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 going to 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.

All right, welcome, today we're going to look at securing your understanding of expressions and equations.

By the end of a lesson, you'll be able to manipulate a simple equation using a bar model.

We're also going to review some of the other skills you may already have, but they're going to be so important when we're looking at bar models and when you're doing further algebra.

Right, some of the keywords that we're going to use today are things that you might have come across before, but it's really important that you are confident when using them.

So to start with an equation.

So an equation is used to show that one number or calculation is equal to another number or calculation.

So equations will have equal signs 'cause we're showing that two things are equal.

Right, an exponent, so an experiment is a number positioned above and to the right of a based value.

It indicates repeated multiplication.

So you might have come across the exponent cubed before, which is a little three.

So in this example we have here we've got five cubed, and that means five times five times five.

The exponent in that case is the three.

We're also going to talk about operations that are commutative today.

So an operation is commutative if the values it is operating on can be written in either order without changing the calculation.

So as an example, two times three is the same as three times two.

So multiplication is commutative.

It doesn't matter which way round I write the values two and three, the calculation I'm doing is the same.

So we're going to start by looking at reviewing our priority of operations skills and then we're going to move on to looking at bar models.

So priority of operations, have a think about what you already know and then we'll talk through how this is going to apply when we're thinking about our algebra skills.

When we evaluate calculations, we follow the priority of operations.

This is really important because when we have a calculation written down, we want every mathematician who tries the calculation to come up with the same answer, not have disputes about which way around the operations need to go.

So it's really, really important that we have a clear consensus about what the priority of operations are.

So in mathematics, multiplication and division are evaluated before addition and subtraction.

Again, there's a bit of a misconception.

Sometimes people think that division has to be done before multiplication.

That's not the case.

Multiplication and division are inverses of each other, so they have the same priority of operations.

However, they do come before addition and subtraction in our priority of operations.

So if we have this example, three times four plus two times four, the multiplication has the priority.

So we calculate the multiplication and what I like to do is I like to write down the answer to the calculation underneath.

So the three times four gives me 12 and the two times four gives me eight.

And notice that I haven't changed the order that I've written them.

They've stayed in the order that they're in in the calculation, and then I'm able to add those together.

So 12 plus eight is 20.

Let's do an example with division.

So again, multiplication and division doesn't actually matter which way round I do them.

So I'm just going to work left to right.

So I'm going to do my three times five.

Again, I'm going to write that down underneath.

That's 15, and I can do my four divided by two, which is two, and I'm subtracting those two things.

So 15 takeaway two is 13.

So we can then use brackets to change the priority of operations.

If we don't want that multiplication to be evaluated before the addition and subtraction, we can use brackets to change the priority of operations.

For example, we've got three multiplied by four plus two, and I've put that in a bracket, multiplied by four.

I want to calculate the four plus two, okay? 'Cause I prioritise that by bracketing it.

So four plus two is six.

And notice I keep that in the same place in the calculation.

I just write the answer to that bit underneath, multiply by four.

Now I can multiply these in any order.

So whichever way you find easiest.

I've chosen to do the three multiplied by four to get 12 and then 12 times six is 72.

Let's look at an example with division then.

So three times four divided by two times two, but I put that two times two in a bracket.

So again, priority of operations, let's evaluate that bit in the bracket.

So two multiply by two is four.

And again, I'm just going to write that underneath, okay? We're grouping those together and saying we've got two times two as essentially like a single entity.

So rather than writing two times two, I'm going to write that as four because that's something I can calculate.

Again, we can do these in either order.

So I'll show you both.

We could do three multiplied by four is 12, and then 12 divided by four is three.

Alternatively, we could do the division first.

So we do three multiplied by and then do that division.

So four divided by four is one.

So three times one is three.

And you'll see that we get the same answer either of the time.

And that's because multiplication and division are inverses of each other and they're evaluated at the same time in our priority of operations.

So what we're going to do now then is I'm going to show you how to do maybe a calculation with a couple of different operations in it and then there's going to be one on the right hand side for you to have a go at yourself.

So watch carefully what I do and then use the same skills when you give that a go yourself.

