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 key words 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.

Hi everyone and welcome to this lesson from the unit "Arithmetic Procedures with Fractions." And in this lesson we'll be looking at dividing a fraction by a fraction.

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 fraction.

So the key words that we'll be looking at today is something called a reciprocal.

And a reciprocal is the multiplicative inverse of any non-zero number, and any non-zero number multiplied by its reciprocal is always equal to 1.

For example, if we have 5 and 1/5, the reciprocal of 5 is 1/5, and the reciprocal of 1/5 is 5.

And this is because 5 multiplied by 1/5 is 1.

Another example, 2/3 and 3/2.

The reciprocal of 2/3 is 3/2, and the reciprocal of 3/2 is 2/3.

And this is simply because if you multiply 2/3 by 3/2 equals 1.

A non example would be 4 and 0.

4.

If you multiply 4 by 0.

4, it doesn't equal 1.

So that means 4 is not the reciprocal of 0.

4, or 0.

4 is not the reciprocal of 4.

Today's lesson will be broken into three parts.

First of all, we'll be looking at dividing two fractions.

Second, we'll be looking at the reciprocals.

And third, we'll be looking at division by multiplying by the reciprocal.

So let's have a look at the first part.

We'll be dividing by two fractions.

Now there are a few different ways to think about division.

For example, 8 divide by 2 we know the answer is 4, and this could be 'cause we are sharing 8 by 2.

We could also say, well, 8 is grouped into 2s, so we have 4 groups of 2.

Or we could also use the unitizing method whereby we have 8 wholes, 2 is our unit, and then we count how many units are in that 8.

You can see there are 4.

Essentially, we're finding out how many times the divisor fits into the dividend, and we can show division of fractions using this simple concept.

So let's visualise this question in a bar model.

We're going to divide 2/5 by 3/7.

So first things first, I'm going to identify what does 2/5 look like.

Well, here's our square.

I'm going to start identifying 2/5.

So you can see I've decided to shade it this way.

I have 5 equal parts and 2 are shaded.

Now let's focus on the 3/7.

I'm going to use horizontal lines for this.

So our horizontal lines, you can see I have my sevenths here and this represents 3/7.

So now what we're going to do is say, "Well, how many 3/7 fit into 2/5?" Well, let's have a look at that green area right here.

What I'm going to do is I'm going to count.

I'm gonna pull out this 3/7, which is that green area, and we're going to count how many times does it fit into that 2/5, which we know is the blue area.

So lets count.

One, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14.

So what we've just worked out is how much of the 3/7 fits into 2/5.

And it's 14 out of our 15, and you can see that from that green rectangle on the right hand side.

There are 14 out of 15 that fit into our 2/5.

This is quite a tricky concept.

So let's see if we can work together and work out another division of fractions.

So now let's work together to work out the answer to this example.

You can press pause at any time if you need a bit more time.

We're going to be looking at 2/3 divided by 3/4.

So first of all, I want you to draw and divide a big rectangle into thirds.

So you can see I've done mine like this.

Now we're going to identify 2/3 from our rectangle.

Well, highlighting 2/3, I have this.

What we're going to do is we're going to look at 3/4.

So let's look at that 3/4.

I'm going to use horizontal lines.

Use horizontal lines and make four rows, and let's identify 3/4 from that.

So I have it here.

All I'm going to do now is pull out and extract that green rectangle and just move it right next to our bar model.

So it's this green rectangle here.

I'm just going to copy it and move it over here.

And hopefully you can spot this green rectangle represented the 3/4 of our entire shape.

Now let's count because we want to ask ourselves how much of the 3/4 will fit into the 2/3.

So let's count.

Remember the 2/3 is the purple bit.

One, two, three, four, five, six, seven, eight.

So that means 2/3 divided by 3/4 is 8/9.

This was a great little activity.

If you want to rewatch this part again, please do.

But now what we're going to do is we're going to do the task.

Work out the area to the following, and you can use these templates for the area model if it helps.

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.

Work out the answer to the following and you can use these templates for the area model if it helps.

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

Great work.

Let's look at question three.

Question three removes the area model, and I'd love you to draw your own area model on squared or gridded paper to work out the answer to 3/4 divide by 4/5.

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

Well done.

So let's see how you got on.

I'm going to look at question one.

For question one, it's 3/5 divided by 2/3.

So you can see the purple section represents 3/5.

Then on the right hand side you can see that represents the 2 rows out of the 3, which is our 2/3.

Then let's simply count.

One, two, three, three, four, five, six, seven, eight, nine.

So that means 9/10 is our answer.

3/5 divided by 2/3 is equal to 9/10.

Massive well done if you got that one right.

For question two, let's identify our 3/7, which is here.

Then looking at those horizontal rows, you can see I've pulled out 3/5 of our horizontal rows.

So let's count how much of our 3/5 goes into 3/7.

One, two, three, three, four, five, six, seven, counting all the way along.

