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 the multiplication of mixed numbers with whole numbers.

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

So by the end of this lesson today, you should be able to say that I can solve problems multiplying fractions, whether they're improper fractions or proper fractions, and mixed numbers by a whole number.

We've got four key words in this lesson today.

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

Are you ready? The first word is represent.

Your turn.

The second word is estimate.

Your turn.

The third word is mixed number.

Your turn.

And the last one is improper fraction.

Your turn.

Thinking about what these mean, then, to represent something means to show something in a different way.

We can estimate to find the value that is close enough to the right answer.

And usually that would involve some sort of thought or calculation to get us there.

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

And an improper fraction is a fraction where the numerator is greater than the denominator.

This lesson today is gonna be broken down into two cycles.

The first one is going to be estimating calculations, and the second cycle will be converting to improper fractions.

Let's get started with the first cycle.

We've got three pupils joining us today to support us with our learning.

We've got Jun, Izzy, and Laura.

Let's see how we get on.

So our lesson starts with year five and the fact that they're completing a sponsored walk for their charity.

They're going to be raising money for every mile that they walk.

Some of the children took part in the sponsored walk, and we can see them here in this table.

We have Izzy, Jun, and Laura, and this table shows us the distance that they walked, as well as the amount that they earned per mile.

So Jun's now asking, "I wonder who's earned the most amount of money, then?" He thinks that we need to multiply the distance travelled by the amount of money per mile.

Izzy's saying, let's have a look at hers first, then.

She's saying that she walked 4 3/4 miles, and for each mile she got paid three pounds.

We can represent this as a multiplication equation here.

We've got 4 3/4 multiplied by 3, and that would be equal to the total amount of money that Izzy made.

So we can start to think about roughly what this answer might look like.

Izzy's saying that 4 3/4 is nearly 5 wholes.

So we could say 5 multiplied by 3 is equal to 15.

So we know it's gonna be roughly somewhere around the 15-pound mark.

Izzy's saying that because she rounded up to 5 from 4 3/4, we haven't actually earned 15 pounds.

We've earned just under 15 pounds, haven't we? Because we actually walked a little less than the five miles that we rounded it up to.

Let's have a think about Jun, then.

Jun said that he walked 5 1/4 miles, and for each one of his miles that he walked, he added four pounds.

So again, we can represent this as a multiplication equation.

And we can write that as 5 1/4 multiplied by 4.

Jun's now saying that 5 1/4 is actually 5 and a bit wholes, isn't it? And it's just a little bit more than five.

So we could round it down to that five, and we could say 5 multiplied by 4 is equal to 20.

So we know Jun's total amount is gonna be roughly around 20 pounds.

Of course, Jun actually rounded down here.

He actually walked further than five miles, didn't he? He walked 5 1/4 miles.

So he would've earned a little bit more than what we've estimated.

And here's Laura's example.

Laura walked 3 1/2 miles, and for each one of her miles, she would've raised six pounds.

So again, we can represent that as a multiplication equation.

3 1/2 multiplied by 6.

So we could round 3 1/2 up to 4 wholes, and we could do 4 multiplied by 6 would be equal to 24.

So Laura thinks that she might have earned roughly around 24 pounds.

However, because she rounded up, then we're looking for a total that would be just under 24 pounds.

So what do you think? Izzy says she earned just under 15 pounds.

Jun says he earned just over 20 pounds.

And Laura is saying that she earned just under 24 pounds.

Laura says, "I've earned the most!" What'd you think about that? Well, Jun's now saying, "Well, that actually, Laura, is not strictly true.

That's only an estimation of what we earned.

It's not the total amount that we've earned, is it?" So we can be more accurate in this estimation.

And maybe we could do that now.

So let's use a partitioning strategy now to calculate this.

We've got 5 1/4 multiplied by 4.

We can partition 5 1/4 into its whole and fractional parts and multiply both of those by 4.

So 5 multiplied by 4 is equal to 20, and 1/4 multiplied by 4 is actually 4/4, or 1 whole.

So that would give us a total of 21 pounds when we recombined that.

Let's have a look at Laura's.

Laura walked 3 1/2 miles, and she earned six pounds for each mile that she did.

So we can partition 3 1/2 into its whole and fractional parts again.

That would be 3 1/2.

We can multiply both of those by 6.

So 3 multiplied by 6 is equal to 18, and 1/2 multiplied by 6 is equal to 6/2.

We know that 6/2 is equivalent to 3 wholes, and therefore we can say 18 plus 3 is equal to 21.

