 triangle. Pascal's triangle, but put your pencils down. I'm going to show it to you and then we're going to write it out. Pascal's triangle, it's a series of numbers, Nick, in the shape of a triangle and it starts out like this. At the top of your triangle, you put a one. And then all down the sides of the triangle, you continue to put ones. But the middle number comes from the two numbers above it added together. This will be a one. This will be a one. What number is going to go right here? Three. What number is going to go right here? This will be a one. By the way, in a second, you're going to draw your own and you may notice I've learned you want to be very wide. If you're not very wide, you're going to get really cramped in a hurry and you're not going to be able to draw it neatly or nicely at all. What's going to go here? Four, six. This is Pascal's triangle and it goes forever. You can draw as much or as little as you want to. But when people drew this, they started to notice all sorts of patterns. So are you ready? Let's draw a nice neat Pascal's triangle where it says the generation of Pascal's triangles. I'm going to go one, one, one. One, two, one. One, three, three, one, one, four, six, four, one, one, five, ten, ten, five, one. I'll just do the first six rows. This may not yet seem very profound, but actually it's hugely profound. So I'd like you, when you do your triangle, stop at the one, five, ten, ten, five, one row. What row number is that? Count carefully. Row number six. And row number six, what's the second number of row number six? Five. Row number five, what's the second number of row number five? Four. In other words, there is a way to glance at this and tell what row you're in. You have to add one to the second number. If you don't feel like counting all over the top. I'm actually going to make a little note here in a different color. I'm going to put a little arrow here and I'm going to say this is row six. What are some of the patterns that people have spotted? Well, the first one is fairly obvious. If you go down this row or you go down this row, what's the next number down here? Six, then. Okay, that's a pretty obvious one. Did anybody see any other ones? By the way, it's called Pascal's Triangle because Blaise Pascal, a mathematician, wrote a book on it, but he did not discover it. In fact, there is an ancient two and a half thousand, sorry, two and a half thousand year old inscription, I think, from China and it has Pascal's Triangle on it. People have been fascinated by this. First of all, it's easy to draw. Megan, it's kind of nerdily fun to draw. We get to add and make a triangle, but there's all sorts of patterns. Let me show you one. What do you see? What's one plus two? Sorry, where are you seeing one plus two plus three plus four? Which row? Which diagonal? Third diagonal? This one here? What's one plus three? What's one plus, sorry, what's three plus six? Nine? What's six plus 10? 16? Did you just say four, nine and 16? I've heard the numbers four, nine and 16 on a list before. What numbers do four, nine and 16 belong to? Really? What numbers do four, nine and 16 belong to boys and girls? Tally, not square roots. I got to be fussy. The other way, the perfect squares, the perfect squares. In other words, the perfect squares are hidden in here. Further down, perfect cubes show up as well. All sorts of weird stuff starts to show up, but what does that have to do with us? What row number? Can you on your calculator go six, choose zero? All of you. Yes, but we're going to do a bunch more. By the way, I like Kim's, isn't it just one? Yes, six, choose zero is one. Okay. What's six, choose one? Oh, you know what? Instead of six, let's try five, choose one. What's five, choose two? Try it. Ten, really? No. What's five, choose three? Is it also 10? What's five, choose four? What's five, choose five? In fact, row six has the five chooses. Here's how I remember that. It has the five chooses. And the reason, by the way, how many numbers are there in this row? The reason for five chooses you need six items is we have to include zero as a choice. Okay? This here is actually four, choose zero, four, choose one, four, choose two, four, choose three, four, choose four. And if you ever wondered, hey, how did they do this a hundred years ago with no calculators? They had in the back of their book trig tables. Remember, we talked about that for a trick, but they would have had a great big Pascal's Triangle probably going up to about row 20. And they would have limited their questions to 20 objects or less, but you would have had all the chooses there. Or really in a pinch, in 15 minutes, you could generate it all yourself by doing some fairly simple in your head edition. This is not counting. This is zero, choose zero. One choose zero, one choose one. Two choose zero, two choose one, two choose two. Three choose zero, three choose one, three choose two, three choose three. See it? Now we're going to try and write that algebraically. So are you ready? What's the relationship to N choose R? If we're in row six, this is the five here, how does that work? What's the relationship to N choose R? I'm going to call this row six and I'm going to call this term one, term two, term three, term four, etc. So if I want to find the fifth term in row 12, the fifth term in the 12th row, now look up for a second. Here is the first term in the sixth row. What is that? What choose what? So I'm just going to write this down. You don't need to write this down, but you said first term, sixth row was five choose zero. Is that right? What would the third term in the sixth row be? Five choose two. So can you spot the pattern? If I want the 12th row and the fifth term, it's going to be 11 choose four. One less, one less. And on your test battalion, as a multiple choice question, I'm going to say to you, tell me the 16th term in the 19th row, Pascal's Triangle. By the way, what is the fifth term of the 12th row? Go, what is it? Sorry? 