Lesson video

Hello.

My name's Mr. Tazzyman and I'm 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.

We'll begin with the outcome then.

By the end of the lesson, we want you to be able to say, I can add and subtract fractions with the same denominator in a range of contexts.

Here's our key words.

I'm gonna 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 you can say it back.

Are you ready? My turn, minuend, your turn.

My turn, subtrahend, your turn.

My turn, numerator, your turn.

My turn, denominator, your turn.

Okay, let's see what those words actually mean in case we're not sure.

The minuend is the number being subtracted from, and a subtrahend is a number being subtracted from another.

There can be more than one.

There's an example at the bottom there with a simple equation that reads seven subtract three is equal to four, and in that equation, seven is the minuend and three is the subtrahend.

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.

Okay then, let's look at the structure for today's lesson.

We're adding and subtracting fractions with the same denominator in a range of contexts so we're gonna begin with some worded problems. Then after that we're gonna look at equations and inequalities.

Let's start with worded problems. We've got Sam and Laura here.

They're two friends who are gonna help us along the way by discussing the maths on the slides.

They'll provide us with some hints and tips.

They'll respond to prompts and they might even help us with some of the solutions.

Hi Sam.

Hi Laura.

Ready to go? Are you ready? All right, let's start then.

On sports day, Laura was in lots of races.

She brought in a water bottle that was 7/8 full.

She drank 6/8 of the bottle.

She topped up the bottle with 3/8 more water.

What fraction of water is in the bottle now? Okay then, Laura starts by saying, "I'm gonna represent this problem using a bar model." It's always a good idea to represent a problem if you're not sure about the structure.

"That will help to understand what's happening." What might it look like? We've got one full bottle at the top as the whole.

"The question uses 1/8, so I'll use those as the parts." There they are.

7/8 full.

There it is.

She drank 6/8, so we've taken away 6/8 and then put 3/8 more back in.

So we put 3/8 more in.

I have four lots of 1/8 in the bottle, which is 4/8.

"I'm gonna write this problem as an equation." So we may know the solution now, but it's important for us to look at it using different representations, that will help strengthen our understanding.

So it's good that Sam is offering to write us an equation.

What might it look like though? Well, we've got 7/8 full, so we start with 7/8.

Drank 6/8, so take away 6/8.

That's our subtrahend.

Then we put 3/8 in, so we're gonna add 3/8 and add in.

"The denominator will stay the same because it's the unit," so we know that we've got eights as our denominator.

"I have a mix of addition and subtraction here.

I know that seven subtract six add three is equal to four, so I know that 4/8 is the solution." She you got the same solution that Laura did using a different type of representation, an equation.

Okay, let's check your understanding so far.

Below is the same problem with different values.

Can you write it out as an equation? Pause the video here, read it carefully and look for those different values and have a go.

I'll be back shortly.

Welcome back.

How did you get on? Did you manage to write this as an equation? Well, let's see.

You should have written 9/10 subtract 7/10 add 5/10 equals 7/10.

On sports day, Laura was in lots of races.

She brought in a water bottle that was 9/10 full, so that was our first fraction written down.

She drank 7/10 of the bottle, so we needed to subtract 7/10.

She topped up the bottle with 5/10 more water, so we needed to add 5/10.

That should have equaled 7/10 altogether.

Okay, ready to move on? Sam is making a juice cocktail in a jug.

First, she pours in 6/10 of pineapple juice.

There it goes.

Then she adds in 2/10 of orange juice.

She pours out 3/10 of the mixture into a glass.

What fraction of juice is left in the jug? Lots going on here.

We need to break this down a bit.

"I am gonna represent this problem using a number line," says Laura.

"That will help me to understand what's happening,." So she draws out the line.

This question uses 1/10, so I'll use those as the intervals.

Remember, the intervals are the gaps between the marks on the number line.

Just be careful if you're using a number line and you're not sure about that.

There they are.

So first of all, we've got 6/10 of pineapple juice.

We've added in 2/10 of orange juice to get 8/10 altogether.

Then we've subtracted 3/10, which has been poured into the glass.

We finished with 5/10.

There are 5/10 of juice cocktail left in the jug.

Sam's gonna have a go now.

Can you guess what Sam might be using as a representation? "I'm gonna write this problem as an equation," Sam says.

What might it look like? What do you think? Can you think of one? Well, we start with 6/10.

Then we're adding 2/10 to that.

Then we are taking away 3/10, which is poured into the glass.

We know that we've got a mix of addition and subtraction here.

The denominator will stay the same because it's the unit.

So Sam writes in the denominator.

"I know that six added to two take away three equals five, so I know that 5/10 is the solution." There it is.

Okay, time to check your understanding again.

We've got a true or false.

When adding and subtracting fractions with the same denominator, the denominator stays the same.

True or false? What do you think? Pause the video and I will give you the solution shortly.

