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 unit, we are looking at part, part whole and additive structures.

By the end of this lesson, you will be able to say that you can create stories to match structures presented in a model.

We have two key words in this lesson today, and I'm going to say them, and I'd like you to say them back to me.

Ready? My turn.

Represent.

Your turn.

My turn.

Addends.

Your turn.

Great job.

Let's take a look at what these words mean.

To represent something means to show something in a different way, and we're going to be thinking a lot about that in today's lesson.

Addends are any numbers added together.

So in the equation, 3 plus 2 is equal to 5, 3 and 2 are the addends because they are the numbers that are being added together to make the sum of 5.

In our lesson today, we're going to be creating stories to match structures presented in different models, and we have two cycles today.

We're going to start by looking at the same model but different stories, and then we're going to look at how we might adjust models.

So if you're ready, let's get started with the first cycle.

In this lesson today, you are going to meet Jun and Jacob, and as always, they're going to be asking us some questions and helping us with our learning.

Let's get going.

Jun and Jacob are thinking about a story that represents this bar model.

Take a good look at the bar model.

What do you see? What do you notice? Hmm, that's right, Jun.

The bar model here is composed of three addends, and these can be added together to make the whole.

The whole at the moment is unknown.

We don't know what the whole is.

So our three addends are, this time, in a unit of measure.

They're in minutes.

So we have 30 minutes, 20 minutes, and 55 minutes.

So what kind of stories are we going to have that represent this bar model? Well, we're going to have stories that relate to time, aren't we? Because minutes are a unit of measure that focus on time.

So let's see what Jacob comes up with.

So he thinks he could say that he's watched three different TV programmes.

The first programme was 30 minutes long, the second was 20 minutes long, and the last one was 55 minutes long.

And the question would be how long did I watch TV for altogether? What do you think? Does that problem represent and match the bar model? It absolutely does, doesn't it? He's asking how long he's watched TV for altogether, so we don't know the whole, and he said that he had three TV programmes and they matched the three parts, or addends in the bar model.

Let's see if Jun can come up with an example.

"What about," he says, "I used the oven to cook some potatoes for 30 minutes then some parsnips for 20 minutes and then some chicken for 55 minutes? How long was the oven on for?" What do you think about that one? Hmm, I'm not sure, and Jacob agrees with me.

That would only work if you cooked them one after the other instead of at the same time.

Now, think back to a dinner that you've seen cooked or you might have cooked yourself.

It's very unlikely that you would cook everything one after the other.

It's much more likely that you'd put multiple things in the oven at the same time.

So Jun's story does use the three parts, but I'm not sure that the whole is the one that we're looking for.

You could change the question, however, to thinking about the longest possible time that it would take to cook the meal.

So imagine you had a really tiny oven that actually did only fit one thing in.

What is the longest possible time that it could take? That's a really nice tweak, Jacob, to think about Jun's question.

"Okay," says Jun, "let's try and calculate that." So this is what Jun would do to find the unknown whole.

He said he would do 30 plus 20 plus 55, and he knows that 30 plus 20 is equal to 50.

So we now have 50 plus 55.

I know that 50 plus 50 is 100, so we have a total of 105.

Remember, that's 105 minutes because we know the units of measurement that we're talking about.

"That's 105 minutes," says Jun.

But remember that if we're talking about minutes, we can also talk about other units of measure.

We know that in 1 hour there are 60 minutes, so we can partition 105 minutes into 60 and 45.

So we can say that 105 minutes is also 1 hour and 45 minutes.

Jacob says, "That's interesting, Jun, I liked your strategy," but actually Jacob would've worked it out differently.

I wonder if you would've worked it out differently.

Jacob says he knows that 60 minutes is an hour, and so we know that the whole is 1 hours and 45 minutes like Jun said.

So we can still think about 30 plus 20 plus 55, but we can partition it slightly differently.

If we think about this as 30 plus 15 plus 60 or we know that 60 is equal to 1 hour, so we can add 30 and 15 together, which is 45.

So we know 45 plus 60 gives us our 1 hour and 45 minutes.

You might have thought of it like that as well.

Which of the two strategies did you prefer? Jun's strategy was to sum the addends from left to right, and then he partitioned the sum into 60 minutes for an hour and the remaining minutes.

So Jun's strategy was to add up the three parts, 20, 30 and 55, and then partition 105 into 60 and 45 to find that it was 1 hour and 45 minutes.

Jacob's strategy was a redistribution strategy.

So he redistributed 5 from 20 and put it onto the 55 to make that whole 60 minutes, and then he could see that we had 30 and 15 minutes, which he added together to make the remaining 45 minutes.

