Lesson video

Right, well done for loading this video today.

My name is Ms. Davies and I'm gonna help you as you work your way through this lesson.

Please feel free to pause bits and rewind bits to help you as you're learning this new topic.

Anything that I can do to help, I'll try to add in as we work our way through.

I really do hope there's bits of these algebra that you're really, really gonna enjoy.

Lots of chances for you to explore things and to develop your algebraic reasoning and thinking skills.

Right, let's get started then.

Today's lesson is gonna be on algebraic terminology.

That means there's gonna be lots of terms, lots of words we're gonna use today.

So by the end of the lesson, you'll be able to correctly identify, term, coefficient, factor, product, expression, formula, identity, and equation.

That might seem a lot at the moment, but some of these you'll have used before and we're gonna talk through each one individually, so don't worry.

So we're gonna look at the keywords now, but these are gonna come up throughout the lesson.

So if you're not a hundred percent sure about them at the moment, don't worry, you might wanna look back at them once you've reached that part of the lesson.

So an identity is an equation that holds true for all values of the variables.

The identity symbol is used, so it's this one here with kind of three lines.

And that symbol is used to show that two expressions are equivalent and therefore form an identity.

A formula is a rule linking sets of physical variables in context.

Note that the plural of formula is formulae.

A numerical coefficient is a constant multiplier of the variables.

That was the numerical coefficient.

The coefficient in general is any of the factors of a term relative to a given term.

So if I had AXE, I might say that A is the coefficient of X.

Equally, I could say that X is the coefficient of A.

We're mostly gonna be sticking to numerical coefficients today.

And the last one for the moment, a constant.

So a constant is a term that does not change.

It contains no variables.

And we'll look at some examples later on.

We're gonna start by identifying coefficients.

So below is an expression.

You may have come across expressions before.

And this expression is made up of three terms. So there's our expression and then those are the terms. So X and Y are our variables in this case.

So we say that the coefficient of X is three, the coefficient of Y is five, and it's a useful word to know so that you can refer to that value, okay? So the coefficient of X is three and the coefficient of Y is five in this example.

So terms which have a fixed value because they don't contain variables are called constants.

Some examples of constants then, so you can have positive integers like three, you can have negative integers like negative eight, you can have decimal values, 2.

7, you can have fractional values like seven eighths, is any value that is fixed, that doesn't change its value depending on the value of the variable.

So some things that aren't constant, anything that has a variable in it.

So X, 4A, negative N, A squared, 12TXY, none of those things are constants, 'cause they're all going to change their value depending on the value of the variable.

If we look back at the example we had before, the constant term in this case is the seven, because that is always going to have a fixed value of seven, it doesn't change depending on the variable in the same way that 3X would change, 5Y would change.

So they're not constants.

Seven in this case is a constant.

Right, let's come back to this idea of coefficients then.

So what do you think the coefficients of X and Y are in this expression? I've got Y minus 3X.

I'm gonna let you have a think about that one and then come back and look at the answers.

Alright, well done.

So we're gonna think about this idea of subtracting a term.

It's the same as adding the negative term.

So just like all numbers and when you're dealing with negative numbers, we know that adding a negative and subtracting are the same thing.

So one subtract three is the same as one add negative three.

So we can write Y subtract 3X as Y add negative 3X.

So our terms are Y and negative 3X.

Because of this, the coefficient of X is negative three.

Let's look at the Ys then.

You may be happy with this already, but when we have exactly one lot of a variable, we often don't write the one for efficiency, if that's something you've come across before.

However, it does still exist.

So where it says Y, that's another way of writing one lot of Y, or 1Y.

So the coefficient of Y is one, we just don't write it.

It's really important not to think that there isn't a coefficient, 'cause there is, it's a coefficient of one, 'cause there's one Y.

Right, lots of new terms there.

So let's see how confident you are with those.

So can you fill in the correct word in each statement? The expression you're describing is 9A plus 13, subtract 2B, you've got some words that are on the bottom to help you.

