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## Episodes

- S1 E1 - The Power of a Mathematical PictureOctober 20, 201634minProfessor 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 key to astounding feats of mental calculation.#Science & MathematicsFree trial of The Great Courses Signature Collection or buy
- S1 E2 - Visualizing Negative NumbersOctober 20, 201629minNegative 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.Free trial of The Great Courses Signature Collection or buy
- S1 E3 - Visualizing Ratio Word ProblemsOctober 20, 201629minWord 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.Free trial of The Great Courses Signature Collection or buy
- S1 E4 - Visualizing Extraordinary Ways to MultiplyOctober 20, 201630minConsider 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 times negative is always positive.Free trial of The Great Courses Signature Collection or buy
- S1 E5 - Visualizing Area FormulasOctober 20, 201630minNever 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.Free trial of The Great Courses Signature Collection or buy
- S1 E6 - The Power of Place ValueOctober 20, 201633minProbe 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 fractional bases.Free trial of The Great Courses Signature Collection or buy
- S1 E7 - Pushing Long Division to New HeightsOctober 20, 201629minPut 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.Free trial of The Great Courses Signature Collection or buy
- S1 E8 - Pushing Long Division to InfinityOctober 20, 201630min"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 sums with the visual approach.Free trial of The Great Courses Signature Collection or buy
- S1 E9 - Visualizing DecimalsOctober 20, 201632minExpand 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!Free trial of The Great Courses Signature Collection or buy
- S1 E10 - Pushing the Picture of FractionsOctober 20, 201630minDelve 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 of two, can't possibly be a fraction.Free trial of The Great Courses Signature Collection or buy
- S1 E11 - Visualizing Mathematical InfinitiesOctober 20, 201630minPonder 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 the counting numbers. In fact, find an infinite number of them!Free trial of The Great Courses Signature Collection or buy
- S1 E12 - Surprise! The Fractions Take Up No SpaceOctober 20, 201629minDrawing 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 the start of the weirdness.Free trial of The Great Courses Signature Collection or buy
- S1 E13 - Visualizing ProbabilityOctober 20, 201631minProbability 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 in the 17th century.Free trial of The Great Courses Signature Collection or buy
- S1 E14 - Visualizing Combinatorics: Art of CountingOctober 20, 201634minCombinatorics 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.Free trial of The Great Courses Signature Collection or buy
- S1 E15 - Visualizing Pascal's TriangleOctober 20, 201632minKeep 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.Free trial of The Great Courses Signature Collection or buy
- S1 E16 - Visualizing Random Movement, Orderly EffectOctober 20, 201631minDiscover 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.Free trial of The Great Courses Signature Collection or buy
- S1 E17 - Visualizing Orderly Movement, Random EffectOctober 20, 201631minStart 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 a strip of paper for its width to equal the Earth-Moon distance.Free trial of The Great Courses Signature Collection or buy
- S1 E18 - Visualizing the Fibonacci NumbersOctober 20, 201634minLearn 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!Free trial of The Great Courses Signature Collection or buy
- S1 E19 - The Visuals of GraphsOctober 20, 201630minInspired 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.Free trial of The Great Courses Signature Collection or buy
- S1 E20 - Symmetry: Revitalizing Quadratics GraphingOctober 20, 201631minThrow 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 in textbooks.Free trial of The Great Courses Signature Collection or buy
- S1 E21 - Symmetry: Revitalizing Quadratics AlgebraOctober 20, 201628minLearn 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 all along without realizing it.Free trial of The Great Courses Signature Collection or buy
- S1 E22 - Visualizing Balance Points in StatisticsOctober 20, 201630minVenture 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.Free trial of The Great Courses Signature Collection or buy
- S1 E23 - Visualizing Fixed PointsOctober 20, 201633minOne 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.Free trial of The Great Courses Signature Collection or buy
- S1 E24 - Bringing Visual Mathematics TogetherOctober 20, 201632minBy 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. But don't panic! This is a test that practically takes itself!Free trial of The Great Courses Signature Collection or buy

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