1
UNIT FOUR
THE TEACHING OF INTEGERS
MEANING OF INTEGERS AND THEIR USES IN EVERYDAY SITUATION
The classification of numbers starts from natural numbers (counting numbers), whole numbers,
integers, rational and irrational numbers and real numbers. This extension came as a result of
man’s development. The real number is not even the limit; there is still another, the complex
numbers, Natural numbers are the numbers we use in counting.
An integer is a set of numbers which consists of a set of positive whole numbers, zero and the set
of negative whole numbers.
i.e. Integers = {positive whole numbers} ⋃{0} ⋃{negative whole numbers}
The set of integers makes it possible to work with the operation of subtraction. In this unit we
will learn about how the child would be able to add, subtract, multiply and divide in the domain
of the set of integers.
Meaning of Integers (Directed Numbers)
With the knowledge of whole numbers, numbers can be matched with points on the number line
as shown below. See fig 1
Fig 1 Whole numbers on the number line.
To get the meaning of a negative number, let us consider things which can be measured in two
opposite directions.
Consider the numbers matched with points on the number line in fig 2.
Fig 2
If we want to consider the point A, 3 units from 0 and point B, also 3 units from 0: How do we
distinguish point A from point B?
To make the points clear, we would say A is 3 units right of 0 and B is the point 3 units left of 0.
0 1 2 3 4 5 6 7 8 9 10
0 1 2 3 4 5 6 7 8
B A
2
Mathematically, we write 3 units right of 0 as +3 and 3 units left of 0 as -3. With this our number
line will look like as in fig 3
The set {…..-4, -3, -2, -1, 0, 1, 2, 3, 4, ……..} is called directed numbers of integers. The subset
{1-, -2, -3,-4,………..}, of the integers is called negative integers whilst the subset {1, 2, 3, 4,
….} is the positive integers.
Activity 1
Ask two pupils to come forward and stand side by side, while one moves forward let the other
move backward. Represent the movement made by the two pupils on a number line.
The name of the collection of all the numbers on the number line above is integer.
Practical and real examples of Directed Numbers
The following are some practical examples of directed numbers which may be used to help
children get the meaning of negative numbers.
(a) Debts (-) and Credit (+)
(b) Height below (-) and above (+) Sea Level
(c) Temperature below (-) and above (+) Freezing Point.
Let us describe the real situations where negative numbers are used.
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
Fig 3 Signed numbers on the number line.
-4 -3 -2 -1 0 1 2 3 4 5 6
3
Debts and Credit: A man with no money and no debts is better than the one with no money and
who owes say ¢10. But the man with no money and no debts is worse than one with no debts but
having say ¢5. We may call the wealth of ¢5 as + 5, the debt of ¢10 as -10 and no debt as 0. This
shows that −10 < 0 < 5.
Height below and above sea level: The height of a land is usually measured from sea level. A
height of hill 100 meters is one whose top is 100 meters above sea level. This may be indicated
by + 100. If a fish is 150 meters below sea level, we could state its position as -150 above sea
level. The land above the sea is considered as being positive, the sea is the zero point and the
land below being negative.
A point at sea level may be indicated by 0. A point -50m above sea level is higher up than one at
-150m above sea level but less than the point at sea level −150 < −50 < 0 < 100.
Temperature Scale (Use of Thermometers): On a thermometer, the temperature of a freezing
water is 00C. Temperature which is colder than 00C are written with a minus sign, for example a
temperature of 50C below freezing point may be written as -5
0C and a temperature 40C above
freezing point +40C. This means −5 < 0 < 4
A weather thermometer, indicates temperatures below 0o
by negative numbers, so on the number
line numbers below or left of zero are negative numbers. The graphs of -4 and 4 are the same
distance from 0, but in opposite directions such a pair of numbers is called opposites. Thus -4 is
opposite 4 and 4 is opposite -4. Distance from zero to a number on the graph is its absolute
value.
Thus 4 and -4 have the same absolute value of 4. So |+4| and |-4| = 4
Note that there are two uses of the signs – and +. We can use them to denote operations and they
can be used to denote directions to the left or right respectively.
To differentiate the use of the signs, we write directed numbers with a bracket around or write
the sign at the top like (-2), (+4) or -2 or +
4.
The main ideas in this lesson were:
Positive and negative whole numbers constitutes the sets of integers
Some practical instances of integers are
– Land above the sea or below the sea
– I have money or owe money
– Activities of people before Christ was born and after the death of Christ (ie. BC and
AD)
– The use of the thermometer in the temperature zone.
4
Operations on Directed Numbers
Operation on directed numbers seem more difficult than that of counting numbers. Concrete
models must be used to link actions with the operations. We can employ.
(i) The idea of having or owing
(ii) The idea of movement along the number track.
(iii) The idea of movement along the number line with directed numbers (set of arrow
lengths).
This activity uses the idea that directed numbers can be seen as one dimensional vector. To
illustrate the operations of directed numbers, we mag employ concrete materials like:
a) Coloured Counters which are Cut out of Cardboards to Represent Positive and
Negative Integers
–
(i) Coloured counters (ii) Walking on the number track
Fig 5. Models of operation
The model of the coloured counters uses the idea of owing or having.
(a) Drawing Number Tracks on the Floor and Children Moving Along it. See fig 5 (ii)
In this model the operation of addition is represented by the action walk forwards,
operation of subtraction is represented by the action walk backwards.
A positive sign on the directed numbers means face towards the right and a negative sign
means face towards the left.
(b) Set of Arrow Length on the Number Line
The arrow lengths are of lengths 1 unit, 2 units, 3 units up to 10 units corresponding to
lengths of units on the number line.
Different colours must be used for positive arrow and negative arrow

