WEBVTT mathematics/geometry/pyo 00:00:00.000 --> 00:00:01.500 Welcome back to Educator.com. 00:00:01.500 --> 00:00:08.000 This next lesson is on inductive reasoning. 00:00:08.000 --> 00:00:17.200 For inductive reasoning, we deal with what is called conjectures; a conjecture is an educated guess. 00:00:17.200 --> 00:00:35.800 When you look at several different situations, or maybe previous experiences, to come up with a final conclusion, then that would be inductive reasoning. 00:00:35.800 --> 00:00:42.500 When you have repeated observations, or you look at patterns, those things would be considered inductive reasoning. 00:00:42.500 --> 00:00:56.900 Basically, you are just looking at past experiences--anything that will lead you to some sort of conclusion is inductive reasoning. 00:00:56.900 --> 00:01:06.900 Looking at patterns: if I have 4, 8, 16, 32, and I need to use inductive reasoning 00:01:06.900 --> 00:01:13.300 to find the next several terms in the sequence, well, I can just see how these numbers came about, 00:01:13.300 --> 00:01:19.600 and then I can just apply the same rule to find the next few numbers. 00:01:19.600 --> 00:01:30.700 So, for this, 4, 8, 16, 32...how did you get from 4 to 8, 8 to 16, and 16 to 32? 00:01:30.700 --> 00:01:42.900 Well, it looks like this was multiplied by 2; that means I would have to multiply by 2 to get my next answer. 00:01:42.900 --> 00:02:01.000 So, if I multiply this number by 2, then I am going to get 64; if I multiply this by 2, then I am going to get 128; and so on. 00:02:01.000 --> 00:02:14.600 Now, for this next one, I have to see...well, here is a triangle, a square, triangle, triangle, square, square...what will be my next shape? 00:02:14.600 --> 00:02:34.800 Well, it went from 1 triangle, 1 square, to 2 triangles, 2 squares; so my conjecture will be that it will be 3 triangles, and then 3 squares. 00:02:34.800 --> 00:02:48.900 Again, we are looking at patterns, or looking at some kind of repeated behavior, to come up with a conclusion (or what will happen next). 00:02:48.900 --> 00:03:04.600 Now, a few more problems: if we make a conjecture about this, if AB = CD and CD = EF, what can I conclude? 00:03:04.600 --> 00:03:18.400 Well, if AB is equal to CD, if I draw AB, here is AB; and here is CD; 00:03:18.400 --> 00:03:37.500 so, if I see that this and this are the same, CD = EF, so this equals this; isn't it true that, if this equals this and this equals this... 00:03:37.500 --> 00:03:49.600 doesn't that mean that AB will equal EF?--so that will be my conjecture: AB = EF. 00:03:49.600 --> 00:04:15.700 Make a conjecture, given points A, B, and C: let me just draw out a coordinate plane. 00:04:15.700 --> 00:04:31.900 A is (-1,0); it is right here; B is (0,2); it is right there; C is (1,4), which is right there. 00:04:31.900 --> 00:04:59.600 If I look at this, A, B, and C line up; so my conjecture would be that points A, B, and C are collinear, because they are on the same line. 00:04:59.600 --> 00:05:06.200 Counter-examples: just because you come up with a conjecture, you come up with some kind of conclusion, 00:05:06.200 --> 00:05:13.300 based on what you see in your observations, based on the patterns, and so on, doesn't mean that it is going to be true. 00:05:13.300 --> 00:05:22.400 Just because something happens 3, 4, 5, or 6 times in a row doesn't mean that it is going to happen again the next time. 00:05:22.400 --> 00:05:31.000 Conjectures are not always true; and to prove that it is not always true, you have to provide a counter-example. 00:05:31.000 --> 00:05:35.900 And a counter-example is the opposite of what you are trying to prove. 00:05:35.900 --> 00:05:42.700 If you are trying to prove that something is true--let's say you saw something a few times, 00:05:42.700 --> 00:05:49.500 and so you conclude--you make a conjecture--that the next time, it is going to happen again; 00:05:49.500 --> 00:05:54.000 you can't prove that it is going to happen again just by showing that it happened. 00:05:54.000 --> 00:06:03.900 You cannot prove something just by giving an example, because it might not happen the following time. 00:06:03.900 --> 00:06:12.700 You might not be able to find a counter-example in order to prove that that is not true. 00:06:12.700 --> 00:06:16.000 A counter-example is the opposite of what you are trying to prove. 00:06:16.000 --> 00:06:24.900 Let's say, for example, that the first five cars you see today are black; does that mean that all cars are black? 