Lesson video

Hi there.

I'm Mr. Tazzyman.

Today, I'm gonna be teaching you a lesson from a unit that is all about solving problems that feature two unknowns.

You've probably come across these kinds of problems lots of times in the past, but you may not have known that that's what we would categorise them as.

We're gonna be looking at the structures that underlie some of these problems to really help you to understand what's going on mathematically.

So, make sure you're ready to learn.

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 use a bar model to represent spatial problems with two unknowns." Here's the one keyword that we're gonna be thinking about today, represent.

I'd like you to repeat it back, please.

Okay, what does it mean though? To represent something means to show something in a different way.

Here's the outline then for today's lesson, we're gonna start by representing spatial problems. Then we're gonna move on to solving spatial problems. In this lesson, you're gonna meet Aisha and Sam.

They're gonna be helping us along the way.

Hi, Aisha.

Hi, Sam.

They're gonna be discussing some of the maths problems at hand.

Okay, let's get started then.

This pattern is made from two different-sized squares.

What is the side length of each type of square? So we've got the pattern drawn there, and we've got some dimension labels as well, 34 centimetres and 42 centimetres.

Aisha says, "We can see that this shape is made up of one larger square in the middle and three smaller squares that are the same." "We are looking for the length of one of these squares," says Sam, "And the length of the larger square.

Let's represent the problem with the drawing.

This will help us to see the problem.

The 42 centimetre length is made up of two lengths of the smaller square and one length of the larger square." "Let's record that as an equation." Aha, good thinking, Aisha.

'Two s plus l is equal to 42 centimetres.

The two s represents two lots of the smaller square's length." That makes sense.

"The L represents one lot of the larger square's length." "Let's do the same for the 34 centimetre length.

The 34 centimetre length is made up of one length of the smaller square and one length of the larger square." "Let's record that as an equation.

S plus l is equal 34 centimetres.

Aisha says the s represents one lot of the smaller square's length.

The l represents one lot of the larger square's length." "We can also represent these equations as bar models.

The first bar model has two lots of s and one lots of l.

This is equal to 42 centimetres.

So you can see there there are two parts labelled s and one part labelled l, and the part labelled l is slightly longer than the parts labelled s.

The total of 42 centimetres has been written alongside.

The second bar model has one lot of the s and one lot of l.

There it is, and the label of 34 centimetres, again, has been put alongside.

Using bar metals helps us to line up like bars to help show the difference more clearly.

If we line up the bars so they are under a like bar, we can see the difference.

Did you see that movement? L has been lined up underneath the l from above.

We can see more clearly now that there is a difference of one lot of s.

There it is.

Can you see that? It's time to check your understanding then.

This pattern is made from two different sized equilateral triangles.

What is the side length of each type of triangle? Now, your task is to write two equations to represent this problem, so we're not looking for you to solve it just yet.

We just want the representation.

Okay, pause the video and give it a go.

Welcome back.

Let's mark the equations that you wrote then.

Here's what we had.

Three s plus l is equal to 17 centimetres, and the s in this represents the smaller triangle side.

The l represents the longer triangle side.

We also had s plus l is equal to 11 centimetres.

Next part then.

You need to draw a bar model to represent this problem this time.

Again, we're not solving it yet, but we'd like you to draw a bar model.

You might wanna make use of the equations that we just went through to help.

Pause the video and give it a go.

Welcome back.

Here's the bar models we had.

The top bar had one part labelled l and one smaller part labelled s with 11 centimetres alongside.

The other bar had one part labelled l and three parts labelled s with the total of 17 centimetres alongside.

Is that what you've drawn? Something like that, I hope.

Here's your first practise task then.

You've got to represent each problem with a bar model and a pair of equations.

So, again, we're not looking for you to necessarily work out the length of each side here, we just want you to represent each of these with a bar model and a pair of equations.

Pause the video here and I'll be back in a little while with some feedback.

Good luck.

Welcome back.

Let's look at a then.

We had this lovely pattern that had been comprised of different sized squares, two different sized squares.

We had some dimensions labelled too, one of 32 centimetres and one of 17 centimetres.

So, the first equation might have been four s plus l is equal to 32 centimetres.

The s was representing the smaller square sides and the l representing the longer square sides.

So you can see this one has been taken from the vertical dimension, along which you had one large square and four smaller squares.

The next equation was s plus l is equal to 17.

That's because you can see the 17 centimetre dimension label and that runs along one smaller side and one longer side of the square.

Here's the bar model then.

One part labelled l, one part labelled s with 17 centimetres.

The other bar was one part labelled l and four parts labelled s with 32 centimetres.

Here's B then this one was made up of squares and isosceles triangles.

We had dimensions of 63 centimetres and another dimension label of 42 centimetres.

The first equation might have been two s plus t is equal to 63 centimetres, where s stands for the side of a square and t stands for the side of a triangle.

