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.

The outcome for today's lesson is that we want you by the end to be able to say, "I can use repeated addition of a unit fraction to form 1." We've got some keywords here that you might expect to hear during the lesson.

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 I'll say your turn, and it'll be your go, 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, here's an explanation of all of those keywords.

A unit fraction is a fraction where the numerator is 1.

A non-unit fraction is a fraction where the numerator is greater than 1, makes sense, doesn't it? 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 what we're gonna expect to be learning about then in today's lesson.

First of all, we're gonna think about the numerator, the denominator, and the whole, and secondly, we're gonna look at a strategy game using unit fractions to form 1.

Let's start with the first part.

In this lesson, there's a couple of friends that are gonna help us along the way, we've got Alex and Aisha.

These two are gonna be chatting about the maths, giving us some prompts, and giving us perhaps some hints and tips to get the correct answers.

Okay then, let's see where they started.

Alex and Aisha look at a hexagon with triangles printed onto it.

"The triangle could fit into the hexagon six times." "So each triangle is worth one-sixth, a unit fraction." Why is that a unit fraction? Well, the numerator is 1.

"There are three printed triangles." "So there are three lots of one-sixths." They've been written on in fraction notation onto the green triangles.

"Let's add them together," says Alex.

One-sixth added to one-sixth added to one-sixth is equal to three-sixths.

This shows there are 3 one-sixths in three-sixth.

Alex shows the equation on a number line One one-sixth is equal to one-sixth.

Two one-sixths are equal to two-sixths.

Three one-sixths are equal to three-sixths.

This non-unit fraction is made up of unit fractions.

Okay, let's check your understanding on that.

True or false, a non-unit fraction is made up of unit fractions.

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

Welcome back, what did you think? That was true.

Okay, but whenever we have a truth like that, we need to think about the justifications for our truth as well.

So let's justify your answer.

Was it that we can count in unit fractions and end on non-unit fractions, or was it that unit fraction is in the phrase non-unit fraction, so it must be true? Pause the video and decide which of those justifications you think is best.

Welcome back.

A was the best justification.

We can count in unit fractions and end on non-unit fractions.

Okay, then let's move on.

Alex and Aisha practise counting in unit fractions using different representations.

One one-sixth.

One-sixth.

Two one-sixths.

Two-sixths.

Three one-sixths.

Three-sixths.

Four one-sixths.

Four-sixths.

Five one-sixths.

Five-sixths.

Six one-sixths.

Six-sixths.

The whole is completely shaded.

One whole is six-sixths.

Here's another representation and they do the same thing again.

One one-fifth, one-fifth.

Two one-fifths, two-fifths.

Three one-fifths.

Three-fifths.

Four One-fifths, four-fifths.

Five one-fifths, five-fifths.

Once again, the whole is shaded completely.

One whole is five-fifths.

Here's a third representation to look at.

One one-quarter, one-quarter.

Two one-quarters.

Two-quarters, Three one quarters, three-quarters.

Four one-quarters, four-quarters.

That forms the whole.

One whole is four-quarters.

Alex and Aisha compare their representations after counting.

There's all three of them.

What do you notice? So have a look at all of those representations, What do you notice, what do you see? What's the same and what's different between them? "I have noticed that all of them are one whole." "I notice that the denominator and numerator are the same." Okay then, time to check your understanding.

True or false, when the numerator and denominator are the same, the fraction is equivalent to one whole? Pause the video and decide whether you think that's true or false.

Welcome back, that was true.

So when the numerator and denominator are the same, the fraction is equivalent to one whole, but now it's time to justify our answer, and here are two justifications.

A, the parts of the numerator cancel out the parts of the denominator, so it is one whole, or B, if the numerator and denominator are the same, then all of the equal parts are shaded making one whole.

Decide which of those you think justifies our statement.

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

Welcome back, B was the best justification here.

If the numerator and denominator are the same, then all of the equal parts are shaded making one whole.

Alex and Aisha both show their understanding of six-sixths in different ways.

"I'll use a repeated addition equation." There you go.

One-sixth added to one-sixth, added to one-sixth, added to one-sixth, added to one-sixth, added to one-sixth is equal to six-sixth, "I'll create a number line." One-sixth added to one-sixth, added to one-sixth, added to one-sixth, added to one-sixth, added to one-sixth, is equal the six-sixths.

Both of these are ways of representing their understanding.

"Six-sixths is one whole.

"I'll show this as well.

Hmm, she's changed the equation slightly there.

She's put another expression in.

Six-sixths is equal to 1.

