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Hi, everyone.
My name is Ms. Coo, 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 large numbers in standard form.
It's under the unit standard form, and by the end of the lesson, you'll be able to write very large numbers in standard form and appreciate the real-life context where this format is usefully used.
And we'll be looking at a couple of keywords, first of all, exponential form.
And when a number is multiplied by itself multiple times, it can be written more simply in exponential form.
For example, two multiplied by two multiplied by two is exactly the same as two to the power of three, or two cubed.
We'll also be using 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.
For example, 2.
4 times 100 is exactly the same as 2.
4 times 10 times 10.
Or three add bracket, four add 10, close bracket is the same as three and four in our brackets, add on 10.
Division does not follow the associative law.
For example, 48 divided by two in brackets divided by six is the same as four.
48 divided by in brackets six divided by two is 16.
They are not the same, so it does not follow the associative law.
We'll also be looking at the keyword standard form, but I'm not going to tell you what that is right now.
It will be defined in our lesson.
Our lesson will be broken into four parts.
First of all, we'll be looking at what is standard form, and then we'll move on to writing large numbers in standard form.
Then we'll move on to standard form to ordinary numbers.
And finally we'll look at standard form on the calculator.
So let's make a start looking at what is standard form.
Well, I want to give you some calculations first, and all of these calculations represent the same integer.
Can you work out what that integer is? Press pause if you need more time.
Well done.
Well, hopefully you've worked it out to be 530.
All of these calculations equate to 530, and there are an infinite number of different ways to write 530, but only one of them uses standard form.
I want you to have a look at these calculations.
Can you work out what the integer is? Well done.
It's 8,450.
Same again, there are an infinite number of ways to write 8,450, but only one uses standard form.
Let's see if we can come up with a standard approach so we can identify how to write a number in standard form.
How many different ways do you think you can write this number? It's massive, 57 trillion, 300 billion.
See if you can write it in as many different ways as you can.
Press pause for more time.
Well done.
Well, here are just some examples.
You could have done this.
You could have done this or this or lots of different ways, but only one uses standard form.
It's this one.
This is the only one which writes the number 57 trillion, 300 billion in standard form.
What do you think standard form means? You're going to have a look at that calculation and have a think.
So given this definition, can you see why only one of these is in standard form? Well, hopefully you can spot it's because in standard form, you must start with a number which is represented as A, which is in between one and 10, including one, not including 10, multiplied by a power of 10.
So hopefully you can spot with all those different calculations, they are not written in standard form.
Now what I want you to do is think, why is writing in standard form so useful? Well, if we were given this huge number, why do you think it would be useful to write it in standard form? Well, it can be useful because this number takes too long to write.
Because of all those zeros, there is a possibility of making errors when copying all those digits down.
The great thing about standard form, it's so much shorter and concise, and we don't lose any accuracy of the number either.
So it's a great way to write really big numbers.
What I want you to do is I want you to have a look at this quick-check question, which of the following is written in correct standard form, and I want you to explain why the others are incorrectly written.
See if you can give it a go.
Press pause for more time.
Well done.
Let's see how you got on.
These are correct because that starting number which we represented as A is a number in between one and 10, including one, not including 10.
9.
8, yep, that's between one and 10, not including 10, and 1.
2456 is in between one and 10, not including 10.
B is not in standard form.
If you notice, 89.
2 is greater than 10.
We need it to be a number in between one and 10, including one but not including 10.
For C, 0.
016, this is incorrect because we need our number in between one and 10.
This 0.
016 is less than one.
Well done if you got this.
Now let's have a look at a place value chart.
Lucas says, "I can see a nice way to convert a number into standard form using a place value chart." He says, "98,400 in standard form is 9.
84 times 10 to the four." Can you explain how Lucas is able to convert the number so quickly using our place value chart? Have a little look.
Well done.
So let's have a look at this explanation.
Well, the powers of 10 in exponential form is given in the table column heading.
So you can see here that the first significant figure of nine is under the column heading of 10 to the four.
So if nine will be representing our integer, that means the 0.
8 and the 0.
04 will be representing our 10 to the three and our 10 squared.
9.
84 times 10 to the four is exactly the same as 98,400.
And you can double-check this by multiplying 9.
84 by 10 by 10 by 10 and by 10 again, and it will give you 98,400.
Using a place value chart and looking at that first significant figure is a really quick, nifty way of identifying a number in standard form.
Great work, everybody.
So now let's move on to a quick check.
Using the place value chart, 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.
