Lesson video
In progress...Loading...
Hi, there.
My name is Ms. Lambao.
I'm really glad that you 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 multiplying numbers in standard form, and this is in the unit standard form calculations.
By the end of this lesson, you'll be able to appreciate the mathematical structures that underpin multiplication 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 1 but less than 10, and N is an integer.
Exponential form, we will 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 it's 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 split into two learning cycles.
In the first one, we will look at multiplying numbers in standard form, and in the second, we will look at applications of standard form.
Let's get going with that first one.
Like I said, we're going to look here at multiplying numbers in standard form.
Lucas and Andeep are working on this question without calculator.
Calculate 3.
2 multiplied by 10 to the power of 4 multiplied by 1.
4 multiplied by 10 squared.
Give your answer in standard form.
Lucas says, "I'm going to convert them into ordinary numbers and use the area model." Andeep says, "Okay, I'm gonna think about it for a moment as I think there might be a more efficient method." Can you think of a more efficient way than Lucas' method? Here's Lucas' method.
He's writing each of the standard four numbers as ordinary numbers.
3.
2 multiplied by 10 to the power of 4 is 32,000, 1.
4 multiplied by 10 squared is 140.
The calculation he needs to do therefore is 32,000 multiplied by 140.
He's now written this in expanded form and used the commutative and associative laws to rearrange the expression.
Now, he needs to calculate 32 multiplied by 14, and he's going to do this with the area model.
Remember, this area model doesn't have to be to scale, it's just a representation of what the area would be.
10 multiplied by 30 is 300, 10 multiplied by 2 is 20, 4 multiplied by 3 is 120, and 4 multiplied by 2 is 8.
So if I were to find the area of each of those rectangles, those would be my areas.
Now, to find the total area, I find the sum of those four, and that will give me an area of 448.
We go back now.
We get 448 multiplied by 10,000 because 1,000 multiplied by 10 is 10,000.
We write that then as a number.
So we do 448 multiplied by 10,000.
That's 4,480,000.
And then we need to write it back into standard form because that was what the question asked for our answer to be in.
Can you notice a more efficient way now? Andeep says, "The power of 10 part has used the multiplication law for powers.
Also, 4.
48 is the product of 3.
2 and 1.
4.
Therefore, we could have just multiplied the numbers and the powers of 10 separately.
So I was right, there is a much more efficient way." I wonder if you spotted that too.
Of course, you did.
So Andeep is right, Lucas' method works but Andeep's method is more efficient.
From now on, we will use Andeep's method because we're all about efficiency.
Throughout this lesson, you may only use a calculator when you see this symbol.
So when you see this calculator symbol, you may use a calculator, otherwise it's going to be a non-calculator question.
Calculate 3.
2 multiplied by 10 cubed multiplied by 2 multiplied by 10 to the power -5, giving our answer in standard form.
We're going to rearrange using the commutative law.
We're gonna put our number parts together and our powers of 10 together.
We're gonna calculate the number part.
3.
2 multiplied by 2 is 6.
4.
Now, we'll deal with the powers of 10, and we use the multiplication law for indices with the powers of 10, and we add the exponents, 3 add -5 is -2.
The answer to this question is 6.
4 multiplied by 10 to the power of -2.
I think you'll agree, Andeep's method is much, much more efficient.
It would've taken us much longer to write those as ordinary numbers than do the multiplication and then convert it back into standard form.
Let's take a look at this one now.
1.
2 multiplied by 10 to the power of 5 multiplied by 1.
1 multiplied by 10 to the power of -3.
We need to give our answer in standard form.
So again, we are rearranging using the commutative law.
We will then calculate the number part, 1.
2 multiplied by 1.
1 is 1.
32.
Then we'll deal with the powers of 10 part using the multiplication law for indices.
So we add the exponents, 5 add -3 is 2.
So our answer is 1.
32 multiplied by 10 squared.
We'll do one more together, and then I know you'll be ready to have a go at the one on the right hand side independently.
We're going to calculate 1.
4 multiplied by 10 to the power of -1 multiplied by 3 multiplied by 10 to the power of -3, and giving our answer in standard form.
First step, rearrange, calculate the number part, 1.
4 multiplied by 3 is 4.
2.
Power of 10 part, we add the exponents because we are multiplying.
We're going to add -1 and -3, giving us 10 to the power -4.
Over to you now.
Pause the video, give this one a go.
Remember, there's no calculator icon on the screen, so you must not use your calculators.
You must use your knowledge of multiplication law for indices and your multiplication of numbers.
Pause the video, and then when you've got your answer, come back.
Super work.
Let's check that then.
We rearrange it, then we calculate the number part, 4 multiplied by 1.
2 is 4.
8.
We're multiplying that by, and now, we'll deal with the powers of 10.
