Lesson video

Hello, thank you for joining me today.

My name is Ms. Davis and I'm gonna help you as we work through this lesson.

I'm really looking forward to working alongside you.

There's lots of exciting things that we are going to have a look at.

I really hope there's bits that you enjoy, and bits that you find a little bit interesting.

Maybe there'll be some things you haven't seen before as well.

Let's get started then.

Welcome to this lesson on securing understanding of equality.

By the end of the lesson, you would appreciate how to maintain equality between 2 statements.

There are several keywords that we're going to use today.

Pause the video and just check that you are happy with those definitions before continuing.

We're gonna start by looking at maintaining equality in bar models.

Bar models can be used to represent expressions with variable terms, constant terms or both.

So here are 5 different bars, all showing expressions with either a variable term, constant terms, or both variable and constant terms. The first one could represent 4x + 5, y + 1, 7 + 8, 2 + 2 + 2 + 2 + 2, and then a + b + a + b, or that's equivalent to 2a + 2b.

Those are all expressions.

When the bars are equal length, we can write an equation.

So you may have seen a bar model before like this because the top and the bottom bars are the same length.

We can write this as 4x + 5 is equal to 10 + 8.

This bar model represents the prices in pounds of two competing bike hire companies.

When a bike is rented for h hours.

The top one represents a company that charges 2 pounds for the number of hours + 10 pounds standard charge.

That's why we've got two hs, 2 lots of the number of hours + 10.

The second company charges four pounds per hour + 4.

That's why we've got four hs because it's 4 times the number of hours + 4.

If you're not sure about that, pause the video and just read through and make sure you're happy with what the bar model is representing.

Because the number of hours a bike is rented for is variable, we could have drawn the bar model like this instead.

Again, we've got 2 lots of the number of hours + 10, and then 4 lots of the number of hours + 4.

Can we write an equation for either bar model? Pause the video, what do you think? Well then, she said, "No, we can't write an equation because the length of the top and the bottom bars are not equal." So bar models can be used in different ways in mathematics.

We can use them to represent expressions, we can use them to do work with things like ratio.

We can also use them to form equations, but it's important to know when we can do that, and when we can't do that.

For what number of hours would the company's cost the same? I don't need an answer.

I just wanna know, can we draw a bar model for this situation? What do you think it would look like? Right, we need the companies to cost the same, so we need the 2 hs + 10 to be the same as the 4 hs + 4.

Can we now write an equation for this situation? Yeah, we can 'cause the bars are the same length.

We are now looking for a specific value of h, which makes the two companies equal.

So like we said before, bar models can be used in lots of contexts.

We can only write an equation, if we know that the bars are equal, if there's some sort of statement of things being the same.

Our equation could be 2h + 10 = 4h + 4.

The bar model below shows the expressions 8 + 2, and 7 + 3.

Those expressions are equal, they both equal 10.

So we can write this as 8 + 2 = 7 + 3.

Because both equal sides of an equal sign are equal, we can also write that the other way around as 7 + 3 = 8 + 2.

When talking about equations, it's useful to be able to refer to the different sides of the equation.

So we refer to the expressions as the left and right-hand side.

So in 8 + 2 = 7 + 3, the 8 + 2, that whole expression would be referred to as the left-hand side of the equation.

The 7 + 3, the whole expression could be referred to as the right-hand side of the equation.

Just be aware of that notation as we move through.

Let's now have a look at what would happen if I add 1 to the left-hand side of this equation.

What happens to my values? Pause the video.

What do you think? Well done if you notice that 8 + 2 + one is now 11.

This is no longer an equation.

As I haven't maintained the equality of both sides of that original equation.

I can no longer write an equal sign between those 2 things because they're not equal anymore.

You can see now in my bar model, if I've added 1, they're no longer equal.

Have a look at what I've written this time.

What have I done differently? Have I maintained equality? What do you think? Well done if you spotted that, yes, this time because I've added 1 to both sides of my equation, they've remained equal.

The left-hand side now = 11 and the right-hand side = 11.

You can also see in the bar model I've added 1 to the top bar and the bottom bar, and now the bars are the same again.

