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.

Right, welcome to our lesson on representing a generalised number.

By the end of this lesson, you'll be able to use a letter to represent a generalised number.

And we'll talk more about what that means in a moment.

So our keywords are really important today that you're using the correct one and you understand what we mean.

So take some time to read through these.

So an unknown is a quantity that has a set value, but it's represented by a symbol or a letter.

So an unknown does have a value, but potentially we don't know what it is or we're gonna work out what it is and we're gonna represent it by a symbol or a letter.

A variable is slightly different.

A variable is a quantity that can take on a range of values.

We might also represent it with a letter, but it doesn't necessarily represent a set value that you're gonna work out.

It could take on a range of values.

Both unknowns and variables are often denoted by a lowercase letter and you'll see that it's generally typed in italics.

So we're gonna start with this first idea of representing an unknown as a letter.

We can use letters to represent unknown numbers in a scenario.

So we're gonna think about this scenario.

Sofia has some money in her pocket.

She buys a can of lemonade.

The lemonade costs her a pound.

She now has five pound in her pocket.

We can form an equation from this scenario using a letter to represent the unknown amount of money.

So we don't know how much money Sofia has in her pocket.

So we're gonna use a letter to represent that.

We do know how much the can of lemonade costs and how much she then had.

So we might write this as A, subtract one equals five.

I've used the letter a to represent the amount of money that Sofia has in pounds and sometimes it's really important to state what that letter is representing.

So I said where a represents the amount of money in pounds.

We can use any letter to represent an unknown.

I chose a in the previous example, but you could have chosen any letter to represent that unknown.

Some letters do have specific meanings in mathematics.

It's not something you need to worry about at the moment, but you might come across some of these in further topics where a specific letter has a specific meaning.

It might mean that as you develop your mathematics, there's some letters you will choose not to use.

You will notice that when I'm writing algebra, there's certain letters I use more often than others.

Why don't you pause the video and have a think about any letters which we might prefer not to use to represent a number.

As I said previously, we can use any letter.

It's generally not a good idea to use the letter O as it resembles the number zero and they're often too close in people's handwritings or on computer programmes to differentiate between the two.

So generally, you'll not see the letter O used to resemble a number.

You might have said x.

You might have said that x looks a lot like a multiplication sign.

We actually regularly use the letter x, but notice that we style it in italics so that it doesn't resemble a multiplication sign.

You might wanna practise try drawing some x with the style that you've got on the screen so that it doesn't look the same as your multiplication sign.

It's really useful as you're learning to write mathematics that you get used to writing down letters in a way where they don't look similar to each other.

So for example, the letters U and the letters V.

In your handwriting, you might wanna practise making your Us look different to your Vs so you don't get them confused when you're using them as unknowns.

So we can use letters in bar models.

So you may have used the bar model before.

We can now put letters into our bar models to help us represent equations.

So Sofia is thinking of a number.

She adds 15, then subtracts three, and she gets an answer of 18.

So let's think about how we can represent this in a bar model.

So Sofia's unknown number, I'm gonna represent as r.

I don't know the size of the number, so I don't know the size to draw my bar.

I know that when I add 15 then subtract three, that's gonna give me an answer of 18.

So now I've got a bar represented below, showing Sofia's letter or Sofia's number that we're representing with a letter.

Again, it doesn't matter what letter you use to represent Sofia's unknown number.

I have chosen r in this case.

So let's see if we can work the other way around.

Let's think about equations that we can write from this bar model.

You might wanna pause the video and see how many you can come up with before watching for my answers.

So the one I went with first is the one we had before with Sofia's scenario.

We've got r plus 15 minus three equals 18.

There's some others that you might have had as well.

You might have said that that top bar and the bottom bar are the same size.

So that means r plus 15 is equal 18 plus three.

You might have thought about how you could calculate r and say that r is 18 plus three minus 15 or you might have had that 15 is 18 plus three subtract r.

It is possible that you came up with other ways around of writing these.

Because addition is commutative, there are some that you might have had that looks slightly different to mine.

Well done if you found any variations.

Right, let's check to see if you're happy with this then.

So this is Sofia again.

I'm thinking of a number, I subtract 10, then add one and my answer is two.

