Lesson video

Hello, my name is Mr. Tazzyman, and I'm really looking forward to teaching you a lesson today from the unit based on understanding and representing multiplicative structures.

Get yourself sat comfortably and ready to learn 'cause it's time to start.

Here's the outcome for the lesson then.

By the end, we want you to be able to say I can explain which is the most efficient factor to partition to solve a multiplication problem.

These are the key words that you're going to hear during this lesson.

I'll say them, I want you to repeat them back to me just so that you know what they sound like.

I'm gonna say my turn, say the word, and then I'll say your turn, and you can repeat it back.

Okay? All right then, let's go for it.

My turn.

Multiples.

Your turn.

My turn.

Partition.

Your turn.

My turn.

Distributive law.

Your turn.

Here's what each of those words means then.

A multiple is the product of a number and an integer.

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

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

For example, four times 3 is equal to two times three plus two times three, which is equal to 12.

And in that example, the very first factor in that expression at the beginning has been partitioned.

Four has been partitioned into two and two.

All right, here's the outline on the lesson then.

We're gonna start by looking at the distributive law, then we're gonna think about what the most efficient factor might be.

Aisha and Alex are here.

They're gonna help us along the way by discussing some of the maths prompts, maybe even answering some of the questions.

Hi Aisha.

Hi Alex.

Okay, ready to start? Let's go.

Aisha and Alex compare multiplication equations.

They've got an array you can see there that they're going to be writing multiplication equations for.

There are two factors multiplied together to create the product.

The three represents the number of groups, and two represents the number in each group.

If we group horizontally, the two now represents the number of groups and the three represents the group size.

So I can see two times three equals six.

Aisha and Alex compare these two and they're thinking about what's the same and what's different.

Hmm.

In both equations, three and two are factors and six is the product.

The factors can be written in a different order, but they will still represent either the number of groups or the group size and have the same product.

So factors can be swapped around, but the product will be the same.

I remember learning that before.

It's called commutativity.

Do you remember that? Okay, let's check your understanding so far then.

In the two equations representing the array below, label the factors as group size or number of groups.

Okay, pause the video here and give that a go.

Welcome back.

On that first equation, four represented the number of groups, and two was the group size, and in the second equation, two was the group size, and four was the number of groups.

Those factors have swapped their positions, but they still represent the same thing in that array.

If you like, you can think of it as the number of groups being the purple boxes you can see and the group size being how many counters are in each of those boxes.

Aisha and Alex look at another array.

There are four groups of six in this array.

I'd write this as 4 times 6 is equal to 24.

We could also make use of the distributive law by partitioning a factor.

"Okay, show me," says Alex, I'll partition six into three and three and draw a line through the array to show it.

So Aisha has decided she's going to partition the six into three and three.

Watch where she draws the line.

There it is.

"How does that change the written equation?" Asks Alex.

Really good question.

Now we have four groups of three twice.

The equation is written like this.

4 times 3 plus 4 times 3 is equal to 24.

If you look at that array, you can see that where Aisha has drawn the line, you've now got two separate arrays almost, and both of those arrays could be stated as four times three in expression form.

I'll try this, but instead of partitioning six, I'll partition four.

So Alex is not gonna partition the six, he's chosen to partition the factor four.

Four can be partitioned into one and three.

Here's the line.

Aha.

So his line is horizontal.

Now there is one group of six added to three groups of six.

So the equation below reads 1 times 6 plus 3 times 6 is equal to 24.

Aisha and Alex compare the equations.

What's the same and what's different? So you could do that too.

What do you notice that's true of both of those sets of equations and what is different about both of those sets? Hmm.

Have a look.

Let's see what they thought.

Aisha says the product was the same for all of these equations.

Alex says, we chose a different factor to partition when using the distributive law.

Aisha chose six.

Alex chose to partition the four.

You can see it shown there.

Alex and Aisha revisit the previous array.

I think we could look at this array differently.

It also shows six groups of four.

Can you see how the grouping has swapped from being horizontal to being vertical? That now means that the factors have swapped their roles.

Let's use the distributive law by partitioning a factor again to see if we can create different equations.

I'll partition six into three and three again and draw a line through the array to show it.

There's the line.

This time there are three groups of four, twice, 3 times 4 plus 3 times 4 is equal to 24.

