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Hi everyone, my name is Ms. Ku, and I'm really happy you've chosen to learn with me today.
In today's lesson, it might be easy in some places and tricky in others, but I am here to help.
You will come across some new keywords, and maybe some keywords you've already come across before.
I hope you enjoy 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 recurring decimals, and by the end of the lesson, you'll be able to divide the numerator of a fraction by its denominator, and know that the result is an equivalent recurring decimal.
So let's have a look at some keywords.
Now, a terminating decimal is one that has a finite number of digits after the decimal point, for example, 92.
2, we have one digit out of the decimal point.
193.
3894, this has four decimal places, four digits after the decimal point.
Now, a non-example would be 1.
9 with a little dot above the nine.
That means there's an infinite number of nines there, 1.
9999 going on forever, and we also have pi.
Pi is a great number to represent a non-terminating decimal.
Your calculator will probably give you five to 10 decimal places, but actually, there are an infinite number of decimal places.
A recurring decimal is one that has an infinite number of digits after the decimal point, and the recurring digits after the decimal point are represented using dots, and a dot is placed above the first digit that recurs and the last digit that recurs.
For example, 0.
3, you'll notice the dot is above the three.
This means the three is the only digit that recurs, so the decimal is 0.
3333 going on forever.
0.
17, now, you'll notice there's a dot above the one and the seven, so this is telling you both the one and the seven recur, so our decimal is no 0.
17171717, and so on and so forth.
Next, 0.
473, you'll notice a dot above the four and a dot above the three.
Now, remember, the dot is placed above the first digit that recurs and the last digit, so it is telling you that 473 all recur, so that means the decimal equivalent is 0.
473473473, so on and so forth.
Today's lesson will be broken into two parts.
We'll be looking at recognising recurring decimals, and also using a calculator.
So let's have a look at recognising recurring decimals.
Notation in mathematics is so important, and we use symbols and notation because it's just easier to read and understand, and it's also more concise, and takes up less space.
They can be used to represent complex concepts, and it allows mathematical ideas to be communicated more effectively than words.
What's important to recognise is that we use special notation to show recurring decimals.
Remember that dot identifies which digits are recurring.
A dot is placed above the first digit that recurs, and a dot is placed above the last digit that recurs.
We don't want to be writing things like 1.
232323 all the time.
We want to be concise and efficient mathematicians, so what we do is it's much more easier to write it as 1.
23 with the dot above the two and the dot above the three, indicating that two and three are the only digits that recur.
Now let's see if we can check some understanding.
I want you to match correctly the notated recurring decimal with its decimal form.
Pay attention to where those dots are, and remember the dot is placed at the beginning and the end of the recurring digits.
See if you can give it a go, and press pause if you need more time.
Well done, so let's see how you got on.
Well, 0.
45 with the dot above the four and the five means both the four and the five are recurring, so the decimal equivalent is 0.
454545, so on and so forth.
Next, 0.
165 with a dot above the one and the five, this tells us every single digit between the first digit with the dot and the last digit with the dot will recur, so the decimal equivalent is 0.
165165165, so on and so forth.
Next, we have 0.
165, but the dot is above the six and the five only, so this means the six and the five are the only digits that recur, So that means the decimal equivalent is 0.
1656565, so on and so forth.
Next, 0.
165, but the dot is above the five.
Now, the dot being above the five only indicates the five is the only one that recurs, so that means the decimal equivalent is 0.
16555 going on forever.
Lastly, we have 0.
45, but the dot is above the five, indicating that five is the only one that recurs, so a decimal equivalent is 0.
4555 going on forever.
Well done if you've got that one right.
Now we know how a recurring decimal is represented in decimal form using the dot, and we also need to recognise recurring decimals when in fractional form, so let's do some short division to determine if the fraction gives a recurring decimal or not.
We're going to look at 1/3.
Is 1/3 equivalent to a recurring decimal? Well, 1/3 is the same as one divided by three, so you know I'm going to use short division, three is the divisor, so we put that outside, and the one is on the inside of this bus stop method.
How many threes go into one? Well, that's none.
Remember those trailing zeros? We have to put our decimal place and a trailing zero.
Well, we didn't deal with the one, so that means we're saying how many threes go into 10, which is three.
Three times three is nine, but there's one remaining, another trailing zero, and that remainder of one.
How many threes go into 10? Well, it's three, 'cause three times three is nine, with a remainder of one.
Put another trailing zero with the remainder of one, and we know it's three.
You can see we have our recurring decimal, so that means 1/3 is 0.
3 with that little dot above the three, indicating that three goes on forever, so we know yes, 1/3 is a recurring decimal.
So now let's do a question.
I'm going to do the first question, and I'd like you to do the second question.
We're going to be using short division to work out the decimal of the following fraction, and I want you to give your answer using the correct notation.
Well, let's have a look at 1/12th.
1/12th means one divided by 12, so remember the 12 is the divisor, and that goes on the outside, and we ask ourselves some questions.
How many times does 12 go into one? Zero, but we have to put our decimal place and that trailing zero, but we still haven't dealt with that one.
