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Hi everyone, my name is Miss Coo, and I'm really happy that you're joining me today to look at standard form calculations.

I really enjoy this lesson because there are so many real-life applications.

I hope you enjoy it too.

So let's make a start.

Hi everyone, and welcome to the lesson on checking and securing understanding of writing large numbers in standard form, and it's under the unit standard form calculations.

And by the end of the lesson you are able to write very large numbers in the form of A times 10 to the n, where 1 is less than or equal to A, which is less than 10, and appreciate the real-life context where this format is usefully used.

We'll be looking at the keyword standard form.

And standard form is when a number is written in the form A times 10 to the n, where 1 is less than or equal to A, which is less than 10, and n is an integer.

We'll also be looking at the associative law.

And 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.

Multiplication and addition, use the associative law, division and subtraction do not.

Today's lesson will be broken into two parts.

We'll be looking at conversions of standard form and ordinary numbers first, and then ordering large numbers in standard form.

So let's make a start looking at conversions of standard form and ordinary numbers.

When numbers are in standard form, they follow this template: A multiply by 10 to the n, where 1 is less than or equal to A, which is less than 10, and n is an integer.

When a number is written in this way we know they're in standard form.

For example, I want you to identify which of the following are not in standard form.

Well done.

Well hopefully you spotted it's these three.

This is because the A number is not in between 1 and 10.

Remember it can include the 1, but it cannot include the 10.

Now writing an ordinary number in standard form can be easily done using a place value chart.

For example, here's 150 billion.

What do you think it would be in standard form? Well, it'll be 1.

5 times 10 to the 11.

And this is because we make the first significant figure a 1, and then the remaining significant figures are the respective decimal places.

And here you can see the power of 10 is indicated in the column heading of the first significant figure.

So now let's do a quick check.

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

Let's write 82,000 in standard form.

Well, putting it in our place value chart you can see we have 82,000 here.

Now let's look at that first significant figure.

Well, the first significant figure is 8, so I'm going to put my 8 here, and then the respective significant figures are our decimal places.

So that means I have 8.

Then we identify the column heading of the first significant figure, which is 10 to the 4.

So that means 8 to 2,000 in standard form is 8.

2, times 10 to the 4.

Now what I want you to do is write 6,560 in standard form.

You're very welcome to use a place value chart if it helps.

Let's see how you got on.

Well, putting in 6,560, our first significant figure is 6, and then the respective decimal places are 5 and 6.

Then looking at the column heading of our first significant figure, it's 10 to the 3.

So that means 6,560 written in standard form is 6.

56 times 10 to the 3.

However, without a place value chart, how do you think we can find the exponent of 11 looking at our 150 billion? Well, it's basically the number of multiplication of 10 needed for the digit in the ones column to be in the correct column.

So let's count.

So you can see my first significant figure of 1.

I need to count how many jumps has it made to get to the first significant figure from 150 billion.

Well, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and 11.

So that means we have now identified our number of 150 billion to be 1.

5 times 10 to the power of 11.

Because there were 11 multiplication of 10 needed from the ones column.

And you can see a 1.

5 satisfies that criteria for A.

In other words, one is less than or equal to A, which is less than 10.

Lucas and Sam want to practise a little bit more.

So Lucas said, "Okay, let's try 290,000." Sam says, "Great, I'm going to write the number "without the place value chart." which you can see here.

Lucas replies, "We know in standard form "it must have an starting number between 1 and 10, "including 1 but not including 10." So Sam says, "So it must be 2.

9, "so he's going to put it here." Note at how he's kept all the place values in line.

Then Lucas says, "How many jumps or multiplication of 10 do we use "for the digit 2 in the ones column "to be in the correct position?" And Sam counts, one, two, three, four, and five." So that means there are five jumps.

So the answer in standard form must be 2.

9 times 10 to the power of 5.

Lucas says, "Well done." And they did it without a place value chart too.

So now what I want us to do without the place value chart is we're going to write 345 million in standard form.

I'm going to do the first question and then I want you to try the second question.

So to do this, I'm going to write my 345 million and then I'm going to identify my number which satisfies that criteria of A.

So it must be 3.

45.

Then I'm going to simply line this up.

In other words, you can see how lining up the digits in the ones column.

Then I can count how many multiplication of 10 does it take for that 3 to be in the correct position.

One, two, three, four, five, six, seven, eight.

So that means there were eight multiplication of 10 from the ones column.

So that means our answer is 3.