Welcome back.

That was true.

When adding and subtracting fractions with the same denominator, the denominator stays the same.

Can we think about why? Well, here are two justifications, because the numerator is on top so you do that first and ignore the other part or the denominator is the unit that is being added and subtracted so it stays the same.

Which of those do you think is the best justification? Choose one.

Pause the video and I'll be back with the solution shortly.

Welcome back.

Well, the bottom justification was best here.

You shouldn't ever really need to ignore any part of a fraction.

You should definitely understand what it means, and the second part, part B gives us a clue on that.

It says that the denominator is the unit that is being added and subtracted so it stays the same.

Equations can tell stories.

Sam decides to use the same equation to write a new story.

"First, a bus was 6/10 full." There's the bus, 6/10 full.

"Then, at the next stop, 2/10 of the full bus got on." There they are, and we've got an equation forming as well here.

"Then, at the next stop, 3/10 of the full bus disembarked." Disembarked means got off.

"Now, the bus is 5/10 full." Okay, time to check your understanding.

Can you spot the mistake in this equation story? 7/8 subtract 4/8 plus 2/8 is equal to 5/8.

That's the equation, but here's the story.

One full pack of pencils contains eight pencils.

There are 7/8 in the pack on yellow table.

Four children use a pencil for art, two children use a pencil for geography.

Now there are 5/8 of the pack left.

What's the mistake here? Pause the video and be a detective.

Try to find it.

Welcome back.

Did you find the mistake? Well, it was here.

"Each pencil is 1/8 of a pack.

This part of the story would be subtraction, not addition.

To make it addition, two pencils would need to be put back into the pack." Okay, here's your first task then.

We've got some worded problems about paint.

Laura is painting a mural on the school wall.

She wants to use the colour teal, so she needs to mix it.

She mixes paint into a measuring jug, which has a scale in tenths.

First she pours in 4/10 of blue.

Then she pours in 2/10 of green.

To finish off, she pours in 1/10 of yellow.

She carefully mixes the paints to make teal.

How much teal paint has she made? Represent the problem, write the equation and solve it.

Here's number two, another worded problem.

Sam is helping to paint the mural and wants the colour indigo, so more mixing is needed.

Sam mixes paint into another measuring jug.

First, 6/10 of blue paint followed by 3/10 of red.

She uses 2/10 of the indigo and leaves the rest.

How much indigo paint is left over? Write this as an equation and solve it.

Then for number three, something slightly different, we've got solve the fraction equations and then turn them into a fraction story.

So you can see there's some missing fractions there.

You need to find what they are and then consider them a story and write your version.

Okay, pause the video here and I'll be back in a little while once you've completed those with some feedback.

Welcome back.

Let's start with number one where we had to write a representation out and do an equation.

Here's a bar model with the colours on.

You've got 4/10 that are blue, 2/10 that are green and 1/10 that's yellow.

If we write that as an equation, we can see that altogether 7/10 has been made.

7/10 of the jug.

2/10 added to 4/10 added to 1/10 equals 7/10.

Note again that the denominator hasn't changed because that's the unit that we're using.

Here's number two, the equation should look like this.

6/10, that's the blue paint added to 3/10, that's the red paint, subtract 2/10, which is the indigo that Sam uses, leaving you with 7/10 of indigo left.

Here's number three, solve the fraction equations then turn them into a fraction story.

So we've got a couple of fraction stories here, but remember you've probably written some very different contexts.

The first one was 5/15 and Sam said, "I make a squash drink.

I add 12/15 water and 3/15 cordial.

Then I drink 10/15.

There are 5/15 left." And for B we had 14/20.

Laura said.

"A bus is 10 twentieths full.

At the next stop, 6/20 gets on, but 2/20 get off.

Now the bus is 14/20 full." Okay, pause the video here because you might wanna share some of the different contexts that you came up with with one another.

All right, it's time for the next part of the lesson then.

We're gonna move on to looking at equations and inequalities.

So sit up, be ready to listen because we've got some more thinking to do.

Sam looks at a fraction equation with some missing numerator.

3/9 is equal to 8/9 subtract something/9 subtract something/9.

What do you think? What could we put into those question mark boxes.

Sam says, "I'm gonna use counters for this.

Each counter can be worth 1/9.

I will start with eight counters to make 8/9, which is the minuend." There they are.

We've got eight, lots of 1/9.

That's the minuend.

"I know that I will have 3/9 after subtraction." She's coloured in those 3/9 in yellow.

"This is the difference." Labelled it.

"That means that the subtrahend is 5/9, but there are two subtrahends, so this will need to be partitioned." Can you see the two subtrahends there in the equation, and they are things that have the missing numerators.

"I know that four plus one equals five, so the numerators could be four and one," and she's used colour again and labels to show us those two different subtrahends.

Laura says, "I think there are more solutions than just four and one.

