Lesson video

Hi, everyone, my name is Ms. Ku and I'm really happy that you're joining me today to look at standard form calculations.

I really enjoy this lesson because there are so many real-life applications.

I hope you enjoy it too, so let's make a start.

Hi everyone and welcome to this lesson on checking and securing understanding of writing small numbers in standard form.

And it's under the unit Standard Form Calculations.

By the end of the lesson, you'll be able to write very small numbers in the form of A times 10 to the n, where one is less than or equal to A, which is less than 10, and appreciate the real-life context where this format is usefully used.

Now we'll be looking at the keyword 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, two multiplied by two multiplied by two is equal to two 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 three parts.

We'll be looking at conversions of standard form and ordinary numbers.

Then we'll be ordering small numbers in standard form and then ordering with standard form.

So let's make a start.

Now we recognise that using standard form is convention for writing very big or very small numbers where A is multiplied by 10 to the n, where one is less than or equal to A, which is less than 10.

And we know n is an integer.

Now remember, this means the A value must be between one and 10.

It can include one, but it can't include 10.

And n being an integer, just remember an integer can be a positive or negative whole number.

So what I want you to do is I want you to think why is it better to rewrite 0.

000 000 000 562 Why is it better to write it as 5.

62 times 10 to the negative 10? Have a little think.

Well, for obvious reasons, because it takes so long to write and also so long to say as well.

There's a possibility of making errors when copying all the digits.

It's shorter and more concise when it's written in its standard form, and accuracy is not lost either.

So not only is it short and concise, but it's also accurate.

So having a standard approach when writing any number using powers of 10 in exponential form is important to have.

For example, we have this tiny number which we've written as 5.

62 times 10 to the negative 10.

We have 0.

0025 can be written as 2.

5 times 10 to the negative three.

A non example would be 0.

0041 written as 41 times 10 to the negative two.

This is not written in standard form because remember that A value must satisfy that criteria of A being between one and 10, including one but not including 10.

So what I want you to do is have a look at these and identify which of the following is correctly written in standard form.

See if you can give it a go, press pause for more time.

Great work, let's see how you got on.

Well, the second one is written in standard form.

This first one is not written in standard form as 981 needs to satisfy that criteria of A.

In other words, one has to be less than or equal to A, which is less than 10.

The second one we know is in standard form, but the third one is not in standard form, as remember, standard form uses multiplication, not division.

And the last one, well, 124.

56 needs to satisfy that A value again.

Well done if you've got this.

Now Lucas spots something very important using a place value table.

He says, "I can see a nice way to convert a tiny number "from standard form using a place value chart." So he sees 0.

00098 as 9.

8 times 10 to the negative four.

Can you explain how Lucas is able to convert the numbers so quickly using a place value chart? Have a little think.

Well, hopefully you've spotted that the powers of 10 in exponential form is given in the table column heading for the first significant figure.

In other words, the first non-zero digit.

So you can see our nine here, and it's our first significant figure for our A value.

So the nine is under the column heading of 10 to the negative four.

And notice that is the power of 10 in our standard form.

Now let's use a place value chart again.

What do you think this number would be in standard form? Have a little think.

Well, it'd be 6.

19 multiplied by 10 to the negative five.

So place value charts are very powerful, but let's see if we can convert this tiny number into standard form without a place value chart.

Well, without a place value chart, we look at the digits six, one, and nine and then we create that number which satisfies that A value.

In other words, it has to be in between one and 10, including one and not including 10, so it must be 6.

19.

Then from here, we know that our standard form should look like this so far, 6.

10 times 10 to the power of something.

Then what we're going to do is count how many multiplications of 1/10 do we use to get 6.

19 to that 0.

0000619.

So it would be one, two, three, four, and five.

So that means 6.

19 multiplied by five lots of 1/10 gives us that tiny number.

But let's rewrite this in standard form.

In standard form, it'd be 6.

19 times 10 to the negative five, which is this tiny number here.

