M A T H E M A T I C S

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1 M A T H E M A T I C S Grade 6 Mathematics Frameworks Unit 2 Fun and Games Student Edition

2 TABLE OF CONTENTS Unit 2 FUN AND GAMES Overview...3 Enduring Understandings...3 Essential Questions...4 Key Standards and Related Standards...4 Selected Terms and Symbols...6 Tasks Notes on Back to School! Notes on Arrays, Factors, and Number Theory Notes on You are the Teacher! Give em homework! Notes on Culminating Task: Three Number Theory Challenges JUNE 24, 2007 Page 2 of 17

3 Grade 6 Mathematics Fun and Games: Extending and Applying Number Theory OVERVIEW: In this unit, students gain a deeper understanding of concepts and applications of number theory. In Grade 5 Mathematics, students studied classification of counting numbers into subsets with distinguishing characteristics such as odd and even numbers and prime and composite numbers. Students also developed a strong foundation for understanding and applying multiples and factors. They will extend this concept to include greatest common factors and least common multiples which in turn will build a deep understanding of the Fundamental Theorem of Arithmetic. This unit is a building block to students deeper understanding of rational numbers which students will study extensively in unit 3 of the 6 th grade mathematics framework. Instruction should include the representation of these numbers and their relationships to other concepts, such as multiplication and division using diagrams, charts, tables, multiple number lines, and explanations. Number theory is a topic that begs for students to reason, discuss, make sense of and justify their thinking. This can be accomplished by students playing games that are based on number theory, working and debating with their peers and in sharing ideas through a teacher-facilitated whole class discussion. Students also should be provided with opportunities for revisions. In order to demonstrate mastery of the learning in this unit, students will explain the Fundamental Theorem of Arithmetic to a friend who has been absent for the unit and solve a puzzle involving factors, multiples and prime numbers. By the conclusion of this unit, students should be able to demonstrate the following competencies: Students should be able to find all the factors of a number by constructing or drawing arrays, thus proving the presence of 1 as a factor of all numbers. Students should be able to list all the factors of any given number and discuss how they know that the number is prime, composite or neither (the number 1 is neither prime nor composite). Students should be able to determine the greatest common factor of two or more numbers and offer situations in which it would be useful to know common and greatest common factors of two or more numbers. ENDURING UNDERSTANDINGS: Factors and multiples are related in ways that are similar to the way that multiplication and division are related. All natural numbers greater than one are either prime or can be written as a unique product of prime factors. The number 1 (one) is always a factor of any number. JUNE 24, 2007 Page 3 of 17

4 ESSENTIAL QUESTIONS: When or why would it be useful to know the factors of a number? When or why would it be useful to know the multiples of a number? What features does a number have if the number is prime? What role does the number 1 have when you are finding factors of any number? How can I use the array model to represent factors of a number? How applicable are multiples in everyday life? How applicable are factors in everyday life? How can I use models to represent multiples of a number? STANDARDS ADDRESSED IN THIS UNIT Mathematics standards are interwoven and should be addressed throughout the year in as many different units and activities as possible in order to emphasize the natural connections that exist among mathematical ideas. KEY STANDARDS: M6N1. Students will understand the meaning of the four arithmetic operations as related to positive rational numbers and will use these concepts to solve problems. a. Apply factors and multiples. b. Decompose numbers into their prime factorization (Fundamental Theorem of Arithmetic). c. Determine the greatest common factor (GCF) and the least common multiple (LCM) for a set of numbers. RELATED STANDARDS: M6P1. Students will solve problems (using appropriate technology). a. Build new mathematical knowledge through problem solving. b. Solve problems that arise in mathematics and in other contexts. c. Apply and adapt a variety of appropriate strategies to solve problems. d. Monitor and reflect on the process of mathematical problem solving. M6P2. Students will reason and evaluate mathematical arguments. c. Develop and evaluate mathematical arguments and proofs. d. Select and use various types of reasoning and methods of proof. M6P3. Students will communicate mathematically. a. Organize and consolidate their mathematical thinking through communication. JUNE 24, 2007 Page 4 of 17

