Lesson video

Right, well done and thank you for loading the video for this lesson.

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

Feel free to pause things, rewind things, so that you are really comfortable with the ideas that we're exploring and I really hope there's bits and pieces that you find really enjoyable and really interesting as we work through.

Right, let's get started then.

All right, welcome to the lesson on substituting particular values.

In this lesson you'll learn how to substitute particular values into a generalised algebraic statement.

We're gonna do that to find a sense of how the value of an expression changes.

So a substitute means to put in place of another.

You might have come across this word in other meanings in the English language.

In algebra, substitution can be used to replace a variable with a value or to replace multiple variables with multiple different values.

So we're gonna start by evaluating expressions.

We can evaluate an expression for a given value of its variables by using this idea of substituting.

So I'd like you to consider the expressions below.

What would the values be if x equals six? If you've got an idea, you might wanna pause the video and write them down.

Otherwise we'll talk through them together on the next slide.

So let's start with the idea of three x.

Three x doesn't have a value at the moment until we give x a value.

So we're gonna say that x equals six.

What that means is three x is three times x, so we're gonna do three times six.

We're substituting that six for an x.

Notice how I'm writing my working out down the page and I'm showing where I've substituted the value of six in using brackets.

I do that for several reasons.

One is it makes it really clear which is the value you're substituting in.

It also helps when I'm working with negative numbers and if I'm using a calculator that we'll look at later on in this lesson, I can use my arrow keys to go back and change that six into a different value that I can substitute for x.

Finishing that one off then three x has a value of 18 when x is six.

If we do the same for x plus seven, so x is gonna be six we'll now looking at the value of x plus seven.

So we're gonna do six plus seven, which is 13.

We do the same for x minus two, we're gonna do six minus two, which gives us four.

With x squared, I'm gonna do six squared, which gives me 36.

And our last one x divided by three.

When x is six, we're gonna do six divided by three, which gives us two.

So we've evaluated those five expressions when x is six.

It's important to remember our priority of operations when substituting.

So we're gonna stick with evaluating when x equals six, but I've got five x plus one and five brackets x plus one.

And let's look at how they're going to be different.

Five x plus one means five lots of x, add one.

So I'm gonna do my five lots of x, which is 30 add one which is 31.

When I've got a bracket round my x plus one that is grouping my x plus one together.

If I'm replacing an x with a six, then I can do six plus one to work out the value of my bracket.

The value of my bracket is seven and then have five times seven, which is 35.

I just want to draw our attention to using exponents when we're substituting.

So I've got 10 lots of x squared and I've got 10 in a bracket all squared.

So this one here, remember the exponent only acts on the letter or the number that it is after.

So that squared only applies to the x in that first case.

So if we do six squared, that is 36 and then we've got 10 lots of six squared, so 10 lots of 36, which is 360.

With this one here, the squared applies to the whole of 10x.

So it's useful to work out the value of 10x first if we can.

If x is six, then 10x is 60.

We're then doing 60 squared, which gives us 3,600.

Right, we're gonna use a slightly more complicated expression here.

I'm gonna show you how to do it on the left-hand side and then you are going to have a go yourself on the right-hand side.

Watch the steps that I'm doing, pay particularly attention to the order that I'm doing things in.

So I'm gonna substitute a for four.

And remember I'm gonna put my brackets around my four to remind me that that is the value I've substituted in.

I've written the rest of the expression just as it's written in the question.

So I've got two lots of four squared and I know that means two times.

So I've got two times four squared, subtract six, all over 13 and then I'm gonna look at that exponent.

So remember that squared only applies to the four.

So I'm gonna to evaluate that.

So Four squared is 16, so I've got two multiplied by 16, subtract six over 13.

And again, I'm working down the page, I'm making it really clear which bits I'm evaluating.

I can then do my multiplication, remembering my priority of operations.

So I've got 32, subtract six over 13 and then that gives me 26 over 13.

Finally, I can evaluate my division to get an answer of two.

Right, time for you to give this one a go.

Have a look at the steps I've done on the left-hand side and see if you can apply that to this question.

I'm hoping guys that you looked at that expression to start with and maybe thought it looked a little bit complicated.

It's got quite a few elements, but if you follow the steps of substitution, then actually it doesn't end up being very complicated at all 'cause you are just inputting that value and following your priority of operations.

So we're gonna start by squaring our three to get our value for a squared.

So a squared is nine, so we've got five times nine plus one over two.

Five times nine is 45, plus one is 46, we've got 46 over two.

It gives us 23.

You should be really proud of yourself, especially if you've got your working out written really nicely down the page like mine.

We're gonna try another one then.

So we're gonna evaluate for a equals four.