We're using the multiplication law, which means we need to sum the exponents so that 10 to the power 6 multiplied by 10 to the power -2.
The sum of 6 and -2 is 4, so 10 to the power of 4.
Now, we'll take a look at this one, 2 multiplied by 10 to the power of 5 multiplied by 20,000 multiplied by 1.
22 multiplied 10 to the power of -3.
We're gonna write any numbers that are not in standard form into standard form.
20,000 in standard form is 2 multiplied by 10 to the power of 4.
So you'll notice, I've replaced the 20,000 with its equivalent standard form.
Now, we're going to rearrange using the commutative law.
Then we're gonna deal with the number part, 2 multiplied by 2 multiplied by 1.
22 is 4.
88.
We're gonna multiply that by, and we're going to deal now with the powers of 10 by adding the exponents.
We're finding the sum of the exponents, 5 add 4 is 9 add -3 is 6.
So it's multiplied by 10 to the power of 6.
Now, let's check that on our calculator.
So get your calculator ready.
We enter the calculation into our calculator, but we notice here, the calculator gives us the answer as an ordinary number, but we were asked to give our answer in standard form.
What is 4,880,000 in standard form? Yep, you're right.
It's 4.
88 multiplied by 10 to the power of 6.
So our answer was correct.
We've checked it on our calculator.
Next question, 1.
65 multiplied by 10 to the power of -3 multiplied by 0.
00000, I hope I said the right number of zeros, 2 multiplied by 2 multiplied by 10 to the power of 7.
Again, we're going to write any numbers that are not in standard form into standard form.
Then, we are going to use the commutative law to rearrange.
Then, we'll calculate the number part, and 1.
65 multiplied by 2 multiplied by 2 is 6.
6.
Then, we'll deal with the powers of 10 part, and we're going to find the sum of the exponents, <v ->3 add -3 is -9 add 7 is -2.
</v> So our answer is 6.
6 multiplied by 10 to the power of -2.
And we'll check this one also with our calculator.
So let's check.
This is the calculation I've put into my calculator, and this is the answer that I've got.
What is 33 over 500 in standard form? We're going to use our calculator to change the fraction to a decimal.
We do that, it gives us a 0.
066.
What is 0.
066 in standard form? That's right, it's 6.
6 multiplied by 10 to the power of -2.
We can now see that our answer is right.
Well, of course, it was right.
Now, we'll take a look at this one.
So we've got some algebra here.
We've got A, B, and C, and we've got them written in standard form.
And we're asked to evaluate A, and then in brackets, B add C.
We need to substitute A, B, and C into the expression we are evaluating, which gives us this.
Now, we need to deal with the bit in the brackets first, order of operations is what we consider the brackets first.
I need to add those two together.
In order to add those two together, I need to make sure that the exponents are the same.
So I need to rewrite them so the exponents are the same, and I write them with the highest exponent the same.
So 3 multiplied by 10 to the power of 3 is the same as 0.
03 multiplied by 10 to the power of 5.
And if you need to, you could double check that yourself.
Now, we're going to deal with the bit in the brackets first.
Because the exponents of the powers of 10 in the bracket are the same, we can add the number parts.
1.
8 add 0.
03 is 1.
83, and we know this will be multiplied by 10 to the power of 5.
Remember, that's why we made those exponents the same so that we could do that.
Now, I'm going to deal with my number parts, 2 multiplied by 1.
83 is 3.
66.
And then I'm going to use the multiplication law for indices on my powers of ten, 4 add 5 is 9.
Here, if you need to remember, you could write it out using the commutative law with your number parts first and your powers of 10 second.
We'll take a look at this one together and then you've got one to do independently.
Evaluate A bracket B plus C, substitute in our values.
We then need to make sure that our powers of 10 have the same exponent.
Remember to go, it's easier if you go for the higher exponent, which is 6.
So we rewrite it as this.
Then we can add together 2 and 0.
0021, giving us 2.
0021 multiplied by 10 to the power 6.
Then multiply your number parts, and then use the multiplication law for indices.
So we end up with 6.
0063 multiplied by 10 to the power of 4.
Now, your turn.
Pause the video, give this one a go, no calculator please.
And then when you've got your answer, come back and we will check that for you.
Well done.
Let's check your answer.
So you should have this.
Then we're going to make sure that inside the bracket, we have matching powers of 10 and choose the highest exponent, which is 4.
Then we're going to add those together, and then we're gonna finish off by doing the multiplication by multiplying the number parts and the powers of 10 separately.
We end up with 9.
84 multiplied by 10, <v ->3 add 4 is 1.
</v> Remember not to write that exponent of one.
The mass of Saturn is 5.
7 multiplied by 10 to the power of 26 kilogrammes.
The mass of Mars is roughly 0.
0011 times the massive Saturn.
