Geometry: An Interactive Journey to Mastery

Course No. 1033
Professor James S. Tanton, Ph.D., Princeton University
The Mathematical Association of America
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Course No. 1033
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Course Overview

Inscribed over the entrance of Plato’s Academy were the words, “Let no one ignorant of geometry enter my doors.” To ancient scholars, geometry was the gateway to gaining a profound knowledge of the world.$1#$ Today, geometry’s core skills of logic and reasoning are essential to success in school, work, and many other aspects of life.

Like other math fields, geometry teaches us how to think. It leads students to uncover new truths based on already established ideas and facts. It requires us to test and examine the conclusions of others. It teaches us to lay out our thinking clearly, describing each step so that others can follow along and verify our results.

This systematic way of thinking is essential in many fields. Drawing conclusions from experimental data is the basis of scientific discovery. Our justice system depends on compelling evidence to render a verdict in a court of law. And we use logical reasoning in everyday conversations to win friends over to our point of view.

In fact, the great Greek scholar Euclid demonstrated just how much you can do with logic. He worked out that basically all of geometry stands on just 10 core principles. You can build the rest using straightforward, logical reasoning.

In short, geometry is among the great intellectual feats of humankind. However, geometry goes far beyond being just an intellectual exercise. Its real-world applications extend to navigation, architecture, engineering, physics, technology, and even art.

  • Botanists use the geometry of triangles to estimate the heights of trees.
  • Astronomers use an understanding of ellipses to describe the orbits of planets.
  • Quantum physicists use the mathematics of rotation to explain aspects of subatomic physics.
  • Architects use principles of symmetry to develop aesthetically pleasing buildings.
  • Engineers use the properties of parabolas to design headlights and satellite dishes.

With its powerful blend of intellectual accomplishment and practical application, it’s no wonder that most schools consider geometry a core subject. Yet as award-winning Professor James Tanton of The Mathematical Association of America shows in Geometry: An Interactive Journey to Mastery, geometry can be an exciting adventure at any age. Those who will benefit from his 36 clear and accessible lectures include

  • high school students currently enrolled in a geometry class;
  • their parents, who seek an outstanding private tutor for their students;
  • home-schooled students and others wishing to study high school geometry on their own;
  • collegestudents who are struggling with math requirements and who need to strengthen their grasp of this fundamental subject; and
  • anyone curious about the intellectual challenge of logic and reasoning that underlies mathematics, the sciences, and our technological world.

Professor Tanton’s excellent teaching style makes the course ideal for those students who have ever believed they’re “not good at math” or have had challenges understanding geometry in the past.

A Different Way to Learn Geometry

Even students who have done well in other math courses such as algebra can sometimes find geometry a challenge. More so than algebra and other equation-based math, geometry places particularly strong focus on making logical inferences from facts and building a story of reasoning. Plus, geometry involves a more visual approach—working with shapes and patterns from the real world.

Many geometry courses begin by teaching the results of geometric thinking—by listing a set of beginning rules first. But how can one build the foundations of a house without first having a sense of what the house should be? Professor Tanton encourages students to start by playing with ideas of the mind (and acts of the hand!) to develop a feel for geometric rules and a context for those rules.

In Geometry: An Interactive Journey to Mastery, Professor Tanton guides students as they build an understanding of geometry from the ground up. With this approach, the instruction focuses on the intellectual play of the subject and its beauty as much as its utility and function. Students begin with elementary building blocks like points, lines, and angles and observe how those basic units interact.

From a clear understanding of the fundamental principles, students use logical reasoning to expand their understanding of geometry. Like building a house brick by brick, each new discovery stands upon the others—without any sudden or confusing jumps.

In the first part of the course, students

  • develop an intuitive context for thinking about terms like point, line, angle, plane, and flat;
  • grasp how to create logical proofs; and
  • uncover the three deep and fundamental assumptions of geometry—the Pythagorean theorem, the parallelism postulate, and the similarity principles.

In the second part, students

  • study common geometric shapes and their properties (such as triangles, polygons, and circles);
  • explore the intersection of geometry and algebra;
  • examine the basics of trigonometry; and
  • learn how to calculate areas.

