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.

Okay then, here's the outcome for the lesson.

By the end, we want you to be able to say, I can add fractions with the same denominator.

Here are the key words that you are gonna expect to see and hear during the lesson.

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

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

Ready? My turn, units.

Your turn.

My turn, denominator.

Your turn.

My turn, numerator.

Your turn.

Okay.

Let's look at what those words actually mean.

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

So if I was to count in apples, I could say one apple, two apple, three apple, four apple, five apple.

Apple would be the unit.

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.

Here's the structure of today's lesson on adding fractions with the same denominator.

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

We'll start with adding and unitizing.

Here's two friends that are gonna help us along the way.

We've got Sam and Sophia.

Hi guys.

They are gonna help us by chatting through some of the mathematics that we think about, giving us some hints and tips and maybe even helping to improve our understanding of adding and unitizing an equations in generalising.

Okay then, let's get started.

What have you got first for us guys? Sam puts down three counters and Sophia puts down two.

What addition calculation does it show? We have added three counters to two counters.

If we add up the counters, we get a sum of five.

Three and two are the addends.

There they are.

Five is the total or sum, which is formed by adding the addends.

There we go.

Here is a bar model showing the parts and whole.

What addition calculation does it show.

This is the same as the counters.

It shows that three plus two is equal to five.

It also shows that two plus three is equal to five.

Addition is commutative.

So Sophia has swapped around the addends in the equation that she's just said, but the total and sum remain the same.

We could also use five is equal to three plus two.

It doesn't matter which way, equals means has the same value.

The same bar model can be used to represent several other addition contexts.

So this is where we start to bring in the story behind the equation.

Three guinea pigs and two guinea pigs, three guinea pigs and two guinea pigs added together, that makes a herd of five guinea pigs.

Ah, look at them.

Aren't they cute? Three flowers and two flowers added together that makes a bouquet of five flowers.

Three pairs of shoes and two pairs of shoes added together, that makes a pile of five pairs of shoes.

This is an interesting one, isn't it? Because actually there are 10 shoes there altogether, but the unit that we're using is pair and there are only five pairs of shoes.

Three pairs added to two pairs, which equals five pairs.

Three guinea pigs plus two guinea pigs equals five guinea pigs.

Three flowers plus two flowers equals five flowers.

Five pairs of shoes is equal to three pairs of shoes plus two pairs of shoes.

What's the same? What's different between these three equations? They all have the same structure.

They all show three added to two is equal to five.

The units are different for each of them though the unit is guinea pigs in the first calculation, it is flowers and pairs of shoes in the second and third.

Okay, your turn.

Let's see if you understand that concept of units.

You've got to identify the units in each of these additions.

You can see there's a blank space on the right hand side for each of the three equations for you to fill in with a word that shows us what the unit is.

Pause the video here and I'll be back in a minute to give you some feedback.

Welcome back.

Did you manage to spot the units? Well, let's reveal them.

Three bananas plus two bananas equals five bananas.

The units are bananas.

Three fingers plus two fingers is equal to five fingers.

The units are fingers.

Three hats plus two hats is equal to five hats.

The units are hats.

A watermelon is cut into eight equal slices.

Sam eats three of the slices and Sophia eats two of the slices.

How many slices have been eaten altogether? The unit is slices of watermelon.

Well done, Sam, for recognising that straight away.

That's an important part of the learning here.

We have added two melon slices to three melon slices.

If we add them, we get a sum of five slices.

Two slices and three slices are the addends.

There they are.

Five slices is the sum.

There we go.

Sam and Sophia look at the question differently.

It's always good to revisit questions and consider them in new ways.

That's what mathematicians will do.

The watermelon has been cut into eight equal slices.

That means that each slice is 1/8 of the watermelon.

Now the unit we are using is different.

Instead of slice it's 1/8, so we are adding three 1/8 and two 1/8.

Giving a sum of five 1/8.

That's 5/8.

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

Sam and Sophia decide to challenge themselves.

I ate 3/8 of the watermelon.

You ate 2/8 of the watermelon.

Altogether we ate 5/8 of the watermelon.

Yes, that's equivalent to five 1/8.

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

We can compare them to help our learning.

Sam and Sophia use a number line representation to start with.

The watermelon was divided into eight equal slices, so the number line needs intervals of 1/8.

I ate three slices.

That's three lots of 1/8, which is 3/8.

There's that first jump, that represents what Sam has eaten.

You ate two slices, so that's two lots of 1/8, which is 2/8.

There's the jump that represents what it was that Sophia ate.

Altogether, that's 5/8 of a watermelon.

This time Sam and Sophia showed the same addition on a bar model.

They have slightly different versions.

Let's take a look.

What's the same and what's different? So compare those bar models.

They both show the same calculation but slightly differently.

What do you notice? Well, Sam says, "We have both shown the addends, the fraction I ate and the fraction you ate." I have shown each addend as a non-unit fraction.