To maintain equality, if a value is added to one side of an equation, it must be added to the other side of the equation.

Let's try with some variable terms. I would write that equation as a + 5 = 17.

What would we have to do to maintain equality in this bar model now? Look at what I've done.

What would we have to do to maintain equality? Well I dunno if you said that we'd have to add a to the other bar as well.

Let's have a look at what this would look like in an equation.

So we could write that as a + 5 + a = 17 + a.

We've added a to both sides.

Or we could simplify that by writing it as 2a + 5 = a + 17.

Let's have a look at some other operations and see whether they maintain equality.

What do you think to this one? Well I dunno if you said no.

Different values are added to each side.

C is a variable term.

We don't know what c equals.

It could equal 3 but it might not.

So adding c and adding 3 is not the same thing.

What do you think to these operations? Would they maintain equality? Yes, the same values have been added to both sides of the equation.

We've added a b and we've added a 5, so the bars are still the same length.

What do you think to this one? No, different values have been added.

In fact we haven't drawn our bar model properly because it's a bit misleading.

Adding -6 is the same as subtracting six, so really our 2 bars should look like this.

The top bar is a + 5 + 6.

The bottom length is 17, subtract 6 which is now smaller.

You can see that the length of the bar is shown by the arrows are no longer the same.

A different value was added to each bar.

What about this time? Do you think those operations would maintain equality? And then I'll show you the bar model.

Yes, the same value has been added to both sides.

So you can add a negative value, as long as you add the same one to both sides of the equation you will maintain equality.

Time for a check.

I'd like you to complete the bar models to maintain equality, off you go.

Let's have a look at our answers.

So we've added 3 to the top bar, so we need to add 3 to the bottom bar.

The second one we've added y to the bottom bar, so we need to add y to the top bar.

And C, we've added x to the top bar so we need to add x to the bottom bar as well.

All right, let's return to 8 + 2 = 7 + 3 that we were working with before.

What operation have I done to the top bar this time? Pause the video.

What do you think? I think you could have come up with two answers.

You could have said that we've added 8 + 2, or you might have noticed this is the same as doubling the length of the bar.

What could I do to maintain equality? Pause the video.

Think about what you would do.

So we could add 8 + 2 to the bottom bar, or we could double the bottom bar, which is the same as adding 7 + 3, so that's also an option.

So if we doubled the top and bottom bars, let's look at what that looks like as an equation.

So we have doubled the entire left-hand expression, so I've put an 8 + 2 in a bracket and doubled it, and we've doubled the entire right-hand expression.

So again, I've put 7 + 3 in a bracket and doubled the whole thing.

We need to use brackets as it's the whole expression that's been multiplied by 2.

And remember the convention is to write the multiplier in front of the brackets.

Just to check, 8 + 2 is 10.

So I've got 2 lots of 10.

7 + 3 is 10, so I've got 2 lots of 10, on the right-hand side.

Both sides of the equation = 20, so we have maintained equality.

What happens if we only double one term in the expression? What about if I double the 8? Can we maintain equality by doubling one term on the other side? What do you think? This is really important.

No, we can not maintain equality by doubling one term on the other side.

We can try it out.

If we double the 7, the left-hand side is now equal to 18, but the right-hand side equals 17.

They are no longer equal.

Can we maintain equality by doubling the entire expression on the other side? No, and this is really important.

When we're multiplying an expression, we need to multiply the whole expression on each side of the equal side.

It doesn't work to just multiply one term in the expression, it needs to be the whole expression.

This is the same with expressions with variable terms. What operation have we done to the top bar this time? Well, if you said we multiplied by 3, I can do the same to the bottom bar.

So this looks like 3 lots of 2p + 2 = 3 lots of q + 7.

If we group like terms in our bar model, so if we group all the variable terms together, and all the concepts together, we can see how every term has been multiplied by 3.

That means we can also write that as 6p + 6 = 3q + 21.

Remembering that it's every term in the expression that's been multiplied by 3.

We can do the same for division.

Let's return to our 8 + 2 = 7 + 3.

If we divide the left hand of our equation, let's see what happens.

We need to divide the whole expression by 2.

This is the same as dividing each term by 2.

