Lesson video

Hello, Mr. Robson here.

Welcome to more maths, great choice to join me.

Let's get started with some quite beautiful algebra.

So, learning outcome for today.

That is that we can recognise when we can simplify before multiplying expressions by their respective terms. Key words: A term is a single number or letter or the product of numbers and/or variables.

Each term separated by operators, + and -.

An expression contains one or more terms where each term is separated by an operator.

For example, 5x + 3 is an expression with two terms, 5x and 3.

Two parts to this lesson.

First, we're gonna simplify with matching expressions, and then we're gonna simplify with matching multipliers.

Let's get started with matching expressions.

I'd like you to expand and simplify.

Pause this video.

In the first expression, we should have got 5x and 5 from the first bracket, 4x and 8 from the second bracket.

Those like terms, the x terms, simplify to 9x.

The constant terms simplify to 13 giving us 9x and 13.

In the second case, 9x and 9.

In the third case, 9x and 14.

We're okay with that skill? Super, but could it be done differently? What's the same and what's different about these expressions? Pause this video.

So what's the same, what's different? Well, there are the expressions being multiplied by, I can see multipliers of 5, multipliers of 4.

But most importantly, what's the same in this expression? Did you notice we didn't have a bracket containing x + 1 and a bracket containing x + 2? We had two brackets that both contained the same expression.

This is gonna be hugely useful, noticing when we've got the same expression.

In this case, the same expression, x + 1, is in both of those brackets.

So, does that mean we could have done this question differently? Absolutely, how we could have done it differently.

It comes down to something you'll be very familiar with, the notion that five metres and four metres is nine metres, five hundreds and four hundreds gives us nine hundreds.

If I said you've got five of something and four of that same thing, you have nine of that thing.

You know that, you are very familiar with that, and that's gonna be useful because it'll resonate into our algebra today.

5 lots of x + 1 and 4 lots of x + 1 gives us 9 lots of x + 1.

So, previously when I said expand and simplify, that was a lot of work.

What have we done here? Well, it appears like we've simplified and then we could expand.

Well, that's interesting.

5 lots of x + 1, we can model that just to prove this truth.

4 lots of x + 1, put those together, 9 lots of x + 1.

Can you see what's happened here? 5 lots of something, 4 lots of something, gives us 9 lots of something, where that something's kilometres, centimetres, hundreds, tens, or x + 1s.

In this case, 5 lots of x + 1, 4 lots of x + 1, 9 lots of x + 1.

So you might see it written like this: 5 lots of and 4 lots of gives us 9 lots of x + 1.

We can then expand the bracket.

We've simplified first, and then expanded.

That was much quicker than expanding both brackets and getting 5x + 5 + 4x + 4, and then gathering the x terms to 9x, gathering the constant terms to 9, and coming out with 9x + 9.

It gives you a choice.

I like to simplify as quickly as possible and keep my life as simple and efficient as possible, but you could expand and then gather the like terms. It's nice to know that we have both options available.

I'm gonna do an example and then ask you to do an example, just so you know you've got the truth and the structure of this.

I'm gonna ask you to draw a bar model to show why your example will be different to mine.

In my case, I'm gonna draw a bar model that justifies that 3 lots of bracket x + 4 and 6 lots of bracket x + 4 gives me 9 lots of bracket x + 4.

I'll start by drawing my 3 lots of x + 4.

I'm gonna draw my 6 lots of x + 4, put them together to demonstrate my 9 lots of x + 4.

Your turn, I'd like you to draw a model to show why 2 lots of bracket x + 5, 4 lots of bracket x + 5 is the equivalent to having 6 lots of bracket x + 5.

Pause this video.

I hope you drew x and a 5 twice, representing 2 lots of bracket x + 5, and x and a 5 four times representing 4 lots of bracket x + 5, and then I hope you put those two sets together to give you 6 lots of x + 5, beautiful.

And how much more simple is your expression now? 6 lots of x + 5, beautiful.

X - 7, does that matter? We've got a -7 term.

No, not at all, as long as it's the same term.

We're dealing with x - 7s.

We've got 5 lots of them and 2 lots of them.

This time, I'm not gonna draw a model.

I've got 5 lots of something and 2 lots of something, that gives me 7 lots of that something.

In this case, 7 lots of the bracket, x - 7, finished? That's right, what now? The command at the top said: Simplify and then expand.

