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Hi, everyone, my name is Miss Coo, and I'm really happy to be learning with you today.

In today's lesson, we'll be looking at something called standard form, and standard form, in my opinion, is one of the best topics in mathematics.

The reason why is because we look at really big numbers or really small numbers.

For example, when do we use really big numbers? Maybe in things like engineering or astronomy, and when do we use really small numbers? Perhaps in things like microbiology or looking at protons or electrons.

Hopefully you'll enjoy this lesson and find it really interesting on how we efficiently write these really big and really small numbers.

I really hope you enjoy the lesson, so let's make a start.

Hi, everyone, and welcome to this lesson on checking and securing understanding of multiples of 10, and it's under the unit Standard Form, and by the end of the lesson, you'll be able to factorise multiples of 10 to the power of N in order to simplify multiplication and division of both integers and decimals.

We'll be looking at this key word, exponential form, and when a number is multiplied by itself multiple times, it can be written more simply in exponential form.

For example, 2 multiplied by 2 multiplied by 2 written in exponential form is 2 to the power of 3.

10 multiplied by 10 multiplied by 10 multiplied by 10 multiplied by 10 multiplied by 10 is written as 10 to the power of 6 in exponential form.

So you can see why we use exponential form as far more efficient.

We'll also be looking at the keywords commutative.

Now an operation is commutative if the values it's operating on can be written in either order without changing the calculation.

For example, 3 add 4 is exactly the same as 4 add 3.

Another example would be 7 multiplied by 2 is exactly the same as 2 multiplied by 7.

A non example would be 10 subtract 2.

This is not the same as 2 subtract 10.

So we have addition and multiplication follow the commutative law, but division and subtraction do not.

This lesson will be broken into two parts.

We'll be looking at using powers of 10 to simplify calculations, and then we'll be using equivalent fractions to simplify calculations, so let's make a start.

Well, here's our Gattegno chart, and what I want you to do is I want you to have a look at this product pair, and what I want you to do is look at the product of the pair of these numbers highlighted, 3 and 7, and the product is 21.

I want you to have a look at this Gattegno chart and find more pairs of numbers on the chart which produce a product of 21.

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

Great work.

Let's see how you got on.

Well 0.

3 multiplied by 70, this would've given us a product of 21, but how did it work? Well, if you notice, the original number was in the third column, which was 3, and the note 0.

3 is one below the original number.

I'm looking at the 7, which was our original number.

If you go one above that 7 in the 7th column, that gives us our 70.

So 3 multiplied by 7 gave us our product to 21, and 0.

3 multiplied by 70 also give us a product to 21.

So why does this give us a product of 21 still? Well, moving down a row on the Gattegno chart divides a number by 10, and moving up a row multiplies a number by 10.

So which of the numbers on this chart have a product of 21? Well, let's see if we can spot those pairs.

Well, you could have 0.

03 times 700.

You could have 30 multiplied by 0.

You could have 300 multiplied by 0.

07.

There are so many different examples which give a product a 21.

Really well done.

So let's explore this in more detail.

I'm gonna look at 30 multiplied by 0.

7, and we're going to use the associative law.

So I'm going to rewrite 30 as 3 multiplied by 10, and I'm gonna rewrite 0.

7 as 7 multiplied by 0.

1, so I've illustrated it here.

We still have our 30, and we still have our 0.

I'm just applying that associative law.

Now what I'm going to do is rewrite this calculation using our commutative law.

So you can see I've put my integers, 3 multiplied by 7, together, and I've got my multiplication of 10 and multiplication of 0.

1 here as well.

Now I'm gonna simplify.

3 times 7 is 21, and the 10 multiplied by the 0.

1 gives us 1.

Thus we have 21 multiplied by 1, which is 21.

So 30 multiplied by 0.

7 gives us 21, and we've illustrated this using the associative law and the commutative law.

You to do is do a quick check question, and I want you to show using the associative law, commutative law, and powers of 10 why the product is 21.

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

Well, first of all, I'm going to use the associative law.

I'm going to change 300 into 3 multiplied by 100, and I'm going to change 0.

07 into 7 multiplied by 0.

01.

From here, I'm going to use the commutative law and put my integers of 3 and 7 to the left, and then I'm gonna put my powers of 10 just to the right there.

From here, I'm going to multiply 3 by 7, which is 21, and 100 multiplied by that 0.

01 gives us 1, and we end up with 21 multiplied by 1, which is 21.

