Lesson video

Hello, My name is Mr. Tazzyman, and today I'm gonna be teaching you a lesson from a unit that is all about solving problems featuring two unknowns.

You might have come across these kinds of problems previously, but in this unit we are really going to look at comparing their structures and trying to solve them using things like trial and improvement and maybe bar models where we can.

So make sure you're ready to learn and let's go for it.

Here's the outcome for today's lesson then.

By the end, we want you to be able to say, I can explain why there is sometimes only one solution to a problem.

These are the key words that you might expect to hear during the lesson.

It's really important that you know and understand them.

The words are some and difference.

I'm gonna get you to repeat them back to me.

I'll say "My turn," say the word, and then I'll say "Your turn," and you can repeat it back.

Ready? Okay.

My turn, sum, your turn.

My turn, difference, your turn.

Okay, we can say them, but do we know what they mean? You might well do, but let's look at a quick recap just to double check.

The sum is the total when numbers are added together, the difference is a result of subtracting one number from another, how much one number differs from another.

Okay, let's look at the outline of the lesson now then.

To begin with on this lesson in which we're explaining why there is sometimes only one solution to a problem, we are going to look at some sum and difference problems. Then we're gonna move on to looking at some sum and multiple problems. In the lesson, we're gonna be joined by Izzy and Jun.

They're gonna be discussing a lot of the problems that you face in this lesson.

They're gonna help us with their learning.

They're gonna give us some hints and tips and they're gonna explain some of their thinking to us.

Hi Izzy.

Hi Jun.

Ready to start? Are you ready to start? Okay, let's get learning.

A and B represent numbers.

The sum of A and B is 30, and the difference between them is 8.

What numbers do A and B represent? So we've got a couple of really good clues there.

Let's see what Izzy and Jun think.

Izzy says, "This is just like the gold and silver stars problem we saw before." Jun says, "Let's represent this problem using a bar model to help us." So they start with two bars, one labelled A and one labelled B.

A and B are numbers.

We know that A is 8 more than B, but hang on, they haven't put that on the bar model.

Hmm.

What could they do? Well, they put an extra part onto A labelled 8 because they know that A is 8 more than B.

You can see now that the difference between 8 and B is 8, and the sum of both A and B is equal to 30.

How will they record that on their bar model? Like this.

There's the total, 30.

"Let's solve this now then," says Jun, 'cause of course drawing a bar model helps us to think about a problem, but it doesn't necessarily solve it.

It helps us to see the structure of it.

We can subtract 8 from the whole, which will leave us with 22.

They subtract 8 from the bar, 30 subtract 8 is equal to 22, so they change the total to 22.

This is now equivalent to two equal bars.

Because the bars are equal in size, we can divide 22 by 2 to find the value of each bar.

22 divided by 2 is equal to 11.

So the value of B is 11.

Now they've labelled part of A as 11 as well, but the value of A can't be 11.

So what do they do next? To find the value of A, we need to add the 11 with the extra 8.

Aha, so now they bring back the extra 8 and that's why their total has changed back to 30.

11 added to 8 is equal to 19.

So the value of A is 19.

We can now check to see if our values for A and B are correct.

How might they check? It's really important to do so, isn't it? But how might they do it? We know the whole is 30 and A and B sum to this.

So if A equals 19 and B is equal to 11, let's add these together to check.

Aha, good thinking, Izzy.

19 added to 11 is equal to 30.

"We did it," says Jun.

Yes, you did, really nicely worked out, and it shows you that some problems the answer won't just appear straight away, you need to make sure that you model it first to think about the structure before you start solving.

Well done you two.

Now it's your turn.

We're gonna check your understanding with a similar type of problem.

Solve the following problem, A and B represent numbers, the sum of A and B is 100 and the difference between them is 20.

What numbers do A and B represent? Okay, I'd advise using a bar model here and thinking about the problem we've just solved because it has a very similar structure.

Pause the video here and give it a go.

Welcome back.

Here are two bars, one labelled A and one labelled B.

But of course we know that we need to put something extra onto A because the difference between A and B is 20.

So there's the 20, the total was 100.

We can start subtracting 20 from that total, 100 takeaway 20 is equal to 80.

80 divided by 2 is equal to 40 so we can label the bars that remain as 40 because they are both equal size.

Now we can add the 20 back on.

We end up with 40 added to 20 which is equal to 60.

A is equal to 60 and B is equal to 40.

Now you might be tempted to move straight on there, but actually we need to do something very important.

