Lesson video

Hi everyone, my name is Ms. Ku 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 subtracting 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 subtraction of numbers represented in standard form.

Keywords, well definitely looking at the word standard form.

And standard form is when a number is written in the form A times 10 to the n, where one is less than or equal to A, which is less than 10, and n is an integer.

We'll also be looking at exponential form.

And when a number is multiplied by itself multiple times, it can be written more simply in exponential form.

For example, 2 multiply by 2 multiply by 2 is equal to 2 cubed.

This is in exponential form.

We'll 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 pairs of values are grouped and we can group using brackets, for example, multiplication and addition, use the associative law, subtraction and division, do not.

Today's lesson will be broken into two parts.

We'll be looking at subtracting numbers in standard form first and then using technology second.

So let's have a look at subtracting numbers in standard form.

Now this is the distance from Manchester to Sheffield and it's approximately 61,000 metres.

And the approximate distance from London to Manchester is 3.

4 times 10 to the 5 metres.

So how do you think you could calculate the distance in metres from London to Sheffield? Have a little think.

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

So let's have a look at what Aisha means.

Well, if we had 3.

4 times 10 to the 5, converting it into an ordinary number gives us 340,000 metres.

Then she's subtracting the 61,000 metres to give 279,000 metres, which then written in standard form is 2.

79 times 10 to the 5 metres.

Jacob on the other hand says he likes to convert both so they have the same powers of 10 and then subtract.

For example, 3.

4 times 10 to the 5 metres is obviously in standard form and 61,000 metres is 6.

1 times 10 to the 4 metres.

Now, identifying that highest exponent is easier when subtracting.

So keeping the exponent to 5, we have 3.

4 times 10 to the 5.

Subtract our 0.

61 times 10 to the 5.

Then working this out, keeping that power of 10 the same, we have 2.

79 times 10 to the 5.

Notice how it's exactly the same answer.

Aisha says she likes this method because writing out really big numbers or really small numbers can cause errors.

But she does ask, why is it not 2.

79 times 10 to the 0? Because they're subtracting.

Do you think you can explain why? Well Jacob explains, using a place value chart.

Here is 3.

4 times 10 to the 5 and here is 0.

61 times 10 to the 5.

From the place value table, you can see that when subtracting, it won't change the number to a magnitude of 10 to the 0.

Jacob then explains, when the powers of 10 are the same, we simply add or subtract the numbers.

So let's work out the following, giving your answers in standard form.

1.

4 times 10 to the 8, subtract 28,000,000.

And 3.

56 times 10 to the 4, take away 4,500.

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, converting them both to standard form and then identifying that highest exponent, we can do our subtraction more easily.

1.

4 take away a 0.

28 gives us 1.

12 and we keep that power of 10 to be 8.

So the answer is 1.

12 times 10 to the 8.

For B, well converting them both into standard form, identifying the highest exponent is 4, then we can simply subtract 0.

45 from 3.

56 giving us 3.

11 times 10 to the 4.

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

Here's my working out all giving the same answer.

Well done if you've got these.

Jacob explains that this approach can also be applied to small numbers too.

So he's going to subtract this tiny number 0.

000034 from 6.

9 times 10 to the negative 4.

So let's confer them into standard form first.

Here are our numbers both written correctly in standard form.

Jacob then asks, what's the highest power of 10? Aisha recognises it's negative 4.

So that means we're going to convert both numbers so they have the power of 10 to be negative 4.

So our numbers become 6.

9 times 10 to the negative 4 and then we're subtracting 9.

34 times 10 to the negative 4.

As the power to 10 are the same, we simply subtract giving her 6.

56 times 10 to the negative 4.

Alternatively, Aisha says I can convert to ordinary numbers and then convert back to standard form.

So here's her working out where she's converted both numbers into an ordinary number, subtract it, and then converted it back into standard form.

Both approaches are fine.

Using standard form is a little bit more efficient, but they both give the right answer.

