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Hi everyone.
My name is Ms. Ku 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 problem solving with standard form under the unit standard form.
By the end of the lesson you'll be able to use your knowledge of standard form to solve problems and we'll be looking at these keywords.
So let's have a look at standard form.
Now, standard form is when a number is written in the form of A times 10 to the power of N, where A is a number in between 1 and 10, including 1 and not including 10 and N is an integer.
They will 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 the pairs of values are grouped.
And we can sometimes use brackets for this.
For example, 2.
4 multiplied by a hundred, this is exactly the same as 2.
4 times 10 times 10.
Also 3 add bracket 4 add 10 close bracket is the same as 3 add 4 in our brackets add our 10.
A non example would be using division.
48 divided by 2 in bracket divide by 6 gives us 4.
And this is not the same as 48 divided by bracket 6 divided by 2 as this equals 16.
Addition and multiplication follow the associative law.
Division and subtraction do not.
In our lesson today, it'll be broken into two parts.
The first part we'll be looking at not quite standard form numbers and the second part we'll be looking at using standard form in problems. So let's make a start on not quite standard form numbers.
When numbers are in standard form, we know it's in this form, A times 10 to the power of N where A is a number in between 1 and 10, including 1 and not including 10 and N is an integer.
So what I want you to do is have a look at which of the following are not in standard form.
See if you can have a little think.
Well hopefully you can spot the ones which are not in standard form are in yellow here.
And this is because the A number is not in between 1 and 10, including 1 and not including 10.
67 is greater than 10.
0.
76 is less than 1.
And 149 is greater than 10, so they're not in standard form.
Now we can change these numbers into standard form using our knowledge of powers of 10.
For example, 67 times 10 to the power of 11.
This is the same as 67 times by eleven 10s all multiplied by each other.
Then we can rewrite this as an ordinary number and then we can convert it into standard form.
So it's 6.
7 times 10 to the 12.
But this approach does take a long time.
So let's see if we can use the associative law to help.
We'll be looking at 67 times 10 to the 11.
Well let's have a look at the 67 first.
we need to convert 67 into a number which satisfies our value of A, which is a number in between 1 and 10, including 1 and not including 10.
So I'm going to use 67 as 6.
7 times 10.
You might notice now we have our number, 6.
7, which satisfies that value of A.
And then multiplying it by 10 gives us 67.
So here I've rewritten my 67 using the associative law and we still have our 10 to the power 11.
Now we know that exponents indicates how many times the base has been multiplied by itself.
We can actually change 10 times 10 to the 11 into 10 times 10 to the 12 because we have 12 multiplications of 10.
Thus we've changed 67 times 10 to the 11 into the correct standard form, 6.
7 times 10 to the 12.
So using the associative law allows us to quickly write a number which is not quite in standard form into standard form efficiently.
What I'm going to do now, I'm going to do a check question and then from there I'd like you to do another check question on your own.
Let's write the following in standard form.
78 times to 10 to the 23.
Well 78 can be written as 7.
8 times 10.
This is what we want because we need a number in between 1 and 10, including 1 but not including 10.
So I have my 7.
8.
So now I know 78 can be written as 7.
8 times 10.
Then I still have my 10 to the power of 23.
Remember using our laws on exponents, this is exactly the same as 7.
8 times 10 to the 24.
Now let's have a look at 2,390 times 10 to the 4.
Well I'm going to rewrite 2,390 so I have a number between 1 and 10, but still using those digits 2, 3, and 9.
That would be 2.
39 times 10 times 10 times 10 as this is the same as 2,390.
Multiplying by 10 to the 4, let's use our knowledge on exponents.
How many tens do I have all multiplied by themselves? Well it's seven.
So hopefully you can spot the answers that I've done on the left, to correctly convert them into standard form, I had to add to my exponents because my A value is too big.
78 was 7.
8 times 10.
So we added an extra 1 to our exponent of 23.
So 78 times 10 to the 23 became 7.
8 times 10 to the 24.
2,390 was written as 2.
39 times 10 to the 3.
So we added that exponent of 3 to the 4 to make it 10 to the 7.
Thus, 2,390 times 10 to the 4 became 2.
