Prime Numbers and Number Patterns

 

Wherever there is number, there is beauty. - Proclus (410-485 A.D.) Proclus's statement represents the spirit of anyone who has once experienced the thrill of numbers and their amazing patterns. Prime numbers are at the heart of numbers and that pattern: to predict the next prime, still eludes us all. In the reading this week, Schoenfeld describes mathematics as: searching for patterns. The NCTM Standard 2 sets the purpose of patterns, functions, and algebra in mathematics education at all grade levels. Mathematics instructional programs should include attention to patterns, functions, symbols, and models so that all students understand various types of patterns and functional relationships; use symbolic forms to represent and analyze mathematical situations and structures; use mathematical models and analyze change in both real and abstract contexts. This week focuses on patterning with numbers: how to build a pattern, how to recognize that pattern, and how to use symbols to represent and/or analyze the pattern.

Assignment Exercises Primes Finding a Pattern Arithmetic and Geometric Sequences

Readings Bello & Britton, Sections 4.1&4.8 Excerpt from Alan Schoenfeld's Article: LEARNING TO THINK MATHEMATICALLY: PROBLEM SOLVING, METACOGNITION, AND SENSE-MAKING IN MATHEMATICS

Web Resources These web resources have links to primes and different types of number patterns including Magic Squares.

Assignment

1) Post to the discussion forum either:

a) a number pattern puzzle that you have created using the Hundred Numbers Chart

or

b) a prime number puzzle ( you might want to check out the Prime Puzzles and Problems website for lots of suggestion that you might want to modify for your classroom!)
(http://www.primepuzzles.net/puzzles/index.html)

or

c) Visit the Number Pattern Sheet created by Mark Warner for Teaching Ideas for Primary Teachers site and create a series of number patterns appropriate to your classroom. Be sure to include grade level on your posting!
(http://teachingideas.co.uk/maths/nopattern/numberwsheet.htm)

Click here to go to the discussion forum (if you are not logged onto Blackboard, log in to bb.wpunj.edu and click on Communication and choose the Discussion Board.

2) Complete the following exercises.

 

Exercises

1) Using the Sieve of Eratosthenes: a) How many primes are there between 2 and 200 b) Why is special about the prime number 2? c) How many twin primes are there between 2 and 200 (twin primes are separated only by one number)? d) How many prime triplets are there (prime triplets are separated from each other by only one number)? e) Why can you stop crossing out numbers when you reach the square root of 200? (http://www.win.tue.nl/math/dw/ida/alge/c1s4p2al1ex.html)

2) p. 189, Ex. 38, 43, 57, 60

3) pp. 190-191, 68, 70, 71, 80a, 81a, 82a, 83, 84

4) In 1644, Mersenne developed a theory about prime numbers which are called Mersenne primes. What is that theory? The top 10 Mersenne primes are found at: The Largest Known Primes (http://www.utm.edu/research/primes/largest.html). Can you find the two smallest.?

5) 1) Find both an explicit formula and a recursion formula for the following sequences and use your formula(s) to find the 12th term: a) 3, 6, 9, 12, . . . b) 2, 8, 32, 128, . . . c) 1, 3, 6, 10, 15, 21, . . .

6) Find the next two terms of the sequence 14, 17, 50, 25, . . . What are some possible answers?

7) Divide 1 by 7. Let be the nth digit in the decimal expansion of 1/7. (Hint: You will have to use long division here, since most calculators stop too soon to see the pattern!) Find using the pattern you discovered.

8) Find the Concentric Rings and the Lightening Bolts patterns on the Hundred Chart.

9) The Crossing the River Problem

10) pp. 262-263, Ex. 2, 7, 22, 26, 37, 38, 41, 42,51, 52, 53, 54

11) This Sequencer applet writes out arithmetic and geometric sequences that you define and then graphs them. However, how does it work? Test the applet with both an arithmetic and a geometric sequence and then compare the two graphs. What is the difference between the graphs of an arithmetic and one of a geometric sequence? (http://www.shodor.org/interactivate/activities/sequencer/index.html)

Web Resources

Coloring Remainders in Pascal's Triangle - an applet which uses colors patterns to help visualize patterns in Pascal's triangle
http://www.shodor.org/interactivate/activities/pascal2/index.html

Eratosthenes Sieve - use the Sieve to find primes up to 200!
http://www.win.tue.nl/math/dw/ida/alge/c1s4p2al1ex.html

Fibonacci chart problem worksheet b- developed by Dr. Griff Elder, Department of Mathematics, University of Nebraska at Omaha
http://www.unomaha.edu/~wwwelder/ops/5.html

Fun With Mathematics - excellent sections on Fibonacci numbers, perfect numbers, and Mersenne primes
http://www.netcom.com/~hjsmith

Integers Containing Many Embedded Primes = Mike Keith's fun puzzle about primes
http://users.aol.com/s6sj7gt/primeval.htm

The largest known primes - the top ten primes, Mersenne primes, Sophie Germain primes, twin primes - all with links
http://www.utm.edu/research/primes/largest.html

Magic Squares, Magic Stars & Other Patterns - a large site on magic squares and related number patterns
http://www.geocities.com/CapeCanaveral/Launchpad/4057/magicsquare.htm

Math Forum's Introduction to Prime Numbers - a good starting introduction
http://www.forum.swarthmore.edu/dr.math/faq/faq.prime.num.html

Mutsumi Suzuki's excellent MAGIC SQUARES - a large site, mostly magic squares with some magic stars and lots of links
http://www.pse.che.tohoku.ac.jp/~msuzuki/MagicSquare.html

Number Patterns at the Annenberg/CPB Math and Science Project Site - Two simple number patterns with handouts: How Many Valentines? and Mystery Operations
http://www.learner.org/teacherslab/math/patterns/number.html

Number Pattern Problems - several good downloadable number pattern problems, Crossing the River came from this site!
http://smard.cqu.edu.au/Database/Junior/Algebra/Number_Patterns/index.html

Number Sequence Lesson plans for Grades 2 - 5
http://www.dpgraph.com/janine/mathpage/patterns.html

Patrick De Geest's Palindrome numbers - a large site on palindromes including palindromic primes and circular primes!
http://www.worldofnumbers.com/

Perfect Number Journey - a site on perfect numbers and presents the relationships of perfect numbers with Mersenne primes
http://home1.pacific.net.sg/~novelway/MEW2/lesson1.html

Prime Number Resources - The Electronic Frontier Foundation's page on primes - another excellent starting place
http://www.eff.org/coop-awards/prime-info.html

The Prime Pages - lots of primes, a history of primes and links to prime web sites
http://www.utm.edu/research/primes/

Prime Puzzles and Problems - Carlos B. Rivera's site is an advanced prime number theory site, but lists lots of references to prime puzzles and problems
http://www.primepuzzles.net/

Suzanne Alejandre: Magic Squares - presents magic squares as a way of teaching math
http://forum.swarthmore.edu/alejandre/magic.square.html

Venn Diagrams for GCD and LCM
http://students.ou.edu/Y/Elaine.Young-1/GCD_LCM.html

  

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