-5 -4 -3 -2 -1 0 1 2 3 4 5
5
Fig 6 Arrow length
Two arrow lengths are used for an operation. For the operation of addition we lay the tail of the
second arrow at the tip of the first arrow. The answer is found at the tip of the second arrow. For
the operation of subtraction, we lay the tip of the second arrow on the tip of the first arrow and
the answer is found at the tail of the second arrow.
A positive directed arrow must point right and a negative directed arrow must point left. The
diagram in fig.7 illustrates the above information.
For addition
For subtraction
Fig 7
The use of Adding Machine
A simple adding machine consists of two cut out strips of cardboard, each a little over 20cm long
and about 2cm wide. See fig. 8
Fig 8 Adding machine
+
1
st
+
2
nd

Ans
Ans
2
nd

1
st +
–
1
st
+
2
nd
– +
+
2
nd
1
st

Ans
Ans
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
6
For the operation on integers, we use the two number strips A and B. To find the sum -4 + 6, we
find the first number -4 on the strip A. We then set the -4 on A against 0 on B. We look for the
second number 6 on B. The number on A which is directly above 6 on B will be our answer. Fig.
9 illustrates the process. The diagram indicates that 2 is the answer i.e. (-4) + 6 = 2
A
Set zero against Second number
First number on A
Fig 9 the use of adding machine.
To use the adding machine for the operation of subtraction we look at subtraction as the inverse
of addition. For example 6 – 2 can be expressed as what is x, if 2 + x = 6? This indicates addition
and we can add using our adding machine.
For 2 + x = 6, the first number is 2, the second number is x and the answer is 6. Fig.10 illustrates
the operation.
Comparing 2 + x = 6 ⇔ 6 − 2 = x
For 6 – 2 = x first number is 6, second number 2 and answer is x. Thus for the subtraction
problem of 6 − 2 , we set the second number 2 on strip A against 0 on strip B. We look at the
first number 6 on strip A. The number on B which is directly on 6 is the answer. This gives 4 as
the answer i.e. 6 – 2 = 4
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
B
First Number Sum
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
First Number
Set zero on B Answer
Fig. 10
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Example 1.
Illustrate how you can use various materials to solve these problems
(i) (-6) + (+3) (ii) (-3) + (-5)
Solution:
(a) Using the idea of owing on
(i) (-6) + (+3)
-6 means owing say 6 cedis, +3 means having say 3 cedis
ie. (-6) + (3) implies you owe 6 cedis and you pay 3 cedis therefore your debt is left
with three cedis
Left
Paid
∴ (−6) + (+3) = −3
ii) (-3) + (-5)
-3 means owing say 3 cedis, (-5) means owing another 5 cedis. This gives a total debt of 8
cedis
-3
and
8
-5
∴ (−3) + (−5) = −8
iii) Using number track
(i) (-6) + (+3)
Here we start from -6. Face right and walk forward 3 places. We land at -3.
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2
∴ (−6) + (+3) = −3
(ii) (-3) + (-5) = -8
With this problem we start from -3 face left and move forward 5 places. We will land at -8
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3

∴ (−3) + (−5) = −8
iv) Using an arrow length
(i) (-6) + (+3)
The first arrow is -6 and the second arrow is +3
Lay the first arrow from point 0 pointing, left. The tail of the second arrow is then placed at
the tip of the first arrow. The tip of the second arrow will be on-3
-6 +3
9
∴ (−6) + (3) = −3
(iii) (-3) + (-5)
The first arrow length is -3 the second arrow length is -5
Lay the first arrow from 0 to point left. The tail of the second arrow is then placed at the tip of
the first arrow.
The tip of the second arrow length will be on -8
∴ (−3) + (−5) = −8
Example 2
Illustrate with diagrams how you will solve
(i) (-3) – (+4) (ii) (-3) – (-4)
Solution
(a) Using number truck
(i) (-3) – (+4)
-3 -5
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4
-6 1
st
2
nd
+3
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2
-5 -3
10
Start from -3 face right and walk back wards 4 steps and this will land on -7.

