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My name's Miss Lambell.
I'm really glad that you've decided to join me today to do some maths.
I hope you'll enjoy it, and of course you'll enjoy it.
Let's get going.
Welcome to today's lesson.
The title of today's lesson is Dividing Numbers in Standard Form, and this is within the unit Standard Form Calculations.
By the end of this lesson, you'll be able to appreciate the mathematical structure that underpins division of numbers represented in standard form.
Quick recap as to what standard form is.
Standard form is when a number is written in the form A multiplied by 10 to the power of n, where A is greater than or equal to one, but less than 10, and N is an integer.
Exponential form, we'll be referring to in today's lesson also, this is when a number is multiplied by itself multiple times, and it can be written more simply in its exponential form.
We will also refer to the commutative and associative laws.
The commutative law states that you can write the values of a calculation in a different order without changing the calculation.
The result is still the same.
It applies for addition and multiplication.
The associative law states that it doesn't matter how you group or pair values, i.
e.
which we calculate first, the result is still the same.
It also applies for addition and multiplication.
Today's lesson is in two learning cycles.
In the first one, we will look at dividing numbers in standard form, and in the second one, applications of standard form.
Let's get going with that first one, dividing numbers in standard form.
We're going to calculate eight multiplied by 10 to the power of eight divided by two multiplied by 10 to the power of four, and we need to give our answer in standard form.
Sam says, "When I think about any division, "I think of them as a fraction "because I'm really good at simplifying fractions." Jacob's response is, "That's a really good point, Sam.
"I had not thought about that, "but you're right, "if we can write any division as a fraction, "of course, simplifying the fraction "will answer the division question." Oh yeah, that's right, isn't it? Because when we are simplifying a fraction, we're dividing, so therefore it's going to give us the same result.
Therefore, we're going to rewrite this calculation as a fraction.
Eight multiplied by 10 to the power of eight over two multiplied by 10 to the power of four.
We're then going to divide the number parts.
Eight divided by two, which we know is four.
We're now going to consider the powers of 10, and we're going to use the division law for indices.
10 to the power of eight divided by 10 to the power of four is 10 to the power of four.
Remember, we subtract the exponents.
The answer to this question then is four multiplied by 10 to the power of four.
Let's take a look at this one.
We're gonna rewrite the division as a fraction.
We're then going to divide the number parts, 3.
9 divided by three is 1.
3.
We're then going to multiply that by 10 to the power of three divided by 10 to the power of negative four.
Here, remember, we are subtracting the exponents.
Three subtract negative four gives us seven, so 10 to the power of seven.
We'll do the one on the left together, and then you can have a go at the one on the right hand side.
Firstly, we're going to write our division as a fraction.
We're then going to deal with the number part, 1.
44 divided by 1.
2 is 1.
2.
Remember, we're multiplying.
If we look at the numerator and the denominator, multiplication, we're going to multiply by the result of 10 to the power of negative four divided by 10 to the power of negative two, which is 10 to the power of negative two.
You are subtracting the exponent, negative four subtract negative two is negative two.
Here's one for you to have a go at on your own, no calculator.
Pause the video, and then come back when you're ready to check your answer.
Well done, let's check that answer.
Hopefully, of course you did, you rewrote your division as a fraction.
8.
4 divided by four is 2.
1.
10 to the power of negative one divided by 10 to the power of negative three is negative one, subtract negative three and that's two.
So multiplied by 10 squared.
Did you get it right? Well done.
Now let's take a look at this one.
We need to calculate three multiplied by 10 to the power of five multiplied by 120,000 divided by two multiplied by 10 to the power of negative three.
We're going to write any numbers not in standard form into standard form, so 120,000 is 1.
2 multiplied by 10 to the power of five.
We're then going to rewrite the division as a fraction.
We end up with this calculation.
Calculate the numerator.
So we're gonna calculate just the numerator at the moment.
So we're gonna do three multiplied by 1.
2, that's 3.
6.
And 10 to the power of five multiplied by 10 to the power five is 10 to the power of 10.
So we end up with 3.
6 multiplied by 10 to the power of 10 over two multiplied by 10 to the power of negative three.
