Lesson video

Hi everyone, my name is Ms. Ku and it's great to have you learning with me today.

In today's lesson, it might be tricky or easy in parts, but I will be here to help.

You also might come across some vocabulary that you are familiar with or maybe not familiar with, but don't worry, we'll learn together.

It's great to have you here so let's make a start.

Hi everyone, in today's lesson we'll be looking at the priority of operations with positive and negative integers, decimals and fractions.

and this is under the unit of arithmetic procedures with fractions.

By the end of the lesson, you'll be able to calculate using the priority of operations, using brackets, powers, exponents, and reciprocals with positive and negative integers, decimals and fractions.

Let's have a look at some key words that we'll be referring to in our lesson.

First of all, the additive inverse.

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

For example, eight is the additive inverse of negative eight.

As you sum these together, it gives zero negative 4.

5 is the additive inverse of 4.

5.

It's because if you add these together, the sum to give zero.

We'll also be looking at an operation is commutative if the values it's operating on can be written in either order without changing the calculation.

For example, negative three add four add 10, is exactly the same as 10 adds that negative three add our four.

Today's lesson will be broken into three parts.

We'll be looking at reviewing the priority of operations first, then powers and roots of fractions, and then looking at calculating efficiently.

So let's make a start by reviewing the priority of operations.

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

This diagram illustrates that priority of operations.

Starting at the top we have brackets, sometimes they're explicit, sometimes they're implicit.

Then we move on to roots and exponents.

Following that, we have multiplication and division.

And then finally we have addition and subtraction.

This hierarchy of operations applies to all positive and negative integers, fractions and decimals.

Now when using the priority of operations, a preferred method for the row of addition and subtraction is simply to use addition with those additive inverses.

For example, if you had three quarters subtract two fifths add three eighths, subtract one 10th.

Now looking at the subtraction of two fifths, I'm going to replace that with the addition of the additive inverse.

So that means we are adding that negative two fifths and the same with the subtraction of one 10th.

We're going to use the addition of negative one 10th.

So this means the calculation is exactly the same as three quarters add that negative two fifths, add three eighths, add that negative one 10th.

From here, let's sum our positives.

Well, our positives are three quarters and three eighths.

Identifying a common denominator of eight and equivalent fractions, we have six eighths are three eighths, which gives us an answer of nine eighths.

Summing up those negatives, we have negative two over five, add on negative one 10th.

Identifying equivalent fractions with a common denominator, we have negative four-tenths add negative one 10th, thus giving us a final answer of negative five tenths, which can be simplified to negative one half.

Then we simply add our nine eighths and our negative one half.

Once again, looking at our equivalent fractions with a common denominator, thus giving us nine eighths, adds that negative four over eight, giving us a final answer of five eighths.

So a preferred method for the row of addition and subtraction is simply to use the addition with those additive inverses.

Let's have a look at a check.

I want you to work out the following question, giving your answer in its simplest form.

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

Well done, so let's see how you got on.

Well first of all, we have the subtraction of two thirds and the subtraction of 17 over 30.

So let's use the addition of those additive inverses.

So that means we have nine over 10, add the negative two thirds, add the five over six, add our negative 17 over 30.

Then summing our positives.

Remember to use those equivalent fractions with that common denominator.

We have 27 over 30 add 25 over 30, which gives us 52 over 30.

Now let's sum those negatives.

We have negative two thirds add negative 17 over 30.

Identifying that common denominator using our equivalent fractions gives us negative 37 over 30.

So summing up our 52 over 30 and our negative 37 over 30 gives us 15 over 30, which simplifies two one half.

A huge well done if you got that one right.

Now when using the priority of operations, the row with multiplication division can be done in any order, but we also know that the division can be written as the multiplication by the reciprocal.

For example, three-fifths add one half times three fifths divided by three sevenths.

So from let's change that division of our fraction to the multiplication of the reciprocal.

Now I've made everything multiplication.

So I have three fifths add one half, multiply by three fifths, multiply by our seven over three.

Now remember from our priority operations, multiplication comes before addition.

So let's work out this multiplication.

We have one multiplied by three multiplied by seven over two multiply by five multiplied by three, thus giving us three fifths add seven tenths, giving us the final answer of six tenths add seven tenths to be 13 over 10, which is one and three tenths.

