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Hello, my name's Mrs. Hopper, and I'm excited to be working with you in this lesson from the unit on comparing fractions using equivalents and decimals.

I love fractions and I hope you do too.

And I hope during this lesson, you'll be able to see and understand a little bit more about them and how they link to decimals.

So if you're ready, let's make a start.

So in this lesson, we are going to be solving problems using fraction decimal equivalents.

We've got one keyword and that's, "Equivalent".

So I'll take my turn, and then it'll be your turn.

So my turn, equivalent.

Your turn.

I'm sure you know what equivalent means, but let's just remind ourselves, 'cause it's a really important word and it'll be important to us today.

Numbers are equivalent if they have exactly the same value, and it's going to be really important to remember that today.

We're going to be creating values that are exactly the same, that are equivalent, so that we can substitute them in our equations and help us to solve our problems. There are two parts to our lesson today.

In the first part, we're going to be looking at fraction and decimal equivalents that are greater than one.

And in the second part, we're going to be solving problems using fraction decimal equivalents.

So let's make a start.

And we've got Sam and Lucas helping us in our lesson today.

So, let's have a look at these number lines.

You can use fraction and decimal equivalents for divisions between any whole numbers, however large.

Here, we've got our zero to one number lines, which you may be very familiar with, and we've got them labelled in half for our top line, and quarters for our bottom line.

But what if we change the start and endpoints? Ooh.

What if we go from 20 to 21, or 102 to 103? All those parts of a whole are still the same, whether we're going between zero and one, or 20 and 21.

So we've got 20, 20 and a half, 20 and two halves, or 20, 20.

5 and 21.

And then for our bottom number line, we're still counting in quarters, but we're not counting from zero to one this time.

We're counting from 102 to 103.

So we've got 102 and a quarter, 102 and two quarters, 102 and three quarters, and 102 and four quarters or 103.

And then if we're thinking about decimals, we've got 102, 102.

25, 102.

5, 102.

75, and 103.

It doesn't matter where we are in our number system on a number line, we can still divide the gap between whole numbers into fractions and decimals.

So what are the missing values on these number lines? We're going to look at them together, but you might want to pause and have a look yourself first.

What can you see? Well, we haven't got a complete top or bottom line anywhere, have we? So in this top example we've got two, and then on the top of our line we've got two and two halves.

Well, two and two halves is equivalent to three, isn't it? So we can put the three in.

So now we know that we've got a number line going from two to three, divided into two equal parts.

And when we divide that difference of one into two equal parts, we're going to end up with halves.

So we're going to end up with two, two and a half, two and two halves on the top, and 2.

53 on the bottom.

What about the lower of the two number lines? Well, we've got an eight and two quarters in the middle, and then we've got an 8.

75 on the underneath of the line.

So it's looking like quarters, isn't it? Eight and two quarters is the same as 8.

58 and a half, and 8.

75 is the same as eight and three quarters, so counting in steps of a quarter.

So the end of our number line is eight and four quarters, or nine.

And then if we go backwards from the middle of our number line, we've got eight and a quarter, or 8.

25, and then eight.

So our number line was going from eight to nine, counting in quarters as a fraction on the top and as a decimal on the bottom.

Okay, so how can you work out the steps along this number line? Again, we've got fractions on the top, and we've got decimals on the bottom.

I wonder what you think.

You might want to have a think yourselves before we see what Sam and Lucas are going to discuss.

So Sam says, "There are numbers between nine and 10." She's seen nine and one fifth and 9.

8, and that number line carries on.

So there are definitely numbers between nine and 10.

She also says, "I think we are counting in fifths, as 9.

8 is equal to nine and four-fifths." Nine and eight 10ths.

Eight 10ths is equivalent to four fifths.

Now Lucas says, "I think the numbers will go past 10 if we are counting in fifths." I think he's probably right, isn't he? He says, "This number must be nine if we're counting in fifths," and we've got nine and one fifths.

This number must be nine.

So let's count in fifths from nine.

