Lesson video

Hello, my name is Mrs. Hopper.

And I'm looking forward to working with you in this lesson.

This lesson comes from our unit on the order of operations.

I wonder if you've come across that before.

Any ideas what it's about? Well, let's get going with this lesson and explore more about the order of operations.

In this lesson, we are going to be explaining which order to complete the multiplication and addition or subtraction elements of a problem.

So we're going to be combining multiplication with addition and subtraction.

And we're going to be learning how to write this as an equation so that we know we're going to get the right answer.

So if you're ready, let's make a start.

There are four key words, well three key words in a phrase in our lesson today, we've got efficient, equation, brackets, and order of operations, which is the title of our unit.

So I'm gonna take my chance to say them and then it'll be your turn to practise.

Are you ready? My turn, efficient, your turn.

My turn, equation, your turn.

My turn, brackets, your turn.

My turn, order of operations, your turn.

Well done.

I'm sure they are mostly words that you're familiar with.

I'm sure you always work hard to be efficient in the way that you work.

Maybe not even just in maths.

Let's look at what the words mean.

So working efficiently means finding a way to solve a problem quickly whilst also maintaining accuracy.

An equation is used to show that one number, calculation, or expression is equal to another.

The idea of equality is really important in an equation.

Brackets are symbols used in pairs to group things together.

And the order of operations is a set of rules that tells you which operations have priority over others in an equation.

And by the operations we mean things like addition, subtraction, multiplication, division, and so on.

There are two parts to our lesson today.

In the first part, we're going to be solving problems efficiently.

And in the second part we're going to be focusing in on brackets.

And Jun, Aisha, and Sam are with us in the lesson today.

Sam is having a party, lucky old Sam.

She's put some cakes out on plates for her guests to eat when they arrive.

Jun says, how many cakes have you put out altogether, Sam? Sam could count the cakes one at a time.

I'm not sure that's very efficient.

She says, one, two, three, four, five, oh, she says, this will take a long time, what else could I try? Aisha has an idea for a more efficient way of finding the total number of cakes.

She spotted something about the way the cakes are organised.

She says the cakes are either in groups of five or three.

She says I could add each group to find the total.

So 5, 3, 3, 3, 5, 3.

So we've got 5 plus 3 plus 3 plus 3 plus 5 plus 3.

Wait, says Sam, I have an idea that could be even more efficient.

Ah, Sam can rearrange the cakes to find the total in a more efficient way.

So she's put all the plates of three cakes together and all the plates of five cakes together, there are four groups of 3 and two groups of 5.

She can represent that with a bar model, four groups of 3 and two groups of 5.

I can write a multiplication expression to represent the groups of cakes, she says.

Four groups of 3 and two groups of 5, 4 times 3 and 2 times 5.

If I calculate the two products, I could add them together to find the total number of cakes.

4 times 3 is equal to 12, 2 times 5 is equal to 10, 12 plus 10 is equal to 22.

There are 22 cakes altogether.

She says, I could write an equation to show how I solved this problem.

4 times 3 plus 2 times 5.

And that's equal to 12 plus 10, which is equal to 22.

Sam has put out 22 cakes altogether.

Let's compare Aisha and Sam's methods.

Aisha added up the plate separately.

5 plus 3 plus 3 plus 3 plus 5 plus 3, and she got some total of 22.

Sam put all the plates with three cakes together, four groups of 3, 4 times 3, and then added on the two groups of 5.

So 12 plus 10 was equal to 22.

Sam's method is more efficient because it involves fewer steps.

It's more efficient to calculate the two products and add them together rather than adding each number separately.

Here are some more cakes that Sam bought for the party.

So we've got some boxes with eight cakes and some boxes with six cakes.

Jun says, show me how you could find the total number of cakes this time.

So here we can see all the chocolate cakes.

So there's 3 lots of 8.

And the vanilla cakes, 2 lots of 6.

And altogether, 3 lots of 8 plus 2 lots of 6 is equal to 36.

Sam says there are 36 cakes altogether this time.

24 plus 12 is equal to 36.

You could write an equation, she says.

3 times 8, for the three boxes of eight chocolate cakes, plus 2 times 6, for the two boxes of six vanilla cakes.

And that's equal to 24 plus 12, which is equal to 36, the total number of cakes.

Sam bought some wrapping paper to make a pass the parcel game for the party.

Can you see she's got some wrapping paper with rolls of four metres of paper and wrapping paper with rolls of three metres of paper.