330. What is the 18th term of the 30th row? What choose what? 29 choose 17. Now I do say memorize that, although if you blank out, you can draw this and figure it out, but it would take you about two or three minutes to derive it down. It's worth memorizing, one less, one less. By the way, what is 29 choose 17? So they do get fairly big eventually. It's funny, when I first did this about eight years ago, I asked for the 12th term of the 14th row, and I still had about half my kids write out the whole triangle. So the next year I said, okay, instead of the 14th row, I'll make it the 18th row. And I still had about 10 of my kids write out Pascal's Triangle. So the question has slowly gotten bigger and bigger every couple of years. I'm like, I want them to use the shortcut. I don't want to see a back page full of an entire triangle or a piece of scrap paper stapled to it where someone's written out 18 rows of Pascal's Triangle. We're trying to find shortcuts. We're trying to count without counting. Right? Then there is a symmetrical pattern. Here is five choose one. What's this? Five choose? Five choose one is the same as five choose four. And we talked about why the other day we said, look, if you're starting with five people, if you're forming groups of one, you're automatically forming groups of four, because that's who's left behind. And you'll have the same number of groups, whether you go one person and leave four behind or four people leave one behind. You can't form a group of one without forming a group of four. You can't form a group of three without automatically forming a group of two, if we're talking about five people. Okay? If we want to generalize that, we would say this, n choose r equals n choose. Can you look at this and write this algebraically, but instead of a one, put an r there, instead of a four, what would you put there? How did we get the four based on n minus r? As it turns out, for what it's worth algebraically, n choose r is the same as n choose n minus r. I don't memorize that. I know this, and I think you guys picked up on that last class. There is also a recursive pattern. Look up, I'm going to erase some of this triangle, and I'm going to write it in terms of choose. So what's this in terms of choose? Five choose what? Five choose zero, five choose one, five choose two. What's this guy? I'm going to erase this and put a five choose three there. You guys don't need to, I just want to spot the pattern. But where did this ten come from? It came from those two added. Oh, what is this in terms of a choose? I'll give you a hint. Four, choose what? Choose zero, choose one. Oh, this is four choose two, and what's this one here? So as it turns out, in terms of a recursive pattern, four choose two plus four choose three equals five choose three. Or five, five, let's go bigger numbers, Mr. DeWitt. So spot this, see if you can spot this next one then. Eight choose five plus eight choose six would be equal to nine choose six. See it? Big deal. Well, Matt, what this is saying is, don't forget how Pascal's triangle is generated. There is a recursive pattern. The next term came from the two above it. The next term came from the two above it. I'll be completely honest. I don't memorize the choose part at all. And in fact, I'm going to do something different this year. I'm not even going to write the choose part here. What you do need to know for Pascal's triangle is not the chooses that this choose this plus this choose this equals this choose this. What you do need to know for Pascal's triangle is that, hey, six and four gave you a ten. You need to know how to generate it numerically, not algebraically. Does that, do you understand the difference in vocabulary? No, you were late. Don't think about running water or water balloons bursting or anything like that. So for the recursive pattern, for what it's worth, this would be four choose two plus four choose three. But you know what, don't write it down. Cross it out. You don't need to know that. You need to know though that like, could you all find the next row for me? Yes. In fact, what Pascal's triangle questions am I going to ask you on the test? I am definitely going to ask you that one there. Find me the 12th term of the 19th row. It will be a multiple choice. Oh, I'll even tell you what the answers will, well for this question here, I'll even tell you what I'll use for my answers. One of the answers will be 12 choose five. One will be 12 choose four. One will be 11 choose five. One will be 11 choose four. They're correct though. There's an even nerdy or cooler use for Pascal's triangle and it comes up in irregular pathway problems. Next page over. Here I have for you two pathway problems. We call this one a regular pathway. We call this one an irregular pathway. It's difficult for me in English to explain what makes an irregular pathway, but can you see with this diagram what I mean by an irregular pathway? There's overlapping. Okay. Yeah. Okay. If we want to solve this, how many pathways if we're always moving right and down? Now the method that I gave you initially, don't write this down for a regular pathway problem. Don't write this down was this. Right, right, right, right, down, down, down, down. And we said that was what factorial over what factorial, what factorial? No. How many letters in this word? Eight factorial over. Four factorial, four factorial, but now by the way, by a nice coincidence that happens to be eight choose four, which is faster to type than eight factorial over four factorial, four factorial. So if this was a two by four rectangle, it would be six choose two or six choose four. You get the same answer no matter what. That method does not work here. By the way, what is the answer? What