Lesson video

Hi, welcome to today's lesson.

My name is Mr. Peters and in this lesson today, we're gonna start thinking about how we can apply our understanding of mixed numbers and fractions to a range of problems that you might meet in your everyday lives.

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

So by the end of this lesson today, you should be able to say that, "I can solve problems involving mixed numbers and fractions." In this session today, we've got one key word we're gonna be referring to throughout.

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

Are you ready? Represent.

Your turn.

So, let's think about what this means, then.

To represent something means to show something in a different way, and we're gonna be thinking about how we can represent these mathematical problems that we have throughout our lesson today.

Today's lesson is broken down into two cycles.

The first cycle is representing problems and the second cycle is calculating problems. Let's get started with the first cycle.

Throughout this lesson today, you'll meet both Sofia and Izzy.

As always, they'll be sharing their thinking and any questions that they have about the maths as we go throughout the lesson.

So, our lesson starts here.

It's cycling proficiency week for Year 5.

Sofia's saying that she wants to cycle every day to school so she can practise and make sure that she's able to earn her badges.

So, let's have a look at how fast Sofia and Izzy cycle to school each day, then.

The distance from Sofia's house to school is 2 1/4 kilometres.

The distance from Izzy's house to school is 1 3/4 of a kilometre.

What questions could we ask based on this information that we've got so far? Izzy's come up with a question.

She says, "I wonder how far we'll have cycled together over the whole of the week." And Sofia's asking, "Hmm, I wonder who'll have cycled the furthest over the course of the week as well." Let's see if we can start to think about how we can represent these questions.

We're gonna start by thinking about how we can represent how far Izzy cycles in one week.

We know that one school week is made up of five days, so we've got a bar which represents the whole school week, and that school week has been divided up into five equal parts.

She says that each day, she cycles both to and from school, so each one of those days has now been divided into two parts.

One of the parts would represent the cycling to school, and the other part would represent the cycling back home from school.

So as we can see on Monday now, on the way to school, Izzy cycles 1 3/4 of a kilometre.

And on the way back from school, she also cycles another 1 3/4 of a kilometre.

She does this for every day of the school week.

So, take a moment now to think.

What do you think the bar at the top represents, then? That's right, Sofia.

The bar at the top will represent the total distance travelled over the course of the week by Izzy.

Hmm, Izzy now thinks she's got a couple of ways she could go about solving this.

Have a quick look for yourself.

Do you think you've got a way now that you could potentially solve this? Izzy also thinks that we could have represented this using unitizing counters.

Let's have a look.

We know that each day, Izzy cycles 1 3/4 kilometres to school and another 1 3/4 kilometres home from school.

So, we can represent each of those journeys as a counter, so each counter represents 1 3/4 kilometres that she cycles.

And we know that we need two of these counters for each day, don't we? Because she cycles to school and from school the same distance.

Can you see the two counters that represent each day? Well done if you can.

At the moment, we've still got the bar model that represents how much Izzy cycles each day.

And now, we're gonna start thinking about how we can change this bar model to represent what Sofia cycles each day.

We know that Izzy cycles 1 3/4 kilometres to school and 1 3/4 kilometres home from school, whereas Sofia is now saying that she cycles 2 1/4 kilometres to school and again, 2 1/4 kilometres on the way home from school.

Have a think.

What's gonna change in our bar model now? That's right, we're going to need to change the value of each of the parts, aren't we? So now, we can see that each one of those journeys represents 2 1/4 kilometres as opposed to Izzy's 1 3/4 kilometres.

Again, we could also represent this as unitizing counters.

You might like to stop the video quickly now and have a quick go at drawing what the unitizing counters might look like to help you represent this problem.

Well done if you did that.

And let's have a look about how we could have represented this using our unitizing counters.

We know that each journey represents 2 1/4 kilometres and we need two of those journeys for each day.

So there we can see on Monday, Sofia cycles 2 1/4 kilometres to school and then 2 1/4 kilometres home from school, and then she does this for every day of the school week.

Well done if you came up with something similar to that.

Right, let's have a think about this one now, then.

How could we represent who travels the furthest? Sofia's asking who is it that travels the furthest? And not only that, by how much more do they travel the furthest? Well, Izzy's saying she doesn't even need to represent this.

She knows straight away who cycled the furthest.

How do you know that, Izzy? Well, she says, "We both made the same number of journeys altogether.

We both cycled a school every day for five days and we did that to and from school." However, Izzy cycled a lesser distance than Sofia did.

So, we know straight away that Izzy is gonna have cycled altogether a lesser distance than what Sofia would've done altogether.

