Lesson video

Hello, there.

How are you today? My name's Ms. Coe, 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're ready to get started, let's get going.

In this unit, we are using our knowledge of part-part-whole to think about additive structures, and by the end of this lesson, you will be able to say that you can use written strategies and known facts to calculate the value of a missing part.

In this lesson today, we have two keywords.

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

My turn, represent.

Your turn.

My turn, addend.

Your turn.

Great work.

Let's think about what these words mean.

To represent something means to show it in a different way.

Addends are any numbers added together.

So, in the equation 3 plus 2 equals 5, 3 and 2 are the addends.

In this lesson today, we're going to be thinking about written strategies, as well as known facts, to calculate the value of missing parts.

Our lesson has two cycles.

We're going to start by thinking about forming equations from problems, and then we're going to think about calculating efficiently.

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

In the lesson today, you're going to meet Andeep and Jacob, and they're going to be helping you with your learning and asking us some questions along the way.

Let's start here by thinking about football.

Football players transfer between teams all over the world.

Now, you might have a favourite football team, and you might know that some of your favourite players might start off playing for your team and then get transferred, get moved.

And this can happen within our country, but it can happen all over the world as well.

And in some cases, when these transfers happen, there is a lot of money involved and we can think about the money involved using a bar model.

The total cost of a transfer can be broken up into four main areas.

A transfer fee, so how much the new team will pay for the new player, the player's wages, so how much the player earns, any bonuses that the player might get, and also an agent's fee.

The player has an agent who is someone who just helps the transfer and helps that process along.

And all of those are added together to make the transfer cost.

You can see there we have a bar model that shows that.

The fee, plus the wages, plus the bonuses, plus the agent's fee all add together to make the transfer cost.

Take a look at our bar model now.

What do you notice? What can you see this time? Well, now we still have the transfer fee, the wages, the bonuses, and the agent's fee, and it still equals to the transfer cost, but we have some numbers in there and we also have a bit of a key.

When we're talking about m, we're talking about million, which is a huge number to think about.

So, let's think about what each number in our bar model represents.

The 15 million pounds represents the total cost of the transfer.

Imagine your favourite football player has been transferred to a new team.

The total cost of that is 15 million pounds.

The 5 million pounds represents the transfer fee.

The 7 million pounds represents the player's wages.

That means the 2.

5 million pounds represents the cost of any bonuses that player might receive.

We have a missing part, there, don't we, shown by the question mark.

We're missing the cost of the agent's fees.

We don't know the cost yet.

So, how can we record that problem as an equation? We might have seen these two strategies before.

We can write this as a repeated subtraction equation, so we can say that 15 million pounds, subtract 5 million pounds, subtract 7 million pounds, subtract 2.

5 million pounds is equal to the missing part.

And we can do that because we know that if we have a whole and we subtract the known parts, we would be left with the unknown parts, because all of the parts, all four parts are equivalent to that whole of 15 million pounds.

So, we can represent this problem using that equation, or we can represent it as a missing addend equation.

We know that the parts sum, add up to, the whole, so we can write 5 million plus 7 million plus 2.

5 million plus something that's unknown value is equal to 15 million.

Now, representing this bar model and this problem in these two ways really can help us think about how we might go about solving the problem and finding the missing parts.

So, if you come across problems like this, it's really worth taking a moment to represent it in these two different ways.

Time to check your understanding.

Which equations match the bar model that you can see there? Take a moment to have a think.

Welcome back.

What did you think? B represents, or matches, the bar model.

We know that the whole is the transfer cost and we have three known parts, the wages, the bonuses, and the agent's fee.

But we have one unknown part which has been represented by a question mark.

So, we can say that the transfer cost is equal to something plus the wages, plus the bonuses, plus the agent's fee, and that equation matches the bar model that you can see there.

We can also say that D is correct.

Remember, we can write this as a repeated subtraction.

We know that the missing part is equal to the whole, which is the transfer cost, subtract the wages, subtract the bonuses, subtract the agent's fee, because what we are left with is that missing part.

Well done if you identified both of those equations that represent that bar model.

Time for another check for your understanding.

This time, I'd like you to write a repeated subtraction equation and a missing addend equation to represent this bar model, so look carefully.

What is the whole, what is the parts, and what is the missing part? Take a moment to have a think.

Welcome back.

