Lesson video

Hello there, my name is Mr. Tazzyman, and today, we're gonna be learning together.

I'm looking forward to it and I hope you are too.

So if you're ready, we can get started.

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

By the end of it, you'll be able to say, I can solve problems involving comparing unit fractions.

Here are the keywords.

I'm gonna say them and I want you to 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 back.

Ready? My turn.

Whole.

Your turn.

My turn.

Part.

Your turn.

My turn.

Denominator.

Your turn.

My turn.

Unit fraction.

Your turn.

Okay, let's also look at the meanings behind these keywords.

The whole is all of a group or number.

A part is a section of the whole.

And the bar model at the bottom of the page there shows that relationship between whole and part.

A denominator is the bottom number in a fraction.

It shows how many parts a whole has been divided into.

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

Here's the outline for today's lesson on solving problems involving comparing unit fractions.

To begin with, we're gonna do some comparing in different contexts, then we're gonna look at some games and puzzles.

Let's get going with the first part.

In this lesson, you will meet Izzy and Andeep, and they're gonna help us by discussing some of the mathematics that will appear in the slides.

They'll give us some hints and tips and some of their thoughts.

Hi, Izzy.

Hi, Andeep.

Okay then, let's see what they start with.

Andeep and Izzy both have a full glass of squash.

The glasses are identical.

They each drink some.

"You have more leftover than I do," says Andeep.

"Yes, you've drunk more of your squash." Let's put an inequality symbol in between our glasses.

Okay, but which one should we use? What do you think? Which inequality symbol needs to go in between those two glasses to show the relationship between their quantities? Andeep says, "I have less squash than you do, so I think it will be less than." I agree.

There's the less than symbol.

They look back at their glasses of squash.

The squash left was a part of the whole glass.

So we could describe the relationship between part and whole using a fraction.

I'll visualise how many parts equal to this one could fill the glass.

Four equal parts, that would fill the glass.

One equal part is one-quarter.

There's the fraction notation, one, a division bar, and four as the denominator.

I'll do the same.

I'll visualise filling the glass.

Didn't take very long, Izzy.

Two equal parts there, so that means that the leftover squash is one-half.

There's the fraction notation again.

So the comparison of leftover squash works with fractions too.

Yes, when the whole is the same, the greater the denominator, the smaller the fraction.

Izzy and Andeep are looking at a fraction wall.

As you go down the wall, the number of parts making the whole increases.

So too does the denominator.

As the denominator increases, the parts get smaller.

The greater the denominator, the smaller the part.

You can see that on the bottom row where we have one-sixth, they are much smaller than in the second row, where we have one-halves.

Who had more, Izzy or Andeep? "I ate one-quarter of the pizza," says Andeep.

"I ate one-half of the pizza," says Izzy.

So what do you think, who had more pizza? There are the fractions of the pizza that they ate.

"I ate more pizza than you," says Izzy.

We can take that context and we can write it using fraction notation and an inequality symbol.

Four slices the size of mine made a whole pizza, so the denominator was four and the numerator was one.

Two slices the size of mine made a whole pizza, so the denominator was two and the numerator one, one-half.

You ate more so your fraction is greater, so one-quarter is less than one-half.

There's the inequality symbol.

Okay, it's your turn to check that you've understood what we've discussed so far.

Which is the greater fraction of a whole pizza? Put the correct inequality symbol between the unit fractions.

Pause the video, have a discussion, which symbol do you think it should be? Welcome back.

The symbol was less than, one-third is less than one-half, and with the pizza slices there you can see that as well.

Izzy and Andeep eat some blueberries.

Who ate most? "I ate one-third of the blueberries," says Andeep.

"I ate one-quarter of the blueberries," says Izzy.

What do you think? Who do you think ate the most blueberries? Andeep thinks about how many blueberries he ate.

There are 12 blueberries altogether.

He puts them in a line to help his thinking.

One-third has a denominator of three, so there needs to be three equal groups.

I ate one of those groups.

I had four blueberries.

Izzy thinks about how many blueberries she ate.

There are 12 blueberries altogether.

Puts them in a line just like Andeep did.

One-quarter has a denominator of four, so there needs to be four equal groups.

There they are, four equal groups.

I ate one of these groups.

I had three blueberries.

