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 3 times table and the 4 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 4 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 1/4.

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 4/4 pieces.

Remember 4/4 is the same as 1.

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

So that means each student got one 1/4 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 1 piece, and then shares the rest equally between the 5 students.

How many pieces will she give to each of her 5 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 3/16 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 16/16.

Then, we subtracted one piece because that was what the Oak teacher wanted, meaning we have 15/16 pieces left, and because we needed to divide it by the five students, we did 15/16 divided by 5, and that gives an answer of 3/16.

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

Here's a pizza, and the pizza is divided into 9 equal pieces.

The Oak teacher takes 1 piece and shares the rest equally between her 4 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 2/9 of a pizza.

Well done if 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.

And this pizza, it's divided into 3 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 3 students? See if you can have a little think.

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

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 2/7." 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 2/9, not 2/7.

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 1/3, because that's what the Oak teacher took, meaning we had 2/3 left.

Because 2/3 wasn't easily divided by three, we decided to divide each piece by three, so that means we had each piece being 1/9.

Then, from here, we could do 6/9 because that was the two pieces, each piece divided into three, divided by three, meaning each student got 2/9.

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 two, 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 three, 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 four, 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.

So let's move on.

For question five, it's Sofia's birthday, and she has four pizzas to split equally between her six friends, but what fraction of a whole pizza does each person receive? See if you can give it a go, and press pause if you need more time.

Well done.

So let's go through these answers.

For question one, a pizza was 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 two, that means each person gets five pieces each, so they get five out of 12.

Well done if you got that one right.

For question two, 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 three, 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 1/4 of the chocolate bar.

Well done if you got this one right.

For question four, 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 1/4 into three pieces, thus making 12 pieces in total.

That means 12 can be divided by six, giving each person two pieces out of the 12, so two out of 12 is the fraction each person gets, which is the same as 1/6.

Well done if you got that one right.

For question five, it's Sofia's birthday, and she has four pizzas to split equally amongst her six friends.

What fraction of a whole pizza does each person receive? Well, you can cut this a number of ways, but really the best way to do it is to cut each pizza into thirds, and the reason for that is because splitting each pizza into thirds means we have 12 pieces in total, and 12 is divided by six.

So that means 12 divided by six is two, meaning each person gets 2/3 of a whole pizza each.

Really well done if 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 a 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 9/10 divided by three.

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

What I want you to do is have a think about what does 9/10 look like? Well, hopefully, you spotted it means that we have a whole bar, we split it into 10 equal parts, and then we highlight nine of those equal parts, thus giving us 9/10.

Then, we're going to divide 9/10 by three.

So I want you to have a look at the bar model, and see, well, when you divide 9/10 by three, what does that look like? If we're dividing 9/10 by three, 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 3/10.

So therefore, 9/10 divided by three is simply 3/10.

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 4/5 divided by two, so hopefully you can spot, I've got my bar model illustrating one, and then I have another bar illustrating the fifths.

So identifying 4/5, I have this, because you can see four out of the five have been shaded.

Now, out of those 4/5, I need to divide it into two groups.

So dividing 4/5 into two groups, I have this.

So now, I've identified the two groups.

This means 4/5 divided by two is 2/5.

Well done if you spotted this.

What I'd like you to do is try this question, 6/7 divided by three.

See if you can use the bar model and shading, and identify what is the answer when you divide 6/7 by three.

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 sevenths, but I was interested in 6/7, so I'm going to shade in here.

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

So that means grouping it into threes, I now have 6/7 divided by three is 2/7 because there are two sevenths within each group of three.

Really well done if 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, and 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 6/7 divide by three, the numerator is six.

Six divided by three is two, so our answer is 2/7.

If you have 9/11 divided by three, nine is our numerator.

Nine is a multiple of our divisor of three.

So nine divided by three is three, so this gives us 3/11.

So let's try with another one.

If you had 16/21 divide by four, 16 is a multiple of four, so we can divide 16 by four.

Gives us 4 over our 21.

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

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 5/6 divide by three, and we've recognised five is not a multiple of the divisor of three, so let's show this with a bar model.

Here's one, and what I want you to do is think about what does 5/6 looks like? Well, hopefully, you can spot 5/6 looks like this.

