Lesson video

Hi, everyone, my name is Miss 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 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 with Positive and Negative Numbers, we'll be converting fractions to terminating decimals, and by the end of the lesson, you'll be able to divide the numerator of a fraction by its denominator, and know the result is an equivalent terminating decimal.

So let's have a look at some keywords, starting with the word prime factors.

Now, prime factors are the factors of a number that are themselves prime.

I've put down an example here, 1,260.

And I've identified the product of 126 and 10 to be 1,260.

Now, I haven't got my prime factors here, so I'm going to look at 126 and identify 2 multiplied by 63 gives the product of 126, and 2 multiplied by 5 gives me the product of 10.

You could choose any numbers really, because eventually, you will always get the same prime factors.

Now, 63 is not a prime, so I'm going to look at it again and identify two numbers that multiply together to give 63.

I'm choosing 7 and 9.

Now, 9 is not a prime number, so I'm going to look at the 9 and identify 9 as the product of 3 and 3.

So now, what I've successfully done, is break 1,260 into the product of its prime factors.

But we will look at this again a little bit more in the lesson, so don't worry.

Let's look at another key word that we'll be using today.

A terminating decimal.

Now, a terminating decimal is one that has a finite number of digits after the decimal point, for example, 92.

2.

This is a terminating decimal as we only have one decimal place after the decimal point, which is a 2.

Another example would be 193.

3894.

This is a terminating decimal, as we have four decimal places.

A non-example would be this.

1.

9 with a little dot above the 9.

That dot above the 9 indicates that that 9 goes on and on forever.

1.

9999, so on and so forth.

Another great example is pi.

If you put pi into your calculator, your calculator will only give you it to, say 10 decimal places, but actually pi has an infinite number of decimal places, so pi is a non-terminating decimal.

Today's lesson will be broken into two parts.

The first part will be recognising terminating and non-terminating decimals, and the second part will be using a calculator.

So let's have a look at the first part where we're recognising terminating and non-terminating decimals.

Short division is a straightforward approach to identify the fraction as a decimal.

This is because the line of a fraction simply means divide.

For example, two fifths actually means 2 divided by 5.

So let's do some short division to determine if the fraction gives a terminating decimal or not.

Let's start with two fifths.

What is two fifths as a terminating decimal? Well, we know two fifths is exactly the same as 2 divide by 5.

So what we're going to do is we're going to divide using short division.

Remember, the divisor of 5 is outside of this bus-stop, some people call it.

So the 5 is on the outside and the 2 is on the inside.

You're constantly asking yourself questions when completing the working out.

The first one would be how many fives fit into 2? Well, it's not.

So what we end up with is a decimal point and a trailing zeroes after it.

Technically there are an infinite number of zeroes there, so we can put as many zeros as we want.

For now, I'm just going to put one and identify the fact that, well, because 5 did not fit into 2, I have to put the 2 here, to identify that we have a remainder of 2.

The next question is how many fives go into 20, which I know is 4.

So that means we know two fifths is a terminating decimal.

So that means we know two fifths is a terminating decimal as it gives us 0.

4.

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

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

We're going to use short division to show the following fractions give a terminating decimal or not.

So let's start with one eighth.

Now, first of all, that line means divide.

So it means 1 divided by 8.

So I'm going to show this bus-stop method.

Remember the divisor, which is 8, goes outside of that bus-stop.

Then you're asking yourself questions.

How many eights go into 1? Well, it's none.

So remember we put our decimal point and then we have those trailing zeros.

Well, because 8 did not fit into 1, we have a remainder of 1.

How many eights go into 10? Well, it's 1, but there is a remainder of 2.

So we put another trailing zero, and then we ask ourselves, how many eights go into 20? Well, it's 2 because it was 16 and there's a remainder here.

So we put another trailing zero, and the remainder is 4.

How many eights now go into 40? Which is 5.

So now we've identified the decimal equivalent to one eighth is 0.

125.

So therefore we know one eighth is a terminating decimal.

