Lesson video

Right, well done for loading this video today to have a go at learning some algebra skills.

My name is Ms. Davies and I'm gonna help you as we progress through this lesson.

So if you have any problems, pause the video.

You can always rewind, look back at things that I've said before.

I really hope that you enjoy some of the things that we are exploring.

Let's get started.

Today's lesson is gonna be about algebraic notation.

By the end of the lesson, you'll be able to use the particular conventions of algebraic notation, and understand that following these aids clear communication.

Essentially we're gonna be looking at different ways of writing things using algebra, and the fact that by writing things in a certain way, it makes it easier for us to read and write algebra and share with other mathematicians.

So there's lots of keywords that we're gonna be using today.

So make sure that you're really comfortable with these, because they will be coming up as we move through the lesson.

So the first one to be happy with is a term.

A term is a single number or letter, or the product of numbers and/or letters.

Each term is separated by the operators plus and minus.

So some examples might be x would be considered a term.

Five would be considered a term.

3y will be considered a term.

ab will be considered a term.

Loads of things that will be called a term as we're working with our algebra skills.

So here we have an expression 3x minus 7.

We've got 3x is a term, negative seven is a term.

Expressions then.

So expressions contain one or more terms, where each term is separated by an operator.

So 3x minus 7 is an expression containing two terms, 3x and negative 7.

Mathematical conventions are notations or processes generally agreed upon by mathematicians.

So they're things, they're ways that we will write things or ways that we will say things with mathematics that we do so, so that we can understand each other.

So they're generally agreed upon about the way we do things in mathematics.

So three sections of today's lesson.

We're gonna start by writing repeated addition.

So in algebra, letters are used to represent either unknowns or variables.

Individual letters or numbers, or letters and numbers multiplied together are called terms. We looked at some examples before, we'll look at some more now.

So 2t, xyz, c squared, negative 30.

All of these things would be terms. Things that wouldn't be terms, 3x + 2y.

That's two terms together.

Five subtract a, again, we've got two terms together.

An operator, so a plus sign is not a term, and an equals sign is not a term.

In algebra, we like to write expressions and equations in the simplest way possible.

So before we talked about conventions, we're talking about trying to write things in the easiest way possible.

This makes them easy to read and easy for other mathematicians to follow our work.

We want to be able to look at something and be thinking the same thing as other people looking at our work.

So certain conventions help mathematicians to write terms, expressions, and equations in the same way, and we're always trying to make things look the easiest and the simplest for us to use.

So if you've got some algebra tiles with you, this might be a good time to get them out and start having a look at what they might represent.

You can also get algebra tiles using the internet, or you can look at what we're doing together.

So this algebra tile represents the number one.

See if you could replicate this picture below, if you have your own algebra tiles.

What value have I represented below? Well, this would be the way that we represent the value four using algebra tiles.

We can add ones together and represent it as a single number.

This is something you've been doing for a very long time in mathematics.

We can do 1 + 1 + 1 + 1 and represent that as the number four.

This is the same with variables, and this is the key point.

If you have your algebra tiles, see if you can find a tile representing x.

This is gonna represent the variable x.

It doesn't have a fixed value.

It can represent any value, and we're gonna call it x.

Again, see if you can get three of those together.

So you've got an x and x and x.

And how would we read that? What value have we shown below? Well, we would write that as 3x.

We've got an x, an x, an x.

We add them together and we get 3x.

Right, time for you to have a little bit of a think.

What value do these algebra tiles represent? Hopefully you agree with me that that is 5x.

You might have written x + x + x + x + x, but we're looking for the simplest way of writing it, so we'd write that as 5x.

Right, how would we show 2x using algebra tiles? If you have algebra tiles, you can put them in front of you, or you can draw this, or you could just think about what it is you would draw.

Brilliant.

Hopefully you've got an x and another x.

We've got x + x, and that's gonna give us 2x.

When we have an expression made up of a single variable added to itself repeatedly, we can write that as a single term.

So here's another example.

I've got n + n + n + n + n + n, I can write that as 6n.

This works with subtraction as well.

So if you've got your algebra tiles, you're now looking for your negative algebra tiles.

So if I've got x + x + x - x.