How many twelves go into 10 now? Well, 12 doesn't go into 10, so we have to put a zero, and now we have a remainder of 10 and another trailing zero.
How many twelves go into 100? Well, it's eight with a remainder of four, because 12 times four is 96, and we wanted 100, so it's a remainder of four.
How many twelves go into 40? Well, it's three because three times 12 is 36, and we have a remainder of four.
Another trailing zero, we have how many twelves going to 40? Well, it's three because three times 12 is 36, and then we still get our remainder of four, so therefore we know 1/12th is 0.
083, and that dot is above the three, indicating it is recurring.
Really well done if you spotted this.
Now I'd like you to try a question on your own.
Using short division, work up the decimal of the following fractions, and I'd like you to use the correct notation.
See if you can give it a go, and press pause if you need more time.
Great work, so let's see how you got on.
1/11 is the same as one divided by 11, so 11 is our divisor.
How many elevens go into one, zero.
We have a decimal place and we have a range of one, and those trailing zeros.
How many elevens go into 10? Well, it's nought again, and we have that remainder of 10 now and those trailing zeros.
How many elevens go into 100? Well, it's nine, 11 times nine is 99, with a remainder of one.
How many elevens go into 10? Well, it's none, and we still have to carry on that 10 over, then we have our trailing zeros, and you might be able to spot a pattern.
We constantly get a remainder of one, then 10, then one, then 10, so that means we have a recurring decimal.
1/11 is exactly the same as 0.
09, where the nought and the nine are recurring.
Really well done if you got that one right.
So short division is a fantastic way to find the decimal form of a fraction whether it's terminating or recurring.
However, if we simply need to recognise if a fraction is equivalent to a recurring decimal, we only use our knowledge of prime factors to identify if it's terminating or recurring.
Remember, a simplified fraction gives a terminating decimal when the denominator has prime factors of two and/or five only.
Therefore, a simplified fraction gives a recurring decimal when the denominator has prime factors of any other prime.
Let's have a look at an example.
One over seven multiplied by 11, this is a recurring decimal.
Look at that denominator.
There are no prime factors of two and or five, so therefore it's not terminating, so it must be recurring.
Three over two times five squared, this is terminating because our denominator only has prime factors of two and/or five.
Next example would be two over 13 times five cubed.
This is a recurring decimal.
It's a recurring decimal because it's not terminating.
Terminating will only have prime factors of two and/or five.
So let's check for understanding.
What I want you to do is put a tick in the correct column to identify if the fraction is equivalent to a terminating decimal or a recurring decimal.
Now remember, a simplified fraction will give a terminating decimal only when the denominator written in its prime factors have bases of two and/or five.
So see if we can give it a go, and press pause if you need more time.
Well done, let's see how you got on.
1/9th is equivalent to a recurring decimal.
1/11th is equivalent to a recurring decimal.
5/36 is equivalent to a recurring decimal, and 3/6, it's actually a terminating decimal.
If you simplify 3/6, it would give you one half.
Really well done if you got that one right.
Let's have a look at a second check question.
Sofia says 2/3 is recurring because 1/3 is a recurring decimal, so that means 2/3 will be a recurring decimal, and Izzy says 3/12 is recurring because one 12th is a recurring decimal, so this means 3/12 will be a recurring decimal.
Can you explain why one person is correct and the other is incorrect? See if you can give it a go.
Well, hopefully you spotted Sofia is correct, but Izzy is not correct.
2/3 is in its simplest form.
3/12 is not in its simplest form and can be simplified to 1/4, and we know 1/4 is equivalent to a terminating decimal.
This was a great question and just reminds you to ensure that you have to simplify your fraction before looking at the denominator as a product of its prime factors.
Now let's have a look at your task.
For question one, you need to match the correctly notated recurring decimal with the decimal form.
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 wants you to use short division to identify the recurring decimal of the following fractions.
See if you can give it a go, and press pause for more time.
Great work, so let's move on to question three.
Question three wants you to circle the fractions which are equivalent to recurring decimals.
See if you can give it a go, and press pause for more time.
Fantastic work, everybody, so let's have a look at these answers.
For question one, hopefully you spotted 0.
7, the dot above the seven is 0.
777 going on forever.
0.
754 with the dot above the seven and the four is 0.
754754754, so on and so forth.
0.
75, the dot above the seven and the five is 0.
757575, so on and so forth.
0.
754 with the dot above the five and four only is 0.
75454 going on forever.
And lastly, 0.
75 just with the dot above the five is 0.
7555, so on and so forth.
Great work if you got that one right.
For question two, you're asked to use short division to identify the recurring decimal of the following.
For A, hopefully you spotted it's 0.
9 recurring, for B, it's 0.
5 recurring, and for C, it's 0.
0416 recurring.
Great work if you've got those right.
Question three, well, let's identify those recurring decimal equivalents.
You should have 6/7, 7/6, 5/12, and 4/11.
I just wanna quickly show you why 3/60 and 9/12 are not recurring decimals.
