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.

Okay, then, let's get started with the outcome.

By the end of the lesson, we want you to be able to say, I can use repeated addition of a unit fraction to form a non-unit fraction.

Here are the keywords that you might expect to hear.

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

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

Ready? My turn, unit fraction.

your turn.

My turn, non-unit fraction, your turn.

My turn, denominator, your turn.

My turn, numerator, your turn.

Okay, let's look at the meanings of each of those keywords.

A unit fraction is a fraction where the numerator is one.

A non-unit fraction is a fraction where the numerator is greater than one.

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 outline then for today's lesson.

You are using repeated addition of a unit fraction to form a non-unit fraction.

And to begin with, we're gonna think about the fact that non-unit fractions are made up of unit fractions.

Then in the second part of the lesson we're gonna look at the repeated addition of unit fractions.

Okay, let's meet a couple of friends.

Here, are Lucas and Laura.

Now, Lucas and Laura are gonna help us today.

They're gonna talk through some of the maths that we're thinking about.

They're gonna respond to some of the prompts, and they might even give us some clues for some of the questions that you might face.

Hi Lucas, hi Laura.

Okay then, let's begin.

A shape is divided into three equal parts and one part is shaded.

"This represents a fraction because it features parts and wholes." "We can write it down using numerals and fraction notation." "So shall I describe the fraction aloud and you can write it down?" There's some teamwork going on here.

Laura's gonna say the fraction aloud, and then Lucas is gonna write it down using fraction notation.

"Yes," he says, "describe the fraction and I'll write the notation.

I'll write it down as you say each stage if you like?" "The whole has been divided into.

." So Lucas has drawn the division bar.

Three equal parts.

Lucas has written three as the denominator.

One of the parts has been shaded.

So Lucas has written the numerator as one.

All done.

What I've written is a symbolic version of what you've said.

A second part of the whole is shaded in.

"This time two of the parts have been shaded in." "Okay, describe the fraction with the change, and I'll write the notation.

I'll write down something for each bit of the description again." "The whole has been divided into.

." "Three equal parts." "Two of the parts have been shaded." "All done!" They compare their notation.

What's the same and what's different? So have a look yourselves.

Look at the fraction notation that Lucas wrote down, and can you see a difference? Can you see something that's the same? Laura says, "Both have been divided into 3 equal parts." You can see the denominator is three for both of them.

"They both have a denominator of three." "The first one has one part shaded and the second has two.

That's why they have different numerators." "A fraction with one as a numerator is a unit fraction." "So what's a fraction with a numerator greater than 1?" "A non-unit fraction." Makes sense.

Okay, time to check your understanding.

I'd like you to sort these fractions into the boxes labelled unit fractions and non-unit fractions.

You can see the fractions there just underneath, and you've got to write those into the correct box.

Are they unit fractions or are they non-unit fractions? Pause the video here and have a go, and I'll be back to reveal the answers shortly.

Welcome back.

Let's see whether you managed to group these fractions into the correct boxes.

These were the unit fractions: 1/2, 1/4, 1/6, and 1/3.

All of them have a numerator of one, which is what makes them a unit fraction.

So that means that 4/5, 3/4, and 5/6 and 2/3 all needed to go in the non-unit fractions box.

All of the numerators here are greater than one.

In an art lesson, Laura and Jacob are experimenting by printing shapes using blocks.

Sounds fun.

Laura has a hexagon and Jacob has a triangle.

"I'll print a hexagon first." There it is.

Nice choice of colour.

"I'll lay a triangle over the top of it." "How many triangles could you fit into my hexagon?" Have a look at the screen and have a think about that.

How many of those green triangles could you lay over the top of the yellow hexagon? "Let's see," says Lucas.

"That was six altogether." "What fraction of the hexagon is one triangle?" "Well, the hexagon fits six.

A denominator of six.

"So one triangle is 1/6 of the hexagon." They create a new print.

Each triangle is still 1/6, but there are three triangles.

"I'm going to count the triangles as fractions: one 1/6, two 1/6, three 1/6." "I'll count them too," says Lucas.

"1/6, 2/6, 3/6." What do you notice? Compare the counting of each of our maths friends.

