 The Teachers' Net is a consortium of teacher circles
throughout the United States. The aim of a Teacher's Circle
is to equip middle school teachers with an effective
problem-solving approach to teaching mathematics. This
style of learning for teachers is based on the Math Circles
environment that originated 150 years ago in Eastern
Europe and has proven to be successful for students around
the world.
 The thrust of the Teacher's Circle program is to present
mathematics as an intellectual endeavor. More than
anything else, mathematics is problem solving. By
presenting interesting and challenging problems as an
incentive, we can endow the task of teaching mathematical
methods and techniques with both a reason and a goal.
Teacher's Circles provide an inspirational experience and
practical resources for middle school mathematics
teachers. Participants in these circles meet regularly with
mathematicians and mathematical scientists for
presentations, practice, and discussions of how to
incorporate problem solving into their classrooms.
An Example
Notice the following:
We reinforce basic computation skills
We ask intriguing and mysterious questions
We connect different concepts
Elegant Solution and adventurous mathematics!
DUE TO TIME CONSTRAINTS WE MUST SHORT
CIRCUIT THE DISCOVERY PROCESS!!!
Consider the numbers
from 1 to 100:
1,2,3,…,73,…,100
 Combine any 2 numbers by
x
y = xy + x + y
 Continue combining numbers until
only 1 number is left.
For Example:
2
50
 = 2 + 50 + (2) (50)
 = 152
 99
1
 = 99 + 1 + (99) (1)
 = 199
 NOW OUR NUMBER LIST BECOMES:
 152, 3, 4, … , 49,51,52,…,98,199, 100
Consider the numbers
from 1 to 100:
1,2,3,…,73,…,100
 Combine any 2 numbers by
x
y = xy+x+y
 Continue combining numbers until
only 1 number is left.
???? QUESTIONS ????
 WHAT FINAL ANSWERS CAN YOU
GET?
 HOW MANY DIFFERENT ANSWERS
CAN YOU GET FOR THE FINAL
NUMBER?
Strategies to SucCeed
PATIENCE
GET DIRTY
i.e. Prepare to do lots of figuring!!!
TRY SOMETHING SIMPLER
TAKE 5 MINUTES & THINK
 4 Minutes To GO!
 3 Minutes To GO!
2 Minutes To GO!
1
Minute To GO!
TIME IS UP!!!!!!!!!!!!!
TRY SOMETHING SIMPLER
 What happens if we change
to ordinary addition?
 In other words, we take two numbers in the list and we
just replace them by their sum and we keep going until
only 1 number remains
 How many answer(s) do we get now?
 What answer(s) do we get?
 WHY?
+ is ASSOCIATIVE & COMMUTATIVE
 WITH JUST + WE GET ONLY ONE
ANSWER NO MATTER HOW WE
COMBINE THE NUMBERS!
 It’s the sum of all the numbers:
 = 5050
IS
 DOES
 xy + x + y
 yx + y + x
 YES
COMMUTATIVE?
x
y
=
y
x?
IS
 (x
Associative?
y)
 =(xy + x + y)
z
z
 =(xy + x + y)z + (xy + x + y) + z
 =xyz + xz + yz + xy + x + y + z
 =xyz + xy + xz + yz + x + y + z
Does
xyz + xy + xz + yz + x + y + z
Have Symmetry?
 (x
y)
z = xyz + xy + xz + yz + x + y + z FROM BEFORE
 x
(y
z)
 =x
(yz+ y + z)
 = x(yz + y + z) + x + (yz + y + z)
 =xyz + xy + xz + yz + x + y + z
 THEY ARE EQUAL!!!!!
SO THERE IS JUST 1 ANSWER
WHAT IS IT?
x
y = xy + x + y
 = xy + x + y + 1
-
1
 = (x+1)(y+1) – 1
 ADD 1 to each, multiply, and then subtract 1
1
 =
2
3
[(2)(3)-1]
 =[(2)(3)(4)-1]
4
3
4
4
5. . .
 =[(2)(3)(4)(5)-1]
 =[(2)(3)(4)(5)…(101)-1]
 =101! - 1
5 . . . .
5. . .
5 . . .
Some of Teachers Circles
Materials Now Available
 Fractions, Ratios , and Percents
 Sudoko and Magic Squares
 Card Tricks
 Cryptography – the math of secret Messages
 Zome and 3 dimensional geometry
 Mathematical Games (Criss Cross and Others)
 Combinations and Permutations
 Exploring Pascals Triangle
 Graph Theory and Matrices
 Tilings
www.theteacherscirlce.org
Why Teachers Math Circles??
 Enrich the mathematical lives of teachers
 Reinforce computational skills but compute for a
reason
 We ask what if in English class – why not in math
class? (Teachers don’t have to know all the answers!!)
 Connect diverse content areas; cover and review
standards efficiently and in context
 Powerful Collaborations – Teachers, Students,
Mathematicians, Administrators, Parents, and the
Community.
Where are Teacher’s Circles??
Established Circles
Starting Circles