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 2p –
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 2p
– 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.