title

Tips to be a better problem solver [Last live lecture] | Ep. 10 Lockdown live math

description

Tips on problem-solving, with examples from geometry, trig, and probability.
Past episodes with integrated quizzes: https://itempool.com/c/3b1b
Full playlist: https://www.youtube.com/playlist?list=PLZHQObOWTQDP5CVelJJ1bNDouqrAhVPev
Home page: https://www.3blue1brown.com
Brought to you by you: https://3b1b.co/ldm-thanks
Huge huge thanks to Ben Eater: https://www.youtube.com/user/eaterbc
And Cam Christensen, creator of ItemPool: https://itempool.com/
Notes by Ngân Vũ:
https://twitter.com/ThuyNganVu/status/1265480770832855040
Mistakes:
50:35, there should be a dx in the integral
54:40, if you notice the mistake here and are inclined to complain, keep watching
------------------
Video timeline (thanks to user "noonesperfect")
0:34 9-Problem Solving Principles/Tip
1:15 Question 1 (Probability)
2:08 Who is Ben Eater?
4:25 Inscribed angle theorem, θL=2*θs
5:58 Tip-1
7:48 Tip-2
8:16 Question 2
9:34 Answer 2
10:29 Tip-3
15:17 Tip-4
22:48 Question 3
25:56 Answer 3 (Marked incorrectly, Answer: Option D)
26:31 Answer 1
27:28 Explanation for Q1, Floor function
30:38 Tip-5
32:53 Tip-6
33:36 Question 4
34:43 Answer 4
36:37 Question 5
38:10 Answer 5
41:48 Probability graph in Desmos
44:08 Revisiting an alternating series sum for ln(2)
47:29 Tip-7
51:08 Tip-8
55:23 Audience questions through tweets
57:28 Tip-9
58:29 Python programming for various probability question
1:04:31 Arts created using Desmos graph tool with mathematical expressions
1:05:54 Thank you, appreciation to the team and all.
------------------
The live question setup with stats on-screen is powered by Itempool.
https://itempool.com/
Curious about other animations?
https://www.3blue1brown.com/faq#manim
Music by Vincent Rubinetti.
Download the music on Bandcamp:
https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown
Stream the music on Spotify:
https://open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u
If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then "add subtitles/cc". I really appreciate those who do this, as it helps make the lessons accessible to more people.
------------------
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detail

{'title': 'Tips to be a better problem solver [Last live lecture] | Ep. 10 Lockdown live math', 'heatmap': [{'end': 297.038, 'start': 244.71, 'weight': 0.873}, {'end': 3900.419, 'start': 3849.569, 'weight': 1}], 'summary': 'Discusses problem-solving principles, geometry, symmetry, trigonometry, and probability, offering insights and tricks to enhance problem-solving skills, emphasizing the application of mathematical concepts and highlighting the significance of recognizing patterns and error rectification.', 'chapters': [{'end': 227.469, 'segs': [{'end': 31.981, 'src': 'embed', 'start': 0.089, 'weight': 0, 'content': [{'end': 1.069, 'text': 'Welcome back, everybody.', 'start': 0.089, 'duration': 0.98}, {'end': 6.071, 'text': "It's hard to define exactly what mathematicians mean when they use the phrase problem solving.", 'start': 1.71, 'duration': 4.361}, {'end': 7.672, 'text': 'However you go about it.', 'start': 6.851, 'duration': 0.821}, {'end': 16.535, 'text': "it's going to involve some notion of approaching puzzles that you've never seen before and still being able to systematically and creatively find some solution to them.", 'start': 7.672, 'duration': 8.863}, {'end': 20.456, 'text': "But that's a weird thing when you think about it, because it makes it a very hard thing to teach.", 'start': 17.415, 'duration': 3.041}, {'end': 26.599, 'text': 'I mean, you can teach someone how to solve one particular problem, maybe even a class of problems, and teach them how to solve another problem.', 'start': 20.856, 'duration': 5.743}, {'end': 31.981, 'text': "But how do you teach someone how to approach a problem that they've never seen before and still make progress on it?", 'start': 27.179, 'duration': 4.802}], 'summary': 'Mathematicians struggle to teach problem-solving for unseen puzzles.', 'duration': 31.892, 'max_score': 0.089, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/QvuQH4_05LI/pics/QvuQH4_05LI89.jpg'}, {'end': 152.524, 'src': 'embed', 'start': 123.978, 'weight': 6, 'content': [{'end': 125.439, 'text': 'what do you think that probability is going to be?', 'start': 123.978, 'duration': 1.461}, {'end': 127.179, 'text': "We'll return back to this later.", 'start': 126.039, 'duration': 1.14}, {'end': 135.321, 'text': "And one thing I want to say is that this whole live quizzing software, it's something that's being built by some friends of mine Ben Eder,", 'start': 127.599, 'duration': 7.722}, {'end': 141.682, 'text': 'who many of you may know because of his YouTube fame, and then another person who used to work with us at Khan Academy, named Cam Christensen.', 'start': 135.321, 'duration': 6.361}, {'end': 147.343, 'text': "And this is just one small little outcropping of what's actually a much deeper product at play.", 'start': 142.242, 'duration': 5.101}, {'end': 152.524, 'text': "And I want to give you a little preview of some of the other things that they've been working on that we'll be developing.", 'start': 148.083, 'duration': 4.441}], 'summary': 'Discussion on live quizzing software built by friends ben eder and cam christensen, with a preview of other products in development.', 'duration': 28.546, 'max_score': 123.978, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/QvuQH4_05LI/pics/QvuQH4_05LI123978.jpg'}, {'end': 207.361, 'src': 'embed', 'start': 173.549, 'weight': 1, 'content': [{'end': 177.95, 'text': 'So as you skip ahead to various different questions, it will skip you to the right point in the video.', 'start': 173.549, 'duration': 4.401}, {'end': 180.671, 'text': "And as the video plays forward, you'll also get to..", 'start': 178.31, 'duration': 2.361}, {'end': 183.972, 'text': "There's just me talking through some problem.", 'start': 182.331, 'duration': 1.641}, {'end': 187.213, 'text': "You'll also get to whatever the appropriate part of the problem is.", 'start': 184.492, 'duration': 2.721}, {'end': 191.655, 'text': 'They often have explanations associated with them and hints and things like that.', 'start': 188.093, 'duration': 3.562}, {'end': 198.817, 'text': "And ultimately, if you look back in just a couple of days, I'm going to fill out like homework and challenge problems.", 'start': 192.375, 'duration': 6.442}, {'end': 204.6, 'text': "So, if you like challenging, problem solving, contest, math type stuff, I'm going to put things that are relevant to the lectures in there,", 'start': 199.098, 'duration': 5.502}, {'end': 205.34, 'text': 'which I think should be fun.', 'start': 204.6, 'duration': 0.74}, {'end': 207.361, 'text': "um, it's not not quite there yet,", 'start': 205.88, 'duration': 1.481}], 'summary': 'The video allows skipping to specific points and will include explanations and hints, with homework and challenge problems to be added in a few days.', 'duration': 33.812, 'max_score': 173.549, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/QvuQH4_05LI/pics/QvuQH4_05LI173549.jpg'}], 'start': 0.089, 'title': 'Principles of problem solving', 'summary': 'Explores teaching problem-solving in mathematics, presenting nine deceptively simple problem-solving principles with specific examples, emphasizing their effectiveness in making meaningful progress in challenging problems.', 'chapters': [{'end': 82.945, 'start': 0.089, 'title': 'Principles of problem solving', 'summary': 'Explores the challenges of teaching problem-solving in mathematics and presents nine deceptively simple problem-solving principles, each illustrated with specific examples, emphasizing their effectiveness in making meaningful progress in challenging problems.', 'duration': 82.856, 'highlights': ["The chapter explores the challenges of teaching problem-solving in mathematics, emphasizing the difficulty of teaching individuals to approach new problems and make progress (e.g. teaching someone how to approach a problem they've never seen before).", 'Nine problem-solving principles are presented, each deceptively simple but effective in making meaningful progress in hard problems (e.g. shockingly meaningful progress in hard problems just by keeping these tips in mind).', 'Each principle is illustrated with specific examples, demonstrating their applicability and effectiveness in problem-solving (e.g. walking through each principle in the context of a specific example).', 'The chapter introduces a hard problem that will be tackled later in the lecture, providing a preview of the type of challenges that will be addressed (e.g. presenting a hard problem to mull over and tackle later in the lecture).']}, {'end': 227.469, 'start': 83.666, 'title': 'Probability of rounding down to an even number', 'summary': 'Discusses the probability of the ratio of two random numbers between 0 and 1 rounding down to an even number, and introduces a live quizzing software being developed by ben eder and cam christensen at itempool.com.', 'duration': 143.803, 'highlights': ['The chapter discusses the probability of the ratio of two random numbers between 0 and 1 rounding down to an even number. The chapter introduces the problem of determining the probability that the ratio of two random numbers between 0 and 1 rounds down to an even number.', 'Introduction of live quizzing software being developed by Ben Eder and Cam Christensen at itempool.com. The live quizzing software being developed by Ben Eder and Cam Christensen at itempool.com is introduced, offering features such as live quizzes, progress tracking, and interactive lectures.', 'Availability of item pool for live polling in classes and beta user opportunities. The item pool offers the opportunity for live polling in classes and is seeking beta users for testing and feedback.']