The Improving Mathematics Education in Schools (TIMES) Project

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Number and Algebra : Module 14Years : 4-7

June 2011

PDF Version of module

Assumed knowledge

(The statement 32 + 54 = 86 is equivalent to the statement 86 − 54 = 32.)

(The statement 6 × 4 = 24 is equivalent to the statement 24 ÷ 4 = 6.)

(For example: 23 + 34 + 17 + 36 = (23 + 17) + (34 + 36) = 40 + 70 = 110 and
37 × 6 = (30 + 7) × 6 = 30 × 6 + 7 × 6 = 180 + 42 = 222)


Traditionally, the term ‘fraction’ was used to describe a part of a whole. The word comes from the Latin frango − I break. In this module, we will take a fraction to mean a non-negative rational number, that is, a number of the form , where n is a positive integer and m is a positive integer or 0. (See Links Forward for further discussion on the definition of a fraction)

Fractions arise naturally in everyday situations involving sharing, cutting up and proportions. For example, the bathtub was one-third full, three quarters of the class walk to school. Fractions between 0 and 1 describe parts of a whole. Fractions extend the whole numbers to a number system in which division by a non-zero number always makes sense. While decimals can be used to represent fractions, many numbers are simpler in fraction notation. For example, compared to 0.142857....

Some computations are much easier if we use fractions rather than decimals.
For example, the fraction A0t3.pdf has decimal equivalent 0.33333... but it is much easier
to find × 12 than to try to find 0.333... × 12.

In this module we will only be concerned with positive fractions and zero.


For the fraction , the top number is called the numerator, the line is called
the vinculum and the bottom number is called the denominator.

Proper fractions and improper fractions

We call a fraction a proper fraction if the numerator is smaller than the denominator.
For example, A0t6.pdf and are proper fractions (A proper fraction x lies in the interval
0 ≤ x < 1 and if a fraction lies in this interval it is proper.)

If the numerator is greater than or equal to the denominator, the fraction is said to be an improper fraction. Examples of improper fractions include A0t8.pdf and . (An improper fraction x lies in the interval x ≥ 1 and any fraction ≥ 1 is improper.)

If the numerator and denominator are equal, then the fraction is equal to 1, so = 1.

Whole numbers can be expressed as fractions, for example, 4 = A0t11.pdf and some fractions are equal to a whole number, for example = 2. Indeed, every whole number can be written as a fraction in infinitely many different ways.

For example: 1 = A0t13.pdf, 2 = A0t14.pdf = A0t15.pdf, 3 = A0t16.pdf = .

The Two Models

There are two main ways to represent fractions.

The number line

The number line has already been used to represent whole numbers.

To represent, for example, fractions with denominator 3, we divide each unit
segment into three equal segments and label the markers as shown.

From 0, the marker for A0t24.pdf is reached by taking four steps to the right each of length .

Area diagrams

We can also represent fractions by shading areas.

The squares below are each considered to have area 1.
The shaded part represents a fraction of the whole.

The shaded areas below represent the fractions A0t26.pdf and .

Each of the two models has its advantages.

The number line is better to use for addition, subtraction and order.

The area model is better to explain multiplication of fractions.

Equivalent fractions

For example, if we mark A0t28.pdf and on a number line, then we mark the same point.

That is, A0t30.pdf and are equivalent fractions.

For example: A0t32.pdf = A0t33.pdf =

For example: A0t35.pdf = A0t36.pdf =

Simplest form

A fraction is said to be in simplest form if the only common factor of the numerator and the denominator is 1. To reduce a fraction to an equivalent fraction in simplest form we use the method of cancelling. For example, when reducing to its simplest form we divide the top (numerator) and bottom (denominator) by the highest common factor of 6 and 8, which is 2.

A0t39.pdf = A0t40.pdf =

We can use cancelling notation to write this process efficiently.

A0t42.pdf = A0g5.pdf =

For some fractions it is more convenient to cancel in stages.

A0g6.pdf = A0g7.pdf =

Exercise 1

a Use a number line to illustrate A0t45.pdf = A0t46.pdf = A0t47.pdf and A0t48.pdf = A0t49.pdf =

b Complete the set of equivalent fractions: A0t51.pdf = A0t52.pdf = A0t53.pdf =

Division by whole numbers

If I had 4 friends over for dinner and ordered three family sized pizzas, how much would each of us get if I shared it equally between the 5 of us? If three pizzas are shared equally among five people, then each one receives of a pizza.

