Lesson video

Hi there, welcome to today's lesson.

My name is Mr. Peters and in this lesson we're gonna be thinking about how we can multiplied a mixed number by a whole number where the fractions bridge over one whole.

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

So by the end of this lesson today you'll be able to say that I can multiply a mixed number by a whole number where the fractions do bridge one whole.

Throughout this lesson today, we've got three keywords.

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

The first one is mixed number, your turn.

The second one is compare, your turn.

And the third one is partition, your turn.

Thinking about what these mean, then.

A whole number and a fraction can be combined to form a mixed number.

We can compare to find out what is the same and what is different.

You can compare numbers or expressions to find out which one is greater or less than the other one.

And finally, partitioning is the act of splitting an object or a value down into smaller parts.

This lesson today is broken down into two cycles.

The first cycle we'll be thinking about recognising expressions that bridge a whole.

And the second cycle we'll be thinking about going on to solve problems including bridging one whole in a range of contexts.

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

In this lesson today, we'll be joined by both Andeep and Alex, who again will share their thinking as well as any other thoughts that they might have about the learning that we're about to think about.

Okay, so let's start here with these two expressions.

We can look at these two expressions and think about how we can compare them using our inequalities.

Have a quick look.

Which expression do you think is largest? Andeep's not sure.

Andeep's thinking, "I'm not sure I'm gonna be able to work this one out straight away.

Maybe it'd be useful if I could calculate each of the expressions to find out which is the largest." Ah, Alex disagrees, though.

He says, "Wait a minute.

I don't think we do need to calculate them to figure this out.

Maybe we can do this by looking more carefully at them." "What do you mean?" Well, let's take that time that Alex has suggested to look carefully at them.

What is it that you notice? That's right, Andeep.

The mixed numbers in both of the expressions are the same, aren't they? Is there anything else that you notice? Well, one of the expressions says 3 lots multiplied by 2 1/4, whereas the second expression shows 2 lots multiplied by 2 1/4.

Hmm, that's different, isn't it? And was there anything else that you noticed? That's right, one of the expressions says it's 3 lots of 2 1/4, and the other expression says it is 4 lots of 2 1/4, isn't it? So how can we use that to help us reason our thinking then? So we could say the expression on the left is 3 lots of 2 1/4 and the expression on the right is 4 lots of 2 1/4, and we know that 3 lots of any positive whole number is always gonna be smaller than 4 lots of any positive whole number where the positive whole numbers are the same in both of the expressions.

So we can use that reasoning then to say that 4 lots of 2 1/4 would be greater than 3 lots of 2 1/4, or 3 lots of 2 1/4 would be less than 4 lots of 2 1/4.

Ah, there you go, Andeep.

You agree, do you now? That was a lot simpler, wasn't it, than thinking about having to calculate all of those straight away.

Let's have a look at another set of expressions.

What'd you notice this time? Ah, well, the mixed numbers this time are slightly different, aren't they? One of them is 2 1/4, whereas the other one is 2 3/4.

Did you notice anything else? That's right, the whole numbers that we're multiplying by are the same this time, isn't it? So what do you think then? Well, we could say that on the left-hand side we've got 3 lots of 2 1/4, and on the right-hand side we've got 3 lots of 2 3/4.

Hmm, so how could we compare these two now then? Well, that's right, 3 lots of 2 1/4 is going to be less than 3 lots of 2 3/4, because each lot on the expression on the right-hand side has a greater amount of 1/4 than the lots on the expression on the left-hand side.

So we can say that 3 multiplied by 2 1/4 is less than 2 3/4 multiplied by 3.

Yes, and that's right Alex, isn't it? If they were square as a chocolate and you like chocolate, then you'd rather have 3 lots of 2 3/4 squares, wouldn't you, rather than 3 lots of 2 1/4 squares.

Here's another example.

Have a look this time, what do you notice? You think this one's slightly trickier this time, do you, Andeep? They're both multiplied by the same whole number, aren't they? However, we're not quite sure which fraction's larger here, are we? Well, we might like to start thinking about partially calculating this to see how we can get on.

So, as Alex has pointed out, he says if you multiply the whole number by the integer in both the examples, we're gonna get the same amounts, aren't we? In the expression on the left-hand side we've got 5, and then we'd have to multiplied that by the 2 wholes, so that would be 10.

And on the right-hand side, we've also got another 2 wholes to be multiplied by 5, so that would be equal to 10.

So at the moment, that's still not gonna help us to decide which one's the largest one, but let's now think about multiplying it by the fractions instead this time.

