Lesson video

Hiya, my name's Miss Lambell.

You've made an excellent choice deciding to stop by and do some math for me today.

I'm really pleased.

Let's get started.

Welcome to today's lesson.

The title of today's lesson is Multiple of 10 and negative exponents.

By the end of this lesson, you will be able to write any integer in a range of forms. We'll be using exponential form a lot in today's lesson.

We need to remind ourselves of what exponential form is.

When a number is multiplied by itself multiple times, it can be written more simply in exponent form.

So for example, two multiplied by two multiplied by two, we could write as two to power of three, which we'll read as two cubed.

Today's lesson, I've split into two learning cycles.

In the first one, we're going to look at negative exponents, and in the second one we will look at writing integers in a range of forms. Let's get going with that first one.

We're gonna be looking at the introduction of negative exponents.

Let's go.

What does four to the power of five mean? An exponent tells us how many identical values must be multiplied together.

So four to the power of five, means four, multiplied by four, multiplied by four, multiplied by four, multiplied by four.

You've seen exponents being used with square and cube numbers.

We've also seen them when we were considering place value.

Here's a reminder of what our place value grid looks like.

So we started off with the column headings, of thousands, hundreds, tens, ones, tenths, hundreds and thousands.

And then we wrote those in the numerical form, and then we looked at those as powers of 10, and then we wrote them in exponential form.

Lucas says, "If we can write 1,000, 100 and 10 in exponent form, surely we can write the others in exponent form too." And Aisha says, "Yes, I think you must be right." One tenth, one hundredth and one thousandth.

We'll just take a look at those three.

Lucas says, "Well, we know that 10 can also be written as 10 to the power of one, 100 as 10 squared, and 1,000 as 10 cubed." Aisha says, "We can rewrite these as.

." One over 10 to the power of one, one over 10 squared, and one over 10 cubed.

Lucas now says, "Exponent form means a repeated multiplication." Yes, Lucas is right.

We know that exponent form means a repeated multiplication.

But wants to know what repeated multiplications have happened here? We know that one over 10 to the power of one is one over 10.

One over 10 squared is one tenth multiplied by one tenth.

And one over 10 cubed is one tenth multiplied by one tenth multiplied by one tenth.

So that answers Lucas's question, as to what repeated multiplication has happened.

Now we could fill in our table like this.

Lucas says, "I was looking at the exponential column headings." He was looking at that bottom row.

Aisha says, "Oh right, did you notice anything?" I'm wondering if you've noticed anything.

I'm gonna pause a moment, and give you an opportunity to think whether you've noticed anything.

Lucas says, "The exponents decrease as you move right, but when you get to the decimal parts, it becomes reciprocal." Aisha says, "The exponents are decreasing by one each time.

Could we write one over 10 to the power of one, as 10 to the power of negative one?" So they've spotted that it's gone 10 cubed, 10 squared, 10 to the power of one, 10 to the power of zero.

So the suggestion is, would it then go 10 to the power of negative one? Here are our numbers, and here are our exponential forms, and we can see it clearly here, can't we? That continuing the pattern, we are going to go 10 to the minus one, 10 to the minus two, 10 to minus three, and 10 to the minus four.

We've got the number form, zero point, or decimal form, 0.

1, then the fractional form, and we now know what that looks like in its exponential form.

Lucas says, "That must mean that one over 100, and 10 to the minus two are equivalent." Aisha says, "That makes sense as 10 squared is 100." I'd like you now to have a go at matching up the fractional forms to the exponential forms. So you're gonna pause the video, you're gonna match them up, and when you've got your answers, you're gonna come back and we'll check those.

Great work.

10 to the power of negative three is one over 1,000.

10 to the power of negative one is one tenth, 10 to the power of negative four is one over 10,000, and 10 to the power of negative seven is one over 10 million.

How did you get on? Well done.

We're now gonna take a look at how all of these things are linked.

We'll start with the integer one.

There's my integer one.

I move the digit one place to the right.

What have I done to the one moving one place to the right? What have I done? I've divided by 10, so I've done one divided by 10.

The one is now in the tenths column.