Division of a whole number by another non-zero whole number always gives a fraction.

If we have 7 pizzas to be shared among 5 friends, then each would receive A0t56.pdf pizzas.
But this is clearly the same as 1 pizza each, plus A0t57.pdf of a pizza. Thus the improper fraction
A0t58.pdf is the same as 1.

When a line segment of length equal to 4 is divided into 3 equal segments, each segment has length 4 ÷ 3 = .

Dividing the whole number m by the whole number n, that is, m ÷ n results in the fraction A0t61.pdf. On the number line, the line segment from 0 to m is divided into n equal parts, each part has length .

For example, 7 ÷ 3 = A0t63.pdfand2 ÷ 7 =

Mixed numERALS

A mixed numeral consists of a whole number plus a fraction, for example 3A0t65.pdf = 3 + .

Every improper fraction can be written as a mixed numeral and vice versa.

For example, A0t67.pdf = 32 ÷ 3 = 10.

In order to write 10 as an improper fraction, we write:

  10 = 10 +
= A0b13.pdf +

Exercise 2

There are approximately 365 days in a year. Express this mixed numeral as an improper fraction.

Comparison of fractions

If two fractions have the same denominator then it is easy to decide which is the larger. It is the one with the larger numerator.

One number is greater than another if it lies to the right of that number on the number line.

Hence we can see from the number line that A0t71.pdf > A0t72.pdf, and A0t73.pdf > .

Using common denominators

If the denominators of two fractions are not equal, it is more difficult to see which of the two fractions is larger. In this case, we can use equivalent fractions.

To compare A0t75.pdf and , first find a common denominator. The lowest common denominator is the lowest common multiple of the two denominators, which in this example is 15.

Now A0t77.pdf = A0t78.pdf and A0t79.pdf = A0t80.pdf and so A0t81.pdf is larger than .

Exercise 3

a Find a fraction which lies between A0t83.pdf and A0t84.pdf. What about A0t85.pdf and ?

b List the fractions A0t87.pdf, A0t88.pdf, A0t89.pdf and from smallest to largest.

Addition of fractions

Addition of fractions is straightforward if the denominators are the same. To find A0t91.pdf + A0t92.pdf,
you take a step to the right of A0t93.pdf from 0 followed by a step to the right of A0t94.pdf to arrive at .
Of course this is exactly the same process as with the whole numbers and the number line.

Thus, when the denominators of the fractions to be added are the same,
we add the numerators.

A0t96.pdf + A0t97.pdf =

When the denominators are different, we use equivalent fractions to express
the fractions using a common denominator and then proceed exactly as before:

A0b21.pdf + = A0b23.pdf +

Addition of Mixed Numerals

We can add mixed numerals together simply by adding together the whole number parts and then adding the fractions.

Here is an example of how to deal with this:

  3A0b31.pdf + 4 = 7 + A0b33.pdf +
= 7 + A0b35.pdf +
= 7 +
= 7 + 1
= 8

Note that the commutative and associative laws for addition have been used in obtaining the result.

Exercise 4

a Find two different fractions that add to give .

b What mixed number when added to 3A0t100.pdf + 4 makes 10?

Subtraction of fractions

Subtracting fractions uses similar ideas to addition of fractions. If the denominators
of the two fractions are equal, subtraction is straightforward.

To find A0t102.pdfA0t103.pdf, move A0t104.pdf units to the left, starting from A0t105.pdf. Thus A0t106.pdfA0t107.pdf = .

When the denominators of the fractions to be subtracted are the same,
we subtract the second numerator from the first.

If the denominators are not equal, we use equivalent fractions to find
a common denominator.

Thus, to find A0t109.pdf, we convert to equivalent fractions with denominator 20.

A0t111.pdfA0t112.pdf = A0t113.pdfA0t114.pdf =

Subtraction of Mixed NumerALS

We can subtract mixed numerals in a variety of ways. Here are two methods.

