Lesson video

Well done for taking the first step and choosing to load this video to learn with today.

We've got lots of exciting algebra to have a look at.

If there's things you've been unsure of in the past, I promise that by the end of this video, you're going to feel more confident with those skills.

My name is Miss Davies, and I'm going to be helping you every step of the way as we work our way through.

Make sure you've got everything that you need to make a good start on this lesson, and then join me, and we'll get ourselves started.

Welcome to this lesson on problem-solving with expressions and formulae.

You'll be using your knowledge of expressions and formulae today to solve problems. Couple of keywords you need to be familiar with.

Pause the video and check over them before moving on.

So we're going to start by looking at forming expressions for area.

We can use the product of two binomials to write expressions for the area of shapes where the side lengths are given as binomial expressions.

For example, here we have x - 1 as one dimension of our rectangle and x + 7 as the other.

Let's look at how we could write an expression for the area of the shape.

Well, the area of a rectangle could be calculated by doing the width multiplied by the length, so we could do x + 7 all multiplied by x - 1.

We need to make sure that our expressions are in brackets because it's the entire length of the shape multiplied by the entire width of the shape that will give us the area.

We could find an equivalent expression by expanding, so we need to multiply the entirety of one binomial by the entirety of the other binomial, so we do x multiplied by x is x squared, x multiplied by -1 is -x, 7 multiplied by x is 7x, and 7 multiplied by -1 is -7.

We can collect like terms to give a final expression of x squared + 6x - 7.

Let's think about this shape.

How could we write an expression for the area of this shape? Well, it's a square, so we could write the expression as 3x + 2, all squared.

That would be an expression for the area of this shape.

Equally, we could write it in another way.

We could write it as 3x + 2 multiplied by 3x + 2.

If we wanted, we can expand those brackets to find the expression of the area in another format, so 3x multiplied by 3x is 9x squared, 3x multiplied by 2 is 6x, 2 multiplied by 3x is 6x, and 2 multiplied by 2 is 4.

Simplifying, we can have an expression that looks like 9x squared + 12x + 4.

All of these are expressions for the area of this shape.

We've just written them in different formats.

Different formats can be useful for different things.

We can do the same for any shape as long as we have an expression for all the necessary dimensions, so here is a triangle.

It has a base of x + 5 and a perpendicular height of 2x + 2.

The area of any triangle could be calculated by multiplying the base by the perpendicular height and dividing by 2.

You can think of that as using the formula area equals 1/2 times the base times the height, so using our binomials, we can write that as 1/2 multiplied by x + 5 multiplied by 2x + 2, or equally, we could write that as x + 5 multiplied by 2x + 2 all over 2.

It's the same thing.

How else could we write this? Okay, well, we could write it in expanded form, so what we could do is multiply the binomials together and then halve, so x multiplied by 2x is 2x squared, x multiplied by 2 is 2x, 5 multiplied by 2x is 10x, 5 multiplied by 2 is 10.

Got those four partial products.

I can simplify, so in total, we need to do 1/2 of 2x squared + 12x + 10.

To halve the expression, we can multiply each term by 1/2 or divide each term by 2, which gives us x squared + 6x + 5, so the area of that triangle could be written as x squared + 6x + 5.

Laura calculated a different way.

Should we see what she did? So Laura's written this, and then her second step is x + 1, x + 5.

Are Laura's first 2 steps correct? What do you think? Yes, they are.

She's written exactly the same first step as we had on the previous slide, but instead of multiplying the binomials together, she's decided to halve the first binomial.

Well that's absolutely fine.

Essentially, we've got three expressions that need to be multiplied, and it does not matter which two she multiplies first.

What could she now do? So if she wanted to, she could now multiply out those two binomials to get x squared + 6x + 5, and you'll notice that that's what we had doing it the other way, so here are all the equivalent expressions we've found so far.

They're all expressions for the area of this triangle.

Time for you to have a go.

Which of these could be expressions for the area of this triangle? There's quite a few to read, so take your time, and then we'll check your answers.