Now, we can do what we were doing in the previous few slides.
We're gonna do 3.
6 divided by two.
Gonna simplify it, which is 1.
8.
And then 10 subtract negative three is 13.
Our final answer is 1.
8 multiplied by 10 to the power of 13.
Let's just recap what we did there.
First step was to write anything not in standard form into standard form.
We then wrote it as a fraction.
Then we dealt with just the numerator, and then we simplified using the numerator and the denominator.
Let's check this on our calculator.
Here's my calculation.
Notice, I've gone back to my original calculation and I've remembered to write my division as a fraction, and we can see that our answer is correct.
Now we'll take a look at this one.
Again, we're gonna write any numbers that are not in standard form into standard form, so that's the 0.
000002.
And that is two multiplied by 10 to the power of negative six.
Now we're going to write the division as a fraction.
Next, we deal with the numerator.
We calculate the numerator.
Three multiplied by two is six.
10 squared multiplied by 10 to the power of negative six, we find the sum of two and negative six, that's negative four.
Six multiplied by 10 to the power of negative four over 1.
2 multiplied by 10 to the power of four.
Now we can simplify.
Six divided by 1.
2 is five, and then 10 to the power of negative four divided by 10 to the power of four, we subtract the powers, giving us 10 to the power of negative eight.
Let's check this on our calculator.
I've gone back.
Remember, I wanna go back to the original calculation.
I know we won't have, but just in case we didn't correctly write 0.
000002 into standard form, always go back to the original calculation, and we can see here that our answer is right.
Sam and Jacob are using their calculator to check their answers to this question.
Why are their answers different? Sam has used the fraction button, and Jacob has written the calculation in a long line.
Without a calculator, I would like you please to answer the question and decide whose answer is correct.
Who did you agree with? Hopefully you agreed with one of them, but the correct answer was three to the power of negative eight.
If you need to pause the video, you can, and you can have a look at the calculations that got us there.
So Sam was right, but why is Jacob's answer wrong? Because multiplication and division have equal priority, the calculator works them out in the order they appear.
If we take a look at the end part of Jacob's calculation that he's put into his calculator, we can see here we're dividing by 1.
6 multiplied by 10 to the power of seven.
Here, the calculator is dividing everything that has gone before by 1.
6, and then multiplying the whole number that they've got there by 10 to the power of seven, instead of divided by 1.
6 multiplied by 10 to the power of seven.
Jacob can enter the calculation as one long line, but he must remember to place the 1.
6 multiplied by 10 to the power of seven in brackets.
Sam says, "Another reason "I write any division as a fraction." Jacob says, "Okay Sam, "you've totally convinced me "that I should be writing all divisions as fractions." I think Sam's feeling a little bit smug there.
Now we'll take a look at this question.
We've got some values of a, b, c and d, and we're going to substitute those into the expression a subtract d over b add c.
we're gonna substitute the values in.
We're then going to rewrite the numerators and the denominators so that they have the same power of 10, and the power of 10, we pick the highest exponent.
Then we can calculate the numerator, five subtract 0.
2 is 4.
8.
And we can only do this, remember, because we've got both powers of 10 matching, so that's still gonna be multiplied by 10 to the power of four.
And then if we look at the denominator, 0.
6 add one is 1.
6, and again, this is because we've made the exponents of the powers of 10 the same.
Now we can calculate this.
4.
8 divided by 1.
6 is three, and we're doing 10 to the power of four divided by 10 to the power of six.
So we subtract the exponents, we end up with an answer of three multiplied by 10 to the power of negative two.
Now, there's quite a lot to unpick there, so if you need to, pause the video, look at each step carefully and make sure you understand where that answer of three multiplied by 10 to the power of negative two has come from.
We'll take a look at this one together, and then you could have a go at the one on the right hand side because I know you're ready for it now.
Substitute in our values of a, b and c.
Then we need to make the powers of 10 the same, choosing the highest exponent, which is four.
Six multiplied by 10 cubed is the same as 0.
6 multiplied by 10 to the power of four.
Deal with the numerator first, and that gives us 3.
6 multiplied by 10 to the power of four, and we know that's over 1.
2 multiplied by 10 to the power of five.
3.
