Lesson video

Hello there.

My name is Mr. Tilston.

I'm a teacher and although I teach all of the different subjects, the one that I enjoy the most is definitely maths.

So it's a real treat and a real privilege to be here with you today to teach you this lesson, which is all about mean averages.

And that might be something you're getting quite confident with already.

So if you're ready to begin, then let's begin.

The outcome of today's lesson is this.

I can explain how the mean changes when the total quantity or number of values changes.

Our key words, we've got two.

If I say them, will you say them back? Are you ready? My turn.

Mean average.

Your turn.

And my turn.

Set of data.

Your turn.

You might have encountered those words very recently, but let's have a little reminder about what they mean.

The mean average is a single number expressing the typical value in a set of data, and it is calculated by finding the total of the set of data and dividing by how many values there are.

So adding first, dividing second.

A set of data is a collection of facts such as numbers, words, measurements, observations, or even just descriptions of things.

Our lesson today is split into two parts or two cycles.

The first is changing the value of the set and the second changing the number of values in the set.

Let's begin by thinking about changing the value of the set.

In this lesson, you're going to meet Lucas, Laura, Jacob, Sam, Jun, and Aisha.

Have you met them before? They're here today to give us a helping hand with the maths, all of them.

Laura, Lucas and their friends are collecting superhero cards.

So we've got a little table there showing that.

So Laura has got 8 cards, Lucas, 3, Jacob, 5 cards, and Sam, 4.

So we have 20 cards in the whole set.

If you add those numbers together, it equals 20.

So all of those cards added together is 20.

So that's our total of set.

How do we then calculate the mean average? What do we do with that number 20? Well, we divide the total of the set of data by the number of values in the set.

So in this case, that's 20 divided by 4 because there are 4 children.

So the mean average is 5.

20 divided by 4 is equal to 5.

Laura says, "5 what?" Well, 5 is the number of cards we would each get if we divided the whole set equally.

If somebody asked about how many cards do you each have, then 5 would be a reasonable answer.

Some have got fewer than 5, some have got more than 5, but 5 is a reasonable answer.

So the mean average is one number to express the typical value of the number in a set.

Lucas has given another pack of 4 cards.

What will happen to the mean average? So Lucas has got 3 cards in the minute and he is given another pack of 4 cards.

What's going to happen? Well, Laura says, "We will need to calculate the mean average again now." Hmm, could do.

Can you think of a different way, a more efficient way? What do you notice? Well, Lucas says, "I'm not so sure we need to." You're right, Lucas.

"My 4 more is 1 more for each of us." Yeah, one for you, one for you, one for you, one for you.

That's 4, if we are sharing equally.

"Oh," says Laura, "so for every 4 cards we add, the mean average will increase by 1." Yes, and we can see that by doing the divisions.

So we started with 20 divided by 4 is equal to 5 when we had 20 cards.

So 5 each and then we had 24, 'cause there's 4 more cards.

So 24 divided by 4 is equal to 6.

So that's 6 cards each.

If I got 4 more cards, the mean average would increase to 7.

Yes it would, and you could keep going.

Or if I lost 4 cards, the mean average would decrease by 1.

Absolutely.

Let's have a little check.

What is the missing mean average here? Can you reason about the missing value without doing the calculations again? Have a good look at that.

Have a good think about that.

If you've got a partner with you, discuss it with them.

Pause the video.

How did you get on? So Laura says, "Jacob and Sam both have 4 more cards.

So the total will increase by 8." For every 4 cards we add, the mean average will increase by 1.

So the mean average will increase by 2.

So there we go.

The total set is now 32, and then 32 divided by 4 is equal to 8.

So well done if you got that.

It's time for some practise.

Number one, the table shows the number of cards each child currently has, the current total and the mean average.

For each scenario below, work out the mean average if.

So, let's have a look at the table to start with.

We've got the name in the left hand column and the number of cards in the right.

So Laura's got 8, Lucas has got 3, Jacob's got 5, Sam's got 4, Jun hasn't got any, Aisha has got 4.

Now if we add all of those numbers together, that gives us the total of the set.

That's 24.

And the mean average is 4, 'cause 24 divided by 6, there's 6 children there, 24 divided by 6 is equal to 4.

Now what would happen if Laura lost 6 cards? So at the minute she's got 8 cards.