Lovely, so the first one cannot be correct because both dimensions have been halved.

You only need to halve one of the dimensions.

The second one, you can see that 2x + 4 has been halved to get x + 2, so we could do x + 2 multiplied by 4x - 6.

With the fourth one down, you can see that we have halved the perpendicular height first and then multiplied by the base.

Then, on the right-hand side, we could multiply the two binomials and then divide it all by 2.

The third one down is almost correct, but that should be a 4 in that binomial if that was to be correct.

Right, we're going to try with a slightly trickier shape.

We're going to have a look at a trapezium, so what we could do is we could split this trapezium into two triangles.

Now I could work out the area of each, so the base is x + 4 and the perpendicular height is x + 5, so I could write that as 1/2 x + 4, x + 5.

The other triangle, B, would be 1/2 x + 6, x + 5.

It's not easy this time to halve either of those binomials, so I'm going to expand the brackets first, so you get x squared + 9x + 20.

Then I need to halve that, so I get 1/2 x squared + 9 over 2x or 4.

5x + 10.

Right, if we do the same for B, again, let's expand the brackets and then multiply by 1/2, so you get 1/2 x squared + 11 over 2, or 5.

5x + 15.

Now we need to add the area of A and B together.

It looks more complicated than it is 'cause a 1/2 x squared and a 1/2 x squared is x squared.

9 over 2 and 11 over 2 is 20 over 2 or 10x and then 10 add 15 is 25.

Aisha thinks she knows an easier way to find the area of a trapezium.

Splitting a trapezium into two triangles and working out the area of both is an absolutely fine way of working out the area of a trapezium, but we can do it in slightly different order.

We can start by adding the parallel sides together and dividing by 2.

We'll have a look at the formula in a moment.

What adding the two parallel sides together and then dividing by 2 does is it finds the middle of the two lengths, so x + 4 add x + 6 is 2x + 10.

Halving that gives us x + 5.

What that does is it finds a value in the middle of x + 4 and x + 6, and it makes sense that x + 5 is halfway between x + 4 and x + 6.

Now that we know halfway between the two parallel sides, we can multiply that by the height, so x + 5 multiplied by x + 5.

That gives us x squared + 10x + 25, which is what we had doing the other way.

What Aisha has done here is she's used the formula for the area of a trapezium.

Area is 1/2 multiplied by a + b where a and b are the parallel sides multiplied by the height, so you can absolutely use that method when you're finding the area of a trapezium.

Okay, Laura wants to try it.

She's got the formula.

Let's look through her working-out, and see if you can spot where she's gone wrong.

Read back through her working.

Can you spot the mistake? Can you help her get the right answer? She's got a bit confused when substituting into her formula.

a and b are the parallel sides, so the two sides she needed to add together and halve is the x + 5 and the 3x + 7.

She's trying to find the middle of those two lengths.

Then she can multiply that all by x + 3.

She's going to give it another go.

Adding the two parallel sides gives 4x + 12.

Halving gives 2x + 6.

She can then do 2x + 6 multiplied by x + 3.

She's 2x squared + 12x + 18.

Well, that's always encouraging as well when somebody else gets the same answer as you.

Time for a practise, then.

Before you get started, I've put some formulae up that could help you, so the area of a rectangle is the length multiplied by the width.

The area of a triangle is 1/2 multiplied by the base multiplied by the perpendicular height, and there's the formula for the trapezium that we've just been using.

Equally, remember you could split the trapezium into two triangles, work them out separately, and add them together, so for this question, I would like you to write a sentence explaining why the area of the shape can be written in each of the following ways.

Then I'd like you to draw a rectangle with an area of 6 + y, 2 + y.

Extra challenge.

Can you write the expression for your area in a different way? Many ways as you can think of would be fantastic.

Having a triangle, now, then, so can you explain why the area of the triangle can be written in that form? And then, can you write the expression for the area in two different ways? Then I would like you to draw and label a triangle with an area of x + 1, 3x - 2, so you draw a triangle and then put on the dimensions so the area could be written as x + 1, 3x - 2.

Fantastic.

