 Lesson two, deductive reasoning. Then I have blank blank. I'm going to tell you what goes in the blank right now is the words deductive reasoning. Deductive reasoning, which Jeremiah is also sometimes called logical reasoning, is the logical process of using true statements to arrive at a conclusion that you know is true. The one says write a logical conclusion which can be deduced from the following statements. All integers are rational numbers, that statement number one. Negative five is an integer, that statement number two. What can you conclude from that logically, reasonably? Sorry, we're doing A. You're on B, and you're right, but let's get there. All integers are rational numbers, negative five is an integer, conclusion. Boston, you're right, say it louder. Negative five is a rational number. How do I know? Well, all integers are rational numbers, and it says that negative five belongs to the integers, which means it automatically belongs to the rational numbers. We call this, you don't need to know this word, but I love logic and reasoning, so this is the nerd within me. We call this a syllogism, where you have two statements and you arrive at a third conclusion. Water freezes below zero degrees Celsius. The temperature is positive six degrees Celsius. Conclusion, Jordan. All triathletes have run a marathon. Mr. Dewick has never run a marathon. Mr. Dewick is over, oh hang on, no, no, no, no, no, let's do that. Not a triathlete. Mr. Dewick, if I had run a marathon, could you assume that I was a triathlete, or is it possible that I could have run a marathon and never done a triathlon? So this one does not work in reverse. You can't say, oh, he ran a marathon, therefore he is a triathlete. Maybe I can't swim. I signed up, let's do this for a second, B, or sorry, D. Now this next one is a very, very famous syllogism, but it's abstract. It says this, if A equals B and B equals C, what else do we know? Taylor, A equals C. Think about that one, ponder that one for a bit. You see it? If A equals B and B equals C, well, if they both equal B, they both equal each other. In fact, this one is used so often in math, it has a fancy name. We'll write it down because we're nerds. It's called transitivity. E, Jordan is taller than Joe. Joe is taller than Boston. Conclusion. Should we model it just to find out? Can you use the names that were given there instead of saying Boston is the smallest? Because I don't know if that's the, I would say that Boston is shorter than, if C times D gave me a positive number, and I know that C was negative, D has to be negative. Look up. I'd like to show you shorter ways to write things, so don't write this down. Don't write this down. That's way too much writing. Do you know how we abbreviate is negative Jeremiah in math? We go like this, is smaller than zero, which is way shorter to write than is negative. But if it's smaller than zero, doesn't that mean that it has to be negative? Well, that's actually a logical conclusion. Deductive reasoning there. You can write is negative. I'm going to do that for the rest of the year because I don't feel like writing out lots of stuff. Given the following, now we're going to try and use this deductive reasoning for a multi-step proof. And this is where Courtney, you want to turn your brain up, turn your brain on. This is where if you get a bit lost, all it means is you're normal. You're looking at proofs. And I got to tell you, as someone with a math degree, I found proofs very, very difficult to wrap my brain around. But I want you to commit to at least trying to follow what's going on. So it's given the following. One plus two plus three, well, that's six. Two plus three plus four is nine. Three plus four plus five is 12. What was the pattern that we noticed if you're adding consecutive integers? What was the three consecutive integers? What was the shortcut for getting the answer? Yeah. Okay, so three at 15, 33, three times negative 24. Is that okay, Jeremiah? Let's make that a negative 24. Sorry? Second one from the bottom? Oh, but are those consecutive? Jeremiah, I did a typo when I was typing in a hurry. Thank you, Jeremiah, for catching two candies for you, but later on, absolutely. Jeremiah is saying, look, this is actually 11 plus 12 plus 13. If I'd done that on purpose, that would have been phenomenal teachings. Sadly, I didn't. You just caught a mistake. So you're saying the answer is actually 36, which it is. Nice catch. But our conjecture is, if we add three consecutive integers, the sum is three times the middle. Prove it. Now, to do a, because it's worked for these ones, but remember, we saw an example a few minutes ago in last day's homework, where that circle pattern worked for the first five, didn't work for the sixth one suddenly. How would I go about proving this? Here is where we're going to do a little bit of algebra. I would let the numbers be those. X minus one, X and X plus one. Now think, first of all, write that down, and then let's think about this for a while. What's the middle number in this group here, Joe? Just read it