Lesson video

Hello.

My name's Mr. Tazzyman and I'm 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.

Let's get started with the outcome then.

I can add on fractions with the same denominator.

So that's what we want you to be able to say by the end of today's lesson.

Here's the key words that we are gonna use.

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

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

Ready? My turn.

Unit.

Your turn.

My turn.

Denominator.

Your turn.

My turn.

Numerator.

Your turn.

Well done.

Now let's see what each of those words refers to.

The unit is one of something that defines the one you are counting in.

A numerator is the top number in a fraction.

It shows how many parts we have.

A denominator is the bottom number in a fraction.

It shows how many parts a whole has been divided into.

Okay then, here's the different sections of today's lesson on adding on fractions with the same denominator.

We're gonna start by adding on and unitising, and then we're gonna move on to looking at equations and generalising.

We'll start with adding on and unitising them.

Here's two friends that are gonna help us along the way by chatting through the maths and by presenting us with some of the different representations.

We've got Lucas and Jun, but to begin with, we're gonna think about some frogs.

You heard right.

Ready to listen? Ready to learn? Let's go.

Some frogs are sitting on a log.

First there were three frogs, then two more frogs jumped on.

Now there are five frogs on the log.

Here is an orchard.

First there were three trees, then two trees were planted and grew.

Now there are five trees in the orchard.

Lucas has a piggy bank.

First there were three pound coins inside.

Then he saved two more pound coins.

Now there is five pounds in his piggy bank.

Lucas and Jun compare the three contexts so far.

We've got our frogs on a log, our trees, and our piggy bank.

And what was the same and what was different about those three contexts? Jun says they each had the same adding on structure.

And Lucas says the units are different for each of them though.

Lucas and Jun use counters to show that the structure was the same.

First there were three.

Then two more were added on.

So the first addend was three.

The second addend was two.

The total or sum was five.

Lucas and Jun turn the contexts into bar models to discuss the difference they found.

They're identical in structure.

The units are different.

They're frogs, trees, and pounds.

Okay, time to check your understanding so far.

I'd like you to identify the units in this adding on context.

First there were three bananas in the fruit bowl.

Then two more were added.

Now there are five bananas in the fruit bowl.

And we've got a stem sentence here that says the units are, and you need to fill in that blank.

Pause the video here and I'll be back to reveal the answer in a moment.

Welcome back.

The answer was bananas.

They were the unit in this context.

Jun collects Battle Robot stickers.

There are nine to collect in a set.

You can see the nine spaces on the page there.

Jun had three, then Lucas gave him two more.

How many stickers does he have now? First, I had three stickers.

Then, I gave you two more.

Now, I have five stickers.

Makes sense.

Seems like a familiar structure too, doesn't it? Jun and Lucas think about the question differently.

Always good to think about things differently.

That's what mathematicians do.

There are nine stickers in the whole set.

That means that each sticker is one-ninth of the whole set.

So the unit can be changed from sticker to one-ninth.

Okay, first you had three one-ninths.

Which is three-ninths.

So they've replaced the first label with three-ninths.

Then, I gave you two more one-ninths.

That's two-ninths.

And again, they've replaced the label with two-ninths.

Now, you have five one-ninths.

So that's five-ninths.

There they go, replacing the label again.

Different representations can help to make the structure of a calculation clear.

And Lucas and Jun decide to challenge themselves.

How many ways can we represent this addition? Let's see.

We can compare them to help our learning.

Lucas and Jun use a number line representation to start with.

Nine stickers make the set.

So each sticker is one-ninth.

We need intervals of one-ninth.

Remember, intervals are the gaps between the marks on the number line.

They've labelled them up.

Zero, one-ninth, two-ninth, three-ninth, four-ninth, five-ninth, six-ninth, seventh-ninths, eight-ninths, one.

First, I had three one-ninths.

That's three-ninths of the set.

We can start there and add on.

Then, you gave me two stickers.

That's two-ninths added on.

So they put a jump of adding two-ninths onto the number line.

Now, I have five stickers.

Five stickers is five-ninths of the set.

This time, Lucas and Jun show the same addition with a bar model.

They have slightly different versions.

What's the same and what's different? Look at those two bar models.

They're both representing that same addition, but in a slightly different way.

We have both shown the addends says Jun.