Well, the first answer is 3.
4 times 10 squared.
Notice how that first significant figure of three is under that column heading of 10 squared, so it's 3.
4 times 10 squared.
For B, you should've had 2.
3 times 10 to the four.
Notice that first significant figure of two under that column heading of 10 to the four.
So we have 2.
3 times 10 to the four.
Well done.
Now it's time for your task.
Using the place value charts, write the following in standard form.
See if you can give it a go, and press pause for more time.
Great work.
Let's move on to question C and D.
Same again, write the following in standard form.
Press pause for more time.
Well done.
Let's move on to question two.
I want you to identify which of the following are not in standard form, and then I want you to explain why.
See if you can give it a go.
Press pause for more time.
Great work, so let's go through these answers.
For question one, let's see what you got.
You should have 8.
1 times 10 to the four and for B, 7.
7 times 10 to the three.
For 1c, you should have 8.
2 times 10 squared.
And for D, you should have 9.
94 times 10 to the four.
For question two, here are our reasons.
For A, this is incorrect because 123 is greater than 10.
B is correct, as it is in standard form.
Notice how the A value is in between one and 10.
For C, we've got 96 is greater than 10, so it's not in standard form.
D is not in standard form, as this is written using a division, not a multiplication.
And lastly, E.
E is not correct because 0.
059 is less than one.
Well done.
Excellent work, everybody.
So let's look at writing large numbers in standard form.
I want to have a look at the distance from Earth to the Sun.
Well, the distance from the Earth to the Sun is approximately 150 billion metres.
So let's use our place value chart to show what this number would look like.
Starting with the place value chart, what I want you to do is add on those extra columns.
What do you think they are? Well, adding on those extra columns, we have a 100,000, millions, 10 millions, so on and so forth.
Now I'm going to add the powers of 10 in exponential form.
Well, we have 10 to the five, 10 to the six, 10 to the seven, 10 to the eight, 10 to the nine, 10 to the 10, and 10 to the 11.
Now, inserting the number 150 billion metres, what do you think the number would be in standard form? Well, hopefully you could spot it's going to be 1.
5 times 10 to the 11, but there must be a more efficient way of identifying a number in standard form without a place value chart.
And to do this, the first significant figure goes in the ones column, and the other digits follow accordingly.
Now what I'm going to do is ask, well, what do I times that 1.
5 by to give me that 150 billion? Well, it's gotta be 10 to the 11, and you can see that because of the multiplications of 10.
But let's do it without a place value chart.
Without a place value chart, how else could you find that exponent of 11? Well, it's the number of multiplications of 10 needed for the digit in the ones column to be in the correct column.
So what I want to do is count.
This would be one multiplication of 10, two multiplications 10, three, four, five, six, seven, eight, nine, 10, and 11.
So there are 11 multiplications by 10 needed from the ones column to get to that 100 billion.
So that means our answer is 1.
5, 10 to the 11 because we've kept our first starting number of A to be in between one and 10, including one, but not including 10.
And we've identified the correct exponent of 11 by counting how many multiplications of 10 are needed.
This is a good example where we can write a number in standard form without a place value chart.
So Lucas says, "Okay, let's try 28,000." Sam says, "Great, I'm going to write the number without the place value chart." So here's our 28,000.
Now Lucas reminds Sam, "We know in standard form it must have a starting number in between one and 10, including one, but not including 10." So Sam says, "Okay, so that means the starting number must be 2.
8 because the first significant figure is two, and the second significant figure is eight.
So that means we're going to put our 2.
8 here." Now what we're going to do is count.
How many jumps of multiplications of 10 do we use for the digit two in the ones column to be in that correct position? One, two, three, four, and five.
So that means we know we have five multiplications of 10 for the digit of two to be in the correct position.
So that means the answer in standard form is 2.
8 times 10 to the five.
Well done, everybody.
So what I'm going to do now is I'm going to do a quick-check question, and I'd like you to do the second question.
So without a place value chart, let's change this huge number into standard form.
We have 956,000,000 Well, to do this, let's identify that number in between one and 10, including one, but not including 10, using the digits nine, five, and six.
Well, it's gotta be 9.
56.
And then to help us visualise, line up the digits in the ones column.
Now we're going to count.
How many multiplications of 10 does it take for that nine to be in the correct position? One, two, three, four, five, six, seven, and eight.
So that means there were eight multiplications of 10 from that ones column.
So our answer is 9.
56 times 10 to the eight.
Now what I want you to do is I want you to try this one.
