Lesson video

Hi everyone.

My name is Miss Kuhl and I'm really happy to be learning with you today.

In today's lesson, we'll be looking at standard form.

And standard form is a great way to write really big numbers or really small numbers in a very efficient way.

I really hope you enjoy the lesson, so let's make a start.

Hi everyone and welcome to this lesson on writing small numbers in standard form under the unit standard form.

And by the end of the lesson you'll be able to write very small numbers in the form of A times 10 to the power of N where A is in between one in 10, including one and not including 10.

And appreciate the real life context where this format is usefully used.

Now we'll be looking at a couple of keywords.

We'll be using the words exponential form.

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

For example, 2 x 2 x 2 is two to the power of three.

The associative law states that a repeated application of the operation produces the same result regardless of how pairs of values are grouped.

And we can group using brackets.

For example, 2.

4 x 100 is exactly the same as 2.

4 x 10 x 10.

Three add, in bracket, four add 10 and close brackets is the same as three add four in our brackets, add 10.

Division does not follow the associative law.

This is because as an example, 48 divided by two in brackets divided by six equals four.

But if you do 48 divided by 6 divided by two in brackets, it gives 16.

They are not the same value.

So division and subtraction do not follow the associative law.

We'll also be using the words standard form.

And standard form is a number which is written in the form A times 10 to the N, where A is in between one and 10, including one, not including 10, and N is an integer.

Today's lesson will be broken into three parts, writing very small numbers in standard form, converting between ordinary and standard form and standard form in the calculator.

So let's make a start.

A decimal has been written in all these wonderful different forms. Can you work out what that decimal is? So you can give it a go and press pause if you need a bit of time.

Well done, let's see how you got on.

Well the decimal is actually 0.

021 and we've written it in lots of different ways, but there are actually an infinite number of ways to write 0.

021, but only one way, using standard form.

and using a standard approach to write very small numbers is so important.

This is how we write that 0.

021 in standard form.

We use standard form because we do not want to write a number written in so many different ways as it can be confusing.

So writing a really big number or a really small number using standard form is convention.

And remember, standard form is written where A, which is between one and 10, including one but not including 10, is multiplied by 10 to the power of N where N is an integer.

And what's important to remember is this emphasises it's got to be between one and 10, including one and not including 10.

And N can be a positive or negative whole number.

So why do you think writing a number using exponential form is more preferred? For example, let's have a look at this tiny number here.

Why do you think we do prefer to write using standard form? Well, it's because writing this tiny number does take a very long time.

Because of all those zeros, there is a possibility of making errors when copying all of these digits down.

Standard form is such a shorter and concise way of writing our number and we don't lose the accuracy of the number either when we write it in standard form.

So having a standard approach when writing any number using powers of 10 in exponential form is really important to have.

For example, we know this tiny number has been written as 5.

62 times 10 to the minus 10.

If we were to write this small number, 0.

0025, it can be written as 2.

5 times 10 to the minus three.

Why do you think this one is not correctly written in standard form? Well it's because this number here, 41, has to satisfy that condition of standard form.

It has to be in between one and 10, including one but not including 10.

And 41 is obviously greater than 10.

Now what I want you to do is have a look at these calculations, which of the following is written in correct standard form and I want you to explain why the others are incorrectly written, so you can give it a go.

Press pause if you need more time.

Great work.

Let's see how you got on.

Well, B is correctly written in standard form.

The first is not written in standard form because 981 needs to satisfy that criteria for A.

As we know B is in standard form, C is incorrect as standard form uses multiplication, we don't use division.

Lastly, D is incorrect because 124.

56 does not satisfy that condition of A.

Lucas spots something with a place value chart.

He says, I can see a nice way to convert a tiny number into standard form using a place value chart.

He recognises 0.

00098 in standard form is 9.

8 times 10 to the minus four.

Can you explain how Lucas is able to convert the number so quickly using our place value chart? So you can give it a go.

Press pause if you need some time.

Well done.

Well, it's because the powers of 10 in exponential form is actually given in the table column heading for the first non-zero digit.

So you can see the first significant figure is nine and nine is under that column of 10 to the minus four.

So he's recognised it should be 9.

8 times 10 to the minus four.

Great spot, Lucas.

So now what I want you to do is see if you can use the following place value charts and what Lucas has spotted to write the following in standard form.

See if you can give it a go.

Press pause if you need more time.

Well done.

Let's see how you got on.

Remember identifying that first significant figure.

So in this case it's one, so it's 1.

56 times 10 to the minus three.

Look at that column heading of our first significant figure.

For B, you should have had 9.