What I started with and what you gave me.

I have shown each addend as a non-unit fraction.

Three-ninths, two-ninths.

You've shown them as unit fractions combined.

I have shown the sum as the whole and the addends as parts.

I have shown the sum as a part of the whole.

So neither of these representations is incorrect.

They're just a slightly different way of showing the same mathematical structure.

This time, Jun and Lucas use a circle.

Can you see both addends and the sum? Nine stickers make the collection, so the circle has been divided into one-ninths.

Here are the addends.

The fraction you had first and what I then gave.

Each addend has been shown as unit fractions combined.

Here is the sum.

What you now have combined.

Again, the sum has been shown as a part of a whole here.

All right, let's check your understanding.

Which representation is showing this addition? So you've got two representations on the right-hand side there and you need to choose which of these representations is showing this.

First, the container was filled with four-tenths of sand.

Then, five-tenths were added.

How much of the container is filled with sand now? Okay, pause the video and have a go at choosing the correct representation.

Welcome back.

The correct representation was the top one.

This bar model shows both addends.

This is a comparison bar model, so if you wanted to compare by maybe saying, well, which of those addends was greater, you might use that comparison bar model.

Alright, time for your first task.

Number one, you need to represent and solve the additions below in two different ways.

Jun has made a rain gauge at school.

It is four-tenths full.

Overnight, it rains heavily and the rain gauge fills another two-tenths.

How full is the rain gauge now? B, Lucas decided to help plant some flowers in the school quiet garden.

One-sixth of the seed has already been planted.

Lucas plants three-sixths more.

How much of the seed has been planted now? Okay, you've got to create two different representations for each of these.

Pause the video here, so you can have a go at that, and I'll be back to show you some of our representations in a little while.

Welcome back.

Jun has had a go at this first one, and he's shown it using a bar model and a circle.

You can see that they're very similar and we've got the two addends there made up of four lots of one-tenth, which is four-tenths, and two lots of one-tenth, which is two-tenths.

Altogether, that's six-tenths.

The rain gauge is six-tenths full.

Here's B.

We had our seeds.

We could have used the number line here starting on one-sixth with a jump of three-sixth to get to four-sixth or we could have used a bar model as you can see here.

How did you manage to do these? What representations did you use? You might like to have a little bit of feedback with one another to compare and contrast.

Pause the video here and try that.

Now it's time for the second part of the lesson, equations and generalising.

Jun and Lucas revisit the Battle Robot stickers.

There's another representation we could use.

We can write it as an equation.

That's a representation.

Okay.

First, you had three stickers, which is three one-ninths.

So that's three-ninths.

Then, I gave you two stickers, which is two one-ninths.

And two-ninths.

Now you have five one-ninths.

Is equal to five-ninths.

They've written an equation and it's true, equations are representations as well.

Seems strange to think about sometimes, but using this funny notation, this strange code is another way of showing the mathematical structure and the solution.

Jun turns this number line representation into an equation adding on a fraction.

The first addend is three-sixths because that's where the addition starts.

So he writes three-sixths down in fraction notation.

The second addend is two-sixths because that is what's being added on.

So he puts down plus and two-sixths.

The sum is five-sixths because that's what the number line finishes on.

So three-sixths plus two-sixths is equal to five-sixths.

He's written his equation.

Your turn.

Can you write the equation represented by this number line? Pause the video, have a go, and I'll be back to show you that equation in a moment.

Welcome back.

Here's what the equation looked like.

One-sixth added to three-sixth is equal to four-sixth.

Of course, it may be that you started with four-sixths 'cause you can start with the sum in equation.

Okay.

Lucas uses an equation to represent and solve a new adding on context.

First, the sunflower is three-tenths of a metre.

Then, over time, the sunflower grew three-tenths of a metre.

How tall is the sunflower now? Be careful here because three-tenths is mentioned twice, but it's slightly different each time.

The metre is the whole and it has been divided into 10 equal parts.

That means that each part is one-tenth.

The first addend is three-tenths of a metre.

The second addend is also three-tenths of a metre.

Now, the sunflower is six-tenths of a metre.

There's the equation.

Okay, your turn.

Can you represent and solve this problem using an equation? Again, pause the video here and have a go at that, and I'll be back to reveal the answer in a moment.