"I'll substitute six-sixths with 1," says Alex, and he does just that.

Aisha links her equation to a real-life example.

"This is like a box of eggs being filled." One-sixth added to one-sixth, added to one-sixth, added to one-sixth, added to one-sixth, added to one-sixth, equals six-sixths.

"At the end, there is one full box." So six-sixths is equal to 1.

Okay, then, another true or false for you to check your understanding.

When the numerator and denominator are the same, the fraction is equivalent to 1, what do you think true or false? Pause the video, and I'll be back to reveal the answer in a moment.

Welcome back, that was true.

When the numerator and denominator are the same, the fraction is equivalent to 1.

Let's pick a justification.

A, one whole has a value of 1 and one whole is formed when the numerator and denominator are the same, or B, if all the parts of the whole are shaded, then the numerator and denominator are equal, so which of those do you think best justifies the statement that when the numerator and denominator are the same, the fraction is equivalent to one? Pause the video and decide.

Welcome back, A was the best justification here.

One whole has a value of 1 and one whole is formed when the numerator and denominator are the same.

B was close, but it didn't touch on the fact that a whole has a value of 1.

It's time for your first task.

For Number 1, you need to match the wholes to the repeated edition equations.

So you've got three equations there, and you've got three representations on the right.

You need to draw a line between the ones that you think match, and then we've got Alex saying, "What does the 1 mean for each?" For Number 2, we've got this.

The number line below represents the repeated edition of a unit fraction totaling 1.

Complete the number line by labelling the numbers and addends.

For Number 3, I'd like you to think about this.

How many ways can you complete the following? Okay, pause the video here, have a go at those, and I'll be back shortly with some feedback.

Welcome back, let's do the answers for Number 1.

We're gonna match the wholes to their equations to begin with.

So we can see that we had the first equation, which was repeated edition of ten-tenths, that matched a pack of 10 felt tips.

The second equation, we had three lots of one-thirds, matched our watermelons, which meant that the pizza matched our bottom equation, four lots of one-quarter.

So what does the 1 mean for each? Well, on the first one it was one pack because it was one pack of felt tips.

On the second, it was one crate because we had crates of watermelon, and on the third one it was one pizza.

Okay, then, let's move on to looking at Number 2.

So we had to complete the number line here.

We were adding on one-seventh with each jump on the number line, and we needed to label the number line with sevenths going up.

One-seventh, two-sevenths, three-sevenths, four-sevenths, five-sevenths, six-sevenths.

You'll notice that there isn't seven-sevenths, that's because seven-sevenths is equal to 1.

Okay, Number 3, how many ways can you complete the following? "Here are some of my solutions," says Aisha, "Three-thirds, five-fifths, ten-tenths.

There are an infinite number of solutions here." As long as the numerator and denominator match, then you can represent 1 in any way you like.

"Yes, so long as the numerator and denominator are equal," says Alex.

"How about this one?" 342, 340ths, ah, wow! Yes, that would work because the numerator and denominator are the same.

Okay, that's the first part of the lesson done.

We're gonna move on now to looking at a strategy game using unit fractions to form 1.

Aisha and Alex play a game using repeated addition of one-sevenths to form a whole.

They take it in turns to place down counters.

Seven counters are equal to 1 whole, so each counter is worth one-seventh.

On their turn, they can choose to either place one-seventh or one-seventh, and another one-seventh, which we know is equal to two-sevenths.

The game is won by the player who places the last one-seventh to complete the whole.

"I'll start, one-seventh." "I'll put down one-seventh and another one-seventh." "Okay, three-seventh so far, made of three one-sevenths.

Hmm, I'll put down one-seventh again." "My turn, I'll use one-seventh and another one-seventh." "I'll put down one-seventh to finish the whole, I win!" Well done, Aisha.

Okay, let's check your understanding.

Look at this game board.

If it was your turn next, what could you do to win? Pause the video and have a chat about that.

Welcome back, what do you think the winning move was? Well, let's have a look.

"Put down one-seventh and another one-seventh to win!" There we go.

Instead of using counters, Aisha and Alex use a number line to play the game.

Oh, good, a change of representation, that always deepens understanding.

"I'll go first this time.

I'll start with one-seventh." "I'll put down one-seventh." "One-seventh and another one-seventh." "One-seventh and another one-seventh." "One-seventh.

Seven one-sevenths are equal to 1, I win," says Alex.

This time, Aisha and Alex write the game as a repeated addition equation, yet another representation, brilliant.

"My turn to start.

