Lesson video

Hello, my name's Mrs. Hopper and I'm really looking forward to working with you in this lesson.

Are you ready to do some maths? This lesson comes from the unit division with remainders.

So we're going to be thinking a bit more about division, how we can represent it, how we can record it, and what it means to have remainders.

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

So in this lesson we're going to represent and interpret division by grouping where there is a remainder and we're going to be using multiplication and addition to think about this.

Well that's interesting using multiplication and addition to think about division.

Ooh, let's explore that a bit further, shall we? We've got three key words in our lesson today.

Groups, remainder and divided.

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

Let's practice them.

Are you ready? My turn, groups.

Your turn.

My turn, remainder.

Your turn.

My turn, divided.

Your turn.

Well done.

Are they words you are familiar with? I'm sure you've talked about groups before.

Let's just remind ourselves what they mean 'cause they are going to be really useful in our lesson.

Grouping is when we divide a number of objects into equal groups.

We know the total number of objects and we know the number of objects in each group, but we do not know how many equal groups there are.

So that's what we're gonna be thinking about today.

A remainder is the amount left over after division when the dividend does not exactly divide by the divisors.

Do you remember the dividend is the number we're starting with and the divisors is the number we are dividing by.

And if when we've done our division there's not an equal number of groups, then the remainder is what's left over.

We'll look more at that in the lesson.

And division means splitting into equal parts or groups.

Something is divided into groups of something and that's a stem sentence we're going to be using today.

There are two parts to our lesson today.

In the first part we're going to describe and represent division as grouping.

And in the second part we're going to be drawing models to represent division.

So let's make a start on part one.

And we've got Jun and Sofia helping us with our learning today.

Sofia has nine counters.

She wants to group them equally.

I wonder how many groups she can make.

What do you think? She says, I will use my multiplication facts to help me.

It's really useful if we have our multiplication facts ready to help us when we're thinking about division.

She knows that nine is equal to three times three, she says.

So I could make three groups of three.

Let's see what that looks like.

Nine is equal to three times three.

One group of three.

Two groups of three.

Three groups of three.

So she can divide her nine into three groups of three.

Let's fill in the stem sentences.

Nine is divided into groups of three.

There are three groups and those stem sentences are going to be really useful helping us to think about our division as our whole number, our dividend divided into groups of, there's our divisors.

And then our answer, our quotient is the number of groups we make.

And as Sofia says, I made three groups.

She says I could represent this as a bar model.

Can you picture what that's going to look like? There we go.

There's nine is her whole and she's made three equal groups of three.

So nine is divided into groups of three.

And there are three groups.

Sofia groups her nine counters in a different way.

Can you see what she's done this time? She says this time I made some other equal groups, but I had some counters left over.

She has got one left over, hasn't she? The leftover counters are called a remainder.

So when we've made our equal groups and we've got some leftover, that is our remainder.

So let's complete the stem sentences to help us write the equation that describes her groups this time.

So we've got, hmm, it's divided into groups of hmm.

But this time our sentence says there are hmm groups and a remainder of, hmm.

So what groups can you see that Sofia has divided her counters into? And how many counters did she start with? Well she started with nine counters again, didn't she? Nine is divided into groups of what groups can you see there? It's two, isn't it? So nine is divided into groups of two.

How many groups can you see? I can see one, two, three, four groups of two.

There are four groups, four times two.

But there's also that remainder.

We've got one more haven't we? One more counter a remainder of one.

So we can say that nine is equal to four times two plus one.

And she says again, I can represent this as a bar model.

So nine are whole is equal to four groups of two plus another one.

So let's think about what each part of this equation represents.

Do you want to have a little think before we share our ideas? We've got nine is equal to four times two plus one.

What do all those numbers represent? Well, nine represents the total number of counters, our whole, and here we've got a part-part.

Well there's lots of parts.

There's five parts, but a part-part-whole model there.

Nine is our whole.

What about the rest of the equation? Well, four times two represents the four groups of two counters.

And you see there at the top we've got our four groups of two? So we can put four groups of two as our parts.

What about this plus one? The plus one represents the remainder of one.

We started off with nine counters in total.

Four groups of two use up eight of the counters, but then we've got a remainder of one.

We've got our extra one.

So we've divided nine into groups of two.

But because nine is an odd number, we can't put all of the nine into groups of two.

We've got one leftover, our remainder of one.

So nine is equal to four groups of two and one remainder.

And we can record that as nine is equal to four times two plus one.

Jun groups the counters in a different way.

And he's preparing to write his equation and fill in his stem sentences.

Do you think you could do that for him? Go on then.

