Lesson video

Hello, there.

I hope you're having a good day.

My name's Ms. Coe.

I'm really excited to be learning with you today in this unit on "Using knowledge of part-whole structures to solve additive problems." If you're ready to get going, let's get started.

In this lesson, we're going to be thinking about parts and wholes and the additive structure, and by the end of this lesson, you'll be able to say that you could identify structures within stories and use knowledge of those structures to create stories.

We have some keywords for this lesson.

I'm going to say them, and I would like you to say them back to me.

Are you ready? My turn.

Represent.

Your turn.

My turn.

Additive.

Your turn.

My turn.

Multiplicative.

Your turn.

Well done.

Let's think about what those words mean.

To represent something means to show an idea or a concept.

Remember that one concept.

One idea can be represented in multiple ways.

Additive means related to addition.

An additive structure is one that is composed of adding parts.

Multiplicative means related to multiplication.

So a multiplicative structure is one that could be composed by multiplication.

Keep an eye out for those words, today.

Our lesson is about identifying structures within the stories and using those structures to create stories, and it split up into two cycles.

In the first cycle, we're going to be thinking about representing structures.

And in the second cycle, we're going to be thinking about types of structures for stories.

If you're ready to begin, let's get started on our first cycle.

In this lesson today, you'll meet Aisha, Sofia, and Alex, and they're going to be helping us with our learning and posing some questions for us to think about.

Let's start here.

Aisha, Sofia and Alex are buying some pick 'n' mix.

They all use the same size bag to put them in.

Now, I love pick 'n' mix.

My favourites are the fizzy sweets.

I wonder, do you have a favourite sweet? Aisha, certainly does.

She says so far she's put cola bottles and fried eggs into her bag.

She's put more fried eggs in than she has cola bottles.

She must prefer the fried eggs.

Now we can represent that using number rods.

Now, you may have seen and used number rods before, but they're just different coloured rods.

So let's have a look at what they do.

Let's say the dark green rod at the top represents the whole bag of sweets, so there we go.

That's our whole.

The pink bar represents the fried eggs.

Remember, Aisha has more fried eggs than she does cola bottles in her bag, and so the red bar represents the cola bottles.

We've made a bar model using the number rods to represent Aisha's sweets.

In fact, that's right.

You may have used number rods, but this reminds me of a bar model, and if you don't have number rods, you might want to draw bar models today instead.

So Aisha's gonna carry on filling her bag, and this time, she's going to add in some fizzy cherries.

Now, they're my favourite.

So now she's got fewer cola bottles than fried eggs, but she's got more fried eggs than she has fizzy cherries.

Remember, the fried eggs are her favourite.

So how will that change the rods that we use to represent it? Hmm.

I wonder what you think.

Well, we're going to use the same rod for the whole.

So the dark green rod still represents the bag of pick 'n' mix sweets, but we still have the fried eggs with the most amount.

So this time, we're going to use the light green rod to represent that we've got fewer cola bottles than we have fried eggs.

So we're going to use the red rod to represent the cola bottles, but we have fewer fizzy cherries than we have cola bottles.

We have a third part, so we're going to use the red rod for the cola bottles, and we're going to need a smaller bar for the cherries.

So we're going to use the white rod for the cherries.

You can see that those three rods are equivalent to the whole bag of sweets that Aisha has.

Let's think about Sofia.

Sofia is going to have cola bottles, fried eggs and fizzy cherries too, but she's going to have the same number of each sweet, so she must like them all equally.

She's going to have the same amount of cola bottles, fried eggs, and fizzy cherries.

How are we going to represent that with the number rods? Hmm.

I wonder.

Well, let's use the same-sized rods.

Remember that Sofia is going to have the same number of each of the sweets.

So this time, we've represented each sweet using a red rod, and we can see that three of those are equal to our whole, which is the whole bag of sweets.

It doesn't matter which way round we label them.

But one red rod represents the fried eggs, one represents the cola bottles and one represents the fizzy cherries.

Let's see what Alex wants.

