Part of the Series: The Power of Mathematical Visualization

One sheet of paper lying directly atop another has all its points aligned with the bottom sheet. But what if the top sheet is crumpled? Do any of its points still lie directly over the corresponding point on the bottom sheet? See a marvelous visual proof of this fixed-point theorem.

Running Time

34 mins

Year

2016

Kanopy ID

2681825

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Visualizing Random Movement, Orderly Effect

Discover that Pascal's triangle encodes the behavior of random walks, which are randomly taken steps characteristic of the particles in diffusing gases and other random phenomena. Focus on the inevitability of returning to the starting point. Also consider how random walks are linked to the "gambler's ruin" theorem.

The Visuals of Graphs

Inspired by a question about the Fibonacci numbers, probe the power of graphs. First, experiment with scatter plots. Then see how plotting data is like graphing functions in algebra. Use graphs to prove the fixed-point theorem and answer the Fibonacci question that opened the lecture.

Visualizing the Fibonacci Numbers

Learn how a rabbit-breeding question in the 13th century led to the celebrated Fibonacci numbers. Investigate the properties of this sequence by focusing on the single picture that explains it all. Then hear the world premiere of Professor Tanton's amazing Fibonacci theorem!

Visualizing Pascal's Triangle

Keep playing with the approach from the previous lecture, applying it to algebra problems, counting paths in a grid, and Pascal's triangle. Then explore some of the beautiful patterns in Pascal's triangle, including its connection to the powers of eleven and the binomial theorem.

Permanent Change - Plastics in Architecture and Engineering

Filmed in the spring of 2011 at the 4th Columbia conference on materials in architecture and engineering.
Plastics have become the most ubiquitous and increasingly permanent materials in construction. The material capabilities of plastics both as a generic material nomenclature and as specific polymers and the processes that underlie them…

Visualizing Ratio Word Problems

Word problems. Does that phrase strike fear into your heart? Relax with Professor Tanton's tips on cutting through the confusing details about groups and objects, particularly when ratios and proportions are involved. Your handy visual devices include blocks, paper strips, and poker chips.

Visualizing Area Formulas

Never memorize an area formula again after you see these simple visual proofs for computing areas of rectangles, parallelograms, triangles, polygons in general, and circles. Then prove that for two polygons of the same area, you can dissect one into pieces that can be rearranged to form the other.

Visualizing Decimals

Expand into the realm of decimals by probing the connection between decimals and fractions, focusing on decimals that repeat. Can they all be expressed as fractions? If so, is there a straightforward way to convert repeating decimals to fractions using the dots-and-boxes method? Of course there is!

Visualizing Probability

Probability problems can be confusing as you try to decide what to multiply and what to divide. But visual models come to the rescue, letting you solve a series of riddles involving coins, dice, medical tests, and the granddaddy of probability problems that was posed to French mathematician Blaise Pascal…

Visualizing Combinatorics: Art of Counting

Combinatorics deals with counting combinations of things. Discover that many such problems are really one problem: how many ways are there to arrange the letters in a word? Use this strategy and the factorial operation to make combinatorics questions a piece of cake.

Visualizing Balance Points in Statistics

Venture into statistics to see how Archimedes' law of the lever lets you calculate data averages on a scatter plot. Also discover how to use the method of least squares to find the line of best fit on a graph.

Visualizing Negative Numbers

Negative numbers are often confusing, especially negative parenthetical expressions in algebra problems. Discover a simple visual model that makes it easy to keep track of what's negative and what's not, allowing you to tackle long strings of negatives and positives--with parentheses galore.

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