Lesson video

Hi there.

My name is Mr. Tazzyman and I'm gonna be teaching you a lesson today from the unit that's all about understanding and representing multiplicative structures.

Are you ready to start? Then let's get going.

The outcome for today's lesson is as follows then.

By the end, we want you to be able to say, I can explain how each part of a multiplication and division equation relates to a story.

Here are the key words that we are gonna be using.

I'm gonna say them and I want you to say them back to me.

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

Ready? My turn.

Dividend.

Your turn.

My turn.

Devisor.

Your turn.

My turn.

Quotient.

Your turn.

This is what each of those words means.

We are gonna be learning a little bit more about that in the lesson later on, but it's worthwhile you having a rough definition just before we get started.

The dividend is the number being divided.

The divisor is a number that will divide the dividend exactly.

The quotient is the results of the dividend being divided by the divisors.

There's an example equation at the bottom there, and I think sometimes they're a little bit easier for us to understand.

10 divided by two is equal to five, and in this equation, 10 represents the dividend, two represents the divisor, and five represents the quotient.

This is the outline then for today's lesson.

We're gonna start by thinking about grouping and then we're gonna move on to looking at sharing.

This is Izzy and Laura.

They're gonna help us today by discussing some of the math that we are thinking about, giving us some prompts and hints and even revealing some answers where they can.

Hi, Izzy.

Hi, Laura.

You ready? Are you ready? Okay, let's go for it.

Izzy and Laura label a division equation, 12 divided by three is equal to four.

12 is the dividend because that's the number we are dividing.

Three is the divider because that's the size of the groups we are creating or the number of groups we are dividing the dividend into.

The quotient is four because that's the result of the division.

Izzy has 14 pieces of fruit and she eats two every day.

After how many days will she run out of fruit? So this is now similar, but it's in a real life context.

This is a division question.

The answer is seven days.

14 divided by two is equal to seven.

14 pieces of fruit is the dividend because that's what you are dividing.

Two is the divisor because that is the group size of fruit for each day.

The quotient is seven days because then the fruit runs out.

The equation tells the story.

Okay, your turn to check your understanding of what we thought about so far.

Read the context and fill in the missing numbers to create a division equation.

Label and explain your choices.

Laura has 20 rubbers, which she puts into piles of five.

How many piles does she have? Pause the video here and give that a good go.

Good luck.

Welcome back.

Let's reveal the answers then.

The first box needed to be the dividend, which was the number of rubbers.

The second needed to be the divisors, the number of rubbers in each pile, and the third needed to be the quotient, the number of piles.

It should have read as 20 divided by five is equal to four.

Is that what you've written? I hope so.

Okay, then let's move on.

Izzy has 15 pencils to divide into pots.

My teacher has asked me to put five pencils into each pot.

How many pots do I need? We know that we need a group size of five, which is a factor, and a total of 15 pencils, which is the product.

We don't yet know the number of groups.

That's the missing factor.

So Laura's put down there an equation with a missing factor.

Something multiplied by five is equal to 15.

"I know my five times table.

The missing factor is three and it means we need three pots," says Izzy.

There they go.

15 pencils into the three pots with five pencils in each pot.

We can swap the factors too because they're commutative.

So five times three is equal to 15 as well.

This equation represents five pencils three times, so it's another way of thinking about that context.

"What about as a division equation?" Oh, that's a good thought, Laura.

15 is the dividend because that was the total number of pencils I divided into pots.

You placed five pencils in each pot, which was the size of each group, so that's the divisors.

The quotient is three because that was the number of pots I needed.

15 divided by five is equal to three.

Okay then, let's check your understanding again.

Below is a context and a division equation.

Is the divisor the group size or the number of groups? Izzy has 21 rubbers.

Her teacher asks her to divide them up into piles with seven rubbers in each.

How many piles of rubbers will there be? Pause the video here and give that a go.

Welcome back.

What did you think? What was the divisors? Was it the group size or the number of groups? The divisors is the group size.

There needs to be seven rubbers in each pile.

This time, Laura has 15 pencils to divide into pots.

"My teacher says, this time, we need three pens per pot.

How many pots will I need?" Let's start with multiplication equations again.

I know that three multiplied by five is equal to 15.

We can use commutativity too, so we know that five times three is equal to 15.

Which factor is the number of pots then? What do you think? Have a look at those multiplication equations.

We're putting 15 pencils into pots.

Which of those do you think is the pots given that we know we need to have three pencils in each pot? We know that three is the size of each group, so five must be the number of pots.

There they are.

"How about the division equation," says Izzy.

15 is the dividend because that's the total number of pencils you were dividing into the pots.