We've now represented this with our bar model.

The top bar is going to represent how far Sofia has cycled and the bottom bar is going to represent how much Izzy has cycled.

We know the second part of Sofia's question was how much more did Sofia's cycle in comparison to Izzy? And we can represent that on our bar model by the difference between the two bars.

And here we go, this arrow represents the difference in size between the two bars.

Take a moment to have a think.

What operations you think you might need to help us identify the difference between how far Sofia travels and how far Izzy travels.

That's right, Izzy.

We would need to use subtraction, wouldn't we? If we were trying to find out the difference between the distance of the girls travelling, then if we subtracted the distance that Izzy travelled from the total distance that Sofia travelled, then we'd be left with the length of the arrow, and the length of the arrow is representing the difference between the distances that these two cycled.

Okay, time for you to check your understanding now.

Sam decides to cycle only from his house to school on three days of the week.

So just to be clear, he's not cycling home from school, he's just cycling from his house to school.

He does that on Wednesday, Thursday, and Friday.

Have a look at the bar models.

Which one of those bar models would help us to represent the total distance that Sam cycles? Take a moment to have a think.

That's right.

It's B isn't it? If you have a look at B, we've got two bars.

The top bar represents the total distance travelled and the second bar has been divided into three equal parts.

Each one of those parts represents the days that he cycled, and we know that on each one of those days he cycled 3 1/2 kilometres to get to school.

So, we're going to need 3 1/2 kilometres for the Wednesday, 3 1/2 kilometres for the Thursday, and another 3 1/2 kilometres for the Friday.

Well done if you've got that.

Here's another quick check.

Sam cycles 1 1/2 kilometres further than Andeep.

Have a look at the bar models again.

Which one of the bar models below helps us to represent this problem? Take a moment to have a think.

That's right, it's B again, isn't it? And why was it B? Well, we know that Sam cycles a further distance than Andeep so we can have a bar to represent Sam's distance, which is longer than the bar that represents how far Andeep has cycled.

And we also know that the difference in how far they cycled is 1 1/2 kilometres.

So we can see that the arrows on B represent 1 1/2 kilometres, that is the difference between how far they each cycled.

Well done if you managed to get that.

Okay, onto our first task for today, then.

What I'd like you to do is have a go at drawing a bar model to represent each stage of this following problem.

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

Okay, welcome back.

Let's have a look at the problem, then.

So, the first stage of our problem shows that Aisha is doing an exercise circuit for her gymnastics.

The exercise circuit has eight stations and each one of those stations last 1 1/2 minutes.

So, let's have a look at how we could represent that on the right hand side.

We've got the bar at the top which represents the total length of the circuit.

Then, we can divide that bar into eight equal parts below because we know there are eight stations in the whole and each one of those stations takes a length of 1 1/2 minutes.

So, each one of those parts now represents 1 1/2 minutes.

Well done if you got that first one.

The second stage says Aisha completes the circuit three times.

How long does she exercise for? So now, we're looking at the length of the total amount of exercise she does altogether, which is gonna be represented by the bar at the top.

And now each circuit, which consists of eight stations, is done three times.

So we've divided our whole amount of time into three parts below, and each one of those parts represents the length of time taken to do all of the eight stations in one of the circuits.

She completes the circuit three times.

We could, if we wanted to, divide each one of those parts into another eight parts and that would show each of the stations within each of the circuits that she was completing.

And then finally, the last section.

After the circuit, she does a five minute cool down.

How much shorter in time was the cool down in comparison to the circuit? Well again, let's have a think.

Which one of the activities was the longest? Was it the length of time taken to do the circuit or the length of time taken to do the cool down? That's right, it was the length of time taken to do the circuits, wasn't it? So, let's have a think.

Which one was the longest, the length of time taken to do the complete exercise or the length of time taken to do the cool down? Yeah, that's right.

It was the length of time to do the exercise, wasn't it? So, we're gonna represent that by the long bar at the top.

And we know that the length of time taken to do the coll down was a lot less, so we've represented that with a smaller bar and now we're looking for the difference in length of time taken between the length of time to do the exercise and the length of time to do the cool down, and we've represented that with the purple arrow as well.

Well done if you've managed to get all of those three.

Okay, moving on to cycle two now.

We're gonna start thinking about how we can calculate these problems. So, let's revisit how we've represented how far Izzy cycles in one week.

We've got the whole at the top which represents the distance that she cycles altogether, and then we've got each part below which represents the number of days that she cycles for, and in each one of those days, it represents the two journeys that she undertakes in order to get to and from school.

