 Okay, picking up from where we left off and it helps me turn the volume on see if you can spot the pattern again Try not to use your calculator. So first of all, what's six times six write down the answer to that What's four times eight? I've picked these numbers carefully. Keep your eye open for a pattern see what you can spot Liam what's nine times nine? What's seven times eleven Courtney? What's four times four? Yep, what's two times six? Anybody spot a pattern yet? Okay, hold it So in this next one the first part of the question Jordan, what's seven times seven? Can you tell me what two numbers I would write here in the pattern that I've been trying to develop here? What times what? Emily, which is what? spot it Shay Okay, so Shay in this next one. I gave you part two of the question. I said 12 times 16 is 192 what did I do here? What times what? Spotted it if you haven't that's fine. No calculator put it away. I don't think you need the calculator to figure out the numbers Taylor Why 14 times 14? Oh? And what is 14 times 14 without a calculator? 196 Taylor, what was the pattern on the left side? I was squaring a number Where did these numbers come from? two less and two more What did you notice about the answer? So if I tell you that 57 times 57 is 3,249 because it is Can you tell me what 55 times 59 is? really 3,245 You know what I'm really saying Here's what I'm really saying If you take a number and Subtract to from it Taylor. That's two less and You take that same number and You add two to it Taylor. That's two more. Oh Let's foil this out. What is x minus two times x plus two? I'll get x squared I'll get plus 2x I'll get minus 2x I'll get Minus 4 Like I get x squared minus 4 now just so my screen's not so cluttered up I'm gonna delete the loops and let's see if this is what Taylor said If you take any number Jordan Subtract to add to and multiply them together Take any number like six subtract to and add to and multiply them together the answer The answer is four less than that number squared four less than that number squared Is that right? What we've just done is we've proven that difference of squares which you guys learned last year in grade That actually works for numbers What's nine times nine? 81 I'm gonna go three less than and three more than Three less than Three more than now. I know the answer is 72, but it's nine less than this number squared in fact if You tell me that 57 times 57 is 3249 I know that three less than Three more than is this answer here, but not minus four minus three times three minus nine Double check me to see if I'm right, but 54 times 60 it is three thousand two hundred and forty is it not? Check me say where it says proof me There's our actual proof we said look if you take any number and you go two less two less and Multiply them together and then Devin. We actually did the foil. Oh, that's the same as the original number squared minus four Just like earlier. We proved that if you did the same thing here Any number one less any number one more than multiply them together. It's the original number squared minus one We talked about two days ago two classes ago the term conjecture. There are some conjectures that seem very simple But have yet to be proved Very famous one is Goldbach's conjecture. I can explain it to all of you right now in 1742 Christian Goldbach proposed the following he said this notice the pattern any even integer bigger than two Can be expressed as the sum of two prime numbers now. What does the word sum mean what mathematical operation? addition, can you give me an Aaron can you give me an even number bigger than two less than 50? Okay He says that you can write that as the sum of two prime numbers What can you write 20 as what plus what? Nine's not prime. Thank you for playing click your consolation prize at the door. Yeah 17 plus 3 it works for that one Let's go a bit bigger. Give me a number. Give me a number Cough Bender give me a number between 20 and 40. That's even I can't hear you 20 30 all right, right that is the sum of two prime numbers 27 oh no 27 is not prime. Oh, yeah 7 plus 23 works Jeremiah give me a number between 40 and 60 and you know what give me a number between 50 and 100 and even even number 76 Little tougher. I think this Three plus what? So I don't think anything goes into 73 view two doesn't go into it five doesn't go into it nine doesn't go in I'm pretty sure 73 is prime Marcus give me an even number between 50 and 100 105 is neither even or between 50 and 100 so strike one strike two give me an even number between 50 and 100 96 okay 93 oh no three goes into 93 though. Oh, I'm pretty sure 91 is prime I don't recall it ever coming up in my time stables. Do you guys? That's how it by the way for me to figure out if a number is prime if it's below 100 I asked did it ever show up on my time stables if it didn't I'm assuming it's prime and You guys know how to tell if two goes into yeah 91 is not prime the 17 go into it Sorry 37 goes into it 7 goes into 91. Oh It does mr. Dewick you're wrong. We need to find a different one. Thank you Johnson can be for you So you know what I think 89 plus 7 works it's a very very simple conjecture Prove it We've been trying since 1742 No one's been able yet to find you know how we did this like let X be the number and thought No one's been able to come up with the way to do that if you do You'll win the fields medal, which is a Nobel Prize for mathematics. It's awarded every four years guarantee you and I said as far as I know as of about two weeks ago It had been this conjecture had been tested for every number up to Million billion trillion one quadrillion six hundred nine trillion There's a guy who's written a piece of software and if you install it on your computer it'll just run on the background very very quietly and Continue trying numbers But because those numbers are so big to do one number now it takes a fair bit of number crunching when you're in the trillions Takes a while I looked around I see about four of you and your calculators trying yeah So conjecture Sonaya don't have to be really tough and confusing to be very tough to prove In fact, this is a fairly simple one any even number you can write it as one prime plus one prime Oh, sorry any even number bigger than two Prove it ha Many people have tried feel free to try You might have spot you may figure something that we heard it. What's your homework? Hold on turn the page. I've seen many a magician perform this mind-reading trick here. So it says this a Number of mentalists perform the following mind-reading trick get your calculators out see if you can do this with me Choose a number from one to nine Marcus that means 11 is no good Multiply that number by nine and then press equals on your calculator Look at the answer that you have in front of you Add both of the digits together. So for example, if you got 15 you would go one plus five and six I'm about to tell you add both of the digits together and Subtract five from that number. So type in the answer that you've got when you add of the digits together minus five and then look at the answer that's in front of you