Lesson video

Hello and welcome to today's math lesson.

I trust you are feeling well.

My name is Dr.

Shorrock, and I'm really looking forward to guiding you through our learning today.

This lesson is from our unit rounding and solving problems with numbers with up to seven digits.

This lesson is called explain how to solve calculations efficiently.

As the title suggests as we move through the learning today, we are going to consider different strategies that we can use to add and subtract.

You can always use a mental or a written strategy, but in certain situations there it is always a more appropriate strategy that we could use.

So we are going to look at when we can decide if we should use mental or written methods.

Now, sometimes new learning can be a little bit tricky, but it is okay because I'm here to guide you and I know if we work really hard together, then we can be successful in our learning today.

Let's get started then, shall we? How can we solve calculations efficiently? These are our key words from our learning today.

We have efficient and precise.

It's always good to practise saying new words aloud, so let's have a go at that together.

My turn, efficient.

Your turn.

Nice.

My turn, precise.

Your turn.

Fantastic.

Being efficient means we can find a way to solve a problem quickly whilst we can also maintain accuracy.

And if we are precise, it means we are being exact.

In this case, it is when numbers are given as exact values and they are not rounded.

Let's get started with our learning today where we are going to start by looking at how we can identify the most efficient strategy to use.

And we have Aisha and Lucas to help us in our learning today.

Lucas researches the average mass of some species of whales and he has presented his data in a table.

Take a look at the table, can you see it has two columns.

The first column are the species of whale, Beluga, Killer, Humpback, Sperm and Blue.

And in the second column we have their average mass given in kilogrammes.

We want to know how much heavy is a Sperm whale than a Humpback whale.

Let's start by representing this as a bar model.

The mass of the Sperm whale is greater, so it is our whole then the mass of the Humpback whale is our part of the whole.

The missing part is our difference.

We can substitute their masses for their name.

So we've put 40,000 is now a whole and 30,000 is a part of the whole.

Using our bar model, we can then form an equation.

We know 30,000 add something, the missing part or the missing addend is equal to 40,000.

Let's rearrange that equation and we know if we subtract the known part from the whole, we will find the missing part.

How would you solve this? Is a mental strategy or a written method more efficient do you think? Well, to decide how to solve this, we need to look at the numbers involved.

The numbers have only one non-zero digit, so these numbers are rounded numbers, they're not precise.

So we can use known facts and unitizing and using a mental strategy would be most efficient this time.

Aisha then researches the area of some European countries.

She presents her data in a table.

We want to know how much larger is Germany then Italy.

So take a look at the table.

Can you see it's two columns again.

The first column are countries, France, Spain, Sweden, Germany, Italy, and the second column are areas given in kilometres squared.

What do you notice about those values that are given? They look very precise, don't they? So let's represent Germany and Italy as a bar model.

Germany is our whole because it is larger and Italy is a part of the whole, we can then substitute their given values for their areas.

Can we form an equation from our bar model now? We know that 301,340 added to something, the missing part or addend is equal to the whole of 357,022.

We can rearrange that equation so that we can subtract the known part from the whole and that will give us our difference.

What about this equation then? The previous equation we solved mentally because the numbers were rounded, they were not precise, but how would you solve this equation when these numbers are really quite precise, aren't they? So do you think a mental strategy or a written method is most efficient? Well, to decide that, let's look at these numbers.

They're precise, aren't they? They're precise values.

So using a written method would be more efficient this time.

Let's compare the two questions.

How much heavy is a Sperm whale than Humpback whale and how much larger is Germany than Italy? What do you notice that is the same and what is different? Well, both questions share the same structure, they are comparing the size of two things, aren't they? And both questions, the size of both things is known and the difference is unknown.

Both questions had a measures context.

What was different though? Well one measure is kilogrammes and the other was square kilometres.

And one question compares rounded averages, but the other compares precise measurements.

One question could be calculated mentally, but the other is less efficient to do mentally and requires a formal written method.

Let's check your understanding with that.

True or false? When finding the difference between 12,000 and 9,000, a mental strategy is more efficient.

Pause the video while you decide if that is true or false.

Press play when you're ready to hear the answer.

How did you get on? Did you say, well that's true, but why is it true? Is it because if we look at the numbers we can see that we can use unitizing and known facts to help 12 - 9.

Or is it true, because a mental strategy should always be used because the written methods take far too long.

Pause the video while you decide which reason, and then when you're ready to hear the answer, press play.

How did you get on? Yes, did you say it must be A because we can see that we can use unitizing and known facts at 12 - 9 will help us.

Well done.

It's your turn to practise now.

For question one, could you identify the most efficient strategy, mental strategy or written method with which to solve these problems? And give reasons for your choice.