Let's just check that's the case.

8 + 2/2 = 10/2, which would be 5.

Let's divide the terms separately.

8 divided by 2 + 2 divided by 2, or the 8 divided by 2 is 4.

2 divided by 2 is 1, 4 + 1 also 5.

Just useful to be aware that you can divide the whole expression by 2, and that's the same as dividing each term by 2.

Make sure that we divide the whole of the right-hand side by 2 as well to maintain equality.

This is the same for expressions with variable terms. So we have the equation p = 12.

If I divide p by 3, I can write that as p over 3 or a p/3, so I can split my bar into thirds.

So there you go, I've got a p/3, or p over 3.

To maintain equality I need to do the same with my right-hand side, which is the same as splitting the bottom bar into 3 'cause I'm doing a 1/3 of 12 or 12 divided by 3.

You could also write that of course as 2b.

So p over 3 = 12 over 3.

We have maintained equality.

I would like you to have a go at matching the bar models to the equations.

Be careful this time because some of them look like they are the same, but it's a different operation that has been applied to both bars.

So take your time, see if you can get those correct.

So the top one we have doubled the top and the bottom.

2 lots of 2x + 3 equals 2 lots of 17.

The second one we've added 2x + 3 to both the top bar and the bottom bar.

That's the third one down.

So the third one down, we've added x + 3 to the top bar and the bottom bar.

And then the bottom one matches up with the bottom equation.

We've multiplied the top and bottom bars by 3.

Well done if you've got all four of those.

Time for a practise.

I would like you to complete each bar model, so it maintains equality.

So the bars in black were the original equation, 2a + 3 = b + 8.

And then the bars in purple, an operation has been performed to one bar.

I want you to maintain equality and fill in the other bar.

Off you go.

Let's have a go at question 2.

The bar models this time are not drawn to scale.

Given that 2c + 18 = 3x + 21, which of the bar models will maintain equality, and therefore should be the same length? It's important here that you are explaining your reasoning.

Give it a go.

Let's look at our answers.

So for the first one, 5 has been added to the top bar, so we need to add 5 to the bottom bar.

For B, a has been added to the bottom bar, so you also need to add a to the top bar.

For C, you had choices.

You could have said that the top bar has been multiplied by 3.

Therefore you can multiply the bottom bar by 3, so you'll have another b + 8 and another b + 8.

Or you could have said that 2a + 3, add 2a + 3 has been added to the top bar.

So you can add 2a + 3 and another 2a + 3 to the bottom bar.

For D, b has been added to the bottom bar, so it also needs to be added to the top bar.

For E, you might have noticed that it is the same thing as doubling 'cause another 2a and another 3 have been added to the top bar, so that's the same as doubling the top bar.

You could have then doubled the bottom bar.

That is one option.

You might also have chosen to add 2a + 3 to the bottom bar as well.

So the one on the screen is just one possible option.

Let's talk about the second set.

Remembering that it was our explanation that was really key here.

So A, no they should not be the same length.

18 is added to the top but 2x is added to the bottom.

B, yes, they should be the same length.

2c has been added to both.

C, yes, 2c + 18 has been added to both, so these should be equal.

D, no, 2c has been added to the top and 3x added to the bottom.

You can't say that the whole expression has been doubled either.

E, yes, both expressions have been doubled so they should be equal.

And the last one, yes, x has been added to both so they should be equal.

Well done, we've talked lots about how to maintain equality in bar models.

Now I'm gonna look at maintaining equality in equations.

So let's apply everything we've talked about so far with equations.

Alex is a years old, Sam is b years old.

How old will they be in 5 years time? How could we write that? Alex's age could be written as a + 5 and Sam's as b + 5.

If Alex and Sam are the same age now, will they be the same age in five years time? Were they the same age five years ago? What do you think? Yes, if they're the same age now.

So if a = b, then a + 5 = b + 5, so they'll both be the same in five years time.

And a subtract 5 = b subtract 5.

So they were both the same age five years ago.

We can write that as in general if a = b, then a + c = b + c.

Let's have a look at how this works.

Here is an equation.

We can add 5 to both sides of the equation and equality is maintained.

