Lesson video

Hello, Mr. Robson here.

Welcome to maths.

You are in a good place, especially because we're doing brackets in equations today.

This should be awesome.

Our learning outcome is I'll be able to appreciate the significance of the bracket in an equation.

Equation being a key word for the lesson.

An equation is used to show two expressions that are equal to each other.

Two parts to the lesson and we're gonna start with representations of equations involving brackets.

Andeep and Laura are playing a number game.

Andeep says, "I'm thinking of a number.

When you multiply it by 2 and add 4, you get 10." Laura says something very similar, "I'm thinking of a number.

When you add 4 and multiply it by 2, you get 10." We both said "Multiply by 2, add 4 and 10.

So we must have the same number, right?" What do you think? Do Andeep and Laura have the same number? Pause and have a little think.

"We don't have the same number.

Order must matter." They conclude.

Andeep's number is 3.

10 - 4 is 6, half of that is 3.

We can check that's right by substituting it back in.

Let's take the number 3 multiply by 2, we get 6, add 4, we get 10.

Andeep's number is indeed 3.

Laura is different.

To find Laura's, we have to do 10 divided by 2.

Undo the multiply by 2.

We have 5 and then we subtract 4 and we get 1.

Laura's number is 1.

We can check that to be true by substituting it back in, so they don't have the same number.

If you think about this algebraically, we can see why not.

Andeep is thinking of a number.

We'll use x as that unknown, multiply it by 2 and add 4 to get 10.

You know that we won't leave the algebra like that.

We'll write it as simply as we possibly can 'cause we like things being written as simply as possible.

We'd write that as 2x + 4 = 10.

In Laura's case, thinking of a number, we use x to represent that unknown.

Let's add 4, multiply by 2 and we get 10.

To simplify that, 4 lots of 2.

We do the multiplication before the addition.

That would simplify to x + 8 = 10.

Have you noticed something amiss? x = 2 would be the solution to that equation, but Laura's number is 1, not 2.

What has gone wrong here? Pause and see if you can spot it.

What went wrong was at this stage here, Laura's unknown number and we add 4.

This next step when we multiply by 2, we need 2 lots of everything.

2 lots of x + 4, which we'd write like that with brackets 2 lots of x + 4 to get 10.

You'll see it written like that but I think this look familiar and you know there's a more simple way to write that.

We would write the 2 in front of the bracket.

2 lots of bracket x + 4 = 10.

When you substitute x = 1 in now, it works.

1 + 4 is 5, multiply that by 2, we get 10.

Really importantly, Laura's equation needed brackets.

These equations can be visually represented with bar models.

On the left, Andeep's equation 2x + 4 = 10.

On the right, Laura's equation 2 lots of bracket x + 4 + 10.

Bar models are very powerful way to represent equations, especially when we're gonna go on to solve them at some point.

In Andeep's case 2x and 4 being equal to 10.

When you substitute in Andeep's number, x = 3, it works, 3 and 3 and 4 makes 10.

Laura's look slightly different.

We need x and 4 and we need 2 lots of it being equal to 10.

Laura's number x = 1, works, when you substitute that in 1 + 4, 1 + 4, add those together, you get 10.

Jun and Jacob are drawing bar models for the equation 4 brackets y + 2 = 20.

Jun draws that bar model.

Jacob draws that one.

They are very different those two models.

Which do you think is correct? Jun's or Jacob's? Pause this video and have a think.

Jun is right.

Jacob's was missing a little something.

Jun correctly drew 4 lots of y + 2 being equal to 20.

y + 2 once, twice, thrice, four times, 4 lots of y + 2.

Jacob drew y and 4 lots of 2 being equal to 20.

He effectively drew that equation.

y + 2 lots of 4 being equal to 20.

That's not the same thing as 4 lots of bracket y + 2 = 20.

True or false for you now.

True or false? x multiplied by 2, plus equals 8 will be the same equation as x plus 3 multiplied by 2 equals 8.

They both say multiplied by 2 plus 3.

They both equal 8.

They're the same thing.

Is that true or is it false? I'd also like you to justify your answer with one of these two statements.

Order matters, this will be two different equations, one of which will need to be written with brackets, or they both have a multiplied by 2 and a plus 3 and make 8 so they're the same equation with the same solution.