Jun, take it away.

Two times four is equal to eight.

So I think I can use the eight counters to make two groups of four and then I will have a remainder of one.

Is that what his representation shows with his counters? Can you see two groups of four and a remainder of one? So nine is divided into groups of four this time.

How many groups are there? There are two groups and a remainder of one.

Great.

Now let's complete the equation.

Nine is equal to two times four for the two groups of four plus one for our remainder.

Perhaps you could say what each part of this equation represents.

Do you think you can do that? You might want to pause and have a think now.

What did you come up with? So nine is our whole, the number of counters we started with.

Two times four represents the two groups of four counters and the one represents the one remainder.

Well done if you've got that.

And it's time to check your understanding.

This time the children of group seven counters, can you complete the stem sentence to describe the groups and then write an equation to represent this? So pause the video, have a go.

And when you're ready for the answers and some feedback, press play.

How did you get on? So we've got seven as our whole and it's been divided into groups of three, isn't it? So seven is divided into groups of three.

How many groups are there and how many left over? What's our remainder? Well there are two groups and a remainder of one.

So remember in our equation what we are equal to is the whole.

So the whole is equal to something.

Our whole was seven and it's equal to two groups of three.

And we can record that as two times three.

So two groups of three plus our remainder, which was one.

So seven is equal to two times three plus one.

Seven is divided into groups of three.

There are two groups and a remainder of one.

Well done if you've got that right.

Now let's use counters to represent this equation.

Can you picture what this is going to be like? How many counters do we need altogether? What have the counters been divided into and what's our remainder? Can you think? Let's have a look.

So what does the 10 represent? Well the 10, the total number of counters.

So we need 10 counters.

There they are.

What about that two times four, what does that represent? Well, two times four.

This time in this lesson we're thinking of two groups of four counters.

That's how we are reading our multiplication.

So two groups of four counters, there they are.

And what about our plus two? What does the two represent? Well the two represents a remainder of two.

There are two counters that don't go into a group of four.

So if we think about our stem sentence, 10 has been divided into groups of four.

There are two groups and a remainder of two.

Well done.

Time for you to have a go.

Can you use counters to represent this equation? Remember to think about what the numbers in the equation represent.

Pause the video, have a go.

And when you're ready for the answer and some feedback, press play.

How did you get on? Did you remember what each part represented? So 11 represents the total number of counters.

So we need 11 counters.

And what about our three? Our three represents a remainder of three.

Well done.

Sofia adds one more counter to this group.

I wonder which part of the equation should be changed.

So do you remember we had seven was equal to two groups of three plus one.

Seven is divided into groups of three.

There are two groups and a remainder of one.

But what if she adds one more counter to this group? Let's add one more in.

So what has to change in our equation? Let's complete the stem sentence to help us find out.

So we started with seven.

What have we got now? We've now got eight is divided into groups of three.

That's right.

How many groups are there? There are still two groups and a remainder of two.

That's right.

Sofia says.

Now there are eight counters, not seven, and I can't make another group of three.

So there are now two counters left over instead of one.

So we could change our equation as well.

Eight is equal to two groups of three plus two.

Well done if you spotted that.

And it's time for you to do some practice, can you write an equation that represents the grouping of the counters? And remember for our multiplications, we are writing, hmm, groups of hmm.

So the number of groups comes first and the size of the group is the second factor in our multiplication.

So you've got four there to write an equation to represent the grouping of the counters.

And then in question two, you're going to use counters to represent the equations shown.

And in question three it says Jun uses 13 counters to make some equal groups with a remainder of three.

What possible groups could he have made? Pause the video.

Have a go at those questions.

And when you're ready for the answers and some feedback, press play.

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

You were going to fill in the equations to represent the counters.

So in A, how many counters have we got altogether? We've got 11, haven't we? And we've made two groups of five plus a remainder of one.

So 11 is equal to two times five plus one.

What about B? How many counters? Well it's 11 again, isn't it? This time 11 is equal to five groups of two plus one.

Our remainder is one.

What about C? Well again, we've got 11 counters, haven't we? 11 is equal to two groups of four this time plus a remainder of three.

And in D it's 11 counters again.

11 is equal to three groups of three plus a remainder of two, three times three is equal to nine.

And we've got two left over.

Well done if you got those right.

And in question two, you were thinking about using counters to represent these equations.

So we've got seven counters and we've made two groups of three, plus a remainder of one.

In B, we've got eight counters again, we've made two groups of three plus a remainder of two.

In C, we've got 10 counters this time, 10 is equal to three groups of three plus one.

