Lesson video

Hi, my name's Ms. Lambeau.

Really pleased that you've decided to join me today to do some Maths.

Come on, let's get started.

Welcome to today's lesson.

The title of today's lesson is Between Numbers and it's in the unit, Comparing and Ordering Fractions and Decimals, Including Positives and Negatives.

By the end of this lesson, you'll be able to appreciate that for any two numbers, there is always another number in between them.

Here are some key words or phrases that we'll be using throughout today's lesson, and so I thought it might be useful to have a quick recap as to what they are.

The first one is about equivalent fractions.

So equivalent, remember, means the same.

So two fractions are equivalent if they have the same value.

A proper fraction is a fraction where the numerator is less than the denominator.

And the lowest common multiple, remember, sometimes abbreviated to LCM, is the lowest number that is a multiple of two or more numbers.

Today's lesson, I'm going to split for us into two separate learning cycles.

In the first one, we will just concentrate on integers and decimals and finding numbers between those.

And then when we're confident with that, we'll move on to looking at finding numbers between two proper fractions.

Let's get started on that first learning cycle.

Let's go.

Here, we have Andeep and Sofia, and they're playing a game.

They have to write down an integer that is between two numbers that are given on a number line.

The winner is the person who writes the highest number.

Here's their number line.

Andeep says, "I've written mine down." Sofia says, "Me too." Can either of them win? I'd like you to think about that question carefully.

Can either of them win? No, they can't both win because they both will have written one.

Why will they both have written one? They will both written one because the game asked for an integer between zero and two, and we know there is only one integer between zero and two, so they've both written one, so therefore neither of them has written the highest number.

No one can win.

They've decided to change the game.

This time, they have to write down any number between two numbers on the number line, so the same thing again, and the winner is the person who writes the highest number.

So the rules of the game are exactly the same, but this time they've changed the numbers on the number line.

We've now got between zero and one.

Andeep says, "I've written mine down and I know I'm definitely going to win." Very confident, Andeep.

Sofia says, "Me too, but I'm not certain I'm going to win." So Sofia has written down a number, but she's not as confident as Andeep.

I'd like you to have a think.

What number do you think Andeep has written considering that he's super confident that he's going to win the game? Andeep has written down 0.

99.

So he says, "So I win." Do you think Andeep has won? Well, let's see what Sofia has written.

She says, "Actually I win, I wrote down 0.

999." So yes, she's right, she did win.

What I want you to do now is I want you to win the game.

I want you to write down a number that would beat Sofia's number.

Here are some examples of some numbers that you may have written, and I'm not gonna read them out because I'll get all tongue tied with saying all of those nines.

Basically, as long as you've written 0.

999 and then something after the nine that isn't zeros, well done, you've won the game.

Play the same game with the same rules here.

Write down a number between 0 and 0.

5 and the highest number wins.

Andeep writes 0.

49999 and Sofia writes 0.

499991.

I want you, please, to write down a number that beats both Andeep's and Sofia's.

What number could you write to win the game? Some examples here, 0.

4999999.

And like I said before, I'm gonna get very tongue tied with those nines, so I'm just gonna let you take a look at those.

So again, as long as you have written 0.

499991 and then something other than zero after the one, you've won the game, well done.

Oh, they're now playing a different game.

They take it in turns to roll a dice and place that digit into one box on the card.

The aim of the game is still to get the highest number.

So they're gonna produce those numbers by rolling a die.

Three.

So Andeep rolls a three.

He decides to put the three in the tenths column.

Sofia rolls a one and she decides that she's going to put the one in the hundredths column.

Wonder if that's where you would've put it if you'd been Sofia.

Let's see what Andeep rolls on his second go.

Andeep now rolls a four.

Where do we think Andeep is gonna put the four? He decides to put it in its hundredths column.

Maybe we could start thinking about Andeep's strategy.

What is he hoping to roll on his last go? Now, let's see what Sofia rolls on her second game.

She rolls a two.

Where do you think Sofia's gonna put that two? She puts it in the tenths column.

Andeep rolls another four.

He's only got one place to put it and that's in the ones column.

I'd like you to think now, can Sofia win? Is it possible for Sofia to win? Yes, she can, can't she? What numbers could she roll to make sure that she's won? She can roll a five or a six.

