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CCSS.Math:

Let's multiply 6 times 7,981. And the way we're going
to do it right now is just to represent or expand
out 7,981 as 7,000 plus 900 plus 80 plus 1. And so multiplying 6 times
7,981 is the same thing as multiplying 6 times 7,000
plus 6 times 900 plus 6 times 80 plus 6 times 1. You'd essentially
distribute the 6. And to help us keep
track of things, let me draw a little
grid right over here. So this is the 6,
and we're going to have to think about what
6 times 7,000 is, 6 times 900, 6 times 80, and 6 times 1. So I'll make a little square for
our rectangle for each of them. Let me do that. So here we go. And so we just need to think
about, what is 6 times 7,000? Well 6 times 7 is 42. So 6 times 7,000 is 42,000. 6 times 900, well once
again, 6 times 9 is 54. So 6 times 900 is 5,400. 6 times 80, well
80 is eight 10s. So 6 times 8 is 48, but since
it's six times 80 or eight 10s, this is going to
be 48 10s, or 480. And then finally, 6 times
1, of course, is equal to 6. So to find what this
product is, we just have to take the sum of
each of these numbers. What 6 times 7,000 is plus
6 times 900 plus 6 times 80 plus 6 times 1. So let's do that
right over here. So it's going to be 42,000
plus 5,400 plus 480 plus 6. And we get, let's see, in the
ones place, we just have a 6. In the tens place,
we just have an 8. In the hundreds
place, 4 plus 4 is 8. In the thousands
place, 2 plus 5 is 7. And then finally, the
ten thousandths place, we still have a 4. So we get 47,886. So this Is equal to 47,886. And what I encourage
you to do is to think about
how this is really underlining what
we're doing here. It's not that
different than what you might have done with the
traditional multiplication techniques. And this is a useful way of
thinking about it because now you really understand
what's going on. And actually, when
you start doing things like this in your head,
at least for myself, this is actually how I try
to tackle the multiplication problem. When someone says 6 times 7,981,
if I was just looking at this and I didn't have any
paper, I would say, OK, what's 6 times 7,000? I'd say, OK, that's 42,000. I'll try to remember that. What's 6 times 900? Oh that's 5,400. Well if I add that to
the 42,000, I get 47,400. Then, what's 6 times 80? 480. Have to add that to the
47,400 to get to 47,880. And then, what's 6 times 1? Well that's 6. Well add that to
the 47,880, which I've been keeping in my brain,
and that's going to be 47,886. So this helps you
understand what's really going on when you
multiply multiple digits, and it's a useful technique for
doing mental multiplication.