Like a few others said on PoA, I wish I had this guy as my math and geometry teacher. His method and style of teaching these principles makes total sense and much simpler to grok and remember,

It's clear the teacher loves his job. I bet the enthusiasm rubs off on his students. CFIs should strive for this IMO.

Fully agree. Which in part is why I enjoy his videos... I'm trying to pick up on his technique and style to improve my own.

For fun, we should ask him to explain some other things like angle of attack or Bernoulli's principle. I bet it would go smoother than a PoA thread about it

Some people can teach and others can't. That seems to be especially true in math. I remember taking a math course in college that I just couldn't comprehend. I withdrew. Later I re-took it - this time with the prof who wrote the book (literally). Crystal Clear. I will never be a math whiz - but I sincerely believe that much of the problem was with the instructors rather than with me. Dave

I wonder if this guy can solve the age old riddle of whether a plane on a treadmill can take off. I bet he can.

I used to stop at a local pub and have dinner after a long day of flying and doodle on a cocktail napkin trying to come up with a way to instantly visualize this very thing while waiting for my order. This was the closest I ever got, practically word for word, step by step. As great as this is, it doesn't satisfy me. Oh, this instructor is superb, no question. The logic is flawless, true enough. But how can one SEE the logic without help from algebra? I still wish I could somehow see it just by folding the corners into the center or some such similar way, but I'm at a loss for how.

I saw where he got the a*2, b*2, and c*2, but I had to watch it two more times to see how he showed that they added up. Like anything, it’s perfectly simple to someone whose mind works the same way as his, but not so much if your mind works differently.

Interesting - This is the explanation I’ve seen in primary school in the 80s. I still remember cutting out the shapes and moving them around like that. In fact I don’t think I even know another way to prove it - what method did you guys use?

I don't recall my schooling including any real explanation or proof of the theorem. Just that if you wanted to know the hypotenuse of a right triangle, this is what you used. In hindsight, the geometry teacher was a multi-decade veteran of our school system, burned out, and not well respected by us teenagers. So he was more there to get through the year and collect some income than trying to ignite sparks of interest in young minds.

Absolutely a proof. In fact it's the one most commonly attributed to Pythagoras himself. Here is it in 5 seconds, which may make it seem more familiar (probably won't play/animate on a phone): https://upload.wikimedia.org/wikipedia/commons/9/9e/Pythagoras-proof-anim.svg

Looking back I came to the same recollection that Mike did below. But I first studied this in the early 1970s so my recollection is a bit faded. I guess you had better teachers or at least a better memory. Or both.

I was lucky and had a teacher like this in high school. I liked him so much, I took (and got A’s in) geometry, algebra 2, trig, and calculus all in two years so I could have him as the instructor for all of them. Then I went to college and failed calc 2 the first time because I had a typical instructor that didn’t actually explain anything. I have to understand 2 things for something to sink in. Why is it like it is, and how can I apply it in the real world; otherwise it’s just noise to me.

Pythagoras may never have existed, kinda like Confusious and the Buddha. I never thought there was anything all that hard about a2+b2=c2, but maybe that's just me. You want hard, try Genetics.

I always wondered why teachers, at least where I went to school, didn’t lead with WHY things like this are so useful. What real world (and very cool) applications just this theory for example gives you. I wish I had a teacher back then that told me “what if you want to cut down a tree, but need to know exactly how long it is, to be sure of how far the tip will end up?” and then tell you you could do it fairly accurately by sighting a 45 deg angle to the top of the tree. As a kid in geometry class, it was hard to see the real world value, they seems to just dive into theory. Nice video! He did a great job showing the relation, not just the actual dry algebraic part of it.

This comes the closest to "seeing" the relationship — by dividing the white area into two smaller ones. Ed Fred's "simpler demonstration" above was the classic illustration in my math books, but it only served as a riddle to me because it doesn't explain "why".

Originally learned it as a kid from my Dad as the 3-4-5 rule. We would be working in a house, and he would have us mark out from the corners 3' and 4' to check for square. I figured it was an old carpenter's trick, only later did I discover it was the Pythagorean theorem.

Pythagoras was a carpenter. When a crew was laying out pier locations for an addition to our house, they did the same thing.