True or false and which justification will you use? Pause now and try this.

I hope you said false, they're not the same thing.

And I hope you justified that with the sentence order matters, this will be two different equations, one of which will need to be written with brackets.

How would that look? x multiplied by 2 plus 3 = 8.

That would give us the equation 2x + 3 = 8.

However, x plus 3, and then multiplying by 2 being equal to 8 would look like that, x plus 3 inside the brackets and then a 2 outside the brackets to show it being multiplied to make 8.

Another check now.

Which of these visual representations matches which equation? You need two equations on the left hand side, 2x + 3 = 8 and 2 lots of bracket x + 3 = 8 and I've given you three bar models, so one of those bar model must not match either of those equations, but can you find the correct bar model to match to those equations? Pause and try that now.

I hope you noticed that 2x + 3 = 8 was the bottom bar model.

In the top row, you see two x's and 3 and it's equal in length to 8.

The second equation, 2 lots of brackets, x + 3 was represented by the top bar model.

Can you see in the top row there? x + 3 twice being equal to 8.

Middle one didn't have a match.

What do we have there? We've got x + 3 + 3 being equal to 8 which is x + 2 lots of 3, x + 6 = 8 is the equation represented by that bar model.

There was no bracket before multiplying by 2, hence only the 3 doubled the x term did not.

Another matching exercise.

Three equations on the left hand side.

Three bar models on the right hand side which matches to which? Pause and see if you can spot it.

We should have matched the top equation 3x + 1 = 15 to the bottom bar model.

You can see in the top row 3 x's and 1 being equal to 15.

The bottom equation x + 1 x 3 = 15.

You can see that in the top bar model, x and three 1's being equal to 15.

The crucial one to spot is the impact of those brackets when x + 1 is in a bracket and being multiplied by 3.

That's that bar model.

You can see x + 1 represented three times in the top row there.

Another quick check now.

Which words match this equation? The equation is 3x + 1 = 15 and I'm giving you a bar model to represent it.

Is that x multiplied by 3, plus 1 = 15? Is it x plus 3, multiplied by 1 = 15 or is it x plus 1, multiplied by 3 = 15? Which one is it? Pause and have a think.

I hope you said option a, that's x multiplied by 3, plus 1 = 15.

How about this one? A different equation.

An equation with brackets.

Which words would match this? 3 lots of in brackets x + 1 being equal to 15.

Is it x multiplied by 3, plus 1 = 15? x plus 3, multiplied by 1 = 15 or x plus 1, multiplied by 3 = 15? Which one is it? Pause and have a think.

I hope you went for option c, x plus 1 being multiplied by 3 to make 15.

I can see x + 1 three times in the top of that bar model.

For all of these equations with brackets that we've seen, the brackets can be expanded.

What do we mean by that? Well, 2 lots of brackets x + 4, I could represent with a bar model x + 4 twice being equal to 10.

We can see in that model two x's and two 4's, 2 lots of x, 2 lots of 4.

Effectively 2 lots of everything inside that bracket.

We'd simplify that to 2x, 2 lots of x being 2x and 8, 2 lots of 4 being 8.

We've expanded the bracket 2 lots of x + 4 to become 2x + 8.

For the equation on the right hand side, 4 lots of y + 2.

You can see y + 2 four times in that bar model.

That helps you to see that 4 lots of y + 2 becomes 4 lots of y, 4 lots of 2, which we'd simplify to 4y + 8.

That's the equation expanded or the expression rather.

4 lots of y + 2 expanded to 4y + 8.

That's their expanded form and there will be times when we want that expanded form, there'll be times when we don't.

Sometimes when we're solving these we will expand the brackets out.

Sometimes it can be quicker and easier not to.

True or false? When we see brackets in an equation we should expand them out.

Is that true or is it false? And can you justify your answer with one of the statements because we can expand brackets when we see them, we should always expand them out or whilst we can expand brackets if we need to, there'll be some moments where it's easier not to.

True or false? And how will you justify your answer? Pause now and have a think.

I hope you said false.

When we see brackets in an equation we should expand them out.

Not necessarily, not always true.

So I hope you went for false.

Why? It should be an option b.

Whilst we can expand brackets if we need to, there'll be some moments where it's easier not to.

Practise time now.

Question one.

