Lesson video

Hi, everyone, my name is Ms. Ku, and I'm really happy to be learning with you today.

It's going to be fun and really interesting lesson, and I'm so excited to be learning with you.

You'll come across some new keywords and maybe some keywords you've already come across before.

Now, we're gonna work really hard today, but I am here to help, and we can learn together.

In today's lesson, from the unit arithmetic procedures with integers and decimals, we'll be looking at the priority of operations with positive integers, and by the end of the lesson, you'll be able to calculate using the priority of operations, including brackets, powers, and exponents with positive integers.

So, let's have a look at some keywords.

In mathematics, an operation is a function which takes an input value to an output value.

Addition, subtraction, multiplication, division, roots, and powers, and many more are examples of operations.

The additive inverse will be used in our lesson today.

And remember, the additive inverse of a number is a number that, when added to the original number, gives the sum of zero.

We'll also use the word commutative, and remember, an operation is commutative if the values it is operating on can be written in either order without changing the calculation.

Our lesson will be broken into 2 parts, whereby we'll be looking at multiplication, division, addition, and subtraction first.

Then, we'll be looking at brackets, roots, and exponents second.

So, let's start with multiplication, division, addition, and subtraction.

The priority of operations is important, as it ensures that everyone can understand and approach a mathematical problem the same way.

Now, this is a lovely little diagram just to show you the priority of operations.

We go from top down.

What I'm going to do is show you an example of how we can use this image to represent the priority of operations.

Well, let's say we have a calculation 4 add 3 times 5.

Using the priority of operations, which operation do you think we apply first? Well, hopefully you can spot, it's multiplication.

We have to apply multiplication first before the addition because it's higher up on that image.

So, let's work out 3 multiply by 5.

Well, 3 multiply by 5 is 15.

So, our calculation is 4 add 15, which is 19.

Let's have a look at another example.

Here, we have 20 subtract 50 divide by 10.

So, using the priority of operations, which operation do you think we apply first, subtraction or division? Well, hopefully, you can spot it's the division, as it comes higher up in that image.

So, 20 subtract 5 divide by 10.

We do the division first, which is 50 divide by 10 is 5.

So, our calculation is 20 subtract 5, therefore giving an answer of 15.

So what do you think happens when we have two of the same operation? For example, I'm going to be looking at 12 divide by 2 and 20 divide by 4.

You notice that we have division occurring in our calculation twice and we have an addition.

So, what do we do when we have two of the same operation? Well, we do them both at the same time.

So that means because we have division before addition, we do the division first, 12 divide by 2 and the 20 divide by 4 is done at the same time.

So, 12 divide by 2 is 6, 20 divide by 4 is 5.

Then, we add them together to give our answer of 11.

Next, so what do you think we do when we have two operations on the same row? Now, for addition and subtraction, using subtraction as the addition of the additive inverse allows every operation to be addition.

For example, if we have 20 add 16 subtract 4.

While using our knowledge on the additive inverse, I'm going to change that subtraction of 4 to the addition of a -4.

Then, we're just using addition.

So 20 add 16 add the -4 gives us 32.

So now, let's have a look at another two operations on the same row, multiplication and division.

For example, if we have 12 multiply by 20 divide by 4, well, they both have the same priority because they're on the same row.

So, let's see if we try the approach where we do multiplication first.

I'm going to do 12 multiply by the 20, which is 240, then divide by the 4 to give me 60.

Next, I'm gonna try a different approach.

Let's do the division first.

20 divide by 4 is 5, 12 multiply by 5 is 60.

So, as they are the same priority, we do either operation first.

A better approach is to write division as fractions, as it makes the divisor clear and reduces errors.

So, let's look at that calculation again.

12 multiply by 20 divide by 4.

Rewriting the division as a fraction is 12 multiply by 20 over 4, and then working that 20 over 4 out gives us 5.

So, we end up with exactly the same answer as 60.

Now, let's have a look at a Check question.

Aisha and Jacob are given this calculation.

10 subtract 6 add 4.

And we need to identify who is correct.

Aisha says 10 subtract 6 is 4, then 4 add 4 is 8.

But Jacob says, "Well, I know subtraction is adding the additive inverse." 10 add the -6 add 4, so I can do this in any order.

10 add the 4 add the -6 is 14, add the -6 is 8.

Who do you think is correct? Press Pause if you need more time.

Both are correct because either can be applied and they work out the correct answer.

Using additive inverse simply means we're using the commutative property of addition.

Really well done if you got that one right.

What about the next Check question? We have the same calculation.

10 subtract 6 add 4.

And you need to have a look and explain why Laura is wrong.

Laura says addition comes before subtraction, so you do 6 add 4 equals 10, and then you can think of it as 10 subtract 10.

See we can find out why she's wrong and explain.

Great work.

So hopefully, you spotted that she's interpreted the 6 to be a positive 6 when the calculation is the addition of the -6.

10 subtract 6 add 4 is equal to 10 add the -6 add 4 because we're using our negative inverse.