}], 'duration': 227.38, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/QvuQH4_05LI/pics/QvuQH4_05LI89.jpg', 'highlights': ['Nine problem-solving principles are presented, each deceptively simple but effective in making meaningful progress in hard problems (e.g. shockingly meaningful progress in hard problems just by keeping these tips in mind).', 'The chapter introduces a hard problem that will be tackled later in the lecture, providing a preview of the type of challenges that will be addressed (e.g. presenting a hard problem to mull over and tackle later in the lecture).', 'Each principle is illustrated with specific examples, demonstrating their applicability and effectiveness in problem-solving (e.g. walking through each principle in the context of a specific example).', "The chapter explores the challenges of teaching problem-solving in mathematics, emphasizing the difficulty of teaching individuals to approach new problems and make progress (e.g. teaching someone how to approach a problem they've never seen before).", 'Introduction of live quizzing software being developed by Ben Eder and Cam Christensen at itempool.com. The live quizzing software being developed by Ben Eder and Cam Christensen at itempool.com is introduced, offering features such as live quizzes, progress tracking, and interactive lectures.', 'The chapter discusses the probability of the ratio of two random numbers between 0 and 1 rounding down to an even number. The chapter introduces the problem of determining the probability that the ratio of two random numbers between 0 and 1 rounds down to an even number.', 'Availability of item pool for live polling in classes and beta user opportunities. The item pool offers the opportunity for live polling in classes and is seeking beta users for testing and feedback.']}, {'end': 576.437, 'segs': [{'end': 297.038, 'src': 'heatmap', 'start': 244.71, 'weight': 0.873, 'content': [{'end': 249.457, 'text': "One, so that you keep your eye out, and you know you're a little bit skeptical of each of the claims that i make.", 'start': 244.71, 'duration': 4.747}, {'end': 255.165, 'text': "and two, so that when i make that mistake and you notice it, um, and you're just, you know, throwing things at the screen, you're getting angry.", 'start': 249.457, 'duration': 5.708}, {'end': 256.867, 'text': "you're clicking that unsubscribe button.", 'start': 255.165, 'duration': 1.702}, {'end': 259.531, 'text': "you can at least quell a little bit of what you're thinking.", 'start': 256.867, 'duration': 2.664}, {'end': 260.351, 'text': 'so just keep that in mind.', 'start': 259.531, 'duration': 0.82}, {'end': 261.853, 'text': 'there will be one very purposeful mistake.', 'start': 260.351, 'duration': 1.502}, {'end': 266.377, 'text': 'Now, before we get to the problem-solving tip, I want to talk about geometry.', 'start': 262.455, 'duration': 3.922}, {'end': 273.141, 'text': 'This is one thing I was hoping to do a little bit more of in this series, but this should be just a fun time to give a little example of it.', 'start': 266.998, 'duration': 6.143}, {'end': 277.643, 'text': 'And in particular, I want to talk about one of my favorite little bits of geometry.', 'start': 274.001, 'duration': 3.642}, {'end': 280.605, 'text': "It's not too simple, but it's also not too hard.", 'start': 278.284, 'duration': 2.321}, {'end': 282.446, 'text': "It's called the inscribed angle theorem.", 'start': 281.045, 'duration': 1.401}, {'end': 288.229, 'text': "It comes up way more than is reasonable for a simple little theorem like this to actually come up when you're solving problems.", 'start': 282.826, 'duration': 5.403}, {'end': 297.038, 'text': 'And basically an inscribed angle of a circle refers to if you have two lines that meet at a point of that circle,', 'start': 289.85, 'duration': 7.188}], 'summary': 'Transcript discusses skepticism, mistakes, and geometry theorem inscribed angle.', 'duration': 52.328, 'max_score': 244.71, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/QvuQH4_05LI/pics/QvuQH4_05LI244710.jpg'}, {'end': 433.72, 'src': 'embed', 'start': 408.793, 'weight': 1, 'content': [{'end': 414.536, 'text': "But moreover, this other point P, that was not just chosen at random, it's defined to be on the circle.", 'start': 408.793, 'duration': 5.743}, {'end': 422.251, 'text': 'And unraveling what that means, it means it is a common distance away and is the same distance away from the center as these other points.', 'start': 415.276, 'duration': 6.975}, {'end': 428.215, 'text': "So I am then inspired to draw a line to add something to the picture and to note that it's the same there.", 'start': 422.731, 'duration': 5.484}, {'end': 433.72, 'text': 'Quite often when you see people solve hard geometry problems, it comes down to adding something to the picture.', 'start': 428.856, 'duration': 4.864}], 'summary': 'Point p is defined on the circle, same distance from center as other points.', 'duration': 24.927, 'max_score': 408.793, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/QvuQH4_05LI/pics/QvuQH4_05LI408793.jpg'}, {'end': 501.027, 'src': 'embed', 'start': 447.31, 'weight': 0, 'content': [{'end': 450.352, 'text': "put a rectangle around this triangle, whatever it is, you're doing.", 'start': 447.31, 'duration': 3.042}, {'end': 457.496, 'text': 'But how do you systematically know what you should add? So in this context, if it relates to the definition of your objects, probably a good idea.', 'start': 450.412, 'duration': 7.084}, {'end': 460.578, 'text': 'And that line actually will be helpful to us.', 'start': 457.956, 'duration': 2.622}, {'end': 463.1, 'text': 'We can start giving a couple things names.', 'start': 461.238, 'duration': 1.862}, {'end': 470.384, 'text': "That's something that again maybe it shouldn't even be described as a tip, but I put it down as number two that when you give things meaningful names,", 'start': 463.58, 'duration': 6.804}, {'end': 472.545, 'text': 'that actually helps you move forward in your problem.', 'start': 470.384, 'duration': 2.161}, {'end': 475.908, 'text': 'And in this context, it might seem like an innocuous thing.', 'start': 473.846, 'duration': 2.062}, {'end': 481.191, 'text': "I'll call this little angle that we formed with our radius alpha and this little angle beta.", 'start': 475.948, 'duration': 5.243}, {'end': 485.034, 'text': 'And just to see if that helps us move forward.', 'start': 482.272, 'duration': 2.762}, {'end': 487.696, 'text': 'you know, recognizing if alpha and beta show up elsewhere.', 'start': 485.034, 'duration': 2.662}, {'end': 492.179, 'text': 'rather than me telling you, I would like you to tell me in the context of our next quiz question.', 'start': 487.696, 'duration': 4.483}, {'end': 498.626, 'text': "So before I answer that probability one for you, I'm going to pull up Question number two for today.", 'start': 492.339, 'duration': 6.287}, {'end': 501.027, 'text': "We've got a diagram, essentially what we just drew.", 'start': 498.706, 'duration': 2.321}], 'summary': 'Systematically name and identify objects to progress in geometry problems.', 'duration': 53.717, 'max_score': 447.31, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/QvuQH4_05LI/pics/QvuQH4_05LI447310.jpg'}], 'start': 227.469, 'title': 'Problem solving and geometry', 'summary': 'Discusses nine problem solving tricks, encourages skepticism and engagement, and provides insights on the inscribed angle theorem in geometry. it emphasizes using defining features and giving meaningful names to elements.', 'chapters': [{'end': 260.351, 'start': 227.469, 'title': 'Problem solving tricks and mistake disclosure', 'summary': "Discusses nine problem solving tricks and the speaker's plan to intentionally make a mistake to encourage skepticism and engagement from the audience.", 'duration': 32.882, 'highlights': ['The speaker plans to intentionally make a mistake to encourage skepticism and engagement from the audience.', 'Discussion of nine deceptively simple problem solving tricks.', 'Encourages the audience to remain skeptical and observant of the claims made.']}, {'end': 576.437, 'start': 260.351, 'title': 'Geometry and problem-solving tip', 'summary': 'Discusses the inscribed angle theorem in geometry and provides problem-solving tips, emphasizing the importance of using defining features and giving meaningful names to elements.', 'duration': 316.086, 'highlights': ['The inscribed angle theorem in geometry is discussed, emphasizing its practical application in problem-solving. The inscribed angle theorem is highlighted as a significant concept in geometry, with practical relevance for problem-solving.', 'Problem-solving tip on using defining features is provided, emphasizing its usefulness in mathematical problem-solving. The importance of using defining features in problem-solving is emphasized, with a personal anecdote of its significance in undergraduate math major experiences.', 'The significance of giving meaningful names to elements in problem-solving is highlighted. The practice of giving meaningful names to elements is emphasized as a valuable strategy in problem-solving.']