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1,2,3,…,73,…,100 CCombine any 2 numbers by xx y = xy + x + y CContinue combining numbers until only 1 number is left.

For Example: 2 50 = (2) (50) = = (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 = x y + 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 COMMUTATIVE? DOES x y = y x ? xy + x + y yx + y + x YES

IS Associative? (x y) z =(xy + x + y) 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 = (x+1)(y+1) – 1 ADD 1 to each, multiply, and then subtract 1

= [(2)(3)-1] =[(2)(3)(4)-1] =[(2)(3)(4)(5)-1] 5... =[(2)(3)(4)(5)…(101)-1] =101! - 1

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

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