6 divided by 1.
2 is three.
And then we're subtracting the powers, four subtract five is negative one, so there's our answer.
Three multiplied by 10 to the power of negative one.
Now it's your turn.
Pause the video.
Good luck with this.
No calculators, and then when you're ready, come back and we will check your answer, although I know it's going to be right.
How did you get on? Well done.
Substitute in your values.
Then make sure that the powers of 10 are the same in the denominator.
Then we can simplify the denominator, 1.
9 subtract 0.
4 is 1.
5.
4.
5 divided by 1.
5 is three, and then we're subtracting the exponents, two subtract three is negative one.
So actually, the answers to both of those questions were exactly the same.
Now you can have a go at task A.
Question number one, so without a calculator, I'd like you to have a go at these questions.
So pause the video, and then come back when you're ready.
Remember, we need to be smug like Sam and write our divisions as fractions.
Good luck with this.
I'll be waiting for you when you get back to check those wonderful answers.
Question number two, so I've given you some values here of a, b, c and d.
And I'd like you to evaluate those different expressions.
Pause the video, and then when you're ready, come back.
Great work, well done.
Let's check those answers then.
Question one A, 2.
34 multiplied by 10 cubed.
B, four multiplied by 10 to the power of negative three.
C, three multiplied by 10 to the power of negative two.
D, 1.
1 multiplied by 10 to the power of negative two.
E, 1.
8 multiplied by 10 to the power of six.
F, 1.
6 multiplied by 10 to the power of negative four.
And G, 1.
4 multiplied by 10 to the power of 14.
Did you get all of them right? Yes, of course you did.
And question two.
A, 1.
5 multiplied by 10 to the power of three.
B, 3.
4 multiplied by 10 squared.
Now, if you've made any errors, pause the video, have a look at my workings and see if you can see where you went wrong and correct yourself.
And part C and D, C, three multiplied by 10 cubed, and D was five multiplied by 10.
Again, if you need to pause the video to take a look at the method, then you can obviously do that.
When you come back, we can move on to the second learning cycle for today's lesson.
And that second learning cycle is applications of standard form.
Let's get going on that.
We're gonna calculate this, and we need to give our answer in standard form.
We're gonna write the division as a fraction, no surprises.
We're then going to calculate the number part, so we're gonna divide 3.
2 divided by four.
I think of that as £3.
20 divided by four people, so I would divide by two, that's £1.
60, and then divide by two again.
That's 80 pence, so it's 0.
8.
We're then going to deal with the powers of 10.
We're gonna subtract, because we're dividing, we're gonna subtract the exponents.
We end up with 10 to the power of four divided by 10 to the power of seven is 10 to the power of negative three.
Is that written in standard form? No, 0.
8 is not greater than or equal to one and less than 10.
We need to write the number part in standard form.
0.
8 is eight multiplied by 10 to the power of negative one, and we're multiplying that by 10 to the power of negative three.
Now, using the multiplication law for indices with powers of 10, we add the exponents, we find the sum of the exponents, we end up with eight multiplied by 10 to the power of negative four.
And this one, we're going to write any numbers not in standard form into standard form.
So 120,000 in standard form is 1.
2 multiplied by 10 to the power of five.
Now what are we gonna do? Yes, of course, we're now gonna write our division as a fraction.
Write our division as a fraction, calculate the numerator.
Three multiplied by 1.
2 is 3.
6.
We're gonna add the exponents, giving us 10 to the power of 10.
And then we're gonna simplify, 3.
6 divided by six is 0.
6.
We're going to subtract the exponents, 10 subtract negative three is 13.
Write in standard form.
So 0.
6 is six multiplied by 10 to the power of negative one.
We're multiplying that by 10 to the power of 13.
So we'd now just need to tidy that up by simplifying 10 to the power of negative one multiplied by 10 to the power of 13, which is 10 to the power of 12.
Let's do the one on the left together, and then you can do the one on the right.
Gonna write as a fraction.
Also, I've written 0.
0003 as three multiplied by 10 to the power of negative four.
We're going to work out what the numerator is.
Then we're going to divide 4.
8 divided by eight, which gives us 0.
6.
We're going to use the division law for indices here.
