Lesson video

Hello there, I hope you're having a good day.

My name is Miss 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 unit, we are focusing on additive structures.

So the outcome for this lesson is that you will be able to use a model to interpret and represent a part-part-whole problem with three addends.

We've got a couple of keywords in this lesson, and I'm going to say them, and I'd like you to say them back to me.

Ready? My turn represents your turn.

My turn addends your turn.

Great job! Let's look at what those words mean.

To represent something means to show something in a different way, and addends are any numbers added together.

So for example, in the equation, three plus two is equal to five, three and two are the addends because they are the numbers that are being added together.

And as you'll see today, there can be more than two addends.

In this lesson today, we're going to be using models to interpret part-part-whole problems with three addends, and we have two cycles.

In the first cycle, we're going to be looking at representing problems; and in the second cycle, we're going to be constructing problems. So if you're ready, let's get started with the first cycle.

In this lesson today, you are going to meet Sofia, Izzy, and Aisha.

And they as always are going to be asking questions and helping us with our learning.

So let's start here.

Izzy, Sofia, and Aisha are planning a shopping trip.

Izzy has been really lucky and she's got some money for her birthday, so she's planning a shopping spree, and she's going to take Aisha and Sofia with her.

Aisha's wondering what Izzy might have her eye on.

I wonder what you would buy if you went on a shopping trip? I definitely would get some shoes.

So Izzy decides that she's going on holiday, and so she needs a new T-shirt, some sunglasses, and a pair of trainers.

And you can see here, the three items that she has decided to buy.

How much would it cost Izzy to buy all of these items? Hmm, I wonder.

Aisha says, "Well, we can represent this using a bar model and three addends." Remember, that's our keyword, addends are just numbers that we are adding together.

So the total cost, how much is going to cost Izzy to buy all three items is the whole, and we don't know what that is yet, we don't know how much it's going to cost to buy all three items. The cost of the T-shirt is one part, which is £15; the cost of the sunglasses is another part, and that was £31; and the cost of the trainers is a third part, which was £39.

So we have our three parts or three addends, but we don't yet know the total amount or the whole.

We can also represent this using a part-part-whole model with three addends.

So again, we don't know what our whole is, but we do know the three parts, of £15, £31, and £39.

These are both ways to represent the same problem.

So let's look more closely at these models.

What does each part of the model represent? Well, the question mark represents the total cost of the items. Remember, we don't yet know how much Izzy spent on these three things.

The £15 represents the cost of the T-shirt, the £31 represents the cost of the sunglasses, and the £39 represents the cost of the trainers.

Izzy has made a really good point.

At the moment the way the models are set out, it could suggest to some people that the parts are equal in size, and we can see that they aren't.

We know that £39 is greater in value than £15, but if we look at our bar model, the three parts are equal in size.

And so some people might think that these numbers are equal in size as well, and we know that they're not.

So Izzy is suggesting that maybe we need to change the size of the parts, so they're a bit more realistic, maybe a bit more proportional in size to the numbers that they represent.

Now we can't actually do that with a part-part-whole model.

When we have a part-part-whole model, the circles stay the same size, they just represent the three parts, but we can change Aisha's bar model, and we can think about how big the parts must be compared to their value.

Let's have a look.

So does this look a bit better? Does this look a bit more proportional with the various sizes of the parts? It's not exact, but the bars are of similar size to the size of the cost.

Now this can be a really useful thing for you to do when you're sketching your own bar models thinking about the proportional relationship of the different numbers.

We know here, for example, that 15 is the smallest value, so it should be in the smallest bar.

£31 and £39 are close together, but £39 is a little bit more than 31, so we'd expect to see the bar for £39 being a little bit bigger.

Time to check your understanding.

Tick the bar models that could represent the following problem.

Sofia decides to buy some nail varnish for £6, a bracelet for £12, and a pair of jeans which costs £25.

How much did she spend altogether? You have three bar models there labelled: A, B, and C.