Which of these equations represents Sofia's number? Fantastic.

If you spotted it, it's that first one.

Note that we don't need to use an s because it's Sofia's number.

Okay, don't fall into the trap of thinking that the letter has to represent the person or the thing that we're calculating.

Any letter can be used.

So it's gonna be that first one in this case.

Great, well done.

Time for you to have go at practising.

There's quite a lot to do in this activity.

So I'm just gonna talk you through it.

So the first one, you've got Aisha, Andeep, and Alex, and they're all thinking of numbers.

You need to write an equation for each number described by those pupils.

Okay, so see if you can write an equation using a letter to represent an unknown.

Question two, you've got a bar model wall.

So just like a normal bar model where things have equal length, that means they're equal.

So the 14 is gonna have the same length of the r plus six, which is gonna have the same length as the a plus a plus two and the same length as the two plus x.

What I want you to do is use the bar model wall to fill in the blanks in these equations.

There may be some that have multiple answers, but most of them have a single letter or a single number as the answer.

Give that a go and then we'll look at it together.

Right, well done.

So remember, any letter can be used.

I've stuck with a at the beginning of the alphabet.

So for Aisha, I've got a, subtract 32, equals 16.

For Andeep, I got a plus five plus 10 minus 11 equals one.

And for Alex, I've written a minus 34 add a 'cause he wants to add on his original number.

So you said add a again equals 22.

There are slight variations of those.

I feel like that's the most common way that you may have written it, but if you've written your additions and your subtractions another way around, that's absolutely fine.

So the bar model wall then.

Your first missing gap should have been r, your second one you should have had two, your third one, you also should have had two.

The last one, there could have been several things you wrote.

So you could have written x, but you might have also noticed that a plus a is the same as 14, takeaway two, or you might have written that as 12, or you could have written it as r plus six.

Takeaway two or x plus two.

Takeaway two.

So there are several ways of writing it.

I think the most obvious one there was the x, but well done if you spotted some of the others.

Fantastic.

We're gonna have a look at the second half of our lesson.

So we've looked at how we can represent unknowns as a letter, we're now gonna look at representing a variable as a letter.

So in some scenarios, the missing number is not a fixed value.

In these cases, we call the missing number a variable.

The word variable obviously comes from this idea of varying, okay? So when something can vary, we can use a variable to represent it.

We use letters in exactly the same way as we represent unknowns.

Let's look at this example.

So Andeep wants to hire a paddleboard.

He sees two companies offering different prices.

Company one charges 11 pounds plus one pound per hour.

So you're always gonna pay 11 pound, but then it's one pound per hour you want to hire the paddleboard.

Company two charges three pound per hour.

So they don't charge a fixed rate.

It's just three pound per hour that you want to hire the board.

I'd like you to pause the video and have a little bit of a think about which is the cheaper option.

You might wanna spend a bit of time here writing some things down.

Maybe there's something you can draw that can help you think about it.

Which of these companies would you pick if you were going to hire a paddleboard and you wanted the cheaper option.

Right, have some thoughts and then we'll come back together and we'll bring our ideas all together.

Right, well done.

Hopefully you've explored that a little bit yourself.

You may have come to this conclusion which is absolutely fine.

Don't worry if you didn't.

So you might have come to the conclusion that the answer to this question will change depending on how many hours he wants to hire the paddleboard.

Well, then if you went further and looked at when one of them was cheaper than the other one, okay? What essentially we're saying is that the amount of time he wants to spend on the paddleboard will change.

Which of these become the better option? So it's going to vary depending on how long we wants to spend on the paddleboard.

I'm gonna put this into a table so we can see how the values change.

Tables are really useful tools at spotting patterns, particularly with sort of variables when we don't know what the value of that letter is.

So I've gone with six hours as a maximum here, but you could have made your table even longer and we're gonna think about how long it'll cost for each of the company.

So company one is gonna be 11 pounds plus one pound per hour.

So for the second, for two hours, it's gonna be 11 pounds plus two pounds.

For three hours, 11 pounds plus three hours and so on.

For company two, it's just three pounds per hour.

Lovely.

