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Int Algebra: Inductive Reasoning

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About this Lesson

  • Type: Video Tutorial
  • Length: 7:49
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 84 MB
  • Posted: 06/26/2009

This lesson is part of the following series:

Int Algebra: Conditional Statements (2 lessons, $2.97)

Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, Intermediate Algebra. This course and others are available from Thinkwell, Inc. The full course can be found athttp://www.thinkwell.com/student/product/intermediatealgebra. The full course covers real numbers, equations and inequalities, exponents and polynomials, rational expressions, roots and radicals, relations and functions, the straight line, systems of equations, quadratic equations and quadratic inequalities, conic sections, inverse and exponential and logarithmic functions, and a variety of other AP algebra and advanced algebra.

Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

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One of the most important things in mathematics, one of the most important lessons that mathematics can really teach us is that of searching for patterns. This really is the fundamental strength of mathematical thinking. The idea that we don't think of the world as in a vacuum, but instead we're thinking about things within context and looking for how things follow, how they fit in, and then once we have a firm understanding of that, trying to then predict what's going to happen next and looking for patterns. This is known in mathematics as "Inductive Reasoning." The idea of looking at evidence and trying to use that evidence to figure out what a pattern is, what will come next, what will happen in the future, predicting what's going to happen, and this is such a powerful tool and this is something that really mathematics can offer us if we embrace it even in our own daily lives. Forget about algebra, which is maybe really easy to do. Take this idea that algebra teaches us and use it no matter what you decide to do.
Let me try to actually show you the idea of inductive reasoning, the idea of searching for patterns with some specific simple examples so you can see this is in action. For example, here's a very, very simple example. I'm going to give you a collection of numbers, a sequence of numbers, and then I want you to see if together we can figure out a pattern for what's producing the sequence. Again, this is not going to be too profound, but let's just start. Here's a sequence of numbers.
Can we see what the pattern is, what the next few terms are going to be? Well, you take a look. Well, the answer is I don't know for sure. Maybe the next one's going to be 15. I don't know. However, if I look at this and look for a pattern, I see 1, 2, 3, 4. It seems that a pattern emerging is that I continually take the previous number and add 1. These are the counting numbers in succession. You're guess would be that the next few terms might be 5, 6, 7, and so forth.
That is inductive reasoning, looking at the evidence and then trying to use that to find the pattern and figure out what's going to happen next. Imagine the power of this in your everyday life if you can really harness this. The reality actually is that math is much easier than life, so it's easy to figure out the pattern here. It's hard to figure out if the person that you're going out with is going to go out with you in a month. That's harder to figure out but if you're really good at inductive reasoning you would know that. Really, stay tuned because all the answers are going to be given.
This is an example. Here's a sequence, 5, 10, 15, 20. Can we see a pattern here? Well, there's a lot of ways of thinking about the pattern and they're all coming out of inductive reasoning. One is to notice that these are just the consecutive multiples of 5, 5 x 1, 5 x 2, 5 x 3, 5 x 4, 5, 10, 15, 20. A good guess might be 25 for the next term, then 30, then 35, and so on, but there are other ways of seeing patterns. You look at this and notice that there's another pattern going on here. Notice that I see a pattern that the last digit is always alternating between 5 and 0. That's a pattern. And then you'll notice that I just have all the counting numbers appearing twice, 1 and then 1, 2 and then 2, 3 and then 3. That's another pattern and if you use that pattern, you would then predict just as you would of with the multiples of 5, the next one would be 4,0. The next one would be 4,5 and so on. There's another pattern that you can see. We're trying to look for any kind of pattern and all patterns. That's what inductive reasoning is all about.
Let's try another one. Here's one, 1, 1, 2, 3, 5. Now this pattern's actually a little bit more tricky. 1, 1, 2, 3, 5 and now what would be the next term here? Well, this is a little tricky to see but let's think about it and see if we can take these terms and figure out how we get to the next one. For example, you'll notice the 2. Well, I can think of 2 in a variety of different ways. One way of thinking of 2 is I just take these two numbers and add them up and I got 2 but is that always true in that pattern? Let's see. Well, here's 3. If I take these two numbers and add them up I do indeed get 3. If I take these two numbers and add them up, I do get 5. The pattern seems to be emerging. That pattern being that I take the two previous people, add them up and that gives me the next person. If that were the case, this next term would then be an 8, followed by a 13, followed by a 21, and so forth. In fact, this sequence of numbers is called the "Fibonacci Numbers." It's a very important sequence.
How about one that's a little more geometric? Now I'm just going to create something here. It's a sequence and remember, we're searching for a pattern. Patterns don't always have to be numbers. In fact, in our lives patterns are not usually numbers. Here's just a pattern of things. Can we look at this and see if we can guess what the next thing should be? Well, if you take a look at this you see here there's a little bit of blob here on the right, then the blob is now up, then the blob is now on the left, then the blob is on the bottom. What would be a good guess as to where the blob's going to be next? Well, it seems as though all I'm doing is taking the square with the blob and turning it 90 degrees around in a counter clockwise fashion. If that were so, a good guess for the next term would be to have the blob back here, and the next term would be to have the blob up here, and so forth.
How about one really abstract example to close up the show, 2x + 1, 4x + 1, 6x + 1, and that's all I give you. Can we find a pattern? Well, now this is harder because now it's not just numbers. There's numbers and variables and stuff but again, we're searching for a pattern. That's what's at the heart of what we're doing here in inductive reasoning. How do I go from here to here? It looks like the only thing I did here was add on another 2x to this. 2x + 1 and if I add 2 more x's I have 4x + 1. If I take 4x + 1 and add on 2 x's I get 6x + 1. If that's really the pattern, a good guess would be what? Well, I add 2 more x's, 8x + 1. The next one would be 10x + 1, then 12x + 1 and so on. That would be a good guess.
Inductive reasoning is a really powerful way, not only looking for patterns in mathematics, but looking for patterns in our own lives.
I'll see you at the next lecture.
The Real Numbers
Conditional Statements
Inductive Reasoning Page [2 of 2]

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