 Here we go. Lesson one. Put your name on it in case you lose it. And the heading is inductive reasoning, conjectures and counter examples. Today is going to be some vocabulary. Do you need to memorize these words, Simon? I'm probably on the test not going to ask you to define them, but I'm going to use them in questions so you want to be able to understand what they mean in a sentence so that you know what the heck I'm asking or talking about. You understand the difference between memorizing and understanding? Okay. The one says, find the pattern. What's 11 times 25? Well, I'm going to tell you what the answer is. It's 275. Double check me, am I right? I don't think I've, did I show you the pattern before or not? No? Okay. What's 11 times 42? I'm going to tell you the answer is 462. Double check me, am I right? Am I? Am I? And I know that 11 times 71 is 781. Double check me, am I right? Am I multiplying these out this fast in my head? No. 11 times 34 is 374. Has anybody spotted the pattern yet? There is a great pattern with the 11 times table when you get to the teens and higher. You all know the pattern for single digit numbers. 11 times 55, 11 times 66. You've all spotted that one. What's the pattern when you get to bigger numbers? Have you seen it yet? If you have, try filling in 11 times 44 on your own. If you haven't, the answer is 484, which it is. I see a few of you squinting. That's okay. I can't teach you how to spot this. All I can say is look and suddenly, aha! That part comes on your own. You know 11 times 63? Tell me what you think the answer to this next one is, Matt. 11 times 63. This is how I know if you spotted the pattern. See, if you can't tell me the answer, well, then you got a luck. Jeremiah, 693. So Jeremiah is on the bandwagon. Let's see if we can get a few more people on. Without a calculator, can somebody tell me what 11 times 81 is? Alex. Sorry. Okay, so Alex is on the bandwagon. You're looking for a pattern here. What's 11 times 27 without a calculator? Has anybody spotted the pattern yet? Taylor. 297. Don't you feel proud and superior right now? Look at me. It's just number tricks. We're going to pass on the next one. The next one, although it fits the pattern, it fits the pattern weirdly and is yucky. Those of you that understand the pattern are going, oh yeah, I can see why that one is a little bit different and we're going to do 11 times 52. Has anybody spotted what 11 times 52 is without a calculator? Sam. Okay. You can't spot it yet. Each one that I'm doing, I'm seeing, it's funny because I can see your eyes. Every time I do one of these, I see about three or four more eyes go, and I realize they just saw it. 11 times 54. Who can tell me what that is without a calculator? Did you do it with a calculator or without a calculator? You spotted it? Okay. Sit up straight and feel superior. And he did. Do you see that? Well, yes, I did. The next one is also a weird one. It fits the pattern, but strange. In fact, the rest of them fit the pattern, but strangely, what's the shortcut for the 11 times table? Once you get to the teens and higher, what was the pattern that we were doing? Can anybody articulate it for me? What's the first digit? In fact, you know what? The first digit is the first digit. What's the last digit? Strange enough, the last digit is the last digit, which makes it fairly easy for me to remember the pattern. Boston, what's the middle digit? What's the fancy word for add? It begins with letter S, and it rhymes with what you say when you're not sure of something. Some, okay? If we're going to do a proof, we want to try and use fancy shmancy words. If you said add, I'd take the marks, but the middle term is the sum of the two digits. Now, why did I skip the weird ones? Well, here, I would go like this. The last digit is definitely a three. The first digit is a nine. What's nine plus three? Two, carry the one. Instead of a nine, it should be a ten. It still works, though. I got to carry a number. Here, 11 times 56, the last digit is a six. The first digit, now I'm not going to write. I'm just going to remember. It's a five. What's five plus six? So it's going to be a one. Carry the one. It's going to be a six instead of a five right there. Now, these aren't as useful today, Jordan, because most of you carry some kind of calculating, but certainly before calculators were around, you memorized all sorts of tricks. This is how we did math. Even now, I have to be honest, I like being able to do my 11 times table in my head. It would be perfect if we had an 11% sales tax, harmonized sales tax, because then I could do all sorts of fancy math in my head and impress the checkover. What's this one going to be? 11 times 84. Four here. There's going to be an eight in the front. Remember that. Eight plus four is what, Alex? So to carry the one that's going to become a nine in the front. Try the last two on your own. Six, five, nine, 17. Is that right? The last digit is a nine. Out of eight, seven, carry the one. Yeah. Inductive reasoning is spotting a pattern and using