# James S. Tanton

### James S. Tanton >

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The Power of a Mathematical Picture

Episode 1 of The Power of Mathematical Visualization

Episode 1 of The Power of Mathematical Visualization

Professor Tanton reminisces about his childhood home, where the pattern on the ceiling tiles inspired his career in mathematics. He unlocks the mystery of those tiles, demonstrating the power of visual thinking. Then he shows how similar patterns hold the…

Visualizing Negative Numbers

Episode 2 of The Power of Mathematical Visualization

Episode 2 of The Power of Mathematical Visualization

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…

Visualizing Pascal's Triangle

Episode 15 of The Power of Mathematical Visualization

Episode 15 of The Power of Mathematical Visualization

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…

Visualizing Probability

Episode 13 of The Power of Mathematical Visualization

Episode 13 of The Power of Mathematical Visualization

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…

Visualizing the Fibonacci Numbers

Episode 18 of The Power of Mathematical Visualization

Episode 18 of The Power of Mathematical Visualization

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…

Pushing Long Division to New Heights

Episode 7 of The Power of Mathematical Visualization

Episode 7 of The Power of Mathematical Visualization

Put your dots-and-boxes machine to work solving long-division problems, making them easy while shedding light on the rationale behind the confusing long-division algorithm taught in school. Then watch how the machine quickly handles scary-looking division problems in polynomial algebra.

Visualizing Decimals

Episode 9 of The Power of Mathematical Visualization

Episode 9 of The Power of Mathematical Visualization

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 Power of Place Value

Episode 6 of The Power of Mathematical Visualization

Episode 6 of The Power of Mathematical Visualization

Probe the computational miracle of place value--where a digit's position in a number determines its value. Use this powerful idea to create a dots-and-boxes machine capable of performing any arithmetical operation in any base system--including decimal, binary, ternary, and even…

Visualizing Combinatorics: Art of Counting

Episode 14 of The Power of Mathematical Visualization

Episode 14 of The Power of Mathematical Visualization

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…

Visualizing Ratio Word Problems

Episode 3 of The Power of Mathematical Visualization

Episode 3 of The Power of Mathematical Visualization

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,…

The Visuals of Graphs

Episode 19 of The Power of Mathematical Visualization

Episode 19 of The Power of Mathematical Visualization

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…

Visualizing Extraordinary Ways to Multiply

Episode 4 of The Power of Mathematical Visualization

Episode 4 of The Power of Mathematical Visualization

Consider the oddity of the long-multiplication algorithm most of us learned in school. Discover a completely new way to multiply that is graphical--and just as strange! Then analyze how these two systems work. Finally, solve the mystery of why negative…

Symmetry: Revitalizing Quadratics Graphing

Episode 20 of The Power of Mathematical Visualization

Episode 20 of The Power of Mathematical Visualization

Throw away the quadratic formula you learned in algebra class. Instead, use the power of symmetry to graph quadratic functions with surprising ease. Try a succession of increasingly scary-looking quadratic problems. Then see something totally magical not to be found…

Visualizing Orderly Movement, Random Effect

Episode 17 of The Power of Mathematical Visualization

Episode 17 of The Power of Mathematical Visualization

Start with a simulation called Langton's ant, which follows simple rules that produce seemingly chaotic results. Then watch how repeated folds in a strip of paper lead to the famous dragon fractal. Also ask how many times you must fold…

Visualizing Balance Points in Statistics

Episode 22 of The Power of Mathematical Visualization

Episode 22 of The Power of Mathematical Visualization

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.

Pushing the Picture of Fractions

Episode 10 of The Power of Mathematical Visualization

Episode 10 of The Power of Mathematical Visualization

Delve into irrational numbers--those that can't be expressed as the ratio of two whole numbers (i.e., as fractions) and therefore don't repeat. But how can we be sure they don't repeat? Prove that a famous irrational number, the square root…

Surprise! The Fractions Take Up No Space

Episode 12 of The Power of Mathematical Visualization

Episode 12 of The Power of Mathematical Visualization

Drawing on the bizarre conclusions from the previous lecture, reach even more peculiar results by mapping all of the fractions (i.e., rational numbers) onto the number line, discovering that they take up no space at all! And this is just…

Visualizing Fixed Points

Episode 23 of The Power of Mathematical Visualization

Episode 23 of 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…

Visualizing Mathematical Infinities

Episode 11 of The Power of Mathematical Visualization

Episode 11 of The Power of Mathematical Visualization

Ponder a question posed by mathematician Georg Cantor: what makes two sets the same size? Start by matching the infinite counting numbers with other infinite sets, proving they're the same size. Then discover an infinite set that's infinitely larger than…

Bringing Visual Mathematics Together

Episode 24 of The Power of Mathematical Visualization

Episode 24 of The Power of Mathematical Visualization

By repeatedly folding a sheet of paper using a simple pattern, you bring together many of the ideas from previous lectures. Finish the course with a challenge question that reinterprets the folding exercise as a problem in sharing jelly beans.…