So we're going to evaluate, so the first bit I'm going to think about is I'm going to think about this calculation that I have grouped together using brackets.

So three multiplied by two is six.

So I'm going to write the calculation out again.

Instead of writing three multiplied by two, I can replace that with six.

Right, division has priority.

So I'm going to do the 12 divided by six, and again, I'm going to write that as a single entity, 12 divided by six, I can write that as two.

So that leaves us with the calculation six plus two, which is eight.

Right, here's one for you to evaluate.

Think about the steps that I've done on the left hand side, see if you can do the same thing on the right.

Pause the video, give it a go and then we'll come and look at it together.

Fantastic, well done if you spotted that we've grouped the two times two with brackets.

So if you can calculate that as four, then you've got 10, subtract four divided by four.

So the division has priority.

So four divided by four is one.

10 takeaway one should give you an answer of nine.

Very well done if you've got that value first time.

Right, now we're going to bring in this idea of exponents.

So an exponent operates on the value before it.

For example, that one we had before, which was five cubed, okay? And that exponent of three only operates on the five, even if it was part of a longer calculation.

What that means is if we want the exponent to operate on something more than just that value, we have to bracket it.

So bracketing a calculation means the exponent operates on the whole calculation in the brackets.

What we could do then is we could calculate the value in the brackets if possible and then we could raise it to the power of the exponent.

For example, if I've got four multiplied by three squared, if I work out the value of three squared, 'cause I know that the exponent only acts on the three, so work out the value of the three squared, which is nine, so four multiplied by nine is 36.

This time I've chosen to bracket my multiplication.

So that means I want the square to operate on that whole calculation, the four times three.

So if I actually work out the value of four times three, that is 12.

So I want that power of two to operate on the 12.

So 12 squared, 144.

So we've got a longer calculation to evaluate this time.

It might a little trickier, but it's really not.

We just carry on applying our priority of operations, okay? And then we should get the same value.

Watch what I do and then there's going to be one for you to have a go at.

So if we look at that section in the middle, we've got six divided by three, which is bracketed, all to the power of two.

So we've got an exponent of two there operating on that whole thing in the bracket.

So the easiest thing to do is to evaluate the bit in the bracket and then we can raise it to the power of two.

So let's start by evaluating the six divided by three, which gives us two, so we can rewrite this calculation.

That's four plus two squared times 10, and now we can actually evaluate what two squared is 'cause remember that exponent only acts on the value before it.

So we've got four plus four times 10, and then we know that multiplication has priority over addition.

So we can calculate that four multiplied by 10.

So we've got four plus 40, which gives us our 44.

All right, there's one for you to have a go at.

Think about the steps that I've done in my example and whether you can apply that to your example, pause the video and give it a go and then we'll see what we got.

Lovely, well done if you realise that the easiest way of doing it is to think about that two times two in the bracket as a single value.

So two times two is four.

So you can rewrite that as 20 minus four squared divided by four.

Again, let's consider that four squared as a value.

So four squared is 16, so we've got 20 minus 16 divided by four.

Priority of operations means we're going to do that division first.

So 16 divided by four is four, 20 subtract four gives us 16.

Very well done if you got to that answer first time.

Right, so now we're going to try and apply that priority of operations to worded rules.

So when we convert worded rules into calculations, we have to consider the priority of operations.

It's not something that only applies when you're doing a lesson on priority of operations.

Priority of operations always applies in mathematics.

Andeep is thinking of a number, I'm thinking of a number, I add five, then multiply it by 10, and it's really clear that he wants to add the five first, then multiply by 10.

In this case I'm going to use a square or an empty box to represent Andeep's number.

If you'd rather use a question mark or something, that's absolutely fine.

So if we use a square to represent Andeep's number, we could write this as square plus five multiplied by 10 and I need to put a bracket around that.

Or I could write it as 10 multiplied by square plus five.

Again, with my brackets round my square plus five.

The brackets are needed as Andeep wants his number to have five added to it first before it is made 10 times bigger.

Right, this time I'm thinking of a number, I multiply it by 10, then add five.

It's really clear this time that he want the multiplication by 10 to happen first, then add five.

