Lesson video

Hello, Mr. Robson here.

You've made the right choice by joining me for a lesson on algebra.

I'm excited.

I hope you are too.

Let's get going.

Our outcome for today is how we'll be able to simplify expressions by collecting like terms. Keywords that we're gonna need today: like terms. They're terms that have the same set of variables and corresponding exponents.

Simplifying: that's to write an expression in a more efficient and compact form without affecting the value of the original expression.

Fully simplifying means the expression cannot be simplified further.

Three parts to today's lesson and we're going to start by collecting like terms of the same variable.

Let's start with something very familiar.

If I said you have three metres and two metres in total, you'd say that's five metres.

If I said that you have three tens and two tens, you'd tell me in total that's five tens.

This is a really important notion in mathematics.

It's gonna be especially important today in algebra.

What we're saying is that three of something and two of that same thing makes five of that thing.

It's really useful in numeracy.

If I said three lots of 1,000 and two lots of 1,000, three lots of five and two lots of five, three lots of 0.

18 and two lots of 0.

18, you could say, "Well, three lots of 1,000, two lots of 1,000, I've got three and two lots of 1,000.

That's five lots of 1,000, that's 5,000." We could do it and turn three times five plus two times five into five fives.

And really importantly with that bottom one, I didn't fancy multiplying by 0.

18 twice when I can just say, "Do you know what? I've got three lots of 0.

18 and two lots of 0.

18, giving me five lots of 0.

18." That's a much easier multiplication when we simplify it first.

It gives us 0.

9.

We can generalise what's going on here and this generalisation unlocks the key to how we simplify like terms. When I say three of something and two of that same thing, three of anything and two of that same thing, well we have two and three lots of that thing, giving us five lots of that thing, so algebraically, 3a and 2a.

We can simplify to 5a.

There might be something unusual in the appearance of the way I've written that at the bottom.

Is that an equal sign? It's not not an equal sign, it's an identity sign.

Whatever the value of a, 3a and 2a is the same.

It's equivalent to 5a.

We call that an identity.

It's true for all values of a.

We simplify the expression by collecting the like terms. They were a terms. 3a and 2a.

We can simplify.

Can we simplify these expressions by collecting the like terms? Yes, we can.

Pause this video and suggest to the person next to you or say out loud yourself what you think we might simplify these expressions to.

3e and 2e.

We have three lots of and two lots of e, giving us 5e.

5x and 2x ,five lots of and two lots of x, giving us 7x.

In the last case, it doesn't matter that there's three terms in that expression.

We've got five lots of and two lots of and 10 lots of x.

In total, we've got 17 lots of x.

We could simplify each of those expressions by collecting their like terms. Algebra tiles or bar models can help us with collecting like terms. 3e + 2e I could represent like that.

5x + 2x I could represent like that.

Now if I start to gather these like terms, I'll take my 2e placed next to my 3e and you can see in total three lots of and two lots of e, making 5e.

Do the same thing for the x.

Put my 2x alongside my 5x, we've got seven lots of the same thing.

Seven lots of x.

Let's check you've got that now.

Can you simplify these expressions by collecting like terms? Pause this video and try to simplify these four expressions.

For the first expression, three lots of x and five lots of x gives us eight lots of x.

Three lots of a and five lots of a is eight lots of a.

Three lots of a and seven lots of a will give us 10 lots of a.

3a + 7a + 6a can be simplified to 16a.

Another example of the use of algebra tiles and bar models, it helps us when we have negative coefficients.

If I take 5g and add a positive g, I can represent it like this.

Five g's, a positive g, I sum that together, 6g.

What is 5g and - 4g look like? Well there's 5g.

There's - 4g.

Four - g's, -4 g's, however you think of it, that's what it looks like.

Then we're looking for pairs, zero pairs to be precise.

A positive g and a - g will give us nothing, so we can remove those tiles and leave us with g.

That explains why 5g subtracts 4g, leaves us with g.

That visual is really useful for helping us understand the difference between when we simplify 5g - g and g - 5g.

Let's start with the top one, 5g and a - g.

There's 5g and a - g.

Look at my zero pair.