Method 1


Method 2

Convert to improper fractions Deal with whole numbers
4A0b41.pdf − 2A0b42.pdf = A0b43.pdf 4A0b49.pdf − 2A0b410.pdf = 3 − 2 + A0b411.pdf
  = A0b45.pdf   = 1 + A0b413.pdf
  =   = 1
  = 1   = 1

When the whole number ‘parts’ of the mixed numerals are large then

For example:

258A0b51.pdf − 254 = 257A0b53.pdf − 254
= 257 − 254 + A0b55.pdf
= 3

Exercise 5

What fraction when added to 4A0t116.pdf gives 6?

Multiplication of fractions and ‘of’

In mathematics, when we are asked, for example, to find of 18 oranges,
we take it to mean that we divide the 18 oranges into three equal parts and
then take two of these parts.

This gives of 18 = 12.

Note that the result is the same as A0t120.pdf = = 12.

Using a number line with ‘of’

This method can be extended. For example, to find A0t122.pdf of A0t123.pdf:
we first calculate A0t124.pdf of and then multiply by 3.

A0t126.pdf of A0t127.pdf can be illustrated by using a number line, by first dividing the interval 0 to into 4 equal parts.

We can see that A0t129.pdf of A0t130.pdf is
and A0t132.pdf of A0t133.pdf is .

Using the area model for multiplication

A diagram to explain how to multiply A0t135.pdf × A0t136.pdf is shown below. The base is divided into three equal intervals (denominators of the first fraction). The height is divided into 4 equal intervals (denominators of the second fraction). The square, which has area 1, is divided into 3 × 4 = 12 (the denominators are multiplied together) rectangles of equal area. Each of the rectangles has area equal to .

The rectangle with side lengths A0t138.pdf and is shaded.

Notice that 2 × 3 = 6 is the number of shaded boxes (the numerators are multiplied together).

  A0t140.pdf× A0t141.pdf = A0t142.pdf =

You get the same shaded region if you first shade A0t144.pdf of the square and then shade A0t145.pdf of the shaded section. This shows that A0t146.pdf of A0t147.pdf gives the same result as A0t148.pdf × A0t149.pdf through the area model. Of course, A0t150.pdf of A0t151.pdf also gives the same result as A0t152.pdf × through the area model.

Here is a diagram which can be used to illustrate A0t154.pdf × A0t155.pdf = A0t156.pdf = 1.

  The dark blue square shows a unit square. It is divided into 8 equal parts.


In summary:

  • To multiply two fractions together, multiply the two numerators together
    and the two denominators together and simplify if possible:

For example, A0t158.pdf × A0t159.pdf = A0t160.pdf =

  • Multiplication of two fractions and the corresponding ‘of’ statement
    give the same answer.

Exercise 6

Use an area diagram to illustrate each of the multiplications.

a A0t162.pdf × A0t163.pdf b 1A0t164.pdf × 1


Let us find A0t166.pdf × .

A0t168.pdf × A0t169.pdf = A0t170.pdf = A0t171.pdf =

We cancelled down the second last fraction to its lowest form, after doing the multiplication.

There is a shorthand way of writing this, which often simplifies the process of multiplication.

A0t173.pdf × A0t174.pdf = A0g16.pdf × A0g17.pdf = A0t175.pdf =

This process is called cancelling. The cancellation can take place because you are doing the same operation to both numerator and denominator which we know gives an equivalent fraction.

Exercise 7

What is the total length of 6 pieces of ribbon if each piece is 8cm?

Multiplying Mixed Numerals

When multiplying two mixed numerals, we can convert both into improper fractions before multiplying. For example,

3A0t178.pdf × 4A0t179.pdf = A0g18.pdf × A0g19.pdf = = 15

Dividing a whole number by a whole number

We have considered dividing a whole number by a whole number in a previous section of this module. It was found that dividing a whole number by a whole number gives a fraction. For example,

4 ÷ 3 = A0t181.pdf and 17 ÷ 5 = A0t182.pdf = 3

The division statement 4 ÷ 3 = A0t184.pdf, has an equivalent multiplication statement to 3 × = 4.

Also note that dividing by 3 gives the same result as multiplying by .

In general, for whole numbers m and n, dividing m by n is the same as multiplying m by A0t187.pdf. Note that is called the reciprocal of n.

