Why Prime Numbers?

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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

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Research Objectives

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

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Background Definition

• What is a Prime Number?
• A positive integer gt1
• A number that has exactly two divisors, 1 and
  itself
• A number that cannot be factored

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Background Applications

• What are some modern uses and applications of
  prime numbers?
• RSA Encryption/Cryptography
• Cicadas
• Factoring

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Background Educational Standards

• 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 4 5
1. Describe and identify the characteristics of even and odd numbers by examining their divisibility by 2 1. Determine the factors of a given number up to 50 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
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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 8
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
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Why is Prime Numbers such a difficult subject to
teach?

• The table to the right shows a list of prime
  numbers less than 100
• 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 20 2,3,4,7,11,13,17,19
Primes between 20 and 40 23,29,31, 37
Primes between 40 and 60 41,43,53, 59
Primes between 60 and 80 61,71,73 79
Primes between 80 and 100 83,89,97
.
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Lesson PlanSubject Algebra 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
notPurpose/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 classmateSummary/Clos
ure 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
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Using the Sieve of Eratosthenes
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Prime Factorization
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Encryption Practice

• 1. Choose a partner
• 2.Pick any prime number lt 20
• Pick a Simple Word to encrypt ( at least 3 but
  less than 7 words
• Using the corresponding Numbers to letters
  (a1,b2.) 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.

• Example
• The Student chooses the word MATH
• Now they choose the prime 7 to encode the word
• M13 A1 T20 H8
• The numbers their partner receive are
• 91 7 140 56

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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.

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Acknowledgements/Thank-Yous

• Mentors Dr. John Bowles and Dr. Caroline Eastman
• RCS Mentor Roxanne Spray
• REU Program and fellow participants
• LS-SCAMP

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Questions???