Once students understand the core principles, they are set loose to play in the third part of the course. Students ponder a range of fascinating and sometimes counterintuitive applications for geometry. They

  • combine two seemingly disparate fields: geometry and probability;
  • dive into the wild world of fractals;
  • investigate conics and their many practical applications;
  • use complex numbers to solve tricky geometry problems; and
  • contemplate spherical and even “taxi-cab” geometry.

Delightful Real-World Examples
A beauty of geometry is its wide variety of fascinating and unexpected applications. Some of the examples students explore in this course include these:

  • Width of a river: You're on a walk and come across a river. Can you estimate how wide it is? See how you can—with no more than a bit of geometry and a baseball cap.
  • Geometry and nature: From the orbits of planets to the shape of your small intestine, geometric shapes appear in some surprising places throughout nature. See how geometry helps us better understand the marvels and mysteries of the world around us.
  • Modern cell phones: Swiping the screen on a cell phone seems to be an ordinary activity. But did you know your phone is actually relying on some clever geometry? Find out exactly what your phone is doing and the mathematics behind it.
  • Works of art: When people think of applications for mathematics, they often mention the fields of science or engineering. But geometry also has its place in the visual arts. See how great artists like M.C. Escher used geometric shapes and principles to create masterpieces.
  • A game of pool: If you're playing pool and want to play a trick shot against the side edge, how do you need to hit the ball? See how you can determine this and more using the reflection principle.

A Teacher of Teachers

Professor Tanton is committed to sharing the delight and beauty of geometry and works with teachers across North America to develop more effective teaching methods for geometry and other math courses.

He is not only a teacher of math, but a teacher of the best ways to teach math. His experience has taught him where students most frequently flounder, which has given him the skills to explain mathematical concepts in a way that removes mental roadblocks to success.

Making each example come to life, Geometry: An Interactive Journey to Mastery engages students in a visual adventure. Professor Tanton uses bright and colorful slides, easy-to-understand whiteboard drawings, and interactive demonstrations to make his explanations crystal clear. And to help students better understand geometric principles, a workbook complete with sample problems and solutions accompanies the course.

Equipped with a firm understanding of geometry, students walk away from the course with the tools and knowledge to continue on to greater challenges in mathematics, school, and life. Your journey into this world of joy and wonder has only begun.