I remove them.

I'm left with 4g.

Example at the bottom, we have a positive g and - 5g.

There's my positive g, there's my - 5g.

We've got one zero pair.

When we remove them, we're left with four - g's or - 4g.

That's how we'd say that.

So we simplified g - 5g to - 4g.

How about this expression? Can we simplify that? What's your instinct telling you? Your mathematical intuition.

Pause this video.

Tell the person next to you, "What do you think? Can we simplify?" We can't.

It's already in its simplest form.

f terms and g terms. They're not like terms so we can't simplify further.

This expression is already in its simplest form.

I'd like to show you something wrong now because it's something I see frequently in the maths classroom.

People want to say 5f and 4g is 9fg.

I've put the five and the four together, I've put the f and the g together.

I think that's 9fg.

This is not true.

This is not a correct use of that identity symbol.

These two expressions are not equivalent.

Why not? I'm gonna justify this with number.

f and g are variables, they could take on any one of an infinite number of numbers.

I'm gonna attribute f the value of 100 and g the value of 10 just to demonstrate why this is wrong.

If I said f's got a value of 105, f would be five lots of 100.

If I said g's got a value of 10, 4g is four lots of 10.

If I were to say that 5f and 4g is the equivalent of 9fg, then I would be saying numerically that five lots of 100 and four lots of 10 would be the same as nine lots of 100 times 10.

Just pause this video and just work out those two sums at the bottom and just see if they are equivalent.

Then now we've got 540 on the left and 9,000 on the right so we'd say they're not equal in the same way that we would say those two expressions are not identical, they're not equivalent.

5f and 4g cannot be simplified to 9fg.

Hang on to that thought, it's going to be useful.

Okay, let's check that you've got this so far.

I'd like to simplify these expressions by collecting the like terms. Pause this video, copy those down, give them a go.

5f and 3f is equivalent to 8f.

5f and -3f is equivalent to 2f.

3f and -5 f is -2f, a bar model with -f tiles would've helped you to see that.

The next expression 3f - 5f + 8f is 6f.

6f + 2g.

What's changed? Well done.

It's already in its simplest form.

We can't simplify any further.

We don't have the like terms with which to simplify any further.

Simplifying by collecting like terms. 4x and 3x is the equivalent to 7x.

Here at Oak, Alex has just said, "So x must be 7." Alex thinks that x equals 7.

Is Alex correct? What do you think? Tell the person next to you or say it aloud.

Is Alex correct? No, Alex is incorrect.

We're not solving an equation here.

We're not evaluating x.

We're simplifying the expression.

In algebra it's really important to pay attention to the command words.

There'll be moments when we're asked to solve.

There'll be moments when we're asked to evaluate, but in this case the command word was simplify.

All we're asked to do is simplify by collecting the like terms so we just leave that expression 4x + 3x as 7x, its simplified form.

True or false now.

The expression 2e + 9f can be simplified further.

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

2e and 9f are not like terms, so the expression cannot be simplified further, or it can be simplified to 11ef.

Pause this video, tell the person next to you or say it aloud to yourself.

I hope you went for false and justified your answer with 2e and 9f are not like terms. The expression cannot be simplified further.

Another check for you.

Sophia has simplified these expressions.

I'd like you to mark her work and correct any errors.

Pause this video, have a look at Sophia's work and check it.

It's a tick on the first one, Sophia.

3d and 10d is equivalent of 13d.

We can simplify.

They're like terms. 4x and 2x, they're x terms. If we drew it out four x's, two x's and gathered them, we'd see six x's.

A visual representation might have helped Sophie there.

6x squared was not correct.

2f + 3f + f is equivalent of 6f.

Sophia gets a tick there.

The bottom one, no.

It's already in its simplest form 3f and 2g we leave as 3f and 2g.

Cannot be simplified further, they're not like terms. Practise time now.

For the first task I'd like to simplify the below expressions by collecting like terms. Pause this video and give those questions again.

For question two, Laura says, "The expression 5j + 8k simplifies to 13jk." Write a sentence to explain why Laura is wrong.

Pause this video, give it a go.

Feedback now.

