Lesson video

Hello, how are you today? My name is Dr.

Shorrock.

I'm really excited to be learning with you today.

You have made a great choice to learn maths with me today and I'm here to guide you through the learning.

Welcome to today's lesson.

Today's lesson is from our unit, Compare and Describe Measurements Using Knowledge of Multiplication and Division.

The lesson today is called: Solve Problems Involving Comparison and Change.

As we move through the learning today, we will be looking at problems where we compare two lengths or describe the change in a length, but we are going to have to decide do we need to multiply because the length has got longer, or do we need to divide because the length has got shorter and we are multiplying by a unit fraction, which means we need to divide.

So today's lesson could be a little bit tricky 'cause it's new learning for you, but don't worry, I am here to guide you through the learning and I know if we work really hard together, then we can be successful.

These are the keywords that we will be using in our learning today.

We've got comparison, change and unit fraction.

Let's practise those words together.

My turn, comparison.

Your turn.

Nice.

My turn, change.

Your turn.

Lovely.

And my turn, unit fraction.

Your turn.

Well done.

Look out for those words as we move through the learning today.

So when a comparison is made, we are determining how different two objects are.

In this case, we're going to think about how many times longer, taller, or deeper objects are than one another.

But a comparison can also be made between just one object before and after a change.

Examples of a change include a change in height of a flower due to growth, or change in depth of a puddle due to rainfall.

A unit fraction is a fraction where the numerator is one.

These are some examples of unit fractions, one half, one quarter and one tenth.

So today we are going to progress our understanding of how we solve problems involving comparison and change.

So let's start by thinking about problems involving comparing lengths.

These are the children who will be helping us in our learning today.

Let's meet Lucas, Sam, Jacob and Izzy.

So let's start by looking at a comparison problem.

Sam has a piece of string that is 35 centimetres long and Lucas has a piece of string that is three times the length of Sam's string.

Can you see why this is a comparison problem? That's right, we've got two lengths of string.

Sam's got a piece and Lucas has, so we are comparing both items and we want to know who's got the longest piece of string and by how much.

So first of all, we need to work out what we know.

What do we know? That's right.

We know that Sam has a piece of string that is just 35 centimetres long.

Can you visualise that in your head? And we know Lucas' string's three times the length of Sam's.

Can you visualise that? Is his string longer or shorter if it's three times the length? That's right.

Because Lucas' string is three times the length of Sam's, that string, his string must be longer.

And the question is asking us to determine how much longer.

So we know Lucas' is longest, but by how much? So let's represent this comparison problem as a bar model.

It's always a really good place to start is representing things in a bar model.

So we know Sam has a piece of string that's 35 centimetres long and Lucas has a piece of string that's three times the length.

So to determine how much longer Luke's string is in Sam's, we need to determine the length of Lucas' string first, don't we? We can't say by how much Lucas' string is longer if we don't know the length of his string.

So this must be the unknown whole because Lucas' string is longer and Sam's string that is 35 centimetres long must be a part because it is shorter.

And we know that string is three times the length, so there are three equal parts.

Each part is 35 centimetres long.

And we know that Lucas' piece of string is longer and we know that it will be equivalent to the length of two of those equal parts.

Let's check your understanding.

Which bar models could represent this comparison problem.

Sam has a piece of rope.

Lucas has a piece of rope that is four times the length of Sam's rope.

How much longer is Lucas' rope than Sam's? Pause the video while you have a look at all four representations and when you are ready to check your answer, press play.

How did you get on? Did you choose representation C? Because Sam has a piece of rope, which is one of the parts, but Lucas' piece of rope is four times the length of Sam's rope.

So Lucas' rope must be equivalent to four equal parts.

Also though, we could do representation D, it's just the bar model which is the other way round where we've represented Sam's piece of rope on the top and the whole on the bottom.

We can now use our bar model to form an equation.

We know Lucas' string is three times as long as the length of Sam's string.

So we can form an equation 35 multiplied by three.

We know we are multiplying by three because we need to find the length of something that's just three times as long.

But how do we do 35 multiplied by three? Well we could use the distributive law to help us.

