Lesson video

Hello, it's Mrs. Marla again, and I'm here to teach you lesson 18 within this fractions topic.

I'm really pleased to be back, I'm sure we're going to have a great lesson together.

So we're going to start by reviewing the practise activity that I sent you at the end of lesson 17.

And this is very much about solving and simplifying, but critically thinking about why you chose your preferred method.

Did you choose method one or did you choose method two? And that's what we're going to look at when we go through my responses anyway, and you can consider whether they were similar to yours.

There was also a challenge and I hope that some of you at least had a bit of time to have a go at that.

So let's have a look.

So I've put my jottings here on the slide and I'm going to talk you through.

Now, 8/9 add 8/9 subtract 1/9 gave us a total of 15/9.

And when I looked at 15/9 and I thought, would I rather simplify now or would I rather convert to a mixed number and then simplify? I don't know why, but I thought I'm going to simplify now, the numbers were, I guess, a little bit bigger and I want to reduce them to a simpler form.

So that's what I did and identified the highest common factor of three, which you can see with my pink jottings.

And then I simplified that fraction to 5/3.

And then it felt a little bit more comfortable for me to work with 5/3 and convert that to a mixed number.

So in my blue jottings, you can see 5/3.

We know that 3/3 will make one whole, and then we will have 2/3 remaining as the fractional part.

So when I simplified that fraction and then converted it to a mixed number, it was 1 2/3.

So here are my jottings for when I solved this calculation with method two and then simplified it.

So we did the same step at the start and we got an answer of 15/9.

And at that point, I then converted that to a mixed number from an improper fraction.

And those are my pink jottings, I had 1 6/9.

I then had to remember that we don't worry about the whole number and we simplify that fractional part.

So I circled the blue, you can see there it's 6/9, and took that across.

And then again, we know that we, the highest common factor is three, so I simplified that fractional part to 2/3.

Then I had to bring back my one and my 2/3 together to give us an answer of 1 2/3.

Did you do it that way? Was that the way you preferred? Which felt most comfortable? I'm hoping that you had an opportunity to explain that to somebody who lives with you or somebody over the phone, well done.

So here was the second calculation that we have to solve, and simplify and consider which method felt more comfortable, which one we preferred.

So 7/10 add 5/10 add 3/10 was equal to 15/10.

Now, when I looked at 15/10, you know, they're multiples, those numbers are multiple of five, they felt quite rounded.

And so I thought I'm going to convert at this point, and I felt comfortable to do that.

So I converted it to a mixed number.

And you can see in my pink jottings, that 15/10, as an improper fraction, converted to 1 5/10.

Again, I knew at this point that I could ignore that one for a moment, and I just had to look at the 5/10 in order to simplify that fractional part.

And it was very obvious to me at this point that five was the highest common factor.

And 5/10 as a fraction, I just know that that's 1/2, so that felt like a really easy step.

So 5/10 simplified by dividing both the numerator and the denominator by five is one half.

And then I brought my one whole back and my fractional parts of a half, and therefore I had my simplified fraction as a mixed number, as one and a half.

Was that the method that you chose or did you choose method one? I would be really interested to know what your choices were and how you explained it to somebody.

Now, you might have been somebody who didn't choose method two and you chose method one.

Here, we can see the same calculation and we can see that the sum of these three fractions is 15/10.

Did you feel at this point that you wanted to simplify by recognising very quickly that the highest common factor was five and then dividing both the numerator and the denominator by five to get three halves or 3/2.

It's funny, isn't it? When we've got two on the bottom, we sometimes don't recognise immediately that it's halves.

So now look at my blue jottings, 3/2, three halves.

And then I have to consider how many halves I could make.

Well, that's one whole and I've got 1/2 left over.

So that would give me one whole and 1/2, which is exactly the same answer as we got using method two, but we got there in a different way.

Was this the way that you did it? Well done, if you thought through both of those calculations and really reasoned as to which method you preferred to use in order to solve and simplify.