This time, I'd like you to match the shapes to the correct expression for that area.

I suggest you give yourself some space to do some working-out to help you.

Off you go, and then we'll look through the answers.

Lovely, so the first one, you might have said something like, "This shape is a square," so the area can be found by squaring one of the side lengths.

Then you might have gone on to say that squaring expression is the same as multiplying that expression by itself, so x - 4, all squared, is equivalent to x - 4 multiplied by x - 4, and then, in order to get x squared - 8x + 16, we can multiply the binomials using the distributive law and simplify by collecting like terms, so when you're drawing your rectangle, we needed a rectangle with dimensions 6 + y and 2 + y, and then equivalent answers anything where you've written equivalent binomials, so you might have gone with y + 6, y + 2.

You might have decided to expand the brackets by multiplying each term by every other term, and that gave you y squared + 2y + 6y + 12, and equivalent expression of y squared + 8y + 12.

Obviously, those three terms can be written in any order.

For our triangle, something like the area of a triangle can be calculated by multiplying the base by the height and dividing by 2 or multiplying the base by the height and then by 1/2.

Lots of different ways you can write an expression for the area of this shape.

See if you've got the same ones as me.

If you've got something different, you might want to check it against a partner.

This was a little bit more challenging because the area was given as x + 1, 3x - 2, and we know the area of a triangle is 1/2 multiplied by the base and multiplied by the perpendicular height, so you needed to double one of those dimensions, so you could have kept the x + 1 and then doubled the 3x - 2 to get 6x - 4 as one of the dimensions, or you could have gone the other way around, kept the 3x - 2 and multiplied the x + 1 by 2 to get 2x + 2.

Obviously, the binomials can be on either dimension, so these four shapes, then.

Let's have a look at the working-out, so we've got 1/2 4x - 4, x + 2, which is 2x - 2 multiplied by x + 2.

Expanding and simplifying gets 2x squared + 2x - 4.

The trapezium, we've got 1/2, and then, if we add the two parallel sides together, that gives us 6x - 2, so it's 1/2 6x - 2 multiplied by x + 1 or 3x squared + 3x - x - 1.

That matches with g.

c 'cause it's a rectangle, just need to multiply the two dimensions together, and you might have noticed that that's going to be double the answer we got for a, so 4x squared + 4x - 8, and that should leave us with e, but let's check.

The interesting thing with this one was when we halved 2x - 2, we got x - 1, and that gave us the difference of 2 squares, so when we multiplied our binomials, we simplified to an expression which only had two terms, x squared - 1.

Fantastic if you matched all of those up, particularly if you're working-out looked really clear, like mine.

Now we're going to put that to the test by looking at the area of compound shapes.

Area of compound shapes can be found by splitting them into shapes whose areas can be found, a little bit like we did with the trapezium before, so this shape here we could split into two rectangles.

Then we need to know the dimensions on both the rectangles.

We've got the dimensions on the larger rectangle, but we're missing a dimension on the smaller rectangle.

To find that dimension, we can look at some of the other lengths that we know, so we've got 3x + 5 as the whole of the left-hand side and 2x + 1 as that section.

If we subtract those two, we'll find the missing length, so 3x + 5 subtract 2x + 1.

Making sure we're subtracting both terms gives us x + 4.

You might just want to check: does x + 4 add 2x + 1 give you 3x + 5? Yes, it does.

Now we can work out the expressions for both areas, so for the first one, x + 2 multiplied by x + 4 gives us x squared + 6x + 8.

Our other rectangle gives us 4x squared + 8x + 3, and then we can combine them for our total area.

Now, 5x squared + 14x + 11.

My advice when you're working through longer questions like this is that you split your working-out up into sections so it's really clear what you're doing at each point.

You could even put little subheadings in and say smaller rectangle area and then larger rectangle area and then total area, and then it makes it really clear for you to follow what you're doing and other people to follow what you've done as well.

In total, the calculation we've done is x + 2 multiplied by x + 4 add 2x + 3 multiplied by 2x + 1.