So let's think about how we would write that.

So I've got square or question mark multiplied by 10, add five.

This time I don't need to put a bracket around it because priority of operations says that the multiplied by 10 apps on square anyway and then we're adding five.

Equally, I could write that as five plus square multiplied by 10 'cause again, that would be telling me to do the square multiplied by 10 and then do five added onto that, okay? 'Cause addition is commutative, doesn't matter which way round I put the five and the square multiplied by 10.

Andeep is now going to use some exponents.

So how could we write these calculations? I'm thinking of a number, I cube it, then multiply by five.

You might want to pause the video and see if you can do this first.

So this is the way I've written it.

So I've got Andeep's number cubed multiplied by five.

Remember the exponent only acts on that number.

Okay, so it's being cubed first and then multiplied by five.

Again, I can write my multiplication either way around because multiplication is commutative.

Right, he's now going to do this one.

I'm thinking of a number, I multiply it by five, then cube it.

Think about how your answer might look different this time.

Give it a go and then come back and see if you've got the same answer as me.

Fantastic, so you might have come up with something like this.

So I've got Andeep's number multiply by five.

Because I want that whole thing to then be cubed, I'm going to put it in brackets and then raise it to the exponent of three.

Again, the multiplication can be written either way around 'cause multiplication is commutative.

It is important that brackets are put in here because we do want the number to be multiplied by five and then that whole thing to be raised to the power of three, okay? So really important that you're using the brackets where you'll need it but not putting the brackets in when you don't want that exponent to apply to anything other than the value it is next to.

Right, time to check see if you are happy with that.

I'm thinking of a number.

I square it, then multiply it by three.

I want you to look at the four things that I have written and decide which of those represent that calculation.

Pause the video and have a go and then see if you've got the same answer as me.

Very well done if you notice that C would be squaring, then multiplying by three.

Even better if you notice that top one is exactly the same thing.

Remember multiplication is commutative, so it doesn't matter which way round I write that multiplication.

That bottom one was there to check that you know that that squared is only applying to the missing number, okay? Which I've represented with a square here, okay? The three itself shouldn't be squared.

I only want my number to be squared and then multiplied by three.

So you shouldn't have squared that three.

Well done if you didn't get caught out with that one.

Brilliant, time for you to give this a go then.

So for these missing numbers, so again you've got four statements with I'm thinking of a number.

You need to match them up with the correct calculation on the right hand side.

Think carefully about the order in which I'm asking you to do things and which of those is going to show that order correctly.

Give it a go and then we'll look at the next step.

Brilliant, let's try some with some exponents in this time.

So exactly the same idea.

We've got some missing numbers on the left that I'm thinking about and you need to match those up with the calculations on the right.

Just like before, think carefully about the order I'm doing those operations to my number and how that looks when writing that as calculation, thinking about the priority of operations in your calculation.

Pause the video, give it a go and then we'll come back for the answers.

Fantastic, well done.

So I'm thinking of a number I subtract for, then multiply by nine.

I want the subtraction to happen first so I need to bracket that.

Doesn't matter which side I put the multiply by nine on because multiplication is commutative.

So that's that third one down.

I'm thinking of a number, I multiply by nine, then subtract four.

So this time I don't need brackets 'cause multiplication has priority anyway.

So I've got nine multiplied by my number, then subtract four.

So that's that second one down.

Third one down, I'm thinking of a number, I multiply by nine, then subtract four, then add four.

This was a bit of an odd case, so I dunno if you spotted this one.

So that one only needs the multiply by nine.

So if you're then going to subtract four and add four, you haven't overall made any difference to that value.

That's that top one.

And then the last one, I'm thinking of a number, I add four first.

Okay, so think about the fact that I want to add four first.

How am I going to show that in my calculation? Multiply that by nine, then subtract four, and then that's that bottom one.

Let's look at that second set.

Brilliant, so some exponents in here.

So the top one, I'm thinking of a number and multiply by five, then square it, then add three.

It's important that I bracket the multiplier by five 'cause otherwise the exponent would only act on square.

I want the exponent to act on the entire five multiplied by my number.