The library near where Lucas lives has about 750,000 books.

At the moment 25,000 books are out on loan.

How many books remain in the library? And part B, there were 34,598 people living in Didcot in 2023.

There were 352,913 people living in Reading at this time.

How many more people lived in Reading in 2023? There is no need to answer these questions.

I just want you to decide the most efficient strategy.

How would you answer them? For question two, could you create two word problems of your own? One that would use a mental strategy to solve and one that would need a formal written method to solve and explain why each strategy is most efficient.

Pause the video while you have a go at both those questions and when you are ready for the answers, press play.

How did you get on? Let's have a look.

For question one, you had to identify the most efficient strategy.

So would you use mental or written methods? Part A was about a library and it says it has about 750,000 books.

So you might have identified that a mental strategy would be more efficient way to solve this problem.

You might have said that you looked at the numbers and it could be seen that the use of unitizing the known fact, so you could have used 50 - 25, or the fact that half of 50 is 25, to calculate the answer.

For part B, you might have identified that a written method would be the most efficient way to solve this problem.

You might have given a reason like by looking at the numbers you could see that they are precise values.

And exchanging would be needed.

So a written method would be more efficient.

For question two, you are asked to create two word problems of your own and explain which strategy you would use.

You might have written two problems like this.

Problem one, Lucas wins 319,799 pounds in a board game and Aisha wins 1,459,807 pounds.

How much do they win altogether? And you might, your second problem might have been the circumference of the Earth is about 40,000 kilometres and that of Uranus is about 160,000 kilometres.

How much greater is the circumference of Uranus? You might then have explained that it would be more efficient to solve problem one using a written method.

And problem two, using a mental strategy because the numbers in problem one were precise, where exchanging would be needed, but those in problem two were values where known facts and unitizing could be used.

How did you get on with those questions? Well done.

Fantastic learning so far.

I can see you are trying really hard.

We're going to move on now and have a look at how we solve problems using the most efficient strategy.

So we've identified that strategy and now we are going to use it.

So let's revisit this data collected by Lucas.

And we want to know on average how much heavier is a Blue whale than a Beluga whale? So let's start by representing this question in a bar model.

We've got the blue whale is our whole because it is heavier and then the Beluga whale is a part and we need to find the missing part.

We can substitute in the values that we have been given.

So let's start by forming an equation from the bar model.

We know we've got 1,400 and we have to add it to something, the other part or the other addend, and then we would find the sum of 140,000.

We can rearrange the equation to subtract the known part from the whole.

And we know this is an additive question because the parts are unequal.

And what do you think, would you use a mental strategy or a written method here? Well, to decide, let's look at the numbers.

They are rounded averages, so we can use partitioning.

Using a mentor strategy therefore would be most efficient this time.

We could partition the 1,400 into 1,400.

We can subtract the 1,000 from 140,000 and then subtract the 400.

140,000 subtract 1,000 is 139,000, then subtract the remaining 400, gives us 138,600.

So on average, a Blue whale is 138,600 kilogrammes heavier than a Beluga whale.

Let's look at this question.

On average, how many times heavier is a Blue whale than a Beluga whale? We've just solved that problem, haven't we? Ah, but Lucas is incorrect.

Why is he incorrect, any ideas? That's because this is a multiplicative question, not an additive question.

It says how many times.

So let's form an equation from this information.

1,400 multiplied by something is 140,000.

So let's have a look at the numbers then we can decide which strategy to use.

The non-zero digits are the same.

We've got digit one and a digit four in both.

So let's use a mental strategy.

140,000 is 100 times greater than 1,400.

So on average, a Blue whale is 100 times heavier than a Beluga whale.

Let's revisit Aisha's data bar model and equations.

If we remember Aisha was comparing the area of Germany and the area of Italy.

Can you solve the calculation to determine how much larger Germany is than Italy? Remember, we decided we would need to use a written column algorithm because the numbers are precise.

So let's lay out our calculation.

Remember to the line the digits up.

Two ones subtract zero ones are two ones.

Two tens subtract four tens, oh, we can't do that, there's insufficient ten, so we need to regroup.

But oh, there's a zero, so we need to regroup from the digit to the left of the zero, the seven.

So we can regroup one 1,000s to give ten 100s then regroup those ten 100s to give one ten.

That gives us twelve tens overall.

Twelve tens subtract four tens are eight tens.

Nine hundreds subtract three hundreds are six hundreds.

Six 1,000 subtract one 1,000 is five 1,000s.

We've got five 10,000 subtract zero, well that's five 10,000s.

And then three 100,000s subtract three 100,000s which are zero 100,000s.

So Germany is 55,682 kilometres larger than Italy.

Let's check your understanding with that.