Izzy thinks she's got a number of ways that she could calculate this.

Take a moment for yourself.

Which ways did you come up with to calculate this? Well, Izzy's saying that she can see that there are 10 journeys altogether.

There are five journeys to school and five journeys from school, so we could say that there are 10 lots of 1 3/4 quarters.

We could write an equation to represent this.

10 multiplied by 1 3/4.

10 multiplied by 1 3/4 is equal to the total distance travelled, so we can use our knowledge of multiplying mixed numbers by whole numbers to help us calculate this.

We could partition our mixed number into 1 and 3/4, and multiply both of these parts by 10.

So, our calculation can now be 10 multiplied by 1 plus 10 multiplied by 3/4.

We know that 10 multiplied by 1 is equal to 10 and 3/4 multiplied by 10 would be equivalent to 7 wholes and an additional 2/4.

Now, to find the total length of distance travelled, we can recombine these two parts, so we've got 10 plus 7 2/4, which is equivalent to 17 2/4.

So that's right, Izzy.

Altogether, you travelled 17 2/4 kilometres to get to school this week.

Sofia thinks she's found a different way of doing that.

Have a look at the annotations she's made on the bar model.

She's put two journeys together to represent both of the journeys that Izzy undertakes on one day.

So, we can say that Izzy travelled two lots of 1 3/4 kilometres each day.

We can represent this as an equation, 2 multiplied by 1 3/4 and again, we could calculate this which would be equivalent to 3 2/4 kilometres.

So each day, Izzy cycles 3 2/4 kilometres and you can see that now represented on our bar model.

Hmm.

What would be the next stage, then? Well, that's right.

Izzy completes this journey five times, doesn't she, in a week? So we're going to need to multiply our 3 2/4 by 5, aren't we? If you want to, take a moment to jot down how you'd go around tackling that.

And that's right.

Altogether, once we've calculated that, we can say that Izzy would cycle 17 2/4 kilometres, which is exactly the same as the strategy that Izzy came up with the first time round.

Nice thinking, you two.

Two different ways of solving the same problem.

So, moving on to how fast Sofia cycled in a week, then.

We're gonna go back to our representation using the place value counters this time, and let's see how Sofia's going to tackle calculating the total distance that she travelled.

She knows that each day would represent two counters because there were two journeys, so she's gonna continue using the same strategy she did before.

She's gonna multiply 2 by 2 1/4 and that will represent the total distance she travels each day.

2 multiplied by 2 1/4 is equivalent to 4 2/4, so we can say that each day, she travels 4 2/4.

Can you see now how we've changed our place value counters to represent this? We no longer have 10 counters, we now only have five counters and each one of those counters represents the distance travelled each day.

Once again, there are five days in the week, aren't there? So Sofia's right, we're going to need to calculate five lots of 4 2/4.

We could partition our mixed number into 4 and 2/4 and multiply both of those parts by 5.

So, 4 multiplied by 5 is equal to 20, and then 2/4 multiplied by 5 would be equivalent to 10 quarters, or 2 2/4.

And then combining that altogether again, we would have 20 plus 2 2/4, which is equal to 22 2/4.

So, Sofia has travelled 22 kilometres and an additional 2/4 of a kilometre altogether.

Right, let's start thinking about the other question, then, we had about finding the difference between how far these two cycled.

Well if we revisit our bar model, we can now replace the language that we've got in the bars with the numbers that they represent.

So, we know that Izzy travelled 17 2/4 of a kilometre, so I'm gonna write that in Izzy's bar.

And we know that Sofia travelled 22 2/4 of a kilometre, so we're gonna write that into Sofia's bar as well.

Now to find the difference here, we know we're going to need to use subtraction, so we're going to need to subtract Izzy's distance from Sofia's distance.

We can represent that as 22 2/4 minus 17 2/4 and once again, we can do 22 minus 17 which would be equal to 5, and then 2/4 minus 2/4 which would be equal to 0.

So, the total difference between how far Sofia cycled in comparison to Izzy was actually five kilometres altogether.

Well done if you managed to get that for yourself.

Okay, let's revisit Sam's problem from earlier on, then.

We know that he cycles on three days of the week and each day, he cycles 3 1/2 kilometres.

What's the total distance that Sam cycles altogether, then? Could you ever go at working that out? That's right, we would need a calculation here of 3 multiplied by 3 1/2.

If we partition the mixed number into 3 and 1/2 and then we can multiply both of these parts by 3.

So, we've got 3 multiplied by 3 plus 3 multiplied by 1/2.