So unfortunately, Laura is wrong.

Now, let's have a look at another Check question.

Izzy and Alex do the same calculation, but in different ways.

Who's correct, and what should they have done differently? 24 divide by 8 times 3.

Izzy says it's 24 divide by 24, which is 1, but Alex says 24 divide by 8 times 3 is 24 over 8 times 3, which is 3 times 3, which is 9.

Who do you think is correct? And for the person who's incorrect, what should they have done differently? Well done.

So hopefully, you spotted Alex is correct.

Izzy could have written the 24 divide by 8 as a fraction and then calculate the result.

So that means she's multiplying by 3.

This would've been a much easier way to minimise errors.

Now, let's have a look at your task question.

So your task question wants you to work out the answers to the following, ensure to show all your working out.

See if you can give it a go and press Pause if you need.

Great work, so let's move on to Question 2.

Question 2, we have 2 students, Sam and Sofia, and they do the same calculation differently.

Who's correct, and can you explain what the person who got incorrect should have done differently? See if you can give it a go and press Pause if you need.

Great work.

So, let's have a look at Question 3.

Question 3 is a fantastic question, and it wants you to insert the operations addition, subtraction, division, and multiplication, so the following calculations are correct.

Remember to use the priority of operations here.

This is a great question.

Please do press Pause, as you will need more time.

Question 4 wants you to fill in the gaps with the operations addition, subtraction, multiplication, and division, so the calculations are correct.

This is a little bit harder, as we have a few more numbers in here.

Take your time with this question and remember the priority of operations.

Press Pause when you're ready.

Great work, everybody.

So, let's go through these answers.

Question 1.

Now, remember the priority of operations for A.

You had to do the multiplication first.

So 12 add our 12 gives us 24.

For B, the priority of operations state we do division first, so it's 100 subtract our 4, which gives us an answer of 96.

For C, remember, it's easier for you to write the division as a fraction.

So it's 30 subtract the 24 over 8, and we know 24 over 8 is 3, so it's 30 subtract 3, giving us 27.

Great work if you got that one right.

For Question 2, we have Sam and Sofia both do the same calculation, but differently, and who is correct? And we have to explain what should that person who got them incorrect done differently.

Well, hopefully, you've spotted Sofia is incorrect, and ideally she should have written 15 divide by 3 as a fraction.

Then writing the 15 over 3 as a fraction, she can then multiply efficiently by 4.

Question 3 is a great question and really does embed that understanding of the priority of operations.

We have to apply the operations addition, subtraction, division, and multiplication to make the following correct.

Huge well done if you've got A, B, C, D, or E.

These were fantastic questions.

Question 4 is really tough as I've added some more numbers here.

Same again, you need to insert the operation addition, or subtraction, division, or multiplication.

Here are the answers for A, B, and C.

Fantastic work if you've got that one right.

Great work, everybody.

So, let's move on to the second part of our lesson, which looks at brackets, roots, and exponents.

Now, we're still using the diagram to represent the priority of operations, but if you look at the image, the use of brackets is at the very top.

And remember, brackets group numbers together and affects the priority of operations.

We do brackets first if they're ever seen or implicitly given in the calculation.

For example, if we have 100 divide by 4 add 6, we have brackets, so we must do this first, which gives us an answer of 10.

So our calculation is now 100 divide by 10, which is 10.

After the brackets, the next operation are roots and powers, and this will require previous knowledge on square and cube roots and evaluating exponents.

So, let's have a look at a question.

Here, we have 12 add 4 add 6 times 5 squared.

Now, remember, we spot our brackets, which is the very top of our priority of operations.

So, which step do you think we do first? Well, well done if you spotted the brackets.

So, working out our brackets first, we have 10.

So our calculation now looks like 12 add 10 times 5 squared.

Now, which step or operation is next? Well, hopefully, you can spot it's the exponent.

So we had to work out the 5 squared, thus giving us the new calculation of 12 add 10 times 25.

Now, what do you think the next operation is? Hopefully, you can see it's multiplication.

So, we're going to do the 10 multiply by 25 next to give me 250.

So, our new calculation looks like 12 add 250, giving us 262.

Let's see if we can do a quick check.

We've got to work out the answer showing all your working out.

So you can give it a go and press Pause if you need.

Great work, so let's see how you got on.

Well, for A, hopefully you've spotted we've got to do those brackets first.

15 subtract 9 gives us our 6.

So that means our calculation is 10 add 24 over 8 add 6.

Now, we work out that division.

So 24 over 8 is our 3, and now we have addition.

So, our final answer is 19.

B.

Well, hopefully, you've spotted we do our brackets first.

So that means our calculation is 8 add 6 times 4 subtract 20.

What's our next operation? Well, it should be multiplication.

So, 8 add our 24 subtract 20.

And remember, we can use our additive inverse here.

So, 8 add 24 add negative 20 gives me my final answer of 12.

Great work if you got that one.