You've shown them as unit fractions combined.

I have shown the sum as the whole.

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

So both of those bar models are shown the same calculation but in slightly different ways.

This time, Sam Sophia use a circle divided into eight equal parts.

There it is.

Can you see both addends and the sum.

Have a look.

Can you identify those on the representation? The watermelon was cut into eight equal slices, so there are eight equal parts.

Here are the add-ins, the fraction you and I each ate.

Each addend has been shown as unit fractions combined.

So you can see that Sam's section has three lots of 1/8 and Sophia's has two lots of 1/8.

Here is the sum what you and I ate combined.

So the sum wraps around 5/8 of the circle.

Okay, let's check your understanding.

Which representation is showing this addition? A pack of plasticine is divided into 10 equal parts.

Sam uses 3/10 and Sophia uses 4/10.

How much of the plasticine do they use altogether? So there you can see there are two different types of bar model being used.

Which of those represents the problem that I just read? Have a think, pause the video and I'll be back to reveal the answer in a moment.

Welcome back.

The top bar model was the one that was showing the addition in this problem.

Sophia says, this bar model shows both addends.

This is a comparison bar model.

So the bottom bar model was showing us the difference between the plasticine that was used by Sam and the plasticine that was used by Sophia, but the top one shows them both side by side added together.

Alright, it's time for your first task.

There's only one question here, but it's gonna be a bit creative.

I'd like you to represent the additions below in two different ways.

One, A, an apple is divided into six equal parts.

Sam eats two parts and Sophia eats three parts.

How many parts have they eaten altogether? So could you represent that in two different ways? B, a cake, (grunts) cake.

A cake is divided into quarters.

Sam eats one quarter and Sophia eats two quarters.

How many parts have they eaten altogether? So again, represent that in two different ways.

If you find that tricky, think back to some of the different representations you've just seen like bar model, number line, or even splitting a circle up into equal parts.

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

I'll be back shortly with some feedback.

Welcome back.

Let's look at the first one, the apple.

Here's a bar model representation of this question and here is a circle divided into equal parts that represents the question.

You can see that both of these representations are showing that calculation in slightly different ways, but each of them has three lots of 1/6 added to two lots of 1/6.

In the first representation, they are showing the sum as part of a whole.

Same in the second representation here as well.

"They've eaten 5/6 altogether," says Sophia.

Okay, let's look at the second one then, the cake.

Here's two representations that you could have used.

In the first one, you've got a bar model in which the sum is shown as the whole.

In the second one we've got a circle split into equal parts again, and there we can see we've got two lots of 1/4 added to one lot of 1/4 and the sum is being shown as part of the whole.

They've eaten 3/4 all together.

Okay then, let's move on to the second part of the lesson, equations and generalising.

Sam and Sophia revisit the melon context.

There's another representation we could use.

We can write it as an equation.

That's a representation and that's true.

It's tricky to think about, but an equation is also a representation of this context of this calculation.

Okay, so you ate three 1/8 of the melon, so that's 3/8.

I ate two 1/8 of the melon, so that's 2/8.

Altogether we ate five 1/8 is equal to 5/8, so they've taken that representation that they used initially and turned it into an equation here.

Okay, your turn to do just that.

Can you represent this context as an equation? An apple is divided into six equal parts.

Sam eats two parts and Sophia eats three parts.

How many parts have they eaten all together? Pause the video and represent that as an equation.

Welcome back.

Let's see whether or not you were able to represent that accurately using an equation.

2/6 is added to 3/6, which is equal to 5/6.

Is that what your equation looks like? It's possible that you started with 5/6 followed by an equal sign.

It doesn't matter which way round you wrote the expressions.

It also doesn't matter which way round you wrote the addends because they are commutative.

Sam and Sophia turn a bar model into an equation.

The hole has been divided into 10 equal parts.

That means each part is 1/10.

They've labelled them all 1/10.

The first addend is four 1/10.

So that's 4/10.

The second addend is five 1/10 plus 5/10.

Altogether, that's nine 1/10 is equal to 9/10.

4/10 plus 5/10 is equal to 9/10.

Okay then, another check for your understanding.

Can you turn this representation into an equation? Just a quick reminder, this is almost the same as the cake question we had in practise A.

(grunts) Cake.

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

Welcome back.

Here's what the equation might have looked like.

1/4 added to 2/4 is equal to 3/4.

Again, you might have started with the whole here.

You might have started with 3/4 and also you could have swapped around the two addends because addition is commutative.

Sam and Sophia list down all the equations they've written so far.

What do you notice about the sums of the equations? What's the same and what's different? Have a good look.

What do you notice about those sums? In each equation, the denominator is the same in the addends and the sum.

There you go.

You can see that the denominators are all the same.

This is because the denominator is showing the addends and some have the same unit.

The numerator changes in value when we add.

You can see it there.

The numerator has changed.

This is because we're adding together certain lots of the denominator, which is the unit.

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

In these equations there were different units, but the same structure.