∴ -3 – (4) = -7
(ii) (-3) – (-4)
Start from -3 face left walk backwards 4 places and this will land on 1.
∴ (- 3) – (-4) = 1
(b) Using arrow length
(i) (-3) – (+4)
First arrow length is -3 Second arrow length is +4
Lay the first arrow length from 0 pointing to the left, the tip of the second arrow length is placed
at the tip of the first arrow and must point right. The answer is found at the tail of the second
arrow length.
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5
-3 +4
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3
+4 -3
11
∴ (−3) − (+4) = −7
(ii) (-3) – (-4)
First arrow length is (-3) Second arrow length is -4
Lay the first arrow length from 0 pointing to the left. The tip of the 2nd arrow must be at the tip of
the first arrow and must point left.
Our answer is at the tail of the second arrow.
∴ (-3) – (-4) = 1
Operations of Multiplication and Divisions on the Directed Numbers.
Multiplication is seen as repeated addition or groups of numbers. Division is represented as the
inverse of multiplication.
Example:
Using the number line illustrate how you will show to a child that (-2) x (4) = -8
Solution
We can illustrate (-2) x (4) as the repeated addition of (-2) + (-2) + (-2) + (-2) or four groups of
-2.
If we take it as repeated addition of (-2) then we model our problem on the number track using
the principles as (-2) x 4 = (-2) + (-2) + (-2) + (-2)
-3 -4
-6 -5 -4 -3 -2 -1 0 1 2 3 4
-3
-4 2
nd
1
st
12
We start from (-2) face left and walk forward 2 places, continue to walk forward 2 places and
then another 2 places forward.
-14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2
We land on -8 ∴ (-2) x 4 = -8
If we consider it as 4 groups of (-2), we model our problem on the number line starting from 0
and make 4 groups of (-2)
Using concrete materials to solve problems like (-2) x (-4) and (-6) ÷ (-2) is not easily
interpreted and hence may seem difficult to the children. To help children see how to solve such
problems, we may apply some pattern.
For example (-2) x (-4). We may generate and use a pattern like
First Part
Second Part
-2 x (+3) = -6
-2 x (+2) = -4
-2 x (+1) = -2
-2 x 0 = 0
-2 x -1 = 2
-2 x -2 = 4
-2 x -3 = 6
-2 x -4 = 8
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2
13
The first part can be easily modelled on the number line but we cannot easily model the second
part on the number line. The child should be helped to see the pattern that each successive
answer is two to the right of the previous answer on the number line or is increased by 2.
For the division like (-6) ÷ (-2) the children should use division as the inverse of multiplication
ie. (-6) ÷ (-2) should be seen as (-2) x = -6, which means how many groups of -2 will be -6
From the number line we see this as 3 groups
∴ (-6) ÷ (-2) = 3
Children should eventually develop the rule that:
(-1) x (+1) = -1 (-1) ÷ (+1) = -1
(-1) x (-1) = 1 (-1) ÷ (-1) = 1
(+1) x (+1) = 1 (+1) ÷ (+1) = 1
(+1) x (-1) = -1 (1) ÷ (-1) = -1
ie. For multiplication and division if the signs are the same the answer is positive but for opposite
signs the answer is negative.
-4 -3 -2 -1 0 1 2 3
-6 -5
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ADDITION OF INTERGERS
We have been using teaching and learning materials like the coloured counters, vector arrows
and sometimes jumps or movement on the number line drawn on the ground to teach operations
of integers. We will consider two other types of materials that are common but not familiar with
every teacher. These are the nomograph and the charged particle model.
i) Using the nomograph to do addition of integers.
Ask your pupils to draw three parallel lines that are the same distance apart in their exercise
books. The scale is 1cm to 1 unit on all the parallel line segments. The outer lines are calibrated
1 cm to 1 unit while the middle line is 1 cm to 2 units. Label the outer lines A and B, the middle
as C, where the result is located.
15
The construction of the nomopgraph is shown below
7 14 7
6 12 6
5 10 5
4 8 4
3 6 3
2 4 2
1 2 1
0 0 0
-1 -2 -1
-2 -4 -2
-3 -6 -3
-4 -8 -4
-5 -10 -5
-6 -12 -6
-7 -14 -7
-8 -16 -8
A B C
+1
16
Your pupils should select any two addends e.g. -2+3
Guide them to locate the first and second addend (-2) on the A line segment and the second
addend (+3) on the line segment B, and then using a transparent straight edge to join the two
addends. The result is then located on the line segment C, where the straight edge intersects C as
+1 shown above.
ii) Using the charged particles to add two or more given integers.
If the integers are more than two you can employ the idea of the associative properties
Pupils draw a circle of about 5 cm radius on a card, show the diameter and label the upper half
with a positive sign (+) and the lower half a negative sign (-).
Pupils then divide the card into two equal parts along the diameter.
Now let us return to the previous, “(-2) + 3”. Pupils select two negative charges and three
positive charges.
Pupils combine the negative and positive charges to form a circle, these circles form neutral
charges and are considered as zeros.
The uncombined pieces represent the sum of addition exercise. For our example, there is one
positive charge that has been isolated. Thus -2 + 3 = + 1
– .
17
SUBTRACTION OF INTEGERS
Consider the task “ 5-3”, the figure 5 is the minuend and the second (3) is the subtrahend, the
result (answer) is the difference. You can help the pupils to use charged particles.
Activity 1
Find (+5) – (+3)
Ask pupil to group five positive charged – particles, and then take away 3 of them.
Take away 3 particles Remaining 2 particles
The result is the two positive charges. So 5 – 3 = 2
Activity 2
Find (−5) – (−3)
Pupils collect five negative charges and take away three of them.
Take away 3 negative particles
Remaining 2 negative particles
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The remaining is negative two charges so (−5) – (−3) = −2
Activity 3
Find −3 – (−5)
Ask pupils to take 3 negative charged particles. It is impossible to remove 5 negative particles
from the 3 negative particles. Pupil model 5 neutral charges and add to the three negative.