Andeep and Izzy write out the problem using fraction notation and an inequality symbol.

One-third, one-quarter, which inequality symbol needs to go between them? It was greater than, one-third is greater than one-quarter, given that the whole was the same.

Andeep and Izzy start with an inequality statement, then they each think of a context where you might see this comparison.

One-sixth is less than one-third.

Andeep thinks, "If I eat one-sixth of a melon and you eat one-third, then you've eaten more." Izzy says, "If I've taken one-third of a pack of balloons and you've taken one-sixth, then I will have more balloons." Okay then, it's time for practise task A.

Number one, I want you to work out who has had more in each of these contexts, and write it as an inequality statement.

A, Andeep eats one-quarter of a cake and Izzy eats one-sixth, who has eaten more? B, Izzy swims one-third of a length.

Andeep swims one-half.

Who has swum further? And C, Izzy and Andeep are both nominated for school council roles.

Izzy gets one-fifth of the votes.

Andeep gets one-quarter.

Who will be elected? Here's number two.

For each of the inequality statements below, write a context where you might see this comparison.

So now it's time for you to be creative.

What context could you think of that would mean that one-sixth was less than one-third, one-half is more than one-fifth, and one-seventh was more than one-10th? Okay then, have a go at these.

Pause the video and I'll be back in a little while with some feedback.

Welcome back.

Let's do number one first, and we'll start with A.

Andeep is the one who's eaten more.

One-quarter is greater than one-sixth.

Here's B.

Izzy has swum further because one-third is less than one-half.

Here's C.

Andeep is gonna get elected.

One-fifth is less than one-quarter.

Now, here's number two.

You obviously might have some different answers for this.

Here's what Andeep came up with.

"If I eat one-sixth of a pineapple and you eat one-third, then you've eaten more." "If I run one-half of a track and you run one-fifth, then I've run further." "If I use one-seventh of the marbles and you use one-10th, then I'm using more marbles." Okay, that's the first part done.

Now let's move on to the second part, games and puzzles.

Ready to do this? Let's go for it.

Andeep and Izzy play a game involving digit cards two to nine.

The game is a version of higher or lower.

"You go first.

I'll draw a digit card at random to be the denominator." "Okay then, it's a seven, so it's one-seventh." "Okay, now you have to decide whether you think the next unit fraction will be greater than or less than." What would you choose? Do you think that it's most likely to be greater or less? Have a think.

Let's see what Izzy chose.

"I think it'll be greater because one seventh is small>" "The denominator is three, so it's one-third, which is greater.

Well done." And you can see she gets a little tally mark next to her score.

"But three is less than seven, so am I wrong?" That's a good question from Izzy.

She's right, three is less than seven.

Andeep explains, "The digit becomes the denominator and the greater that is, the smaller the fraction.

Now we draw another, but you have to decide if it will be greater or less than." So now, she's thinking about what will be greater or less than a third.

"I think it will be less than one-third." It's a four.

"It's a denominator of four, which is one-quarter.

Well done." It's less than, she gets another tally mark on her score.

"Same again, greater than or less than." So this time, she's working from a quarter.

"I think it will be less than one-quarter." "It's a denominator of two, and that's one-half, which is greater than one-quarter." "I was wrong, but I got two right before that." You can see that's given her a score of two.

"So you scored two and now it's my turn to try and score more." "Okay, here's the first one.

It's a three, so it's one-third." "Less than," says Andeep confidently.

It's one-fifth.

He gets a score.

Now he goes from one-fifth.

"Greater than," he says.

It's one-half.

He scores another point.

Now he goes from one-half.

"Less than," he says.

It's one-quarter.

He gets another point.

And now he works from one-quarter.

"Tough choice.

I think I'll go for less than." It's a three.

"One-third is greater than one-quarter, so that's you out." "But I got three right altogether, so I won." He's correct.

Look at his score in comparison to Izzy's.

Okay, let's check your understanding of that game.

Why might this be a bad choice in this game of higher and low for Izzy? So you can see that she's going from one-half and she says, "I think it will be greater than." Hmm, why might that be a bad decision? Discuss it with somebody else and pause the video.

I'll be back in a moment to reveal the reasons.

Welcome back.

