Lesson video

Hi everyone, my name is Miss Coo, 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 checking and securing converting improper fractions to mixed numbers.

And by the end of the lesson, you'll have awareness that fractions in the form of A over B, where A is greater than B, are greater than one and so convert from improper fractions to mixed numbers.

So let's have a look at some keywords to begin with.

A proper fraction is a fraction where the numerator is less than the denominator.

For example, two thirds is a proper fraction as two is less than three.

An improper fraction is where the numerator is greater or equal to the denominator, for example, seven over five, seven is greater than five.

And a mixed number is an improper fraction written as integer parts plus the fractional part where the fractional part is a proper fraction, for example, three and a half.

Today's lesson will be broken into two parts.

The first one will be looking at proper and improper fractions, and the second part of the lesson will be looking at improper fractions and mixed numbers.

So let's have a look at proper and improper fractions.

A proper fraction is a fraction where the numerator is less than the denominator, for example, three eighths.

And it's also important to recognise on a number line, a proper fraction is less than one.

So you can see I've represented three eighths here.

And so it's important to remember a proper fraction is always less than one.

Now an improper fraction is a fraction where the numerator is greater than or equal to the denominator, for example, seven fifths.

And on a number line, you can see seven fifths is greater than one.

So an improper fraction is always greater than or equal to one.

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

Andeep has £6 and he's dividing this between his seven friends.

Izzy says, each person will get more than £1 each, but Sofia says, each person will get less than £1 each.

Without any working out, explain who is correct using your knowledge on proper and improper fractions.

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

Well done.

So hopefully you figured out that Sofia is correct.

Six divided by seven is the same as six over seven, written as a proper fraction, and given that the numerator is less than the denominator, it's clear that the value each student receives will be less than one.

Now let's have a look at a check question.

Here we have a number line and I want you to mark the position of 13 over five on the number line and two over five on the number line.

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 13 over five means we have 13 fifths.

So where would this be on our number line? Well counting up, you should find out that 13 fifths would be here.

Starting from zero and counting up one fifth at a time.

Two fifths would therefore be here, starting from zero and our number line is split into fifths, so therefore two fifths would be here.

Well done if you got that one right.

Now let's have a look at your task questions.

For question one, it wants you to insert the inequalities, equals, greater than, or less than to make the following statements true.

For A, four fifths is what compared to one? For B, eight thirds is what compared to one? For C, 11 fifths is what compared to one? And for D, six times two over three times four is what compared to one? See if you can give it a go and press pause if you need more time.

Great work.

Let's move on to question two.

Question two is a table and it wants you to identify if the following statements are true or false.

So is the value of a proper fraction, is it always less than one? Is the value of an improper fraction, is it always greater than one? Is a fraction in the form of A over B, where A is greater than B, is it called an improper fraction? And is 789 divided by 788, will it give a number greater than one? See if you can work these out and press pause if you need more time.

Well done.

Let's move on to question three.

Question three shows a number line and, without calculating the answer, we need to identify which letter corresponds to which calculation.

You have 5677 divided by 5678.

Which letter do you think this is? We have 5678 divided by 5678.

Which letter do you think this is? We have 5679 divided by 5678.

Which letter do you think this is? And we have 5678 divided by 2880.

So which letter do you think this is? See if you can give it a go and press pause if you need more time.

Now let's have a look at question four.

Question four shows a section of a number line.

Which whole number is the arrow pointing to? Approximately mark two other whole numbers on the number line, and what are these whole numbers? For C, how would your answers change if the denominator was six? And for D, what could the denominator be if there were no whole numbers between the fractions? This is a great question, see if you can give it a go and press pause for more time.

Well done.

So let's see how you got on.

For question one, hopefully you identified four fifths is less than one.

Four fifths is a proper fraction and we know proper fractions are less than one.

For B, eight thirds is greater than one.

It's an improper fraction.

For C, 11 divided by five is the same as 11 over five, which is an improper fraction so is greater than one.

Six multiplied by two over three multiplied by four is equal to one.

Six multiplied by two is 12, over three multiplied by four is 12, so therefore 12 divided by 12 is one.

Well done if you got that one right.

For question two we had to identify if the following statements were true or not.

The value of a proper fraction is always less than one.

This is true.