The fraction equation has the same structure, but the unit isn't guinea pigs or flowers.

What do you think the unit is in our fractions equation here? The unit is 1/8, which is shown by the denominator of eight.

The unit doesn't change, but the number of units does, this is the numerator.

The numerator tells us how many units there are.

If we add together the units, we get a fraction with a greater numerator.

Sophia and Sam use this generalisation to solve some more additions of non-unit fractions.

I know that the denominator is the unit, so it doesn't change.

So we can complete that for all three already.

Oh, that's a good strategy, Sophia.

There they go.

The denominators are in already.

The pair of numerator for each addend here are the same.

I know that two plus four equals six, so that's the numerator.

There we go.

Six for the numerator in each of these solutions.

Okay, can you solve the equation below? 2/10 added to 4/10 is equal to and there's a box with a missing fraction in.

Pause the video and have a go.

Welcome back.

Did you manage to get this? So we know that the denominator remains the same because it's the unit, so it should have been 10 as the denominator.

I know that two plus four equals six, so the numerator is six.

The answer was 6/10.

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

Sophia and Sam solved some more equations.

I know that the denominator is the unit so it doesn't change.

So she's filled in all of those already.

I know that two plus three equals five, so that's the numerator for the first.

The second is the same, but the expressions have been swapped.

Does that matter? No, because both expressions are still equal.

The third one has three addends, but I can still use what I know.

I know that two plus three plus one is equal to six, so I know the numerator is six.

Okay, have a go at this one then.

There are three addends here.

Solve the equation below.

Pause the video and I'll be back to reveal the answer shortly.

Welcome back.

The denominator remains the same.

It's the unit, so we know that that's 10.

I know that two plus four plus three equals nine, so the numerator is nine.

Did you manage to get that? I hope so.

Sophia and Sam decide to think of a context for the equation below with three addends, so you can see they've got an equation that's already been solved.

6/7 is equal to 2/7 added to 3/7 added to 1/7.

They're now thinking of where that context might come up.

So Sam says, in a week, I jog on two days.

I play football on three days, and I swim on one day.

I exercise on 6/7 of the days of the week.

Well done, Sam.

What a great way to stay healthy.

Let's look at a second context that Sophia's came up with.

I mix a fruit cocktail in a jug.

I fill it with 2/7 pineapple juice, 3/7 orange juice, and 1/7 grape juice.

The jug is now 6/7 full.

Do they make sense? I wonder if you could come up with a different context for these.

Okay, here's your second task.

To begin with, you've got to match the representations to the addition.

So you can see that you've got some equations on the right and remember, those are representations too, and you've got some contexts on the left.

Read them carefully and match them correctly.

For number two, you've got to complete these equations by writing in the sum.

Remember, you might find that the sum appears at the start, it doesn't matter because the expressions can be either way around as long as they equal one another.

And number three, write an adding up context in which you might complete these calculations.

So a little bit like the exercise or fruit cocktail example that you've just heard about.

You've got to write a story, almost a context for each of these.

Okay then, pause the video, enjoy those questions, think carefully whilst you do them, and I'll be back soon to give you some feedback.

Welcome back.

Let's start by marking number one where we were matching contexts and equations.

A pizza is cut into eight equal slices.

Sam eats four slices and Sophia eats three slices.

How much of the pizza have they eaten altogether? Well, you might have been able to recognise this straight away by thinking about the denominator.

If the pizza's been cut into eight equal slices, then the denominator had to be eight, so it matched to the bottom equation.

That of course, left only one equation for the other context to match two, but we still need to double check just to make sure.

An apple is divided into six equal parts.

Sam needs two parts and to three parts.

How many parts have they eaten altogether? Well, the denominator is six and the numerator is needed to be two and three, so it did match to the top one.

Okay, here are the completed sums. Two A was equal to 3/5, two B was equal to 5/8, two C was equal 8/9, two D was equal to 3/4, two E was equal to 8/12, and two F was equal to 7/8.

Pause the video here if you need some more time to be able to mark those carefully, and I will be back in a moment to continue the feedback.

Okay, here's number three.

Now I'm hoping you've got all sorts of weird and wonderful contexts here.

Our characters have had a go at this as well.

So Sam said for the first one, I eat 1/5 of pineapple and Sophia eats 2/5.

Altogether, we've eaten 3/5 of the pineapple.

And for B, Sophia said, "I drink 2/6 of the orange juice from the bottle.

Sam drinks 1/6 and another friend drinks 2/6.

Altogether 5/6 of the orange juice has been had.

They're two of the contexts that they came up with, but you might have caught with some different ones.

Pause the video here and maybe share a few, if you can.

Okay, that's the end of the lesson, but here's a quick summary of the things that we've learned today.

The structure of edition is the same for adding fractions.

There are addends and the sum or total.

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

I've really enjoyed that lesson and I hope you did too.

My name is Mr. Tazzyman.

I'll see you again next time.