Lesson video

Hi everyone.

My name is Miss 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 everybody.

From the unit, arithmetic procedures with fractions, we'll be looking at dividing a whole number by a fraction, and by the end of the lesson you'll be able to use mathematical structures that underpin the division of fractions to divide a whole number by a fraction.

So let's have a look at some keywords.

First of all, we'll be starting with a proper fraction and a proper fraction is a fraction where the numerator is less than the denominator.

We'll be also looking at unitizing, and unitizing means treating groups that contain or represent the same numbers of those things as ones or as units.

Today's lesson will be broken into two parts.

The first will be making sense of dividing by a fraction, and the second will be deepening understanding of dividing by a fraction.

So let's make a start by looking at making sense of dividing by a fraction.

Now, it's important to remember division can be written in all sorts of different ways.

For example, how many different ways can you write 8 divide by 2? Well, we could do it as 8 shared by 2.

So here's our 8 shared by 2.

We could do it as grouped into 2.

So here's our 8 and we've grouped it into four 2s.

We could also say 8/2 where the line indicates division.

Another example is using unitizing, for example, 20 divided 5 by unitizing or grouping.

Here's our 20 and I'm going to ask, how many times does 5 fit into 20? Well, you can see 5 fits into 20 four times.

What this means is the 5 is the counting unit.

In other words, we're identifying the 5 as the unit, and 4 is how many times that unit appears in the 20.

So we can see 5 appears in the 20 four times.

This approach can help us understand and represent division of fractions too.

For example, 3 divided by 1/2.

Here we have the 1/2 is the unit.

So let's identify our 3 using a bar model.

Here I've got three 1's to represent our 3.

I'm going to identify, well, how many halves do I have? Remember each 1/2 is our unit.

So we have one, two, three, four, five, six.

So our answer to 3 divided by 1/2 is 6 because we have six units where each unit is the 1/2.

Let's have a look at another example.

6 divided by 2/3.

Well, we're going to look at how many times does 2/3 fit into 6? Here's my 6 and I'm gonna break each one into 1/3.

Now we're going to count how many 2/3 fit in.

Remember our 2/3 is our unit, so that will be one, two, three, four, five, six, seven, eight and nine.

So 6 divided by 2/3 is 9 because 9 lots of 2/3 fit into 6.

Now what I'm going to do is I'm going to do a question and I'd like you to do another question.

Here, we need to work out 4 divide by 2/3, and I'm going to use a bar model help.

I've got my 4 divide by 2/3 is represented as four wholes.

I'm going to split each whole into thirds and then I'm going to say, well, "Our unit is 2/3, so I'm going to count how many 2/3 fit into our 4".

One, two, three, four, five and six.

So that means 4 divided by 2/3 is 6.

See if you can give this one a go.

3 divided by 3/5 and draw a bar model if it helps.

Well done.

So let's see how you got on.

Well, 3 divided by 3/5 we can illustrate this with our three whole 1's, divide them into fifths, and then identifying our unit to be 3/5.

So let's count one, two, three, four, five.

A huge well done if you've got the answer to be 5.

So using bar models is a good way to see how the division by fractions works.

However, if we have bigger numbers, it does become difficult with bar models.

For example, 24 divided by 3/5.

We don't want to be drawing our bar model with 24 ones and then identifying a fifth of each 1 because that's a lot of working out.

So we can work out the division by the fractions easily without our bar models.

So let's have a look at an example and see if we can spot an efficient method.

6 divided by 3/5.

Well, here's our 6 and I'm going to split it into our six 1's.

Then I've broken it into fifths, and the question I'm going to ask is how many fifths are there and how can you identify how many fifths you have in the whole number without drawing the bar model? See if you can look at the question and figure out how many fifths are there without counting in the bar model.

Well, hopefully you can spot there's 30 fifths, but how did we get that? Well, if you multiply the denominator of the divisor by the whole number, you'll get your 30.

In other words, 6 multiplied by the 5 gives our 30 so we have our 30 fifths.

Now what we're going to do is we need to work out how many 3/5 we have in 30 fifths.

So remember we're identifying the 3/5 of the unit.

Here's one, two, three, four, five, six, seven, eight, nine and ten.