Can you check your work along with me? j + j + j is equivalent of 3j.

3j and another j is equivalent of 4j.

3j + 5j + j can be simplified to 9j.

3j + 5j subtracted a j will leave us with 7j, 6k and 4k simplifies to 10k.

Positive 6k - 4k, that will leave us with 2k.

Positive 4k - 6k, that simplifies to - 2k.

And then 4k - 4k, we'd be left with nothing.

Laura's problem now.

The expression 5j + 8k simplifies the 13jk.

How do we explain to her that this is incorrect? Your sentence might have included, "5j and 8k are not like terms so we cannot further simplify.

You leave it as 5j + 8k." Next, simplifying expressions with unlike terms. Fully simplify by collecting the like terms. Here's two expressions, 2a + 3a, 2a + 4b + 3a.

What's the difference between those two expressions? Pause this video, tell the person next to you or say it aloud to yourself.

I hope you said something along the lines of, "In the first expression 2a + 3a, we've got two like terms whereas in the second expression, we've got a terms and b terms. We've got unlike terms." Visualising this is useful.

2a and 3a.

Gather those like terms, we can see we've got two lots of and three lots of a, five lots of a in total.

For the second expression, 2a +4b + 3a looks like that.

Gather the like terms. The a terms, we can gather the a terms together.

Our expression starts to look like that and we can say that we've got two and three lots of a and four b.

We can simplify the a terms to 5a, but the b terms are not a like so we can't simplify.

That expression is 5a + 4b and that's its simplest form.

You might want to copy that down as an example.

When we fully simplify by collecting the like terms, we identify the a terms or like, they become 5a.

The b terms not alike, so our final simplified expression is 5a + 4b.

Okay, let's check you've got that.

I'd like to fully simplify by collecting the like terms in this expression.

Write that one down and give it a go.

We should see seven lots of and three lots of c and a 5d term.

Seven lots of and three lots of c gives us 10c and 5d.

That's our simplified expression.

We've gathered the c terms, that's as far as we can simplify.

What if I had a fourth term? Fully simplify by collecting the like terms 7c + 5d + 3c + d.

Pause this video and give that one a go.

I hope you gathered the c terms and you gathered the d terms. We had seven lots of and three lots of c, five lots of and one lots of d, giving us 10 lots of c and six lots of d, collecting the like terms fully simplifying our expression to 10c + 6d.

Okay, check now.

I'll do the two on the left and I'll give you two on the right to have a go at.

Fully simplify the below expressions: 3a + 6b + 5a + b.

I see three lots of and five lots of a, six lots of and one lot of b, giving us an expression simplified to 8a + 7b.

The second example is a little more complex because we've got exponents.

I hope you recall that we need matching exponents, corresponding exponents to have like terms. So the c squared terms are alike and the c terms are alike, but the c terms and the c squared terms are not alike.

I can see in the expression c squared + 8c, subtract 3 + 3c squared - 5c - 2.

I can see one lot of and three lots of c squared.

I can see a positive 8c and a -5c and then the constant terms. I can see a -3 and a -2.

That's the gathering of the like terms, gathering the c squared terms, gathering the c terms, and gathering the constant terms. To write that more simply, one lot of than three lots of c squared is 4c squared.

eight lots of, subtract five lots of c leaves us with 3c, and then the constant terms -3 and -2 is -5.

There's a fully simplified expression.

4c squared + 3c and -5.

Your turn now.

I'd like you to fully simplify these expressions.

Pause this video, then give them a go.

In the first expression, we had g terms and we had constant terms. I can see three lots of and six lots of g, and in the constant terms I can see a -1 and a positive 6.

We can simplify the g terms to 9g.

Simplify the constant terms to positive 5 giving us a simplified expression, 9g + 5.

The expression below that one I can see two lots of x squared, one lots of x squared, five lots of x and -4 lots of x.

We write them more simply as three lots of x squared and x, one lot of x, but we don't write that one.

It's just more simply written as x.

so our fully simplified expression, 3x squared + x.

Practise time now.

I'd like to fully simplify these expressions by collecting the like terms. Pause this video, take a few minutes to have a go.

