Lesson video

Hi, welcome to today's lesson.

My name is Mr. Peters, and in this lesson today, we're gonna be thinking about how we can solve problems involving using the multiplication of fractions by whole numbers.

If you're ready to get going, let's get started.

So by the end of this lesson today, you should be able to say that I can solve problems involving multiplying proper fractions by whole numbers.

Throughout the session today, we've got four key words.

I'll have a go at saying them, and then you can repeat them after me.

The first word is numerator, your turn.

The second word is denominator, your turn.

The third word is proper fraction, your turn.

And the fourth word is improper fraction, your turn.

So let's think about how we can describe what these words mean in a bit more detail.

A numerator is the number on the top of a fraction, and it shows us how many parts we have.

A denominator is the number on the bottom of a fraction, and this tells us how many equal parts the whole has been divided into.

A proper fraction is a fraction where the numerator is less than the denominator, and an improper fraction is a fraction where the numerator is greater than the denominator.

This lesson today is broken down into two cycles.

In the first cycle, we're gonna be thinking about comparing expressions, and then in the second cycle, we're gonna be thinking about applying fraction by whole number multiplication.

If you're ready, let's get started.

Throughout this lesson today, we've got both Lucas and Sam who'll be sharing their thinking.

So we start our lesson here, thinking about these two expressions, and we're wondering which one of these expressions is greater than the other one.

Sam says that he thinks that 2 multiplied by 5/8 would be the greater expression because we need to multiply the whole number by the numerator, and that would give us a larger numerator in the product.

Lucas thinks differently.

He thinks that 4 multiplied by 1/5 would be the larger expression because we have a larger whole number that we're multiplying by.

Let's have a look at this in a bit more detail.

So we can see both of these expressions here, and we've got images to represent both of these.

On the left hand side, we've got two lots of 5/8.

Each whole has been divided into eight equal parts, and five of them have been shaded each time, so we can see that that would be equal to 10/8 or one whole and two additional eights.

However, for Lucas' example, we can see that we've got four wholes and those wholes have been divided into five equal parts, and one of those parts has been shaded each time.

So we can say we've got four lots of 1/5, or if we were to group 'em together, we actually don't even have one whole, do we? We've got 4/5 of a whole.

4 multiplied by 1/5 is equal to 4/5.

So Sam was correct, wasn't he? We can see that Sam's expression shows that it's greater than one whole, whereas Lucas's expression is less than one whole.

So we can say that 2 multiplied by 5/8 is greater than 4 multiplied by 1/5.

Sam's saying that he didn't even feel that he needed to calculate that to help solve this.

Hmm, what do you mean? Well, Sam's saying that we know that the numerator of the proper fraction multiplies with the whole number.

And we know that two multiply by five would be equal to 10.

At this point, we can see that the numerator is larger than the denominator, isn't it? So we know it's gonna be greater than one whole.

Let's have a look at Lucas's example.

He's multiplying his whole number by the numerator of the proper fraction, and that leaves us with 4/5 as the fraction, doesn't it? And actually 4/5, the numerator is less than the denominator this time so it is less than one whole at this point.

So we can see that the fraction on the left was greater than one whole, and the fraction on the right was less than one whole just by comparing the size of the numerator in comparison to the size of the denominator.

You're right, Lucas.

That was a lot quicker, wasn't it? Let's have a look at another example here.

We've got 5 multiplied by 2/10 and 2 multiplied by 5/6.

5 multiplied by 2/10, well, five multiplied by two would be equal to 10, so that would be equal to 10/10, wouldn't it? And the numerator and the denominator would be the same here so that would be equal to one whole.

Let's have a look at the other example.

This time we've got two multiplied by five, however, the denominator is six.

Two multiplied by five is equal to 10, but the denominator is six, so that would be 10/6.

And that's greater than one whole, isn't it, because the numerator is larger than the denominator.

So Lucas is saying that because his denominator is six, that means the whole has been divided into six equal parts so we only need six parts to make one whole, and he's got 10 parts so that would leave him with four additional parts, meaning it's greater than one whole.

So we can say that 5 multiplied by 2/10 is less than 2 multiplied by 5/6.

Okay, time for you to check your understanding now.

Can you take the expressions that are greater than 2 multiplied by 3/9? Take a moment to have a think.

That's right.

It's B, C, and D.

Can we explain how we knew that? Well, we're working with 2 multiplied by 3/9.

B is 4 multiplied by 3/9.

So the fraction is the same, but we're multiplying it more times, aren't we? We're multiplying it four times rather than two times.

So we know that's definitely gonna be greater.

We've then got 6 multiplied by 2/10.

Well, 2 multiplied by 3/9, 2 multiplied by 3/6, so that would be 6/9.

6 multiplied by 2/10 is equal to 12/10, and that's greater than one whole, isn't it, because the numerator 12 is larger than the denominator 10.