I've simplified, how many x - 7s have I got? 7 of them.

I've written that expression more simply.

Now, we'll expand.

7 lots of x, 7 lots of -7, that'll expand to 7x - 49.

Previously in mathematics you would've seen the command words "expand" and "simplify".

This time round, the command was to simplify and then expand.

In my second example, -2 multiplier.

Hmm, what's that gonna do? The expressions are the same, we're dealing in x + 9s.

I've got 5 lots of them and I subtract 2 lots of them, that's gonna gimme 3 lots of my x + 9.

That's it simplified, expand that, 3 lots of x, 3 lots of 9, 3x + 27.

Your turn, pause this video, simplify, then expand for these two expressions.

Okay, in your first case, you have matching expressions inside the brackets, x - 3, 2 lots of and 3 lots of, that would've given you 5 lots of x - 3, and then when you expand that 5 lots of x, 5 lots of -3,, your expanded expression would reach 5x - 15.

In the second case, you had 8 lots of and you were taking away 3 lots of, that left you with 5 lots of your x + 6.

To expand that bracket, 5 lots of everything inside that bracket, 5 lots of the x, 5 lots of the 6, 5x + 30.

A little check now.

Which one of these can be simplified before expanding? Two expressions, one of them can be simplified before we expand, the other cannot.

Can you see which one can be simplified, and can you explain why the one which cannot be simplified before expanding cannot be simplified before we expand? Pause this video, say your answer aloud to your teacher, to your friend next to you, or just to me at the screen.

It was the second example that could be simplified before expanding.

I hope you spotted, the important thing about the second expression, we have matching expressions within the bracket.

What are we dealing with here? We're dealing with x + 2s, 5 lots of x + 2 and 3 lots of x + 2.

How many lots of x + 2 do you have? In the first expression, we didn't have matching expressions within the brackets, x + 2 is not equivalent to x + 3, so we're not able to simplify before expanded.

If we were simplifying and then expanding, 5 lots of and 3 lots of, that gives us 8 lots of x + 2, 8 lots of x, 8 lots of 2 would give you 8x + 16.

Sofia's asked a really good question.

"The multiplier doesn't have to be a number.

What would happen if there was a variable?" Sofia are you asking, "Instead of 5 lots of x + 2, we had x lots of x + 2, the thing outside the bracket that we're multiplying by is a variable, not a constant?" "Yes, Mr. Robson, that's exactly what I'm asking." In that case, Sofia, that's a brilliant question and it's really important that we all behave like this in the maths classroom and we interrogate maths, we ask "What if," "Will that work?" That's how we behave as mathematicians.

So Sofia's asked a brilliant question.

"What if the multiplier outside the bracket is a variable?" It helps to come back to the thing we do understand when it's a constant.

Our x + 1s in this case are being multiplied by 3, they're being multiplied by 2.

The language is important.

I've got 3 lots of x + 1, and I've got 2 lots of x + 1.

I've got 3 and 2 lots of x + 1, giving us 5 lots of x + 1.

That moment there where we bracketed the 3 and the 2, we have 3 lots of and 2 lots of x + 1.

It just so happens we can simplify 3 and 2 together to make 5.

In this case, wow, that's what Sofia was asking about.

The x + 1 is being multiplied by a, and then in the second bracket, it's being multiplied by b.

So, can we simplify this? Yes, I've got a lots of x + 1 and b lots of x + 1.

I've got a and b lots of x + 1, which we would write like that, a + b lots of x + 1.

Can we simplify the a + b? Correct, we can't, they're not like terms, a terms and b terms are not like terms. So in the case above, 3 + 2 makes 5.

They're constant terms, they're numbers, they're integers we can simplify.

But in the case of a + b, two different terms, we can't simplify that any further.

"Sofia, when the multipliers are not like terms, they cannot be collected.

That is a wonderful contribution to make to this lesson, thank you." Quick little check now.

"Which of these expressions can be simplified before expanding?" Have a look, have a think, pause this video.

So in the top one, we would have 2 + d lots of x - 4, 2 + d, we cannot not simplify 'cause they're not like terms. In the second case, 4a lots of 2x - 4 subtract a lots of 2x - 4, 4a subtract a, we can simplify to 3a lots of 2x - 4.

If we expanded that, 6ax - 12a.