Well, because they're not in their simplified form, and when they're simplified, the denominators only have prime factors of powers of two and/or five, so that means they will terminate.
Just to show you, 3/60 can be written as three over three times two times two times five, which when simplified gives one over two squared times five.
We have a denominator with bases of two and/or five, so therefore it's terminating.
For 9/12, it can be written as three times three over three times two times two, so this is the same as three over two squared when simplified, we have a denominator with bases of two and/or five, so therefore it's terminating.
Fantastic work, everybody.
So let's move on to using a calculator.
Now, scientific calculators are fantastic tools, and I will be referring to the Casio FX-991 ClassWiz, and it allows you to change the format of a number.
For example, let's input the fraction 5/8, and convert it into a decimal form.
I want you to press five, the fraction button, and the eight, and execute, and what you'll see is 5/8 displayed on the calculator screen, but to convert it to a decimal, all you need to do is press format, scroll down to decimal and then press execute, and it'll convert that 5/8 into its decimal form, which is 0.
625.
Now, calculators are fantastic, but do remember they only have a limited number of digits to display.
For example, I want you to write all the digits displayed on your calculator screen when converting 1/6 into a decimal.
Now, does this mean 1/6 is a terminating decimal? As your calculator says it stops at the 10th decimal place, it says 0.
16666, and then we have a seven at the end, so is 1/6 a terminating decimal? Well, no it's not, and it's because the calculator is limited to a 12 digit display, and the final digit is rounded.
It's important to note 1/6 is a recurring decimal, but your calculator is limited to the number of digits it can display on the screen.
So what I'd like us to do is an investigation, because there is no button to press to show a recurring decimal as a fraction on the Casio ClassWiz.
What you need to do is input the repeated digits a certain number of times for the equivalent fraction to appear.
Now I'm going to give you a task.
What is the minimum number of decimal places you need to press to get 0.
166, so on and so forth, displayed as 1/6 on your calculator? I've created a little table here just to help.
So all I want you to do is identify how many decimal places did you input in, what input did you input into the calculator, and what did your calculator tell you? See if you can use this table and investigate how many decimal places do you need to input to change 0.
166, so on and so forth into 1/6 on your calculator.
See if you can give it a go, and please do press pause.
Well done, well, I've just decided to use the following decimal places.
If I put four decimal places in, in other words 0.
1666, I get 833/5000, so that is not 1/6.
If I put six decimal places in, for example, 0.
1666666, unfortunately, that would give me 0.
16666, so it hasn't converted it to 1/6 yet.
What about 10 decimal places? Well, if I put no 0.
1666 so on and so on, just only 10 decimal places, does it convert it to 1/6? No, it wouldn't, it just repeats what I've inputted.
12 decimal places, does that give me 1/6? No, it doesn't, it actually rounds the last decimal place that's seen.
How about 18 decimal places? Yes, if I were to put in my recurring decimal to 18 decimal places, my calculator will convert it to a fraction.
So what we need to remember is to input 18 decimal places of a recurring decimal when the number is less than one, and that's when the calculator will convert it into a fraction.
So now what I'd like you to do is use your calculator to match the fraction to the equivalent decimal, and then I want you to identify is it terminating or not.
See if you can give it a go, and press pause if you need more time.
Well done, let's have a look at your second task.
Match the correctly notated recurring decimal with the decimal form.
Remember you can use your calculator here.
Fantastic work, let's move on to question three.
Question three wants you to write the following recurring decimals as a fraction.
So remember, if the recurring decimal is less than one, you have to put in at least 18 decimal places in there for it to convert it to a fraction.
See if you can give it a go, and press pause if you need more time.
Well done, so let's see how you got on.
For question one, hopefully you spotted 59/64 is 0.
921875, and is a terminating decimal.
17/53 is 0.
320754717, but this is a non-terminating decimal.
Remember, your calculator will only tell you 10 decimal places, so you really do have to look at the fraction, simplify it, and look at that denominator.
355/113 gives us 3.
14159292, which is a non-terminating decimal, so again simplify the fraction and identify the denominator as a product of its prime factors.
For question two, did you match it correctly? Well, 0.
4 recurring is 4/9.
0.
75 recurring is 25/33.
0.
75, but the five is the only one recurring is 34/45.
0.
7 recurring is 7/9.
0.
754 recurring is 754/999, and 0.
754 recurring where the five and the four is the only recurring digits is 83/110.
For question three, let's see how you got on with identifying these recurring decimals as a fraction.
This would be 1121/9000, B would be 3853/1125, and C would be 283/90.
Fantastic work, everybody.
Remember a recurring decimal is one that has an infinite number of digits after that decimal point, and recurring digits after the decimal point are represented using dots, and a dot is placed above the first digit that recurs and the last digit that recurs.
For example, 0.
3 with a dot above the three is showing that three is the only thing that recurs.
0.
17 with the dot above the one and seven is showing the one and seven both recur.
We can identify recurring decimals using short division, prime factors, and spotting relationship with those denominators.
Huge well done today, everybody.
It was great learning with you.