Guide the pupils to remove 5 negatives; combine the negatives and the positives to form neutral
charges. The uncombined charges give the result.
So −3 – (−5) = 2
Activity 4
Find (-5) – (+ 3)
Take away 3 positive particles from 5 negative particles. Let pupils model as follows:
Pupils model five neutral charges and five negative charges and then take away three positive
charges.
Take away
+5
-5
+2
3 neutral charges
19

Rearrange the particles
There are two neutral charges and eight isolated negative charges
Therefore (−5) – (+3) = −8
So −3 – (−5) = −3 + 5 because 5 is the opposite of −5
MULTIPLICATION OF INTEGERS
Activity 1
Find 6 × 2 (Multiplication of positive integers)
Ask your pupils to take six lots of two charged particles

Altogether there are 12 positive charge particles
Activity 2
Find −6 × 2 (Multiplication a negative and a positive)
This can be interpreted as 2 lots of six negative particles

Take away
3333 3
20
Pupils put together the particles and get 12 negative charges. Therefore −6 × 2 = −12
Activity 3
Find −6 × −2 (Multiplication a negative and negative)
This sentence cannot be interpreted as negative six lots of negative 2 or the vice versa. The
product of two negative integers can be arrived at using the intuitive method as shown below.
−6 × 4 = −24
−6 × 3 = −18
−6 × 2 = −12
−6 × 1 = −6
−6 × 0 = 0
−6 × −1 = 6
−6 × −2 = 12
We build on the fact that pupils are able to multiply a positive and a negative integer. A pattern
emerges in which each product is six more than its predecessor. Continuing this pattern yields
the conclusion that the product of two negative integers is positive.
Another method of showing this is the use of the new familiar charged-particle model.
Let us use the model to solve −6 × −2 =
Ask pupil to start with a neutral diagram
Remove six groups of two negatives
Guide pupils to interpret this to mean that we “remove” six groups of two negatives from the
diagram

21
The diagram now contains twelve isolated positive charges
Thus, as a result of this modeling technique, −6 × −2 = 12
The main ideas in this were:
The use of intuitive method to discover that the product of two negative integers is
positive
The use of charged-particle model to find the various situations of product of integers
DIVISION OF INTEGERS
Review the measurement, grouping (repeated subtraction) and partitioning (sharing) approaches
to whole number with your pupils. These were activities that have already been discussed with
the pupils in the primary school.
Activity 1: Guide the pupils to solve 12 ÷ 3
Ask pupils to form a collection of twelve positive charged particles and use sharing approach to
determine the quotient. Ask three pupils to share the twelve positive charged particles equally by
taken one at a time in turns till all are finished. Each pupil counts the number of positive particle
he has. Each pupil had four positive charge particles.
22
So 12 ÷ 3 = 4
Activity 2: Guide the pupils to solve -12÷ 3
Pupils collect twelve negative charged particles and use sharing approach to determine−12 ÷ 3.
This may be interpreted as three pupils sharing twelve negative charged particles. How many
will each get? Ask three pupils to share the twelve negative charged particles equally by taken
one at a time in turns till all are finished. Each pupil counts the number of negative particle he
has. Each pupil had four negative charge particles.
23
Therefore −12 ÷ 3 = −4
We sometimes find it difficult to interpret division types like 12 ÷ −3 and−12 ÷ −3. To solve
such problems, division should be considered as the inverse of multiplication.
12 ÷ −3 = ( ) is then written as −3 × ( ) = 12. From the experience in multiplication of
integers the ( ) gives -4 as the answer. This means 12 ÷ −3 = −4
Solving −12 ÷ −3 becomes −3 × ( ) = −12 so from previous experience 4 is the result.
Therefore −12 ÷ −3 = 4
Division involving zeros
Now let us consider0 ÷ 4 = 0, this can be written as 4 × ( ) = 0. This means that 4 × 0 = 0.
Therefore 0 ÷ 4 = 0.
Consider also a second case where a non-zero integer is being divided by zero 0 × ( ) = 4, there
is no integer that can satisfy this mathematical sentence.
So we conclude that division of any non-zero integer by zero is not possible or undefined.
Give more exercises to your pupils so that they can discover for themselves:
The quotient of two positive or two negative integers is positive.
The quotient of a positive integer and negative integer is negative

1

UNIT FOUR

THE TEACHING OF INTEGERS

MEANING OF INTEGERS AND THEIR USES IN EVERYDAY SITUATION

The classification of numbers starts from natural numbers (counting numbers), whole numbers,

integers, rational and irrational numbers and real numbers. This extension came as a result of

man’s development. The real number is not even the limit; there is still another, the complex

numbers, Natural numbers are the numbers we use in counting.