Why can't she roll a four? Have a think about why she can't roll a four.

Why is it only a five and a six? She can't roll a four because Andeep already has a three in the tenths column and Sofia only has a two in the tenths column.

We'll now let Sofia roll the dice to see if she did win.

Let's see what Sofia rolled on the next roll.

And remember she wants a five or a six to win, and she rolls a three.

So unfortunately Sofia didn't win, Andeep won.

They played the same game, but this time their numbers must also be between one and four.

Sofia rolls the dice, she goes, first, only fair 'cause Andeep went first in the first game and she gets a six.

Can Sofia put the six in the ones box? She can't put it in the ones box because the number has to be less than four.

Now, on the previous game she would've been really happy to roll a six first I'm sure, but here she can't put it in the ones column.

So she decides to put it in the tenths column.

Andeep rolls a five.

Can Andeep put a five in the ones box? No, for exactly the same reason, it has to be less than four.

So we cannot put the five into the one's box.

So Andeep's decided to put it in the tenths box.

Let's have a look how the game progresses.

Two, Sofia now rolls a two.

Where is she gonna put the two? She decides to put it in the hundredths column.

Why do you think Sofia put the two in the hundredths column? Because 2.

6 is still quite far away, sorry, from four.

So she's thought if I put a two in the ones column, it'd be 2.

6.

That's still quite a long way from four, so she decided to hope that she does better on her next roll.

Andeep rolls a three.

He decides to put that in his ones column.

Why do you think he decided to put that in his ones column? He knows he'll be fairly close to four then, doesn't he? 'Cause it's gonna be 3.

5 something, that's fairly close to four.

Sofia now rolls a four.

She puts it in her only remaining box.

Can Sofia win the game? No, because her number is greater than four.

She's got a higher number, but it doesn't satisfy the rules of the game.

The numbers they got had to be between one and four.

Does Andeep need to throw the last die? No, because he knows he's already won 'cause Sofia can't win because her number is already greater than four.

"Which of the following are between 2.

346 and 2.

351?" Pause the video and decide, and come back when you're ready.

Let's take a look.

B, 2.

35 and D, 2.

34608.

Well done, if you identified those two.

"Which decimal is exactly halfway between two and three?" So now, we are not just thinking about any number that is between two numbers, we're specifically thinking about what number is halfway between them.

So here's two and three on the number line.

And I've marked the halfway point.

The distance between the two numbers I'm trying to find the halfway point is one unit, halfway, will be half a unit larger than two and half a unit smaller than three, which is two 2.

5.

I've noticed that halfway is half the sum of the two numbers.

Do you agree with Andeep? The sum of the two numbers? Remember, we add them together to find their sum, that's five, half of that, he says, half the sum, half of five is 2.

5.

It looks like he might be right.

"Which decimal is exactly halfway between 1.

24 and 1.

38?" So again, here's my number line 1.

24, 1.

38 and I've marked halfway point.

The distance between the two numbers, this time is 0.

14.

I need to find halfway, so I'm gonna half that.

(clears throat) Excuse me.

So that's 0.

07.

So it can either be 0.

07 above 1.

24 or 0.

07 below 1.

38.

I'm gonna end up at the same spot.

So this here is 1.

24, add 0.

07, which is 1.

31.

Andeep says, "Does what I've noticed work here?" Do you think Andeep has spotted something that always works or do you think it was just a bit of a fluke? We'll check it out.

We can check it out, can't we? So we checked it on the previous one, we added 2 and 3, 5, and we halved it, 2.

5, and that was halfway between the two.

So let's check this one.

Sum of the two numbers we're finding in the middle of, 2.

62, and then we're gonna half that, is 1.

31, it's the same.

So yes, it does work.

Well done, Andeep, you've spotted something that might save us a little bit of time.

Now, you are ready to have a go at this check for understanding.

I'd like you please to find the decimal that is exactly halfway between 3.

56 and 3.

94, and you can use either of the methods that we've looked at previously.

So you can pause the video now and come back when you've got your answer.

Good luck.

We'll check now.

I've decided to use Andeep's method because I think that is most useful.

So I've added together my two numbers, which is 7.

