Lesson video

Hello and welcome to today's lesson on ratio and proportion.

In today's lesson, we're going to be using multiplication and division to calculate missing values in ratio problems. We're gonna really focus in on how we can identify the multiplication or division calculation or equation that we need to record and that we need to solve in order to solve the problem that we've been set.

So let's have a think about what this lesson's gonna look like.

In the first part of our lesson, we're going to be representing ratio problems using multiplication and division.

We might not actually solve them, but we're really gonna look at what we know about a problem, maybe representation we are more familiar with and how from that can we represent the problem using an equation, a multiplication or a division equation? We're going to use the context of the school quiz team.

So a school quiz team is a mixture of adults and children.

And the school quiz team is made up of two adults and three children.

So you might be familiar with the language of for every.

So we could say that for every two adults there are three children.

For every three children there are two adults.

And in a school quiz team that each team is made up of five people.

So let's have a think about this in terms of multiplication and division.

You might be familiar with a bar model to represent this idea of a ratio.

So the top representing five people in one quiz team and the bottom bar broken up to show the relative sizes of those parts of the whole.

So that we have two adults for every three children.

Now we've adapted the bar here to show that there are three teams taking part in the quiz.

So three teams mean three times as many people taking part.

So 15 people taking part.

Three times as many adults, so six adults, and three times as many children, so nine children.

And our question this time is how can we represent this with multiplication? So here's another representation that you might be familiar with when it comes to multiplication.

And this is an array.

And I hope you can see in here that each row of our array has two adults and three children, so is one quiz team.

And that the three rows show us that we've got three teams taking part.

So those numbers in that bottom bar model, where are they in our representation and what do they look like as multiplications? Let's have a look.

So the part of the array I've put a box around at the moment, shows us the adults involved.

So in each team there are two adults.

We've got three teams. So we've got two multiplied by three to give us six adults.

So we can represent the adults in three quiz teams as two, the original number of adults in one quiz team, multiplied by three for the number of quiz teams we've got.

What about the children? Do you want to pause and maybe think about how we would represent the number of children as a multiplication? We'll do it together in just a moment.

So I've moved my box to highlight the children in our array and we know that in one quiz team there are three children.

We know we've got three teams. So we can see here that we've got three children multiplied by three for the three teams we've got.

So we can represent the number of children as three multiplied by three.

What about the overall number of people taking part? Where's that in our array? But if I highlight the whole of the array, I've got all of the quiz team.

And we can see there that we had five people in one quiz team and we've got three quiz teams. So five multiplied by three will give us our 15 people who are taking part in the quiz when we've got three teams involved.

So it's really important when we are representing our thinking just with symbols, just with the numbers that we are sure about where all those numbers are represented in the bar model and again in the equations.

So let's have a think.

What does the three in each of the equations represent? Well, the three in that top one, two times three equals six represents the fact that there are three teams taking part in the quiz.

And we can see that multiplied by three in all of those three equations.

So the three in each of the equations represents the three teams taking part.

I'm sure you've spotted that one of our equations has got three times three.

So it's got, three is in there twice.

So what about that other three? Well that three there, that first three is the number of children in one team.

And if we look at the other equations that we've written down, the two in that top equation, two times three equals six, the two represents the number of adults there were in one team.

And the five in the five times three equals 15 represents the number of people in one whole team that we've then multiplied by three.

So really important when you're writing down equations that you can explain where those numbers are in the other representations and how those numbers relate to the problem and to the scenario that the context that we're working in.

Okay, so time for you to have a think about this and check that we are happy with where we've got to with representing ratio using multiplication and division.

So we're still thinking about those school quiz teams made up of two adults and three children.

And we want you to use multiplication and division this time to justify whether you think this statement is true or false.

If there are five teams, there will be 15 children.

So you might be able to draw a bar model or you might just think of the numbers have just popped into your head and you can give an answer.

But I want you to write down a multiplication or a division calculation that justifies why you are right in saying that that statement is either true or false.

So you might want to pause the video at the moment to have a go and then we'll have a look at it together.

I hope you decided it was true as well, that if there are five teams there will be 15 children.

But the key here was how we were going to justify it using multiplication and or division.

I think multiplication's gonna help us in this case.

So let's have a think.

There are five times as many teams, so there will be five times as many children in those five teams. So let's think about it using our array.

