Multiplying two 2-digit numbers using partial products

In a previous video, we figured out a way to multiply a two-digit number times a one-digit number. What we did is we broke up the two-digit numbers in terms of its place value. So, the three here in the tens place, that's three tens; this is seven ones.

So, we viewed 37 sixes as the same thing as 30 sixes, three tens times six plus seven sixes, seven times six, and then we added those together to get a total of two hundred and twenty-two. What we'll try in this video is now see what happens if we try to do two digits times two digits.

So, let's try to tackle 37. Instead of six, let's multiply that times 26. So we're trying to figure out 37 26s is one way to think about it. So pause this video and see if you can tackle that and see if you could maybe use a similar method to what we used before.

Well, one way to think about it is you could view this as 37 26s, or you could view this as 37 sixes plus 37 20s. So first, we can do the 37 60s and 37 sixes, which is exactly what we did over here. We said, "Hey, that's the same thing," and we could do it in either order.

We could say, "Hey, let's first think about 30 sixes." So we're going to multiply 30 times 6. 36's is 180. So that right over there is 3 tens, 3 tens times 6. Then we could think about the 7 ones times 6, and so that's going to be 42; that's the 7 ones times 6 or the 7 6's.

Then we can do the same thing with the 20s. We could say, "Hey, what are three tens? What are thirty 20s going to be?" So let's write that over here. Or you could say, "What are three tens times two tens?" Well, that would be six times 10 times 10.

So let me write this down; this is 3 tens times 2 tens. That's what we're going to do now times 2 tens, and that's the same thing as 30 times 20, which is the same thing as 3 times 2 times 10 times 10. Well, that's going to be 600, so we could write that here, 600.

And just to be very clear, we've already thought about 37 sixes; that's these two numbers up here. We have to add that; we still have to add them. But now we're thinking about 37 20s.

So first, we thought about 30 20s, which is 600, and now we could think about seven 20s. So 7 20s is going to be 7 times 2, which is 14. So 7 times 20 is 140. So I'll write that right over here, 140. To be clear, this is 7 ones times 2 tens or seven 20s.

Now we can add it all together to get what the total would be. So, in total, we have— and that's why it's useful to have everything stacked by their place value. We could look at the ones place and say, "Okay, we only have a total of two ones here," so I'll put a two there.

Now, let's see, tens. We have eight tens plus four tens is twelve tens plus another four tens is sixteen tens. Sixteen tens we can also break up as one hundred and six tens. So we could write the six tens here and then put that hundred up here.

So, one hundred plus another hundred is two hundred plus six hundreds is eight hundreds plus one more hundred is nine hundreds. And so there you have it, this is equal to 962.

I really want you to understand what we just did. It might look a little bit complicated, but first we thought about what is 37 sixes—that's where we got these numbers from, and that's what we had done in a previous video.

Then we just thought about, well, what is 37 20s going to be? And that's where these numbers came from. And actually, let me write that down. This whole thing that I'm circling in orange, that is 37 sixes or, let me yeah, I'll write it as 37 times 6, and then this is 37 20s, so 37 times 20.

So if it's not obvious, pause the video or after this video, reflect on this: why this works? Try it with other numbers because if you really understand this, then your multiplication life—and actually your mathematical lives in your future—will only become more and more intuitive.