Four can be partitioned into one and three.

Here's the line.

Now there are six groups of one added to six groups of three.

6 times 1 plus 6 times 3 is equal to 24.

Aisha and Alex compare both sets of equations representing the array.

What's the same and what's different? So you can see there the first array in that left-hand column, which was 4 groups of 6 is equal to 24.

And in the second column you've got 6 groups of 4 is equal to 24.

What's the same and what's different about those jottings? Aisha says the product was the same for all these equations.

We always had the same number of counters.

That's true, I never saw any added in or any taken away.

There were always the same number of counters.

When we swapped the group size and number of groups, the factors in the partitioned expressions also swapped position.

Aisha has eight boxes of eggs.

Each box contains six eggs.

How many eggs has she got altogether? There are eight boxes of eggs, so that's eight times six, but I don't know that times table.

Now some of you might know that times table already, but what I'd encourage you to do is just make sure that you're still engaged in what we're about to talk about.

Because even if you know that times table, it's really important that you understand the multiplicative structures underneath it too.

Neither do I, says Alex.

Let's use the distributive law to help us.

We can partition a factor.

I'm going to partition eight into four and four by rearranging the boxes.

There they are rearranged.

Now we've got two groups of four shown by the fact that there's a row of four followed by another row of four.

Now I'll write out the new equation and solve it.

Eight times six is equal to something, that was the part that Aisha wasn't sure about.

She's partitioned eight into four and four so now she's got a new equation that reads four times six plus four times six is equal to something.

24 plus 24.

She knows that because four sixes, she knows are 24.

24 doubled is 48.

I did it differently by partitioning six instead.

Interesting.

We already know the result, but let's see how Alex managed to get to 48.

How can you? That's the number of eggs, not the boxes.

That's a really good point.

How can he split up the number of eggs rather than the boxes? I'll show you, says Alex.

I imagined cutting each box in half like this.

Aha.

That gave me these equations and a solution.

Eight times six is equal to something.

He cut the boxes in half in his imagination.

So he was partitioning six into three and three.

So now he's got eight times three plus eight times three is equal to something.

Eight times three is 24.

So he's got 24 plus 24 is equal to 48.

Both of these methods worked.

I would never have thought of partitioning the group size, says Aisha, but if the product is identical, it doesn't matter.

Aisha and Alex compare their jottings for efficiency.

Alex says, I preferred my method because I know my threes.

So he was comfortable because he was able to partition the six into threes, and he knows his three times table really well.

Aisha says my method was more efficient for me because I know my six times table really well.

Interesting.

So Aisha preferred to partition eight because she knows her six times table pretty well.

We can choose because either factor can be partitioned.

Okay, time to check your understanding.

Which factor has Alex decided to partition to solve the multiplication seven times four is equal to something? You can see he's already got his answer at the bottom there.

He's ended up with 28.

But which factor did he choose to partition? Have a think about that, pause the video, and I'll be back to reveal the answer in a moment.

Welcome back.

Alex chose to partition the four into two and two.

You can see that in his jottings there.

Aisha makes an observation.

If the products are the same, we could write a longer equation like this.

Wow.

Four times six plus four times six, so that was the way that Aisha solved it, is equal to eight times three plus eight times three.

That was the way that Alex solved it.

And both of those gave 48.

Okay, time to check your understanding again.

Use the jottings below to complete the longer equation underneath, filling in the missing numbers.

Pause the video here and give that a go.

Welcome back.

So the equation, the long one at the bottom should have read seven times two plus seven times two, and that could be got from those left hand jottings is equal to three times four plus four times four, that was on the right, which is equal to 28.

Okay, it's time for your first practise task.

For number one, I want you to look at how a factor has been partitioned and draw in the partition line for each array.

Then complete the missing numbers in the equations.

For number two, complete these equations filling in the missing numbers.

There's a longer equation at the bottom there where you can get two of the expressions from both sets of jottings.

Number three.

Alex has nine boxes of six eggs.

Use the distributive law to calculate how many eggs he has in total.

Okay, pause the video, have a good go at those and I'll be back for some feedback in a little while.

Good luck.

Welcome back.

Let's look at number one to begin with.

So this has been partitioned right through the middle there because the four has been partitioned into two and two and you can see that with the jottings below.

