Lesson video

Hi, my name is Mr. Tazzyman, and I'm really excited to be learning with you today.

Fractions sometimes can be considered quite tricky, but hope that we can have some fun with it and we can boost your understanding.

Okay, let's get going.

Here's the outcome for today's lesson then.

By the end, I want you to be able to say, "I can solve problems by identifying parts "and wholes in a range of contexts." These are the keywords.

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

I'll say, "My turn," say the word, and then I'll say, "Your turn," and you can say it.

Ready? My turn, whole.

Your turn.

My turn, part.

Your turn.

My turn, equal or unequal.

Your turn.

Okay.

Let's see what they each mean.

The whole is all the parts or everything, the total amount.

Apart is some of the whole.

There's a bar model at the bottom there that you can see showing that relationship.

We say that two or more things are equal if they have the same quantity or value.

We say that two or more things are unequal if they do not have the same quantity or value.

Here's the lesson outline.

First of all, we're gonna solve some problems involving Venn diagrams. And secondly, we're gonna look at some visualisation puzzles.

Are you ready to start? Let me introduce some people to you first.

We've got Alex and Sofia.

These two are going to share some of their discussions, and they're gonna respond to some of the prompts on screen, which should help us in our thinking.

Hi Alex, hi Sofia.

All right, are you ready? Let's go for it.

Alex and Sofia look at a Venn diagram, multiples of two, multiples of three.

"What is this?" says Alex.

"It's a diagram to help us sort things," says Sofia.

"This time, it's numbers.

"These ones." There they are, you can see them on screen.

Two, three, four, five, and six.

Important to note, these are numbers, not digits.

We're not gonna combine them.

They are numbers.

"The left circle is for multiples of two," says Alex, and Sofia replies, "The right circle is for multiples of three." That's what those labels are for.

"What about the part where they overlap?" Can you see that? The two circles have overlapped and they created a space in the middle.

"That's for numbers that are multiples of two and three," says Sofia.

"I'm gonna put two into the multiples of two circle." I think that makes sense, doesn't it? There it goes.

"Three goes into the multiples of three circle." Again, I think that makes sense.

"Four is two lots of two, "so it goes into the multiples of two circle." There it goes.

"Five is not a multiple of two or three." "So where should that go?" What do you think? Hmm.

Well, Sofia says, "It goes anywhere outside either circle." Ah, that makes sense.

So she puts it there.

"Six could go in either as it's a multiple of two or three." "So put it in the middle," says Sofia.

There you go.

A completed Venn diagram.

"There, all done," says Alex.

Okay, let's check your understanding of Venn diagrams. Have a look at this one.

You need to identify which two numbers are in the wrong place.

Have a go at that.

Pause the video here, and I'll be back in a little while to reveal those two numbers.

Welcome back.

Let's see which two numbers are in the incorrect place.

Well, firstly it was nine.

It's a multiple of three, but not two.

So it needs to move over to the multiples of three circle.

Next was six, and six is a multiple of two and three, so it needs to move into the middle part.

How did you get on? Did you spot those errors? Okay, let's look at some more Venn diagrams. Alex sorts five representations into a Venn diagram.

Can you see the conditions of each circle there? We've got equal parts that look different, and we've got one quarter in fraction notation.

"The left oval is for wholes divided into equal parts "that look different.

"The right oval is for anything showing one quarter "of the whole.

"I'll sort the representations one at a time." Good thinking, Alex.

Sometimes it's best to separate things and look at them one at a time.

There's the first representation.

Let's see what Alex makes of it.

He says, "This is made up of three equal parts "that are differently shaped.

"One third is highlighted." So it goes in that left oval.

There's the next representation, and Alex says, "This is made up of four equal parts "that are identically shaped.

"One quarter is shaded.

"It goes into the right oval." There's the next one.

Alex says, "This is made up "of two equal parts that are identically shaped.

"One half is shaded.

"This doesn't fit into either oval, so it goes outside.

"This is made up of four equal parts "that are differently shaped.

"One quarter is shaded.

"It goes into the middle because it fits into both.

"This is made up of four equal parts "that are differently shaped.

"One quarter is shaded.

"It goes into the right oval.

"Finished." Well done, Alex.

Okay, let's see if you could have a go at something similar.

