Lesson video

Hi, I'm Mrs. Wheelhouse and welcome to today's lesson, on checking and securing, understanding and expanding a single bracket.

This lesson falls within our unit on algebra manipulation.

Now algebra can be extremely useful, so let's get started with our lesson and see how we're going to be using it today.

By the end of today's lesson, you'll be able to use the distributive law to multiply an expression by a term.

Now just a quick reminder, the distributive law says that multiplying a sum is the same as multiplying each addend and summing the result.

Here's an example.

Our lesson has two parts to it today, and we're going to begin by reviewing expanding a single bracket.

Here's a reminder of the distributive law using numerical values.

We are calculating five lots of 38 by partitioning 38 into 30 and eight.

So remember, we can use an area model to help us understand what's happening here.

So you can see the total length of our rectangle would be 38 and we've broken it into two parts, the 30 and the eight.

So to work out the area of this shape, I could do five lots of 30.

And then I could add onto that five lots of eight.

Well five lots of 30 is 150.

And five lots of eight is 40, meaning the result is 190.

Now we have calculated the product of five and 30 plus eight by multiplying each term in the bracket by the term outside the bracket.

You can also multiply expressions using algebraic terms. And we can use algebra tiles and an area model to see what is happening.

So for example, six lots of x + 3.

This means the expression in the brackets is being multiplied by six.

And we can see we can represent this with an area model.

We have six lots of X and we have six lots of three.

Six lots of x is 6x and six lots of three is 18.

When we multiply out the brackets like this, we call the result the expanded form of the expression.

What about with negative terms? Well, it works exactly the same way.

Remember, subtraction can be written as adding the negative value.

So four lots of x and then four lots of negative five.

And you can see that using our algebra tiles or if you like, you could have just gone straight to using an area model and you may not have needed to draw this at all.

Four lots of x remember is 4x and four lots of negative five is negative 20.

Meaning we can write this as 4x subtract 20.

It's now your turn.

Please write the following in expanded form, pause and do this now.

Did you use an area model? Did you use algebra tiles or did you opt for neither? It doesn't matter as long as you are able to reach the correct expanded form.

What we have here is three lots of three and three lots of negative x.

Three lots of three is nine and three lots of negative x is -3x.

Meaning we can write this as nine, subtract 3x or -3x add 9.

We can use the distributive law to multiply any expression by a term.

Remember, our area models don't have to be drawn to scale.

They're just there to help us make sure that we're multiplying the correct things together.

Let's look at this example.

I want 12 lots of 4x + 50.

So I can draw an area model to help me remember what's being multiplied together and to check that I have done all the multiplication I'm supposed to do.

12 lots of 4x is 48x.

12 lots of 50 is 600, so the expanded form of 12 lots of 4x plus 50 is 48x plus 600.

Notice that I'm using my identity symbol here, and that's because these two expressions are equivalent for every single value of x.

In all of these examples you've seen, we've expanded the brackets by multiplying the term outside the bracket by each of the terms inside the brackets.

We can use the same method when our multiplier is negative.

For example, negative five lots of 7x plus two.

Negative five times seven x is negative 35x.

And negative five times two is negative 10.

Meaning that our expanded form is negative 35x subtract 10.

Sofia says I want to practise examples where everything is negative.

Well, let's think about what happens when we multiply two negative values.

Negative four multiplied by negative 11 is positive 44.

And negative four multiplied by negative 7x gives us positive 28x.

Meaning that our expanded form is 44 add 28x.

Now the expression in the bracket can contain any number of terms. So far we've only looked at brackets that contain two terms. But we might see expressions like this.

I want negative three lots of 2x subtract five subtract y.

Our area model can still be used here.

When I multiply, I have negative three lots of 2x, which is negative 6x, I have negative three times negative five, which is positive 15.

And then negative three times negative y, which is positive 3y.

So our expanded form is negative 6x plus 15 plus 3y.

We can multiply an expression by a fraction as well.

For example, five sixth of 3x subtract 12.

