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, they 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, -3 add 4, add 10 is exactly the same as 10 adds that -3, add our 4.

Today's lesson will be broken into three parts.

We'll be looking at reviewing the priority of operations first, then powers and routes 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 3/4 subtract 2/5, add 3/8 subtract 1/10.

Now looking at the subtraction of 2/5, I'm going to replace that with the addition of the additive inverse.

So that means we are adding that -2/5, and the same with the subtraction of 1/10, we're going to use the addition of -1/10.

So this means the calculation is exactly the same as 3/4 add that -2/5, add 3/8, add that -1/8.

From here, let's sum our positives.

Well, our positives are 3/4 and 3/8, identifying a common denominator of 8 and equivalent fractions, we have 6/8 add 3/8, which gives us an answer of 9/8.

Summing up those negatives, we have -2/5, add on -1/10.

Identifying equivalent fractions with a common denominator, we have -4/10 add -1/10 thus giving us a final answer of -5/10 which can be simplified to -1/2.

Then we simply add our 9/8 and our -1/2.

Once again, looking at our equivalent fractions with a common denominator, thus giving us 9/8, add that -4/8, giving us a final answer of 5/8.

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

Let's have a look and 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 2/3 and the subtraction of 17/30.

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

So that means we have 9/10, add the -2/3, add the 5/6, add on -17/30.

Then summing our positives.

Remember to use those equivalent fractions with that common denominator.

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

Now let's sum those negatives.

We have -2/3 add -17/30.

Identifying that common denominator using our equivalent fractions gives us -37/30.

So summing up our 52/30 and our -37/30 gives us 15/30, which simplifies to 1/2.

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, 3/5 add 1/2 times 3/5 divided by 3/7.

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

Now I've made everything multiplication.

So I have 3/5 and 1/2 multiply by 3/5, multiply by our 7/3.

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

So let's work out this multiplication.

We have 1 multiply by 3, multiply by 7, over 2 multiply by 5 multiply by 3, thus giving us 3/5 add 7/10, giving us the final answer of 6/10 add 7/10 to be 13/10 which is one and three-tenths.

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

We have 2/3 add 1/6 divided by 4/5 takeaway 7/8.

See if you can give it a go and remember the priority of operation.

Press pause if you need more time.

Great work.

So let's see how you got on.

Hopefully you've spotted we have a division of 4/5, so let's replace that by multiplying by the reciprocal.

So we have 2/3, add 1/6, multiply by 5/4, subtract the 7/8.

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

So then we have 5/24.

Look, we have a addition and a subtraction.

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

2/3 add 5/24, add on -7/8.

Making a common denominator of 24, we have 16/24, add 5/24, add on -21/24.

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

21/24, add on -21/24, gives us zero.

Massive well done if you have 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 in simplest form and a mixed number where possible.

See if 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 3/4 multiply by 2/5 gives us 6/20.

Then we have 9/10 and our 6/20, making that common denominator of 10, we have 9/10 and 3/10, thus giving an answer of 12/10 which can be simplified to 6/5 or one and one-fifth.

Well done if you got this one right.

For B, well, you've got a division of 9/10.

So let's use multiplication of the reciprocal.

So we have 11/15 subtract 3/5, multiply by our 10/9.

Using the priority of operations, we do the multiplication first, giving us 11/15, subtract our 30/45.

We can spot we can use a common denominator of 15 thus giving us an answer of 1/15.

For C, we have multiplication and division.

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

So we have 9/20 subtract 3/5 multiplied by that 5/4.

From here, this gives us 9/20 subtract 15/20 which gives us a final answer of -6/20, which then can be simplified to give us -3/10.

Massive well done if you got this one right.

For question, two we had to work out the following using the priority of 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 63/12.

This can be simplified then to 21/4 and as a mix number, five and one-quarter.

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

So we have 9/4, multiply by 4/5, subtract our 3/5 divide by 13/10.

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 36/20, subtract 3/5 multiply by 10/13, giving us 36/20, subtract off 30/65.

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

So we have 117/65, subtract 30/65 giving us 1 and 22/65 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, 2/3 or two-thirds, all squared.

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

This is the same as 2/3 multiplied by 2/3, which we can then work out to be 4/9, therefore, if we were to do the square root of 4/9, this is equal to 2/3.

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 4/9 is being square rooted thus giving us 2/3.

Let's have a look at a quick check.

Aisha and Laura are given a question.

The question states, the square root of 25/36.

Aisha says the answer is 5/36 and Laura says the answer is 25/6.