Which of these could represent the following problem? Take a moment to have a think.

Welcome back.

Did you take a close look at the bar models and carefully discuss which one you think would best represent the problem? C would best represent the problem.

All the bar models do show the correct parts and the unknown whole, but C is proportional.

Now this helps us think about the problem more easily.

So it really helps if we're thinking about how to proportionally size our bars, so that we can think about the problem.

Time for another check.

This time I'd like you to draw a bar model to represent the following problem.

So think really carefully about the proportionality of your bars.

Izzy spent £85, Sofia spent £43, and Aisha spent £15.

How much money did they spend altogether? Draw a bar model to represent that problem.

Welcome back.

How did you get on? Now this is the bar model that I drew.

Remember that yours might have looked slightly different to mine.

For example, your parts may have been in a different order because remember, we can add the parts in any order.

£85 is the largest part, so I represented it with the largest bar.

£15 is the smallest part, so I represented it with the smallest bar.

Remember, this is just one way that you could have represented the problem using a bar model.

Don't worry, if yours looked a bit different to mine, but do think about, was your £85 the biggest bar that you had? Time for your first practise task.

I would like you to draw a bar model to represent each of these four problems. So for example, Sofia spends £38, Izzy spends £20, Aisha spends £60.

How much do they spend altogether? Really think about the different proportions of your bars.

Have fun during those bar models, and I'll see you shortly for some feedback.

Welcome back.

How did you get on? Let's have a look at some of the bar models that I drew.

So for the first one, Sofia spends £38, Izzy spends £20, Aisha spends £60.

How much did they spend altogether? So for my bar model, £60 was the largest bar because it's the largest amount of money.

£38 was the next largest bar, and £20 was the smallest bar.

Now you can see that £38 and £20 are quite close together in size, so I've represented that by the bars being quite close together in value as well.

For the second one, Sofia spent £22, Izzy spent £40, Aisha spent £10.

So again, you can see that £40 is the greatest bar, and £10 is the smallest bar.

Now remember, it doesn't matter which way around your bars have been put, you may have written £40 first and £10 in the middle.

Addition is commutative, which means we can add the parts or the addends in any order, but we are really focusing on thinking about that proportion.

So take a look at your bar model.

Was £40 the largest bar outta those three? For the third one, Sofia spends £80, Izzy spends £80, Aisha spends £82.

I've drawn bars that are very close together, £80 and £80 are the same value, so the bar should be the same size.

£82 is slightly more, and so that bar would be slightly bigger.

And finally, what did you notice for D? That's right, the three addends, the three parts are all the same size, they're all £15.

So that's reflected in my bar model where £15 is repeated three times, and the bars are all the same size because they represent the same amount of money.

Well done if you drew bar models that had a similar proportion to mine.

Now let's move on to the second part of our learning where we're going to be constructing problems. So let's start here.

Take a close look at this bar model.

What do you notice? What can you see? Well, I can see that there is an unknown whole, which is shown by the question mark.

I can see that we have three addends or three parts, and I can see that they've been drawn proportionally.

But unlike earlier bar models, this bar model uses decimal numbers.

Hmm, do you think that's going to make any difference? Well, not really.

Even though these addends are decimal numbers, we can still use the same strategies to think about addends and missing wholes.

So what problems could you come up with for yourself based on this bar model? You might like to talk to a friend about this, you might like to think about what problems you could say that would match this bar model.

But let's see what Izzy and Sofia have to say.

So Izzy says, "Well, does that mean we can use any context as long as it has three addends?" What do you think? Yes, we have three addends here, and we know that there is an unknown whole, we don't know what the whole is yet.

So we can use any kind of context, but we do have decimal numbers, so that might affect the context that we use, the things that we think about.

Sofia has got one, let's see what she has to say.

"How far did I run this week?" she says.

The first time she ran, she ran 2.

4 kilometres, the second time she ran 1.

5 kilometres, and the third time she ran 0.

8 kilometres.

