Transcript math art posters and clock arithmetic

MATH ART POSTERS
AND
CLOCK ARITHMETIC
Laurie Burton
Western Oregon University
MAA PREP
Active Learning Workshop
July 7, 2003
“Algebra Theme” Day
Art Poster “Goals”
In the setting of Clock (Modular)
Arithmetic converse about:
Factors & Primes
GCFS & LCMs “In Action”
Number System Properties
Explore the Beginnings of
Making Math Art Posters
VISUAL
APPETIZERS
Clock Warm-Up Activity
Clock by Joe Spooner
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> = {a, 2a, … }
•What is the size of the set
generated by a? Discuss!
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
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
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
How Do You Make
Math-art Posters?
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!
Cayley Table: Addition on
the 4 O'clock Clock
+
1
2
3
4
1
2
3
4
1
2
2
4
1
2
3
4
1
2
3
4
1
2
3
4
To Make a
4 o’clock (addition)
Math Art Poster
Replace the Numbers
with Patterns
as demonstrated by
these "Colored Numbers"
To Make a Rotation Poster
Make Duplicate Copies
and
Rotate
0º, 90º, 180º & 270º
To Make a
Reflection Poster
Draw Reflected Patterns:
This is much more
challenging and a great
place to explore
mirror images!
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 Zp*..
However, the patterns are still fun!
A few more
Math Art Posters
Which “clocks” are used
What operation is used?
Rotation or Reflection?
I made these posters
using a Java Applet
designed by SDL.
Here are a few samples
of hand drawn and
scanned student work.
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!
Now:
Participant Activity
The beginnings of a
Math Art Poster