Can you put the right word in the right part of the sentence? Give it a go, then come back and check.

Alright, well done.

So the first one, it's the coefficient of A, which is nine.

B is a variable, as is A.

The constant is 13, it's that value that's not going to change.

And 9A is called a term.

There are three terms in this expression, 9A, negative 2B, and 13.

Well then if you've got some or all of those correct, you're already getting to grips with this new language, which is fantastic.

So time to have a bit of a practise.

So for each statement I want you to decide if it is true or false for this expression, as well as writing true or false, see if you can justify your answer.

Give those a go and then we'll look at the next set.

Well done, so the second set of questions, this time I'm gonna give you some criteria.

See if you can write an expression.

Again, there's gonna be lots of different answers for this question, so don't worry if you don't have the same answer to me or the same answer as somebody else.

Your expression just has to fulfil these criteria.

So you need the expression to contain exactly five terms. The coefficient of N.

So it's got to be an N in there somewhere is five.

Two variables have to have negative coefficients.

The variable R has to have a coefficient of one.

The value of the constant, there must be a constant there is positive but smaller than the coefficient of R.

Give that a go and then check your answers against mine.

Right, well done then, let's see if we can bring our answers together.

So the coefficient of Y is three.

That is true, the Y term is 3Y, so the coefficient is three.

There are three terms in the expression, false.

Well done if you noticed that there are four terms, the constant does count, I wonder if that's the one that you forgot if you didn't get that one right.

C, false.

We can't tell that the Xs are bigger than the Ys, because we don't know the value of X and Y.

They're variables, they don't have a fixed value.

So there's no way of saying which of those terms is bigger.

The variable A has a coefficient of zero.

False, it has a coefficient of negative one.

Just because it's not written there doesn't mean it's not there.

Remember when there isn't a coefficient, it means that there's a coefficient of one because it's negative A, the coefficient is negative one.

There's no constant in the expression, that is false.

There is a constant term in that expression.

Doesn't matter which way around we write our terms. So it's not always gonna be the case that the constant is at the end, okay? It can be anywhere within your expression.

And then the last one, the coefficient of X is five.

True, the X term is 5X, so the coefficient is five.

Right, so loads of possible answers for this.

So have a think about the one I've written and see whether yours has the same features.

So one term there has to be 5N, it can be anywhere in your expression, but it needs to be 5N.

If it's somewhere else in your expression, it'll probably say plus 5N.

One term must be R.

Well then if you've written R instead of 1R, the other two should be negative.

They could be any letter that you like, not R or N, 'cause we already have R's or N's, I went with T and S.

And again, they can be any negative value you like.

So I went with negative 3T and negative 4S.

But that doesn't matter which negative values you went for, okay? The constant must be a fraction or decimal between zero and one because it said it needed to be positive but smaller than one, okay? So I went with 0.

5, but you could have come up with any of your favourite fractions or decimals.

Fantastic, now we're gonna have a look at factors of terms. So more great vocabulary that we're gonna use today.

We can find factors of algebraic terms in the same way as you find factors of numerical values.

We're gonna do it by thinking about things that multiply together to give us a product.

So if the product is 2C, what things could we multiply together to get 2C? Have a think and then see if you went with the same one as me.

Right, I wonder if you said the same one as me.

I went with two multiplied by C.

So two multiplied by C is equivalent to 2C.

We are gonna look at that identity symbol there in more detail later on in this lesson.

So don't worry about it for the moment.

The two is a factor, the C is a factor.

So two times C gives us the product of 2C.

Did I miss any though? Are there other factors of 2C? Well done if you spotted this one, there's also one and 2C.

Don't forget that one is a factor of everything.

So when you're writing your factors, you'll always have one and you'll have the the term itself, in the same way as you do when you're writing numerical factors of numbers.

So factors can be any term which multiply to give the required product.

So this includes variables and variables with coefficients as well.

What that means is you can sometimes end up with loads and loads of factors of certain terms. It can be really quite fun to see if you've got them all.