What if she lost 6? What would that do to the total of the set? Now just consider for a second, the number 6 and the significance of the number 6 there.

What about b, if Jun got 12 more cards? And again, think about the number 12, something special about it in this case.

So what if he got 12 more cards? How would that change the total of the set and how would that change the mean average? What if Jacob gave one of his cards to Sam? Hmm.

Think about what would happen then.

D, what about if Lucas and Aisha got 3 more cards each? What would happen to the total? What would happen to the mean average? E, what if Sam got 6 more cards, but he gave two of them to Jacob? How would that change things? And f, what if Lucas got 3 more cards? What would that do to the total? What would that do to the mean average? Have a really good think about those and see what you notice before you dive in.

You might be able to make that very easy for yourself.

Okay, if you can work with somebody else and you're allowed to do that, I always recommend that.

Then you can bounce ideas off each other and help each other out if something goes wrong.

Pause the video and away you go.

Welcome back.

How did you get on? Well, let's have a look at a.

If Laura lost 6 cards, there would be a total of 18 cards instead of 24.

And then the mean average would drop to 3 because 6 of the cards have been lost.

What about if Jun got 12 cards? Now 12 is a multiple of 6.

So there would be a total of 36 cards and then the mean average would rise to 6 'cause every 6 cards there added, the mean average is going to rise by 1 in this case.

See if Jacob gave one of his cards to Sam, well, Jacob and Sam would've different numbers of cards, but the total of the set wouldn't change.

It's not like there's more cards than there were before.

And it's not like there's fewer cards than there were before.

There's the same number just distributed differently.

So the mean average would still be the same.

That doesn't affect it.

And d, if Lucas and Aisha got 3 cards each, well that'd be 6 cards altogether.

So the mean average will be one more 'cause we'd be dividing by 6.

So the new mean would be 5.

And well done if you are able to say that without actually calculating.

And e, Sam got 6 cards and gave two of them to Jacob.

So Sam and Jacob would have different numbers, but the total will only increase by 6.

So the mean average has gone up by 1 because there's 6 children.

F, if Lucas got 3 more cards, well 3 more cards means that each child would get one and a half more if the cards were shared equally.

So the mean average would increase by 0.

5 or 1/2 to 4.

5 cards each.

This would not change the number of cards they could each play within the game.

There would be 3 cards left over.

So there's quite a lot to think about with that last one, wasn't there? Okay, you're doing really, really well and I think you're ready for the next cycle, which is changing the number of values in the set.

It's wet play and the four children are halfway through a game when Aisha comes in and asks to play.

So before we think about Aisha, look at the children playing at the moment.

How many children? There's 4.

"We can put all the cards together and divide them equally again between the 5 of us," she says.

"I hope we have a total number of cards that's a multiple of 5." "Yes, we have 40 cards.

40 divided by 5 is equal to 8.

So we can start again with 8 cards each." So here we go.

Here's Aisha.

She's joining in.

So the number of values now is not 4 anymore, it's 5.

There are 5 children and they've got 8 cards each.

Before Aisha came into play, what was the mean average number of cards when the children were playing in a group of 4? Pause the video.

So when the children were playing in a group of 4, that's the number of values in the set, 4.

While there's 40 cards altogether, that's a total of our set of data of 40 and we're dividing 40 by 4 in this case.

So 40 divided by 4 is equal to 10.

So that was a mean average.

That's the number of cards they got each.

10 cards.

Well done if you said that.

Now Jun comes in and asks if he can join in.

"We can put all the cards together," says Laura, "and divide them equally again, between 6 of us." This time there's going to be 6 players.

"This could be a problem as 40 is not a multiple of 6." It's not, is it? So what's going to happen to the other cards, I think, and what's going to happen to the mean average? What do you think? "What can we do," says Laura.

Well, "There are 6 of us and 40 divided by 6 is equal to 6 remainder 4.

So 6 each and then 4 left over." So 4 that they just maybe won't play with.

Here we go.

Now it's still fair.

They've just got a few cards left over.

That's a good solution to that problem, isn't it? The mean average is not a whole number this time.

What number of cards under 100 could the children have had so that the mean average would've been a whole number when 4, 5 and 6 of them were playing? And is there more than one answer? There's quite a bit to think about there.

So do take your time, read it again if you need to make sure you've understood the question.

And if you can work with somebody else, please do and pause the video.