39 times 10 to the 7.
So when A is too big, we're adding to their exponents.
See if you can give this a go with your questions.
Well done.
Hopefully you spotted it's the same again.
So when your A value is too big, you are adding to the exponent.
960 is 9.
6 times 10 squared.
So the exponent of 2 is added to that 9 to make 10 to the 11.
The 19.
01 can be written as 1.
901 times 10.
So we're adding to our exponent.
Now Lucas makes a really good observation.
He says, "Does it also apply with a negative exponent?" And Sam says, "Good question.
Let's find out." Before Lucas and Sam investigate, what do you think? I want you to have a look at this question and see if you still continue to add to the exponent when A is too large and it, when there is a negative exponent.
What do you think? Well, let's investigate you with Sam and Lucas.
We still need to convert 9,800 into a number which satisfies A, in other words, between 1 and 10, including 1 and not including 10.
This can be written as 9.
8 times 10 times 10 times 10.
Sam recognises we're using the associative law here.
So that continues.
We still have our multiplication of 10 to the negative 8.
And Sam recognises 10 to the negative 8 means we have eight tenths all multiplied by each other because of that negative exponent.
Now what we're going to do is we're going to have a look at what I've highlighted here.
Sam asks, "What do you think happens here when you multiply 10 by 10 by 10 by a 10th and a 10th and a 10th?" Lucas replies, "Well, I see that 1/10th multiplied by 10 is the same as 10 over 10, which is 1.
So that means if we have 3 multiplications of 10 and 3 multiplications of 1/10th, all of these come together to make 1." So everything which has been highlighted in yellow is the same as 1 times 1 times 1.
and then we still have those other multiplications of 1/10th.
So how many multiplications of 1/10th do we have? Well we have 5 multiplications of 1/10th.
So that's why our answer to 9,800 times 10 to the negative 8 is 9.
8 times 10 to the negative 5.
So we are still adding when that A number is too big because you can see we started with the exponent of negative 8 and we wrote 9,800 using the associative law and powers of 10 as 9.
8 times 10 to the 3.
Adding that 3 to the negative 8 gives me the new exponent of negative 5.
Really well done if you found this.
Now what I want you to do is convert the following into standard form.
See if you can give it a go.
Press pause if you need more time.
Great work everybody.
Now it's time for your check.
I want you to convert the following into standard form.
See if you can give it a go and press pause for more time.
Fantastic work everybody.
Let's see how you got on.
Well, you should have all of these wonderful answers.
So remember when your A value is too big, you're adding to the exponent.
Press pause if you need more time to mark these answers.
Now Lucas asks, "What happens when the starting number cannot be made into multiplications of 10? In other words, it's too small.
For example, 0.
87 times 10 to the 5." And Sam says, "Well, we continue to use the associative law, but take a look at what happens next." So let's have a look.
In this question we're looking at Lucas's question.
It says 0.
87 times 10 to the 5.
Now you'll notice our 0.
87 does not satisfy that criteria for A, in other words, it has to be in between 1 and 10 including 1 and not including 10.
So what we have to do is use our knowledge of multiplications of 1/10th.
So I'm going to change 0.
87 into a number where the starting number is in between 1 and 10, including 1 but not including 10.
And that would be 8.
7 times 1/10th.
So my 0.
87 has now been changed into 8.
7 times 1/10th and I still have my 10 to the power of 5 as before.
Then, I'm going to pair up this 1/10th and the 10.
What do you think happens here? Well, Lucas recognises that the 1/10th multiplied by the 10 gives one.
So that means what I've highlighted in yellow becomes 1.
So that means we end up with 8.
7 times 1 times 10 times 10 times 10 times 10, which gives us the final answer of 8.
7 times 10 to the 4.
Sam makes a good observation.
"Because of this we have one less multiplication of 10, so that's why we subtract from the exponent when our starting number is too small." Let's have a little look at these questions.
So let's have a look at this check.
Now which of the following is correct? See you can give it a go.
Press pause if you need more time.
Well, let's see how you got on.
Which of the following are correct? Well, they all are, so really well done if you got this one.
So let's have a look at each one.
Well, the first one, 0.