Find the alphabet letter that matches your numbers So if you have a one that would be an a if you have a two that would be a B If you have a C that would be a three a three that would be a C, etc Think of a country that begins with that letter remember that add five to the number that's on your Calculator the number that you had before you match a letter to it and once you've added five Find the letter that it matches again a being one two three Etc. And think of a mammal that begins with that letter So you should have a country that begins with the first number first letter that we got And you should have a mammal a Mammal is a warm blooded animal and then the punchline most magicians give is they say hey wait a minute There are no elephants Denmark how many of you had that Only one of you I think some of you couldn't follow the math Most what when you do the first one get for me Because you got a D all of you for that first letter got a four Okay What countries begin with D now? It's a little tougher because your teenagers and your stupid in geography. You don't know much What countries begin with D that Dominican Republic is the other obscure one which I was prepared for I might have had that as a backup now We have a German here who said Deutschland his country in German begins with D. We're staying English You know, what are the countries begin with D and by the way, how many people will think of Dominican Republic not that many There's two more Only two and they're really obscure fact. I can't remember the second one Can't remember. What was the first one? These are the D. It's a it's one of the Indonesian small countries that change all the time There's sort of the Democratic Republic of Congo, but that really begins with a C for Congress. Most people are gonna come with Denmark Then it says add five. You know what? When you added five, I think that's wrong. I think this is a typo. I think you should add one Because you're supposed to end up on an E and you're supposed to think of a mammal that begins with an E And if I was really trying to sell you if I was really trying to do this court And here's what I would say think of a mammal big or small That begins with your letter and by emphasizing the word big almost everyone's gonna pick an elephant But I can't even just say think of an animal big or small what animals begin with E Five of them that aren't really obscure elephant What other animals begin with an E fly like and Eagle now if I emphasize the word big that probably takes care of that one There are two more that I know of Ah three more I stand corrected elk Emu which is like an ostrich, but not an ostrich and then how many of you know what an echidna is It's a shrew. It's a like a mini little groundhog mouse. It's across between a mouse and a hamster So a lot of magicians do some modification of this is like wow How'd you read my mind? No, just knowing the odds of crying the odds except it works better If the trick that I copied from the internet didn't have a typo in it I should have run through this myself to make sure that the title didn't work If you're bored for a candy if you prove to me how this trick works If you let the number be X and follow it all the way through this one is fairly tough I'll give you a candy when I see you on Tuesday What did I say the homework was So turn if you would please to lesson two in your book page 76 okay homework Preferably turn a little quicker to page 76 Who am I glaring at who forgot their book again? No, we're doing pretty good. I think so number one This is how we started out last day with little truth statements So Leona lives in a hundred mile house 100 miles is in British Columbia. What can you conclude? number two Here is one of those number puzzles It says complete columns choice one choice to try it for three different numbers and You should notice a pattern in your answer when you're done It may be the same answer or it may look very similar with only slight differences. I haven't tried this one yet So B says make a conjecture based on your answers in columns one two and three with the three ones that you've tried and Then C says use deductive reasoning to complete the column in the general case So they're saying let the number be M and try following this one all the way through algebraically like we did last day for This one here See if you can do it. We used X instead of M, but Matt. It's the same idea So to all good number three There are no answers in the back for these we will go over these next class So the answers for one and two are in the back. These are tough if You can't get these Do not despair Means you're normal See if you can prove the sum of any three consecutive even numbers is Divisible by six now what are consecutive even numbers? six plus eight plus ten or Two plus four plus six by the way, what is two plus four plus six? Twelve the six go into twelve. What is six plus eight plus ten? 24 the six go into 24 Apparently it always will prove it. Here's your hint. I Think I Would let the first number be X What would the second number be then? How do you go from a two to a four? Add two what would the second number be then if the first number is X. What would the second number be? X plus two What would the third number be then not X plus two? How do you go from a two to a six? How do you go from a six to a ten? Try adding those three together gathering like terms and see if you notice anything Okay, same with B except B is going to be any odd numbers. Oh This said consecutive even numbers if I really want to guarantee they're even you know what I should do I should call the number 2x and 2x plus 2 and 2x plus 4 Why will that guarantee that they're even think about it? for B Three odd numbers is an odd number. I would call the odd numbers 2x plus 1 2y plus 1 and 2z Plus 1 there's your three odd numbers. How do I know those are odd? We said last day take an even number and add one to it now. It's odd See what happens if you try adding those three gather like terms. What do you notice? Again, if you're finding this tough normal C says the product of any two odd numbers is an odd number try and prove that 2x plus 1 multiplied by 2y plus 1 will always be odd try foiling that out and see what you notice D says the difference of the squares of two consecutive even numbers is divisible by four Huh? Here's eight Here's 10. Those are two consecutive even numbers What's eight squared eight squared Eight times eight Yeah, no calculator 10 squared What's 100 take away? 64 136 it's saying this number this answer will always be divisible by four does four go into 36 Prove that that will always work for any two consecutive even numbers These are tough We'll spend a lot of time next class talking about these In lamb and a pause there. I'm gonna nuke number four Can you turn back one lesson, please to lesson one inductive reasoning? We went through these want you to make sure didn't finish these last class It's like you hadn't started these make sure you finish those and then the remainder of class last day I gave out an investigation Let me hit pause Because I'm done the lesson part right click