I've got x + x + x, and I've taken an x away, that gives me 2x.

The reason I said to have a look at your algebra tiles is that subtracting a variable is the same as adding the negative of that variable.

So you could also show that by doing x + x + x, like we did before, but instead of subtracting an x, add a negative x.

And actually an x add a negative x is a zero pair, so that would be the same as 2x.

Okay, so it's useful to be able to move between the ideas of subtracting and adding a negative.

Right, you've got four expressions below.

Which of those can be written as 3x? Take a couple of seconds to come up with your answer, and then see if you've got the same as me.

Right, well done if you managed to spot that that second one, x + x + x is 3x.

Especially well done if you also spotted the third one will give us 3x.

We've got x + x + x, subtracting an x and adding an x, and that will give us 3x.

Brilliant.

So a chance to practise this new skill then.

So for the first question, I'd like you to match the expression on the left hand side with the equivalent term on the right hand side.

Give you a couple of minutes to do that, and then we'll look through the next task together.

Right, well done on that first task.

So for this second task, I'd like you to have a go at writing an expression as a single term.

So the first ones I've represented using in algebra tiles, and then D and E, I've represented just using letters for our variables.

So what I'd like you to do is see if you can write each of those as a single term, in a nice simple way, like mathematicians like.

Off you go and then we'll run through the answers.

Well done, guys.

So the top one, a + a, you should have got as 2a.

Second one, a + a + a + a + a is 5a.

Third one, a + a + a is 3a.

Then we had some with some subtraction.

So a + a + a + a + a would give us 5a, but then we're subtracting an a, so we've got 4a.

And the last one was the trickiest to work out, but hopefully by filling in the others that helped you.

You've got a + a, which is 2a.

Then you're taking away an a, which is 1a, taking away another a, which is nothing, adding an a.

So you should end up with a or 1a.

Right, that second set of questions then when you're writing these terms yourself.

So the first one you should have written as 5x.

The second one as 3x.

Well done if you matched up your zero pairs for your third one, and you ended up with x or 1x.

And then D, you should have got 7f.

And the last one, t + t + t, would be 3t, but then you're taking away a t, so you've got 2t.

Well done.

So we've done lots of focusing on writing repeated addition, so hopefully we're really confident with that idea of writing repeated addition in a nice simple way.

We're now gonna look at how we write multiplication in algebra.

So multiplication is the same as repeated addition.

You've probably looked at this before with number.

So 5 + 5 + 5 + 5, I'm sure you're happy with the idea that that's four lots of five.

We've got 5 + 5 + 5 + 5, that's four lots of five.

Lots of can be written as a multiplication, so that is four times five.

The key point with this is all these things that work with your number skills work exactly the same with your algebra skills.

So a + a + a, we could say is three lots of a, or three times a.

We're looking at the fact that that repeated addition is the same as a multiplication.

We can write both of those as 3a.

Remember before when we wrote a + a + a, we wrote that as 3a, didn't we? Okay, so that means three times a can also be written as 3a.

In algebra, we don't write that multiplication symbol where we can help it, 'cause again, we're trying to write things in a nice simple way, and we know that three times a means three lots of a or three a's.

Multiplication is commutative.

So if I've got a multiplied by three, again, I write that as 3a.

It is really important that we write the three first before the a.

That is a mathematical convention that you need to be using confidently.

Okay, we wouldn't write a3.

So 3a is the way that we write it using our mathematical convention.

Again, if you have your algebra tiles, we can show this with our algebra tiles.

So if we have an x across the top and three down the side, we can set ourselves up a multiplication grid.

And then if we do x multiplied by three, you should be able to fill in your multiplication grid, so that you have 3x's.

So x multiplied by three gives us 3x's.

Time to check that you're happy with these conventions then.

So which of those below is the correct way of writing b multiplied by 15? Think about your answer and then come and check if you're right.

Fantastic.

Really well done if you spotted that it's that second one.

Remember where possible, we don't write the multiplication sign to keep things looking nice and concise, and we have to write our numerical value before our variable, again, just to help keep things looking nice and precise.

That shows us that we have 15bs.

So there are also conventions for multiplying different variables together, or the same variable together.

So we're gonna start by different variables together.

