Lesson video

Hi everyone, my name is Ms Coo and I'm so excited to be learning with you today because we're looking at standard form calculations.

In other words, using really big numbers or really small numbers in a real life context.

I'm so excited to be doing this lesson with you.

So let's make a start.

Hi everyone and welcome to the lesson on adding numbers in standard form under the unit "Standard Form Calculations".

And by the end of the lesson, you'll be able to appreciate the mathematical structures that underpin addition of numbers represented in standard form.

We'll be looking at the keywords, standard form and standard form is when a number is written in the form, A times 10 to the n, where one is less or equal to A, which is less than 10 and n is an integer.

Now today's lesson will be broken into two parts.

Firstly, we'll be looking at adding numbers in standard form.

The second one will be using technology.

So let's make a start.

Here we have a diagram clearly not to scale.

Now the distance from Venus to earth is approximately 40 million kilometres and the distance from the sun to Venus is approximately 1.

1 times 10 to the eighth kilometres.

Ignoring the diameter of Venus, how would you find the total distance from the sun to the air? Have a look and think.

See if you can give it a go.

Well, Aisha says she likes to convert the number into an ordinary number and then add and then convert it back into standard form.

So let's have a look what she does.

She gets 1.

1 times 10 to the eight, convert it into 110 million kilometres.

Then she adds the 40 million kilometres giving her 150 million kilometres and then she converts it back to standard form.

Absolutely fine.

Now Jacob says he's gonna convert them both so they have the same powers of 10 and then add.

So we have 1.

1 times 10 to the eight already given in standard form and then he has converted 40 million into standard form four times 10 to the seven.

Now Jacob recognises to add, we need to make sure they both have the same powers of 10.

So he chooses the highest exponent which is eight, and then ensures both numbers have a power of 10 equal to eight.

So 1.

1 times 10 to the eight stays and four times 10 to the seven now becomes 0.

4 times 10 to the eight.

Because we have the same power of 10, we can add our 1.

1 and our 0.

4 to give us 1.

5 times 10 to the power of eight.

Aisha says she likes this method because writing out really big numbers can cause errors, but she does ask why is it not 1.

5 times 10 to the 16? Do you think you can explain why we are not adding those exponents? Well, here is 1.

1 times 10 to the eight kilometres and then here is 0.

4 times 10 to the eight kilometres.

From our place value chart, you can see when you are adding it doesn't change the magnitude to that 10 to the power 16, which is what Aisha was saying.

In other words, when the powers of 10 are the same, we simply add or subtract our starting number.

Now what I want you to do is a quick check.

I want you to work out the following, give your answer in standard form.

So you can give it a go.

Press pause a bit more time.

Well done.

Let's see how you got on.

Well, I'm choosing the approach of writing them in standard form and then looking at the highest power of 10.

So 1.

4 times 10 to the six is in standard form.

28 million can be written as 2.

8 times 10 to the seven.

The highest power is that seven.

So I've converted 1.

4 times 10 to the six to 0.

14 times 10 to the seven.

Because I have the power of 10 the same, I simply sum that 0.

14 and that 2.

8 to give me 2.

94.

Let's have a look at b.

Converting them both to standard form, I have 3.

5, six times 10 to the four and 4,500 is 4.

5 times 10 to the power of three.

Making them both have the same highest power of 10 means 3.

5, six times 10 to the four add 0.

45 times 10 to the four.

I can simply sum those numbers because they have the same power of 10.

Well done.

An alternative approach could have been converting them into an ordinary number and then converting them back.

You would've got exactly the same answers.

And here's my working out just to show you.

Either approach is really good.

Converting into standard form and looking at the highest exponent is a little bit more efficient.

Now let's have a look at another check question.

The question asks Jacob and Andeep to sum 9.

38 times 10 to the 10 and 1 billion, 300 million and give their answer in standard form.

Here's their working out and what I want you to do is have a look at their working out, but I want you to explain their approach and which method do you think you prefer? See if you can give it a go.

Press pause if you need more time.

Well done.

Let's see how you got on.

Well, looking at Andeep's, Andeep is using standard form but he's decided to convert them to the lowest exponent of 10.

In other words, he chose the nine.

As a result, then when he summed the two numbers, he got 95.

1 times 10 to the nine.

Given that 95.

1 is greater than 10, he then had to convert it back into standard form.

He's still got the right answer, it's just a different approach.