Question two, fully simplify by collecting these like terms. It might look really mean this time, but remember, we're looking for corresponding exponents.

Pause this video, give it a go.

Good luck.

Feedback now.

For the first expression, 7x + 2y + 3x + 4y, we gather the x terms to have 10x.

Gather the y terms to have 6y.

Our simplified expression, 10x and 6y.

In the second expression we've got a positive 7x and a -3x, leaving us with 4x.

Positive 2y, positive 4y giving us positive 6y.

Our simplified expression: 4x + 6y.

For c, we had a positive 7x and -3x.

That gives us 4x.

Positive 2y and a -4y, The negatives outweigh the positives.

That gives us -2y.

Our simplified expression for x - 2y.

For d, we've got constant terms. This time we've got x terms, we've got constant terms. The x terms simplified 10x, the constant terms simplified to 6.

10x + 6 was our answer there.

In e, we had x squared terms which gathered to 10x squared, and x terms which we gathered to 6x.

Our simplified expression: you've got 10x squared and 6x.

In that last example I see x squared terms, x terms and constant terms. 7x squared and 3x squared gives us 10x squared in total.

2x and 4x gives us 6x.

1 and 8 gives us 9.

Our simplified expression, 10x squared + 6x + 9.

3x squared -3 + 4x cubed + 7x squared + four - x cubed + 4x.

I'll start with the x cubed terms please.

I see 4x cubed and a -x cubed.

That'll simplify to 3x cubed.

What's next do you think? Let's do the x squared terms. I see 3x squared and 7x squared.

That's 10x squared.

What's coming next? The x terms, I only see one x term and it's 4x.

Finally, the constant terms I see a positive four, a -3 giving us a total of 1.

Our simplified expression: 3x cubed + 10x squared + 4x + 1.

The next one, really deep breath this time.

7a squared but - 3ab + ab squared + 8a squared + 2ab squared -5ab + 9a squared b.

Okay, term by term, I see eight squared b terms. I see nine a squared b at the end.

I see seven a squared b at the beginning, giving me 16 a squared b's.

How about the a b squared terms? Notice the corresponding exponents or lack of corresponding exponents.

The ab squared terms, I see one ab squared and two ab squared give me three ab squareds.

Next, the a squareds and I only see the one a squared term.

Next the ab's, I saw two ab terms. There was a -3ab, a -5ab, giving us -8ab in total.

Finally, the use of simplified expressions.

They're really, really powerful and useful in mathematics.

We want to do things as simply as possible.

If x equals 7 and y equals 3, find the perimeter of this rectangle.

Hopefully you're familiar with the word perimeter.

Perimeter is the distance around the outside of a shape, so I could work out the perimeter of this rectangle.

Given that I know that x has a value of seven and a y has a value of three, I can evaluate those two expressions and calculate the lengths of this rectangle, add them together to get the perimeter.

So, it's a rectangle, so the opposite sides are equal length.

There are two lengths of 2x + 7y, two lengths of 3x + 4y, and I could work out the length of those lengths by substituting in my x value and my y value.

So for that right-hand side, three lots of x becomes three lots of seven, four lots of y becomes four lots of three.

Say 3 times 7 + 4 times 3 equals 21 + 12 equals 33.

For the opposite length I could do the same.

3 times 7 + 4 times 3 equals 21 + 12 equals 33.

For the base, two lots of seven plus seven lots of three equals 14 + 21 equals 35.

And for the top the same.

Two lots of seven plus seven lots of three is 14 + 21 + 35.

And then all I've got to do is add those four lots together.

33 + 35 + 33 + 35 gives me 136.

Can you tell I'm tiring? That was a lot of work.

It was more work than we needed to do.

Was it not easier to go, "I can write a simplified expression for the perimeter before working out each individual length." If I look around the lengths, 3x, 2x, 3x, 2x, gather those x terms, 10x, 4y, 7y, 4y, 7y, gather those y terms, 22y, giving us a simplified expression for the perimeter, 10x + 22y.

10 sevens and 22 threes is 136.

That was so much easier, simpler, quicker, better.

That's what we're seeking.