3 multiplied by 3 would be equal to 9 and 3 multiplied by 1/2 is equal to 3 halves or 1 1/2.

So altogether, we could say 9 plus 1 1/2 would be equal to 10 1/2.

So, Sam cycles 10 1/2 kilometres on the three days altogether when he's going to school.

Okay, and then our final check for today.

We know that Sam cycles 1 1/2 kilometres further than Andeep.

How far does Andeep cycle, then? Well, we now know how far Sam cycles altogether, don't we? Sam actually cycles 10 1/2 kilometres altogether.

So, we know that Sam cycles 10 1/2 kilometres and the difference between how far they cycled is 1 1/2 kilometres.

Hmm.

So, what operation could we use here, then? That's right, we need to find out the distance that Andeep cycles and to do that, we can subtract the 1 1/2 kilometres that was the difference between the two of them from the total amount that Sam cycled.

So now, our calculation is 10 1/2 kilometres minus 1 1/2 kilometres.

10 minus 1 is equal to 9 and 1/2 minus 1/2 would be equal to 0, so the distance that Andeep would have cycled would've been 9 kilometres altogether.

Again, well done if you managed to get that.

Right, onto our final tasks for today, then.

What I'd like you to do is use the images you used from earlier on in the first task to complete and actually solve the problems based on Aisha's exercise routines for her gymnastics.

And then once you've done that, I'd like you to have a go at solving this problem here.

This problem involves Alex's dad and the time it takes for him to travel to London to get to work each month.

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

Okay.

Well, to calculate the length of time it takes for Aisha to complete a circuit, we need to multiply 8 by 1 1/2.

8 multiplied by 1 1/2 is equal to 12, so it takes her 12 minutes to complete a circuit.

Then, we know she completes three of those circuits in the whole session, so we know that if one circuit is equal to 12 minutes and we need three lots of one circuit, which would be equal to 36 minutes.

And finally, the cool down.

We know that the cool down takes five minutes altogether.

The whole session takes 36 minutes, so to find out how much shorter in time the warm down was in comparison to the circuits, we need to subtract the length of time of the warm down from the total amount from the circuits.

So, 36 minus 5 is equal to 31.

So, we can say that the warm down was 31 minutes shorter than the whole exercise, or the whole exercise was 31 minutes longer than the warm down.

Well done if you managed to get all of those.

Okay, and then on to problem two, then.

Alex's dad travels to London for work once a week every month.

The train takes 2 3/4 hours to get from his nearest station to London Paddington.

How many hours does he spend travelling per week, per month, and per year? Well, we know that he travels to London once a week for work, don't we? So, we could represent that as 2 multiplied by 2 3/4.

One journey is 2 3/4 long, so two journeys would be two lots of 2 3/4 hours altogether.

2 multiplied by 2 3/4 hours is equal to 5 1/2.

So in one week, he would travel 5 1/2 hours altogether, assuming there's no delays on the train or he doesn't catch a slightly quicker train.

We know that a month is equivalent to roughly four weeks, so we know he can travel to London four times altogether over a month.

If it takes him 5 1/2 hours to get to and from London each time and he does that four times, then we can represent that as an equation here.

4 multiplied by 5 1/2 hours, and that would be equivalent to 22 hours altogether.

Okay, and then per year, then.

Well, assuming he works for 48 weeks of the year in total, then we can say that he travels for 5 1/2 hours per week and he does this 48 times over the year.

So, we're gonna do 48 multiplied by 5 1/2 hours.

So, 48 multiplied by 5 1/2 is equal to 264 hours.

So we can say that over the course of the year, he spends roughly 264 hours travelling by train to get to London for work and back.

Question D's asking, were there any other questions you could have potentially asked based on the information that we've got here? Sofia's got one.

Last year, he missed two weeks of work due to illness.

So, what was the total distance he travelled altogether there, then? Great question, Sofia.

Well, we know that we could subtract two lots of 5 1/2 hours.

That would be equivalent to the time taken to travel to London for two weeks and back.

So as a result of that, 2 multiplied by 5 1/2 would be equal to 11, and then we can subtract the 11 from the total amount, which was 264.

So, we'd say that he travelled 253 hours together in the whole year.

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

To synthesise what we've learned, we know that multi-step problems can be represented using bar models to help understand the structure of the problems. And we can also apply our understanding of multiplying mixed numbers by whole numbers to a range of multi-step problems. That's the end of our lesson for today.

Thanks for joining me and hopefully you're feeling more confident about how you can represent problems involving mixed numbers, whole numbers, and fractions themselves.

Take care and I'll see you again soon.