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Unit 1 Prime Time OBJECTIVES •Understand relationships among factors, multiples, divisors and products •Recognize and use properties of prime and composite numbers, even and odd numbers, and square numbers •Use rectangles to represent the factor pairs of numbers •Develop strategies for finding factors and multiples, least common multiples, and greatest common factors •Recognize and use the fact that every whole number can be written in exactly one way as a product of prime numbers •Use factors and multiples to solve problems and to explain some numerical facts of everyday life •Develop a variety of strategies for solving problems-building models, making lists and tables, drawing diagrams, and solving simpler problems Mrs. Carol Lange 1 Unit 1 Prime Time Big Ideas Factors and Products Whole Number Patterns and Relationships Common Multiples and Factors Factorizations: Searching for Factor Strings Essential Questions Will breaking a number into factors help me solve a problem? What relationships are revealed with factors? What do the factor and multiples of the numbers tell me about the situations? How can I find the factors of the numbers? How can I find the multiples? What common factors and common multiples do the numbers have? Mrs. Carol Lange 2 Textbook page 4 Unit 1 Prime Time UNIT PROJECT Choose a whole number that you especially like: •Your number must be between 10 and 100 •Record your number •Explain why you chose that number •List three or four mathematical fact about your number •List three or four connections you can make between your number and your world •Add to your notes as you progress through the unit •At the end of the unit you will be creating a project to highlight your number Mrs. Carol Lange 3 Textbook page 5, 70 Unit 1 Prime Time ESSENTIAL VOCABULARY factor common multiples divisor least common multiple multiple common factors prime number greatest common factor composite number factorizations factor pair prime factorizations square numbers exponent even number relatively prime odd number conjecture Venn diagram Mrs. Carol Lange 4 Textbook page 74-79 Unit 1 Prime Time INVESTIGATION ONE Factors and Products •Factors are whole number divisors of numbers: 1,2,5,10 are factors of 10 Every number has a limited number of factors! •Play The Factor Game to practice finding factors, and to see relationships among numbers http://www.PHSchool.com webcode amd-1101 •Proper factors are all the factor of a number except the number itself: 1,2,5 are the proper factors of 10 •Multiples are the products of a whole number and another whole number: the multiples of 5 are 5 (5x1), 10 (5x2), 15 (5x3), and so on. Every number has an unlimited number of multiples. •Play The Product Game to practice finding multiples, and to see relationships among numbers http://www.PHSchool.com webcode amd-1103 Mrs. Carol Lange 5 Textbook page 6-21 Unit 1 Prime Time INVESTIGATION ONE Factors and Products •Prime Numbers are numbers that have exactly two factors: 3,5,7,11,13,17,19…are prime because the only divisors they have are 1 and the number itself http://www.PHSchool.com webcode ame-9031 •The number one is neither prime, nor composite •Composite Numbers are numbers that have more than two factors: 2,4,6,8,9,10,12,14,15…are composite because they have more than two factors (divisors) Mrs. Carol Lange 6 Textbook page 6-21 Unit 1 Prime Time Always support your INVESTIGATION ONE answers with examples! Sample ACE questions and answers: APPLICATIONS: 5) What factor is paired with 6 to give it 54? 9, because 9 x 6 = 54 16) Why is the set of factors of a number not the same as the set of proper factors of that number? Factors are different than proper factors because they do not include the number itself. Factors of 9 are 1,3,and 9; proper factors of 9 are 1,3 Multiple-choice skills practice: http://www.phschool.com webcode ama-1154 Mrs. Carol Lange 7 Textbook page 14-15 Unit 1 Prime Time Always support your INVESTIGATION ONE answers with examples! Sample ACE questions and answers: CONNECTIONS: 33A) In developing the ways in which we calculate time, astronomers divided an hour into 60 minutes. Why is 60 a good choice (better than 59 or 61)? 60 is a good choice for calculating time, because it has many factors (1,60; 2,30; 3,20; 4,15; 5,30; 6,10). 59 and 61 do not have as many factors. 