And again, it doesn't matter how complicated my expression gets, I'm just gonna substitute a four in wherever my a is.

So I've got a hundred subtract three times and then instead of a, I'm gonna write four.

So I've got four plus two all squared.

Now that exponent applies to the whole bracket.

So I'm gonna start by evaluating that bracket.

So if I evaluate the bracket, I get six.

So I've got 100 subtract three times six squared.

That exponent remember, applies to that bracket.

So now that I know that the bracket has a value of six, I can square it.

So 100 subtract three times 36, which gives me 100 subtract 108.

Final answer then is negative eight.

I've just gone one step at a time evaluating and look how neat it all looks.

It's really easy for me to see if I've made a mistake and to follow my working.

Give this one a go yourself then with a having a value of three.

Well done guys.

You showed loads of skills in that question.

Let's just talk ourselves through it.

So we're substituting three in for a and then we want to evaluate that bracket.

So that bracket has a value of four and that enables us to square it nice and easily.

So four squared is 16, so we've got 10 times 16, subtract five.

Remembering my priority of operations.

I've got 160, subtract five, which gives me 155.

Lots of thinking involved there.

Well done guys.

When variables are being substituted for negative values, the process is exactly the same.

If we're using a calculator like we've talked about before, make sure you put a bracket around the value you have substituted.

Let's try 50 subtract three n squared when n is negative five.

So I'm just gonna replace my n for a negative five.

So I've got 50, subtract three negative five in brackets, squared.

Remember negative five squared is positive 25.

So I've got that as 50 subtract three lots of 25 which is 50, subtract 75.

Final answer is negative 25.

If you're using a calculator, you can write that top line in all in one go making sure you've got a bracket around your negative five.

Then when you evaluate it, it'll give your answer in the bottom right-hand corner.

The useful thing about that is you could then use your arrow keys and change the negative five to a different number.

If you then wanted n to have a different value.

Aisha is evaluating the expression 10 subtract three b squared when b is two.

Have a look at her lines of working.

Can you spot any mistakes she has made? Good first spot if you got this one, Aisha should have only squared the two.

What she's done is she's multiplied the three by the two, but remember that squared only applies to the two.

So she wants to square the two to get a value of four and then she can do a multiplication.

You might have also spotted that she ends up with 10 subtract 36 and she writes 26.

We know that that should be negative 26.

It looks like she swapped her values around.

Subtraction's not commutative.

We can't swap the values around as we like.

So for each expression I'd like you to evaluate when a has a value of two.

Give those a go and then we'll look at our answers.

Fantastic work guys.

So we've got the same expressions this time, but I want you to think about their value when a is negative five.

Be particularly careful when you are squaring your negative numbers.

Off you go.

Let's look at our answers then.

So five a when a is two has a value of 10.

10 subtract a has a value of eight.

Four a plus seven, you might have wanted to do it in two steps.

So four a is eight plus seven is 15 and a squared is four.

For the right-hand side, remember we want to square our a, so a squared is four, three times four is 12.

The F, we want to do three lots of a and square the whole thing.

So three times two is six, six squared is 36.

For G again, wants to do this in several steps.

So a squared is two squared, which is four.

Four fours is 16.

So we've got three subtract 16 which gives us negative 13.

Lots going on in this last one, so well done if you did this without any mistakes.

So two a is gonna have a value of four.

So we think about our bracket, that's one plus four which is five.

Our whole bracket is being squared.

So five squared is 25, 10 times 25 is 250.

If we do the same for a is negative five.

So five lots of negative five, negative 25.

B, 10 subtract negative five is 15.

Write it out to help you.

C, four lots of a is gonna give you negative 20.

Add seven, negative 13.

D, negative five squared is positive 25.

Because a squared is positive 25, three times a squared is positive 75.

F, we need to do three a first.

So three lots of negative five is negative 15 square it is positive 225.

Well done if you know your square numbers up to 15 squared.

G, several steps this time.

So a squared is 25, positive 25.

Four times 25 is 100.

So we've got three subtract 100, negative 97.

And the last one, so two lots of a, two lots of negative five is negative 10.

One add negative 10 is negative nine, negative nine squared is positive 81.

10 lots of 81 is 810.

Hopefully you've got some really nice working out to support your answer so that you agree with every step that I did there.

We're gonna now look at substituting to spot patterns.

So what we can do is we can use substitution to investigate expressions.

I'd like you to look at y squared, two y, and y plus two.

We know that the value of these change depending on the value of the variable, that's why they're called variables because they don't have a set value.

The value can change and that will change the expression.

What we can do is we can use a table to help us investigate them.

So if we look at y squared, I've picked some values for y and then I've got some values for y squared.