There's no written exponent for 10, but remember it's one.
One subtract negative two is three.
Then we need to write 0.
6 in standard form, and then we tidy up those powers of 10.
So we end up with six multiplied by 10 squared.
Your turn now.
Pause the video, and when you've got your answer, come back.
Super work, well done.
Let's check that answer.
Of course, you rewrote 0.
02 as two multiplied by 10 to the power of negative two, and of course you wrote your division as a fraction, and then you worked out what your numerator was.
1.
2 multiplied by two is 2.
4.
Find the sum of the exponents, the sum of four and negative two is two.
Then we're going to do 2.
4 divided by six, which is 0.
4.
Two subtract negative four is six.
Rewrite 0.
4 as four multiplied by 10 to the negative one.
And then tidy up those powers of 10 by adding the exponents.
We end up with four multiplied by 10 to the power of five.
Mercury takes approximately 90 days to orbit the sun.
Venus takes approximately six multiplied by 10 to the power of four days to orbit the sun.
How many times longer does it take Venus to orbit the sun? And we need to give our answer to two significant figures.
We're going to take the time it takes Venus and divide that by the time it takes Mercury.
90, however, is not in standard form.
We need to write it in standard form and remember to write our division as a fraction.
Six over nine simplifies to two over three, and 10 to the power of four divided by 10 is 10 to the power of three.
We want our answer to two significant figures, so we need to write 2/3 as a decimal to two significant figures, which is 0.
67.
And we're still multiplying that by 10 to the power of three.
Write 0.
67 in standard form, that's 6.
7 multiplied by 10 to the power of negative one.
And then like I said, we can tidy up, we can simplify 10 to the power of negative one multiplied by 10 cubed, and that gives me 10 squared.
Final answer is 6.
7 multiplied by 10 squared, which is 670 times longer.
Notice here, the question didn't ask us to give our answer in standard form.
A grain of rice has a mass of three multiplied by 10 to the power of negative two grammes.
How many grains of rice are there in one kilogramme? Give your answer as an ordinary number.
How many grammes are in one kilogramme? There are 1,000, but we want that to be written in standard form.
1,000 is one multiplied by 10 cubed.
We need to divide 1,000 grammes by the mass of one grain of rice, which is three multiplied by 10 to the power of negative two.
We're gonna write that as a fraction.
We end up with one third multiplied by 10 to the power of five.
Remember, we're dividing, so we subtract the exponents.
Three subtract negative two is five.
We're going to write it as a decimal to two significant figures.
So 1/3 is 0.
33.
And that's multiplied by 10 to the power five.
Now we're going to write 0.
33 in standard form, and then we're going to use that multiplication law for indices with the powers of 10, giving us 3.
3 multiplied by 10 to the power of four, but the question asked for our answer as an ordinary number.
So we are now going to write that as an ordinary number.
There are approximately 33,000 grains of rice in one kilogramme.
One grain of sand has a mass of 1.
6 multiplied by 10 to the power of negative five grammes.
How many grains of sand will there be in one kilogramme? We take our one kilogramme, and we divide it by the mass of one grain of sand.
We write that division as a fraction.
We then calculate one divided by 1.
6, and 10 cubed divided by 10 to the power of negative five.
We need to subtract the exponents.
Three subtract negative five is eight.
We need to write now 0.
625 in standard form, and then simplify the powers of 10, leading us to 6.
25 multiplied by 10 to the power of seven.
There are approximately then 62,500,000 grains of sand in a kilogramme.
We certainly wouldn't want to check that, would we? Now your turn.
How many grains of sugar will there be in one kilogramme? Pause the video.
No calculator for this one please.
When you're ready, you can come back.
And here are your calculations.
So we've got our kilogramme, and we're dividing it by the mass of one grain of sugar.
We're writing that as a fraction.
We're dividing the number part, 0.
15, and then we're using the division law for indices.
So we're going to subtract the powers.
Three subtract negative two is five.
Rewrite 0.
15 as 1.
5 multiplied by 10 to the power of negative one, and that's multiplied by 10 to the power of five.
Use the multiplication law for indices, we end up with 1.
5 multiplied by 10 to the power of four.