that to apply the pattern to new data. That's the fancy definition. By the way, we haven't proven that this works all the time. I think it does, and I think I've convinced you. To prove it, we would either have to look at every single number between 100, sorry, between 11 and 99 and see if it works for all of them, or come up with some kind of a generic proof. This is actually a very tough proof. I'm not going to do a generic proof, but that's going to be next lesson. Example two, find the pattern. Four comma 12 comma 36 comma 108 comma, what comes next? You may need to calculate for this one. Liam, 324, I think you're right. Liam, what comes after that? What comes after that? 2916? Liam, what is the pattern? What is each term generated from the previous one by times by three? And now that he said it, I saw a bunch. Oh, yeah. The real question isn't just find the pattern. The real question is now that I know the pattern, can I use it as a shortcut? Can I find the 14th term without finding the previous 13 terms? There's our goal. That's going to be our inductive reasoning. And there's probably all sorts of ways to approach this, but Sidney, I tend to approach these kind of in an organized manner. I tend to approach these systematically and carefully. Sorry for those of you who are listening at home on the internet. Okay, you know what I would do? I think I would do some kind of a table, and I think you have room off to the right wheel space right here. And I'm going to make a three column table. The first column is what I'm going to call the location stop. That's too much writing. I want it to stand for location, but do you mind? I'm going to write LOC period for location because I need to fit three columns here. And what I mean by location is this. This is location one, that's location two, that's location three, that's location four, and we want to find location 14. But I want to see if I can figure out how to get to 14 by looking at what's going on for the first three or four, and find a shortcut. So I'm going to write location one, location two, location three, location four. In the next column, I'm going to write the value. I'm going to write value, but you know what? Again, well, maybe value fits in there fair enough. Oh, use black, Mr. Dewey. Be consistent. What's the value at location one? Four. What's the value at location two? 12. What's the value at location three? 36. And what we want to try to do is generalize this so we can find the value at location n. And then if you want n to be 14, we can find an equation and plug and chug. We want the nth term. Now the first, then I'm going to in the third column, write the pattern. The first number is just four. But Liam, how did you find the next one? So you went four times three. How did you find the next one? I'm going to I'm going to write that as basic as possible. Here's what I really think you did. Don't write this down, look up, look up, but don't write this down. I think you took the original one, and you times that by three. Except what's an easier way to write three times three with a lot less work? Not as a nine. I want to keep everything broken down as much as possible so I can spot a pattern. Otherwise, I would just write 36. I don't want big numbers. I heard it here, Alex, what? You know what? Since I put an exponent on this three, Jeremiah, to help me spot the pattern better, I'm going to put the exponent on that three, which is an invisible one that's always there. The next number that we found in location four was 108. How did we get the 108? It was that original four times, do you see why I wrote it this way? Because now the pattern is easier just, okay, three cubed. What would the fifth term be? Without me filling it in here, what would the pattern be for the fifth term? Four times three to the fourth? Is this ever changing? Is the three ever changing? What's changing? How is this exponent related to the term number? You see how we can hopefully spot that, Aaron? I think the fifth term, I put a four there. The sixth term, what would I put there? Ah, the 14th term, what would I put there? Without having to find all the previous ones. Term 14, there's my abbreviation for term 14 equals four times three to the 13th power, which is what on your calculators? Do you get an answer in scientific notation or does it fit? Let's see. It fits, read me the digits one a time then. I heard six three seven seven and then you got quieter, which is six three seven seven two nine two. Is that right? Hey, if you want a general formula, the nth term is four times three to the n minus one power. I think if I wanted to try and do this like a math nerd and explain the pattern, Emily, that's what I would write. You want the 20th term, four times three to the 19th. You want the 80th term, four times three to the 79th. It's the whatever term you want is four times three to the power of one less than the term number that you want. Is that okay, Jordy? Do you see how I spotted that? There was a few of you that wanted me to write a nine there. I said, I