Lesson video

Thank you for joining us in today's lesson.

My name is Ms. Davies and I'm going to be helping you every step of the way today.

We've got lots of exciting algebra to look at, so let's get started.

Welcome to today's video.

We are going to be checking and securing your understanding of using the distributive law with algebraic terms. If this is not something that you are already really confident with, then you are definitely going to be confident by the end of this lesson.

We're going to look at an example of using the distributive law during the lesson.

If you want to spend a little bit more time just checking through it yourself, feel free to pause the video and have a read at these examples.

So we're going to start by multiplying expressions by a term.

So here is a reminder of how the distributive law works using numerical values.

Let's say I wanted to do five multiplied by 38.

I could split my 38 into 30 add eight, and that's the calculation you can see on the screen.

I can show this with an area model.

So I'm going to do five multiplied by 30 add eight.

And I can split that into five multiplied by 30 and five multiplied by eight.

If I think about five multiplied by 30, I've got five lots of 30.

We've represented that with a bar model.

And I can have five lots of eight, again represented by our bar model.

And hopefully you can see now that that gives us five lots of 38 in total.

We can calculate those separately and sum the result.

What we've done is we've calculated the product of five and 38 by splitting a 38 into 30 and eight and multiplying each term in the bracket by each term outside the bracket.

Let's have a look now then with algebraic terms. If you have algebra tiles, you can get those out and use them to explore this idea.

Otherwise you can draw algebra tiles.

As we work through, I'll show you how that's done.

For example, we're going to start with five lots of x plus four.

This means the expression in the brackets multiplied by five.

So what we can do is we can draw a representation of x plus four and multiply that by five, want five lots of that.

If we make five copies, we can then see that in total, we have five lots of x, which is 5x and five lots of four, which is 20.

Just going to bring that back with the area models because we're going to use those loads today.

So we've got five on the vertical height of our rectangle and we split x plus four into x and four.

We can see that five multiplied by x and five multiplied by four gives us 5x plus 20.

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

We're going to just remind ourselves how this works with negative terms. And again, algebra tiles can be really useful to look at these.

So we've got three lots of x minus four, but we know that x minus four can be written as x plus negative four.

So with my algebra tiles we could have x and negative four and we can make three copies 'cause we want three lots of x minus four.

We can see clearly then that we've got three lots of x and three lots of negative four or 3x minus 12.

Let's show this again with our area model.

So we're looking at three lots of x, which is 3x, and three lots of negative four, which is negative 12.

A quick check then.

I'd like you to complete the area model to show the expansion of four lots of 2x plus 10.

Think about what's going to go on each part of your diagram and how you could write that in expanded form.

Off you go.

Fantastic.

So you might have started by writing a 10 in the yellow bar or spitting it into 10 ones.

Or you might have just written the 2x and the 10 underneath the bars to show the length of our area model.

We also need four on the height of our area model and then we can do four lots of 2x, which gives us 8x, and four lots of 10, which gives us 40.

Our final expression is 8x plus 40.

So let's have a go at two lots of three minus x.

Again, we can look at the expression in the bracket, which is three minus x.

We can build that with algebra tiles and make two copies.

Remember three minus x is the same as three add negative x.

We're going to make two copies and we can see that's equivalent to six add negative 2x or six minus 2x.

Using the distributive property and our area models, we've got a height of two and a length of three plus negative x.

We can look at two multiplied by three and two multiplied by negative x.

And that leaves us with six minus 2x, just as we had before with our algebra tiles.

Drawing out those algebra tiles for every question is not always going to be efficient.

What we can do though is use the area models and our distributive property to remind us how the multiplication of a term and expression works.

What using the area model does is it helps us remember to multiply both terms. So let's have a look at 10 multiplied by 3x plus 70.

I'm definitely not going to want to draw 70 ones and I don't really want to multiply 3x by 10 and have to draw all those x's either.

So let's just use our area model.

So I've got 3x plus 70 on one side of my rectangle and 10 on the other length of my rectangle.

I can now show 10 multiplied by 3x plus 70, by multiplying 10 and 3x which is 30x and 10 multiplied by 70, which is 700.

Definitely didn't want to draw out 700 ones.

Let's try when our multiplier is negative.

So we've got negative eight multiplied by 5x plus one.

We need to do negative eight lots of 5x and a negative value multiplied by a positive value gives us a negative value.

So 5x multiplied by negative eight is negative 40x.

Negative eight multiplied by one gives us negative eight.

In total then we have negative 40x plus negative eight or we can write that as negative 40x minus eight.

Aisha says, how will the expanded form of negative eight lots of 5x minus one look different? Let's have a look.

So let's think about what's going to happen when we multiply two negative values.

We'll set up our area model.

Put negative eight on the height, just as we have before.