45 multiplied by 10 to the power of 8.

Now what I want you to do is I'd like you to your check question.

I want you to write 780,000 in standard form.

See if you can give it a go.

Press pause if we need more time.

Great work.

Let's see how you got on.

Well, 780,000, let's use those significant figures of 7 and 8 and make a number in between 1 and 10, including 1 but not including 10.

So it has to be 7.

Notice how I've lined up those digits in the ones column.

Then we'll go into count.

How many multiplication of 10 does it take that 7 to be in the correct position? One, two, three, four, five.

So that means there were five multiplication from the ones column.

So our answer is 7.

8 multiply by 10 to the power of 5.

Really well done if you've got this one right? So understanding how to write a number in standard form is important, as is learning how to switch a number from standard form into ordinary number.

And to do this, it's always nice to know the equivalent exponential form.

So for example, if the question wants you to convert 5.

7 times 10 to the 3 into an ordinary number, well we know 10 to the 3 represents 1,000.

So the calculation is asking you to work out 5.

7 times 1,000, which is 5,700.

You can also use the associative law and work out the calculation in steps.

For example, 5.

7 times 10 to the 3 is the same as 5.

7 times 10 times 10 times 10.

Then doing it in steps, this is the same as 57 times 10 times 10, which is the same as 570 times 10, which gives us the same answer as 5,700.

You could also imagine that place value chart and you can imagine those jumps on multiplication of 10.

So we know the exponent is 3, so the digit of 5 must be in the 10 to 3 column.

So putting this in, we have a 5.

7, and moving our first significant figure of 5 into that 10 to the 3 column gives me 5,700.

Now what I'd like you to do is pair up the ordinary number with the standard form using whatever method you prefer.

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, 5.

1 times 10 to the 3 is 5,100.

1 times 10 to the 5 is 510,000.

1 times 10 to the 6 is 5,100,000.

01 times 10 squared is 501.

And 5.

01 times 10 to the 5 is 501,000.

Really well done if you've got this one.

Now let's have a look at writing a standard form number as an ordinary number.

See if you can give this a go.

Press pause if you need more time.

Great work.

Let's see how you got on.

Well for a, you should have got 7,800; for B 135,000, for C 89,100,000, and for D 521,300,000.

Really well done.

Next I want you to put a tick next to the correct statements.

Where the answer is incorrect, write the correct ordinary number.

See if you can give it a go.

Press pause if you need more time.

Great work.

Let's see how you got on.

This is incorrect, and the correct number should be 56,300,000.

This is also incorrect and the correct number should be 23,000.

This is also incorrect, and the correct number should be 70,500.

But the last one is correct, 7.

02 times 10 to the 6 is 7,020,000.

Great work.

Now it's time for your task.

I want you to write the following as ordinary numbers.

See if you can give it a go.

Press pause if you need more time.

Well done.

Let's move on to question two.

Question two wants you to fill in the table for a missing item and think of those real-life examples.

We have the item in the left column, we have the length in metres in standard form.

Then we have the length in metres written as an ordinary number, and then they want you to convert it into length in centimetres, but in standard form.

And then the last part of the question says, "What do you notice about the length in metres "in standard form and the length in centimetres "in standard form?" And I want you to explain why.

See if you can give them a go.

Press pause one more time.

Great work.

Let's see how you got on with these answers.

Well, for question one, here are all our lovely answers.

See if you can mark them.

Press pause if you need more time.

For question two, the height of Ben Nevis in standard form is 1.

35 times 10 to 3 metres.

So as an ordinary number, that's 1,350 metres, but in centimetres that will be 1.

35 times to 10 to the 5.

Well done if you got this.

The radius of the moon is 1.

7 times 10 to the 6 metres.

As an ordinary number, that's 1,700,000 metres, and given in centimetres that would be 1.

7 times 10 to the 8.

Now the distance from Liverpool to London as an ordinary number is 340,000 metres.

So in standard form that would be 3.

4 times 10 to the 5, and in centimetres that would be 3.

4 times 10 to the 7.

Now can you think of a real-life example where the length would be 9 times 10 to the 3.

In other words, 9,000 metres.

Nice example would be the height of Himalayas.

There's an infinite number of examples you could give there, but as long as it's got a height or length of 9,000 metres, which is 9 times 10 to the 5 centimetres.

But what do you notice about the length of metres in standard form, and the length in centimetres in standard form? Well, the exponent is plus two more for centimetres.