So that table is showing us our different options depending on how many hours he's hiring it for.

So I want you to think in a bit more detail now.

When is that first company cheaper? When is the second company cheaper? And what would happen if we continue that table? You might wanna think about some of the patterns you can see with each of the companies, okay, and see if that helps you get this idea of when the two companies are gonna be cheaper option.

Off you go and then come back and we'll share our answers.

Brilliant.

What you might have noticed that company one is cheaper for six hours.

You might have noticed that company two is cheaper up to six hours.

So for one through to five hours, that company two is going to be cheaper.

If we're carrying that table on, company one will continue to be cheaper because it's going to go 17.

You might have spotted a pattern.

You might have seen it's gonna go 17, 18, 19, 20 with company one.

Whereas company two is going up in threes, isn't it? So it's gonna go 18, 21, 24, 27, and so on.

We can generalise the rule for the cost of the two companies.

So we're gonna use this idea of a variable to generalise the rule.

Think about what it is that varies in this scenario.

So what is it that's changing? What is it that we don't know the value of which is changing which company we might pick.

Right, it's the number of hours that varies.

We are changing the number of hours he might wanna hire the paddleboard for, and this then has a direct impact on the cost.

That means we can use a variable to represent the number of hours.

We can choose any variable.

I'm gonna use x in this case.

So we've got one pound per hour.

So that's gonna represent be represented by x.

Okay, so x is gonna be one if it's one hour, two if it's two hours, three if it's three hours.

And then there's always gonna be a plus 11 'cause you've always gotta pay 11 no matter how long you're hiring it for.

You could write that as x plus 11 or 11 plus x.

Company two is three pound per hour.

So it's just three times the number of hours.

So however many hours he wants to hire it for.

If he multiplies it by three, that'll give him the cost of the paddleboard.

As we've just explored with our scenario, there is no way to tell which is smaller, 3x or x plus 11 'cause it changes depending on what the value of x is going to be.

And that is the quality of a variable.

Is that 'cause it can vary, we can't tell the size of the value we have.

Either three x in this case or x plus 11.

We can also represent the shoes in a bar model.

Notice my bars aren't equal length 'cause I'm not saying that anything is equal.

I'm just comparing these ideas.

So I've got my time in hours as x.

That means company one must be x plus 11 and company two must be three x's.

All right, just stop and have think.

Is this bar model also correct? Pause the video to formulate your answer and then see if you agree with me.

Right, well done if you said something similar to this.

The number of hours can be any size.

Remember, it's a variable.

It doesn't have a fixed size.

That means it could be as big as it is in the top example or it can be smaller like the bottom example.

We also don't know what size it is in relation to 11.

We don't know if he's gonna hire it for more than 11 hours.

It's unlikely, but it's possible, or less than 11 hours.

Okay, so we don't know how big to draw the x in relation to the 11.

In this case, it might not have that much of a use in terms of figuring something out, but bar models are a good way to show the relationship between things.

Right, see you what you think about this statement then.

When you draw a bar model, the variable can be any length.

Do you think that's true or false? And then see if you can justify your answer.

Right, well done if you thought that that was true.

The variable can be any length.

What was your justification? It's this one.

The value of the variable can change.

So the bar is just a representation.

It doesn't have to represent the number, you don't have to measure it and make it a certain length.

It's representing a variable so it can be any length you like as that representation.

Brilliant time for you to have a go then.

So in this scenario, Andeep and Aisha are playing a game.

They each collect two different types of tokens.

They can either get a square or they can get a circle.

This is what they end up with at the end of the game.

So Andeep ends up with three squares and two circles.

Aisha ends up with six squares and a circle.

We don't know who has won at the moment because we don't know what the squares and the circles are worth.

This is where you're gonna come and answer some questions then.

So if Andeep won the game, what would this tell you about the value of the circle compared to the square? So if Andeep is winning with this amount of counters, what is that gonna tell you about the circle and the square's values? If you want a little bit of help with this one, you might wanna draw some things out or you might wanna get some counters and move them around to see if you can work out what's happening with this example.

For question two, how could you write a generalised rule for the number of points Andeep has? So remember the squares are representing the number of points for a square and the circle a number of points for a circle.