In the bottom case, are they the same expression in the brackets, 5 - w, -w + 5? Absolutely, we've got a +5 and a -w, it's the same expression inside each bracket.

So we can simplify before expanding because they're both being multiplied by constants.

We would get 3 lots of 5 - w and 4 lots of 5 - w, giving us 7 lots of 5 - w, which would expand to 35 - 7w.

Practise time now.

"Match the expressions:" I love these tasks.

We've got some very convoluted expressions on the left hand side and I'm gonna ask you to match them to their simplified partner.

Draw some lines, match up those expressions.

Pause this video, give it a go.

(car engine revving) Question two practise: Simplify then expand.

Notice my command words in that order, simplify, then expand.

For that first expression, I'd like you to gather together the x + 6s.

How many x + 6s do we have before you expand that bracket? Pause this video, give it a go.

Question three: Laura simplifies this expression incorrectly.

She thinks 3 lots of bracket x + 5 and 4 lots of bracket x + 5 is the equivalent to 7 lots of bracket 2x + 10.

It's a really common error.

We see a lot of students do this in mathematics, and it's understandable why they do, but it's also understandable to explain why we can't.

I'd like you to show why this is wrong and see if you can write a sentence to explain to Laura the error that she's made.

Pause this video, show why this is wrong, and write a sentence.

Question four: Jacob has been asked to expand and simplify the following.

He has been asked to expand and simplify the following.

That would mean expanding out all those brackets and then simplifying all those terms. Jacob's rightly questioned, "Is that the most efficient way that this can be done?" For this question, I'd like you to explore this with Jacob.

Is that the most efficient way this can be done? I'd like you to do this question in two ways and then make a comparison.

Firstly, could you expand all those brackets and then simplify by collecting the like terms, and then do the same question again, but the second time round, simplify before you expand.

Then I'd like you to just think to yourself, "Which method did I prefer?" Pause this video, give that question a go.

Feedback time now, matching up the expressions.

In the first expression, we're dealing with x + 2s, 5 of them and 3 of them.

That's gonna give us 8 lots of our x + 2s.

In the second expression, we're dealing in x + 3s.

5 of them and 2 of them, that'll give us 7 lots of x + 3.

In the third case, x + 2s, 5 lots of them subtract 3 lots of them, will leave us with 2 lots of that x + 2.

The next example, 5 lots of subtract 2 lots of will give us 3 lots of the matching expression, x + 3.

And the final example, what was different and difficult about this one? The negative: 2 lots of x + 3 minus 5 lots of x + 3 leaves us with -3 lots of x + 3.

Question two: Simplifying and expanding.

7 lots of and 3 lots of x + 6 gives us 10 lots of x + 6 which expands to 10x + 60.

We simplified and then expanded.

7 lots of and 3 lots of that matching expression, that's gonna be 10 lots of the expression x - 6, which we can expand to 10x - 60.

The next one's lovely, 7 lots of something subtract 6 lots of something, leaves us with 1 lot of that something.

In this case, 1 lot of that x + 3.

Notice I didn't write a 1 outside the bracket, it would serve no purpose multiplying by 1, we'll just leave it as x + 3, so we frequently don't write that 1.

In fact, we wouldn't even need to write those brackets.

We've just got x + 3.

How simple was that? In the final example.

Oh, we're multiplying by variable, what does that change? I've got 7x lots of and 3x lots of the expression in the bracket.

That gives me 10x lots of that expression, multiplying out, 10x squared + 60x.

Question three, I asked you to show Laura why this is wrong and explain her error to her.

There's lots of ways we could have done this.

We could expand the brackets to show 3 lots of x, 3 lots of 5, 4 lots of x, 4 lots of 5, and then simplify by gathering the like terms. The expression we started with is equivalent to 7x + 35.

So we can show Laura that what she's got just can't be correct because 7 lots of 2x, 7 lots of 10, would give us 14x + 70, and we know that 7x + 35 is not the equivalent expression to 14x + 70.

So we've shown Laura that something's gone wrong.

Our explanation about her error might have included, "When you add 3 lots of x + 5 to 4 lots of x + 5, you have 7 lots of x + 5, which you write as 7 lots of bracket x + 5.

The expression inside the brackets doesn't change.

Jacob's question, we ask you to do this in two ways: Expand all those brackets out and then simplify by gathering like terms, and then secondly, simplify before you expand.

How did that look? When we expand first, are you ready for this? We get that.

