 So, we ended up using a little bit of trigonometry to set up the grid in the last video. What we're going to do in this video is learn a multiplication table for a 10 by 10 grid. And the reason that I'm doing this video is because someone posted a comment requesting this video to be made. They actually wanted it to be a 12 by 12, but I think 10 by 10 is, well, it's the one that I work with. It's the one that I get everyone to learn. The rest of it sort of flows, 11 by 11, 12 by 12 grid, sort of flows after this, right? And it becomes quite easy to calculate to figure out what it is once you know your 10 by 10 grid, 10 by 10 multiplication grid. And this is, you know, I can't emphasize this enough. I never thought about making this video until the request was put in on all the places I believe it was the video, the ASMR video where I was showing you my grandmother's back camera board, right? But it is something that I ask all my students to know. And most of the time I assume they know this multiplication table. When I'm working with a student or a group of students, if I find out that they don't, you know, they're struggling with the multiplication table, what I end up doing is stopping everything like I put on the full brakes. And we go back and I make sure they know the multiplication table, okay? And the way you're going to learn this is, we're going to go through it right now, but keep this in mind, what you're going to do, what I'm going to ask you to do, what you need to do to learn this table is once we finish generating it, you're basically going to copy it down, right? Then what you're going to do is you're going to take that copy that you have and you're going to try to generate it, you know, by memory or by understanding, really, because multiplication is just an expansion of addition, right? If you wanted to add 15 twos together, you could either go two, two, two, two, plus two, plus two, plus two, plus two, plus two, plus two, plus two. You could write it out 15 times or we come up with a short-hand, you know, a simpler way to do it. We would go two times 15, right? So multiplication is a way for us to do mathematics more easily, right? It's sort of something, a process that came up, right? That we need it because we're doing higher level mathematics. We're getting into more complicated stuff, right? 100 times 100 means 100 plus 100 100 times, right? You're not gonna sit there and write out 100 plus 100 plus 100 plus 100 100 times. You're gonna go 100 times 100, okay? That's the reason why you really need to know multiplication because it makes calculations easier, right? Now, keeping that in mind, this is the way you're going to quotation marks memorize this table or understand this table, learn this table. After we're done doing it, you're gonna take the copy that you have and you're gonna generate it, right? By memory or by generating it by thinking about what they are, right? And you're gonna have this one, the original copy as a reference. The next time you're gonna generate it, you're gonna turn this original copy over and try to generate it by memory or generate it, period, right? Wherever you get stuck, you have a problem, right? You don't know what it is, you can't calculate it. What you're gonna do is you're gonna flip the original over and find out what it is, right? The third time you're gonna do it, you're gonna put the original away on one side. You're gonna take a blank piece of paper and you're gonna generate this thing from beginning all the way to the end, right? Without looking at the original, right? It doesn't make a difference how long it takes you. I'm not at the speed here, right? You have to know how to do it. If you can do it, if it takes you a long time to do it, it doesn't make a difference. All that means is it's in your brain and it's new to you, right? You're gonna use this a lot. It's gonna be everywhere, everywhere that you do mathematics. So I'm not worried about you getting enough practice to do multiplication. It's an understanding of you, sort of a you understanding how to generate it and what it means, okay? So to do the 10 by 10 multiplication table, what we need is a numbers here, right? What we're gonna do is go one to 10 down this way and one to 10 this way, okay? And what we're gonna do is we're gonna put those things on there with posted notes, okay? That way we have our grid set up with the proper axes, right? So what I'm gonna do is I'm gonna do these and should we do it in black or blue? Black or blue? Got two different color pens here. Let's do the axes in blue. So I need one, two, three, four, five, six, seven, all the way to 10, twice, all right? What I'm gonna do is I'm gonna lay these down here. Here comes out one enough, one, sure. We're gonna put on our glasses, does he see? I need to put all my glasses when we close up. See, that's not bad. One, two, three, we're gonna go all the way through. So we're gonna go three, right? And then we're gonna split these guys up right now. So we have our, I guess that's the y-axis, down to 10, right, the vertical. Now we need to go across. Again, we need one to 10. Let's see that, right? Yes, you do. Grab the black pen and we're gonna fill this in, all right? And the way this works is all we do is we match columns with rows and figure out what they are. Now, again, I mentioned this. Multiplication is an extension of addition. I will talk about how that's related to this when we're generating it, right? The one thing you have to really appreciate is that there's a total symmetry in the multiplication table, right? And the symmetry goes diagonally. So everything here, if you go on a diagonal line, oops, 45 degree diagonal line, right? Everything on this side repeats over here. So what we're gonna do is, should we do the top or the bottom? Let's do, I usually do this side first. So let's do this side, okay? So what we're gonna do is we're gonna go do this and then do this up to where it's the same number multiplied by itself. So we're gonna go one times one, two times two, three times three, four times four. So we're doing a perfect squares right away. So we're gonna create the diagonal going across first. Okay. And this is again, this is something that the perfect squares that you should know inside out, right? Super, super