Which area model represents the expression four times 735?
We have five possible answers to this question. And they’re five different area models. To help us answer the question, we need to think about what an area model is. And how it works. And how we can use it to find the answer to a multiplication. We know that the area model is a way of showing a multiplication. And it’s based on a fact we already know, which is that the area of a rectangle is its length multiplied by its width. So we could expect a rectangle that has a width, we could call it its height, of four and a length of 735 to have an area where both numbers are multiplied by each other.
Now, if we were trying to calculate the answer to four multiplied by 735, we might like to split 735 into smaller parts to make it easier. And this is exactly what’s happened to the area models that we can see in our answers. The one large rectangle has been split into smaller rectangles. It’s a way of showing how 735 can be partitioned to help us. All of our area models have a width of four, but not all our area models have a length of 735. For example, we could look at the third one here and say to ourselves, “Well, we can see 735 at the top. This must show four multiplied by 735.” But really, what this area model shows is four times seven, four times three, and four times five. Seven plus three plus five is actually equal to 15. This area model shows four times 15.
So how can we split up the number 735 to make it easier to multiply? Well, we can partition it into its hundreds, tens, and ones. The hundreds digit in 735 is a seven. This has a value of 700. So we need to look for an area model where the first rectangle’s length is worth 700. If we quickly looked through them, we can see that only the first two area models fit this. Our correct answer is going to be one of these. Next, we need to think about the tens digit in our number. If we partition 735 into hundreds, tens, and units, the tens are going to be three tens. And three tens have a value of 30. Which of our two possible answers shows a rectangle where the three is worth 30?
Again, both of them do. So far, both area models look the same. But there’s one difference. And to identify which is correct, we need to think about what the digit five represents in the number 735. But we know it represents five ones or five. The correct area model is the one that partitions 735 into 700, 30, and five. And we can use this model then to find the answer. First, we’d multiply 700 by four and write the answer here. Then, we’d multiply 30 by four and write the answer here. And finally, we’d multiply five ones by four and write the answer here. Then, we add our three products together or the three areas. And this is how we’d find the answer to four multiplied by 735.
The correct area model is the one that shows that 735 has been partitioned correctly. In other words, it’s the one where the width or the height of the rectangle is four. And it shows three separate rectangles with lengths of 700, 30, and five.