Hopefully, you've worked out the answer to be 15 out of 21.

So that means 3/7 divided by 3/5 is 15 out of 21.

We can simplify this further to give us an answer of 5/7.

Massive well done if you got that one right.

Question three, we had to work out the answer by drawing the bar model.

Hopefully, you would've drawn something like this.

On the left hand you can see 3/4.

The right hand side shows you that rectangle illustrating 4/5 of that.

Counting up, you should have identified the answer to be 15/16.

Huge 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 looking at reciprocals.

Now remember, a reciprocal is the multiplicative inverse of any non-zero number, and any non-zero number multiplied by its reciprocal is always equal to 1.

So if we were identifying the reciprocal of 8, what number do we multiply 8 by to give 1? Well, we know 8/8 is 1.

So what fraction gives us 8/8? 8 multiplied by 1/8 had to give us the 8/8 which, as we know, is 1.

So therefore, the reciprocal of 8 is 1/8 because their product is 1.

Or you could say the reciprocal of 1/8 is 8 because their product is 1.

What do you think we multiply 3/4 by to make the product of 1? You can use this to help you out.

1 is equal to 3 times 4 over 3 times 4.

Well, we can break this into two fractions.

We know 3/3 represents 1.

multiplied by 4/4, which represents 1.

The question gives us 3/4.

So that means the other fraction had to be 4/3.

So therefore, the reciprocal of 3/4 is 4/3 because their product is 1.

Really well done if you spotted that.

Now let's do a quick check.

I want you to identify the reciprocal of the following.

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, the reciprocal of 4 is 1/4.

The reciprocal of 5 is 1/5.

The reciprocal of 2/3 is 3/2.

And the reciprocal of 9/11 is 11/9.

Massive well done if you got those right.

A nice little check is to pop into your calculator and multiply those two numbers together and the product will always be 1.

Now let's have a look at another check.

Sams says, "The reciprocal of 3 2/5 is 3 5/2.

Is Sam correct? Explain.

See if you can give it a go.

Press pause if you need more time.

Great work.

So let's see how you got on.

Well, hopefully, you spotted Sam is incorrect because Sam has to convert the mixed number into an improper fraction.

Well done.

Now it's time for your task.

I want you to work out the reciprocals of the following and press pause if you need more time.

Great work.

Let's move on to question two.

Work out the reciprocals of these, and press pause if you need more time.

Well done.

Let's move on to question three.

Using the cards only once, make the following statements true.

The reciprocal of something over something is something and the reciprocal of something over something is something over something.

Remember, you are only allowed to use each card once.

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

Fantastic work.

So let's see how you got on with question one.

Question 1A states, "What's the reciprocal of 8?" It's 1/8.

B, the reciprocal of 10 is 1/10th.

The reciprocal of 7 is 1/7.

The reciprocal of -3 is -1/3.

The reciprocal of 11/3 is 3/11.

The reciprocal of 2/9 is 9/2.

The reciprocal of 4/15 is 15/4, and reciprocal of 1 is 1 because 1 multiplied by 1 is 1.

Great work if you got this one right.

Question two, convert them into an improper fraction first.

So 5 1/2 is 11/2.

Thus the reciprocal is 2/11.

9 2/3 is 29/3.

So the reciprocal is 3/29.

10 3/5 is 53/5.

So the reciprocal is 5/53.

Really well done if you got this one right.

For question three, using the cards once, you should have got this answer.

This was a really sneaky one as you had to identify 10/8 is an equivalent fraction to the reciprocal.

The reciprocal of 4/5 is 5/4, and an equivalent fraction of 5/4 is 10/8.

That was a tricky one.

Well done.

Great work everybody.

So let's have a look at the last part of our lesson where we'll be looking at division by multiplying by the reciprocal.

So let's have a look at these two calculations, 3/5 divided by 2/3 and 3/5 multiplied by 3/2.

Let's use that area model again to do 3/5 divided by 2/3.

Here's our 3/5.

This is our 2/3.

Pulling out that 2/3, let's count how much of our 2/3 goes into 3/5.

Counting up gives you an answer of 9/10.

Now let's have a look at that 3/5 multiplied by 3/2.

Well, it's 3 multiplied by 3 over 5 multiplied by 2 which is 9/10.

So what do you notice? Well, hopefully you can spot dividing by a number and multiplying by the reciprocal of that number gives the same result.

Do you think this will always work? Well, let's have a little look at the understanding of why it works.

For example, we looked at this calculation before, 2/5 divided by 3/7, and we worked out the answer to be 4/15.

So let's break this down a little bit more Here we know this represents our 2/5, and we know this represents our 3/7.

Let's see if we can identify rectangles within our whole.

Well, we have a width of 2 and a length of 5, and we have a width of 3 and a length of 7.

So when we extract that rectangle which represents our 3/7, you might actually spot we have a rectangle which is 5 by 3.