But now we've got a 5x plus a negative one.

So just like before, we've got negative eight multiplied by 5x, so negative 40x.

And now for this part of our area model, we need to do negative eight multiplied by negative one.

But if we multiply a negative value by a negative value we get a positive value.

So negative eight multiplied by negative one is positive eight.

We can write that as negative 40x plus eight.

You could either write those two terms the other way around and write that as eight minus 40x.

Time for a quick check then.

Aisha has tried to expand negative three lots of 20 minus 2x.

Have a look at her area model and her final answer.

What mistakes has she made and what can you do to help her get to the correct answer? Pause the video and give this one a go.

Well done.

One of the mistakes she made was in setting up her area model in the first place.

So the top length should be 20 minus 2x or 20 plus negative 2x.

Then when she comes to do her multiplication, negative three lots of 20 should be negative 60.

She's remembered that it should be a negative value, but she's actually forgotten to do the three lots of 20.

So she should have negative 60 in that first box and then because that should read negative two on top of the second part of the area model, negative three multiplied by negative two is positive six.

So they should be positive 2xs to give positive 6x.

The final answer then would give you negative 60 plus 6x, or 6x minus 60.

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

So let's try x multiplied by x plus five.

We've got there a representation of x plus five.

Since we do not know the value of x, we cannot make x number of copies.

We don't know how many copies to make.

What we can do is we can use an area model to show the product of x and x plus five.

So there we've got x multiplied by x plus five.

We want to work out an expression for that product.

Well first, we need to multiply x and x.

So x multiplied by x is x squared.

If you've got your algebra tiles, you'll want to set this up on your desk.

Then we need to do x multiplied by five and that's going to give us five lots of x.

In total then, we have the expression x squared plus 5x.

The tiles around the edge of the rectangle were just a guide.

So we don't want to include those in part of our product.

So make sure you remove those or draw them separately to your final product 'cause they should not end up as part of our product.

I'm now going to show you an example.

I'd like you to watch carefully what I do and then you're going to have a go on the right hand side.

So we're going to write 4x multiplied by 2x minus three in expanded form.

We do that using an area model.

So I've got a height of 4x and a width of 2x plus negative three.

So let's think about what happens when we multiply 4x by 2x.

So four multiplied by two is eight and x multiplied by x is going to give us x squared.

We end up then with 8x squared and then we need to do 4x multiplied by negative three and we end up with negative 12x.

Our final answer then is 8x squared minus 12x.

I'd like you to have a go at multiplying 2x and 3x minus one.

Off you go.

Let's have a look together.

So, 2x multiplied by 3x, going to end up with 6x squared.

Then we've got 2x multiplied by negative one, so we end up with negative 2x.

Our final answer then is 6x squared minus 2x.

We've written that in an expanded form.

The expression in the bracket can contain any number of terms. We've seen some so far where there's been two terms in a bracket.

We're now going to look at what happens when there's three terms in a bracket.

So we've got x multiplied by 2x plus 15 plus negative y.

We'll look at each of those separately.

So x multiplied by 2x gives us 2x squared.

X multiplied by 15, gives us 15x.

And x multiplied by negative y.

Got a positive multiplied by negative, so it's going to be a negative product.

We write x multiplied by y as xy.

It's useful to write your variables in alphabetical order and then it's easier to spot like terms. In total, we have 2x squared plus 15x minus xy.

Let's have a try together.

So I'm going to show you one on the left hand side and then you are going to give it a go on the right.

So we're going to split our area model into seven, negative 6x and 2z.

We're going to start by doing 5x multiplied by seven, which is 35x.

5x multiplied by negative 6x.

So five multiplied by negative six is negative 30.

X multiplied by x is x squared.

So we're going to have negative 30x squared and 5x multiplied by 2z.

So five multiplied by two is 10, x multiplied by z we can write as xz.

In total, we have 35x minus 30x squared plus 10xz.

Use my example on the left hand side to help you with this one on the right.

Off you go.

Well done.

Might look tricky at first, but if you split it out into those steps it becomes a lot easier.

So let's have a look.

We've got 10x multiplied by 12, which is 120x.

10x multiplied by 5y which is 50xy.

And then we've got 10x multiplied by negative 9x, which is negative 90x squared.

Well done then if you've got that final product of 120x plus 50xy minus 90x squared.

Time to put that all to the test.

So you've got four area models and I'd like you to use them to write those expressions in expanded form.

Come back when you're happy and we'll look at the next bit.

Well done.

You're going to do the same again this time.

Use those area models to help you.

Off you go.

Final practise then.

You've had lots of skills using area models.

If you'd like to use area models again absolutely draw those on the side of your page or you might've started to move away from using the area models to expand these brackets.

Think about what is changing as you move through each question to help you avoid making any mistakes.

