Mrs. Curran

Portfolio Projects Guidelines

To make a portfolio, an artist selects a variety of original work to represent the range of his or her skills.  Each of the portfolio projects that you will be assigned will give you a chance to create a finished product that you will be proud to add to your algebra portfolio. 

These projects will help you develop your ability to present and communicate your ideas.  They will also help you develop your problem-solving and reasoning abilities as you make connections between what you know and what is new.  Your individual insight and creativity will help shape the mathematics you discover.

Let these projects be springboards for further exploration.  Feel free to expand them to include new questions or areas of interest that arise.  Most of all, have fun!!

Requirements:

Each portfolio project must have:

1) A cover page which includes the title, chapter, author’s name (that’s you) and date finished.

2) Original portfolio assignment sheet

3) Completed portfolio assignment with several investigations (not just one or two)

Portfolio Project

Chapter 5

Pursuing Primes

For centuries, people have tried to find formulas that generate prime numbers.  Formulas have been found that work in some cases, but never in all.  In Exercises 2 and 3, you will investigate two of these formulas.  First, though, you should establish an efficient method of testing a number for primeness, which is the purpose of Exercise 1.  Exercise 4 involves a little research.  Exercise 5 is a game you can play that relies on knowing prime numbers and factors of numbers.

1.  Which of the following numbers are prime: 79, 157, 253, 679?  Describe an efficient method of testing whether a number is prime.

2.  A polynomial formula that has been tried as a prime-number generator is x² + x + 17.  show that the formula gives a prime number for x = 0, 1, 2, and 3.  Find a positive integer for which the formula does not produce a prime.

3.  In the seventeenth century, the French abbot, Marin Mersenne,

looked for primes that can be written in the form 2^p – 1, where p

is a prime number.  (In 1975, the postmark pictured at the right

commemorated the discovery of the Mersenne prime 2¹¹²¹³ – 1.)

Find a prime number p less than 20 for which 2^p – 1 is not prime. 

4.  In a book on mathematics or on the internet, find Goldbach’s Conjecture and then demonstrate that it is true for at least the first 10 even numbers greater than 4.

5.  In this solitaire game, your goal is to get the highest score you can by taking numbers according to the rules given below.  Play the game several times and then describe your strategy for choosing the numbers you take.

Begin with a list of the first 20 positive integers.  Take a number that has at least one of the remaining numbers as a factor.  Then eliminate from the list all factors of the number you took.  Continue to take numbers in this way for as long as you can.  Your score is the sum of all the numbers you take.