to me. X, what's one less than X? X minus one, is that how you would write it? Yeah. What's one more than X? In other words, are those three consecutive numbers? They are, I don't know what X is, but I'm telling you, whatever I plug in for X, those are three consecutive integers. Okay? I'm trying to do a general proof that works for any number. So I'll use X's or N's or whatever variable. Usually I use X's, if you like that. Can you read to me? Excuse me. Adam, can you read to me? Oh, I'm missing the word add. That was dumb. If we add three consecutive, can you read to me that first sentence? If we what? Stop. How would I write these three added together? How would I add these together? What would I write algebraically? Wouldn't I write? That's the sum. That's the, if we add three consecutive integers. So far, so good. The weirds so far, but. Oh, I think I can gather like terms here. For instance, I'm noticing X plus X plus X. What is 1X plus 1X plus 1X? And what is negative one plus one? What? They cancel? Why have I, by that statement there, by those two lines, why have I proved our conjecture? What does that last line say? Matt, read me the last line here. No, no, the very last line that I wrote here. Read me the last line right here. 3X equals sum. Oh, what was X? Joe, what did you tell me X was? It was the middle number. Yes? So here's what we've said. If we add three consecutive numbers, that's how I would translate that into math. When I gather like terms, it turns out the sum ends up being three times the middle. Not X, but what was X in this case? The middle. Did I just prove this? Yes. I showed algebraically that if you add any three consecutive integers, should I, any three? When you add them together, it simplifies to three times the middle one. How do I know this is the middle one? Well, it has to be because this one's one less and this one's one more. That must be in the middle. What do three dots mean? You've seen that symbol before? What do three dots mean? Boston, you're going to open your eyes? Take your jacket off. Seriously. Take your jacket off. That's why you're falling asleep. You've wrapped yourself in a blanket. I'm not mad at you. If you take your jacket off in about five minutes, your body temperature will go down, your heart rate will go up, and you'll be more awake. Give it time. What do these three dots mean? And you guys have you seen this in science yet? Three dots like that means, therefore, in conclusion, I'm going to use that because I'm a nerd. Therefore, our conjecture, why is it true, Boston? This conjecture was if we add three consecutive integers, so I algebraically, I wrote the algebraic version of adding any three consecutive integers, and when I simplified it, I got 3x, which was the second half of the statement. The sum is three times the middle number. There's three times the middle number. Turn the page. Are we next page over? I think we are, yes? I have to tell some of you that. Okay. Here's some notation for you. To symbolize an even number, I will write 2n. Why does that number have to be even no matter what? Is that an even number? If I put an even number in there or an odd, you know what? Because 2 goes into it no matter what, which makes it even. To symbolize an odd number, I'll either go 2n plus one, or sometimes you'll see the textbook go like this. Jordan, what we're saying is one more than an even number is automatically odd. Or conversely, Devin, we're saying one less than an even number is also automatically odd. Now, does that make sense? If you're an even number and you go up by one, are you automatically an odd number? No? Yes? The following case? Right? Even plus one is odd. Okay. We're going to use that notation. So if you ever start talking about an even number, just think 2n or 2x or 2y or whatever, two times something. If they ever say we start talking about an odd number, think two times something plus one. Or minus one. I like a plus one better because I hate negatives. Conjecture. If I add two odd numbers, the sum, the answer, must always be is always, well, if you add two numbers, what can you tell? Two odd numbers. What can you tell me about the answer? You're right. Even? Is that correct? Now, don't write this down. I would probably go, well, let's see. Five plus seven is 12. 13 plus 21 is 34. I would try a few odd number examples and I would see if I could spot the pattern. But I think you just reasoned it out. You said, look, two odd numbers when I add them together, they have to be even. I agree. Prove it. Okay. I'm going to let the numbers be 2x plus one and 2y plus one. Write that down and we'll talk about why I just did that. So here's the first number. Here's the second number. Why did I just do that? Well, is that an odd number no matter what? Yeah. Is that an odd number no matter what? And I used an x here and a y here because I want different answers to any two odd numbers. So far so good. A little weird. You with me? I'm right by last year. You with me? Okay. Taylor, can you read to me the conjecture again? Conjecture. Stop. Say that