That means there are approximately 15,000 grains of sugar in one kilogramme.
The population of the UK in 2021 was 6.
73 multiplied by 10 to the power of seven.
In 2011, it was 6.
33 multiplied by 10 to the power of seven.
By what percentage had the population increased? Sam says, "To do this, "we need to convert the population's ordinary numbers "and use a double number line, "ratio table or equation to find the percentage change." Jacob says, "We only need to look at the percentage change "from 6.
33 to 6.
73." Do you agree with Jacob? As the powers of 10 are the same, we can just consider the number parts.
Remember, if they hadn't been the same, we could have made them the same, and then still considered just the number parts.
By what percentage is the population increased? That's the question we are wanting to answer.
I'm going to use a ratio table.
The original population was the population in 2011, and that was 6.
33 multiplied by 10 to the power of seven.
But as the population in the UK in 2021 was also a number multiplied by 10 to the power of seven, we can just compare the number parts.
That's what we've talked about on the previous slide.
We need to work out then what's missing.
What's the missing percentage? What is my multiplier that takes me from 6.
33 to 6.
73? I multiply by 6.
73 over 6.
33.
I need to do the same then to the percentage.
And if I do that, I end up with 106.
3.
The population, therefore, has increased by 6.
3%.
It's your turn now to have a go at task B.
Give your answers in standard form to the following questions.
And this is without a calculator, please.
Good luck.
Pause the video, come back when you're ready, and I'll reveal to you the next question.
Question number two.
In 2024, the Vatican City is the smallest country in the world, both by area and population.
Its population is 5.
3 multiplied by 10, and that's given correct to two significant figures.
The country with the largest population is China, with a population of 1.
4 multiplied by 10 to the power of nine, and again, that's given to two significant figures.
How many times bigger is the population of China than the population of the Vatican City? I'd like you please to give your answer to two significant figures, and you'll notice here on the slide, we can see the calculator, which means you are allowed to use a calculator for this question.
Pause the video, and when you come back, we'll see if you are surprised by how much bigger the population of China is than the Vatican City.
I certainly was.
And question number three, the population of the UK in 2021 was 6.
73 multiplied by 10 to the power of seven.
100 years before, so in 1921, it was only 4.
29 multiplied by 10 to the power of seven.
By what percentage have the population increased? And I'd like you please to give me your answer correct to one decimal place.
Pause the video.
You'll notice here the calculator icon is on the slide.
So yes, you may use your calculator, but make sure you write down all the steps of your working.
I'll be here waiting for you when you get back.
We'll check those answers for you.
Super work, well done.
Now let's check the answers.
Question one A, 5.
6 multiplied by 10 to the power of eight.
B, 2.
5 multiplied by 10 to the power of 12.
C, five multiplied by 10 to the power of negative two.
D, 1.
375 multiplied by 10 to the power of four.
E, 1.
1 multiplied by 10 squared.
Question number two, the answer is 2.
6 multiplied by 10 to the power of seven.
And the workings are there if you need to pause the video and take a look at how I got that answer, if you made an error, but I'm sure you didn't make an error because you've done fantastically well today.
And finally, the population in that 100 years has increased by 56.
9%.
That is a huge increase, isn't it? 56.
9%.
I wonder what the figures will look like in another 100 years time, so in 2121.
Be interesting to see.
Let's summarise the learning that we've done during today's lesson then.
We've looked at numbers written in standard form, and we know that we can divide them using the associative and index laws.
We rewrite any division as a fraction.
Let's be smug like Sam.
Rewrite any division as a fraction.
And there's an example there of what we did.
Remember, any number that is not written in standard form, write into standard form first.
Then you are going to write it as a fraction.
Then you are going to simplify and make sure your answer is in the form asked for in the question.
Is it asked for in standard form? Is it asked for as an ordinary number? And notice here, we needed to change 0.
6 multiplied by 10 cubed because 0.
6 is not greater than or equal to one and less than 10.
Therefore, the answer was not in standard form.
Superb work on today's lesson.
I'm really glad that you decided to join me.
Fantastic choice, and hopefully I will see you again very, very soon to do some more maths.
Take care of yourself, goodbye.