We have changed the expressions on both sides of the equal sign, but because we've added the same value to both, they're still equal to each other.

We can simplify by collecting like terms. So if you look at the constant terms, <v ->7 + 5 is -2.

</v> So I can write the left-hand side as 2x - 2.

On the right-hand side, 3 and 5 are constant terms. 3 add 5 is 8.

So the right-hand side is y + 8.

Remember we can only collect terms if they are alike.

We can add any value to the left and the right-hand side, and as long as we add the same value to the expressions on both sides, the expressions will remain equal.

You can see that this time I've added 5.

2 and again I can collect the like terms and I get 2x + 3.

2 = y + 13.

2.

That must still be true 'cause we've added the same value to both sides of the equation.

This is true for any value we wish to add, including negative values and variables.

How would I maintain equality this time? Pause the video.

What do you think? Yeah, of course I've got to add -5 to the other side as well.

We can collect like terms, so I get 2x - 12 = y - 2.

Have a look at this one.

How could I maintain equality this time? Well done, if you spotted that I've added the expression y - 2, I could do that in 2 steps.

I could add y to both sides of the equation and then add -2, or I can just add the whole expression.

So if we do that now I get 2x - 7 + y - 2.

And again you might choose to collect like terms and rearrange.

Let's have a check then? Can you fill in the blanks to maintain equality? Let's have a look at our answers.

So A, you should have + 5 for the right-hand side.

B, you need to have subtract 2, or add -2 to the right-hand side.

C, you need to add another a and add a 3.

If you've collected right like terms and written that as 2a + 3 absolutely fine.

And D, you need to add a B and add a -6.

So you could write that left-hand side as a + b - 6, well done.

Alex is a years old, Sam is b years old.

They both have a brother who is twice their age.

How could we write the ages of their brothers? Pause the video.

What do you think? We could write Alex's brother's age as the expression 2a, and Sam's as 2b.

If Sam and Alex are the same age now, will their brothers be the same age? Yes, of course.

If a = b, then 2a = 2b.

We can multiply both sides of an equation by the same value to maintain equality.

So we can write this in general, as if a = b, then ac = bc.

Let's look at an example.

Remembering that we need to use brackets 'cause we're multiplying the whole expression.

And we could expand the brackets if we wished to give an expression without brackets.

If we do that, make sure you multiply each term in the bracket by the multiplier.

Alex says, "So I can multiply my equation by any value I like." Have a look at the right-hand side.

What value do you think Alex has tried to multiply by? He's tried to multiply by 0.

1 or at least that's what it looks like.

He has gone wrong somewhere though.

Where has he gone wrong? Pause the video.

See if you can explain it.

He hasn't multiplied the whole of the left-hand side or the whole of the right-hand side.

He's only chosen to multiply the variables.

That doesn't work.

We tried it earlier with the 8 + 2 = 7 + 3.

Let's have a look.

He's gonna give it a go.

Has he maintained equality this time? Yes, he's now managed to maintain equality by multiplying both sides of the equation by the same thing.

"Writing the step with brackets really helps!" And I agree with Alex here.

I wouldn't skip that step out in your working.

Write the step in with the brackets, and then if you want to expand those brackets you can do.

Sam, I have written this equation.

Can I divide it all by the same value and maintain equality? Well let's try, let's try dividing the equation by 4.

So 4x + 1 divided by 4 = 2y + 6 divided by 4.

If we want to at this stage we can divide the individual terms by 4.

So we could write that as 4x divided by 4 + 1 divided by 4 = 2y divided by 4 + 6 divided by 4, which looks complicated at the moment but some of these might simplify the left-hand side that's the same as saying x + a 1/4.

It's 4x divided by 4 is x.

On the right-hand side you could write that as a 1/2 of y, or y over 2 + 3 over 2, or 1.

5.

"Isn't divided by 4 the same as multiplying by a 1/4.

Does this give the same answer? Well let's try it.

So a 1/4 lots of 4x + 1 = a 1/4 lots of 2y + 6.

If we expand the brackets, then yeah, we get exactly the same thing as we did before.

Sam is correct, divided by 4 and multiplying by a 1/4 are the same thing.