Again, the area model can be used to help here.

Five six of 3x can be written as 15x over six.

And five six of negative 12 is negative 60 over six, but we should try to simplify as much as possible.

15 over six can be written as five over two.

So what I actually have is five over two lots of x.

And negative 60 divided by six evaluates to negative 10.

And here is my expanded form.

Time for a quick check.

Aisha asks the computer programme to find the product of negative 15, lots of 3y squared, subtract 12x add six.

And the answer she receives is 45y subtract 180x plus 90.

Why must that be incorrect? And what do you think has happened to cause this error? And then part B, please show what the correct answer should be and use an area model to support your reasoning.

Pause and do this now.

Welcome back.

So why must that be incorrect? Well each term is being multiplied by a negative, so the sign of each term should change.

The x term should now be positive, and the other two terms should be negative.

And in fact, notice that the first term should have been negative 45 times y squared, not just y.

Now, Aisha may have mistyped the equation or the programme did not understand the way she inputted the negative multiplier or the exponent.

For part B, asked you to show the correct answer with an area model.

So we should have negative 45y squared, positive 180x and then negative 90.

So we should have had negative 45y squared plus 180x subtract 90.

Well done if you corrected this.

The area model can also help to make sense of multiplying by a variable.

For example, to expand the bracket X lots of x plus five.

I can use my algebra tiles around my area model.

So I have x lots of x which is equivalent to x squared.

And then x lots of five, which is 5x.

So the expanded form is x squared plus 5x.

The tiles around the edge of the rectangle were a guide to the lengths we were multiplying and are not part of the product, so I could get rid of them in the end.

Let's use an area model to write 4x multiplied by 2x subtract -3 in expanded form.

Well 4x multiplied by 3x is eight lots of x squared.

And 4x multiplied by negative three is negative 12 lots of x.

So my result is 8x squared subtract 12x.

And Jacob says, I like this method.

Will it always work? And yes, there's Andeep, we're just using the distributed property.

I use area models all the time.

Now the same method can be applied to any expression no matter how complicated it might look.

Algebra tiles will not always be helpful here, but we can still use an area model to calculate all the terms. Let's have a look at this expression.

It looks very complicated.

But here's our area model and we can see everything we have to multiply together here.

So let's work out each of the products.

Three quarters of xy multiplied by 5x, gives us 15 over four lots of x squared multiplied by y.

We've then got three quarters of xy multiplied by negative four fifths x squared.

Which is negative 12 over 20x cubed y.

And then finally three quarters of xy multiplied by 8xy cubed gives us 24 over four lots of x squared y to the four.

Now we can do some simplifying here.

15 over four lots of x squared y.

I'm going to leave as it is, but the 12 over 20 can be simplified to three fifths.

So I've got subtracting three fifths of x cubed y and then 24 divided by four is six.

So we're adding 6x squared y to the four.

Let's do a check.

First I'm going to do one and then it'll be your turn.

I'm going to write 6x lots of seven y subtract 3x squared, subtract 2xy in expanded form.

Now remember an area model can be drawn to be helpful if you need it.

I'm going to draw one here.

So I have 6x lots of 7y.

And then 6x lots of -3x squared and then 6x lots of -2xy.

So 6x multiplied by 7y is 42xy.

6x multiplied by - 3x squared is negative 18x cubed.

And then 6x multiplied by negative 2xy is -12x squared y.

Writing this out means the expanded form is 42xy subtract 18x cubed, subtract 12x squared y.

Remember the terms can be in any order.

It's now your turn.

Write three y lots of 8y squared subtract 5x squared y plus 9z in expanded form.

Remember you can draw an area model if it helps.

Pause and do this now.

Welcome back.

Let's see how you got on.

So using my area model 3y multiplied by 8y squared is 24y cubed.

3y multiplied by negative 5x squared y is negative 15x squared y squared.

and 3y multiplied by 9z is 27yz.

Meaning in expanded form I have 24y cubed.

Subtract 15x squared y squared and then adding 27xz.