It's worth pointing out that Jacob's method does not always give the answer in correct standard form when subtracting.

For example, 1.

2 times 10 to the 5, subtract 98,000.

Now converting them both into standard form, we have 1.

2 times 10 to the 5 and we're subtracting 0.

98 times 10 to the 5.

We have both of them having the same powers of 10.

So we can simply subtract our numbers.

1.

2 subtract our 0.

98 gives us 0.

22.

Now this is not written in standard form.

0.

22 times 10 to the 5 is not in standard form.

So we're going to have to convert it into standard form to give 2.

2 times 10 to the 4.

But why is it in some calculations when we subtract, it gave us the answer in standard form, there was no need to convert.

But in other calculations, we do need to convert it back into standard form.

Have a little think.

Well, it's simply down to the fact that when the powers of ten are the same and the difference between those numbers is less than one, that means we do need to convert the answer back into standard form.

So it's worth pointing out.

Now what I want you to do is I want you to work out the following.

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, working out the first part, I've converted them both into standard form and then I converted them so they both have a power of 10 to be negative 5.

Then I've simply subtracted 0.

28 from 4.

6 giving me 4.

32 times 10 to the negative 5.

For the B, I've converted them both into standard form, I've subtracted, and then I have 0.

95 times 10 to the negative 4.

Notice how that is not in standard form.

So I had to change it into standard form to give me 9.

5 times 10 to the negative 5.

Now, you can also convert them into an ordinary number and subtract and then convert them back into standard form.

Both methods are absolutely fine.

Great work everybody.

So now it's time for your task.

See if you can work out the following, giving your answer in standard form and as an ordinary number.

See if you can give it a go.

Press pause if you need more time.

Well done.

Let's move on to question two.

Well question two says, work out the following, giving your answer in standard form and as an ordinary number.

See if you can give it a go.

Press pause if you need more time.

Well done.

Question three is a lovely little pyramid.

Using addition and subtraction, work out the missing numbers in the pyramid in standard form and the number above is made from the sum of the two bottom numbers.

See if you can work it out and press pause if you need more time.

Well done.

Let's move on to question four, a lovely real life context question.

A virus that infects a bacterium is called a bacteriophage.

And the size of the bacteriophage ranges from 2 times 10 to the negative 8 metres and 9 times 10 to the negative 7 metres.

Now the escherichia coli bacteria is approximately 0.

000002 metres.

I want you to work out the maximum and minimum approximate differences in sizes between the bacteria and the bacteriophage.

Give your answer in standard form.

This is a lovely real life question.

See if you can give it a go and give a bit of thought into how to find the maximum and minimum differences.

Well done.

Let's see how you got on.

Well, for question one, here are our answers.

For one a, we have 170,000, which is 1.

7 times 10 to the 5.

For b, we have 3,230,000, which is 3.

23 times 10 to the 6.

For c, we have this huge number, 48,100,000, which is 4.

81 times 10 to the 7.

And for d we have 40,000, this is 4 times 10 to the 4.

Mark them and press pause if you need.

For question two, here are answers.

For a, we have 0.

0342, which is 3.

42 times 10 to the minus 2.

For b, we have 0.

00085, which is 8.

5 times 10 to the minus 4.

For c, we have 0.

0435, which is 4.

35 times 10 to the negative 2.

And for d, we have 0.

006, which is 6 times 10 to the negative 3.

Mark them and press pause if you need.

For question three, here are our answers all given in standard form.

Excellent.

Well done if you got this.

Mark them and press pause if you need.

And for question four, this is a great question as you had to work out the maximum and minimum differences.

In other words, the maximum difference is found by the escherichia coli bacteria subtract absolute smallest length of a bacteriophage.

So that means it's 0.

000002 subtract 2 times 10 to the negative 8.

This will give you the maximum difference which is 1.

98 times 10 to the negative 8.

Now the minimum difference would be the escherichia coli bacteria subtract the absolute largest length of that bacteriophage.

