Lesson video

Hi, everyone, my name is Miss.

Coo.

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 keywords and maybe some keywords 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^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 9.

That means that 9 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 1, fractions greater than 1, and then writing the denominator in exponential form.

So let's make a start on writing fractions less than 1.

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 tenths, hundredths, thousandths, 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 7 thousandths here, and we also have 50 thousandths, or we could say 5 hundredths.

And we have 2 tenths or you could say 200 one-thousandths.

So that means our final answer for the decimal 0.

257 as a fraction is 257 over 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 4891 over 10 000.

And looking at each digit, you can see we have 1 ten-thousandths, we have 9 one-thousands, 8 one-hundredths, and 4 tenths.

So that gives us the answer of 4891 over 10 000.

Now what do you think 0.

32 is as a decimal? See if you can use the place value chart if it helps.

Well hopefully you've spotted, as a decimal, 0.

32 is 32 over 100, as we have 2 one-hundredths and 3 tenths.

But we can actually simplify this a step further.

So simplifying, we can identify 32 to be 8 times 4, and 100 to be 25 times 4.

And remember using our knowledge on the fact that 4 over 4 is 1, this means we know 32 over 100 can be simplified to give 8 over 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? If you look at Andeep's, he wrote 824 over 1000, can be written as 412 times 2 over 500 times 2.

Then he's identified 412 times 2 to be 206 multiplied by 2, multiplied by 2.

And the 500 multiplied by 2, he's identified it to be 250 times 2 times 2.

And then simplifying further, Andeep's identified that it's 103 over 125.

And if you look at it, you can see 2 times 2 times 2 in the numerator and 2 times 2 times 2 in the denominator is the same as multiplied by 1.

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

So Jun has looked at 824 over 1000 and identified that 8 is a factor.

So from here, 824 divided by 8 is 103 and 1000 divided by 8 is 125.

So that means he's written 0.

824 as 103 over 125.

So who do you think is correct? Well hopefully you could spot that 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 8, 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 that value chart, you can see I have 0.

78.

Then I can identify this is the same as 78 over 100 as I have 8 hundredths and 7 tenths.

From here I can simplify.

So I notice I have a common factor of 2, so 39 multiplied by 2, over 50 multiplied by 2.

Remember that 2 over 2 is the same as 1, so therefore I have simplified my fraction to be 39 over 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 6 thousandths, 1 hundredths and no tenths.

So that means the fraction is 16 over 1000.

Simplifying this further, I can spot a highest common factor of 8.

So therefore 8 over 8 is the same as 1.

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

Great work if you've 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.

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

Well done.

Let's move to question 2.

Question 2 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 2 decimal places and 0.

560 is three decimal places.

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

Can you explain who's correct and work out the fractional equivalent to the decimals? Well done.

So let's move on to question 3.

Question 3 says write the fraction that is halfway between 0.

96 and 0.

88.

And 3b wants you to write the fraction which is a quarter of the way between 0.

42 and 0.

84.

See if 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 1, you were asked to convert the following into simplified fractions.

0.

65 is 13 over 20, 0.

34 is 17 over 50, 0.

12 is 3 over 25, 0.

542 is 271 over 500, 0.

864 is 108 over 125, 0.

1248 is 78 over 625.

Well done if you got any of those right.

For question 2, 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 over 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 over 25, and the fraction which is a quarter of the way between 0.

42 and 0.

84 is 21 over 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 over 25, 0.

12 is 3 over 25, so the middle is easily seen as 6 over 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 2 is 0.

24, which is 6 over 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 halve it and add it to the 0.

12 and convert this to a fraction.

Well, the difference between 0.

36 and 0.

2 is 0.

24.

0.

24 divided by 2 is 0.

12 and then adding it to that 0.

2 gives us 0.

24, which is 6 over 25.

So therefore all three student methods work.

Great work, everybody.

So we've done fractions less than 1.

Now let's have a look at fractions greater than 1.

Now we know how to convert decimals which are less than 1 into a fraction and we need to use the same skills with decimals greater than 1.

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 3 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 over 100 simplified.

So taking out that highest common factor of 2, 7 over 50 multiplied by 2 over 2, which we know is 7 over 50 multiplied by 1, which gives us the simplified fraction to be 7 over 50.

So that means we now know 3.

14 is equal to 3 and our 7 over 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 mixed number.

Well to do this, let's partition again.

We know 12.

35 is the same as 12 add that 0.

35 and we can actually change that 0.

35 into a proper fraction, 35 over 100.

I'm going to identify my highest common factor of 5 and then cancelling down gives me 7 over 20.

So now I know 12.

35 is the same as 12 and 7 twentieths.

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 over 20 and we also have our 7 twelfths.

So adding them makes 247 over 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 5 add 0.

68.

Looking at our 0.

68, this can be simplified to 17 over 25.

Then we have our mixed number, we have 5 and 17 over 25.

Writing this as an improper fraction, well, look at the denominator 25.

So we need to convert 5 into an improper fraction with a denominator 25.

So it has to be 125 over 5, which represents our integer 5, add our 17 over 25 gives us our answer of 142 over 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 2.

Question 2 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 mixed number equivalent to 3.

141, 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 and 9 twentieths, and as an improper fraction, it's 49 over 20.

As a mixed number, 8.

125 is 8 and 1 eighth, as an improper fraction, it's 65 over 8.

As a mixed number, 8.

288 is 8 and 36 over 125, as an improper fraction, it's 1036 over 125.

9.

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

Great work if you got this one right.

For question 2, were you able to convert the following decimals into these mixed numbers? The first one was 3 and 141 over 1000.

The next was 3 and 283 over 2000.

And the final one, which was so hard, 3 and 17699 over 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 tenth, second decimal place is 1 hundredths, 1 thousandths, and then 1 over 10 000.

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

Well, 1 over 10 is still 1 over 10, 1 over 100 is actually 1 over 10 squared, 1 over 1000 is one over 10 cubed, and 1 over 10 000 is 1 over 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 it's equivalent to 671 over 1000, but in exponential form it's simply 671 over 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 charge 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 2 and that 0.

457.

So that means we know 0.

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

Putting this together, we have a mixed number of 2 and 457 over 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 2.

Question 2 wants you to write the following decimals as a fraction where the denominator is in 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 3.

Question 3 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 out the power of the exponent.

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

Great work if you got this one right.

For question 2, we had to write the following decimals as a fraction where the denominator is in exponential form.

Question 1a, we should have 1257 over 10 to the four, for b, 3567 over 10 cubed, and for c, this was a tough one, 23 over 10 to the five.

Remember that place value chart could have helped you there.

For question 3, 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 3,167 over 1000, which is 3167 over 10 cubed.

Well done if you got that one right.

For b, you should have got 0.

0357 which is equal to 357 over 10 to the four.

For c, 0.

15 is the same as 3 twentieths, which is the same as 15 over 100, which is the same as 15 over 10 squared.

And for d, it'll be 0.

784 is the same as 98 over 125, which is equivalent to 784 over 1000, which is 784 over 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.