Which of these did you prefer? Well, both strategies are absolutely fine, and this will depend on efficiency.

So for me, personally, it was easier to think about adding the addends first and then using 105 because I am quite confident in partitioning 105 into 60 and 45.

You might have preferred to look for the 60 minutes first.

That's absolutely fine.

Time to check your understanding.

Create your own story for the following bar model.

So look carefully at the three addends or parts.

What story could you come up with.

You might like to tell a friend? Take a moment to have a think.

Welcome back.

Now, of course there are lots and lots of different stories that you could have come up with for this bar model as long as you had three parts, one of 25, one of 60 and one of 15.

So Jun came up with a money context.

He said, "I have 25 pounds in my wallet, 60 pounds in my bank account," and he got another 15 pounds for his birthday.

How much money does he have altogether? So that uses the three parts, the three addends, and we're asking how much altogether.

So we're asking for that unknown whole.

Your context may have been measures, it may have been another unit such as people going to a show.

It absolutely doesn't matter as long as you thought carefully about those three addends.

Let's do another check for your understanding.

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

So again, we have three addends and an unknown whole.

Read the three worded problems really carefully and decide which of them could be represented by the bar model.

Take a moment to have a think.

Welcome back.

What did you think? So A and C could be represented by the bar model.

Let's take a closer look as to why.

A says Jacob spends 72 pounds on a pair of boots, 28 pounds on a guitar strap and 120 pounds on a pair of speakers.

How much does he spend altogether? So we can clearly see here that we have the three addends, 72, 28 and 120, and we're asking for the unknown whole.

C is a fishing context, so our unit is fish.

Jun goes sea fishing each month.

He catches 72 fish in August, 120 fish in September and 28 fish in October.

How many does he catch altogether? So I can see that this time the addends in the worded problem are in a different order to the ones in the bar model.

Does that matter? No, we know that addition is commutative, and we can add those parts in any order.

The important thing was that we have a context where we've used the three parts, 72, 28 and 120.

Why does B not work? Well, we've got Alex walks for 120 minutes, he sits down for 28 minutes, then walks for another 72 minutes.

How much time does he spend walking? Now, in this instance, we do not have three parts that make an unknown whole.

Alex walks for 120 minutes but then he sits down, so he's not walking, so the 28 is not part of the whole, which means that this worded problem does not match the bar model.

Well done if you spotted that.

Time for your first practise task.

I would like you to use the bar models that you can see here to create your own stories.

And Jacob is setting you a bit of a challenge.

Can you create one where one context would work for all three bar models? Now, look carefully at those three bar models.

I can see that we have multiples of 100, we have decimal numbers and then we have a mix of two-digit numbers there.

So we've got some different numbers, but is there a context that would work for all three bar models? Pause the video here, and I'll see you shortly for some feedback.

Welcome back.

How did you get on? Now, remember that you're creating your own stories, so your stories may have looked a bit different to ours.

For the first one, we've got 7,400, 1,000 and 2,500 as our parts.

So Jacob has come up with a context around money, but this time using rupees, which is an Indian currency.

"On our holiday to India, dad took 7,400 rupees for spending money, I took 1,000 rupees and my sister took 2,500 rupees.

How much spending money did we have altogether?" Now, we can see that that fits the bar model because we have our three parts, and we're looking for an unknown whole.

For the second bar model, we have decimal numbers.

So this is an example Jacob came up with, and he chose the context of money.

He said, "I spent 7 pound 40 on a cinema ticket.

I upgraded my seat to a premium ticket for an extra 1 pound.

I also bought a tub of popcorn for 2 pound 50.

How much did my trip to the cinema cost?" So remember, 7.

4, if we're thinking about money, can be read as 7 pounds and 40 pence.

So we still have the same three addends here.

We've just said them in a slightly different way.

You might have chosen a money or a measures context for this one.

And for the last one, we've also got a money context, but this time we're thinking about slightly bigger values.

"I had 74 pounds in my bank account.

I was given 10 pounds by my aunt for my birthday.

My mum and dad also gave me 25 pounds to spend.

How much money do I have altogether now?" So this time we've used whole pounds, and we've used money context again to think about these numbers.

Did you notice that the context was the same for all three problems that Jacob came up with? He used a money context, and just because the unit of money was different in the first one, rupees instead of pounds, it's still the same context.

You may have used another measures context or you might have chosen money as well.

Well done if you thought of some interesting problems for these three bar models and managed to come up with a context that worked for all three sets of numbers.

Let's move on to the second learning cycle, which is adjusting models.

This time Jun has created a story and he has drawn his own bar model to represent it.