Division of a fraction by a whole number

The idea of performing a division by multiplying by the reciprocal can also be used to divide a fraction by a whole number. For example, A0t189.pdf ÷ 6 means that we take the fraction and divide it into 6 equal parts.

This can be illustrated by drawing a unit square divided into fifths. Shade (shown in
blue below) and further divide the square into 6 equal horizontal strips.

The purple shaded area A0r1.pdf represents A0t192.pdf = .

It can be seen that to divide A0t194.pdf by 6 , we multiply by the reciprocal of 6.

A0t196.pdf ÷ 6 = A0t197.pdf × A0t198.pdf = A0g21.pdf =

In general, when a fraction A0t200.pdf is divided by a whole number n, the result is
the same as when A0t201.pdf is multiplied by the reciprocal of n, .

Exercise 8

Use a number line to consider each division and give an interpretation of each division in terms of ‘how many’.

a 1 ÷ A0t203.pdf, 3 ÷ A0t204.pdf, 1 ÷ A0t205.pdf, 3 ÷

b 1 ÷ A0t207.pdf, 1 ÷ A0t208.pdf, 3 ÷

Dividing a fraction by a fraction

We will look at this in two ways. First we will repeat the division of the previous section.

Approach 1 (common denominator) The division A0t210.pdf ÷ 6 can be written as A0t211.pdf ÷ .

The division is now expressed in fifths. That is, 3 fifths divided by 30 fifths = A0t213.pdf = .

Approach 2 (a little algebra) A0t215.pdf ÷ 6 is the number x that when multiplied by 6 gives .

6x = A0t217.pdf. Multiply both sides by 5, 5 × 6x = 3, 30x = 3. Therefore x = .

A new question: A0t219.pdf ÷ = ?

Approach 1 (common denominator) The division A0t221.pdf ÷ A0t222.pdf can be written as A0t223.pdf ÷ .

The division is now expressed in twentieths.
That is, 5 twentieths divided by 12 twentieths = .

Approach 2 (a little algebra) A0t226.pdf ÷ A0t227.pdf is the number x that when multiplied by A0t228.pdf gives .

A0t230.pdfx = A0t231.pdf. Multiply both sides by 4, 4 × A0t232.pdfx = 1, 12x = 5. Therefore x = .

Division and reciprocals (Approach 3)

The reciprocal of a fraction is the fraction obtained by swapping the numerator
and denominator.

Hence the reciprocal of A0t234.pdf is A0t235.pdf. Dividing by A0t236.pdf is the same as multiplying by its reciprocal .

Thus, A0t238.pdf ÷ A0t239.pdf = A0t240.pdf × A0t241.pdf = .

In general, when a fraction A0t243.pdf is divided by a fraction A0t244.pdf, the result is the same as when A0t245.pdf is multipled by the reciprocal of A0t246.pdf, that is multiplied by .

For example, A0t248.pdf ÷ A0t249.pdf = A0g22.pdf × A0g23.pdf =

Note that, A0t251.pdf × A0t252.pdf = = 1.

Dividing Mixed Numerals

As with multiplication, to divide two mixed numerals we convert both to improper fractions.

3A0t254.pdf ÷ 4A0t255.pdf = A0t256.pdf ÷ A0t257.pdf = A0t258.pdf × A0t259.pdf =

exercise 9

A plumber has a piece of pipe which is 7 metres in length and has to cut it into 5 equal pieces. How long must each piece be?

Links forward

An understanding of fractions is crucial before meeting decimals.

The techniques and ideas that are used in the four operations on fractions are needed when numbers are replaced with pronumerals. Hence algebraic competence and progress in school mathematics relies heavily on fluency with fractional arithmetic.

Many algebraic formulas used in both mathematics and science involve fractions.
For example, the volume of a sphere V = A0t262.pdfπr3 and Ohm’s law I = , involve fractions.

All fractions have either terminating or recurring decimal expansions and conversely every number that has a terminating or recurring decimal representation is a fraction.

Probability calculations rely heavily on fractions, as do problems involving percentages, rates and ratios.

In this module a fraction is a positive rational number - negative rational numbers are also fractions. The term fraction is used in a number of ways. A surd like A0g24.pdf is sometimes referred to as a fraction, as is an algebraic expression such as . Techniques similar to those developed in this module can be applied to surds and algebraic expressions.