241 times 10 to the minus two because our first significant figure is nine, and look at that column heading, it's under the column heading of 10 to the minus two.

So using a place value chart is an excellent way to visualise the number and convert to standard form, but it does take a while to draw.

So using a place value chart first, what I want you to do is have a look at this number and what do you think this number would be in standard form? Well hopefully you spotted it's gonna be 6.

19 times 10 to the -5.

So now what I'm gonna do is I'm going to see if we can take away the place value chart and get the same answer.

So removing our place value chart, here's our tiny number again.

Now remember, we want to write it in standard form.

So to write it in standard form, we look at those significant figures, six, one and nine and we want to create a number which satisfies that criteria for that A.

So what do you think it would be? Well, it would be 6.

19.

So using six as our first significant figure, one as our second significant figure and nine as our third significant figure, ensuring we create a number in between one and 10.

So we know now the standard form should start to look something like this, 6.

19 times 10 to the something.

So let's say we can find out what that exponent is.

Well, we can do this by counting how many multiplications of 1/10th we use in order to change our 6.

19 into that tiny number, 0.

0000619.

So let's count.

All I've done here is highlight that first significant figure of six.

We're going to count how many multiplications of 1/10th does it take for that six in the ones column to go to the six in the correct column.

It'll be one, two, three, four, five multiplications of 1/10th.

In other words 6.

19 multiplied by five lots of those 1/10ths does give us the same answer of 0.

0000619.

So writing this in standard form means it's 6.

19 times 10 to the -5 because we had five of those multiplications of 1/10th.

In other words, five of those jumps.

So let's summarise what we just did.

We know the first starting number has to satisfy that criteria for A, we also counted the number of multiplications of 1/10th needed from that one's column.

Well done, so let's try another one with Lucas and Sam.

Lucas says, "Let's try 0.

0045." Sam says, "Great, we're going to do it without a place value chart.

So writing our number, we have it here." Remember we need to identify that starting value in standard form.

So using the significant figures four and five, can you create a number which is in between one and 10, including one and not including 10? Well, it has to be 4.

5.

So let's line up all those digits in the ones column so we have it here.

All I've done is highlight that significant figure of four.

Then we're going to count how many multiplications of 1/10th does it take? One, two, three.

So that means our answer is 4.

5 times 10 to the -3.

Great work, everybody.

So let's move on to a quick check.

I'm going to do the first one and I'd like you to do the second one.

Without a place value chart, we're going to write 0.

00065 in standard form.

Well first of all, using the digits six and five, we're going to create that number which satisfies A.

So it's 6.

5.

Notice how I've lined up all my ones together.

Now from here, we're going to count how many multiplications of 1/10th are there and I've highlighted the significant figure of six in each of our numbers.

And then we're going to count one, two, three, four.

It's taken four multiplications of 1/10th for that six in the ones column to be in the correct position.

So that means the answer is 6.

5 times 10 to the -4.

Because there were four multiplications of 1/10th from that one's column.

Now without a place value chart, I want you to write the following in standard form, see if you can give it a go and press pause if you need more time.

Well done, let's see how you got on.

Well using the digits one, five and six to make a numberwhich satisfies A, we should have 1.

56.

I'm going to identify the significant figure of one, which I've done here.

And we're gonna count how many multiplications of 1/10th has it taken for our digit of one in the ones column to be in the correct position? One, two, three, four, five and six.

So that means there were six multiplications of 1/10th from the ones column.

So the answer is 1.

56 x 10 to the -6.

Great work if you got this one right.

So understanding how standard form works allows us to convert an ordinary number into a standard form and vice versa.

So what do you think 9.

8 x 10 to the -5 is as an ordinary number? So you can give it a go.

Press pause if you need more time.

Great work, let's see how you got on.

Well first of all, you could choose to visualise a place value chart.

In other words, 9.

8 x 10 of the -5, I've illustrated it in my place value chart so I can write my number quite easily, 0.

000098.

Another way you could have done it is by recognising there are five multiplications of 1/10th and then applying a step-by-step process while 9.

8 x 1/10 x 1/10 x 1/10 x 1/10 x 1/0 is the same as 9.

8 x 0.

00001.

And then you can work out the answer to be exactly the same as what we did before.

There are lots of different ways that you could work this out, but what's important is that you recognise that negative exponent indicates how many multiplications of 1/10th there are.

These two are really good efficient ways, but let's see if we can spot another shortcut.

Now, Laura says she knows a shortcut.

She says "For small numbers, the negative exponent tells you it's small and the number tells you how many zeros to put in." Sophia says, "Oh I see." So that means 1.