An integer is a set of numbers which consists of a set of positive whole numbers, zero and the set

of negative whole numbers.

i.e. Integers = {positive whole numbers} ⋃{0} ⋃{negative whole numbers}

The set of integers makes it possible to work with the operation of subtraction. In this unit we

will learn about how the child would be able to add, subtract, multiply and divide in the domain

of the set of integers.

Meaning of Integers (Directed Numbers)

With the knowledge of whole numbers, numbers can be matched with points on the number line

as shown below. See fig 1

Fig 1 Whole numbers on the number line.

To get the meaning of a negative number, let us consider things which can be measured in two

opposite directions.

Consider the numbers matched with points on the number line in fig 2.

Fig 2

If we want to consider the point A, 3 units from 0 and point B, also 3 units from 0: How do we

distinguish point A from point B?

To make the points clear, we would say A is 3 units right of 0 and B is the point 3 units left of 0.

0 1 2 3 4 5 6 7 8 9 10

0 1 2 3 4 5 6 7 8

B A

2

Mathematically, we write 3 units right of 0 as +3 and 3 units left of 0 as -3. With this our number

line will look like as in fig 3

The set {…..-4, -3, -2, -1, 0, 1, 2, 3, 4, ……..} is called directed numbers of integers. The subset

{1-, -2, -3,-4,………..}, of the integers is called negative integers whilst the subset {1, 2, 3, 4,

….} is the positive integers.

Activity 1

Ask two pupils to come forward and stand side by side, while one moves forward let the other

move backward. Represent the movement made by the two pupils on a number line.

The name of the collection of all the numbers on the number line above is integer.

Practical and real examples of Directed Numbers

The following are some practical examples of directed numbers which may be used to help

children get the meaning of negative numbers.

(a) Debts (-) and Credit (+)

(b) Height below (-) and above (+) Sea Level

(c) Temperature below (-) and above (+) Freezing Point.

Let us describe the real situations where negative numbers are used.

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

Fig 3 Signed numbers on the number line.

-4 -3 -2 -1 0 1 2 3 4 5 6

3

Debts and Credit: A man with no money and no debts is better than the one with no money and

who owes say ¢10. But the man with no money and no debts is worse than one with no debts but

having say ¢5. We may call the wealth of ¢5 as + 5, the debt of ¢10 as -10 and no debt as 0. This

shows that −10 < 0 < 5.

Height below and above sea level: The height of a land is usually measured from sea level. A

height of hill 100 meters is one whose top is 100 meters above sea level. This may be indicated

by + 100. If a fish is 150 meters below sea level, we could state its position as -150 above sea

level. The land above the sea is considered as being positive, the sea is the zero point and the

land below being negative.

A point at sea level may be indicated by 0. A point -50m above sea level is higher up than one at

-150m above sea level but less than the point at sea level −150 < −50 < 0 < 100.

Temperature Scale (Use of Thermometers): On a thermometer, the temperature of a freezing

water is 00C. Temperature which is colder than 00C are written with a minus sign, for example a

temperature of 50C below freezing point may be written as -5

0C and a temperature 40C above

freezing point +40C. This means −5 < 0 < 4

A weather thermometer, indicates temperatures below 0o

by negative numbers, so on the number

line numbers below or left of zero are negative numbers. The graphs of -4 and 4 are the same

distance from 0, but in opposite directions such a pair of numbers is called opposites. Thus -4 is

opposite 4 and 4 is opposite -4. Distance from zero to a number on the graph is its absolute

value.

Thus 4 and -4 have the same absolute value of 4. So |+4| and |-4| = 4

Note that there are two uses of the signs – and +. We can use them to denote operations and they

can be used to denote directions to the left or right respectively.

To differentiate the use of the signs, we write directed numbers with a bracket around or write

the sign at the top like (-2), (+4) or -2 or +

4.

The main ideas in this lesson were:

Positive and negative whole numbers constitutes the sets of integers

Some practical instances of integers are

– Land above the sea or below the sea

– I have money or owe money

– Activities of people before Christ was born and after the death of Christ (ie. BC and

AD)

– The use of the thermometer in the temperature zone.

4

Operations on Directed Numbers

Operation on directed numbers seem more difficult than that of counting numbers. Concrete

models must be used to link actions with the operations. We can employ.

(i) The idea of having or owing

(ii) The idea of movement along the number track.

(iii) The idea of movement along the number line with directed numbers (set of arrow

lengths).

This activity uses the idea that directed numbers can be seen as one dimensional vector. To

illustrate the operations of directed numbers, we mag employ concrete materials like:

a) Coloured Counters which are Cut out of Cardboards to Represent Positive and

Negative Integers

–

(i) Coloured counters (ii) Walking on the number track

Fig 5. Models of operation

The model of the coloured counters uses the idea of owing or having.

