Lesson video

Hello there.

How are you today? My name's Miss K, and 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 are ready to get started, let's get going.

This unit focuses on additive structures.

By the end of today's lesson, you'll be able to say that you can use knowledge of additive structures to solve problems. We only have one keyword in this lesson today.

I'm going to say it and I'd like you to say it back to me.

My turn, represent.

Your turn.

Let's take a look at what that word means.

To represent something means to show something in a different way, and you'll be thinking about different representations for additive structures today.

In today's lesson, we're going to use knowledge of our additive structures to solve different problems, and we're going to have two cycles in our learning today.

We're going to start by spotting structures, so thinking about additive structures, and then we're going to focus in on statistics.

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

In today's lesson, you are going to meet Lucas and Laura, and they are going to be helping our learning and asking us some questions along the way.

So let's start here by thinking about representations for mathematical stories.

I'm going to show you some bar models.

I want you to think about which of them best represents this story.

Laura went on a three-day fishing trip with her dad.

On Friday they caught eight fish, on Saturday they caught 36 fish, and on Sunday they caught 14 fish.

How many fish did they catch altogether? Does this bar model best represent the story? What about this one? Or this one? What do you think? Well, Laura thinks that we can dismiss the last bar model immediately, and I can see why.

The last bar model has four parts or four add-ins, and actually, we only need three parts to represent each day.

We can see that we have Friday, Saturday, and Sunday.

That's the information that we have.

And so we only need three parts on our bar model.

So she's right.

The last bar model does not represent this problem, so we can get rid of that one.

Let's look at the other two more closely.

Laura thinks it could be either of those two bar models.

What do you think? Lucas actually disagrees and thinks the second bar model best represents this problem.

What do you think? Do you think he's right? Why do you think he's chosen that second bar model over the first one? Because they all have the same values in.

So what makes the second one different? Well, the first bar model has each part of the same size, but the second one is more proportional to the numbers.

So as you can see in the second bar model, the smallest value is eight, and that's been represented by the smallest part or bar, and the largest value is 36, and that's been represented by the largest part.

So I think Lucas is right.

This bar model is the one that best represents the problem.

Now, there's nothing wrong with the first bar model, but when we're representing additive structures, it can be really, really helpful to think about the relative size of the different parts, especially if we're thinking about adjusting problems or we're thinking about more than two add-ins.

So it's a really good habit to get into, thinking about the proportionality of your bars when you draw your own bar models.

Let's take another look at another problem.

Which bar model could best represent this story? Lucas is dividing 1.

5 litres of water between three bottles.

Take a moment to have a close look at those three bar models.

Which best represents that story? Hmm, I wonder.

Well, Laura says this time it can't be the first bar model because it's only two parts, and we need three bottles or three parts because Lucas is dividing something between three bottles, so we need three parts, so it can't be the first one.

Laura's absolutely correct.

What's about the other two? Laura thinks that it must be the last one because it's been divided equally into three bottles.

So if we look closely at the last bar model, we can see that 1.

5 litres, we have three equal parts of 0.

5 or half a litre.

What do you think? Do you agree with Laura? Well, actually, Lucas points out that the problem we've got doesn't say whether he divided the water equally or unequally.

It's not clear.

It just says that he is dividing 1.

5 litres of water between three bottles.

It's not saying equally.

So therefore, it could be either of those bar models because they both have three parts that sum to 1.

5 litres.

So sometimes we don't know the relative size of the parts because in this example it's not clear what the size of the parts are.

As long as the parts sum to 1.

5 litres, then our bar model could represent that problem.

Good thinking, Lucas.

Time to check your understanding.

Tick the bar model that best represents the following problem.

Laura shares a 120-gram bar of chocolate with three of her friends.

Take a moment to have a think.

Welcome back.

Which bar model do you think best represented this problem? Well, it could be B or C.

Let's think about why.

Laura is sharing her chocolate bar with herself and three friends, so there are four people, which means our bar model needs to have four parts.

A does not have four parts.

It has three parts.

We're also not told how Laura shares that 120-gram bar of chocolate.

It may be that she decides to share fairly and so each person gets an equal share, or she may decide to keep more for herself.