24 x 10 to the -3 has a negative exponent, so she knows it's small, and the number in the exponent is three.

So that means she puts three zeros, 0.

00124.

And Laura's shortcut does really work.

But what I want you to do is explain why does Laura's shortcut work for small numbers? Well, the reason why Laura's shortcut works is because when a number is correctly written in standard form, the negative exponent represents the number of divisions by 10.

So, using Laura's shortcut will give you the correct answer from converting a number in standard form into an ordinary number.

Now what I want you to do is I want you to do this check, write the following as an ordinary number, so you can give it a go.

Press pause if you need more time.

Great work.

Let's see how you got on.

These are all the answers.

You are free to either use a place value chart, use the associative law, or perhaps you liked Laura's little cheat whereby you are spotting the negative exponent tells you it's small and the numerical exponent tells you how many zeros to put.

So well done if you've got this one right.

Great work everybody.

So let's move on to your task.

Question one says, using place value charts, write the following in standard form, see if you can give it a go.

Press pause every need more time.

Great work.

Let's move on to question two.

Question two says which of the following are not in standard form? I want you to explain why.

Give it a go.

Press pause if you need more time.

Fantastic work, everybody.

So let's move on to question three.

Without a place value chart, I want you to write the following in standard form.

Give it a go.

Press pause for more time.

Great work.

Let's move on to question four.

Question four, without a place value chart, write the following in standard form.

Well done.

Let's go through these answers.

Well, for question one, you should have had these answers.

Excellent work.

Let's move on to question two.

These are the reasons why they are not written in correct standard form, well done if you've got these right, press pause if you need more time to mark them.

For question three, here are our answers written correctly in standard form, great work, and press pause if you need more time to mark.

And for question four, here are our answers written from standard form into ordinary form.

Press pause if you need more time to mark them, well done.

We've done a lot of work in this cycle.

Well done, everybody.

That was a lot of work in the last part of our lesson.

So let's move on to the second part of our lesson where we'll be converting between ordinary and standard form.

Before we do this, what I want you to do is give some real life examples where we're using very small numbers, have a little think.

Well, there's lots of examples.

I'm going to give an example of biology or microbiology where we're looking at organism cells, et cetera.

You might have thought about microchips and microchips are virtually part of every electronic device across the globe.

You may have even thought about something called nanotechnology.

So nanotechnology uses technology but on a very, very tiny scale.

Really good work if you've got any real life examples where you're using very small numbers.

So now what I'm gonna ask is give some examples of where we're using very large numbers.

Have a little think, press pause if you need more time.

Here are some examples, but there are many more.

Astronomy is a nice one where we're using massive measurements.

Another example would be global figures, maybe populations, distances, currencies.

Another example would be construction and engineering.

There are many examples out there where we're using very large numbers.

And for the purpose of this learning cycle, large numbers will be those greater than one and small numbers will be those less than one.

So let's have a look at two calculations.

How do we know which one is a big number and which one is a small number? Have a little look and a think.

Well, 8.

36 times 10 to the four has a positive exponent.

So that means it's a large number, 8.

36 x 10 to the -4 has a negative exponent, so therefore it's a small number.

You can see using our column headings when we have a positive exponent, it refers to a big number.

And when we have a negative exponent, it refers to a small number.

Let's have a quick check.

Some pupils have worked out the answers to these questions.

I want you to identify if they're correct or not and explain why, so you can give it a go.

Press pause if you need more time.

Well done.

Well, the first one is correct, so I'm quite happy with that one.

But the second one is incorrect.

This is because the exponent is negative.

So that means our answer should be a small number.

The third one is also incorrect and the last one, well, this is also incorrect.

It's incorrect because the exponent is positive.

So that means our answer should be a big number.

Let's have a look at another check.

Some pupils have worked out the answers to these questions.

I want you to identify if they are correct or not and explain why.

So you can give it a go, press pause for more time.

Well, let's have a look at the first one.

Well first of all, this is correct.

Really happy to see this because what they've done is created a number, 7.

8 using the digits seven and eight and it does satisfy that criteria for A.

And then they've correctly counted the number of multiplications of 10 to make 7.

8 into 780,000.

Now for the second working out, unfortunately this is not correct, they've created a number 92.

3 and that does not satisfy the criteria for A, they've also counted the number of multiplications of 10 to make 92.

3 into 92,300.

Therefore, they have not written 92,300 in standard form.

Great work, so now what I want you to do is I want you to convert the following into standard form.

Remember, pay attention to that exponent.

If the exponent is negative, that means we're talking about a small number and if the exponent is positive, then we're talking about that big number.