00:06:24.900 --> 00:06:32.100 That would be a conjecture; the conjecture would be that, since you saw five cars that are black... 00:06:32.100 --> 00:06:43.100 my conjecture would be that the next car that I see will be black; and that might not be true. 00:06:43.100 --> 00:06:53.400 In order to prove that whatever you concluded, your conjecture, is not true, you are going to provide a counter-example. 00:06:53.400 --> 00:06:58.400 A counter-example would be to show an example of it not being true. 00:06:58.400 --> 00:07:07.300 Now, let's go over a few examples of this: the first one: Any three points will form a triangle. 00:07:07.300 --> 00:07:19.800 If I have three points like this, I know that it is going to form a triangle. 00:07:19.800 --> 00:07:31.300 Is this conjecture true? It could be true, but just because I gave an example of it being true does not make this conjecture true, 00:07:31.300 --> 00:07:37.900 because I know that three points will not always form a triangle. 00:07:37.900 --> 00:07:46.300 And so, what I can do to prove that this is not true--to prove that it is false: I can give an example of when this is not true. 00:07:46.300 --> 00:08:01.500 And that would be a counter-example: so three points that do not form a triangle...there are three points; they don't form a triangle. 00:08:01.500 --> 00:08:13.200 This is my counter-example: by giving an example of when this is not true, I am proving my conjecture false. 00:08:13.200 --> 00:08:22.100 So, this conjecture...sometimes it could be true, but it is not always true. 00:08:22.100 --> 00:08:32.300 By showing an example of when it is not true, a counter-example--that is when you are proving the conjecture false. 00:08:32.300 --> 00:08:39.000 The square of any number is greater than the original number: well, that could be true, but it is not always true. 00:08:39.000 --> 00:08:43.500 To show that it is not always true, I need to provide a counter-example. 00:08:43.500 --> 00:08:51.000 Let's say I have some numbers: let's say 2--if I square it...I am saying the square of any number, 00:08:51.000 --> 00:08:56.600 so if I take a number, and I square it, then it is going to be greater than this original number. 00:08:56.600 --> 00:09:05.900 If I square this, then it is going to be 4; well, this is greater than this number, the original number. 00:09:05.900 --> 00:09:13.200 What if I have 0? If I square it, what do I get? 0. 00:09:13.200 --> 00:09:23.700 If I have, let's say, 1/2, and I square this, I get 1/4. 00:09:23.700 --> 00:09:31.500 Well, is 1/4 greater than the original number, 1/2? No, 1/4 is smaller than 1/2. 00:09:31.500 --> 00:09:41.800 So, for this example, this is true; but this one is not true; this is false, and this is false, for this being greater than the original number. 00:09:41.800 --> 00:09:54.000 0 is not greater than 0, and 1/4 is not greater than 1/2. 00:09:54.000 --> 00:10:02.600 Here is a counter-example, and here is a counter-example, because these two examples show that, 00:10:02.600 --> 00:10:10.500 if you square the number, then the answer is not going to be greater than the original number. 00:10:10.500 --> 00:10:23.000 Counter-example, counter-example: by showing the counter-examples, I am proving that this conjecture is false. 00:10:23.000 --> 00:10:37.500 Find the pattern and the next two terms in the sequence: 15 to 12, 9 to 6...the pattern here that I see is subtracting 3. 00:10:37.500 --> 00:10:54.700 So, from here to here, I subtract 3; subtract 3; subtract 3; if you subtract 3 again, then you get 3; you get 0, -3, and so on. 00:10:54.700 --> 00:11:06.800 This one: 1 to 2, 2 to 6, 6 to 24, and so on--this one seems a little tricky, but you just have to look at it. 00:11:06.800 --> 00:11:23.300 Here, if I...let's see...1 to 2: I can either add 1; I can multiply by 2; here I can add 4, or I can multiply by 3; 00:11:23.300 --> 00:11:30.800 here I can add something bigger...for this one, it doesn't seem like that would be the pattern; 00:11:30.800 --> 00:11:45.800 so here it is multiplied by 4; here, multiply by 5; so then, for the next number, I can multiply by 6. 00:11:45.800 --> 00:12:01.400 If I multiply this by 6, then I will get...let's see: 120 times 6 is 0, 12, 720. 00:12:01.400 --> 00:12:08.300 And the next one...you can just multiply by 7. 00:12:08.300 --> 00:12:16.300 This one right here: I have a square; then I have this shape in there; and then I have another one. 00:12:16.300 --> 00:12:30.400 The next pattern will be...so then, here is the next step; that is up to there. 