So this was the challenge, and I really hate that some of you had the opportunity to have a go at this.

Ahmed says, "To simplify a fraction, you just halve the numerator and halve the denominator." Is Ahmed's statement always true, sometimes true or never true? Explain your answer.

I love a always true, sometimes true or never true question.

You can really get your teeth into it and provide lots of examples to prove your thinking.

Now, here I have copied across one of our previous questions, and I want you to have a look at 15/10 and how we've simplified it.

Have a look at the numerator and the denominator there.

We've got 15 and 10, could I halve both of those numbers? Could I halve 15 and could I halve 10 in order to simplify? Well, I could halve 10 because that would give me a denominator of five, but I couldn't halve 15, why? Why couldn't I halve 15? Well, in this context of fractions and simplifying, we couldn't have a numerator of seven and a half.

So we've proved here that it's not always true.

We've got an example that's proved that it is sometimes true.

Well, I know there's been lots of examples in previous lessons where the highest common factor has in fact been two, we've divided by two, which as we know is the same as halving.

So we've got an example there again, which shows that it's sometimes true.

An example of that would be eight 8/10.

So the highest common factor there in order to simplify it is two.

So we know that it's not never true, we know that it's not always true, and we've proved here that it's sometimes true.

Well done, if you had a go at that challenge.

So moving on to today's lesson.

Now, lesson 17 was about simplifying fractions when we just added or subtracted them.

Today, we're going to move on, we are still going to be simplifying fractions, but today this is going to be after we have multiplied a fraction by a whole number.

And so we're going to have to draw on some of our prior learning.

We have to note before, we're going to be using that today in today's lesson.

So, multiplying a fraction by a whole number.

We've got a calculation here.

It's 1/8 multiplied by five, and we're going to start off by looking at something that offended mindset.

Charlie says, "When I multiply a fraction by a whole number, I multiply both the numerator and the denominator by the whole number to find the product." Now think about that.

This is what Charlie therefore did.

One is the numerator multiplied by five is equal to five.

Eight, as the denominator, is multiplied by five and is equal to 40.

So 1/8 multiplied by five is equal to 5/40.

I'm going to ask you to pause the video and just take a look at that.

Try and access that prior learning, consider whether Charlie is right or not.

And also look at 5/40 and think to yourself, does that make sense, 5/40, if I'm multiplying 1/8 by five? Pause the video now.

So, 1/8 multiplied by five, Charlie said was equal to 5/40.

And when I think about the whole being divided into 40 parts, and only having five of them, and when I've multiplied 1/8 by five, it just doesn't feel quite right? Now, let's remember that these lessons have been about simplifying, so I'm going to ask a question.

Could we simplify this fraction? Could we express it in its simplest form by using the strategies that we know? Well, to simplify a fraction, we know that we need to identify the highest common factor.

And looking at 5/40, five is a factor of 40 and therefore, that's the highest common factor.

So what I'm going to do now is I'm going to divide 5/40, five and 40, by five, and that gives me 1/8.

What do you notice about that? 1/8, well, it was the number that we started with wasn't it, it was the fraction we had at the start.

And we know that when we multiply a fraction by a whole number, the product is going to be bigger, it's not going to be the same.

And therefore, Charlie's strategy, the method of multiplying a fraction by a whole number was incorrect.

So we're now going to look in the next slide at how we do multiply a fraction by a whole number just to remind us.

So let's remind ourselves about how we multiply a fraction by a whole number.

We're going to carry on working with a familiar fraction.

And we're going to use a story that you used in lesson 15, I think, about an apple that has been divided into eight equal parts.

So if my apple has been divided into eight equal parts, what fraction have I divided my apple into? That's right, I've divided it into 1/8, and 8/8 make the whole apple.

Now the story goes that through the day I eat and slices of this apple that I've divided into 1/8s.

And I can represent this on this model here.