5 b. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others. c. Use the language of mathematics to express mathematical ideas precisely. M6P4. Students will make connections among mathematical ideas and to other disciplines. a. Recognize and use connections among mathematical ideas. b. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole. c. Recognize and apply mathematics in contexts outside of mathematics. M6P5. Students will represent mathematics in multiple ways. a. Create and use representations to organize, record, and communicate mathematical ideas. b. Select, apply, and translate among mathematical representations to solve problems. c. Use representations to model and interpret physical, social, and mathematical phenomena. JUNE 24, 2007 Page 5 of 17

6 SELECTED TERMS AND SYMBOLS: The following terms and symbols are often misunderstood. These concepts are not an inclusive list and should not be taught in isolation. However, due to evidence of frequent difficulty and misunderstanding associated with these concepts, instructors should pay particular attention to them and how their students are able to explain and apply them. Arrays: rectangular arrangements that have equal numbers in the rows and columns. Decompose: The process of factoring terms and numbers in an expression. Exponent: The number of times a number or expression (called base) is used as a factor of repeated multiplication. Also called the power. Factor: When two or more integers are multiplied, each number is a factor of the product. "To factor" means to write the number or term as a product of its factors. Fundamental Theorem of Arithmetic: Every integer, N > 1, is either prime or can be uniquely written as a product of primes. GCF: Greatest Common Factor: The largest factor that two or more numbers have in common. Identity property of multiplication: A number that can be multiplied by any second number without changing the second number. The Identity for multiplication is 1. LCM: Least Common Multiple: The smallest multiple (other than zero) that two or more numbers have in common. Multiple: A number that is a product of a given whole number and another whole number. Prime factorization: The expression of a composite number as a product of prime numbers. Prime number: A positive number that is divisible only by itself and the number one. Square number: A number that is the product of a whole number and itself. This is also known as a Perfect Square. Composite number: A composite number is a number that has factors in addition to one and itself. You may visit or to see definitions and specific examples of many terms and symbols used in the seventh-grade GPS. JUNE 24, 2007 Page 6 of 17

7 TASKS: The collection of the following tasks represents the level of depth, rigor and complexity expected of all sixth grade students to demonstrate evidence of learning. These tasks or tasks of similar depth and rigor should be used to demonstrate evidence of learning. JUNE 24, 2007 Page 7 of 17

will give a free album download to each person

He will also give a backstage pass to each fourth

Which person will receive the free album download,

8 Back to School! Part 1: Music You and your friends have tickets to attend a music concert. While standing in line, the promotion states he will give a free album download to each person that is a multiple of 2. He will also give a backstage pass to each fourth person and floor seats to each fifth person. Which person will receive the free album download, backstage pass, and floor seats? Explain the process you used to determine your answer. Backstage P A S S MUSIC Gift Card VIP JUNE 24, 2007 Page 8 of 17

9 Part 2: School Supplies The Parents Teachers Association (PTA) at your school donated school supplies to help increase student creativity and student success in the classroom. Your teacher would like you to create kits that include one package of colored pencils, one glue stick, and one ruler. When you receive the supplies, you notice the colored pencils are packaged 12 boxes to a case, the rulers are packaged 30 to a box, and glue sticks are packaged 4 to a box. Colored Pencils 12/box Rulers 30/box Glue 4/box 1. What is the smallest number of each supply you will need in order to make the kits and not have supplies left over? Explain your thought process. 2. How many packaged rulers, colored pencils, and glue sticks will you need in order to make the kits? Explain the process you used to determine how many packages are needed for each supply. JUNE 24, 2007 Page 9 of 17

10 Extension School Lunch The Yearbook club at your school is sponsoring a fall festival to kick off the annual yearbook drive. The club sponsor has asked for your help in determining what food items to sell and how much of each item he needs to buy. Write a budget report supporting your decision. JUNE 24, 2007 Page 10 of 17

11 Arrays, Factors, and Number Theory Create, draw or shade all possible arrays for the numbers o Label all of the dimensions of the arrays, which are the factors of each number. o Look for patterns in the arrangements, factors, or drawings. o Describe the patterns or observations that help you see the factors, prime numbers, composite numbers and square numbers. o In the numbers 1-20, label the prime, composite, and square numbers. o Describe all the things you notice about the arrays and patterns, but especially discuss what you notice about the number 1. No. Arrays Facts Factors Patterns, observations JUNE 24, 2007 Page 11 of 17

