Why Prime Numbers An evaluation of prime numbers

Why Prime Numbers? An evaluation of prime numbers: Their use and teaching methods William S. M. Dunn South Carolina State University Mentor: Dr. Caroline Eastman

Research Objectives To understand analyze prime numbers and their applications n Find out the teaching standards and expectations for students to learn about prime numbers n Construct a feasible lesson plan in order to teach prime numbers in an appropriate learning environment n

Background: Definition n What is a Prime Number? v. A positive integer >1 v. A number that has exactly two divisors, 1 and itself v. A number that cannot be factored

Background: Applications n What are some modern uses and applications of prime numbers? v. RSA Encryption/Cryptography v. Cicadas v. Factoring

Background: Educational Standards n What are the Educational Standards and expectations for learning about Prime Numbers? Grades 3 -5: Number and Operations Standard: Understand numbers, ways of representing numbers, relationships among numbers, and number systems. Expectation G: Describe classes of numbers according to characteristics such as the nature of their factors. 3 1. Describe and identify the characteristics of even and odd numbers by examining their divisibility by 2 4 1. Determine the factors of a given number up to 50 5 1. Identify a number as prime, composite, or neither 2. Explain the characteristics of prime numbers and composite numbers 2. Determine common multiples of pairs of whole numbers each of which is less than or equal to 12 3. Determine the least common multiple of two whole numbers

Background: Educational Standards cont. Grades 6 -8: Number and Operations Standard: Understand numbers, ways of representing numbers, relationships among numbers, and number systems. Expectation F: Use factors, multiples, prime factorization, and relatively prime numbers to solve. 6 7 1. Solve problems using prime factorization, common multiples, and common factors and then explain the reasoning used 1. Apply primes, composite, factors, multiples, and relatively prime numbers in a variety of applied and mathematical situations and explain the reasoning used 8

Why is Prime Numbers such a difficult subject to teach? n The table to the right shows a list of prime numbers less than 100 n Looking at the first few primes, shown above, it is noticeable that prime numbers become less and less frequent. However, any fears that the prime numbers may eventually die out are unnecessary. There is in fact an infinity of primes. Despite this limitless supply of primes identifying primes is not as straight forwards as might be expected. Primes less than 2, 3, 4, 7, 11, 13, 1 20 7, 19 Primes between 23, 29, 31, 37 20 and 40 Primes between 41, 43, 59 40 and 60 Primes between 61, 73 79 60 and 80 Primes between 83, 89, 97 80 and 100

Subject: Algebra Lesson Plan Homework: Students will have to create their own sieve in order to find the first 40 prime numbers. They also will be given a list of numbers to not only factor but tell whether the number is prime or not Purpose/Objective of the Lesson: The purpose of the lesson is to give knowledge of prime numbers. The students will be able to recognize and find prime numbers. They will also be able to use prime numbers in problem solving situations such as factoring, and simple encryption. Class Activity Guided Practice: 1. Notes on Prime Numbers and uses 2. Examples of Using the Sieve of Eratosthenes 3. Factoring Examples 4. Learning about Encryption Independent Practice: 1. Worksheet on Factoring 2. Practice Using the Sieve 3. Encryption practice with a classmate Summary/Closure: With a review period to ensure understanding I will end the section with a test or quiz focusing on newly learned techniques for finding and using prime numbers

Using the Sieve of Eratosthenes

Prime Factorization

Encryption Practice n n n 1. Choose a partner 2. Pick any prime number < 20 Pick a Simple Word to encrypt ( at least 3 but less than 7 words Using the corresponding Numbers to letters (a=1, b=2…. ) multiply each letter by the prime number picked and show partner the numbers Your partner will have to factor the numbers to find the letters, prime number picked, and the mystery word. n Example: n The Student chooses the word MATH n Now they choose the prime 7 to encode the word n M=13 A=1 T=20 H=8 The numbers their partner receive are: 91 7 140 56 n

Conclusion With an open-ended research objective, I have come to the conclusion that prime numbers will remain and always be a difficult subject to teach for some of the following reasons: § There is an infinite number of primes, and everyday there is a new one discovered. ( the largest known to date is 4, 053, 946 digits long) § No real formula to find all primes § The subject area is somewhat advanced for the young minds that it is exposed to.

Acknowledgements/Thank-You’s n Mentors: Dr. John Bowles and Dr. Caroline Eastman n RCS Mentor: Roxanne Spray n REU Program and fellow participants n LS-SCAMP

Questions? ? ?