really want to spot Courtney. What's going on? I'm going to keep everything as broken down as possible. And I think most of you by the time I've written three of them, spotted the pattern, okay? When we reach a conclusion by looking at a numerical pattern or a visual pattern, we call this inductive reasoning. What we just did was inductive reasoning. When we use inductive reasoning, we make an educated guess. We spot the pattern. The fancy name for this guess is called a conjecture. You may have heard that term before. It's from logic and reasoning. You make a conjecture and then you try and prove or disprove the conjecture. Proving a conjecture very, very, very, very tough. Disproving a conjecture very easy. We'll talk about that a bit later. So, here's our official definition of conjecture. Conjecture, a conclusion, a generalization, or an educated guess which is arrived at by inductive reasoning. It's also sometimes called, and I started with the letter P, premise. Premise, the following, prove or disprove the premise. So, P-R-E-M-I-S-E. We'll usually use the term conjecture, but I looked through some of the notes and some of the handouts and a few of them occasionally used premise. So, example three. How many matchsticks will you need to build the 100th shape? How would I try and use inductive reasoning to come up with a conjecture that will let me state how many matchsticks I'll need without having to build 100 of these shapes? Any suggestions? Shea, what about 100? Oh, you think I need 100 matchsticks? How many matchsticks did I use to make this shape? I don't think matchsticks are shaped like a triangle. Okay, now you, by the way, it's my fault because I should have put dots on the end but I was doing it. How many matchsticks did I use for the first one? You know what? Let's write that down. How many matchsticks did I use for the second one? Let's write that down. How many matchsticks did I use for the third one? Oh, I recognize this pattern of numbers. Let's do one more to confirm that we spotted the correct pattern because it might go haywire all of a sudden but I think if four in a row work we can satisfy ourselves that the pattern holds. How many matchsticks did I use in the next shape? Is it one, two, three, four, five, six, seven, eight, nine? Okay, instead of doing a vertical column like we did right here, since I wrote the numbers horizontally, this is location one. This is location number what? The second term, the third term, the fourth term. We want to try and generalize this for the nth term and then see if we can plug in a 100 for the n and get an answer. So here's the question. Can you come up with an equation that when I plug in a one I get a three, when I plug in a two I get a five, when I plug in a three I get a seven, when I plug in a four I get a nine and this I can't help you with. All I can say is think and be clever. Say that louder. She's right and the fact that she spotted it so quickly I'm impressed. Louder. Take the location double it and add one. Does that work? If you double the one and add one you get three. If you double the two and add one do you get five? If you double the three and add one do you get seven? The number of matchsticks? If you double the four and add one do you get nine? Now we haven't proved that this works all of the time but did it work for our first four? Then we'll use inductive reasoning. I think it'll work for the rest. Let's write this out as an algebraic nerd. So if I want to find term 100 you're telling me I take that 100, Devin and do what to it? Double it. I'll put a two in front because that's usually where we're used to seeing it to. And then add one and I'm going to argue you don't even need a calculator. What is two times 100 plus one? Is that okay Boston? If I wanted to generalize this I would say this Boston the nth term is equal to two times n plus one. And now if you want to find term 50 put a 50 there. If you want to find term 70 put a 70 there. If you want to find term 3,492,563 put a 3,492,563 there. Why you would? I don't know but okay. And I'm impressed. I thought it would take a lot longer for one of you to spot that. Devin I'm impressed. Candy for you later. I said it was very very tough to prove a conjecture it is but Sydney it's very easy to disprove a conjecture. All you need is one thing. A counter example. A counter example. A counter example is an example that shows that a conjecture is false. If you can find one exception to your rule it's not a rule. You need to revise the rule. You need to revise or modify the conjecture. To disprove a conjecture all you need is one uno, eins, un, ik, one, counter example. And there have been many a mathematical theory that looked very very good and then somewhere in the quadrillions they found a counter example. It doesn't work. Try to find a counter example for each of the following conjectures. If we can't we'll write down true. We'll say the conjecture is true. So the first one every prime number is odd. First of all what's a prime number? You guys did this way back in