(a) Drawing Number Tracks on the Floor and Children Moving Along it. See fig 5 (ii)

In this model the operation of addition is represented by the action walk forwards,

operation of subtraction is represented by the action walk backwards.

A positive sign on the directed numbers means face towards the right and a negative sign

means face towards the left.

(b) Set of Arrow Length on the Number Line

The arrow lengths are of lengths 1 unit, 2 units, 3 units up to 10 units corresponding to

lengths of units on the number line.

Different colours must be used for positive arrow and negative arrow

-5 -4 -3 -2 -1 0 1 2 3 4 5

5

Fig 6 Arrow length

Two arrow lengths are used for an operation. For the operation of addition we lay the tail of the

second arrow at the tip of the first arrow. The answer is found at the tip of the second arrow. For

the operation of subtraction, we lay the tip of the second arrow on the tip of the first arrow and

the answer is found at the tail of the second arrow.

A positive directed arrow must point right and a negative directed arrow must point left. The

diagram in fig.7 illustrates the above information.

For addition

For subtraction

Fig 7

The use of Adding Machine

A simple adding machine consists of two cut out strips of cardboard, each a little over 20cm long

and about 2cm wide. See fig. 8

Fig 8 Adding machine

+

1

st

+

2

nd

Ans

Ans

2

nd

1

st +

–

1

st

+

2

nd

– +

+

2

nd

1

st

Ans

Ans

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

6

For the operation on integers, we use the two number strips A and B. To find the sum -4 + 6, we

find the first number -4 on the strip A. We then set the -4 on A against 0 on B. We look for the

second number 6 on B. The number on A which is directly above 6 on B will be our answer. Fig.

9 illustrates the process. The diagram indicates that 2 is the answer i.e. (-4) + 6 = 2

A

Set zero against Second number

First number on A

Fig 9 the use of adding machine.

To use the adding machine for the operation of subtraction we look at subtraction as the inverse

of addition. For example 6 – 2 can be expressed as what is x, if 2 + x = 6? This indicates addition

and we can add using our adding machine.

For 2 + x = 6, the first number is 2, the second number is x and the answer is 6. Fig.10 illustrates

the operation.

Comparing 2 + x = 6 ⇔ 6 − 2 = x

For 6 – 2 = x first number is 6, second number 2 and answer is x. Thus for the subtraction

problem of 6 − 2 , we set the second number 2 on strip A against 0 on strip B. We look at the

first number 6 on strip A. The number on B which is directly on 6 is the answer. This gives 4 as

the answer i.e. 6 – 2 = 4

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

B

First Number Sum

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

First Number

Set zero on B Answer

Fig. 10

7

Example 1.

Illustrate how you can use various materials to solve these problems

(i) (-6) + (+3) (ii) (-3) + (-5)

Solution:

(a) Using the idea of owing on

(i) (-6) + (+3)

-6 means owing say 6 cedis, +3 means having say 3 cedis

ie. (-6) + (3) implies you owe 6 cedis and you pay 3 cedis therefore your debt is left

with three cedis

Left

Paid

∴ (−6) + (+3) = −3

ii) (-3) + (-5)

-3 means owing say 3 cedis, (-5) means owing another 5 cedis. This gives a total debt of 8

cedis

-3

and

8

-5

∴ (−3) + (−5) = −8

iii) Using number track

(i) (-6) + (+3)

Here we start from -6. Face right and walk forward 3 places. We land at -3.

-8 -7 -6 -5 -4 -3 -2 -1 0 1 2

∴ (−6) + (+3) = −3

(ii) (-3) + (-5) = -8

With this problem we start from -3 face left and move forward 5 places. We will land at -8

-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3

∴ (−3) + (−5) = −8

iv) Using an arrow length

(i) (-6) + (+3)

The first arrow is -6 and the second arrow is +3

Lay the first arrow from point 0 pointing, left. The tail of the second arrow is then placed at

the tip of the first arrow. The tip of the second arrow will be on-3

-6 +3

9

∴ (−6) + (3) = −3

(iii) (-3) + (-5)

The first arrow length is -3 the second arrow length is -5

Lay the first arrow from 0 to point left. The tail of the second arrow is then placed at the tip of

the first arrow.

The tip of the second arrow length will be on -8

∴ (−3) + (−5) = −8

Example 2

Illustrate with diagrams how you will solve

(i) (-3) – (+4) (ii) (-3) – (-4)

Solution

(a) Using number truck

(i) (-3) – (+4)

-3 -5

-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4

-6 1

st

2

nd

+3

-8 -7 -6 -5 -4 -3 -2 -1 0 1 2

-5 -3

10

Start from -3 face right and walk back wards 4 steps and this will land on -7.

∴ -3 – (4) = -7

(ii) (-3) – (-4)

Start from -3 face left walk backwards 4 places and this will land on 1.

