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The Power of a Mathematical Picture

Part of the Series: The Power of Mathematical Visualization

Part of the Series: 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

Part of the Series: The Power of Mathematical Visualization

Part of the Series: 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 the Fibonacci Numbers

Part of the Series: The Power of Mathematical Visualization

Part of the Series: 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…

Visualizing Probability

Part of the Series: The Power of Mathematical Visualization

Part of the Series: 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 Extraordinary Ways to Multiply

Part of the Series: The Power of Mathematical Visualization

Part of the Series: 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…

Surprise! The Fractions Take Up No Space

Part of the Series: The Power of Mathematical Visualization

Part of the Series: 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 Mathematical Infinities

Part of the Series: The Power of Mathematical Visualization

Part of the Series: 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…

Visualizing Decimals

Part of the Series: The Power of Mathematical Visualization

Part of the Series: 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…

Visualizing Pascal's Triangle

Part of the Series: The Power of Mathematical Visualization

Part of the Series: 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…

The Power of Place Value

Part of the Series: The Power of Mathematical Visualization

Part of the Series: 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 Area Formulas

Part of the Series: The Power of Mathematical Visualization

Part of the Series: The Power of Mathematical Visualization

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…

Visualizing Ratio Word Problems

Part of the Series: The Power of Mathematical Visualization

Part of the Series: 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,…

Pushing the Picture of Fractions

Part of the Series: The Power of Mathematical Visualization

Part of the Series: 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…

Visualizing Combinatorics: Art of Counting

Part of the Series: The Power of Mathematical Visualization

Part of the Series: 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 Balance Points in Statistics

Part of the Series: The Power of Mathematical Visualization

Part of the Series: 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 Long Division to Infinity

Part of the Series: The Power of Mathematical Visualization

Part of the Series: The Power of Mathematical Visualization

"If there is something in life you want, then just make it happen!" Following this advice, learn to solve polynomial division problems that have negative terms. Use your new strategy to explore infinite series and Mersenne primes. Then compute infinite…

Visualizing Orderly Movement, Random Effect

Part of the Series: The Power of Mathematical Visualization

Part of the Series: 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 Fixed Points

Part of the Series: The Power of Mathematical Visualization

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…

Symmetry: Revitalizing Quadratics Algebra

Part of the Series: The Power of Mathematical Visualization

Part of the Series: The Power of Mathematical Visualization

Learn why quadratic equations have "quad" in their name, even though they don't involve anything to the 4th power. Then try increasingly challenging examples, finding the solutions by sketching a square. Finally, derive the quadratic formula, which you've been using…