Lesson video

Hello, my name's Mrs. Hopper and I'm really looking forward to working with you in this lesson on how to apply the distributive law to multiplication.

I wonder if you've come across the distributive law before.

Well, if not, we're gonna learn all about it and how it can really help us to be efficient when we're thinking about multiplications.

So if you're ready, let's make a start.

In this lesson, we're going to be using knowledge of the distributive law to calculate products using our known times table facts.

You might have been thinking about the distributive law and how we can link addition and multiplication together to help us to make problems easier to solve, so let's look and see how that works using products, using our times table knowledge.

So we've got three key words, well, one word and two key phrases.

So we've got partition, distributive law, and partial product today.

So I'll take my turn to say them and then it'll be your turn.

Are you ready? My turn, Partition.

Your turn.

My turn, distributive law.

Your turn.

My turn, partial product.

Your turn.

Excellent, I wonder if you've come across those words before.

Let's check what they mean 'cause they're going to be really useful to us in our lesson today.

I'm sure you've come across partitioning before.

Partitioning is the act of splitting a whole object or value down into smaller parts.

I'm sure you've partitioned numbers before, maybe using your knowledge of place value.

Let's see if we're going to be doing that today with our numbers.

The distributive law says that multiplying a number by a group of numbers added together is the same as doing each multiplication separately.

So, for example, we've got 4 X 3 here, but we can partition our 4 into 2 + 2 and we can do 2 X 3 + 2 X 3.

So we can split it apart to make our multiplication easier or we can find the common factor, add the other factors and make our multiplication simpler that way.

And a partial product is any of the multiplication results we get leading up to an overall multiplication result.

So if we did 2 X 3 + 2 X 3, we'd get two partial products of 6 and we'd add them together to equal 12.

Let's have a look at how those words are gonna help us in our learning today.

There are two parts to our lesson.

In the first part we're going to be looking at arrays and grids, and in the second part we're going to be looking at jottings and equations.

So let's make a start with some arrays and grids.

And we've got Alex and Jacob helping us in our lesson today.

So they're looking at this multiplication.

Alex says, "I find this a tough question to answer because I don't know my 12s very well." So he's not that confident with his 12 times table.

He might know his 9 times table, but he's not so confident with this fact.

Jacob says, "Me neither.

I wonder if we can use the distributive law." Ooh, this is interesting.

He says, "We could partition 12 and use times tables that we know." "Okay," says Alex, "Let's start by analysing 12." He's gonna look carefully at 12 and see what he can do with it.

Can you think what he might do with it? Ah, what's happened to 12 here? Alex says, "This shows 12 as 10 and 2, so it has been partitioned." So it's been split into two parts and the parts combined are equal to the whole, so 12 can be partitioned into 10 and 2.

Jacob says, "We can partition 12 in other ways, but I think this would be useful." I think he's right, there are lots of other ways we could partition 12 using our addition facts, but 10 and 2 gives us 2 times tables that we know really well, doesn't it.

Let's just check, though.

Can you tick the representations below that show 12? Pause the video, have a look and when you're ready for some feedback, press play.

Which ones showed 12? Did you get it? That's right, they all do.

We've got our 10 frame that we were looking at just before with the 10, and then another one with the 2, we've got some base 10 blocks there, we've got a 10 block and 2 1s, and then we've got a gattegno chart showing a 10 and 2, and 10 and two = 12, so all of those representations show 12 partitioned into 10 and 2.

Alex says, "Let's just use the counters for now." So he's got his counters in his 10 frames.

And he's taken the 10 frames away for just the counters, so we've got 9 X 12 but we've partitioned our 12 into 10 and 2.

There they are together, equaling 12.

And Jacob says, "We can use this representation to make an array." The expression is 9 X 12, so we need 9 rows of 12, so let's put those in.

Wow, that's a lot of counters, isn't it.

Alex says, "This shows 9 X 10 and 9 X 2." He's looking at the colours of the counters.

And Jacob says, "This array is showing 9 X 12." What do you think? Alex says, "I think we're both correct, I think they are too." "Yes," says Jacob, "but you've used the distributive law." Alex has used the fact that he can partition 12 and have 9 X 10 and 9 X 2, and he's shown that with the different coloured counters.

He says, "Let's label the array to help." So we know we've got 9 rows, and we know we've partitioned our 12 into 10 and 2.

Let's check our understanding.