So she has 60 grammes and her three friends get 20 grammes each.

It's not clear in the problem, so either B or C could represent the problem.

Well done if you reasoned it like that.

Time for your first practise task.

For question one, I would like you to circle the bar models that could represent the following problem.

Lucas, Laura, and Andeep all have some mobile data left on their phones.

Altogether, they have 11.

7 gigabytes of data between them.

There are six bar models there.

Which ones could represent that problem? Have a careful look at the parts and the whole and circle the ones that you think could represent that problem.

For question two, we have some blank bar models and I'd like you to match the bar model to the story that has the same structure.

For example, Laura has four bags of sweets that all have the same mass.

You have four different bar models there.

Which bar model has the same structure as that worded problem? Good luck with those two tasks, and I'll see you shortly for some feedback.

Welcome back.

How did you get on with those two tasks? So the first task asks you to think really carefully about this problem and circle the bars that could represent the problem.

So we have Lucas, Laura, and Andeep, and they all have some mobile data left on their phones and they have 11.

7 gigabytes together.

So how could that look as a bar model? So we know from the question that we have three parts.

I can see that one of these bar models has four parts, so it's definitely not that one.

I also know that they all have some mobile data left.

So there's an example on the bottom row that has a zero as one of the parts.

They all have some data, so therefore one of the parts can't be zero.

So that leaves me with four other options.

It could be this one.

I have three parts.

The three parts sum to 11.

7.

It could also be this one.

We're not told if they have the same amount of data or not, but if they have the same amount of data, they would have 3.

9 gigabytes of data each, which would sum to 11.

7.

It could also be this one.

We can see that we have three parts and they sum to 11.

7.

So it could also be that option to represent the problem.

Why could it not be the middle one on the top row? We have three parts.

We have a whole of 11.

7.

Ah yes, good spot.

The three parts sum to 12, not 11.

7, so we can't have that as an option.

Well done if you circled those three bar models.

For question two, I asked you to match the bar model to the story that had the same structure.

So Laura has four bags of sweets that all have the same mass.

So we have four parts here and we all have the same mass, so that means the parts have to be equal, so it matches to that one.

Jacob divides 240 grammes of flour equally between three bowls.

Now the word equally there is a massive clue.

Equally means they're going to have the same parts, and we have three parts here because we have three bowls, so that one matches to that bar model.

Lucas has a turn on the computer for 20 minutes.

Jun gets the same amount of time.

In the last 20 minutes, both Laura and Alex share the remaining time equally.

Well, have four children here, so I need four parts, and I can see that Lucas and Jun get the same amount of time and then Laura and Alex get the same amounts of time as well, but it's a shorter amount of time than Lucas and Jun.

So that goes to the first bar model.

And then finally, let's just check, Andeep buys a sandwich, some melon bites, and a drink.

The sandwich was the most expensive item.

So we have three parts here where one of the parts is bigger than the other parts, which matches to that bar model.

Well done if you connected all the bar models to the stories.

So we've been thinking about stories and additive structures, and now we're going to move on to thinking about statistics.

Let's go.

In London, like in some other cities, you can hire bicycles.

And so this table shows the number of bikes hired in London using a cycle hire scheme across one week.

So you can see there's some really big numbers there.

Clearly, lots of people choose to hire a bike in London.

It's a really good way to get around the city.

Lucas is wondering, what bar models could we create from this information? Hmm, I wonder what bar models you can visualise.

Well, Laura thinks that we could find the sum of any of the days combined.

We could also think about specific days, so we can think about the number of bikes hired at the weekend.

And to do this, we could draw a bar model.

So we can see that on Saturday there were 21,582 bikes, and on Sunday there were 18,359 bikes.

So we've drawn a bar model here with those two bits of data.

We don't know yet the total, we don't know the sum, but we do know that to find that out we need to add together those two values.

We only need two parts because we've only got two days.

We've taken some of the data and we're just using two of those days, and we can add together and find out that the total number of bikes hired over the weekend was 39,941.

That is a lot of bikes in one weekend.

And we can show that by changing the whole to reflect how many bikes were hired over the weekend.

We could also find out the number of bikes hired on more than two days.

So we could think about how many bikes were hired on Monday, Tuesday, and Wednesday.