I'm going to read you his story.

Look really carefully at the bar model.

Do you think it matches? There are 120 pupils in Key Stage 2.

68 pupils have packed lunch, 32 pupils have the meat option and 30 pupils have the vegetarian option.

And that is the bar model that Jun has drawn to represent his story.

Ooh, Jacob's saying, "Are you sure that's right, Jun?" What do you think? Jacob says he thinks the numbers are wrong.

Should we take a closer look? Ah, you're right.

If we add those addends up together, it actually makes 130, not 120.

So 68 plus 32 plus 20 is equal to 130, and yet we've said the whole is 120.

Ooh dear, what's gone wrong there? "We could change it by adding 10 to the whole," says Jun.

So we could just change the whole to make 130 rather than 120.

I wonder if there's a problem with that though.

What do you think? "Well, actually," says Jacob, "you could do that, but there are actually 120 pupils in Key Stage 2, so the whole has to be 120.

We can't just imagine 10 more pupils into Key Stage 2, so the whole has to stay at 120." So that means that other parts of the problem that Jun has said must be incorrect.

So we might need to change the amount in each part.

How could we do that? Well, as Jun has pointed out, we know that the parts add up to 10 more than they should.

So we have 10 too many pupils in the parts altogether because they add up to 130 at the moment, and they should add up to 120.

So we have 10 too many.

What could he do? "Well," he said, "actually, it turns out that there were only 20 pupils having the vegetarian option." So at the moment we've got 30 for that part, but actually Jun has recounted and found that only 20 pupils have that vegetarian option.

So if we change that part to 20, then our bar model makes sense.

They are equivalent.

In this case, 68 plus 32 plus 20 is equal to 120.

And that's right, Jacob.

We also need to adjust the size of the bars slightly because at the moment, it looks like the number of pupils that have the meat option and the number of pupils that have the vegetarian option are the same.

And actually we know now that fewer people have the vegetarian option.

So Jacob's just gonna adjust those bars slightly to make it more proportional.

If we continue to think about lunch, we then learned that six pupils actually went out before lunch to go to a sports event.

You might have done that before, you might have actually missed your lunch sometimes to go to a really exciting football event or a netball event.

But how might our bar model change now? Let's take a look.

Well, we now know that there are 114 pupils altogether having lunch in school because six of them have gone, so we need to adjust our whole from 120 to 114.

We know that there are six children missing, so we've got six fewer in our whole, but what does that mean to our parts? Well, we don't actually know what type of lunch those six children were having, so we can't say for certain which part we need to subtract six from, which gives us different options.

So they could have had packed lunch, and originally, 68 pupils had packed lunch, so we could change that part to 62.

If all of the children who left, all six children, had packed lunch, that means there would be six fewer children.

So instead of having 68 for packed lunch, we'd have 62 for packed lunch.

And if we look at our bar model now, the parts sum to the whole, 62 plus 32 plus 20 sums to equals 140.

However, they could have had the meat option.

So at the moment, 32 pupils had the meat option for hot dinner, but we could change that to 26.

So if all six children who left to go to the sports event had a hot dinner meat option, there would be six fewer children.

And again, this bar model is now correct.

The whole is 114.

68, 26, and 20 add up to 114.

Or all six children could have had the vegetarian option.

So we know that 20 children originally had the vegetarian option, but if there were six fewer, there would be 14 children having the vegetarian option.

So we can see that there are lots of ways that our parts can change if we need to adjust the whole by six, "But also, says Jacob, "there could have actually been a mix of each option." So we don't know that all of them chose the same option.

So we could have had two of them have packed lunch, two have the meat option and two have vegetarian options.

We just don't know.

But in that case, we could reduce each of the parts by two, and we would still have our whole of 114.

66 plus 30 plus 18 is equal to 114.

We could have done it a different way.

We could have had five children with packed lunch and one with vegetarian.

There are lots of ways you could have done it.

Time to check your understanding.

66 children have packed lunch.

32 have the meat option and 18 have the vegetarian option.

However, four children are away from school that day.

Tick the options to show how the bar model could be adjusted.

So we have the bar model there representing the original problem.

How could the problem be adjusted? Take a moment to have a think.

Welcome back.

What did you think? Well, we know that the whole should change from 120 to 116 because 4 children are away, which means the whole has been reduced by 4.

The packed lunches could go down by 4, so they could change from 66 to 62 if all 4 children who are away had the packed lunch option.

The meat option could also change from 32 to 28 because that is 4 fewer children having the meat option.

However, D is incorrect.

Can you think about why? That's right.

The current vegetarian option is 18, and if we changed it to 24, that would be more children.

