Question Video: Using Area Models to Understand How to Solve Multiplication Problems Using the Distributive Property and Subtraction Facts Mathematics • 3rd Grade

Chloe is learning about different strategies to multiply. She drew this to help her calculate 9 × 4. What expression is missing in 4 × 9 = (_) − (4 × 1)? What is 4 × 9?

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Video Transcript

Chloe is learning about different strategies to multiply. She drew this to help her calculate nine times four. What expression is missing in four times nine equals what take away four times one? What is four times nine?

There are lots of different methods or strategies that we can use to multiply numbers together. And in this question, Chloe is using one of them. We’re told that she wants to find the answer to nine times four. And we can see that she’s drawn a rectangle on squared paper to help her. Now, before we move on and start to answer the questions, let’s take a moment to look at this rectangle and understand what Chloe is doing here to try and work out the answer to this multiplication.

The first thing that we can notice is that Chloe’s drawn one large rectangle and then split it up into smaller parts. So, it seems like she started off drawing a rectangle that’s four squares tall and 10 squares wide. But wait a moment. If we do a four-by-10 rectangle or an array like this, we’d expect it to show the multiplication four times 10 or 10 times four, not nine times four. Why would Chloe use a rectangle like this to help work out the answer to nine times four? Well, Chloe knows two things that can help her.

Firstly, she knows what 10 times four is. She can remember it really quickly. But importantly, she knows how to use this to help. She knows that 10 times four is only one lot of four more than nine times four. And that’s why her diagram shows this big area nine times four here, this is the answer she’s trying to find, and then one lot of four at the end here that she doesn’t need any more. In other words, she knows that nine times four is four less than 10 times four. Let’s cross out four counters to show these on our array too.

Now that we can see what Chloe is doing in breaking up 10 times four into different parts, let’s complete the first part of our question. What expression is missing in four times nine equals what take away four times one? Well, as we’ve just said, to find the answer to nine times four or four times nine as it’s written here, Chloe’s first going to use her knowledge of the 10 times table to help. Because nine, of course, is very close to 10. She’s going to find the answer to four times 10 and then take away four times one. The missing expression is four times 10.

In the final part of the question, we’ve just got to do what Chloe was doing. We need to use her method to find the answer. What is four times nine? We know that four times 10 is equal to 40. And we need to take away four times one which, of course, is four. And 40 subtract four is 36.

This question’s being really interesting to answer because although we know we often can split a much harder multiplication into two easier multiplications and add those two multiplications together, in this question, what we’ve done is start with an easier fact that we know already and split it up so that we have to take away to find the answer that we’re looking for. It’s still a very good example of what we call the distributive property of multiplication where we can partition a multiplication fact to help us.

To find the answer to nine times four or four times nine, Chloe realized she could use the answer to four times 10 to help her. She just needed to work out 10 lots of four take away one lot of four or, in other words, four times 10 take away four times one. The expression that was missing in that first sentence was four times 10. So, four times nine is the same as 40 take away four, which equals 36.

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