Multiplying 3-digit by 2-digit: comparing standard algorithm to partial products

0:01- [Instructor] So what we're gonna do in this video is try to figure out what 312 x 23 is. And we're gonna do it two different ways simultaneously to realize that they're really just doing the same thing. So the first way that we're going to do it, I would call that the standard way.

0:18Some people would call it the standard algorithm. So let's do it here, 312 x 23. So that's what we're gonna do here on the left. And then here on the right, we're going to do what's known as partial products. And actually we're gonna do a few partial products even in the standard way.

0:34But over here, let me just write it, 312 x 23. Now either technique, what we're going to think about is breaking up these numbers and multiplying each of the components times each other. So in the standard algorithm or the standard way, you start with the once place over here and you could first multiply it times the one place above.

1:00So 3 x 2 = 6. Then you could multiply 3 x this one, it's really 3 x 10. So you get 30. And then you could multiply 3 x this 300, which would be 900. So 3 x 312 is 936. Now in the partial products world, we just break that up. First, we say what is 3 x 2? Well that is going to be 6.

1:27What is 3 x 10? Well, that is going to be 30. And what is 3 x 300? Well, that is going to be 900. So notice, here I just added up the 930 and 6 altogether to get 936 when I'm doing it the standard way. And with the partial products, I've just kept them separate 3 x 2, 3 x 10, 3 x 300.

1:55Now if we do it the standard way, we would go to the 10s place on this bottom number, this 20. Now, since we're dealing with a 10s place, that's why we put this zero here while we're doing this standard way of doing it and we say, "Okay, what's 2 x 2?" We would get 4.

2:14Now notice when I write it down, it's actually 40 and that makes sense 'cause I really did 20 x 2, which is 40. Then I could say, what is 2 x 1? Well, that is just 2, but notice this 2 is in the 100s place and that makes sense because 20 x 10 is 200. And then last but not least, if I multiply the 20 times the 30, I could say that that is going to be 6,000.

2:39You might just say 2 x 3, but it's really 20 x 30, which is why the six is in the 1000s place. Now if I were to do it with the partial products technique, I would say 20 x 2 is 40. I would say 20 x 10 is 200, and I would say 20 x 3 is 6,000. So once again, I've just broken it all up.

3:04I have more partial products, while here, I only have two partial products, is one way to think about it. But then in either case, I just wanna add up all the partial products. On the standard technique, 6 + 0 is 6. 3 + 4 is 7. 9 + 2 is 11. I regroup one of those ones.

3:221 + 7 plus is 6. I get 7,176. Here, I do the same thing. Look at the one's place I got a little bit uneven, so I'm gonna make these lines to just make sure I add up the right places. So I get a six here, 6 + 0 + 0 + 0 plus a bunch of zeros. Then I have 3 + 4, I get a 7 here.

3:42Then I have 9 + 2. I have a one, regroup of one. 1 + 6 is 7, 7,176. And either way, in the partial products, I guess the pure partial products, I break up everything. I get six different partial products here while in the standard algorithm or the standard technique, I essentially take the partial products of 3 x 312 and 20 x 312.

4:05So I only have two partial products, but hopefully you see that these are very similar techniques.