So we've shown using the associative law, commutative law, and powers of 10 why 300 multiplied by 0.

07 is 21.

I want you to do another check question.

Same again.

Still showing that associative law, commutative law, and powers of 10, I want you to show why the product is 21.

See if you can give a go.

Press pause if you need more time.

Great work.

Let's see how you got on.

Well, 30,000 can be written as 3 multiplied by 10,000, and 0.

0007 can be written as 7 multiplied by 0.

0001.

Same as before, I'm going to use the commutative law just to rewrite this calculation and put my 3 multiplied by 7 to the left and my powers of 10 to the right.

Working this out, I have 21 multiplied by 1, thus showing why 30,000 multiplied by 0.

0007 is equal to 21.

Now what I wanted to do is have a look at this exponential form a little bit more, and writing multiplications of 100, 10,000s, a millions, et cetera, et cetera can take time.

So using exponents to show this multiplication is far more efficient.

What I want you to do, I want you to fill in this table to show the multiplications of 10 in exponential form.

I have kindly done the first one for you, and I want you to fill in the rest.

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

Great work.

Let's see how you got on.

Well, 1 is exactly the same as 10 to the zero, 10 is exactly the same as 10 to the 1, 100 is the same as 10 squared, 1,000 is 10 cubed, 10,000 is 10 to the 4, 100,000 is 10 to the 5, and one million is 10 to the 6.

Really well done if you spotted this, and do you spot a sequence using those exponents? So using powers of 10, let's write this calculation.

2,000 multiplied by 4,000.

Well, first things first, I'm going to use the associative law again.

I'm gonna rewrite 2,000 as 2 multiplied by 1,000 and 4,000 as 4 multiplied by 1,000.

So I've simply illustrated it here.

From here, I'm going to rewrite this using the commutative law.

So I'm going to put my powers of 10 to the right-hand side and my 2 and 4 to the left-hand side.

Now from here, I'm going to simplify.

2 multiplied by 4 is 8.

1,000, multiplied by 1,000 is one million.

Now from here I can actually start to rewrite that million in exponential form with a base of 10.

So think back to that table.

How did we rewrite a million in exponential form? Well, it was 10 to the 6, so my calculation is 8 times 10 to the 6.

So what I've successfully done is I've rewritten 2,000 multiplied by 4,000 as 8 times 10 of the 6 using our knowledge on that associative law and commutative law.

Now I want you to see if you can do the same.

I want you to write this calculation using powers of 10.

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

Great work.

Let's see how you got on.

Well, I'm going to rewrite 300 as 3 times 100, and I'm gonna write 3,000 as 3 times 1,000.

Then putting all my powers of 10 just to the right-hand side, that means I've got 3 multiplied by 3 multiplied by 100 multiplied by 1,000.

This gives me 9 multiplied by 100,000.

Remember that table and the powers of 10 in exponential form.

We know 100,000 is the same as 10 to the 5.

So that means 300 multiplied by 3,000 is 9 times 10 to the 5.

Well done if you got this one right.

I'm gonna give you another check, and what I want you to do is have a look what these two pupils did.

Two pupils were given this calculation and were asked to write it in powers of 10.

They were both given 800 multiplied by 200, but both got different answers.

Lucas got 16 times 10 to the 4, and Sam got 1.

6 times 10 to the 5.

Now we need to have a look at their working out and identify who's correct, and can you explain why? So if you can give it a go, and press pause for more time.

Well done.

Well, let's see how you got on.

Both of them are correct because both of them did use powers of 10.

This is a really good example to show how you can write the same calculation but in different ways using powers of 10.

16 times 10 to the 4 is exactly the same as 1.

6 times 10 to the 5.

So all Sam did was look at that 16, use the associative law, and write 16 as 1.

6 times 10.

Then from there, he simply multiplied the 10,000 by that extra 10, which is why it's 10 to the 5.

Well done if you spotted this.

Now let's have a look at our last check before we move on.

I want you to identify which one is the odd one out and explain how you know.

See if you can give it a go.

Press pause one more time.

Well done.

Let's see how you got on.

Well, you can work out the actual answer of each of them and find the odd one out.

I'm going to do 2.

5 times 3 times 10 to the 4.

Well, this is the same as 7.

5 times 10 times 10 times 10 times 10, which is 75,000.

5 times 10 to the 4 is the same as 7.