I'd like you to write an equation and draw a bar model for Andeep and Laura's number problems. Andeep says, "I'm thinking of a number.

When you multiply it by 5 and add 10, you get 35." Laura says something very similar but you know it's different and your equation will look different.

Laura says, "I'm thinking of a number.

When you add 10 and multiply by 5 you get 35." So draw a bar model for Andeep, draw a bar model for Laura and write their respective equations.

Pause and do that now.

Question two.

Write equations for the bar models that Jun and Jacob have drawn.

If you can see more than one way to write the equation, you should do so.

Pause and do that now.

Feedback time.

Write an equation and draw a bar model for Andeep and Laura's number problems. So Andeep is thinking of a number, the unknown number will call x when you multiply it by 5 and add 10, you get 35.

So x multiply by 5 + 10 = 35, but I hope you didn't leave it written like that.

I hope you simplified it to 5x + 10 = 35.

In Laura's case, her unknown number had 10 added to it before multiplying by 5 to get 35 and you know we won't leave it written like that.

We'd write it more simply as 5 lots of x + 10 being equal to 35.

In terms of a bar model, Andeep should have five x's and a 10 in the top row and it being equal to 35, whereas Laura's will have 5 lots of x + 10.

Your bar models should look like those.

For question two, writing equations for Jun and Jacob's bar models.

I spotted that June's bar model on the top row is 3 lots of y + 5.

Now you wouldn't leave it written like that.

You'll write it more simply as 3 bracket y + 5 = 30.

I said if you can see more than one way to write these equations, you should, and you can see more than one way.

I can see in that top row three y's and three 5's, so I could call it 3 lots of y + 3 lots of 5 being equal to 30.

We will of course simplify that to 3y + 15 = 30.

Jacobs, I can see y + 3, one, two, three, four, five times.

So, y + 3 in brackets multiplied by 5, which we'll write more simply as 5 lots of bracket y + 3.

You can also see the expanded version.

I can see 5 lots of y, I can see 5 lots of 3, but we wouldn't leave it written like that.

We'd write it as 5y + 15 = 30.

Most commonly, you'll only see or write these in these simplified forms. Sometimes we'll want it in the bracket.

Sometimes we'll want that bracket expanded out.

Onto the second half of the lesson now.

Effectively using brackets in equations.

Izzy is expanding a bracket and spots that she has made an error.

She turns 4 lots of bracket y + 2 = 20 into 4y + 2 = 20.

Can you point out her error before she corrects it? Well spotted, it was the 2.

What went wrong? Well, she didn't multiply every term inside the bracket by 4.

4 lots of bracket y + 2 is y + 2 + y + 2 + y + 2 + y + 2.

I can see there 4 lots of y and 4 lots of 2.

That would become 4y + 8, 4y + 8, not 4y + 2.

Another really important common error to look out for.

It's important that we multiply both sides of an equation by the same value at the same time.

Here's an equation that you'd be very familiar with.

8 + 2 = 7 + 3, number 1's to 10.

10 on the left hand side, 10 on the right hand side.

If I multiply both sides but by different multipliers, this will all go wrong.

2 lots of my left hand side, 3 lots of my right hand side.

So I now have 2 lots of 10 on the left hand side, 3 lots of 10 on the right hand side, 20 = 30.

No, it doesn't.

We've lost equality.

We no longer have an equation.

When we're manipulating equations, it's really important that we multiply both sides of an equation by the same value.

If I do that, multiply both sides by the same value, make the multiplier 2.

I have 2 lots of 10 on the left hand side, 2 lots of 10 on the right hand side, 20 = 20.

Here are, we still have an equation.

We maintained equality.

It's also important, we multiply every term inside the brackets in order to maintain equality.

We saw in Izzy's example a moment ago, she had a bracket on one side of the equation and she didn't multiply every term inside the bracket and it went wrong.

Let's have a look at what happens when we multiply both sides by 2.

We're gonna multiply every term inside the bracket.

So on the left hand side, 2 lots of 8 + 2 will become 2 lots of 8 and 2 lots of 2.

On the right hand side, 2 lots of bracket 7 + 3, that's gonna become 2 lots of 7, 2 lots of 3.

I've multiplied every term inside the bracket to maintain equality.

