Lesson video

Hi, everyone.

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

Today's lesson will consist of some keywords that you may or may not have come across before, but I will be here to help.

You might find it tricky or easy in parts, but we'll learn together.

Really excited to be learning with you.

So let's make a start.

In today's lesson under the unit, "Arithmetic procedures with fractions," we'll be looking at checking and securing, dividing a fraction by a whole number.

And by the end of the lesson, you'll be able to use the mathematical structures that underpin the division of fractions to divide a fraction by a whole number.

So let's have a look at some keywords.

Well, the keywords in the lesson today, we'll be looking at the lowest common multiple.

And the lowest common multiple is the lowest number that is a multiple of two or more numbers.

For example, the lowest common multiple of 3 and 4, is 12.

If you list the three times table and the four times table, the lowest number that's in both of them is 12.

The lowest common multiple of 4 and 10, is 20.

If you list the four times table and the 10 times table, the lowest multiple in both of these numbers is 20.

In today's lesson, it'll be broken up into two parts.

The first part, we'll be looking at context for dividing fractions, and the second part, we'll be looking at dividing any fraction by a whole number.

So let's make a start by looking at context for dividing fractions.

Now, we use fractions in everyday life, as fractions are really useful when dividing quantities such as money, time, or food.

And what's important to remember is we're not even aware that we use fractions and division when we are doing these everyday life questions.

And here are some examples of how we've divided a fraction by a whole number.

Let's look at a brownie.

So here, you can see a brownie, and what we want to do is equally share it between four students.

What fraction does each student receive? So hopefully you've identified it's a quarter.

The 4 identifies how many equal pieces we've split our brownie into, and the 1 of the numerator identifies how many pieces we've given to each student.

Now we can also write it like this.

Well, we started off with a brownie, which is actually 4 over 4 pieces.

Remember 4 over 4 is the same as 1.

Then we divide 4 over 4 by 4, because we want four equal pieces.

So that means each student got one quarter each.

We'll be summarising our calculations a lot in the lesson just so you can see the overview of the calculations.

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

Here, we've got a chocolate bar and it has 16 equal pieces.

The Oak teacher takes one piece and then shares the rest equally between the five students.

How many pieces will she give to each of her five students? Have a look at the picture and see if it helps you.

Well, hopefully you've spotted each student would actually get three pieces.

This would be for one student, another student, another student, another student, and another student.

So from the five students, each student got three sixteenths of the chocolate bar.

I'm going to summarise this again.

So here's our chocolate bar.

Remember, it's broken into 16 pieces.

So a whole would be 16 over 16.

Then we subtracted one piece because that was what the Oak teacher wanted, meaning, we have 15 over 16 pieces left.

And because we needed to divide it by the five students, we did 15 over 16 divided by 5, and that gives an answer of 3 over 16.

Now let's have a look at a quick check question.

Here's a pizza.

And the pizza is divided into nine equal pieces.

The Oak teacher takes one piece and shares the rest equally between her four students.

What fraction does each student receive from the whole pizza? 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, if the teacher took one piece, we need to look at the remaining pieces and divide it equally between the four students.

So hopefully you've spotted one student will get two pieces, another student will get two pieces, another student will get two pieces, and another student will get two pieces.

So that means, from the entire pizza, each student gets two ninths of a pizza.

Well done.

Hope you got that one right.

Sometimes when we're dividing, the divide is not easily seen or calculated.

So let's have a look at a pizza again.

In this pizza, it's divided into three equal pieces, and the Oak teacher takes one piece, but the remaining pieces will be equally divided amongst three students now.

How do you think this can be divided equally amongst three students? See if you can have a little think.

This is quite tough because two pieces is not an easy number to divide by 3.

So what we're going to do, is we're going to split those two pieces into six pieces like this, and you can see it makes it so much easier because now, we can divide this amongst our three students.

Laura looks at it and says, "Now each student gets two sevenths." Have a little look and I want you to explain, is Laura correct? Well, hopefully you spotted she's incorrect.

The whole pizza needs to be divided equally.

So that means for the pizza to be divided equally, there are nine parts in total.

So that means each student got two ninths, not two sevenths.

So remember, when you're working out a fraction, always look at the whole.

In this case, the whole pizza is divided into nine parts.

So let's summarise what we just did.

Well, we started off with a whole pizza and it was divided into three equal parts.

Then we subtracted one third because that's what the Oak teacher took, meaning, we had two thirds left.

Because two thirds wasn't easily divided by 3, we decided to divide each piece by 3, so that means we had each piece being one ninth.