Where would this representation go in the Venn diagram below? The two circles have different conditions.

The left circle has equal parts that look different, and the right circle has one eighth.

Pause the video and see if you can place the representation correctly.

Welcome back.

Which part of the Venn diagram do you think it needed to go into? Well, Alex says, "This whole has been divided "into eight equal parts, differently shaped.

"One equal part has been shaded showing one eighth.

"The representation fits into both ovals, "so it goes in the middle." Did you get that? Okay, let's move on and look at some more Venn diagrams. Sofia looks at a completed Venn diagram.

She has to work out what condition each oval has.

"I'm gonna compare the left "and right oval to begin with," says Sofia.

"I'll ignore the middle where they overlap for now.

"All the representations on the left are 2D "and divided into equal parts.

"On the right, they're all 3D and split into unequal parts.

"It can't be 2D and 3D though, because you can't have both." And what she means is, there wouldn't be a middle section if it was 2D and 3D, or at least, there wouldn't be anything in the middle section because you can't be both.

"Let's look at the middle.

"The shapes of 3D.

"So the right oval must be 3D.

"The left oval must be equal parts." Sofia looks at a new Venn diagram.

She has to create a representation for each part.

She's got one third in the left oval and groups of people on the right oval.

"The left oval has to show one third, "so one equal part selected "with three equal parts altogether.

"I'll use cubes to represent one third." There they are.

She's put three cubes together and one of them is a different colour to show that that's been selected.

"The right oval is about groups of people.

"There needs to be more than one group, "but they don't have to be an equal in number." She's put in two groups.

One has four people in, and the other has three.

"A group of three and a group of four.

"The middle needs to be groups of people showing one third." And there they are.

She's drawn six people in total, but she's separated those six people into pairs, so she's divided the group into three equal parts, and then she's drawn a ring around one of the parts to show that that's been selected, so a numerator of one.

Okay, let's move on.

It's time for your practise.

For number one, you need to sort each of the representations below into the correct part of the Venn diagram shown, and the conditions on the Venn diagram, well, on the left oval, it's different shaped parts, and on the right oval, it has to be one quarter.

You've got A, B, C, and D, so four different representations to put into the Venn diagram.

For number two, it says, "Can you work out the missing conditions "for each oval on this Venn diagram?" So the Venn diagram's actually been completed, but you need to say, what does the left oval have and what does the right oval have? Number three, you need to create a representation to place in the left, middle, and right parts of this Venn diagram.

The conditions are one sixth for the left oval, and same shaped parts on the right oval.

Sofia gives you a tip there.

"It's useful to start with the part for the same shape." Okay, pause the video here and have a go at those tasks.

Enjoy them, think carefully, and I'll be back in a little while to give you some feedback.

Good luck.

Welcome back.

Let's have a look at the answers.

Be ready to mark.

A went into the right-hand column.

That representation was showing one quarter, and all the parts were the same shape.

B went into the middle.

You can see that the square part that had been coloured differently was one quarter of the whole, but all of the parts were differently shaped.

C went into the left-hand side.

There were different shaped parts, but it was showing one half, not one quarter.

And D went on the right-hand side.

This was showing one quarter and all of those parts were the same shape.

How did you get on? Well, let's look at number two now.

For this one, we had to try to figure out what the conditions for each of the ovals were.

The left oval was lines, and if you look at all three representations inside that oval, you can see that they're all lines.

The right oval was one half.

And again, look within just that oval, all of those representations are showing one half.

Did you manage to get that? I hope so.

Let's move on to looking at number three for feedback.

Here, you had to create a representation to put in the left, middle, and right parts of the Venn diagram.

Sofia's gonna share hers.

There's her first representation, showing one sixth, and you can see that the oblong that's a different colour would fit into the whole six times, so that means it's one sixth.

There's her representation in the middle.

She's showing one sixth, but all of the parts are the same shape.

She will have started with that shape, one of those, and she will have repeated it six times to create the whole.

And here is her last representation.

These are the same shape because it's cubes, but this is showing one quarter of the whole, not one sixth.

I hope you got on well with those representations as well.

Are you ready to move onto the second part of the lesson? We're gonna look at some visualisation puzzles.