So the calculation simplified would be 30/5 divided by 3/5 is 10, and using the numbers in the question is a much more efficient method than using a bar model.

So let's see if we can work out this calculation efficiently.

14 divided by 2/3.

Well, we know if we do 14 times the 3, which is the divisor of the fraction we have 42 thirds, so that means 14 is equivalent to 42/3.

Then we simply do 42/3 divided by 2/3, which gives us 21, because we're asking ourselves how many 2/3 go into 42/3 and it's simply 21.

So let's have a look at a check.

Here you are asked to work out 24 divided by 3/5.

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

Well done.

So let's see how you did.

Well, hopefully you've spotted 24 multiplied by the 5 is 120.

So that means we have 120 fifths.

120/5 is equivalent to 24.

Then we're dividing 120/5 by 3/5.

So how many 3/5 go into 120/5? Well it's 40.

Huge well done if you got this one right.

Now let's move on to your task.

I want you to use bar models to show the division of the following.

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 match the question with the calculation with the answer.

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

Really well done.

Let's move on to question three.

Circle which of the following gives an answer of 4? See if you can give it a go and press pause for more time.

Well done.

So let's go to question four.

Question four wants you to put the following in ascending order, smallest to largest, and question five, see if you can create three of your own questions when dividing an integer by a proper fraction, giving an answer of 10.

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

Amazing work everybody.

So let's go through our answers.

For question one, using our bar model, hopefully you can spot the answer to 1a is 9.

For B, using our bar model, how many 2/5 are in 2? Well, identifying our unit as 2/5 we've got one, two, three, four and five.

So our answer is 5 there.

6 divided by 3/4? Well, our unit is 3/4, so let's count.

Here's a 3/4, another 3/4, so on and so forth.

So that means we get our total answer to be 8.

Well done if you got this one right.

For question two, we had to match the question with the calculation with the answer.

4 divided by 2/3 is the same as 12/3 divided by 2/3, which gives us 6.

12 thirds is exactly the same as 4.

We've simply counted how many thirds are in four, which is 12/3.

6 divided by 3/4 is the same as 24/4 divided by 3/4, which is 8.

12 divided by 3/4 is 48/4 divided by 3/4, which is 16.

8 divided by 2/3 is 24/3 divided by 2/3, which is 12.

Great work.

For question three, which ones give us an answer of 4? Well it was 3 divided by 3/4, 12 divided 3/1, and 4 divided by 4/4.

For question four, putting them in ascending order, smallest to largest.

So let's work out each one first.

For 3 divided by 1/10 we have 30/10, which is our 3, divided by 1/10 which gives us 30.

10 divided by 2/5 is the same as 50/5 divided by 2/5, which is 25.

3 divided by 1/4 is the same as 12/4 divided by 1/4, which is 12.

And 10 divided by 2/3 is the same as 30/3 divided by 2/3, which is 15.

Thus our answer is, in ascending order, 3 divided by 1/4, 10 divided by 2/3, 10 divided by 2/5, and 3 divided by 1/10.

Well done if you got this one right.

Great work everybody.

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

So not every division by a fraction will give an integer answer.

So let's look at when the fraction does not fit a whole number of times using our bar models, for example, 3 divided by 2/5.

I'm going to show this with our bar model.

So here's our 3 and now I'm going to show our fifths.

Now we know our unit is 2/5, so we're going to count how many times does our unit, 2/5, fit into 3? One, two, three, four, five, six, seven whole times.

However, we have a little bit on the end of it.

So this is our unit, but it's not a full unit.

Can you see what fraction of the unit fits into that last part of our bar model? Well, hopefully you can spot it's 1/2 a unit, so that means 3 divided by 2/5 is 7 1/2.

Because 2/5 has fit in seven whole times and a half a time.

This can then be converted into an improper fraction, which is 15/2.

Let's have a look at a quick check.

I want you to use the bar model and identify what do you think the answer to 4 divided by 3/5 is? See if you can give it a go.

Press pause if you need more time.

Well hopefully you can spot it's going to be six full times.

So each unit is 3/5 and look at that.

It's 2/3 of our unit, so that means it's 6 and 2/3.