}], 'duration': 348.968, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/QvuQH4_05LI/pics/QvuQH4_05LI227469.jpg', 'highlights': ['Discussion of nine deceptively simple problem solving tricks.', 'The inscribed angle theorem in geometry is discussed, emphasizing its practical application in problem-solving.', 'Problem-solving tip on using defining features is provided, emphasizing its usefulness in mathematical problem-solving.', 'The significance of giving meaningful names to elements in problem-solving is highlighted.', 'The speaker plans to intentionally make a mistake to encourage skepticism and engagement from the audience.', 'Encourages the audience to remain skeptical and observant of the claims made.']}, {'end': 956.291, 'segs': [{'end': 750.347, 'src': 'embed', 'start': 719.673, 'weight': 0, 'content': [{'end': 724.734, 'text': "you know about the inscribed angle theorem, but I really want you to think about this from a beginner's mind, right?", 'start': 719.673, 'duration': 5.061}, {'end': 729.976, 'text': "If you were just approaching this and you didn't necessarily already know about it, What would you have done to find that solution?", 'start': 724.774, 'duration': 5.202}, {'end': 732.017, 'text': 'And what principles can you take away as you do that?', 'start': 730.076, 'duration': 1.941}, {'end': 735.259, 'text': 'Because that helps us as we start to get to harder and harder geometry setups.', 'start': 732.277, 'duration': 2.982}, {'end': 741.042, 'text': "So, in this context, once I have these three equations recognizing that there's an alpha prime here and one here,", 'start': 736.08, 'duration': 4.962}, {'end': 744.144, 'text': "there's a beta prime here and one here I might think about canceling them out.", 'start': 741.042, 'duration': 3.102}, {'end': 750.347, 'text': "So I'm going to, you know, add this top equation and maybe I've subtract off the other two equations.", 'start': 745.065, 'duration': 5.282}], 'summary': "Exploring beginner's approach to inscribed angle theorem for geometry problem solving.", 'duration': 30.674, 'max_score': 719.673, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/QvuQH4_05LI/pics/QvuQH4_05LI719673.jpg'}], 'start': 576.858, 'title': 'Symmetry and trigonometry in geometry', 'summary': "Explores symmetry in geometry problems, demonstrating its utility in making connections and drawing conclusions. it covers properties of isosceles triangles, angle measurements in radians, geometric manipulations, and trigonometric identities, encouraging a beginner's problem-solving approach.", 'chapters': [{'end': 735.259, 'start': 576.858, 'title': 'Symmetry in geometry problems', 'summary': "Discusses the use of symmetry in geometry problems, using examples to demonstrate how it helps in making connections and drawing conclusions, with key points including the properties of isosceles triangles and the application of angle measurements in radians, as well as encouraging a beginner's mindset for problem-solving.", 'duration': 158.401, 'highlights': ['The properties of isosceles triangles are leveraged to demonstrate symmetry in the given geometry problem, showing that the angles alpha and beta are equal within the triangles, which creates a basis for further deductions.', 'The application of angle measurements in radians is used to express the sum of angles in the triangles, with the sum of alpha angles equating to pi radians and the sum of beta angles equating to 2 pi, providing quantifiable data for analysis.', "Encouragement is given to approach geometry problems with a beginner's mindset, emphasizing the importance of finding solutions and principles without prior knowledge, which aids in tackling harder geometry setups.", 'The inscribed angle theorem is mentioned, highlighting the significance of considering alternative approaches and principles in problem-solving, particularly for audience members familiar with advanced concepts.']}, {'end': 956.291, 'start': 736.08, 'title': 'Geometric manipulations and trigonometric identities', 'summary': 'Discusses geometric manipulations involving canceling out terms and deriving the relationship between squaring and doubling angles in trigonometric identities, which can be unexpectedly useful in geometry puzzles and graph manipulations.', 'duration': 220.211, 'highlights': ['Deriving the relationship between squaring and doubling angles in trigonometric identities The chapter discusses the relationship between squaring and doubling angles in trigonometric identities, emphasizing its unexpected usefulness in geometry puzzles and graph manipulations.', 'Geometric manipulations involving canceling out terms The chapter covers geometric manipulations, including canceling out terms to simplify equations and derive results, demonstrating the process with practical examples.', 'Usefulness of the relationship in various geometry puzzles and graph manipulations The chapter highlights the unexpectedly useful nature of the relationship between squaring and doubling angles in various geometry puzzles and graph manipulations, underscoring its practical applications.']}], 'duration': 379.433, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/QvuQH4_05LI/pics/QvuQH4_05LI576858.jpg', 'highlights': ['The properties of isosceles triangles are leveraged to demonstrate symmetry in the given geometry problem, showing that the angles alpha and beta are equal within the triangles, which creates a basis for further deductions.', 'The application of angle measurements in radians is used to express the sum of angles in the triangles, with the sum of alpha angles equating to pi radians and the sum of beta angles equating to 2 pi, providing quantifiable data for analysis.', 'Deriving the relationship between squaring and doubling angles in trigonometric identities The chapter discusses the relationship between squaring and doubling angles in trigonometric identities, emphasizing its unexpected usefulness in geometry puzzles and graph manipulations.', 'The inscribed angle theorem is mentioned, highlighting the significance of considering alternative approaches and principles in problem-solving, particularly for audience members familiar with advanced concepts.']}, {'end': 1440.889, 'segs': [{'end': 1150.508, 'src': 'embed', 'start': 1126.993, 'weight': 2, 'content': [{'end': 1134.383, 'text': 'So cosine squared refers to this length, a portion of the hypotenuse of our right triangle.', 'start': 1126.993, 'duration': 7.39}, {'end': 1138.308, 'text': 'if that hypotenuse had a length of one which in our unit circle it always does,', 'start': 1134.383, 'duration': 3.925}, {'end': 1146.823, 'text': 'and incidentally you can show very similar reasoning that the sine of theta is this other, uh other portion of that hypotenuse.', 'start': 1138.308, 'duration': 8.515}, {'end': 1150.508, 'text': 'and this gives you a nice sort of a clever proof of the pythagorean theorem.', 'start': 1146.823, 'duration': 3.685}], 'summary': "Cosine squared and sine of theta represent portions of the unit circle's hypotenuse, providing a clever proof of the pythagorean theorem.", 'duration': 23.515, 'max_score': 1126.993, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/QvuQH4_05LI/pics/QvuQH4_05LI1126993.jpg'}, {'end': 1241.648, 'src': 'embed', 'start': 1210.836, 'weight': 4, 'content': [{'end': 1214.518, 'text': "It means that if we take that diameter of the circle, if it's 180 degrees,", 'start': 1210.836, 'duration': 3.682}, {'end': 1222.739, 'text': "it's just drawing out a diameter and we have an inscribed angle with lines that hit either end of that diameter,", 'start': 1214.518, 'duration': 8.221}, {'end': 1224.88, 'text': 'then this angle is necessarily half of that.', 'start': 1222.739, 'duration': 2.141}, {'end': 1231.263, 'text': 'So what it means is that we can put a right triangle inside a circle, and whenever you do that,', 'start': 1225.541, 'duration': 5.722}, {'end': 1234.745, 'text': 'the hypotenuse of the right triangle is exactly the diameter of that circle.', 'start': 1231.263, 'duration': 3.482}, {'end': 1235.965, 'text': "It's a very cute fact.", 'start': 1235.085, 'duration': 0.88}, {'end': 1241.648, 'text': "If you wanted another proof of it, that's not just via the inscribed angle theorem.", 'start': 1236.746, 'duration': 4.902}], 'summary': 'In a circle, an inscribed angle is half the measure of the intercepted arc.', 'duration': 30.812, 'max_score': 1210.836, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/QvuQH4_05LI/pics/QvuQH4_05LI1210836.jpg'}, {'end': 1323.096, 'src': 'embed', 'start': 1283.05, 'weight': 0, 'content': [{'end': 1287.532, 'text': 'that could also be a little side homework problem if you wanted to chase around the relevant angles,', 'start': 1283.05, 'duration': 4.482}, {'end': 1294.716, 'text': "but I think that's a very beautiful way to think about Thale's theorem that you reflect everything 180 degrees and you necessarily conclude it must be a rectangle,", 'start': 1287.532, 'duration': 7.184}, {'end': 1296.277, 'text': 'which