S plus t is also equal to 42 centimetres.

Thankfully, the sides of the triangle that were in each of these dimension labels were the same sides, because of course, if it's an isosceles triangle, there's a possibility that one of the sides will be longer.

The bar model might have looked like this, one part labelled t, one part labelled s is equal to 42 centimetres, and one part labelled t and two parts labelled s is equal to 63 centimetres.

Let's move on to the second part of the lesson then, solving spatial problems. Which representation do you think is most helpful when trying to solve the problem? So we've got three different representations here.

We've got our drawing, we've got our equations, and we've got our bar models.

Which do you think was the most helpful? Aisha says, "I prefer to annotate on the drawing itself.

The 42 centimetres is two lots of the smaller square's length and one lot of the longer square's length." The 34 centimetres is only one lot of the smaller square's length and one lot of the longer square's length.

It's one lot of the smaller square's length less.

That means the difference between the two lengths is one lot of the smallest square's length.

42 centimetres subtract 34 centimetres is equal to eight centimetres.

That means the length of the smaller square side is eight centimetres.

We now know that these two lengths are equal to eight centimetres, so we can subtract the two lots of eight centimetres from 42 centimetres to find the length of the larger square's side.

So the larger square's side is 26 centimetres.

Solved." 42 centimetres subtract 16 centimetres is equal to 26 centimetres.

Sam says, "I prefer to use a bar model to help me see the structure of the problem.

We can see clearly that the difference between the two lengths is one lot of the small square's length." If you look at the bars, you can see the extra part is s, the small square side.

"So we can subtract the shorter length from the larger length to find this difference." 42 centimetres subtract 34 centimetres is equal to eight centimetres.

Sam relabels the bar model.

"That means each of the smallest squares have a side length of eight centimetres.

Now I can subtract the eight centimetres from the 34 centimetre length to find the length of the largest square." 34 centimetres subtract eight centimetres is equal to 26 centimetres.

So as we showed earlier, the larger square's length was 26 centimetres." They got the same answer, but they used a different model to approach the question.

Okay, your turn.

If s represents the side length of the smaller equilateral triangles and l represents the side length of the larger equilateral triangle, what is the value of s? Pause the video here and give that a go.

Welcome back.

You can see that the difference between the two bars was two parts labelled s, two smaller sides of a triangle.

17 centimetres subtract 11 centimetres is equal to six centimetres.

The difference between the two total lengths was six centimetres.

So six centimetres divided by two is equal to three centimetres, and because that difference was also two smaller sides, we can now say that each of the smaller sides was three centimetres long.

S is equal to three centimetres.

Did you manage to get that? I hope so.

Okay, another turn for you.

If three centimetres represents the side length of the smaller equilateral triangle and l represents the side length of the larger equilateral triangle, what is the value of l? Pause the video and give that a go.

Welcome back.

L centimetres plus three centimetres is equal to 11 centimetres.

11 subtract three is equal to eight centimetres, so l is equal to eight centimetres.

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

Solve each problem.

For a, it's composed of two types of square.

It might look quite familiar.

And for b, it's composed of squares and is isosceles triangles.

Again, it might look quite familiar.

For number two, you've got to create your own shape problem, including two different-sized shapes and two unknown side lengths that can be calculated.

Pause the video here and have a go at that task.

I'll be back with some feedback in a little while.

Good luck.

Welcome back.

Let's look at 1 a to begin with.

We had our bar model, which we drawn from practise task A.

You can see it there.

The top bar is one larger part labelled l and one smaller part labelled s, totaling 17 centimetres.

The bottom bar was one larger part labelled l and four smaller parts labelled s, totaling 32 centimetres.

The difference then between them was three lots of s, and the difference in terms of centimetres was 32 centimetres subtract 17 centimetres, which is equal to 15 centimetres.

That meant that each part s was worth five, and that's because we knew that three of them were worth 15 centimetres.

15 centimetres divided by three is equal to five centimetres, so s was equal to five.

17 centimetres subtract five centimetres is equal to 12 centimetres.

We've used the top bar there to work out the length of l.

L is equal to 12 centimetres.

Here's b then.

Again, we've taken the bar model from our first practise task.

We know that s is equal to 21 centimetres.

63 centimetres subtract 42 centimetres is equal to 21 centimetres.

42 centimetres subtract 21 centimetres is equal to 21 centimetres.

So t is equal to 21 centimetres.

Okay, here's number two, and this is what we came up with.

You can see there was a pattern of big squares and small squares, and some dimension labels again.

One of 18 centimetres and one of 22 centimetres.

"Here is an example that I made," says Sam.

All right, it's time to summarise the learning then.

Problems involving shapes can have the same structure as questions where the sum and the difference is known.

Sum and difference problems involving shape can be represented using a bar model or equations.

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

Maybe I'll see you again soon.

Bye for now.