00:12:30.400 --> 00:12:37.700 And then, my next one will be to draw a square within that square, like that. 00:12:37.700 --> 00:12:48.700 And the next one would be to do the same thing; and you are going to draw another square inside. 00:12:48.700 --> 00:12:56.400 OK, so you are given a statement, and you need to come up with a conjecture. 00:12:56.400 --> 00:13:00.800 The given statement is that angle 1 and angle 2 are adjacent. 00:13:00.800 --> 00:13:07.200 "Adjacent" means that the angles share a side and a vertex. 00:13:07.200 --> 00:13:23.000 So, if I draw angles 1 and 2, they are adjacent; then what can I conclude? 00:13:23.000 --> 00:13:36.700 Well, I conclude that angles 1 and 2 (since I know that "adjacent" means that they are next to each other) share a side; 00:13:36.700 --> 00:14:02.400 so, angles 1 and 2 share a side and a vertex, because they are adjacent. 00:14:02.400 --> 00:14:15.400 The next one: the given statement: line m (here is line m, and this is a line) is an angle bisector of angle ABC. 00:14:15.400 --> 00:14:27.200 Here is an angle, ABC; and line m is an angle bisector--"bisector" means that it cuts in half. 00:14:27.200 --> 00:14:41.100 So, this line cuts this angle ABC in half; that means that these two parts right here are the same--they are congruent. 00:14:41.100 --> 00:14:46.800 If I label this angle 1 and this angle 2, what can I conclude? 00:14:46.800 --> 00:14:55.500 If this line is an angle bisector, then these two parts right here, angle 1 and angle 2, I can say are congruent. 00:14:55.500 --> 00:15:02.100 Or I could say that the measure of angle 1 equals the measure of angle 2. 00:15:02.100 --> 00:15:09.500 Or I can say that angle 1 is congruent to angle 2. 00:15:09.500 --> 00:15:18.500 That would be my conjector, since line m is an angle bisector of angle ABC. 00:15:18.500 --> 00:15:32.300 The next example: Decide if each conjecture is true or false; if true, then explain why; and if it is false, then we have to give a counter-example. 00:15:32.300 --> 00:15:46.700 Given that WX = XY, the conjecture is that W, X, and Y are collinear points. 00:15:46.700 --> 00:15:59.700 The conjecture is saying that if I have WX...here is point W, point X, and point Y...W, X, and Y are collinear. 00:15:59.700 --> 00:16:08.100 WX = XY; the conjecture is that these are collinear; is that always true? 00:16:08.100 --> 00:16:11.400 Can you think of an example of when it is not true? 00:16:11.400 --> 00:16:36.800 Well, what if you have WX, XY...WX is equal to XY; this still applies here, but it doesn't prove that the conjecture is true. 00:16:36.800 --> 00:16:45.500 This is a counter-example, because this is an example of when my conjecture is false. 00:16:45.500 --> 00:16:52.000 This one I know...this conjecture is false. 00:16:52.000 --> 00:16:59.600 The next one: Given that x is an integer, the conjecture is that -x is negative. 00:16:59.600 --> 00:17:13.600 So, if x is an integer (some integers are 2, 0, and -2; so these are x), the conjecture is that -x is negative. 00:17:13.600 --> 00:17:24.700 If I make this negative, -2 is -x--is it negative?--these would be -x. 00:17:24.700 --> 00:17:35.300 -2: this is true; this is an example of the conjecture; what about 0? 00:17:35.300 --> 00:17:47.400 If I make it negative x, then that will be -0, which is just 0; and that is not a negative, so this is false. 00:17:47.400 --> 00:17:56.200 And then here, -2: if I make it -x, then it is -(-2); well, that is a positive 2. 00:17:56.200 --> 00:18:03.300 So, does that show that -x is negative? No, because this is positive 2. 00:18:03.300 --> 00:18:17.200 So, in this case, this one is false; if x is an integer (these numbers right here), and you make those integers negative, 00:18:17.200 --> 00:18:22.900 then the answer is going to be negative: in this case, this works. 00:18:22.900 --> 00:18:32.100 If you make this number negative, it is not a negative; if you take the negative of this number, it becomes positive. 00:18:32.100 --> 00:18:42.300 So, -x is positive in this case; so this one is also false, and then here is my counter-example, right here. 00:18:42.300 --> 00:18:54.900 This one and this one are both counter-examples, because they show that this is not true; it is false. 00:18:54.900 --> 00:19:00.000 Well, that is it for this lesson; thank you for watching Educator.com--see you next time!