5.

I then find half of 7.

5, which is 3.

75.

So that is halfway between 3.

56 and 3.

94.

Now with that calculator, if you'll find it a little bit tricky to do half of 7.

5, I would think of it as money and I would find half of £7 first, that's £3.

50 and half of 50 pence is 25 pence, £3,50, add 25 pence is £3,75.

Although here, I might let you use a calculator.

Now, we're ready for our first task.

You're going to write down at least three decimals between the following pairs of numbers.

So notice it says at least three, you don't have to stop at three.

You can give me more.

And the more you write down, the more impressed I'm going to be.

Pause the video and when you come back, we'll check your answers.

Good luck.

Well done.

We'll now move on and look at question number two.

So question number two.

Here, I've given you some number lines and I've covered up a digit with a cloud.

Your job is to decide what digit the cloud is covering.

Each arrow is pointing to the halfway point.

Pause the video.

Good luck with this.

You'll smash it, I know.

Come back when you've got your eight answers.

Good luck.

Pause the video now.

Great work.

Now, let's check our answers.

So here, I've given you some examples of decimals that would be acceptable, but remember, there are an infinite number of decimals that we could have written.

Question number two.

Now this time, I can only accept one answer because this time we were looking for an exact point, the halfway point.

So in the first one, we were missing a 5, 0.

75, the second one was an eight, I'm going down, so that was 2.

88.

In the third one, I was missing a 5, 5.

55.

And then the bottom one on the left hand side, was missing a 1, 6.

195.

Starting at the top of the right hand, sorry, at the top of the right hand column, we were missing a 0, it was 10.

95.

And the second one we were missing a 2, 4.

62.

On the third one we were missing a 2, 14.

25.

And then the final one we were missing a 5, 0.

0054.

How did you get on with those questions? I hope you did well.

Maybe if you didn't get any answers right, you could pause the video and see whether you can work out where you went wrong.

Now, let's move on.

We're now going to look at finding numbers between two proper fractions.

And remember we said a proper fraction is a fraction where the numerator is less than the denominator.

Andeep and Sofia are sticking with us to help us through this.

We're gonna find a fraction between 1/4 and 3/4.

Andeep says, "I think it's 2/4." And I'm choosing half, Sofia says.

Have Andeep and Sofia, both chosen a fraction that is between 1/4 and 3/4? They have both chosen a fraction between 1/4 and 3/4.

They have chosen equivalents of the same fraction.

1/2 is equivalent to 2/4.

Andeep and Sofia are now trying to find a fraction between 1/5 and 2/5.

Hmm.

Andeep says, "I think 3/8." And Sofia says, "I'm choosing 11/25." This is a much harder problem, isn't it? So let's have a think about how we're going to work through this.

'Cause it's not obvious here to see if they're right, is it? It is not obvious.

What could we do to check? We could use equivalent fractions and a common denominator.

So here are the four fractions we had on the previous slide.

We were trying to find a fraction that was between 1/5 and 2/5.

And then we have Andeep's answer of 3/8 and Sofia's answer of 11/25.

We need to find a common denominator of all of the fractions.

Prime factors of the denominators are.

Already, as prime factors, 5 is, 25 is 5 multiplied by 5, and 8 is 2 multiplied by 2, multiplied by 2.

The lowest common multiple of 5, 25 and 8 is 2 multiplied by 2, multiplied by 2, multiplied by 5, multiplied by 5, which is 200.

We've done finding lowest common multiple in a previous lesson.

So if you need to, you could go and re-find that lesson if you need to recap on what I've just done there.

We're going to change them all now then so that they have a common denominator of 200.

And again, we've done this in previous lessons, so if you need to go back and recap that, you can.

I'll talk you through the first one and then I'll go through the others a little bit swifter.

So here we want the denominator to be 200.

5 multiplied by what makes 200, that's 40.

Remember not to change the value, we cannot change the value of one fifth because we've got those equal symbols.

We're saying they're equivalent so therefore the only thing we can multiply by is 1, So the numerator must also be 40 because 40 over 40 is 1.

That means we're multiplying by one so we haven't changed the value of 1/5.

1 multiply by 40 is 40.