One-seventh and another one-seventh." There's what it looks like.

"One-seventh," says Alex.

"One-seventh and another one-seventh." Oh, it's getting close now.

"One-seventh and another one-seventh." "Ah, seven one-sevenths added together are equal to 1.

I win, says Alex." Oh, he's good at this, isn't he? Aisha makes a conjecture about the game, and remember that a conjecture is when you make a statement about something without much evidence, but you think it might be true, so it needs testing.

"I don't think you can win if you start with one-seventh and another one-seventh." "I think you can," says, Alex, what do you think? If you started with one-seventh and another one-seventh, how much would that be? And could you win from then on? "Let's try," says Alex.

"One-seventh and one-seventh." So he's gone first, and he started in the way that Aisha says you can't win.

"One-seventh," says Aisha.

"One-seventh." "Another one-seventh." "One-seventh and another one-seventh.

Ah, seven one-seventh added together are equal to 1.

I win." Alex has won again, and he's managed to prove the conjecture was false.

"Wow, well done! You can win if you start with two-sevenths." They play again.

"One-seventh." "One-seventh and another one-seventh." "Another one-seventh." "You've already won." How does Alex know that? "How do you know?" What do you think? Is Alex right? Can he say for certain that Aisha's already won? "There are three one-sevenths left to form seven one-sevenths.

If I use one one-seventh, you can use two one-sevenths." There it is.

"If I use two one-sevenths, you can use one one-seventh." There it is.

"Either way, you will put down the seventh one-seventh.

Seven one-sevenths are equal to 1, so you win." Aisha and Alex decide to create some questions about the game.

They list them down on sticky notes.

"How can you win this game?" "Is it best to go first or second?" "What happens if you use a different denominator?" "When do you know if you've won?" Can you think of any questions that you'd like to ask about this game? Okay, it is time for your second task.

For Number 1, we've got a game situation.

Alex and Aisha are part way through playing.

It's Aisha's turn.

There are three one-sevenths already.

What should Aisha should do? Should she play one one-sevenths or two one-sevenths? Explain what you think will happen for either option in the table below, so you need to do an explanation here for each of those choices.

For Number 2, it's time to play the game with a learning buddy.

Whilst playing, think about the questions posed by Alex and Aisha.

What are your thoughts? Can you consider any other questions and discuss them? Okay, pause the video here, and enjoy those practise tasks.

I'll be back later for some feedback.

Welcome back, let's look at Number 1.

Here's an explanation for what might happen if Aisha plays one-seventh.

If she plays one-seventh, she can win on her next go.

Alex will have to play one or two-sevenths, so she will be one-seventh or two-sevenths from the whole.

Looks like Aisha could win from this position then.

If she plays two-sevenths, then Alex can win by playing two-sevenths.

In total, that will be seven one-sevenths, which is equal to 1 whole.

Aisha's got a tough choice to make there then.

Okay, pause the video here, to look at that closely if it was a bit tricky to understand the first time.

Okay, then, let's look at Number 2.

I hope you enjoyed playing the game.

Here are some of the thoughts about the questions that Aisha and Alex asked.

How can you win this game? Well, make sure that you can place the fourth one-seventh, leaving three one-sevenths left.

Then your opponent will have to put you in reach of the whole on the next go, whether they place one-seventh to form five-sevenths or two-sevenths to form six-sevenths.

Is it best to go first or second? If you go first and play one-seventh, then you can win by being the player to play the fourth one-seventh.

If you go first and play two one-sevenths, your opponent can win by playing two one-sevenths and being the person who lays the fourth one-seventh, it's all about being the person who lays that fourth one-seventh.

What if you use a different denominator? Well, this game will work for any number of different denominators, but the strategy of leaving your opponent with three unit fractions to form the whole will always work.

When do you know if you've won? If the other player has three one-sevenths to add to form the whole, then you know you've won.

If you go first and place one-seventh to start off with, then you have one, provided you play correctly and respond to what your opponent place is.

Okay, I hope you had some other questions you asked, and maybe you did a bit of mathematical thinking to explain them.

Here's a summary of today's learning.

Non-unit fractions are made up of unit fractions.

A whole is made up of unit fractions.

When the numerator and denominator are the same, the non-unit fraction is equivalent to one whole or 1.

A whole can be formed through the repeated addition of a unit fraction.

This can be represented in many ways, including a number line or an equation.

My name is Mr. Tazzyman, I've enjoyed learning with you today, and I hope you've enjoyed it as well.

I'll see you again next time, bye-bye for now!.