This was a really useful example where simplified expression made our lives quicker and easier.

Another example for you now, Sam's sister works in a restaurant.

They earn 10 pound per hour plus tips.

Here are expressions for four days earnings: Day one, five hours of work and eight pound and tips.

Day two, three hours of work and 15 pound and tips.

Day three, six hours and five pound and tips.

On the fourth day, six hours and 12 pound and tips.

So we could work out what Sam's sister is earning.

Five lots of 10 + 8 is 58.

Three lots of 10 + 15 is 45.

Six lots of 10 + 5 is 65, 6 lots of 10 + 12 is 72, and I just need to add together the 58, the 45, the 65, the 72 and get 240 pounds.

Are you sensing my tiredness again? That was a lot of work.

Was it necessary or could we have done things differently? What do you think we want to do? There's Sam's sisters earnings again.

Five hours and eight pound and tips, three hours and 15 pound and tips, et cetera.

Which of these show a simplified expression for Sam's sister earnings? Is it a, 8h + 20 + 12h + 20? Is it b, 20h + 40? Or is it c, 5h + 3h + 6h + 6h + 8 + 15 + 5 + 12? Pause this video, see if you can identify which show a simplified expression.

I hope you identified a and b.

They were both simplified expressions.

Now you might be looking at c and going, "I'd take that one as well because I can see five hours, three hours, six hours, six hours, 8 pound in tips, 15 pound in tips, 5 pound in tips, 12 pound in tips." c had the same value, it's the same expression, but the question was very specific.

A simplified expression.

c's no simpler, it's just commuted, so we're adding it in a different way, but it's the same thing.

Not simplified.

But if you look at c, five hours and three hours is eight hours, six hours and six hours is 12 hours.

Eight pound in tips, 12 pound in tips, that's 20 pound in tips.

15 pound in tips, five pound in tips, that's 20 pound in tips.

You can see that c could have been simplified a little to turn it into the expression a, 8h + 20 + 12h + 20.

That is where we're starting to simplify.

b is utopia for us.

It's fully simplified.

When I add the eight hours and the 12 hours, it's 20 hours.

When I add the 20 pound and tips to 20 pound and tips, it's 40 pound in tips.

Here we've fully simplified.

a is a simplified expression, but it can be simplified more.

b is the fully simplified expression.

Practise time now.

Sam's sister works in a restaurant.

They earn 10 pound per hour plus tips.

Here are expressions for five days worth of earnings.

Use a simplified expression to calculate their earnings.

I'm gonna repeat that instruction because it's important.

I didn't say calculate their earnings.

I said use a simplified expression to calculate their earnings.

Pause this video.

Give this question a go.

Don't forget to use a simplified expression.

Question two, a scalene triangle with lengths 6x - y 9x - 2y, 5x - 2y, and we're told that x has a value of six and y has a value of four, which means we can calculate the perimeter.

But I'm not asking you just to calculate the perimeter.

I'm gonna ask you very specifically, again to use a simplified expression to calculate the perimeter.

Use a simplified expression, you know that's going to make this task quicker.

Pause, give it a go.

Okay, feedback now.

When we add together all those worked hours, we have 24 hours.

Adding the h terms, collecting the like terms gives us 24h, 24 hours worked.

We add the constant terms, adding those tips, 50 pound in tips.

And then at 10 pound an hour we can simply say, "Well, 24 hours at 10 pound an hour plus 50 pound in tips becomes 290 pounds." Wonderfully quick, simple, easy when we use a simplified expression.

A scalene triangle now.

Let's sum the x terms. Simplify the x terms, 6x, 5x, 9x, that's 20x.

The y terms, -y, -2y, -2y, that's -5y.

And then we're substituting in an x value of six, a y value of four.

We want 20 lots of six and -5 lots of four.

That's 120.

<v ->20 giving us a total of 100.

</v> We're at the end now.

To summarise, we can simplify expressions by gathering together like terms and we simplify because it makes our mathematics more simple and efficient and we like that.

We mathematicians, we like things simple and efficient.

Anyway, I've thoroughly enjoyed this lesson.

I hope you have too and I hope I see you again very soon for some more mathematics.