Ah.

So actually, Laura also earned 21 pounds.

So was Laura correct, then? No, she wasn't, was she? And I wonder why that was, then.

Well, Jun's amount was 5 1/4 miles, wasn't it? And we rounded that down to five miles.

So we actually would've lost a little bit of the money that we estimated with here.

So actually, our estimation would've been slightly smaller than the amount that Jun actually earned.

And Laura's, on the other hand, well, we rounded her 3 1/2 miles up to 4 miles.

And that's actually got a half a mile difference.

So we've rounded up by half a mile there up to four miles, which is actually quite a lot once it's been multiplied several times.

So maybe our estimation of Laura's wasn't the best.

Maybe it was a little bit overestimated.

Okay, time for you to check your understanding now.

Can you tick the values that would give a good estimate for this equation here? I'll give you a moment to have a think.

That's right, it could have been b or c.

We've got 3 2/6 multiplied by 2.

Well, we know that 3 2/6 is actually a little bit more than 3.

So if we rounded it down to 3, then three multiplied by 2 is equal to 6, which gives us b.

However, once we multiply 2/6 by 2, actually, that's gonna get us nearer to a whole.

So therefore we could also say that seven would be a useful estimate as well, because 3 multiplied by 2 would be equal to 6.

And then we're gonna be near to the next whole 'cause we figured that out working with the fractions slightly.

Okay.

And the next one, then.

Can you give your best estimation for this equation here? So we know that 4 1/9 is equal to 4 and a bit.

So we round it down to four.

And then we can multiply 4 by 4, which is equal to 16, altogether.

Okay, onto our first task for today, then.

What I'd like you to do here is complete the table for some of the other children in the class who've completed the sponsored walk.

In one of the columns, I'd like you to estimate how much money that they've earned.

And in the final column, I'd like to actually calculate how much money they earned.

And once you've done that, have a look at these expressions here.

What I'd like to do is use your estimation skills to help you to order these from the smallest to the largest.

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

Okay, let's see how you got on.

So Aisha walked 1 mile and 2/10 of a mile altogether, and she earned 7 pounds for every mile.

We know that 1 2/10 is just over 1, so we can estimate hers as 1 multiplied by 7, which would be equal to 7 pounds.

And actually, once we've calculated that, Aisha earned a total amount of 8.

40 pounds.

Well done if you've got that one.

Alex walked 2 miles and 8/10 of a mile, and he earned 4 pounds for every mile that he did.

2 8/10 is nearly 3 miles, so we could round it up slightly, and we'd say that 3 multiplied by 4 is equal to 12.

So 12 pounds is a slight overestimate.

And we can then work it out exactly to find out what he had.

And he earned 11.

20 pounds.

So I think our estimate was actually quite good there.

That was not far off, was it? Sophia walked 2 1/4 miles, and for each one of those miles she earned 5 pounds.

2 1/4 miles is 2 and a little bit more, isn't it? So we can round it down to two.

2 multiplied by 5 is equal to 10, so that means our estimate would be 10 pounds.

So another good estimate there, I'd say.

And for Sam, Sam walked 2 miles and 250/1000 of a mile.

For each one of those, you add eight pounds.

So we can say that 2 250/1000 is actually just over 2.

So we can say 2 multiplied by 8 is equal to 16, which is our estimate.

And then when you calculate that, we've got 18 pounds altogether.

Well done if you managed those.

Okay.

And then estimating these ones here.

I'm gonna place them in the correct order for you here, and you can tick them off if you manage that for yourself.

We know that the largest one is 4 2/4 multiplied by 5.

The second largest one was 5 1/4 multiplied by 4.

We can see that the whole number and the integer, actually, 4 and the 5, multiply, which have to make the 20.

And actually, the fraction in 4 2/4 is larger than the fraction in 5 1/4.

Therefore we can say that 4 2/4 multiplied by 5 would be larger than 5 1/4 multiplied by 4.

And then for the smallest two, well, they're actually both 4 multiplied by 4 here, aren't they? But one of them is 4 3/4 multiplied by 4, and one of them is 4 2/4 multiplied by 4.

Of course, we know multiplying 4 3/4 by 4 would give us a greater amount than 4 2/4 multiplied by 4.

Well done if got that too.

Okay, that's the end of our first cycle now.

Moving onto our second cycle, converting to improper fractions.

Izzy's saying that she used a different strategy for calculating the total amount.

She said instead of using the mixed number, she could have converted the mixed number into an improper fraction to help her here.

So let's have a look here.

