Lesson video

Hi everyone.

My name is Ms Ku.

I hope you enjoy the lesson today and I'm really happy you've chosen to learn with me.

There may be some easy or hard parts of the lesson, but don't worry, I am here to help.

You'll also come across some new key words and maybe some key words you've already come across before.

I do hope you'll like the lesson.

So let's make a start.

In today's lesson from the unit comparing and ordering fractions and decimals, positive and negative numbers, we'll be converting terminating decimals to fractions.

And by the end of the lesson you'll be able to appreciate that any terminating decimal can be written as a fraction with a denominator of the form 10 to the power of n.

Now, just to recap on some keywords, a terminating decimal is one that has a finite number of digits after the decimal point.

For example, 92.

2 is a terminating decimal because we only have one digit after the decimal point.

193.

3894 is a terminating decimal because we have four digits after the decimal point.

A non-example would be 1.

9 with that little dot above the nine, that means that nine is recurring, so it's 1.

999 going on forever.

So that means it's not a terminating decimal.

Another nice non-example is pi.

Pi is not a terminating decimal.

If you put it in your calculator, it might give you pi to 10 decimal places say, but actually pi has an infinite number of decimal places.

So remember a terminating decimal is one that has a finite number of digits after the decimal point.

Our lesson today will be broken into three parts.

We'll be looking at fractions less than one, fractions greater than one, and then writing the denominator in exponential form.

So let's make a start and writing fractions less than one.

Given terminating decimals have a finite number of digits after the decimal point, we can use a place value chart to help us identify the decimal as a fraction.

And just to remind you, a decimal number is a number that has parts that are not whole and the place value chart splits whole numbers into tens, hundreds, thousands and so on.

So let's have a look at writing the decimal 0.

257 as a fraction.

So let's write this in fractional form.

Well, we have seven thousandths here and we also have 50 thousandths, or we could say five hundredths and we have two tenths or you could say 200, one thousandths.

So that means our final answer for the decimal 0.

257 as a fraction is 257/1000.

And what we've successfully done is write a terminating decimal as a fraction.

What do you think 0.

4891 is as a fraction and you can use this place value chart if it helps.

Well hopefully you've spotted, putting our number into our place value chart.

It's going to be written as 4,891/10,000.

And looking at each digit, you can see we have one ten thousandths, we have nine thousandths, eight hundredths, and four tenths.

So that gives us the answer of 4,891/10,000.

Now what do you think 0.

32 is.

See if you can use the place value chart if it helps.

Well hopefully you've spotted 0.

32 is 32/100 as we have two one hundredths and three tenths.

But we can actually simplify this a step further.

So simplifying, we can identify 32 to be eight times four and 100 to be 25 times four.

And remember using our knowledge on the fact that 4/4 is one, this means we know 32/100 can be simplified to give 8/25.

So a terminating decimal can be written as a fraction with a power of 10 denominator and the number of the decimal places will help you select an appropriate power of 10 for the fraction which can be simplified.

So let's have a look at a quick check.

Here Andeep and Jun both do the following working out, who has the correct method? You look at Andeep's, he wrote 824/1000 can be written as 412 times two over 500 times two.

Then he's identified 412 times two to be 206 multiply by two, multiply by two, and the 500 multiply by two.

He's identified it to be 250 times two times two and then simplifying further, Andeep's identified that it's 103/125.

And if you look at it, you can see two times two times two in the numerator and two times two times two in the denominator is the same as multiply by one.

So he's got the answer of 103/125.

So Jun has looked at 824/1000 and identified that eight is a factor.

So from here, 824 divided by eight is 103 and 1000 divided by eight is 125.

So that means he's written 0.

824 as 103/125.

So who do you think is correct? Well hopefully you can spot the both are.

They're both completely different ways.

However, they are both correct.

Andeep's is a nice way to embed the understanding of cancelling down and prime factors.

And Jun's is a nice way of identifying the highest common factor, which is eight, and then identifying the simplified fraction from there.

So now let's have a look at another check question.

We're asked to convert the following terminating decimals to simplified fractions and we can use the place value chart to help.

I'm going to do the first question and then I'd like you to do the second question.

So what we need to do is convert 0.

78, which is our terminating decimal into a fraction.

So putting it in my place value chart you can see I have 0.

78.

Then I can identify this is the same as 78/100 as I have eight hundredths and seven tenths.

From here I can simplify.

So I notice I have a common factor of two.

So 39 multiply by two over 50 multiply by two.

Remember that 2/2 is the same as one, so therefore I have simplified my fraction to be 39/50.

Now let's see if you can try a question, see if you can write the terminating decimal 0.

016 as a simplified fraction and use the place value chart if it helps.