So now let's have a look at a check.

We have two thirds add one sixth, divided by four fifths takeaway seven eighths.

See if you can give it a go and remember the priority of operations, press Pause if you need more time.

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

Well, hopefully you've spotted we have a division of four fifths, so let's replace that by multiplying by the reciprocal.

So we have two thirds, add one six, multiply by five over four, subtract our seven over eight.

Now we're going to use our priority of operations and spot we do the multiplication before the addition.

So then we have five over 24.

Look, we have a addition and a subtraction.

So let's use our knowledge on the additive inverse.

Two thirds add five over 24, add on negative seven over eight, making a common denominator of 24.

We have 16 over 24.

Add five over 24, add on negative 21 over 24.

From here we can work this out to give us a final answer of zero.

21 over 24, add on negative 21 over 24 gives us zero.

Massive well done if you've got this one right.

Now it's time for your task.

See, you can give these a go and press Pause if you need more time.

Well done.

So let's move on to question two.

Question two wants you to work on the following using the priority of operations.

Giving your answer its simplest form and a mixed number where possible.

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

Great work.

So let's go through these answers.

Question one, hopefully you spotted we do the multiplication first.

So that means three quarters multiply by two fifths gives us six over 20.

Then we have nine-tenths add our six over 20 making that common denominator of 10.

We have nine-tenths and three-tenths, thus giving an answer of 12 tenths, which can be simplified to six fifths or one and one fifth.

Well done if you've got this one right.

For B, well you've got a division of nine-tenths.

So let's use multiplication of the reciprocal.

So we have eleven-fifteenths, subtract three over five, multiply by our 10 over nine.

Using the priority of operations, we do the multiplication first, giving us eleven-fifteenths, subtract our 30 over 45.

we can spot, we can use a common denominator of 15, thus giving us an answer of one over 15.

For C, we have multiplication and division.

So let's change that division into a multiplication of the reciprocal.

So we have nine 20ths, subtract three-fifths multiplied by that five over four.

From here this gives us nine over 20, subtract 15 over 20, which gives us a final answer of negative six over 20, which then can be simplified to give us negative three over 10.

Massive well done if you've got this one right.

For question two, we had to work out the following using the priority operations where we have mixed numbers.

Well first things first, let's convert those mixed numbers into improper fractions.

Then using our priority of operations, we apply the multiplication first.

Then we're going to identify a common denominator, which is 12 and sum our fractions to give us 63 over 12.

This can be simplified then to 21 over four and as a mix number five and one quarter.

For B, let's convert those mixed numbers into an improper fraction.

So we have nine over four, multiply by four over five, subtract our three fifths divided by 13 over 10.

Now from here now we have a division of a fraction, so let's change it to the multiplication of its reciprocal.

So then we have 36 over 20, subtract three fifths multiply by 10 over 13, giving us 36 over 20, subtract off 30 over 65.

Now we can simplify this a touch more and then identify a common denominator which is 65.

So we have 117 over 65, subtract 30 over 65 giving us one and 22 over 65 as a mixed number.

Massive well done if you've got that one right.

Great work everybody.

So let's move on to the second part of our lesson using powers and roots of fractions.

Now the next operation, we'll be looking at with fractions are roots and powers and this will require some previous knowledge on square and cube roots and evaluating using exponents.

Similar to negative numbers, we use brackets or implicit brackets to show the root or exponent of a fraction.

So let's begin with notation first.

For example, two over three or two thirds all squared.

Notice the brackets here to illustrate it's the whole fraction being squared.

This is the same as two thirds multiplied by two thirds, which we can then work out to be four ninths.

Therefore, if we were to do the square root of four ninths, this is equal to two thirds.

Notice those implicit brackets, the radical or that square root sign is big and covers the whole fraction.

So this means the entirety of the four ninths is being square rooted, thus giving us two thirds.

Let's have a look at a quick check.

Aisha and Laura are given a question.

The question states the square root of 25 over 36.

Aisha says the answer is five over 36 and Laura says the answer is 25 over six.

Who is correct and can you explain how they made their error? See if you can give it a go and press Pause if you need more time.

Great work.

So let's see how you got on.

