Lesson video

Hi there.

Mr. Robson here.

Delighted that you've made the choice to learn maths with me today.

I love maths.

I love teaching maths.

I'm so looking forward to working with you today.

Okay, let's go.

The outcome for today's lesson, I'd like you to be able to express place value column headings using both exponential and fractional representations.

Now there's gonna be some words in that statement that are very familiar to you, place value column headings.

You'd have seen those before a lot.

You'd be very familiar with them.

But maybe exponential and fractional representations might be slightly new to you.

So don't worry about that.

We're gonna go into that in quite a lot of detail.

So the keywords, exponential form, they're gonna form keywords.

When a number is multiplied by itself multiple times, it can be written more simply in exponential form.

An example of that would be, we could say 2 times 2 times 2, but in exponential form, we would say 2 to the power of 3.

That just makes the number a lot easier to write.

That's 2 times 2 times 2 written as 2 to the power of 3, that's its exponential form.

Our second keyword, fractional form.

What do we mean by fractional form? A number's in its fractional form when it's displayed as one integer divided by another integer.

Integer's a word that I hope you're very familiar with.

An example of a fractional form would be 3 over 4, 3 being an integer, 4 being an integer, 3/4 being a number in fractional form.

So the lesson outline.

We're gonna start by looking at place value headings expressed in exponential form and then the second half of the lesson, we'll look at when they're expressed in fractional form.

Let's start with the exponential form.

I'd like you to pause this video and consider for a few seconds what do these numbers have in common? I wonder what you said.

I wonder if anybody suggested that they're multiples of 10? Multiples of 10, well, 100's a multiple of 10, 1,000's a multiple of 10, a million is a multiple of 10.

10 itself is a multiple of 10, but 1? No, 1 not a multiple of 10.

So there's something else going on here.

These numbers aren't multiples of 10.

They're powers of 10.

What I mean by powers of 10, powers are when a number is repeatedly multiplied by itself.

So 100 being 10 times 10, 1,000 being 10 times 10 times 10, and so on up to a million, which is 10 times 10 times 10 times 10 times 10 times 10.

Whew, you'll notice I've omitted 10 and 1 from that list.

That's for a reason.

We'll deal with those shortly.

As a reminder, exponential form, when a number is multiplied by itself multiple times, it can be written more simply in exponential form.

So does that mean we could do that with these numbers? Yes, we can add a third column.

Instead of its expanded form or its numerical form, we could write it very concisely in exponential form.

10 times 10, 100, we'll write that as 10 to the power of 2.

1,000, 10 times 10 times 10, that's 10 to the power of 3.

The next row, 10 to the power of 4, 10 to the power of 5.

You guessed it, 10 to the power of 6.

But what's going to go in those two positions? Pause this video and tell the person next to you or perhaps just say it aloud to yourself.

I wonder what you said.

Maths is logical, it's sensible.

It has a order.

You want it to be 10 to the power of 1 and 10 to the power of 0, 10 to the power of 6, 5, 4, 3, 2, 1, 0.

It should be 10 to the power of 1 and 10 to the power of 0.

And it is.

That might please you.

It's logical.

It has order.

And that's what maths is, isn't it? But why, in order to explain those top two, the third one's all right.

100, 10 times 10, that's 10 to the power of 2, but 10 being 10 to the power of 1, 1 being 10 to the power of 0, they take some considering.

What do you think I would write in those two positions? Pause the video.

Have a think.

Tell the person next to you or just say it aloud to yourself.

In order to explore what's gonna go in those spaces, I'm gonna communicate the other numbers in a slightly different way.

Rather than thinking of 100 as 10 times 10, I'm gonna think of 100 as 1 multiplied by 10 twice.

It still makes 100.

1,000, 1 multiplied by 10 three times, 1 multiplied by 10 four times to get 10,000, 1 multiplied by 10 five times to get 100,000, 1 multiplied by 10 six times to get a million.

That explains why we can express 10 as 10 the power of 1, because it's 1 multiplied by 10 just the once.

And then how would you make 1? Well, you would take 1 and not multiply it by 10 at all.