So when y is zero, y squared is zero when y is one, y squared is one, when y is two, y squared is four.

And I'm looking at that pattern for y squared.

I can do the same when y has a negative value.

I could pick non integer values as well.

What do you notice about the value of y squared for different values of y? Can you come up with a sentence or two explaining what you can see? For a start, y squared is always positive.

We know that if you square a negative number you get a positive number.

Therefore you can't have y squared as negative.

You might have also looked at how the y squared seems to get bigger in the positive y direction and y squared seems to get bigger as y gets smaller in the negative direction.

Let's investigate what happens with two Y then.

We can think about some negative values as well.

Alright, what do you notice about the value of two y for different values of y? You don't have to stick with the values I went with.

You can explore what happens with other values as well.

Off you go.

Lovely, loads of things you could have spotted.

So you could have said as y increases two y increases for positive values if we look at that zero, one, two, three, four.

Two y seems to increase.

You might have noticed that they seem to increase in twos as well.

It goes two, four, six, eight.

You might have looked to the negative values, okay? And seen that y is getting smaller a lot, two y is getting smaller, a lot quicker than y is.

What would two y be if y is 2.

5? Fantastic, two y would have a value of five.

Don't forget we can have non integer values for our variables.

Let's explore what happens with y plus two.

Again, look at the values I've come up with.

Maybe come up with your own values and what can you tell me about this expression y plus two.

Again you might have said something like as y increases y plus two increases.

You might have said that y plus two is always bigger than y.

You might have spotted that the values for y plus two increases in ones.

What would y plus two be this time when y is 2.

5? We'll look at that together.

Y plus two would have the value of 4.

5.

So this time I would like you to pick a value to substitute into all three of these expressions and see if you can find a value that makes y squared have the biggest value.

Can you find a value that makes y plus two have the biggest value, and can you find a value that makes two y have the smallest value? Play around, write some things down, see if you can spot any patterns that help you.

Lovely, we can use some of those things we explored in our tables before.

Any value greater than two will give y squared the biggest value.

Any value less than negative one will also give y squared the greatest value.

'Cause remember a negative number squared is a positive number.

Any value between negative one and two will give y plus two the biggest value and then for two y, any negative value and two y will have the smallest value.

Last challenge.

Can you substitute the same number into all three expressions and get the same value? Off you go.

Yeah, nice guys, well done.

Hope you spotted they all have the same value when y equals two.

Right, we're gonna apply this to our formula skills.

So here is a formula for converting measures of temperature where C represents the temperature in degrees Celsius and F represents the temperature in degrees Fahrenheit.

So F equals 1.

8 C plus 32.

What would we do to convert 10 degrees Celsius into Fahrenheit? Well we just substitute 10 into our formula.

So we've got 1.

8 times 10 plus 32, which gives us 18 plus 32, which gives us 50.

That means that 10 degrees Celsius is 50 degrees Fahrenheit.

Right, why don't you try the same for zero degrees, one degree, two degrees, see if you can see any patterns.

If we try zero degrees, we get 32.

One degree, 33.

8.

Two degrees, 35.

6.

If you are using calculator, remember you can use your arrow keys to go back, delete, and then change the value in the bracket.

So some patterns you might have spotted, you might have spotted that they're going up in 1.

8s.

Okay, see where that is in our formula.

As the temperature in degrees Celsius increases by one, the temperature in degrees Fahrenheit increases by 1.

8.

We're looking at a relationship now between Celsius and Fahrenheit.

True or false in the formula P equals four L where P is the perimeter and L is the length, P will always be even.

What do you think? See if you can justify your answer as well.

Well done if you spotted that was false.

If you only start with integer values, you might have thought that multiplying by four always gives an even number.

But remember L can be a decimal, odd and even only applies to integers.

So you're gonna give this a go yourself now.

So I would like you to complete the table with the words always or sometimes or never.

The easiest way to do this is to try different values for u.

So for question A if u is a positive odd number, so think about some positive odd numbers.

Is u squared always even? Then if u squared is a positive odds number still is u squared always positive or is it sometimes positive or is it never positive? And what about two u plus 10? Try some different values for u and can you tell me whether it is always less than 10, sometimes less than 10 or never less than 10? This is gonna require quite a lot of thinking and I do suggest you write your answers down as you go to help you explore these values.

Off you go.

Why don't for playing around with those numbers then guys? So if u is a positive odd number, u squared is never even.

An odd multiplied by an odd cannot give you an even value.

U squared is positive, always.

Two u plus 10 is less than 10, never.

When u is a positive even number.

An even multiplied by an even is always gonna be even.

So always.

Always going to be positive and two u plus 10 cannot be less than 10.

