Prime Numbers

A Multiple Intelligence Lesson Plan designed by Rebecca A. Zeier

(source: http://primes.utm.edu/programs/music/listen/)

**Objectives:**

* Students will review the difference between prime and composite numbers.

* Students will be able to express a positive integer as a product of prime numbers (aka. The Fundamental Theorem of Arithmetic)

* Students will explore some patterns within the prime numbers and contrast them with the primes’ inherit irregularities.

* Students will engage in an activity to HEAR the irregularities of the prime numbers in and of themselves.

* Increase learning by increasing involvement by addressing two most powerful senses: sight and sound

Day 1:

__Getting Started:__

* overhead puzzle to get student’s mind started

__Review:__

* review difference between prime and composite numbers

__Introduction of Prime Factorization:__

* explain relevance – The Fundamental Theorem of Arithmetic

* explain the how to do it (most common is the tree diagram)

__Tree activity:__ (sight)

* students will create a prime factorization tree of different positive integers.This may be done individually or in small groups.

__Discussion:__

* ask students to present their trees and explain how they came to create their tree the way the did

* note the different ways to factor a number but anyway it is done, there is still a very unique prime factorization.

* Discussion on why the number 1 is not a prime number nor a composite number.See if the students can make any conjectures (hint:1 can be written to any power and still equal one … thus it fails the fundamental theorem of arithmetic that says every positive integer has ONE UNIQUE prime factorization)

__Homework:__

* see attached

Day 2:

__Exploration of Patterns:__

* There are quite a few patterns that the prime numbers
have when used together.For example:the sum of 5 consecutive prime numbers is a prime number.The sum of 7 consecutive prime numbers is a prime number.Does this pattern always hold true?Students
can test these patterns out and make conjectures on their own.Playing with numbers is always fun!More
patterns can be found at the following website: http://www.geocities.com/~harveyh/primes.htm

__Irregularities of Primes:__ (sound)

* List the first 20 prime numbers on the chalkboard.Ask the students if they can figure out (without already knowing what it is) what the next prime number is.Is there a pattern to how the prime numbers are listed?(hint:NO!!!)

* Sometimes it is hard to believe that there is no real pattern.There is an infinite amount of prime numbers however it takes years to figure out what the next one is.We will use a musical representation to HEAR the inherit irregularities of the prime numbers (this keys into the multiple intelligence theory).

* How do you do it?Simply by using modular arithmetic.List the first 20 prime numbers on the chalkboard.Pick a modular 1 to 9.Divide the prime numbers by that fixed modulo and list their remainders (note:it helps to figure out all the mods ahead of time to double check the work the students do).

* I, the teacher, listed several guitar chords on the board and asked students to assign a chord to the different remainders (example:if the remainder was 1 I might play a D chord … if the remainder was 2 I might play a C chord … and so on)

* After listing all the chords to the different remainders, I sit down and play the guitar chords that the students listed.Students will hopefully hear the irregular pattern of the guitar chords and they will hear that the chords don’t fit together just right.

* Some students are not as musically inclined as others.Some cannot tell the difference between 2 chords.We make up for this by using some instruments that have very apparent sounds to them … percussion instruments.

* We again pick a new modulo to convert the prime numbers to.We assign the percussion instruments to each remainder (percussion instruments I used were the drum, cymbals, maracas, whistle, harmonica, tambourine … we also added clapping hands, stomping feet, etc).I handed out the instruments to students and when I pointed to the remainder, the students were to play their instrument.

* Students get very involved and are having fun that they want to try several different modulo until time has run out!

__NCTM Standards:__

Numbers and Opperations Standard:

* Understand numbers, ways of representing numbers, relationships among numbers, and number systems.

* Use prime factorization to solve problems

Representation Standard

* Create and use representations to organize, record, and communicate mathematical ideas.

* Use representations to model and interpret physical mathematical phenomena

Communications Standard

* use of representations to help communicate thinking

Connections Standard

* understand how mathematical ideas interconnect and build on one another