math 8. I know it's a while ago. What is a prime number? Is 8 a prime number? Okay a number that has, I'll use the math language first of all. You ready? A number for whom it has as divisors itself and one. A number that only has two things that go into it. One and itself. So is 21 a prime number? Why not? 23 is that a prime number? Okay so here is the statement. Every prime number is odd. True or find a counter example to just prove it? Taylor. Ah two, in fact two, only two and one go into it. Two is prime and two is actually unique in that it's the only non-odd prime number. It's the only even prime number. So we're going to write down here false and there's our counter example. Two. How could I modify that into a true statement? Every prime number greater than two is odd. That's true. Because after two there's no more even prime numbers because every other even number two goes into it. So if this was a theory that you were trying to prove, you would now revise your theory to reflect that counter example. B. The square root of a number is always, you know what? Here's my first typo. Cross out the word larger and put the word smaller. The square root of a number is always smaller than the original number. True or false and if false provide a counter example. Well let's try some in our head. What's the square root of 25? Five. Is five smaller than 25? Okay seems to be. What's the square root of 10? Decimal. What is the square root of 10 on your calculator? Is that smaller than 10? Okay seems to be. What other numbers if I was trying to do a reasonable proof or trying to find a counter example? What other things might I try if I was trying to approach this systematically? Ah what's the square root of one? What is one smaller than one? Did you just find a counter example? What about the square root of zero? What's the square root of zero? Okay so supposing I modified this and don't write this part down because I'm going to muck up the diagram quite a bit but supposing I said is always smaller than but I went like this or the same size as. Except there's a better way to write or the same size as it's shorter or equal to that's through the math way of writing the same size it. So suppose I said to you every square root is always smaller than or equal to the original number. Is that true if not find a counter example? In other words is there a number that when you take the square root of it you get a bigger answer? Yeah? Ah decimals so she went to you numbers I would always try one and zero because they're a little bit weird and if those don't work I always go to fractions and decimals. So let's get out our calculator and let's go square root of and let's put in a decimal. Now Alex's suggestion was 0.25. What do you get? Ah is that bigger than when you started from? So as it turns out we could not revise this one to make it true. The square root of a number could be bigger than smaller than or the same size as your original answer and if it could be bigger than or smaller than or equal to there's no pattern there that just means it can be anything. Good catch. See when I cube a number the answer is always larger than the original. True or false and if false give me a counter example. Yep what if I remove one and zero what if I rewrote this and I said besides one and zero if I cube a number the answer is always larger than the original number. What are you checking right now Boston? Ah see so now you're using inductive reason you've seen another counter example from the previous question and you just picked up now a third strategy I'll try ones I'll try zeros decimals I'll try negatives right what else might I try? Probably those are it for now I might try weird numbers like pi because other numbers when you cube a decimal do you get an answer that's bigger smaller oh so if I go point two to the third power I get point zero zero eight which is smaller so you know what false I'm going to say one and zero or numbers between zero and one. Decimals. D multiplying a number by one always results in the original number true or false and if false provide a counter example let's see negative five times one did that give me my original number back her suggestion was negatives I think negatives work at least that didn't provide a counter example to me I multiplied it by one and I got my original number back so right now I'm leaning towards true or is there another counter example is there a number that when you multiply it by one you don't get the original number back let's see my original number is zero times one oh you know what actually zero works it does give me zero back so we tried negatives they seem to work we tried zero what about one is one times one one okay that you know what I think true now I haven't proved that it's true Sydney but my gut instinct is anytime you multiply something by one you get the same answer I can't think of an exception fractions no I think we're good decimals I think we're good in fact have you seen me often say there's an invisible one in front of everything but we just never