We know to calculate the total amount that Izzy needed, she needed 4 3/4 multiplied by 3.

And we could then go on to convert the 4 3/4 miles into an improper fraction.

We know that one whole is the equivalent of 4/4.

So 4 wholes would be the equivalent of 16/4.

And then we've got an additional 3/4, haven't we? So now that we've partitioned the mixed number and the whole and the part are now representing fractions, we can now multiply both of these by the integer that we started with.

So we can calculate 16/4 multiplied by 3 and then add on the 3/4 multiplied by 3.

16/4 multiplied by 3 is equal to 48/4.

And 3/4 multiplied by 3 is equal to 9/4.

So 48/4 plus 9/4 is actually altogether equal to 57/4.

So we could say that Izzy earned 57/4 of a pound.

Does that make a lot of sense? It doesn't really, does it? And that's right, Laura.

Actually, leaving it is an improper fraction hasn't really helped us to understand the total amount that Izzy has raised, does it? So we're better off converting this back now to a mixed number.

If we were to do this, we could say that 57/4 would be equivalent to 14 wholes and a quarter.

Hmm.

And the quarter of a pound we know would be 25 pence.

So the total amount altogether would be 14.

25 pounds that Izzy earned.

Let's have a look now at what Laura earned.

So if we'd have a look at a similar strategy again, then, we could partition the 3 1/2, couldn't we, into its whole and its fractional parts, and then convert the whole into an improper fraction.

So we have 3 wholes, which is equivalent to 6/2, and then we've got the additional 1/2.

Now we can multiply each of these by the amount that we need to.

We've got 6/2 multiplied by 6, and then we've got an additional 1/2 multiplied by 6.

6/2 multiplied by 6 is equal to 36/2.

And then 1/2 multiplied by 6 is equal to 6/2.

And then to bring these back together, then, the total amount would be 42/2.

Again, 42/2 of a pound doesn't really help us to really understand how much Laura has earned, does it? Laura is saying, "42/2 sounds like 42, 50 Ps," which actually, it does a bit.

So what is 42, 50 Ps, then? Well, again, we need to convert it to a mixed number to find out, don't we? 42/2 is equivalent to 21, so we know that Laura has earned 21 pounds altogether.

Nice one, Laura.

Okay, time for you to check your understanding again.

Can you tick the improper fraction that is equal to 4 1/2? Take a moment to have a think.

That's right, it's b, isn't it? 4 wholes is equivalent to 8/2, and then we've got an additional 1/2.

Such is equal to 9/2.

And another quick check here.

Can you calculate 3 1/4 multiplied by 6? Give your answer as an improper fraction this time.

Okay, well, 3 wholes is equivalent to 12/4, and we've got one more additional quarter.

So we can multiply both of these by the 6.

12/4 multiplied by 6 plus 1/4 multiplied by 6.

That gives us 72/4 plus 6/4, which is equal to 78/4 altogether.

Well done if you got that.

And onto our final tasks for today now, then.

Can you complete each equation and give your answer both firstly as an improper fraction and then convert it to a mixed number? And then once you've done that, have a go at this task here.

Using the digits one to nine only once, what I'd like you to do is find as many different ways as you possibly can to complete this problem here.

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

Okay, there we go.

I've placed all the answers in the boxes for you here.

I'm gonna give you a moment now to go through and tick them off.

Well done if you've got all of those.

Did you notice anything as you went along? That's right, the mixed numbers that we were multiplying on the left-hand side were increasing by 1/4 each time, weren't they? Which therefore meant that the product was increasing by 5/4 each time, 'cause we were multiplying that 1/4 by 5.

And then on the right-hand side, we were multiplying it by one more lot each time, weren't we? The first one was three lots, the second one was four lots, and the last one was five lots.

What impact did that have on the answer? That's right, we were increasing each answer by 2 2/3 each time.

Okay.

And here's a solution that you may have found for this problem here.

We've got 7 multiplied by 1 2/5, and that's equal to 49/5 altogether.

Well done if you came up with that one or a different one.

If you haven't already, make sure you check with other people who've also had a go at this problem here to see if you've come up with similar or different solutions as well.

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

To summarise what we've learned, you can estimate the size of a product by rounding the mixed number to the nearest whole, then multiplying it by the integer.

And you can leave the product in the form of an improper fraction.

However, it is better and the correct mathematical convention to convert it to a mixed number, as this will help us to better understand the context of the answer that we have found for the problems that we're solving.

Thanks for joining me again today.

Take care, and I'll see you again soon.