Lesson video

Hello, Mr. Robson here.

Very wise decision to join me for some maths today, like terms, algebra.

I love it, I hope you will too.

Let's get going.

So our outcome for today.

We'll be able to identify like terms in an expression and generalise an understanding of unitising.

Key words, unitising is one of them.

Unitising means treating groups that contain or represent the same numbers of things as ones or units.

like terms are terms that have the same set of variables and corresponding exponents.

There's some very difficult language on the screen right now, but don't worry, I'll soon put unitising into context for you and we'll soon see lots of variables and you'll see clearly what they are.

And at the end of the lesson we'll come to exponents.

Three parts to this lesson on like terms. We'll identify like terms with just one variable and then we'll move on to like terms with multiple variables before finishing with like terms containing exponents.

I'm excited, so let's get going with identifying like terms of just one variable.

Let's start with something you're very familiar with.

Here are some numbers.

I'd like you to put them into groups.

I'm not going to be any more specific about how you should group them.

I'm just going to ask you to group them.

Pause this video and think about which groups you would put them into.

So here at Oak this is how Aisha grouped them.

300 with 3,000.

200 with 2,000.

500 with 5,000.

She's put things I have three of together, three hundreds, three thousands, I have three of them.

Same for the twos, same for the fives.

You might have grouped them the same way as Aisha.

Jacob did something different.

This is how Jacob grouped them, 200 with 300 with 500.

2,000 with 3,000 with 5,000.

So what's different about the way Jacob grouped them? Pause this video and say your answers that question to the person next to you or maybe say it aloud to yourself.

You might have said he unitised the hundreds and he unitised the thousands.

Whether I have two of them, three of them or five of them, I have hundreds in that first group.

Whether I have two of them, three of them or five of them, I have thousands in that second group.

We might say he's unitised the thing that we have.

Here are some measures.

Again, you'd be very familiar with measures.

I'd like you to put them into groups and again, I won't be any more specific about how you might group them.

Just pause this video and think about how you want to group them.

Aisha grouped them like this.

Four centimetres with four kilometres.

Eight centimetres with eight kilometres.

Nine centimetres with nine kilometres.

Whether they are centimetres or kilometres, I have four of them in that first group.

Whether they're centimetres or kilometres, I have eight of them in that second group.

Nine for that third group.

Jacob did something different again, Jacob grouped them like this.

Four centimetres with eight centimetres with nine centimetres, four kilometres with eight kilometres with nine kilometres.

What's different about the way Jacob grouped them? Pause this video, say it to the person next to you or say it aloud to yourself.

Again, we might use that word unitised, and in this case we're really close to the actual use of the word unit.

We're talking about centimetres, we're talking about kilometres.

They're units of measure.

But unitising we can use as a sense of grouping things.

Previously Jacob grouped the hundreds together, he grouped the thousands together.

Here he's grouped the centimetres together and he's grouped the kilometres together.

Whether I have four of them, eight of them or nine of them, they're still centimetres.

Whether I have four of them, eight of them or nine of them, they're still kilometres.

Ah.

This is a lesson on algebra.

Not on number, not on units of measure.

Here are some algebraic terms, 7a, 5x, 7y.

5y, 5a, 7x.

They're algebraic terms. I'd like you to put those into groups.

Pause this video, group them.

Again, you might have grouped them like Aisha.

5a, 5x, 5y.

Whether it's an a, an x or a y term, I have five of them.

Whether it's an a, an x or a y term, I have seven of them.

That's how Aisha grouped them.

Jacob again did something different.

5a with 7a, 5x with 7x, 5y with 7y.

Jacob's grouped them into like terms in the same way he unitised the hundreds and put the hundreds together, the same way he unitised the centimetres and put the centimetres together, here he's looked at the a's and put the a's together.

5a and 7a are our a terms. 5x and 7x are our x terms. It doesn't matter that we have five x's in one of those terms and seven x's in the other term, we're still talking about x's, the x terms he's grouped together.

This is what we call like terms and you'll see the variables the same in each grouping.

a's with a's, x's with x's, y's with y's.

So we would say in algebra, "5a and 7a are like terms. The coefficients are different but the variables are the same".

There's another tricky word there which you need to know.

Coefficients.

In 5a, 5 is the coefficient of a.

In 7a, 7 is the coefficient of a, the variables being the a.

"5x and 7x are like terms. The coefficients are different but the variables are the same".

