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Sal introduces lattice multiplication. Created by Sal Khan.
Video transcript
I'm going to do a couple of lattice multiplication examples in this video, and in the next one we'll try to understand why it worked. Let's say we're trying to multiply twenty-seven times forty-eight. What you do is you write down your twenty-seven. The two and the seven are going to get separate columns and you write your forty-eight down the right-hand side, and then you draw a lattice. This is why it's called lattice multiplication. So the two is going to get its own column. The seven is going to get its own column. The four is going to get its own row, and the eight is going to get its own row. Now the fun thing about lattice multiplication is you get to do all of your multiplication at one time, and then you can finish up the problem with all your addition. You don't have to keep switching gears by carrying and all of that. Although there is carrying, but it's all while you're doing the addition step. So we're almost done with our lattice. We actually have to draw these diagonals here. We'll understand in the next video why these diagonals even work. Just like that. And now we're ready to multiply. Seven times four is twenty-eight. Seven times four is equal to twenty-eight. So you write down a two and an eight just like that. Two times four is equal to eight. So you write down a zero, eight, just like that. Then you have seven times eight. Seven times eight is equal to fifty-six. So we write down a five and a six. And then finally, two times eight is equal to sixteen. You write down a one and a six, just like that. And we're done with all of our multiplying. Now we're ready to add. So what you do is you go down these diagonals that I drew here. So this first diagonal, which is really the ones diagonal, you just have a six sitting here. So you write a six just like that. Then we move over to the next diagonal. This diagonal with the six, five, and eight in it. That's our tens diagonal. So eight plus five is thirteen. Thirteen plus six is nineteen. So you write your nine right here in the tens place, and now you carry the one in nineteen up there into the hundreds place, because this isn't just nineteen, it's actually one hundred ninety. It's nineteen tens. Anyway, you carry your one. You have one plus two is three. Three plus eight is eleven. Eleven plus one is twelve. You write the two in your hundreds place and you carry the one into your thousands place. One plus zero is one, so you just have a one in your thousands place, just like that. And you get our answer. Twenty-seven times forty-eight is equal to one thousand two hundred ninety-six. Let's do a harder problem. One that requires more digits, just to show that this will work for any problem. Let's say we had five thousand four hundred seventy-nine times-- let's do a three-digit number-- Times seven hundred eighty-seven. So just like we did in the last time, we make four columns. One for the five, the four, the seven, and the nine. We'll have five thousand four hundred seventy-nine and then times seven hundred eighty-seven. So they each get their own row. Seven hundred eighty-seven. Looks like that. Then we have to draw our lattice. Draw the lattice. Each of these guys get their own column. Draw the columns just like that. And then each one of these characters get their own row. One row for the seven. One row for the eight. One row for this other seven. Then we have to draw the diagonals. Draw it like that. One diagonal, two diagonals, three diagonals, four diagonals. I think you get the idea. And then we have just one, two more diagonals. We're ready to multiply now. So it's nine times seven. I won't do it on the side here. We know our times tables. Nine times seven is sixty-three. Seven times seven is forty-nine. Four times seven is twenty-eight. Five times seven is thirty-five. Let me switch colors arbitrarily. Nine times eight is seventy-two. Seven times eight is fifty-six. Four times eight is thirty-two. Five times eight is forty. I'll switch colors again. Nine times seven-- we saw that before. It's sixty-three. Seven times seven is forty-nine. Four times seven is twenty-eight. And then five times seven is thirty-five. We're done with all of our multiplying. Now we can switch our brains into addition mode. Let me find a nice suitable color for addition. Maybe a pink will do for addition. So we start at our ones place. Just have a three there, so you write the three in your ones place. You go to the tens place. Two plus six is eight. Eight plus nine is seventeen. Write the seven in the tens place, carry the one into the hundreds place. I wrote a one really small here. One plus three is four. Four plus seven is eleven. Eleven plus six is seventeen. Seventeen plus four is twenty-one. Twenty-one plus eight is twenty-nine. Write the nine in the hundreds place and carry the two. Two plus six is eight. Eight plus nine is seventeen. Seventeen plus five is twenty-two. Twenty-two plus two is twenty-four. Twenty-four plus two is twenty-six. Twenty-six plus five is thirty-one. Carry the three. Three plus four is seven. Seven plus eight is fifteen. Fifteen plus three is eighteen. Eighteen plus zero is eighteen. Eighteen plus three is twenty-one. Write the one, carry the two. Two plus two is four. Four plus five is nine. Nine plus four is thirteen. Write the three, carry the one. One plus three is four. And we're done! It's that easy. Well, there's two advantages here. One is, we got to do all of our multiplication at once. And then we got to do all of our addition at once. The other advantage is it's kind of neat and clean. When you just do it the traditional way with carrying and number places, it takes up a lot of space. But notice, we did our whole problem in a nice, neat and clean area like that and we got our answer. Our answer is four million three hundred eleven thousand nine hundred seventy-three. There you go. Now in the next video, we are going to try to understand why this worked.