I can do 30 multiplied by three, three threes are nine.

So 30 multiplied by three is 90 and then we can do five threes, which are 15.

And then we can recombine the 90 and 15 to get 105.

So the length of Lucas' string is 105 centimetres.

Let's check your understanding.

Can you tell me which equation is represented by this bar model? Pause the video, maybe find someone to have a chat about this with.

And when you're ready for the answer, press play.

How did you get on? Did you choose option C? Because we've got one part is 70 centimetres and there are four equal parts.

So it must be 70 centimetres multiplied by four.

Well done.

So let's revisit the question to ensure that we have answered it.

Always a good thing to do.

We had to find out who has the longest piece of string.

Well we know that's Lucas 'cause his was three times as many.

But actually the whole question is asking us by how much.

So how much longer is Lucas' piece of string than Sam's.

So we need to find the difference between the two lengths and we haven't done that yet.

So Sams string is 35 centimetres long and we've calculated Lucas' to be 105 centimetres long.

And we can represent this as a bar model.

105 centimetres is the length of Lucas' piece of string and this is our whole.

And 35 centimetres is the length of Sam's string.

So to find an unknown part, we need to subtract the known part from the whole.

105 centimetres, subtract 35 centimetres.

Well I'm going to take the five off first to give 100, then subtract the 30.

So that's 70 centimetres.

So Lucas' string is 70 centimetres longer than Sam's string.

Let's look at another example.

In their P.

E.

lesson, the teacher asks the children to run or walk for one minute.

Izzy runs 200 metres in the one minute.

Jacob walks one quarter times the distance that Izzy runs.

How much further does Izzy run than Jacob walks? First let's work out what we know.

Good idea Izzy.

So we know that Izzy runs for 200 metres and we know Jacob walks one quarter times the distance.

Can you visualise that? Does he walk a shorter or a longer distance if he's walking one quarter times the distance that Izzy ran? Thank you Sam.

Because Jacob walks a fraction, so one quarter times of what Izzy ran, he will have walked a shorter distance than she ran.

And the question is asking us to determine how much further Izzy travelled than Jacob.

So let's represent this comparison problem as a bar model.

Izzy runs 200 metres in the one minute.

This this must be the whole amount because Izzy runs the greater distance.

One quarter times tells us that the whole has been divided into four equal parts and we want one of those parts.

And how much further? Well that will be the difference between the part and the whole.

So we can now use our bar model to form an equation.

We know the whole is 200 metres and that's the length we are comparing to.

And we are multiplying by one quarter because that is how many times the distance Izzy ran than Jacob walked.

Once we formed our equation, we can calculate, oh, how do we multiply by unit fraction to calculate the length that Jacob walked.

Do you know? That's right from the bar model we can see that the whole has been divided into four equal parts.

So when we multiply by a unit fraction of one quarter, it is the same as dividing by the denominator of the unit fraction, four.

So 200 metres multiplied by one quarter is the same as 200 metres divided by four.

Let's check your understanding.

True or false, 150 times one fifth is equal to 150 divided by five.

Is that true or false? And then why? Is it A, when you multiply the product is larger.

So multiplying by one fifth cannot be the same as dividing by five.

Or is it B? Dividing by five is the same as splitting the whole into five equal parts.

Each part is one fifth times the whole.

Maybe find someone to have a chat about this with.

Pause the video and when you're ready to go through the answers, press play.

How did you get on? Did you decide that that was true? And why is it true? Because dividing by five is the same as splitting the whole into five equal parts and each part is one fifth of the whole.

Here's another check for you.

Look at this bar model.

Can you form two equations from it? Pause the video while you write them down and when you are ready, press play.

How did you get on? Did you see that the whole had been divided into five equal parts? So that is the same as multiplying by one fifth and multiplying by one fifth is the same as dividing by five.

Well done.

Let's revisit our equation.

We know that Izzy ran 200 metres and Jacob walked one quarter times a distance and we know multiplying by one quarter is the same as dividing by four.

So to calculate 200 metres multiplied by one quarter, all we have to do is divide 200 metres by four.

How do we divide by four? Do you know a way? Ah, that's right.

