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Hi there.
My name is Mr. Tilstone.
I'm a teacher.
I hope you're having a lovely day.
I'm having a great day, and it's about to get even better because I get to teach you a lesson about probably my favourite part of maths, which is fractions.
Today we're going to be looking at converting from improper fractions to mixed numbers, and you might already have some experience with that and confidence with that.
So let's see if we can take you even further.
So, if you're ready, I'm ready.
Let's begin.
The outcome of today's lesson is this: "I can explain how an improper fraction "is converted into a mixed number." You might have some experience of using quarters and fifths.
Our key words.
We've got two.
My turn.
Mixed number.
Your turn.
My turn.
Improper fraction.
Your turn.
You may well already know what those words mean.
Hopefully you do.
But let's have a reminder just in case.
A mixed number is a whole number and a fraction combined.
So it's got a whole number part and a fractional part.
And an example would be 1 1/2.
Can you think of a different example? Can you think of a situation from real life where you would use mixed numbers? And an improper fraction is a fraction where the numerator, that's the top number, is greater than or equal to the denominator, that's the bottom number.
For example, 5/3 or 9/8.
Can you think of a different example of an improper fraction? And, again, can you think of a situation from real life where we would use improper fractions? You may remember that mixed numbers can be expressed as improper fractions and vice versa.
There's an equivalence.
Our lesson is split into two parts.
The first will be expressing improper fractions as mixed numbers, and the second using a generalisation.
Let's begin by expressing improper fractions as mixed numbers.
In this lesson, you will meet Aisha and Jun.
Have you met them before? They're here today to give us a helping hand with the maths.
Aisha is given 21 counters.
Hopefully you've got some counters at hand that you can use today.
She labels each counter as one eighth, and you might have some counters that are already labelled one eighth.
So here are her 21 counters.
There are 21 one-eighths, and we can write that as an improper fraction, 21/8.
21/8 is an improper fraction.
How, though, could you convert it into a mixed number? Hmm.
Aisha rearranges her counters into groups that have a value of one.
So how many counters would be in each group? This will support her to convert 21 eighths into a mixed number.
What would you put in one group then to make one whole? Our unit is eighths, so we will be thinking about groups of eight.
So we can put them into groups of eight.
There are eight eighths in one whole.
So when we've got eight eighths, we've got one whole, and I think we've got a few wholes there.
Aisha rearranges her counters into groups that have a value of one.
So here's one whole group.
That's one group of eight one-eighths.
This is another one group of eight one-eighths.
What have we got now then? We've got two wholes.
And then we've got some leftover.
We've got a remainder if you like.
So we've got five one-eighths leftover or 5/8.
What can you see there? Can you see a mixed number? There are two groups of eight eighths in 21 eighths and five more eighths.
We say two and five eighths, and we write it just like that.
That's 2 5/8.
That's our mixed number.
Aisha summarises her work in a table.
So the improper fraction that she's just looked at is 21/8.
The prompt question and the answer are as follows.
The question is: How many groups of eight-eighths are there in 21 eighths? And the answer: There are two full groups of eight-eighths and five more eighths left over.
That gives a mixed number of 2 5/8, and they're equivalent.
So the improper fraction 21/8 is equal to the mixed number 2 5/8.
Jun is also given 21 counters.
Not one eighths though.
He's got one sixths.
He labels each counter as one sixth, and you may have some one sixth counters in your classroom.
There are 21 one-sixths.
We write that as 21/6.
21/6 is an improper fraction.
It's equal to or greater than 1.
How would you convert it into a mixed number? Let's think about what we did before.
We put them into groups.
What would the group size be this time? Would it be eight again? No.
Think about the sixths.
Jun rearranges his counters into groups that have a value of one.
This will support him to convert 21/6 into a mixed number.
One unit is six, so we will be thinking about groups of six this time because six sixths equals one whole.
There are six sixths in one whole.
Jun rearranges his counters into groups that have a value of one.
So that is one.
Six sixths is equal to 1.
We had a few more though, didn't we? Got another one.
Another six sixths is equal to another 1.
What have we got so far then? How many wholes? Two.
And then we've got another group of six sixths.
How many wholes have we got? Three.
We've got some counters still left though.
Got 3 one-sixths left over.
Okay, what mixed number can you see there? How many wholes, and what's the fractional part left over? There are three groups of six sixths in 21 sixths and three more sixths.
So we say three and three sixths, and we write it just like that.
That's 3 3/6.