What do you think about that problem? Do you think that it is a three addend problem? Well, yes, it is.

So we have the three addends, 2.

4, 0.

8, and 1.

5.

And Sofia has used the unit of measurement of kilometres, which is something we might use decimal numbers for.

She's asked us to add together her three runs to find the total length that she ran.

So that is absolutely perfect, and Izzy agrees.

It works really well for this context.

Her turn, what is she going to come up with? So Izzy says, she baked a cake, which needed 2.

4 kilogrammes of flour, 1.

5 kilogrammes of butter, and 0.

8 kilogrammes of sugar.

Has she got a three addend problem? Well, yes, she has.

How much do these ingredients weigh altogether? So Izzy is asking us to find the sum or the total of 2.

4, 0.

8, and 1.

5.

Instead of Sofia's length question, this time Izzy has used the units of measure, kilogrammes.

So she's asking us a question about mass, but it is still a three addend problem with an unknown whole.

I think that's a great question, Izzy.

And Sofia agrees.

Ah, nice! A measuring problem to do with measuring mass.

So both of these examples so far have been to do with measuring, length and then mass.

I wonder if we can think of any other problems. Izzy's clearly on a roll.

So this time she says, I spent £2.

40 on a pasty, 80 pence on a packet of crisps, and £1.

50 on a drink.

How much did I spend altogether? Again, this is an addend problem, but did you notice how I said the numbers slightly differently? This time we have a money context.

So 2.

4 would be said as £2.

40 or £2, 40 pence.

0.

8, well, that's £0.

8 or we could say 80 pence because there are 100 pennies in one pound, and 1.

5 is said as £1.

50.

Even though we've changed the way we say these numbers slightly, we're still using the same three addends.

And Izzy is asking us, how much does this cost altogether? So we're still looking for that unknown whole.

And Aisha's right, she did well to buy a pasty for £2.

40.

I think they're much more expensive where I live.

Aisha has got one more example for us.

I wonder what context she'll use.

Aisha said, she drank 2.

4 litres of water on Monday, on Tuesday she drank one and a half litres of water, and on Wednesday she drank 0.

8 litres of water.

How much water did she drink over those three days? Again, did you notice the context there? This time we're thinking about another units of measure, litres.

And did you notice how I might have said some of those numbers slightly differently? Ah, yes.

So we can say 1.

5 as one and a half, 0.

5 and one-half are equivalent.

So Izzy finds it really interesting how we've mixed up fractions and decimals there, and it's absolutely fine to do that.

In the context of liquid, so using litres, you might well say one and a half litres rather than 1.

5 litres.

As long as you're using equivalent fractions and decimals, then you've still got three addends.

They're the same addends, and you're still finding the unknown whole.

Really nice examples there.

Well done! Time to check your understanding.

Tick the problems that could be represented by the bar model.

So you can see the bar model there, and we have three decimal addends.

Which of these problems represent the bar model? A, I spent £1.

20 on some chocolate, 20 pence on a lolly, and 80 pence on a drink.

B, I glued together a 1.

2 metre length of wood with a 20 centimetre piece of wood and a 0.

8 metre piece of wood.

C, I put 1.

2 kilogrammes of butter into a bowl, I then added 2 kilogrammes of oats and another 0.

8 kilogrammes of sugar.

Take a moment to have a think.

So the three parts in our bar model are 1.

2, 0.

2, and 0.

8, and they are our three addends.

So which of these problems represented that? A and B did.

Let's have a closer look at why.

A, I have £1.

20, 20 pence, and 80 pence.

Now remember that £0.

2, we can say as 20 pence, £0.

8, we can say as 80 pence.

So here we've still got the same three addends, it's just the unit of measure we've used is slightly different, and so we've changed the numbers slightly.

For B, we have 1.

2 metres, 20 centimetres, and 0.

8 metres.

So we've done a very similar thing here with the 20 centimetres and 0.

2.

In the context of metres, 0.