means that this is a right angle.', 'start': 1294.716, 'duration': 1.561}, {'end': 1298.38, 'text': 'Now, for our purposes, what does that mean?', 'start': 1297.077, 'duration': 1.303}, {'end': 1300.846, 'text': "Well, we've got a right triangle sitting here.", 'start': 1298.781, 'duration': 2.065}, {'end': 1301.568, 'text': "that's from zero.", 'start': 1300.846, 'duration': 0.722}, {'end': 1304.134, 'text': "We've got one of the points here, another point on the circle.", 'start': 1301.728, 'duration': 2.406}, {'end': 1306.439, 'text': "Let's inscribe that in a separate circle.", 'start': 1304.555, 'duration': 1.884}, {'end': 1314.89, 'text': "So I'm going to take a copy of that triangle, but I'm going to, instead of making the hypotenuse a radius of the circle,", 'start': 1307.964, 'duration': 6.926}, {'end': 1318.392, 'text': "I'm going to make that hypotenuse a diameter of the circle.", 'start': 1314.89, 'duration': 3.502}, {'end': 1323.096, 'text': 'So this is still going to be an angle of theta sitting right here.', 'start': 1319.493, 'duration': 3.603}], 'summary': "Applying thale's theorem and inscribing a triangle in a circle.", 'duration': 40.046, 'max_score': 1283.05, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/QvuQH4_05LI/pics/QvuQH4_05LI1283050.jpg'}], 'start': 956.291, 'title': 'Geometric representation and theorems in trigonometry', 'summary': "Discusses the geometric representation of cosine squared, leveraging symmetry to show that it refers to a length, a portion of the hypotenuse of a right triangle, providing a clever proof of the pythagorean theorem and emphasizing the concept of double projecting based on angle symmetry. it also explores thale's theorem and demonstrates the inscribed angle theorem with a right triangle inside a circle, connecting it to the length l in terms of theta.", 'chapters': [{'end': 1167.406, 'start': 956.291, 'title': 'Geometric representation of cosine squared', 'summary': 'Discusses the geometric representation of cosine squared, leveraging symmetry to show that it refers to a length, a portion of the hypotenuse of a right triangle, providing a clever proof of the pythagorean theorem and emphasizing the concept of double projecting based on angle symmetry.', 'duration': 211.115, 'highlights': ['The chapter emphasizes the geometric representation of cosine squared as a length, a portion of the hypotenuse of a right triangle, providing a clever proof of the Pythagorean theorem.', 'It discusses leveraging symmetry to show that cosine squared refers to a length, a portion of the hypotenuse of a right triangle, which has a length of one in the unit circle.', 'The chapter emphasizes the concept of double projecting based on angle symmetry to illustrate the geometric representation of cosine squared.']}, {'end': 1440.889, 'start': 1167.886, 'title': "Inscribed angle theorem and thale's theorem", 'summary': "Discusses the relationship between an angle and two times that angle, explores thale's theorem and demonstrates the inscribed angle theorem with a right triangle inside a circle, connecting it to the length l in terms of theta.", 'duration': 273.003, 'highlights': ['The inscribed angle theorem is demonstrated using a right triangle inside a circle, revealing that the hypotenuse of the right triangle is the diameter of the circle, leading to the conclusion that it forms a rectangle, implying a right angle. The demonstration of the inscribed angle theorem with a right triangle inside a circle clarifies that the hypotenuse of the right triangle is the diameter of the circle, leading to the conclusion that it forms a rectangle, implying a right angle.', 'The relationship between the hypotenuse of the large right triangle and the length l in terms of theta is explored, with the hypotenuse having a length of 1 in the context of a unit circle. The exploration of the relationship between the hypotenuse of the large right triangle and the length l in terms of theta clarifies that the hypotenuse has a length of 1 in the context of a unit circle, prompting the search for an expression for l in terms of theta.', 'The concept of reflecting a point through the origin and the center of the circle is introduced to leverage symmetry and demonstrate the inscribed angle theorem. The introduction of reflecting a point through the origin and the center of the circle to leverage symmetry for demonstrating the inscribed angle theorem provides an alternative approach beyond the traditional theorem.']}], 'duration': 484.598, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/QvuQH4_05LI/pics/QvuQH4_05LI956291.jpg', 'highlights': ['The chapter emphasizes the geometric representation of cosine squared as a length, a portion of the hypotenuse of a right triangle, providing a clever proof of the Pythagorean theorem.', 'The chapter emphasizes the concept of double projecting based on angle symmetry to illustrate the geometric representation of cosine squared.', 'The inscribed angle theorem is demonstrated using a right triangle inside a circle, revealing that the hypotenuse of the right triangle is the diameter of the circle, leading to the conclusion that it forms a rectangle, implying a right angle.', 'The relationship between the hypotenuse of the large right triangle and the length l in terms of theta is explored, with the hypotenuse having a length of 1 in the context of a unit circle.', 'The concept of reflecting a point through the origin and the center of the circle is introduced to leverage symmetry and demonstrate the inscribed angle theorem.']}, {'end': 2235.904, 'segs': [{'end': 1698.679, 'src': 'embed', 'start': 1668.758, 'weight': 1, 'content': [{'end': 1670.88, 'text': 'and the idea is that each point is as likely as another.', 'start': 1668.758, 'duration': 2.122}, {'end': 1679.106, 'text': "Or more specifically, a given range of points of a certain size should have a given probability that's independent of where that range showed up.", 'start': 1671.16, 'duration': 7.946}, {'end': 1680.487, 'text': "It's only dependent on its size.", 'start': 1679.166, 'duration': 1.321}, {'end': 1684.629, 'text': "So you might have in the back of your mind the idea that we've chosen two points.", 'start': 1681.407, 'duration': 3.222}, {'end': 1686.911, 'text': "They're each somewhere between 0 and 1.", 'start': 1684.909, 'duration': 2.002}, {'end': 1698.679, 'text': 'And just to give an example of what we mean by uniform distribution, the probability that x sits between 0.3 and 0.5,', 'start': 1686.911, 'duration': 11.768}], 'summary': 'Uniform distribution assigns equal probability to all points in a given range.', 'duration': 29.921, 'max_score': 1668.758, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/QvuQH4_05LI/pics/QvuQH4_05LI1668758.jpg'}, {'end': 1924.78, 'src': 'embed', 'start': 1889.744, 'weight': 0, 'content': [{'end': 1890.745, 'text': 'Things, they run away from you.', 'start': 1889.744, 'duration': 1.001}, {'end': 1896.388, 'text': 'Even your objects sometimes get tired of math class and want to play truant now and then.', 'start': 1892.102, 'duration': 4.286}, {'end': 1898.331, 'text': 'But he has to stay.', 'start': 1897.55, 'duration': 0.781}, {'end': 1900.955, 'text': 'Whether he likes to or not.', 'start': 1899.993, 'duration': 0.962}, {'end': 1906.124, 'text': "I'm absurd.", 'start': 1905.224, 'duration': 0.9}, {'end': 1909.047, 'text': "So let's say this is our x coordinate.", 'start': 1906.765, 'duration': 2.282}, {'end': 1912.95, 'text': 'x can fall anywhere between 0 and 1 with uniform probability.', 'start': 1910.028, 'duration': 2.922}, {'end': 1915.412, 'text': 'y can fall between 0 and 1.', 'start': 1913.791, 'duration': 1.621}, {'end': 1924.78, 'text': "So when we have a pair of numbers, something like 0.2, what is that, maybe like 0.8, a pair of numbers, it's just a single point in this diagram.", 'start': 1915.412, 'duration': 9.368}], 'summary': 'Transcript discusses the concept of coordinates and probability.', 'duration': 35.036, 'max_score': 1889.744, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/QvuQH4_05LI/pics/QvuQH4_05LI1889744.jpg'}, {'end': 2123.315, 'src': 'embed', 'start': 2093.045, 'weight': 3, 'content': [{'end': 2095.525, 'text': "And let's take a moment to think about why that's the case.", 'start': 2093.045, 'duration': 2.48}, {'end': 2100.427, 'text': 'You can do so just with a pile of examples and just see which ones seem to fall in the region or not.', 'start': 2096.306, 'duration': 4.121}, {'end': 2107.029, 'text': "But let's see if we can understand this in a way that lets us make progress onto the other even numbers.", 'start': 2100.767, 'duration': 6.262}, {'end': 2113.171, 'text': "So when we say that it rounds down to zero, what we're basically saying is that that ratio sits somewhere between zero and one.", 'start': 2107.729, 'duration': 5.442}, {'end': 2115.732, 'text': "And it's awkward to think of x divided by y.", 'start': 2113.972, 'duration': 1.76}, {'end': 2117.933, 'text': 'we kind of like to think of y in terms of x.', 'start': 2115.732, 'duration': 2.201}, {'end': 2123.315, 'text': 'so if I multiply everything by y which is okay to do with these inequalities because y is always positive,', 'start': 2117.933, 'duration': 5.382}], 'summary': 'Analyzing data to understand the ratio between x and y for progress onto other