See if you can use what you've learned about using letters to write a generalised rule.

How could you write a generalised rule for the number of points Aisha has? Give those three a go and then we'll look at the next set together.

Well done, lots of good thinking in that one.

So question four, what value could the tokens have if the players drew the game? So if Andeep's points and Aisha's points are the same, what value could the tokens have? Again, you might want to use tables, you might wanna use counters to help you work this one out.

Question five, if Aisha wins the game when the circle is worth five, so we're gonna try this with the circle being worth five, what could the square be worth to make Aisha win? And then if the circle's worth five that Andeep wins, what could the square be worth? So again, we're gonna keep changing that value of the square to see if we can make Aisha win, and then make Andeep win.

Right, try your best with those ones and then come back for the answers.

Brilliant.

There's lots of ideas to explore behind that.

So hopefully, you found some fun bits with working out what these squares and these circles could be, particularly if you thought a little bit outside the box with some of your values, and I'll look at what I mean by that in a moment.

So what I think has really helped with this one is if you group the drawings together.

So I've put Andeep's three squares in one circle together.

And then he has an extra circle, doesn't he? Aisha also has three squares in one circle, and then she has three squares left over.

So I've kind of ruled out the fact that they've each got three squares in a circle, and then I'm focusing on what extra they've got.

So Andeep has an extra circle, Aisha has an extra three squares.

So for Andeep to win, the extra circle he has must be worth more than the extra three squares that Aisha has.

So that was quite an interesting concept there.

Well done if you managed to put that into words similar to me.

So for Andeep to win, one circle must be worth more than three squares.

I'm looking at the relationship between the circle and the squares.

Brilliant, a generalised rule then.

Again, this is bringing together a lot of the ideas you've been thinking about in algebra.

So well done if you managed to get something similar to me.

I've used a and b just 'cause it's at the beginning of the alphabet.

So I've written this as three a plus two b, where a represents the points for a square and b represents the points for a circle.

In terms of Aisha's rule then, you'd write that as six a plus 1b or six a plus b where we only have one other variable.

We often don't write the one just for efficiency.

So if you wrote six a plus b, it's the same thing as writing six a plus 1b.

One of them is not wrong.

It's just the idea of six a plus b is slightly quicker to write.

Again, I've made it clear what the a and b represent here.

So well done if you added that to your answer.

So what value could the tokens have if the players drew the game? Well, it's any value where one circle is worth the same as three squares.

If you think back to what we drew for that first question, it'll be easy to see if they are the same, then that extra circle Andeep has has to be the same as the extra three squares Aisha has.

Loads of examples you could have come up with this.

So you could have had a square as one and a circle as three.

You could have had a square as two and a circle as six, a square as three and a circle of nine, or you could have gone with negative numbers.

A square is negative two, a circle is negative six.

You could have been really creative and come up with some decimal values.

So a square is 0.

5, a circle is 1.

5.

As long as your circle is three times your square, then these two players are going to draw.

Right, and lastly, if Aisha wins the game when the circle is worth five, what could the square be worth? There's lots of options for this.

If you start to enter your values, you're looking for anything greater than one.

So two would work, three would work, four would work.

You have to be a bit careful if you try to use non-integer values.

It's around 1.

66 recurring.

So if you went with a non-integer value bigger than 1.

6666, you're fine.

Okay? Otherwise, use those integer values, two, three, four.

For Andeep to win the game, any integer less than two.

So one would work, zero would work, negative three would work.

Again, you've gotta be careful with the non-integer values.

So anything less than 1.

66 recurring again.

So 1.

5 would've worked, 0.

5 would've worked, anything like that.

Brilliant, let's look at what we've learned today then.

So a letter can be used to represent an unknown value in a scenario.

A letter can also be used to represent a variable number in a scenario.

We know that we can use letters to represent unknowns and variables and we can write equations for those.

We know that bar models can help us represent a scenario with an unknown or with a variable.

Brilliant, there was lots packed into that lesson today.

I hope you found some of it enjoyable.

I hope you had a chance to explore a little bit of unknowns and variables and learn some new language that you can bring into your mathematics.

Thank you very much.

It'd be really nice to see you again soon.