Transcript 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 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

Background: Definition

What is a Prime Number?
A positive integer >1
A number that has exactly two divisors, 1 and itself
A number that cannot be factored
Background: Applications

What are some modern uses and
applications of prime numbers?
RSA Encryption/Cryptography
Cicadas
Factoring
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
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?

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 2,3,4,7,11,13,17
20
,19
Primes between 23,29,31, 37
20 and 40
Primes between 41,43,53, 59
40 and 60
Primes between 61,71,73 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





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.

Example:

The Student chooses the word
MATH

Now they choose the prime 7
to encode the word

M=13 A=1 T=20 H=8
The numbers their partner
receive are:
91 7 140 56

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

Mentors: Dr. John Bowles and Dr. Caroline
Eastman

RCS Mentor: Roxanne Spray

REU Program and fellow participants

LS-SCAMP
Questions???