Then from here, we could do six ninths because that was the two pieces, each piece divided into 3, divided by 3, meaning, each student got two ninths.

So now let's have a look at your task.

Here is a pizza and it's split into 12 equal pieces.

A teacher takes two pieces and the remaining is given equally between two students.

What fraction of the whole pizza does each student receive? For question 2, "There are 14 crayons in a box.

Two are lost and the rest are shared equally between a table of four students.

What fraction of the full crayon box does each student receive?" See if you can give these a go, and press pause if you need more time.

Well done.

So let's move on.

For question 3, "A chocolate bar has 24 equal pieces and the teacher eats six pieces and shares the rest between three students.

The teacher says, 'Everyone has had the same fraction of the chocolate bar.

' Is the teacher correct?" For question 4, "A pizza is cut into four equal pieces, and there are six people who need to share the pizza equally.

How should the pizza be cut so that everyone receives an equal piece? And what fraction does each person receive?" See if you can give this a go, and press pause for more time.

Great work.

Well done.

For question 1, "A pizza is split into 12 equal pieces and a teacher takes two pieces, and the remaining is given equally between two students.

What fraction of the whole pizza does each student receive?" Let's have a look at our pizza.

Removing two pieces means there are 10 pieces left, so 10 pieces out of the 12.

Now, because there's 10 pieces, when we divide this by 2, that means each person gets five pieces each.

So they get five out of 12.

Well done.

You got that one right.

For question 2, "There were 14 crayons in a box.

Two were lost, and the rest were shared equally between four students.

What fraction of the full crayon box does each student receive?" Well, if there were 14 crayons and two were lost, that means there are 12 left.

12 divided by those four students means each person gets three each.

So it was three out of the 14, because there were 14 crayons originally in the box.

For question 3, "The chocolate bar had 24 equal pieces.

Now the teacher eats six and shares the rest between the three students.

And then the teacher says, 'Well, everybody's had an equal share of the chocolate bar.

' So is the teacher correct?" Well, let's have a look.

Removing six pieces means there are 18 pieces left.

And out of those 18 pieces, it's divided amongst three students.

So each student gets six pieces.

So that means each person got a quarter of the chocolate bar.

Well done.

Hope you got this one right.

For question 4, "A pizza is cut into four equal pieces, but there are six people who need to share the pizza equally.

How should the pizza be cut so everyone receives an equal piece? And what fraction does each person receive?" Well, at the moment the pizza's cut into four pieces, but we need to divide it amongst six people.

So that means we need to cut each quarter into three pieces, thus making 12 pieces in total.

That means 12 can be divided by 6, giving each person two pieces out of the 12.

So 2 out of 12 is the fraction each person gets, which is the same as one sixth.

Well done.

Hope you got that one right.

Great work, everybody.

So let's move on to the second part of our lesson where we'll be dividing any fraction by a whole number.

So drawing pictures of brownies, cakes, and pizzas is good pictorial way to represent the division of fractions by an integer, but it can take some time.

So let's move on to bar models to show it in a more quicker and efficient way.

For example, let's look at nine tenths divided by 3.

I'm going to use a bar model to illustrate 1 whole here.

What I want you to do is have a think about what does nine tenths look like.

Well, hopefully you spotted it means that we have our whole bar, we split it into 10 equal parts, and then we highlight nine of those equal parts, thus giving us nine tenths.

Then, we're going to divide nine tenths by 3.

So I want you to have a look at the bar model and see.

Well, when you divide nine tenths by 3, what does that look like? If we're dividing nine tenths by 3, we're creating three groups.

So what's in each group? Well, hopefully you can spot we have three equal groups here.

And in each equal group we have three tenths.

So therefore, nine tenths divided by 3 is simply three tenths.

Next, let's do a check question.

I'll do the first question, I'd like you to try the second one.

Same again, we'll still be using bar models.

We're going to be working out four fifths divide by 2.

So hopefully you can spot I've got my bar model illustrating 1, and then I have another bar illustrating the fifths.

So identifying four fifths, I have this, because you can see four out of the five have been shaded.

Now out of those four fifths, I need to divide it into two groups.

So dividing four fifths into two groups, I have this.

So now I've identified the two groups.

This means four fifths divided by 2, is two fifths.

Well done if you spotted this.

What I'd like you to do is try this question, six sevenths divided by 3.

See, we can use the bar model, and shading and identify what is the answer when you divide six sevenths by 3.

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

Well done.