And so my question is, at the end of the day, what proportion of the apple had I eaten if the parts, the equal parts that I had eaten, have been represented here in blue? Let's take a look.

So here I have a representation of the proportion of the apple that I have eaten over the day.

And if I wanted to calculate how much of the apple I had eaten, and what fraction, then I could add 1/8 five times, 1/8 add 1/8 add 1/8 add 1/8 add 1/8.

And that's called repeated addition, because I'm repeatedly adding one of the fractional parts that I've eaten and how many times I've eaten it, it's five times.

Now, let's say that I didn't want to use addition.

What other way could I express the amount of the apple that I had eaten? I want you to pause the video and have a think.

So I don't want to use addition.

What other way could I express the amount of apple that I have eaten? Okay, I'm sure that many of you looked at those 1/8s, realised that there was five of them, and said that actually I could use multiplication, I could multiply 1/8 by five, 1/8 multiplied by five.

Now, did you write it like that? Or did you write it another way? And that's okay if you did, some of you may have written five multiplied by 1/8, is that okay? Do they mean the same thing? Have a pause of the video and just try to remind yourself about what it's called, that rule, that law, and multiplication, where we can change the order of how we present in this case, the fraction, and the whole number.

Okay, did you remember? So this is called the commutative law, and in multiplication ,with whole numbers, we can change the order of the whole numbers.

And in this case, we can change the order that represents the whole number and the fractional part.

So all these ways of expressing that representation are okay, they will work, because repeated addition, 1/8 adding it five times is equal to five multiplied by 1/8 which is also equal to 1/8 multiplied by five.

So here's my model again.

And I think it's probably quite obvious that 1/8 multiplied by five is equal to 5/8.

By the end of the day, I'd eaten 5/8 of the apple, that was the proportion or the fraction that I'd eaten.

Now, let's go back to what my friend Charlie said.

Do you remember that he said, we multiply both the numerator and the denominator by the whole number, and he got 5/40 as well.

We can see here that that's not right.

And indeed at the start, it didn't really feel right.

We know from our prior learning that this is the generalisation, that when we multiply a fraction by a whole number, the numerator of the fraction is multiplied by the whole number, but the denominator stays the same.

So have a look at this calculation here, that's exactly what we can see.

One, is the numerator, and we multiply that by five, equals five.

Have a look at the denominator that is eight.

And then here, in our calculation at the bottom, with the green eight, we can see that the 1/8 has stayed the same.

So 1/8 multiplied by five is equal to 5/8, and we've used the model and the generalisation to help us understand that and solve that equation.

So we have reminded ourselves about how we multiply a fraction by a whole number, but let's remember that all of these lessons have been about simplifying fractions.

So, now that we have our product of 5/8, my question is, can I simplify this fraction? So I want you to draw on the knowledge that you've got, pause the video and have a think or talk to somebody next to you about whether you could simplify 5/8 or whether it's in its simplest form.

Pause the video now.

What do you think? So I think you will have recognised that we cannot simplify this fraction.

We cannot identify a highest common factor other than one, that will divide into both five and eight.

So if we divided it by one, we would just get 5/8.

So 5/8 is in its simplest form, we cannot simplify this fraction any further.

Okay, so we've reminded ourselves about multiplying fractions, and we've now considered again, simplifying them.

We're going to move on to a new example.

Now, this example here is 2/10 multiplied by four.

And as we progress with this lesson, I want you to consider what is different about this calculation to the previous one? Pause the video and just have a think.

So, what did you come up with? I think it's quite obvious that we multiplied by four here, and previously we were multiplying by five, our denominator is different.

But what I want to draw your attention to is in the last example, we had 1/8, which was a unit fraction, whereas here we've got a non-unit fraction we've got 2/10.

So, and that's a little bit different, and this model here might help us to understand.

We have got 2/10, and we're going to multiply that by four.

So using this model, and using the generalisation that we looked at with the previous question, I'd like you to pause the video and see if you can calculate 2/10 multiplied by four.