Now, can we divide our 5/6 by three easily? Well, hopefully you can see it can't be divided into three easily.

So let's divide each of these 1/6 by three.

So now, we can see we have 15 out of 18 pieces, which still represent our 5/6.

So can 15/18 be divided by three? Well, yes, they can.

We can group them in this way, so we have three equal sized groups, thus 5/6 divided by three is 5/18.

Really well done if you got this.

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

Well, we couldn't divide 5/6 by three easily.

So what we did was we looked at each sixth, and divided each one by three, thus giving us 18 equal parts, but the 5/6 represented 15 out of the 18.

Then, we could divide the 15 by the 18, thus giving us 5/18.

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 one whole, and then we broke it into four equal parts.

Then, from our four equal parts, we've identified 3/4, but 3/4 couldn't be divided by two easily.

So what we did was we divided each 1/4 by two, and you can see that in our bar model.

Now, dividing each 1/4 by two has now meant I have eighths.

I actually have 6/8, which is exactly the same as 3/4.

6/8 can be divided by two, which then gives me a final answer of 3/8.

A huge well done if 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 7/10 divided by two.

Now, we can recognise the numerator is seven, and we have the divisor of two.

Seven is not a multiple of two.

So let's use our knowledge on equivalent fractions.

What equivalent fractions should we use for 7/10 to make the division easier? Well, you're looking for a common multiple of seven and two.

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

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

You could use 28/40.

That is equivalent to 7/10.

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

You could even use 140/200.

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 as a multiple of the divisor, and that makes the whole calculation much easier.

So let's have a look at the 7/10 divided by two again.

So we're writing 7/10 as an equivalent fraction with a numerator of 14.

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

Now, I'm going to divide 14/20 divided by two, which is much easier, which is 7/20.

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 3/5 divided by four, and what we need to do is show all our working out.

Well, hopefully, you can spot three, which is our numerator, and four is our divisor.

Three is not a multiple of four.

So we need to think about a lowest common multiple of three and four.

Well, hopefully, you've spotted 12.

12 is the lowest common multiple of three and four.

So if I multiply our 3/5 by 4/4, that means I have my 12/20.

12/20 divided by four is so much easier, as that simply gives us 3/20.

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 5/8 divided by three.

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 five and that divisor of three.

Well, five is not a multiple of three.

So let's identify the lowest common multiple of five and three, which is 15.

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

Now, we can divide 15/24 by three, giving us 5/24.

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 6/7 divide by four, showed this working out, and got 3/14.

Jacob got 6/7 divide by four, did this working out, and got 6/28.

Whose 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 six and four, 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 if 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 two, 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 three.

Question three 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 four.

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

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

Well done.

So let's move on to the last question.

Question five shows the answer, but we don't know the question.

See if you can work out what is the question when the answer is 2/15.

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

Great work, everybody.

So let's see how you got on.

For question one, hopefully you've spotted 12/19 divided by four is 3/19, 12/19 divided by three is 4/19, and 12/19 divided by two is 6/19.

Well done if you got that one right.

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

So here are the answers.

24/25 divided by six is 4/25.

21/31 divided by seven is 3/31, 9/11 divided by three is 3/11, and 10/19 divided by five is 2/19.

Very well done if you got that one right.

For question three, we had to identify the working out, given Jacob spilled ink all over his work.

For A, you have 3/4 divided by two.

You spot that three is not a multiple of two.

So multiplying 3/4 by 2/2 is our working out here, thus giving us the equivalent fraction of 6/4, which then can be divided by two to give me 3/4.

5/8 divided by four.

Well, hopefully you spotted five is not a multiple of four.

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

Great work if you got this one.

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

Here, you'll spot the numerator of five is not a multiple of four.

So we're going to write an equivalent fraction, multiplying 5/9 by 4/4, giving us 20/36, which can be divided by four, giving us 5/36.

For B, six is not a multiple of five.

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

This can then be divided by five to give us 6/55.

For C, you have 10/13 divided by four.

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

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

Really well done if you 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.

Well done.

For question five, let's go through these questions.

Hopefully you've got these questions all to give the answer of 2/15.

Fantastic work if you got this one 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.