Let's see how you got on.

Well, you can see the bar model is split into sevens, but I was interested in six sevenths, so I'm going to shade in here.

Then I'm gonna group it into threes because I'm dividing by 3.

So that means grouping it into threes, I now have six sevenths divided by 3, is two sevenths, because there are two sevenths within each group of three.

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

So let's move on to another check question.

But here I want you to have a look at the questions that we've just done.

I want you to see if there's an easier way to divide the fraction by the integer without bar models.

Have a little think.

Well, hopefully you can spot when the numerator of the fraction is a multiple of the divisor, the division of the fraction by the integer can be quickly worked out.

So let's have a little look.

If you had six sevenths divided by 3, the numerator is 6.

Six divided by 3, is 2, so our answer is two sevenths.

If you have nine elevenths divided by 3, 9 is our numerator, 9 is a multiple of our divisor, a 3.

So 9 divided by 3, is 3, so this gives us three elevenths.

So let's try with another one.

If you had 16 over 21 divided by 4, 16 is a multiple of 4, so we can divide 16 by 4.

Gives us 4 over 21.

Now, in the next one, you'll need to divide the 5, which is our numerator, into 3.

This is a little bit trickier because the numerator of the fraction is not a multiple of the divisor.

But let's see how we tackle this question.

Well, if we have to do five sixths divided by 3, and we've recognised 5 is not a multiple of the divisor of 3.

So let's show this with a bar model.

Here's 1, and what I want you to do is think about what does five sixths looks like.

Well, hopefully you can spot five sixths looks like this.

Now, can we divide our five sixths by 3 easily? Well, hopefully you can see it can't be divided into 3 easily.

So let's divide each of these one sixth by 3.

So now we can see we have 15 out of 18 pieces, which still represent our five sixths.

So can 15 over 18 be divided by 3? Well, yes they can.

We can group them in this way.

So we have three equal size groups, thus, 5 over 6 divided by 3, is 5 over 18.

Really well done if you got this.

But let's summarise what we've just done.

Well, we couldn't divide five sixths by 3 easily.

So what we did was we looked at each 6, and divided each one by 3, thus giving us 18 equal parts.

But the five sixths represented 15 out of the 18.

Then we could divide the 15 by the 18, thus giving us five eighteenths.

Now what I want you to do is have a look at a check question.

I've done part of it for you.

I'd like you to fill in the gaps.

Have a look at the bar model and the question, and see what you can figure out.

Press pause if you need more time.

Great work.

So let's see how you got on.

Looking at our bar model, we illustrated 1 whole, and then we broke it into four equal parts.

Then from our four equal parts, we've identified three quarters.

But three quarters couldn't be divided by 2 easily.

So what we did, was we divided each quarter by 2, and you can see that in our bar model.

Now dividing each quarter by 2, has now meant I have eights.

I actually have six eighths, which is exactly the same as three quarters.

Six eighths can be divided by 2, which then gives me a final answer of three eighths.

A huge well done.

Hope you got that one right.

Bar models are fantastic.

But removing the bar model and using equivalent fractions can make the calculation much easier.

So let's have a look at seven tenths divided by 2.

Now we can recognise the numerator is 7, and we have the divisors of 2.

Seven is not a multiple of 2.

So let's use our knowledge on equivalent fractions.

What equivalent fractions should we use for seven tenths to make the division easier? Well, you're looking for a common multiple of 7 and 2.

You could use 14 over 20, as that is equivalent to 7 over 10.

The numerator is 14, which is a multiple of 2.

You could use 28 over 40, that is equivalent to seven tenths.

You might notice the numerator is 28, and that is a multiple of 2.

You could even use 140 over 200.

There are an infinite number of equivalent fractions to use.

But what is really important, is recognising that you're writing an equivalent fraction with a numerator of the fraction is a multiple of the divisor, and that makes the whole calculation much easier.

So let's have a look at the seven tenths divided by 2 again.

So we're writing seven tenths as an equivalent fraction with a numerator of 14.

To do this, I'm simply multiplying the numerator and the denominator by 2 over 2, because we know 14 is the lowest common multiple 7 and 2.

Now, I'm going to divide 14 over 20, divide by 2, which is much easier, which is 7 over 20.

Using the lowest common multiple of the numerator of the dividend and the divisor, means you can divide more efficiently.

Let's have a look at another check question.

I'm going to do the one on the left and I'd like you to do the one on the right.

We have three fifths divided by 4.

And what we need to do, is show all our working out.