I needed three pencils in each pot, which was the size of each group, so that's the divisor.

The quotient is five because that was the number of parts needed.

15 divided by three is equal to five.

Izzy and Laura compare the pencils and pots, so we've got three times five is equal to 15.

That represents both of those images.

These are two different images from one equation.

You can see they've been labelled there.

Yes, because the factors play a different role in each.

In my image, three is the number of pots and five the number of pencils.

In mine, three is the number of pencils and five, the pots.

Two equations there where the factors of swap positions because of the fact that they're trying to signify the different roles they're playing.

Next, they compare the related multiplication equations, so now they've used commutativity to swap the position of those factors.

However, the factors still have the same roles.

The factors are swapped, but the product is the same.

Each factor still represents the same part of the image though, three pots and five pencils per pot for me and vice versa for you.

This shows factors are definitely commutative.

Finally, they compare the related division equations.

We've got 15 divided by five is equal to three and 15 divided by three is equal to five.

The numbers are the same in all the equations.

They have different positions and roles though.

The missing factor became the quotient each time.

So you can see there, the ones with the boxes are the missing factors from where they were worked out originally, and they've ended up becoming the quotient each time.

The product is the same as the dividend in the matching division equation, and again, you can see that the product was 15 for all of the multiplication equations, and the dividend in both of those related division facts was 15, so they matched.

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

For each of these, complete the equations that represent a number of pencils filling in the missing numbers.

So you can see the boxes there.

They each require numbers to be filled in.

You've got A, B, and C.

For number two, we've got some worded problems. For each of the following context, write two multiplication equations and a division equation.

Laura is asked to divide up 20 paintbrushes.

Her teacher wants her to put four paintbrushes in each pot.

How many pots will she need to use? B, Izzy is sorting through the PE cupboard.

She finds 50 spare tennis balls.

Her teacher asks her to put them into buckets with 10 balls in each.

How many buckets will she need? Pause the video and give those a good go.

I'll be back shortly for some feedback.

Welcome back.

Let's mark number one first.

Here's A.

Three times two is equal to six.

Two times three is equal to six.

Six divided by two is equal to three.

Here's B.

Four times three is equal to 12.

Three times four is equal to 12.

12 divided by three is equal to four.

Here's C.

Five times four is equal to 20.

Four times five is equal to 20, and 20 divided by four is equal to five.

Pause the video to give yourself a chance to mark those carefully.

Okay then, let's do number two.

We'll start with the three equations.

Five times four is equal to 20.

Four times five is equal to 20, and 20 divided by four is equal to five.

In each equation, the five represents the number of pots she needs.

The four represents the number of paintbrushes per pot, and the 20 represents the number of paintbrushes in total.

Here's B.

Five times 10 is equal to 50, 10 times five is equal to 50.

50 divided by 10 is equal to five.

In each equation, the five represents the number of buckets she needs.

The 10 represents the number of tennis balls per bucket, and the 50 represents the number of spare balls in total.

Let's go to the second part of the lesson then, sharing.

So far, the quotient has represented a number of groups.

We know this as grouping.

Could the quotient also represent the size of the group? You mean when we already know how many groups there are, but we don't know how many there are in each group? Yes.

That's sharing, isn't it? Yeah.

We do that all the time with friends or siblings.

Do you do that? Do you share things with friends or siblings? I bet you can do that quite quickly.

Okay, let's check your understanding then of sharing or grouping.

Which of the following is a sharing question and which is a grouping question? Laura has 20 sweets.

She divides them with four other friends.

How many sweets does each of them get? Laura has 20 sweets.

She piles them into separate groups of four.

How many groups will there be? Pause the video here and decide which of those is sharing and which is grouping.

Welcome back.

The first one was sharing and the second one was grouping.

You could almost see that within the question or by thinking about what they were being asked to do.

Izzy has a punnet of 12 strawberries to share between four friends.

Let's begin by finding missing factors again.

The product is 12 because that's the total number of strawberries, and one of the factors is four because that's the number of friends.

So they've got an equation there with a missing factor.

Four multiplied by something equals 12.

Let's use commutative as well.

The factors can swap position.

Something multiplied by four is equal to 12.

I know my four times table and four times three is equal to 12, so the missing factor is three.

They each have three strawberries.

There they are.

The strawberries have been sorted into groups.

Doesn't that look like a bar chart or maybe even a pictogram? Let's complete a division equation as well.

The product is the dividend for the matching division equation.

The known factor is the divisor and the missing factor is the quotient.

12 divided by four is equal to three.

So there were 12 strawberries altogether shared between four friends, meaning they had three strawberries each.

Yes, that's what each of the numbers in the equations represents.