And we know we have fewer children because four children are away.

Well done if you reasoned about that.

Another check for your understanding.

Spot the mistake with this bar model.

Jun says, "I shared a tub of strawberries with Jacob and Alex.

There were 24 altogether, and we had an equal amount." So he's thought up that problem, and he's drawn a bar model to represent it.

What's wrong with the bar model? Take a moment to have a think.

Welcome back.

Did you spot the mistake Jun had made? Well, the whole was correct because there were 24 strawberries, and we know that we shared the amount equally with him and two people, so we needed three equal parts.

But what did you notice about the size of the parts? That's right, Jacob.

You each had eight strawberries and not nine strawberries.

Three lots of 9 is not equal to 24.

It needed to be eight strawberries for the bar models to be correct.

Well done if you spotted that.

Time for your second practise task.

For question 1, I'd like you to draw two bar models for each story.

So thinking about the original and then the change or the adjustment that's happened.

So this is A, Jacob decides to buy a pear for 40 pence, a magazine for 3 pounds 99 and a bottle of water for 1 pound 49.

How much did Jacob expect to spend altogether? So draw a bar model for that scenario.

At the till the magazine was actually priced for 3 pound 49.

How much did Jacob actually spend? So think about how your bar model would need to adjust for that problem.

For B, Laura goes on a run from her house to the harbour, which is 1.

4 kilometres, from the harbour to the school, which is 1.

3 kilometres, and then back to her home, which is 1.

2 kilometres.

How far does she run altogether? So draw a bar model to represent that, and then think about how it needs to be adjusted for when she gets home she looks at her smartwatch, and it says the distance from her house to the harbour was actually 1.

8 kilometres.

How far did she actually run altogether? And then for question 2, I'd like you to draw a bar model to represent this story.

For A, at a banquet, 40 people had been invited with a seating plan of five tables of eight seats to sit around.

However, 10 people did not turn up to the banquet.

Draw three possible bar models to show how this might have affected the seating plan.

Good luck with those tasks, and I'll see you shortly for some feedback.

Welcome back.

How did you get on? So for the first question, we knew that there was 40 pence, 3 pound, 99 and 1 pound 49.

Notice how we thought about the proportion of the bars that we've drawn, and we can see that the total there, how much Jacob expected to spend, was 5 pounds and 88 pence.

But remember, at the till, the magazine, which was 3 pound 99, was actually priced at 3 pound 49.

So we've adjusted the bar model where one of our parts is a little bit smaller than we anticipated, and so, therefore, the whole, the total amount that he spent has been adjusted as well.

For B, again we have the expected run here.

We've got 1.

4 kilometres, 1.

3 kilometres and 1.

2 kilometres.

Notice how in these bars they're quite similarly sized because 1.

4, 1.

3 and 1.

2 are very close together, and we can see that Laura thinks she's run 3.

9 kilometres altogether, but, as we know, her smartwatch actually changed 1.

4 kilometres to 1.

8 kilometres.

It was a little bit further.

So we've adjusted one of the parts of the bar model, which means we've had to increase our whole as well.

We've adjusted one of the parts by 0.

4 kilometres, so we've had to adjust the whole by 0.

4 kilometres as well.

Well done if you drew both of those bar models and identified the parts and the wholes.

So for question 2, we started off with 40 people, and we had five tables with eight seats around each table.

Now, we know that five groups of 8 is equal to 40, so our first bar model would've had five equal parts each with a value of 8 because that describes the tables.

However, 10 people didn't actually turn up to the banquet, so we had 10 people not there, so we know that our whole now is 30.

And I asked you to find three different ways to adjust the model to cope with this idea that 10 people were not at the banquet.

Here are just some of the ideas that we came up with.

So if there are 30 people at the banquet, we could have had five groups of 6, so 5 times 6 is 30, so, therefore, we could have had five tables with 6 people on them.

So we could have reduced each table size by two or we could have kept four tables of eight and then had too much smaller tables, so two tables of 3, and that would have a whole of 30.

Or we could have been super rebellious and actually said, "Well, we need more tables," because if we had six tables with 5 people on each one, then that would've also made 30.

So we could have just changed the whole seating plan around to cope with these 10 people not here.

I'm sure you came up with lots of ways to adjust the bar model to cope with the change in the story.

We've come to the end of the lesson, and we've thought really carefully about creating stories to match structures presented in a model.

Let's summarise what we've learned.

A story and a bar model can both represent the same mathematical structure.

For example, we can have a problem with three or even more addends.

The same structure of a bar model can represent different stories, and bar models can be adapted to any changes that occur throughout a mathematical story.

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