Are you ready? Nine, nine and one fifth, nine and two fifths, nine and three fifths, nine and four fifths, nine and five fifths, which is equivalent to 10.

10 and one fifth, 10 and two fifths, 10 and three fifths, 10 and four fifths, 10 and five fifths, which would be equivalent to 11.

Now let's count in the decimal steps.

Sam says, "Let's count backwards from 9.

8 to check that this number at the beginning is nine." And remember we're counting in fifths, and a fifth is equal to 0.

So let's count backwards from 9.

8, 9.

6, 9.

4, 9.

nine.

So yes, we were right, it is nine.

And Lucas says, "Now let's count on from 9.

8 to check that this number is 11." Our 10 and five fifths is equal to 11, isn't it? So let's count on from 9.

8, 10, 10.

2, 10.

4, 10.

6, 10.

8, 11.

We were right.

And Sam says, "You can see the fraction and decimal equivalents on the number line." Nine and three fifths is equivalent to 9.

They have exactly the same value, so they are at exactly the same point on the number line.

And Lucas says, "The fraction and decimal equivalents are in the same position on the number line," because they are the same number, just written in two different ways.

Time to check your understanding.

What are the steps along this number line? We've again got fractions on the top and we must have decimals on the bottom, I imagine.

So we're given three and 10 10ths on the top.

What do we know about three and 10 10ths? And we're given five on the bottom.

Pause the video, see if you can label all these missing numbers with both fractions and decimal equivalents.

How did you get on? What did you reckon? Well, Sam says, "There are 20 divisions in the number line, and I think," she says, "We are counting in 10ths between three and one 10th, and five and one 10th." So we've definitely got three somewhere, and we are definitely ending just above five, and we've got that 10 10ths, which is a clue as well.

So she's going to put three and one 10th, and five and one 10th onto the number line.

So let's count up, see if we get to three and 10 10ths.

She's put her 10ths in and yes, we do.

Counting from three and one 10th, we do hit three and 10 10ths where it was marked on the number line, and then counting on from three and 10 10ths, or four, we count all the way up to five and one 10th on the number line.

Lucas says, "Now we can count in the equivalent decimal steps to check," so from 3.

1 all the way to 5.

And if we count all that way, yes, we are counting in steps of a 10th, and our fractions and our decimals match up.

They are equivalent, and at the same point on the number line.

Time for you to do some practise.

Can you label the steps on these number lines using fraction and decimal equivalents? So you've got one, two on this slide, and you've got three and four on this slide.

So pause the video, have a go at completing those four number lines, and we'll get back together for some answers and feedback.

How did you get on? So how many steps are there between the whole numbers, and how would you represent the steps as a fraction and a decimal? So on number one, we've got number lines with three and four on them, and we seem to have one step between three and four.

So one whole divided into two equal parts would give us a step of a half.

So, we are counting in halves.

So we've got two and a half, three, three and half, four, four and a half.

And equivalent decimals would be 2.

5, three, 3.

5, four, and 4.

For our second number line, we've got the whole numbers five and six, but we've got four equal steps to get from five to six, so we must be counting in quarters.

So our top line, we can mark those as fractions, counting in quarters.

And on the bottom line, we know that one quarter is equivalent to 0.

25.

So we're counting in steps of 0.

25.

What about number three? This time we have the whole numbers of seven and eight on our number line, but this time we had five equal steps to count in.

So we were counting in fifths.

So there's our top number line marked in one fifths, and our bottom number line is going to be marked in steps of 0.

2, because we know that 0.

2 is equivalent to one fifth.

So there, we have our number line.

And what about for number four? Well, we've got nine and 10, but we've got 10 equal jumps between nine and 10.

So we must be counting in 10ths.

So we can fill in the top of our number line with 10ths, counting back two steps below nine, and two steps beyond 10.

So we've got it in fractions on the top, and then in decimals on the bottom of our number line.

I hope you were able to use those whole numbers and how many steps there were in between to work out the fraction or decimal step that we were counting in on those number lines.

And on into the second part of our lesson, where we're going to be solving problems with fraction decimal equivalents.

