Tom Davis – MathCircles.org

Pick’s Theorem

By:

Aimee S. Johnson, Joshua M. Sabloff, Tom Davis

Topic(s):

Supporting Resources:

Facilitator Guides, Handouts, Virtual Tools

Austrian mathematician Georg Pick first stated this theorem in 1899. However it wasn’t brought to broad attention until 1969. In this exploration, participants will use rates of change to aid them in discovering Pick’s famous formula by finding a relationship between the area of the figure, the number of perimeter...

Visualization in Algebra

By:

Tom Davis

Topic(s):

Algebra / Arithmetic, Geometry, Problem Solving / General

Supporting Resources:

Facilitator Guides

Many topics in mathematics can be made much clearer when symmetric aspects are made clear or when nice alternative visualizations are possible. When this occurs, it helps both the student and the teacher. We will examine visualization and symmetry in a very general way by means of a set of...

Introduction to Diophantine Equations

By:

Tom Davis

Topic(s):

Algebra / Arithmetic, Number Theory

Supporting Resources:

Facilitator Guides

Diophantine equation – an equation whose roots are required to be integers. In this article we will only touch on a few tiny parts of the field of linear Diophantine equations. Some of the tools introduced, however, will be useful in many other parts of the subject. Suppose that dolls...

KenKen

By:

Tom Davis

Topic(s):

Mathematical Games, Problem Solving / General

Supporting Resources:

Facilitator Guides, References

Kenken is a puzzle whose solution requires a combination of logic and simple arithmetic skills. The puzzles range in difficulty from very simple to incredibly difficult. Students who get hooked on the puzzle will be forced to drill their simple addition, subtraction, multiplication and division facts.

Big Numbers

By:

Tom Davis

Topic(s):

Number Theory

Supporting Resources:

Facilitator Guides

Using the standard arithmetic operations (addition, subtraction, multiplication, division and exponentiation), what is the largest number you can make using three copies of the digit “9”? While this would be a pretty large number, with a little cleverness you can do far better. This session explores VERY large numbers!

Catalan Numbers

By:

Tom Davis

Topic(s):

Combinatorics

Supporting Resources:

Facilitator Guides

Suppose you have n pairs of parentheses and you would like to form valid groupings of them, how many groupings are there for each value of n? How many “mountain ranges” can you form with n upstrokes and n downstrokes that all stay above the original line? What about counting...

Coloring

By:

Tom Davis

Topic(s):

Geometry, Problem Solving / General

Supporting Resources:

Facilitator Guides

Consider a 9 × 9 chessboard, and you wish to cover as much of it as possible using figures shaped like the one to the right, where each of the four squares is the same size as the squares on a chessboard. The pieces can be rotated or flipped over....

Conway’s Rational Tangles

By:

Tom Davis

Topic(s):

Algebra / Arithmetic

Supporting Resources:

Facilitator Guides, Lesson Plan, Photos & Videos

What do adding positive and negative fractions have to do with tying knots? In this entertaining lesson, students will use ropes to explore and identify mathematical operations that untangle knots and lead to new thinking. Simple operations of twists and rotations circle back to practicing the addition of positive and...

Practical Probability: Casino Odds and Sucker Bets

By:

Tom Davis

Topic(s):

Combinatorics, Probability and Statistics

Supporting Resources:

Facilitator Guides

Gambling casinos are there to make money, so in almost every instance, the games you can bet on will, in the long run, make money for the casino. However, to make people gamble, it is to the casino’s advantage to make the bets appear to be “fair bets,” or even...

Activity Database