So what if we multiplied each of the fractions by the integers? We know how to do that, we can multiplied the numerator of the fraction by the integer, can't we? And then the denominator would stay the same.

So, 5 multiplied by 2/12 would be equal to 10/12, and 5 multiplied by 3/11 would be equal to 15/11.

Hmm, what do you notice now? Well, that's right, the product for the top one is a proper fraction, isn't it? It's 10/12, isn't it? Whereas the product of the bottom example is an improper fraction.

The numerator is greater than the denominator, isn't it? Ah, so actually what does that tell us then? Well, we know that an improper fraction is greater than one whole, isn't it? So that must be larger than the proper fraction above it.

So if that's larger, we don't need to calculate to find out exactly how large the expression on the right-hand side is now.

We just know that when we multiplied the fractional part by the whole number, we've got a larger part.

So we know that one of them is an improper fraction and one of them is a proper fraction, and we know that improper fractions are larger than proper fractions because an improper fraction has a numerator that is greater than the denominator, which means they're larger than one whole.

So, 15/11 is equal to 1 whole and an additional 4/11.

Hmm, so because when we multiplied the whole number of the mixed number by the integer, this gave us the same amount, didn't it? The 5 multiplied above the 2 was equal to 10.

We can now see that when we multiplied it by the fractional part of the mixed number, that one of them gives us a product greater than a whole and one of them gives us a product which is less than a whole.

So altogether we can say that 5 multiplied by 2/12 is less than 5 multiplied by 2 3/11, isn't it? Because when we multiplied the integer of 5 by 3/11, that gave us a product greater than one whole.

And in comparison to when we multiplied the other expression, the 5 multiplied by the 2/12, that gave us a product that was less than one whole.

And we know that something greater than one whole is gonna be a larger number than something that is less than one whole.

Okay, time for you to check your understanding now.

Can you check the correct symbol to compare these expressions? Take a moment to have a think.

That's right, 2 multiplied by 3 4/5 is greater than 2 multiplied by 3 4/50.

And can you explain why? Well, they've both got the same whole number to multiplied the mixed number by, haven't they, which is 2.

However, 3 4/5 is a greater fraction than 3 4/50, isn't it? 4/5 is larger than 4/50, therefore 2 lots of 3 4/5 is going to be greater than 2 lots of 3 4/50.

Well done if you managed to get that.

And here's another check.

Can you tick the expressions that are greater than 5 multiplied by 1 2/10? Good luck.

That's right, it's A, B, and D this time.

And how can you explain that you know this this time? Well, let's look at A.

A is 5 multiplied by 1 9/10.

Well, they're both multiplied by the same integer, aren't they? But 1 9/10 is greater than 1 2/10, so this one is definitely true.

Having a look at B, again, it's multiplied by the same in integer again of 5, and this time the mixed number is 2 2/10.

2 2/10 is greater than 1 2/10, therefore 5 multiplied by 2 2/10 would be greater than 5 multiplied by 1 2/10.

And then finally, D, well, we've got the same mixed number here, haven't we? We've got 1 2/10.

However, this time we're not multiplying it by the same integer, are we? The example in the question is multiplying it by 5 and the example in D is multiplying it by 6.

So, 6 lots of something is always gonna be greater than 5 lots of something if that something is the same amount, so we can say that 1 2/10 multiplied by 6 is greater than 1 2/10 multiplied by 5.

Okay, time for us to check our understanding now.

What I'd like to do is look at these expressions here and group them into expressions that do bridge one whole and which do not bridge one whole.

Once you've tackled that, what I'd like to do is use your inequalities just like we've done so far to compare these expressions, telling me which expression is greater than or less than the other one or whether they're equal to each other.

Good luck with those two tasks and I'll see you back here shortly.

So, 3 multiplied by 4/6 would bridge one whole.

We know that 3 multiplied by 4 is equal to 12, so 3 times 4/6 is equal to 12/6 and that gives us an improper fraction, which therefore would be greater than one whole.

2 multiplied by 100 2/6 would not bridge one whole here.

2 multiplied by 100 is 200 and 2 multiplied by 2/6 is 4/6, so that one does not bridge one whole.

Here's another example that doesn't bridge one whole.

2 1/4 multiplied by 3.

2 multiplied by 3 is equal to 6 and 1/4 multiplied by 3 is equal to 3/4, so that does not bridge one whole.

And here's another example.

2 multiplied by 3 2/5.

