WEBVTT mathematics/geometry/pyo
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Welcome back to Educator.com.
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This next lesson is on inductive reasoning.
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For inductive reasoning, we deal with what is called conjectures; a **conjecture** is an educated guess.
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When you look at several different situations, or maybe previous experiences, to come up with a final conclusion, then that would be inductive reasoning.
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When you have repeated observations, or you look at patterns, those things would be considered inductive reasoning.
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Basically, you are just looking at past experiences--anything that will lead you to some sort of conclusion is inductive reasoning.
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Looking at patterns: if I have 4, 8, 16, 32, and I need to use inductive reasoning
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to find the next several terms in the sequence, well, I can just see how these numbers came about,
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and then I can just apply the same rule to find the next few numbers.
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So, for this, 4, 8, 16, 32...how did you get from 4 to 8, 8 to 16, and 16 to 32?
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Well, it looks like this was multiplied by 2; that means I would have to multiply by 2 to get my next answer.
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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.
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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?
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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.
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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).
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Now, a few more problems: if we make a conjecture about this, if AB = CD and CD = EF, what can I conclude?
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Well, if AB is equal to CD, if I draw AB, here is AB; and here is CD;
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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...
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doesn't that mean that AB will equal EF?--so that will be my conjecture: AB = EF.
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Make a conjecture, given points A, B, and C: let me just draw out a coordinate plane.
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A is (-1,0); it is right here; B is (0,2); it is right there; C is (1,4), which is right there.
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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.
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Counter-examples: just because you come up with a conjecture, you come up with some kind of conclusion,
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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.
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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.
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Conjectures are not always true; and to prove that it is not always true, you have to provide a counter-example.
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And a counter-example is the opposite of what you are trying to prove.
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If you are trying to prove that something is true--let's say you saw something a few times,
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and so you conclude--you make a conjecture--that the next time, it is going to happen again;
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you can't prove that it is going to happen again just by showing that it happened.
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You cannot prove something just by giving an example, because it might not happen the following time.
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You might not be able to find a counter-example in order to prove that that is not true.
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A counter-example is the opposite of what you are trying to prove.
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Let's say, for example, that the first five cars you see today are black; does that mean that all cars are black?
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That would be a conjecture; the conjecture would be that, since you saw five cars that are black...
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my conjecture would be that the next car that I see will be black; and that might not be true.
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In order to prove that whatever you concluded, your conjecture, is not true, you are going to provide a counter-example.
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A counter-example would be to show an example of it not being true.
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Now, let's go over a few examples of this: the first one: Any three points will form a triangle.
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If I have three points like this, I know that it is going to form a triangle.
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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,
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because I know that three points will not always form a triangle.
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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.
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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.
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This is my counter-example: by giving an example of when this is not true, I am proving my conjecture false.
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So, this conjecture...sometimes it could be true, but it is not always true.
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By showing an example of when it is not true, a counter-example--that is when you are proving the conjecture false.
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The square of any number is greater than the original number: well, that could be true, but it is not always true.
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To show that it is not always true, I need to provide a counter-example.
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Let's say I have some numbers: let's say 2--if I square it...I am saying the square of any number,
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so if I take a number, and I square it, then it is going to be greater than this original number.
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If I square this, then it is going to be 4; well, this is greater than this number, the original number.
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What if I have 0? If I square it, what do I get? 0.
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If I have, let's say, 1/2, and I square this, I get 1/4.
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Well, is 1/4 greater than the original number, 1/2? No, 1/4 is smaller than 1/2.
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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.
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0 is not greater than 0, and 1/4 is not greater than 1/2.
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Here is a counter-example, and here is a counter-example, because these two examples show that,
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if you square the number, then the answer is not going to be greater than the original number.
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Counter-example, counter-example: by showing the counter-examples, I am proving that this conjecture is false.
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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.
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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.
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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.
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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;
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here I can add something bigger...for this one, it doesn't seem like that would be the pattern;
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so here it is multiplied by 4; here, multiply by 5; so then, for the next number, I can multiply by 6.
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If I multiply this by 6, then I will get...let's see: 120 times 6 is 0, 12, 720.
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And the next one...you can just multiply by 7.
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This one right here: I have a square; then I have this shape in there; and then I have another one.
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The next pattern will be...so then, here is the next step; that is up to there.
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And then, my next one will be to draw a square within that square, like that.
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And the next one would be to do the same thing; and you are going to draw another square inside.
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OK, so you are given a statement, and you need to come up with a conjecture.
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The given statement is that angle 1 and angle 2 are adjacent.
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"Adjacent" means that the angles share a side and a vertex.
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So, if I draw angles 1 and 2, they are adjacent; then what can I conclude?
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Well, I conclude that angles 1 and 2 (since I know that "adjacent" means that they are next to each other) share a side;
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so, angles 1 and 2 share a side and a vertex, because they are adjacent.
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The next one: the given statement: line *m* (here is line *m*, and this is a line) is an angle bisector of angle ABC.
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Here is an angle, ABC; and line *m* is an angle bisector--"bisector" means that it cuts in half.
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So, this line cuts this angle ABC in half; that means that these two parts right here are the same--they are congruent.
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If I label this angle 1 and this angle 2, what can I conclude?
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If this line is an angle bisector, then these two parts right here, angle 1 and angle 2, I can say are congruent.
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Or I could say that the measure of angle 1 equals the measure of angle 2.
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Or I can say that angle 1 is congruent to angle 2.
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That would be my conjector, since line *m* is an angle bisector of angle ABC.
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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.
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Given that WX = XY, the conjecture is that W, X, and Y are collinear points.
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The conjecture is saying that if I have WX...here is point W, point X, and point Y...W, X, and Y are collinear.
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WX = XY; the conjecture is that these are collinear; is that always true?
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Can you think of an example of when it is not true?
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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.
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This is a counter-example, because this is an example of when my conjecture is false.
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This one I know...this conjecture is false.
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The next one: Given that x is an integer, the conjecture is that -x is negative.
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So, if x is an integer (some integers are 2, 0, and -2; so these are x), the conjecture is that -x is negative.
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If I make this negative, -2 is -x--is it negative?--these would be -x.
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-2: this is true; this is an example of the conjecture; what about 0?
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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.
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And then here, -2: if I make it -x, then it is -(-2); well, that is a positive 2.
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So, does that show that -x is negative? No, because this is positive 2.
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So, in this case, this one is false; if x is an integer (these numbers right here), and you make those integers negative,
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then the answer is going to be negative: in this case, this works.
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If you make this number negative, it is not a negative; if you take the negative of this number, it becomes positive.
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So, -x is positive in this case; so this one is also false, and then here is my counter-example, right here.
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This one and this one are both counter-examples, because they show that this is not true; it is false.
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Well, that is it for this lesson; thank you for watching Educator.com--see you next time!