So to identify factors, we can write them as multiplications, just like we did before.

So we think of 18C.

What I like to do is I like to start with the number one to make sure I don't forget and then I go through all my numerical factors.

So one times 18C, two times 9C, three times 6C and so on, okay? Making sure so you'll see that I did three times 6C and the next numerical factor is six, six times 3C, okay? Because four isn't a factor of 18 and five isn't a factor of 18, I know that I've got them all.

Factors can also have exponents.

So what are the factors of 4C squared? So again, I like to start by just writing, choosing the numerical factors.

So one times 4C squared, two times 2C squared, and four times C squared.

So those are the three numerical factors of four.

Then you can think about your factors which are variables, okay? So for example, C's also gonna be a factor, 4C squared is four times C times C.

So C must be a factor, so C times 4C and then if C's a factor, then two C must also be a factor.

So I've got 2C times 2C.

So sometimes it's say if you're trying to make sure that you've got 'em all, you might wanna think all about your constant factors first, and then thinking about the variables and the possible coefficients your variables could have.

So we can also have factors which are the product of multiple variables, okay? I'll show you what that's like with 25PQ.

So just like before, I started with my constant factors, so one times 25PQ, five times 5PQ, and 25 times PQ.

Right, now I'm just gonna look at P, okay? So if I just take P as a factor, I've got P times 25Q, I can then have 5P as a factor, so 5P times 5Q or I could have 25P as a factor, 25P times Q and then I just wanna check whether I've got Q as a factor somewhere.

Yes I have, okay? And I've got PQ as a factor somewhere, yes I have.

Right, have a go at this question then.

So which of these are factors of 24AB squared? You don't have to try and come up with all the factors, I just want to know that which of these four are the factors or could be factors of this value.

There will be loads of others as well for this one, okay? So give that a go, we'll check it together.

Right, well done.

So 8AB and B are both factors.

AB squared is A times B times B, so A squared is not a factor.

14 is not a factor of 24, so although A is a factor of AB squared, 14 is not a factor of 24.

All right, now you're gonna have go at practising for yourself.

So you are gonna fill in the missing factor so that each product is 6AB.

So I've got one term for you, you just need to fill in the other term.

Once you've done that, you're gonna write down all the factor pairs of 5N squared, see if you can get them all without repeating any.

Come back and then we'll have a look at the second set.

Brilliant, well done for question one and two.

So question three, you're gonna decide whether each statement is true or false.

Again, see if you can think about a reasoning as you are doing that.

And then question four is my favourite question, so you're going to have about changing one thing to make the statement correct.

So it's currently incorrect and you need to try and change one thing so it is then correct.

If you are already confident with this, maybe see if you can do that in some different ways.

There are a couple of different ways that you could change something and then make that statement correct.

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

Fantastic, so we're gonna fill in the missing factor.

So the first one, you should have got one.

Second one, three.

Third one, 2AB.

And the fourth one, six.

And that first column's doing all the constant factors.

Right, middle column you should have B, to make A times 6B, then 3B, then 3A, then 6A.

Final column you should have B, 3A, 2A, and then A.

Notice there's some repetition there, okay? The middle column and the end column are actually the same.

I've just written them a different way round.

Alright, for your factor pairs of 5N squared, you should have one and 5N squared, 5N and N, five and N squared.

Looking at question three then.

So three is a factor of 3X, that is true.

10X is a factor of 5X, that is false.

4X is a factor of 8XY, that's true, 4X times 2Y would give you 8XY.

And XY is a factor of 4X squared, that is false.

Right, couple of different ways you can do this bottom one.

So what you could have done is you could have changed the 3A to a 3AB, and then 3AB times 4B is 12AB squared.

Or you could have changed the 4B to a 4B squared, so 3A times 4B squared is 12AB squared.

Or you could have changed the product to just 12AB, 'cause 3A times 4B is 12AB.

Right, well done with that one.

Let's have a look at the next section of our lesson.

Right, now we're gonna have a look at all those exciting new words.

