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 knowledge. Its core skills of logic and reasoning are essential to success in school, work, and many other aspects of life. Yet sometimes students, even if they have done well in other math courses, can find geometry a challenge. Now, in the 36 innovative lectures of Geometry: An Interactive Journey to Mastery, Professor James Tanton of The Mathematical Association of America shows students a different and more creative approach to geometry than that usually taught in high schools. Like building a house brick by brick, students learn to use logical reasoning to uncover fundamental principles of geometry, and then use them in fascinating applications."
Geometry: An Interactive Journey to Mastery
Experience an intuitive and fun approach to learning the subject of geometry in this course taught by an award-winning educator.
Rated 5 out of
5
by
mathenator from
Love This!
I'm a teacher and using all his ideas to present to my class. So glad I came across this!
Date published: 2023-11-21
Rated 5 out of
5
by
MarvinMartian from
Nice review of Geometry, great explanations!
I am viewing lesson 8, and remember who I loved Geometry 50 years ago in high school.
Two minor things bugged me. In the example of construction, I believe it was parallel lines, he used a protractor. I hope when he digs deeper he will be teaching compass & straight edge. In lesson 8 it bugged me that in discussing similarity he didn’t mention that one was impossible. It was the triangles with angles of 90, 40 & 50 the lengths were similarly scaled as he mentioned, but the hypotenuse was shorter than either of the legs of the right triangle.
Now back to finish up what is generally a wonderful course.
Date published: 2023-05-28
Rated 5 out of
5
by
hnbrgm from
Excellent teacher, learned so much
Tanton is an inspiring dynamic teacher. His progress through the material is flexible and makes geometry much easier to understand. Pity I can only find 2 courses by him.
Date published: 2023-03-19
Rated 5 out of
5
by
WatchingU from
Great reintroduction after many decades
Dr Tanton does indeed teach a great course worthy of the name. I know a lot of math, but have drifted far away from simple geometry over the years. So it was nice to see an expert introduction to the basic concepts, which reconnected some unused synapses to excellent effect. It is a pleasure to see how this mathematician approaches these problems, in contrast to a physical scientist or perhaps an engineer. Bravo!
Date published: 2022-10-27
Rated 5 out of
5
by
Jeff Towers from
Thoroughly enjoyed the course.
I already knew geometry and had no need to study it other than to dangling problems that stuck in my mind, like how to prove angles cannot generally be trisected with straight edge and compass. But it is foreseeable that my son would struggle in the future with it, and so I was interested in having him preview geometry.
This course was a dream come true for that purpose. My real objective was that he would love the material and become familiar with a few of the topics. I loved the course for many reasons, it has motivating puzzles, I've felt critical of curricula seeming for the reasons this prof also does, and it seems to connect all the dots with nice lines of thought. I think you can love geometry after a certain level of mastery, and I hope we'll get there in a review of the material.
So I do need to understand what makes this course tick. Reading the reviews, it seems we can divide the audience into two groups: those that want to study geometry, and those that have to take it as a means to an end. This latter group seems to have trouble. The course is really designed to move people to love the material, and use that as a basis for mastery. This can be difficult when there is an ingrained disbelief in this approach. I am firmly in the "love it" camp and I am interested in how we move the hardened captives into the captivated group.
I wonder if there is room for another dozen lectures extending the material into any missed nooks and crannies from Hilbert's Geometry and the Imagination. Also, I would like to see functions on the real line extended to their complex counterparts, and maybe something about soap bubbles tieing back to the earlier lecture.
Date published: 2022-08-28
Rated 5 out of
5
by
jackF from
Very Enjoyable Presentation
My expectations were minimal but the reality was amazing. I found it very useful and unexpected. The everyday connections to geometrical principles are extremely well done. I specially liked the demonstration of the complex number "i"
Date published: 2022-08-17
Rated 5 out of
5
by
chrisp228 from
Brilliant all around!
This is how you teach. The professor brings together and ties together ideas at just the right time. He's a happy 'bloat' and the goodays might get to you but this professor is brilliant, absolutely brilliant. Of all my courseware at 'the Great Courses' this guy is tops followed just behind by the guy that does Algebra. The logic and reasoning with a hint of joviality makes this a fun course. This course is highly recommended and I can't wait to see his Visualization course too. I need this guy to help me with differential equations, too......
Date published: 2022-06-30
Rated 5 out of
5
by
patman from
Hooked with Measuring height of Pyramid
First few minutes... I was hooked. Admittedly, I'm not a total beginner, but it's been 50 years or more since I took Geometry. So, for me, this is a review. His methodology is a bit "snappy", quick, but I like it, personally. And he seems to actually love the subject matter. That's a huge plus.
Date published: 2022-06-09
Overview
About
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 excellence in math teaching. Professor Tanton is the author of a number of books on mathematics including Solve This: Math Activities for Students and Clubs, The Encyclopedia of Mathematics, and Mathematics Galore! Professor Tanton founded the St. Mark's Institute of Mathematics, an outreach program promoting joyful and effective mathematics education. He also conducts the professional development program for Math for America in Washington, D.C.
Trailer
01: 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.
33 min
02: 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.
28 min
03: 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
28 min
04: 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.
31 min
05: 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.
29 min
06: Distance, Midpoints, and Folding Ties
Learn how watching a fly on his ceiling inspired the mathematician Rene Descartes to link geometry and algebra. Find out how this powerful connection allows us to use algebra to calculate distances, midpoints, and more.
29 min
07: 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!
35 min
08: 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.
29 min
09: 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.
34 min
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.
31 min
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.
29 min
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.
33 min
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.
31 min
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.
30 min
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).
30 min
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.
32 min
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.
28 min
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.
32 min
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.
29 min
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.
33 min
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.
28 min
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).
31 min
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.
32 min
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.
30 min
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.
30 min
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.
29 min
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.
31 min
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.
32 min
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.
29 min
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.
28 min
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.
30 min
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.
31 min
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.
29 min
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.
32 min
35: Complex Numbers in Geometry
In lecture 6, you saw how 17th-century mathematician Rene 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.
32 min
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.
32 min
