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.