That's why we end up with 10 to the power of 0 representing 1 and 10 representing 10 to the power of 1.

I just wanna check that you've understood some of what I've said so far.

I'll ask you to pause this video and think about this question.

Exponential form is a way of writing repeated what? Is it addition of the same value, subtraction of the same value, multiplication of the same value, or division of the same value, press Pause.

That's right, exponential form is a way of writing repeated multiplication of the same value.

It's exponential form, which enables us to write the number 1 million as 10 to the power of 6 rather than 10 times 10 times 10 times 10 times 10 times 10.

Did I say the right number of 10s? We just say 10 to the power of 6 and then we know.

Let's check you've understood this.

What is the value of 1,000 in exponential form? Is it 10 to the power of 4, 10 to the power of 3, 10 times 3 or 10 times 10 times 10? Pause this video and write down which one you think it is.

That's right, it's 10 to the power of 3.

It's not 10 to the power of 4.

Some of you might've wrote that, because there's four digits there and you want to write 10 to the power of 4 to represent four digits.

That's not the case.

10 to the power of 3 represents 1,000.

Not 10 to the power of 4.

What about the difference between 10 to the power of 3 and 10 times 3? They look really similar, but they're very different things.

10 times 3 is 30.

You know that.

You've known that for a long time.

10 to the power of 3 is very, very different.

Powers are powerful.

That multiplies 10 by itself three times, 10 times 10 times 10, thus making 1,000.

So why's it not the fourth answer? Why can't I have ticked 10 times 10 times 10? Well, it's because of the keyword, exponential form.

10 times 10 times 10 has the value of 1,000.

It's just not in exponential form.

It's only 10 to the power of 3, that's the only point at which it's an exponential form.

Okay, on with our explanation.

The purpose of getting you to turn a million into 10 to the power of 6 or 1 into 10 to the power of 0 is to just make a little more simple our place value chart.

You'll have seen this before.

It'll be very familiar to them.

Place value being, when we communicate numbers, is that in the ones column, the 10s column, the 100s number, the 100s column, the 1,000s column? You'll have seen this.

Have you seen it represented like this though? Instead of writing a 1 followed by six 0s, we'll write 10 to the power of 6.

Instead of writing 100,000, we can write 10 to the power of 5, 10 to the power of 4, 10 to the power of 3, 10 to the power of 2, 10 to the power of 1, 10 to the power of 0.

How much more clear and concise is it to write 10 to the power of 6 instead of a million? 10 to the power of 5 instead of 100 thousand? We like, we mathematicians, we like to be clear and concise in our communication.

Expressing these place value headings in exponential form allows us to be really clear and concise, especially, imagine numbers beyond this table.

Imagine when you get to the billions, the trillions.

Well, I don't wanna write all those 0s and neither do you.

That's why exponential form here is super useful.

Okay, let's check that you were paying attention there.

I've written the same chart, but I've left a few moments blank.

Pause this video, take a moment, and write down what have I missed? There you go, for the millions, I omitted 10 to the power of 6.

For 10,000, I omitted 10 to the power of 4, and then the two peculiar ones we explored earlier.

The 10s column, we call 10 to the power of 1, the 0, the ones column, sorry, we call 10 to the power of 0.

The ones column is 10 to the power of 0.

Next, what's the value of each digit in this number? Wait on one minute, Mr. Robson.

8,352 broken down into 8 in the 1,000s column, 3 in the 100s column, 5 in the 10s column, two in the ones column.

We know that.

We've known that for years.

I know, I know.

But can we express these in exponential form? Instead of saying 8,000 or eight 1,000s, we could say 8 times 10 to the power of 3.

Instead of saying three 100s, we could say 3 times 10 to the power of 2, and so on, and so on.

Your turn, what's missing from the composition of this number? See if you can fill in that third row.

Pause this video.

Write down what you think goes in those gaps.

Four 1,000s we could represent us four lots of 10 of the power of 3.

30, three 10s, that's three lots of 10 of the power of 1, and then five ones, that's five lots of 10 of the power of 0.

Okay, working forwards in mathematics, taking a number and breaking it down into the value of each digit is interesting, but working backwards in mathematics, well, that's when things get really interesting at times.

