Visual representations of decimal multiplication

So we have here on this number line that we've now marked off with the tenths, and you can see that this is three tenths. Here we can think about this as a multiplication of a decimal. And so what is this representing? I'll give you a hint: it's representing something times three tenths. So pause the video and try to think about that.

Well, let's see. We are going one times three tenths, two times three tenths, three times three tenths, and then four times three tenths. So what's represented here is four times three tenths. And so what is this going to be equal to? Well, you can see you go from three tenths to six tenths to nine tenths, and then you could do this as twelve tenths. But twelve tenths is the same thing as one, one and two tenths. So you could view this as one point two, one and two tenths.

Let's do another example now. Actually, I'll do it on the same number line. If we wanted to represent three times three times 0.2, what would that look like on this number line and what would this be equal to? So I'll put a little equal sign here. Pause this video and see if you can figure that out.

All right, so let's think about where two tenths is. This is 1 ten, 2 ten is right over there; this is 0.2, and we're going to multiply it times 3. So we're going to multiply it times 1, then we're going to multiply it times 2. That takes us to 4 tenths, and then we're going to multiply it times 3 to get us 2 six tenths, 0.6. So it's 6 tenths, just like that.

Now you can also visualize two tenths as parts of a whole. So for example, this represents two tenths. I have this whole; the square is a whole. It's split into ten equal columns here, and we have two of them filled in. So this represents two tenths. So if you have three times two tenths, well this is one times two tenths, this is two times two tenths, and this is three times two tenths.

And so how many tenths do we now have? Well, we have one, two, three, four, five, six tenths, which is exactly what we have here: six tenths.

Let's do one more example that gets a little bit more involved. So here we're told to multiply. It says you may use the model shown to help find the product, and this is a screenshot from the exercise on Khan Academy. So pause this video and see if you can figure out what this is.

All right, so they're saying fifty-two hundredths times 3, and they have fifty-two hundredths depicted right over here, and then they have it depicted three times. So the total number of hundredths depicted here, that is fifty-two hundredths times 3, because we have fifty-two hundredths here, another fifty-two hundredths, and then another fifty-two hundredths.

So how many hundredths is that going to be? Well, you could view this as fifty-two times 3, and that will give you the number of hundredths we have. So let's think about this. So if we were to just say fifty-two times 3, well this is going to be 2 times 3, which is equal to 6, and then 5 tens times 3 is 15 tens, which is the same thing. We could either just write it as 15 tens or that's 105 tens.

But either way, if I have fifty-two of something and I multiply that by 3, I now have 156 of that something, and here the something is hundredths. So if I say fifty-two hundredths times three, that's going to be 156 hundredths.

And how do we represent 156 hundredths? Well, there are a couple of ways to think about it. If this is the ones place, this is the tenths place, this is the hundredths place. Well, we would write the 6 there, the 5 there, and the 1 there. So you could recognize this as, "Hey look, a hundred hundredths!"

Let me color code it: a hundred hundredths is the same thing as a whole, and I'll circle that in red. Fifty hundredths is the same thing as five tenths, and of course, six hundredths is the same thing as six hundredths.

So this is going to be equal to, we could say one point five six, or you could view this as a hundred fifty-six hundredths. So you could view this as a whole, which is a hundred hundredths, and five tenths, which is fifty hundredths, and six hundredths.