Lesson Video: Distributive Property of Multiplication Mathematics • 3rd Grade

In this video, we will learn how to use the distributive property of multiplication to multiply numbers up to 10 × 10 by breaking the numbers apart.

15:19

Video Transcript

The Distributive Property of Multiplication

In this video, we’re going to learn how to use what we call the distributive property of multiplication. We’re going to use what we find out to help make multiplying two numbers together a lot easier by breaking one of those numbers apart. Do you ever find that some multiplication facts are easier to remember and use than others? In this video, we’re going to look at something we’re gonna do to help us with those trickier multiplication facts. We’re going to look at how we can break up a fact into the some of smaller, hopefully easier, facts.

There’s a little phrase that comes up again and again in maths, and it’s so helpful. It’s the theme of this video really, and that phrase is “I may not know something, but here’s what I do know, and here’s how I can use it to help!” And in this video, the way that we’re going to be using multiplication facts that we do know to help find facts that we don’t know is by applying something that we call the distributive property of multiplication, which might sound complicated, but in actual fact it’s quite straightforward. Let’s go through an example.

Let’s imagine that we’ve been asked to find the answer to six times eight. This is one of those multiplications that can be quite tricky to remember. Here’s what six times eight looks like as an array. To help us find the answer, we could make this calculation a lot easier by breaking one of the numbers up. Let’s imagine that you find multiplying numbers by eight quite tricky, so we’ll pick that number as the number we want to break up.

Now, there’s more than one way we could partition the number eight. But we need to be careful here because we want to make our calculation easier. If we’re not sure about our seven times tables facts, there’s no point partitioning eight into seven and one. What about five and three? Let’s get two colored pencils and color in our array so that it shows six times five and six times three. Now, the first thing to check is, can you see that the array is just the same size as it was before? We’ve just colored it in differently.

The orange part shows six times five. The red part shows six times three. That’s six times eight altogether. We’ve broken up six times eight into two facts that are easier for us to deal with. Six times eight is the same as six times five plus six times three. Perhaps we’d better find out the answer then. We know that six times five is 30. Six threes or three sixes are 18. And if we add together 30 and 18, we get the total 48. And so, we know that six times eight is 48.

But what if we’d have split up the calculation a different way? Does the distributive property work however we split up a number? Well yes, it does work. We just need to choose numbers that are easier for us to use. If we take eight again, we could think to ourselves, “Wow, I know my four times table really well, and I know that four plus four is eight. So, I know that six times eight is equal to six times four plus six times four.” And because you’re so good at your four times table, you know that six times four is 24. So, the answer is going to be 24 plus 24, which is the number we’re expecting, 48.

And it doesn’t even matter which number we partition. Let’s keep the eight this time and let’s split up the six. How about if we split up the six part of our calculation into four and two? Our array would look like this. And we could say that the sum of four times eight and two times eight are going to be the same as six times eight. Now, I guess for this to make things easy, you’ve got to know what four times eight is, don’t you? Four eights are 32. Two eights is a little easier. That’s 16. And 32 plus 16? You guessed it.

So although the title distributive property of multiplication might sound complicated, it’s actually quite easy to understand. We’re using facts we already know to help find facts we don’t. And we can do this by breaking up one of the numbers in the multiplication into easier chunks. It’s a great strategy to use if you want to make a multiplication easier. Let’s have a go at putting it into practice.

Use the model to help you multiply. Write the multiplication expression that is missing. Eight times five equals what plus three times five which equals 25 plus 15 which equals 40.

This question is really interesting because although it’s all about finding the answer to a multiplication, we’re actually given the final product at the end. Eight times five equals 40. We don’t need to find this final answer. Instead, we need to find a multiplication expression that’s missing in our working out. And to do this, we need to understand what’s going on here. What method are we using to try to find out what eight times five is? As well as all the written calculations, we’re given a model and we’re told to use it to help us multiply. Let’s look at it carefully to see why it’s being drawn like this.

Firstly, we can see that the rectangle is eight squares tall and five squares wide. And although we can’t see the individual squares inside the rectangle, we know how many it would cover. It’s the same as eight times five. So, the whole of this rectangle represents the multiplication that we’re trying to find out. But we can see something else about this rectangle. It’s been split up. Here, at the bottom, we have a rectangle that it’s three squares tall and five squares wide. This represents the calculation three times five, and the rectangle above it is five squares tall.

Well, we say rectangle, but it’s also five squares wide. This is a square. And we could see a question mark in the middle of this part of the diagram. What calculation does this part represent? Five squares tall and five squares wide means that it represents five times five. And this diagram shows us that we don’t need to know the fact eight times five to find the answer to eight times five. We could split it up into two easier facts: five times five and three times five. Let’s complete that multiplication expression then. Five times five plus three times five. And we can see we got the right answer because we can follow through with the working out. Five times five is 25; three times five is 15. And if we add the two together, we get 40.

We’ve used what we call the distributive property of multiplication here. We’ve broken apart a multiplication fact into the sum of other multiplication facts. Eight times five is the same as five times five plus three times five. Our missing multiplication expression is five times five.

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.

Look at Ethan’s work. He used eight equals five plus three to write eight times three as the sum of simpler products. Eight times three equals five times three plus three times three. Find another sum of products that is equal to eight times three. Two times six plus six times three, three times three plus six times three, two times three plus six times three, or two times two plus six times three.

One of the useful things about multiplication is something called the distributive property. This means that if we’re like Ethan and we come across a multiplication that perhaps we don’t know the answer to, we can split it up into the sum of simpler products, in other words, two multiplication facts that are a little bit easier that we can add together to find the fact that we want to know.

Now we can see that the multiplication fact that Ethan wants to find out is eight times three. But perhaps he’s not great with his eight times table because he split up the eight part of eight times three into five and three. And because five and three make eight, instead of working out eight times three, he just needs to work out the answer to five times three and three times three and then add the two together.

And this is what we can see in his number sentence. Eight times three equals five times three plus three times three. Here’s what eight times three might look like as an array. And by splitting the number eight into five plus three, we can see that Ethan split up the larger calculation into smaller ones that are hopefully easier to work out.

Now, this question doesn’t ask us to use Ethan’s method to find the answer to eight times three. In a way, we’re asked to use his method but differently. We need to find another sum of products that’s equal to eight times three. Now, the first thing we can say about what Ethan’s done is he split up the number eight, but he’s kept the number three in his calculation. Can you see he’s working out both five times three and three times three? Both of these simpler products are still multiplying by three, aren’t they? It’s only the eight that’s been split up.

And if we look at our possible answers, only two of them could be correct. Only this one here and this one here show two products that involve multiplying by three both times. We’ve got three times three plus six times three, and then we’ve got two times three plus six times three. The answer has got to be one of these two. So, we need to ask ourselves, “Ethan’s split up the number eight into five plus three. How else could we split this number up?”

For example, eight is the same as four plus four. Then, we get the calculation four times three plus four times three. But that’s not one of our possible answers. How else could we split up the number eight? What about six and two? Then we could answer the question by finding the sum of six times three and two times three. Now, if we look carefully at our possible answers, we can see that one of them shows this addition. The multiplications may be the other way around. But we know we can add two values together in different orders and they’d still make the same answer. We know that eight equals two plus six. And so, we also know that eight times three can be solved by working out two times three plus six times three.

What have we learned in this video? We’ve learned how to use the distributive property of multiplication to multiply numbers by breaking one of the numbers apart.

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