So what if I gave you this number? Eight lots of 10 of the power of 3, two lots of 10 to the power of 2, seven lots of 10 to the power of 1, and six lots of 10 to the power of 0 and said, what is that number? Well, what is that number? Pause the video.

Say to the person next to you or say it aloud to yourself.

It's 8,276, you would've seen at the bottom of the screen that I included a place value chart because it really helps.

It helps us to see that the 8 times 10 of the power of 3 is in that column.

Two lots of 10 to the power of 2 in that column.

Seven lots of 10 to the power of 1, six lots of 10 to the power of 0.

When we put everything in their respective place value columns, we get the number 8,276.

He's changed something.

Mr. Robson's changed something.

Pause the video.

Tell the person next to you or say aloud to yourself.

What is it I've changed in this example? Pay particular attention.

Well done, they're not consecutive powers of 10.

10 to the power of 5, 10 to the power of 4, 10 to the power of 3, 10 to the power of 0.

I've missed a couple of columns.

This number is 827,006.

An 8 in the 10 to the power of 5 column, a 2 in the 10 to the power of 4 column, a 7 in the 10 to the power of 3 column.

Then nothing in the 10 to the power of 2 column, nothing in the 10 to the power of 1 column, and a six in the 10 to the power of 0 column, giving us 827,006.

If you spotted that, well done.

Okay, just to check that you've understood what we've been through in the last few minutes, what is this number? Eight times 10 to the power of 6 plus 2 times 10 to the power of 5 plus 7 times 10 to the power of 4 plus 6 times 10 to the power of 3.

Pause this video, write down that number.

Let me throw a spanner in the works.

Alex gave it a go and Alex wrote 8,276.

Did you write that? Did you write something different? Alex is wrong.

What's Alex done wrong? If you spotted that the real number was 8,276,000, well done.

Could you communicate what Alex has done wrong? Can you see why he wrote what he wrote? And can you see why 8,276,000 is in fact correct? Can you pause this video and write a sentence, try and explain the error that Alex made? There are lots of ways that you could've explained it, that Alex has just written the digits 8, 2, 7, and 6 in order.

He hasn't considered the place value.

Whereas if you rightly wrote 8,276,000, you were considering correctly that it's not an 8,000, it's an 8 in the 10 to the power of 6 column.

It's 8 million, eight 10 to the power of 6s, well done.

Okay, time for a practise task.

In the first task, I'd like you to fill in the blanks in this table.

1, 10, 100, 1,000 10,000, 100,000, million.

Fill in the blanks in the column labelled exponential form.

I'd like to pause this video and take a moment to write that down.

Question two and question three.

Question two, I'd like you to fill in the blanks.

The number 372,500.

372,500, that's not a 2.

That's a 2 in the 10 to the power of 3 column.

So what would I write to express the value of that three, that seven and that five using exponential form? Question three, I'd like you to write those numbers out as ordinary numbers, not in their exponential form.

You pause this video and give it a go.

Okay, time for some feedback.

I gave you 100 is 10 to the power of 2 in the exponential form column.

I hope you filled in the rest as 10 to the power of 0, 10 to the power of 1.

I gave you 10 to the power of 2, 10 to the power of 3, 10 to the power of 4, 10 to the power of 5.

10 to the power of 5, 1 million being 10 to the power of 6.

At the other end of the scale, 1 being represented by 10 to the power of 0.

How about filling in the blanks for 372,500, how would I express the value of those digits in exponential form? Well, I hope you wrote that that 3's in 10 to the power of 5 column, 7 that's in the 10 to the power of 4 column, the 2 I gave you in the 10 to the power of 3 column, and the five is in the 10 to the power of 2 column.

Okay, how 'bout those numbers? 2 times 10 to the power of 3 plus 5 times 10 to the power of 2 plus 8 times 10 to the power of 1 plus 4 times 10 to the power of 0, that would be 2,584.

The bottom two questions look ever so slightly similar, but ever so slightly different.

If you spotted the difference between these two, well done.

The first one's 4,830,000.