So expressions, equations, formulae and identities.

So with this bar model, we can write an equation for X, you've probably come across equations before.

So we can write this as X plus four equals 10, and that is an equation.

We can actually use the bar model to see that that's X equals six in this case.

The reason why we looked at that is so you can look at this one.

What do you notice about this bar model? Have a bit of a think, pause the video and then we'll have a look at it together.

Right, we can write an equation, X plus X equals 2X, but X can have any value and this will always be true.

I can't suddenly tell you that X is five, okay? Or X is 10 or X is six, 'cause actually any value for X will work, X plus X is always 2X, no matter what X is.

We can say that X plus X and two X are identical expressions.

They're just essentially different ways of writing the same thing.

So when two expressions are identical, regardless of what X is or whatever variable we're using, they form an identity, 'cause what we're saying is those two expressions are identical.

So X plus X and 2X is an identity, not an equation.

What that means is it should be written with an identity symbol, that's that symbol with the three lines, okay? So actually I'm gonna change those equations because they're not equations, they're actually identities.

So really when you're writing something like this, you should be using the identity symbol because X plus X is always 2X no matter what the value of X is.

Okay, let's see if you've got your head around what an identity is.

So have a read of these three.

Which of these correctly shows an identity? Off you go.

Awesome, if you spotted that it's that second one.

That top one isn't an identity.

Two times T is not always equal to four.

Two times T is only equal to four when T is two, okay? It shouldn't have an identity symbol.

It's wrong that it's written with an identity symbol.

The second one though is an identity.

It's true for any values of R and Y.

It's just another way of writing R times Y and RY.

The different ways of writing the same thing.

The bottom one, those two expressions do form an identity, but I didn't write my identity symbol, so that's not written correctly.

So that should be written with an identity symbol in, 'cause A plus A and two A are identical expressions.

Fantastic, so we've looked at expressions and equations and we've just learned at what an identity is.

So the last of our key vocab to look at today is formulae.

A formula is a rule linking two or more physical variables in context.

So we're applying this to something in real life or in a context.

What formulae do is they show how things are related to each other.

So it's a relationship between two or more physical things.

You might have seen this one before.

A equals B times H or A equals BH, is the formula for finding the area of a rectangle, so that's what the A is representing, given the base and the height.

Here are some other formulae you may have heard of, okay? It's not a problem if you haven't, see if you can look at those and recognise any.

Are there any other formulae you know? Lovely, so just having a look at those then.

So this first one, the volume of a cuboid is the length times the width, times the height.

So it might have been something you've used before.

Don't worry if you haven't.

This one really well done if you recognise this one.

This is actually a formula shown the relationship between degrees celsius and degrees fahrenheit.

Two different ways of measuring temperature, okay? Again, not something I expect you to have seen before, but is one that can sometimes be useful when you're converting between temperatures.

E equals MC squared often gets referred to in popular culture, it's a relationship between energy and mass and it's attributed to Einstein's Theory of Special Relativity, okay? Not expecting you to know it or to use it, it is just another example of a formula that does say come up quite a lot in pop culture references.

And S equals D divided by T is a formula showing the relationship between speed, distance, and time.

This is a formula that you will use at some point in your mathematics education.

In all these cases, knowing the value of some of the variables, if I tell you what some of the letters represent in terms of what value they are, you'd then be able to calculate the other variables.

So I would like you to have a go at sorting these into equations and expressions.

So you've got six algebraic statements there.

Can you sort them? So you've got equations and expressions, off you go.

Brilliant, so your equations are given to you on the left, your expressions are those three on the right.

The equations tell us the expressions are equal.

So a good way to spot if something's an equation is to look for that equal sign and then it's either equation or it's a formula depending on the context.

Expressions are just collections of terms representing a value, they don't have equal signs, okay? So well done if you've got all of those six in the correct place.

Okay, so formulae contain more than one variable.

Do you think that is true or false? Think about justifying your answer as well.

Brilliant, that is true.