Jun and Laura look at a shape with a shaded part, and Alex says, "What fraction "of the whole is the shaded part?" Sofia says, "Well, the whole hasn't been divided "into equal parts, so how can we tell?" "We'll have "to visualise the shaded part filling up the whole." "What does that mean?" "Imagine the shaded part multiplying to cover the whole." There it goes.

"There would be four equal parts, "so the denominator would be four." "We started with one shaded part, "which gives a numerator of one." "It's one quarter." Okay, it's your turn.

What fraction of the whole is the shaded part here? Have a look at it and try and use some visualisation to figure it out.

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

Welcome back.

Let's help you out.

We're gonna do some visualisation.

Is that the kinda thing you thought about? You can see, there are eight triangles there.

Only one part selected though, so as Sofia says, "Eight of these parts fit into the whole, "so the shaded part is one eighth." Is that what you got? I hope so.

Okay, let's look at some more visualisation puzzles.

Alex and Sofia look at a second puzzle with three differently shaped shaded parts of the whole.

"This looks tricky.

"Different shaped parts on the same whole." "Let's stick to our visualisation and be systematic.

"Start with the oblong." It's one eighth.

Did you see that? There were eight of those oblongs fitting into the whole.

"Now the square," says Sofia.

Four of them.

That means it's one quarter.

"I'll do the triangle," says Alex.

Four of them.

That means that that's one quarter as well.

"Puzzle complete," says Sofia.

Sofia and Alex look at another similar puzzle.

"I'll do the triangle at the top.

"It fits into the whole four times." You can see that there, so that's one quarter.

"I'll do the triangle on the right.

"It also fits the whole four times." There you go.

So it's one quarter.

"I'll do the smaller triangle," says Alex.

"It fits in eight times, so it's one eighth." Alex and Sofia solve another visualisation puzzle.

This time, it's a 3D object.

And Alex says, "What fraction does this represent?" "I'm going to imagine the cubes filling up the cuboid.

"The whole has been divided into three equal parts "so the denominator is three." "One equal part is shaded in, so the numerator is one." "It's one third." Okay, your turn.

What fraction of the whole is the shaded part? Imagine filling up that whole with cubes that size.

That's gonna help you out, because that will give you the denominator.

Okay, pause the video, have a go at visualising, and I'll be back in a moment to reveal the answer.

Welcome back.

Okay.

You might have visualised, and it might have looked something like this.

"The part fits into the whole four times, "so it is one quarter." Alex and Sofia look at another visualisation puzzle with 3D objects.

This time, there are two shaded parts.

"What fraction of the whole do the two parts represent?" Sofia says, "I'm gonna start with one part "and ignore the other.

"Three parts would fit into the whole, "so this is one third." And there's that fraction notation to help us out.

Alex's turn.

He says, "I'll do the other part.

"Four parts would fit into the whole, so it is one quarter." And there's the fraction notation.

Okay, your turn to solve some puzzles like these.

Number one, work out the fraction of the whole for each of these parts.

Number two, work out the fraction of the whole for each of these parts.

Pause the video here, have a go at those two, and I'll be back in a little while for some feedback.

Welcome back.

Let's look at number one first one.

1a was one eighth, because you could fit eight of those triangles into the whole.

1b was one half.

Same reasoning, you could fit two of those triangles into the whole.

You might have had to cut the triangle in half, but you still could.

And c was one eighth.

Pause the video here to make sure that you've got those answers down, and to discuss anything should you need to.

Okay, let's look at number two.

We had two different parts to complete here.

There's the first part.

We've used visualisation to fill the whole, and it fills it six times.

That means it's one sixth.

Now let's look at the second part.

We can put that in twice into the whole.

That means that it's one half.

How did you get on? Did you manage to solve those visualisation puzzles? I hope so.

Here's a summary of the learning today.

Problems involve reasoning in different ways.

Venn diagrams require you to analyse things, reason, and then sort them into different ovals or circles with certain conditions.

This sort of reasoning can be applied to your understanding of fractions and wholes and parts.

Unequal parts can be represented on the same whole.

In 2D and 3D contexts, visualisation is a useful method for working out what fraction of a whole that one part is.

I really enjoyed that today, and I hope you did.

It's always good fun to work with Venn diagrams to do some reasoning, and then also to do some visualisation.

I hope to see you again soon in another maths lesson.

My name is Mr. Tazzyman.

Bye-Bye for now.