It's now time for your first task.

Question one, use the area models to write these expressions in expanded form.

Pause and do this now.

Question two, write these expressions in expanded form using the area models if you need to.

Pause and do this now.

Question three, put each term in a space to make two pairs of equivalent expressions.

You can only use each term once.

Pause and do this now.

Welcome back, time to go through the answers.

Question one, for 1a, you should have expanded form of 24x plus 96.

In B, the expanded form is 24x, subtract 96.

In C, the expanded form is negative 24x add 96.

And in D, the expanded form is 24x subtract 96.

Question two, you had to write the following expressions in expanded form and use the area models if you wish.

For 2a, you should have 6x squared plus 2x.

2b, you should have 6x cubed plus 2x squared y.

2c, negative 6x cubed y subtract 2x squared y squared plus 10x squared y cubed.

2d, you should have 4x squared y subtract 12 fifths of y squared, or you could have written that as a decimal value.

So in other words, instead of the 12 fifths, you could have 2.

4.

Question three, feel free to pause the video to check that you've put your terms where I put mine.

Remember terms can be either way round in the bracket and terms could be written either way round in the expanded form.

Well done if you got this right.

It's now time for the second part of the lesson and that's on expanding and simplifying with multiple brackets.

We can expand multiple brackets in the same expression.

For example, we could expand the first bracket, which gives us 14x squared plus 49x, and then we expand the second bracket.

The second bracket expands to give us 12, subtract 9x.

We would then group together the like terms. And simplify by adding those.

So we end up with 14x squared plus 40x plus 12.

We can expand expressions which have one bracket expression subtracted from another, like in this example, expanding the first bracket gives us 3x squared subtract 5x.

Now we need to be careful here.

We're going to isolate the second bracket.

In other words, 2x multiplied by 2x subtract one, is equivalent to 4x squared subtract 2x.

Because I'm subtracting the product, I'm then going to write this in brackets so that I'm subtracting all of 4x squared takeaway 2x.

Remember, it's important that we're subtracting the whole expression.

So just to make sure I can see this with my algebra tiles, I've got here 3x squared, subtract 5x, remember I'm taking away the following, 4x squared subtract 2x.

Now this is the same as adding the additive inverse of the expression.

So I switch my algebra tiles around.

So let's combine this, 3x squared added to negative 4x squared gives me negative x squared.

And negative 5x add 2x gives me negative 3x.

Without the tiles, you could write this as follows.

So considering it as two separate expressions, expanding each set of brackets.

So the first one would give me 3x squared subtract 5x and the second set is negative 4x squared add 2x.

And then again, I can write the expressions together and then simplify.

Let's do a check.

I'm going to expand and simplify first and then it's your turn.

When I expand the first bracket, I reach 30x squared - 15x.

Remember I'm going to add negative three lots of 2x plus three and that's to make sure that I get my signs correct.

Well, negative three times 2x is negative 6x and negative three times three is negative nine.

I'm then going to collect and simplify with my like terms. So my result is 30x squared subtract 21x subtract nine.

It's now your turn.

Please expand and simplify the following.

Pause the video and do this now.

Welcome back.

How did you get on? Well, the first bracket expands to give 8x squared plus 14x.

And now let's expand the second bracket.

I've got negative five times 4x is negative 20x and negative five times three is negative 15.

Let's now gather the like terms. This results in 8x squared subtract 6x subtract 15.

Well done if you got that right.

Now this time I'm going to have three bracket expressions and I'm going to need to expand each of them.

The first thing I've did is I've made sure that I'm subtracting the result of 3x lots of 4x subtract one.

I'm going to expand each of the brackets now.

So the first bracket gives me 2x squared plus 6x.

When I expand the second bracket, I've got 12x squared subtract 3x and then my bracket on the end I have nothing to multiply by.

But I kept the brackets in because I wanted to make sure I was taking away each of the terms. So I've got 2x squared + 6x, subtract 12x squared subtract 3x subtract 5x and subtract four.