How many parts would we need in our bar model? That's right.

We'd need three parts because we have three bits of data here.

We'd need three parts to represent the three days, Monday, 20,604, Tuesday, 25,005, and Wednesday, 26,063.

Notice how we've put them in a different order.

Does that matter? Absolutely not.

We know that addition is commutative, so we can add the parts in any order, which means we can put them into our bar model in any order.

As long as we've got three parts and those three parts properly reflect the data, that's absolutely fine.

And we can add those together to find that the total number of bikes hired on Monday, Tuesday, and Wednesday was 71,672, and we can put that into our whole.

We could also find out the total number of bikes all week.

Goodness me, I think that's going to be a very large number.

So in that case, we would need seven parts because a week is seven days.

So we'd need all seven parts there, and you can see that we've put all seven parts in and we're trying to find the whole, which is the total bikes hired, and we could add up all of those numbers to find the whole.

So the total number of bikes hired across the whole week was 159,363, and we can change our whole to show that.

Time to check your understanding.

Draw a bar model to calculate how many bikes were hired on Thursday, Friday, Saturday, and Sunday.

Think carefully about how many parts you need for your bar model.

Take a moment to have a think.

Welcome back.

So we have Thursday, Friday, Saturday, Sunday.

That is four days, so we'd need four parts to our bar model.

I've carefully copied over the data.

It can be really easy, can't it, to make some mistakes? So take time to copy over the correct numbers and then we can add those four numbers together to find that the total number of bikes was 87,691.

Well done if you drew that bar model and well done if you summed those numbers to get that total.

Time for your second practise task.

In this task we're going to be focusing on data in this bar graph.

So this bar graph is showing the web visits to amazon.

com from July 2023 to December 2023.

It's worth taking some time to think about what this graph is telling you.

You might want to annotate it with some information, but we can see the months along the x-axis, which is the horizontal axis, from July to December, and we can see that the website visits are in billions going up the side.

Take a moment to think about what the scale is telling you.

Once you've spent a bit of time looking at that graph, draw a bar model for each of the following questions and use the bar model to help you answer them.

So for example, question A says, how many website visits were there in July and August? So there are four questions there.

So all of that data can be found by looking closely at the graph.

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

Welcome back.

How did you get on? The first thing I would've done is looked really closely at this graph, and I would've annotated it to tell me what the value of each bar was.

So for example, I can see that the scale on the y-axis, the vertical axis, I can see that I have 0, 1, 2, 3, and I can see that there are 10 divisions in between each one, which means that each division represents one tenth.

Now in this case it's 1/10 of a billion.

So for example, the bar for July is at 2.

6 so I can say that there were 2.

6 billion visits to amazon.

com in July 2023.

I would've annotated that graph before thinking about the answers to the questions.

So the first question was, how many website visits were there in July and August? Once I'd annotated the graph, I'd be able to see that there were two parts, one with a value of 2.

6, and one with a value of 2.

5.

If I find the total, add those together, that is 5.

1 billion people visiting amazon.

com in July and August.

B asked us to find out how many website visits were there in July, August, and September.

So this time I needed three parts to my bars, 2.

6, 2.

5, and 2.

3.

Now, I would've used my total from a and added 2.

3.

So 5.

1 billion plus 2.

3 billion is equal to 7.

4 billion.

It's a lot of people, lots of visits.

Then we've got how many visits were there from July to December.

So that means I would need six bars to represent each of the months, and if I found the sum of all of that data, it would be 15.

1 billion visits to amazon.

com.

Question D is a little bit different.

Why do you think that most website visits occurred in December? So if we look at our graph, we can see that the most were in December.

Why is that? Hopefully you thought about the time of year.

December has lots of festive holidays, and so more people are more likely to go to somewhere like amazon.

com to buy presents or even just look for presents or make a wishlist.

Well done if you thought about that.

So we have come to the end of our lesson thinking, about using our knowledge of additive structures to solve problems and we've thought about worded problems and we've also thought about statistics.

Let's summarise our learning.

Identifying the structure of a problem can help make a problem easier to solve.

We can still use bar models to represent the structures of stories in a range of different contexts, including measures and statistics.

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