We need to check that our values work.

A plus B is equal to 100, and 60 plus 40 is equal to 100.

So our values do work, this is a solution.

Speaking of solutions, is your turn to find some.

For your first practise task, you've got to solve the following problems. For number 1, two planks of wood have a combined length of 2 metres.

Plank A is 30 centimetres longer than plank B.

What is the length of both plank A and B? For number 2, Izzy and Jun have £90 in total.

Jun has £18 more than Izzy.

How much money do both Izzy and Jun have individually? For number 3, Izzy's brother is 29 years younger than her dad.

The sum of their ages is 77 years.

How old is Izzy's brother and Izzy's dad? And Jun says, "What is the same about all of these problems?" That gives you some thinking to do at the end.

All right, pause the video here and give those a go.

I'll be back soon with some feedback, enjoy.

Welcome back.

Let's go through some of the answers then.

We are gonna look at making sure that we've used modelling as we do it to reveal the structure of each problem.

We've got plank A and plank B drawn as bars here, but we know that plank A is 30 centimetres greater.

In total there were 200 centimetres.

If we then subtract the extra 30 centimetres from that, we end up with 170 centimetres.

170 centimetres halved or divided by 2 is equal to 85 centimetres.

So we can label each of those bars 85 centimetres because they're equal in size.

But of course plank A needs the 30 centimetres added back on.

85 centimetres plus 30 centimetres is equal to 115 centimetres.

So plank A is equal to 115 centimetres and plank B is equal to 85 centimetres.

Are we finished? No.

We need to make sure we check.

Plank A plus B is equal to 200 centimetres, 115 centimetres plus 85 centimetres, the values that we worked out plank A and B to be, is equal to 200 centimetres, it works.

Pause the video here if you need to catch up with some of that marking.

Welcome back.

Let's do number 2 then.

Same sort of start, two bars drawn out, one labelled Jun and one labelled Izzy this time.

Jun has £18 more because the difference between the two is £18.

In total they have £90.

If we subtract 18 from £90, we get £72, and £72 is equal to those bars combined.

If we halve £72, then we get £36.

Each of those bars is equal size so we can label each bar £36.

Now let's add back on that £18.

There it goes, and we've got 36 added to £18 is equal to £54.

So we know that Jun had £54 and Izzy had £36, but let's just check.

Jun plus Izzy is equal to £90, £54 plus £36 is also equal to £90, and they were the values that we managed to calculate.

So they are correct, that's our solution.

Pause the video again here if you need to catch up with that marking.

Let's do number 3 then, the last question in this first practise task.

We have Izzy's dad and Izzy's brother.

Izzy's dad is 29 years older.

In total, the number of years that they have between them is 77.

If we subtract 29 from 77, we get 48.

48 divided by 2 is equal to 24.

So we can label each of those equally sized bars as 24.

If we then add back on 29 to give us a total of 77 as we know that's what their total is, we have 24 added to 29, which is equal to 53.

So Izzy's dad is 53 and Izzy's brother is 24.

If we added those two scores together, the number of years that is, we know we'd get 77 years and if we add 53 and 24, that's also equal to 77.

So we know that those ages are accurate.

There's our solution.

Pause the video here if you need to catch up with that marking.

Okay, Jun also asks what's the same about these problems. You can see them all drawn out there.

All these problems share the same structure even though they use different contexts like money or measurements.

When the sum and difference of the unknown numbers is known, there is only one solution.

All right, it's time for us to move on to the second part then, some sum and multiple problems. The sum of two numbers is 24, one number is three times the size of the other number.

What are the two numbers? Izzy says, "This problem is slightly different from the structure of the problems we've just looked at.

Jun says, "Let's represent this problem using a bar model to help us." So they start with two bars, again, A and B.

Let's call the unknown numbers A and B.

We know that A is three times the size of B, but hang on, their bar doesn't show that.

Ah, now it does, A is three times the size of B and the sum of both A and B is equal to 24.

So they've written the total in there.

I can see that four equal parts sum to 24.

So Jun is looking at the bar for A and the bar for B and he's totaling the number of equal parts that that constitutes.

There are four of them there.

In A, the bar for A, there are three equal parts, and in B there's another equal part so altogether there are four.

That means 4 multiplied by something is equal to 24.

4 multiplied by 6 is equal to 24.

Jun knows his times tables.

So now they label each of those equal parts with 6.

So we can see that B is equal to 6 because B only had one of those equal parts B equals 6, and A is three lots of 6 which is 18.