Thank you Sam.

To divide by four we can halve and halve again.

So I've got 200 metres divided by four.

I could take 200 metres half it and half it again.

So divide by two and divide that answer by two, which is 100 metres divided by two, which is 50 metres.

Is there another way? That's right Izzy.

We could also divide 20 by four instead of 200, and make the quotient 10 times bigger.

20 divided by four is five and then make five 10 times bigger, 50.

Either way we get 50 metres.

Jacob walked 50 metres and that is one quarter times the distance that Izzy ran.

So let's revisit the question to ensure that we answered it.

Ah, the question is asking us how much further Izzy runs than Jacob walked.

So we need to find the difference between their two lengths.

We know Izzy runs 200 metres and we calculated Jacob walked 50 metres.

So let's represent this as a bar model model.

The whole is 200 metres.

The part that Jacob walks is 50 metres and we need to find the difference between the two.

To find an unknown part, we need to subtract the known part from the whole.

200 metres, subtract 50 metres is 150 metres.

So is he runs 150 metres further than Jacob walks.

Time for you to practise now.

Could you, for question one, solve these problems by identifying what we know, representing the information as a bar model and forming an equation.

For question A, Jacob kicks a football 36 metres, Sam kicks the ball three times as far.

How much further does Sam kick the ball than Jacob? For part B, a frog jumps 40 centimetres, a toad jumps one 10th times the length at the frog jumps.

How much further can a frog jump? For question two, where possible I'd like you to work in this pair or a small group.

I'd like you to write your own comparison problem related to length.

And then represent it as a bar model, identify the calculation and solve it, checking that the numbers you have used in your problem result in a whole number.

Then I'd like to sort questions and have a go at solving your new question and then ask for feedback from the question writer.

Pause the video While you have a go at both of those questions and when you're ready to check your answers, press play.

Shall we see how you got on? So we had to identify the key information first.

So Jacob kicks that football 36 metres and Sam kicks ball three times as far.

We can then represent that information in a bar model.

So the whole amount is Sam is an unknown whole.

Sam kicks ball three times as far.

Jacob's is a part and one part is 36 metres, and we know we need three equal parts.

So we can form an equation to solve.

36 metres multiplied by three is 108 metres.

So Sam kicks the ball 108 metres.

But we need then to work out how much further Sam kicks the ball than Jacob.

We need to find the difference.

The whole is 108 and we need to subtract the part, which is 36 metres, 108 metres, subtract 36 metres is 72 metres.

So Sam kicks ball 72 metres further than Jacob.

For part B, you had a question about a frog and the key information the frog jumps 40 centimetres and the toad jumps one 10th times that length.

And we need to work at how much further a frog jumps.

So we're going to represent this in our bar model.

A frog jumps 40 centimetres, that is our whole, and the tow jumps one 10th times that distance.

So my whole has been divided into 10 equal parts.

The distance that the toad jumps is equal to one of those parts.

So we've got 40 centimetres, we're going to divide by 10, which is four centimetres.

The toad jumps four centimetres.

But we needed to work at how much further the frog jumped.

So we're going to find the difference by subtracting a known part from the whole.

40 centimetres subtract four centimetres is 36 centimetres.

So the frog jumps 36 centimetres further than the toad.

And for question two, you were asked where possible to work in a pair or small group and to write your own comparison problem.

You might have written a problem like this: Sam jumps 120 centimetres, Izzy hops one quarter times the distance.

How much further does Sam travel than Izzy? You might have checked that your problem was appropriate by representing it in a bar model and checking the calculation like this.

So Sam travelled 120 centimetres, that is my whole.

Izzy hops one quarter times the distance.

So the whole has been divided into four equal parts and the distance Izzy hops is equivalent to one of those parts.

We then had to calculate the distance that Izzy hopped.

So 120 centimetres divided by four is 30 centimetres.

And then we can work out how much further Sam travels than Izzy by subtracting the known part from the whole.

120 centimetres, subtract 30 is equal to 90 centimetres.

So Sam travelled 90 centimetres further than Izzy.