Have a look at this array and can you label it in the same way, filling in those missing numbers? Pause the video, have a look at the array and when you're ready for some feedback, press play.

What did you see this time? How many rows have we got and how many are in each row? Well, there are six in each row and then we had 10 red counters across and 3 blue counters.

So we've got an array showing 6 X 10 and 6 X 3, or 6 X 13.

Back to Alex and Jacob's array.

How can we use this array to help us to calculate 9 X 12? Well, Jacob says, "This is 9 X 12 partitioned into 2 partial products." Ah, these were the bits we were going to do on the way to solving our whole equation.

Now we only need to know the 10 and the 2 times tables, because we can work out 9 X 10 and 9 X 2.

Alex says, "Can we use boxes to make it clearer and create a grid?" I think that's a good idea.

There's a lot of counters there, aren't there? Let's think about how we could represent this more simply.

So they've drawn boxes around the 2 sets of counters, and these two arrays are the partial products so we can clearly see 9 X 10 in one box and 9 X 2 in the other.

"Now we can work out the value of the two arrays, and we can use our known times table facts and then add them together." And Alex says, "These are times table facts I can recall." "I know the times table 9 X 10 = 90," so we can replace all those counters with the number 90, and 9 X 2 = 18 so we can replace all those counters with an 18.

Now, to add together 90 and 18, the partial products.

And he says, "I know that 90 + 18 = 108, so 9 X 12 = 108," because we've combined 9 X 10 and 9 X 2.

Time to check your understanding.

This was the one you looked at earlier and you filled in the missing numbers.

Can you complete the partial products by filling in the missing numbers using times tables that you know, and then combine them to get the product of 6 and 13? Pause the video, have a go, and when you're ready for some feedback, press play.

How did you get on? Well, our partial products are the number of counters in our 10 X 6 part, and in our 6 X 3 part.

So 10 X 6 + 60 and 3 X 6 = 18.

So now we can combine our partial products of 60 and 18, 60 + 18 = 78 so 6 X 13 = 78.

So let's have a look at 12 X 9.

"Uh-oh," says Alex, "Let's start arranging the counters again." But Jacob says, "We already know this one." What's Jacob spotted? Ah, he says, "Remember, multiplication is commutative so the factors can swap position and the product will stay the same." So 12 X 9 = 9 X 12.

"Oh, I see," says Alex, "so we've just worked out that 9 X 12 = 108 so it will have the same product." 9 X 12 = 12 X 9 and the other way round, but they both have a product of 108.

And Alex is just reflecting on what he did, so he sold 9 X 12 using distributive law by partitioning the 12.

He said, "These are the three stages that helped me to understand 9 X 12." So first of all he thought about 12 partitioned into 10 and 2, and he made two arrays of counters, 9 X 10 counters and 9 X 2 counters.

Then we drew the boxes around them, just to help us to see what we were looking at, and then we were able to replace the counters with our known multiplication facts, 9 lots of 10 = 90 and 9 lots of 2 = 18.

And when we combine those together, we have 9 X 10 + 9 X 2.

Our common factor there is 9, and 10 + 2 = 12.

So we had 10 lots of 9 + 2 lots of 9, which = 12 lots of 9.

So when we added the partial products we had a total of 108, which is the product of 9 and 12.

And Jacob said, "Yes, by using the distributive law we were able to calculate two partial products from our known times tables, and then combine them." Alex says, "Do we need an array every time?" Let's have a look, we've got 7 X 12 here.

He says, "I think the counters will take too long.

I feel confident enough just to use the boxes." Hold on Alex, let's have a look at how that will look 'cause that will be a lot of counters.

So let's have a think, the counters would go into these boxes, wouldn't they, in their arrays, let's think what would be represented in each one.

He says, "We'll use the distributive law by partitioning 12 into 10 and 2 again," 'cause he's confident with his 10 and 2 times tables.

So 12 will be partitioned into 10 and 2, and we know we've got 7 of each of them.

And if you imagine the rows of counters in there, we'd have 10 X 7 and 2 X 7.

And Jacob says, "I know the times table 7 X 10 = 70, And I know that 7 X 2 = 14." So if you can picture those counters in there, we'd have had 7 rows of 10 and 7 rows of 2, we'd have had 70 counters all together and then 14 counters all together.