How did you get on? This time, I know that the whole is 15 million pounds and I have three known parts, 5 million pounds, 2.

5 million pounds, and 1.

5 million pounds, and I have a missing and unknown part.

If I was going to write a repeated subtraction equation, I would start with my whole, which is 15 million pounds, and I would subtract each of my known parts.

So, 15 million pounds, subtract 5 million, subtract 2.

5 million, subtract 1.

5 million is equal to the unknown part.

If I was going to write a repeated addition equation, then I would sum my known parts and my unknown parts and it would be equal to the whole, so I could write 5 million plus 2.

5 million plus 1.

5 million plus something, my unknown part, is equal to 15 million.

Well done if you wrote both of those equations for that bar model.

Time for your practise task, I would like you to represent each of these problems as a repeated subtraction equation and a missing addend equation.

This time we're thinking about cola and we're thinking about the total sales.

We have cola, diet cola, and an unknown something which is equal to the total sales.

How would you write that as repeated subtraction and a missing addend equation? And the second one you might notice is very similar, but this time, we have numbers attached to it.

Again, we're thinking about millions of pounds, here.

So, can you write a repeated subtraction and a missing addend equation for that one? And for Question 3, we have a worded problem.

Represent this problem as a repeated subtraction and a missing addend equation as well.

Good luck with those three tasks, and I will see you shortly for some feedback.

Welcome back.

How did you get on? You may have noticed that the first two bar models are the same, apart from the fact that one just has cola, diet cola, and unknown, and the other one has values ascribed to those parts and the whole.

For the first one, we can say that the total sales subtract the number of cola sold subtract the number of diet cola sold is equal to the missing part, now, the missing part could be anything.

It could be lemonade, it could be caffeine-free cola, it doesn't matter.

We can also write a missing addend equation.

We can say that the cola sales plus the diet cola sales plus the unknown is equal to the total sales.

Now, you may have realised that for your second bar model, you could write very similar equations.

But this time, we're just substituting in the numbers.

We can say that 30.

02 million subtract 13.

75 million subtract 9.

52 million is equal to the missing value that we don't know.

Or we can say 13.

75 million plus 9.

52 million plus something is equal to 30.

02 million, which is the whole that we're looking for.

Well done if you wrote those equations for that bar model and well done if you spotted the connections and relationships between the two.

For Question 3, we're still thinking about cola, but this time, we have a worded problem: Over the last three years, diet cola has increased in sales.

The total sales over the last three years has been 120.

45 million pounds.

In 2021, sales were 26.

3 million pounds.

In 2022, they were 47.

5 million pounds.

What was the value of sales in 2023? Remember, we were representing this as repeated subtraction and a missing addend equation.

You may have drawn a bar model to support your thinking.

We can say that 120.

45 million subtract 26.

3 million subtract 47.

5 million would give us our missing part.

Or we can write a missing addend equation, 26.

3 million plus 47.

5 million plus something is equal to 120.

45 million.

Well done if you thought really carefully about those three problems and wrote two different equations to represent them.

Let's move on to the second cycle of our learning, where we're thinking about calculating efficiently.

Let's return to thinking about our footballers and our football transfers.

Remember that there are four parts that make up the transfer fee for one footballer.

This was the breakdown of costs for one footballer's transfer.

How can you calculate the agent's fee, which is the missing part? Look carefully at that bar model.

What do you notice? What are the known parts? What is the unknown part? Andeep is going to use a repeated subtraction strategy to find the missing part.

He is going to subtract each known part from the whole using his number facts.

He starts with 15 million, which is the whole, and he's going to start by subtracting 5 million.

He knows that 15 subtract five is equal to 10, so 15 million subtract 5 million is equal to 10 million.

He's then going to subtract 2.

5 million, and he knows that 10 subtract 2.

5 is 7.

5, so 10 million subtract 2.

5 million is equal to 7.

5 million.

Now, he's probably having to make jottings here to keep up with that, but that's absolutely fine.

He's working out using his number sense and it's absolutely fine if you need to keep some jottings to keep track of what's happening.

What's he got left to subtract? That's right, 7 million.

He knows that 7.

5 million subtract 7 million is equal to 0.

5 million, or half a million.

I think that's a really efficient strategy, Andeep.

Well done.

But Jacob has a strategy that he says uses much less subtracting.