So basically this tiny number is 6.

19 times 10 to the negative five.

We have 6.

19 satisfying that A value, in other words between one and 10, including one but not including 10.

And we have the number of multiplications of 1/10 needed from that six in that ones column to be in the correct position.

So what I'm going to do, I'm going to do a quick check question.

I'll do the first one and I'd like you to do the second one.

What we're gonna do is convert 0.

00065 into standard form.

Well, to do it, let's think about those digits, six and five.

And we need to make a number which satisfies that criteria for A.

So it's got to be our 6.

5.

Now looking at that six, how many multiplications of 1/10 does it take for this first significant figure of the six in the ones column to be in the right position? Well, let's count, one, two, three, and four.

So that means there were four multiplication of 1/10 from the ones column.

So that means the number written in standard form is 6.

5 times 10 to the negative four.

Now what I want you to do is I want you to convert this tiny number into standard form.

So if you can give it a go.

Press pause if you need more time.

Great work, let's see how you got on.

Well, using those digits, one, five and six, hopefully you created 1.

56.

Focusing on that one, how many multiplications of 1/10 did we do? One, two, three, four, five, and six.

So that means our answer is 1.

56 times 10 to the negative six, well done.

So without a place value chart, what I want you to do is write the following in standard form.

Take your time, press pause if you need.

Well done, let's see how you got on.

Well for a, it should have been 3.

54 times 10 to the negative seven.

For b, 1.

45 times 10 to the negative three.

For c, 5.

63 times 10 to the negative 10.

For d, 4.

86 times 10 to the negative two.

And for the last one, 6.

31 times 10 to the negative 10, well done.

So understanding how standard form works allows us to convert an ordinary number into standard form and vice versa.

For example, what do you think 9.

8 times 10 to the negative five is as an ordinary number? Have a little think.

Well, there's a couple of different ways in which you could change the standard form number into an ordinary number.

You could visualise a place value chart.

So you can see, using those column headings in our place value chart, this is our tiny number, 0.

000098.

You could also visualise that 10 to the negative five as being multiplications of 1/10.

In other words, we have 9.

8 being multiplied by five lots of 1/10.

So then my answer is 9.

8 multiplied by that 0.

00001, which gives me my final answer here.

Now Laura says she knows a shortcut, and she says, "For small numbers, the negative exponent tells you "it's small and it tells you how many zeros to put." And Sophia says, "Oh, I see, "so 1.

24 times 10 to the negative three "has a negative exponent," so she knows it's a small number.

And, "The number in the exponent is three, "so there are three zeros, 0.

00124." Can you explain why Laura's shortcut is correct and it does work for small numbers? See if you can give it a go.

Press pause if you need more time.

Well, it's simply because when a number is correctly written in standard form, the negative exponent is the number of divisions by 10.

So Laura's shortcut is an excellent little shortcut that works.

Now what I'd like you to do is I'd like you to convert the following into an ordinary number.

See if you can give it a go, press pause for more time.

Well done, so here are our answers.

Press pause if you need more time to mark them.

Very well done, great work everybody.

So now it's time for your task.

What I want you to do is convert the following into standard form or an ordinary number.

See if you can give it a go, press pause if you need.

Great work, let's move on to question two.

Question two is a lovely little joke hidden in these calculations.

What you need to do is identify if the calculation is correct in each row.

And if it's correct, I want you to highlight the word connected to that question.

And when you finish, it should reveal a joke.

I hope you enjoy it, see if you can give it a go.

Press pause if you need.

Well done, let's see how you got on.

Well for question one, here are all our answers.

Press pause if you need more time to mark.

Great work, and for question two, here is the joke.

How many monsters are good at maths? None, unless you "Count" Dracula.

I think that was quite funny, personally.

Right, so let's move on to the second part of our lesson.

So the second part of our lesson, we'll be looking at ordering small numbers in standard form.

