Lesson video

Hi everyone.

My name is Ms. Coon.

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 in 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 results in an equivalent terminating decimal.

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

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

I've put on 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 two multiplied by 63 gives the product of 126, and two multiplied by five 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 seven and nine.

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

So now what we'll have 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 two.

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 nine.

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

1.

9999 so on and so forth.

Another great example is pie.

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 the fraction simply means divide.

For example, 2/5 actually means two divided by five.

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

Let's start with 2/5.

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

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

Remember, the divisors of five is outside of this bus stop, some people call it.

So the five is on the outside and the two 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 two? Well, it's naught.

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 five did not fit into two, I have to put the two here to identify that we have a remainder of two.

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

So that means we know 2/5 is a terminating decimal.

So that means we know 2/5 is a terminating decimal as it gives us no 0.

4.

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

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

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

So let's start with 1/8.

Now, first of all, that line means divide, so it means one divided by eight.

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

Remember the divisors, which is eight, goes outside of that bus stop.

Then you're asking yourself questions.

How many eights go into one? Well, it's naught.

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

Well, because eights did not fit into one, you have a remainder of one.

How many eights go into 10? Well, it's one, but there is a remainder of two, so we put another trailing zero and then we ask ourselves, how many eights go into 20? Well, it's two because it was 16, and there's a remainder here.

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

How many eights now go into 40, which is five? So now we've identified the decimal equivalent to 1/8.

It's 0.

125.

So therefore we know 1/8 is a terminating decimal.

Now let's have a look at B.

Well, we know the question it's asking goes one divide by three.

So the divider goes outside of this bus stop.

Then you're asking yourself questions.

How many threes go into one? Well, it's naught.

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

As we haven't dealt with that one, so to speak, we say, "Well, how many threes go into 10?" which is three, and now we have a remainder of one.

How many threes go into 10? And we still have that remainder of one? How many threes go into 10? Well, you might notice that we're constantly getting a remainder of one, thus identifying three goes into 10 three times still giving us that remainder of one.

So that means we know 1/3 is not 0.

333 three going on forever.

So that means we know 1/3 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, 7/20 means the calculation is seven divided by 20.

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

So that means we end up with 0.

35 is 7/20, therefore 7/20 is a terminating decimal.

Well done.

Have you got this one right? 1/12, well, one divided by 12 means the divisors is outside of that bus stop, so to speak.

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

08333 going on forever.

You might notice we constantly get a remainder of four, therefore we know 1/12 is 0.

08333 going on forever, identifying that 1/12 is not a terminating decimal.

Well done.

Have 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 3/10 is as a decimal? Well, if we put it in our place value chart 3/10 means we have a three in the tenths column, so that means it's 0.

3.

So we've converted 3/10 to the decimal 0.

3.

Now what I want you to do is have a look at 231/1,000.

What do you think that is as a decimal? Well, hopefully you've spotted we have 1/1,000.

We have 30/1,000 we say 3/100 and we have 200/1,000 or we can say 1/10.

So that means 231/1,000 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, 3/5.

How can we make this denominator of 10? Well, if I multiply 3/5 by 2/2, remember 2/2 is equals one, this makes 6/10.

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

So I know my answer is no 0.

6 because 6/10 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.

37/25.

Well, can we make that denominator 100? Well, we multiply 37/25.

by 4/4.

Remember 4/4 is the same as one.

We get 148/100.

Using a place value chart or by simply looking at the values, you can see 148/100 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 3/8.

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

And to do this, we're going to multiply the 3/8 by 125/125.

This means we have 375/1,000.

I have 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, 100, 1,000, 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 13/5 You could use short division.

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

well, 13/5 I can multiply it by 2/2, thus making 13 multiplied by 2/5 multiplied by two, which is 26/10.

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 31/25.

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 4/4, giving me 124/100, which is 1.

24.

So I know 31/25 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, 11/910.

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 basis of two and/or five.

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

So let's have a look at 11/910 and identify the numerator, 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 two multiplied by two multiplied by seven multiplied by 13.

Now looking at those prime factors, do we only have basis of two and/or five when it's simplified? Well, no.

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

So let's have a look at a check.

We need to identify if 51/32 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 two and/or five as its basis.

Well, the numerator 51 is the product of three multiplied by 17, and the denominator 32 is the product of two multiplied by two multiplied by two multiplied by two multiplied by two.

We can't do any more simplifying from here.

So that means looking at that denominator, we only have bases of two, so therefore we know it is a terminating decimal because, remember, when the denominator has prime factors with basis of two and/or five, it is a terminating decimal.

What I want you to do is try this question.

See, we can identify 71/44 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 is a product of its prime factors.

it's two multiplied by two multiplied by 11.

Now, do we only have base factors of two and/or five? 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 two and/or five.

Let's have a look at another check.

Laura says 3/15 is not terminating because the denominator of 15 has three as a prime factor.

However, 3/15 is a terminating decimal.

Can you explain what Laura needs to do first before deciding if the fraction is terminating? So 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 three and the denominator as the product of three and five, you can see we can cancel out the 3/3, thus making 1/5, and we know 1/5 is a terminating decimal.

Well done.

Have 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, 17/60 is 17 over two multiplied by two multiplied by five multiplied by three.

B, which is 1/64 is 1 over two multiplied by two, multiplied by two multiplied by two multiplied by two and multiplied by two.

And 23/88 is 23 over two multiplied by two multiplied by two multiplied by 11.

Therefore, the only fraction which has a denominator of which prime factors being two and/or five is 1/64.

Well done.

Have 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 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 33/132 is not equivalent to a terminating decimal.

And he shows his working out and he says the prime factors of the denominator are two, three, and 11.

Can you explain why the 33/132 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 three.

Question three 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, you should have 0.

35.

For B, 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 two, 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, given goes one over two multiplied by two, which is a quarter.

For question three, 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 two.

So therefore they're all factors of the power of 10.

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

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

You should have 3/2, 12/15, 11/10, and 7/40 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 5/8 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.

So 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 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 two.

Question two 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 one, hopefully you've identified 0.

1294 is 647/5,000.

13.

444 is 3,361/250.

0.

39484 is 9,871/25,000.

And 0.

856 is 107/125.

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

This was a great little puzzle, so let's see how you got on.

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

156, which is 39/250.

For the next one, it will be 539/2,000.

We have 6.

4595 is our 12,919.

and our last one was 15.

245 is 3,049/200.

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 basis of two and/or five 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.