| Meaning and Models | Key Ideas | Addition | Subtraction | Multiplication | Division |

1. Five out of nine means five ninths. The five is the numerator and the nine is the denominator,
2.

There are 5 equal parts, so we shade 2 parts

3.

To shade of the following shape, cut the shape into 8 equal pieces and shade one piece

4. She is not correct because although she divided the rectangle into seven pieces, the pieces are not equal.
5. There are 11 marbles altogether. Seven of them are blue, so the fraction of blue =
6. To shade of this shape, we must shade 3 out of every 5 squares. There are 15 squares altogether, i.e. 3 lots of 5 squares, so we must shade 3 lots of 3 squares, i.e. 9 squares must be shaded.

7.

For every group of six apples, five apples were eaten. First, put the 24 apples into groups of six.

There are 4 groups of apples. If 5 apples from each group are eaten, there will be one apple from each of the four groups left. This means there will be 4 apples left.

8.

I want two out of every three smiley faces stickers. First, put the eighteen smiley faces stickers into groups of three.

I receive two from each of these six groups of three. I receive 12 smiley faces stickers.

9.

For , the rectangle is divided into 6 equal parts and one part is shaded.

For the rectangle is divided into 4 equal parts and three parts are shaded.

From the above diagram it can be seen that there is more green than orange. Hence, is bigger than

10.

Each hexagon is divided into 6 equal pieces. This gives 12 equal pieces altogether. is one piece out of 6 or 2 pieces out of 12. Since we have 12 pieces here, we must shade 2 pieces in order to shade

11(a) There are twelve hexagons altogether, eight of which are coloured yellow. Therefore the fraction coloured yellow =
11(b)

means 8 yellow for every 12, or 4 yellow for every 6, or 2 yellow for every 3

= = .
In simplest form the fraction is

 8 out of 12 hexagons are coloured yellow 4 out of 6 hexagons are coloured yellow 2 out of 3 hexagons are coloured yellow
12. There are nine ninths in one whole. The shape is divided into 9 equal pieces and all 9 pieces are required to give the whole.
13. Any integer can be written as a fraction by putting the integer over a denominator of one. 8 =
14.

A. 2 B. C. - 2 D. E.

2 and - 2 are integers and so can be written as fractions. is written as a fraction already. = 2 which can be written as a fraction. cannot be written as a fraction. Answer B

15.

We can answer this question without referring to the number of eggs because we are told that the eggs are partioned into 3 parts and that Sophie receives 1 of these parts. Therefore Sophie receives one third, , of the eggs.

An alternative explanation follows:

Twenty-seven eggs are partitioned in 3 parts , means there are 9 eggs to each part.

Sophie gets one part, therefore Sophie gets 9 eggs. 9 eggs out of 27 eggs is . This is the same as getting 1 egg for every three eggs, i.e. .

16. Five fifths in one whole plus another fifth = six fifths =

1. 1.To find fractions that are equivalent to another fraction, just multiply the numerator and the denominator by the same number or divide the numerator and the denominator by the same number.
 x 2 x 3 x 4 x 2 x 3 x 4

is equivalent to all of the following fractions and more ...

2.
 x 2 x 3 x 4 x 2 x 3 x 4

is equivalent to all of the following fractions and more ...

3.

a)

 x 2 x 3 x 4 x 2 x 3 x 4

 x 5 x 6 x 7 x 5 x 6 x 7

The fractions in the boxes are

The sequence goes as follows,

b)

 x 3 x 4 x 3 x 4

 x 5 x 6 x 5 x 6

The fractions in the boxes are

The sequence goes as follows,

4.

a)

 x 5 x 5

b)

 x 5 x 5

c)

 x 12 x 12

5.

a)

Multiples of the denominator 5 are:

5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60…

Multiples of the denominator 6 are:

6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66…

Multiples of the denominator 12 are:

12, 24, 36, 48, 60, 72, 84…

Multiples of the denominator 4 are:

4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60…

The smallest number common to all these denominators is 60, so make equivalent fractions of the given fractions with denominator 60

Ordering these fractions from smallest to largest gives

That is,

b) Multiples of the denominator 8 are:

8, 16, 24, 32, 40, 48, 56, 64, 72…

Multiples of the denominator 6 are:

6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66…

Multiples of the denominator 3 are:

3, 6, 9, 12, 15, 18, 21, 24, 27, 30…

Multiples of the denominator 4 are:

4, 8, 12, 16, 20, 24, 28, 32, 36, 40…

The smallest number common to all these denominators is 24, so make equivalent fractions of the given fractions with denominator 24

Ordering these fractions from smallest to largest gives

That is,

c)
Multiples of the denominator 9 are:

9, 18, 27, 36, 45, 54, 63, 72…

Multiples of the denominator 4 are:

4, 8, 12, 16, 20, 24, 28, 32, 36, 40…

Multiples of the denominator 3 are:

3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36…

Multiples of the denominator 12 are:

12, 24, 36, 48, 60, 72, 84…

The smallest number common to all these denominators is 36, so make equivalent fractions of the given fractions with denominator 36

Ordering these fractions from smallest to largest gives

That is,

 1 2 3 4 5 6

 1 2 3 4 5 6

1.

The yellow shading (Y) represents and the blue shading (B) represents . The rectangles with the double shading of purple (Y,B) represent of .

There are 10 out of 24 rectangles with double shading, so x =

 Y,B Y,B B Y,B Y,B B Y,B Y,B B Y,B Y,B B Y,B Y,B B Y Y Y Y Y Y

2.
a)
b)
c)

First we change the mixed number to an improper fraction.

The number 2 has 10 fifths and another fifth gives us 11 fifths. We can also work this out by multiplying 2 x 5, which gives us 10 and adding 1 - so we have 11 fifths, 11 over 5.

Then we do the multiplication,

d)

e)
f)

6 goes into 66 11 times and 6 goes into 30 5 times, so we can reduce the improper fraction by a factor of 6 to give the answer of 11 fifths.

3.
a)
b)
c)
d)
e)
f)