again. Okay. I'm going to write this algebraically. If I want to add two odd numbers, add means is going to be a plus sign. Here's my first odd number. Here's my second odd number. There is adding two odd numbers. Somehow I want to see if I can do some arithmetic, some algebra and end up with an answer that satisfies this conjecture. End up with an answer that I know has to be even. No matter what x or y I put in there. Hmm. Oh. Can I gather like terms? Courtney, see the little positive one right there? See the little positive one right there? What's positive one plus one? So here's what I have. What can you tell me about this? Hmm. Hmm. Can this be odd? Why not Jeremiah? You're right. I agree with you. You know what? I agree with you. Here's how I would do one more step. I would factor. Remember factoring from last year? There is a GCF in all three of these terms. What's the greatest common factor in all three of these terms? I would write this as two and then it's going to be bracket x plus y plus one. And believe it or not, Sydney, we're done. You may not realize it, but we are. Sydney, at the top of the page, I wrote an even number can be represented by. What did we say that we could represent an even number by at the very, very top of the page there? Louder. Two times something. I'm going to replace. You know what? I'm going to replace the letter n with time something. So what's any even number? Two times something. Look what I have here. Don't I have two times something? You know how I know? Because there's a two in front of everything. Doesn't that mean it has to be even? This is Jeremiah. This is how I would prove what you were saying. Look, I can see it. I don't know how to prove it. I would say, well, if I define every even number as two times something, why don't I try writing what I think is even as two times something? And even though Sydney, this is an ugly something compared to what I wrote at the top of the page, I'm still going to argue that it matches what we wrote at the top of the page. It's two times something. That has to be even since it's multiplied by two. That has to be even. If you're tough, if you're finding these awkward, if you're finding this weird, you're normal. Don't feel bad, but just don't shut down. Odd plus odd has to be even. Why? Because if I do a generic odd number and another generic odd number and I add them together algebraically, as it turns out, when I simplify, I get a two in front of everything, which means it's got to be even because two goes into everything. Let's try this one. You need your calculators for this. Get your calculators out. If you don't have a calculator, now's the time to fess up because I'm going to make fun of you if you don't have one. Get your calculator. It says choose a number. So all of you right down, right now, pick a number. Pick a number on your calculator. I'm going to pick 17. You all have different numbers on your calculator? It says add five. So plus five and then press equals. Then it says double the result times by two. Press equals. Then it says subtract four minus four and press equals. Divide by two. Subtract the number that I started with. I started with 17. I got three as an answer. What did you guys get? What? Oh, how could you do something different? What? Huh? You got three? You got three? Seven? And you followed the instructions and hit equals after each step? And did you get three? Oh, you got three. Oh, I thought you said you got something different. Sorry? 23. What did you start out with? 20. And you hit equals. Enter after each step? Okay. Yes. And if you don't do that, it's going to do it properly with bed mass. I don't think they want you to do it properly with bed mass. They want you to do it all in this order. Okay. Let's try seven. 20 plus five. Enter equals times two equals minus four equals. Divided by two equals minus my original number. Yeah, I'm getting three back. You have to follow the instructions that are in front of you right on the paper. About six inches from your nose there. That, those ones? Prove it! What did you use as your number? Yeah, you. 562. Ah, she's trying to see if she can game the system with a big number. What did you use? Okay. 566. What did you use? Have we proved it for every single number out there? How many numbers are there out there? Think Buzz Lightyear. How many numbers are there out there? Infinity. Can I try them all? No. You know what? Instead, I'm going to try it with one number. And you know what I'm going to let that number be? What have we done for our generic any number so far? X. Jordan, write this down. I'm going to let the number be X. Can I regate the word number with a number sign? Is that okay? I'm going to. And Sidney, what we're going to do is we're going to walk through this step by step by step, except instead of doing it on our calculator to a number, we're going to do it to the X. So, step one says choose a number. I did. Step two says add five. How would I add five to X? How would I write that algebraically? Five X? Is that adding or is that timesing? That's