A formal construction of the rational numbers is given in the module, The Real Numbers. Negative fractions are discussed in the module, Fractions and the Index Laws in Algebra.


In general, the ancient civilisations avoided fractions by giving special names to parts of various measures. We still do this today. For example, instead of saying 1 hours we say
1 hour 13 minutes.

Egyptian fraction notation was developed in the Middle Kingdom of Egypt (2080−1640 BC), altering the Old Kingdom’s Eye of Horus numeration system.

With few exceptions they wrote all their fractions as unit fractions, that is, with numerator 1. To write the unit fractions used in their Egyptian fraction notation the Egyptians placed the hieroglyph above the numeral for the denominator.

For example A0t266.pdf =

There were separate symbols for some common non-unit fractions such as A0t267.pdf and
but most fractions, as we know them, were expressed as a sum of unit fractions.

For example A0t269.pdf can be written as A0t270.pdf + A0t271.pdf + A0t272.pdf + .

Note that if p is an odd number, then A0t274.pdf = A0t275.pdf + A0t276.pdf + A0t277.pdf + gives a formula for expressing fractions of this form as a sum of unit fractions.

Exercise 10

a Prove the identity A0t279.pdf = A0t280.pdf + A0t281.pdf + A0t282.pdf + A0t283.pdf where p is a non-zero whole number and use it to write as the sum of four different unit fractions.

b Prove the identity A0t285.pdf = A0t286.pdf + A0t287.pdf and use it to split into unit fractions.

The Rhind papyrus (1650 BC) and other ancient sources have been used to explore the methods the Egyptians used in calculating with Egyptian fractions.

While the Egyptians used fractions as objects − we do not know exactly what they thought of them. Up until the Hellenistic period the Greeks preferred to think of fractions in terms of ratios and proportions. They did not tend to think of them as we do, and certainly not as points on the number line.

Hindu mathematicians are believed to be the first to indicate fractions with numbers rather than words. Brahmagupta (c. 628) and Bhaskara (c. 1150) were early Hindu mathematicians who wrote fractions as we do today, but without the bar (vinculum). They wrote one number above the other to indicate a fraction.

The next step in the evolution of fraction notation was the addition of the horizontal fraction bar. This is generally credited to the Arabs who used the Hindu notation, then improved on it by inserting this bar in between the numerator and denominator, which was later named the vinculum. Later on, Fibonacci (c.1175-1250), was the first European mathematician to use the vinculum as it is used today.



a For A0t290.pdf = A0t291.pdf, draw an interval from 0 to 1. Mark A0t292.pdf and on it. Divide each of the three subintervals formed into two equal parts. The interval from 0 to 1 is now divided into subintervals of length one sixth.

For A0t295.pdf = A0t294.pdf , draw an interval from 0 to 1. Mark A0t295.pdf , A0t296.pdf and on it. Divide each of the four subintervals into 3 equal parts. The interval from 0 to 1 is now divided into subintervals of length one twelfth.

b A0t298.pdf = A0t299.pdf = A0t300.pdf =



a - infinitely many possible answers

- infinitely many possible answers

b A0t305.pdf, A0t306.pdf , A0t307.pdf,


a A0t309.pdf + A0t310.pdf = A0t311.pdf or A0t312.pdf + A0t313.pdf =

b 2




a One unit square divided into five equal columns and 6 equal rows to form 30 rectangles of equal area. Shade 3× 1 = 3 rectangles.

b A square 2 units by 2 units consisting of 4 unit squares. Divide each unit square into five equal columns and six equal rows. Shade 8 × 7 = 56 rectangles.


51 cm


How many halves in 1? (Number line with 0, 1 and one half marked)
How many halves in 3? (Number line with 0, 1, 2 and 3 and the halves marked)
How many quarters in 1? (Number line with 0, 1 and the quaters marked)
How many quarters in 3? (Number line with 0,1, 2 and 3 and the quarters marked)

How many thirds in 1?
How many sevenths in 1?
How many sevenths in 3?


1 m or 1.58 m


A0t319.pdf = A0t320.pdf + A0t321.pdf + A0t322.pdf +

A0t324.pdf =A0t325.pdf +


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