So we've got a multiplied by b, we write that ab or ba.

Multiplication is commutative, so it doesn't matter which way around you write it.

However, the convention is to write them alphabetically.

In this case, it's not wrong.

Like before I said that you wouldn't write b15, 'cause it's not clear what it is you're trying to say.

In this case, writing ab or ba, neither of them is wrong, but the convention is to write them in alphabetical order.

This again allows us to spot easily when terms are the same.

Right, I'm gonna let you take a second to see if you can answer this question.

Are these terms identical? What do you think? I'm hoping what you're saying is that is really hard to tell.

You might spend some time going through matching up the variables and seeing if I've written the same thing.

Right, what about now? Are these terms identical? What have I done differently? Yeah, you've probably seen then that what I've done differently this time is I've written them alphabetically.

It's really easy to see now that the terms are identical.

Okay, you might wanna check, or you might just believe me that these are exactly the same thing that I wrote on the first line, but because I've put the variables in alphabetical order, it's really easy to see that they're identical terms. So when we're multiplying different variables and numbers, the number is written first and then the variables, in alphabetical order.

So I've got e X d X x X 5, and I said I'm gonna write the number first, so that's five, and then the variables in alphabetical order, so 5dex.

If we have a calculation with more than one number multiplied together, we can actually calculate the product of the numbers.

So again, we're trying to write things in the easiest way possible.

So where there's things we can multiply together to get a simpler answer, then we can calculate that product.

So here I've got b X 5 X a X 2.

I'm just gonna rewrite that, so I'm gonna group my numerical values together and my variables together.

And then I can actually calculate 2 X 5 and give a complete value for 2 X 5.

So 2 X 5 is 10ab.

Useful thing to remember is that multiplying any term by one does not change its value.

So generally, we choose not to write it.

You might have seen this already.

So if I've got B X 1 X a, I could write that as 1ab.

But the general convention in mathematics is that we don't write the one where there is only one of a term.

So time for you to give this a go then.

So can you rewrite each of these terms to follow mathematical convention? Okay, so think about all those rules and conventions we've talked about.

Can you rewrite these so they're perfect for a mathematician to have a look at? Give it a go and see if you get the same answers as me.

Well done.

So that first one, we've got our numerical value first, which is great, and then we want to write our variables in alphabetical order.

So 12txy would follow mathematical convention.

For that second one, again, we do an alphabetical order, so pqr.

And remember where there's a multiplier of one, we don't generally write one of a number.

Last one you had a little bit more to do.

So well done if you grouped the numerical values together and then worked out the product.

So the product of 10 X 3 X 2 is 60, and then we're writing our variables in alphabetical order, so 60cr.

Right, so let's think about how we can write a multiplication in a more efficient way.

So here we're multiplying seven by itself multiple times.

You've got 7 X 7 X 7 X 7.

Because we're using the same number and multiplying it by itself, we can use exponents.

So 7 X 7 X 7 X 7 can be written as 7 to the power of 4.

So we've got a base of 7 and an exponent of 4.

The same thing applies when we multiply a variable by itself.

So just like you did with number, we do exactly the same with our variables.

So a multiplied by a would be written as a squared.

a multiplied by a multiplied by a, what are you thinking this one's gonna look like? a cubed, that's right, well done.

And if we carry that pattern on, so a X a X a X a, you'd write that as a to the power of 4.

That pattern's gonna carry on just the same way it does when you're working with numerical values.

So we can show these with algebra tiles, and it's useful to be really confident with algebra tiles now, so you can use them when you're getting onto some trickier algebra.

So again, let's put a multiplication grid together.

So we've got X across the top, X down the side, and we're gonna do X multiplied by X.

See if you can find that algebra tile.

Don't worry if you don't have them.

As I say, you can use them online or you can have a look at the ones I've got here.

What do you think then that these algebra tiles represent? Pause and have a think and then see if you're right.

Well done if you wrote 3x squared.

So, what we have here is we've got three lots of x squared.

We'd write that as 3x squared.

So we've got a X a X a X b X b.

So we've got some variables that are the same, but some variables that are different this time.

So how do you think we're gonna write that one? So we can write the a variables using exponents.

So a X a X a is a cubed.