timesing. How would I add five? Even easier. Yeah. Five plus X. I like the variable to come first. All right. What's step three? We know what does the step say. Step three. Step three. Step three. Double the result. That would look like this. Two times X plus five. Except I can now front door, get rid of brackets. What does that simplify to? Two X plus 10? What's step four? Math now. Okay. So how would I do that? I would take my number, which is plus 10, and I would subtract four. When I gather like terms, what does two X plus 10 minus four simplify to? I'll give you a hand. Two X and a positive six. Okay. What was step five, Boston? Okay. Step five is take this two X plus six and divide it by two. Now, Boston, what that really means is divide the two X by two and divide the six by two. Yeah. What was step six, Boston? I've scrolled down. Ah, so I'm at X plus three. Subtract X. What is X plus three? Take away X. What does the answer have to be no matter what infinity of numbers you plug in? Five hundred and seventy works just fine. A trillion works just fine. What's the answer simplify to? Three. There's our proof. I think something like this might be an okay test question in my mind. First of all, because Corny, I kind of like these number puzzles where it's like, oh, what's the answer? My birthday. Yay. I think asking you to walk through a proof like that because it is kind of step by step algebraic. This I would feel would be okay on a quiz or a test. Prove why the answer is the answer that you got. Is that okay? There are way more famous ones and tougher ones and bizarre ones than just this one. But there's our proof. We're going to finish with example five, I think. No, example six and example five are the same thing. You know what? We may even finish the whole lesson. I won't give you any homework today, though. Example five says, see if you can spot the pattern. I'd like you to try not to use your calculator. Put your calculators down and away. Yes, Boston, you too. It should. You know what? I would argue I don't need to. I trust my algebra. I mean, I've convinced myself. You ready? This is kind of a series of questions. And if you pay attention, if you spot the pattern, you may actually be able to predict the upcoming questions. So what I'd like you to do first of all is what's six times six? Write down the answer. And I'm going to freeze the screen. And then right across from six times six, what's five times seven? And write down the answer. Write across from it. Yes? What's nine times nine? And then write across from it. What's eight times 10? Yeah, keep it to yourself. What's four times four? And then what's three times five? Have you spotted yet how these two, each line is similar? If you spotted it, if I said to you D, seven times seven, first of all, what's seven times seven? Can you predict what this question would be over here? Have you spotted the pattern yet? Shay, except I've gone small number, big number. And what's the answer? Have you noticed any pattern between your answers yet? Oh, if I told you the second question was nine times 11, and the answer was 99, what's the first question? Yeah? And what's the answer? So if I tell you this without a calculator, if 50 times 50 is 2,500, what's 49 times 51? 24.99, is it? Are we right, Shay? Sorry, you reached for your correct double check. Yes, you are. Okay. Oh, proof. Sorry. I want to prove that this pattern always works. You know what I really want to prove? Let the first number be X. So instead of a six times six, X times X. How would I write X times X? I heard it. X squared. So I'm going to try and line up my columns. This column here is X squared. Where'd the five and the seven come from based on the six? How did I get the five and the seven? Where'd the eight and the ten come from based on the nine? How did I get an eight and a nine? How did I get the eight? Okay, so if I do that with an X, how would I write this? Is that one minus our generic number? And what I write this is that one plus our generic number? In other words, if our number is X squared, one minus one plus, you said yes? Here's our number. One less, one more. Foil that out. We've done foil. Foil it out. X minus one times X plus one. When you foil it out. Remember foil? We did this at the beginning of the year of the review. Boom, boom, boom. I don't want to put the red lines on this one because we're going to clutter things up. If you foil that out though, what do you get? You don't get X squared. You don't get 2X. Don't you get X squared minus one when you foil it out? Look up. Here's the English to this math statement that we've wrote. But once you've written that down, look up. Jeremiah, here's what it's saying. If you square a number, one less than the number times one more than the number is one less than the number squared. Is that actually what was happening? Isn't that how we did 49 times 51? We said it's the square minus one. It's the difference of squares. You called it, I think, last year in grade 10. How much more am I planning on doing here? We're going to pause for a second.