is eight choose four or what is eight factorial all over four factorial, four factorial? What do you get? 70? Write that down. So I'd like you to write down here eight choose four, 70. The problem is the factorial shortcut only works for regular pathways, not for irregular ones. So I want to show you a different way to get the answer. And somewhere along the way, maybe you might even all of you start to smile with your inner nerd because it's grade three math to solve a grade 12 question. But to do that, I need you to put your pencils down. And what I would like you guys to do is rotate your page 45 degrees so that A is vertically right above the B. Spin your page. I can't do it up here, but spin like so that so that the A is right above the like spin like that. Boy, that was a dumb move. Okay. Y'all done that? Rio? No, you haven't spun it yet. Spin it so that spin tilt your paper. So the A is right above the B. I'm serious. All of you. I'm serious. And then watch. So I can't do this. I got to turn my head and it makes it way, way tougher. But watch. I'm going to put a one there. I'm going to put a one there and a one there. I'm going to put a one there. You know what I'm going to put right here? Two. A one there. I'm going to put a one there. You know what I'm going to put right here? Three. Three. One. One. What's going to go here? Four. Six. Four. One. Let me zoom in. What's going to go right here? Don't write this down yet. Watch. Put your pencils down because you're going to do this yourself afterwards for practice. What's going to go right here? Five. What about here? What about here? What about here? What about here? Here? Here? Here? Pascal's Triangle. Now I would never do a regular one this way, but the nice thing is this will work for the weird one next to it when the factorials won't. So try this on your own now. And by the way, do you see why I said tilt the paper? It's way easier to work straight down. Trust me. Trust me. Try filling this in on your own without copying mine out. And then I'm going to try the great big one. And then I'm going to pull it off the screen and let you try the great big one on your own. That means my little corner of three ladies over there, when I do the great big one, you're not going to write anything down while I'm doing it. Then you'll try recreating it afterwards. Come on. That is nerdly cool though that it works. Now Jasmine, I'm not going to prove to you that it works for the irregular ones, but we're just going to say it worked for this. It will work for this as well. I really wish I could rotate this about 45 degrees. Sadly I can't. Except this is the way I've done this. I've done it as a JPEG. So no. In my word, original file, yes. And that's what I used to do, but I'm not desperate enough to try and fix that right now. Let's try the next one. Look up. Watch. Ready? Don't write this down. Just watch and see if you can derive this with me. One. What's going to go here? Yeah, by the way, what I'm really doing is saying, oh, and right here, there's an invisible number right there. What? Zero. I'm going zero plus what's it? One. One. One. Two. One. And always go one complete row at a time. Don't try and work ahead down one side. You'll screw up. What's going to go here? Here. Here. Here. It's going to go here. Four. What's going to go here? What number is right here? It's invisible. Zero and a 10. What number is going to go here? The 10 just slides down. 20. 10 plus zero. 10. 10 plus zero. 10. 30. 30. 10. 40. 60. 40. 100. 100. 200. Try it on your own. You want to go to the bathroom and miss this? Come on. About 10 more minutes, kiddo. What you have to be very careful of is don't make a dumb math mistake because that will end domino through the rest of your work. There is going to be on the written section of your test an irregular pathway problem because I have to admit that is very, that is way high cool on my nerdy list of, wow, who knew that Pascal's Triangle could solve pathway problems by adding. By the way, what we're really saying is because these are choices, you have that many choices to get to there, that many choices to get to there, that many choices to get to there, that many choices to get to there, so if we keep adding all the way down, we'll end up with a total number of choices to get to there. That's why it works. Can you do me a favor, please, and open your workbooks to page, page, it's lesson seven. Page 418. On page 418, we have a bunch of little pathway questions. Page 418, we're there. This here is a regular pathway problem because even though there's two different rectangles, can you see there's no overlapping squares? This is an irregular pathway problem as is this because you have missing or overlapping squares, so it's very tough for me to give you a definition, but I'm hoping you can see the difference between that and that. This one we can use a shortcut for. Now to do this one, what we're going to do is we're actually going to temporarily in our minds chop it right there. We're going to solve this as two separate rectangles. So this first rectangle, it would be right, right, right, down, down, down, down. How many letters in this word? Seven. So you could either go seven factorial over three factorial, four factorial, or you could say that just happens to be seven choose three, which is less typing. It's also seven choose four, so it doesn't matter whether you went down, down, down, down, down, right, right, right first. This rectangle here, I'll change colors, would be right, right, down, down. How many letters in this word? In fact, this one's going to be four choose two. What's the total? Listen closely, listen closely, listen closely. Starting here, I have to go through here and through here. What does and mean? Multiply. Incommodatorics. By the way, you can also solve that one with Pascal's triangle. That's faster, right? But