bother writing ah that's another way of saying that statement so I think I'm leaning towards that e multiplying a number by a negative always gives you a negative Sam false convince me false our counter example have I showed you my abbreviation for the word negative because I don't like writing out neg8 my abbreviation for the word negative is a minus sign in a ve you know what my abbreviation for positive is a plus sign in a ve negative times negative equals positive you see Madison I can't go minus times minus equals plus because to me I that that could be a subtraction or a negative sign so I when I want it that's what I say Mattis wow where'd that come wow your last name I don't know how that came from sorry Emily um I I would not write that if I want the word negative in my mind I write a little ve on the end of it you don't have to but it works for me f multiplying by zero always results in zero true or false and a false by the counter example what do you think Matt as a matter of fact that has a special name we actually call it the zero principle we use it all the time to solve what are called quadratic equations which we'll get to later this year but basically don't write this down preview of coming attractions don't write this down if I tell you that a times b equals zero what can you tell me that one of these numbers had to have been one of them had to be zero zero principle it says look I don't know what this one is but this one was a zero or if this one it wasn't the zero I know this one was a zero g says can we modify conjecture d so that's a true statement it already is okay example five says try to find a counter example for the following conjecture every polygon polygon is a fancy word for a closed shape with straight line sides not a circle every polygon with three or more sides has more sides than diagonals what are diagonals those are diagonals those are diagonals how many sides does this have count three how many diagonals how many ways can i go from corner to corner here it's a trick question actually in a triangle there are none no diagonals how many sides does this have count how many diagonal lines show up when i connect the corners two the statement says every polygon with three or more sides has more sides than diagonals so far it seems to be true but is two examples enough to convince me i don't think so because triangles and rectangles or quadrilaterals are nice shapes what's the next shape i should try how many sides five okay one two three four let's try that again one two three four five there there's a five sided shape corner there corner there corner there corner there so how many sides five how many diagonals how many corner to corner connections can i make well there's one two any others what oh right here three any others right corner oh four five does every polygon with three or more sides have more sides than diagonals now we were lucky we found a counter example after three examples and as i mentioned before there are plenty of theorems in math where that counter example didn't show up for such a long time that they assumed it was true until they found out it was false and then they had to rewrite the books nowadays we usually use computers to check these in a hurry because we're kind of lazy so what are your keywords those inductive reasoning spotting a pattern conjecture making a hypothesis or premise and a counter example the best way to disprove a conjecture next class we're talking about we'll talk about how do you prove a conjecture beyond a shadow of a doubt by coming up with a generalization that fits and works for every single situation tough homework is that but first can you open your books please to page to page 63 okay page 63 investigation part one we're not going to do i did some number patterns with you instead part two is looking at a visual pattern i just want to see if you can spot the pattern so it says bruno is investigating a fabric design for new furniture the original design is a trapezoid and then the pattern goes like this then like this this is not complete how would i complete this sketch here what's the pattern this is harder to articulate in english but what's the visual pattern can you see what's going on here what is going on jordan oh yeah okay so if i was going to you're right if i was going to continue this i got a zoom in because it's so small but if i was going to continue this it would be little tiny trapezoid let's see trapezoid right there right there right there right there right there right there right there and right there how many trapezoids did i draw the first time one how many trapezoids did i add the second time two how many did i add the third time Four, how many did I add the fourth time? So I heard you go one, two, four, eight. Can you tell me how I'm going to add next? What number pattern goes one, two, four, eight, 16, 32? Powers of two. OK, I can figure out what's going on here. But we're not going to do the rest of that one. Inductive reasoning, if you want to, if you turn the page, you can underline inductive reasoning. But I gave you the same definition, and conclusions, and conjectures. There's the nice little definition of conjecture. But I put that in your lesson. Look at page 65. There's a more complicated number pattern. I'm not going to worry about that at all. Turn to page 66. Says Jerome. I want to do example four, and then I'll turn you loose. Jerome infestigated the following number patterns. So what Jerome did is he wrote down consecutive integers. 