We have a 4 in the 10 to the power of 6 column, an 8 in the 10 to the power of 4, 10 to the power of 5 column, and a 3 in the 10 to the power of 4 column.

Now pay close attention to the powers on the bottom question because that moves the digits eight and three into very different positions.

The four remains in the 10 to the power of 6 columns, it's still 4 million and, but the 8 has moved to the 10 to the power of 3 column, and the 3 has moved to the 10 to the power of 0 column.

How did you do? Did you make any errors? It's not a problem if you did.

I made plenty of errors when I was learning maths at your age.

Maybe just pause the video and consider what have you written that's different to what's on the screen and how is what's on the screen correct? That would be a useful thing to do if you've made any errors.

Okay, that's place value headings expressed in exponential form, what's next? Let's look at what we mean when we say fractional form.

I'd like to start by following this pattern, 1, 10, 100, 1,000, 10,000, 100,000.

Mr. Robson, is this not the first half of the lesson? Haven't we done this already? Okay, I was just checking that you were paying attention.

What's happening here? We're multiplying by 10 each time.

1 times 10 is 10, multiply that by 10 to get 100, multiply that by 10 to get 1,000, repeatedly multiplying by 10.

What if I turn this around? 1,000, 100, 10, 1, what's happening now? Pause this video and tell the person next to you or say it aloud to yourself.

Well spotted, we're dividing by 10 each time.

Okay, what's next? If this pattern continues, what's coming next below that one? Pause the video, write it down.

What do you think's coming next? 1 divided by 10, you might have written 0.

1, divide that by 10, you might have written 0.

01.

You may have written something slightly different.

I'll come to that in a moment.

When we follow this pattern and go beyond one right the way down to 0.

001, all of a sudden we're not talking about integers anymore.

We're talking about non-integers.

Now you might have written something slightly different when that pattern continued.

1 divided by 10 is not 0.

1, 1 divided by 10 is also 1 divided by 10, which we would write as 1/10.

And then when you divide that by 10, 1/100, and when you divide that by 10, 1/1,000.

So you might have written them in decimal form, you might have written them in fractional form.

Now it's my hope that you'll go on and do A-level maths in the future, in which case, fractional form is the best way to communicate.

So getting used to communicating in fractional form is a really powerful skill.

Just wanna check that you've understood what I've said in the last minute or two.

So a true or false for you.

0.

01 and 1/100 have different values, true or false? How might you justify your answer? Would you say, well, one is a decimal, one is a fraction, or would you say they both have the value of 1/100 of 1? I'll give you one minute to think about this.

Pause the video.

I hope you said false.

0.

01 and 1/100 have different values, now that's false.

It's fair to say that one is written in decimal form and one is written in fractional form, but that's not a reason why they'd have different values.

They have the exact same value.

They both have the value 1/100 of 1.

Okay, how does this look in our place value chart? In the first half of the lesson, we saw the ones column be replaced by 10 to the power of 0, the 10s column to 10 to the power of 1, the 100s columns to 10 to the power of 2.

When we start to express place value headings in fractional form, it's quite literal.

The 10s column is 1/10, the 100ths column is 1/100, and the 1,000th column is 1/1,000.

Isn't maths wonderfully logical sometimes? As a reminder, a number's in its fractional form when it's displayed as one integer divided by another integer, like 1 over 10, 1 over 100, 1 over 1,000, et cetera.

I just wanna check that you've got all that.

Pause the video and write down the little moments I've missed from this place value chart.

I hope you wrote 100ths in that column and then represented it as 1/100 in fractional form.

I hope you've replaced that space with a TH to represent 1,000ths.

That's 10s of 1,000ths.

And then that last column, 100s of 1,000ths.

That's 1 over 100,000.

That's that column expressed in fractional form.

So place value you're very familiar with.

If I said 8 100s, eight lots of 100, eight lots of 10 to the power of 2, you'd put an eight there, but you'd know it's not finished.

We can't just write that eight there and leave it.

We put in the zeros as place value holders.

So what if I don't say eight 100s? I say eight 100ths.

Well, we put the eight in the 100ths column.

That makes sense.

