1. MATH ART POSTERS AND CLOCK ARITHMETIC Laurie Burton Western Oregon University MAA PREP Active Learning Workshop July 7, 2003 “Algebra Theme” Day

2. Art Poster “Goals” In the setting of Clock (Modular) Arithmetic converse about: FFactors & Primes FGCFS & LCMs “In Action” FNumber System Properties FExplore the Beginnings of Making Math Art Posters

3. VISUAL APPETIZERS

8. Clock Warm-Up Activity Clock by Joe Spooner

9. Connecting Clocks to Factors Question: On any n o’clock clock, n = 2, 3, 4, … If you start with a number, a & you generate all of the multiples of a to form the set generated by a = {a, 2a, … } What is the size of the set generated by a? Discuss!

10. GCFs & LCMs We know GCF (A, N) X LCM (A, N) = A X N thus LCM (A, N)/N = A/GCF (A, N) This is an excellent exercise to really get students to think about Greatest Common Factors & Least Common Multiples

11. Clock Arithmetic in the Abstract Algebra Course In the Abstract Algebra course: We explore groups in the Dihedral setting and then we use clock arithmetic to continue the study of number properties

12. Clock Arithmetic in the Abstract Algebra Course This allows us to study  The Identity Property  The Inverse Property  The Associative Property  The Commutative Property and  Closure In a setting other than the familiar real numbers

13. Understand the basics of clock arithmetic Create numerical codes on grids using clock arithmetic Replace numbers with patterns. Using these patterns: Color/draw on grids (directly or reflecting) Cut & paste onto colored backgrounds to make Math Art Posters! How Do You Make Math-art Posters?

14. Cayley Table: Addition on the 4 O'clock Clock

15. To Make a 4 o’clock (addition) Math Art Poster Replace the Numbers with Patterns as demonstrated by these "Colored Numbers"

17. To Make a Rotation Poster Make Duplicate Copies and Rotate 0º, 90º, 180º & 270º

19. To Make a Reflection Poster Draw Reflected Patterns: This is much more challenging and a great place to explore mirror images!

21. You Can Also Create A Multiplication Table For Clocks  Of course, for “clocks” which contain the additive identity, using the operation of multiplication does not form a group.  This is a great “non-group” example to work with in the class which leads the students to explore Z p *..  However, the patterns are still fun!

23. A few more Math Art Posters Which “clocks” are used What operation is used? Rotation or Reflection?

29. I made these posters using a Java Applet designed by SDL. Here are a few samples of hand drawn and scanned student work.

33. There are many more images linked to my web page. Also (described on the handout) a web page full of links & activities for clock art posters. You are welcome to use these!

34. Now: Participant Activity The beginnings of a Math Art Poster