33B) If you could select another number to represent the number of minutes in an hour, what would be a good choice? Why? I’d pick a number like 100, because it has a lot of factors (1,100; 2, 50; 4,25; 5,20; 10,10) Mrs. Carol Lange 8 Textbook page 18 Unit 1 Prime Time Always support your INVESTIGATION ONE answers with examples! Sample ACE questions and answers: EXTENSIONS: 43) What is the best move (in the Factor Game) on a 49-board? Why? 47 is a good first move, because it is the greatest prime number. You will get 47 points, and your opponent will get only 1. Mrs. Carol Lange 9 Textbook page 19 Unit 1 Prime Time INVESTIGATION TWO Whole-Number Patterns and Relationships •Factor Pairs are two whole numbers that are multiplied to get a product factor pairs of 10 are 1,10 and 2,5 Remember: every number has a limited number of factors! •Square Numbers are numbers that result from multiplying the same number times itself. (4 x 4 = 16, 2 x 2 = 4) •Odd Numbers are numbers that do not have two as a factor. •Even Numbers are numbers that have two as a factor. •A conjecture is a best guess or prediction based on an observed pattern •A Venn diagram uses circles to group things that belong together Mrs. Carol Lange 10 Textbook page 22-36 Unit 1 Prime Time Always support your INVESTIGATION TWO answers with Sample ACE questions and answers: examples! APPLICATIONS: 8) What type of number has an odd number of factors? Give examples Square numbers have an odd number of factors, because one factor pair is always the same. (16: 1,2, 4, 8, 16) 15) How can you determine whether a sum of several numbers, such as 13 + 45 + 24 + 17 is even or odd? If all the numbers in the ones place are even, or if there is an even number of odd numbers, than the sum is even. For example: The sum of 3 + 5 + 4 + 7 will be odd, because there are three odd addends. Problem 15 help: http://www.phschool.com webcode ama-1215 Mrs. Carol Lange 11 Textbook page 31 Unit 1 Prime Time Always support your INVESTIGATION TWO answers with examples! Sample ACE questions and answers: CONNECTIONS: 28) Allie’s eccentric aunt, May Belle, hides $10,000 in $20 bills under her mattress. If she spends one $20 bill every day, how many days will it take for her to run out of bills? It will take 500 days. I made a chart for 5 days, and saw that it would take 5 days to spend $100. 100 goes into 10,000, 100 times, so 100 times 5 days equals 500 days. Be sure to explain any strategy you use! Mrs. Carol Lange 12 Textbook page 33 Unit 1 Prime Time Always support your INVESTIGATION TWO answers with examples! Sample ACE questions and answers: EXTENSIONS: 37) For any three consecutive numbers, what can you say about odd numbers and even numbers? Why? For any three consecutive numbers, either two of the numbers are odd (4,5,6) or two of the numbers are even (5,6,7). It depends on what the first number is. Mrs. Carol Lange 13 Textbook page 34 Unit 1 Prime Time ORDER OF OPERATIONS Order of Operations is a rule developed by mathematicians that determines in what order you solve an equation. 1) Solve all operations in parentheses first. 2) Solve all exponents 3) Multiply/divide left to right 4) Add/subtract left to right Do ALL multiplication before you add or subtract! Solve these: 7+6–3x2= 7 (multiply 3 x 2 first, then add 7 + 6 , then subtract 6 from 13) 32 – 2 x 3 + 6= 9 (Solve 32 first to get 9, then multiply 2 x 3 to get 6; 9 – 6 + 6 = 9) Mrs. Carol Lange 14 Textbook page 26 Unit 1 Prime Time INVESTIGATION THREE Common Multiples and Common Factors •Common Multiples are multiples that appear on the list of multiples for two (or more) different numbers. Some multiples of 5 are 5, 10, 15, 20, and some multiples of 10 are 10, 20, 30. Common multiples for 5 and 10 are 10, and 20 because they are on both lists. Numbers have an infinite number of common multiples. The smallest number on both lists is the Least Common Multiple. •Common Factors are factors that appear on the list of factors for two (or more) different numbers. Factors of 18 are 1,18; 2,9; 3,6 and the factors of 12 are 1,12; 2, 6; 3,4 so common factors are 1, 2, 3, and 6. the Greatest Common Factor is 6. Mrs. Carol Lange 15 Textbook page 37-48 Unit 1 Prime Time Always support your INVESTIGATION THREE answers with Sample ACE questions and answers: examples! APPLICATIONS: 6) List the common multiples from 1 to 100 for each pair of numbers. Then find the least common multiple for each pair. 20 and 25 20: 20, 40, 60, 80, 100 25: 25, 50, 75, 100 The LCM is 100. It is the smallest multiple on that appears on both lists. Mrs. Carol Lange 16 Textbook page 42 Unit 1 Prime Time Always support your INVESTIGATION THREE answers with Sample ACE questions and answers: examples! CONNECTIONS: 36) 3 x 5 x 7 = 105 