Alex likes fizzy cherries and fried eggs, but cola bottles are his favourite, so he likes them the most, so he is going to have more of them, but then he likes the eggs and the cherries about the same.

So he's going to have the same amount of cherries and eggs.

Hmm, I wonder how we're going to represent that.

Maybe you want to play with some number rods and see if you can represent it.

Remember, the whole is still the same as everyone else's.

We have the same size bag, so we're going to keep the whole as the dark green rod.

So we need the same number of fried eggs and fizzy cherries.

So we're going to use the white rod here to represent that we've got the same amount of both, but we know that Alex loves his cola bottles.

So the cola bottles needs to represent the most, so it needs to be a larger bar.

So we're going to use the pink rod to represent the cola bottles like that.

We can see that those rods are equivalent to the whole.

And Sofia's saying, well, actually now she wants more cola bottles.

I'd still want more fizzy cherries, but that's fine.

Everyone's different.

Time to check your understanding.

Which of these models a, b, and c could represent the following story? A fruit bowl has the same number of apples and bananas, but fewer oranges.

Take a moment to have a think.

Remember if you have number rods, you could try making these yourself and saying the story aloud to see which one fits.

Welcome back.

Well, b, fits that story.

I wonder why.

Let's think about this.

A fruit bowl has the same number of apples and bananas.

So we needed an example that had two rods of the same size, and we can see that b does that.

Hmm, but so does a.

So why is it b and not a? Well, we have fewer oranges than we do apples or bananas.

So the third rod, the final rod needs to be smaller than the two of equal size, which is why it's b.

We've used the light green rod for the apples and bananas because there are the same number of apples and bananas, but there are fewer.

There's a smaller number of oranges.

So the white rod represents the oranges.

Well done if you've got that and you reasoned it that way as well.

Another check for your understanding.

Look at the number rods there.

Which of these stories could the number rods represent? Does the fruit bowl, according to the representation contain, a, the same number of oranges and apples, but fewer bananas, b, more apples than oranges and bananas, c, more apples than bananas, and the same number of oranges as bananas, or d, the same number of apples, oranges, and bananas.

Take a moment to have a think.

Welcome back.

Well, let's have a look at those number rods.

I can see that there are two parts of equal value and one part that has a larger value.

So that means that we could have b or c for this representation.

Let's look at b, more apples than oranges and bananas.

Well, if the red rod represented the apples, this statement would be true.

There would be more apples than oranges and bananas.

Or for c, more apples than bananas and the same number of oranges as bananas.

That could also be true.

The same number of oranges and bananas could be represented by the two white rods, and we need more apples than bananas, which could be represented by the red rod.

Well done if you identified both of those.

Time for your first practise task.

This one looks at marbles.

Now, you may have played with marbles before.

They are glass spherical objects that you can roll and they come with different patterns.

So in our story today, we have stripy, spotty and swirly marbles.

I'd like you to use number rods to represent each story.

So for example, a, says there are more stripy marbles than spotty marbles.

There are more spotty marbles than swirly marbles.

Can you use the number rods to represent that story? Remember, if you don't have number rods, you could sketch a bar model instead.

Pause the video here and have a go at making those four models.

Welcome back.

How did you get on? Now, remember, your bars may have looked different to ours, but you need to think about the proportion of the bars.

So for the first example, there are more stripy marbles than spotty marbles.

There are more spotty marbles than swirly marbles.

Now, we have a whole, which is all of the marbles together, and we've chosen to represent that with a dark green rod.

There are more stripy ones than spotty ones.

So we needed a bar that had more stripy than spotty, but we also know that there are more spotty ones than swirly ones.

So that meant that the bar that we used for the spotty marbles had to be smaller than the stripy marble bar, but bigger than the swirly marble bar.

So you may have put something like this.

We've used the light green rod, the red rod, and the white rod to represent those marbles.

We've stuck to the same whole.

So we've used the dark green rod for the second question.

There are fewer spotty marbles than swirly marbles.

There are more stripy marbles than swirly marbles.