You can see within it, c terms and ch terms, which I can simplify.

Let me gather those c terms and gather those ch terms, it's -4c and 8ch.

So, I could gather together the 2 + h terms and the 2 - h terms, leaving myself with 3c lots of 2 + h and -5c lots of 2 - h, which we could then expand, and then I've only got four terms to simplify.

I'm gonna reach the same expression, obviously, 8ch - 4c.

What do you think? Which method did you prefer? You could have said "I prefer expanding first, I like to see all the terms written out and I find that easy to simplify," you're welcome to think that.

You might think, "I'd prefer simplifying first.

If I simplify first there's less terms when I expand, therefore I'm less likely to make a mistake." Both opinions are fine.

We're always seeking the most efficient, most accurate route.

Part two of our lesson: Simplifying with matching multipliers.

Notice the change in language.

Matching expressions, we were just dealing with, now we're gonna look at matching multipliers.

What if I said "Simplify these:" 6 lots of x + 10, mm-hmm, 4 lots of and 2 lots of, that's gonna be 14 lots of x + 2, 4 lots of and 10 lots of.

Can't do the bottom one, can we, can we? No, they're not matching expressions, x + 2, x + 4, they're not matching expressions, so that one cannot be simplified, or can it? Going back to number quite often helps us to understand the structure of algebra.

If we just consider this pattern of numbers, this multiplication, addition, if we think of that for a moment, that'll help us understand the structure of what's happening in the algebra.

10 lots of 1 + 2 and 10 lots of 3 + 4.

Well, that's 10 lots of 3, it's 10 lots of 7.

That's 30 and 70, that's 100.

Look at that, hmm? Did I have to do it that way, or could I have thought it's 10 lots of 3 and 10 lots of 7? Or is it 3 lots of 10 and 7 lots of 10, 3 lots of and 7 lots of 10? Well, that then just gives us 10 lots of 10.

We're talking about how many 10s have we got, wasn't that quicker to do it that way? But right from the very start, when we start thinking about how many 10s have we got, I've got one of them, and I've got two of them and I've got three of them, and I've got four of them.

So, we turn the expression, 10 lots of bracket 1 + 2 and 10 lots of bracket 3 + 4 into.

How many 10s are there? 10 lots of bracket 1 + 2 + 3 + 4, it was ten 10s, it was as simple as that.

So, can we do that with the algebra? If we can turn that expression with number into 10 lots of 10, can we do it with algebra? Instead of, typically when we see this in algebra, we're thinking, "Well, I expand that bracket, 10 lots of x, 10 lots of 2, the second bracket, 10 lots of x, 10 lots of 4." Let's turn that language around.

That first bracket is x lots of 10 and 2 lots of 10.

That second bracket is x lots of 10 and 4 lots of 10.

When we think of it like that, we get: x + 2 + x + 4 lots of 10.

So we can simplify that expression to that.

I do hope you are looking at the screen now going, "Well, the bracket will simplify even further, sir," it absolutely will.

We can gather together the x terms, we can gather together the constant terms to 2x + 6.

So we turn that first expression into how many lots of 10 have we got? I've got 2x + 6 lots of 10.

We can expand now to 20x + 60.

So, we didn't have matching expressions inside the brackets.

Instead, we had matching multipliers outside the bracket.

We were still able to simplify.

So what do I see? X + 3, x + 2 inside my brackets, they're not matching expressions, but the multiplier is 5.

I've got a common multiplier of 5 in both brackets.

So that first bracket gives me x lots of 5 and 3 lots of 5.

The next bracket gives me x lots of 5 and 2 lots of 5.

How many lots of 5 do I have? I've got x + 3 + x + 2 lots of 5.

Can I simplify? Absolutely, and I add the like terms inside my bracket, becomes 2x + 5.

I've got 2x + 5 lots of 5.

I can expand that now to 10x + 25.

Your turn, pause this video, give it a go.

I hope you wrote: I have X and 7 and x and 1 lots of 4.

I can simplify that to 2x + 8 lots of 4, which will expand to 8x + 32.

Let's check you've got that.

"Which is the correct simplification for this expression? 8 lots of bracket x + 7 plus 8 lots of bracket x + 6." Pause this video, take your pick.

It was b, 8 lots of bracket 2x + 13.

What would happen if the multipliers had the same value but different signs, the same value but different signs, what does that mean? It means this.