Off you go.

Well done.

Pause the video and have a look at the area models.

If you're not sure how these final expressions came to be.

You should have 6x plus 18, 6x minus 18, negative 6x plus 18 and 18 minus 6x.

Remember the terms can be written in either order.

Just make sure that we've got our positive and our negative values the right way around.

And for two, we've got x squared plus 10x 2x squared plus 20x, 2x squared plus 10x, and 4x squared plus 20x.

Fantastic.

Pause the video and just have a look through three.

Think about as I said before, what's changing between each question and see if you can spot how that's changed the answer.

Once you're happy with that, we'll move on to the next part of our lesson.

Well done.

So what we're going to do now, is we're going to use all those skills to look at multiplying and simplifying multiple expressions.

So all the skills that we already have, we're just going to have a look at some examples with more terms, more brackets, and more things to do.

So we can expand multiple brackets in the same expression.

For example, five times 2x plus seven plus three times x minus three.

All we need to do is expand the first bracket, so that we've got five lots of 2x plus five lots of seven which is 10x plus 35.

And don't forget our second bracket is still there.

So we've got plus three lots of x minus three.

If you make sure you do these in steps really clearly, don't try and do too much at once and present your working down the page, you are less likely to make any mistakes.

Now we can expand the second bracket.

So we've got plus three lots of x, and we've got three lots of negative three.

So we're going to add 3x and we're going to add negative nine.

Then we can group our like terms together.

So 10x and 3x are like terms because they've got the same variable and corresponding exponent.

So 10x plus 3x and we've got 35 add negative nine.

In total we've got 13x plus 26.

We've just written that in a simpler way.

Andeep wants to write the expression 2x multiplied by five minus y plus 7y multiplied by x plus one in expanded form.

It looks really complicated at the moment, but we have the skills to do this.

We just have to apply it step by step.

Andeep reckons that they will not be able to simplify the expression, as there won't be any like terms. Let's have a look at this.

So we've got 2x multiplied by five which is 10x 2x multiplied by negative y, which is negative 2xy.

7y multiplied by x, which is 7yx, but remember, writing them in alphabetical order might help us spot like terms. So 7yx can be written as 7xy and 7y multiplied by one is 7y.

Notice then that we have got some like terms. Negative 2xy add 7xy is 5xy.

So Andeep was incorrect this time.

There was a pair of like terms. Not as easy to spot to start with, but once we'd multiplied out our terms and we'd written the variables in alphabetical order, it was easier to see.

Let's do the same where we're subtracting one bracketed expression from the other.

So again, we can start by multiplying out this first bracket.

X multiplied by 3x is 3x squared.

X multiplied by negative five is negative 5x.

So you've got 3x squared minus 5x.

Now we're going to subtract this second expression.

So let's just look at the 2x multiplied by 2x minus one.

So just isolate that second bracket and that gives us 4x squared minus 2x.

So what we're going to do is we're going to subtract 4x squared and negative 2x.

So we've got 3x squared minus 5x and we're subtracting 4x squared minus 2x.

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

So 3x squared minus 4x, subtract 4x squared minus 2x is the same as adding negative 4x squared plus 2x.

If you're using algebra tiles, you can turn your algebra tiles over to get the additive inverse.

So we can now see that we're adding 3x squared to negative 4x squared to give us negative x squared and we're adding 2x to negative 5x, which gives us negative 3x.

Let's look at how we might do that without algebra tiles.

What we can do is we can separate our two brackets and we can expand each separately.

X multiplied by 3x is 3x squared.

X multiplied by negative five is negative 5x.

And then we can have a look at the second bracket as negative 2x lots of 2x minus one.

Negative 2x multiplied by 2x is negative 4x squared.

Negative 2x multiplied by negative one is positive 2x.

So remembering now that we can put that together and collecting like terms. And that gives us the same expression we had doing it the other way.

I'd like you to just pause the video and take a second.

Which method did you like the best? Can you see how the two methods are the same? And how to avoid making mistakes with those negative values.

Make sure you're happy before you move on.

Aisha has tried to expand and simplify this expression.

She's happy expanding the first bracket, but doesn't know what to do with that second bracket.

Pause the video.

What do you think Aisha needs to do? You might even want to give this a go before we look at it together.

Let's have a look then.

So expanding the first bracket, we've got 21x minus 28x squared.

And then just like Aisha says, we've left the second bracket 'cause she's not really sure what to do.

Well hang on, we're subtracting the whole of that expression.

so we're subtracting 5x minus six.

So all we need to do is make sure we subtract both values.

Make sure we're subtracting 5x and subtracting negative six.

Collect like terms like normal, so 21x minus 5x and then we've got negative 28x squared, subtract negative six, which gives us 16x, subtract 28x squared plus six 'cause subtracting negative six is the same as adding the additive inverse, so adding six.