What would happen to the division equation if we swapped our missing factor and our known factor around? The image would change.

I wonder how.

What do you think? So let's imagine that instead of doing 12 divided by four equals three, we were doing 12 divided by three equals four.

What would the image look like? How would it be different? Let's start by swapping around the factors in the multiplication.

So instead of four times three is equal to 12, we've got three times four is equal to 12.

Not much difference there apart from the position of the factors.

Same in the second row.

Three multiplied by four equals 12 turns into four multiplied by three equals 12.

Again, the product is the same.

Not much change apart from the position of the factors, but because of commutativity, we know that, that was gonna happen.

Now we can swap the divisor in quotient too as they are the matching parts of the division.

We've got 12 divided by four equals three has become 12 divided by three is equal to four.

So there are three friends this time with four strawberries.

There it is, the new image.

We haven't got as many friends, but we've got one more strawberry each for those friends.

What do you notice? I'll compare them and have a look.

The numbers are the same in all the equations.

They have different positions and roles though.

Even though we're sharing.

the missing factor became the quotient each time again.

The product is the same as the dividend in the matching division equation.

Okay, then time to check your understanding.

True or false? The product of a multiplication is the same number as the dividend of a matching division fact.

Decide whether you think that's true or false and I'll be back in a little while to give you the feedback.

Pause the video here.

Welcome back.

That statement was true, but let's think about why.

Here's two justifications and I'd like you to consider which one you think to be the best.

Was the justification A, dividend and product are the same thing, or B, multiplication and division are inverse processes.

Pause the video and decide which you think is best.

Welcome back.

B was the best justification here.

A, dividend and product are the same thing, is not true.

They are related, but they're not the same thing.

and multiplication and division are inverse processes.

Okay, it's time for your second task.

For each of these, I want you to complete the equations that represent a number of strawberries, filling in the missing numbers.

So you've got some boxes there which need numbers put into them, numbers that match to the images you can see there.

Number two, for each of the following context, write two multiplication equations and a division equation similar to what you did in the first task, but this time we've got sharing context rather than grouping context.

For A, Laura is asked to divide up 30 paintbrushes.

Her teacher gives her five pots to use.

How many paintbrushes should go in each pot? For B, Izzy is sourcing through the PE cupboard.

She finds 100 spare tennis balls.

Her teacher gives her five buckets to divide the spare balls into evenly.

How many balls will she place in each bucket? For number three, for each context, decide which division equation represents the question best.

A, Laura is asked to divide 24 pairs of scissors between six class tables.

She works out that she needs four pairs of scissors on each table.

Is that best represented by 24 divided by six is equal to four, or 24 divided by four is equal to six? B, Izzy has 20 pens.

She needs to divide them into groups of four, so she works out that she will fill five empty parts.

Is that best represented by 20 divided by four is equal to five, or 20 divided by five is equal to four? Have a think about each one.

Good luck.

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

Okay, let's mark number one to begin with.

We had some missing numbers in these equations that were describing the representations above them.

For A, three times two is equal to six, two times three is equal to six.

Six divided by three is equal to two.

For B, there were more strawberries here.

Five times four is equal to 20, four times five is equal to 20, and 20 divided by five is equal to four.

Let's do two then.

We'll start with A.

The three equations would've read six times five is equal to 30, five times six is equal to 30, and 30 divided by five equals six.

Just a quick reminder, it doesn't matter what order you've written those equations in, as long as you have those three.

In each equation, the five represents the number of pots she needs.

The six represents the number of paint brushes per pot, and the 30 represents the number of paint brushes in total.

Here's B then.

Five times 20 is equal to 100, 20 times five is equal to 100.

100 divided by five is equal to 20.

In each equation, the five represents the number of buckets she needs.

The 20 represents the number of tennis balls per bucket, and the 100 represents the number of spare balls in total.

Here's number three, then.

We were choosing which division equation represented the context best.

In A, it was 24 divided by six is equal to four.

This is a sharing question, so the divisor represents the number of groups.

For B, it was 20 divided by four is equal to five.

This is a group in question, so the divisor represents the size of the group.

Okay, let's summarise our learning today.

A division equation has three parts, the dividend, the divisors, and the quotient.

For grouping questions, the divisors represents the size of each group, and for sharing questions, the divisors represents the number of groups.

Multiplication equations with missing factors can be used to solve division questions where the known factor is the divisor, and the product is equivalent to the dividend of the related division fact.

The different parts of related multiplication and division equations represent a context.

My name's Mr. Tazzyman.

I really enjoyed that.

I hope you did too, and I might see you again in the future in another maths lesson.

Bye for now.