We now know that we are representing the number 0.

1.

But we can also see that that's the same as one divided by 10.

That's how that one got into the tenths column.

We know that that is also equivalent to one tenth, and we also know that we couldn't write any division as a fraction, and we also now know that that is one multiplied by 10 to the negative one.

Because we've continued those exponent forms down into our decimals now.

I'm gonna try by looking at some simple decimals first, at how we can write numbers using the negative exponents.

So we're gonna start with 0.

007.

Here's my place value grid.

I'm going to put in my number, 0.

007.

We know that that is seven lots of one thousandth, because it's in the thousandth column, and there are seven of them.

So seven multiplied by one thousandth.

We know that that is equivalent to seven thousandths, but we also now know that we could write that as seven multiplied by 10 to the negative three, because we know the thousandths column is the 10 to the negative three column.

Let's now try 0.

0002.

Place value grid, let's place our number in, 0.

002.

I'm not sure I said enough zeros there, it's 0.

0002.

And we can see this time it is in the 10 thousandths column and there are two of them, which is two over 10,000, which is also the same as two, multiplied by 10 to the negative four.

I'd like you now to write 0.

09 in expanded exponential form.

Pause the video and when you've got your answer come back.

Well done.

Put our 0.

09, and we can see 0.

09 is the same as nine multiplied by one hundredth, which is nine hundredths, which is the same as nine multiplied by 10 to the minus two.

Previously, you will have used a place value grid to write numbers in their expanded form, and previously we'll have looked at integers.

8,349, writing that in its expanded form, we would write that as eight multiplied by 10 cubed.

Add three multiplied by 10 squared, add four multiplied by 10 to the power of one, add nine multiplied by 10 to the power of zero.

That is how we write it in expanded form.

Let's just check that with one more.

This one, it's going to be two multiplied by 10 to the power of four.

Add five multiplied by 10 squared, add eight multiplied by 10 to the power of one, add six times 10 to the power of zero.

Now you should be used to doing that.

It may have been a while ago, which is why I thought it would be an idea to just double check that we can remember that from previous learning.

What is the number? Two multiplied by 10 to power of six, add nine multiplied by 10 to the power of four, add five multiplied by 10 squared, add eight multiplied by 10 to the power of zero.

And Lucas writes as his answer, 2,958.

What mistake has Lucas made? Pause the video and come back when you can tell me what mistake he's made.

What did you come up with? Let's see if you've got the same as me.

He has just written the digits 2, 9, 5, and 8, and he's not considered the place value of each digit.

It should be 2,090,508.

We can see it's two multiplied by 10 to the power of six.

That two needs to be in the millions column.

Add nine multiplied by 10 to the power of four.

So the nine needs to be in the 10 thousands column, five multiplied by 10 to the power of two, or 10 squared, I should say, should be in the hundreds column, and then eight multiplied by 10 to the power of zero should be in the ones column.

We're now gonna extend this into decimals.

Let's place 0.

0487 into our place value grid.

Now, just as we did with the integers, we can do the same thing now, because we've got an exponent form of our decimals.

So this is gonna be four multiplied by 10 to the negative two, add eight multiplied by 10 to the power of negative three, add seven multiplied by 10 to the negative four.

Let's try another one.

Into our place value grid.

This is going to be one multiplied by 10 to the negative three, add four multiplied by 10 to the negative five.

Hopefully you can see how I've written these in their expanded form.

I'd like you now to match the decimal to its correct expanded form.

So pause the video, when you've got your answers, come back and we'll check those.

Good luck with this.

Great work.

Now let's check those answers.

So the first one, 0.

023, matches with the fourth option, two multiplied by 10 to the negative two.

Add three, multiplied by 10 to the negative three.

0.

203 was the third option, two multiplied by 10 to the negative one, add three multiplied by 10 to the negative three.

0.

0023 was the second option, two multiplied by 10 to the negative three.

Add three multiplied by 10 to the negative four.

0.

00203 matched with the bottom one, two multiplied by 10 to the negative three.

Add three multiplied by 10 to the negative five, and then no surprises, 0.