What did you think? Well, Andeep explains here, "The digits are two to nine, so there aren't any denominators that you can create to make a greater unit fraction." So you can't get anything greater than one-half as a unit fraction.

Okay, Izzy introduces Andeep to a puzzle called futoshiki.

Each row and column must feature the numbers one to three.

The numbers must also obey the greater than and less than symbols that lie between the spaces.

So in a sense, each row and column has to have one to three, but it doesn't matter what order they go in unless you see some inequality symbols.

They have to be obeyed.

Could you solve this, do you think? Let's see how Andeep and Izzy get on.

Izzy says, "On the top row, I know that the left box is three.

It is greater than the other boxes of that row, so it must be.

The middle must be two, and the right box is one following the inequalities." There they go, they're in.

"The right box of the middle row must be three.

It is greater than the middle box.

There is already a three in the first column, so three can't go in the left box." There it goes.

"The middle box can't be two because there is already a two in that column," so it has to be one.

"The left box must be two then." Now we're getting somewhere, Izzy.

"The bottom right box must be two as the other numbers are already in that column.

Looking at the columns, I can see where three and one now go." It's completed.

Andeep wants to try the same futoshiki, but use unit fractions instead.

Tricky, but a really good challenge.

He uses one-half, one-third, and one-quarter.

Can you solve it? "On the top row," says Andeep, "I know that the left box is one-half because that's greater than the others.

I also know that the middle box is one-third and the right box is one-quarter, the smallest.

On the next row, the right box is one-half because it's greater.

It also can't go in the first column.

That already has one-half.

The middle must be one-quarter because it can't be one-third, that's in the column already.

The left box must be one-third, as that's the only fraction left in that row.

If I look through the columns, I can complete the bottom row." One-half, one-third, one-quarter, one-third, one-quarter, one-half, one-quarter, one-half, one-third.

I was reading down the columns there to help out.

Okay, it's your turn to have a go at some of these games and puzzles.

Number one, have a go at playing higher or lower using two to nine digit cards, or digit dice and re-roll one or zero.

The fraction wall below can be used to help.

For number two, you've got to solve these futoshiki using the fractions one-quarter, one-third, and one-half.

You can see on B that you've been helped out a little bit because one-half is already there, but be careful, there are two solutions for B.

For C and D, it's the same thing.

Again, look at C, you've got one-third already put in to help you out.

Try this for number three, a four-by-four futoshiki using the fractions one-fifth, one-quarter, one-third, one-half.

Okay, lots of enjoyable games and puzzles to have a go out there.

Pause the video and I'll be back in a little while for some feedback.

Enjoy.

Welcome back.

Here was the game between Izzy and Andeep.

Andeep got three and Izzy one with a score of five.

How did you get on? Did you manage to win? And if you didn't, did you enjoy it anyway? All right, here's number two.

A, there are the solutions.

I'll read them across from left to right, One-half, one-third, one-quarter, one-quarter, one-half, one-third, one-third, one-quarter, one-half.

Pause the video so you can mark those carefully.

Okay, let's look at B.

There are two solutions here.

One-half, one-quarter, one-third, one-third, one-half, one-quarter, one-quarter, one-third, one-half, or one-quarter, one-third, one-half, one-third, one-half, one-quarter, one-half, one-quarter, one-third.

Again, pause the video so you can carefully mark these.

And here is C and D.

For C, we had one-quarter, one-third, one-half, one-half, one-quarter, one-third, one-third, one-half, one-quarter.

And for D, one-third, one-half, one-quarter, one-quarter, one-third, one-half, one-half, one-quarter, one-third.

Pause the video to accurately mark these.

All right, here was the mega one, the four by four.

One-half, one-fifth, one-third, one-quarter, one-quarter, one-half, one-fifth, one-third, one-third, one-quarter, one-half, one-fifth, one-fifth, one-third, one-quarter, one-half.

Again, pause a video so you can accurately mark these.

It's time to summarise our learning.

Problems involve reasoning in different ways.

Worded problems require you to fully understand the context and apply your understanding.

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

Similarly, reasoning is needed to justify choices made whilst playing higher or lower using denominators, or while solving puzzles like futoshiki featuring unit fractions and inequalities.

I hope you've enjoyed that lesson, and I'll see you again soon on another maths video.

My name is Mr. Taziman, bye for now.