Did you manage to persevere there and find an answer? Let's see.

Well, Laura says, "We need a number of cards that's a multiple, 4, 5, and 6." So any number that's a multiple 4, 5 and 6.

Lucas says, "It must be a multiple of 10 as multiples of 5 with 5 in the ones are odd numbers and multiples of 4 and 6 are even." That's good deduction.

Yes.

So 60 works.

It's 4 lots of 15, 5 lots of 12 and 6 lots of 10.

So you could play with 60 cards and share them equally.

"I think that's the only answer less than 100." So do I, Lucas.

Well done if you got that.

So we can see that if 4 of them are playing, each has 15 cards.

If 5 of them are playing, each has 12 cards.

And if all 6 are playing, each has 10 cards.

So 60 is a number of cards that works.

Laura and Lucas are investigating mean averages.

There are packs of cards with a different number of cards in each box.

So we can see one's got 18, another's got 26, another's got 34, another pack's got 42, another pack's got 51, and their final pack's got 49 cards.

They need to decide which boxes they could use to play the game with 4, 5 or 6 children.

They have to find a total where the mean average for 4, 5, or 6 children playing is a whole number.

Well, let's have a look.

Lucas says, "I've chosen the first 3 boxes of cards." I wonder why.

18 plus 26 plus 34 is equal to 78.

"I know it's a multiple 6 as it's an even multiple of 3." So Laura's thinking about her tests of divisibility.

That is true.

"It has a digit sum of 15." So when we add those digits together, it equals 15.

7 plus 8 is equal to 15.

"Yes," as Lucas, "if the digit sum is a multiple of 3, the number is a multiple of 3." "It isn't a multiple of 5 and it isn't a multiple of 4.

78 plus 2 is equal to 80." "78 divided by 6 is equal to 13.

So the mean average would be 13," if we combine those cards together.

"If 6 of us play with those packs, we will start with 13 cards each and none leftover.

So that works." Good work, Lucas.

Let's do some practise.

Have a go at the investigation.

So number one, decide which boxes could be used to play the game with 4, 5, or 6 children.

Find a total number of cards with a mean average for 4, 5, or 6 children playing is a whole number.

So think about your test of divisibility as we just saw there.

And number two, what if 7 and 8 children wanted to play? Can you find totals using more than one pack that will work for them? Now, before you start this, just realise there are lots, lots and lots of possible answers here.

So if you've got one answer, don't stop there.

Keep going and find as many as you possibly can.

This is very open-ended.

Once again, if you can work with a partner, please do.

Pause the video.

Good luck with that.

And away you go.

Welcome back.

How did you get on with that investigation? Did you get lots of possible answers? On number one, the last two boxes, total 100, which is a multiple of 4 and 5.

So 4 children would start with 25 cards each.

The mean average and 5 children would start with a mean average number of 20.

The first 4 packs give a total of 120, which is a multiple of 4, 5, and 6 giving mean averages of 30, 24 and 20.

So the cards could be shared out equally.

And number two, what if 7 and 8 children wanted to play? Can you find totals using more than one pack that will work for them? So 98 is a multiple of 7.

So they could use the packs of 26 and 72 added together, the mean average, starting number of cards would therefore be 14.

Meanwhile, 232 is a multiple of 8.

So they could use all the cards except the pack of 18.

The mean average starting number of cards would then be 29.

Very well done if you were successful with that task.

There was a lot to think about there, a lot of arithmetic to do, perhaps a lot of perseverance.

So well done to you.

We've come to the end of the lesson.

Today, we've been explaining how the mean changes when the total quantity or number of values changes.

So in the first cycle, we looked at the total quantity changing and the second, the number of values changing and how that's affected the mean average.

If the number of values in the set of data stays the same and the total increases as in the first cycle, then the mean average increases.

If the number of values in the set of data stays the same and the total decreases, then the mean average decreases.

If the number of values in the set of data changes, the mean average has to be recalculated.

Very well done on your accomplishments and your achievements today.

You've been a superstar and you deserve a pat on the back.

Go on, treat yourself and say, "Well done, me." I really do hope I get the chance to spend another math lesson with you at some point in the very near future.

But until then, have a fabulous day, whatever it is you've got in store and remember to be the best version of you that you can possibly be.

You can't ask for more than that.

Take care and goodbye.