Okay, how did you get on? So from the model, we can see that we've got 2/10 four times, 2/10 multiplied by four.

And when I look at that generalisation, I'm reminded that the denominator stays the same when I multiply the numerator by the whole number.

And so here we go, 2/10 multiplied by four, remembering that numerator multiplied by the whole number, and remembering that the denominator stays the same.

Here is my workings out, my method, two multiplied by four is equal to eight.

And then therefore, 2/10 multiplied by four, keeping that denominator the same, is equal to 8/10.

2/10 multiplied by four is equal to 8/10.

I'm sure you got that right, well done.

So the next step, 2/10 multiplied by four is equal to 8/10, can I simplify this fraction? I'm sure you've been really confident with this now.

So I would like you to pause the video and have a go at simplifying.

How did you get on? Well, I think that was quite an easy one, wasn't it? We can identify that the highest common factor of eight and 10 is two.

So if I divide the numerator and the denominator by two, that gives me 4/5, and 4/5 is this fraction in its simplest form, well done.

So let's have a look at this next example, six multiplied by 2/9.

Now, remember I was going to ask you as we go through this lesson, to think about what was different about each calculation.

And obviously, the numbers are different here, but if you noticed anything else, pause the video and just have a think, maybe look back at the previous calculations, what's different about this one? So what did you come up with? Did you notice that the order in which the whole number and the fraction has been presented has changed? And in this case, we've got the whole number first multiplied by the fraction.

But previously, that was the opposite way round, but we know that that doesn't matter.

We know about the commutative law of multiplication that it doesn't matter in which order you present, in this case, the whole number and the fraction.

So 2/9 multiplied by six is exactly the same as six multiplied by 2/9, we're going to have a go at solving that there, we're going to use this model here which will help us to understand, and we're going to use the generalisation that we've learned.

So in a moment, I'm going to ask you to solve this, but before you dive into solve it, I'd first of all, I'd like you to predict what the size of the product is going to be, and maybe how that's different to the calculations that we've solved previously.

Okay, have a go now.

Okay, so six multiplied by 2/9.

Using the generalisation and using this model, I can now calculate the product to this multiplication.

So, I'm remembering that rule that I multiply only the whole number and the numerator, and the denominator stays the same.

So when I do that, six multiplied by two is equal to 12.

And therefore, keeping that denominator the same, six multiplied by 2/9 is equal to 12/9.

I asked you in the last slide, to predict about the products and what kind of size it would be, did you predict that it would be greater than a whole? And therefore 12/9 is an improper fraction, isn't it? Well done, I'm sure you spotted that.

So remember that this lesson is all about simplifying fractions after we have multiplied a fraction by a whole number.

So we know that the product here is 12/9, can I simplify this fraction? What was the most efficient way? Now, we know from lessons 17, that you've been taught two methods, so I'd like you to pause the video in a moment and have a go at simplifying this method, and thinking carefully about which method you would prefer to use, have a go now.

So I looked at our improper fraction of 12/9, they're quite small numbers, and I could see immediately that they were both in the three times table, and therefore, the highest common factor is three.

So I'm going to simplify at this point.

So 12 and three, that nine divided by three is equal to 4/3.

Again, I can see that this is still an improper fraction, and I need to convert it to a mixed number.

Looking at this model here, I can see my equivalent fractions, they both occupy the same proportion of the whole 12/9 is equivalent to 4/3.

But I want to convert 4/3 now into a mixed number.

I can see here that I've got my whole, this is made up of 3/3, and I've got 1/3 remaining.

And therefore, 4/3, which is an improper fraction, if I convert that to a mixed number, is one whole and 1/3.

Now, I wonder whether you used a different method to me, I wonder whether you used method two, which we learnt about in lesson 17.

So we can see here that 2/9 multiply by six is equal to 12/9.

And we can see it's an improper fraction, and we can see our 9/9, making up that one whole, with the 3/9 leftover.