1, 2, 3, 5, 6, 7, negative 6, negative 5, negative 4, negative 12, negative 10, sorry, negative 11, negative 10. And he added them up. And then he asked, can I spot a pattern? Or let me put it this way. How do I know that 21 plus 22 plus 23 is 66? How do I do it so fast in my head? How do I know that, give me a number between 1 and 20, Alex. 17 plus 18 plus 19 is, oh, you gave me a tough one there, 54. Anybody spot it yet? I'll give you one hint. Has something to do with that number. And now look at the answer. Jordan, it turns out if you have to add three consecutive numbers and you're looking for a shortcut, it's the middle one times three. Prove it, much tougher. In fact, we are going to prove this one in the next lesson. We're going to do a generalized pattern and we're going to prove that when you're done, you end up with three more than the middle. So it says use inductive reasoning to make a conjecture of three consecutive integers. The answer equals the middle times three. Inductive reasoning, Jeremiah, can be used to make conjectures, but you can't use it to prove that it always works. There may be a counter example that we haven't thought of to prove that a conjecture is true next lesson. We're going to have to use logical or deductive reasoning. Ignore example five. What did I assign from your homework? So number one, this is a fractal. Number one is called the Mickey Mouse Fractal. Can you tell me why it's called the Mickey Mouse Fractal? Yes, mathematicians actually do have a sense of humor somewhere. So there's original. Sketch one, sketch two. After drawing two more sketches and making a table, Carrie reached the conclusion that she would have to add 128 more circles for her seventh sketch. It says complete sketches three and four below. You know what? I'm going to say don't complete sketch four. I'd like to see if you can spot the pattern from three. I'm lazy. But then see if you can fill in what would go here. Spot the pattern. So number one. And then B says use deductive or inductive reasoning to determine the total number of circles needed to sketch one. What you want to do is you want to add one more column here and call it the total number of circles. Not many you added, but how many there are grand total. And see if you can spot a pattern. You probably will. Did I assign number two? Yeah. Pascal's triangle, a very, very famous triangle. So number two. What you want to try and figure out is how they're generating this. I'll give you your first hint. It's always ones down the side. So there's going to be a one there and a one there on the next row. What you really want to try and figure out is where did that 15 come from? Where did that six come from? Where did that 10 come from? Where did that 20 come from? Think about it. I think you'll get it. Now I'll give you a hint later on. OK. Did I assign number three? I don't think I did. Did I? Sorry? Oh, three B and C, because I'm going to pretend that you can figure out the pattern from the first three. You may need to draw one more. But I'm trying to see the whole point of inductive reasoning is we don't want to write them all out. We want to spot the pattern. Did I assign four? Yep. Five? Yep. Six? Yep. Seven? Seven is an extension of what we did with diagonals. That's a bit tricky. Well, you try it. Did I assign eight? Now, how many of you watched the movie that came out or read the book that came out a couple years ago called the Da Vinci Code? Have you watched the movie or read the book? He talks about the Fibonacci sequence of numbers. It's a very famous series of numbers. 1, 1, 2, 3, 5, 8, 13, 21, spot the pattern. What comes next? How do we get a 3? How do we get a 5? See it? Why does the 34 come next, Emily, whose name I got right this time? You're right. What comes next after that? 55? How is each number found, Emily? Add the previous two. 1 plus 2, 3. 2 plus 3, 5. 3 plus 5, 8. 13 plus 21, 34. 34 plus 55. That's your next one. Fill those in. So A is good. And then if you read the instructions for B, they're a bit tricky, but very, very puzzle them out. You'll note Fibonacci numbers, there's a gazillion patterns, Taylor. Books have been written on the Fibonacci sequence. They show up in nature all over the place as well. Sunflower seeds, spiral, and Fibonacci numbers, they show up all over the place. So see if you can figure out B and C and D. Did I assign number nine? Go. OK. And I didn't assign 10? No. So you got 15 minutes. Was that a little better? I hope that lesson was a bit clearer. I hate teaching lousy. I hate it. Like fingernails on the blackboard, hate it.