So we've looked at fractions and decimals greater than one, but we can also use fraction and decimal equivalents when working with measures.

Sometimes we give measures in decimal values, and sometimes we might give them in fractions as well.

So what would these measurements be if we wrote them as mixed numbers? So a whole number and a fraction.

You might want to have a think, and then we're gonna to look at them together.

So let's look at 2.

Lucas says, "The 0.

8 makes me think of 10ths or fifths," and he says that even numbers of 10ths can be written as fifths.

Well, a fifth is 0.

2, and it's also two 10ths.

So 0.

2, 0.

4, 0.

6, 0.

8 represent even numbers of 10ths.

So 2.

8 is two and eight 10ths, or two and four fifths.

And both of those, we can say, are two and eight 10ths or two and four fifths of a kilometre.

What about 4.

25 metres? Well, Lucas says, "I know that 0.

25 is equal to one quarter.

So 4.

25 metres must equal four and a quarter metres." What about 12.

5 kilogrammes? Lucas says, "I know that 0.

5 is equal to one half." So 12.

5 kilogrammes is equal to 12 and a half kilogrammes.

And you might have heard someone talk about 12 and a half kilos, which is another way of saying kilogrammes as well.

So being able to go between fractions and decimals for measurements is really useful.

Right, we've got some boxes sitting on scales here, and they're sitting on different types of scale.

So we have to look at those carefully.

So what is the total mass of the boxes as a decimal and as a mixed number? We're gonna need to look at these scales properly, aren't we? So let's look at box one.

Lucas says, "The scale is divided into four equal parts.

So each part must be 0.

25 of a kilogramme, or a quarter of a kilogramme." So, box one must have a mass of 1.

25 kilogrammes.

We've got a scale there, going from zero to two, so one must be in the middle.

And those smaller divisions therefore represent a quarter step each time.

So 1.

25 kilogrammes.

Sam's having a look at box two.

What do you spot here? Well, Sam says, "The scale starts at 2.

5," so it doesn't start with a whole number.

Halfway, she says, "Must be three kilogrammes," because we're going from 2.

5 to 3.

So halfway must be three kilogrammes.

So, box two must have a mass of 2.

75 kilogrammes.

We are still counting in quarter kilogrammes here, but our scale doesn't go from two to three, it goes from 2.

5 to 3.

So box two must have a mass of 2.

75 kilogrammes.

We need to find the total mass of the box as a decimal and as a mixed number.

So let's have a look.

So the total mass of the boxes is 2.

75 plus 1.

25, or two and three quarters plus one and a quarter.

And whichever way we look at that, it's equal to four kilogrammes.

We've got two and one as our wholes, which give us a total of three, and then three quarters add one quarter or 0.

75 and 0.

25 give us another whole kilogramme.

So the total mass of the boxes is four kilogrammes.

Here's another problem to think about.

Sam is walking Snowy the dog this weekend.

Altogether, they've walked six and a half kilometres.

On Saturday, they walked 3.

75 kilometres, so how far did they walk on Sunday? Oh gosh, there's lots of information there.

I think I might want to draw a bar model here.

So we know that altogether they walked six and a half kilometres, so that is our whole distance.

And we know that on Saturday they walked 3.

75 kilometres.

What we don't know is how far they walked on Sunday, which is the missing part.

So to find a missing part, I subtract the known part from the whole.

So we've got to subtract 3.

75 kilometres from six and a half kilometres.

Well, Sam says, "I know that 0.

75 is equal to three quarters," so that's three and three quarter kilometres, and we can write that in.

Now we've got them both as fractions, we've got to subtract that three and three quarters, but Sam says, "I can partition three quarters into one half and one quarter." So six and a half kilometres, subtract three and a half kilometres is equal to three kilometres, but we've got another quarter to subtract.

And three subtract one quarter is equal to two and three quarters.

So our missing distance that they walked on Sunday is two and three quarter kilometres.

And there it is.

So again, by working out how to convert from, in this case, a decimal into a fraction, we were able to do our subtraction.

If we converted six and a half into 6.

