Video: Investigating Division and the Distributive Property Using Area Models

Liam and Isabella both had 24 squares. Isabella made a group of 20 and a group of 4. They divided their groups in half and still had the same amount of squares. What number is missing? 24 ÷ 2 = (_ ÷ 2) + (4 ÷ 2).

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Video Transcript

Liam and Isabella both had 24 squares. Isabella made a group of 20 and a group of four. They divided their groups in half and still had the same amount of squares. What number is missing? 24 divided by two equals what divided by two plus four divided by two.

In this problem, we’re shown that partitioning numbers can help us divide them. And we use something called the area model because we’re looking at the area of rectangles. Rectangles are a good way of showing how this works. So to begin with, we’re told that Liam and Isabella both had 24 squares. Then, we’re told that Isabella made a group of 20 and a group of four. In the diagram that follows, we can see 24 squares made in two different ways. The first group of 24 is shown as one rectangle. The 24 squares are altogether. It’s a rectangle showing four rows of six. And then, we’re shown 24 made up of a group of 20, that’s four rows of five, and a group of four.

We can see now whose shapes must be whose. We know that Isabella split her group into a group of 20 and a group of four. So hers must be the groups on the right. And Liam’s must be the group on the left. He hasn’t split it up at all. It’s important that we notice that in between both groups of squares, we can see an equal sign. Both children have the same number of squares. 24 all in one group is the same as 20 plus four. Next, we’re told that the children divided their groups in half. This is the same as dividing by two. So Liam did have one group containing 24 squares. But now he’s divided that group into two. And we can see this from the diagram. There’s a dotted line that’s cutting his shape in half.

Now, remember, Isabella had two groups, a group of 20 and a group of four. So when she cuts her groups in half, she needs to divide her group of 20 by two. And then, she needs to divide her group of four by two as well. Now, this expression has three different operations in it. So to help us see what’s going on, we can use brackets or parentheses. First, we’ve got 20 divided by two and then four divided by two. And again, on the diagram, we can see that the area of both shapes has been split in half using the dotted line. So the children both split their groups in half. And then, we’re told that they still had the same amount of squares.

Now, of course, this doesn’t mean that they still had 24 squares. They didn’t still have the same amount that they began with. What this means is that they still had the same amount as each other. And we can see this on the diagram. Liam has the answer to 24 divided by two. And this is labelled as 12. Although we can’t see the individual squares, this is the same as two rows of six. If you remember, he did have four rows of six. And now, that’s been halved. And although Isabella split her 24 squares into two groups, we can still see that she has the same amount as Liam. She’s halved her group of 20. 20 divided by two equals 10. And she’s also halved her group of four. Four divided by two equals two. And, of course, the value of 10 plus two equals 12. And we can see that if we put Isabella’s groups back together again, it would look exactly the same as Liam’s group.

Now, we have the part of the question that needs answering. What number is missing? 24 divided by two is equal to what divided by two plus four divided by two? What the question is getting us to think about is that if we can’t work out 24 divided by two. We can split 24 into two other numbers that we can divide by two. And then add the answers together. And it’ll still give the same answer. Liam worked out 24 divided by two. But Isabella split 24 into 20 and four. She then found the answer to half of 20 and half of four and added them together. 24 divided by two equals 20 divided by two plus four divided by two. The missing number is 20.

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