23 matched with the top one, two multiplied by 10 to the negative one, add three multiplied by 10 to the negative two.

Now you are ready to have a go at this task.

I'd like you to fill in the gaps.

We've got table which has decimal fraction and exponential form, and you need to fill in the missing two given the one that I've given you.

Pause the video now, and when you've got your answers come back.

well done.

And question number two, what mistake has Aisha made? The question said write 0.

0003 in expanded exponential form.

Aisha writes, 0.

0003 equals three multiplied by 10 to the negative three, because there are three zeros after the decimal point.

I'd like you to write down for me, what mistake Aisha has made.

Pause the video, and I'll be waiting when you get back.

Great work.

And now question three, you're going to write each of the following in the alternative format.

So I've given you decimals, and I've given you exponential form.

So I'd like you to fill in the gaps please.

When you're done, pop back, and we'll check the answers for the entirety of task A.

Here we go then, question one, you're gonna pause the video now, and you're gonna check your answers for me.

Great, and question number two, the three is in the 10 thousandths column, 10 to the negative three is one thousandth, not one ten thousandth.

The answer should be, three multiplied by 10 to negative four.

And question number three, again, here, it's gonna get a little bit awkward I think if I try and read the answers out.

So I'm gonna ask you to pause the video, check your answers carefully, and then come back when you're ready.

Great work.

How did you get on? Well done.

Now we'll move on to the second learning cycle.

We're going to look at writing intEgers in a range of forms. How many different ways can you write 6,300? Pause the video, write down as many different ways as you can, the more the better.

And when you get back, I look forward to seeing what you've come up with.

How many did you get? Wow, that is impressive.

So for example, I could write it as 63 multiplied by 100, keeping the 63, how else could I write it? So I want to start with this.

I want my calculations to start with 63.

What else could I write? I could write it as 63 multiplied by 10, multiplied by 10.

So I could split the hundred into 10 multiplied by 10.

I could write a hundred as 10 squared, so I could write it as 63 multiplied by 10 squared.

What about if I change the 63 to 6.

3? Can you think of some ways that I can write that calculation now, to make it equivalent to 6,300? I could do 6.

3 multiplied by a thousand, which is 6.

3 multiplied by 10, multiplied by 10, multiplied by 10, which I could write as 6.

3 multiplied by 10 cubed.

I could write it as 63 over 10 multiplied by 10 cubed.

So write in the 6.

3 or 63 over 10.

There are actually an infinite number of ways of writing any integer, and you may have come up with some of those, and you may have come up with some others that I've not written.

Which of the following are alternative ways of writing 27,000? Pause the video, decide which of these, no calculators, decide which of these are alternative ways of writing, 27,000.

And when you've made a decision on all nine, come back and we'll see if you are right.

Okay, how many did you decide were alternative ways? There are actually six that are and three that aren't.

Let's take a look at the ones that are, and the ones that aren't.

So the first one is an alternative way.

I'm gonna go horizontally, I'm gonna go across the page.

Top middle is correct, and the last one, top right is correct.

Onto the middle row, got incorrect, incorrect and incorrect.

And then the bottom row are correct, correct and correct.

We'll take a look now at some of the more challenging ones.

So we'll look at some of these in a little bit more detail.

Some of them will be more obvious.

So for example, 27 multiplied by 10 cubed.

I know you know that's equivalent to 27 multiplied by a thousand, which is equivalent to 27,000, but some of the others are not so obvious.

And we'll take a look at those now.

And those ones we'll look at are, 2,700,000 multiplied by 10 to the minus 2, 27 over 10 multiplied by 10 to the power of 4, 270 over 10 multiplied by a hundred thousand, and 2,700 over 10 squared, multiplied by 10 cubed.

Those are the four that we're going to look at in a little bit more detail now.

Let's start with this one, 27 over 10 multiplied by 10 to the power of four.

We can rewrite this as 27 divided by 10, multiplied by 10,000.

We know that a fraction is another way of writing a division.

27 over 10 is equivalent to 27 divided by 10, and 10 to the power of four is equivalent to 10,000.

27 divided by 10 is 2.

