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Teacher’s Name: Mary Anne Bertola Subject: Mathematics Grade: 6 Date: October 9, 2008 Topic: Prime Factorization Essential Questions/Big Ideas: How are numbers related to one another? How can we determine if a number is prime or composite? Learning Objectives: VA SOL – Number and Number Sense 6.3 The student will a) find common multiples and factors, including least common multiple and greatest common factor; and b) identify and describe prime and composite numbers; and c) identify and describe the characteristics of even and odd integers. NCTM Standards – Number and Operations Standard for Grades 6-8 In grades 6-8 all students should – Use factors, multiples, prime factorization, and relatively prime numbers to solve problems. Students will be able to define factor, prime number, composite number, and prime factorization. (knowledge) Given a number, students will be able to generate a prime factorization for that number. (application) Students will be able to compare prime and composite numbers. (analysis) Students will understand the relationship between divisibility rules and prime factorization. (evaluation) Student and Teacher Activities with Estimated Time Blocks: Time Warm-up/Homework Check – 15 minutes Teacher (what the teacher will do) Display warm-up at front of class for students to see While students are completing warm-up questions, determine attendance by checking homework for completion Lead class discussion of warm-up answers Once warm-up is completed, recite answers to homework problems Lead class discussion of homework answers and Students (what the students will do) Settle down at beginning of class and get out homework from previous night Copy down tonight’s homework from front of class Answer warm-up questions Correct warm-up answers and ask questions if appropriate Correct homework answers and ask questions Lecture: Prime Factorization – 15 minutes Gizmo: Finding Factors with Area Models – 15 minutes (student centered instruction) Closure Activity – 5 minutes (formative assessment) answer student questions Supply definitions for factor, prime number, composite number, and prime factorization Provide examples for finding the factors of a number, determining if the number is prime or composite, and finding the prime factorization of the number Establish how divisibility rules are related to prime factorization Help clarify student understanding Display Finding Factors with Area Models Gizmo on projector Review with students how the prime factorization is on the left and the area model is on the right Select various numbers to show how prime factorization can be modeled through area of rectangles Select individual students to choose a number and change the rectangles in the area model section (interactive learning) Hand out assessment questions after the activity Collect assessment questions Display exit card questions on the board for students to answer Hand out homework worksheet if appropriate Copy notes regarding the definitions for factor, prime number, composite number, and prime factorization (including examples) Participate in discussion of how divisibility rules are related to prime factorization Ask questions if appropriate Participate in gizmo Help select various numbers to show how prime factorization can be modeled through area of rectangles Complete assessment questions worksheet with a partner after activity is completed Hand in assessment questions Answer exit card questions prior to leaving class Write down any questions regarding the lesson on the exit card to be discussed tomorrow Put away homework worksheet and complete for tomorrow’s lesson Materials Needed for Lesson: Gizmo: Finding Factors with Area Models Projector Computer with internet access Resources: (2008). Finding factors with area models. Retrieved October 11, 2008, from Explore Learning Website: http://www.explorelearning.com/index.cfm?method=cResource.dspView&ResourceID=2 18 Bailey, R. et. al. ( 2005). Mathematics applications and concepts: Course 1. New York: McGraw-Hill Glencoe. Ferrini-Mundy, J. et. al. (2000). Principles and standards for school mathematics. Reston, VA: The National Council of Teachers of