Good knowledge of times tables again.

And the sum of both A and B is equal to 24.

"Let's check it says Jun." Good idea.

A plus B is equal to 24, and 18 plus 6, the values that we worked out for A and B is also equal to 24.

This is correct, well done Izzy, well done Jun.

There we go.

We solved it.

The sum of two numbers is 24, one number is one-third times the size of the other number.

What are the two numbers? Interesting.

There's a slight change in the wording here.

Read that middle part again.

One number is one-third times the size of the other number.

"How does this change our problem," says Jun.

"I don't think it does, look!" Let's call the unknown numbers A and B again.

In this last example, A was three times the size of B.

Another way of saying that is that B is one-third times the size of A, both A and B still sum to 24 as well.

So this bar model represents both of these questions.

There are still four equal parts that sum to 24, 4 multiplied by something is equal to 24.

4 multiplied by 6 is equal to 24.

So again, we can see that B is equal to 6 and A is three lots of 6 which is 18.

Again, the sum of both A and B is equal to 24.

Let's check it again.

Good thinking, Jun.

A plus B is equal to 24, A, if it's equal to 18, and B, if it's equal to 6, those added together is also equal to 24.

"That makes so much sense now," says Jun.

Okay, let's check your understanding then.

The sum of two numbers is 100, one number is three times the size of the other number.

What are the two numbers? Use a bar model here if you can, pause the video and give it a go.

Welcome back.

Here's the bar model structure.

Four lots of something is equal to 100.

100 divided by 4 is equal to 25.

That means we can label each part 25.

So B is equal to 25 and A is equal to 25 multiplied by 3 which is equal to 75.

Time to check it.

We know A plus B is equal to 100, and 75 plus 25 is equal to 100, it works.

Have a go at this one then.

A slight change here.

The sum of two numbers is 100, one number is one-quarter the size of the other number.

What are the two numbers? Pause the video and give this one a go.

Be careful when you are drawing out your bar model, there is a slight difference.

Welcome back.

Here's what the bar model might have looked like.

B is one-quarter the size of A here, a total 100, so five equal parts can be seen.

Five multiplied by an unknown is equal to 100.

Let's use the inverse to work that out.

100 divided by 5 is equal to 20.

So we can label each of the parts 20.

That means that B is equal to 20, and A is equal to 20 multiplied by 4, which is equal to 80.

A plus B is equal to 100, 80 plus 20 is equal to 100.

So we've checked it and it's correct.

Okay, it's time for your practise task then, the second one.

Here you've got to solve some more problems. Number 1, Jun earns £10 doing odd jobs on the weekend.

He earned 4 times as much on Saturday as he did on Sunday.

How much did he earn each day? For number 2, Izzy's garden has an area of 78 square metres made up of grass and a patio.

The area of the patio is one-third times the size of the area of the grass.

What is the area of the grass? For number 3, the combined mass of two mass of suitcases is 22.

5 kilogrammes, one suitcase is half the mass of the other.

How many kilogrammes do the suitcases weigh? And Jun asks again, "What is the same about all of these problems?" Here's number 4 then.

You've got to work out the value of each angle in this triangle.

Here's number 5, the ages of Alex and his two brothers sum to 18 years.

Alex is two years younger than Alfie, Alfie is two years younger than Sam, how old are Alex, Alfie and Sam? How many years is it until their ages total 54? Some really nice problems there for you to get to grips with.

Use bar modelling where you can, pause the video here and I'll be back in a Little while with some feedback.

Good luck.

Welcome back.

Let's start with number 1.

You might have drawn a bar model that looks something like this.

Saturday and Sunday have been labelled and you can see that Saturday is four times the size of Sunday.

£10 was earned in total, there are five equal parts, so we know that 5 multiplied by something is equal to £10.

£10 divided by 5 is equal to £2.

So each part was worth £2.

That meant on Sunday he earned £2, and on Saturday he earned £8 because 2 multiplied by 4 is 8.

If we check it, Saturday plus Sunday is equal to £10, 8 plus £2 is equal to £10.

Okay, let's move on to the next one.

Here you might drawn a bar model that looked like this with grass and patio.

The patio was one-third times the size of the grass, which if you look at it in the other way is that the grass was three times the size of the patio.

Altogether that totaled an area of 78 metres squared.

4 multiplied by something is equal to 78 metres squared.

And that's there because we can see there are four equal parts in our bar model.

Each equal part works out as 19.

5 metres squared and you get that by dividing 78 metres squared by 4.