5, and then tried to subtract 3.

75, we might have ended up needing perhaps a column method, with lots of exchanging and regrouping to do.

So in this way, thinking of our mixed numbers as fractions, so converting our decimal into a fraction, helped us perhaps to solve the equation more efficiently, or it did for Sam.

Maybe for you, the decimal would've been better.

Whichever is better for you is the way you should convert.

Time to check your understanding.

Now, we don't want the answer this time, but Lucas has a question for you.

He's pouring juice into a jug, and there is one and three quarter litres in the bottle.

He pours 0.

8 litres into the jug, and he wants to know how much is left in the bottle.

And his question is, "Is it best to convert the fraction into its decimal equivalent or the decimal to a fraction?" We don't want to know the answer at the moment.

He's trying to work it out.

He wants to know which conversion would be best.

So pause the video, have a think about that, and we'll come back and talk about your thoughts.

So what did you think? Well, perhaps this time, converting the fraction to its decimal equivalent is better.

It's easier to calculate with the decimals than with the fractions.

If we calculated with the fractions, we'd be thinking about subtracting fifths or 10ths from quarters and that wouldn't be so easy.

But if we convert our three quarters into a decimal, then we are dealing with 10ths and 100ths, aren't we? So we will be able to calculate much more efficiently.

So on this occasion, turning our fraction into a decimal is possibly the more efficient way to look at this.

And one and three quarter litres is equal to 1.

75 litres as a decimal.

Time for you to do some practise.

You've got some problems to solve.

So in question one, you're going to write these measurements as their decimal equivalents.

In question two, you are going to order the numbers from the smallest to the largest.

So think about whether you are going to convert the fractions into decimals, or the decimals into fractions.

For question three, you're going to put the correct symbols in between these numbers.

They're measurements, aren't they? So are they less than, greater than, or equal to each other? And then you're going to solve these two problems, and you're going to decide whether it's best to convert both values into fractions or into decimals.

You might recognise question four.

And question five is again about Sam walking Snowy the dog, but this time over three days.

So pause the video, have a go at those questions, and then we'll come back for some answers and feedback.

How did you get on? So for question one, we wanted you to write these measurements as their decimal equivalents.

So we know that one fifth is equal to 0.

2, so three fifths must be equal to 0.

So 10 and three fifths kilometres will be equal to 10.

6 kilometres.

They are equivalent.

What about three and eight 10ths of a litre? Well, one 10th is equal to 0.

1, so eight 10ths is equal to 0.

So three and eight 10ths of a litre is equivalent to 3.

8 litres, written as a decimal.

What about four and two thirds of a kilogramme? Yes.

Well, one third is that strange one, it's equal to 0.

33333, and two thirds is equal to 0.

66666.

So four and two thirds kilogrammes written as a decimal is 4.

66666 carrying on forever, possibly rounding to a seven kilogrammes.

But in fact, it's more accurate to say four and two thirds of a kilogramme.

The 0.

66 as a decimal is never completely accurate, because as we've said before, three is not a factor of 100.

Question two, you were asked to order these numbers from the smallest to the largest.

I wonder which way you converted.

We converted the fractions to decimals, because 1.

44 was possibly not so easy to express as a fraction.

I mean it is 44 100ths, but not as easy to compare with the others perhaps.

So a quarter is equal to 0.

25, and four 10ths is equal to 0.

So we can convert those mixed numbers.

So one and a quarter is equal to 1.

25 and one in four 10ths is equal to 1.

And now, we can order those numbers, because they're all written as decimals, and there they are written in the correct order.

For question three, you had to put the correct symbols between these numbers.

So we had two fifths of a metre and 2.

5 metres.

Well, I hope you spotted there that we'd use the same digits, but we know that 2.

5 metres is a whole number of metres, it's two metres and another half a metre.

So it's very definitely greater than two fifths of a metre, which is less than a whole metre.

Now two and two thirds of a litre and 2,500 millilitres.

Oh, we've gotta think carefully here.

2,500 millilitres is equivalent to two and a half litres.