∴ (- 3) – (-4) = 1

(b) Using arrow length

(i) (-3) – (+4)

First arrow length is -3 Second arrow length is +4

Lay the first arrow length from 0 pointing to the left, the tip of the second arrow length is placed

at the tip of the first arrow and must point right. The answer is found at the tail of the second

arrow length.

-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5

-3 +4

-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3

+4 -3

11

∴ (−3) − (+4) = −7

(ii) (-3) – (-4)

First arrow length is (-3) Second arrow length is -4

Lay the first arrow length from 0 pointing to the left. The tip of the 2nd arrow must be at the tip of

the first arrow and must point left.

Our answer is at the tail of the second arrow.

∴ (-3) – (-4) = 1

Operations of Multiplication and Divisions on the Directed Numbers.

Multiplication is seen as repeated addition or groups of numbers. Division is represented as the

inverse of multiplication.

Example:

Using the number line illustrate how you will show to a child that (-2) x (4) = -8

Solution

We can illustrate (-2) x (4) as the repeated addition of (-2) + (-2) + (-2) + (-2) or four groups of

-2.

If we take it as repeated addition of (-2) then we model our problem on the number track using

the principles as (-2) x 4 = (-2) + (-2) + (-2) + (-2)

-3 -4

-6 -5 -4 -3 -2 -1 0 1 2 3 4

-3

-4 2

nd

1

st

12

We start from (-2) face left and walk forward 2 places, continue to walk forward 2 places and

then another 2 places forward.

-14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2

We land on -8 ∴ (-2) x 4 = -8

If we consider it as 4 groups of (-2), we model our problem on the number line starting from 0

and make 4 groups of (-2)

Using concrete materials to solve problems like (-2) x (-4) and (-6) ÷ (-2) is not easily

interpreted and hence may seem difficult to the children. To help children see how to solve such

problems, we may apply some pattern.

For example (-2) x (-4). We may generate and use a pattern like

First Part

Second Part

-2 x (+3) = -6

-2 x (+2) = -4

-2 x (+1) = -2

-2 x 0 = 0

-2 x -1 = 2

-2 x -2 = 4

-2 x -3 = 6

-2 x -4 = 8

-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2

13

The first part can be easily modelled on the number line but we cannot easily model the second

part on the number line. The child should be helped to see the pattern that each successive

answer is two to the right of the previous answer on the number line or is increased by 2.

For the division like (-6) ÷ (-2) the children should use division as the inverse of multiplication

ie. (-6) ÷ (-2) should be seen as (-2) x = -6, which means how many groups of -2 will be -6

From the number line we see this as 3 groups

∴ (-6) ÷ (-2) = 3

Children should eventually develop the rule that:

(-1) x (+1) = -1 (-1) ÷ (+1) = -1

(-1) x (-1) = 1 (-1) ÷ (-1) = 1

(+1) x (+1) = 1 (+1) ÷ (+1) = 1

(+1) x (-1) = -1 (1) ÷ (-1) = -1

ie. For multiplication and division if the signs are the same the answer is positive but for opposite

signs the answer is negative.

-4 -3 -2 -1 0 1 2 3

-6 -5

14

ADDITION OF INTERGERS

We have been using teaching and learning materials like the coloured counters, vector arrows

and sometimes jumps or movement on the number line drawn on the ground to teach operations

of integers. We will consider two other types of materials that are common but not familiar with

every teacher. These are the nomograph and the charged particle model.

i) Using the nomograph to do addition of integers.

Ask your pupils to draw three parallel lines that are the same distance apart in their exercise

books. The scale is 1cm to 1 unit on all the parallel line segments. The outer lines are calibrated

1 cm to 1 unit while the middle line is 1 cm to 2 units. Label the outer lines A and B, the middle

as C, where the result is located.

15

The construction of the nomopgraph is shown below

7 14 7

6 12 6

5 10 5

4 8 4

3 6 3

2 4 2

1 2 1

0 0 0

-1 -2 -1

-2 -4 -2

-3 -6 -3

-4 -8 -4

-5 -10 -5

-6 -12 -6

-7 -14 -7

-8 -16 -8

A B C

+1

16

Your pupils should select any two addends e.g. -2+3

Guide them to locate the first and second addend (-2) on the A line segment and the second

addend (+3) on the line segment B, and then using a transparent straight edge to join the two

addends. The result is then located on the line segment C, where the straight edge intersects C as

+1 shown above.

ii) Using the charged particles to add two or more given integers.

If the integers are more than two you can employ the idea of the associative properties

Pupils draw a circle of about 5 cm radius on a card, show the diameter and label the upper half

with a positive sign (+) and the lower half a negative sign (-).

Pupils then divide the card into two equal parts along the diameter.

Now let us return to the previous, “(-2) + 3”. Pupils select two negative charges and three

positive charges.

Pupils combine the negative and positive charges to form a circle, these circles form neutral

charges and are considered as zeros.

The uncombined pieces represent the sum of addition exercise. For our example, there is one

positive charge that has been isolated. Thus -2 + 3 = + 1

– .

