. a. On which days will MacKenzie clean the girls room? b. On which days will Lacey clean the girls room? c. Is Lacey s plan fair to each girl? d. Is there a day on which the room would not be cleaned? Write a problem that uses prime and composite numbers in the solution. Exchange problems with a classmate and solve each others problems. 13. Research prime numbers. Why might people be fascinated by them? I can solve problems involving multiples. CHAPTER 4: Factors and Multiples 99