Work out an estimate for the mass of Mars.
Give your answer in standard form.
We need to do 5.
7 multiplied by 10 to the power of 26 multiplied by 0.
0011.
We can see now, this is a very similar question to what we've previously been doing.
We're going to write 0.
0011 in standard form.
Then we can rearrange using that commutative law.
Then we can calculate the number part which gives us 6.
27 and the powers of 10 part, which gives us 10 to the power of 23, the sum of 26 and -3 is 23.
Without a calculator, I'd like you to calculate the answer to this question.
I've given you working out and I've given you answers.
Please decide which of these is correct.
Pause the video, and when you've made your decision, come back.
What did you decide? Well, the correct answer was the final one, D.
What was wrong with the first one? It was, 0.
003 is not 3 multiplied by 10 to the power of -2, it's 3 multiplied by 10 to the power of -3.
Also, even using that incorrect value, <v ->2 and 5, the sum of those is not 2, it's 3.
</v> There were two mistakes in that one.
B, what was the mistake here? The mistake here was the sum of -3 and 5, the exponents of the powers of 10 is not equal to -1.
And C, again, that exponent there was wrong for 0.
003.
And then here, again, that exponent was wrong.
The sum of -2 and 5 is 3.
So the correct answer is D.
Well done if you identified that.
Now, I'd like you to have a go at these questions without a calculator, remember.
Pause the video, and then when you're ready, you can come back.
Super work.
And now, question two.
Pause the video, give it a go, and come back when you're ready.
Super.
And question number three.
Pause the video again, and then come back when you're ready.
Let's check those answers.
I'm gonna ask you to pause the video, check your answers, and then when you're ready, you can come back and we'll check question number two.
Question number two.
Again, pause the video, check your answer and come back.
And finally, question number three.
Pause the video, check your answer, and then we'll be ready to move on to learning cycle two.
How did you get on with those? Amazing, well done.
Let's now then move on to applications of standard form.
Lucas and Andeep again.
"Andeep, you are an absolute legend.
That method is so much quicker and more efficient.
I've used it on my homework and it saved me so much time." Lucas is really, really grateful for Andeep sharing his much more efficient method.
Andeep says, "Thanks, Lucas.
Let's see if we've got the same answers." They're now gonna check the answers of their homework.
Lucas and Andeep agree on all of their answers except one.
We will take a look at their answers to this question, 3 multiplied by 10 to the power of -2 multiplied by 5 multiplied by 10 to the power of 5.
Give your answer in standard form.
Here's Lucas' work and his answer.
And here's Andeep's work and an answer.
Pause a video and check through each step of each of their workings.
Whose answer is correct? Lucas' is incorrect and Andeep's is correct.
Lucas says, "Why is mine wrong? 15 multiplied by 10 cubed equals 15,000, and 1.
5 multiplied by 10 to the power of 4 is 15,000." Lucas is right.
They do both give an answer of 15,000.
Why is his answer wrong? Andeep says, "We had to give all of our answers in standard form." Why is Lucas' answer not in standard form? "Yours is not in standard form as 15 is not greater than or equal to 1 and less than 10." Lucas says, "Of course, I must watch out for that.
Thanks, Andeep." Which of the following are written in standard form? Any that are not, I'd like you please to write them so that they are.
Pause the video, and then have a go at these four questions and then come back when you're ready.
Let's take a look them.
A was not in standard form.
If it's written into standard form, I'm gonna write 45.
6 in standard form and then I'm going to use the multiplication law for indices, giving me 4.
56 times 10 to the power of 4.
B was in standard form, C was in standard form, and D wasn't.
So we rewrite 2,345 in standard form, which is 2.
345 multiplied by 10 cubed, and we're multiplying that by 10 to the power -6.
So use the multiplication law for indices and find the sum for the 3 and -6, which is -3.
Calculate 2 multiplied by 10 to the power of 4 multiplied by 500,000 multiplied by 3 multiplied by 10 to the power of -1.
Give your answer in standard form.
Write any numbers not in standard form into standard form, which gives us this.
Rearrange using the commutative law.
Calculate the number part and calculate the power of 10 part.
This is not written in standard form.
We need to write any part not in standard form in standard form.
And then you use a multiplication law for indices with the powers of 10.
30 is 3 multiplied by 10.
Now, we need to use the multiplication law for indices.
Remember, if there isn't an exponent, it actually is an exponent of 1, the sum of 1 and 8 is 9.
So the answer to this question was 3 multiplied by 10 to the power 9.
And we can check this on our calculator.
Again, notice it doesn't give us in standard form, it gives us it, to us in ordinary number.
What is 3 billion in standard form? It's 3 multiplied by 10 to power of 9.
So we know that our answer is right.
Andeep is answering this question.
Can he use a calculator? Yes, he can.
The calculator icon is on the slide.