Mathematics, Inc. Schroder, K. et. al. (2001). Mathematics standards of learning for Virginia public schools. Richmond, VA: Commonwealth of Virginia, Board of Education. Notes on Prime Factorization I. Warm-Up 1. Tell whether each number is divisible by 2, 3, 4, 5, 6, 9, or 10. Then classify each number as even or odd. Remember to explain your reasoning. a. 60 2: Yes; the ones digit, 0, is divisible by 2 3: Yes; the sum of the digits, 6, is divisible by 3 4: Yes; the number formed by the last two digits, 60, is divisible by 4 (trivial) 5: Yes; the ones digit is 0 6: Yes; the number is divisible by both 2 and 3 9: No; the sum of the digits, 6, is not divisible by 9 10: Yes; the ones digit is 0 The number is even. b. 17, 256 2: Yes; the ones digit, 6, is divisible by 2 3: Yes; the sum of the digits, 21, is divisible by 3 4: Yes; the number formed by the last two digits, 56, is divisible by 4 5: No; the ones digit is not 0 or 5 6: Yes; the number is divisible by both 2 and 3 9: No; the sum of the digits, 21, is not divisible by 9 10: No; the ones digit is not 0 The number is even. II. Definitions When two or more numbers are multiplied, each number is called a factor of the product. A whole number that has exactly two unique factors, 1 and the number itself, is a prime number. A number greater than 1 with more than two factors is a composite number. Every composite number can be expressed as a product of prime numbers. This is called a prime factorization of the number. A factor tree can be used to find the prime factorization of a number. We can find factors of a number by applying divisibility rules to that number. Is it possible to have numbers which are neither prime nor composite? Concept Summary: Number Prime Composite Neither prime nor composite Definition A whole number that has exactly two factors, 1 and the number itself. A number greater than 1 with more than two factors. 1 has only one factor. 0 has an infinite number of factors. Examples 11, 13, 23 6, 10, 18 0, 1 III. Examples 1 × 7 = 7 The factors of 7 are 1 and 7. 1 × 6 = 6 and 2 × 3 = 6 The factors of 6 are 1 and 6, and 2 and 3. Tell whether each number is prime, composite, or neither. o 28 The factors of 28 are 1 and 28, 2 and 14, and 4 and 7. Since 28 has more than two factors, it is a composite number. o 11 The factors of 11 are 1 and 11. Since there are exactly two factors, 1 and the number itself, 11 is a prime number. Find the prime factorization of 54. 54 Write the number that is being factored at the top. 54 2 × 27 Choose any pair of whole number factors of 54 3 × 18 2 × 3 × 9 Continue to factor any number that is not prime 3×2×9 2×3×3×3 3×2×3×3 Note: Except for the order, the prime factors are the same. The prime factorization of 54 is 2 × 3 × 3 × 3 or 2 × 33 . IV. Exit Card 1. The state of Kentucky has 120 counties. Write 120 as a product of primes. Answer: 120 = 2 × 2 × 2 × 3 × 5 or 23 × 3 × 5 2. Give an example of a number that is not composite. Answer: Based on student input. Example: any prime number 3. Write down any questions you have regarding today’s lesson. Name: _________________________ Assessment Questions: Finding Factors with Area Models Gizmo 1. Which of the following is not a prime number? a. 13 b. 15 c. 17 d. 19 2. The number 105 is: a. a prime number. b. a composite number. c. a prime factorization. d. None of the above. 3. Which of the following is not a prime factorization? a. 2 • 2 • 5 • 7 b. 22 • 5 • 7 c. 2 • 2 • 7 • 5 d. 2 • 2 • 5 • 8 4. What is the prime factorization of 36? Name: _________________________ Homework Worksheet: Prime Factorization 1. Tell whether each number is prime, composite, or neither. Explain your reasoning. a. 56 b. 114 c. 291 2. Find the prime factorization of each number. a. 102 b. 55 c. 126 3. Animal Cheetah Antelope Lion Coyote Hyena Speed (mph) 70 60 50 43 40 Animal Rabbit Giraffe Grizzly Bear Elephant Squirrel a. Which speed(s) are prime numbers? Speed (mph) 35 32 30 25 12 b. Which speed(s) have a prime factorization whose factors are all equal? c. Which speeds have a prime factorization of exactly three factors? 4. To find the volume of a box, you can multiply its height, length, and width. The measure of the volume of a box is 357. Find its possible dimensions. l h w