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

Well done.

So hopefully you've spotted 0.

016 looks like this on our place value chart, you can see we have six thousandths, one hundredths and no tenths.

So that means the fraction is 16/1000.

Simplifying this further, I can spot a highest common factor of eight, so therefore 8/8 is the same as one.

So I can cancel my fraction of 16/1000 into 2/125.

Great work if you got this one right.

Now, let's move on to your task.

What you need to do is convert the following into simplified fractions.

You can use a place value chart if it helps.

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

Well done.

Let's move to question two.

Question two says Laura and Jacob are given 0.

56 and 0.

560 to convert into a fraction.

Laura says the answers will be different as 0.

56 is two decimal places and 0.

560 is three decimal places.

Jacob says it is the same decimal, so it'll be the same fraction.

Can you explain who's correct and work out the fractionally equivalent to the decimals.

Well done.

So let's move on to question three.

Question three says write the fraction that is halfway between 0.

96 and 0.

88 and 3b want you to write the fraction which is a quarter of the way between 0.

42 and 0.

84.

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

Well done.

So let's move to the third part on question c.

Three students created their own method to find the middle fraction.

Izzy said, "I converted 0.

12 and 0.

36 into fractions and then you can see the middle easily." Aisha says, well I'm going to add 0.

36 with 0.

12 and then half the answer and then find the fraction of the decimal.

Alex says, I'm going to work out the difference between 0.

36 and 0.

12 and then half it and add it to 0.

12, I'll then convert it into a fraction.

Do all three methods work and explain? Well done.

So let's go through these answers.

For question one, you were asked to convert the following into simplified fractions.

0.

65 is 13/20, 0.

34 is 17/50, 0.

12 is 3/25, 0.

542 is a 271/500, 0.

864 is 108/125, 0.

1248 is 78/625.

Well done if you've got any of those right.

For question two, are the terminating decimals of 0.

56 and 0.

560, will they give the same fraction? Yes, they would.

0.

56 and 0.

560 are equal as they are both equivalent to 14/25.

For question 3a, we needed to find out the fraction which is halfway between 0.

96 and 0.

88.

Well hopefully you've spotted it's 23/25 and the fraction which is a quarter of the way between 0.

42 and 0.

84 is 21/40.

Well done if you got that one right.

Question 3c, let's have a look at each student separately.

We're going to start with Izzy.

Now Izzy said we're going to convert 0.

12 and 0.

36 into fractions.

So you can see them here and then we can spot the middle easily.

Well 0.

36 is 9/25, 0.

12 is 3/25.

So the middle is easily seen as 6/25.

Aisha says let's add them and then half the answer then find the fraction of the decimal.

So adding 0.

36 with 0.

12 gives 0.

48, 0.

48 divided by two is not 0.

24, which is 6/25 the same as what Izzy got.

Now let's try what Alex said.

Alex says, "Work on the difference between 0.

36 and 0.

12 and then half it and add it to the 0.

12 and convert this to a fraction.

Well, the difference between 0.

36 and 0.

12 is 0.

24, 0.

24 divided by two is 0.

12 and then adding it to that 0.

12 gives us 0.

24, which is 6/25.

So therefore all three student methods work.

Great work everybody.

So we've done fractions less than one.

Now let's have a look at fractions greater than one.

Now we know how to convert decimals which are less than one into a fraction and we need to use the same skills with decimals greater than one.

For example, let's convert 3.

14 into a fraction.

Do you think you can see how we do this? Well, you can partition the decimal from the integer, the three and the 0.

14 because we know this together makes 3.

14.

Then we can use our skills to convert 0.

14 into a fraction.

Well, 0.

14 is 14/100, simplified, so taking out that highest common factor of two, 7/50 multiply by 2/2, which we know is 7/50 multiplied by one, which gives us the simplified fraction to be 7/50.

So that means we now know 3.

14 is equal to three and 7/50.

So now let's have a look at a check.

I'm going to do the first question on the left and then I'd like you to do the question on the right.

The question on the left says we need to convert 12.

35 into a mix number.

Well to do this, let's partition again, we know 12.

35 is the same as 12, adds that 0.

35 and we can actually change that 0.

35 into a proper fraction.

35/100.

I'm going to identify my highest common factor of five and then cancelling down gives me 7/20.

So now I know 12.

35 is the same as 12 7/20.

Now we're asked to change it to an improper fraction.

So we need to look at that denominator of 20 and identify what is 12 as an improper fraction with a denominator of 20.

Well 12 is 240/20 and we also have our 7/20.

So adding them makes 247/20.

See if you can give it a go and convert 5.

68 into a mixed number and an improper fraction.

Great work.

So let's see how you got on.

Well, 5.