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36 lectures
 |  Average 30 minutes each
  • 1
    Geometry—Ancient Ropes and Modern Phones
    Explore the origins of one of the oldest branches of mathematics. See how geometry not only deals with practical concerns such as mapping, navigation, architecture, and engineering, but also offers an intellectual journey in its own right—inviting big, deep questions. x
  • 2
    Beginnings—Jargon and Undefined Terms
    Lay the basic building blocks of geometry by examining what we mean by the terms point, line, angle, plane, straight, and flat. Then learn the postulates or axioms for how those building blocks interact. Finally, work through your first proof—the vertical angle theorem. x
  • 3
    Angles and Pencil-Turning Mysteries
    Using nothing more than an ordinary pencil, see how three angles in a triangle can add up to 180 degrees. Then compare how the experience of turning a pencil on a flat triangle differs from walking in a triangular shape on the surface of a sphere. With this exercise, Professor Tanton introduces you to the difference between flat and spherical geometry x
  • 4
    Understanding Polygons
    Shapes with straight lines (called polygons) are all around you, from the pattern on your bathroom floor to the structure of everyday objects. But although we may have an intuitive understanding of what these shapes are, how do we define them mathematically? What are their properties? Find out the answers to these questions and more. x
  • 5
    The Pythagorean Theorem
    We commonly define the Pythagorean theorem using the formula a2 + b2 = c2. But Pythagoras himself would have been confused by that. Explore how this famous theorem can be explained using common geometric shapes (no fancy algebra required), and how it’s a critical foundation for the rest of geometry. x
  • 6
    Distance, Midpoints, and Folding Ties
    Learn how watching a fly on his ceiling inspired the mathematician René Descartes to link geometry and algebra. Find out how this powerful connection allows us to use algebra to calculate distances, midpoints, and more. x
  • 7
    The Nature of Parallelism
    Examine how our usual definition of parallelism is impossible to check. Use the fundamental assumptions from the previous lectures to follow in Euclid’s footsteps and create an alternative way of checking if lines are parallel. See how, using this result, it’s possible to compute the circumference of the Earth just by using shadows! x
  • 8
    Proofs and Proof Writing
    The beauty of geometry is that each result logically builds on the others. Mathematicians demonstrate this chain of deduction using proofs. Learn this step-by-step process of logic and see how to construct your own proofs. x
  • 9
    Similarity and Congruence
    Define what it means for polygons to be “similar” or “congruent” by thinking about photocopies. Then use that to prove the third key assumption of geometry—the side-angle-side postulate—which lets you verify when triangles are similar. Thales of Ionia used this principle in 600 B.C.E. to impress the Egyptians by calculating the height of the pyramids. x
  • 10
    Practical Applications of Similarity
    Build on the side-angle-side postulate and derive other ways of testing whether triangles are similar or congruent. Also dive into several practical applications, including a trick botanists use for estimating the heights of trees and a way to measure the width of a river using only a baseball cap. x
  • 11
    Making Use of Linear Equations
    Delve deeper into the connections between algebra and geometry by looking at lines and their equations. Use the three basic assumptions from previous lectures to prove that parallel lines have the same slope and to calculate the shortest distance between a point and a line. x
  • 12
    Equidistance—A Focus on Distance
    You’ve learned how to find the midpoint between two points. But what if you have three points? Or four points? Explore the concept of equidistance and how it reveals even more about the properties of triangles and other shapes. x
  • 13
    A Return to Parallelism
    Continue your study of parallelism by exploring the properties of transversals (lines that intersect two other lines). Prove how corresponding angles are congruent, and see how this fact ties into a particular type of polygon: trapezoids. x
  • 14
    Exploring Special Quadrilaterals
    Classify all different types of four-sided polygons (called quadrilaterals) and learn the surprising characteristics about the diagonals and interior angles of rectangles, rhombuses, trapezoids, and more. Also see how real-life objects—like ironing boards—exhibit these geometric characteristics. x
  • 15
    The Classification of Triangles
    Continue the work of classification with triangles. Find out what mathematicians mean when they use words like scalene, isosceles, equilateral, acute, right, and obtuse. Then, learn how to use the Pythagorean theorem to determine the type of triangle (even if you don’t know the measurements of the angles). x
  • 16
    "Circle-ometry"—On Circular Motion
    How can you figure out the “height” of the sun in the sky without being able to measure it directly with a ruler? Follow the path of ancient Indian scholars to answer this question using “angle of elevation” and a branch of geometry called trigonometry. You learn the basic trig identities (sine, cosine, and tangent) and how physicists use them to describe circular motion. x
  • 17
    Trigonometry through Right Triangles
    The trig identities you explored in the last lecture go beyond circles. Learn how to define all of them just using triangles (expressed in the famous acronym SOHCAHTOA). Then, uncover how trigonometry is practically applied by architects and engineers to measure the heights of buildings. x
  • 18
    What Is the Sine of 1°?
