Lesson video

Hello, my name's Mr. Tazzyman, and I am going to be teaching you a lesson from this unit today all about the composition of non-unit fractions.

Sometimes, fractions can feel as if they're a little bit tricky, but actually, if we think carefully, listen well, and look at what's in front of us, then we should be able to overcome any difficulties.

I hope you enjoy.

Here's the outcome for today's lesson then.

By the end, we want you to be able to say, I can substitute a fraction representing a whole to solve subtraction problems. Here's the keywords that you're going to see and hear during this lesson.

I'm going to say them and I want you to repeat them back to me.

So I'll say my turn, say the word, and then you can repeat it.

Are you ready? My turn, substitute.

Your turn.

My turn, expression.

Your turn.

Great.

Now let's see what those words mean in case we didn't know already.

Substitute means to put in place of another and it can be used to replace a fraction with another fraction of equal value.

An expression contains one or more values where each value is separated by an operator.

Two or more expressions with the same value can be separated by an equals sign to create an equation.

There's an example at the bottom which might be a bit easier to understand.

There's an equation which reads four plus seven equals 11.

That's an equation, but the expression is one side of that.

The example has been circled there.

Four plus seven.

That's one expression.

11 hasn't been circled, but it is also an expression.

Two or more expressions go together to make an equation.

Okay.

Let's have a look at the structure of the lesson now.

Firstly, we're going to be thinking about substituting the fraction representing the whole.

Secondly, we're going to look at reasoning by substituting the whole.

And all of this will help us to substitute a fraction representing a whole to solve subtraction problems. We'll start with the first part.

And we've got two friends to help us, Lucas and Sofia.

Hi, Lucas.

Hi, Sofia.

They're going to be chatting through some of the maths that we learn about, giving us some hints and tips and maybe even prompting some of our own discussion.

Ready to begin? Let's go for it! Sofia has been set a tricky challenge.

Four-quarters subtract seven-eighths is equal to something, something.

I'm finding this really tricky because the denominators are different.

I can't do four take away seven because I would go below zero.

I think we need to find a way of substituting one of the terms in the expression.

And when it says terms, it means one of the parts of it.

So we've got four-quarters or seven-eighths.

They're both terms in that expression.

What do you mean? We need to replace a fraction with another of the same value.

Let's think about what we know about fractions and wholes first.

Okay.

It's fine not to know straight away I suppose? Yes, of course! And that's true for everyone who's a good mathematician.

You're not always going to know straight away, but as long as you're prepared to be ready to explore and to investigate and to learn, then that's okay.

Being a mathematician means not knowing sometimes.

They start with some counting of fractions to help them warm up their brains a bit.

Zero.

One-sixth.

Two-sixths.

Three-sixths.

Four-sixths.

Five-sixths.

Six-sixths.

One.

Hang on! That's not right.

What has Lucas spotted? Have a look at that number line.

Is there a mistake there? If the denominator and numerator are the same, the fraction is equivalent to one whole.

That means six-sixths is equivalent to one whole.

So Lucas starts again.

Zero.

One-sixth.

Two-sixths.

Three-sixths.

Four-sixths.

Five-sixths.

One.

So you can see that Lucas has counted carefully and he's realised that when he gets to six-sixths, that's the same as one whole or one.

Okay, let's check your understanding so far.

With a learning buddy, practise counting to one using the fractions below and taking it in turns.

Pause the video here and have a go at that.

Welcome back.

Did you enjoy counting? Here's Lucas, and he says, we practised counting in one-fifths.

One-fifth.

Two-fifths.

Three-fifths.

Four-fifths.

Five-fifths or one.

Is that what you got up to? Did you have a similar kind of thing? I hope so.

All right.

Let's move on.

Lucas cuts a small tray bake cake into quarters to share between himself, Sofia, Jun, and Aisha.

Oh, that looks tasty.

One quarter.

One quarter.

One quarter and one quarter.

Sorry, Lucas.

I don't like this flavour cake.

No worries! But I've already cut it into quarters.

How can I turn that into one-thirds to share? What do you think? How could Lucas make sure that he's sharing the cake equally between himself, Aisha, and Jun? Because we know that Sofia doesn't want any.

You could cut it into one-thirds vertically.

Oh, yes! We wouldn't lose any cake.

One-third, one-third, and one-third.

Sofia turns the cake into a representation instead.

Probably not as tasty, but it's going to help us to think about it.

We started with one whole.

Then we changed the whole to be four-quarters.

So we substituted it.

Then we decided to use thirds instead.

We substituted four-quarters for three-thirds.

I think this representation shows another fraction.

It does.

It shows twelve-twelfths as well.

You can see them there.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12.

12 equal parts.

That's twelve-twelfths.

Okay, let's check your understanding.

