 In this video, I'm going to talk about making a conjecture. Conjecture is something that you believe to be true based on inductive reasoning. In a previous video, I talked about inductive reasoning that was identifying a pattern if you want to go watch that one. In that previous video, inductive reasoning is basically you come to some conclusion based on prior experience is how I simplify inductive reasoning. Making a conjecture is that conclusion based on inductive reasoning. What we're going to do is we're going to look at a couple of different examples of using inductive reasoning to try to come up with a conclusion, to try to come up with what the next logical type of answer would be for these two different problems. Complete each conjecture. Looks like we're just completing the sentence type of example. Here we go. The sum of two positive numbers is what? Well, I don't have a word bank or anything like that to go with, so let's think about this. Two positive numbers, one really, really great thing to do here is to just try a couple of examples. See what they're talking about. What logical conclusions do they want you to come to? The sum of two positive numbers, the sum of two positive numbers, when we talk about positive numbers, we're talking about numbers like one and two. Here's the sum of two positive numbers, that would be one plus two is three. What are they trying to get me to see here? Some other positive numbers, 10 and 25, that makes 35. What else? Two positive numbers, the sum of two positive numbers, so let's try one and 52, which is going to give me 53. When I take the sum of two positive numbers, when I add two numbers together, so here's two add numbers, I get three, two numbers, I get 35, two numbers, I get 53. All these answers, three, 35 and 53, what's the one thing that they have in common? So one thing that they have in common. Now, as I look at that, I might think to myself, well, three, 35, 53, those are all odd numbers, aren't they? Those are all odd numbers? Well, that's actually kind of right. In other words, I do have a three, which is odd, a 35, which is odd, and a 53, which is odd. Interesting enough, 53 and 35 are kind of transposed numbers, interesting. But I can't really say that you're always going to get an odd number because I can come up with an example of two plus four equals six. There's an even number, so I can't really say that when I add two positive numbers, I'm always going to get an odd number, because there's an example of an even number right there, so what else do they all have in common? Three, 35, 53, and six, what do they all have in common? Well, you can get a hint by what they're giving you. The sum of two positive numbers, so the thing is 53, 35, and six, they're all positive numbers. So the thing is when I keep adding positive numbers together, they just get bigger and bigger and bigger. They keep getting more and more positive, bigger and bigger positive numbers. So the conclusion I can come to is, the conjecture that I can make is the sum of two positive numbers is positive, positive, or you could say the sum of two positive numbers is going to be a positive number. I guess that's another way that you can say that. So again, notice that what we did there, we had to come up with examples. We had to use all these examples here to help us make this conjecture. All these examples here, this is what we call prior experience. I have to use all of this to try to figure out what conclusion I'm trying to come to, what conjecture I'm trying to make. Okay. All right. The next example, we'll go on to the next example, the number of lines, the number of lines formed by four points, no three of which are collinear is something. Okay, so now hold on a second, we got a little bit of vocab here, okay? So a number, the number of lines, okay, so we're counting lines, formed by four points, okay, so I have four points. Now I think this next line tries to tell me how to arrange those points. Four points, no three of which are collinear, collinear, what have I heard that before? Collinear is a vocabulary word that we used in geometry, collinear means points that are on the same line, collinear, points that share, co-align-linear, collinear, that's where that word comes from. Okay, so one more time. The number of lines, so we're counting lines, formed by four points, no three of which are on the same line that are collinear is something. Okay, so let's, you know what the thing is, is that reading through this is kind of confusing, so what I'm going to do is I'm going to draw myself a picture. I mean lines, points, collinear, these are all, these are all words that we use for pictures, so I'm going to draw myself a picture. So I got four points, so here's a point, here's a point, okay, so I can't go down and make a line, because no three of which are collinear, so here's another point here, and let's put our other point right here, try not to make straight lines, okay, so what I want to do is I want to, the number of lines that are formed by four points, okay, so what I need to do is I need to start at a point and I need to make lines, okay, so I want to try to figure out how many lines are made, so I'm about the only way to do that is to actually draw lines, so I'm going to start here, here's a line here, so there's one, here's two, and there's three, not bad, not bad, but the thing is, is that you might say okay that's only three, but that's only starting from one of those points, I need to continue on with the other points to see how many more lines can be made, so there's three lines right there, so I'm going to go to the next point, so from here, I can go, so I'm going to have one, two, three, so now I'm at four, four, and then five, five lines, now I can't go back this way, because I've already made that line, so now I've got five lines, one, two, three, four, and five, I have five lines, so let's go to the next point, okay, so I got five lines, I can't go back because I've already made those lines, so I have five, and then the only place I can go right here to make six, okay, and I really can't go anywhere else, so I go to this point, well that's made, that one's made, that one's made, that's about it, okay, I can't really make any more, make any more lines, so let's count them all, one, two, three, four, five, and six, so the number of lines formed by four points, no three of which are cool, and here is six, six lines, okay, it looks like what I've made there is actually, you really want to get technicals, a quadrilateral with the diagonals that are created, that's a little bit of vocabulary for another time, anyway that is making a conjecture, again a conjecture is a conclusion that is made using inductive reasoning, an inductive reasoning is using experience, prior experience, logic, and reasoning to come to some conclusion, okay, so a conjecture is that conclusion, so our conjectures in this point, in this example, we were figuring out the positive numbers from the sum of two positive numbers, sum of two positive numbers is always going to be positive, that's a conjecture, and then the number of lines formed by four points, no three of which are collinear is six lines, with this little picture down here we were able to determine that you can create six lines from that, okay, that is making a conjecture, hopefully this, hopefully this video is informative.