Visual understanding of regrouping decimals

What we're going to do in this video is explore place value involving decimals, and in particular, we're going to think about how you can regroup value from one place to another. This is going to be very useful later in your life when you start doing some more arithmetic with decimals.

So let's first think about what this number is right over here. Each square represents a 1 or represents a whole. So what number is this? Well, you could see that we have three whole ones, so we could write three. Then we have, right over here, a whole that is divided into ten equal bars, these vertical bars, and so each of these are a tenth. Four of them are shaded in, so it's three ones, four tenths.

In this part right over here, we've divided a whole into a hundred equal sections; it's a 10 by 10 grid, and we can see that one, two, three, four, five, six, seven of them are shaded in. So this represents seven hundredths. This is three point four seven, or three and forty-seven hundredths. Now what we're going to do is explore how are there other ways you can put the value into the different places.

Let me set up a little table here. I don't think I'm going to need all that space, but we have, in this first example, our ones place, our tenths place, and our hundreds place. I had to curve up a little bit, and what we just did is say hey, this was pretty straightforward. This was three ones, four tenths, and seven hundredths.

But are there other ways that we could look at it? For example, is there a way of reimagining this? Instead of three ones, we have two ones and still have seven hundredths. So how many tenths would we have in order for it to be the same value? Pause this video and think about that.

All right, we'll do it together, and to help us, I will put what we had here just now. Now what's different? Instead of having three ones, we now have two ones. So we could say that these are our two ones right over here, our two ones. We have our seven hundredths right over here, so essentially we would have to express all of this in terms of tenths.

How would we express it all in terms of tenths? Well, this what used to be a 1 is equivalent to 10 tenths. Let me make that very clear. I could see—let me see if I could shade this in with that green color. So there you go, I'm gonna show you with the green color. Then let me draw a bunch of lines here to make it very clear.

So I'm just gonna hand-draw it: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. Did I do that right? 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. I didn't draw it as straight as I need to; they really should be 10 equal sections. But it was hand-drawn, so I think you understand.

Notice I took the exact same value, but I have regrouped this one right over here into tenths. So how many tenths are here? Well, now I have 10 plus 4 tenths. So now I have shaded in fourteen tenths. That was interesting!

Let's do another example. Now let's imagine another scenario. Let's imagine a scenario where we have three ones again, but this time instead of having seven hundredths, we have 27 hundredths. So in that circumstance, how many tenths would we have? Pause this video and see if you can work it through.

All right, well let's get that same number again and now let's think about how we might have to regroup between the places. So we have our three ones, so that was just like the first case right over there. So we have our three ones, but now we have 27 hundredths.

So in addition to these seven hundredths, we have to find another 20 hundredths someplace. Well, the most natural place to go would be right over here, and hundreds is the same thing as two tenths. So what we want to do is convert these two tenths into hundredths.

Let me actually just shade it in a little bit. So I'm going to convert this right over here to hundredths. And so they did that right: one, two, three, four, five, six, seven, eight, nine, ten. There you have it. This plus this gives us our twenty-seven hundredths.

So how many tenths do we have left? We have two tenths, two tenths. So in this situation, we regrouped from the tenths place and expressed them as hundredths. Two tenths became 20 hundredths added to the seven hundredths that you already have, and we now have 27 hundredths. Hopefully, that makes sense.