And, for at that point, you might have converted 12/9 into a mixed number and then decided to simplify it.

Remember, we don't have to simplify whole numbers, so we're just focusing on that fractional part of 3/9.

And it's quite easy to say that three is a factor of nine and therefore, that's the highest common factor.

So when we divide 3/9, the numerator and the denominator by three, we know that this is equal to 1/3.

And therefore, 12/9, when we convert that to a mixed number, then we simplified it, it is 1 1/3, just the same as the answer in the previous slide, but we've used a different method.

So whichever one you used, whichever one felt more comfortable, well done.

Let's move on to our next example.

Here, we have a whole number multiplied by a fraction.

Our fraction is 1 1/6, and we're going to multiply that by three.

Now, what do you notice about what's different with this calculation? Pause the video, and have a think and see if you can explain that.

So did you notice? So here we have got a mixed number multiplied by a whole number, we've got one whole and 1/6.

And I'm going to use apples, real apples to show you how we're going to calculate this.

So here is my one whole apple, and this apple here, you can see, I have divided into six equal parts.

And here, is 1/6 of my apple.

So I've got one whole apple, and I've got 1/6 of an apple.

And I want to multiply that by three.

So I'd like you to pause the video and either draw or predict what I'm going to do when I solve this calculation, off you go.

So did you pause, did you have a go? Right, so here is my one whole apple and I'm going to multiply that by three.

There are my three whole apples.

Here is my 1/6 of an apple, and I'm going to multiply that by three.

There we go, so I've got my three, if you can see them, one two, three, 3/6 of an apple.

It's a bit tricky to see, but I hope you can see that, that we've got three lots of one whole and three lots of 1/6.

And what I did was I multiplied the wholes and the fractional parts separately, is that what you got? Great.

So apples are quite fun, not the easiest to show on a video when you're at home making this, but it might be easier to have a look at this mixed number multiplied by a whole number, using some models.

So remember that with the apples, I multiplied the one whole by three, and then the 1/6 of the apple by three.

And that's what these models are going to show here.

So here is a area to represent one whole, and I'm going to multiply that by three.

I got one whole multiplied by three is equal to three wholes.

And then over here, I've got my 1/6 of an apple represented in this model.

And I'm going to multiply this 1/6 by three.

There we go, we've got 1/6 multiplied by three, which is equal to 3/6.

Three wholes add 3/6 is equal to 3 3/6.

So once we have multiplied the wholes and multiplied the fractional parts, we need to add those together.

Well done.

So let's remember that this lesson is all about simplifying fractions after we have multiplied a fraction by a whole number.

So 3 3/6 of an apple, can I simplify this fraction? Can it be simplified further? I'd like to pause the video now and use your knowledge to simplify this fraction, off you go.

So, what did you come up with? You will have remembered, I'm sure, that we don't simplify whole numbers so we can just leave the three for the moment and focus in on the fractional part of 3/6.

Looking at three and six, I can see that three is the highest common factor.

And therefore, I'm going to divide both the numerator and the denominator by three.

Doing that, that will give me an equivalent fraction of 1/2.

So bringing back my three, I know that three and 3/6 is equivalent to 3 1/2.

So just to finish off with this example, I want to look at this model here, to look at the equivalence between 3 3/6 and 3 1/2, the fraction in its simplest form.

On the left, we can see the three wholes and on the right, we can see the 3/6, but we can very much see in that model why it is equivalent to 1/2.

Well done, great work.

So here's your practise activity for today that I'd like you to have a go at before the next lesson.

I'd like you to complete these calculations and express the product in the simplest form.

Now, these are all calculations where you have to multiply a fraction by a whole number, but then you have to simplify them as well.

I hope when you look at these calculations that you can see how they relate to the examples that we've looked at in the lesson today.

So use all of your knowledge and have a good go at those.

Well done for all of your hard work today, I've really enjoyed teaching you, and I hope to see you again soon.