So that's definitely a larger product.

And the last one, 5 multiplied by 1/5.

Well, 5 multiplied by 1/5, five multiplied by one is equal to five so the numerator of the product would be five, and the denominator is also five, and they're equal to each other so that must mean it's equal to one whole and 2 multiplied by 3/9 is less than one whole so that also means that 5 multiplied by 1/5 is greater than 2 multiplied by 3/9.

Okay, true or false, then.

4 multiplied by 5/10 is less than 6 multiplied by 2/12.

Take a moment to have a think.

That's right.

It's false, isn't it? And let's have a think about the reasons why for that.

That's right.

The expression on the left is 4 multiplied by 5/10.

That would be 20/10 where the numerator is larger than the denominator.

Whereas on the right hand side, we've got six multiplied by two would be equal to 12, and 12/12 is equal to one whole.

So we can see that the expression on the left hand side is greater than the expression on the right hand side.

Okay, time for us to do some practise now.

What I'd like you to do is use these inequalities to compare the expressions.

You can work down the columns, working on column A first and then on column B.

Once you've tackled that, I'd like you to think about this problem here where we're gonna use the digits naught to nine only once, and I want you to think about how many different ways you can find a solution for this problem as well.

Good luck with this and I'll see you back here shortly.

Okay, let's go through these then.

The first one, 4 multiplied by 2/10 is less than 6 multiplied by 2/10.

4 multiplied by 3/10 is equal to 4 multiplied by 3/10.

4 multiplied by 2/10 is equal to 2 multiplied by 4/10.

Hmm, I wonder why that one is.

Well, that's right.

If you multiply the whole number by the numerator, they both end up giving us eight as the numerator, don't they? So 8/10 would be equal to 8/10, and the bottom one, 4 multiplied by 5/10 is greater than 5 multiplied by 2/10.

For B, we can see that 6 multiplied by 1/6 is equal to 3 multiplied by 1/3.

That's because both the numerator and the denominator are equal to one another and therefore, they're both equal one whole.

3 multiplied by 4/12 is less than 3 multiplied by 4/9.

Again, if you multiply the whole number and the numerators here, they actually both give us 12, but the denominator on the left hand side is 12, and the denominator on the right hand side is 9.

On the left hand side, 12/12 is equal to one whole, and on the right hand side, 12/9, where the numerator is larger than the denominator, so that means it's greater than one whole.

So we can reason it like that.

The next one, 52 multiplied by 2/100 is greater than 20/100 multiplied by 5.

Hmm, or 20/100 multiplied by 5.

That's equal to 100/100.

So again, that's equal to one whole, and 52 multiplied by 2/100, we know that 50 multiplied by 2/100, so 52 multiplied by two is greater than 100.

So 52 multiplied by 2/100 is a greater expression.

And then finally for the last one, we can see that 7 multiplied by 5/10 is less than 70 multiplied by 50/100.

The reason for that is that 5/10 is equal to 50/100, but the whole number is 10 times larger, isn't it? We're not multiplying by 7, we're multiplying by 70 on the right hand side, so therefore that would be larger.

Okay, and for task two, then here's one example you might have come up with.

Did you notice anything as you went through them? That's right.

The denominators needed to stay the same, didn't they? And the numerator and the whole number needed to multiply together to create the same number on either side.

So 4 multiplied by 6 equal to 24 and 3 multiplied by 8 is also equal to 24.

So that 24/9 is equal to 24/9.

Well done if you managed to get that.

Okay, onto task two now.

Applying fraction by whole number multiplication.

A class teacher has bought several bars of chocolate into school and the teacher has chosen a group of 10 children.

The bar of chocolate looks like this, and it can be divided into different sizes.

We can have 1/2 of the chocolate bar as a size, we could have 1/3 of the chocolate bar as a size, and we could also have 1/4 of the chocolate bar as a size.

The teacher places all of the different sizes she makes with the chocolate bar into the bag.

Each pupil at a time comes up and dips their hand blindfolded into the bag and chooses a size of chocolate.

They then take that piece of chocolate and take it to a table, which is either for children who picked out half of the chocolate bar, there's a table for the children who picked out 1/3 of the chocolate bar or a table who picked out 1/4 of the chocolate bar.

The table that has the most amount of chocolate altogether will be the table that gets to eat their chocolate.

So have a look here after the first round.

This is how it played out.

We can see that two children picked out 1/2 a bar of chocolate.

We can see that five children picked out a 1/3 of the bar of chocolate, and we can see that three children picked out 1/4 of the bar of chocolate.

So the question is, which table one, which table has the most chocolate? Let's all think about Jean and Sophia's table.

They've got two halves of chocolate.

We can record this as 2 multiplied by 1/2, and we know that 2 multiplied by 1/2 is equal to 2/2 or one whole, isn't it? And there we go.