Now let's have a look at B.

Well, we know the question is asking us 1 divide by 3.

So the divisor goes outside of this bus-stop.

Then you're asking yourself questions.

How many threes go into 1? Well, it's none.

Don't forget that decimal point, and we have those training zeros.

As we haven't dealt with that 1 so to speak.

we say, "Well, how many threes go into 10?" Which is 3, and now we have a remainder of 1.

How many threes go into 10? And we still have that remainder of 1.

How many threes go into 10? Well, you might notice that we're constantly getting a remainder of 1.

Thus identifying 3 goes into 10 three times, still giving us that remainder of 1.

So that means we know one third is 0.

333 going on forever.

So that means we know one third is not a terminating decimal.

Now what I want you to do, I want you to try some questions on your own.

See if you can use short division to show if the following fractions give a terminating decimal or not.

Press pause if you need more time.

Well done.

So let's have a look at what you did.

Well, 7 over 20 means the calculation is 7 divided by 20.

The divisor is 20, which goes outside of this bus-stop.

So that means we end up with 0.

35 is 7 over 20.

Therefore 7 over 20 is a terminating decimal.

Well done if you got this one right.

One twelfth.

Well, 1 divided by 12 means the divisor is outside of that bus-stop, so to speak.

So that means we've worked it out to be 0.

0833 going on forever.

You might notice we constantly get a remainder of 4, therefore we know one twelfth says 0.

8333 going on forever, identifying that one twelfth is not a terminating decimal.

Well done if you got that one right.

I just want to remind you that short division is a really good method, but sometimes the denominator can be made into a power of 10, and this means we can work out the decimal equivalent easily without division and a place value chart can help us out.

For example, what do you think the fraction three tenths is as a decimal? Well, if we put it in our place value chart three tenths means we have a three in the tenths column.

So that means it's 0.

3.

So we've converted three tenths to the decimal 0.

3.

Now what I want you to do is have a look at 231 over a thousand.

What do you think that is as a decimal? Well, hopefully you've spotted we have one one thousandths.

We have 30 one thousands or we say three one hundreds and we have 200 one thousands or we can say one 10th.

So that means 231 over a thousand as a decimal is 0.

231.

So when the denominator is a power of 10, therefore we know it is a terminating decimal.

However, sometimes the denominator is not given as a power of 10.

So we need to use our knowledge on equivalent fractions to convert the denominator to a power of 10.

For example, three fifths.

How can we make this denominator of 10? Well, if I multiply three fifths by 2 over 2, remember 2 over 2 is equal to 1, this makes 6 over 10.

I now have a denominator which is a power of 10.

So I know my answer is 0.

6 because six tenths is 0.

6, and I also know it's a terminating decimal.

So converting our denominator to a power of 10 can help us convert the fraction into a decimal easily, and it can also identify if the fraction is a terminating decimal.

Now let's have a look at another example.

37 over 25.

Well, can we make that denominator a hundred, where we multiply 37 over 25 by 4 over 4, remember 4 over 4 is the same as 1, we get 148 over a hundred.

Using a place value chart, or by simply looking at the values, you can see 148 over a hundred is the same as 1.

48.

So we have the decimal equivalent and we also know it's a terminating decimal.

So now let's have a look at three eighths.

We need to change that denominator, so it's a power of 10.

And to do this, we're going to multiply the three eighths by 125 over 125.

This means we have 375 over 1000.

I've made the denominator a power of 10.

Then we can convert it to a decimal, which is 0.

375.

So having the ability to convert a denominator to a power of 10, 10, a hundred, a thousand, 10,000 maybe, will enable you to identify if the fraction is a terminating decimal and work out the actual fraction in its decimal form.

So let's have a look at a question.

I'm going to do the first part.

And I'd like you to do the second part.

We're asked to work out the decimal value of the fraction 13 over 5.

You could use short division.

For me, I'm going to look at making that denominator a power of 10.

Well, 13 over 5, I can multiply it by 2 over 2, thus making 13 multiplied by 2 over 5, multiplied by 2, which is 26 over 10.