important, super powerful. You're not using it a lot, right? So what we ask ourselves is what's one times one? That's obvious, that's one. One times one is one. Now we could go one times all these numbers and just this column is gonna be, all these guys just transpose over, right? Because one times anything is itself, right? Where that thing was. But we're gonna do the diagonal first. Two times two, and that's the way you read this, right? You go, okay, what do I wanna multiply two by, right? I wanna multiply two by, I can multiply by one, two. I can multiply by anything this way, right? So we read the numbers this way, we multiply them this way. Conversely, we could read the numbers this way and multiply them by this way. Because multiplication is, what's that thing called? There's a word for it where it doesn't make a difference which order you do it in. If you go two times three, it's the same thing as three times two, right? I should have looked this up, I forget what the term is for it, right? So what we're gonna do is do the diagonal first. Two times two, it means two plus two, right? Two times two is four. We're gonna go three times three. Again, multiplication is an extension of addition, right? So what that really is, is three plus three plus three. It's easier to go three times three, right? Three times three is nine. Four times four is 16. Five times five is 25. Six times six is 36. Seven times seven, 49, 49. If you have a hard time doing these multiplications, remembering these multiplications, remember, we put out two videos using hand tricks to learn your multiplication table. One of them was for the nines, how you multiply the nines, which is really easy. And another one was a hand trick where you can multiply six, seven, eight, nine, 10 with six, seven, eight, nine, 10, right? So when you get up into the higher numbers, if you have a hard time remembering this stuff, there is a video out there that I put out, you know, two videos showing you the trick for just using your hands to learn this multiplication table, right? Eight times eight, right? You connect those guys up, 64, right? Nine times nine is 81. 10 times 10 is 100. So we've got 81, 100. Now, the way it works is this side, this line here, oops, how do we do this? This line here mirrors this, is the mirror line. So everything here would flip over this way, right? So what we're gonna do is we're gonna generate the bottom part first. One times anything is whatever the number is, right? So one times two is two. One times three is three. One times four is four. So all we're gonna do is just copy these numbers down here and that's what we're gonna do. Seven, that's easy enough. Sheets on this one, let's grab a new stack. We could go one times two is two, right? But I don't wanna go on that side yet. I wanna generate these guys first, okay? So what we're gonna do, we're gonna go three times two. Three times two is the same thing as three plus three, So if we're going down this column, if we're going this way, all we're doing is we're adding three every time. So three times two is gonna be six. Three times three is nine, which basically means three plus three plus three, right? This is basically an arithmetic sequence, right? We haven't talked about this stuff. It's basically an arithmetic sequence, right? The next one would be 12. The next one would be 15. The next one would be 18, right? You're adding three every time on the rows. On the columns, you're adding whatever that guy is. Four plus two is six. The next one is gonna be eight, right? Well, four times two is eight is the same thing as two times four. If we're multiplying, we're gonna go four. We're going this way. You're adding four every time. Eight, 12, 16, 20, 24, 28, keep on going, right? If you don't wanna learn it initially as a multiplication, learn it initially as an addition, right? If you're going down a column, you're adding whatever the top of the column tells you to add. If you're going across a column, across a row, you're adding every jump is adding whatever this number is. The trick is to learn this whole thing and know how to, instead of starting from here to figure out what, what are these numbers? Six, where is this? Six, what do we have? One, two, three, four, five, six. Six times seven is, you don't wanna go six times one is six and then add another six, 12, 18. You don't wanna do that. You wanna go directly from six times seven, right? You wanna know exactly what that is without having to do every step, right? That means you've learned your multiplication table. To generate it, use the addition if you want, if it helps you out, but slowly learn how to jump around this table everywhere, right? So what we're gonna do is, we're gonna continue with our rows here without crossing this over, okay? Four times two is eight. Four times three is 12. Four times three means add four together three times, four plus four is eight, plus four is 12. Four times four, 16. Adding four fours together, you get 16. Let's go to the fifth column, or the number five column. Let's put the posted notes here. Five times one was five. Five times two is 10. Five times three is 15. Five times four is 20. Oops, 20, do this, kill this. If I was doing this with the chalk, like the language of mathematics, I would have to go over the way I'm doing it here. With posts, I know it's easy, I just remove, replace, right? So this is 15, this is 20. Five times five is 25, right? Six, six times two is 12. Put the posted notes up, it'll stick these up. Six times two is 12, six times three is 18. Six times four is 24. Six times five is 30, right? And then six times six is 36, right? Let's do the whole thing. Let's do, let's do seven. Seven times one, seven. Seven times two, 14. Seven times three, 21. Seven times four is at seven to 21, you get 28, right? Seven times five at seven to 28, you get 35. Seven times six is 42. Seven times seven is 49. Seven, seven's added together, gives you 49, right? Let's do the eight column, or eight row. Eight times two, 16. Eight times three, 24. Eight times four, 32. Eight times five is 40. Eight times six is 48, you just add eight more to 40, right? Eight times seven is 56, you add eight to 48, right? These ones are usually the ones that people have the hardest time with, okay? Let's do the nine column, or nine row, right? And for the nine rows, remember we had, we learned that hand trick, right? Where you hold up your hands and bend your finger over and figure out what the single digit, single numbers are times nine, right? So for nine, we got nine times two is 18. Nine times three is 27. Nine times four is 36. Nine times five, right? You hold out five, you got 45, right? 