So in this case, the spotty marbles would be represented by the white rod because there are fewer than the swirly marbles and there are more stripy ones than the swirly ones.

So we'd need the biggest bar for the stripy marbles.

Now, you may have noticed that this one has very similar rods, just in a different order.

For c, there are more swirly marbles than stripy marbles, but there are the same number of stripy and spotty marbles.

So because there are the same number of stripy and spotty marbles, we needed to use the same-sized rod to represent the stripy marbles and the spotty marbles.

And then we needed a bigger rod to represent the swirly marbles.

So we chose to use two white rods and a pink rod.

And finally, for d, there are the same number of stripy, spotty and swirly marbles.

So this time we needed to use three bars of equal size to represent those.

Well done if your sketches or your number rods look like these.

So let's move on to the second cycle of our learning today, which is looking at types of structures.

Let's go back to thinking about pick 'n' mix sweet.

If we gave a value to the number of each sweet, what is the total number of sweets in the bag? So this time, we're going to use our representation with our number rods, but we're going to give a value to the number of each sweet.

Let's have a look.

So what we've done here is we've sketched a bar model based on our number rods.

We don't yet know the whole, but we can see that we have 12 fried egg sweets, 6 cola bottle sweets, and 2 fizzy cherry sweets.

And that makes our whole bag.

What is the total number of sweets in the bag? How could we work it out? Well, Aisha says that we can find the total number of sweets by adding up each part.

We can find the sum of those parts of the total.

So we can do 12 plus 6 plus 2, which is equal to 20.

So our whole, our total is 20 sweets.

Now, hopefully, you did that calculation mentally.

You may have seen that 6 plus 2 is equal to 8.

And then I can see add number complement to 20.

I know that 12 plus 8 is equal to 20.

And that's right, Aisha, we can call this an additive structure.

The only way to find that whole or that total is by addition.

So this is an example of an additive problem is an additive structure because we had to add each part to find the whole or total.

Hmm, I wonder if you notice something slightly different about the representation here.

What if we were to give a number to each of these sweets? Again, we dunno the whole, but Sofia has counted the sweets and there are 6 fried egg sweets, 6 cola bottle sweets, and 6 fizzy cherry sweets.

I wonder how we can find the whole this time.

Well, we can add the parts, Aisha, you're absolutely right.

So we can do 6 plus 6 plus 6, which is equal to 18 sweets.

Hopefully, you've spotted a slightly different way that we could approach this problem.

So this means that this structure is also additive because we can add the parts to find the total or the sum.

However, I wonder if you've spotted something about the parts.

Ah, I should spotted it.

We can also find that in total number of sweets by using multiplication.

I wonder why we can use multiplication now when we couldn't earlier.

It's because the parts have equal value.

We have three lots of, or three groups of sweets with the same number of sweets in each group.

We have six three times.

So we can say that 6 multiply by 3 is equal to 18.

Because we can use repeated addition or multiplication to solve this particular problem, we can say that it is a multiplicative structure.

Or wonder if you can say that, it is a multiplicative structure.

Absolutely.

When the groups are of equal size, we can use multiplication to find the answer and you may find that to be more efficient than using addition.

What about this example then? I can see that we have the same number of fried eggs and fizzy cherries, but we have more cola bottles.

What if we gave a number to each sweet type? So this example, we're going to say that we have 3 fried eggs, 3 fizzy cherries, and we have 14 cola bottles.

Remember, Alex really liked his cola bottles.

So to find the total number of sweets here, we can use a combination of multiplication and addition.

We could just add the parts, but you might find it more efficient to use multiplication and addition in combination.

So we have two lots of three sweets, two groups of three sweets, which is 2 multiplied by 3.

And then we need to add an additional 14 sweets to find the total number of sweets.

So we can describe this as a combination of additive and multiplicative structures because although we can just use addition, it's a bit more efficient to think about the multiplication first.

We know that 2 multiplied by 3 is 6.

And then I have another complement to 20 here.

14 plus 6 is equal to 20.

So there are 20 sweets altogether.

Let's have a closer look at these structures.