We've now got seven times two plus seven times two is equal to 28, which means 14 plus 14 is equal to 28.

For the second one, there's a slightly different version of the partitioning.

This time the seven has been partitioned, so the line goes there.

It's been partitioned into three and four.

You've got three times four plus four times four is equal to 28.

12 plus 16 equals 28.

Pause the video here for some extra time to mark accurately.

Welcome back.

Here we go.

Number two.

Complete these equations filling in the missing numbers.

So we had eight times six is equal to 48, and using the distributive law jottings that are underneath you would've got that by doing eight times two plus eight times four is equal to 48, which gives 16 plus 32 is equal to 48.

For the second one then, a different factor was partitioned.

This time the eight was partitioned, so we ended up with eight times six is equal to 48 because two times six plus six times six is equal to 48, which gives 12 plus 36 is equal to 48.

Now we have this big equation at the bottom that should have read eight times two plus eight times four is equal to two times six plus six times six, which is equal to 48.

Pause the video here for some extra time to mark those accurately.

Here's number three then.

This is the way that Alex decided to do it.

He says I chose to partition nine into three and six, so he ended up with three times six plus six times six is equal to 54.

That gave him an addition of 18 plus 36, which is equal to 54.

You may have done this differently.

When you're using the distributive law, it's up to you how you choose to partition one of the factors.

If you wanna pause the video here and compare some of your different answers, that might be a really useful thing to do because you can explain any differences as well.

Okay, I'll be back in a little while to be able to carry on with the second part of the lesson.

Pause the video here.

Let's move on then.

Now we're gonna start thinking about the most efficient factor.

We touched upon this a little bit already, but we're gonna really start to examine the numbers now to see which one is best to solve multiplication problems. Oak Academy is organising a rounders competition across year four.

How many players will there be in total? We can't answer this yet.

We don't have enough information.

What else do we need to know? Says Alex.

What information is missing? So what else do you think you need to know in order to be able to get that question correct? Aha.

There will be nine teams of eight children competing.

We need to calculate nine groups of eight, so it's multiplication.

I don't know nine times eight, so I'll use the distributive law.

I'll draw it as an array to help us visualise it.

There's the array.

Nine lots of eight.

Wow! That's a big array.

Let's partition, quick! They write down an equation to help.

Nine times eight is equal to something.

I'm going to partition the nine into three and six.

There's the line, and he's relabeled the array to help him to think about it.

Now he's got three times eight plus six times eight is equal to something.

I know that three times eight is equal to 24 and six times eight is equal to 48.

So 24 plus 48 is equal to 72.

Here's Aisha's method.

I'm going to partition the eight into four and four.

Now she's got nine times four plus nine times four is equal to something.

She knows that four times nine is 36, so she can find the product.

36 plus 36 is equal to 72.

I think there's a way of using subtraction.

Interesting.

Watch this, says Alex.

What did Alex do? Look at his jottings there.

What do you think he's done? How has he managed to arrive at 72? Hmm.

I see, says Aisha, you changed the factor nine to 10 because that's easier to work with.

Then you subtracted one group of eight because you had one group too many.

Okay, which method do you think was more efficient? There are the three methods.

We've got two in which one of the factors was partitioned and then the final one where Alex has used a multiple of 10 rather than a multiple of nine in order to make it an easier subtraction.

Aisha says I prefer the second method because I can quickly calculate the fourth multiple of nine by doubling and doubling again.

Have you ever done that? Really good way of multiplying something by four is to double it and then double it again.

I like the last method because I can find the 10th multiple using place value and the subtraction involved number pairs to 10.

Interesting.

So he knew he could do 80 subtract eight easily because he knew his number pairs, eight plus two equaled 10.

Both are preferred methods partitioned different factors, so efficiency is partly personal choice.

That's good, isn't it? You get to choose which factor you want to partition.

Okay, time to check your understanding then.

Which of these methods is most efficient for finding the product of 12 and 4? Explain your reasoning.

So you've got two sets of jottings there.

Both of them have given the same product, but we're looking for you to decide which one you think is most efficient.

Pause the video and have a go.

Welcome back.

Aisha says, I think the addition equation is more efficient because I'm more comfortable adding.

I find this subtraction more efficient, says Alex, because I know my fives really well and that's the starting point.