I wonder what he's going to do.

Jacob says that we can use our known facts to add the parts together.

Remember, another strategy when we're finding a missing part is to add the known parts together and then subtract that from the whole.

So, he's going to add together 5 million and 7 million, which is 12 million, and then he's going to add 2.

5 million, which gives him a sum of 14.

5 million, so he knows that the three parts that we know add up to 14.

5 million.

Now, remember, once we've added those up, we can subtract that from the whole, and actually, this case, we know that the three parts add up to 14.

5, so therefore, we know that we can do 15 million subtract 14.

5 million, which gives us 0.

5 million.

Hmm, was that easier? Was that easier for you to think about? Andeep's right.

Either method works well when we're working with those numbers.

Sometimes, efficiency comes down to preference.

You may have preferred Andeep's strategy or you might have preferred Jacob's strategy.

This is a different breakdown for a different footballer's transfer.

Have a close look at that bar model.

What's the same? What's different? What do you notice? How can you calculate the agent's fee this time? We still have three known parts and one unknown part, but those numbers do not look as straightforward as they did in the last example.

And I think Andeep's right.

If he uses his strategy of subtracting each part each time, he's going to need to do three separate calculations.

So, actually, this time, it might be easier to add the known parts together first.

Rather than trying to do 94.

6 subtract 45.

2, then 33.

47, which I know is going to need lots of regrouping, what might be easier to add those three known parts together first, and then subtract that from the whole.

And sometimes, when we need to add three parts together, it may be most efficient to use a written strategy.

Look carefully at that written strategy.

Remember that we have numbers with a different number of decimal places.

For example, 45.

2 has one decimal place, 33.

47 has two.

So, it's a really good strategy to align your decimal points so that we make sure that we're adding everything together properly.

If we add those together using our written method, remember, we need to regroup sometimes as well.

In our tenths column, we will need to regroup 10 tenths for one 1.

If we add those together, we can see that the total of the three parts is 82.

17 million.

Now, we can subtract 82.

17 from 94.

6.

Now, I think this is going to be easier, because although we have to do some regrouping, the numbers are closer together, so I might want to estimate.

I know that 94 subtract 82 is equal to 12, so I'm looking for an answer that's around 12 million.

Again, sometimes when we're working with numbers like this where we know there will be regrouping, it may be more efficient to use a written strategy.

You may have also counted on to find the difference.

Remember, it's really important that we align our digits carefully, and you might want to add a placeholder where we have different numbers of decimal parts.

Here, we've written 94.

60 because that helps us to think about the regrouping that we will need when we're subtracting.

Let's take a look at that written subtraction.

We know we need to regroup, and now we can think about the rest of the calculation.

The agent's fees were 12.

43 million, and I'm really glad that I decided to do that quick estimate, because I expected it to be around 12 million and my answer is very close to 12 million.

That's a really good trick as well, to think about, what do you expect your answer to be? Because that way, you can check the reasonableness of it.

We've used written strategies, here, we've added first, and then subtracted those three parts once we've added them up from the whole to find our missing part is 12.

43 million.

That's probably more efficient than trying to do it mentally or trying to do multiple subtraction equations.

And Andeep's right.

I think that strategy is much more efficient than his would've been.

Time for you to have a go yourself.

Calculate the value of the missing parts.

Now, what I'd like you to do before you launch straight into calculating is have a really careful look at these numbers.

Which strategy would you use? Would you subtract first, or would you sum the parts and then subtract that from the whole? Do you need to use a written strategy, or can you use mental strategies? Remember, you can do some jottings if you need to.

Spend a moment thinking about the value of the parts and the whole and which strategy you are going to find most efficient.

Take a moment to have a think.

Welcome back.

Which strategy did you use? Let's see what Andeep did.

He started by subtracting 10 million from 25 million, and that left him with 15 million.

He's now got to subtract 3.

5 million and 1.

5 million, and so he actually decided to add those together.

He could quickly and easily see that 3.

5 and 1.

5 is equal to five, so 3.

5 million plus 1.

5 million is equal to 5 million.

He added those two remaining parts together, and then he subtracted that from 15.

Remember, he's already subtracted 10 from 25, so he subtracted a further 5 million from 15 million, which left him with 10 million.

The missing value is 10 million pounds.

