Lesson video

Hi there.

My name is Mr. Tazzyman, and I'm really looking forward to teaching you this lesson today from the unit all about the composition of non-unit fractions.

Some people can find fractions a bit tricky sometimes, but I hope, with what I'm about to show you, I can make sure that you finish this lesson feeling really confident and ready to tackle any fractions that come your way.

Okay.

We're gonna get started.

Here's the outcome, then.

By the end of the lesson, we want you to be able to say, "I can add fractions with the same denominator and generalise the rule." The key words are listed here.

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

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

Are you ready? My turn, generalisation.

Your turn.

My turn, denominator.

Your turn.

My turn, numerator.

Your turn.

And here's what those words mean.

A generalisation is a statement or rule that applies correctly to all relevant cases.

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, then.

We're gonna start by looking at generalising, and then we're gonna use the generalisation for reasoning.

We'll start with generalising.

And here are two friends that we're gonna meet along the way.

Hi, Aisha.

Hi, Izzy.

They're both gonna help us here by discussing some of the maths prompts that we see on screen and giving us some hints and tips when it comes to working out some of the concepts in this lesson.

Let's start, then.

A watermelon is cut into 10 equal slices.

Aisha eats three of the slices and Izzy eats two of the slices.

How many slices have been eaten altogether? We have added two melon slices to three melon slices.

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

Makes sense? Two slices and three slices are the addends, five slices is the sum.

Aisha and Izzy look at the question differently.

Always a good idea.

Can you see something different? The watermelon has been cut into 10 equal slices.

"That means that each slice is 1/10 of the watermelon." They've been labelled there.

"Now the unit we are using is different.

Instead of slice, it's 1/10." "So we are adding three 1/10 and two 1/10, giving a sum of five 1/10." There's the sum.

"That's 5/10." Aisha and Izzy use a number line representation.

"The watermelon was divided into 10 equal slices." "So the number line needs intervals of 1/10." And remember, an interval is the gap between the marks on the number line.

They're all the same size, but it's the gap that we're considering when we say interval.

"I ate three slices." "That's three lots of 1/10, which is 3/10." So they put that jump on the number line.

"You ate two slices." "So that's two lots of 1/10, which is 2/10." They put that jump on the number line.

"Added up, that's 5/10 of the watermelon." You can see that it's been highlighted there.

They use the number line to form an equation.

Remember, an equation is another type of representation that you can use.

"Can we turn this number line into an equation?" "I think so.

We have two addends, 3/10 and 2/10." 3/10 plus 2/10.

"The total is 5/10." Is equal to 5/10.

Okay.

It's your turn then to check that you've understood what we've done so far.

I'd like you to represent this number line as an equation.

Pause the video here, have a go, and I'll be back in a little while to reveal the equation.

Welcome back.

Here's the equation that this number line was representing.

4/10 added to 3/10 is equal to 7/10.

You can see there are two jumps on the number line, and those two jumps are the addends, 4/10 and 3/10.

And then the total has been shown by drawing around it a shape, 7/10.

Izzy has helped to grow a sunflower.

I like sunflowers.

They bring a lot of happiness to the world, don't they? First, the sunflower was 3/10 of a metre.

Then, over time, the sunflower grew by 3/10 of a metre.

How tall is the sunflower now? We've got ourselves a maths question here.

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

That means that each part is 1/10.

The first addend is 3/10 of a metre," so she writes down 3/10.

"The second addend is also 3/10 of a metre." Added to 3/10.

"Now the sunflower is 6/10 of a metre," so she writes the equal sign and 6/10.

That equation is a representation of this context, and it's given us the solution of 6/10.

Your turn.

Can you represent and solve this problem using an equation? It's a sunflower problem again.

First, the sunflower was 2/10 of a metre.

Then, over time, the sunflower grew 4/10 of a metre.

How tall is the sunflower now? Pause the video and write that out as an equation with the total.

Welcome back.

Let's reveal the equation.

2/10 plus 4/10 is equal to 6/10.

The 2/10 represents the first part of the context, the 4/10 represents the then part of the context, and the 6/10 is our now part.

Did you get it? Time to move on, then.

Aisha and Izzy list 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? So have a look here at all of those equations.

Compare the numerators, compare the denominators, what do you notice? Well, Aisha says, "In each equation, the denominator is the same in the addends and the sum." She's drawn boxes around to show us.

You can see that they're all tenths.

"The numerator changes through addition." And again, she's drawn boxes.

In the first equation, you've got 3 and 3 is equal to 6, then 3 and 2 is equal to 5, 4 and 3 is equal to 7, and 2 and 4 is equal to 6.

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

That's a good example of a generalisation, and one that we are gonna use going forward.

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

Izzy challenges Aisha's generalisation.

"So far ,we have only shown this generalisation using tenths.

All the denominators are 10." There they are.

"Is this generalisation true for other denominators?" That's a good thought, Izzy.

It's always worth trying to prove generalisations or conjectures incorrect, that way you can learn more.

Izzy tests the generalisation using other fractions.

Ann apple is divided into six equal parts.

Izzy eats two parts and Aisha eats three parts.