    So far, you’ve seen how to calculate the sine, cosine, and tangents of basic angles (0°, 30°, 45°, 60°, and 90°). What about calculating them for other angles—without a calculator? You’ll use the Pythagorean theorem to come up with formulas for sums and differences of the trig identities, which then allow you to calculate them for other angles. x
  • 19
    The Geometry of a Circle
    Explore the world of circles! Learn the definition of a circle as well as what mathematicians mean when they say things like radius, chord, diameter, secant, tangent, and arc. See how these interact, and use that knowledge to prove the inscribed angle theorem and Thales’ theorem. x
  • 20
    The Equation of a Circle
    In your study of lines, you used the combination of geometry and algebra to determine all kinds of interesting properties and characteristics. Now, you’ll do the same for circles, including deriving the algebraic equation for a circle. x
  • 21
    Understanding Area
    What do we mean when we say “area”? Explore how its definition isn’t quite so straightforward. Then, work out the formula for the area of a triangle and see how to use that formula to derive the area of any other polygon. x
  • 22
    Explorations with Pi
    We say that pi is 3.14159 … but what is pi really? Why does it matter? And what does it have to do with the area of a circle? Explore the answer to these questions and more—including how to define pi for shapes other than circles (such as squares). x
  • 23
    Three-Dimensional Geometry—Solids
    So far, you’ve figured out all kinds of fun properties with two-dimensional shapes. But what if you go up to three dimensions? In this lecture, you classify common 3-D shapes such as cones and cylinders, and learn some surprising definitions. Finally, you study the properties (like volume) of these shapes. x
  • 24
    Introduction to Scale
    If you double the side-lengths of a shape, what happens to its area? If the shape is three-dimensional, what happens to its volume? In this lecture, you explore the concept of scale. You use this idea to re-derive one of our fundamental assumptions of geometry, the Pythagorean theorem, using the areas of any shape drawn on the edges of the right triangle—not just squares. x
  • 25
    Playing with Geometric Probability
    Unite geometry with the world of probability theory. See how connecting these seemingly unrelated fields offers new ways of solving questions of probability—including figuring out the likelihood of having a short wait for the bus at the bus stop. x
  • 26
    Exploring Geometric Constructions
    Let’s say you don’t have a marked ruler to measure lengths or a protractor to measure angles. Can you still draw the basic geometric shapes? Explore how the ancient Greeks were able to construct angles and basic geometric shapes using no more than a straight edge for marking lines and a compass for drawing circles. x
  • 27
    The Reflection Principle
    If you’re playing squash and hit the ball against the wall, at what angle will it bounce back? If you’re playing pool and want to play a trick shot against the side edge, how do you need to hit the ball? Play with these questions and more through an exploration of the reflection principle. x
  • 28
    Tilings, Platonic Solids, and Theorems
    You’ve seen geometric tiling patterns on your bathroom floor and in the works of great artists. But what would happen if you made repeating patterns in 3-D space? In this lecture, discover the five platonic solids! Also, become an artist and create your own beautiful patterns—even using more than one type of shape. x
  • 29
    Folding and Conics
    Use paper-folding to unveil sets of curves: parabolas, ellipses, and hyperbolas. Study their special properties and see how these curves have applications across physics, astronomy, and mechanical engineering. x
  • 30
    The Mathematics of Symmetry
    Human aesthetics seem to be drawn to symmetry. Explore this idea mathematically through the study of mappings, translations, dilations, and rotations—and see how symmetry is applied in modern-day examples such as cell phones. x
  • 31
    The Mathematics of Fractals
    Explore the beautiful and mysterious world of fractals. Learn what they are and how to create them. Examine famous examples such as Sierpinski’s Triangle and the Koch Snowflake. Then, uncover how fractals appear in nature—from the structure of sea sponges to the walls of our small intestines. x
  • 32
    Dido's Problem
    If you have a fixed-length string, what shape can you create with that string to give you the biggest area? Uncover the answer to this question using the legendary story of Dido and the founding of the city of Carthage. x
  • 33
    The Geometry of Braids—Curious Applications
    Wander through the crazy, counterintuitive world of rotations. Use a teacup and string to explore how the mathematics of geometry can describe an interesting result in quantum mechanics. x
  • 34
    The Geometry of Figurate Numbers
    Ponder another surprising appearance of geometry—the mathematics of numbers and number theory. Look into the properties of square and triangular numbers, and use geometry to do some fancy arithmetic without a calculator. x
  • 35
    Complex Numbers in Geometry
    In lecture 6, you saw how 17th-century mathematician René Descartes united geometry and algebra with the invention of the coordinate plane. Now go a step further and explore the power and surprises that come from using the complex number plane. Examine how using complex numbers can help solve several tricky geometry problems. x
  • 36
    Bending the Axioms—New Geometries
    Wrap up the course by looking at several fun and different ways of reimagining geometry. Explore the counterintuitive behaviors of shapes, angles, and lines in spherical geometry, hyperbolic geometry, finite geometry, and even taxi-cab geometry. See how the world of geometry is never a closed-book experience. x