How many different fractions are shown as being equivalent to the whole here? Remember, when the numerator and denominator are the same, the fraction is equivalent to one whole.

Look at the representation.

Maybe have a chat with someone near you about it.

And pause the video.

I'll be back with some feedback soon.

Welcome back.

You could have had three-thirds here and you can see the thirds are vertical.

They're the columns, if you like.

Could have had five-fifths.

Those are the rows.

They're horizontal.

They're going across.

You could also have had 15 fifteenths because there are 15 boxes here that have all been put together to make this one whole.

15 equal parts, 15 fifteenths.

Lucas turns the cake example into an equation.

And remember, an equation is another type of representation.

It might not be as visual, but it is a way of writing down what we can see.

I started with one whole cake.

So he's written one.

I cut it into quarters to start with.

One is equal to four-quarters.

Then I cut it into thirds instead.

One is equal to four-quarters is equal to three-thirds.

I also ended up with twelfths.

One is equal to four-quarters, which is equal to three-thirds, which is equal to 12 twelfths.

What do you notice? The numerators and denominators are the same.

The whole hasn't changed but the numbers have.

The value is still one, but all of those numbers are different.

The numerators and denominators match, though.

That is crucial.

Each of them could be substituted by the others.

I think this substitution might help me with my tricky problem.

Sofia revisits her tricky challenge.

Four-quarters subtract seven-eighths is equal to something, somethings.

I know that four-quarters can be substituted by one because they are equivalent.

So now we've got one subtract seven-eighths is equal to something, somethings.

I can also substitute one for eight-eighths because they're equivalent.

This is starting to look a little bit easier all of a sudden.

Eight-eighths subtract seven-eighths is equal to we don't know yet.

Now the denominators match so the solution will also be in eighths.

Oh, good thinking.

Denominator stays the same because that's the unit that we're calculating with.

I know that eight subtract seven is equal to one so the numerator will be one.

Well done, Sofia.

Wow! Well done, says Lucas.

Okay, here's the first task for you.

For number one, I want you to write three different fractions that are represented in the images below.

You've got A and B.

For number two, can you draw a representation to show the following equation? Use the boxes below as starting points.

Little clue here.

You might want to think about doing one of the denominators horizontally and one of them vertically.

And number three, solve the following by substituting and finding the missing fraction.

So remember here that you can substitute any of those fractions where the numerator and denominator are the same, with one.

And you can also substitute one with any other fraction you'd like where the numerator and denominator match.

Okay, pause the video here and have a go.

Enjoy.

Think about it and I'll be back soon with some feedback.

Welcome back.

We're going to start with one A, and this is what it might have looked like.

Four-quarters, three-thirds, and 12 twelfths.

If you look closely, you can see that the four-quarters are the rows.

They've been shown horizontally.

Three-thirds are the columns.

They've been shown vertically.

And altogether, you've got 12 equal parts there.

So 12 twelfths.

I think this also shows two halves.

Wow, that's good thinking.

Wonder where it shows two halves? Aha! There is a horizontal line running through the middle.

So we've also got two halves being shown here.

Did anybody spot that? Here's one B.

Six-sixths, five-fifths, and 30 thirtieths.

Wow.

You can see that the six-sixths have been shown vertically, the five-fifths have been shown horizontally, and together, they've made 30 thirtieths.

This one shows halves as well.

This time vertically.

Can you see that with the colours and the arrows pointing at the line that runs down the middle of this whole? Okay, number two.

Can you draw a representation to show the following equations? We've got the boxes already.

So if we split the first one vertically into thirds.

Remember, you might have done that horizontally.

And then the second one, we can split horizontally, fifths.

Okay, let's look at B.

Six-sixths.

We've chosen to do that vertically.

You might have done it horizontally.

And then we've got four-quarters, which we've done horizontally.

Here's number three then.

For the first one, we had three-thirds subtract four-tenths is equal to six-tenths.

You could have substituted three-thirds with 10 tenths, and that would've meant that you could complete it much more easily.

For B, we had six-ninths.

You can substitute seven-sevenths with nine-ninths.

So you'd have nine-ninths subtract three-ninths is equal to six-ninths.

For C, we had five-sixths.

You could have substituted 12 twelfths for six-sixths.

Six-sixths subtract one-sixth is equal to five-sixths.

And then for D, we had six-eighths.

If you'd substituted six-sixths for eight-eighths, you could have had eight-eighths subtract two-eighths is equal to six-eighths.

Pause the video here to discuss any of those and to make sure that you're up to date with your marking.

Okay, then, the first part of the lesson is complete.

Now let's move on to reasoning by substituting the whole.

Sofia compares two expressions with an unknown symbol between them.

I'm going to start by calculating each expression.

Let's substitute the wholes so that they match the denominator of the subtrahend.

And remember, the subtrahend is what it is that you are subtracting.