Writing this as an improper fraction, it's 20/3.

Really well done If you've got this one.

What I'm going to do now is another check question, but I'm going to do the one on the left and I'd like you to do the one on the right, and using our bar model, I want you to work out the answer to the following, 3 divided by 2/3.

So you can see I've split my 3 into three 1's.

Then I've identified my thirds and I'm going to count how many 2/3 there are.

So simply identifying the unit as 2/3.

Here's one, two, three, four.

Now I have a half a unit, so that means the answer is 3 divided by 2/3 gives us an answer of 4 1/2.

Or you could write it as 9/2.

See if you can give the second one a go and press pause if you need more time.

Great work.

So let's see how you got on.

2 divided by 3/4.

Well hopefully you can spot we've done our two whole 1's.

I've identified each whole as quarters, and now we're gonna identify the unit of 3/4, find out how many times did it fit into 2.

One, two and 2/3.

So that means 2 divided by 3/4 is 2 2/3.

Massive well done if you got that one right.

Now let's move on to your task.

For question one, I want you to use the bar models to work out the answers to the following.

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

Well done.

So let's move on to the second question.

Question two says, "Izzy and Laura have done the following bar models for the question 3 divided by 2/7.

Both have made an error with their bar model".

What I want you to do is explain where they've made their error and then amend the bar model to help work out the correct answer.

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

Great work.

So let's move on to question three.

Using squared or gridded paper, draw your own bar models to work out the answers to 5 divided by 2/3, 7 divided by 3/4, and 5 divided by 3/5.

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.

Going through our answers, let's start with question 1a.

2 divide by 3/5.

Well from the bar model, our unit is 3/5.

So let's count how many 3/5 go into 2.

One, two, three and 1/3.

So that means it's 3 1/3.

For question B, 3 divided by 4/5.

So let's identify our unit, which is 4/5.

One, two, three and you might spot, we have 3/4 of our unit, so that means it's 3 3/4.

One divide by 2/5, so that means our unit is 2/5.

One, two and then we have a half a unit.

So that means it's 2 1/2.

Great work if you got this one right.

For question two, let's start with Izzy.

Well, where did Izzy make her mistake when she thought the answer was 9? The question said 3 divided by 2/7.

Well, Izzy has divided each 1 into sixths rather than sevenths, and then Izzy's actually calculated 3 divided by 2/6 which is 9.

So that's where her mistake was.

She didn't do it in sevenths, she did it in sixths.

Next, let's have a look at Laura.

Well, Laura started to correctly use the unit 2/7, but then incorrectly changed the unit to be 7/7.

Remember the unit was 2/7, so she needed to find the fraction of the 2/7 not the 7/7.

Okay, so amending the bar model to help work out the answer to 3 divided by 2/7 we're going to amend Laura's.

So all I've done here is simply amend what she identified to be that final unit, which we know the unit is 2/7.

So if you look at it, what fraction of our unit is that remaining amount? Well, it's a half a unit, so that's why 3 divided by 2/7 is 10 whole units and a half a unit.

Really well done if you got that one right.

Question three says, using square paper or gridded paper, draw your own bar models to show the answers to the following.

So for 3a, I'm hoping your bar model looks something like this.

Here we have our 5 whole units dividing into thirds.

Then identifying each unit to be 2/3, one, two, three, four, five, six, seven and a half a unit.

So the answer is 7 1/2.

For B, 7 divided by 3/4.

Let's identify our seven and then break it into quarters.

Now we're identifying our unit to be 3/4.

So let's count.

One, two, three, four, five, six, seven, eight, nine and we have 1/3.

So that means it's 9 1/3 of a unit.

For question C, 5 divide by 3/5.

Well, here's our 5 and dividing into fifths we have this.

Each unit is 3/5, so let's count.

One, two, three, four, five, six, seven, eight and 1/3.

Huge, well done if you've got this answer.

Great work, everybody.

So in summary, using bar models are good ways to see how the division by fraction works.

And unitizing means treating groups that contain or represent the same number of things as ones or units.

Using the unitizing method alongside bar models helps deepen that understanding of how we get the answer when dividing a whole number by a fraction.

A huge well done everybody.

It was great learning with you.