Here are some matched standard form with some items. We have the length of a strawberry, the width of a fingernail, the thickness of a human hair, the width of a microchip, and the radius of an atom.

Hopefully you can see these lengths are going in descending order, starting with the largest, going to the smallest.

Now removing the imagery, how do you identify that these standard forms are in descending order? I want you to have a little look, have a little think.

Well, hopefully you can spot it's because the exponents are in descending order.

We have negative two, then a negative five, negative six, and then negative 10.

And in a situation where the exponents are the same, we order the A number.

Now what I want you to do is put the following in ascending order, smallest to largest.

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, we should have 4.

4 times 10 to the negative four, then 4.

6 times 10 to the negative four, 5.

6 times 10 to the negative two, 1.

2 times 10 to the negative one, and then one times 10.

Next, we have to put these in ascending order.

So we're going to start with 6.

798 times 10 to the negative six, then 6.

9 times 10 to the negative six, then 6.

45 times 10 to the negative two, 6.

1 times 10 to the negative one, and then 6.

798 times 10.

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

Sometimes a number is not quite given in standard form.

So there's a need to convert them into standard form using the associative law.

For example, we have 890 times 10 to the eight.

Why is this not in standard form? Well, quite obvious, because our starting value is not in between one and 10, including one, not including 10.

So what we have to do is use the associative law to rewrite this calculation.

8.

9 times 10 squared is exactly the same as our 890.

Then we can simply sum those powers of 10 to give me my number written in standard form, 8.

9 times 10 to the 10.

Sam notices a really interesting point.

"When the A number is too big, "we add to the power of 10." Let's have a look at a different example.

What we have, 0.

0023 times 10 to the seven.

Why do you think this is not in standard form? Well, it's because our A number, again, does not satisfy that criteria of being in between one and 10, including one but not including 10.

So rewriting the 0.

0023 using the associative law is the same as 2.

3 times one over 10 cubed.

Then using our knowledge on laws of indices, from here, we can write it in standard form as 2.

3 times 10 to the four.

Really well done if you got this.

Sam notices another good point.

He says, "When the A value is too small, "we subtract from the powers of 10." Good spot, Sam.

Now what I want you to do is spot these mistakes and write the correct answers.

Have a little think, press pause if you need.

Well done, let's see how you get on.

Well, the first one you can see is not correctly written in standard form.

So we need to add three to the power of 10.

Negative six and three is equal to negative three.

So the correct answer is 3.

4 times 10 to the negative three.

This person has unfortunately incorrectly summed the three and the negative six.

For the second part, well, same again, we notice the 0.

00092 is less than one and it's not in standard form.

So that means we need to subtract four from the power of 10.

So it's negative five, subtract our four, which gives us a power of negative 10.

So the answer is 9.

2 times 10 to the negative nine.

This person unfortunately hasn't correctly subtracted four from the negative five.

Now what I want you to do is convert the following into standard form.

See if you can give it a go, press pause for more time.

Well done, let's see how you got on.

Here are all our answers.

Press pause if you need more time to check.

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

Aisha says she cannot put these in ascending order because they're not in standard form.

Explain why she can put them in order and order these numbers in the form they are given.

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, Aisha can convert them all into standard form and then order.

Alternatively, she can convert them into an ordinary number and then order from there.

But it is important to use this as a working-out so in order to order what the question originally gave us.

So let's use this working-out.

I'm gonna convert them all into standard form first.

So looking at 41 times 10 to the negative five, I know I can rewrite this using the associative law, and I know 0.

021 times 10 to the six can be rewritten again using the associative law.

Using this, I can correctly convert all of these numbers into standard form.

Then, it's really easy for me to put them in ascending order.

Here in ascending order, I should have my 41 times 10 to the negative five.

Then my 0.

00439, then my 0.

92, my 0.

021 times 10 to the six, and finally, 24,000.

So notice how I've used my working-out to help me order what was originally given in the question.

