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Multiplying 1-digit numbers by multiples of 10, 100, and 1000

Lindsay finds a pattern from multiplying 1-digit numbers by multiples of 10, 100, and 1000.

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Video transcript

- [Voiceover] Let's multiply four times 80. So we can look at this a few ways. One way is to say four times, we have the number 80. So we have the number 80 one time, two times, three times, four times. Four times we have the number 80. And we could do this computation, add all of these, and get our solution. But let's look at it another way. Let's try to stick with multiplication. And one way we can do that is to break up this 80. We know a pattern for multiplying by 10, so let's try to break up this 80 to get a 10. So if we have four times, and instead of 80, let's say eight times 10. Because 80 and eight times 10 are equal; those are equivalent; so we can replace our 80 with eight times 10. And then we have this times 10 back here which is super helpful 'cause there's a nice neat pattern in math that we can use to help us with the times 10 part. So let's start to solve this. Four times eight is 32. And then we still have 32 times 10. And then we can use our pattern for multiplying by 10, which is that anytime we multiply a whole number times 10 we take that whole number, in this case 32, and we add a zero to the end. So 32 times 10 is 320. And there's a reason that pattern works; we went into it in another video, but here just real quickly, 32 times 10 is 32 tens. And we can do a few examples. If we had, say, three times 10, that would be three tens, or a 10 plus another 10 plus another 10, which equals 30: our whole number with a zero on the end. If we had something like 12 times 10, well, that would be 12 tens. And if we listed out 10 12 times and counted 'em up, there would be 120, it would add up to 120, which again is our whole number with a zero on the end, or 12 with a zero on the end. So we can use that pattern here to see that 32 times 10 is 32 with a zero on the end. Let's try another one. Let's do something like, let's say 300 this time, we'll do hundreds instead of tens, times six. 300 we can break up, like we did with 80 in the last one, and we can say that 300 is 100 three times, or 100 times three. And then we still have our times six after that. So these two expressions, 300 times six and 100 times three times six, are equivalent 'cause we replaced our 300 with a 100. And then from here we can multiply. And let's start with our one-digit numbers. Let's multiply those first. Three times six is 18. And then we still have 18 times 100, or 18 hundreds, so we can write that as 18, and then to show hundreds we'll put two zeros on the end, or 1800. Just like up here, just like we saw that 300 is equal to three times 100, or our three with two zeros on the end, same thing here. 18 times 100 is 18 with two zeros on the end, or 18 hundreds. So 300 times six equals 1800. Let's try another one, but this time let's go even another place value and try thousands. Something like seven times 7,000. So like in the previous ones we're gonna break up our thousands. 7,000 is the same as seven times 1,000. 1,000 seven times. And we still have our times seven in the front here to bring down. And again, we can multiply our single digits first, our one-digit numbers. Seven times seven is 49. And then 49 times 1,000 is going to be 49,000, which we can write as 49, and this time, maybe the pattern's becoming clear, we're going to have three zeros on the end, so it'll be a 49 with three zeros, or 49,000. Just like up here, seven times 1,000 was a seven with three zeros, 49 times 1,000 is a 49 with three zeros, or 49,000. Let's look at this as a pattern. If we show this as a pattern, let's do something like nine times 50. And then in another one let's do nine times 500. And one last one we can do nine times 5,000. I encourage you to pause here and see if you can work these out. See if you can come up with solutions for these three expressions. Now we can work them out together. Nine times 50 will be the same as nine times five times 10, 'cause we broke up our 50 into a five times 10. And then if we multiply across, nine times five is 45, and to the end we're going to add one zero. The pattern for times 10 is to add one zero. We can keep going here. Nine times 500 will be nine, and then times five times 100. 500 is five hundreds, just like 50 was five tens. Multiplying across, nine times five still equals 45, but this time we will add two zeros to the end, or 4500. And finally, nine times 5,000 will be nine times five times 1,000; because 5,000 is five thousands, or 1,000 five times. Working across, nine times five, 45. And this time we add three zeros, so 45,000. So when we multiply each of these expressions we can see the only thing that changed was the number of zeros on the end. So the pattern: anytime we multiply a whole number times 10 we add one zero to the end of our number; anytime we multiply a whole number times 100 we'll have two zeros; and times 1,000 we'll have three zeros. And once we know that pattern, we can use it to help us with questions like this where initially we don't see a 10; a 100; or a 1,000; but we can get one; we can break up or decompose these numbers to get a 10 or 100 or 1,000 to help us solve the problem.