It's absolutely fine to write your calculation all in one go, but I suggest that you then do your steps in sections working down your page.

Alex reckons he knows another way to do it.

Let's see if that is the case.

We could also see this is one big rectangle subtract a smaller rectangle.

Ah, I wonder if you can see what he means now.

There's a large rectangle subtract a smaller rectangle does get us the same compound shape.

What are the dimensions of these rectangles? Pause the video.

See if you can work them out.

Okay, we've already done most of the hard work.

The larger rectangle, we can see the dimensions.

The smaller rectangle, we need to work out the length, so that's 2x + 3 subtract x + 2, making sure we're subtracting both terms, so 2x + 3 - x - 2 gives us x + 1, so that's x + 1.

The other dimension we've already worked out is x + 4.

We can now expand and simplify.

Remember, Alex said we could find the largest rectangle and subtract the smaller rectangle, so the larger rectangle is 6x squared + 19x + 15.

Smaller rectangle's x squared + 5x + 4, and we need to make sure we're subtracting every term, so subtracting x squared gives us 5x squared.

19x subtract 5x is 14x, and 15 subtract 4 is 11.

I wonder which method you preferred.

Pause video.

Check you're happy with both.

Have a think about which way you would do it if you had the choice.

Time for a check.

I'd like you to work out the missing dimensions on this compound shape.

Off you go.

So for that smaller horizontal length, we can do 5x + 7 subtract 2x + 3.

Making sure we subtract both terms, we get 3x + 4.

For the longer vertical length, I wonder if I caught you out with this one, we actually need to add the two shorter lengths together.

We need to add 3x + 2 and 3x - 1, and that'll give us that full height, so that gives us 6x + 1.

How would you think we could calculate the area of the shaded shape below? There's no dimensions at the moment, so if I told you some of the dimensions, what could you do to work out the area of the shaded bit? What do you think? So what we could do is subtract the smaller rectangle from the larger rectangle.

That's definitely going to be quicker than trying to split it up into multiple shapes, so I've given you some dimensions now, so we can find the larger rectangle subtract the smaller rectangle.

We can expand both of our brackets and collect like terms. Now we've got 2x squared subtract x squared 19x subtract x.

Then here's where you've got to be careful.

We've got 9 subtract -2, so we have x squared + 18x + 11 'cause 9 subtract -2 is 11, so this time, the diagram shows two squares, one inside the other.

How can we write an expression for the area that is shaded? Let's do it together, so we know the shapes are squares, and we know the length of the side of each.

We can then work out the area of both squares and subtract them.

Let's try it, so we've got x + 12, all squared, minus x + 7, all squared.

We get x squared + 24x + 144 subtract x squared + 14x + 49, and this time, we end up with a linear expression because we've got an x squared subtract an x squared, we end up with 10x + 95.

Okay, what about if I told you the area is 120 square units? Let's see if we can work out the value of x, so if the shaded area is 120, we can now form an equation.

10x + 95 = 120.

If we add the additive inverse of 95, we've got 10x + 95 add negative 95 equals 120 add negative 95, making sure we're adding the same value to both sides of our equation.

The whole point of doing that is that we now have a 0 pair on the left-hand side, so that gives us 10x = 25, which gives x = 2.

5.

If we wanted, at this point, we could substitute that back in and check our answer, so we've got 2.

5 add 7, so that's 9.

5, and 2.

5 add 12, so that's 14.

5, and just check where the 14.

5 squared subtract 9.

5 squared does give you 120.

Your go this time.

The area of the shaded shape is 250 square units.

Which of these would be a suitable equation for the area? Give it a go.

There was a lot of reading to do there.

Well, I don't know if you spotted.

It was the third one.

Be the area of the smaller shape subtract the area of the smaller shape.

The problem with a was that we had the wrong dimensions multiplied together.

b, we were adding them.

We need to subtract to find the shaded area, and d, we can't subtract the dimensions first before finding the area.

We don't end up with a rectangle that we can then find the area of.

Time for you to have a go.