How do I know I've still got equality? 'Cause 2 lots of 8 is 16, 2 lots of 2 is 4, 2 lots of 7 is 14, 2 lots or 3 is 6 and that makes 20 = 20.

I still have an equation because I multiplied out those brackets correctly.

What if we didn't multiply out correctly? Start with that exact same simple equation, multiply both sides by 2 and then if I make the accidental mistake not multiplying every term, I do 2 lots of 8 but not 2 lots of 2, I do 2 lots of 7 but not 2 lots of 3.

Can you see those errors? I didn't multiply every term inside the bracket, so now we no longer have an equation.

We've lost our equality.

That was a deliberate error to demonstrate that this is no longer an equation.

Make sure you multiply every term inside the brackets when you're manipulating equations.

Quick check, you've got that now.

When manipulating equations, which of these are important? That we multiply both sides of the equation by the same value? If we're expanding brackets, we multiply every term inside the bracket or we draw a visual representation? Which of those are important? Pause and have a think.

I hope you said option a is important.

It's really important, we multiply both sides of the equation by the same value to maintain equality in our equation.

I hope you said b, is really important.

We must multiply every term inside the bracket if we're expanding brackets.

For c, that's all for debate really, we don't have to do that, but it can support our understanding of what's going on with the mathematics.

Be your choice whether you use a visual representation to support your learning in algebra.

Izzy is experimenting with multiplying both sides of the equation she is solving.

She is solving y + 1 = 3y + 11 and she adds a -y to both sides.

She adds -11 to both sides and she finds a solution, y = -5.

Pretty standard equation solving there.

So she asks the question, "Will I get the same solution if I multiply both sides by 10 or will I get -50 as a solution instead?" What does she mean by that? Well, if she didn't start with y + 1 = 3y + 11, she started with 10 lots of bracket y + 1 = 10 lots of bracket 3y + 11.

Will that affect the solution? What do you think? Is it gonna be the same solution? Why it cause -5 or will it be y = -50, <v ->5 multiplied by the 10?</v> What do you think? Pause and tell the person next to you.

I hope you said it won't matter at all.

The fact she's multiplied both sides of the equation by 10.

Despite multiplying both sides, we can see through substitution that the solution is still -5.

If I substitute this, y = -5 into that equation.

So instead of 10 lots of bracket y + 1, 10 lots of bracket -5 + 1 and substituting on the right hand side as well.

I can simplify that to 10 lots of -4 on left hand side, 10 lots of bracket -15 + 11 on the right hand side, which would simplify to 10 lots of bracket -4 on the right hand side.

Can you see we've still got equality by using the solution, y = -5, <v ->40 = -40, still got equality, still an equation,</v> it didn't matter.

We could multiply both sides of that equation by anything and it will still give us the same solution.

Why have I told you that? Why have I bothered to demonstrate to you that we can multiply both sides of an equation? When would we need to do that? It's useful in moments like this.

1/2x + 10 = 4x - 4.

Okay, we can solve this.

Let's add +4 to both sides 'cause that will make it a little more simple.

1/2x + 14 = 4x.

What do we do next? Let's add -1/2x to both sides because then we'll isolate the unknown term on one side, 14 = 3 1/2x.

Where do we go from here? I need to divide through by 3 1/2, so 14/3 1/2 = x, is that the simplest where I can write that answer? No, x = 4, 14 divided by 3 1/2 is 4x = 4.

So, it works, we can solve it that way, but is there something we could have done to make this more efficient to solve? I hope you are screaming at the screen now.

Yes, absolutely.

We could multiply both sides of that equation.

It was this which caused the problem throughout the 1/2x, but I hope you notice the multiplicative inverse of a 1/2x is to multiply by 2, so multiply both sides of the equation by 2.

When we do that, 2 lots of a 1/2x is just x, so we multiply out those brackets.

We've now got the equation in the form x + 20 = 8x - 8.

In this form, it's much more simple to solve and we mathematicians like simplicity in our lives.

From here, we can manipulate it to 7x being equal to 28, x is equal to 4.

Oh, look the exact same solution but far more easily achieved.

Let's check you've got that.

What might be a useful next step in solving this equation? 7x - 1 = 1/5x + 3.

Looks tricky in its current form, but one simple step is gonna make it look a whole lot easier.