So here you can see each row in my array represents one team.

I've got five teams. So in one row, in one team there are three children and I've got five teams. So I've got three children multiplied by five.

I've got three times five children, and three times five is equal to 15.

So there are my three times five children equaling 15.

So what do the three, the five and the 15 in our equation represent? Well, the three represents the number of children in one quiz team.

The five represents the number of quiz teams, and the 15 represents the number of children in the five quiz teams. So I've got my original number of three children multiplied by five because I've got five times as many teams and my answer, my product of 15 represents the number of children in those five quiz teams. So another opportunity for you to use multiplication and division to represent the way that you would solve these problems. You are asked here to write down the multiplication and division equations that you would need in order to solve these problems. So question one says if there are 40 people in the quiz, there will be how many teams? Question two, if there are 24 adults taking part, there will be what number of teams and what number of children? And question three says, if there are 27 children taking part, there will be what number of teams and what number of adults? So you can see that in each of those problems we've changed a different element of our ratio context, a different variable.

In question one, we are looking at the total number of people in the quiz and thinking about the number of teams. In the second question, we are looking at the total number of adults taking part and working out how many teams and children there will be.

And in the third part, we're looking at the total number of children taking part and working out how many teams and therefore how many adults will there be.

So what we just want you to do is to record the equations that you would use in order to solve these problems. Pause the video and have a go and then we'll look at them together.

Okay, so number one said, if there were 40 people in the quiz, how many teams would there be? So if there are 40 people in total, I know that there are five people in each quiz team.

So I need to know what 40 divided by five is.

How many fives are there in 40? How many lots of five are there? And 40 divided by five is equal to nine.

The key bit was that I knew I was going to have to divide 40 by five.

So there's my division in order to solve question one.

Question two told us that there were 24 adults taking part and we had to work out how many teams there would be and therefore how many children there would be.

So what do we know? We know there are 24 adults.

Well, in one team there are two adults.

So how many teams can I make from 24 adults? So I need to take my 24 adults and divide them into twos.

How many twos can I get outta 24? And 24 divided by two is 12.

So therefore I know that there are going to be 12 teams. So I need then to think, well I could, I've got three children in one team, but I've got 12 teams, so I've got to multiply three by 12 to find out how many children and there are going to be 36 children.

Three times 12 is equal to 36.

Question three told us that there were 24 children taking part.

So how many teams and how many adults? So this time I've got to take my 24.

It's 24 children this time.

So I've got to divide it into groups of three children to find out how many teams I can make.

And I know that they've gotta be three children in each team.

So 24 divided by three is equal to eight.

So therefore I know I can make eight teams from those 24 children.

And if I'm making eight teams then I've got to have eight times as many adults.

I know there are two adults in one team.

So two multiplied by eight is 16, and that will give me the 16 adults.

Okay, it's your turn to do some practise now.

So your task here is to think about sandwiches.

So for every cheese and tomato sandwich you need, you need two slices of bread, three slices of cheese, and four slices of tomato.

And you've got four problems there.

Four questions.

And it's asking you to represent the problems using multiplication and division equations.

So A says how many sandwiches can you make with 26 slices of bread? How many slices of cheese and tomato will you need? Which equations are you solving in order to answer those questions? B says, how many slices of cheese do you need to make a hundred sandwiches? Again, what's the equation that you will be solving in order to answer that question? And we want the equations to solve the problems for C and D.

So in C, if you have 28 slices of tomato, how many slices of cheese do you need? And D, a pack of cheese has 25 slices, how many sandwiches can you make? So recording those multiplication and division equations that you will need in order to solve those problems. How did you get on? Let's have a look at these solutions together and really think about the equations that we've written down.

So A asked us, how many sandwiches can you make with 26 slices of bread? While I know that each sandwich needs two slices of bread, so I need to divide my 26 into twos, divide it by two in order to find out how many sandwiches I can make.

So 26 divided by two is the equal to 13.

So I know that you can make 13 sandwiches from 26 slices of bread.

But the question also wants me to work out how many slices of cheese and tomato you'll need to make all those sandwiches.

So I now know that I'm making 13 times as many sandwiches as my bar model shows.

So I'm going to need to multiply my slices of cheese by 13 and my slices of tomato by 13.

So let's think about cheese first.