U is negative.

Then u squared is even, sometimes.

U squared is positive, always, and two u plus 10 is less than 10, always, if u is a negative value.

If u is greater than five u squared can be even, it can also be odd.

U squared is always going to be positive and two u plus 10 is less than 10, never, if u is greater than five.

And the last one, if u is less than one, u squared is even, sometimes.

U squared is positive, always, and two u plus 10 is less than 10, sometimes.

Okay, if u is between zero and one then it's going to be greater than 10.

But if it's any of our negative values then it'll be less than 10.

Right, well done.

Let's look at our last section of our lesson.

So we're gonna try and substitute with multiple variables.

This is a formula for working out the volume of a cuboid.

We're gonna have a go at evaluating the volume when the length is five, the width is three and the height is four.

So lwh means L times W times H.

So when L is five, W is three and H is four.

We're gonna multiply those together.

We get our answer as 60.

We do need to be careful when we've got negative values.

So we're gonna look at a minus b.

When a is two and b is three to start with.

Then we're gonna look at when a and b are negative.

So a subtract b if a is two and b is three, we've got two, subtract three which is negative one.

When a is negative two and b is three.

So we've got negative two, subtract three and that's negative five.

Notice that I'm writing this underneath with my values in brackets so that I'm not getting in a muddle.

For a is two and b is negative three.

We've got two, subtract negative three which gives me five and the last one negative two, subtract negative three.

We think about a number line, think about starting on negative two.

If you're subtracting negative three, you're gonna go up the number line and get to one.

Okay, so now we're gonna have a go at substituting with multiple variables together.

So we're gonna do the first one together and then you're gonna have a go at doing a similar one on your own.

So we've got three ab, subtract two c when a is negative five, b is negative two and c is negative six.

So I want you to have a look through my working out and then I'm gonna talk through what I did at each stage.

Just have a look first.

Okay, so I've started by substituting in my values using brackets to separate them.

So I've got three times negative five 'cause that's a times negative two 'cause that's b, and then we're subtracting two lots of c, so that's two times negative six.

Right, I can choose any of those multiplications to do first I'm gonna do three times negative five, but it doesn't matter.

So three times negative five is negative 15.

So I can replace the three times negative five with a negative 15 and I'm just gonna write the rest out as it is.

So now I've got negative 15 multiplied by negative two and subtract two lots of negative six.

Again, I can do either of these multiplication, I'm gonna start with negative 15 times negative two.

That gives me positive 30 and then I've got to subtract and I've still got to do two times negative six.

Right, two times negative six remember is negative 12.

So I've got 30 subtract negative 12 or 30 add 12, which is 42.

Right, I would like you to have a go at this one on the right-hand side.

So five a, subtract two bc.

Think about each step of your working, set it out like mine.

Off you go.

Right, well done.

You should have five multiplied by three, subtract two multiplied by negative three, multiplied by four.

Well done with all your negative number skills.

So check that we're happy with that.

True or false, the value of xyz will always be positive.

See if you can justify your answer.

Well done if you spotted that that's going to be false.

If two of them were negative and then one was a positive, then it would be a positive value.

But three negative values multiplied together would give a negative value.

Equally, if just one of them was negative, then that would be a negative answer.

Lovely, time for you to have a practise then.

I'd like you to evaluate each of these expressions with a as three, b as negative one, c as negative two and d as 10.

They do get progressively harder.

I suggest that you focus on writing your working out down the page.

Give yourself plenty of space.

Good luck and I hope there's some nice ones in there to challenge you.

Hopefully you found that as a really nice brain workout.

We're gonna have a look at our answers.

So A, you should have 29.

B, you should have 29.

C, you should have 39.

D, we've got 300.

Make sure you square just the 10.

E, we should have 900.

You're squaring three times 10 that time.

F, being careful with your negative numbers see how I've bracketed them to help me, you should have 15.

Then you should have zero.

Then H, you've got 99.

I, was a really tricky one.

Have a look at my working out there where I've substituted in using a bracket.

You should have an answer of six.

And J, you get 11 over 10 or 1.

1.

You could have written that as one and a 10th as well.

If you found some of those last ones a little bit tricky, do not worry.

The more you practise working with your negative numbers, the easier it gets.

The process of substituting though should be quite straightforward.

It's just then making sure that you don't make mistakes with your arithmetic.

Right, I hope you feel like your brain had a really good workout today.

We have looked at evaluating an algebraic statement by substituting.

We've looked at spotting patterns by substituting different values and then we got onto algebraic statements with multiple variables and if we know all the variables, we can then evaluate that statement.

Thanks for joining me today in that lesson and I look forward to seeing you again.