68 is the same as five add 0.

68.

Looking at our 0.

68, this can be simplified to 17/25.

Then we have our mixed number.

We have 5 17/25, writing this as an improper fraction.

Well look at the denominator 25.

So we need to convert five into an improper fraction where the denominator 25.

So it has to be 125/5, which represents our integer five.

And our 17/25 gives us our answer of 142/25.

Huge.

Well done.

If you got that one right.

Now let's have a look at your task.

What I want you to do is convert the following decimals into mixed numbers and improper fractions.

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

Great work.

So let's move on to question two.

Question two shows that pi is a special symbol and represents a non-terminating decimal.

3.

141592654 going on, so on and so forth.

And there is no fractional equivalent to the exact value of pi.

But what we are asked to do is work out the mix number equivalent to 3.

14 to 3.

1415, and to 3.

141592.

This is a tough question, see even give it a go.

Great work everybody.

So let's go through our answers.

2.

45 as a mixed number is 2 9/20 and as an improper fraction, it's 49/20.

As a mixed number 8.

125 is 8 1/8.

As an improper fraction it's 65/8.

As a mixed number 8.

288 is 8 36/125.

As an improper fraction it's 1036/125.

9.

72 as a mixed number is 9 18/25, as an improper fraction it's 243/25.

Great work if you got this one right.

For question two, were you able to convert the following decimals into these mixed numbers? The first one was 3 141/1000.

The next was 3 283/2000.

And the final one, which was so hard, 3 17,699/ 125,000.

That was a tough one.

Fantastic work everybody.

So let's look at the third part of our lesson where we're writing the denominator in exponential form.

Now we can also write a terminating decimal as a fraction where the denominator is given in exponential form.

And a place value chart is really useful here in showing how we write the denominator in exponential form.

For example, here's our place value chart and we know our first decimal place is 1/10.

Second decimal place is 1/100, 1/1000, and then 1/10,000.

So that means let's see if we can rewrite it in exponential form.

Well 1/10 is still 1/10.

1/100 is actually 1/10 squared.

1/1000 is 1/10 cubed or 1/10,000 is 1/10 to the four.

So all we're doing now is recognising the equivalent exponential form.

Now what do you think 0.

671 is as a fraction, but where the denominator is in exponential form.

Well let's put in our value into our place value chart and hopefully you can spot its equivalent to 671/1000, but in exponential form it's simply 671/10 cubed.

Let's have a look at a quick check question.

Are you able to write the decimal 2.

457 as a fraction where the denominator is in exponential form? You can use this place value chart to help.

Great work.

So let's see how you got on.

Well, here's my number insert into my place value chart.

So we know using our partitioning method, we have two and that 0.

457.

So that means we know 0.

457 is exactly the same as 457/1000, which is the same as 457/10 cubed.

Putting this together, we have a mixed number of 2 457/10 cubed.

Well done if you got that one right.

Now let's have a look at your task.

What I'd like you to do is identify what the exponent is in each of these questions.

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

Great work everybody.

So let's move on to question two.

Question two wants you to write the following decimals as a fraction or the denominator is an exponential form.

You can use that place value chart if it helps.

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

Great work.

So let's move on to question three.

Question three is a really good tough puzzling question.

I want you to fill in the gaps so we have some information given.

See if you can work out what those missing digits are in those gaps.

Really well done if you get any of these correct.

Great work.

So let's go through our answers for question one.

Work on the power of the exponent.

Well, we should have 4567/10 to the four, 17/10 to the five and 3001/10 to the three.

Great work if you got this one right.

For question two, we had to write the following decimals as a fraction with a denominator is an exponential form.

Question 1a, we should have 1257/10 to the four.

For b, 3567/10 cubed.

And for c, this was a tough one, 23/10 to the five.

Remember that place value chart could have helped you there.

For question three, this was a great puzzling question.

So let's see how you got on.

For a, it would've been 3.

167.

So that means as a fraction it's 3167/1000, which is 3167/10 cubed.

Well done if you got that one right.

For b, you should have got 0.

0357, which is equal to 357/10 to the four.

For c, 0.

15 is the same as 3/20, which is the same as 15/100, which is the same as 15/10 squared.

And for d, it'll be 0.

784 is the same as 98/125, which is equivalent to 784/1000, which is 784/10 cubed.

That was a great question.

Fantastic work today everybody.

Remember, a terminating decimal is one that has a finite number of digits after the decimal point and a terminating decimal can be written as a fraction with a power of 10 denominator.

This can be seen from the place value chart.

Given terminating decimals have a finite number of digits after the decimal point, we can use place value charts to help us identify the decimal as a fraction.

A huge well done everybody.

It was great working with you.