Write 780,000 in standard form.
See if 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, I'm going to write my 780,000, and let's have a look at those digits seven and eight.
We need to make a number in between one and 10, so it has to be 7.
8.
Now let's line up those digits in the ones column.
So let's count carefully.
How many jumps or multiplications of 10 are there? One, two, three, four, and five.
So that means there were five multiplications of 10 from the ones column.
So our answer is 7.
8 times 10 to the five.
Well done if you got this one right.
Now it's time for your task.
Without a place value chart, I want you to convert the following into standard form.
See if can give it a go.
Press pause if you need more time.
Great work.
Let's move on to question two.
Well, question two says three pupils were given this huge number and asked to convert it to standard form.
Identify where they made their error and give the correct answer.
See if you can give it a go.
Press pause one more time.
Great work, everybody.
Let's move on to question three.
Question three says, "Explain why standard form is important." See if you can give it a go.
Take your time and press pause.
Well done.
Let's see how you got on.
Here are all our answers to question one.
Huge well done if you got this one right, especially F.
Remember to identify one as your first significant figure, and then from there, you had to put the 1.
002.
That was a tough one.
Really well done if you got this one right.
Well, for question two, let's see how you got on.
Well, hopefully you spotted the first part of standard form must have a number between one and 10, including one and not including 10.
In other words, 905 does not satisfy this.
For the second question, it does not begin with 9.
5.
It should have 9.
05 as our starting value for A.
And for the third one, unfortunately this pupil incorrectly counted the number of multiplications of powers of 10.
There are 11 multiplication of 10, not 10 multiplications of 10.
So what's the correct answer? Well, the correct answer is 9.
05 times 10 to the 11.
Really well done.
For question three, explain why standard form's important.
Well, we write numbers in standard form because it's easier, concise, and more efficient.
It also reduces errors, as it would take too long to write huge numbers, and there would be more room for error when writing a number in its ordinary form.
Standard form also allows any number to be written using one standard approach.
In other words, the starting number of A has to be in between one and 10, including one and not including 10, and n is an integer.
Finally, when writing in a concise form, it's easier to use in calculations.
Well done.
Okay, everybody, so let's have a look at the third part of our lesson, standard form to an ordinary number.
Understanding how to write a number in standard form is important as is how to write a number from standard form to an ordinary number.
What I'm going to do is recap on our exponential form.
Knowing this exponential form really does help.
For example, if someone were to ask you to convert 2.
3 times 10 to the three into an ordinary number, well, we know 10 to the three represents 1,000.
So that means the calculation basically means 2.
3 times 1,000, which is 2,300.
So knowing the exponential form really does help.
What I'm going to do is I'm also going to show you how to write a number from standard form to an ordinary number using the associative law, and we're going to do it in steps.
For example, 2.
3 times 10 cubed again, this means 2.
3 times 10 times 10 times 10.
Well, I'm going to do it in steps.
This is the same as 23 times 10 times 10, which is the same as 230 times 10, which gives me 2,300.
You can use the associative law and steps.
You can also imagine the place value chart and those jumps on multiplications of 10.
Given that the exponent is three, so that means the digit of two must be in the 10 to the three column.
So there's my 2.
3, and I'm going to simply move it across so my first significant figure of two is in that 10 to the three column.
That's another way.
So there are lots of different ways in which you can convert a standard form number into an ordinary number.
What's important is that you find an efficient method for you.
So what I want you to do is I want you to pair up the ordinary number with the standard form number.
See if you can give it a go.
Press pause if you need more time.
So let's see how you got on.
Well, here are all our answers.
A nice way to check to see if you're right is looking at the ordinary number and checking to see, does it convert back into the correct standard form matched here.
That would be another nice little technique.
Well done if you got this one right.
Now it's time for another check.
I want to see if you can write the following numbers in standard form as an ordinary number.
I'd also like you to check by converting that ordinary number back into standard form, just making sure that it does give you the question.
See if you can give it a go.
Press pause for more time.
Well done.
Let's see how you got on.
Well, we should have got 7,800, 135,000, 89,100,000, and finally, 521,300,000.
Well done.
Next, I want you to have a look at this check.
I want you to put a tick next to the correct statements.
Where the answer is incorrect, I want you to write the correct ordinary number.
See if you can give it a go.
Press pause if you need more time.
Well done.
Well, the first one is not correct.
5.
63 times 10 to the seven does not give this value.
The correct number is 56,300,000.
For B, this is also incorrect.