This is problem-solving, so it does take a little bit longer than some of the more straightforward questions, so give yourself space to show you're working, and you need to remember to feel really proud of yourself when you get to a final answer.

Even if it's not 100% correct, having some lines of working is a really good start.

For each of these, I'd like you to write a simplified expression for the area of the shape.

Give it a go.

Well done.

For 3a, I'd like you to find a simplified expression for the area of the shaded shape, and then I've given you the area of the shaded shape, and I'd like you to use that to find the value of x.

Off you go.

For 4a, I would like you to calculate a simplified expression for the area of this shaded shape.

I then told you that the area is 99 square units.

Can you use this to find the value of x? Off you go.

And finally, this shape is made of four identical rectangles sitting around the outside of a shaded square.

I'd like you to find a simplified expression for the area of one rectangle.

Then a simplified expression for the area of the whole shape, so that's the whole square made up of the shaded square and the four rectangles, and finally, a simplified expression for the area of the shaded square.

Give that a go.

Come back when you're ready for the answers.

Fantastic.

I hope you found some challenge with some of those questions.

I have chosen to split the shapes into two rectangles this way.

There are other ways to do this.

You should end up with a simplified expression of 2x squared + 22x + 48.

Well done.

I've decided to split this one vertically this time.

In order to find the missing length I need, I need to do 5x + 4 subtract 3x + 2, so the missing length that I want is 2x + 2.

Then I can find the area of my two rectangles, and I get 9x squared + 42x + 26.

Take some time to check back through your calculations.

It's often just a small mistake with some of our multiplications, and we just need to go back through and check.

This time, an area of the shaded shape.

We could do the area of the whole rectangle subtract the area of the smaller rectangle.

That gives us an area of 23x + 80.

Again, spend some time just checking back through, particularly if you've made a mistake with subtracting some of your negative values.

19x subtract -4x is the same as 19x add 4x, so just be aware of that, and then we can form an equation 'cause we know it's equal to 149 square units, so add the additive inverse of 80 and then divide by 23.

There's three 23s in 69, so x must be 3.

Again, you could substitute that in to check whether you get the right shaded area, and 4, so this was definitely easier if you found the area of the entire rectangle and subtracted the rectangle within the shape.

That gives you an expression of 27x - 63.

Again, pay attention to my working-out if you think you made a mistake somewhere.

If that's equal to 99, you can do this in several ways.

I can see a common factor of 9, so it'd be easier to divide through by 9 first, so that gives me 3x - 7 = 11.

Can you see how my numbers are a lot easier to use now? That gives me 3x = 18 and x = 6, and final question, so for a, we need to do 2x + 3 multiplied by x + 2, which gives us 2x squared + 7x + 6.

The area of the whole shape, the easiest thing to do is to find one length, which would be 3x + 5, adding those two together, and then 3x + 5, all squared, gives us 9x squared + 30x + 25.

Make sure you haven't missed any of your partial products, and then there's options.

I'm going to show you two different ways.

What we could do is we could find the area of all four rectangles.

We've shown that one rectangle is 2x squared + 7x + 6, so we multiply that by 4.

We get 8x squared + 28x + 24.

Then we can work out the large square that we worked out in b, subtract the four rectangles, so 9x squared + 30x + 25 subtract 8x squared + 28x + 24.

That gives us x squared + 2x + 1, and I was really glad to see that there was another way of doing it 'cause that means I can check if I get the same answer.

The length of the shaded square is actually 2x + 3 subtract x + 2, so that gives me a dimension of x + 1, so if I do x + 1 multiply by x + 1, I get x squared + 2x + 1, which was the same answer we got doing it the other way, which is always nice to see.

Well done for all your hard work in that problem-solving lesson.

What we've seen today, then, is when lengths of the shape are given as expressions, we can use the distributive law to write an expression in expanded form.

Couple of times, we saw that that allowed us to solve an equation.

We found that the area of compound shapes can be found by adding or subtracting the areas of known shapes, and we just had lots of fun playing around multiplying binomials together and checking our answers by trying things different ways.

Thank you for joining me, and I'd love to see you again.