And for D, 13 counters is equal to four groups of three plus one for our remainder.

And for question three, Jun uses 13 counters to make some equal groups with a remainder of three.

What possible groups could he have made? So what was our whole, it was 13 wasn't it? But we've got to remember that he had a remainder of three.

So I wonder if we should put those to one side so that we're just looking at the equal groups part.

That's a good idea.

So 13, subtract 3 is equal to 10.

So there must be 10 counters in equal groups.

So how could Jun have done that? Those are our three that are the remainder and there are our 10, so what can we see? Well, we could have one group of 10 with a remainder of three.

We could have two groups of five with a remainder of three.

Oh, can we have five groups of two? Ah, there's a problem here.

Why is five groups of two not a possibility? Can you see why? What do you notice about that remainder? That's right, there'd be another group of two and we must have a remainder of three.

So we can't make another group of two from 13 and still have a remainder of three.

Well that's the problem, we can make another group of two from our 13.

So we don't only have a remainder of one.

So the number we are dividing by the size of our group has got to be greater than three.

Otherwise we would be able to make another group and so we wouldn't have a remainder of three.

That's something we're going to explore more as we go on I think.

And on into the second part of our lesson, we're going to be drawing models to represent division.

So here are some different representations.

Let's see if we can write the equation that each model represents.

So we've got a part-part-whole model where it's got four parts, hasn't it? And we've got a bar model.

What can you see in each case? Can you see some equal groups and a remainder? Jun says, let's use the stem sentences to help us.

Good idea, Jun.

So here were our stem sentences.

Hmm is divided into groups of hmm, there are hmm groups and a remainder of hmm.

So we're going to be able to tell our groups because they're going to have the same number in each and the sort of odd one out will probably be our remainder.

Well it will be in these cases.

So let's have a look.

So here in our part-part-whole model with lots of parts, seven is our whole and it's divided into groups of two.

That's right.

We've got groups of two, haven't we? How many groups are there? Well there are three groups of two and there's a remainder of one.

So you can see there in the representation that we've got three groups of two and one.

And then we can represent that with an equation.

Seven is equal to three times two.

Three groups of two plus our one remainder.

Let's have a look at our bar model and 10.

So let's think about the stem sentences.

What's our whole this time? It's 10, isn't it? And can you see what we've divided it into groups of? Groups of three.

Well done.

So 10 is divided into groups of three.

How many groups are there? There are three groups and a remainder of one.

We've got that odd one, haven't we? So our equation would be 10 is equal to three times three.

Three groups of three plus one.

And it's time to check your understanding.

Can you write an equation to represent each model? Use the stem sentences to help you.

So we've got a part-part-whole model with three parts and we've got a bar model there.

Can you see what size of group we're dividing into each time? Pause the video, have a go at completing the stem sentences and writing an equation.

And when you are ready for some feedback, press play.

How did you get on? So let's look at the part-part-whole model.

So what's our whole, it's 11, isn't it? And can you see that there's two groups of five there and a one? So 11 is divided into groups of five.

There are two groups and a remainder of one.

So what's our equation going to look like? 11 is equal to 2 times 5.

Two groups of five plus one are remainder.

And then for the second one in our bar model, what's our whole list time? Well, it's 16, isn't it? So 16 is divided into groups of five.

It's groups of five again, isn't it? How many groups are there? There are three groups and a remainder of one.

So our equation will look like this.

16 is equal to 3 times 5 plus 1.

Three groups of five plus one.

Well done if you've got those right.

The children each draw a model to represent the counters.

So we've got nine counters, two groups of four and one remainder.

So we've got it as a bar model, as an equation.

Oh, and we've got a part-part-whole model there.

So which child has represented the groups correctly? Can we use the stem sentence to help us find out? So Jun's drawn a bar model and Sofia has drawn a part-part-whole model.

Who's got it right? Let's think about the stem sentence.

So what's our whole, it's nine, isn't it? And looking at the counters, what groups have they divided into? Divided into groups of four.

That's right.

So how many groups of four are there? There are two groups.

And what's the remainder? There's one, one counter left over.

Oh, oops, says Sofia.

There should be two groups of four and a remainder of one.

I think she's just taken the numbers from the equation, hasn't she? She's not thought about what the part-part-whole model represents.

So she needs two groups of four and one.

And Jun says my model was correct.

He has got two groups of four and his one remainder, but he says to Sofia, well done for noticing your error.

Absolutely.

It's really useful to notice our errors to explain what we've done and then we can learn and we won't make the same mistake twice, we hope.

Sofia writes this equation.

Let's represent it as a part-part-whole model.