Jun says, can I show you how I would find the total length of the wrapping paper? Go for it Jun.

So there are five rolls of four metres and there are six rolls of three metres.

So he's drawn those and represented them as bars.

5 lots of 4 and 6 lots of 3.

5 times 4 is 20, 6 times 3 is 18, 18 plus 20 is 38.

5 times 4 plus 6 times 3 is equal to 20 plus 18, which gives us a total length of wrapping paper of 38 metres, as Jun says.

Great work, Jun, that was very efficient, says Sam.

I think he's used her method, hasn't he? Time to check your understanding.

Which bar model could you use as a representation to help find the total length of this wrapping paper? We've got some roll with two metres and some rolls with six metres.

So is it bar model A, B, or C which represents the total length of this wrapping paper? Pause the video, have a think.

And when you're ready for the answer and some feedback, press play.

How did you get on? So we've got four rolls of two metres and three rolls of six metres.

So which bar model represents that? It's B, isn't it? Four rolls of two metres and three rolls of six metres.

Time for another check.

Can you use the bar model to help you to complete the equation to find the total length of this wrapping paper? So pause the video, fill in the gaps.

And when you're ready for the answers and some feedback, press play.

How did you get on? So we can see that we've got 4 times 2 plus 3 times 6.

4 times 2 is equal to 8, plus 3 times 6 which is equal to 18, 18 plus eight is equal to 26.

So the total length of the wrapping paper is 26 metres.

Let's look at these boxes of cakes again.

How could you answer Jun's question this time? He says, how many more chocolate cakes than vanilla cakes are there? Oh, we could draw a bar model again, says Sam.

3 times 8 and 2 times 6.

And you could write a multiplication expression for each of the groups of cakes just like we had before, 3 times 8 and 2 times 6.

But this time we need to find the difference.

How do we find the difference? Ah, we're looking for that gap, aren't we? The gap between the end of the 2 times 6 and the end of the 3 times 8.

So what calculation do we need to do? That's right.

This time we need to subtract the number of vanilla cakes from the number of chocolate cakes.

This time our whole in a way is the number of chocolate cakes, that's the larger value.

We need to subtract the part that's vanilla cakes to find the part that's the difference.

And you can write this equation to show you how you solve the problem.

3 times 8 subtract 2 times 6.

So this time it's equal to 24 subtract 12, which is equal to 12.

There are 12 more chocolate cakes than vanilla cakes.

Let's look again at Sam's wrapping paper.

How could you answer Jun's question this time? Jun's asking, how much more wrapping paper is in the box of four metre rolls than in the box of three metre rolls? We could draw a bar model again.

You might want to pause and have a think about what the bar model in the equation model look like before Sam and Jun share their ideas.

Let's look at Sam's bar model.

There are five rolls of four metres and there are six rolls of three metres.

So we've got 5 times 4 and 6 times 3.

And we know from last time that the 5 times 4 is greater.

So we can write a multiplication expression for each box.

5 times 4, five lots of four metre rolls.

And 6 times 3, six lots of three metre rolls.

This is another difference problem.

We want to find out how much more wrapping paper is in the box of four metre rolls than three metre rolls.

So we're interested in finding that gap, that difference.

That means we need to subtract the total length in the three metre box from the total length in the four metre box.

You can write an equation.

So 5 times 4 subtract 6 times 3.

So that's 20 subtract 18 which is equal to 2.

So the total length of paper in the four metre box is two metres longer than the total length of paper in the three metre box.

Time to check your understanding.

Which bar model could you use as a representation to solve this problem? Jun has three packs of 15 balloons.

Sam has four packs of 10 balloons.

Who has the most balloons, and by how many? So which bar model would you use as a representation to solve the problem, A, B, or C? Pause the video, have a think.

And when you're ready for some feedback, press play.

So what were you thinking about? This time, we're finding the difference between three lots of 15 and four lots of 10.

So which bar model represents that? It's C, isn't it? We've got 3 lots of 15 in our bottom bar and 4 lots of 10 in our top bar.

And we can see if we work out those values that 3 lots of 15 will have a greater value, so we're looking for that difference.

So can we use the bar model to help complete the equation? Time to check again.

Pause the video, fill in the gaps in the equation.

And when you're ready for the feedback, press play.

Okay, so this time we knew we had 3 times 15, and we are subtracting 4 times 10.

Because we have three lots of balloons in packs of 15 and four lots of balloons in packs of 10.

So we've got 45 and we're subtracting 40.