That's our mixed number.
Jun adds his work to Aisha's table.
So the improper fraction this time is 21/6.
What was the prompt question before? It was: How many groups of eight-eighths are there in 21 eighths? Won't be that this time, will it? This time it's: How many groups of six-sixths are there in 21 sixths.
And what will the answer to that be? How many groups of six-sixths are there in 21 sixths? Well, 6, 12, 18.
That's three.
There are three full groups of six-sixths and then three more sixths left over or remainding.
There's our three wholes.
There's our three sixths.
So our mixed number is 3 3/6.
Let's do a little check.
Let's see how you're getting on.
By using or drawing counters to help, express the improper fraction 21/10 as a mixed number, and complete the table.
So there's our improper fraction, 21/10.
Our prompt question is this: How many groups of ten-tenths are there in 21 tenths? And then we've got a stem sentence for you to fill in.
There are mm full groups of ten-tenths and mm more tenths.
Pause the video, and give that a go.
Let's see.
There are two full groups of ten-tenths because 10, 20; that's two.
And then one more tenth.
That gives us a mixed number of 2 1/10.
It's time for some practise.
Number one.
Using counters, express these improper fractions as mixed numbers.
If you don't have physical counters, you can draw little circles.
So we've got 21/4, 21/5, 21/9.
There's a link between those three examples, isn't there? 17/2, 13/6, and 41/7.
28/10, 34/3, and 55/8.
So we've got a wide range of denominators this time.
Remember to use that prompt question: How many groups of mm are there in mm? So think about that denominator.
Number two.
Using counters, express these improper fractions as mixed numbers.
What do you notice? Can you explain what you notice? So 21/7 is equal to what? And 24/8 is equal to what? You'll hopefully notice something a little bit different about those examples compared to the ones in number one.
Okay, pause the video.
Good luck with that, and I'll see you soon for some feedback.
Welcome back.
How are you getting on? Are you starting to feel confident about expressing improper fractions as mixed numbers? You've done it with a wide variety of denominators there.
Let's see how we're getting on.
Knowing your times tables is a real help here I think.
So number 1.
21/4.
This is what 21 quarters would look like.
How many whole groups have we got? We've got five whole groups and then one quarter leftover.
So 5 1/4.
21/5 is equal to 4 1/5.
21 ninths is equal to 2 3/9.
17 halves? Well how many times does 2 go into 17? It doesn't go into 17, but it goes into 16 eight times with one half left over.
So that's 8 1/2.
13/6 is equal to 2 1/6.
41/7 is equal to 5 6/7.
28/10? Well, two lots of 10 are 20 with eight tenths left over.
So that's 2 8/10.
34/3 is equal to 11 1/3.
And 55/8 is equal to 6 7/8.
And then what did you notice about these examples? Well, 21/7 is equal to 3; three wholes.
There's no extra fractional part.
It's not a mixed number.
It's just a whole number.
24/8 is also equal to 3.
There is no fractional part, just a whole number.
You might have noticed that when these improper fractions were expressed as a mixed number, they were both equal to three wholes.
So they weren't mixed numbers at all.
They were just whole numbers.
You might have then explained that this is because, in both cases, the numerator is a multiple of the denominator.
21 is a multiple of 7, and 24 is a multiple of 8.
So when the numerator is a multiple of the denominator, the fraction is equivalent to a whole number.
Hey, you're doing very, very well.
Let's move on to the next part of the lesson.
That's using a generalisation.
Let's look at this improper fraction.
It's got something missing.
What's missing? The denominator's missing? We don't know it.
What do we know? We know the numerator.
So it's 20 something.
Could be 20/2, 20/3, 20/4.
Could be anything.
How would we express that as a mixed number? Well, we can't.
We can't determine the mixed number from this.
There's not enough information.
The denominator is missing, and this would tell us the unit that we are working with, and then we could express it as a mixed number.
And let's look at this improper fraction.
So it's an improper fraction not a proper fraction? What's missing this time, and what's known this time? How could we express it as a mixed number? Well, we know the denominator this time.
It's 3.
We don't know the numerator.
We can't determine the mixed number from this.
There's not enough information.
The numerator's missing, and this tells us how many of the unit we already have.
To convert an improper fraction to a mixed number, we determine how many groups of the denominator can be made out of the numerator by dividing.
So you've been using division skills so far.
So that's something divided by 3, whatever the number is.
It could be, for example, 13.
It could be 20.
