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Hi everyone, my name is Miss Ku and I'm really happy to be learning with you today.
It's going to be a fun, interesting, and challenging lesson in parts, but don't worry, I am here to help.
You will come across some new keywords and maybe some keywords you've already come across before.
We're going to work really hard today, but I am here to help and we can learn together.
In today's lesson from the unit arithmetic procedures with integers and decimals we'll be looking at dividing with decimals and by the end of the lesson you can generalise and fluently use written division strategies to calculate accurately with decimals.
Now let's have a look at some keywords, and don't worry, we'll use these keywords often in our lesson.
Division is the inverse operation to multiplication and the dividend is the number we are dividing.
The divisor is the number by which the dividend is divided and the quotient is the integer result of division.
For example, 30 divided by six is equal to five.
30 is the dividend because this is the number we're dividing.
Six is the divisor because this is what we're dividing the dividend by.
And five is quotient as it's the integer result of division.
Today's lesson will consist of two parts.
We'll be looking at division written as fractions and then we'll be looking at dividing using decimals.
So let's start with division written as fractions.
Operations can be written and notated in different ways.
For example, see if you can think of the different ways in which five multiplied by six can be written.
Press pause if you need more time.
Well, hopefully you can spot five multiplied by six can be written as five lots of sixes.
So, I'm going to show you in an array.
Another way we can write five multiplied by six is using a different array where it's six lots of five.
So we have two different ways of representing five multiplied by six.
Another different way is simply saying the product of five and six, where the product is the result of multiplication.
So there's lots of different ways we can show the operation multiplication.
Now, let's have a look at division.
Division can also be written in different ways too.
For example, have a little think about the different ways in which you could write eight divided by two.
Press pause if you need more time.
Another way eight divided by two could be written is eight shared by two.
So you can see eight shared by two gives us four and a four.
Another way is eight grouped into twos.
So all I'm going to do is group into twos, so I have four groups of twos.
Another way is using a fraction and the line indicates division, and this is an important way to recognise division can be represented using a fraction line.
So writing a division as a fraction makes calculating answers from division much easier.
And given we know a simple fraction is where the numerator and denominator are integers, making the divisor an integer enables us to divide more easily.
So, let's have a look at an example.
1.
2 divided by 0.
4.
That's quite scary because we have decimals, so we know we can write this using a fraction, 1.
2 divided by 0.
4.
Now what I'm going to do is change that denominator into an integer, because dividing by 0.
4 or writing a fraction with a denominator of 0.
4 isn't very nice.
So, we're going to change the denominator from a decimal into a nice integer by multiplying both the numerator and denominator by 10.
This is using our work on equivalent fractions.
So that means my 1.
2 over 0.
4 gives me 12 over four.
I can divide 12 by four quite easily, which gives us three.
So writing the calculation as a fraction and then making that denominator a whole number makes the calculation much easier.
So let's see if we can check our understanding.
All I want you to do is fill in the blanks so the fraction is equivalent to the calculation.
We have 2.
4 divided by 0.
4.
How do you think we can represent these using fractions and our knowledge of equivalent fractions? So you can give it go and press pause if you need.
Well done, so hopefully you spotted 2.
4 divided by 0.
4 can be represented as a fraction of 2.
4 over 0.
4.
We can also use our knowledge on equivalent fractions to say, well we don't like that denominator of 0.
4, multiplying the numerator and denominator by 10 gives us 24 over four.
Now, we have another equivalent fraction, but we have our decimals, so instead of 2.
4 it's 0.
24, so therefore the denominator would have to be 0.
04.
What about the numerator of 240? What did we do to 2.
4 to make the numerator 240? Well, we multiplied the numerator and denominator by 100, so that means our denominator must be 40.
What did we do to our numerator 2.
4 to give 1.
2? Well, we divided by two, so we do the same to our denominator to give us 0.
2.
You can use any one of these equivalent fractions and you can probably spot, well to make a denominator of two, our numerator must be 12.
And 12 divided by two is six, this is the easiest calculation to work out our answer.
Well done if you got that one right.
Now let's have a look at another check question.
You have to match the question with the equivalent calculation and the simplified fraction and answer.
I've done the first one for you, 3.
5 divided by 0.
1 is the same as 3.
5 over 0.
1, which is the same as 35 over one, working out to be 35.
See if you can work out the rest.
Press pause if you need more time.
Well done, so let's have a look how you got on.
Well, 4.
8 divided by 0.
2 is 4.
8 over 0.
2, and changing this into a nice equivalent fraction with an integer denominator is going to be 48 over two because we've multiplied both the numerator and denominator by 10.
48 over two is 24.
Well done if you got that one right.
Next, 480 divided by 0.
2.
Well, as a fraction, 480 over 0.
2.