28 times 10 to the 5 can be written as 2.
8 times 1/10th.
That gives us the 0.
28.
And then you might notice we have those five 10s.
So I'm multiplying by 10, multiplying by 10, multiply by 10, multiply by 10 and multiply by 10.
Now I've highlighted in yellow a pair of 1/10th multiplied by a 10.
Thus giving us 1.
That means I end up with 2.
8 times 10 to the power of 4.
You can imagine as ignoring what I've highlighted in yellow, as you know that's a multiplication of 1.
Let's have a look at B.
Well 0.
0034 times 10 to the 7.
Let's look at that 0.
0034.
Well that's the same as 3.
4 times 1/10th times 1/10th times 1/10th.
And all I've done is pair it with a multiplication of 10, another multiplication of 10, another multiplication of 10.
Just like I said before, imagine what I've highlighted in yellow all just becomes a multiplication of 1.
Thus you end up with 3.
4 times 10 to the 4.
Next you've got this tiny number multiplied by 10 to the power of 3.
So let's have a look.
That's the same as 9 multiply by 7 lots of 1/10th.
And then we have 3 lots of multiplication of 10.
Same as before, I've coloured in yellow those 3 multiplication of 1/10th and 3 lots of multiplication of 10.
So you can imagine what I've coloured in yellow to be 1, thus leaving us with 9 times 10 to the negative 4.
And D 0.
4 times 10 squared or this is the same as 4 times 1/10th times 10 times 10.
Highlighting those pairs again and ignoring what I've highlighted in yellow as we know that's a multiplication of 1, leaves me with 4 times 10.
If you want to use a highlighter pen in the same way I've done just to really help you visualise what happens here, please do.
Great work if you've got this one right.
So Lucas now says, "Does it also apply with a negative exponent?" And Sam says, "Great question.
Let's find out." But I want you to have a little think before we start to investigate.
What do you think? We'll have a look at this question and see do you continue to subtract from the exponent when A is too small and there is a negative exponent? Well, let's investigate.
We noticed 0.
075 does not satisfy that criteria for A.
So let's use our associative law to identify 0.
075 to be 7.
5 times 1/10th times 1/10th and this is being multiplied by our 10 to the negative 8.
Oof, we've got a lot of 1/10ths there.
So now let's count up how many 1/10ths we have.
We have 8 multiplication of 1/10th.
Think back to our fractional and exponential form.
When we're using 1/10th, it means we have a negative exponent of 10.
So that means my final answer is 7.
5 times 10 to the negative 10.
Now let's have a quick check.
Which of the following are correct and if incorrect, write the correct answer.
See if you can give it a go.
Press pause for more time.
Well done.
Let's see how you got on.
0.
28 does not satisfy that criteria for A, so we need to write it as 2.
8 times 1/10th and we already have a multiplication of 10 to the negative 2.
So yes, it's correct because our multiplication of a 10th and the multiplication of 10 to the negative 2 makes an exponent of negative 3.
Well done.
For B, 0.
0034 times 10 to the negative 5.
That is not the same.
So what I'm going to do is just recap the fact that 3.
4 times 1/10th times 1/10th times 1/10th gives us our 0.
0034.
So I have 3 lots of 1/10th all multiplied by each other and then I have a 10 to the negative 5 as well.
So that means, in total, I have 8 multiplications of 1/10th, which is illustrated by that negative exponent of our 10.
Well done if you got this one right.
For C, we have this tiny answer.
Is it correct? No, it's not.
Rewriting that 0.
0000009 into we have 9 multiplied by 7 lots of 1/10th.
So if we have 7 lots of 1/10th all multiplied by each other and then we have another 23 1/10ths multiplied by each other, that means in total we have an exponent of negative 30 for our base of 10.
Great work if you got this one right.
And for D, well this is not correct.
So rewriting our 0.
4 so it satisfies that A, we have 4 times 1/10th.
That 1/10th that multiply by 10 gives us 1 which is the same as 10 with an exponent of 0.
Well done if you got this one right.
Well done everybody.
So now it's time for your task.
I want you to write the following in correct standard form.
Remember, if our A value is too big, you are adding on the necessary multiplications of 10 to the exponent.