I can see a 5, but in the case of the second bracket, the multiplier is -5.

Can't be done, can be done, or it can be done? But it comes down to understanding that -5, that's 5 lots of -1.

So, I can leave 5 bracket x + 3 as it is and then consider that second bracket.

Instead of -5 lots of, I'm gonna call it 5 lots of -1 lots of.

Now, the -1 next to the bracket means we can multiply the x term and the 2 term by -1 and turn that into 5 lots of -x and -2.

Now, we haven't got different signs on that 5, they're both positive 5.

So we can simplify in the way we were simplifying a moment ago.

Writing the negative multiplier as the product to the positive multiplier and -1 allows us to rewrite the expression.

So, in my example, 5 lots of x + 3, I can leave that positive.

My second bracket was being multiplied by -5, so that's 5 lots of -1.

So, I'm gonna multiply the contents of the bracket by -1.

X by -1 becomes -x, 2 by -1 becomes -2.

I now have matching positive multipliers.

So, how many lots of 5 do I have? I've got x lots of 5, 3 lots of 5, <v ->x lots of 5, -2 lots of 5,</v> which I write like that.

Can I simplify? Boy, can I simplify? X subtract X, well, that's a joy, that's a negative, that's a zero pair, that becomes nothing.

So I just end up with 3 - 2 being 1.

That bracket simplified to 1.

So you're telling me my answer is 1 lot of 5? Absolutely I am, it's 5.

That complicated looking expression simplified down to 5.

Over to you now, your turn, pause this video, give it a go.

Step one: 4 lots of bracket x + 7, we leave the same, but that -4 we need to deal with.

<v ->4, that's 4 lots of -1.

</v> I'm gonna multiply the contents of the bracket by -1, x multiplied by -1, 1 multiplied by -1, changes the content of that bracket, the expression of that bracket, to -x - 1.

How many lots of 4 do we have? I've got x lots of, and 7 lots of, and -x lots of, and -1 lots of 4.

Can that simplify? Absolutely, and it's beautiful, x - x, nothing.

7 - 1, 6, 4 lots of 6, 24.

Practise time now, another matching task.

I'd like you to match the expressions on the left hand side to the simplified expressions on the right hand side.

Pause this video and match them up.

Question two, I'd like you to simplify and then expand each expression.

That first case, how many 10s do we have? Simplify, and then expand.

Pause, and give these questions a go.

Question three: Is Aisha correct? Aisha said, "You cannot simplify as the multipliers have different signs." If you think "Yes," you are gonna need to expand and then simplify.

If you think that Aisha's not correct, you can simplify and then expand.

Pause this video, give it a go.

Feedback time.

For the first expression, we've got, x and 3, and x and 7 lots of 5, that's 2x + 10 lots of 5.

In the second case, x and 3, and x and 3 lots of 5, that's 2x + 6 lots of 5.

In the next case, 2x and 7, and x and 3 lots of 5, that's 3x + 10 lots of 5, x + 3 + x + 7, that's 2x + 10 lots of 3.

2x + 7 + x + 3, that's 3x + 10 lots of 3.

Next, simplifying and then expanding each expression.

How many lots of 10 have we got? 2y and 6 lots of 10.

By the time we've simplified, expanding to 20y + 60.

The second case, so we've got 2y and -8 lots of 2.

Expand that, 4y -16.

I've got y + 1 + y + 5 lots of x, that's 2y + 6 lots of x.

Expand that to 2xy + 6x.

In the bottom case, how many 3d squareds have we got? I got a + b + 5b + c lots of 3d squared.

So that simplifies to a + 6b + c lots of 3d squared, and that will expand to 3ad squared + 6bd squared + 3cd squared.

So, is Aisha correct? No, just because we've got a positive a multiplier and a -a multiplier, we need to think of -a as a multiplied by -1.

When we do that, we think of multiplying the contents of the bracket by -1, x multiply by -1, 7 multiply by -1, and now we've got both positive a multipliers.

How many lots of a have we got? X + 3, -x - 7 lots of a.

Simplify that bracket, x - x is nothing, 3 - 7, I've got -4 lots of a, done.

So just because the multipliers have different signs doesn't mean we can't work with it and simplify before we expand.

To summarise, before we expand and simplify multiple expressions, we should ask ourselves, "Can we simplify before we expand?" I hope to see you again soon for more mathematics.