We don't like that decimal denominator, so what do we have to do to change that 0.
2 into an integer? Simply multiply both the numerator and denominator by 10 to give me 4800 over two, which works out to be 2400.
Next, 4.
8 divided by 0.
02.
Well you can see the fraction, 4.
8 over 0.
02.
We don't like that decimal denominator, so let's multiply by 100, and multiplying by 100 gives us 480 over two which is 240.
Massive well done if you got that one right.
Let's have a look at another check question.
Laura says eight divided by two, 80 divided by 20, and 800 divided by 200 all give the same result.
Is Laura correct and explain why? See if you can give it a go and press pause if you need.
Well done.
So let's see how you got on.
Well yes, she is correct because writing as a fraction they are all equivalent.
Eight over two, multiplying our numerator and denominator by 10 is the same as 80 over 20.
Multiplying our numerator and denominator by 10 again gives us 800 over 200.
So all of these are equivalent.
Well done if you got that one right.
Let's have a look at another check question.
Here is a calculation, 26.
4 divided by 0.
8 and it gives us 33, but we're asked to work out the answer of 26.
4 divided by 0.
08.
Aisha says well the answer is 330 because the divisor is 10 times smaller, so the quotient will be 10 times bigger.
And Laura says, the answer is 3.
3 because the divisor is 10 times smaller, so the quotient will be 10 times smaller.
Who do you think is correct? And explain.
Well done.
So hopefully you spotted Aisha is correct because using equivalent fractions and simplifying each time gives us the answer of 330.
So 26.
4 over 0.
08.
Don't like that decimal denominator, so let's simply multiply by 100 to give a nice integer of eight, but we do multiply by 100 on the numerator as well to give us 2640.
Now I'm going to simplify our fraction a little bit to give us 1320 over four, and simplifying again gives us 660 over two, giving me a final answer of 330.
This is a great question and really does give you an opportunity to practise writing equivalent fractions and dividing.
Now let's have a look at your task.
Question one wants you to fill in the blanks so the fraction is equivalent to the calculation.
See if you can give it a go and press pause if you need.
Well done.
So let's move onto question two.
Question two wants you to work out the answer to the following and you must show your working out.
See if you can give it a go and press pause if you need more time.
Great work.
So let's have a look at question three.
Question three wants you to shade in all the equivalent fractions to 4.
8 divided by 0.
4 and it will reveal a lovely little word for you.
See if you can give it a go and press pause if you need.
Great work, so let's go through our answers.
For question one, we needed to fill in the blanks so we have equivalent fractions to the calculation.
Well 16.
8 divided by 0.
8 is the same as 16.
8 over 0.
8.
16.
8 divided by 0.
8, well we have a decimal denominator, how do we make that an integer of eight? Well we multiply the top and the bottom by 10 to give you 168 over eight.
16.
8 divided by 0.
8, well the 16.
8 has now been changed into 1.
68, so that means the equivalent fraction would be 1.
68 over 0.
08.
Next, you've got enough information on the screen to help you with this equivalent fraction.
Something over four, well given the fact we know 168 over eight is our equivalent fraction, that means we can simply halve the numerator and denominator giving me 84 over four.
You can use this equivalent fraction to help you out and to help you identify 42 over two.
Well done if you got that one right.
Question two wants you to work out the answers to the following and please do show your working out.
Well we know this is 46.
2 over 0.
2, we don't like that denominator of 0.
2, multiplying the numerator and denominator by 10 gives us 462 over two to give me a final answer of 231.
Well done if you got that one right.
12.
4 divided by 0.
4, well we don't like that 0.
4, but let's write it as a fraction.
Well this will be the same as 12.
4 over 0.
4, which is 124 over four, simplifying is 62 over two to give me 31.
For C, 4.
2 divided by 0.
12, writing as a fraction and then simplifying each time gives me a final answer of 35.
Well done if you got that one right.
Let's have a look at question three.
Did you identify all the equivalent fractions to 4.
8 over 0.
4? Well, hopefully you did and you revealed the word, hi.
Great work if you got this one right.
Fantastic work so far, so now let's have a look at dividing using decimals.
Well, not all division of decimals will give integer results, but writing the divisor as an integer will enable us to calculate more efficiently.
For example, 1.
8 divided by 1.
5.
As a fraction, we know this is 1.
8 over 1.
5 and we also know we don't like that decimal denominator, so we can change to 18 over 15.
Now we're going to look at that fraction, 18 over 15, and simplify as it might be tricky to divide 18 by 15.
So, simplifying a step further gives us six over five and six over five is much easier to divide as it's easy to divide by five.
So, knowing our divisor is five and we have a dividend of six, let's see if we can work out six divided by five.
Five goes into six exactly once, but please note that when you have a remainder, you have a number of trailing zeroes to work out that decimal answer.