Well done if you found that missing part and well done if you found an efficient strategy like Andeep's to work that out.

Time for another practise task.

I would like you to draw a bar model for each of these problems and then solve it using your preferred strategy.

Remember, you may find it easier to subtract the parts, or you may find it easier to sum the parts and then subtract from the whole.

Remember that you can write different equations to help you see the different ways of doing it, and you may need a mental strategy or a written strategy.

It's worth really interrogating those numbers to see which one will work most efficiently for you.

For example, Question 1, "The total cost of 4 pears, 3 bananas, and a bag of nuts is 4.

15 pounds.

The pears all together cost 1.

75 pounds and the bananas cost 97 pence.

How much do the nuts cost? Really think about those numbers.

Is there anything you can do to them to make them easier to work with? Draw a bar model first and then think about what is the most efficient strategy to solve the problem.

You have Question 2, there, and then Question 3 as well, which is thinking about time.

Good luck with those three questions.

Think carefully about the strategy you want to use, and I'll see you shortly for some feedback.

Welcome back.

How did you get on? Did you look really closely at the numbers and think about what strategy was most efficient for you? This is the bar model for Question 1.

We know that the whole is 4.

15 pounds We know that the one part is 1.

75 pounds, another part is 97 pence, and we don't know how much the nuts cost.

Remember, your parts might have been drawn in a different order to mine.

That's absolutely fine, because addition is commutative.

Which strategy would you use here to calculate the missing parts? Remember, you might have chosen a different strategy to me, and again, that's absolutely fine.

I think it is more efficient here to sum the known parts first and then subtract from the whole, so I would do 1.

75 pounds plus 97 pence.

Now, I know that 97 pence is 3 pence away from 1 pound, so I could round that to the nearest pound and then adjust my answer.

I know the two parts together cost 2.

72 pounds, and then I could use a written strategy to calculate 4.

15 pounds minus 2.

72 pounds, which is equal to 1.

43 pounds.

So, the missing part was 1.

43 pounds, the nuts cost 1.

43 pounds.

Remember, you might have chosen a different strategy.

That's just the most efficient strategy for me.

Question 2, Jacob has a voucher for 14,500 Nitro points for his game.

He buys an outfit for 3,750 points and a car for 7,420 points.

How many points does he have left to spend? This time, our whole was 14,500 and we had two known parts, 7,420 and 3,750, and we're finding our unknown parts.

Again, I think for me it was more efficient to add together those two parts.

I know that 7,000 and 3,000 make 10,000, and so I can use my other mental strategies there to sum those two numbers.

I know that the total of those two parts is 11,170, and then I can subtract that from 14,500.

I think if I subtracted the parts separately, I would've had to have done more written subtraction.

So, I now know that 14,500 subtract 11,170 is equal to 3,330.

He has 3,330 points left to spend.

Let's look at Question 3.

Jun, Jacob, and Sofia are completing an escape room.

They have two tasks to complete in 45 minutes.

The first task took them 20 minutes and 15 seconds, and they finished the second task with 42 seconds to spare.

How long did it take them to complete the second task? Hmm, I've got mixed units, here.

That's something to think about.

I know that the whole escape room took 45 minutes and I know that the first task took 20 minutes and 15 seconds, and I know that they had 42 seconds at the end.

The missing part is the unknown, there.

Again, I think an addition strategy is useful, here, and I can add together the seconds.

I know that 20 minutes and 15 seconds plus 42 seconds, well, 15 and 42 make 57, so 20 minutes and 57 seconds.

And then, I can subtract that from 45 minutes.

Now, I know that 57 seconds is 3 seconds away from a minute.

I would've rounded 20 minutes and 57 seconds to the nearest minute, which is 21 minutes, and then I would've adjusted my answer.

So, they took 24 minutes and 3 seconds to complete the second task.

Well done if you found all three missing parts and if you really thought carefully about the most efficient strategy for you.

We've come to the end of our lesson where we've been using written strategies, as well as known facts, to calculate the value of missing parts.

Let's summarise our learning.

You can represent a missing part problem as a repeated subtraction equation or a missing addend equation.

You can find a missing part by either subtracting each known part from the whole, or by adding known parts together and then subtracting them from the whole.

When we have more complex numbers, it's sometimes best to use a written method to add those known parts together, and then subtract them from the whole.

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