Formulae show a relationship between more than one variable.

So that is that second justification.

So there has to be more than one variable because we are looking for that relationship between two or more things.

Lovely, let's see if you've got your terminology spot on.

So you've got equation, expression, identity and formula down the left hand side.

See if you can match them up with their definitions.

When you've done that, have a go at writing your own example for each, okay? You could use some of the examples that you've learned today or even better if you can come up with one of your own.

Off you go.

Brilliant, so now we're gonna think about identities.

So which of these equations are identities? None of them have the identity symbol at the moment, okay? So you are just looking at whether they should be identities, okay? And then can you rewrite those ones with an identity symbol? So the ones that should be identities, can you write with an identity symbol? Off you go and come back to the answers.

Lovely, well done.

So an equation is a mathematical statement show two or more values are the same for certain values of the variables, showing that two things are equal.

An expression, one or more terms representing a value connected by the operations, plus or minus, okay? So an expression is just representing a value, okay? It is not showing that things are equal.

An identity is a mathematical statement showing that two values are always the same regardless of the value of the variables.

That's the important thing with identities.

Doesn't matter what X is or N is, whatever letter you're using, okay? Those things will always be the same.

And last, a formula is a mathematical statement showing the relationship between two or more physical variables.

Amazing if you've got all four of those matched up correctly.

There's so many examples for these, I'll talk you through mine, okay? And you might want to check yours against a partner if you have that opportunity.

So an equation should have an equal sign and only be true for certain values of the variable.

So 3A plus four equals 10 is an equation, that's only true for certain values of A, and you could work those out if you wanted to.

An expression, any set of terms added, subtracted or divided should not have an equal sign.

So I've gone with 3A plus four, I've got two terms connected with an addition symbol, that is an expression A formula, this is quite tricky to do without giving a context.

So I've gone with 2A equals 3B plus C.

So I've got a couple of variables and if I knew what B and C was, I could work out A, if I knew what A and B was, I could work out C it's a relationship between those variables.

And the last one, an identity.

So I've gone with A times A is the same thing as A squared.

There's loads of different ones you could come up with.

Don't forget to use that identity symbol.

Lovely, so which of these are identities? That first one is an identity.

A plus A plus A plus A is the same thing as 4A.

So make sure you rewrite that with the correct symbol.

B plus three equals eight isn't.

It's only true when B is five.

C multiply by C is C squared.

Yeah, that's true, that's always the case.

So that's an identity.

Make sure you rewrite it with the identity symbol.

D times 2E is 2DE, yeah those two things are always the same.

So that should be an identity.

On the right hand side, X equals 3F plus 2G.

No, that value of X is gonna change depending on what F and G is, okay? It can't always be true for any value of F and G.

That X is gonna actually change depending on F and G.

H plus H minus five equals negative 10 plus H.

No, that's only true for certain values of H, I've worked it out for you, that's only true when H is negative five, okay? It's not something where the left hand side and the right hand side always have to be true of no matter what H is.

Three times K is the same thing as K plus K plus K, yes it is, so that's an identity.

And last one, two times M equals P plus P plus P plus P.

No, that's not always true.

It is true when P is half of M, okay? But two lots of M doesn't have to be P plus P plus P plus P, it's all gonna depend on the values of M and P.

Brilliant, I hope you enjoyed learning all that new terminology, okay? And that that's something you can bring in when you are talking about your mathematics.

So words such as term, coefficient and constant can be used to describe expressions.

Factors and products can be found for algebraic terms. You did lots of that today, finding factors in exactly the same way as you did with numbers.

Two expressions that are always true for any value of the variable form an identity.

And we learned how to use that identity symbol today, okay? Again, a nice challenge for you to try and use that identity symbol where it is correct in your mathematics, 'cause lots of people will use an equal sign, when really it should be that identity symbol.

And lastly, formula show a relationship between multiple variables in context.

Right, fantastic guys, I hope you enjoyed that as much as I did.

I'll look forward to seeing you again.