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, we want you to be able to say, I can identify equal parts in a whole when they do not look the same in 2D shapes.

Here are the keywords.

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

My turn, whole.

Your turn.

My turn, part.

Your turn.

My turn, equal or unequal.

Your turn.

Okay.

Let's see what each of those words means.

The whole is all of a group or number.

A part is a section of the whole.

You can see the bar model at the bottom there that's showing us 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 outline for today's lesson then.

We're gonna start by looking at the fact that equal parts can look different and then we're gonna solve some reasoning puzzles.

Let's get going with the first part.

Here's two friends we'll meet along the way, Laura and Jun.

Both of those are gonna help us out today by discussing some of the maths prompts, revealing some of the answers and probing us to think a little bit more deeply about mathematics.

Laura and Jun look at a statement.

Parts of a whole can only be equally sized if they are the same shape.

Laura says, "I think that's false because you can divide wholes in different ways." Jun says, "I think that's true because different shaped parts would have a different area." "I'll prove it to you," says Laura.

Laura uses two identically sized pieces of paper of different colour to prove her idea.

"I'll start by folding one of them into two equal parts which shows one-half." There she goes, one-half.

"Then I'll lay the one-half on top of the other piece of paper to make a whole".

There it is.

So the other part is one-half as well.

"Both have the same shape, so I was correct," says Jun.

"But what if I rotate the folded paper," says Laura.

And she does.

"Both parts are still one half because I haven't changed their size, but they are differently shaped.

I can do it in lots of different ways" "Every time the two parts are differently shaped but have equal area," says Jun.

Okay, time to check your understanding.

I want you to have a go at Laura's proof yourself using two pieces of paper.

Pause the video here, have a go at that and I'll be back in a little while.

Welcome back.

Did you manage to show that parts can be differently shaped but have equal size? Here's the first step and there's the last step.

Did you manage to complete that? Did you find lots of ways? Okay.

Jun thinks of another way to prove using four squares.

"Each of these squares is identical and is a whole.

I will split them up into four equal parts but in different ways." There's the first way, there's the second way, there's the third way, and there's the fourth way.

"Each part is one-quarter of the whole.

Now I'll look at each part by itself." So you can see there, he's removed all the other parts from the whole.

He's just got parts that are worth one-quarter.

"Then I'll combine them to make a whole with four equal parts differently shaped." There's his new shape, his new whole, and each part is worth one-quarter.

Laura takes Jun's proof further.

"I'll use one of Jun's ways of dividing the whole square into four equal parts." She's chosen the way in which the whole has been divided into four oblongs.

"Each part is one-quarter of the whole square.

I'll draw a line through the middle of one part and then cut it.

Then I'll rotate the cut part.

Now the whole square has been divided into four differently shaped equal parts.

Here are all the steps shown together." So she starts with those four oblongs.

She draws a line, she cuts it, she rotates it, and then she's managed to show four differently shaped equal parts.

Line, split, rotate.

Jun looks at a proof using skipping ropes rather than shapes.

"I'll get three equal length skipping ropes." There they are.

"Each skipping rope can be one part of a whole.

I'll lay them all horizontally to start with to show the whole." There they are.

"Each skipping rope is one-third of the whole." And Jun used fraction notation there to show that.

You can see the denominator is three because there are three equal parts making up the whole and each of those parts is a numerator of one, one-third.

"If I change the shape of each skipping rope, they will still be one-third." One third, one third, and one third.

"The length of each part is still equal but their shapes are different." Okay, it's time to check your understanding.

Each skipping rope is equal length.

They're connected together to make a whole.

What fraction of the whole is each skipping rope? So look closely at the picture there.

Look at the skipping ropes, pause the video and I'll be back in a while to reveal the answer.

Welcome back.

What did you think? What fraction of the whole was each skipping rope? Well, Jun says, "The whole has been divided into four equal parts, so each part is one-quarter." There's the fraction notation.

And you can see the four skipping ropes there.

Okay, time for your first practise task.

Number one, draw in the missing steps for the proofs below, making use of Jun's differently shaped parts, which were each one-quarter of the whole square.

So for a, you can see you've already got the shapes making the whole in the first part.

Then you need to find a way to put a line in, to split the shape where you've drawn the line, rotate part of that shape and you should end up with the configuration you see at the end, four squares making up that whole.

Same again for b.

This time you are turning that triangular part, which is one-quarter to begin with, into a rectangular part by the end.

Number two, have a go at completing Jun's skipping rope proof.

You could replace skipping ropes with something like string.

Can you think of any other proofs like this? Okay, pause the video here.

Have a go at those tasks.

Enjoy them.

Make sure you do lots of thinking, and I'll be back in a little while for some feedback.

Welcome back.

Here's the answers for number one, A.

There's the first step.

You can see that a line has been drawn across that triangular part.

That triangular part's then been split.

The triangular tip of it has been rotated to create the square quarter part that you can see there.

For B, similar sort of thing.