even numbers.', 'duration': 30.27, 'max_score': 2093.045, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/QvuQH4_05LI/pics/QvuQH4_05LI2093045.jpg'}, {'end': 2210.189, 'src': 'embed', 'start': 2180.322, 'weight': 2, 'content': [{'end': 2189.429, 'text': 'So, with that, maybe you can start to think about the harder variant, which is when is it that x, divided by y, rounds down to be two?', 'start': 2180.322, 'duration': 9.107}, {'end': 2192.632, 'text': "And again, I don't want to answer it.", 'start': 2190.47, 'duration': 2.162}, {'end': 2193.473, 'text': 'I want you to answer it.', 'start': 2192.632, 'duration': 0.841}, {'end': 2203.16, 'text': 'When is it that x, divided by y inside our unit square of points x, comma y rounds down to two?', 'start': 2194.934, 'duration': 8.226}, {'end': 2210.189, 'text': "And we've got four possible geometric regions that this could correspond to A, B, C and D.", 'start': 2203.641, 'duration': 6.548}], 'summary': 'Exploring when x/y inside unit square rounds down to 2', 'duration': 29.867, 'max_score': 2180.322, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/QvuQH4_05LI/pics/QvuQH4_05LI2180322.jpg'}], 'start': 1441.669, 'title': 'Trig application and probability problem', 'summary': 'Covers the application of the inscribed angle theorem to relate angles in trigonometry and explains a probability and ratio problem with a 50% chance of an even number in live quiz scenarios.', 'chapters': [{'end': 1521.058, 'start': 1441.669, 'title': 'Inscribed angle theorem application', 'summary': 'Discusses the application of the inscribed angle theorem to relate a single angle to twice that angle, resulting in a non-trivial relationship in trigonometry between the cosine squared and the cosine of 2 theta.', 'duration': 79.389, 'highlights': ['The application of the inscribed angle theorem to relate a single angle to twice that angle allows us to derive a non-trivial relationship in trigonometry between the cosine squared and the cosine of 2 theta.', 'The inscribed angle theorem provides a way to relate a single angle to twice that angle, leading to a relationship between the cosine squared and the cosine of 2 theta.', 'The insight into the non-trivial relationship in trigonometry between the cosine squared and the cosine of 2 theta is a beautiful aspect of the application of the inscribed angle theorem.', 'The discussion explores the multiple instances where the inscribed angle theorem suspiciously shows up, highlighting its versatile application in various contexts.']}, {'end': 2235.904, 'start': 1524.215, 'title': 'Probability and ratio problem', 'summary': 'Discusses a probability and ratio problem, explaining the concept of uniform distribution, and using geometric visualization to solve the problem, ultimately leading to a live quiz on different scenarios of rounding ratios, with a 50% probability of getting an even number.', 'duration': 711.689, 'highlights': ['The probability question involves choosing two random numbers from the range 0 through 1 and guessing the probability that the ratio of these numbers rounds down to an even number, with 1380 participants getting the answer wrong. The initial probability question involves choosing two random numbers from the range 0 through 1 and guessing the probability that the ratio of these numbers rounds down to an even number, resulting in 1380 participants getting the answer wrong.', 'The concept of uniform distribution is explained, where each point is as likely as another, and the probability is independent of where the range showed up but dependent on its size. The concept of uniform distribution is explained, emphasizing that each point is as likely as another, and the probability is independent of where the range showed up but dependent on its size.', 'Using geometric visualization by considering x and y as coordinates in a two-dimensional space to solve the problem, leading to a 50% probability of getting an even number. The approach of using geometric visualization by considering x and y as coordinates in a two-dimensional space is highlighted, ultimately leading to a 50% probability of getting an even number.', 'The method of asking a simpler variant of the problem to gain a foothold and understanding the boundaries of the regions in a geometric way to determine the rounding ratios. The method of asking a simpler variant of the problem to gain a foothold and understanding the boundaries of the regions in a geometric way to determine the rounding ratios is emphasized as a problem-solving strategy.']}], 'duration': 794.235, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/QvuQH4_05LI/pics/QvuQH4_05LI1441669.jpg', 'highlights': ['The inscribed angle theorem relates a single angle to twice that angle, deriving a non-trivial relationship in trigonometry.', 'The versatile application of the inscribed angle theorem is explored in various contexts.', 'The concept of uniform distribution is explained, emphasizing equal likelihood and size-dependent probability.', 'Geometric visualization leads to a 50% probability of getting an even number in the probability problem.']}, {'end': 3009.23, 'segs': [{'end': 2342.817, 'src': 'embed', 'start': 2313.789, 'weight': 3, 'content': [{'end': 2321.371, 'text': "It's saying that x, divided by y, is greater than or equal to two if it's rounding down to that, but it's not greater than three.", 'start': 2313.789, 'duration': 7.582}, {'end': 2322.211, 'text': "so it's less than three.", 'start': 2321.371, 'duration': 0.84}, {'end': 2328.152, 'text': "And again it's a little bit awkward to think of this ratio.", 'start': 2324.171, 'duration': 3.981}, {'end': 2335.154, 'text': "so let's write that as two times y is less than or equal to x, which is less than or equal to three times y.", 'start': 2328.152, 'duration': 7.002}, {'end': 2342.817, 'text': "Now quite often we don't think of y as a function of x, we think of x as a function of y, if that makes you feel more comfortable.", 'start': 2336.633, 'duration': 6.184}], 'summary': 'X/y >= 2, x/y < 3. express as 2y <= x < 3y, x as a function of y.', 'duration': 29.028, 'max_score': 2313.789, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/QvuQH4_05LI/pics/QvuQH4_05LI2313789.jpg'}, {'end': 2399.314, 'src': 'embed', 'start': 2374.541, 'weight': 1, 'content': [{'end': 2382.797, 'text': "so it'll be a line like this that describes part of the boundary of our region, And the other line is when y is equal to x thirds.", 'start': 2374.541, 'duration': 8.256}, {'end': 2392.087, 'text': "So we know we actually have to be above this line that I'm about to draw, where one of these represents x over two,", 'start': 2383.898, 'duration': 8.189}, {'end': 2394.549, 'text': 'and one of these represents x over three.', 'start': 2392.087, 'duration': 2.462}, {'end': 2399.314, 'text': 'And then pardon the intrusion into the space of my last inequality.', 'start': 2395.611, 'duration': 3.703}], 'summary': 'The region boundary is defined by y=x/2 and y=x/3.', 'duration': 24.773, 'max_score': 2374.541, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/QvuQH4_05LI/pics/QvuQH4_05LI2374541.jpg'}, {'end': 2512.287, 'src': 'embed', 'start': 2493.175, 'weight': 4, 'content': [{'end': 2504.744, 'text': 'X fourths, Y equals X fifths, and this little sliver of area gives us all of the times that our ratio X over Y rounds to be four.', 'start': 2493.175, 'duration': 11.569}, {'end': 2510.046, 'text': "And we're going to have to add infinitely many of these, so that gives us sort of another phase of challenge to the problem.", 'start': 2505.725, 'duration': 4.321}, {'end': 2512.287, 'text': "I've drawn this all out in Desmos.", 'start': 2510.866, 'duration': 1.421}], 'summary': 'X fourths equals x fifths, leading to ratio x over y rounds to four. infinitely many of these need to be added, presenting a challenge.', 'duration': 19.112, 'max_score': 2493.175, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/QvuQH4_05LI/pics/QvuQH4_05LI2493175.jpg'}, {'end': 2642.128, 'src': 'embed', 'start': 2610.348, 'weight': 5, 'content': [{'end': 2614.652, 'text': "This next one, what's the distance between these two points? Well, it's a fourth minus a fifth.", 'start': 2610.348, 'duration': 4.304}, {'end': 2617.054, 'text': "That's the distance between these, given how they were defined.", 'start': 2614.832, 'duration': 2.222}, {'end': 2619.597, 'text': 'So we have a fourth minus a fifth.', 'start': 2617.555, 'duration': 2.042}, {'end': 2623.301, 'text': 'And in general, we have this kind of oscillating sum.', 'start': 2621.239, 'duration': 2.062}, {'end': 2632.221, 'text': "sixth minus the seventh, where we have all the reciprocals of the natural numbers, but we're adding that up infinitely many different times.", 'start': 2624.736, 'duration': 7.485}, {'end': 2635.824, 'text': 'okay, and we want to know what that sum happens to be, and from here?', 'start': 2632.221, 'duration': 3.603}, {'end': 2642.128, 'text': 'uh, the the problem solving tip associated with this will seem a little bit strange,', 'start': 2635.824, 'duration': 6.304}], 'summary': 'Finding the distance between reciprocals of natural numbers, such as 1/4 minus 1/5, leads to an oscillating sum that needs to be calculated infinitely.', 'duration': 31.78, 'max_score': 2610.348, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/QvuQH4_05LI/pics/QvuQH4_05LI2610348.jpg'}, {'end': 