5a and 7a, like terms. 5x and 7x like terms. It's really important to understand what are not like terms. 5a and 5x are not like terms. They have the same coefficient, 5, but the variables are not the same.

a and x are different variables so they are not like terms. Let's check you've got that.

I'd like you to group these algebraic terms into like terms remembering that like terms have the same variable.

Pause this video and group away.

I hope you grouped them like this.

You group the b terms together, the e terms together and the z terms together.

2b with 7b, 3e with 4e and 4z with 5z.

Those groups have the same variables.

They are like terms. Wow! That looks different.

But is it different? The command I'm asking you to follow is the exact same.

I'm saying group these into like terms remembering that like terms have the same variable.

Is it not the same task I asked you to do? It might look a little different but there's a lot of familiarity.

Again, pause this video and give those groupings a go.

It wasn't any different.

We had a fractional coefficient, a negative coefficient, in the term of z, no coefficient.

Well it's a coefficient of one.

We just don't write 1z.

The one won't change the value of that z, so we just write z alone but it was no different.

Like terms are like terms. b terms are b terms whether it's 7b, 2b, 5b, half of b.

They're b terms. e terms. You could have 4e, 400e, -400e, in this case we had -3e, they're still e terms. And in the final case z with 4z because the z variables are the same, they're z terms. Visual representations are really useful in mathematics and they especially help us to solidify our algebraic understanding.

So bar models are really powerful here in representing terms or expressions.

I could represent x like so, 2x like so, 3x like so.

I think you know what I'm going to draw next for 4x.

And your prediction for 5x? You're not happy are you? Rightly so.

What's wrong with my representation of 5x? Pause this video, tell the person next to you, or say it aloud to yourself.

It's there isn't it? I changed the size of the bar and you're not comfortable with that.

You're right to not be comfortable with that.

It should look like that.

It's really important that I'm making each bar the same length or size.

It helps us to see the likeness of the terms, whether we have five x's, four x's, three x's.

You can see the thing that we're talking about is the same, x's.

Your turn now.

I'd like you to pause this video and draw bar models to represent each of the below terms and expressions.

For 2a, I hope you drew that.

For 4b, I hope you drew that.

For 3a I hope you drew three a's keeping them the same size as the a's you used earlier, and for 6b, six b's of the same size and the same size as the b's you communicated earlier and then 2a and 4b and 3a would look like so.

Practise time now.

First question.

Complete this sentence.

"Like terms are terms that have the same blank".

What word are you gonna put in that blank space? It's one of our keywords today.

Question two, I'd like you to group these terms into like terms. 100h, e, 11e, <v ->f, -7e,</v> 5f, 5h.

Group them into like terms. For question three I'd like you to circle the unlike term.

Which is not like the others? I'd like you to write a sentence to justify your answer.

Can you explain why the one you've circled is an unlike term? Pause this video.

Give those questions a go.

Feedback now.

To complete that first sentence, we needed the word variable.

"Like terms are terms that have the same variable." By understanding that we can do question two and group these into like terms. We should have grouped the e terms together.

e with -7e, with 11e.

<v ->f with 5f would group the f terms together</v> and 5h with 100h would group the h terms together.

For question three we had to circle the unlike term, writing a sentence to justify our answer.

The terms were x, -8x, 8y, 8x.

The unlike term was 8y.

Some people might accidentally think this is an odd one out question and circle negative -8x because it's the only one that's negative.

I didn't ask you about coefficients, I asked you about like terms. I can see three x terms and one y term, in which case the y term was the one that was unlike.

"Your justification might have included 8y was the only term with a different variable.

In all three other cases the variable x was the same".

Okay, like terms with multiple variables, How's that going to be different? Again, like terms, the definition doesn't change.

They are terms which have the same variable.

So are these like terms? a, ab, abc.

What's your gut instinct tell you? Pause this video, tell a person next to you, or say it aloud to yourself.

Do you think these are like terms? They're not.

The confusion comes in that they contain, they all contain the variable a, but they don't have the same variables.

a, a times b, a times b times c.

Are they the same thing? No.

So these are not like terms. There's a few ways I might justify this to you.

I'm gonna bring in geometry for a moment and say what if a, b and c were lengths? Well, a would just be one length of a on its own.

What would ab be? I hope you appreciate that this is a multiplied by b, and if I did that I could take a rectangle, multiply a by b, and use ab to define the area of that rectangle.

So what geometric representation might I use for abc? Well it would be a multiplied by b multiplied by c.

So we could create a cuboid and have abc representing the volume of that cuboid.