I think Aisha just needs to have a little bit more faith in herself.

We've got all the skills, we just need to apply this to different questions.

We can do this for any bracketed expression.

Have a look at this one here.

Andeep is not sure.

Multiplying both expressions by different decimals is going to take some time.

Aisha though has got her confidence back.

She reckons she can see an easier way.

Pause the video.

What do you think Aisha has spotted? How can we use this to our advantage? Good spot if you saw that both brackets are the same and thought about how that meant that we could unitize the brackets and the multipliers can be added together.

So if we've got 4.

6 lots of eight minus 3x and we're adding 5.

4 lots of eight minus 3x, in total we're going to have 10 lots of eight minus 3x and that's going to make our question a little bit easier.

Let's have a look then.

We've got 10 lots of eight minus 3x as we've said previously, which gives us 80 minus 30x.

That's definitely a lot simpler than what we started with.

Time to have a check then.

I'd like you to have a look at these four questions and see if you can spot the mistake.

If we can spot the mistake in other people's work, it helps us spot our mistakes in our own work.

Off you go.

Let's have a look then.

First one, we did not expand our brackets properly.

We made a mistake.

We need to do five multiplied by six and seven multiplied by four.

With the second one, well done if you spotted that should be a negative 55.

With C, we should have positive four 'cause negative four multiplied by negative one is positive four.

And there's a couple of mistakes in the last one.

Seven times 12 is 84 not 94.

And if we're subtracting negative 5x, that's the same as adding the additive inverse, so adding 5x.

What you might have spotted also with that last one, is if you've got seven lots of 12 minus 5x and you're subtracting one lot of 12 minus 5x, you can actually write that as six lots of 12 minus 5x.

That might make it easier not to make those mistakes with negative values.

It's also useful to have a second way of doing it to check our answers.

I'm hoping that what you saw from that then is that by presenting our work really neatly down the page, it was easy to spot those mistakes.

And that's something I'd like you to give a go as you try these yourself.

So time to give these a go.

For each one I would like you to expand and simplify where possible.

Take your time, being careful with your negative and your positive values.

Think about how you're going to write your answers down the page so that you can check back for any mistakes.

Give that a go.

Come back when you're ready for the next bit.

A little bit of a challenge.

So all four of these expressions are equivalent to each other.

It's a little bit of a puzzle.

A, B, C, and D are all equivalent.

I would like you to fill in the missing terms. I suggest to start, you have a go at writing A in another form.

And that might help you see what you need to fill in in B and C and D to make them the same.

Have a go at that puzzle.

When you want to know the answers, come back Superb.

Just like before, I'd like you to pause the video and have a read through these answers.

Follow step by step, see if you can check back through your work and spot anything that you didn't quite get right.

Once you're happy, we'll look at that puzzle together.

Fantastic.

So let's have a look at this puzzle.

If we expand and simplify the top expression, we get 7x minus 21 plus 8x plus 26.

And that simplifies to 15x plus five.

If we have a look at our next expression, let's focus on the x terms to start with.

We're going to have two multiplied by 3x, that's 6x and three multiplied by 3x, that's 9x.

And 9x plus 6x gives us 15x, which is what we wanted.

So now we're going to work out how to get this plus five.

Well two multiplied by four is eight.

We need the next term to be negative three.

So what do we multiply three by to get negative three? It's negative one, well done if you got that.

Let's try the next one.

And we can do this without doing B.

So if you made a mistake on B, that does not mean that you're going to have made a mistake on C.

So let's have a look at our x terms again.

So we want to get get 15x.

At the moment, we've got negative 5x.

So that means our other x term needs to be 20x.

So that 20x subtract 5x is 15x.

In order to get 20x, five multiplied by 4x gives us 20x.

So that bracket there needs to be 4x minus one.

And finally, there's a few ways to do this last one.

I'm going to have a look at that 15x plus five.

And you might be able to see a relationship between the 3x plus one that I've got in my bracket and the 15x plus five.

We can factorise 15x plus five into five lots of 3x plus one.

Well now looking at our expression on the left hand side, we've got seven lots of 3x plus one, minus some lots of 3x plus one, gives us five lots of 3x plus one.

That means that remaining multiplier must be two.

Seven lots of 3x plus one minus two lots of 3x plus one is five lots of 3x plus one.

There was lots of tricky manipulation of algebra in that last bit.

So I hope you're really proud of yourself for giving that a go and for all the correct answers you got.

Well done today.

Some of that we've probably seen before and we're now really confident that we remember how that worked.

But we also extended our knowledge today as well and had a look at some trickier puzzles.

So be really proud of all the hard work that you put in and I look forward to seeing you again.

Maybe we can use these skills in another context.