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 5/6.

Aisha thought the question asked 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 root 36.

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

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

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

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 5 squared over 7 is equal to 25/7.

The 5 is the only value which is squared.

For 5/7, all squared, it's 25/49.

The entire fraction is being squared.

For 5/7 squared, it's 5/49 because the 7 is the only one which is squared.

Lastly, the square root of the entirety of 25/49 is 5/7.

Well done if you've got this one right.

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

For example, 3/4 subtract 2/5, all squared.

So we know we work out the exponent first, 3/4, subtract the 2/5, multiply by the 2/5, which is 3/4 subtract our 4/25.

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

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

Let's have a look at another check question.

Jacob has 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 2/3, all cubed, gives us 8/27, and the square root of 4/25 is 2/5.

Remember the priority of operations, we do multiplication first, so that means the 8 multiply by two is 16, 27 multiply by 5 is 135.

Then let's see if we can identify a common denominator.

Using our knowledge on equivalent fractions, we have 75 /135 subtract our 16/135, gives us a final answer of 59/135.

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 wants 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 for 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 3/5 squared is 9/25.

Remember our priority of operations, do our multiplication first to give us 8/15 and it's still adding that 9/25.

Replacing the subtraction of 8/15 with its additive inverse, we have 9/25 add that -8/15.

Identifying a common denominator with our knowledge on equivalent fractions gives us 90/150, add our 54/150, add our -80/150 giving us the final answer of 64/150 or simplified to be 32/75.

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 way to our root first.

So let's work out the cube root of 8/27, which is 2/3.

So that means we have 9/10 multiplied by 4/5, subtract 2/3, multiply by 2/3.

Using our priority of operations, let's do our multiplication first, giving us 36/50, subtract 4/9, then our common denominator giving us a final answer of 124/450 which can be simplified to 62/225.

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 25/4.

Using our priority of operations, let's do the square root to 25/4 first, giving us 2/3, add 5/2, multiply by 3/5.

Multiplication is next.

So we have 2/3, add 15/10.

Identifying a common denominator of 30 gives us a final answer of 13/6.

Massive well done if you got that one right.

Positive 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 3/4 to be 0.

75 and 1/5 to be 0.

2.

Then using his knowledge on exponents, 3/10 has been converted into 9/100.

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

Then using the priority of operations, 0.

75 times 0.

2 is 0.

150.

Then adding, using our knowledge on additive inverses gives 3 add our -0.

15, add our 0.

09 thus giving us a final answer, 2.

94.

Now Sophia is keeping with fractions.

So she's recognised 3/10 all squared gives us 9/100.

Then using the priority of operations, spotting the multiplication of fractions gives us 3/4 multiplied by 1/5 is 3/20.

Then using our knowledge on additive inverses.

So we're adding each number, gives us 3, add on 9/100, add on -3/20, and using our knowledge on a common denominator and equivalent fractions, we sum them together to give 294/100, which simplifies to 147/50, which is the same as 2 and 47/50.

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 2/3 to 0.

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

When using recurring decimals, for example, 2/3, 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, 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 2/5, all squared, gives us 4/25.

The square root to 49/100 is 7/10.

Using multiplication.

Next we have 28/250.

Identifying those equivalent fractions with that common denominator works our answer to be 1 and 111/125.

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 9/10, add 2.

4/0.

8, multiply by 2/3.

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 24/8, doing the multiplication.

Next, we have 9/10, add 48/24 giving us a final answer of 2 and 9/10.

Great work if you've got that one right.

For B, let's get rid of that 3 divide by 5 and that 0.

3 divided by 4 by writing it as a fraction.

So it gives us 15, add 3/5 multiply by 0.

3/4.

That 0.

3/4 doesn't look very friendly, so let's convert it to the equivalent fraction 3/40.

Using our multiplication, next we have 15 add 9/200, giving us 15 and 9/200 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 3/4, multiply by 7/9.

Multiplication is next, to give us 2 subtract 21/36.

Using the common denominator of 36 gives us an answer of 51/36 which is 1 and 5/12.

For B, we've spotted we have an exponent of 2, and we also have a division of decimals.

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

81/100 subtract 2.

5/0.

5, multiplied by 2.

2.

5/0.

5 isn't very pretty so let's change it to an equivalent fraction, which is 25/5.

Using our knowledge on the priority of operations, doing the multiplication next, and then working out our answer to be -9/19/100.

That was a really tough question.

Great work if you 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 as 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.