Next, Aisha says she likes to convert the number into an ordinary number first and then order.

So let's have a look at a question.

Here, it wants us to put the following in ascending order.

Why do you think Aisha's approach might not be the most efficient? Have a little think.

Well, it's because the numbers are just incredibly small and that means it could potentially lead to a lot of errors when writing these really small numbers into its ordinary form.

Now it's time for your check.

What I want you to do is insert the following in the correct place, show your working-out, and don't forget to put the numbers which were given in the question in the correct order.

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, I've converted them all to standard form first and that means I have my two times 10 to the negative 13, 450 times 10 to the negative 12, this tiny number, 4.

5 times 10 to the negative nine, and 1.

2 times 10 to the negative eight.

Well done if you've got this.

Great work everybody.

Now it's time for your task.

I want you to put the following in their correct position.

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 is a little bit harder.

Same again, put them in their correct position.

See if you can give it a go, press pause for more time.

Great work, let's have a look at question three.

Probably one of my favourite questions.

Match up the average lengths to the different living creatures.

Notice how we're given different units here and some are not given in standard form as well.

Great question, little hint, make sure you convert them all to the same units and then convert them into correct standard form in order to order.

Great question, see if you can give it a go.

Well done, let's see how you got on with these answers.

Well, these are our answers in the correct position.

Mark them, press pause if you need.

For question two, here are our answers in the correct position.

Mark them, press pause if you need.

For question three, these are the correct lengths for the correct living creatures.

And the reason being is because I've chosen to convert them all into millimetres.

Now I've taken them into their ordinary form just so you can see, on average, a flea is about 1.

2 millimetres, an ant is about two millimetres, a cockroach is about 41 millimetres, a tarantula, on average, is 130 millimetres, and a snake is 1,000 millimetres.

Well done if you got this.

Okay, great work everybody.

So now let's have a look at ordering with standard form.

How do we know when one number is big and when another number is small, looking at our standard forms? Well, the positive exponent indicates a large number and you can see that on our place value table.

And the negative exponent indicates the small number.

Knowing when a number written in standard form is going to be a big number or a small number is really important and is indicated by that exponent.

Let's have a look at a check.

Some pupils have worked out the answers to these questions.

Identify if they are correct or not and explain why.

See if you can give this a go.

Press pause if you need more time.

Well done, let's see how you got on.

Well, the first one is correct.

The second one, unfortunately, is not correct.

It's incorrect because the exponent is negative, so our number should be small.

The third one is incorrect because the standard form is written incorrectly.

0.

562 does not satisfy that criteria for A.

And the last one, this is incorrect as the exponent is positive, so therefore the number should be big.

Well done if you got this.

Great work everybody.

Now it's time for your task.

Here are the average lengths of some other living creatures.

Match the correct measurement with the correct living creature.

I do love these questions, real-life application of standard form.

Notice how some units are given in metres, millimetres, and centimetres, as well as the fact that they're not written in standard form.

See what you can do, great question.

Press pause for more time.

Well done, let's see how you got on.

Well, these are the correct lengths for the correct creatures.

I've chosen to convert them all into metres, only because I think it visualises those average lengths of those living creatures a little bit better.

So you can see the ant is 0.

002 metres.

The cockroach is 0.

041 metres.

Remember, we've looked at the average length before as well.

The frog is 0.

08 metres.

A cat is, on average, 0.

45 metres.

And a lovely blue whale is 23 metres in length.

Great work everybody.

So in summary, we know standard form is the convention to write very large or very small numbers and we use standard form in a number of industries.

Once numbers are written in standard form, we can order using the exponents.

The bigger the exponents, the larger the number.

For example, nine times 10 to the 10 is greater than nine times 10 to the six.

Or two times 10 to the negative five is greater than two times 10 to the negative nine.

And converting all calculations to standard form, or sometimes to an ordinary number, does allow ordering to be more efficient.

Great work, everybody.

It was wonderful learning with you.