Now Jacob is using a similar approach to Andeep but he converts both numbers so they both have the highest power of 10.

In other words, he's chosen the exponent of 10.

This automatically converts it correctly into standard form when adding.

Now Jacob says this is the same with small numbers too, so he is going to add 6.

9 times 10 to the negative five and 0.

00034 and convert to standard form first.

So here's my numbers in standard form.

6.

9 times 10 to the negative five and 3.

4 times 10 to the negative four.

He asks Aisha, which exponent of 10 is the highest and Aisha recognises its negative four.

So we're going to convert both numbers so the power of 10 is that negative four.

So we have 0.

69 times 10 to the negative four and our 3.

4 times 10 to the negative four.

Adding 0.

69, 3.

4 and keeping that power of 10 the same, that means we get 4.

09 times 10 to the negative four.

Alternatively, we can convert them into an ordinary number and then back into standard form.

So here's the working out Aisha prefers.

Converting them into ordinary numbers, summing them, and then converting back into standard form.

Notice how they both give exactly the same answer.

Now what I want you to do is work out the following, giving your answer 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, if you were using standard form, the highest power would've been negative five.

Summing those numbers, we would've got 4.

88 times 10 to the negative five.

For b, you may have chosen the highest power to be negative five again.

So summing the 0.

14 and our 4.

5 gives us 4.

64 times 10 to the negative five.

Alternatively, you may have converted them into an ordinary number and then back into standard form.

Both methods are absolutely fine.

Using the standard form approach is a little bit more efficient.

It is important to recognise that sometimes it does not give us the answer in correct standard form when adding, using our powers of 10.

For example, 5.

9 or 10 to the five add our 620,000.

Converting them both into the same power of 10, we know we keep the power of 10 the same and then we sum our 5.

9 or 6.

2.

But this is not in standard form.

12.

1 times 10 to the five is not in standard form.

So what we have to do is convert it into correct standard form and why is it before we were able to do it and there was no need to convert it, but now there are some questions where when you add you do need to convert the answer back into standard form? Why do you think that is? Well, it's because when the powers turn the same and you sum the two numbers and they sum to a number greater than 10, that's when you're gonna have to convert the final answer into standard form.

Well done if you spotted this.

Now it's time for your task.

I want you to work on the following, giving your answer in standard form.

So you can give it a go.

Press pause if we need more time.

Well done.

Let's move on to question two.

Question two, work out the following, giving your answer in standard form Great work.

Let's look at question three.

Question three is a really lovely question whereby we're looking at the mass of different creatures given in grammes and kilogrammes and even milligrammes.

See if you can work out the total mass of the elephant and giraffe and the total mass of all three animals.

See if you can give it a go.

Press pause if you need more time.

Well done.

Let's move on to question four.

Another wonderful real life context question.

Here, we know the human body has around about between 206 and 213 bones, and the smallest bones in the human body are actually located in the middle ear and these bones are called the malleus, the incus, and the stapes.

Now the average width of the malleus is given in millimetres.

The average width of the incus is given in centimetres and the average width of the stapes is given in metres and in standard form.

The question wants you to work out the total width of the malleus, incus and stapes in metres and in standard form.

Lovely little real life context question here.

So if you can give it a go, press pause if you need more time.

Well done.

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

Here are our answers, mark them.

Press pause if you need.

For question two, here are our answers.

Press pause if you need more time.

Question three, here are our answers.

We've converted them into standard form and then I've correctly summed them.

Massive well done if you got this one right.

And for question three, same again, converting them all into standard form with the same units.

And then we can sum giving us an answer of 1.

04 times 10 to the negative two metres.

Really well done.

Great work, everybody.

Now it's time for the second part of your lesson using technology.

Now scientific calculators can be fantastic tools as long as we correctly input.

For example, using a calculator, I want you to put this calculation into your calculator and then tell me what does the calculator output.

Well, the calculator outputs a number in standard form, and this is because the number is too big or it can be too small for the calculator to display fully.

The calculator automatically converts the answer to standard form.

As a result, it's important to know how to convert from standard form back into an ordinary number.

So if it was an ordinary number, 2.

05 times 10 to the 12 is this huge number here.

So you can see why your calculator uses standard form.

To input a number using standard form, it can be done a couple of ways.

We can use the "power of" button or we can use the "power of 10" button.

Now this button is sometimes called the exponential base of 10 button.

I'll be calling it the "power of 10" button in this lesson.