Well, hopefully you spotted neither of them are correct as the correct answer is five-sixths.

Aisha thought the question as for the square to 25, get that answer and put it over the 36 and Laura thought that the question was 25 over the square to 36.

So it's important to recognise the correct mathematical notation.

The radical or the square root is of the entire fraction, 25 over 36.

So that means we had to do the square root of the whole 25 over 36.

Let's have a look at another check.

Looking at our notation again, I want you to match up the calculation on the top row with the correct answer on the bottom row.

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

Well done.

So let's see how you got on.

Well, hopefully you've spotted five squared over seven is equal to 25 over seven.

The five is the only value which is squared.

For five over seven all squared it's 25 over 49.

The entire fraction is being squared.

For five over seven squared, it's five over 49 because the seven is the only one which is squared.

Lastly, the square root of the entirety of 25 over 49 is five over seven.

Well done if you've got this one right.

Now, it's important to recognise we use our knowledge and square and cube numbers to work out the root or exponent of a fraction.

For example, three quarters subtract two fifths all squared.

So we know we work out the exponent, first three quarters subtract the two fifths, multiply by the two fifths, which is three quarters, subtract our four over 25.

Then from here, identify our common denominator using equivalent fractions.

We have 75 over 100, subtract 16 over 100, thus giving us an answer of 59 over 100.

Let's have a look at another check question.

Jacob spilt ink all over his work.

Can you figure out what is underneath each ink blot? See if you can give it a go and press Pause if you need more time.

Great work.

Let's see how you got on.

Well remember the priority of operations.

We have to identify the exponent and the roots first.

So that means two over three all cubed gives us eight over 27, and the square root of four over 25 is two over five.

Remember the priority of operations, we do multiplication first, so that means the eight multiply by two is 16, 27 multiply by five is 135.

Then let's see if we can identify a common denominator using our knowledge on equivalent fractions.

We have 75 over 135 subtract our 16 over 135 gives us a final answer of 59 over 135.

Great work everybody.

So let's move on to your task.

Here, you need to fill in the blanks to complete the calculation.

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

Well done.

Let's move on to question two.

Question two, want you to work out the answer to the following giving your answer in its simplest form.

See if you can give it a go and press Pause one more time.

Well done.

Let's move on to question three.

Question three states that Lucas has made an error.

Find the error and correctly complete the calculation.

Well done everybody.

Let's move on to these answers.

For question one, hopefully you spotted let's do our exponent first.

That means we have three over five all squared is nine over 25.

Remember our priority of operations, do our multiplication first to give us eight over 15 and it's still adding that nine over 25.

Replacing the subtraction of eight over 15 with its additive inverse, we have nine over 25 adds that negative eight over 15.

Identifying a common denominator with our knowledge on equivalent fractions gives us 90 over 150.

Add our 54 over 150, add our negative 80 over 150, giving us the final answer of 64 over 150 or simplified to be 32 over 75.

This was a great question, well done if you got that one right.

For question two, you need to work out the answer to the following, giving your answer in its simplest form.

Hopefully you can spot we do our root first.

So let's work out the cube root of eight over 27, which is two over three.

So that means we have nine-tenths multiplied by four fifths subtract two thirds multiply by two thirds.

Using our priority of operations, let's do our multiplication first, giving us 36 over 50 subtract four ninths.

Then our common denominator giving us a final answer of 124 over 450, which can be simplified to 62 over 225.

Well done if you got this one right.

For question three, did you spot the error? Well, hopefully you can spot it right here.

Lucas needed to convert the mix number into an improper fraction first.

So doing the calculation correctly, let's convert that six and one quarter into an improper fraction.

So we have the square root of 25 over four.

Using our priority of operations, let's do the square root to 25 over four first giving us two thirds, add five over two, multiply by three fifths.

Multiplication is next.

So we have two thirds add 15 over 10.

Identifying or common denominator of 30 gives us a final answer of 13 over six.

Massive well done if you've got that one right.

fantastic work everybody.

So let's have a look at the last part of our lesson.

Calculating efficiently.

Now we're going to use a combination of fractions and decimals with positive and negative integers.

Converting fractions to decimals is an option.

For example, Andeep and Sophia both do the same question and both are correct.