Could you look at the table and answer the questions efficiently? How much heavier was the Diplodocus than the T-Rex? And how many times the mass of the T-Rex was the Diplodocus? Pause the video while you have a go at those and when you are ready to go through the answers, press play.

How did you get on? Let's have a look at the first question.

How much heavier? So this is an additive relationship.

We can form an equation, we've got 6,000 add something is 24,000.

We can rearrange the equation.

We need to subtract the known part from the whole and we can use known facts here 24 - 6 is 18.

So 24,000 - 6,000 is 18,000.

What about how many times the mass of the T-Rex was the Diplodocus? So here we can form an equation 6,000 times something is equal to 24,000.

Here we can use our known times table facts.

We know six fours are 24, so 6,000 fours must be 24,000.

So we can use a mental strategy with actually more efficient for both of those problems. Your turn to practise now.

Using the data recorded in the table, could you answer these questions using the most efficient method? A, on average, how much heavier is a Humpback whale than a Killer whale? B, on average, how many times heavy is a Sperm whale than a Killer whale? For question two, using the data in this table, could you answer these questions using the most efficient method.

Part A, how much larger is the area of France than the area of Sweden? And part B, what is the combined area of Spain, Germany, and Italy? Pause the video while you have a go at those questions and when you are ready to go through the answers, press play.

How did you get on? Let's have a look.

So for question one A, you had to work out on average, how much heavy is a Humpback whale than a Killer whale? That's an additive relationship and you could have represented this in a bar model before then forming an equation.

Our whole is the larger amount, 30,000 and one of our parts is 4,000.

4,000 added to the other part would be 30,000.

We can rearrange that equation and subtract the part from the whole 30,000 subtract 4,000.

And you might have realised that you could use a mental strategy as the most efficient strategy because we can use our known facts.

30 subtract four is equal to 26.

So 30,000 subtract 4,000 must be equal to 26,000.

So on average, the Humpback whale is 26,000 kilogrammes heavier than the killer whale.

For part B, on average, how many times heavier is a Sperm whale than a Killer whale? You might have spotted that this was a multiplicative problem and formed an equation.

4,000 times something is 40,000.

We can use our times table facts here, can't we? We know four tens are 40, so 4,000 tens must be 40,000.

So on average the Sperm whale is 10 times heavier than the killer whale.

For question two, we could use this in different data set recorded in the table and find out how much greater is the area of France than the area of Sweden.

You might have spotted this was an additive problem and represented this in a bar model before forming an equation.

450,295 added to the missing part would give the sum of 643,801.

We can rearrange the equation and subtract the known part from the whole.

You might have realised that using a written method would be most efficient this time because the values are precise and we might need to do some regrouping.

So we can set up our column algorithm.

One one subtract five ones, well there's insufficient ones so we need to regroup.

We can't regroup from the zero because there's nothing there, so we need to regroup from the digit to the left of the zero.

11 ones subtract five ones are six ones.

Nine tens subtract nine tens, well that's zero tens.

Seven hundreds subtract two hundreds are five hundreds.

Three 1,000 subtract zero 1,000s are three 1,000s.

Four 10,000 subtract five 10,000s, there are insufficient 10,000s, so we need to regroup.

14 10,000s subtract five 10,000s are nine 10,000.

And then five 100,000 subtract four 100,000 leaves us with one 100,000.

So the area of France is 193,506 kilometres square greater than the area of Sweden.

For part B, what is the combined area of Spain, Germany, and Italy? Well, you might have spotted that this was an additive problem, and represented it in a bar model, noting that there are three addends this time.

We can then form an equation.

You might have noted that using a written method would be most efficient again because the values given are precise and there might be some regrouping.

We can then set up our column algorithm with three addends.

Zero ones add two ones add zero ones, two ones.

Seven tens add two tens add four tens is 13 tens, so we need to regroup.

Double three is six add the regrouped hundred, we've got seven hundreds.

Five 1,000s plus seven 1,000s plus one 1,000 leaves us with 13 1,000s, so we need to regroup.

Five 10,000s add the regrouped 10,000 is six 10,000s.

And then five 100,000s add three 100,000s add three 100,000s, well double three is six.

Add the five is 11, so we've got 11 100,000s, so we need to regroup.

So the combined area of Spain, Germany, and Italy is 1,163,732.

Fantastic learning today.

I can see you really deepened your understanding on how we can solve calculations efficiently.

We know that fluent calculation requires flexibility to move between mental and written methods and supports us to be efficient.

Looking at the numbers within the calculation supports us to know whether to use a mental strategy or a written method.

And mental strategies work best when known facts can be used and written methods work best when the numbers given are precise or when regrouping is involved.

So well done today.

You should be really proud of yourselves.

I've had great time learning with you and I look forward to learning with you again soon.