2 metres is equivalent to the same as 20 centimetres because there are 100 centimetres in one metre.

So that problem represents perfectly as well.

Why doesn't C work? Hmm.

Well, this time we've got 1.

2 kilogrammes, but then we've got two kilogrammes.

0.

2 kilogrammes and two kilogrammes are not equivalent, they're not the same.

So we can't say that the bar model represents that particular problem because the addends are different.

Well done if you spotted that.

Time for your second practise task.

For question one, I'd like you to write three of your own problems to represent this bar model.

So as you can see, we have three addends: 250, 105, and 395, and a missing whole.

I really want you to challenge yourself here.

Can you write three very different problems using those three addends? Can you think about different context or different units of measure that you might use that would fit with those three addends? For question two, I'd like you to first complete the bar model to choose your own numbers that will be suitable for that bar model.

So think about the proportionality of those bars and what numbers would go there.

Remember, you can use integers, whole numbers, or you can use decimals.

Then I'd like you to write three of your own problems that represent the bar model that you've made.

Think carefully about what you might notice about the size of the parts in this bar model.

Welcome back.

How did you get on? Did you think really carefully about the different context you could use for different problems? Now remember that you could have come up with loads of different problems, and I'm sure you are much more creative than I was, but let's see what our three Oak children came up with for this bar model.

Aisha went for a measures context.

So she said, I used 250 grammes of coconut milk, 105 grammes of spinach, and 395 grammes of pineapple to make a smoothie.

How much did the ingredients weigh altogether? So let's just check.

Our three parts, our three addends are 250, 105, and 395, and we're finding the unknown whole.

Yep, Izzy's problem absolutely works with that.

She has her three parts, and she's used the context of gramme, so units of mass there.

Well done, Izzy.

Sofia said, 250 people watched the show in the morning, 105 people came to the afternoon show, and 395 people came to the evening show.

How many people came altogether? So different context to Izzy, but we're still using those same three addends this time in the context of people coming to a show.

Let's see what Aisha said.

Aisha went for money context.

She said, "I saved £250 two years ago, last year I saved £105, and so far this year I've saved £395." Wowsers, Aisha! "How much money have I saved?" So again, we've got three different context, different units, so grammes, people, and pounds.

But we've used those same three addends, and we are looking for the unknown whole.

I wonder what context you came up with.

Now let's look more closely at the bar model that we have here.

You may have spotted that the largest bar was half the value of the whole.

You may have also spotted that the two smaller bars were equal in size.

So we have two smaller parts that are equal in size and one larger part, which is about half the value of the whole.

So hopefully the problems you came up with will reflect that proportion found in the bar model.

Now remember, you could have chosen any values that you like for this bar model.

So your questions could have been all sorts of different things in all sorts of different context.

But let's take a look at what our Oak children said.

Izzy went for a measures context again.

So she said, I mixed 30 millilitres of red paint with 15 millilitres of both, blue and green paint.

How much paint did I use altogether? So we can see here that she's chosen the largest value of 30 and then two lots of 15 millilitres, blue and green paint.

Sofia says, on a nature walk, half of the class went kestrel spotting, seven children went bug hunting, and the rest went rock climbing.

How many children were in the class? So Sofia has really carefully thought about the proportion of the bars.

Half of the class went kestrel spotting, which would be the larger bar, and then one of the smaller bars would be the bug hunting.

So that's a really interesting way of thinking about this bar model.

Let's see what Aisha said.

"I read three books this week, the first two books both had 90 pages, and the final book has 275 pages.

How many pages did I read altogether?" So again, Aisha's recognised that those two parts have the same value.

I wonder what problems and context you came up with.

That brings us to the end of our lesson where we've been using models to interpret part-part-whole problems with three addends.

Let's summarise what we've learned.

You can use a bar model to represent problems with three addends, and we know that identifying the structure of a problem can help us to solve problems when the context are less familiar.

Thank you so much for all your hard work today, and I look forward to seeing you again soon.