So when we talked earlier about unitising and centimetres and kilometres were different, in terms of these units, well one is a length, one is an area, one is a volume, they're very different units.

So these are not like terms. a and ab are not like terms, ab and abc are not like terms and a and abc are not like terms. So are any of these like terms? 3x, 2xy, 9xyz, 9z, 5y, 5xy? We've seen a lot of not like terms. Can you spot any like ones in there? Pause this video, see if you can spot them.

To help me understand what like terms are I like to try and think about what is it that's actually varying in each term? In the term 3x the 3 remains constant, the x varies.

In the term 2x, the 2 remains constant, the xy varies.

In the term 9xyz, the 9 remains constant, the xyz varies.

I can highlight the things that vary in those bottom three terms as well.

By identifying the things that vary can you see when the things that vary are alike? I hope you spotted that there's a pair of matching variables, xy.

We have 2xy in that term, 5xy in that term.

Whether you've got two of them or five of them, we have xy's.

We have the same terms. So we can say 2xy and 5xy are like terms. They share the same variable, that variable being xy.

So are any of these like terms? 4ab, 4abc, 5ac, 5bc, 6ab, 6bc, 6abc? Pause this video, see if you can spot any of that are like terms. So grouping them into matching variables is a really good way to spot the like terms. We could have grouped them like this, 4ab with 6ab, 5bc with 6bc, 4abc with 6abc, leaving 5ac as the only one that didn't have a pair.

There was no other like term, there was no other ac term, whereas in the other cases we could pair them up with like terms. Right, let's check that you've got that.

True or false.

5de and 2de are like terms? Is that true? Is it false? And I'll ask you to justify your answer by adding one of these two sentences.

Sentence A, "They have matching variables".

Sentence B, "One is a 2 and one is a 5 so they're not alike".

Pause this video and declare that true or false and add your justifying sentence.

Okay, I hope you went for true.

5de and 2de are like terms. They have matching variables.

In this case, de is the matching variable.

It doesn't matter whether we've got 5 of them or 2 of them.

We're talking about the variable de.

They're de terms, like terms. Time for some practise now.

I enjoy matching tasks.

I'd like you to match the like terms in the left column with like terms in the right column.

Pause this video, see if you can pair them up.

Question two, "Izzy says 4de and 7bc are like terms because in both terms there's two letters being multiplied together".

I'd like you to write a sentence justifying why Izzy is wrong.

Pause this video, write me a sentence.

Feedback now.

Matching the like terms. 2xy would match with 4xy, matching up the xy terms. 3xyz matches with 6xyz.

They're xyz terms. 4xz matches with 5xz, 5yz matches with 2yz, and 6y and 3y to finish.

In terms of Izzy, well she's telling the truth.

In both terms there's two letters being multiplied together but she's wrong about them being like terms. Why? Perhaps your sentence included, "Whilst both terms do have two variables multiplied together, those variables do not match hence they are not like terms." Okay, so that was like terms with multiple variables.

Next, identifying like terms containing exponents.

Exponents.

I wonder what they are? Let's have a look.

Here are some measures, put them into groups.

Again, I won't be any more specific about how you group them.

I would just like you to pause this video and think about how you want to group them.

Here at Oak, this is how Aisha grouped them.

Two centimetres with two centimetres squared, five centimetres squared with five centimetres cubed, and seven centimetres with seven centimetres cubed.

Whether she had centimetres or centimetres cubed, she's got seven of them so she's grouped those together.

Whether she's got centimetres squared or centimetres cubed, she's got five of them, she's grouped those together.

I think you can guess what's coming next.

Jacob did it differently.

Jacob grouped two centimetres with seven centimetres, two centimetres squared with five centimetres squared, five centimetres cubed with seven centimetres cubed.

What's different about the way Jacob grouped them? We might say Jacob's unitised them again.

He's put centimetres with centimetres, centimetres squared with centimetre squared, centimetres cubed with centimetres cubed.

He's unitised them.

So if centimetres, centimetres squared and centimetres cubed are not the same thing, are these the same thing? x, x squared, x cubed? What does your mathematical gut instinct tell you? Your intuition.

Pause this video.

Tell a person next to you or say it aloud to yourself.

They're not the same thing, they're not like terms. x is just x multiplied by itself once.

x squared is x multiplied by itself twice.

x cubed is x multiplied by itself three times.

They are not the same thing hence they're not like terms. I like algebra tiles.

I hope you have a set of algebra tiles maybe in your classroom? That's my algebra tile for x.