So let's have a look at how Andeep tackled this question.

Andeep looked at the question and preferred to use decimals.

So recognising this, he's recognised three quarters to be 0.

75 and one fifth to be 0.

2.

Then using his knowledge on exponents, three-tenths has been converted into nine over 100.

From here he's converted every single number into a decimal.

Then using the priority operations, 0.

75 times 0.

2 is 0.

150.

Then adding, using our knowledge on additive inverses gives three add our negative 0.

15 add our 0.

09, thus giving us a final answer of 2.

94.

Now Sophia is keeping with fractions.

So she's recognised three tenths all squared, gives us nine over 100.

Then using the prior of operations, spotting the multiplication of fractions gives us three quarters multiplied by one fifth is three over 20.

Then using our knowledge on additive inverses.

So we're adding each number gives us three, add on nine over 100, add on negative three over 20, and using our knowledge on a common denominator and equivalent fractions, we sum them together to give 294 over 100, which simplifies to 147 over 50, which is the same as two and 47 over 50.

So both methods work and both methods are correct.

But let's do another question and somehow one answer is less accurate than the other.

I want you to have a look and see if you can identify who is less accurate and how did it happen? See if you can give it a go and press Pause if you need more time.

Well, hopefully you've spotted Andeep's answer is less accurate because of the rounding of two thirds to 0.

667, it's made and deeps answer less accurate than Sophia's.

When using recurring decimals for example, two thirds, the fractional form ensures greater accuracy.

So let's move on to our task and we're looking for that accuracy.

So that's why we're going to continue to use fractions.

See if you can give this a go, fill in the blanks to complete the calculation.

Well done.

So let's move on to question two.

Question two wants you to work out the answer to the following, ensuring to give your answer as a mixed number or in its simplest form.

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

Well done, let's move on to question three.

Here, I want you to work out the answer to the following, ensuring to give your answer as a mixed number and in its simplest form.

Give it a go and press Pause for more time.

Well done.

So let's go through these answers.

For question one, hopefully you've spotted the priority of operations state.

We can do the exponent and the root at the same time.

So two fifths all squared gives us four over 25.

The square root to 49 over 100 is seven over 10.

Using multiplication next we have 28 over 250.

Identifying those equivalent fractions with that common denominator works our answer to be one and 111 over 125.

Great work if you got that one right.

For question two, let's see how you did.

Well first things first, let's convert the division of decimals into a fraction.

So we have nine-tenths add 2.

4 over 0.

8, multiply by two-thirds.

We have a fraction with a decimal denominator, so it's not very nice.

So let's see if we can convert it to a nice equivalent fraction to 24 over eight.

Doing the multiplication next, we have nine-tenths add 48 over 24 giving us a final answer of two and nine-tenths.

Great work if you've got that one right.

For B, let's get rid of that three divide by five and that 0.

3 divided by four by writing it as a fraction.

So it gives us 15 add three fifths multiply by 0.

3 over four.

That 0.

3 over four doesn't look very friendly.

So let's convert it to the equivalent fraction three over 40.

Using our multiplication next, we have 15 add nine over 200, giving us 15 and nine and 200 as our final answer.

For question three, we want our answer as a mixed number in its simplest form.

So hopefully you spot.

Let's do the root first.

Giving us two subtract three quarters multiply by seven ninths.

Multiplication is next to give us two subtract 21 over 36.

Using the common denominator of 36, gives us an answer of 51 over 36, which is one and five twelfths.

For B, we've spotted we have an exponent of two and we also have a division of decimals.

So let's say we can write it a bit nicer.

81 over 100 subtract 2.

5 over 0.

5 multiplied by two.

2.

5 over 0.

5 isn't very pretty.

So let's change it to an equivalent fraction, which is 25 over five.

Using our knowledge on the priority of operations, doing the multiplication next, and then working out our answer to be negative nine over 19 over 100.

That was a really tough question.

Great work if you've got that one right.

Well done everybody.

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

And this applies to positive and negative integers fractions and decimals writing division as a fraction and the multiplication of the reciprocal as well as the addition of the additive inverses can make calculations easier and calculations can be written in decimal form, but can lose accuracy when using those recurring decimals.

A huge well done everybody.

It was great learning with you.