Three slices of cheese multiplied by 13, 'cause I'm making 13 times as many sandwiches, three times 13 is equal to 39.

So I'm going to need 39 slices of cheese to make my 13 sandwiches.

What about the tomato? Well I need four slices for one tomato, so I'm going to need 13 times as many to make my 13 sandwiches.

So four multiplied by 13 is equal to 52.

So I'm going to need 52 slices of tomato to make 13 sandwiches.

B asked how many slices of cheese you need to make a hundred sandwiches.

So this time I know I'm making a hundred times as many sandwiches, so I'm going to need a hundred times as many slices of cheese.

So if I need three slices of cheese for one sandwich, I need three times a hundred slices of cheese, which means 300 slices of cheese in order to make my 100 sandwiches.

You need a hundred times as many slices of cheese as for one sandwich.

C asked us what would happen if we had 28 slices of tomato, asked us how many slices of cheese we'd need.

Can we work that out straight away? I think we might need to think about the number of sandwiches we can make with 28 slices of tomato.

So I know that I can make a sandwich with four slices of tomato.

So I need to divide my 28 into fours to find out how many sandwiches I can make.

28 divided by four is equal to seven.

So I know that I can make seven sandwiches with 28 slices of tomato, but how many slices of cheese do you need? Well I need seven times as many slices of cheese.

So I need three slices of cheese multiplied by seven to make seven sandwiches.

So I need 21 slices of cheese.

Three multiplied by seven is equal to 21.

So you need 21 slices of cheese to make seven sandwiches.

And finally, D said a pack of cheese has 25 slices.

How many sandwiches can you make? Well I know that I need three slices of cheese to make one sandwich.

So I've got to divide 25 into groups of 3, 25 divided by three.

I'm not so sure I'm going to get a whole number answer here because 25 is not a multiple of three.

24 is, three times eight is 24.

So I know that 25 divided by three will be eight with a remainder of one.

So I'm going to be able to make eight sandwiches with 25 slices of cheese, with one slice left over.

So I'll make eight complete sandwiches, but I'll have one slice of cheese left over.

So hopefully that first part of the lesson has meant that you're feeling more confident about identifying the multiplication and division equations that we need in order to be able to solve ratio problems. So we're gonna move into the second part of our lesson and really use those multiplication and division equations in order to help us to solve some more problems. So let's get into some more ratio.

So we've got Aisha and Alex and they are sharing marbles.

And the ratio is that for every two marbles Aisha has, Alex has three marbles.

So we've got a table here to collect some values.

So we're still working on that basic ratio that for every two marbles Aisha has, Alex has three marbles giving us a total number of marbles of five each time they sort of share them out.

So could we complete this table to show how many marbles they each get and the total number of marbles each time.

So we could fill in this table.

So let's have a go.

You might want to pause the video and have a go at filling it in yourself.

So here's a completed version of the table.

So what do you notice? Are there any patterns that you spot? If we look at the first row with Aisha's marbles, I hope you can all see that we've sort of got the two times table there, haven't we? And if we think about it every time they share another set of marbles, Aisha will get two more than she had last time.

So yeah, she's always gonna have a multiple of two marbles and her number of marbles will increase by two each time.

But I really want you to focus in on the idea of a multiple of two marbles.

We're thinking about multiplication and division in order to solve these problems. Let's look at Alex's marbles.

What do we spot there? Well hopefully you can spot the three times table in there because Alex has three marbles each time the marbles are shared out.

So every time they share them again, he gets another three marbles.

And the bottom row shows us the total number of marbles.

And every time they share the marbles out, five marbles in total are shared out.

So if we share another set of five, there'll be five more.

So you can see that that total number of marbles is always going to be a multiple of five.

And so it's really important to spot the multiplication links between these values within the table.

So here is our incomplete table and we're going to look at a series of problems. And what we're going to identify for each problem is which column of the table is needed to solve each problem.

So which bit of the table are we looking at in order to solve the problem? So here's our first one.

If they share 15 marbles, how many marbles does Alex get? I'm gonna tell you now, we're not gonna answer these questions right now.

We are just gonna have a look and see which bit of the table would we need in order to solve the problem.

So the information we've got here is that they share 15 marbles.

So can we see a total of 15 in the table? There we go.

It's in that column, isn't it, that third column along.

So that's the column that we're going to need.