The b variables using exponents will give us b squared.

So as a single term, it's a cubed times b squared, or a cubed b squared.

So which is the correct way of writing n X n X n? Pause the video and give it a go.

Superb.

Hopefully you got that one as the bottom one, n X n X n is n cubed.

It's really, really important that you're not confusing that with 3n.

So don't worry if you've got that wrong this time, but just see if you can understand the difference between the two now.

So 3n is n + n + n, or three lots of n.

n cubed, it's n multiplied by n multiplied by n.

Time for you to have and practise then.

So in each row, you need to circle the odd one out.

So three of them will be the same expression, and then one of them will be an odd one out.

When you've done that, so when you've identified the odd one out, write the remaining expressions as a single term.

So I want you to focus on row two, so that's B.

Once you've identified your odd one out, write the remaining three terms as a single term, 'cause you'll notice that none of those four options are a single term at the moment.

Then I want you to do the same as row four or letter D.

You're gonna write, pick your odd one out, and then you're gonna write that, the remaining ones as a single term.

Give those a go and then come back.

Right, well done.

We're gonna have a look at those odd one out then.

So the first one, 13 + a is the odd one out.

For B, you've got 2 X b X 3 is the odd one out.

For C, you've got 2cd is the odd one out.

For D, you've got a 3e is the odd one out.

And for E, you've got f X g X g is the odd one out.

Writing them as a single term then, so a is already done for you, it's 13a.

Row B, 6b squared would be the simplest way of writing that as a single term.

And for row four, row D, then you're gonna have e cubed, okay, not to be confused with that 3e.

So well done if you picked that one as the odd one out and wrote that the correct way as e cubed.

Fantastic.

The last bit of the lesson we're gonna look at today then is reading and writing expressions.

We're gonna bring all those ideas we've looked at together.

So we're gonna start by looking at fractional notation.

So we can use fractional notation to represent division.

So a divided by 4, which is probably the way you've seen it written before, will be written as a over 4.

In fractional notation, the entire numerator is divided by the entire denominator.

So that's gonna help us with some of our priority of operations.

So if we look at these two expressions below, we've got a plus 3 divided by B.

Remember, priority of operations say the 3 divided by b has priority, subtract four.

And then I've bracketed, I've grouped the first two and the second two terms together.

So I want to do my a + 3 and my b - 4, and I wanna divide those two terms. Okay, so how do you think we would write those two using our fractional notation? Doesn't matter if you're wrong at the moment because it's possibly the first time you've seen this.

Why don't you give some things a go, write some things down, and see whether you agree with my answers.

Right, well done if you gave that a really good go.

So if we look at the first one, remember, priority of operations say that our 3 divided by b is grouped.

Okay, it's only the 3 that I'm dividing by b.

So using my fractional notation, it's the 3 that I'm having as a numerator and b as a denominator.

Let's think about how this might differ to the second one.

Because I want the entire a + 3 to be grouped together, I wanna divide that by the entirety of b - 4, then they become my numerator and my denominator separately.

I don't actually need brackets here, because the convention of fractional notation means that the entire numerator is divided by the entire denominator.

So they're grouped just by the fact that they're on the numerator together, or are part of the denominator together.

Okay, so fractional notation groups the numerator and the denominator anyway, so you don't need those brackets.

So we can bring all these conventions together to form expressions for scenarios.

So Izzy is x years old.

Her dad is three times her age.

Her aunt is two years older than her dad.

Her sister is half her aunt's age.

How can we write an expression for each person's age? I suggest you give this one a go first, and then you come and see whether we've got the same thing.

Think about all the conventions that we've talked about so far.

Pause the video and then come back.

Right, well done for giving that a go.

So her dad is three times her age.

So we're gonna do Izzy's age multiplied by three, and remembering our convention, we write that as 3x.

Her aunt is two years older than her dad, so her dad is 3x, we've just said.

If her aunt is two years older, we're gonna add two.

So we've got that as 3x add two.

And her sister is half her aunt's age, so we're gonna have her aunt's age, which is 3x plus two divided by two.

Notice that I've had to put brackets around it the way I've written it, otherwise priority of operations would say that the two is grouped, and I want the 3x plus two first, then divided by two.