2 multiplied by 3, again, is equal to 6, and 2 multiplied by 2/5 would be equal 4/5, so that wouldn't bridge one whole.

And another example.

8 1/5 multiplied by 4.

Well, 8 multiplied by 4 is 32.

And then if we look at the fractional part, 1/5 multiplied by 4, that would be equal to 4/5, so it doesn't go onto bridge a whole, therefore that would stay in the left-hand column.

Here it looks like one more example which would go in the left-hand column.

21 multiplied by 1 1/22.

Hmm, what did you notice? We know we need to look at the fractional part of the mixed number and see how that's gonna multiply with the integer to find out whether it would bridge a whole.

And we've got 21 multiplied by 1/22.

Well, that gives us 21/22, doesn't it? Therefore, that does not bridge one whole because the whole would be 22/22.

And if you're bridging that, it'd have to be greater than 22.

And the last two examples, then.

We've got 9 multiplied by 4 1/8.

Well, looking at the fractional part of that, 9 multiplied 1/8 would be 9/8, so that gives us an improper fraction, therefore that would need to bridge one whole.

And the last one, 21 multiplied by 1 5/7.

Hmm, 21 multiplied by 5/7, that's gonna give us 105/7, which obviously is a rather large improper fraction.

So, as a result of that, that would most definitely bridge one whole.

Okay, and now I'll walk through these examples here and allow you to tick them off to see how you got on.

Well done if you managed to get all of those.

Okay, and moving on to our second cycle now, solving worded problems. So we start here with Andeep who's drinking 1 1/2 glasses of water for breakfast every day.

That's a really great healthy start to your day, Andeep.

What a great decision to do that.

So, how much water does he drink for breakfast each week is our question here.

So let's express this as an equation then.

We could write this as 1 1/2 multiplied by 7.

There are seven days in the week, which is why we have a 7 in our equation.

And he drinks 1 1/2 glasses of water on each one of those days, doesn't he? We can now go through the process of partitioning the 1 1/2, can't we? So we've gonna partition the 1 1/2 into a whole number and its fractional part, and we need to multiply both of those parts by 7.

So, 1 multiplied by 7 is equal to 7, and then 1/2 multiplied by 7 would be equal 7/2.

Hmm, what do you notice here? That's right, we've got the whole number now and we've also got an improper fraction on the right-hand side.

We know that 7/2 is equivalent to 3 wholes and an additional 1/2, so altogether we can combine this together to make 10 1/2 glasses of water that he drinks over the course of the week for his breakfast.

Well done if you managed to get that.

Here's a different example.

Alex is painting his bedroom with his parents.

He uses 1 3/4 tins of paint for each wall.

And in his bedroom he has 4 walls that are all roughly the same size.

How many tins of paint is he going to need to use altogether then? So once again, let's express this as an equation.

We can express this as 1 3/4 multiplied by 4.

The 1 3/4 represents the number of tins of paint he uses for each wall and the 4 represents a number of walls that he has to paint.

So once again, partitioning our mixed number into its whole and fractional parts.

The whole would be 1 and the fractional part would be 3/4.

We can multiply both of these by 4.

So, 1 multiplied by 4 is equal to 4, and 3/4 multiplied by 4 is equal to 12/4.

We can see we've got an improper fraction again here, which we now need to convert into a mixed number.

So we can use our understanding of converting improper fractions into mixed numbers to know that 12/4 is equivalent to 3 wholes.

And we can now recombine these parts together.

4 plus 3 is equal to 7, so we've got 7 wholes, 7 tins of paint altogether.

4 plus 3 is equal to 7, so we needed 7 tins of paint altogether for Alex to complete painting his whole bedroom.

Here's one more example for us to think about.

It takes Mercury 58 Earth days and another 6/10 of an Earth day to spin completely on its axis.

How many Earth days does it take for Mercury to spin on its axis three times? Well, let's express this as an equation then, shall we? We know it takes Mercury 58 Earth days and another 6/10 of an Earth day to spin once on its axis.

However, we want it to spin on its axis three times, so we're gonna need 3 lots of this amount, aren't we? So we've now got 58 6/10 multiplied by 3.

We can now partition this mixed number, can't we, into its fractional part and its whole part.

And we need to multiply both of these by 3.

So the whole number part of our mixed number was 58.

58 multiplied by 3.

Well, we can do that, and that's 174.

And then we've got 6/10 and we can multiply that by 3 as well.

That would give us 18/10.

You may have noticed now that 18/10 is obviously an improper fraction, so we need to convert this into a mixed number.