Then from here we ask ourselves, "Well, how much of this rectangle fits into this rectangle?" And you can see how these calculations now work together.

So working this out, 2/5 multiplied by 7/3 gives us the answer of 14/15.

This is a wonderful illustration of using area models to demonstrate the division of fractions.

So simply multiplying by the reciprocal of the divisor gives us the same answer, and division can be rewritten as multiplying by the reciprocal.

This works all of the time and can make calculations, particularly with fractions, more efficient.

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

I'll do the one on the left, and you'll do the one on the right.

Let's be efficient and use our knowledge of reciprocals.

2/5 divided by 4/9.

We're going to show all our working out.

Well, we already understand that when dividing by the fraction, we're multiplying by the reciprocal.

So 2/5 divide by 4/9 is exactly the same as 2/5 multiplied by 9/4, thus giving us the answer to be 2 multiplied by 9 over 5 multiplied by 4, giving us a final answer of 18/20, which can then be simplified.

Identifying our highest common factor of 18 and 20, which is 2, this simplified fraction gives us the answer of 9/10.

Massive well done if you got this.

Now what I want you to do is try the question on the right.

See if you can give it a go and show all you're working out.

Well done.

So let's see how you got on.

Well, we know 5/6 divided by 3/4 is the same as 5/6 multiplied by the reciprocal of 3/4, which is 4/3.

Working this out, 5 multiplied by the 4 over 6 multiplied by the 3 gives us 20/18.

Simplifying gives us a final answer of 10/9.

You can write this as a mixed number to give you 1 1/9.

Massive well done if you got that one right.

So now let's have a look at another check question, but now we'll be looking at area.

The question wants us to write the answer as a mixed number.

Here we have a rectangle.

We don't know our length, but we know our width is 3/8 and we know the area is 4/5.

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, we know the formula for the area of rectangle is length multiplied by width.

Therefore, to work out the missing length, we need to calculate area divide by width, which gives us that length.

So it's 4/5 divided by 3/8.

Remember we're dividing by our fraction, so we're going to multiply by the reciprocal.

4/5 multiplied by 8/3 gives us an answer of 32/15, which I can then simplify as a mixed number into 2 2/15.

Massive well done if you got this one right.

Now let's have a look at your task.

First of all, see if you can work out the answers to the following, simplifying your answer where necessary.

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

Well done.

Moving on to question two.

We have some areas of shapes, and you need to work out those missing lengths.

Shape A and B are rectangles, but shape C is a triangle.

So make sure you know your formula for the area of a triangle.

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

Well done.

Let's go through these answers.

Well for 1A, we're dividing 3/7 by 4/9.

So 3/7 multiplied by the reciprocal of 4/9 is 3/7 multiplied by 9/4, giving you the answer of 27/28.

B, 9/10 divide by 8/11.

The reciprocal of 8/11 is 11/8.

So that means it's 9/10 multiplied by 11/8, giving us an answer of 99/80.

99/80, you could also write it as a mixed number to be 1 19/80.

For C, 3/4 divided by 5/12.

Well, multiplying by the reciprocal, we have 3/4 multiplied by 12/5, giving us 36/20.

I can spot we can simplify this a touch more, thus giving us 9/5 or 1 4/5.

Well done if you got this one right.

For question 2, here are some areas of some shapes, and you're asked to work out the missing lengths.

So remember the formula for the area of rectangle is length times width.

Rearranging this, that means we do the area divided by the width will give us the length.

1/4 divided by 3/8 is the same as 1/4 multiplied by 8/3, giving us 8/12, which we know simplified is 2/3.

So our length is 2/3 of a metre.

For B, we have the area of a rectangle is a length multiplied by width is the area.

Rearranging, so the area divided by the width is our length.

Now you might spot we have a mixed number, so let's convert it to an improper fraction.

This will be 11/4, multiplied by that reciprocal, which is 6/5.

This then gives us 66/20, which is 3 6/20.

I spot that 6/20 can be simplified a touch more to give us 3 and 3/10.

So our length is 3 and 3/10.

Now let's have a look at that triangle.

Now remember, the formula for the area of the triangle is base multiplied by the perpendicular height divided by 2.

So rearranging this, we have to multiply the area by 2 and then divide by that 3/7.

So the area is 5/7 multiplied by 2 is 10/7.

Then we're going to divide that 10/7 by 3/7.

Well, remember when we're dividing by the fraction, we're multiplying by the reciprocal.

So 10/7 multiplied by 7/3 gives us 70/21, which is 10/3.

Writing this as a mixed number, we have 3 1/3.

That was a tough question.

Well done if you got that one right.

So in summary, we know the division of a fraction by a fraction can be seen using the area model, and the reciprocal is the multiplicative inverse of any non-zero number, and any non-zero number multiplied by its reciprocal is always equal to 1.

However, it is easy to calculate the division of a fraction by a fraction by simply multiplying the dividend by the reciprocal of the divisor.

A huge well done.

It was great learning with you today.