Then add three, I could have put add three at the end, but I've written it as three add.

'Cause again, addition is commutative, so it doesn't matter which way round I write that.

So that's that bottom one.

Second one, I'm thinking of a number, I square it, then add three, then multiply by five.

So that's that top one.

Again, I only want to square my missing number, but I want to add three and then multiply the whole thing by five.

So I need to group my missing number squared plus three before I then times it by five using brackets.

So that's the top one.

Third explanation down, I'm thinking of a number, I square it.

Then multiply by five, then add three.

No brackets needed this time because our priority of operations works in the order that I've written it.

So that's that second one down.

And the last one, I'm thinking of a number, I add three, then square it, then multiply it by five.

So again, I know that I want to group my missing number and my add three.

Because I want my squared to apply to that whole thing, I bracket it and put the squared on the outside of the bracket and then multiply by five.

Again, I could have just put times five on the end.

I've chosen to write it as five times.

Because multiplication is commutative, it means the same thing.

So that bottom explanation matches with that third calculation down.

Right, loads of things to think about there.

But if you are really confident with that priority of operations, that's going to be super helpful as you move through all your number work and your algebra work.

Okay, so well done, let's look at the next bit together.

Brilliant, so now we're going to have a look at using bar models.

Brilliant, so below we've got a section of a number line.

We've got 216, 231, 268 and we've got them joined with a red dash section and a blue dotted section.

We can work out the length of the red dash section by calculating 231 minus 216.

That'll give us the difference between those two values, and therefore that gap between the 216 and 231 on our number line.

We can then write that as an equation.

So what I would be saying is red equals 231 subtract 216.

Right, think about then how we could write an equation for the length of the blue dotted line.

Right, you might have put something like this, blue equals 268 minus 231.

What I want to think about now is other facts that we could write about these values.

If we know that red is 231 minus 216, what other statements can we make? Right, well, we could say that 231 is the 216 add the red line.

You might want to trace it with your finger to show that that does give us the 231.

You might also say that 231 is the red line, add the 216, addition's commutative.

So we can add those either way around.

We could also say that to get to the 216, you'd need to do the 231, subtract the red section.

Again, you might want to trace that with your finger to check that you agree with me.

So let's take your statement that blue equals 268 subtract 231 and think about what other facts we can write.

You might want to pause and see if you can get them all down before you look at the answers.

So 268 is 231 plus blue, 268 equals blue plus 231 and 231 is 268 minus blue.

Well done if you've got all three.

Now we can do exactly the same thing even if we don't have any values.

So we've got a new section of the number line.

We've still got a red dash section and a blue dotted section, but this time we've haven't got any numbers.

We've got a green post-it note, a purple post-it note and an orange post-it note.

Even though we don't know any numbers, we can still write an equation.

So we think about this red line, the red line can be calculated by doing the purple number subtract the green number.

We don't know what the numbers are, but we can write that equation.

Red equals purple subtract green.

These words are representing numbers.

What other equations can we write from this? So just like we did before, we've got purple equals green plus red, purple equals red plus green and green equals purple minus red.

All these equations, this is really important, all these equations show a relationship between purple and red and green.

They are showing the same relationship.

We're just writing them different ways round and that can be helpful depending on what we is we're trying to calculate.

Right, time for you to have a go then.

So which of these is not an equation linking the purple, the blue and the orange? So we've got the purple post-it note, the orange post-it note and the blue dotted line linking them.

And what I want to know is which of these is not an equation linking the purple, blue and orange.

Read them carefully, see what you come up with and then check your answers.

Fantastic, well done if you spotted that it is the bottom one that is incorrect.

You might want to trace with your finger to show that the blue plus the purple is going to give you the orange.

Equally, if we do the orange subtract the purple, we're finding the difference between the two and that'll be the blue.

It's that bottom one that is not correct.

So bar models then can be used to represent equations.

Equations, remember, tell us when two things have the same value.

It tells us when two things are equal.

That means on a bar model, if two things are equal, they need to have equal length.

If we have this example, so we've got the value of a square equals the value of a triangle, add three could be represented by this bar model.

Notice the length of the square bar is the same as the length of the triangle bar add three because those two things are equal.