Welcome back.

There's the equation.

Two-tenths is added to five-tenths, which is equal to seven-tenths.

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

Lucas and Jun list down some of the equations they've written so far.

What do you notice about the sums of the equations? What is the same and what's different? Hmm, tricky one on this.

Try comparing all the numerators and then compare the denominators.

What do you notice? Jun says in each equation, the denominator is the same in the addends and the sum.

And he's highlighted them for us to see.

You can see in that first equation, it's always ninths.

In the second, it's always sixths.

And in the third, it's always tenths.

The numerator changes through addition.

You've got three added to two equals five, three added to two equals five, and three added to three equals six.

Lucas justifies the generalisation by thinking about the denominator and numerator.

In these adding on contexts, there were different units but the same structure.

Do you remember these? This Battle Robot context has the same structure, but the unit is different.

The unit is one-ninth, which is shown by the denominator of nine.

The unit of one-ninth doesn't change, but the number of units does.

The numerator tells us how many units of one-ninth there are.

If we add together units of one-ninth, we get a fraction with a greater numerator.

When adding on fractions, the numerator changes, but the denominator is the same.

There's quite a lot to that, but hopefully by looking at these representations and thinking about all of your learning so far, thus starting to make sense.

Okay, it is time for your second practise task.

For number one, I'd like you to draw the equations onto the number line using the adding on structure and then solve them.

For number two, for each of the worded problems below, draw a representation, write them out as an equation and solve them.

For number three, you've got to write an adding on context in which you might complete these calculations.

So you are almost thinking about a story behind these calculations because they've already been sold.

Okay, pause the video here and have a go at those.

I'll be back in a little while with some feedback.

Welcome back.

Let's start with number one, A.

We had one-fifth added to two-fifths, so you needed to start off by labelling your number line carefully using fifths.

Then you had a jump from one-fifth of two-fifths and that would've ended up on three-fifths.

So the answer was three-fifths.

One-fifth added to two-fifths is equal to three-fifths.

Alright, let's look at B.

So for B, we needed to make sure that the number line was labelled using ninths.

We started on two-ninths added four-ninths on, and we ended up then with six-ninths being our solution.

I hope you got both of those.

Let's move on to number two.

So for each of these worded problems, we had to draw a representation and write them out as an equation and solve them.

Here's a representation version of A.

First, the sunflower was four-tenths of a metre.

So they've started on four-tenths.

Then over time, the sunflower grew by two-tenths of a metre.

That's where there's a jump of two-tenths.

And how tall is the sunflower now? Well, it finished on six-tenths.

There's the equation, four-tenths added to two-tenths is equal to six-tenths.

Did you get that? I hope so.

Here's B.

We chose to use a bar model to represent this one.

Remember, there are a number of different representations that you could have used, and if yours doesn't match the one that we've chosen, that doesn't make it wrong.

It's just a matter of choice.

Okay, so first one-fifth of a cup of orange cordial is poured into a glass.

You can see that's been represented on the bar model with one-fifth.

Then, three-fifths of a cup of water is poured into the same glass.

We've got our three-fifths of water there.

What fraction of a cup of squash is in the glass now? Well, here's our equation.

One-fifth added to three-fifths is equal to four-fifths.

Let's look at number three then.

We've got a couple of contexts here that we've chosen, but again, this was about you making sure that you've chosen one for yourself, but it's got to demonstrate that you understand what the equation is.

Jun, for the first one said that the sunflower was one-fifth of a metre.

It grew two-fifths of a metre, and now it is three-fifths of a metre.

For the second one, Lucas said the sandpit is one-sixth full of sand.

I add in two-sixths more.

Now it is three-sixths full of sand.

Okay, you might wanna pause the video here and share a few of your context to look at what's the same and what's different.

It's time for the end of the lesson then.

Here's a summary of the things that we've learned.

The structure of addition is the same for adding on fractions.

There are addends and a sum or total, and the context features first, then, and now stages.

When adding fractions with the same denominator, the denominator is the unit and so stays the same in the sum as well.

The numerator tells us how many units there are.

The numerators of the addends are added together to create the numerator of the sum.

My name is Mr. Tazzyman.

I've really enjoyed learning with you today.

Hope to see you again in the future.

Bye-bye.