And in fact we can see there, we've got it filled in.

Alex gets nine marbles.

And maybe we can look back to that first column, the original ratio to see how our new values linked to that.

How have we varied the total number of marbles and how does that mean that Alex's number of marbles changes? Let's have a look at another problem.

If Aisha gets 18 marbles, how many does Alex get? Can you see the column in the table where Aisha has 18 marbles? I'm sure you saw it, that last column there.

Aisha has 18 marbles.

We haven't got the values filled in for that, but we're going to have a look and see at how the table can support us.

But that's the area of the table we are looking at.

If we completed that column, we'd know that when Aisha gets 18 marbles we could work out how many Alex gets and indeed how many marbles in total if that's what we want to know.

What if Alex gets 18 marbles, how many does Aisha get? So same number of marbles but different person getting them this time.

So which column would help us to solve that problem? Now I'm sure you spotted it.

There's the column where Alex has 18 marbles.

So if we use the values in our table, thought about those multiples of three and two, we would be able to work out how many Aisha gets and we will go on and do that later in this part of the lesson.

If Aisha gets 18 marbles, hang on, we've had this before, haven't we? This time though, how many did they share all together? So I'm still looking at that column where Aisha gets 18 marbles, but this time I'm going to be interested in the value in the bottom of that column.

In the bottom row, the total number of marbles that they get.

And again, another example where we're going to work from what we know about a part of the ratio to try and work out the whole.

If Alex gets 18 marbles, how many did they share altogether? So which column? Yeah, it's that column with Alex with 18 marbles again.

But in the previous problem, we wanted to know how many Aisha got.

This time we want to know how many they got all together, how many they shared altogether.

Let's now think really about those multiplication and division problems to solve them and find our answers.

So if they share 15 marbles, how many does Alex get? So we identified that that was the column we needed.

Those are the values that were important because we found the column where there were 15 marbles altogether.

So how are we going to work using multiplication and division and all division in order to work out how many marbles Alex gets? Well let's go back to that original information, for every two marbles Aisha has, Alex has three marbles and they've shared a total of five.

What's happened to that total number of marbles? Well it's been multiplied by three, hasn't it? There are three times as many marbles, five multiplied by three is equal to 15.

So we know that there are three times as many marbles in total, and that means that Alex must also have three times as many.

There are three times as many marbles in total so Alex must have three times as many.

So we'll take Alex's original number of marbles which was three and multiply them by three.

Three multiplied by three equals nine.

So Alex must have nine marbles.

So let's look at another problem.

If Aisha gets 18 marbles, how many did they share all together? So we've gone from knowing about the whole and working out a part.

Now we know one part, Aisha's part of the marbles, and we're trying to work out the whole from that.

So we've identified that column where Aisha has 18 marbles.

So let's look back to that original column that we got from our original ratio, for every two marbles Aisha has, Alex has three marbles.

But she hasn't got two marbles now she's got 18 marbles.

So she's got nine times as many marbles.

Two multiplied by nine is equal to 18.

She's got nine times as many marbles.

So if Aisha has nine times as many marbles, there must be nine times as many all together.

So our original value of the total of marbles was five.

We're going to multiply that by nine.

Five multiplied by nine is equal to 45.

So they've shared 45 marbles all together.

So here are three more of those problems that we looked at.

Can you identify the part of the table that is useful? And can you, by using that stem sentence, and thinking about how many times more either Alex has got or Aisha has got or they've got altogether, use that stem sentence to help you to find the multiplication or division that is needed to solve the problem.

So how did you get on then? Our first question said if Aisha has 18 marbles, how many does Alex get? So we know we're looking at that final column and we're going to work out how many times the number of marbles Aisha has got so that we can work out how many times the number of marbles Alex will have.

So Aisha has nine times as many marbles.

So Alex will have nine times as many marbles.

We know that two multiplied by nine is equal to 18.

So that's Aisha's marbles.

Three multiplied by nine is equal to 27 so Alex will get 27 marbles.

So if Alex gets 18 marbles, how many does Aisha get? So this time we're looking at Alex's marbles which have gone from three up to 18.

So what is it that we've multiplied three by to equal 18? Six, three times six is equal to 18.

So Alex has six times as many marbles.

So Aisha will have six times as many marbles Two multiplied by six is equal to 12 so Aisha will get 12 marbles.