They've substituted both of those wholes with four-quarters, which of course has the same value, but it's going to help them to think about these expressions.

The denominators are the same so I can compare the numerators.

I know four take away one is equal to three so the first expression is equal to three-quarters.

There it is.

I know four subtract two is equal to two so the second expression is equal to two-quarters.

There it is.

What symbol do you think will go in the middle then? Three-quarters is greater than two-quarters.

Greater than.

Lucas compares two expressions.

We can start by calculating each expression again.

I don't think we need to here, says Lucas.

What do you think? Let's substitute the wholes to match the denominator.

So they've substituted them again so they've got four-quarters.

Snap! The expressions now match.

That means the missing symbol is equals.

There we go.

Okay, time to check your understanding.

We've got a similar kind of problem here.

Can you work out the missing symbol? Pause the video and have a go.

Welcome back.

How did you get on? You might start it by substituting the wholes.

So you had four-quarters for each of them.

Calculate the expressions and compare them.

Four-quarters subtract three-quarters is equal to one-quarter.

Four-quarters subtract one-quarter is equal to three-quarters.

So we knew that in between, you needed to have a less-than symbol because one-quarter is less than three-quarters.

Did you get that? Okay, let's move on.

Sofia and Lucas need to complete the inequality statement using the digit cards, which can be reused.

Let's substitute the wholes again, says Lucas.

Notice that they've substituted them with a numerator and denominator that are the same but that also match the other denominators in the question.

Five-fifths and five-fifths.

The first expression needs to be less than the second.

Sofia's getting that from the symbol in between, the less-than symbol.

So I need to have a larger subtrahend in the first expression.

I'll try a two as the numerator in the first expression.

Then a one for the numerator in the second.

I'll calculate the expressions, says Lucas.

I know that five subtract two is equal to three so the first expression is equal to three-fifths.

I know that five subtract one is equal to four so the second expression is equal to four-fifths.

Three-fifths is less than four-fifths, so it works! Well done, Sofia.

Well done, Lucas.

I think there are more solutions than this.

Ah, interesting, Lucas.

What do you think? Do you think there are more solutions? Hmm, maybe that's something to test out.

Let's see if we can use what you think for your check for understanding.

One denominator has already been completed.

Which other card would you use to complete this? And you can see you've got the same inequality statement there at the bottom.

But this time, there's only one incomplete numerator and one's been done for you.

Nine-ninths subtract three-fifths is less than eight-eighths subtract something-fifths.

Pause the video here and have a go.

Welcome back.

Let's see how you got on then.

You could have used a one or two here.

So to explain that, if you'd replaced nine-ninths and eight-eighths with five-fifths, then you had an expression which you could have calculated.

However, you needed to make sure that the numerator in the second expression that you were choosing was less than the numerator in the first.

The subtrahend had to be less than three.

So only two or one was available to you.

Okay, it's time for your second task.

For number one, you're going to compare these expressions and write the symbols that go between them.

Remember to use what we've learned so far.

Substitute some of those wholes for fractions that are easier to use which match the denominator in the question already.

For number two, use the digit cards below to complete both of the following comparisons of expressions.

You can only use each card once.

How many solutions can you find? says Sofia.

Okay, pause the video here, have a go at those, and I'll be back in a little while for some feedback.

Welcome back.

Let's look at number one first.

Ready to mark? So one A, you could have replaced those wholes with 10 tenths.

That would've given you seven-tenths as the first expression.

Six-tenths as the second.

So seven-tenths is greater than six-tenths.

Now let's look at B.

The wholes could have been replaced with seven-sevenths.

That meant that you had four-sevenths on one side and two-sevenths on the other.

It was a greater-than symbol.

Let's look at C.

Replacing the wholes with five-fifths, you end up with two-fifths on one side and four-fifths on the other.

So it's a less-than symbol.

And finally, D.

Replace the wholes with eight-eighths and seven-sevenths.

So you end up with zero on either side.

So it should be an equal sign in between.

Pause the video here if you want to discuss any of those further or you need a bit more time to mark.

Okay, number two.

Use the digit cards below to complete both of the following comparisons of expressions.

There were several different combinations.

One and two, four and three.

One and three, four and two.

One and four, three and two.

Two and three, four and one.

Two and four, three and one.

Three and four, and two and one.

Did you get all of those? I hope so.

Pause the video here to mark carefully.

Welcome back.

Here's a summary of what it is that we've learned.

A whole can be represented as a fraction by using the same numerator and denominator.

A fraction that represents a whole can be substituted for another fraction that represents a whole.

This can be useful when subtracting a fraction from a whole that has been represented using a different denominator.

With this knowledge, you can solve several reasoning problems involving missing numbers or inequalities.

My name is Mr. Tazzyman.

I've enjoyed learning with you today and I hope you have as well.

I'll see you again next time.

Bye-bye.