2678.384, 'src': 'embed', 'start': 2654.497, 'weight': 0, 'content': [{'end': 2663.099, 'text': 'that this alternating sum one minus a half plus a third, minus a fourth plus a fifth on and on actually equals the natural log of two.', 'start': 2654.497, 'duration': 8.602}, {'end': 2667.781, 'text': 'And the way this actually came about.', 'start': 2665.64, 'duration': 2.141}, {'end': 2671.622, 'text': "it's such a weird procedure it's worth just walking through again really quickly,", 'start': 2667.781, 'duration': 3.841}, {'end': 2678.384, 'text': "because it's a bizarre thing that you're not gonna be able to just stare at this formula and then immediately see that this is how you're gonna solve it,", 'start': 2671.622, 'duration': 6.762}], 'summary': 'The alternating sum equals the natural log of two, achieved through a bizarre procedure.', 'duration': 23.887, 'max_score': 2654.497, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/QvuQH4_05LI/pics/QvuQH4_05LI2654497.jpg'}, {'end': 3014.435, 'src': 'embed', 'start': 2988.937, 'weight': 6, 'content': [{'end': 2993.86, 'text': "it's the most frustrating, but it's the most real of all the problem solving tips that there can be,", 'start': 2988.937, 'duration': 4.923}, {'end': 3001.003, 'text': "which is that true problem solving comes down to a kind of pattern recognition, and there's no two ways around it.", 'start': 2993.86, 'duration': 7.143}, {'end': 3003.985, 'text': 'You just have to do a lot of practice and expose yourself to a lot.', 'start': 3001.564, 'duration': 2.421}, {'end': 3009.23, 'text': 'Now, I would bet that when we did talk about this infinite series a couple lectures back,', 'start': 3005.386, 'duration': 3.844}, {'end': 3012.953, 'text': "you wouldn't have thought that that's a thing that you're going to be using in a probability question one day.", 'start': 3009.23, 'duration': 3.723}, {'end': 3014.435, 'text': "But that's just how these things go.", 'start': 3013.294, 'duration': 1.141}], 'summary': 'Problem solving requires pattern recognition through extensive practice and exposure to various scenarios.', 'duration': 25.498, 'max_score': 2988.937, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/QvuQH4_05LI/pics/QvuQH4_05LI2988937.jpg'}], 'start': 2236.925, 'title': 'Understanding x/y ratio and pattern recognition', 'summary': 'Discusses understanding the x/y ratio and probability, illustrating how to determine the probability of a ratio falling within certain ranges using graphical representations, and applying calculus to determine the sum of an infinite series. it also explores the significance of recognizing patterns in mathematical problem-solving and emphasizes the role of pattern recognition in grasping complex concepts, which can be cultivated through extensive reading and thinking about problems.', 'chapters': [{'end': 2770.977, 'start': 2236.925, 'title': 'Understanding x/y ratio and probability', 'summary': 'Discusses understanding the x/y ratio and probability, illustrating how to determine the probability of a ratio falling within certain ranges by using graphical representations, and concluding by applying calculus to determine the sum of an infinite series.', 'duration': 534.052, 'highlights': ['Illustrating the x/y ratio and probability using graphical representations The speaker explains using graphical representations to determine the probability of the x/y ratio falling within certain ranges, demonstrating how to visualize the regions where the ratio rounds to specific numbers.', 'Applying calculus to determine the sum of an infinite series The speaker demonstrates applying calculus to determine the sum of an infinite series, specifically illustrating the procedure of integrating a series of terms related to x to power n divided by n to find the sum.', 'Explaining the concept of x/y ratio and its relation to inequalities The speaker explains the concept of the x/y ratio and its relation to inequalities, emphasizing the process of explicitly writing out the inequality and discussing the implications of the x/y ratio.']}, {'end': 3009.23, 'start': 2771.537, 'title': 'The power of pattern recognition', 'summary': 'Discusses the significance of recognizing patterns in mathematical problem-solving, emphasizing the role of pattern recognition in grasping complex concepts and solving problems, which can be cultivated through extensive reading and thinking about problems.', 'duration': 237.693, 'highlights': ['The key to problem solving lies in pattern recognition and exposure to a wide range of patterns, which can be developed through extensive reading and deep contemplation of problems.', 'The ability to recognize geometric series and leverage the natural log in integrals is crucial in solving complex problems, as demonstrated in the evaluation of the infinite series resulting in the natural log of two.', 'The chapter emphasizes the importance of extensive reading and thinking about problems in developing the ability to recognize patterns and solve problems effectively, highlighting the role of practice and understanding the deeper principles behind problem-solving techniques.', 'Understanding the concept of pattern recognition and its role in problem-solving is presented as a fundamental and essential aspect of mathematical proficiency, emphasizing the need for extensive exposure to various patterns and pensive contemplation of solutions.']}], 'duration': 772.305, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/QvuQH4_05LI/pics/QvuQH4_05LI2236925.jpg', 'highlights': ['Applying calculus to determine the sum of an infinite series The speaker demonstrates applying calculus to determine the sum of an infinite series, specifically illustrating the procedure of integrating a series of terms related to x to power n divided by n to find the sum.', 'The ability to recognize geometric series and leverage the natural log in integrals is crucial in solving complex problems, as demonstrated in the evaluation of the infinite series resulting in the natural log of two.', 'Understanding the concept of pattern recognition and its role in problem-solving is presented as a fundamental and essential aspect of mathematical proficiency, emphasizing the need for extensive exposure to various patterns and pensive contemplation of solutions.', 'Illustrating the x/y ratio and probability using graphical representations The speaker explains using graphical representations to determine the probability of the x/y ratio falling within certain ranges, demonstrating how to visualize the regions where the ratio rounds to specific numbers.', 'The key to problem solving lies in pattern recognition and exposure to a wide range of patterns, which can be developed through extensive reading and deep contemplation of problems.', 'Explaining the concept of x/y ratio and its relation to inequalities The speaker explains the concept of the x/y ratio and its relation to inequalities, emphasizing the process of explicitly writing out the inequality and discussing the implications of the x/y ratio.', 'The chapter emphasizes the importance of extensive reading and thinking about problems in developing the ability to recognize patterns and solve problems effectively, highlighting the role of practice and understanding the deeper principles behind problem-solving techniques.']}, {'end': 3374.346, 'segs': [{'end': 3086.976, 'src': 'embed', 'start': 3057.325, 'weight': 2, 'content': [{'end': 3063.77, 'text': "At this point, when you've done the problem and you've got your nice, elegant solution and you want to draw a box around it, you're not done.", 'start': 3057.325, 'duration': 6.445}, {'end': 3070.764, 'text': 'Just always, always principle number eight here always gut, check your answer, okay?,', 'start': 3065.1, 'duration': 5.664}, {'end': 3073.626, 'text': "Because there's gonna be some little mistake that happens all the time.", 'start': 3070.784, 'duration': 2.842}, {'end': 3076.389, 'text': "The great problem solvers aren't the ones who just never make little mistakes.", 'start': 3073.686, 'duration': 2.703}, {'end': 3079.471, 'text': "They're the ones who have some way of recognizing what those mistakes are.", 'start': 3076.749, 'duration': 2.722}, {'end': 3086.976, 'text': "So in this context, let's say I, I just wanna see numerically what my answer turns out to be right?", 'start': 3080.372, 'duration': 6.604}], 'summary': 'Always check your answer for little mistakes. great problem solvers recognize mistakes.', 'duration': 29.651, 'max_score': 3057.325, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/QvuQH4_05LI/pics/QvuQH4_05LI3057325.jpg'}, {'end': 3166.115, 'src': 'embed', 'start': 3140.406, 'weight': 3, 'content': [{'end': 3145.267, 'text': "The way to do it is to be able to systematically know when you've made them.", 'start': 3140.406, 'duration': 4.861}, {'end': 3150.069, 'text': 'So always gut check your answer, have like two different perspectives that can give you a reasonability check.', 'start': 3146.108, 'duration': 3.961}, {'end': 3155.751, 'text': 'In this context if it inspired us to go and look a little bit more carefully at how we were applying things.', 'start': 3150.429, 'duration': 5.322}, {'end': 3159.733, 'text': "this sum isn't quite the alternating sum that converges to natural log of