So we can put our decimal point here and we can have our trailing zero.
So five fits into six once, but we have a remainder of one, then we ask ourself, well how many times does five go into 10? Well it's two.
So that means the answer to six divided by five is 1.
2.
In other words, the answer to 1.
8 divided by 1.
5 is 1.
2 and all we've done is simply write our calculation as a fraction, simplified, and then we've chosen a denominator which is nice and easy to divide.
So now let's have a look at a check question.
Andeep and Jun are given the same calculation, 2.
88 divided by 1.
2.
Jun works it out to be 2.
88 over 1.
2, then he writes the equivalent fraction 28.
8 over 12 which is the same as 14.
4 over six, which is the same as 7.
2 over three.
He works out the answer using short division to be 2.
4.
Andeep looks at 2.
88 divided by 1.
2, writes the equivalent fraction of 2.
88 over 1.
2, looks at the equivalent fraction of 288 over 120 and works out the answer there to be 2.
4.
Who do you think is correct? Well hopefully you can spot they are both correct.
Which method do you think is easier? Jun's method is more efficient as we have a single-digit divisor, which makes the division much easier.
Sometimes it's possible, but sometimes it's not.
What was fantastic is how we have so many different equivalent fractions and then we can choose which one is more efficient.
Now let's move onto another check.
I'll do the first part and you can do the second part.
We've got 96.
3 divided by 1.
5 and we need to work out this answer ensuring I show all my working out.
Well, I know 96.
3 divided by 1.
5 is the same as 96.
3 over 1.
5 which is equivalent to 963 over 15.
Now I can simplify this a touch more to give me 321 over 5.
I like that denominator of five, so I'm now going to divide.
321 divided by five, working this out we're going to ask ourselves some questions.
How many fives go into three? Well, they don't, so that means I have a remainder of three.
How many fives go into 32? Well it's six with a remainder of two.
How many fives go into 21? Well it's four.
Remember we have those trailing zeroes, so we put our decimal point and our zero.
How many fives went into 21? It's four and we have a remainder of one.
So then we ask ourself, how many fives go into 10? Which is two.
So the answer to 96.
3 divided by 1.
5 is 64.
2.
Now let's see if you can try a question.
I want you to work out 5.
52 divided by 1.
2 and see if you can show all your working out.
Press pause if you need more time.
Great work, so let's see how you got on.
Well, I've written these equivalent fractions, you can choose any to divide, but I like 27.
6 divided by six because then I have a single-digit divisor.
Working this out, 27.
6 divided by six gives me 4.
6 and you can see the working out here.
Thus, the final answer to 5.
52 divided by 1.
2 is 4.
6.
Well done if you got that one right.
Let's have a look at another check question.
Here Lucas has spilled ink all over his maths work.
Can you figure out the missing number? See if you can give it a go and press pause if you need.
Well done, so let's see how you got on.
Well what was our original calculation if we know the equivalent fraction was 48.
4 over 1.
6? Well we knew it had to be 1.
6.
Then, let's identify our equivalent fraction with a denominator of 16.
48.
4 over 1.
6, to make a denominator of 16 you had to multiply by 10.
How did we get a denominator of four? Well hopefully you spotted we're diving the numerator and denominator both by four.
Then, 121 over four, well this is our calculation.
Using short division, worked it out to be 121 over four to be 30.
25 and that would be our final answer.
Great work if you got that question right.
So now let's move onto your task.
Here you have three questions and you have the answers, but you need to show your working out for the following calculations.
See if you can give it a go and press pause if you need.
Great work, so let's move on to question two.
Question two uses the numbers on the cards to complete the divisions and you may use each card only once.
Can you find more than one way this can be done? Something divided by something is four.
Something divided by something is 0.
4.
Something divided by something is 40.
Remember you can only use each card once.
See if you can give it a go and press pause if you need more time.
Great work.
Well for question one, let's go through our answers.
Well there are lots of different equivalent fractions here but I'm going to choose five over eight as the calculation to divide.
So showing this with short division gives me a wonderful answer of 0.
625.
Well for B the answer's 1.
2 and using our knowledge and equivalent fractions and division I've done this working out to give me my 1.
2.
For C the answer is 8.
4.
For me, here are my equivalent fractions and this is the division.
Well done if you got that one right.
So moving on to the next question, there's quite a few different ways in which you can only use each card once to give the answer.
So I'm going to show you one way here.
I'm going to show you another way now.
And another.
And another.
And another.
So there's quite a few different ways here, but remember, it was important to only use each card once.
Well done and great work today.
So remember writing a division as a fraction makes calculating answers from division much easier.
Given we know a simple fraction is where the numerator and denominator are integers, making the divisor an integer enables us to divide more easily.
A huge well done, it was great learning with you today.