You might have then swapped questions and had a go at solving a new question before asking for feedback from the question writer.

So you might have got your answer marked with a tick.

Well done.

Fantastic progress in your learning so far about comparing length and solving problems related to comparisons.

We're now going to move on and look at how we can solve problems involving changing lengths, so when one thing changes.

Let's look at this problem.

Izzy is meant to cycle five kilometres to school.

Unfortunately.

Oh dear, something bad is going to happen, I can tell.

She has a puncture when she's at one half times the length of this five kilometre distance.

How far has she got to push her bike until she gets to school.

First let's work out what we know.

Good idea, Izzy.

What do we know? Well we know you were meant to cycle five kilometres to school.

Can you visualise that? I'm seeing a straight line of her journey to school in my head at the moment.

The puncture happens when Izzy is one half times the full five kilometre length into the journey.

Can you visualise that? I can see halfway on my line that's in my head at the moment.

We are being asked to find the distance Izzy still has to travel and I can see that in my head.

It's a bit that happens after the puncture.

We can represent this in a table and a bar model to help us.

Good ideas Sam.

So my table, we know that Izzy was meant to cycle five kilometres, but what happened? Well she only cycled one half times the five kilometre distance.

So we can represent that with an arrow and multiplying by one half and we know we need to find the remaining distance.

We can also represent this in a bar model.

The whole journey was meant to be five kilometres and we know that puncture happened halfway into her journey.

So the whole has been divided into two equal parts and the distance that she actually did cycle is one of those parts which is equivalent to half of the whole.

Let's check your understanding.

Which bar model represents this table.

Pause the video while you have a look at the table and the different representations.

And when you are ready to check your answer, press play.

How did you get on? Did you say, well it can't be a because that's only showing something that has been made twice as large.

B, I can see there are three equal parts so that's definitely correct.

C, the 13 centimetres is the longer bar and we know that that can't be correct because we are multiplying the 13 by three.

And D, well, there are four equal parts and we only need three times as many because the table is showing we're multiplying by three.

Well done.

So we can use our representations to form an equation.

So we've got my bar model and the table and we can see from both of them that five kilometres is our starting amount and we need to multiply by one half.

And when we multiply by a unit fraction, it is the same as dividing by two.

And we can see that on our bar model because the whole has been divided into two equal parts.

So five kilometres multiplied by one half is the same as five kilometres divided by two.

Let's then use our equation to solve the problem.

Now, Sam is telling us to calculate one half times that five kilometres.

We can divide five kilometres by two or just half five kilometres.

Ah, thank you Izzy.

I was wondering how we were going to half five kilometres.

It will be easier to convert five kilometres to 5,000 metres.

So 5,000 metres multiplied by one half is the same as 5,000 metres divided by two.

How can we do 5,000 metres divided by two though? How can we half 5,000? Oh good idea Sam.

Let's partition the 5,000 into 4,000 and 1000 metres to help us and then we can half both of those separately and then recombine them.

We know half of four is two, so half of 4,000 must be 2000 metres.

And we know half of 1000 is 500 metres.

And if we recombine them, half of 5,000 metres is 2,500 metres.

So half of five kilometres is the same as halving 5,000 metres which is 2,500 metres.

So 5,000 metres multiplied by a half is 2,500 metres.

2,500 metres is one half times the length of five kilometres.

And all those equations are different ways of representing the same thing.

So Izzy had cycled 2,500 metres to school before the puncture.

So this is equivalent to two kilometres 500 metres.

And it was halfway, so she still had the same amount left to go.

Let's check your understanding.

Solve this equation using partitioning to help.

You've got 7,000 metres multiplied by one half.

So complete the statements there.

Whilst you write that down, pause the video and when you are ready to go through the answers, press play.

How did you get on? Did you realise that it would be easier to partition 7,000 into 6,000 and 1000? 6,000 metres divided by two is 3000 metres.

1000 metres divided by two is 500 metres.

Then we can recombine 7,000 metres divided by two, must be the same as 3,500 metres.

Well done.

So let's revisit our question just to make sure we have answered it.

Oh, we haven't yet have we.

We've got to determine how far she's still got to push her bike until she gets to school.