What's the same and what's different between them? Lucas says, "We counted in different ways." "Yes, my counting shows that each part is 1/6.

"My counting shows the running total of those one-sixths." So Lucas is actually keeping a running total, which means he, as he goes, is adding up all the one-sixths that have gone before, and he's giving the total of them.

"So there are three one-sixths in three-sixths." They create another new print.

What do you notice? Hmm, looks a bit different to the ones that we've had previously.

Why is that? The triangles are arranged differently, but each is still 1/6.

"I know this because they've not changed size." One one-sixth, two one-sixths, three one-sixths.

That was the same as last time; the counting is the same.

"I'll count them again as well, but in my way," says Lucas.

And remember, Lucas was the one who was keeping a running total as he counted.

"1/6, 2/6, 3/6." "My final number is the non-unit fraction, 3/6." The numerator in 3/6 is greater than one, so it must be a non-unit fraction.

Make sense? "Yes," says Laura, "Mine shows how many unit fractions there are." "It doesn't matter where the 1/6 are placed." "There are still three 1/6 in 3/6." Okay, your turn.

We've got a counting diagram here with both the different types of counting that we just heard from Lucas and Laura.

One of them where you're counting the number of unit fractions and the other where you're keeping a running total by saying the non-unit fractions.

Your job is to label the missing parts on this counting diagram.

Okay, pause the video and have a go, and I'll be back with some feedback shortly.

Welcome back.

How did you get on? I hope it made sense.

Let's see what the labels were.

We had one 1/6, which is 1/6, and two 1/6, which is 2/6.

Did you get that? Okay, let's carry on with the lesson.

Laura makes a conjecture.

A conjecture is when you make a statement based on limited evidence or information.

She's saying what she thinks is true, but she knows that she doesn't have enough evidence yet to say for sure.

That's what a conjecture is.

Well, here's hers.

"I think that any non-unit fraction is made of unit fractions." She writes down the conjecture so she can keep it in front of her while she starts to experiment.

Lucas says, "Let's test it out using different contexts and denominators." Okay, we've got a diagram in front of us here.

The whole is made up of seven equal parts." "So the parts are all 1/7." "Let's count the shaded parts! You first." They're gonna do their different counting styles again.

One 1/7, two 1/7.

"Your turn, count the running total." "1/7, 2/7." "So, 2/7 is made of two lots of 1/7." "It proves your conjecture." They've got some evidence.

Okay, they're gonna need a little bit more than just one piece though, to make sure that they can say the conjecture is true.

Some pandas.

Looks like some of them have been having a party.

"There are 10 pandas altogether here." "So each panda is one-tenth of the whole group." "Four of them are wearing hats, so let's count them." "I'll go first then: one 1/10, two 1/10, three 1/10, and four 1/10." "1/10, 2/10, 3/10, 4/10." They've used their different counting styles again.

Lucas has kept that running total, whilst Laura has said how many unit fractions there are.

"4/10 of the group are wearing hats." They're great hats as well.

"Each panda is 1/10 and there are four wearing hats." "So 4/10 is made of four lots of 1/10." It proves the conjecture again.

More evidence, fantastic.

Okay, we've got a glass here.

I wonder what they're gonna do with this.

"Let's fill this glass up 1/4 at a time with squash." "Okay, and then we can count at the same time as we go." So this time they're not gonna take it in turns to count.

They're gonna try and count together.

"I'll start with one 1/4, one 1/4." "1/4." "And another 1/4, so two 1/4." "2/4." "Another 1/4, so three 1/4." "3/4." Let's stop there.

We don't want anything to overflow, and if they put in another quarter, they'll have a full glass and 1/4 after that, well, it would be overflowing and nobody wants a spillage.

"Okay, the glass is 3/4 full." "Which is three lots of 1/4." More proof, more evidence.

Fantastic! Okay, it's your turn to do some practise tasks now.

For number one, what I'd like you to do is complete the labels for the following representations with the two types of counting.

We have the counting of the non-unit fraction and the counting of the unit fraction.

For number two, I'd like you to draw or make your own representations with counting labels for each of the following non-unit fractions: For A, there's 2/5, B, is 3/5, and C is 3/6.

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

I'll be back in a little while to give you some feedback.

Good luck.