See if you can give it a go.
Press pause if you need more time.
Great work.
Let's move on to question two.
Question two wants you to write the following in correct standard form.
You might notice our A value is too small.
See if you can give it a go and press pause one more time.
Well done.
Let's move on to question three.
Question three is a mix where the A value is too big or the A value is too small.
So you've got to use your knowledge of when to add to the exponent and when to take away.
See if you can give it a go.
Press pause for more time.
Well done.
Well let's go through these answers for question one.
Well you should have all of these wonderful answers.
So remember when your A value is too big, you're adding to the exponent.
Press pause if you need more time to mark these answers.
Well done.
Let's move on to question two.
Well for question two you should have had all of these wonderful answers.
You might notice the A value is too small, so that means we're subtracting from the power.
Great work if you've got this.
Press pause if you want time to mark the answers.
Really well done.
Let's move on to question three.
This was a lovely mix and you need to know when to subtract from that exponent of 10 given A is too big or too small.
So let's see how you got on.
Here are all the answers and please press pause if you need more time to mark them.
Great work.
Excellent work everybody.
So let's move on to using standard form in problems. We use many prefixes associated with standard form every day.
For example, kilo means 10 to the power of 3.
Can you match the name of the number with the corresponding powers of 10? See if you can give it a go.
Press pause for more time.
Great work.
So let's see how you got on.
Well a million is 10 to the 6, which is exactly the same as the prefix, mega.
A billion is 10 to the 9, which is exactly the same as the prefix giga.
And tera is 10 to the 12 and micro is 10 to the minus 6.
Well done if you've got this one right.
But now what I want you to do is I want you to think of some real life examples where you've heard or even used some of these prefixes.
See if you can give it a go and press pause for more time.
Well, I've heard people talk about a million pounds.
I've heard people talk about populations on earth.
For example, in 2024 it's approximately 8 billion.
And while a good example of where we use the word mega is a USB flash drive, which has a memory of 64 megabytes.
Tera is another good example.
An external hard drive has one terabyte of memory.
And micro, well you may have heard the word micro used in the words microchips, which are found in most electronic devices.
There's an abundance of different examples there where you may have heard or used these prefixes.
Lucas says, "Why do we say 64 megabytes instead of 64 times 10 to the 6 bytes or 6.
4 times 10 to the 7 bytes?" And Sam says, "It just sounds better." Which it most certainly does.
Although standard form is not sometimes used, the convention is to use the prefix and then from there we can use it to convert standard form and use it where needed.
For example, let's have a look at this check.
I want you to work out how many bytes there are in the following and give your answer in standard form.
If you want to flick back and see what the exponent of 10 was for giga or the exponent in 10 was for tera, you're very welcome to.
See if you can give it a go.
Press pause one more time.
Well done.
Let's see how you got on.
Well, 128 gigabytes is 128 multiplied by 10 to the 9.
This is not in standard form, so let's rewrite this in standard form.
1.
28 times 10 times 10 times 10 to the 9.
Rewriting this in proper standard form, it gives us 1.
28 times 10 to the 11 bytes.
Well done if you got this.
30 terabytes, well we know tera means 10 to the 12.
30 does not satisfy that criteria for A.
Remember it has to be in between 1 and 10, including 1 and not including 10.
So we have to use 3 times 10 times 10 to the 12.
So rewriting this in a standard form gives us 3 times 10 to the 13 bytes.
You can see why we write 128 gigabytes rather than 1.
28 times 10 to the 11 bytes.
Or, 30 terabytes instead of 3 times 10 to the 13 bytes.
It just sounds a lot better.
Now we use words like billions and trillions in everyday life, but sometimes it is hard to imagine what a billion or million actually looks like.
For example, I want you to have a little think.
If you stacked a million pennies one on top of another, how high do you think it would be? So you can give it some thought.
Well, do you think it'll be taller than Big Ben? Yes, it would be taller than Big Ben.
A million stacked pennies will be taller than Big Ben.
What about London's tallest building, The Shard? Do you think if you stacked a million pennies it'll be taller than London's tallest building? Yes, it will be.
A million stacked pennies will be taller than The Shard.