7.

And then we're going to multiply that by 10,000, giving us 27,000.

We can see now that that one was equivalent to 27,000.

This was one that was incorrect.

Let's take a look at why.

We can rewrite this as 2.

7 multiplied by a thousand, because 10 cubed is equivalent to 1,000.

2.

7 multiplied by 1,000 is 2,700.

So that's why that one was incorrect.

This one that was correct.

Let's take a look at this one.

We can rewrite this as 2,700 divided by a hundred, multiplied by a thousand.

2,700 over 10 squared, is equivalent to 2,700 divided by a hundred, and multiplied by 10 cubed, is equivalent to multiplied by a thousand.

2,700 divided by a hundred is 27.

And then we multiply that by a thousand.

We can see now that this one is clearly equivalent to 27,000.

And finally, 2,700,000 multiplied by 10 to the negative two.

We know that 10 to the negative two is the same as one over 10 squared.

So we've got 2,700,000 multiplied by one over 10 squared.

We know that one over 10 squared is equivalent to one hundredth.

We know we could rewrite that as 2,700,000 divided by, or over a hundred, giving the final answer of 27,000.

Hopefully now we've been through some of those more challenging ones, you can see why they were, or were not equivalent to 27,000.

I'd like you to have a go at this one now.

Which of the following are alternative ways of writing 340? Pause the video, go through that process that I've just done with those examples, and when you've got your answers, you can come back.

Which ones did you decide? A, let's take a look is correct.

3.

4 multiplied by 10 squared is equivalent to 3.

4 multiplied by a hundred, which we know is 340.

B is also correct.

340 over 10 is equivalent to 340 divided by 10.

And then I'm gonna multiply that by 10.

My divide by 10 and multiply by 10 are inverses, so they're gonna cancel each other out to leave me with 340.

Or I could do 340 divided by 10, which is 34, and multiply that by 10 to give me 340.

C was also correct, 34,000, and it would be divided by 100, because we're multiplying by 10 to the negative two.

And that gives me 340, and the final one was incorrect.

34,000 divided by 100, because 10 squared is 100, divided by 10, because 10 to the negative one is the same as one over 10.

And I can see here that I'm dividing by a thousand, giving me the answer of 34.

Superb work, we are nearly there.

Just this final task to have a go at.

Identify the incorrect alternative in each row, record the letter and rearrange them to make a word.

In the first column, I've given you a number, and then I've given you four alternative ways of writing that number.

However, one of them is not an alternative way.

You need to decide which is not the alternative way.

You're going to record the letter, that will give you eight letters, which you can then rearrange to make a word.

Now, if you cannot work out what the word is, that is not a problem.

The most important thing is that you work these out without a calculator, and you are confident as to which are the ones that are alternatives, and which is that red herring, which is that one that isn't.

Pause the video, and when you come back we'll see whether you've managed to rearrange to make the mystery word, good luck.

How did you get on? Did you manage to work out the word? Well done.

Here we go, let's check those answers.

So 3,000, the incorrect one was alternative one, giving us the letter N.

74,000, it was alternative three giving us the letter X, 120,000 was alternative three giving us an E.

450 was alternative four.

Giving us the letter O, 8 million was the second alternative giving us letter P, 628,000 was the third alternative giving us letter N.

32,490 was alternative one giving us the letter E, and 12,400,000 was alternative four, giving us the letter T.

And if you rearrange those letters, you end up with the word exponent.

Well done.

Now let's summarise what we've done in today's lesson.

Exponential column headings in a place value chart, can be extended to decimal parts.

10 to the negative two is equivalent to 0.

01, which is equivalent to one hundredth.

Integers can be written in a range of forms, and there's an example there.

450 could be written as 4,500 multiplied by 0.

1, or 4,500 multiplied by one tenth, or 4,500 multiplied by 10 to the negative one, or even 450 over 10 cubed, multiplied by 10 cubed.

That is not an exhaustive list of how we could write 450.

One of the main things from today's lesson is remembering that we've now added to our place value chart, those negative exponential column headings.

Well done, and I look forward to seeing you again soon.

Goodbye.