How many parts have they eaten altogether? I will model this on a number line and an equation.

The first addend is 2/6, so there's a jump on the number line of 2/6.

And Izzy has also written down the fraction notation underneath: 2/6.

"Then it's 3/6," so she's done another jump on the number line and added on 3/6.

"Altogether, there have been 5/6 eaten," so it's equal to 5/6.

"The denominator is the same." The generalisation is true for other fractions as well.

And although she's only tested tenths and sixths, I'm sure you'll see through this lesson that it is true for all the denominators.

Okay, your turn.

What is the missing number in the equation below? Have a look, think about our generalisation, and I'll be back in a little while, after you pause the video, to give you some feedback.

Welcome back.

What number do you think went into the box? It was 8 because the denominator stays the same when we're adding fractions.

"When adding fractions, the numerator changes, but the denominator is the same." Aisha challenges her own generalisation.

Oh, good.

Another challenge.

"So far, we have only shown this generalisation with two addends.

"Is this generalisation true for addition calculations with three addends?" Good thinking, Aisha.

Aisha revisits the melon context to test three addends.

"Let's imagine that another person had eaten two slices as well.

That would make a 3rd addend of 2/10.

Altogether, that would be a total of 7/10.

Each slice is still 1/10, so the denominator stays the same." That means that the generalisation is still true, but it's good of Aisha and Izzy to have tested it out just in case.

Okay, have a look here.

What mistake has been made in the calculation below? Pause the video, think about it, and then return when you are ready for some feedback.

Welcome back.

Did you spot it? "The mistake is to add together both the denominator and the numerator of the addends.

The denominator of the sum should be the same as the addends.

In this case it should be an 8 because the unit is 1/8.

When adding on fractions, the numerator changes, but the denominator is the same." Okay, it's time for your first task.

You've got to complete these equations by filling in the missing numerator or denominators.

Can you see all the question marks? They're the parts that you need to complete.

And for question two, for each of the following, work out how many ways you can complete them.

Can you see the question marks again? Those are the parts that you can change, and you can put in numbers that you want to use.

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 look at number one first.

So for A, you should have put fifths in for those denominators.

For B, it should have been 3 for the first numerator and then sixths for the denominators.

For C, the denominators should have been ninths and the missing numerator was a 2.

For D, the missing numerator was a 3 and the denominators were twelfths.

And for E, the missing numerator was 14 and the denominators were fifteenths.

For F, the missing denominators were twentieths and the missing numerator was 18.

These can be quite tricky sometimes, but when you look at it, you're thinking about using the inverse to calculate some of the missing numerators because we know that the numerators are the parts that add together and the denominators stay the same.

All right.

Pause the video here for extra time in case you need it for marking.

Welcome back.

We're on to number two now.

We had to work out how many ways we could complete these.

So for A, the numerator pairs could have been 1 and 3, 3 and 1, or 2 and 2, and that's because all of those are equal to 4.

The denominator was staying the same no matter what.

The numerator pairs for B were 1 and 4, 4 and 1 , 2 and 3, and 3 and 2.

They all added together to make five.

And for C, there was 1 and 5, 5 and 1, 4 and 2, 2 and 4, and 3 and 3.

They all added together to make 6.

You could also have chosen to use for A, 0 and 4; for B, 0 and 5; and for C, 0 and 6, because as a pair they do create the numerators.

However, 0/7, 0/7, and 0/7 are all equal to 0/7.

However, 0 wasn't including the answers here because 0/7 sevens is equal to 0.

But if you've included them, they're still correct.

Now it's time for the second part of the lesson: using a generalisation for reasoning.

Aisha and Izzy play correct or incorrect.

They look at some matching representations and equations and discuss whether they are correct.

So we've got here 1/2 added 1/2 equals 2/4.

What do you think? Do you think that equation and representation match? "The representation shows two 1/2," says Aisha.

"So does the expression with both addends." "Two lots of 1/2 make a whole." "The denominator shouldn't change." "The sum in the equation is incorrect." "Because this is a whole, the numerator and denominator should be the same." Here's another one.

4/10 plus 3/10 is equal to 7/10.

And there's the representation.

What do you think? Do you think that's correct or incorrect? "The equation follows our generalisation." "The numerators are added and the denominator the same." "The equal parts making the addends aren't adjacent in the representation," meaning they're not next to one another, they're all jumbled up.

"They don't have to be.

There are still 4/10 and 3/10." You can count them and see them there.

Here's another one.

2/4 plus 2/4 equals 4/8, and there's the representation underneath.

What do you think? Do they match? Well, "The equation is incorrect," says Aisha.

"The denominator should be 8 for the addends as well." That's because the whole in the representation has been split into eight equal parts, but the addends are showing that the whole had been split into four equal parts, so they were incorrect.

What about this one? 3/9 plus 2/9 is equal to 5/18.

What do you think? Have they got the correct number of parts? "The total is incorrect.

The denominator should be the same." "Yes, it should be 5/9." You can see in that representation it's been split into nine equal parts.

And five of those parts have been shaded, three parts and two parts, five of those parts, but they should be 5/9.