Here is his answer and calculator in display.
Is Andeep's answer right? No.
What mistake has Andeep made? He has forgotten to press the right arrow after input in the first exponent of 4.
We can see that the multiplied by 76,000, et cetera is still part of the exponent.
We need to make sure that we press the right arrow.
I'd like you to use your calculator to calculate the correct answer.
Remember to press the right arrow after you input an exponent.
The correct answer was 4.
47 multiplied by 10 to the power of 3.
The question asked for our answer in standard form to three significant figures.
Did you get that? Well done.
One sheet of paper is 9 multiplied by 10 to the power of -3 centimetres thick.
An exercise book contains 40 sheets of paper.
How thick is it in millimetres? What calculation will we need to do to find the thickness of the exercise book? We need to take the number of sheets of paper and multiply it by the thickness of one sheet of paper.
We're then going to write anything that is not in standard form in standard form, rearrange using the commutative law, calculate the number part and calculate the power of 10 part.
Then anything that is not written in standard form, we write into standard form.
And again, use the multiplication law for indices with the powers of 10, giving us 3.
6 multiplied by 10 to the power -1.
And that was centimetres, remember.
But if we go back to the question, the question wanted to know how thick is it in millimetres? So we need to take our 3.
6 multiplied by 10 to the power of -1 and multiplied by 10 because there are 10 millimetres in a centimetre, giving an answer of 3.
6 millimetres because 10 to the power of -1 multiplied by 10 is 1.
A grain of rice has a mass of 3 multiplied by 10 to power of -2 grammes.
What is the mass of 1 million grains of rice? Give your answer in standard form.
What is 1 million in standard form? It's 1 multiplied by 10 to the power of 6.
We're going to take the mass of 1 grain of sand and we're going to multiply it by 1 million.
Rearrange, using the commutative law.
And then find the product to the number parts and use the multiplication law for indices, we end up with 3 multiplied by 10 to the power of 4.
And remember, this was in grammes.
We must give the units of our answer.
We'll do this one together, and then you can have a go at the one on the right hand side.
1 grain of sand has a mass of 1.
6 multiplied by 10 to the power of -5 grammes.
What is the mass of 1 million grains of sand? Here's our calculation.
Rearrange, 1.
6 multiplied by 1 is 1.
6.
The sum of negative 5 and 6 is 1.
Remember, don't write that exponent of 1.
So it's 1.
6 multiplied by 10 grammes.
Now, your turn.
Pause the video, work out for me what the massive 1 million grains of sugar is.
Good luck.
And then when you are ready, you can come back and we'll check your answer, although I know it'll be right.
Super work, well done.
Now, let's check that answer.
6.
5 multiplied by 10 to the power of -2 was the mass of 1 grain of sugar.
Multiply that by 1 multiplied by 10 to the power of 6 because that's 1 million grains of sugar.
Rearrange, and then calculate.
The sum of -2 and 6 is 4.
So our final answer is 6.
5 multiplied by 10 to the power of 4 grammes.
Now, I'd like you to have a go at this task.
So please, would you pause the video, have a go at these.
Remember, there's no calculator icon on the screen, so therefore, no calculators please.
And then when you're ready, come back and we'll check those answers for you.
Good luck.
Question number two.
You are only allowed to use a calculator for part A.
Pause the video, answer these questions, come back when you are done.
Now, we can check our answers.
One A, 3.
4 multiplied by 10 to the power 8.
B, 9.
6 multiplied by 10 to the power of -1.
C, 1.
25 multiplied by 10 to the power of 6.
D, 1.
1 multiplied by 10 to the power of -2, and E, 7.
2 multiplied by 10 to the power of 6.
Question two.
A is 3.
97 multiplied by 10 to the power of 5, B, 1.
4 multiplied by 10 to the power of 5, and C, 6.
85 multiplied by 10 to the power of 4.
How did you get on with those questions? Superb.
As long as you didn't use a calculator, that is.
You didn't.
Brilliant, well done, apart from part A.
Let's summarise now then what we've done in today's lesson.
We know that numbers written in standard form can be multiplied using the associative and index laws, and there's an example there for you to look through.
Remember, if not all numbers are in standard form, make sure you write them all in standard form, then rearrange so that you've got your numbers together and your powers of 10 together, and then you can calculate the number part, you can calculate the powers of 10 part using that multiplication law for indices.
But just remember, it's important to check that your answer is given in the form asked for.
So for example here, if I'd left my answer as 30 multiplied by 10 to the power of 8, unfortunately, that wouldn't be right because the question wanted my answer in standard form, and we know that that first number has to be greater than or equal to 1 and less than 10, and 30, clearly is not.
Well done on today's lesson, really enjoyed working through this with you, and I look forward to seeing you again really soon.
Take care of yourself and goodbye.