I could say it more simply as eight 100ths.

But if you put the zeroes in the place value positions, you can see in decimal form 0.

08.

You're perfectly welcome to communicate that as 0.

08, but like I said, we like to communicate using fractions, so eight 100ths in it's fractional form is a really powerful way to communicate that number.

Okay, so value of each digit in this number, 0.

352.

That's 0.

3, it's 0.

05 and 0.

002.

That's a three In the 10ths column, it's a five in the 100ths column, it's a two in the 1,000th column.

Oh, and I can express it like that in fractional form A 3 in the 10ths column, A 5 in the 100ths column, a 2 in the 1,000th column.

And I could simplify that to 3/10s, 5/100, 2/1,000.

That's the value of those digits expressed in their fractional form.

I'd like to check that you followed that.

I'd like you to write the value of the digits six and three in this number in their fractional form.

The number being 0.

4603, and I'll give you a starter.

That 4 is 4/10.

What's the value of the 6 and the 3? Pause this video and write those two numbers down.

Okay, did we look and say, well, I've got a 6 in the 100th column.

I've got nothing in the 1,000th column and then a 3 in the 10,000th column.

If you did write that, lovely.

If you didn't write that and you wrote something different, just pause the video, have a little read, consider why is it three 10,000ths in that position? It's really important to analyse your errors in maths and it's okay to make them.

Next a little task for you and it's our last task for this lesson.

I'd like you to start question one, write the value of each non-zero digit in fractional form, the number 0.

01204.

It's the digit one.

It's not got a value of one.

What's its value in fractional form? And then question two, I'd like you to write that number in decimal form, 4 plus 4/10 plus 1/100 plus 7,000th plus 700,000ths.

What does that look like in decimal form? Pause this video.

Write those two down.

Question three 300 is bigger than 30.

Yep, there's more zeros.

That's why 0.

03 is bigger than 0.

3.

Okay, I see why Jun has made that misunderstanding.

Do you think you could write a sentence to justify why Jun is wrong? You might want to make reference to fractional form in your explanation.

Pause this video and write a sentence to explain why Jun is wrong.

Okay, let's go through some answers.

The value of each non-zero digit in fractional form in the number 0.

01204, well, I see nothing in the ones column.

I see nothing in the 10ths column.

I see a 1 in the 100th column.

I see a two in the 1,000th column, and then nothing in the 10,000ths column, so that form must be in the 100,000ths column.

Question two, write this number in decimal form.

Well, that's a 4 followed by a 4 in the 10ths column, 1 in the 100ths column, a 7 in the 1,000th column, nothing in the 10,000th column, and a 7 in the 100,000ths column, giving us the number 4.

41707.

How did you do? Again, it's fine to make errors, just pause, and look and can you understand why what is on the screen is correct/ Next, how did we get on explaining to Jun justifying why that statement is incorrect? It's, well, it feels correct.

There's more zeros, that makes numbers bigger.

Notice I'm not comfortable with that explanation.

A nice way to justify why Jun is wrong is to use fractional form.

Your explanation might have included that 0.

3 is greater than 0.

03 because 0.

3 is a 3 in the 10ths column, whereas 0.

03 is a 3 in the 100ths column.

We're then comparing 3/10 to 3/100.

10ths are bigger than 100ths.

Would you like 1/10 of a pizza or 1/100 of a pizza? Well, assuming I'm hungry, I'll take the 10th please.

10ths are bigger than 100ths, so 3/10 is greater than 3/100, and this is the joy of fractional form.

Is that not more logical to explain it in that way? Okay, I hope you've enjoyed today's lesson.

I have, to summarise, place value headings like 1,000s, 100s, 10s, and ones can be written in exponential form.

For example, 1,000 is 10 to the power of 3, 100s is 10 to the power of 2, 10s is 10 to the power of 1, and ones is 10 to the power of 0.

And then non-integer place value headings in our place value chart, 10ths, 100ths 1,000ths, they could be written in fractional form, 1/10, 1/100, 1/1,000, et cetera.

Thanks for paying attention.

Thanks for all your hard work today.

I look forward to seeing you in lessons soon, bye now.