Could also be drawn the other way up.

There's no problem with that 'cause we can say that the triangle plus three is the square.

Again, we don't know the value of the triangle and the square so they can be any size for the moment, it doesn't matter.

Bar models can represent equations with subtraction.

So if we have a look at an example of that one.

So if we have the value of the square is the value of the triangle, subtract three this time.

So there's the value of the square and I want it to be the same as the value of a triangle.

So I'm going to draw a triangle, subtract three.

So if I need to take three off the triangle and that'll give me the square.

Okay, we can write that together as a bar model.

Think about what other equation we can write for this bar model.

You might want to pause and have a think yourself before you check your answers.

Well done if you've got all of these.

When we only have one missing value, bar models can be really useful to help us calculate the missing values.

For example, I've got 24 equals something subtract three.

So there's my something.

I know that when I subtract three from it, I get 24.

The bar model makes it really clear to see how we're going to calculate that missing value.

So the length of the 24 plus the length of the three will give us the length of the missing value, which we represented with a square.

That means that that missing value will be 27.

Right, which bar model correctly shows the equation? The value of the square equals 15.

Subtract the value of a triangle.

Have a look at those, see if any of those seem to fit that equation.

Fantastic, well done if you spotted that that first one is going to give us the correct equation.

Time for you to have a go yourself.

So for each bar model, you've got three choices.

You need to circle any which match the bar model.

There may be more than one in each of the rows.

Give that a go and then we'll look at the second set.

Well done, for this second set then, we're going to look at it the other way around.

So for each equation you are going to have a go at drawing a bar model.

Then see if you can write three other equations that could be represented by your bar model.

Remember, there are different ways of writing a bar model.

So if you've got slightly different picture to me or to somebody else, that doesn't necessarily mean that you are wrong.

Then for the bar models which only have one solution, so look at your bar models once you've drawn them and see if there's any that you think will just have one solution, calculate that missing value.

Can you tell me then what that missing value is going to be? Give those a go and then come back for the answers.

Fantastic, guys, right, that first one then, you've got two correct equations.

15 subtract seven equals missing value or 15 equals the missing value add seven.

The second one, again, you've got two correct answers.

You've got three equals 10, subtract the value of the square and that last one, the value of the square equals 10 subtract three and the last one, well done if you spotted that all three of those equations fit that bar model.

Right, remember what I said, even if your bar model looks slightly different, it doesn't necessarily mean it's wrong.

So what you need to have is you need to have 12 as your longest bar and then you need to have your 10 and your bar with your square in it adding to the same length as that 12, okay? So something similar to mine, you could have had the 12 on the top bar and the 10 and the square making up the bottom bar, that's fine.

Three other equations, 12 equals square plus 10, 10 equals 12 take away square, square equals 12 take away 10.

For the second one, so again, you want your square and your 24 to equal the same length as your 30.

Doesn't matter if your 30 is on the bottom or the top.

And then your three equations, 30 equals square plus 24, 30 equals 24 plus square, or square equals 30 take away 24.

Last one, so we need the 11 and the triangle to add to the same length as the square.

We don't know the size of the square or the triangle, so it doesn't matter how big or what proportions you've drawn those in.

Again, you could add the square at the top.

We just need to make sure the 11 and the triangle add up to the square.

Your answers then, square equals 11 plus triangle, square equals triangle plus 11, triangle equals square minus 11.

And for last bit, the bar models that only have one solution, so that was A and B.

Well done if you spotted that the square must represent two and then for B, the square must represent six.

What I'm really hoping is that you're seeing that the bar models can be a really useful solution, a really useful way of spotting what those values must be.

Right, fantastic work today then, guys.

So hopefully you've seen that calculations can be evaluated following the standard priority of operations.

Even if you've seen priority of operations before, I'm hoping that you're really secure now with how you can use that.

We know that equations can be formed from scenarios and they can also be represented in a bar model.

We know that several equations can be written from a single bar model.

When one value is not known, a bar model is really useful to help us calculate it.

Fantastic, well done today and I really hope you choose to join us for some more learning as you progress with your mathematics.