And Alex says, "I know that 70 + 14 = 84, so 7 X 12 = 84." And it's useful to think that we can always work out our 12 times table by partitioning our 12 into 10 and 2, and adding together the partial products that we know really well.

7 X 12 = 84.

He says, "I've seen this strategy before, called the grid model," or the grid method, and you might have done some multiplication like this.

It's a really useful way to see how the numbers you are multiplying together can be partitioned to help you to multiply more easily.

Time for you to do some practise.

Can you match the arrays to the expression that represents them by drawing a line between them? And for question two, can you fill in the missing numbers on the arrays and then use them to calculate the product? So you're going to fill in the missing numbers, work out your partial, and then combine them to make the product overall.

And then in question three, it's kind of over to you, there are some multiplication here and you are going to think about how you would partition them to create partial products to work out the answers more easily.

These may be times table facts that you do know, but it's useful to use facts that we do know when we're learning to think about them in a new way.

And for question four, you're going to find the product for each equation and you're going to use the distributive law by calculating partial products in the grid model.

So this time you're going to draw the grids for yourself as well.

Pause the video, have a go at these four questions and when you're ready for the answers and some feedback, press play.

How did you get on? So in A you had to match the arrays to the expressions that represent them.

So let's have a look at what the first one was.

So we had four rows and we had 8 in each, so we had 4 X 8.

So straightforward, no partitioning to do there, really, because 4 X 8, we probably know our 4 times table, and we can work things out by perhaps doubling, finding 8 X 2 and then doubling it for 8 X 4.

What about the middle one? Well, we've got five rows there, we've got 10 columns of red counters and 1 column of blue counters, so we've got 5 X 10 and another 5, or 11 X 5, which is the same as 5 X 10 and another 5.

And what about the last one? I think we've got 10 and 2 again and we've got 6, so we've got 6 X 10 and 6 X two, which is 6 X 12 altogether.

So there we go, our 6 X 12 has been partitioned and we've used the distributive law to collect together 6 X 10 and 6 X 2.

So 11 and 12 were both partitioned using the distributive law.

Ah, and there are those 2 calculations we were looking at, so can we fill in the missing numbers and calculate them using partial products? So for the first one we could see that 11 had been partitioned into 10 and 1, and we were multiplying them both by 5.

10 X 5 = 50, 1 X 5 = 5, and we can combine those 50 + 5 = 55.

And I expect you knew that 5 X 11 was 55, but it's useful to see numbers that we know in a way that we perhaps haven't arranged them before, and using a method that we haven't used before.

And in B we had 6 X 12.

This time we partitioned our 12 into 10 and 2 and we had 6 lots of 10 and 6 lots of two.

6 X 10 = 60, 6 X 2 = 12, 60 + 12 = 72, so 6 X 12 = 72.

For question three you were going to think about the partitioning for yourself this time.

So we've got 6 X 11.

So we can partition the 11 into 10 and 1, and then we've got 6 X 10 = 60 and 6 X 1 = 6, and 60 + 6 = 66 so 6 X 11 = 66.

And then 12 X 5, well, we can partition our 12 into 10 and 2 and we've got 5 X 10 and 5 X 26, which gives us 50 + 10 which = 60, 12 X 5 = 60.

And finally for question four, you were drawing the grids for yourself or the boxes.

So 7 X 11, we possibly know this is 77.

Let's prove it using the distributive law.

So we can draw our grid to show that 11 can be partitioned into 10 and 1, we've got 10 X 7 and 1 X 7, which when we add them together = 77, so 7 X 11 = 77.

And for 12 X 4 we could partition our 12 into 10 and 2, so that gives us 4 X 10 and 4 X 2, partial products are 40 and 8 and 40 + 8 = 48, so 12 X 4 = 48.

Well done if you've got all those right.

And on into the second part of our lesson, we're looking at jottings and equations.

So we've calculated 7 X 12 = 84.

Alex says, "I think there's a quicker way than using the grid model." He says, "I can use jottings." Let's have a look at what he does.

And Jacob says, "Show me this by redoing this calculation then." He says, "I'll start with the expression and then write an expression for each partial product added." So our first grid is representing 7 X 10 and our second part of the grid is representing 7 X 2, so we can rewrite 7 X 12 as = 7 X 10 + 7 X 2.

And this is the distributive law, we can partition one of our factors and multiply each of those parts by our common factor, so we know we are multiplying by 7.