So remember the first one was an additive structure, the second was a repeated addition or we can say, it's a multiplicative structure.

And the third one was a combination of additive and multiplicative structures.

Hmm, what do you notice about those structures? What's the same? What's different? Well, Alex has noticed that if all of the parts are unequal, it is an additive structure.

So in the first model, the number rods that make the parts are all different sizes.

They are unequal.

So we can only find the answer there using an additive structure.

If all of the parts are equal, it is a repeated addition or multiplicative structure.

So you can see in the second bar model that we have three equal parts and no unequal parts.

So we can use multiplication, it's a multiplicative structure.

Or if there are some parts that are equal and some parts that are not equal, then it has both an additive and multiplicative structure.

You can see in the third model that we have two equal parts and one unequal parts.

So we can use a combination of addition and multiplication to solve that problem.

It's really worth keeping an eye out for these structures in your everyday maths.

When can you use addition? When can you use multiplication? Or when can you use a combination of both? You can identify these structures within lots of different problems. So let's think about angles, which is something you might have encountered recently.

Three angles in an equilateral triangle, sum or have a total of 180 degrees.

What is the size of each angle? Hmm, I wonder is this an additive structure, a multiplicative structure, or a combination of addition and multiplication? Which structure would help us represent that problem? We know that if it's an equilateral triangle, it means the internal angles for that equilateral triangle are equal.

They are the same size.

So that means, we have three equal parts.

So we can represent this problem as a multiplicative structure.

Three lots of something is equal to 180 degrees.

Or we can use division, so we can think about it as 180 divided by 3, divided into three equal parts is equal to something and we can solve that using our understanding of multiplicative structures.

We know that something multiplied by 3 is equal to 180 degrees, so therefore, we can think about it as 180 divided by 3 is equal to 60.

So each part, each angle within an equilateral triangle is 60 degrees and we can use the multiplicative structure to solve that problem.

Let's think about a different problem.

Two packets of biscuits cost 3 pound each and one box of chocolates cost 5 pound 50.

What is the total cost? Hmm? Which structure would help us represent this problem? Is it an additive structure, a multiplicative structure, or a combination of additive and multiplicative? Hmm, I wonder.

Well, in this case, it's a combination of additive and multiplicative.

We have two packets of biscuits, which cost three pound each.

So we have two equal parts of three pounds, and then we have our box of chocolates, which costs 5 pound 50.

It costs more.

What is the total cost? Well, in this case, we can use addition and multiplication to help us solve this problem.

We can do two lots of three pounds, so 2 multiply by 3 pounds, which is 6 pounds, and then we can add the 5 pound 50, 6 pounds plus 5 pound 50 is equal to 11 pounds 50.

So the total cost in this case is 11 pound 50.

And we've used addition and multiplication to help us find that answer.

Times check your understanding.

Look at these three structures made with number rods.

Which of these shows a multiplicative or a repeated addition structure? Take a moment to have a think.

Welcome back.

That's right, it's c.

And that's because when we have a multiplicative structure, we are looking for a problem that has equal parts and only equal parts, and c is the only one that has only equal parts.

Well done if that's what you got.

Another check for your understanding.

Alex buys three packets of stickers, each with 5 stickers in.

He also buys a bonus pack which has 20 stickers in.

Which of the bar models made with number rods represents this problem? Is it a, b, or c? Take a moment to have a think.

Well, let's think about this problem, we have three packets each with five stickers, so that's three, lots of five or three equal groups.

And then we have a bonus packet, which has got more stickers in.

So b is the correct representation here.

We have three equal parts plus one larger part.

And you may have said that this is a combination of additive and multiplicative structures.

Time for your practise task.

For question one, I would like you to draw a bar model or use number rods to represent the structure of each of these problems. So for example, a is the internal angles of a regular pentagon.

Hmm, pentagons.

How many internal angles do they have? Are 108 degrees each.

What is the sum of the angles? And you have b and c to think about as well, which have different contexts.