It could be something that's a multiple of 3, like 30.
And, in this case, it's 20 divided by something.
We don't know what that denominator is, but that's what we would do to turn it into a mixed number.
This gives us a whole number part.
The remaining part is the fractional part.
Let's use the generalisation to express this improper fraction as a mixed number.
So we've got 22/7 this time.
How many groups of the denominator can be made out of the numerator? So, in this case, how many groups of 7 can be made out of 22? How many whole groups? And is there anything left over? 22 divided by 7 is equal to 3 remainder 1 because 7, 14, 21.
That's three with one leftover over.
This gives us a whole number part.
The remaining part is the fractional part.
So 3 remainder 1 is the same 3 1/7 in this case.
The 3 is the whole number part, and the remainder 1 is the 1/7 that we've got in addition to the whole number part.
Let's have a little check.
Use a generalisation to express the improper fraction as a mixed number.
I'll do one more example, and we'll leave that up there, and then you have a go yourself.
So 28/3 is equal to something.
Find out how many groups of the denominator can be made from the numerator by dividing.
So, in this case, we're doing 28 divided by 3.
28 divided by 3 is equal to 9 remainder 1.
9 times 3 is 27, and there's 1 left over.
So you can use your times tables as well to help here.
This is nine full groups of three-thirds and one more third.
That makes that 9 1/3.
9 remainder 1 in this case is the same as 9 1/3.
Okay, see if you can do the same thing with 32 1/3.
Find out how many groups the denominator can be made from the numerator by dividing.
Pause the video, and off you go.
How did you get on? How did you do that? Did you count in threes and get as close as you could to 32? Or did you use a times tables fact that was just before 32? Hmm.
Well, 32 divided by 3 is equal to 10 remainder 2 because 3 lots of 10 are 30.
So that's 10 with 2 left over.
So 10 remainder 2.
And we can turn that into 10 full groups of three-thirds and two more thirds.
So that's 10 2/3.
That's our mixed number.
And if you got that right, very well done.
You're on track, and you're ready for some final practise.
Number one.
Use the generalisation to match these improper fractions to their equivalent mixed number.
And you may see that we've got two different denominators there.
And number two, Jun expresses 19 fifths as a mixed number.
Jun shows his workings like this.
Find out how many groups of the denominator can be made from the numerator by dividing.
That's 19 divided by 5 equals 3 remainder 3.
He says this is three full groups of five fifths and three more fifths.
So he thinks 19 fifths is equal to 3 3/5.
Jun has made a mistake.
That is not correct.
Can you spot the mistake and correct the mistake? Righteo! Pause the video, and away you go.
Welcome back.
How did you get on? Do you think you've cracked this? Well, let's see.
23/2 is equivalent to 11 1/2, 2 lots of 11 to 22 and then 1 left over; one half left over.
So 11 1/2.
83/9 is equivalent to 9 2/9.
9 lots of 9 are 81 and two ninths left over.
11/2 is equivalent to 5 1/2.
5 lots of 2 are 10 with one half left over.
23/9 are equal to 2 5/9, and 59/9 are equal to 6 5/9.
And then Jun wasn't right there, was he? 19/5 is not equal to 3 3/5.
But where did he go wrong? He used a good method.
He had the right idea.
You might have spotted that Jun divided incorrectly.
He made a little mistake.
Then you might have corrected his division like this.
19 divided by 5, in fact, is equal to 3 remainder 4 not 3 remainder 3.
So we can express a mixed number as so.
That's 3 4/5.
Well done if you said that.
We've come to the end of the lesson.
I've thoroughly enjoyed this lesson.
I hope you have too, and I hope you're feeling a lot more confident about converting improper fractions into mixed numbers.
An improper fraction can be written as a mixed number of equal value, and you can do that with any improper fraction.
You can turn any improper fraction into an equivalent mixed number using the skills that we practised today.
And this generalization's helpful.
The generalisation: First, we find out how many groups of the denominator can be made from the numerator by dividing can help to convert improper fractions to mixed numbers.
And instead of dividing, you might like to use your times tables facts as well and then see what extra bit you've got left over; what remainder? So there's a very strong link here between dividing with a remainder and improper fractions converted to mixed numbers.
Once again, you have been fantastic.
I hope I get the chance to spend another maths lesson with you in the near future and that we can do all of this again.
I hope you have a great day whatever you've got in store, and, whatever you do, be the best version of you.
Try as hard as you possibly can, and be successful.
Take care and goodbye.