So she was meant to cycle five kilometres and we've calculated that she had a puncture at 2,500 metres into the journey.

And we can represent this as a bar model.

So the whole journey is five kilometres or 5,000 metres and we know she had a puncture after 2,500 metres.

So to find the unknown part, we need to subtract the part we know from the whole.

5,000 subtract 2,500 is 2,500 metres.

So Izzy has to push her bike another 2,500 metres.

And what do you notice about this? Did you notice something? Ah yes, thank you Sam.

Me too.

I noticed that the distance Izzy had cycled before the puncture and the distance that she had to push her bike afterwards were the same.

Why is that? Do we know? Ah, thank you.

Yes, it's because Izzy's puncture was at one half times the distance.

So there were two equal paths to the total journey.

Time for you to practise now.

Question one, could you draw a line to match the multiplication equation to its division equivalent and then solve it? For question two, could you solve these change problems by identifying what we know, representing the information in a table and bar model then forming an equation? So for question A, Lucas was growing some cress.

He measured its height shortly after it sprouted to be four millimetres.

Over the next week, the cress grew to seven times the height of when Lucas first measured it.

How much taller is the cress now than when it was first measured.

And for part B, a candle was 270 millimetres tall.

After it had burned for a few hours, it was one third times its original height.

How much shorter is the candle now? Pause the video while you have a go at both questions and when you are ready for the answers, press play.

Shall we see how you got on? For question one, you were asked to draw a line to match equivalent equations.

Seven kilometres times one half is the same as 7,000 metres divided by two.

700 centimetres multiplied by one quarter is the same as 700 centimetres divided by four.

700 metres times one half is the same as 700 metres divided by two.

And seven kilometres multiplied by one quarter is the same as 7,000 metres divided by four.

You were then asked to solve the equations.

We now know seven kilometres multiplied by one half is the same as 7,000 metres divided by two and that is equal to 3,500 metres or three kilometres, 500 metres.

For the second equation, 700 centimetres multiplied by one quarter is the same as 700 centimetres divided by four.

So we could halve and halve again and we would get 175 centimetres.

700 metres times one half is the same as 700 metres divided by two or 700 metres halved, which is 350 metres.

And then seven kilometres multiplied by one quarter is the same as 7,000 metres divided by four, which is 1,750 metres, which is equivalent to one kilometre, 750 metres.

For question two you were asked to solve some change problems. So Lucas was growing some cress and he measured its height shortly after sprouted to be four millimetres.

So that's our key bit of information.

The cress grew to seven times the height and we need to find how much taller the cress is now.

So I can represent this in the table and a bar model.

The starting height is four millimetres and we are representing it multiplying by seven 'cause it grew to seven times its height.

And my bar model has got seven equal parts.

We can then form an equation.

We've got four millimetres, we are multiplying it by seven, which is 28 millimetres.

And then we have to find out how much taller the cress is now, so we need to subtract the known part from the whole.

28 millimetres, subtract four millimetres is 24 millimetres.

So the cress is now 24 millimetres taller than when it was first measured.

For part B, the question about the candle.

We've got a candle is 270 millimetres tall and it's now one third time its original height and we have to work out how much shorter it is now.

So I can represent this in a table or bar model.

The whole is 270 millimetres and we need to divide the whole into three equal parts because we are multiplying by one third.

We can then form an equation.

270 divided by three is 90, so the candle is now 90 millimetres tall.

But we need to find out how much shorter the candle is now.

270 millimetres, subtract 90 millimetres is 180 millimetres.

So the candle is now 180 millimetres shorter than its original height.

How did you get on with both of those questions? Well done.

Fantastic progress today on your ability to solve problems involving comparison and change.

We know comparison and change problems can be represented visually in tables and/or bar models to help solve the problem.

We know in a problem where we are finding a whole number times a length, we know we'll be multiplying that length by the whole number.

And we know that in a problem where we are finding a unit fraction times the length, we know we will be dividing the length into the same number of equal parts equivalent to the denominator.

Brilliant learning today, you should be very proud of yourselves and I look forward to learning with you again soon.