So let's have a look at inputting 7.

6 times 10 to the six into our calculator, and we're going to use the "power of" button.

Firstly, you put in 7.

6 into your calculator, multiply by 10, and then you press that "power of" button and you input six.

And what should come up on your screen is 7.

6 times 10 to the 6.

Then pressing, "okay" or "execute", the output should be 7,600,000.

Now let's input the same number but using that "power of 10" button.

Well to do it you press 7.

6, the "power of 10" button, and then six and pressing "okay", it'll give you exactly the same answer.

So using either button works fine.

What I'd like you to do, I'd like us to do a check.

And then from there I'd like you to do a question on your own.

A formula for a perimeter of a shape is given as a+b+c.

We know a is 3.

4 times 10 to the 12, b is 1.

4 times 10 to the nine, and c is 9.

8 times 10 to the 11, all given in centimetres.

We're asked to work up the value of P, giving our answer 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, all we need to do is write down what we input into our calculator.

So we're summing the value of a, b and c.

Our calculator does everything else.

So inputting it into the calculator.

Here are each button I've pressed.

I've decided to use the "power of 10" button.

You may have used the "power of" button.

Either one is fine because you'll still get the same output, 4.

3814 times 10 to the 12.

Now it's time for your question.

In a space station, there's a sub frame and it's an isosceles triangle.

You have to work out the total perimeter of the sub frame.

Give your answer 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've got on.

Well in putting this into the calculator, notice how I've used the "power of 10" button.

Here are the individual buttons I've pressed and then my output is simply 13,340.

But notice how the question wanted it in standard form.

So converting it back into standard form, I have 1.

334 times 10 to the four centimetres.

Really well done if you got this one right.

So let's have a look at small numbers.

For example, let's use our calculator and find out what our output is when we divide 24 by 250,000.

Can you put it into your calculator and tell me what does your calculator output as a decimal? Well, the calculator will output 9.

6 times 10 to the negative five.

It won't output the answer as a decimal.

This is because when numbers are too small, the calculator automatically converts it into standard form.

So let's input a number in standard form using those buttons.

Again, we can either use the "power of" button or the "power of 10" button.

Here, I'm going to input it using 7.

6 times 10 to the power of negative six.

And that gives me an answer as a fraction.

And what I want you to do is I want to change the format from a fraction into a decimal.

Well, it won't convert it into a decimal.

It keeps the answer in standard form because once again, the answer is too small.

So therefore it's really important you're able to convert standard form into an ordinary number.

So in this case, 7.

6 times 10 to the negative six, what is that as an ordinary number? Well, it's 0.

0000076.

Well done.

Let's have a look at another check.

I want you to write the following as a decimal.

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, we should have had these tiny numbers.

Great work if you converted these.

Let's have a look at another check.

Same again, we're going to use our calculator.

Write the following as an ordinary number and 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, we should have had these numbers and then converting it into standard form and an ordinary number, we have this.

Really well done if you've got this one right.

Great work everybody.

Using a calculator, write the following as an ordinary number and 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 get on with question two.

A lovely real life context question.

Here are the approximate areas of the major oceans, Pacific, Atlantic, Indian, Arctic, and the Southern.

For a, what is the total surface area of the Arctic and the southern ocean and I want you to give your answer in standard form.

For b, approximately two oceans sum to make a third ocean.

Name the three oceans and show your working out.

And lastly, work out the total surface area of these major oceans giving your answer 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, here are all our answers to question one.

Press pause if you need more time.

And for question two a, or the total surface area of the arctic and the southern is 3.

4 times 10 to seven kilometres squared.

And for b, those two oceans that sum together to make a third ocean, well the Atlantic plus the Indian is approximately Pacific.

And working this out, you should have got 8.

7 times 10 to the seven and our 73 million is equal to our 1.

6 times 10, the eight kilometres squared, which is approximately the same as that Pacific area.

So well done.

Lastly, the total surface area, adding all of these together gives us 3.

6 times 10 to the eight.

Very well done if you've got this.

Great work, everybody.

So just remember there are a couple of different approaches when adding very large or very small numbers.

One approach is to convert an ordinary number and then back to standard form.

A more efficient approach would be converting using the highest powers of 10.

Scientific calculators have that "power of" button or the "power of 10" button to efficiently input standard form numbers.

However, it is important to know how to convert from standard form into an ordinary number.

Really well done.

Great work.

It was wonderful learning with you.