two.", 'start': 3155.751, 'duration': 3.982}, {'end': 3166.115, 'text': "In particular, we added the one, but then we also add one half, and it's only after that that we start alternating.", 'start': 3160.632, 'duration': 5.483}], 'summary': 'Systematically check answers, consider multiple perspectives to ensure reasonability, and reevaluate application to achieve accurate results.', 'duration': 25.709, 'max_score': 3140.406, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/QvuQH4_05LI/pics/QvuQH4_05LI3140406.jpg'}, {'end': 3231.809, 'src': 'embed', 'start': 3197.6, 'weight': 0, 'content': [{'end': 3201.222, 'text': 'that is the thing that equals the natural log of two,', 'start': 3197.6, 'duration': 3.622}, {'end': 3207.204, 'text': 'which in turn implies that that remainder of the sum looks like one minus the natural log of two.', 'start': 3201.222, 'duration': 5.982}, {'end': 3210.711, 'text': 'Okay, so all of that.', 'start': 3208.53, 'duration': 2.181}, {'end': 3211.371, 'text': 'what does that tell us??', 'start': 3210.711, 'duration': 0.66}, {'end': 3218.134, 'text': "When we plug it into our original expression, it's saying that the actual answer should not be 1, half the natural log of 2..", 'start': 3211.751, 'duration': 6.383}, {'end': 3220.695, 'text': "That didn't even pass our basic reasonability test.", 'start': 3218.134, 'duration': 2.561}, {'end': 3231.809, 'text': "Instead, it'll be 1 half of 1 plus 1 minus the natural log of 2, which is just 2 minus the natural log of 2.", 'start': 3221.495, 'duration': 10.314}], 'summary': 'The sum equals 2 minus the natural log of 2.', 'duration': 34.209, 'max_score': 3197.6, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/QvuQH4_05LI/pics/QvuQH4_05LI3197600.jpg'}, {'end': 3296.965, 'src': 'embed', 'start': 3268.843, 'weight': 4, 'content': [{'end': 3277.007, 'text': "know we've got one wedge here that's covering 0.5, and then this other one covers, well, uh, about one sixth, half of a sixth,", 'start': 3268.843, 'duration': 8.164}, {'end': 3281.129, 'text': "basically because this length was a half minus a third, which makes it a sixth, and then it's a triangle.", 'start': 3277.007, 'duration': 4.122}, {'end': 3283.13, 'text': 'so one half base times height.', 'start': 3281.129, 'duration': 2.001}, {'end': 3284.43, 'text': "so that's about a 12 or 0.083.", 'start': 3283.13, 'duration': 1.3}, {'end': 3287.552, 'text': 'and then the rest of it.', 'start': 3284.43, 'duration': 3.122}, {'end': 3289.133, 'text': "you know it's not going to fill a huge amount.", 'start': 3287.552, 'duration': 1.581}, {'end': 3290.293, 'text': 'something around 0.65 seems pretty reasonable.', 'start': 3289.133, 'duration': 1.16}, {'end': 3296.965, 'text': 'And so that you know we could call that good on our gut, check that this is probably the correct answer.', 'start': 3292.782, 'duration': 4.183}], 'summary': 'The wedge covers 0.5 and one sixth, resulting in a total area of 0.65, which seems reasonable for a triangle.', 'duration': 28.122, 'max_score': 3268.843, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/QvuQH4_05LI/pics/QvuQH4_05LI3268843.jpg'}], 'start': 3009.23, 'title': 'Probability and problem solving', 'summary': 'Delves into the application of alternating sums in probability, yielding the answer as 1/2 of the natural log of two, and highlights the significance of error recognition and rectification in problem-solving.', 'chapters': [{'end': 3040.934, 'start': 3009.23, 'title': 'Probability and alternating sums', 'summary': 'Explores the unexpected use of alternating sums in probability questions, leading to the answer being one half of the natural log of two, with a clean and clear expression derived from adding up areas.', 'duration': 31.704, 'highlights': ['The answer to the final question is one half of the natural log of two, derived from the alternating sum, showcasing unexpected application in probability (1)', 'The expression involves a very clean and clear natural log expression, showcasing the elegance of the solution (2)', 'The alternating sum unexpectedly appears in a probability question, highlighting its unforeseen relevance and application (3)']}, {'end': 3374.346, 'start': 3043.536, 'title': 'Effective problem solving techniques', 'summary': 'Emphasizes the importance of systematically checking for mistakes in problem-solving, illustrating with a probability problem and highlighting the significance of recognizing and rectifying errors to achieve precision.', 'duration': 330.81, 'highlights': ['The significance of systematically checking for mistakes in problem-solving is emphasized, with an illustration using a probability problem, to achieve precision and perfection.', 'The importance of recognizing and rectifying errors to approach perfection in problem-solving is highlighted, emphasizing the need for systematic error checks and reasonability tests.', 'Illustrating the process of recognizing errors in a probability problem, emphasizing the significance of reasonability tests and numerical verification to ensure precision in problem-solving.', 'The analogy of intentionally making mistakes to test understanding, and the importance of recognizing and rectifying errors in problem-solving are highlighted, emphasizing the significance of systematic error checks.', "The speaker's personal anecdotes about intentional mistakes and humorous remarks about filming are mentioned, providing lighthearted commentary amidst the technical discussion."]}], 'duration': 365.116, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/QvuQH4_05LI/pics/QvuQH4_05LI3009230.jpg', 'highlights': ['The answer to the final question is one half of the natural log of two, derived from the alternating sum, showcasing unexpected application in probability (1)', 'The expression involves a very clean and clear natural log expression, showcasing the elegance of the solution (2)', 'The alternating sum unexpectedly appears in a probability question, highlighting its unforeseen relevance and application (3)', 'The significance of systematically checking for mistakes in problem-solving is emphasized, with an illustration using a probability problem, to achieve precision and perfection.', 'The importance of recognizing and rectifying errors to approach perfection in problem-solving is highlighted, emphasizing the need for systematic error checks and reasonability tests.', 'Illustrating the process of recognizing errors in a probability problem, emphasizing the significance of reasonability tests and numerical verification to ensure precision in problem-solving.', 'The analogy of intentionally making mistakes to test understanding, and the importance of recognizing and rectifying errors in problem-solving are highlighted, emphasizing the significance of systematic error checks.', "The speaker's personal anecdotes about intentional mistakes and humorous remarks about filming are mentioned, providing lighthearted commentary amidst the technical discussion."]}, {'end': 4032.843, 'segs': [{'end': 3902.661, 'src': 'heatmap', 'start': 3840.743, 'weight': 1, 'content': [{'end': 3849.569, 'text': "shared with me the fact that they were doing like an art contest among some students and And I just wanted to showcase some of I'm not sure if these are the winners or the finalists of the art contest,", 'start': 3840.743, 'duration': 8.826}, {'end': 3857.215, 'text': 'but basically in various different categories of students who are, I think it was like 12 through 14, 15 through 17, or something like that,', 'start': 3849.569, 'duration': 7.646}, {'end': 3860.537, 'text': 'using like mathematical graphs to try to draw pictures, okay?', 'start': 3857.215, 'duration': 3.322}, {'end': 3868.163, 'text': "So keep in mind, what I'm about to show you are mathematical graphs that someone wrote just with an analytic description,", 'start': 3860.577, 'duration': 7.586}, {'end': 3871.866, 'text': 'and they were just prompted to create something artistic from that.', 'start': 3868.163, 'duration': 3.703}, {'end': 3880.071, 'text': 'Okay, so one of my favorites and I think this one was from someone named Carrie is two giraffes.', 'start': 3872.768, 'duration': 7.303}, {'end': 3882.892, 'text': "that just from an artistic standpoint it's actually quite lovely.", 'start': 3880.071, 'duration': 2.821}, {'end': 3887.174, 'text': 'And then to think through like, actually mathematically describing everything involved here.', 'start': 3883.692, 'duration': 3.482}, {'end': 3894.336, 'text': "it's such a beautiful blend of well like the creative side of things, the artistic side with aesthetics and everything, and the analytic side.", 'start': 3887.174, 'duration': 7.162}, {'end': 3897.738, 'text': 'Another that was, this is just genuinely insane.', 'start': 3895.577, 'duration': 2.161}, {'end': 3900.419, 'text': 'This is by Cazzini.', 'start': 3897.758, 'duration': 2.661}, {'end': 3902.661, 'text': 'is a Bézier night.', 'start': 3901.179, 'duration': 1.482}], 'summary': 'Art contest featured students using mathematical graphs to create artistic images, including two giraffes and a bézier night.', 'duration': 61.918, 'max_score': 3840.743, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/QvuQH4_05LI/pics/QvuQH4_05LI3840743.jpg'}, {'end': 3921.9, 'src': 'embed', 'start': 3880.071, 'weight': 0, 'content': [{'end': 3882.892, 