The value of an improper fraction is always greater than one.

Well it's false because it can also equal one.

A fraction in the form of A over B where A is greater than B is called an improper fraction.

It's true.

789 divided by 788 will give a number greater than one.

Yes, it's true because writing 789 divided by 788 as a fraction, we have a numerator which is greater than the denominator, so it's an improper fraction, so it's greater than one.

Question three wants us to identify which letter corresponds to the calculation.

Remember we're not allowed to calculate the answer.

Well hopefully you spotted for 5677 divided by 5678 is letter D.

This is because, if we were to write it as a fraction, our numerator is less than the denominator, so it's a proper fraction so we know it'll be less than one.

For B, we have 5678 divided by 5678.

This must be letter A, as the numerator is equal to the denominator, thus meaning it equals one.

For C, we have 5679 divided by 5678.

Well this is letter C, we have an improper fraction as the numerator is greater than the denominator.

It's only a little bit greater than the denominator, so that's why we can see it's only a little bit more than the one.

And for D, 5678 divided by 2880.

And we know this to be letter B.

Well done if you got this one right.

For question four, it shows a section of a number line.

And on the number line you can see 21 over five and 26 over five.

And part A says, which whole number is the arrow pointing to? Well it had to be 25 over five, because 25 divided by five is five.

Now part B says approximately mark two other whole numbers on our number line and identify what these whole numbers are.

Well hopefully you spotted 20 over five would give us the integer value of four, which would be around about here, and 30 over five would give us the integer value of six which is around about here.

Well done if you got that one right.

Next, part C says, how would your answers change if the denominator was six? Well, if I was to change my denominator of six, immediately this whole number for part A would be 24 over six, which would give me four, and the two other whole numbers would be 18 over six and 30 over six, which would give us three and five, which I plotted around about here and here.

So for question D, it says what could the denominator be if there were no whole numbers between the two fractions? Well, any integer which does not have one of its multiples between 21 and 26.

For example, it could be seven.

The denominator could be seven, so I'm going to change it here, because if it's seven then we know the first fraction would be a whole number, it'd be three, and there would be no more integers before the second fraction.

But what we do know is it couldn't be eight.

The reason why we know the denominator couldn't be eight is because 24 is a multiple of eight, and because 24 is a multiple of eight, 24 divided by eight is three, so that means three would lie between the two fractions.

We also know it could be nine because there are no multiples of nine between 21 and 26.

Two is before the first fraction and three is after the second fraction.

Well done if you got that one right.

Well done and let's move onto the second part of our lesson where we'll be changing improper fractions to mixed numbers.

Now remember a mixed number is an improper fraction written as integer part plus the fractional part where the fractional part is a proper fraction.

For example, three and a half is a mixed number.

This means we have three whole ones and a half.

And it's important to recognise that the denominator indicates how many equal parts make a whole.

For example, let's have a look at the fraction 12 over five, what does this mean? Well, because the denominator is five, this is telling us that we split each whole into five equal parts.

So I'm gonna show with an example diagram.

If each pentagon represents one whole, this shows we have 12 over five is shaded.

I'm also gonna show you on a number line.

As we've split each one whole into five equal parts, this is one whole, this is one whole, and this is one whole, and we've split them into five equal parts.

And 12 fifths would be where the arrow's indicated.

But, the number line shows that we have two whole ones and two fifths.

It's also indicated here with the example diagram.

We have two whole pentagons shaded as well as two fifths.

So, how can we identify the mixed number from the improper fraction? Well, looking at our denominator, we know we have five fifths would make one whole, so we can partition our fraction.

In other words, 12 over five is exactly the same as five over five add five over five and two over five, which is the same as one add one add two fifths.

Summing these together makes our two and two fifths.

So what we've done is identify our improper fraction as a mixed number.

And drawing diagrams or number lines are methods to visually represent the mixed number, but using knowledge that the denominator identifies how many parts are a whole is a quicker and easier way.

So now let's consider multiples of the denominator to make it more efficient.

Well, you could also partition 12 over five to be 10 over five add two over five.

This makes it more efficient giving us two and two fifths, which is our mixed number.

And this is a much more efficient way.

Now, let's have a look at a check question.

By filling in the gaps, show the conversion from an improper fraction to a mixed number.