Now, it's important to remember the use of brackets does group numbers together and affects the priority of operations, but sometimes the brackets are invisible or implicitly given.

So, what do I mean by this? So let's have a look at example to explain.

25 over 4 add 1.

You can clearly see our brackets here, but actually this is exactly the same as 25 over 4 add 1 without the brackets.

The continuation of the line indicates the implicit brackets.

So sometimes, you don't always see the brackets, but they are implied implicitly.

Let's have a look at another example.

Well, the square root of 12 add 4.

Same again.

We have a continuation of the line, and this indicates implicit brackets.

So that means it's the same as square root of 12 add 4 in our brackets, and it's important to recognise those invisible or implicit brackets in a question.

So, let's have a look at quick Check question.

Here we have Jun, and Jun did some working out.

12 subtract 1 plus 15 over 2, and he says it's the same as 12 subtract 1/2 add 15 over 2.

Then, he works it out to be 12 subtract 0.

5 add 7.

5.

Unfortunately, Jun made a mistake.

What should he have done? See if you can give it a go and press Pause if you need.

Great work.

So hopefully, you've spotted we have implicit brackets.

You can see that continuation of the line.

You should have grouped the 1 and 15 together first, as we have those hidden brackets.

So, let's look at the correct working out.

Here, we have the implicit brackets.

Then, working this out, this gives us 16 over 2.

Well, we know this is division, so that means 16 divide by 2 is 8, and that gives me a final answer of 12 subtract 8, which is 4.

So, we do have brackets here, but they're invisible or implicit.

So remember that continuation of that line.

Let's have a look at another Check question.

Andeep was given a different question and did some working out, as you can see here.

Where did he make his mistake and what should he have done differently? And also, can you work out the correct answer? See if you can give it a go and press Pause if you need.

Great work.

So, let's see how you get on.

Well, first of all, we do have implicit brackets because you can see the continuation of that line.

So, we should have grouped the 16 and the 9 first to give the square root of 25.

The correct working out is shown here, grouping together the 16 and the 9 because of that continued line gives us a square root to 25.

Then, we have a root.

So we apply the root first to give me 5, then we apply the division to give me 20.

So, the final answer is 15 add 20, which is 35.

Great work if you've got this question right.

Now, let's have a look at your practise questions.

Question 1, I want you to work out the answers to the following, ensuring you show you are working out.

Remember those implicit brackets and the continuation of those lines.

See if we can give it a go and press Pause if you need.

Great work.

So, let's move on to Question 2.

Question 2 is a great question, and now, not only are you using the operations addition, subtraction, multiplication, division, but I also want you to insert brackets where appropriate.

So, where do you insert these brackets and/or operations to make the following calculations? This is a great question.

See if you can give it a go.

Great work.

So, let's move on to Question 3.

Question 3 is one of my favourite questions.

Use exactly four 4s, but no other numbers, together with of the operations multiplication, division, addition, root, subtraction, exponents, and/or brackets to write calculations.

Given your answer of 1, and then 2, and 3, all the way up to 20.

An example of 8 has been given for you.

For example, 4 divide by 4 times 4 add 4.

Remember the priority of operations.

You do the division first.

4 divide by 4 is 1, then 1 times 4 is 4, add the 4 gives us our 8.

So, that's how we got our answer 8 using four 4s.

This is a great question, and you certainly will need more time, so press Pause when you're ready.

Great work, everybody, so let's see how you got on.

We'll go through these answers for Question 1.

So let's do our brackets first, and then I'm going to do my exponent.

So 6 subtract 2 is our 4, and our exponent 3 squared is 9.

Now, we're going to do our multiplication to give 24 add 36, which is 60.

Great work if you got that one right.

For B, we have implicit brackets because they're that extended line.

So, it's the same as 100 subtract the square root of 64 divided by the 2 squared.

So, let's work out the root and exponent to give me 8 and 4.

Thus, our calculation is 100 subtract 8 divide by 4.

You can write 8 divide by 4 as a fraction, 8 over 4, giving you 2.

So, our answer is 100 subtract 2, which is 98.

Well done if you got that one right.

For Question 2, you had to insert the operation, as well as brackets.

A huge well done if you've got any of these right, as they were really tough.

Here's the answer to A, B, C, and D.

Great questions, and a huge well done if you've got that one right.

For Question 3, there are so many examples of how to get each number, but I'm just gonna show you a few just in case you didn't get these numbers.

See if you got any of these answers, and press Pause if you need to have a look at these calculations a little bit more.

So, let's have a look at how we made the numbers 11 to 20.

Well, same again.

There are plenty of different calculations out there.

Press Pause if you want to have a little look at how I've worked out these answers.

Great work.

This is one of my favourite questions, and well done if you've got any of these right.

Great work today.

So remember, the priority of operations is important, as it ensures that everyone can understand and approach a mathematical problem in exactly the same way.

The diagram illustrates the first priority at the top to the least priority at the bottom.

A huge well done for the work you've done today.

It was tough, and it was great learning with you as well.