'text': "that just from an artistic standpoint it's actually quite lovely.", 'start': 3880.071, 'duration': 2.821}, {'end': 3887.174, 'text': 'And then to think through like, actually mathematically describing everything involved here.', 'start': 3883.692, 'duration': 3.482}, {'end': 3894.336, 'text': "it's such a beautiful blend of well like the creative side of things, the artistic side with aesthetics and everything, and the analytic side.", 'start': 3887.174, 'duration': 7.162}, {'end': 3897.738, 'text': 'Another that was, this is just genuinely insane.', 'start': 3895.577, 'duration': 2.161}, {'end': 3900.419, 'text': 'This is by Cazzini.', 'start': 3897.758, 'duration': 2.661}, {'end': 3902.661, 'text': 'is a Bézier night.', 'start': 3901.179, 'duration': 1.482}, {'end': 3907.125, 'text': 'So I definitely wanted to highlight this on behalf of the team over at Desmos,', 'start': 3902.921, 'duration': 4.204}, {'end': 3912.13, 'text': "where each one of the curves here is described according to something that's called a Bézier curve.", 'start': 3907.125, 'duration': 5.005}, {'end': 3913.772, 'text': 'Very useful for computer graphics.', 'start': 3912.37, 'duration': 1.402}, {'end': 3916.835, 'text': "It's a kind of cubic parametric term.", 'start': 3914.032, 'duration': 2.803}, {'end': 3921.9, 'text': "And they just recreated a starry night in a way that's completely beautiful, I think.", 'start': 3917.295, 'duration': 4.605}], 'summary': 'Desmos team created a beautiful bézier night using mathematical curves for computer graphics.', 'duration': 41.829, 'max_score': 3880.071, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/QvuQH4_05LI/pics/QvuQH4_05LI3880071.jpg'}, {'end': 3981.79, 'src': 'embed', 'start': 3940.551, 'weight': 2, 'content': [{'end': 3947.715, 'text': "so the the amount that things have changed in terms of when someone's noodling off with their graphing calculator in class and what they can do, uh,", 'start': 3940.551, 'duration': 7.164}, {'end': 3948.436, 'text': 'truly next level.', 'start': 3947.715, 'duration': 0.721}, {'end': 3954.482, 'text': 'And then, at the very end, I just want to say again a highlighted thank you to Ben Eater and to Cam,', 'start': 3949.216, 'duration': 5.266}, {'end': 3959.828, 'text': "who've been extremely helpful with the whole series in ways that's like hard to even articulate properly.", 'start': 3954.482, 'duration': 5.346}, {'end': 3961.65, 'text': 'Eater in particular.', 'start': 3960.669, 'duration': 0.981}, {'end': 3966.936, 'text': "I mean he's let me borrow a lot of his equipment and helped out with like figuring out live footage type stuff,", 'start': 3961.65, 'duration': 5.286}, {'end': 3968.258, 'text': "because that's not something I usually do.", 'start': 3966.936, 'duration': 1.322}, {'end': 3972.522, 'text': 'which is not even to mention the work on the live stats and live quizzes.', 'start': 3969.058, 'duration': 3.464}, {'end': 3977.066, 'text': "so if you aren't already familiar with his channel, it's simply named Ben Eder.", 'start': 3972.522, 'duration': 4.544}, {'end': 3980.89, 'text': 'like 100%, check it out, definitely subscribe to it.', 'start': 3977.066, 'duration': 3.824}, {'end': 3981.79, 'text': 'try some of the projects.', 'start': 3980.89, 'duration': 0.9}], 'summary': 'The advancements in graphing calculators have been remarkable, with significant contributions from ben eater and cam.', 'duration': 41.239, 'max_score': 3940.551, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/QvuQH4_05LI/pics/QvuQH4_05LI3940551.jpg'}], 'start': 3374.366, 'title': 'Mathematics course and python for probability analysis', 'summary': 'Discusses launching a full course on combinatorics and the importance of learning programming, and explains using python for probability analysis, including generating random numbers, calculating ratios, and empirical verification with practical examples using numpy and matplotlib.', 'chapters': [{'end': 3500.947, 'start': 3374.366, 'title': 'Mathematics course and combinatorics', 'summary': 'Discusses the possibility of launching a full course on a separate channel, with a focus on combinatorics, derangements, inclusion and exclusion principles, and the relation of e to counting problems, as well as the importance of learning programming for mathematical problem solving.', 'duration': 126.581, 'highlights': ['The possibility of launching a full course on a separate channel with a focus on combinatorics, derangements, and inclusion and exclusion principles. The speaker expresses interest in offering a full course on a separate channel, potentially focusing on combinatorics, derangements, and inclusion and exclusion principles.', 'The relation of E to counting problems and the importance of learning programming for mathematical problem solving. E is related to counting problems, and learning programming is essential for mathematical problem solving.', 'The importance of coming at problems from two different angles, mathematically and computationally, to enhance problem solving. Approaching problems from both mathematical and computational perspectives can lead to better problem solving by providing different insights.']}, {'end': 4032.843, 'start': 3501.547, 'title': 'Using python for probability analysis', 'summary': 'Explains the use of python to generate random numbers, calculate ratios, empirical verification of mathematical concepts, and data visualization, with a practical example using numpy library and matplotlib for numerical simulations and empirical validation.', 'duration': 531.296, 'highlights': ['Using numpy library to generate random numbers and calculating ratios. Demonstrates the use of numpy library to generate random numbers and calculate ratios, showcasing the practical example of using Python for probability analysis.', 'Empirical verification of mathematical concepts through Python simulations. Illustrates the process of empirically verifying mathematical concepts through Python simulations, including the use of floor function and mean calculations for proportion analysis.', 'Data visualization using matplotlib for empirical validation. Shows the use of matplotlib for data visualization, specifically creating histograms to visually represent the empirical validation of mathematical concepts.', 'Recognition of mathematical art created using Desmos. Acknowledges the artistic and analytic blend in mathematical art created by students using Desmos, highlighting specific artworks and the significance of mathematical graphing.', 'Expression of gratitude to contributors and announcement of upcoming content. Expresses gratitude towards contributors and announces future content, including problem-solving topics and the release of homework challenges associated with the series.']}], 'duration': 658.477, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/QvuQH4_05LI/pics/QvuQH4_05LI3374366.jpg', 'highlights': ['The relation of E to counting problems and the importance of learning programming for mathematical problem solving.', 'The possibility of launching a full course on a separate channel with a focus on combinatorics, derangements, and inclusion and exclusion principles.', 'Using numpy library to generate random numbers and calculating ratios.', 'Empirical verification of mathematical concepts through Python simulations.', 'The importance of coming at problems from two different angles, mathematically and computationally, to enhance problem solving.']}], 'highlights': ['The chapter emphasizes the importance of extensive reading and thinking about problems in developing the ability to recognize patterns and solve problems effectively, highlighting the role of practice and understanding the deeper principles behind problem-solving techniques.', 'The speaker plans to intentionally make a mistake to encourage skepticism and engagement from the audience.', 'The inscribed angle theorem relates a single angle to twice that angle, deriving a non-trivial relationship in trigonometry.', 'The chapter discusses the probability of the ratio of two random numbers between 0 and 1 rounding down to an even number. The chapter introduces the problem of determining the probability that the ratio of two random numbers between 0 and 1 rounds down to an even number.', 'The chapter emphasizes the geometric representation of cosine squared as a length, a portion of the hypotenuse of a right triangle, providing a clever proof of the Pythagorean theorem.', 'The inscribed angle theorem is mentioned, highlighting the significance of considering alternative approaches and principles in problem-solving, particularly for audience members familiar with advanced concepts.', 'The properties of isosceles triangles are leveraged to demonstrate symmetry in the given geometry problem, showing that the angles alpha and beta are equal within the triangles, which creates a basis for further deductions.', 'The concept of uniform distribution is explained, emphasizing equal likelihood and size-dependent probability.', 'The ability to recognize geometric series and leverage the natural log in integrals is crucial in solving complex problems, as demonstrated in the evaluation of the infinite series resulting in the natural log of two.', 'The relation of E to counting problems and the importance of learning programming for mathematical problem solving.']}