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Lattice multiplication

Sal introduces lattice multiplication. Created by Sal Khan.

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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 27 times 48. What you do is you write down your 27. The 2 and the 7 are going to get separate columns and you write your 48 down the right-hand side, and then you draw a lattice. This is why it's called lattice multiplication. So the 2 is going to get its own column. The 7 is going to get its own column. The 4 is going to get its own row and the 8 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. 7 times 4 is 28. 7 times 4 is equal to 28. So you write down a 2 and an 8 just like that. 2 times 4 is equal to 8. So you write down a 0, 8 just like that. Then you have 7 times 8. 7 times 8 is equal to 56. So we write down a 5 and a 6. And then finally, 2 times 8 is equal to 16. You write down a 1 and a 6 just like that. And we're done 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 1's diagonal, you just have a 6 sitting here. So you write a 6 just like that. Then we move over to the next diagonal. This diagonal with the 6, 5, and 8 in it. That's our 10's diagonal. So 8 plus 5 is 13. 13 plus 6 is 19. So you write your 9 right here in the 10's place and now you carry the 1 in 19 up there into the 100's place because this isn't just 19, it's actually 190. It's nineteen 10's. Anyway, you carry your 1. You have 1 plus 2 is 3. 3 plus 8 is 11. 11 plus 1 is 12. You write the 2 in your 100's place and you carry the 1 into your 1,000's place. 1 plus 0 is 1, so you just have a 1 in your 1,000's place just like that. And you get our answer. 27 times 48 is equal to 1,296/ Let's do a harder problem. One that requires more digits, just to show that this'll work for any problem. Let's say we had 5,479 times-- let's do a three-digit number. Times 787. So just like we did in the last time, we make four columns. One for the 5, the 4, the 7, and the 9. We'lll have 5,479 and then times 787. So they each get their own row. 787. 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 got their own row. One row for the 7, one row for the 8, and one row for this other 7. Then we have to draw the diagonals. Draw it like that. One diagonal, two diagonals, so three diagonals, four diagonals. I think you get the idea and than we have just one, two more diagonals. We're ready to multiply now. So it's 9 times 7. I won't do it on the side here. We know our times tables. 9 times 7 is 63. 7 times 7 is 49. 4 times 7 is 28. 5 times 7 is 35. Let me switch callers arbitrarily. 9 times 8 is 72. 7 times 8 is 56. 4 times 8 is 32. 5 times 8 is 40. I'll switch colors again. 9 times 7-- we saw that before. It's 63. 7 times 7 is 49. 4 times 7 is 28. And then 5 times 7 is 35. We're done 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 1's place. Just have a 3 there, so you write the 3 in your 1's place. You go to the 10's place. 2 plus 6 is 8. 8 plus 9 is 17. Write the 7 in the 10's place, carry the 1 into the 100's place. I wrote a 1 really small here. 1 plus 3 is 4. 4 plus 7 is 11. 11 plus 6 is 17. 17 plus 4 is 21. 21 plus 8 is 29. Write the 9 in the 100's place and carry the 2. 2 plus 6 is 8. 8 plus 9 is 17. 17 plus 5 is 22. 22 plus 2 is 24. 24 plus 2 is 26. 26 plus 5 is 31. Carry the 3. 3 plus 4 is 7. 7 plus 8 is 15. 15 plus 3 is 18. 18 plus 0 is 18. 18 plus 3 is 21. Write the 1, carry the 2. 2 plus 2 is 4. 4 plus 5 is 9. 9 plus 4 is 13. Write the 3, carry the 1. 1 plus 3 is 4. And we're done. 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 4,311,973. There you go. Now in the next video we're going to try to understand why this worked.