Remember, are you looking for additive structures, multiplicative structures, or a combination of both? For question two, I'd like you to match the problem to the correct structure.

So for example, the first question says, four children eat five strawberries each.

How many strawberries do they eat altogether? Is that repeated addition or multiplicative? Is it a combination of additive and multiplicative? Is it an additive structure or is it an unknown structure? Can you match those up? And for question three, I'd like you to make up your own story for each structure.

So you have three different structures there.

Can you make up your own story and you can be as creative as you like, but really think about the proportion and the relationship between those parts? Good luck with those three tasks.

Enjoy being creative in your own problems and I'll see you shortly for some feedback.

Welcome back.

How did you get done? So for question one, I asked you to draw a bar model to represent the structure of each problem.

The internal angles of a regular pentagon are 108 degrees.

Now a pentagon has five interior or internal angles.

So we needed five equal groups because they are regular, so that all the same size.

So your bar model should have looked a bit like mine, 5 equal groups of 108, and we don't know the sum of the angles.

The second question, Sofia's dad buys two books and a travel pillow online.

Each book costs 7 pounds 99, and the travel pillow cost 10 pounds 99.

How much did he spend altogether? So in this representation we had two equal groups of 7.

99 and a larger part of 10.

99.

So that's a combination of additive and multiplicative structures.

For c, in October, there was 171 millimetres of rainfall.

In November, there was 120 millimetres of rainfall, and in December, there was 180 millimetres of rainfall.

What was the total amount of rainfall across these three months? Now, I can see that there are three parts, but the parts are all different in size.

So this is an additive structure and your bar model should have looked something like this.

Now, let's think about matching the problems to the structures.

Four children eating five strawberries each is a multiplicative structure because each child ate the same number of strawberries.

Sam does three pieces of homework.

She spends 10 minutes practising her times tables, 20 minutes reading and five minutes practising her spellings.

But I can see three unequal parts there.

So that is an additive structure.

Alex's dad goes through a run three times in a week is an unknown structure.

We don't know if the runs were equal length or equal time or different times.

We need more information to decide if it's an additive, multiplicative or a combination of those structures.

And finally, Aisha's family take two suitcases weighing 22 kilogrammes each, and a bag for hand luggage that weighs five kilogrammes.

What was the mass of all of their belongings? This is a multiplicative and additive combination structure because we have two equal groups of 22 kilogrammes and we have an additional five kilogrammes.

Well done if you're able to match all of those problems. Finally, for question three, I hope you had fun at making up your own story.

Now, remember, your stories may have been very different to ours, but you need to think carefully about what the representation is telling you.

So for the first example, this is an additive structure because we had three unequal parts.

So your story should have also had three unequal parts.

We decided to go for scoring goals.

Jacob scored three goals in the first match, two goals in the second, and one in the third.

How many did he score altogether? You can see that our problem represents those three unequal groups.

The second example had three equal-sized groups.

So whatever your problem was, you needed to have three groups with an equal value.

We decided to talk about piano practise.

Izzy plays piano for 45 minutes each day on Monday, Tuesday, Wednesday.

How many minutes does she play piano for? So an hour example, we had three equal groups of 45 minutes, which could be represented by that bar model.

For the third one, we had a combination of additive and multiplicative structures.

We had two parts of equal value and one larger part.

So we went for, Andeep buys two bottles of milk and a box of cereal.

One bottle of milk costs 1 pound 50, and the box of cereal costs 3 pound 40.

How much does Andeep spend all together? So we had two parts there, which were the bottles of milk and a larger part that was the box of cereal.

Well done if you chose a multiplicative and addition combination for your problem.

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

So we've been thinking about different structures within stories and we've used that knowledge to create our own stories.

Let's think about what we've learned today.

When we solve problems, we can use mathematical structures to help us.

When the size of the parts are unequal, this is an additive structure.

When the size of the parts are equal, this is a repeated addition or multiplicative structure.

And when there are some parts that are equal and some parts that are unequal, this is a combination of both additive and multiplicative structures.

Thank you so much for your hard work today and I hope to see you again soon.