"The representation is correct." "I agree, even though the whole is an unusual shape." "I wouldn't choose this representation myself." Wow, look at this one.

3/9 plus 2/9 equals 5/9.

What do you think? Well, "The equation is correct." "Yes, the numerate have been added and the denominator is the same." "There's something wrong about the representation, though." "The whole hasn't been split into equal parts." 1/12 plus 2/12 plus 5/12 equals 8/12.

There's the representation.

What do you think? "The equation representation are correct." "I agree," says Izzy.

You can see that the whole has been split into 12 equal parts, and we've got 1/12, we've got 2/12, and we've got 5/12 represented by the different colours.

Altogether, that is equal to 8/12.

What about this one? 1/12 plus 2/12 plus 5/12 equals 2/12 plus 5/12 plus 1/12.

Hmm, not sure about this one.

"These are two additions.

Where is the total?" says Aisha.

"An equation doesn't always show a total.

It shows us two or more expressions that are equal to one another.

The representation shows us the first expression." "I see.

I will rearrange this representation to make the second expression." There it is rearranged.

2/12 to begin with, 5/12 in the middle, and 1/12 at the end.

Okay, let's check your understanding.

True or false? Fraction addends are commutative.

Pause the video here, decide whether you think that's true or false.

Welcome back.

The first one was the best justification.

Fractions behave in the same way as whole numbers, so can be rearranged in an addition equation.

Aisha looks at an equation with some unknown numbers.

"I'm gonna start with what I know," she says.

"One of the expressions is complete," and she circled it.

"I'll calculate this first." 2/7 plus 3/7 is equal to 5/7.

"The two missing numbers must total 5.

I know that 1 plus 4 equals 5, so I'll use these." 1/7 plus 4/7.

Izzy says, "I think there are more ways.

We need any two numerator that total 5.

So it could be 2 and 3.

Also, addends are commutative.

So it could be these as well": 3 and 2, 4 and 1, or 1 and 4.

"Hang on.

What about zero?" And we mentioned this in the first half of the lesson.

"You can have 0/7.

That's equivalent to 0.

So it could be 0 and 5 or 5 and 0 as well." There they are, the other solutions.

Izzy looks at a new missing number problem, this time it features an inequality.

12/18 added to something-eighteenths is less than 17/18.

"I'm gonna find all the solutions here," says Izzy.

"I know that 12 plus 5 equals 17, so it can't be as much as 5.

That leaves 1, 2, 3, and 4.

I'll try each." 12/18 plus 1/18 equals 13/18, which is less than 17/18.

That's correct.

12/18 plus 2/18 equals 14/18, which is less than 17/18.

That's correct.

12/18 plus 3/18 is equal to 15/18, which is less than 17/18.

That's correct as well.

12/18 plus 4/18 is equal to 16/18, which is less than 17/18.

That's also correct.

"You've forgotten about zero again," says Aisha.

"I know that 12 plus 0 is less than 17." so she's written out the equations and the inequality: 12/18 plus 0/18 equals 12/18, which is less than 17/18.

That's correct.

Don't forget zero.

Okay, time for your second practise task.

Using the generalisation when adding fractions with the same denominator, the numerator changes, but the denominator stays the same, spot the mistake in the matching equation and representations below.

For number two, for each of the missing number problems below, find all the possibilities.

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

Welcome back.

Let's start with 1 A.

We had 2/5 added to 1/5 is equal 3/10, and then we had a representation underneath.

Did you spot any mistakes in this? Izzy said, "The denominator of the sum is incorrect.

The unit is 1/5 here, not 1/10." Okay, let's look at B.

We had 2/6 added to 4/6 is equal to 6/12.

"The denominator of the addends is incorrect.

The representation shows that the whole has been divided into 12 equal parts, so all the denominators should be 12." Let's look at number two, then.

We'll start with A.

You could have had 2 and 1 or 1 and 2, 0 and 3 or 3 and 0.

And that's because the numerator needed to total 3 in order to make sure that these two expressions were equal and that the equation was balanced.

Let's look at B.

Here we had to make sure that the numerators totaled 9.

So we could have had 0 and 9 or 9 and 0, 1 and 8 or 8 and 1, 7 and 2 or 2 and 7, 6 and 3 or 3 and 6, 4 and 5 or 5 and 4.

Let's look at C now.

Here you could have had 0, 1, 2, 3, 4, or 5, and that's because we needed to make sure that the sum of the numerators were less than 24.

Then D.

0, 1, 2, 3, 4, 5 or 6.

And here we had to make sure that the numerator was less than 7.

We knew we were adding 3 to them and we knew that the total of that addition needed to be less than 10.

Okay.

That brings us to the end of the lesson.

I hope you enjoyed it.

Here's a summary.

The structure of addition is the same for adding fractions.

There are addends and a sum or total.

By analysing addition equations featuring fractions, you can generalise that when adding fractions with the same denominator, the numerator changes but the denominator is the same.

This generalisation can be used to reason about the addition of fractions.

My name's Mr. Tazzyman, and I really enjoyed that lesson.

I hope you did too.

And I'll see you again soon.

Bye bye.