What about this mountain, UK's tallest mountain Ben Nevis? Do you think if you stacked a million pennies it'd be taller than Ben Nevis? Yes, it would be taller than Ben Nevis.
It's really hard to imagine, but a million stacked pennies has a height of approximately 1,500 metres and this is taller than UK's tallest mountain Ben Nevis.
Really hard to imagine, but we use words like millions and billions so easily.
But it is so difficult to imagine what a million looks like or even what a billion looks like.
Now what I want you to do, I'd like you to use your calculator and I want you to work out how many days is a million seconds.
See if you can work it out and press pause for more time.
Great work.
Let's see how you got on.
Well, 1 million is 1 times 10 to the 6.
So we have our million seconds here.
We know if we divide by 60, it gives us minutes.
We have 16666.
6 recurring minutes.
Now what we're going to do is divide by another 60 to give us the hours, which gives us 277.
7 recurring hours.
Now if there's 24 hours in a day, that means it's 11.
57 to two decimal places days.
So a million seconds is approximately the same as 11 and a half days.
Sometimes some things are so big or so small we can't visualise them.
For example, the radius of the Earth is approximately 6 million metres.
And the radius of Jupiter is approximately 60 million metres.
So how many times bigger is Jupiter than Earth? Given the fact that the Earth's radius is approximately 6 times 10 to the 6 metres and Jupiter's radius is 6 times 10 to the 7 metres, we can see using standard form, Jupiter's radius is 10 times bigger than Earth.
Now these numbers are just so big, it can be hard to imagine.
I do like this picture as a small little reminder of how little Earth is compared to our neighbouring planets.
Great work, everybody.
Now it's time for your task.
For question one, how many years are 1 billion seconds? And I do want you to use a calculator and also use 365 days as a standard year.
For question two, a flash drive comes in 16 megabytes, 64 megabytes and 128 megabytes.
Now a video is made and it has 21,700,000 bytes.
Given that 1 times 10 of the 6 is 1 megabytes, what is the minimum flash drive you need to put on this film? See if you can give it a go and press pause if you need more time.
Great work.
Let's have a look at question three.
Question three says the radius of Mercury is approximately 2,500 kilometres, but the radius of Neptune is approximately 25 million metres.
How many times bigger is Neptune than Mercury? See if you can give it a go.
Press pause for more time.
Great work.
Let's have a look at question four.
I'm looking at these planets again and I've given you the approximate radius in kilometres.
And what I want you to do is write any two of your own questions, including the use of standard form.
See if you can give it a go.
Press pause for more time.
Fantastic work everybody.
Let's go through these answers.
Well, 1 billion seconds is written as this.
Divide by 60 gives us the minutes.
Divide by 60 gives us the hours.
Divide by 24, give us the days.
And divide by 365 gives us the years.
That means 1 billion seconds is approximately 31.
7 years.
Astonishing and it's so hard to imagine.
Question two, what's the minimum flash drive? Well, 21,700,000 is 2.
7 times 10 to the 7.
This is the same as 21.
7 times 10 to the 6, which we know that 10 to the 6 is the same as mega.
So that means it's 21.
7 megabytes.
So the minimum flash drive needed is 64 megabytes.
Well done if you've got this one right.
You may have shown it in a different way.
So for question three, the radius of Mercury is 2,500 kilometres.
So I'm going to convert it into metres and then rewrite it in standard form.
The radius of Neptune is 25 million metres.
So rewriting this in standard form, I have this.
So Neptune, as you can see, using our standard form is 10 times bigger than Mercury.
Well done.
And for question four, you've got freedom to write any two questions of your choice as long as you include the use of standard form.
I've said write all of these radii in metres using standard form.
And then another question would be, put these planets in ascending order using standard form.
Great work everybody.
So in summary, when a number is not quite written in standard form, we can use the associative law to write it in standard form.
For example, 960 times 10 to the 9 is exactly the same as 9.
6 times 10 to the 11.
Or 0.
0034 times 10 to the 7 is exactly the same as 3.
4 times 10 to the 4.
Problems involving real life applications of extremely large numbers or small numbers can be solved using standard form.
Great work.
It was fun learning with you.
