The Power of Mathematical Visualization

Professor Tanton speaks about the ceiling tile pattern that inspires his career in mathematics; he unlocks the mystery of those tiles, demonstrating the power of visual thinking; how similar patterns hold the key to feats of mental calculation.

Negative numbers are often confusing; discovering a visual model that makes it easy to keep track of what is negative and what is not; allowing one to tackle long strings of negatives and positives with parentheses galore.

Professor Tanton's tips on cutting through the confusing details about groups and objects, particularly when ratios and proportions are involved; learning through handy visual devices, which include blocks, paper strips and poker chips.

Considering the oddity of the long-multiplication algorithm; discovering a new way to multiply that is graphical; analyzing how these two systems work; solving the mystery of why negative times negative is always positive.

Seeing visual proofs for computing the areas of rectangles, parallelograms, triangles, polygons in general and circles; proving that for two polygons of the same area, one can dissect one polygon into pieces that can be rearranged to form the other.

Probing the computational miracle of place value, where a digit's position in a number determines its value; using this idea to create a dots-and-boxes machine capable of performing any arithmetical operation in any base system.

Using a dots-and-boxes machine to solve long-division problems; making them easy while shedding light on the rationale behind the long-division algorithm; how the machine quickly handles scary-looking division problems in polynomial algebra.

Learning to solve polynomial division problems that have negative terms; using the new strategy to explore infinite series and Mersenne primes; computing infinite sums with the visual approach.

Probing the connection between decimals and fractions; focusing on decimals that repeat; whether they can be expressed as fractions; whether there is a straightforward way to convert repeating decimals to fractions using the dots-and-boxes method.

Delving into irrational numbers, which cannot be expressed as the ratio of two whole numbers and therefore do not repeat; how people can be sure they do not repeat; proving that the square root of two cannot possibly be a fraction.

Pondering what makes two sets the same size; matching the infinite counting numbers with other infinite sets; discovering an infinite set that is infinitely larger than the counting numbers; finding an infinite number of them.

Drawing on the conclusions from one's look at infinite sets; reaching even more peculiar results by mapping all the fractions onto the number line; discovering that they take up no space at all.

Probability problems can be confusing as one tries to decide what to multiply and what to divide; visual models come to the rescue; solving riddles involving coins, dice, medical tests and a probability problem posed to mathematician Blaise Pascal.

Combinatorics deals with counting combinations of things; discovering that many such problems are really one problem of how many ways there are to arrange the letters in a word; using this strategy as well as the factorial operation.

Playing with the approach from combinatorics; applying it to algebra problems, counting paths in a grid and Pascal's triangle; exploring some of the patterns in Pascal's triangle; its connection to the powers of eleven and the binomial theorem.

Discovering that Pascal's triangle encodes the behavior of random walks; focusing on the inevitability of returning to the starting point; considering how random walks are linked to the "gambler's ruin" theorem.

Starting with a simulation called Langton's ant; how repeated folds in a strip of paper lead to the dragon fractal; asking how many times one must fold a strip of paper for its width to equal the Earth-Moon distance.

How a rabbit-breeding question in the 13th century leads to the Fibonacci numbers; investigating the properties of this sequence by focusing on the single picture that explains it all; the world premiere of Professor Tanton's Fibonacci theorem.

Probing the power of graphs; experimenting with scatter plots; seeing how plotting data is like graphing functions in algebra; using graphs to prove the fixed-point theorem; returning to the Fibonacci question.

Throwing away the quadratic formula; using the power of symmetry to graph quadratic functions with ease; trying a succession of increasingly scary-looking quadratic problems; seeing something that is not found in textbooks.

Learning why quadratic equations have "quad" in their name, even though they do not involve anything to the fourth power; trying increasingly challenging examples; finding the solutions by sketching a square; deriving the quadratic formula.

Venturing into statistics to see how Archimedes' law of the lever lets one calculate data averages on a scatter plot; discovering how to use the method of least squares to find the line of best fit on a graph.

One sheet of paper lying directly atop another has all its points aligned with the bottom sheet; whether a crumpled sheet's points still lie over the corresponding points on the bottom sheet; seeing visual proof of this fixed-point theorem.

Bringing together many mathematical principles by repeatedly folding a sheet of paper using a simple pattern; concluding with a challenge question that reinterprets the folding exercise as a problem in sharing jelly beans.