A difference of opinion there, but that's okay.

Oak Academy participates in an inter-school rounders competition.

There are eight schools participating.

How many players are there in total? We can't answer this yet.

We don't have enough information.

It's happened again.

What else do we need to know? What do you think? What information is missing? What do they need to know to be able to calculate the total number of players? Aha, there we go.

Each school needs a squad of 15 players.

We need to know how many players there were in each squad.

This is another multiplication question.

15 times eight is equal to something.

Time for the distributive law again.

I agree, but which factor should we partition? Which factor should they choose? So you've got 15 or eight.

Which would you choose to partition to make this a more efficient calculation, to make it one that you would find easier and therefore quicker? 15 says Alex, because I don't know that times table.

No, probably not.

Good idea.

I know my 10 and 5 times tables well.

So they take 15 and they partition it into 10 and 5.

Giving them 10 times 8 plus 5 times 8 is equal to something.

10 times 8 is equal to 80.

5 times 8 is equal to 40.

If you combine those two, you get 120.

So there are 120 players in total.

Okay, let's check your understanding.

Which factor would you choose to partition in this multiplication using the distributive law? 16 multiplied by 6 is equal to something.

Remember, you're just choosing a factor here.

Pause the video and give it a go.

Welcome back.

Aisha says I'd partition the 16 because I don't know that times table.

That makes sense.

There it is partitioned into eight and eight.

Eight times six plus eight times six gives 48 plus 48 is equal to 96.

Now remember you might not have gone on to solve that.

This is only there just to demonstrate that we can make 16 easy to calculate from.

Okay, time for the second practise task.

For number one, you need to solve the following equation in two different ways using the distributive law.

Partition the first factor, and then in the next solution, partition the second.

So they've got both of them down there, but you've got the instruction above just to remind you.

And lastly, it says which did you find the most efficient? For number two, solve the following worded problem in two different ways using the distributive law, partition the first factor and then in the next solution, partition the second.

There are seven rounders teams of nine players in a local league.

How many players are playing in the league altogether? Then number three, below are the jottings that Aisha has written to show the distributive law being used to find the product of eight and three.

What mistake has been made? And here's number four.

On a school residential, all the children in year four have an apple as a snack.

There are 17 packs of 6 apples.

There are 90 children in year four.

Do they have enough apples? Use the distributive law to help solve the problem.

All right.

Good luck with those.

Pause the video and I'll be back in a little while with some feedback.

Welcome back.

Let's look at number one first.

So we decided to partition six into two and four for the first one, two times seven plus four times seven is equal to 42 because two sevens is equal to 14, four sevens is equal to 28, and those two combined give 42.

For the second one, we decided to partition the seven into three and four, which eventually gave 42.

There it is.

Now you might have decided to partition them differently.

If you've got the correct product, then it's likely you partitioned correctly.

Aisha says there were other ways to partition.

Alex says I found the first most efficient because I could use doubling.

I see what you mean Alex.

Can you see that the first expression of multiplication was two times seven and the second was four times seven.

So the second part, 28 was double the 14.

That's useful.

Here's number two then.

The question was asking for the product of seven and nine.

Here's the first example in which seven was partitioned into five and two.

So we had five times nine plus two times nine.

So that gave 45 plus 18, which was equal to 63.

And here's the second version where nine was partitioned into three and six.

Seven times three plus seven times six gives 21 plus 42, which is equal to 63.

There were 63 players.

You might have partitioned differently.

Okay, here's number three.

What mistake has been made? Aisha has partitioned accurately but then added one rather than eight, which is the product of eight times one.

It should be 16 plus 8, which equals 24.

Here's number four then.

Alex and Aisha decided to partition 17 into 10 and 7.

So they ended up with 10 times 6 plus 7 times 6.

That's 60 plus 42, which is equal to 102.

They do have enough apples.

In fact, they will have 12 spare because 90 plus 12 is equal to 102.

You might have partitioned differently, says Aisha again.

Okay, let's summarise all of our learning today.

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

This means a factor can be partitioned to solve multiplication problems more efficiently.

It is important to consider which factor to choose to partition so that it enables you to solve a problem more efficiently.

My name's Mr. Tazzyman.

Hope you enjoyed that.

Maybe I'll see you again soon.

Bye for now.