So it's 0.

000002 subtract on 9 times 10 to the negative 7.

This gives us 1.

1 times 10 to the negative 6.

Really well done if you've got this one.

Great work everybody.

Now it's time to look at the second part of our lesson using technology.

Now when a number is too big or 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.

And the input of a number using standard form can be done a couple of different ways.

So we can use the power of button or we can use the power of 10 button.

Remember it's sometimes called the exponential base of 10 button.

I'm going to call it the power of 10 button in this lesson.

So let's input 7.

6 times 10 to the 6 using the power of button.

Here are the buttons that I press.

And from here, I simply press OK or EXE.

And the answer will be outputted as 7,600,000.

So I'm also going to show you how to input it using that power of 10 button, 7.

6 times 10 to the 6.

So to input it using that power of 10 button, 7.

6, the power of 10 button, and then 6.

Few less buttons and it gives us the exact same answer.

Now what I'm going to do is I'm going to do a quick check.

I'm going to look at a formula for a perimeter of a shape and it's given as P is equal to a plus b subtract c.

a is 9.

4 times 10 to the 15 millimetres, b is 8.

7 times 10 to the 14 millimetres, and c is 4 times 10 to the 13 millimetres.

And we'll asked to work out the value of P, giving your answer in standard form.

See if you can give it a go and press pause if you need more time.

Well done.

Let's see how you got on.

Well, inputting this into our calculator can be done in this way.

Notice how I've used that power of 10 button, then my calculator has outputted the answer as 1.

023 times 10 to the 16.

So now let's have a look at small numbers.

For example, I want you to input 24 divided by 250,000.

What does your calculator output as a decimal? Well, the calculator does not output the answer as a decimal.

It outputs the answer in standard form, 9.

6 times 10 to the negative 5.

When the number is too small, the calculator automatically converts to standard form.

So let's input 7.

6 times 10 to the negative 6.

Either using the power of button or the power of 10 button, I've decided to use the power of button and that means I get this fraction.

And what I want you to do is press the format and change it into a decimal.

What happens? Well, it won't convert it into a decimal.

It keeps the answer in standard form because the answer is so small.

Therefore it's important you are able to convert standard form to an ordinary number.

So let's look at 7.

6 times 10 to the negative six.

What do you think that is as an ordinary number? Well, it'd be 0.

0000076.

Well done.

And let's have a look at a quick check where you need to convert the following standard forms into a decimal.

See if you can give it a go.

Press pause one more time.

Well done.

Let's see how you got on.

Well, we should have had these decimal equivalents.

Really well done for working this out.

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.

Now let's have a look at another check.

Same again, still using our calculator, but I want you to write the following as an ordinary number and in standard form.

See if you can give it a go.

Press pause one more time.

Well done.

Let's see how you got up.

We should have had these answers in standard form and as an ordinary number.

Well done if you've got this.

Press pause if you need more time to mark.

Great work.

Now it's time for your task.

I want you to use a calculator and write the following as an ordinary number and in standard form.

See you can give it a go.

Press pause for more time.

Well done.

Let's move on to question two.

Question two is a great question.

We have a, b, and c given and you have to fill in the missing letters to make the calculation correct.

See if you can give it a go.

Press pause if you need more time.

Well done.

Let's go through these answers.

Here are answers given in standard form and in ordinary number.

Well done.

Press pause if you need.

And for two, here are the missing letters.

Fantastic work if you work this out.

Press pause if you need.

Excellent work everybody.

I really do hope you've enjoyed the lesson because we've looked at some real life context questions with standard form.

Well done everybody.

So there are a couple of different approaches when subtracting very large or very small numbers.

One approach is to convert to an ordinary number and then back to standard form, but a more efficient approach would be converting using the highest power.

Scientific calculators are of course fantastic and they have multiple buttons to insert the power.

You can use the power of button or the tent, the power button to efficiently input standard form numbers.

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

Fantastic work, everybody.

It was great learning with you.