5 times 10 times 10 times 10 times 10, which is 75,000.

25 times 3 times 10 to the 3.

Well, that's 75 times 10 times 10 times 10, which is 75,000 again.

15 times 5 times 10 to the 3.

Well, that's 75 times 10 times 10 times 10, 75,000 again, and lastly, 2.

5 times 10 to the 3.

That's 2.

5 times 10 times 10 times 10, which is 2,500.

So which is the odd one out? Well, it's the last one.

Using your knowledge on that associative law really does help here.

Well done if you worked this one out.

Great work, everybody.

So now it's time for your task.

I want you to match the calculation with the calculation using exponential form.

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 wants you to do the same again.

Match the calculation with the calculation using exponential form.

See if you can give it a go.

Press pause if you need more time.

Well done.

Let's move on to question three.

Question three wants you to fill in the missing numbers and make sure to show your working out.

Press pause for more time.

Great work.

Let's move on to question four.

Question four wants you to put a tick next to the calculations which are correct and create two of your own calculations using powers of 10.

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

Great work, everybody.

Now let's move on to our answers.

For question one, hopefully you've matched the calculation with the calculation using exponential form.

Great work.

For question two, hopefully you've matched these calculations with the calculation in exponential form, and for question three, I'm gonna show a little bit of working out here.

All I've done is use the associative law, identify all my powers of 10, and, using that exponential form, I've got my answer to be 9 times 10 to the power of 5.

For B, same again.

I'm going to use our knowledge on the associative law, and, using my knowledge on powers of 10, I know my answer is 75 10 to the 3, but I can also look at that 75 and use the associative law again and recognise 75 is 7.

5 times 10.

So that means I have 7.

5 times 10 times 10 to the 3.

So it's 7.

5 times 10 to the 4.

Massive well done if you got that one right.

For question three, same again.

Let's use that associative law and identify those powers of 10.

So I have 60 times 10 to the 5, but looking at that 60, I can use the associative law again and identify it's 6 times 10.

So that means I have 6 times 10 times 10 to the 5, which gives me 6 times 10 to the 6.

Really well done if you got this one right.

For question four, put a tick next to the calculations which are correct.

Here are the correct calculations, and there are an infinite number of examples you could've given.

Just ensure your calculation equals to 24,000.

I've just put two here.

You can use a calculator to check if you want.

Great work.

Excellent work, everybody.

So let's move on to the second part of our lesson, which uses equivalent fractions to simplify calculations.

Now we've looked at multiplying large integers by factorising multiples of 10 to the power of N in order to simplify multiplication, and we can use the same process when multiplying any number using knowledge of decimal and fractional equivalents.

So we already know the following powers of 10 for large numbers.

For example, 10 to the zero is 1, 10 to the 1 is 10, 10 squared is 100, so on and so forth, but what about those non-integers? I want to see if you can fill in the fractional form of the following decimals.

See if you can give it a go.

Press pause for more time if you need.

Well done.

Let's see how you got on.

Well, 0.

1 is 1 over 10, 0.

01 is one over 100, and 0.

001 is 1 over 1,000.

So now what I'm going to do is use our knowledge on exponential form.

So we have 1 over 100.

How did we write 100 in exponential form? Well, it was 10 squared, so that means 1 over 100 or 0.

01 is the same as 1 over 10 squared.

What about 1 over 1,000 or 0.

001? Well, we know 1,000 can be written as 10 to the 3.

So that means 1 over 1,000 can be written as 1 over 10 cubed.

It's such an efficient method to recognise the equivalent fractions written in exponential form.

Well done.

Now what I'm gonna do is look at another calculation using non-integers, for example, 3.

2 times 4.

We're still gonna continue using that associative law so we can write the non-integers as integers using a multiplication of powers of 10.

So let's have a look at 3.

2 and 4.

Well, 3.

2 can be written as 32 multiplied by 9.

Notice how we've written 3.

2 as an integer multiplied by a power of 10.

1 can be written as 41 times 0.

We've written 4.

1 as an integer, 41, multiplied by a power of 10, 0.

Now we're going to recognise our fractional equivalence.

So remember what 0.

1 is as a fraction.

Well, it was 1/10.

So this calculation is the same as 32 multiplied by 1/10 multiplied by 41 multiplied by 1/10.

Using our commutative law again, I'm going to group together our powers of 10 on one side and my integers on the other side, 32 multiplied by 41 multiplied by 1/10 multiplied by 1/10.