I'll show you on the counters and list down the solutions." Three and two.

Two and three, one and four.

Laura and Sam discuss one more possibility.

"What about zero? Can we use zero as a numerator?" Says, Sam.

What do you think? Do you think it's okay to use zero as a numerator? "Yes, we can.

I know that zero plus five is equal to five, so the numerator could be zero and five." There they are plunked in and it works.

"Yes, or five and zero because zero plus five is equal to five." So they've swapped around.

Okay, let's check your understanding.

A fraction with zero as a numerator is equivalent to zero.

True or false.

Pause the video and have a think.

Welcome back.

Well, that was true, but here are some justifications for it because it's always important for us to be able to explain why.

A, because both the fraction and the number have the numeral zero in them.

Or B, the numerator of zero tells us that none of the parts are selected, shaded, or in use.

Pause the video and decide which justification you think is correct.

Okay then, it was the second one.

The numerator of zero tells us that none of the parts are selected, shaded, or in use.

Ready to move on? Laura looks at an inequality comparison with a missing number.

2/10 added to 3/10 is more than 9/10 subtracts something/10.

What do you think it might be? "I know that when adding fractions with the same denominator, the denominator stays the same.

The denominator is just the unit.

In this equation, it is 1/10.

I need to focus on the numerators.

I'll write a similar inequality comparison using ones instead of 1/10." two added to three is more than nine subtract an unknown.

"I know that two plus three equals five, so I can substitute the first expression." There we go.

Five is more than nine subtract an unknown.

"The second expression has to be less than five, so it must be equal to four or less.

I know that nine subtract five equals four, so five could be a solution for the missing numerator too." Brilliant reasoning, Laura.

"I think you could have had six, seven, eight, or nine as well," says Sam.

"I agree.

All of those subs create a difference that is less than five." Okay, check your understanding then.

What's the mistake here? Have a look.

Pause the video and think.

Welcome back.

What did you think the mistake was? Well, Sam says, "The first expression is equal to 5/10.

If the numerator was four, then the second expression would also be 5/10.

That would mean that this was an equation rather than an inequality comparison." Both of those expressions are the same.

Laura and Sam look at the sorting diagram.

"The table has rows which are horizontal." "Yes, and columns that are vertical." "The row headings are different criteria." "So are the column headings." "In the top left box we need a green fruit." "In the top right we need a red fruit." "Bottom left needs to be a green vegetable." "Bottom right needs to be a red vegetable." Sam looks at another sorting diagram.

"I need to sort these expressions into correct boxes.

1/9 and 2/9 is equal to 3/9.

3/9 is less than 5/9 and it's addition." So it goes in the box where the headings are less than 5/9 and addition.

"9/9 subtract 1/9 is equal to 8/9.

8/9 is greater than 5/9 and it was subtraction." Which box do you think it's gonna go in? Point.

It's going in this box here because it is greater than 5/9 and it featured subtraction.

"7/9 subtract 6/9 is equal to 3/9." Which box do you think this is gonna go in? Let's see what Sam does.

"3/9 is less than 5/9 and it was subtraction." It goes there.

I think you probably know where the last expression goes, but let's just check anyway.

"Lastly, 5/9 and 4/9 is equal to 9/9.

9/9 is greater than 5/9 and it was addition." There it goes.

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

Fill in the missing numbers in the equations below and find all the possibilities.

Number two, find all the possibilities for the missing number in these inequality comparisons.

And number three, sort the expressions into the correct boxes in the sorting diagram.

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

Welcome back.

Let's look at some answers.

Be ready to mark.

For 1A, the combinations were three and zero, two and one, one and two, zero and three.

Note the use of zero, it still counts.

We had to make sure here that we were subtracting from 7/8, numerators that totaled three altogether, so all of these combinations together made three.

Let's look at B.

Here we had zero and four or four and zero, one and three, two and two and three and one.

On this one, you had to ensure that the combination of those two numerator in total made four.

Here's number two then, possibilities for 2A, four, three, two, one, or zero.

So you had to make sure that the combination of those numerator didn't exceed six because if it did, then the inequality would no longer be accurate.

For B, we had six or seven.

Okay, let's look at this sorting diagram.

8/15 added to 2/15 went into this box because it was addition and it was greater than 7/15.

9/15 subtract one 15th went into this box because it was a subtraction that was greater than 7/15.

4/15 takeaway 2/15 was less than 7/15 and was a subtraction.

And lastly, 1/15 added to 3/15 was an addition that was less than 7/15.

Okay, we've reached the end of the lesson.

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

When adding or subtracting fractions with the same denominator, the denominator stays the same and the numerator can be used for the calculations.

Fraction, additions, and subtractions can appear in many different contexts and reasoning can be applied to them in the same way that it is for whole numbers.

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

I hope you have as well, and I hope it prompted you into lots of great mathematical thinking.

Until the next time.

Bye-Bye.