When I write that using fractional notation though, I no longer need the brackets, because the fact that they're grouped together on the numerator means it's all of that that's then divided by two.

So I've got 3x plus two as a numerator, divided by two as a denominator.

Right, well done if you managed to get any of those without my help.

Okay, so let's have a look at this one together and then you can give it a go.

So Laura is n years old.

Her dog is half her age.

Her sister is a year younger than the dog.

Read that through again and then decide which expression represents the age of Laura's sister.

Fantastic.

There's lots going on with that question.

So well done if you noticed that it was c.

So we want to half the age of Laura, so the dog's age is written as n/2, and then her sister is a year younger than the dog, so that's n/2 subtract one.

Fantastic.

Now a good chance for you to bring all that knowledge together and have a practise yourself.

So for each of these sentences, you need to match them with an expression.

The letters in this case don't matter.

I've just picked random letters to use.

So you can see how we use lots of different letters in algebra.

Okay, so for each of those sentences on the left, I want you to match those with the expressions on the right.

Fantastic.

Give it a good go and then come back.

Lovely.

I hope you found that there was some nice easy ones in that last questions, and maybe some that made you think as well.

We're just gonna have a look at some of the fractional notation and some of the exponents.

So same ideas you have before, you've got some sentences on the left and you've got some expressions to match them to on the right.

Be particularly careful about your fractional notation, and making sure that you are grouping your numerator or your denominator as necessary.

Give it a good go and we'll look at the answers together.

Well done, guys.

So lots going on in that first question, so I'm going to talk you through it.

So the first one, any number add five, I've used the letter k, so k + 5, that's that third one down.

Any number multiplied by five.

So I've used e to be any number, so I've written that as 5e.

That's the fourth one down.

C, five lots of a number added to four lots of another number.

So I need two different letters for my two different numbers.

And then I've got five lots and four lots, so that should be the bottom one.

5y plus 4z.

D, a number added to four, then multiplied by five.

Remembering if we want the four to be added first and then the whole thing to be multiplied by five, we need to bracket the plus four.

So I've used G in this case, so this is five bracket G plus four.

Notice I don't put a multiplication sign between my five and my bracket, just writing the five next to the bracket tells me I'm multiplying my five by the value that's in the bracket.

E, a number multiplied by another number.

Okay, so I just need two different variables together, multiplied together.

So I've got wx in that case.

And the last one, a number multiplied by five then added to four.

So 5n + 4, that's that top one.

Right, let's look at the second set together.

So again, loads going on in this one.

So the first one, any number add itself.

Really well done if you spotted this.

If I had a + a, so a add itself, that's 2a.

Any number multiplied by itself, this time I've used the letter d.

So d X d gives you d squared.

Okay, really, really good if you spotted that distinction between adding something to itself and multiplying something by itself.

It's a really useful one not to get mixed up.

Right, a number divided by another number.

So this time we've got h and l, so h divided by l, that's that bottom one.

A number divided by two more than another number, okay? So you've gotta be careful with how you're reading this, and this is why algebra mathematics is better than words in that sense, is that it sometimes depends on how you read something as to how you would then write it.

Okay, so we're gonna move away from the words and move into more of the algebra, so that we don't have that confusion.

So a number divided by two more than another number is t divided by r plus two, so that top one on the right hand side.

A number added to two, then divided by another number.

So that is s plus 2 divided by p.

So that's that fourth one down.

And the last one, a number divided by two, then added to another number.

So we want to make sure the divide by two is separate to the plus m, because we're gonna add m to the whole value of u divided by 2.

Brilliant, well done.

There was so much to think about there.

Okay, make sure you spend some time just looking back over your answers, making sure you know which ones match up with which, so that you can use this algebraic notation in the future.

So we've looked at today expressions involving repeated addition can be written as a single term.

We've talked loads about how these conventions help us read and write expressions in a consistent way.

We know that different variables multiplied together are written in alphabetical order without a multiplication sign.

We've talked about using exponential form for a variable multiplied by itself.

And we've used fractional form to represent division.

You are now experts of using algebra and reading algebra.

So I hope you feel like you've learned something, and that you can now show off your algebra skills.

Thank you for learning with us and I hope to see you again.