Grouping the like terms leaves you with a result of negative 10x squared plus 4x subtract four.

It's now your turn please and simplify the following.

Welcome back.

Now you may have done this in a slightly different way to me, but as long as you followed correct steps, you'll get to the same answer.

Expanding the first bracket gives me 20x plus 24.

The second bracket I've left there because I'm just subtracting everything inside the bracket.

There's no multiplying to do first.

And then the second bracket, I've grouped it so that I'm subtracting the result of 2x lots of 3x plus five.

So I'm subtracting all of 6x squared plus 10x.

So this gives me the following, 20x plus 24 subtract 4x subtract negative five subtract 6x squared subtract 10x.

Collecting like terms and simplifying leaves a result of 6x plus 29 minus 6x squared.

Remember, your terms could be in a different order as long as the correct signers with each term you have this right.

Aisha says, I like to look for opportunities to simplify before I expand.

Which of these do you think Aisha could simplify before she expands the brackets? It's the first one, and that's because the brackets are identical.

So we can treat those brackets as like terms and add the coefficients before we do any expanding.

In the third expression because the coefficients are the same, it is possible for us to add the expressions in the brackets and then multiply the result by two fifths.

Let's consider both methods.

So for the first one, remember the terms inside the brackets were the same, so we treated the bracket expressions as like terms. This means what I'm actually doing is saying two fifths of 2x plus one, add three fifths of 2x plus one.

Well, that's the same as saying one lot of 2x plus one.

Well, that was certainly easier than expanding and then simplifying.

I agree Aisha, for the second one, we are going to be treating the two fifths as our like terms, and we're going to look at the sum of the two bracketed expressions.

So we can write this as 2x plus one plus 3x subtract one lots of two fifths.

Well that leaves us with 5x multiplied by two fifths or just 2x.

Again, that looks a lot easier than expanding and then simplifying So looking for common factors can reduce the amount of manipulation needed, but sometimes we'll see expressions where there are no common factors so we can't do this step first.

Time for a quick check.

Which of these is equivalent to the following expression? Pause the video and make your choice now.

Welcome back.

You should have picked B, the three.

So the coefficient for each bracket was the same, and therefore we could sum the two expressions in brackets and say it's that multiplied by three.

Well, 7x plus 2x is 9x and four subtract six is negative two So we end up with 9x subtract two lots of three.

Time for our final task.

For question one, please expand and simplify where possible.

Pause and do this now.

Question two, part A.

Please show that the follow expression can be written as a value multiplied by 3x subtract one and then part B.

Write your answer in expanded form.

3a, show that that expression can be written as an expression multiplied by two sevenths.

And then part B, write your answer in expanded form.

Pause and do this now.

Welcome back, answer on the screen for question one.

Please pause the video and take a moment to go through and check your answers with the ones that you can see here.

Pause and do this now.

Question 2a, show that the following expression can be written as a value multiplied by 3x subtract one, and you should have got that it's half multiplied by 3x subtract one 'cause you have three eighths plus a quarter minus an eighth.

To write this expanded form, we've got half multiplied by 3x, so that's three over 2x and then a half multiplied by negative one.

So we're taking away a half.

for question 3a, you had to show that that expression could be written as another expression multiplied by two sevenths.

So by summing what was in each of the brackets, you should end up with two sevenths of 14x subtract 21 Part B, you had to write your answer in expanded form.

Well, two sevenths of 14 is four, so we have 4x and then two sevenths of 21 is six, so subtracting six.

Well done if you've got these all right.

It's now time to sum up what we've learnt today.

The distributive law can be used to multiply an expression by a term.

The expression can contain any number of terms. The term can be negative, fractional and contain variables with exponents.

Expressions containing multiple brackets can be expanded and simplified.

And if either the term or expression are identical, these can be combined.

And if they can be combined, they do tend to make things that little bit simpler, but it's not always possible.

Well done, done a great job today checking and securing understanding of expanding a single bracket.

I look forward to seeing you for more lessons in our algebra manipulation unit.