So 7 X 12 = 7 X 10 + 7 X 2, which = 70 + 40, and then he's going to calculate the sum of his partial products, which is 84, and that's the product overall.

So 7 X 12 = 84, but we've calculated by using the distributive law, partitioning our 12 into 10 and 2, multiplying each of those by 7 to create our partial products, and then adding them together.

So it's the same as the grid, but this time we've written out the expressions rather than the grid.

Time to check your understanding now.

Can you fill in the missing numbers in the equation below using the model to help you? So can you turn that grid model into Alex's new way of recording this using jottings? Pause the video, have a go and when you're ready for some feedback press play.

How did you get on? So what is it we were calculating? We were calculating 6 X 13 and we partitioned the 13 into 10 and 3.

So we had 6 X 10 + 6 X 3 which = 60 +6 18, which = 78, so 6 X 13 = 78.

And you can see our grid areas there represented by 6 X 10 and 6 X 3 in our equation in our jottings.

So we've got our array, we've got our grid, and we've got our equation, all representing 7 X 12.

Which is the most efficient method? Well, Alex says, "The array helps me to visualise but it takes too long." So he likes the array because it's helped him to see what's happening when we partition and find those partial products, so it's a really important step on the way but it does take quite a long time if we've got a lot of equations to solve.

But Jacob says, "I like the grid method 'cause it's visual." He likes to be able to see what's happening with his partial products, and that's absolutely fine.

And he says, "It helps my thinking so I don't make errors." He can really see clearly what's going on there, he's not just relying on numbers to represent what's happening.

Alex says, "I like the equation because it's much quicker." Well that's great Alex, as long as you know exactly what that equation is representing.

So some of you may still feel that the grid helps you to see what's happening, but it's really important that you can see where the grid links to the equation so that you know, like Jacob says, be able to tell if you've made a mistake.

Alex says, "I'm going to solve this using equations and a jotting." And Jacob says, "Yes, I'll do the same, and then we can compare." Okay, let's have a look and see how they get on.

So we've got 12 X 12 this time.

So 12 X 12, Alex says, = to 12 X 10 + 12 X 2.

So he's partitioned his 12 into 10 and 2, and he's multiplied the other 12 by the 10 and by the 2.

So that gives him two partial products of 120 and 24, and when he adds them together he gets the answer of 144, 12 X 12 = 144.

Ah, we've got another one here.

12 X 12 = 12 X 6 + 12 X 6, is that right? Is that the same? Well it is, isn't it, because what Jacob's done is partitioned one of his 12s into 6 + 6, so he's got 12 X 6 plus 12 X 6, and he knows that 12 X 6 = 72 so he is got 72 + 72, which is also = of course to 144.

So they both worked out that 12 X 12 = 144.

What's the same and what's different? Well Alex says, "We both got the same product and both used the distributive law." But Jacob said, "Yes, you partitioned 12 into 10 and 2, whereas I used 6 and 6." And Jacob says, "That gave us different partial products with the same sum." 120 + 24 = 144 and 72 + 72 = 144.

Alex says, "I don't fully understand, let's draw an array to help me." I think that would help us to see it clearly, wouldn't it? So there are the two jottings or equations written out.

So there's Alex's array and there's Jacob's array.

We've got our 12 partitioned into 10 and 2, giving us 120 and 24, but here we've got our 12 partitioned into 6 and 6, giving us partial products of 72 and 72.

"Now I see," says Alex, "We partitioned differently, creating different partial products." So although Alex likes the equations, the grids there and the counters really helped him to see what was going on in the two different ways of calculating 12 X 12.

Who was the most efficient, though, do you think? Hmm, that's an interesting question, isn't it.

Alex says, "I preferred mine because I'm confident with multiples of 10." And 2 I hope as well, Alex.

And Jacob says he liked his because he could use the skill of doubling, he knew 12 X 6 and so he could double it, 12 X 6 and another 12 X 6.

So each one would be efficient depending on where your strengths lie in your times table knowledge.

Time for you to do some practise, can you match the equations and jottings with the array to showing the partial products? For question two, can you use jottings and equations to solve the following with the distributed law? And there's an example there to help you.

In fact, it's Alex's way of calculating 12 X 12.

And for question three, complete the table showing the 12 times table calculated using the distributive law.

We've got the first row has been completed, 4 X 12 is in there.

You're going to get your partial product expressions, your partial product totals and add them together, and then find the overall product.