So two and a half is smaller than two and two thirds.

So two and two thirds of a litre is greater than 2,500 millilitres.

It would be roughly 2666.

66666 millilitres if we converted it into millilitres from its litres.

And what about our final one? Four and a quarter kilogrammes and 4,250 grammes? Well, let's think about that grammes to kilogrammes conversion.

We've got 4,000 grammes, which is four kilogrammes, and 250 grammes, which is a quarter of a kilogramme.

Ah, so those are equivalent.

Four and a quarter kilogrammes is equal to 4,250 grammes.

I hope you got all of those right.

And finally, let's solve these problems. So this was the problem that we looked at as a check for understanding earlier in the lesson.

Lucas is pouring juice into a jug.

There's one and three quarter litres in the bottle.

He pours 0.

8 litres into the jug.

How much is left in the bottle? So we decided that it was better this time to convert the one and three quarter litres into 1.

75 of a litre.

And then we knew that that was the whole, and we had one part to subtract to find our missing part.

So we had to do 1.

75, subtract 0.

Well, 1.

75, the 0.

75 is a little bit less than 0.

8, isn't it? So we had to go through our one litre, and take away another five 100ths of a litre.

So our answer was 0.

95 litres.

And as we said, there were values in 10ths and quarters.

So it's best to convert the fraction into the equivalent decimal to do the calculation, 'cause adding and subtracting 10ths and quarters as fractions is not as easy as doing it in their decimal equivalents.

And finally, "Sam walks Snowy the dog for three days.

On day one, they walk 3.

5 kilometres.

On day two, they walk three quarters of a kilometre further than day one, and in total they walk 10 kilometres.

How far do they walk on days two and three?" Oh, lots are going on here, isn't there? So on day one, they walk 3.

5 kilometres, and on day two, they walk three quarters of a kilometre further than day one.

So here are our three days and our whole, so we know they walk 10 kilometres in total.

On day one, they walk 3.

5 kilometres.

On day two, they walked 3.

5 plus three quarters of a kilometre.

And we don't know how far they walked on day three.

So let's think about day two.

So you could convert the decimals into fractions, because halves and quarters are quite easy to add and subtract mentally.

So let's do that.

So on day two they must have walked three and a half kilometres plus three quarters of a kilometre.

And we know that three quarters is one half and one more quarter.

So they've added on a half and a quarter.

So on day two, they walked four and a quarter kilometres.

So if they walked three and a half kilometres on day one, and they walked four and a quarter kilometres on day two, then we can add those together and we can actually go back to the calculation we had; three and a half plus three and a half plus three quarters, and that's equal to seven and three quarters of a kilometre, 'cause three and a half plus three and a half is equal to seven.

And then, they walked an additional three quarters of a kilometre.

So, we now know how far they walked on day one and day two together.

So on day three they must have walked 10 kilometres.

Subtract that seven and three quarter kilometres.

Our whole is 10, our known part is seven and three quarters.

So we've got to subtract the known part from the whole.

And if we do 10 subtract seven, we get three.

And if we do three subtract three quarters, we get two and a quarter left.

So they walked two and a quarter kilometres or 2.

25 kilometres on day three.

So now, we know how far they walked on each of the days, and that their total was 10 kilometres.

And we've come to the end of the lesson.

We've been solving problems using fraction decimal equivalents, and exploring fraction and decimal equivalents greater than one.

So what have we learned about? We've learned that sometimes it is easier to convert the fractions to their decimal equivalents, and sometimes it's better to convert the other way from decimals to fractions.

When you have to calculate with fractions, it can be easier to calculate with the decimals if the fractions are not from the same family, such as that family of quarters or the family of thirds, or maybe the family of 10ths and fifths.

Converting fractions to equivalents with 10 or 100 as the denominator can make them easier to convert.

But knowing some by heart is really useful.

I hope you've enjoyed playing around with fraction and decimal equivalents today and solving some problems. I'm really grateful for all the work you've put into it and all your mathematical thinking, and I hope I get to work with you again soon.

Bye bye.

I've finished the video