But this time the line was drawn vertically down.

The shape part was split, rotated to create the oblong quarter that you can see there.

Okay, number two.

Well, hopefully you had a good go at being able to complete Jun's skipping rope proof.

Laura says she thought of rope fencing around the playground.

People change its shape as they touch it but the parts don't change length.

And Jun thought of making toy racing tracks where curves and straits have the same length.

What did you think of? Okay, are we ready to move on to the second part of the lesson? We are gonna look at some reasoning puzzles now.

Let's go for it.

Jun and Laura look at a shape with a shaded part.

"What fraction of the whole is the shaded part?" Asks Laura.

"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?" Says Jun.

"Imagine the shaded part multiplying to cover the whole." "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." Brilliant.

Here's another example.

What fraction of the whole is the shaded part here? "Let's visualise again," says Jun.

"The whole has been divided into three equal parts so the denominator is three." "One equal part is shaded so has a numerator of one." "It's one-third." Tricky one this.

"What fraction of the whole is the shaded part?" "Let's visualise again.

But this time we will have to rotate as well." "The whole has been divided into eight equal parts so the denominator is eight." "One equal part is shaded so it has a numerator of one." "It's one-eighth." Okay, let's check your understanding.

Time for you to visualise.

What fraction of the whole is this shaded part.

Have a go.

Pause the video and I'll be back in a little while to reveal the answer.

Welcome back.

What did you think? What fraction of this whole was the shaded part? Well, if you visualised, you'd have noticed that there were four equal parts.

It's one-quarter.

Jun and Laura look at a puzzle with three different shaped shaded parts of the whole.

"How can we work out each of these coloured parts?" Asks Laura.

"I think we already have.

They are the same as before.

Let's visualise each." "But our visualisations will bump into the other parts, won't they?" "No, they can go over the top because it's our imagination." What a powerful thing that is.

"Okay then.

Here it goes." So Laura started with that triangular part and she's visualised it so she can tell how many of those triangular parts would fit into the whole.

The triangular part is one-eighth because eight of them fit in.

There we go.

Now it's time for the square part.

And you can see four of those square parts would fit into the whole.

Square part is one-quarter.

Lastly, the oblong.

You can see that three of those fit into the whole.

That means that the oblong part is one-third.

The puzzle has been completed.

"We solved the puzzle," says Laura.

Jun and Laura look at a second puzzle with four differently shaped shaded parts of the whole.

"This looks tricky.

They're all differently shaped." "Let's stick to our visualisation and be systematic.

Start with the oblong," says Laura, One-eighth.

Did you see that the oblong would fit into the hole eight times? "Now the square," says Jun.

It fits equally into the whole four times, what does that make it? It makes it one-quarter.

"I'll do the greater triangle," says Laura.

The biggest one.

That also fits into the whole four times, so that means that it's one-quarter.

Jun says, "I'll do the smaller triangle." That fits into the whole eight times, so it's one-eighth.

What do you notice about those parts? Laura says, "I notice that the square and the greater triangle are differently shaped equal parts." You can see they're both one-quarter.

"So too are the oblong and the smaller triangle," says Jun.

Both of those are one-eighth.

Okay, it's time for your practise task, the second one.

For number one, I want you to write the fraction notation for each of the shaded parts of the whole below.

Which parts are differently shaped equal parts? So you can see you've got A, B, and C.

For number two, again, write the fraction notation for each of the shaded parts of the whole below.

Which parts are differently shaped equal parts? For number three, I want you to try to design your own puzzles which have differently shaped parts.

Have three or four parts to calculate.

And use the template below to help.

Try it on a partner when you are finished.

Okay.

Enjoy the tasks, get thinking.

And I'll be back in a little while to give you some feedback, so pause the video here.

Welcome back.

Let's look at these first ones.

A was one-eighth because that triangular part can fit into the whole eight times.

B was also one-eighth because that oblong can fit into the whole eight times.

And C was one-third because that oblong can fit into the whole three times.

Jun said he noticed that the triangular part and the oblong part are differently shaped equal parts.

They're both one-eighth of the whole, but they have different shapes.

Here's number two.

A was one-sixteenth because that square fits into the whole 16 times.

B was one-quarter.

C was one-eighth.

And D was one-quarter.

The triangular part and the greater oblong part are differently shaped equal parts, both of them would worth one-quarter.

Okay, here's number three.

Did you try any of your puzzles on a partner? This was Laura's puzzle, and Jun has completed it.

One-sixteenth, one-quarter and one-half.

I hope that you've had fun creating those puzzles and trying them out on one another.

Here's a summary of what we've learned today.

A whole is made up of many parts.

Parts can be equal or unequal.

Equal parts can look different but still be equal.

A whole can also have differently shaped parts of different sizes.

You can use visualisation to work out what fraction of a whole a part is.

I really enjoyed learning with you today and I hope you did too.

I also hope that I'll be able to see you again in another math lesson soon.

My name is Mr. Taziman.

Goodbye for now.