We can then label each of the parts.

We now know that the patio is 19.

5 metres squared.

The grass is three lots of 19.

5 metres squared, which is equal to 58.

5 metres squared.

Patio plus grass is equal to 78 metres squared.

So let's check it with the values we found and we can see that they total 78 metres squared as well.

For number 3 then you might have drawn a bar model that looks something like this.

We've got suitcase 1 and suitcase 2.

The second suitcase is half the mass of the other Altogether, their mass combined is 22.

5 kilogrammes.

We can see three equal parts in our bar model.

So 3 multiplied by something is equal to 22.

5 kilogrammes.

22.

5 kilogrammes divided by 3 is equal to 7.

5 kilogrammes.

So now we can label each of our equal parts, 7.

5, 7.

5, and 7.

5.

So suitcase 2 is equal to 7.

5 kilogrammes and suitcase 1 is equal to two lots of 7.

5 kilogrammes, which is equal to 15 kilogrammes.

Let's check that.

Suitcase 1 and 2 added together give a mass of 22.

5 kilogrammes and so does 15 kilogrammes added to 7.

5 kilogrammes.

It works.

Here's number 4 then.

Nice to see some angles in there, but really we could still draw this as a bar model.

You can see that we've got three rows of bar models this time.

We've got two of the angles that are the same, both labelled x, and then the angle at the top is three lots of x.

The total of the internal angles within a triangle is 180 degrees, and that's a known fact that you needed.

Five lots of something is equal to 180 degrees.

We've done that because there are five equal parts in our bar model.

180 degrees divided by 5 is equal to 36 degrees.

So we can label each of the equal parts.

That means that 3x is equal to 108 degrees, x is equal to 36 and the other x is equal to 36.

But let's just check that 3x plus x plus x is equal to 180 degrees, 108 degrees plus 36 degrees plus 36 degrees is also equal to 180 degrees.

Brilliant.

Here's number five then.

Quite a complex question in terms of structure.

Let's start by bar modelling.

We've got Alex.

We've got Alfie who we know is two years older than Alex or Alex is two years younger than Alfie.

And then we've got Sam, who we know is two years older again, altogether they total 18 years.

So if we put that into an equation, we know we've got those three equal parts which are unknowns as far.

So we've got 3 multiplied by something is added to 3 multiplied by 2, and that is equal to 18.

Remember the order of operations here, we can then simplify that equation by completing the second multiplication expression.

So now we have 3 multiplied by an unknown.

Add 6, which was the product of 3 multiplied by 2 is equal to 18.

Now we can look at it in this way.

3 multiplied by an unknown is equal to 18 subtract 6, so three multiplied by an unknown is equal to 12.

12 divided by 3 is equal to 4.

Now we know our unknown and we can label each of those equal parts on the bar model.

That means that Sam is equal to 8 because that's the total of 4 and 2 and 2, Alfie is equal to 6 because that's the total of 4 and 2, and Alex equal to 4 because there's only one of those equal parts that we've just calculated the unknown value of.

We can check that we've definitely got our answer correct by adding 8 and 6 and 4 to give us 18.

The second part of that question asked how many years is it until their ages total 54.

Well, we've got Alex, Alfie, Sam, and he's still 2 years older than Alfie, who's 2 years older than Alex.

Altogether, now we need to think about their total being 54.

We can use our equation again, but this time there's some slight changes because the total is 54.

We can still simplify the equation in the same way by using our order of operations and working out that second multiplication expression first, we end up with 3 multiplied by something is equal to 54 subtract 6.

So 3 multiplied by something is equal to 48.

48 divided by 3 is equal to 16.

So each of those equal parts on this occasion, because our original total has changed, are equal to 16.

Let's label them and use those to work out the ages of Sam, Alfie, and Alex if their ages total 54, Sam would be equal to 20, Alfie would be equal to 18 and Alex would be equal to 16.

If you add all three of those together, you get a combined total of 54.

So those answers work.

"It would take 16 years," says Izzy.

So these were the structures of each of these problems. Jun said, "All these problems share the same structure even though they use different contexts like money or measurements.

When the sum and the multiplicative relationship between the unknowns is known, there is only one solution." Let's summarise today's learning then.

When the sum and difference of the unknown numbers is known, there is only one solution.

When the sum and the multiplicative relationship between the unknowns is known, there is only one solution.

My name's Mr. Tazzyman, hope you enjoyed that lesson.

Maybe I'll see you again soon.

Bye for now.