But what is that simple step? Is it to multiply both sides of the equation by 2? To multiply both sides of the equation by 3 or to multiply both sides of the equation by 5? What do you think? Pause and tell a person next to you.

I hope you said option c.

Multiply both sides of the equation by 5.

Why 5? Well, it's the multiplicative inverse of 1/5.

x5, the inverse, puts it in a format which is much easier to solve.

When we expand out those brackets on the left hand side, we have 35x - 5 and on the right hand side x + 15.

In that form, it's so much easier to solve.

It is also useful for solving rational equations.

Multiplying both sides of this equation is gonna make it look far more simple.

The multiplicative inverse of divide by x is to multiply by x, so multiply both sides of the equation by x algebraically, that's what you'll write.

Why do we do that? 'Cause on the left hand side, we've got one divided by x multiplied by x.

That'll just leave us with 1 on the right hand side.

Once we've got it in the form 1 = 3x, it's much more simple to solve.

Let's try another example.

1/x = 2/5.

The inverse of divide by x and the inverse of divide by 5 is to multiply by 5 and x or multiply by 5x.

So, multiply both sides by 5x, we expand that bracket, we get 5x/x = 10x/5.

The left hand side cancels down 5x/x is just 5.

The right hand side will cancel, 10/5 is 2, that becomes 2x.

And then in that form much more simple to solve.

This is where multiplying both sides of the equation by the same thing is a really powerful strategy when working with equations.

Another quick check.

What might be a useful next step in solving this equation? 14 = 3/y.

There's one option for you.

Option b, there's option c.

Which do you think is a useful next step? Pause and tell the person next to you.

Did you say b? Multiply both sides of the equation by y, if you did, absolutely that's a really good next step.

To multiply by y is the inverse of divide by y.

It'll put in a format which is much easier to solve.

Multiply the left hand side by y, we'll get 14y, multiply the right hand side by y, we'll just be left with 3.

We can solve from there.

Part a was not correct because did you notice only the right hand side was multiplied by y and we know we need to multiply both sides by the same thing to maintain equality.

Practise time now.

I'd like you to spot the errors in these manipulations of equations.

Once you spot them, put a little circle around them and then write a sentence to explain each error.

Pause and do that now.

Question two.

I'd like to fill in the missing parts of these manipulations of equations.

Three equations for you.

Just a few things missing.

Can you fill those gaps incorrectly? Pause and try this now.

Some feedback.

Question one, part a.

Spot the error.

The error was there.

Why is that an error? Because the multiplication only happened on one side of the equation and we know if we wanna multiply the left hand side by 4, we must also do the same thing to the right hand side.

That's what went wrong in the first one.

For b, there's the error.

A very common error, and I hope you spotted it.

All terms inside the bracket need to be multiplied.

2 lots of y is 2y, but 2 lots of 3 is 6, 2 lots of 4y is 8y but 2 lots of -6 is -12.

We need to multiply every term inside those brackets.

For part c, there's the error.

Both sides need to be multiplied by the same term.

You can't multiply one side by 4 unless you're multiplying the other side by 4.

They have to be the same multiplier.

For part two.

Fill in the missing parts of these manipulations.

For part a, I hope you noticed we'd multiply both sides by 4.

The inverse of 1/4 to multiply by 4.

Once expanded, we get 8e + 12 = e.

That form much easier to solve from there.

For part b, I hope you spotted we'd multiply both sides of the equation by 2 1/2d, d divided by 2.

What's the inverse of dividing by 2? Multiplying by 2.

Once expanded, we'll end up with the equation, d + 8 = 6d - 2.

Part c, little trickier.

Essentially the same principle.

What's the multiplicative inverse of divide by f, multiply by f and then when you expand those brackets, we get 7 on the left hand side, 18f on the right hand side.

It's in a much easier form for us to solve from there.

As the end of the lesson now.

In summary, brackets are significant in equations.

If manipulated incorrectly, they can cause us to lose equality.

But if manipulated well they can help us to transform complex looking equations into simple ones.

For example, 2e + 3 = 1/4e.

If we multiply both sides through by 4, we'll turn that equation into 8e + 12 = e.

A far easier form to work with.

Hope you enjoy today's lesson.

I hope to see you again soon for more mathematics.

Goodbye for now.