So you can give it a go and press pause if you need more time.

Great work.

Let's move on to question two.

Question two gives you some numbers written in standard form and you need to write it as an ordinary number.

See if you can give it a go, press pause if need more time.

Question three, four and five talks about some real life applications of standard form.

So a microchip has a width of 2 x 10 to the -6 metres.

Write this as an ordinary number.

For question four, Mars is approximately 225 million kilometres away from the earth, write this as standard form.

And five, the radius of an atom is approximately 1 x 10 to the -10 metres.

Write this as an ordinary number.

See if you can give it a go, press pause if you need more time.

Let's move on to the next question.

A hint and anagram has been given.

The first has been done for you.

See if you can fill in the rest.

Well done, everybody.

So let's move on to these answers.

Well, for question one, you should have had all of these numbers written in standard form, excellent.

For question two, you should have had all these ordinary numbers written from standard form.

Well done.

For question three, here's our answers.

Fantastic work everybody if you got this one, especially for question four where you had to convert kilometres into metres and then convert it into standard form.

Great work.

Let's move on to the next question.

Here are all the words that we were looking for and here are them all located on our word search.

Great work.

Great work everybody, so let's have a look at standard form on the calculator.

Well, we know scientific calculators are fantastic tools as they do so many things without the user requesting.

For example, using a calculator, let's input this calculation.

24 divided by 250,000.

What does the calculator output as a decimal? You can give it a go.

Well the calculator will output 9.

6 x 10 to the -5.

So when a number is too big or too small for the calculator to display an answer, the calculator automatically converts the answer to standard form and the output in standard form is simple.

So let's see how we can input something.

Well, we're going to go through the main operations with our fx-570/991 Casio ClassWiz.

And the input of a number using standard form can be done a couple of ways.

We can use the power of button located here where we can insert the base and the exponent.

We can also use the power of 10 button, which is here, whereby it sets the base to be 10.

And then you simply input the exponent.

This button is sometimes called the exponential base of 10 button.

So let's input 7.

6 x 10 to the -6.

Using that power of button, simply put in 7.

6 times 10.

Then access that power of button.

Automatically, it sets the base to be 10.

Then input the exponent to be negative six.

From here, you'll notice when you press okay or execute, it'll give you it as a fraction.

But remember we want it as a decimal.

So I want you to press that format and change it to a decimal.

What happens? Well what you should get is it will not convert it into a decimal.

It'll give you the answer in standard form and it keeps the answer in standard form because the answer is so small.

So therefore it's important you are able to convert from standard form into an ordinary number because your calculator won't always give you those tiny numbers as a decimal.

So now I want you to look at that number, 7.

6 x 10 to the -6, and tell me what is it as an ordinary number? Well done.

It should be 0.

0000076, great work.

We can also use that power of 10 button by inputting 7.

6, press that power of 10 button.

Remember it sets the base of 10 and all you have to do is put in the exponent.

And if you put in the exponent of negative six and press execute, same again, you'll need to convert it into a decimal by accessing format and select decimal.

The output will be the same so you'll need to change it from standard form into a decimal.

Well done.

So what I want you to do is have a look at this quick check question.

I want you to convert the following outputs from the calculator into a decimal.

See if you can give it a go.

Press pause if you need more time.

Well done.

Let's see how you got on.

Well, here are our answers.

So it's so important that you recognise that calculators sometimes don't give you those tiny answers as a decimal.

It'll give you it as standard form and you'll need to convert it into decimals.

Well done.

Now it's time for your task.

I want you to write the output as a decimal of the following.

See if you can give it a go.

Press pause if you need more time.

Well done.

Let's move on to question two.

And for question two, I want you to investigate, when does the number become so small that the calculator converts into standard form? See if you can investigate.

Press pause for more time.

Great work.

Let's move on to question three.

I want you to fill in that missing information.

Give it a go.

Press pause for more time.

Fantastic work everybody.

Let's go through these answers.

Here are our decimal equivalents.

Excellent work.

For question two, any number when written in standard form has an exponent less than or equal to negative three.

Well done if you spotted this.

And for the third part, here's our missing information.

Great work, everybody.

So in summary, it's clear there is a need for standard form in real life.

So it's important that incredibly large or small numbers can be written in standard form and vice versa, and just remember, A times 10 to the N is the way in which we write standard form when A is a number between one and 10, including one and not including 10 and N is a positive or negative whole number.

And remember, when the exponent N is positive, the overall number is large.

And when the exponent N is negative, the overall number is small.

Great work, everybody.

It was wonderful learning with you.