17

SUBTRACTION OF INTEGERS

Consider the task “ 5-3”, the figure 5 is the minuend and the second (3) is the subtrahend, the

result (answer) is the difference. You can help the pupils to use charged particles.

Activity 1

Find (+5) – (+3)

Ask pupil to group five positive charged – particles, and then take away 3 of them.

Take away 3 particles Remaining 2 particles

The result is the two positive charges. So 5 – 3 = 2

Activity 2

Find (−5) – (−3)

Pupils collect five negative charges and take away three of them.

Take away 3 negative particles

Remaining 2 negative particles

18

The remaining is negative two charges so (−5) – (−3) = −2

Activity 3

Find −3 – (−5)

Ask pupils to take 3 negative charged particles. It is impossible to remove 5 negative particles

from the 3 negative particles. Pupil model 5 neutral charges and add to the three negative.

Guide the pupils to remove 5 negatives; combine the negatives and the positives to form neutral

charges. The uncombined charges give the result.

So −3 – (−5) = 2

Activity 4

Find (-5) – (+ 3)

Take away 3 positive particles from 5 negative particles. Let pupils model as follows:

Pupils model five neutral charges and five negative charges and then take away three positive

charges.

Take away

+5

-5

+2

3 neutral charges

19

Rearrange the particles

There are two neutral charges and eight isolated negative charges

Therefore (−5) – (+3) = −8

So −3 – (−5) = −3 + 5 because 5 is the opposite of −5

MULTIPLICATION OF INTEGERS

Activity 1

Find 6 × 2 (Multiplication of positive integers)

Ask your pupils to take six lots of two charged particles

Altogether there are 12 positive charge particles

Activity 2

Find −6 × 2 (Multiplication a negative and a positive)

This can be interpreted as 2 lots of six negative particles

Take away

3333 3

20

Pupils put together the particles and get 12 negative charges. Therefore −6 × 2 = −12

Activity 3

Find −6 × −2 (Multiplication a negative and negative)

This sentence cannot be interpreted as negative six lots of negative 2 or the vice versa. The

product of two negative integers can be arrived at using the intuitive method as shown below.

−6 × 4 = −24

−6 × 3 = −18

−6 × 2 = −12

−6 × 1 = −6

−6 × 0 = 0

−6 × −1 = 6

−6 × −2 = 12

We build on the fact that pupils are able to multiply a positive and a negative integer. A pattern

emerges in which each product is six more than its predecessor. Continuing this pattern yields

the conclusion that the product of two negative integers is positive.

Another method of showing this is the use of the new familiar charged-particle model.

Let us use the model to solve −6 × −2 =

Ask pupil to start with a neutral diagram

Remove six groups of two negatives

Guide pupils to interpret this to mean that we “remove” six groups of two negatives from the

diagram

21

The diagram now contains twelve isolated positive charges

Thus, as a result of this modeling technique, −6 × −2 = 12

The main ideas in this were:

The use of intuitive method to discover that the product of two negative integers is

positive

The use of charged-particle model to find the various situations of product of integers

DIVISION OF INTEGERS

Review the measurement, grouping (repeated subtraction) and partitioning (sharing) approaches

to whole number with your pupils. These were activities that have already been discussed with

the pupils in the primary school.

Activity 1: Guide the pupils to solve 12 ÷ 3

Ask pupils to form a collection of twelve positive charged particles and use sharing approach to

determine the quotient. Ask three pupils to share the twelve positive charged particles equally by

taken one at a time in turns till all are finished. Each pupil counts the number of positive particle

he has. Each pupil had four positive charge particles.

22

So 12 ÷ 3 = 4

Activity 2: Guide the pupils to solve -12÷ 3

Pupils collect twelve negative charged particles and use sharing approach to determine−12 ÷ 3.

This may be interpreted as three pupils sharing twelve negative charged particles. How many

will each get? Ask three pupils to share the twelve negative charged particles equally by taken

one at a time in turns till all are finished. Each pupil counts the number of negative particle he

has. Each pupil had four negative charge particles.

23

Therefore −12 ÷ 3 = −4

We sometimes find it difficult to interpret division types like 12 ÷ −3 and−12 ÷ −3. To solve

such problems, division should be considered as the inverse of multiplication.

12 ÷ −3 = ( ) is then written as −3 × ( ) = 12. From the experience in multiplication of

integers the ( ) gives -4 as the answer. This means 12 ÷ −3 = −4

Solving −12 ÷ −3 becomes −3 × ( ) = −12 so from previous experience 4 is the result.

Therefore −12 ÷ −3 = 4

Division involving zeros

Now let us consider0 ÷ 4 = 0, this can be written as 4 × ( ) = 0. This means that 4 × 0 = 0.

Therefore 0 ÷ 4 = 0.

Consider also a second case where a non-zero integer is being divided by zero 0 × ( ) = 4, there

is no integer that can satisfy this mathematical sentence.

So we conclude that division of any non-zero integer by zero is not possible or undefined.

Give more exercises to your pupils so that they can discover for themselves:

The quotient of two positive or two negative integers is positive.

The quotient of a positive integer and negative integer is negative

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