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.