And this is because given we know 100 centimetres is one metre, the length in centimetres in standard form has to be 10 squared times bigger than the length in metres.

Really well done if you got this.

Great work everybody.

So let's have a look at ordering large numbers in standard form.

When a number is written in standard form, we're able to compare and use them in calculations more efficiently and effectively.

For example, here are some items given in ascending order.

We have the height of the Statue of Liberty, 9.

3 times 10 metres.

We have the length of a football pitch, 1.

1 times 10 squared metres.

We have the height of The Gherkin in London, 1.

8 times 10 squared metres.

We have the height of Mount Fuji, which is 3.

8 times 10 cubed metres.

And we have the latitude of the Earth, which is 1.

1 times 10 to the 5 metres.

But removing the imagery, how do you know that these standard forms are in ascending order? Well, it's because the exponents are in ascending order.

You can see 1, 2, and 3 and 5.

Now if the exponents are the same, then we simply look at the number and order the number from there.

So our 1.

1 is less than our 1.

Sometimes a number is not quite given in standard form, so there's a need to convert them into standard form using the associative law.

For example, why is this number not in standard form? Well, it's not in standard form because the starting number must be greater than 1 or equal to 1, but less than 10.

This is 890.

So what we need to do is use the associative law and rewrite the calculation.

890 can be written as 8.

9 times 10 squared.

And this is still being multiplied by our 10 to the 8.

Then we now can simplify our number because we know 10 squared times 10 to the 8 gives us 10 to the power of 10.

So now we have our number in standard form.

Sam makes a really good observation.

"When the A number is too big, we add to the power of 10." Now let's have a look at another example.

Why do you think this one is not in standard form? Well, it's because our A number again does not satisfy that criteria.

It has to be greater than or equal to 1 but less than 10.

So we have to use the associative law and rewrite the calculation.

We know our 0.

0023 is exactly the same as 2.

3 times our 1 over 10 cubed.

So now what I can do is I can simplify this: 1 over 10 cubed multiplied by our 10 to the 7, gives me the 10 to the 4.

So now we have our number in standard form.

Once again, Sam has made an excellent summary.

When the A number is too small, we subtract from the power of 10.

So what I'd like you to do is I want you to write the following in standard form.

See if you can remember those hints that Sam has given or you can use the associative law.

See if you can give it a go.

Press pause if you need more time.

Well done.

Let's see how you got on.

The 78 can be written as 7.

8 times 10, which then is multiplied by 10 to the 23, thus giving us an answer of 7.

8 times 10 to the 24.

Next 2,390: well that's far too big, so let's rewrite it as 2.

390 times 10 times 10 times 10, which is still being multiplied by that 10 to the 4.

Writing it in standard form gives me 2.

39 times 10 to the 7.

960 times 10 to the 9, not in standard form.

So let's rewrite it as 9.

6 times 10 times 10 times 10 to the 9, which then gives us 9.

6 times 10 to the 11.

19.

01 times 10, not in standard form.

So rewriting that 19.

01 as 1.

901 times 10, still multiplying by the 10, gives us 1.

901 times 10 squared.

Really well done if you have this.

Now what I want you to do is another check question and identify which of the following is correct.

See if you can give it a go.

Press pause if you need more time.

Well done.

28 can be written as 2.

8 times 1/10.

Now multiplying that by a 10 means that I only have 2.

8 times 10 to the 4.

Here I've rewritten my 0.

0034 as 3.

4 times 1/10 times 1/10 times 1/10, and then multiplying that by a 10 and a 10 and a 10 means I'm just left with 3.

4 times 10 to the power of 4.

Next I have this tiny number and writing it as powers of 10 you can see I'm left with four lots of 1/10.

In other words, 9 times 10 to the -4.

And for the last one, I'm rewriting 0.

4 as 4 times one 1/10, which is then multiplied by a 10 and a 10, which gives me 4 times 10.

Really well done if you got this.

"Now, given different representations of numbers, "what approach should you use "to put the following in ascending order?" Have a little think.

Well, Aisha says she likes to convert the number into an ordinary number and then order.

So all she's done here is convert each one into an ordinary number, identify the magnitude of each number, and then she's ordered the question given.

But Lucas says he likes to write each number in standard form as it's much more efficient.

So the 45 times 10 of the 5 is 4.

5 times 10 to the 6.

390,000 is 3.

9 times 10 of the 5.

4 times 10 to 4 doesn't need to change.

12 times 10 of 7 is 1.