See if Sofia's got it right this time.

So 19 is equal to 3 times 6 plus 1.

So what's the size of our group in here? Three times six.

Three groups of six plus one, right? So our whole is 19 and Sofia says 19 has been divided into groups of six.

There are three groups of six and a remainder of one.

So our equation tells us there are three times six, three groups of six and a remainder of one.

Jun represents the same equation as a bar model.

He says 19 has been divided into groups of six.

There are three groups of six and a remainder of one.

Is his bar model, correct? He says My bar model also represents this as I still have three groups of six and a remainder of one.

It doesn't really matter where the one is.

We've had the one sort of at the end of the bars, haven't we? But it doesn't matter where it is.

His bar model still shows three groups of six and a remainder of one.

Sofia divides some counters into equal groups with the remainder.

The remainder is more than one, but less than six.

Hmm.

Let's see how many counters she could have started with.

Jun's gonna help us think this through.

He says 3 times 7 plus 1 is equal to 22.

So she must have started with more than 22.

If the remainder's got to be more than one, then the whole has got to be more than three times seven plus one.

And he knows that 3 times 7 plus 1 is 22.

So our whole must be more than 22.

Three times 7 plus 6 is equal to 27.

So she must have started with less than 27.

Yes.

'Cause her remainder is more than one, but less than six.

So her starting number must be somewhere between 22 and 27.

Let's check.

Jun says she could have started with 23 counters.

23 is equal to 3 times 7 plus 2.

That would work.

24 counters.

24 is equal to 3 times 7 plus 3.

So that works.

And 25 counters, 25 is equal to 3 times 7 plus 4.

That works three groups of seven.

And a remainder that's greater than one, but less than six.

And 26 counters works.

26 is equal to 3 times 7 plus 5.

Well done, Jun, I think you found all the possibilities.

You found all the possible whole numbers of counters that Sofia could have started with.

And it's time for you to do some practice.

Can you write an equation that represents each model? So you've got two bar models and two part-part-whole models with more than two parts.

In question two, draw a model to represent the equations.

Try to use more than one type of part-part-whole model.

So you could do part-part-whole models or bar models.

So pause the video, have a go at questions one and two.

And when you're ready for some feedback, press play.

How did you get on? So here we were writing equations to match our representation.

So in A, we've got a whole of 19, three parts of six and a one.

So we can say that 19 is equal to 3 times 6 plus 1.

Three groups of six plus a remainder of one.

What about B? We've got 20.

Oh, can you see we've still got three groups of six? Well, if 19 was three groups of six and a remainder of one, then 20 must be equal to three times six plus two mustn't it.

It's one more.

It's not another group of six, but our remainder has increased by one.

What about C? Well now we've got 25.

So we've now got more than four groups.

4 times 6 is equal to 24.

So 25 is equal to 4 times 6 plus 1.

And can you see what's happened in D? We've still got four groups of six, but this time our whole is 26, it's one more.

So our group that isn't six, our remainder has increased from one to two.

26 is equal to 4 times 6 plus 2.

I hope you spotted the connection between A and B and the connection between C and D.

Let's have a look at question two.

This time we'll be going to draw some models.

So 17 is equal to 2 times 8 plus 1.

Two groups of eight plus one.

So we could draw a bar model.

B says 25 is equal to 3 times 8 plus 1.

Oh, can you see we've gone from two times eight plus one to three times eight plus one? So we can show that with a bar model, three groups of eight and a remainder of one.

What about C? 42 is equal to 5 groups of 8 plus 2.

5 times 8 is equal to 40 and a remainder of 2.

Whoa and then we've drawn a part-part-whole model with lots of parts there.

Can you see our remainder of two sneaking in just after the first eight? And D says 45 is equal to 5 groups of 8.

Well that's three more than 42, so our remainder's going to be three more.

There's a remainder of five, so five times eight plus five.

And we've shown it there again with a part-part-whole model.

Well done if you've got those right.

You may have drawn a part-part-whole model for A and B and bar models for C and D as well.

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

We've been representing division by grouping with multiplication and addition equations.

What have we learned about? We've learned that objects can be divided into equal groups.

Sometimes with a remainder.

We can use multiplication facts to solve grouping problems. In fact, our times tables are really important when we're thinking about grouping and division.

And we can use multiplication and addition to show when a set of objects has been divided into equal groups with the remainder.

We've used multiplication and addition, but we've also used some really useful representations to help us think about what's going on.

Thank you for your hard work in this lesson.

I hope you've enjoyed it and I hope I get to work with you again soon.

Bye-bye.