So 45 subtract 40 is equal to 5.

So Jun has the most balloons and he has 5 more balloons than Sam.

And it's time for you to do some practise.

Can you draw a bar model to represent each problem? You've got four problems there.

So part one is to draw a bar model.

Part two, same problems, but solve each problem efficiently and write an equation to show this.

You can look back at your bar models from question one to help you.

Pause the video, have a go at questions one and two.

And when you're ready for some feedback, press play.

How did you get on? So let's look at the bar models first.

So in A, a class is organised into three teams of five children and two teams of four children.

How many children are in the class? So we've got three teams of five and two teams of four.

In B, I buy nine pencils for 50 p each and four erases for 75 p each.

How much money have I spent altogether? So we've got 9 lots of 50 and 4 lots of 75.

There's our bar model.

For C, I have three packs of 10 stickers and you have two packs of 20 stickers.

How many more stickers do you have? So three packs of 10 stickers, two packs of 20 stickers, and we're looking to find that difference marked by the bracket.

So for D, Sam runs 10 kilometres a day for five days and Aisha runs seven kilometres a day for seven days.

Who has run further and by how much? So we've got 5 lots of 10 and 7 lots of 7 and we are looking for that difference.

So that was question one to draw the bar models.

Let's move on to question two and solve each problem efficiently and write an equation.

So our class is organised into three teams of five and two teams of four.

How many children are in the class? So we've got 3 lots of 5, 3 times 5, plus 2 times 4, two teams of four children.

So that's 15 plus 8 which is 23 children.

There are 23 children in the class.

In B, we were buying nine pencils for 50 pence each and four erases for 75 pence each.

How much have I spent altogether? Well that's 9 times 50 plus 4 times 75.

Well 9 lots of 50 p is 4.

50 pounds.

and 4 lots of 75 p is equal to 3 pounds.

So 4.

50 pounds plus 3 pounds is equal to 7.

50 pounds.

I've spent 7.

50 pounds altogether.

In C, I have three packs of 10 stickers, you have two packs of 20 stickers.

How many more stickers do you have? So this is a difference problem.

So 2 times 20 subtract 3 times 10, so that's 40 subtract 30 which is equal to 10.

You have 10 more stickers than me.

And for D, Sam runs 10 kilometres a day for five days, Aisha runs seven kilometres a day for seven days.

Who has run further and by how much? So we've got 10 times 5 which is equal to 50 and 7 times 7 which is equal to 49.

So Sam has run further than Aisha.

10 kilometres times 5 subtract 7 kilometres times 7 is equal to 50 kilometres subtract 49 kilometres, which is equal to 1 kilometre.

So Sam has run 1 kilometre further than Aisha.

Well done if you worked all of those out.

And onto part two of our lesson, thinking about brackets.

Sam bought some small gifts to put in her party bags.

There were nine guests at the party.

Sam wanted to give one robot toy and one lollipop to each guest.

How much did she spend? Aisha says, I could find out how much she spent on the robots first, and then add the cost of the lollipops afterwards.

She could do that.

9 times 3 is equal to 27.

Now I can multiply the cost of a lollipop by the number of guests, she says that was the robots.

9 times 1 pound is equal to 9 pounds.

Finally, I can add together the cost of the robots and the lollipops.

27 plus 9 is equal to 36.

So Sam spent 36 pounds altogether.

You could write this as one equation.

9 times 3 plus 9 times 1 is equal to 36.

Jun can see a different way to calculate how much Sam spent.

Jun says, I could find out how much she spent on one guest and then multiply by the number of guests.

If a robot costs 3 pounds and a lollipop costs 1 pound, Sam must have spent 4 pounds on each guest because 3 plus 1 is equal to 4.

So 3 pounds plus 1 pound is equal to 4 pounds.

Now I can multiply the cost of the items by the number of guests.

4 pounds multiplied by 9 for the nine guests is equal to 36 pounds.

Sam spent 36 pounds.

Of course it's the same total value as Aisha found.

Jun tries to write this as an equation.

9 times 3 plus 1 is equal to 36.

Sam says, I don't think that's right.

I get a different answer to you and I solve that equation.

9 times 3 plus 1 is equal to 27 plus 1, which is equal to 28.

That's not right, is it? Well, it's not the same as Jun's answer.

And Jun knows that he wants the answer to be 36 pounds.

Oh, that's because I want you to do the 3 plus 1 first, he says.

Should I write it this way if I want you to do the 3 plus 1 first? 3 plus 1 times 9 is equal to 36.