This is equivalent to 1 whole and 8/10.

So now we can recombine those parts altogether to make 174 plus 1 whole and 8/10 would be equal to 175 days and an additional 8/10 of a day.

So altogether, for Mercury to spin on its axis three times, it would take 175 Earth days and additional 8/10 of an Earth day.

Rounded to the nearest whole day, that would be 176 Earth days.

Well done if you managed to get that as we went along.

Okay, time for us to check our understanding again now quickly.

Which expression represents the following problem? Alex's mum runs 3 1/4 kilometres every Sunday for 4 weeks.

How many kilometres does she run altogether? Take a moment to have a think.

That's right, we could represent that as B or C, couldn't we? B shows us how we could partition 3 1/4 into its whole and fractional part, and we'd know that both of those need to be multiplied by 4.

So 3 is the whole number part, and that could be multiplied by 4.

And we'd need to add that to the 1/4, which would also need to be multiplied by 4.

So that is how B would be a correct expression for this problem here.

And then have a look at C.

What did we notice about that? Well, actually, we just rotated the multiplication expressions within the whole expression itself.

So you can see in B we had 3 multiplied by 4 plus 1/4 multiplied by 4.

Well, actually, now we've just got 1/4 multiplied by 4 plus 3 multiplied by 4.

So we rotated each of those multiplication parts of the expression, and that therefore would also represent the process of multiplying 3 1/4 kilometres by 4.

And here's another quick true or false for us.

A product can be left recorded as an improper fraction, true or false? Okay, and that is false, okay? And have a look at these two justifications here.

Which one of these helps you to justify our reasoning? That's right, it's A, isn't it? It's actually the correct mathematical convention to express an improper fraction as a mixed number.

Often, leaving something as an improper fraction doesn't really help us make sense of the context within which we've been looking at the numbers, so it's often best to leave it as a mixed number.

So, to finish off our task for today, what I'd like you to do is have a go at solving these worded problems here.

We've got both problem A, B here, and we've got problem C and D here.

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

Earth takes 365 days and 1/4 of a day to orbit the Sun.

How many days old will Andeep be on his 10th birthday? Well, we know that 365 days and an additional 1/4 of a day is the length of time it takes Earth to orbit the Sun, which is equivalent to 1 year, and we are looking for the age of Andeep after 10 years, so we need to multiply 365 1/4 by 10.

Once you've calculated this, this would give you 3,652 days and an additional 1/2 of a day.

So Andeep, on your 10th birthday, you'll be 3,652 days old with an additional 1/2 a day as well.

Question B, a string of outdoor fairy lights are 2 3/4 metres long.

If we use 5 sets to cover the garden fence, how much of the garden fence will be covered altogether? Well, we can represent that as 2 3/4 and we need that 5 times, don't we? And that would altogether give us 13 3/4 metres of fence altogether.

Question C, a chef uses 2 1/2 onions to make a lasagna that feeds 8 people.

How many onions would be needed to feed 24 people? Well, if we need 2 1/2 onions for 8 people, we actually don't need 8 people or 16, we need 24 people, that's 3 times the amount of people that we need, therefore we need 3 times the number of onions.

So 2 1/2 multiplied by 3, that would be equal to 7 1/2 onions.

And then for question D, each class uses 3 1/2 bag fulls of apples for snack time on each day.

And on the Monday they use 21 bags of apples altogether.

How many classes are there in the school? This is slightly different now, isn't it? We know the number of bags that each class uses and we know the total number of bags that the classes use altogether, but we don't know how many classes there are.

So we can represent this as an equation of 3 1/2 bags of apples multiplied by an unknown number of classes is equal to 21 bags of apples altogether.

Hmm, how could we tackle this? Well, 3 multiplied by 6 is equal to 18, and if you do 1/2 multiplied by 6, that would give you an additional 3, so 18 plus 3 would be equal to 21, therefore there are 6 classes altogether in the school.

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

Let's have a think about what we've learned.

Expressions including the multiplication of a mixed number by a whole number can be compared by thinking about when the fractional part of the mixed number is multiplied by the integer, whether this part gives us a product which is greater than or less than one whole.

When multiplying a mixed number by a whole number, we can partition the mixed number into wholes and fractional parts and multiply them separately.

And it's the correct mathematical convention for improper fractions to be left as a mixed number.

Brilliant, that's the end of our learning for today.

Hopefully you're feeling a lot more confident about multiplying a mixed number by a whole number or an integer.

Take care and I'll see you again soon.