Different ways of writing expressions can be useful at different times.

So you might decide one of those looks neater than the other.

You might decide one of those is more useful to you depending on what you are then going to do with your equation.

Generally fractions are easier to work with than decimals.

So you'll see that I've left them as fractions rather than converted to decimals.

Alex and Sam have written a new equation each.

Alex says, "I think we have written the same equation." Is he correct? Yeah, a 1/5 of x and x divide by 5 are equivalent.

We're gonna try and multiply them by 3.

Let's see what happens.

So multiplying the whole of the left-hand side, so a 1/5 times 3 is 3/5, so we've got 3/5x + 1 x 3, which is 3.

Multiply the right-hand side by 3 and we get 18.

Let's do the same with Sam's and see if it's the same thing.

So we've got 3x over 5 + 3 = 18.

Every term's been multiplied by 3, and these things are the same.

3/5x and 3x over 5 are the same thing.

Let's try and multiply them by 5.

So multiplying every turn by 5.

5x over 5 or 5 over 5x.

It's the same as 1x isn't it? 5 over 5 is 1.

So that simplifies to 1x + 5 = 30, and Sam's is exactly the same.

Notice that multiplying by 5 gave us expressions with integer constants and integer coefficients of x this time.

It might be useful later on when you're doing work with algebra to think about what operations give you integer values.

Time for a check, which of these maintain equality? What do you think? Well done.

There was lots there.

Let's have a look at it.

It's A, D, and E.

A.

Both sides have been multiplied by 7.

Let's look at B.

A 1/4 of 6 is not 4, so we haven't multiplied both sides by a 1/4.

C, the right-hand side has been multiplied by 3.

The left-hand side has not.

The one, the constant term of -1 would have to be multiplied by 3 as well.

D, yes, both sides have been multiplied by 9.

Well done, if you spotted that one.

And E, yes, both sides have been divided by 2.

F, no, the entire left-hand side has not been divided by 2.

It was only the x term.

We also need to divide the constant term by 2, if we're dividing the right-hand side by 2.

There was loads of new information there, so time to have a bit of a practise.

I would like you to fill in the blanks in each equation.

so that they maintain equality.

Take your time and then we'll look at the answers.

Well done, so for a, because the left-hand side we have added 8.

The right-hand side we need to add 8.

For the second part, because the left-hand side we have subtracted 2 from the original, we need to subtract 2 from the original, so we get b - 2.

For b, we should have a + r + 5 on the left, then underneath a + r + 3 on the left.

And C, there are options here if you've seen that it's been multiplied by 3, you could have multiplied the left-hand side by 3 as well.

You might have said that the right-hand side we've added 2b.

So if you wrote the left-hand side as a + 2b, that is another way of doing it as well.

For D, we have subtracted an x or added a -x, so we need to do the same to the right-hand side.

For E, we've multiplied the right-hand side by 8, so we need to do the same to the left-hand side.

Make sure you've got your expression in a bracket, so 8 bracket 3x + 4.

For F, you might have noticed that we have added -5, so you can write that as y + one add -5.

If you've simplified and written that as y - 4, absolutely fine.

For G, we have divided by 5, so 10 divided by 5 or you could have written that as 2.

H, there are options for h.

So if you thought about how 10 triples to get 30, you can multiply the left-hand side by 3.

That's either 3 bracket p + q, or 3p + 3q.

The other option is you could have seen that we've added 20 to the right-hand side to get from 10 to 30.

So you could have written that as p + q + 20.

So 2 different operations get from 10 to 30.

You do either to the left-hand side to maintain equality.

And finally p + q = 10.

We have doubled the left-hand side, so you could double the right-hand side and get 20.

Or you could say that we've added p + q to the left-hand side, so you can add p + q to the right-hand side.

Well done for all your hard work today.

Quick summary of what we've looked at.

If two expressions in a bar model are equal in length, then they're equal in value.

That means we can write an equation.

Operations could be performed on the bars to maintain equality, and the same thing we can do with an equation.

Adding the same value to the left and the right-hand side, maintains equality, multiplying the left and the right-hand side by the same thing, maintains equality.

Fantastic work today and I look forward to seeing you again.