No, that's still not correct, says Sam.

Let's find out more about this.

Many years ago, scientists across the world were working on new discoveries and they wanted to share their findings.

When they looked at each other's work, they realised that they were recording their results in different ways.

Different scientists could look at the same equations and calculate different solutions.

They decided on a set of rules for writing equations so that everyone would record their findings in the same way and there would only be one possible solution.

These rules are called the order of operations.

This diagram shows the order of operations.

The operations at the top have the highest priority, so that means they get done first.

So right at the top, we've got brackets.

The next layer down, we've got something called exponents, and they're things like squares or cube numbers.

You might have come across squared numbers.

Numbers written with a little 2 above them to show, say 2 squared is 2 times 2, 3 squared is 3 times 3.

And there are other things that you'll learn about as you go on with your maths.

And we call those exponents.

And they are the next layer down from brackets.

Then comes the layer of multiplication and division.

And finally the lowest priority is addition and subtraction.

So in the order of operations, multiplication comes before addition.

It's more important.

So let's look at Jun's equation again.

3 plus 1 times 9.

Ah, the order of operations, he says, means I need to multiply first and then add.

When we multiply 1 and 9, the product is 9.

That means that this equation is equal to 12.

3 plus 9 is equal to 12.

But brackets are part of the order of operations too.

Bracket have the highest priority.

That means any part of the equation in brackets will be calculated first.

Jun says, I want you to calculate 3 plus 1 first.

Shall I write it inside brackets so it has the highest priority? Ah, he's rewritten his equation.

3 plus 1 is now in brackets.

3 plus 1 multiplied by 9.

That's right, well done, Jun.

Now I will calculate 3 plus 1 first because it is inside brackets, then I will calculate 4 times 9.

So 3 plus 1 is equal to 4, 4 times 9 is equal to 36.

That's the answer that Jun needs us to get because it's the total cost of those gifts for the party bags.

So let's just check your understanding.

Think back to that triangle.

Which of these has the highest priority in the order of operations? Is it A, addition and subtraction, B, multiplication and division, or C, brackets? Pause the video, have a think.

And when you're ready for some feedback, press play.

Could you remember? This is something that we need to remember.

It's brackets.

Brackets have the highest priority in the order of operations.

If you visualise that triangle, they're right at the top.

Let's look at another problem that you might solve by adding before you multiply.

Jun's mum is training for a triathlon.

Oh, well done to Jun's mum.

Every day she runs two kilometres and cycles three kilometres.

How far has she travelled altogether in one week? Let's do the 2 plus 3 first to calculate how far she travels in one day, suggests Jun.

Ah, then we can multiply by 7 because there are seven days in one week, says Sam.

Jun's mum has travelled 35 kilometres altogether.

Let's write that as an equation, says Jun.

2 plus 3, and we're going to do that first so he is put it in brackets, that's the total distance she travels in one day.

And then we're going to multiply that by the number of days, by 7.

So the 2 plus 3 is in brackets, so this part will be calculated first.

2 plus 3 is equal to 5.

5 times 7 is equal to 35.

Let's explore how this equation would be solved differently without the brackets.

Well, in the order of operations, multiplication has priority over addition.

So when there are no brackets, you would multiply before you add.

So then we'd have 2 plus 3 times 7, which is 21, which is equal to 23.

And that's not the same distance that Jun's mum had travelled in her training over the week.

Time to check your understanding.

Can you match each equation to the correct solution? Think carefully about the order of operations, what has priority? Pause the video, have a go.

When you're ready for the answers and some feedback, press play.

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

So that is the correct matching, but why is that the case? When there are no brackets, you multiply before you add.

So the top equation becomes 3 plus 5 times 8, 3 plus 40, which is equal to 43.

When there are brackets, you calculate the part of the equation inside the brackets first.

So this time we're going to do the 3 plus 5 first.

3 plus 5 is equal to 8.

So we get 8 multiplied by 8, which is equal to 64.

Well done if you've remembered that the brackets take priority.

Now let's look at a problem that you might solve by subtracting before you multiply.

Aisha wants to buy some T-shirts that are 8 pounds each.

She has a coupon that will take 2 pounds off every item she buys.

Aisha buys five T-shirts.

How much does she spend? Aisha says we could do 8 subtract 2 first to find the cost of one T-shirt.

So she's going to take the 2 pounds off her 8 pounds.

Then we can multiply this by 5, says Sam, because you've bought five T-shirts.