I know this to be a terminating decimal with a value of 2.

6.

What I'd like you to do is see if you can work out the decimal value of 31 over 25.

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

Hopefully you can spot we can change that denominator into a power of 10.

I've simply multiplied by 4 over 4, giving me 124 over a hundred, which is 1.

24.

So I know 31 over 25 is a terminating decimal with a decimal value of 1.

24.

If we only need to determine if the fraction is equivalent to a terminating decimal or not, then we can use another approach.

For example, 11 over 910.

Is this a terminating decimal? You might not want to do short division here to identify if the fraction is equivalent to a terminating decimal or not.

So what we're going to do is we're gonna look at simplifying the fraction and then writing the numerator and denominator as a product of its prime factors.

The reason for this is because the denominator must only have prime factors of bases of 2 and or 5.

They are the only prime factors of 10, and we want powers of 10 as our denominator.

So let's have a look at 11 over 910 and identify the numerator as the product of its prime factors, and the denominator as a product of its prime factors.

Well, we know 11 is a prime, so I can leave that, but 910 can be identified by 2 multiplied by 2, multiplied by 7, multiplied by 13.

Now looking at those prime factors, do we only have bases of 2 and or 5 when it's simplified? Well, no.

Therefore we know 11 over 910 is not a terminating decimal.

So let's have a look at a check.

We need to identify if 51 over 32 is equivalent to a terminating decimal.

So firstly, let's identify the numerator as a product of its prime factors and the denominator as a product of its prime factors and then we can simplify and look to see if the denominator only has 2 and or 5 as its bases.

Well, the numerator 51 is the product of 3 multiplied by 17, and the denominator 32 is the product of 2 multiplied by 2, multiplied by 2, multiplied by 2, multiplied by 2.

We can't do any more simplifying from here.

So that means looking at that denominator, we only have bases of 2.

So therefore we know it is a terminating decimal, because remember, when the denominator has prime factors with bases of 2 and or 5, it is a terminating decimal.

What I want you to do is try this question.

See if you can identify 71 over 44 as an equivalent terminating decimal.

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

Great work, so let's see what you did.

Well, the numerator 71 is a prime number, so that means it's simply 71.

44 as a product of its prime factors is 2 multiplied by 2, multiplied by 11.

Now, do we only have base factors of 2 and or 5? No, we don't.

So therefore it is not a terminating decimal.

This is a great question and it really does work on your knowledge on prime factors as well as recognising, in its simplified fractional form, the denominator has to have prime factors of 2 and or 5.

Let's have a look at another check.

Laura says, "Three fifteenths is not terminating because the denominator of 15 has 3 as a prime factor.

However, 3 over 15 is a terminating decimal." Can you explain what Laura needs to do first before deciding if the fraction is terminating? See if you can give it a go and press pause if you need.

So hopefully you spotted she needs to simplify the fraction first, identifying the numerator as 3 and the denominator as the product of 3 and 5, you can see we can cancel out the 3 over 3, thus making one fifth.

And we know one fifth is a terminating decimal.

Well done if you got that one right.

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

I want you to identify if the following fractions are equivalent to terminating or non-terminating decimals.

Use your knowledge on prime factors.

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

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

Well, 17 over 60 is 17 over 2 multiplied by 2, multiplied by 5, multiplied by three.

B, which is 1 over 64 is 1 over 2, multiplied by 2, multiplied by 2, multiplied by 2, multiplied by 2 and multiplied by 2.

And 23 over 88 is 23 over 2 multiplied by 2, multiplied by 2, multiplied by 11.

Therefore, the only fraction which has a denominator with prime factors being 2 and or 5 is 1 over 64.

Well done if you got that one right.

Now it's time for your task.

Without using a calculator, can you calculate the decimal value of the following fractions? Remember those two methods that we've looked at? You can use short division or you can use your knowledge on making the denominator a power of 10.

And you can imagine that place value chart if it helps.

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

Well done.