Well, hopefully you can spot 3, which is our numerator, and 4 is our divisor.

Three is not a multiple of 4.

So we need to think about a lowest common multiple of 3 and 4.

Well, hopefully you've spotted 12.

12 is the lowest common multiple of 3 and 4.

So if I multiply our three fifths by 4 over 4, that means I have my 12 over 20.

12 over 20 divided by 4 is so much easier, as that simply gives us three twentieths.

A huge well done if you spotted this one.

See if you can give the next one a go.

You have to work out the answer to five eights divided by 3.

Ensure you show all your working out.

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've looked at that numerator of 5 and that divisors of 3.

Well, 5 is not a multiple of 3, so let's identify the lowest common multiple of 5 and 3, which is 15.

Now to make an equivalent fraction with a numerator of 15, we're going to multiply by 3 over 3, thus giving us 15 over 24.

Now we can divide 15 over 24 by 3, giving us 5 over 24.

Huge well done if you got that one right.

Let's have a look at another check question.

Lucas and Jacob are both doing this question.

Lucas did six sevenths divided by 4, showed this working out and got three fourteenths.

Jacob got six sevenths divided by 4, did this working out and got 6 over 28.

Who's working out is correct? See if you can give it a go, and press pause if you need more time.

Great work.

So hopefully you've spotted they're both correct.

But Lucas has just used the lowest common multiple of 6 and 4, which is 12, and then gave the answer in its simplest form.

Jacob is still correct, but the answer is not in its simplest form.

Both Lucas and Jacob's answer are equivalent and they're both correct.

Well done.

Hope you got that one right.

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

I want you to match the questions and the answers.

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

Well done.

Let's move on to the next one.

Question 2, you need to work out the answers to the following, and you can draw bar models if you need.

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

Fantastic work.

Let's move on to question 3.

Question 3 shows Jacob has used equivalent fractions to work out his answer, but he spilled ink all over his work.

Can you identify what his missing numbers were? See if you can work it out, and press pause for more time.

Really well done.

So let's have a look at question 4.

Question 4 wants you to work out the answers to the following, giving your answer in their simplest form.

See if you can give it a go, and press pause for more time if you need.

Well done.

For question 1, hopefully you've spotted 12 over 19 divided by 4 is 3 over 19.

12 over 19 divided by 3, is 4 over 19.

And 12 over 19 divided by 2, is 6 over 19.

Well done.

Hope you got that one right.

For question 2, you could use bar models if you like.

So here are the answers.

24 over 25 divided by 6, is 4 over 25.

21 over 31 divided by 7, is 3 over 31.

9 over 11 divided by 3, is 3 over 11.

And 10 over 19 divided by 5, is 2 over 19.

Very well done.

Hope you got that one right.

For question 3, we have to identify the working out given Jacob's spilled ink all over his work.

For A, you spot that 3 is not a multiple of 2, so multiplying by 2 over 2 is our working out here, thus given us the equivalent fraction of 6 over 4, which then can be divided by 2 to give me three quarters.

Five eighths divided by 4, well, hopefully you spotted 5 is not a multiple of 4.

So let's multiply by 4 over 4 thus giving us an equivalent fraction of 20 over 32 to be divided by 4, giving us the final answer of 5 over 32.

Great work if you got this one.

For question 4, you had to show your working out and you can use bar models if you like.

Here, you spot the numerator of 5 is not a multiple of 4, so we'll going to write an equivalent fraction multiplying 5 over 9 by 4 over 4, giving us 20 over 36, which can be divided by 4, giving us 5 over 36.

For B, 6 is not a multiple of 5.

So identifying our lowest common multiple, which is 30, we can multiply 6 over 11 by 5 over 5 to give 30 over 55.

This can then be divided by 5 to give us 6 over 55.

For C, you have 10 over 13 divided by 4.

10 is not a multiple of 4, so let's identify that lowest common multiple, which is 20.

I'm going to multiply 10 over 13 by 2 over 2, thus giving us 20 over 26, which can be divided by 4, to give us 5 over 26.

Really well done if you've got these answers.

You may have got an equivalent fraction of these, but as long as you simplify and get these answers, you know you're right.

Great work, everybody.

So in summary, we use fractions in everyday life as fractions are useful when dividing quantities such as money, time, or food.

And sometimes we can divide a fraction by an integer by simply looking at that numerator of the dividend.

And sometimes we need to use an equivalent fraction to divide the fraction by an integer and make the calculation easier.

A huge well done.

It was great learning with you.