12 Part II: 1. Using your previous work from Arrays, Factors and Number Theory, list all the factors in order from least to greatest, for each number No. Arrays Factors JUNE 24, 2007 Page 12 of 17

13 2. Choose any two numbers from your list of What factors are in both lists? 3. What is the largest factor that they have in common? 4. Try this on several other pairs of numbers from Can you do it for 3 of the numbers? Try it for 3 numbers. 6. When would it be useful to know the common factors or the greatest common factor of two or more numbers? 7. What advice would you offer to a friend who was having trouble finding all the factors of any number? JUNE 24, 2007 Page 13 of 17

14 You are the Teacher! Give em homework! Your teacher s favorite method for assigning homework problems is to assign the factors or multiples of some of the number of problems in your book. For example, he might say, On page 78, out of the 32 problems that are there, do the problems that are the factors of 24. On another day, he might say, On page 84, out of the 32 problems that are there, do the problems that are the multiples of 3. In which case would you do more problems? Explain how you figured it out. Suggest another use of factors, multiples, or primes to your teacher to use when assigning problems. Explain why you chose this method. JUNE 24, 2007 Page 14 of 17

15 Culminating Task Three Number Theory Challenges Part I: Hi Mike! Let me tell you about the Fundamental Theorem of Arithmetic! The fundamental theorem of arithmetic states that every natural number greater than one is either prime or can be written as a unique product of prime factors. What does this mean? Refer to the work you did in previous problems to help you explain the fundamental theorem of arithmetic to your friend, Mile, who has been absent. Be sure to include the following terms: factor, multiple, divisible, prime, composite, prime factorization and exponents.. JUNE 24, 2007 Page 15 of 17

16 Part II: Secret Number Juanita has a secret number. Read her clues and then answer the questions that follow: Juanita says, Clue 1 My secret number is a factor of Can you tell what Juanita s secret number is? Explain your reasoning. 2. Daren said that Juanita s number must also be a factor of 120. Do you agree or disagree with Daren? Explain your reasoning. 3. Malcolm says that Juanita s number must also be a factor of 15. Do you agree or disagree with Malcolm? Explain your reasoning. 4. What is the smallest Juanita s number could be? Explain. 5. What is the largest Juanita s number could be. Explain. 6. Suppose for Juanita s second clue she says, Clue 2: My number is prime. 7. Can the class guess her number and be certain? Explain your answer. 8. Suppose for Juanita s third clue she says, Clue 3: 15 is a multiple of my secret number. 9. Now can you tell what her number is? Explain your reasoning. 10. Your secret number is 36. Write a series of interesting clues using factors, multiples, and other number properties needed for somebody else to identify your number. This task is from Balanced Assessment for the Mathematics Curriculum, Middle Grades Assessment Package 2 Berkely, Harvard, Michigan State, Shell Centre, Dayle Seymour Publications, Copyright 2000, pages JUNE 24, 2007 Page 16 of 17

Part III: Slammin Lockers Georgia Performance Standards Framework Georgia Middle School has 100 students with lockers numbered 1 through 100.

17 Part III: Slammin Lockers Georgia Performance Standards Framework Georgia Middle School has 100 students with lockers numbered 1 through 100. One day, Sally walks down the hall and opens all the lockers. Eric goes behind her and closes all the lockers with an even number. Then, Jane changes the situation of the lockers with numbers that are multiples of 3. This means that a closed locker is opened and an open locker is closed. If this pattern continues FOR ALL 100 STUDENTS, which lockers will remain open after the 100 th student walks down the hall? Explain your thinking giving details, and using both appropriate mathematical models and language What if there were 500 students and 500 lockers? What if there were 1000 students and 1000 lockers? Can you find a rule for any number of students and lockers? Explain why your rule works. JUNE 24, 2007 Page 17 of 17

Exemplar 6 th Grade Math Unit: Prime Factorization, Greatest Common Factor, and Least Common Multiple

Exemplar 6 th Grade Math Unit: Prime Factorization, Greatest Common Factor, and Least Common Multiple Unit Plan Components Big Goal Standards Big Ideas Unpacked Standards Scaffolded Learning Resources