Actually, Pythagoras was quite the Renaissance man, a veritable da Vinci. Unfortunately little of his work survives in an attributable form. Pythagoras of Samos[a] (c. 570 – c. 495 BC) was an ancient Ionian Greek philosopher and the eponymous founder of Pythagoreanism. His political and religious teachings were well known in Magna Graecia and influenced the philosophies of Plato, Aristotle, and, through them, Western philosophy. Knowledge of his life is clouded by legend, but he appears to have been the son of Mnesarchus, a seal engraver on the island of Samos. Modern scholars disagree regarding Pythagoras's education and influences, but they do agree that, around 530 BC, he travelled to Croton, where he founded a school in which initiates were sworn to secrecy and lived a communal, ascetic lifestyle. This lifestyle entailed a number of dietary prohibitions, traditionally said to have included vegetarianism, although modern scholars doubt that he ever advocated for complete vegetarianism. The teaching most securely identified with Pythagoras is metempsychosis, or the "transmigration of souls", which holds that every soul is immortal and, upon death, enters into a new body. He may have also devised the doctrine of musica universalis, which holds that the planets move according to mathematical equations and thus resonate to produce an inaudible symphony of music. Scholars debate whether Pythagoras developed the numerological and musical teachings attributed to him, or if those teachings were developed by his later followers, particularly Philolaus of Croton. Following Croton's decisive victory over Sybaris in around 510 BC, Pythagoras's followers came into conflict with supporters of democracy and Pythagorean meeting houses were burned. Pythagoras may have been killed during this persecution, or escaped to Metapontum, where he eventually died. In antiquity, Pythagoras was credited with many mathematical and scientific discoveries, including the Pythagorean theorem, Pythagorean tuning, the five regular solids, the Theory of Proportions, the sphericity of the Earth, and the identity of the morning and evening stars as the planet Venus. It was said that he was the first man to call himself a philosopher ("lover of wisdom")[c] and that he was the first to divide the globe into five climatic zones. Classical historians debate whether Pythagoras made these discoveries, and many of the accomplishments credited to him likely originated earlier or were made by his colleagues or successors. Some accounts mention that the philosophy associated with Pythagoras was related to mathematics and that numbers were important, but it is debated to what extent, if at all, he actually contributed to mathematics or natural philosophy. Pythagoras influenced Plato, whose dialogues, especially his Timaeus, exhibit Pythagorean teachings. Pythagorean ideas on mathematical perfection also impacted ancient Greek art. His teachings underwent a major revival in the first century BC among Middle Platonists, coinciding with the rise of Neopythagoreanism. Pythagoras continued to be regarded as a great philosopher throughout the Middle Ages and his philosophy had a major impact on scientists such as Nicolaus Copernicus, Johannes Kepler, and Isaac Newton. Pythagorean symbolism was used throughout early modern European esotericism and his teachings as portrayed in Ovid's Metamorphoses influenced the modern vegetarian movement.

Pythagoras as I learned it. The story goes that a Sioux chief had a son who needed a wife, and three men in the tribe offered their daughters, along with dowry offers for the chief. One dowry offer was for 20 beaver skins; they were soft and luxurious. Another dowry offer was for 10 buffalo hides; they had many uses because of their toughness. The third dowry offer, however, was for a single hippopotamus hide, which was rough and scratchy and, because of its extreme toughness, of no practical use. But the daughters being offered with the first two dowries were rather plain, while the daughter being offered with the third dowry was quite attractive. So the chief accepted the third dowry, proclaiming "The squaw of the hippopotamus is equal to the sum of the squaws of the other two hides"

My Dad was a carpenter before he worked at Hiller. He could barely do advanced math, but could use a carpenter's framing square from memory. He could compute the run and rise of a flight of stairs by simply laying it on the floor. He showed me how pieces of wood, cut into right triangles, could be cut to support those stair steps. By laying them down on the floor just like the teacher in the video did, I could see the relationship of the area of the pieces.

Also if his father wasn’t going around engraving seals...probably Pythagoras had to hold them, seals don’t stand still for engraving you know.

Maybe my Neanderthal brain missed it, but why is that example better than just saying you can find the length of any of the sides of a right triangle by manipulating the formula a^2 + b^2 = c^2? I mean, I get the further application of the math if you happen to have 4-identical right triangles and want to find the area/lengths of any of the pieces. I just don't know why it was better than just using it to find a missing value for a right-triangle. I will say it's one of the few theorems from geometry I've ever had cause to use in real life.

That's pretty handy for when you only have two walls/sides. Not being a carpenter, I had to think for a minute about what your Dad was doing. What I had previously been aware of as a carpenter's trick is simply measuring the diagonals, and seeing that they are the same. That's a quick and easy practical application of Pythagoras' Theorum. But that obviously only works where you have four walls/sides.

The idea is to get the students interested in and invested in STEM, which is an area that we are falling behind in. Not just give them a formula to cut a board. Or we could just keep giving our best jobs to highly-educated immigrants under H-1B visas and the like (assuming for the moment that this video is of a US classroom). https://www.engineeringforkids.com/about/news/2016/february/why-is-stem-education-so-important-/ "According to the U. S. Department of Commerce, STEM occupations are growing at 17%, while other occupations are growing at 9.8%. STEM degree holders have a higher income even in non-STEM careers. Science, technology, engineering and mathematics workers play a key role in the sustained growth and stability of the U.S. economy, and are a critical component to helping the U.S. win the future. STEM education creates critical thinkers, increases science literacy, and enables the next generation of innovators. Innovation leads to new products and processes that sustain our economy. This innovation and science literacy depends on a solid knowledge base in the STEM areas. It is clear that most jobs of the future will require a basic understanding of math and science. Despite these compelling facts, mathematics and science scores on average among U.S. students are lagging behind other developing countries." Emphasis added, LOL

I have a lot of respect for that. My dad was a tool and die maker at Republic Aviation, Fairchild-Hiller, Boeing. Man could do anything with his hands. Doubt he graduated high school. Also a WW2 combat vet (unfortunately no-one knew about PTSD back then but that is another story - sorry, got something in my eye). This looks interesting:

I have no problem with the way the information was presented, nor the impact that a great teacher has on maintaining the interest of his students (especially in STEM fields). My point was that I feel like I was missing some "connection" that was to be made with the example. I was hoping someone might expound on why the example of relating the area of the empty space and the area of the right triangles was critical to the theorem itself, or if it was just an illustration on how to use a2xb2=c2 to solve a slightly more difficult scenario. This was the image which I was more familiar with: which shows that a square with 4-equilateral sides of "a", and a square with 4-equilateral sides of "b", then the sum of those equals a square with 4-equilateral sides of "c".