That's right, that's an equation, says Aisha.

8 subtract 2 because we've got to do that first, the 8 pounds subtract the 2 pounds, so we're gonna put that in brackets, and then multiply by 5.

8 subtract 2 is in brackets, so this part will be calculated first.

8 subtract 2 is equal to 6 and 6 times 5 is equal to 30.

So Aisha spends 30 pounds in total.

Let's compare this solution to the same equation without the brackets.

Including brackets in an equation can change the solution.

It's really important to record your equation correctly for this reason.

If there were no brackets, we'd do 2 times 5 first, this is 10, so we'd have 8 subtract 10.

Oh, and 8 subtract 10 is equal to negative 2.

And that's not how much you spent, is it? So getting the brackets right and recording the equation correctly is really important.

In the order of operations, multiplication has a priority over subtraction.

When there are no brackets, you multiply before you subtract.

And it's time for you to do some practise.

In question one, you're going to read each problem carefully and decide which part of the problem you would complete first.

Write the letter of the problem into the correct column in the table.

Are you multiplying first, adding first, or subtracting first? And you've got four problems there, A, B, C, and D.

Then you've got the same four problems. You're going to solve each problem.

And for question three, again the same problems, you are going to write the equation that shows how you solve each problem.

And remember that if you added or subtracted first, you will need to use brackets.

Pause the video, have a go.

And when you're ready for the answers and some feedback, press play.

How did you get on? Let's do the sorting first.

So in A, Sam has 100 pounds.

She buys eight games costing 9 pounds each.

How much money does she have left? So she buys eight games costing 9 pounds each.

So we'll multiply first and then we'll subtract.

So A goes in the multiply first column.

In B, Aisha wants to buy eight bunches of flowers that are labelled at 6 pounds.

Today there is 1 pound off every bunch.

So how much will it cost? So we want to subtract 1 pound from the 6 and then multiply by 8.

So we're going to subtract first.

So B will go in the subtract first column.

In C, Jun has eight sheets of 10 stickers and six sheets of seven stickers.

How many stickers does he have in total? Well, we need to do 8 times 10 and 6 times 7 and then add them together.

So we're going to multiply first.

So C goes in the multiply first column.

And in D, Sam has a 20-minute violin lesson and a 15-minute piano lesson each week.

How long does she spend in music lessons over six weeks? Well, the efficient way here would be to add up the total amount of time she spends and then multiply by the number of weeks.

So we'd add first.

So D will go into the add first column.

So thinking about all of that, can we solve the problems? So for A, Sam has spent 72 pounds so she has 28 pounds left.

For B, Aisha's flowers will cost 40 pounds.

For C, Jun has 122 stickers in total.

And for D, in six weeks, Sam spends 210 minutes or 3 1/2 hours in music lessons.

Now let's look at the equations we use to calculate those answers.

So Sam has 100 pounds, she buys eight games.

So this time we'll be going to multiply first.

So it's 100 subtract 8 times 9.

And this equation doesn't need any brackets because you can multiply first to solve it.

In B, Aisha wants to buy the eight bunches of flowers, but there's a pound off every bunch, so she wants to subtract first.

So this equation does need brackets.

6 subtract 1 and then multiply by 8.

We're gonna solve it by subtracting first and then multiplying.

In C, Jun has eight sheets of 10 stickers and six sheets of seven stickers.

How many stickers does he have in total? Well he's going to do his multiplication first and then add.

So 8 times 10 plus 6 times 7.

This equation does not need brackets because you can multiply first to solve it.

And finally in D, it was Sam and her-20 minute violin and 15-minute piano lessons.

So this time we wanted to add together the length of the music lessons and then multiply by 6.

So 20 plus 15 does need to be in brackets because we're going to solve this by adding first and then multiplying.

And when we do that addition, multiply it by 6, we end up with 210 minutes.

Well done if you've got all of those right and well done for discussing the use of brackets.

And we've come to the end of our lesson.

We've been combining multiplication with addition and subtraction.

What have we been thinking about? Well, when solving problems that combine multiplication with addition or subtraction, you can think about the context of the problem and choose which element of the problem to complete first.

When you write your solution as an equation, you need to know the order of operations.

Multiplication has priority over addition and subtraction.

If you solved a problem by adding or subtracting first, you need to use brackets to show that this part of the equation needs to be calculated first.

Thank you for all your hard work and your mathematical thinking.

I hope I get to work again with you soon, maybe on some more order of operations.

But for now, bye-bye.