Okay, welcome back, let's get some feedback.

We'll start with number one.

There's the first representation with the labels completed: One 1/6 was 1/6, two 1/6 was 2/6, three 1/6 was 3/6.

Now, let's look at the second one.

We had one 1/5 was 1/5, two 1/5 was 2/5.

Okay, pause the video here so you can have a little bit of time to mark those and discuss any misconceptions or any other issues.

So here's number two.

I hope you've got lots of wonderful representations and creations in front of you.

I'm gonna show you what our characters came up with.

A was 2/5.

So here's an example: One 1/5, 1/5, two 1/5, 2/5.

Here's an example for B with 3/5.

Same kind of diagram, but this time, instead of having 2/5 shaded in, there was 3/5 shaded in.

Here's C, 3/6.

This time our characters decided that they were gonna use cubes, and they had six cubes making the whole, which meant that each cube was worth 1/6, and they shaded three of them in.

One 1/6, 1/6, two 1/6, 2/6, three 1/6, 3/6.

Fantastic.

Okay, we've done the first part of the lesson, and now it's time to move on to looking at the repeated addition of unit fractions.

Here we go.

Laura and Lucas decide to use a number line to represent their counting.

A good idea.

It's always worthwhile using different representations to strengthen your understanding.

There's that yellow hexagon again and there are some of the triangles.

Six triangles would fit into the hexagon.

Each triangle is 1/6.

So the number line should go to 6/6.

There we go, they've labelled their number line.

Let's count together this time.

One 1/6, 1/6.

Two 1/6, 2/6.

Can you see what's happening on the number line each time? Three 1/6, 3/6.

"I think we could write this as an equation using repeated addition." Okay, time to check your understanding of the number line so far.

What's the mistake in the number line below? Why do you think the mistake has been made? They've got their representation there and then they put the jumps on the number line.

Pause the video here and discuss this with somebody near you.

Welcome back, let's see what our characters thought, and whether that was similar to what you thought.

Laura says "The addends on the number line aren't adjacent." That means they're not right next to one another.

"They tried to follow the pattern of triangles." So you can see that it goes, green triangle, no green triangle, green triangle, no green triangle, green triangle.

It makes a nice symmetrical pattern as an art print, but is it right on a number line? The position of parts doesn't matter on a number line, so that's where they went wrong.

Laura turns the number line into an equation with repeated addition.

"I have added together 1/6 three times.

So my first addend was 1/6.

Then I added another 1/6, then I added a third 1/6.

The total was 3/6." Laura and Lucas look back at the pandas in hats.

I think they're still having a good time.

"I'm gonna make this an equation straight away without a number line." "Okay, there are 10 pandas altogether in a group." "Each panda is 1/10 of the whole." "We know that a non-unit fraction is made up of unit fractions." "Let's find the pandas with hats." There they are, four of them, so we are adding together four lots of 1/10." "Here's the equation using repeated addition." And there it is.

You can see the equation there: Four lots of 1/10, and that makes 4/10.

The non-unit fraction is 4/10.

Now, they're looking at that squash again.

"This glass was filled with 3/4 of squash." "Let's write out the repeated addition equation here and compare." So you've got 1/4, added to 1/4, added to 1/4, is equal to 3/4.

Or 3/4, is equal to 1/4, added to 1/4, added to 1/4.

These equations look slightly different.

What's the same? What's different? Have a think.

"We both have an expression with repeated addition." "Yes, three lots of 1/4 have been added together.

Our expressions are in a different order.

I've started with 3/4." "That doesn't matter so long as the expressions are equal." Okay, let's check your understanding again.

Show this fraction as an equation using repeated addition.

Pause the video, write it down, and I'll be back shortly to reveal what it should look like.

Welcome back.

Let's see what this equation should look like.

There it is: 1/5 added to 1/5 added to 1/5 is equal to 3/5.

And it was like that because we had a whole that had been split into five equal parts.

That meant that the denominator was five.

Each of those shaded parts was worth 1/5 of the whole, and there were three of them.

So we needed to add together three lots of 1/5, which is equal to 3/5.

Lucas and Laura play a game called fifths with two other friends, Alex and Andeep.

They take it in turns to guess what non-unit fraction has been created from unit fractions of 1/5.