So let's go through the second question.

Second question says, Lucas says, "33 over 132 is not equivalent to a terminating decimal." And he shows his working out and he says, "The prime factors of the denominator are 2, 3 and 11." Can you explain why the 3 over 132 is equivalent to a terminating decimal? See if you can give it a go and press pause if you need more time.

Well done.

So let's go through question 3.

Question 3 says, "Jacob has discovered all these denominators will give a terminating decimal." Now what I want you to do is see if you can explain why.

Why will all these fractions give a terminating decimal? For our last question, you have to circle which fraction will give a terminating decimal.

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

Well done.

So let's go through our answers.

For question 1A is you should have 0.

35 for B, no 0.

625, for C, 0.

12, for D, 0.

235, for E, 12.

54, and for F, 0.

1875.

Well done.

Remember, you could have used a couple of different methods to find the decimal equivalent.

For question 2, well, hopefully you spotted he needed to simplify first.

Identifying the numerator as a product of its prime factors allows us to cancel down further, giving us 1 over 2 multiplied by 2, which is a quarter.

For question 3, well, we had to explain why all of these will give terminating decimals.

Well, because if you look at the denominators, they all have a prime factor base of 2, so therefore they are all factors of the power of 10.

A nice little way to show this is one half, 1 over 2 squared, 1 over 2 cubed, and so on and so forth.

So for question 4, let's identify all these fractions which equate to terminating decimals.

You should have 3 over 2, 12 over 15, 11 over 10 and 7 over 40.

Well done if you got those right.

Great work everybody.

So let's move on to using a calculator.

Now, the scientific calculators are fantastic tools and the Casio fx 991 classwiz will allow you to change the format of any number.

For example, let's input the fraction 5 over 8 and convert it into a decimal form.

Well, to do this, you simply press 5, find that fraction button and the 8, and simply press "execute." What you'll notice is it's still in fractional form and we want it in decimal form.

So what we need to do to convert it to a decimal is press "format," then scroll down and press "execute." Because you'll see we can change the format into decimal form, which is what we want.

So changing it into decimal form gives us 0.

625.

Scientific calculators are fantastic devices as they allow us to interchange between fractional and decimal form.

So using a calculator, I want you to work out the equivalent decimals.

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

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

Well, for A, it's 0.

1238, for B, it's 2.

225, for C, it's 1.

25 and for D, it's 0.

2475.

Well done.

Now just to remind you, the Casio classwiz also allows you to change a terminating decimal into a fraction as well.

So how do you think we'd change 0.

8392 into a fraction? Well, all we need to do is press 0.

8392 into our calculator and then press "execute," and our calculator will automatically convert it into fractional form.

If it doesn't convert it immediately, go to that format button and press "standard." And from here it will convert it back into that fractional form, but usually it will always give it in the fractional form.

Now for our task.

I want you to use your calculator to match the fraction to its terminating decimal.

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

Well done.

Let's move on to question 2.

Question 2 says, "Find the missing values on the calculator displays." See if you can give it a go and press pause if you need more time.

Well done, so let's go through our answers.

Well, for question 1, hopefully you've identified 0.

1294 is 647 over 5000.

13.

444 is 3,361 over 250.

0.

39484 is 9,871 over 25,000 and 0.

856 is 107 over 125.

Now let's work out the missing values on the calculator displays.

This was a great little puzzle.

Let's see how you got on.

Well, for the first one, it's 0.

156, which is 39 over 250.

For the next one it will be 539 over 2000.

We have 6.

4595 is our 12,919, and our last one was 15.

245 is 3049 over 200.

So in summary, a terminating decimal is one that has a finite number of digits after that decimal point.

And short division is a really good straightforward approach to identify the fraction as a decimal.

But we can also identify if a simplified fraction gives a terminating decimal when the denominator has a product of its prime factors of bases of 2 and or 5 only.

We can also use a scientific calculator to identify the decimal equivalent to a fraction and vice versa.

A huge well done today.

It was great learning with you.