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Your professor

James S. Tanton

About Your Professor

James S. Tanton, Ph.D., Princeton University
The Mathematical Association of America
Dr. James Tanton is the Mathematician in Residence at The Mathematical Association of America (MAA). He earned a Ph.D. in Mathematics from Princeton University. A former high school teacher at St. Mark's School in Southborough and a lifelong educator, he is the recipient of the Beckenbach Book Prize from the MAA, the George Howell Kidder Faculty Prize from St. Mark's School, and a Raytheon Math Hero Award for...
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Geometry: An Interactive Journey to Mastery is rated 4.3 out of 5 by 62.
Rated 5 out of 5 by from great review for an old mind I am retired teacher and military. I had geometry in high school 1965, and wanted a review, to keep my 73 year old mind alert. It is a great course, which I wish was available back in '65'. (sounds like pioneer days, saying that) It got the old cogs working. I thoroughly enjoyed it
Date published: 2020-08-04
Rated 5 out of 5 by from Geom The instructor is great he uses tools to help explain the topic wish I had this course when I went through high school geom he definitely gave me great ideas totally different when I learned it high school gym teacher taught me it
Date published: 2020-07-21
Rated 5 out of 5 by from Beautiful A frequent descriptive used by Professor Tanton when he has made a particularly satisfying observation as to some mathematical phenomena. And why not, as his obvious love of math in general and geometry in particular comes out in a delivery that generates a high level of energy and a vocabulary filled with superlatives. As has been pointed out by other reviewers TTC math courses come in two flavors: those heavy on proofs and that take a rigorous approach to the subject and those that often have the word “joy” in the title. Everything about Dr. Tanton and this course puts it firmly in the second category. Although I have taken a few TC math courses, I recently decided to work my way through TTC math catalog as a refresher—to reacquaint myself with what had been long forgotten or never learned. I really thought that while I would remember and revisit much that had been either not used or forgotten, that there would not be very much in high school math that I had not known at one time. In this course I was pleased that there were areas that not only had I not studied in high school, but that had not been covered in college either. For example the material in lecture 33: “The Geometry of Braids” was entirely new to me. Ditto for large parts of lecture 34: “Figurate Numbers”. I’m not really sure that this course alone would prepare anyone to comp geometry or pass an exam, but it likely would inspire a prospective student with enough enthusiasm to pay close attention in the classroom and with enough background that she could really understand and appreciate what was going on. Perhaps the course material would be helpful, although according to several other reviews it contains some errors, but as I was unable to download the guidebook onto my devices. I’m deducting a bar from “content” because of this. I’ve encountered this on a few “Great Courses Plus” offerings and have no idea why. Professor Tanton is the kind of teacher that most of us only rarely encountered. It is nice to know that they are still teaching. Some might find that he goes a bit too fast, but for me it was a part of his overall love for the subject and desire to share it all with us. I would add that he often gives asides in his lectures on the entomology of words and other things that spring to his mind. As an example of his teaching approach, his lecture on how to cover the greatest area with a fixed perimeter is simply delightful. He uses the legend of Queen Dido’s founding of Carthage as the springboard to the subject and just takes off from there. Thankfully he kept her tragic affair with Aeneas out of the classroom, perhaps to keep such sordid matters far from innocent high school students—or at least best left for those studying literature. On ‘Y Professor.
Date published: 2020-06-08
Date published: 2020-05-25
Rated 5 out of 5 by from Very easy to understand I bought this for my granddaughter who was having trouble with geometry. It really helped.
Date published: 2020-05-18
Rated 4 out of 5 by from A nice overview of geometry I teach HS geometry and bought this to brush up on my knowledge. Most of it is too high for what I do. The professor does a good job of explaining the lessons thought. It may go well if you like geometry or are in college.
Date published: 2020-01-05
Rated 5 out of 5 by from Love math review I have purchased four other math courses from Algebra 2 to calculus II. Enjoy refreshing my math and learning from great Professors.
Date published: 2019-12-13
Rated 5 out of 5 by from Antidote to Neuropenia My last encounter with Geometry was over 56 years ago. After a lifetime in Medicine and I have decided to wake up my comatose geometric memories in my brain and to make new ones . Why? Because aging is an carpet bombing by the 'penias': osteopenia that melts your bones, sarcopenia that shrinks your muscles, and neuropenia (I made this up!) that diminishes your mind, so I work out to keep my bones and muscles from fading away and I take courses like Dr. Tanton's 'Geometry' to keep my brain from getting shrink wrapped by the aging process. Although the workbook gets mixed reviews, it is well worth working in. Feeling 'the burn' is key to muscle growth in a weight workout. Take this curse and work out in the work book and your brain will feel 'the burn'. So why? To feel the brain burn which is the antidote to neuropenia. Lastly, for all you non-septuagenarians out there: Aging doesn't start when you are old, it starts when you are young. 'Geometry' is a course for all ages as well as the aged.
Date published: 2019-06-20
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