45. Nine times six, you add nine to this, right? So 54. Nine times seven is 63. Nine times eight is 72. Nine times nine is 81. The 10 is easy. You just add zeros to the end of those numbers. Desi system is amazing. I think it's called the desi system anyway, with multiples of 10. I actually found out today, the house that we're in right now is from 1915, right? So it's a really old house, solid house, but squeaky, okay? So 10, multiply all those numbers by 10. It's just a zero added at the end of each one of those numbers, right? 10 times 20, 10 times two is 20. 30, 40, 50, 60, 80, 90. That's the bottom half of the multiplication table, right? All that happens now, if you want to generate the other half, is you can just flip this to the other side. If you can visualize it anyway, because for multiplication, there's a word for it that it doesn't make a difference if you go one number times another number or this number times that number. The answer is the same, right? Two times three is six. Three times two is six. It's the same number, right? Five times four is 20. Four times five is 20. Doesn't make a difference. So that's what happens. That's why this thing mirrors this, right? So if we had five times four, five times four is 20. Well, four times five is also 20, right? Four times five is also 20. Let's see what else we got. Let's go eight times two. Eight times two is 16, right? Well, two times eight is 16. Two times eight is 16. Nine times three is 27. Well, three times nine is 27. Three times nine is 27. Let's go for a number here. I don't know. Let's go eight times five is 40. Well, five times eight is 40. That's one random way you can generate a jump, right? The other way is you can just go across. Let's do the number third, the third row. Let's say you want to generate the third row. You can do the jumps if you want. You can do multiplication if you want, if you've memorized it, if you've learned it. Or you can go across on the rows and just add three to the number before it, right? Three plus three is six, plus three is nine, plus three is 12. And 12 also appears here as well, right? 12 plus three is 15. Where's 15 appear? 15's right there. So what you end up having is the three row is going to be the same as the three column, which is super neat, right? If you go across the three row, what's the number here after 15? Three times six is 18, right? Three times six is 18. What's the next number here? The next number here is 21. What's the next number? 24. What's the next number? 27, and the next number is three. Cemetery again, right? The rows are the same as the columns. If you go down, the numbers repeat, right? Again, it's a mirror. Let's go down, I don't know. Let's do the six, okay? So let's lay it out, put our sticky mark, stickies here, and paper here, so we can write it all out. So what do we got? This is six, and this is six, right? So all I gotta do is copy these guys down. This is 42. This guy's 48. This guy's 54. This guy's 60. Should we do the rest? If you're practicing this, you should try to see if you can generate this. The number eight column only has two missing gaps, so I'll say three missing gaps, I guess, but so let's fill this one up. What do we got? We want the eighth column, so we can just go down to the eighth row. Well, one times eight is eight. That was easy. After 24, between 24 and 40, between 24 and 40 is 32. That's what goes up here, 32. Between 48 and 64, between 48 and 64 is 56. 56. So seven times eight is 56. That's the same thing as eight times seven is 56. The one row is easiest. Whatever that was there, just plop style, right? Two, three, four, seven, eight, nine, 10. The 10th column is easy. They're just going up by 10. So 10, 20, 30, 40, 50, 60, 70, 80, 90. Easy. Let's do the ninth column. Why not? It's got how many holes? One, two, three, four, five holes. We can fill this one up easy. The ninth column is the same thing as the ninth row, right? Nine times two is 18. That's the same thing as two times nine, 18. What else we got? After 27, we got 36 and then 45, right? 36, 45. We have 54 and we got 63 and 72, right? So 63 and 72, right? Seven times nine is 63. Eight times nine is 72, right? You should be able to jump on all the numbers anywhere on this table. Let's fill in these four. What do we got? We have, this is the sixth column and we want after 18, 24, 30, right? Six times four is 24. Four times six is 24, 30. Six times five, five times six is equal to 30. All we got left is two links here. I'll do those ones, right? What is that? That's the seventh column. Let's go down the seventh row. We got seven, then we need 14. It's between seven and 21, right? Seven, 21, that's gotta be 14. After 21, we have two blanks, right? The seventh column, the seventh row. 21, 28, 35, 28, 30. That's four, two. Two times two, two times one is two. Two times two is four. You just add two every time, right? Four, six, eight, 10, 12. Which also happens to be what the columns are, right? I think we have it all. Looks like it. Super important for you to know your multiplication table. It's your ABCs of mathematics, right? What mathematics is pretty powerful compared to natural languages. So there's a few more alphabets, letters, rudimentary stuff that you need to learn, such as moving around an equal sign, how to multiply and divide and subtract, how to deal with fractions, right? Exponents of radicals. But this is step number one to learn your multiplication table. I hope this helped, okay? If you wanna know how we set up this grid, we used a little trigonometry that was a previous video. In the next video, what we're gonna do is we're gonna take down all these numbers and I'm gonna show you a little 10 by 10 math puzzle game that a student showed me a few years ago, a long time ago, and I like that it was fun to do, fun to play. We just wanted to meditate and chill, okay? That's it for now. I'll see you guys in the next video.