So altogether, they actually had one whole chocolate bar on their table.

That's quite a lot.

I wonder if one of the other tables can be able to beat that.

Here are the group of five children and they said they had 5/3.

We can record that as an equation, 5 multiplied by 1/3.

And let's find out how much they had altogether.

Well, they had 5/3 of chocolate, didn't they? Hmm, what'd you notice? That's right.

The numerator is larger than the denominator, isn't it? So we know that they've got more than one whole of a bar of chocolate, haven't they? How much have they got exactly? Well, we know we can convert 5/3 into one whole and 2/3, and there we go.

So actually the third table are the table with the most amount of chocolate at the moment.

It looks like they're gonna win.

And here's the last table.

There was three of them here and they picked out 1/4 each.

Again, we could represent this as 3 multiplied by 1/4.

And we know that that's equal to 3/4 of a chocolate bar.

So they haven't got as much, have they? So it looks like the table who each picked out 1/3 of the chocolate bar are gonna be the winners.

And there we go, Sam.

You're right.

It looks like you are the winners and you get to eat your chocolate bar.

Okay, time for you to check your understanding now.

Could you write an equation to represent the amount of chocolate that's on this table here? Take a moment to have a think.

That's right.

There were five children and they each picked out 1/4 of the chocolate bar, didn't they? So we could represent that as 5 multiplied by 1/4 and say that is equal to 5/4 or 1 1/4.

Well done if you've got that.

And onto our final task for today then.

What I'd like to do is using the game, can you, A, find a way for the 1/4 table to win? You might like to find more than one way to do that.

For task B, can you find a way for each table to have an equal amount, and is it possible to have an equal amount with 10 players? For task C, what I'd like to do is find a way where the 1/2 table has the most, the 1/3 table has the second most, and the 1/4 table has the least.

And then for task D, do the inverse of that.

Find a way for the 1/2 table to have the least, the 1/3 table to have the second most and the 1/4 table to have the most.

Once you've done that or have a go at changing the game as if there were 12 players this time, and if the 1/4 table had 1 2/4 of a bar of chocolate, how much could the other two tables potentially have? Good luck with this task and I'll see you back here shortly.

Okay, let's go through it then.

So finding a way for the 1/4 table to win, we could say that table 1/2 had 2 multiplied by 1/2, that was equal to a whole.

Table 1/3 could have 1 multiplied by 1/3, so one person picked out a third, and that was equal to 1/3.

And then for table 1/4, we could have five people picking out 1/4 that would be equal to 5/4, which is greater than one whole, which is what table 1/2 had.

To question B, hmm.

No, it wasn't.

It was impossible to find an equal way with only 10 players playing the game.

You may have adjusted this and started to add a few more extra players to see how many players you would need to make it possible to have an equal amount, and well done if you did.

Sam realised he could do that.

He said if we had nine players, it would be possible because each table could have one whole chocolate bar.

Two people could pick up 1/2, three people could pick up 1/3, so that's five players, and then four people could pick out 1/4, so that would be nine players.

For table C, here's a way you could have organised it.

Take a moment to have a look and see if you've got something similar to that.

And then for task D as well, here's another way you could have organised it.

Did you manage to come up with something similar to that as well? Well done if you did.

Make sure you share your ways with anybody else around you who may have come up with something different to you.

And then for task two, if 12 players were now playing and table 1/4 had 1 2/4 of a chocolate bar altogether, how much would the other tables have? Well, we need to know how many children picked out 1/4 to make 1 2/4.

Well, that would be equal to six.

6 multiplied by 1/4 would be equal to 6/4 or 1 2/4.

That leads us with six more players available left to play, doesn't it? So you could divide that however way you wanted to.

You could put three on one table and three on another table, you could put four on one table, two on one table, you could put one on a table and five on a table, or you could put zero on one table and six on a table.

We settled for having two on the 1/2 table, which made one whole chocolate bar altogether, and then we settled for four on the 1/3 table, which left us with 4/3, or that would be equal to one whole and one additional 1/3.

Hmm, who got the most chocolate there then? That's right.

It would be table 1/4, wouldn't it? 1 1/2 is greater than 1 1/3.

1 2/4, which is equal to a half is greater than 1 1/3.

So table 1/4 would've been the winners and got the most chocolate that time.

Well done to them.

Okay, and that's the end of our lesson for today.

Hopefully you've enjoyed yourself and hopefully you didn't have too much chocolate if you tried to replicate the game for yourself.

So just to recap what we've learned.

You can compare expressions that include a whole number multiplied by a proper fraction by considering the size of the numerator in comparison to the size of the denominator.

And you can compare your knowledge of multiplying a proper fraction by a whole number to compare different amounts in real life.

Thanks for joining me again today.

Hopefully you enjoyed this lesson and again, you're feeling really confident by multiplying whole numbers by proper fractions.

Take care and I'll see you again soon.