Each of them can choose to put in 1/5 or nothing, which they show with a raised finger for 1/5 or a closed fist for nothing.

There you go.

This would be 2/5, because there are two fingers raised, so two lots of 1/5.

There it is as our repeated addition equation.

Player one starts by counting aloud the maximum number of 1/5 that there could be starting with themselves and pointing at the other players.

One 1/5, two 1/5, three 1/5, four 1/5.

Then they say, ready, steady.

and instead of go, they estimate the non-unit fraction they think will appear.

At the same time, all the players choose to put in 1/5 or zero.

Ready, steady.

3/5.

The object of the game is to be out.

The last person remaining loses.

We've got here: 1/5 + 1/5 + 1/5 = 3/5.

"I said 3/5," says Lucas.

What came up matches what he estimated.

"I was correct.

I'm out." So, Lucas has managed to escape.

One 1/5, two 1/5, three 1/5.

Ready, steady, 1/5.

1/5 added to 1/5 is equal to 2/5.

Bad luck, Laura.

"This is 2/5, I stay in." One 1/5, two 1/5, three 1/5.

Ready, steady, 2/5.

1/5 added to 1/5 is equal to 2/5.

"I got it right," says Alex, "I'm out." One 1/5, two 1/5.

Ready, steady, 1/5.

"No need for any addition here." "I'm out," says Andeep, "I lose but I really enjoyed playing," says Laura.

Great attitude, Laura.

Okay, it's time for your practise task.

In the first question, you need to match the representations and equations by drawing a line between each column.

You can see the representations on the left-hand column and the equations on the right.

Draw a line between the ones that match.

For number two, you're gonna turn each of the following representations into an equation featuring repeated addition.

You've got A there, and B, with the pandas again.

And Lucas says, "What do you notice?" So you're gonna need to write down an explanation of what you notice when you compare A and B answers.

For three, you need to complete the missing numbers in these equations.

Think carefully about them, and look at what you've already got and what you know.

For number four, I want you to have a go at playing fifths with some learning buddies.

And Laura asks a really good question: "Can you use other denominators?" Does it have to be fifths? Could you play with a different type of denominator? Okay, pause the video here and have a go at those.

I hope you enjoy them.

I'll be back shortly with some feedback.

Welcome back, ready for some feedback? Ready to mark? Okay, we'll start with number one, and it was quite a tricky one in the first part of number one.

The top representation matched the bottom equation, which was 3/6 = 1/6 + 1/6 + 1/6.

Now, it was a little bit cheeky in the sense that not all of the shapes you could see in that hole was 1/6, but the three green ones that were shaded in were all 1/6 of the shape.

You could have fit six of those green rectangles into the whole.

The second representation was 1/5 + 1/5 + 1/5 = 3/5.

And the last one then, was 1/8 + 1/8 + 1/8 = 3/8.

Okay, if you need to discuss any of those further, pause the video here.

I'll be back in a little while to finish off the rest of our feedback.

Let's look at number two then.

So here we had two A.

The equation was 1/5 + 1/5 + 1/5 = 3/5.

Now let's think about the pandas.

It was the same: 1/5 + 1/5 + 1/5 = 3/5.

So what do you notice? Well, although the representations are very different, they are both showing the same fraction and have the same equation.

Here's number three, complete the missing numbers in these equations.

For A, we have 4/5 is equal to 1/5 + 1/5 + 1/5 + 1/5.

There were four lots of 1/5 there.

For B, we had 2/3 is equal to 1/3 added to 1/3.

Okay, let's look at number four then, which I hope you enjoyed.

Our characters had a little go.

They played with some different denominators and found that it worked, "But we had to use five or more so we didn't reach the whole." All right, that brings us to the end of this lesson.

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

Non-unit fractions are made up of unit fractions.

This could be shown using two different methods of counting and represented on a number line.

This can also be represented by using repeated addition of unit fractions to create a non-unit fraction.

The position of the parts within the whole does not affect how the addends are written on a number line or in a repeated addition equation.

My name is Mr. Tazzyman, and I have really enjoyed learning with you today.

I hope you enjoyed playing the game, and I hope that you feel really confident about fractions.

I'll see you next time.

Bye-bye.