And for question four, complete the table once more but this time partition the factors differently each time to give you a different partial product.

And, again, we've done the first one for you.

This times 4 X 12 is 4 X 6 + 4 X 6.

How else can you think about partitioning 12 in order to give you different partial products? Pause the video, have a go at your questions and when you're ready for some feedback, press play.

How did you get on? Let's look at question one.

You had to match the equations and jottings with the array showing the partial products.

So in the top one we've got 8 rows and we've got 10 lots of 8 + 2 lots of 8, so we've partitioned our 12 into 10 + 2.

10 lots of 8 + 2 lots of 8 = 12 lots of 8, and that's 80 + 60 which = 96.

So that matches with the top set of equations.

And in the bottom one we've got the same number of counters, but this time we've got 8 rows but we've partitioned them into 6 and 6.

So we've still got 8 X 12, but this time we've got 6 lots of 8 + 6 lots of 8, which gives us our 12 lots of 8, and that's 48 + 48 which = 96.

In question two you were going to use jottings and equations to solve the following using the distributive law, so A is 11 X 6.

So we can partition the 11 into 10 and 1, so we've got 10 lots of 6 + 1 lot of 6 which = 11 lots of six, that's 60 + 6 which = 66.

11 X 6 = 66.

And then we've got 12 X 6.

Well, we know that 12 X 6 = 10 X 6 + 2 X 6, 10 + 2 = 12, so we've got partial products of 60 and 12 which = 72.

But as Jacob says, "You might have partitioned differently, so have different partial products." So you might have partitioned 11 into 5 and 6, and then you'd have had 5 lots of 6 which is 30 + 6 lots of 6 which is 36, and 30 + 36 = 66; so you might have partitioned differently.

Let's look at question three.

So, again, you might have partitioned differently, this is how we partitioned to get our partial products to find our product overall.

So 5 X 12 we used 5 X 10 + 5 X 2, so we partitioned our 12 into 10 and 2.

We did the same for 6 X 12, 6 X 10 + 6 X 2, 7 X 10 + 7 X 2, 8 X 10 + 8 X 2 and 9 X 10 + 9 X 2.

So can you see that our partial products were 40, 50, 60, 70, 80 and 90 for our 10s, and then 8, 10, 12, 14, 16, and 18.

So we added an extra 10 and an extra 2 each time, which is another 12 each time, which is how our times table works, isn't it, and those are the products for the 12 times table.

How might we have partitioned differently to use the distributive law? Well, in the first one we said 4 X 12 = 4 X 6 + 4 X 6, so we partitioned our 12 into 6 + 6, which gave us partial products of 24 + 24 which = 48.

And again, we've done the same, we've partitioned 12 into 6 + 6, so we've got + X 5 + 6 X 5 or 5 X 6 + 5 X 6 which = 30 + 30, which is 60.

Do you remember Jacob liked using this for the doubling, didn't he.

Same again for this one, this time, though, we've got 6 X 6 + 6 X 6, 36 + 36 = 72.

7 lots of 6 + 7 lots of 6, so that's 12 lots of 7 altogether, or 14 lots of 6.

Again, we've partitioned our 12 into 6 + 6, so we've got 6 X 8 + 6 X 8 which = 12 X 8, 48 + 48 = 96.

And we've partitioned our 12 again into 6 + 6, so 9 lots of 6 + 9 lots of 6 or 6 X 9 + 6 X 9 gives us 12 X 9, that's partial products of 54 and 54 which = 108.

But as I say, you may have partitioned your 12 in different ways.

And we've come to the end of our lesson.

We've been using knowledge of the distributive law to calculate products using known times tables.

So what have we learned about today? Well, we've learned that the distributive law says that multiplying a number by a group of numbers added together is the same as doing each multiplication separately.

So we were partitioning one of our products to create partial products using timetables that we knew.

When faced with tricky times table, one of the factors can be partitioned to split the multiplication into 2 partial products which are known timetables, and that can be really useful to help us when we're calculating.

And this can be represented using an array or a grid model, and also calculated with these.

But a quicker method is to use jottings and equations, but that only works when we are really confident that we know what we're doing.

So those arrays and grid models, especially the grid models we can draw out, can really help us to see what's going on.

Thank you for all your hard work and your mathematical thinking today, and I hope I get to work with you again soon.

Bye-Bye.