If Alex gets 80 marbles, how many did they share altogether? So we're still looking at that same jump from the three to 18, six times as many marbles.

So that means they must have six times as many marbles altogether.

Originally there were five marbles altogether, so five multiplied by six is equal to 30.

So if Alex gets 18 marbles, then they will have shared 30 marbles all together.

Time for some practise.

So Aisha is using this recipe to make 12 pancakes.

So for 12 pancakes she needs two eggs, three spoons of sugar and six spoons of flour.

So you are going to complete this table and explain how you calculated the missing values and remembering that we are thinking about using multiplication and division.

So there's a second part to task two, and when you filled in your table, you're going to use the table to identify the equations needed to solve these problems. Okay, how did you get on with filling in your table and how were you using those ideas of multiplication and perhaps division as well in order to fill in the missing values? Well I filled in the table here and I've now started to identify those multiplications.

So if I look at that second row, moving on from three spoons of sugar, we knew there were six spoons of sugar.

What's happened? Well we've got two times as many spoons of sugar, so there's two times as much of everything else.

So there's four eggs, 12 spoons of flour and we will make 24 pancakes twice as much as in the first row where we had three spoons of sugar.

What about this third row then? Let's think about the flour this time 'cause we had the flour there.

So we knew that originally we were using six spoons of flour.

Now we are using 18 spoons of flour.

We've got three times as many spoons of flour.

So we can see that there will need to be three times as much of everything else.

Two times three equals six, three times three equals nine, and 12 times three equals 36.

So we'll make 36 pancakes.

So we knew this time about the spoons of sugar.

We had three in our original recipe.

This time we're using 15, five times as many.

So three times five is equal to 15.

So we need five times as much of all the other ingredients and we will make five times as many pancakes.

This time we knew about how many pancakes, original recipe made 12.

This time we've made 120.

That's 10 times as many pancakes.

So we're going to need 10 times as much of everything else to make those 120 pancakes.

This last one's an interesting one, isn't it? Because this time we've not increased the value, we've decreased it, we've made it smaller.

Instead of using two eggs, we've used one egg.

So we have halved our recipe, so we've multiplied by a half or divided by two.

So there's half times as many eggs.

So there's a half times as much of everything else.

So for the second part of our task, we were going to use the table to help identify the equations to solve the problems, and we may have to add some extra rows to this table.

So the first problem was that Aisha needs to make enough so that she and three friends can have six pancakes each, and we needed to know how many spoons of sugar she would need.

Okay, so that's four friends.

So that's Aisha and three others, so four friends and they're each having six pancakes.

So I need six pancakes multiplied by four, which is 24 pancakes.

So 24 pancakes is two times as many as our original 12 pancakes.

So she will need two times as many spoons of sugar.

So in order to make enough pancakes so that she and her three friends can have six each, she's going to need six spoons of sugar.

The second part was to say that Aisha has a dozen eggs, 12 eggs.

How many spoons of flour will she need if she uses all the eggs? So our original recipe used two eggs, 12 eggs is six times as many as two.

So she'll need six times as many spoons of flour.

Six multiplied by six is equal to 36.

So she's going to need 36 spoons of flour.

What do the two sixes represent in this equation? One six represents the original six spoons of flour and the other represents the fact that we've got six times as many because we are using a dozen eggs.

We could also think about this using division.

If we've got 12 eggs, if we are going back to our original number of eggs of two, we would divide by six.

And so 36 divided by six would equal six as well.

So we can think about it in terms of division and multiplication.

So for the Summer Fair, our final problem, Aisha's class needs to make 300 pancakes.

How many eggs will they need? Well there's various different ways we could go around this, couldn't we? We need 300 pancakes.

And so we can see here that if we put in a row with 300 pancakes, we've got to find a way of relating that to something else we know.

So we had a row in there which talked about making 60 pancakes.

And 60 pancakes is 300 divided by five.

So we could think about a division by five here, but also we can think about multiplying by five.

If we know what we need to make 60 pancakes, we can multiply by five to work out how many eggs we'll need to make 300 pancakes.

So we're going to need five times as many eggs.

So we will need 50 eggs in order to make 300 pancakes.

Thank you for all your hard work in this lesson today.

I hope you are feeling more confident that you can use multiplication and division to represent ratio problems and use calculations to solve problems with missing values.