Now, like I said, I'm going to put the others up in steps, but I'm not going to talk it through because you are super confident with this because we have done this multiple times in the lessons leading up to this.

Now we can clearly see that 1/5 is 40/200 and 2/5 is 80/200.

So is Andeep's fraction between the two? Andeep's fraction is between them because 75 is between 40 and 80.

And we've changed the denominator to be the same.

What about Sofia's? Sofia's fraction is not between 1/5 and 2/5 because she's gone above the 80, which was the upper limit for 2/5.

"Which of the following fractions are between 1/3 and 2/3?" Now you can pause the video to decide which of those are between 1/3 and 2/3.

Good luck.

Now let's check.

You should have 1/2 and 4/9.

Did you identify both of those? Did you impress me with your workings that I know that you didn't use a calculator or even worse, just guess? You did, brilliant.

"What fraction is halfway `between 4/11 and 5/11" So now we are homing in again to not just looking at any fraction that is between them, but what's halfway.

So using our number line, just as we did with the decimals in the previous learning cycle, why can't we just find the middle of four and five? I'm gonna pause while you think about that question a moment.

What would give us a decimal numerator halfway between 4 and 5 is 4.

5, and we know for a proper fraction that we cannot have decimal parts as numerator or denominators.

We could however, use equivalent fractions.

What would that look like? We've got 4/11 and 5/11.

I'm going to multiply both of those by one.

I'm going to use two over two.

So I got 8/22 and 10/22.

Now, can you see, which number is halfway between the two or which fraction is between the two? It's 9/22.

Halfway between 8/22 and 10/22 is 9/22.

What fraction is halfway between 4/7 and 2/3? Here's our number line.

This time, I'm going to change them into a common denominator, which is 21 because the lowest common multiple of 7 and 3 is 21.

There are my calculations.

So it's 13, 12/21, and 14/21.

The middle would be 13/21.

Gonna do one more together and then I'm confident you'll be ready to have a go at one independently.

So again, which fraction is halfway between? 3/13, multiply both by 2/2.

So that gives me 6/26 and 8/26.

So halfway is 7/26.

Right, you give it a go now.

Pause the video, come back when you're ready.

Great work.

Hopefully, you decided to change them into fractions with a common denominator of 22.

You may have chosen something different and then we can find that the halfway point is 17/22.

Now, we can have a go at Task B.

So in question number one, I'm looking for you to write me down, please, a fraction that is between the two fractions or the number and the fraction that I've given you.

It doesn't have to be halfway, just any fraction that is between the two.

Pause the video now.

Good luck, come back when you're ready.

Super work, well done.

Now, we're going to make that little bit more challenging because this time I am gonna ask you specifically for the fraction that is halfway between the two.

Now, these do get a little bit harder as they go down, but stick with it.

Remember, you've got all of the skills to be successful at this and I know you can be.

You can pause the video now and I'll look forward to seeing you when you come back with those answers.

Let's check in then.

Let's check those answers.

Question number one, we've got some examples here.

So these are examples of fractions that are between them.

So I'm not going to read them out.

You may have something different.

What you could do is you could pause the video, you could change them all into decimals to make it easier to compare them with your calculator.

Question number two.

Now, here you have to have these answers because there can't be more than one point that is exactly halfway between two fractions, A, 3/5, B, 13/20, C, 19/16 D, 7/24, and E, 33/80.

Now, you may have an equivalent to those.

If you've got something different, you could use the fraction button on your calculator to simplify your answer and to see whether it simplifies to the ones I've given you on the screen because those are all in their simplest form.

Now we can review our learning from today's lesson.

It is always possible to find a number between two numbers.

We can end up with lots and lots and lots of digits after the decimal point, but it's always possible.

To find halfway between two numbers, you find the sum of the numbers and half that.

For example, halfway between 0.

26 and 0.

37, we add those two together and then we half it, to give us 0.

315.

Often when finding infractions between two fractions or halfway between two fractions, it may be helpful to find equivalent fractions.

So for example, in the question we looked at with 8/11 and 9/11, we converted them so that the denominator was common and it was 22, and then we could find the halfway point between them.

Thank you so much for joining me today, I've really enjoyed it, and I hope to see you again very soon.

Bye.