Then I'm going to simplify, 32 multiplied by 41 multiplied by 1 over 100.

Now we have a calculation using integers and powers of 10, 32 multiplied by 41 multiplied by 1 over 100, and we can also use our knowledge on exponential form by rewriting it as 32 multiplied by 41 multiplied by 1 over 10 squared.

So what's really important to recognise is that this calculation can be written in a number of different ways using our knowledge and multiplying by fractions.

So this is what we know so far.

We know 3.

2 multiplied by 4.

1 can be written as 32 multiplied by 41 multiplied by 1 over 100.

We also know it can be written as 32 multiplied by 41 multiplied by 1 over 10 squared.

Now using our knowledge and multiplication of fractions, we also know this can be written as 32 over 1 multiplied by 41 over 1 multiplied by 1 over 100, which is the same as 32 multiplied by 41, rolled over, 1 multiplied by 1 multiply 100, which then creates another equivalent, which is 32 multiplied by 41 over 10 squared.

And there are an infinite number of different ways of writing the same calculation.

Here are just a few which can help us decide which calculation to use to work up the answer.

2 times 4.

1 is exactly the same as 32 times 41 times 1 over 100, which is exactly the same as 32 multiplied by 41 multiplied by 1 over 10 squared, which is exactly the same as 32 multiplied by 41 over 100, which is exactly the same as 32 multiplied by 41 over 10 squared.

There's an infinite number of different ways, but which one do you think can help us work out the calculation more efficiently? I'll leave that one with you.

What I want you to do now is do a couple of check questions with me.

True of false.

01 and one over 100 have different values, true or false? And I want you to justify this answer by saying it's A, one is a decimal, one is a fraction, or B, they both the value 1/100 of 1.

See if you can give it a go.

Press pause if you need more time.

Great work.

Hopefully you've spotted it's false because they both have the same value of 1/100 of 1.

Well done.

Let's have a look at another check.

Which of the following are equivalent to the calculation 5.

6 multiplied by 2.

7? See if you can give it a go.

Press pause if we need more time.

Great work.

Let's see how you got on.

Well, I'm going to use the commutative law and associative law and show all this wonderful working out, and I'm gonna see if I can spot our equivalent calculations.

Well, I definitely know it's A, I also know it's B, and I also know it's C.

It can't be D because the denominator is 1,000.

Well done if you got this one right.

Excellent work, everybody.

So now it's time for your task.

What I want you to do is identify the equivalent calculations to 3.

4 times 9.

See if you can give it a go.

Press pause if you need more time.

Well done.

So let's move on to question two.

Question two wants you to identify the equivalent calculations to 0.

4 multiplied by 3.

See if you can give it a go.

Press pause if you need more time.

Great work.

Let's move on to question three.

Question three wants you to put a tick next to the calculations which are correct and create two of your own correct calculations using powers of 1/10.

Give it a go.

Press pause for more time.

Fantastic work, everybody.

Let's move on to question four.

Question four is blank.

So you can create your own calculation and six equivalent calculation used in the power of 10.

See if you can give it a go.

Press pause one more time.

Great work.

Let's look at these answers.

Well, for question one, hopefully you've identified these equivalent calculations.

Well done.

For question two, here are our equivalent calculations.

Really well done.

For E, this required you to calculate the answer to 4 multiplied by 38, which was 152.

That was a little sneaky there, so well done if you got this one right.

Oh.

For question three, here are our correct calculations, and for those calculations where you needed to create your own calculation using powers of 1/10, there are an infinite number of calculations.

Just remember to ensure your calculation does equate to 0.

036.

Here are two examples.

You could have chose anything really, so well done if you got this one right.

For question four, there are an infinite number of answers, but ensure to check with the calculator or any written methods.

Very well done.

Great work, everybody.

So in summary, we've looked at multiplying large integers by factorising multiples of 10 to the power of N in order to simplify multiplication.

For example, 2,000 times 4,000 is the same as 2 times 10 to the 3 multiplied by 4 times 10 to the 3, which we know is 8 times 10 to the 6.

We can use the same process when multiplying any number using knowledge of decimal and fractional equivalents.

For example, 3.

2 times 4.

1 is the same as 32 times 41 times one over 100, which is the same as 32 times 41 times 1 over 10 squared, which is exactly the same as 32 times 41 over 10 squared.

Fantastic work, everybody.

It was great learning with you.

I've finished the video