The correct number would be 23,000.
For C, this is also incorrect.
The correct number is 70,500.
Answer D is correct.
Well done if you've got this.
Great work, everybody.
Now it's time for your task.
What I want you to do is 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 says, "Laura teaches Alex and Andeep how to change a number from standard form into an ordinary number." She says, "Nine times 10 to the three is a nine, and stick on three zeros.
That means the answer is 9,000." Andeep says, "Okay, so 8.
3 times 10 to the five is 8.
3, and if I stick on five zeros, that means the answer is 8.
3," and then he's stuck on five zeros.
And Alex says, "I think the answer is 803," and then he's stuck on his five zeros.
I want you to explain why Laura's method works sometimes but not all the time.
And why is it not an effective way to convert a number from standard form into an ordinary number? See if you can give it a go, and press pause for more time.
Well done.
Let's see how you got on with these answers.
Well, for question one, here are all our answers converting standard form into an ordinary number.
Well done if you've got this.
For question two, well this is simply because Laura's method only works when the number being multiplied by the powers of 10 is a single-digit integer.
Laura's method ignores the importance of place value and the multiplication of powers of 10.
Great work, everybody.
Now let's move on to the last part of our lesson, which is standard form on the calculator.
Scientific calculators are fantastic tools.
For example, using a calculator to input this calculation, 82,000,000 multiplied by 25,000, can you tell me what does the calculator output? See if you can give it a go, and press pause if you need more time.
Well, let's see how you got on.
Hopefully you spotted inputting that massive calculation gives you 2.
05 times 10 to the 12.
So when a number is too big or too small for a calculator display fully, the calculator automatically converts the answer to standard form.
The output is in standard form.
It's so simple to see.
You can see it on the screen, 2.
05 times 10 to the 12.
So let's see how we could input it.
Well, we're going to use the Casio fx-570991 ClassWiz.
And to input a number using standard form, it could be done a couple of different ways.
First of all, you can use the power-of button.
You can see it here.
So this button allows you to insert a base and the exponent.
Alternatively, you can use the power-of-10 button.
So you can see it here.
It automatically puts the base of 10 in, and you insert the exponent.
This button sometimes called the exponential base of 10 button.
Now let's input 7.
6 times 10 to the six, and we're going to use that power-of button.
So to do it, input 7.
6 times 10.
Press that power-of button, and you'll see the 10 is now as the base.
Then you simply input the six to represent the exponent.
So it should look like this on the calculator screen.
Now by simply pressing OK or execute, it'll output this value.
So what we've done here is we've inputted a number in standard form using the power-of button.
Now we're going to use that power-of-10 button.
So to do this, what we're going to do is input 7.
6, press that power-of-10 button, and then insert the six.
It should look exactly the same as the previous calculation.
And press OK or execute, and it will output exactly the same answer.
You got two ways to input a number in standard form.
What I'm going to do is I'm going to do a quick-check question and say, "Laura inputs this calculation 6.
13 times, and then she uses the power-of-10 button, presses five, and executes.
What do you think the output would be?" Why don't you try it in your calculator and find out? Well, hopefully you could spot that Laura has used two multiplications, so the output gives an error.
So in other words, she didn't need to put in the extra multiplication, as the power-of-10 button already includes that in its programming.
Well done.
Now let's have a look at this calculation that Laura puts in.
Can you explain the output? Why is it so different than what Laura has inputted? Well, Laura did not input it in standard form, and the calculator will automatically convert it into standard form, as that's what it's programmed to do.
So now we're going to move on to our task by simply using our calculator.
I want to see if you can identify the missing information based on what's on these calculator screens.
See if you can give it a go.
Press pause for more time.
Well done.
Let's move on to question two.
I want you to investigate.
When does the number become so big that the calculator converts it into standard form? See if you can give it a go.
Press pause for more time.
Great work, everybody.
Let's move on to those answers.
Well, here are our missing values.
Massive well done if you got this one correct.
And for question two, how did your investigation go? Well, any integer on the screen which has more than nine digits will be converted into standard form.
Well done if you investigated this.
So in summary, we write a number in standard form because it's easier, concise, and more efficient to write numbers in this way.
If we do not write large numbers in standard form, it would take too long to write huge numbers, and there would be more room for error when writing a number in its ordinary form.
Lastly, standard form allows any number to be written using one standard approach, A multiplied by 10 to the n, where A is a number in between one and 10, including one, not including 10, and n is an integer.
Great work, everybody.
It was wonderful learning with you.