Use this fact to find each product. a) 9 x 5 x 7 = 105 x 3 ( 315) because 9 is 3 x 3. You need to include one more factor of 3. b) 3 x 5 x 14 = 105 x 2 (210) because 14 is 7 x 2. You need to include one more factor of 2. c) 3 x 50 x 7 = 105 x 10 (1050) because 50 is 10 x 5. You need to include one more factor of 10. d) 3 x 25 x 7 = 105 x 5 (525) because 25 is 5 x 5. You need to include one more factor of 5. Mrs. Carol Lange 17 Textbook page 45 Unit 1 Prime Time Always support your INVESTIGATION THREE answers with Sample ACE questions and answers: examples! EXTENSIONS: 38) Ms. Santiago has many pens in her desk drawer. She says that if you divide the total number of pens by 2, 3, 4, 5, or 6, you get a remainder of 1. What is the smallest number of pens that could be in Ms. Santiago’s desk? The smallest number of pens in Ms. Santiago’s desk is 61. You need to find the least common multiple of 2, 3, 4, 5 and 6, and add 1 (the remainder) to it. Mrs. Carol Lange 18 Textbook page 45 Unit 1 Prime Time INVESTIGATION FOUR Factorizations: Searching for Factor Strings •Factorizations are strings of factors of a number. All numbers can be written as strings of factors. For example: 50 can be written as 2 x 25, or 2 x 5 x 5. •All numbers are products of prime numbers. The strong of factors that is made up of all prime numbers is called the Prime Factorization of a number. Prime Factorizations can be found by dividing each factor by a prime number, or by using a factor tree to identify factor pairs. •Exponents tell you how many times a number is used as a factor. For example, 2 x 2 x 2 is written in exponent form as 23. •Number pairs whose greatest common factor is one, are called relatively prime. Relatively prime numbers can be multiplied together to find their least common multiple. Mrs. Carol Lange 19 Textbook page 49-60 Unit 1 Prime Time Always support your INVESTIGATION FOUR answers with Sample ACE questions and answers: examples! APPLICATIONS: 13) To indicate multiplication, you can use a raised dot symbol. For example, 3 x 5 = 3 . 5. Find the prime factorization of 312 using raised dot symbols. 312 6 52 3 2 2 26 2 13 3 . 2 . 2 . 2 . 13 = 312 or 3 . 23 . 13 = 312 Mrs. Carol Lange 20 Textbook page 56 Unit 1 Prime Time Always support your INVESTIGATION FOUR answers with Sample ACE questions and answers: examples! CONNECTIONS: 32) What is my number? a) My number is a multiple of 2 and 7. Start by listing multiples of 2 and 7: 14, 28, 42, 56, 70, 84, 98 b) My number is less than 100 but greater than 50. Using the list from question a, eliminate multiples less than 50: 56, 70, 84, 98 c) My number is the product of 3 different prime numbers. Find the prime factorization of each of your multiples. 56 (2 x 2 x 2 x 7); 70 (2 x 5 x 7); 84 (2 x 2 x 3 x 7); 98 (2 x 7 x 7). Only 70 is a product of three different prime numbers. Mrs. Carol Lange 21 Textbook page 58 Unit 1 Prime Time INVESTIGATION FIVE THE LOCKER PROBLEM Students in a school create a problem by opening 1000 lockers. •student 1 runs down the row of lockers and opens every door. •student 2 closes the doors of all the even number lockers. •student 3 changes the state (opens or closes) the doors of the lockers of multiples of 3. •Student 4 changes the state of the doors of lockers that are multiples of 4. •Student 5 changes the state of every 5th door. When all the students have finished, which locker doors are opened? Discuss strategies you can use to solve the problem. Make a conjecture about the answers! NEED HELP? http://www.PHSchool.com webcode amd-1501 Mrs. Carol Lange 22 Textbook page 61-64 Unit 1 Prime Time Always be able to END OF UNIT REVIEW provide an explanation for your For your Unit Test, know how to: work! •Find factors and factor pairs of a number •Find common factors and the greatest common factor of two numbers •Find multiples of a number and least common multiples of two or more numbers •Use exponents and Order of Operations •Find a factor string and prime factorization of a number Use your study guide from the back of the parent letter! Mrs. Carol Lange 23 Textbook page 6-60 Unit 1 Prime Time SPECIAL NUMBER PROJECT For your special number project you need to choose one of the following projects: •Collage •Comic strip •Narrative story In your project, you need to include as much information about your special number as possible. You must include the following vocabulary words: factor multiple prime composite factor pair common factor common multiple even number odd number prime factorization square number Venn diagram Mrs. Carol Lange 24