See if you can give this a go and press pause if you need.

Great work, so let's see how you got on.

Well for part A, well hopefully you spotted it's got to be 12 over six because our 12 over six add our five over six is the same as 17 over six, and our 12 over six represents our two.

For B, hopefully you spotted we have nine thirds add one third, makes our 10 thirds.

Well we know nine thirds represents three, so our answer is three and one third.

For C, how can we partition 57 over 50? Remember we can use multiples of 50.

Well, 50 over 50 add our seven over 50 would give us one and seven fiftieths.

For D, 234 over 55 is 220 add 14 over 55 which is four and 14 over 55.

Well done if you got that one right.

So partitioning the fractions into wholes and proper fractions is not always the most time efficient method.

For example, would you want to partition 239 over two into its whole numbers and proper fraction? So let's see if we can find a more efficient way.

Improper fractions can be converted to mixed numbers using the knowledge of the dividend, quotient, remainder, and divisor.

So let's have a look at 22 over seven.

What we're doing is we're identifying the divisor as seven and the divided as 22.

And we're saying, right, well how many times does seven go into 22? And this is our quotient and we can see it's three whole times, but there is remainder, there's a remainder of one and our divisor is still seven.

So to convert an improper fraction into a mixed number, we identify a quotient, a remainder, and the divisor still remains the same.

So let's have a look at another check question where you want to match the improper fraction with the correct mixed 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.

So first of all, 13 over two is six and a half.

Two goes into 13 six times with a remainder of one and the divisor is still two.

15 over four is three and three quarters.

Four goes into 15 three times with a remainder of three over four.

10 thirds is three and one third.

Three goes into 10 three times with a remainder of one and a divisor of three.

11 over two is equal to five and one half, as two goes into 11 five times with a remainder of one and the divisor is two.

Lastly, we know 14 over three is the same as four and two thirds.

Three goes into 14 four whole times with a remainder of two and the divisor is still three.

Well done if you got this one right.

Now let's have a look at another check question.

Jun says 46 over four is equal to 11 and a half.

Is he correct? And can you explain? Well done, well hopefully you can spot he is correct because 46 over four is equal to 23 over two, and converted, this gives us 11 and a half.

Now it's time for your task.

So I want you to write the mixed number of the following improper fractions.

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

Well done.

So let's move on to question two.

Question two wants you to fill in the blanks.

You have your improper fractions and you need to convert it into a mixed number.

See if you can work it out.

Well done, so let's move on to the last question.

Here it wants us to put the numbers in ascending order, smallest to largest.

Converting to a mixed number may help.

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 one, 13 over three is the same as four and one third.

For B, 20 over seven is the same as two and six sevenths.

For C, 104 over five is the same as 20 and four fifths.

And for D, 50 over three is the same as 16 and two thirds.

For question two, let's see how you got on.

15 over six is the same as two and a half.

91 over six is the same as 15 and one sixth.

For C, we have 142 over four can be simplified to give 71 over two, which then can be converted into 35 and one half.

For D, 134 over eight can be simplified to 67 over four, which can then be converted into 16 and three quarters.

For question three, converting them to a mixed number may help.

So you can see we have a proper fraction which is nine over 11, so that must be the smallest as it's less than one.

Now I'm gonna convert the rest.

17 over two is the same as eight and a half.

Nine over two is the same as four and a half.

So from this I know one and two thirds is the next smallest number.

From there, I can see four fifths is the next smallest, followed by nine over two, and the 17 over two.

So converting to mixed numbers has really helped us out.

For B, we have some proper and improper fractions here.

So we're gonna have to do some converting.

So looking at our proper fractions, we have two thirds and four fifths.

Well two thirds is the smallest fraction, followed by four fifths.

So now let's convert.

Seven over two is the same as three and a half.

19 over three is the same as six and one third.

And 211 over 50 is the same as four and 11 fiftieths.

So putting them in order, we have this.

Massive well done if you got this one right.

So in summary, a proper fraction is a fraction where the numerator is less than the denominator and an improper fraction is a fraction where the numerator is greater than or equal to the denominator.

Remember, a mixed number is an improper fraction written as an integer part plus the fractional part where the fractional part is a proper fraction.

Well done, it was great learning with you today.