I can represent x multiplied by x by putting two algebra tiles there and then considering it's that area there, that's x squared.

In the case of x cubed, I would have to lay them out as a three dimensional object of length x and that would represent x cubed.

Look at those visual representations.

Are they the same thing? Absolutely not! Just as centimetre, centimetre squared and centimetres cubed are different things, x, x squared and x cubed are different things.

They are not like terms. We'd say that the variables do not have the same exponent.

Exponent's not a word we see frequently.

What does it mean? It's the two or the three in x squared and X cubed, the power to which it's raised.

We might say x is the base and three is the exponent.

In the case of x you've got an exponent of one, but again it doesn't affect the value so we don't tend to write it.

Those are exponents, and in this case, because they're not the same exponent, we don't have like terms. Okay, a matching exercise now just to check that you've got that.

I'd like to match up the like terms. On the left we have 5x, 5x squared, x cubed, 2x to the power of 8.

On the right, 13x, 5x to the power of 8, <v ->6x cubed and 3x squared.

</v> Can you match the like terms? Pause this video, do some matching.

First up, we could have matched 5x with 13x, our x terms. Next, matching 5x squared to 3x squared.

They're our x squared terms. x cubed matches to -6x cubed.

It doesn't matter that we've got a negative coefficient, they're still our x cubed terms. And finally, it doesn't matter how big that exponent gets, x to the power of 8, they're our x to the power of 8 terms. So are these like terms? a squared b, ab squared? When I look at them I can see a lot that is alike.

An a, a b, an exponent of two.

Does that make these like terms or do you think they're not like terms? What's your mathematical intuition telling you? Pause this video.

Tell a person next to you or say it aloud to yourself.

Are these two like terms? They're not.

a squared b and ab squared are not like terms. In the case of a squared b we have a multiplied by a multiplied by b.

It's the a which is squared.

In the case of ab squared, we have a multiplied by b multiplied by b.

It's the b that is squared making them not like terms. a squared b and ab squared are not like terms, the exponents are not the same.

The exponents are not the same in a squared b, the exponent for a is two, but in ab squared the exponent for a is one.

It's not the same exponent so they're not like terms. So with that in mind, are you able to identify that we have a pair of like terms here? Can you spot them? Pause this video, see if you can find the matching pair.

I hope you identified 7c squared d and 2c squared d as the pair of like terms? Saying these aloud often helps.

If I look at the three and unmatch that pair, 7c squared d, 3cd squared, 2c squared d.

It's a little easier to see the c squared d terms and match them as a pair of like terms. Wow! (chuckles) Identify the pair of like terms. If you can do the last one, you can do this one.

We're looking for the same variables raised to the same exponent.

There's a pair in here.

Have fun spotting them.

Pause this video.

Did you find them? We were looking for the e squared f cubed terms. We had 3e squared f cubed and e squared f cubed.

Just one of them.

They were the pair of like terms. No other pair existed in that group.

We have no other matching exponents, matching variables.

Okay, your final task now.

Question one.

I'd like you to fill in the blank with a keyword.

"In order for terms to be like terms their bases and their respective blank need to match".

Which keyword are you gonna put in that blank space? For question two, identify the pair of like terms from those three terms, and then in question three I'd like you to justify is this a pair of like terms? 2a to the power of x, b to the power of y.

6a to the power of x, b to the power of y.

Are they like terms? Whether you think it's true or not I'd like you to justify your answer.

Pause this video, give this task a go.

Feedback now.

Question one.

In order for terms to be like terms the bases and their respective exponents need to match.

In question two, identifying the pair of like terms, we should have identified the x squared y cubed terms as a like.

7x squared y cubed, and 10x squared, y cubed are a pair of like terms. In the middle case Y has an exponent of one.

Without an exponent of three it couldn't possibly be a like term to the other two.

Is this a pair of like terms? This is a beautiful bit of maths and in the future you are gonna learn how truly beautiful and wonderful this is in all sorts of ways.

For now, your understanding of like terms, matching bases and exponents, your explanation should have read yes, they are like terms. "Whilst a, b, x, and y are all variables, a is consistently raised the power of x and b is consistently raised the power of y.

The same exponents make these like terms. The end of the lesson now.

In summary, "Unitising is a useful mathematical technique to group things.

In algebra like terms are terms that have the same set of variables and corresponding exponents.

For example, 4b and 6b are like terms. 3x squared and 5x squared are like terms".

I hope you enjoyed this lesson as much as I did and I hope to see you again for more maths very soon.