Pascal would also get you there. What's the answer? Seven choose three times four choose two. How many different pathways are there for that one? Two hundred and ten? Is there, I keep hearing two and then I'm missing the second number. Two hundred and ten, I know there's more than two, okay, two hundred and ten. What about 3D? Do you notice we're starting at the back of the page and finishing at the front of the page? This one would look like this, right, right, right, down, down, forward. How many letters in this word? Grand total. Six factorial. How many Rs? Three factorial. How many downs? Two factorial. Do those two add to six? Then I can't use choose as a shortcut. The reason I could use choose as a shortcut here is because three plus four added to seven and that was seven choose three. This is not six choose three. This is six factorial over three factorial, two factorial, although I would go 12 on the bottom on my calculator. What is six factorial divided by 12? How many pathways? 60. So both of these, I was able to use factorial notation because they were regular pathway problems. Ah, the fun ones, the irregular pathway problems. Do you want to try A on your own or do you want to do it together with me or do you want me to do it with you with you not writing anything and then you try and recreate? Okay, let's do that. Ready? So I'll do this with you. If you want to try this solo, just ignore me and see if we get to the same spot. Now we're going from the train station to the football stadium. By the way, one nice thing about pathway problems is you get the same answer starting here and going this way and sometimes one direction is easier than the other. Same answer. One, one, one, one, two, one, one, three, three, one, one, four, six, four, five, ten, ten. Oh, what number is right here? It's invisible right here, not a one. What number is right here? It's invisible, zero, right? It's going to be four plus, you know what? A four is going to go there again because that's the confusing part for most kids, right? 15, 20, 14. What number is right here, Kim? It's invisible. So the 15 plus zero is going to give me a 15. 15 and 20 is going to give me a 35. 20 and 14 is going to give me a 34. 15 and 35 is 50, 69, 119. Now, if you didn't get 119, what you want to do is work your way backwards along the triangle and where you first spot, you see it, sorry, work your way frontwards along the triangle. First spot you see a number different from me is where you make your mistake. Is that okay? We can even nick with this method cut out squares for a park, like in B. Oh, by the way, did you guys rotate your book 45 degrees so that the train station you think I'm joking? Trust me, I've learned kids make way fewer mistakes because you're adding vertically, which is what you're used to from elementary school. So we're going to start the train station up here. 1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1. Now we're getting to the weird stuff. 1, 5, 10, 10, 5, 1, 1, 6, 15, 20, 15, 6, okay, okay. 1 plus 6, 7, 6 plus 15, 21, nothing, nothing, nothing, nothing, 20 plus 15, 35, 21, 28, what's going to go right here, Kim? 21 plus nothing, right? There's a zero here. So 21 is just going to drop down like a domino, and the 35 is just going to drop down like a domino, and 35 and 21, 56, 28 and 21, 49, 21 and 35, because both those line bleed here, 56, I think I just did that, didn't I? 35 and 56, 86, 91, 49 and 56, well 56 and 50 would be 116, so one less is going to be 105, why'd I go what? I can't because the 49 and the 91 don't join, right? I'm always just going to the two right above me diagonally. It's going to be the 56 and the 91. 56 and 91, 91 and 50 is going to be 141 plus 647. Final answer, 105 and 147, 247 is the final answer, 252, isn't that nearly cool? Pascal's triangle, I can get a little bigger if you want me to still, you sure? Uh-huh. It's always an empty spot next to Rio if you need to. By the way, these ones, those of you that tend to slough on the homework, these ones need practice because they seem easy, but you're going to find little weird curveballs along the way, and you'll figure them out once you hit them, but you don't want to be figuring them out for the first time when the quiz are done. In other words, I'm kind of saying for those of you that never do the homework tonight. Brent, can I make it smaller, Brent? Skip example four. What's your homework? First of all, turn back one lesson, please, to lesson six, to page 414. This is some Pascal's triangle questions, and all I'm going to assign right now is two and three from this unit, from this lesson. Find the first three terms of the ninth row. Ninth row is going to be, I think it's going to be 8 choose 0, 8 choose 1, and 8 choose 2. One of the last three terms in the 16th, I'll let you think about those, then turn ahead one lesson, please, to page 419. 1a, that's a regular one. You can use factorial shortcuts. 1b, that's two regular ones. Solve them each individually using factorial shortcuts, and then multiply your answers together because you got to go through one and the other. 2c, that is an irregular one. Pascal's triangle. 1c, sorry, 1c, 1c, that's an irregular one, so Pascal's triangle. I'll give you one 3d one. Now I've done the three dimensional ones with you. They've never showed up on the provincial, but I don't know why, because I think they're nerdily cool, so I've covered it just in case. I'm going to skip e. F is good. Now, do any, F is made up of three big rectangles. See them? Do they overlap at all? So you can do each one separately, multiply, multiply, multiply. You can do this one and this one and this one using factorial annotations, but then let's do the irregular ones. g is good, h is good, i, j, and k. Pathway problems and Pascal's grade three math solves grade 12 questions. You also have the take-home quiz.