Here it's quite clear that we can order according to those powers of 10, giving us exactly the same answer.

So which method do you prefer? Have a little think.

Now what I'd like you to do is apply that method to the following question.

This is a nice little check.

Have a little think, press pause if you need.

Well done.

Well for me, I'm gonna convert each number into standard form.

So this is 5.

6 times 10 to the 6.

4 times 10 to the 5.

7 times 10 to the 4.

3 times 10 to the 6.

So putting them in ascending order, I have this.

For part b, converting them all to standard form I have 4.

1 times 10 to the 23.

2 times 10 to the 23.

7 times 10 to 25.

And 5.

3 times 10 to the 21.

Putting them in order, I have this.

Now given that second question, and given what Aisha says about converting it to an ordinary number and then ordering, and what Jacob said about converting to standard form first, given these numbers in the check, has your preference changed? And explain.

Well, given these numbers are so big, converting to an ordinary number is not the most efficient approach and can lead to errors, especially when you have over 20 zeros there.

Well done.

So let's move on to another check.

"Order these cards correctly "between the inequality symbols." 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, for me, I'm gonna convert them all to standard form.

It's just much easier and efficient.

So that means I've got 9.

6 times 10 to the 6.

1 times 10 to the 6.

82 times 10 to the 6.

9 times 10 to the 8.

9 times 10 to the 6.

Putting them in order, remember to order what the question has given you.

The use of converting them to standard form just makes it so much more easier and efficient.

Very well done.

Great work everybody.

So now it's time for your task.

I want you to put the following in the correct position.

See if you can give it a go.

Press pause if you need more time.

Well done.

Let's have a look at question two.

Here are the populations of different cities or towns in the U.

from 2024.

List them in ascending order.

See if you can give it a go.

Press pause if you need more time.

Well done.

Let's have a look at question three.

Question three is a great question.

It's probably one of my favourite questions.

Here's a list of distances from each planet to the sun.

We have some massive numbers here given in centimetres, metres, kilometres.

And the question wants you to identify the corresponding length with each planet or with each letter.

See if you can give it a go.

Take your time.

Well done.

Let's see how you got on.

Well, for question one, I've converted them all to standard form and then I've put them in the correct order.

For question 1b, same again, converted them to standard form, and I put them in the correct order.

If you need to press pause and compare, please do.

For question two.

Writing them all in the correct standard form, I have these populations which I then can use to order the cities and towns in ascending order.

Really well done if you got this.

Now for question three.

First of all, let's look at this massive number given in centimetres.

Well, first things first, converting them all to the same unit is important.

So converting this to metres gives me this huge number and then converting it to kilometres gives me a little bit of an easier number, 228 million kilometres.

Now I'm going to rewrite this in standard form to give me 2.

28 times 10 to the 8 kilometres.

So now I have one length, which is 2.

28 times 10 to the 8 kilometres.

Next 1.

43 times 10 to the 12 metres.

Well that's given in metres, so let's convert this into kilometres.

So I have 1.

43 times 10 to the 9 kilometres.

Next, I have 1.

085 times 10 to the 8 kilometres.

That's correctly in standard form and in the right unit, so I'm going to leave it there.

And then I have this massive number given in metres.

Converting it to kilometres, I have 150 million kilometres, which is 1.

5 times 10 to the 8 kilometres.

So I'm going to put my length here.

Finally, I have this huge number given in kilometres, so I'm going to rewrite it in standard form to give me 4.

5 times 10 to the 9 kilometres.

Now I have all my lengths written correctly in standard form and correctly written in kilometres, so I can see which planets are closest or furthest from the sun.

This means this has to be Neptune 'cause it's the furthest from the sun.

This has to be Saturn as it's second furthest from the sun.

This must be Mars as it's the third furthest from the sun.

This must be Earth.

And this must be Venus.

Huge well done if you got this one right.

Excellent work everybody.

I really do hope you've enjoyed the lesson because we've looked at some real-life context questions with standard form.

"We know standard form is the convention to write "very large or very small numbers." "And we use standard form in a number of industries.

"And once numbers are written in standard form, "we can order using the exponents." Remember, "The bigger the exponent, the larger the number.

"For example, 9 times 10 to the 10 "is greater than 9 times 10 to the 6.

"Another example would be 2 times 10 to the -5 "is greater than 2 times 10 to the -9.

"Converting all calculations to standard form "is more efficient than converting to an ordinary number." Really well done everybody.

I hope you've enjoyed the lesson.

I've finished the video