◄UNIT 1: Reading, writing, comparing and calculating whole numbers up to 1 000 000UNIT 3:Prime factorisation and divisibility tests►

UNIT 2:Addition and subtraction of integers

  • UNIT 2:Addition and subtraction of integers

    2.1 Location of positive and negative numbers on a number line

    Let us do the activity below.

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    Tip:
    When a number is positive it is located on the right side of 0. A negative

    number is located on left side of 0.

    Practice Activity 2.1

    1. Make these number cards.

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    Locate them on the number line below.

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    2. Look at the number lines below. Write the integers represented by the

    letters

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    2.2 Comparison and ordering of integers

    (a) Comparing integers using a number line

    Activity 2.3
    Draw a number line. Use it to compare the following. Tell the integer
    that is greater. Tell the integer that is smaller. Explain your answer.
    (i) –3 and +2                (ii) –4 and +4
    (iii) –5 and –2              (iv) +1 and +6

    Tip:
    Integers on the right side of 0 are greater than those on the left. Positive

    numbers are greater than negative numbers.

    Example 2.2
    Use a number line to compare –5 and +5.

    Solution

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    +5 is greater than –5

    A number to the right is greater than a number to the left on a number line.

    Practice Activity 2.2
    1. Study the number lines below. Tell which integer is greater in each
    number line.
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    2. Use a number line to compare the following. Which one is greater?
    (a) +2 and –2     (b) +4 and –4         (c) –1 and +5

    (d) +1 and +5     (e) 0 and –6            (f) 0 and +9

    (b) Ordering integers and comparing integers using <, > or =

    Activity 2.4
    • Draw a number line. Have written paper cutouts for –10, –9, –8, –7,
    –5, –4, –3, 0, +3, +4, +5, +6, +7, +8, +9, +10. Fix them on the number line.
    • Use the integers position to arrange –10, –7, 0, +7, +2 from;
    (i) Smallest to largest
    (ii) Largest to smallest
    (iii) Use < or > or = to compare

    (a) –10 ___ –7         (b) +2 ___ –7

    Tip: We use > for ‘greater than’, < for ‘less than’ and = for ‘equal to’.
    Ordering numbers can mean to arrange numbers from the smallest to
    the largest. It can also means to arrange numbers from the largest to the
    smallest. Arranging/ordering numbers from the smallest to the largest is
    called ascending order. For example +4, +5, +6, +7.
    Ordering/arranging numbers from the largest to the smallest is called

    descending order. For example; +4, +3, +2, +1, 0.

    Look at the example below.

    Example 2.3
    1. Use > or < or = to compare the integers given below.
    (a) +3 _____ +1     (b) –6 _____ +2        (c) +5 _____ 5
    2. (a) Arrange in ascending order: +3, –4, 0, +6

    (b) Arrange in descending order: +6, +1, –1, –3, +2

    Solution
    1. (a) +3 > 1. On a number line 3 is farther to the right side than 1 from
    0.
    (b) –6 < +2. On a number line –6 is to the left side while +2 is to the
    right side of 0.
    (c) +5 = 5. It is the same point on number line.
    2. (a) In ascending order, start from the smallest to the largest:
    –4, 0, +3, +6
    (b) In descending order, start from the largest to the smallest:

    +6, +2, +1, –1, –3

    Practice Activity 2.3
    1. Use >, < or = to compare the integers given below.
    (a) –10 ____ +3       (b) –15 ____ 0         (c) +3 ____ +3
    (d) –6 ____ +4         (e) +6 ____ 0           (f) +4 ____ –2
    2. Use >, < or = to compare the integers below. Discuss your answers.
    (a) –5 ____ +1         (b) +7 ____ +9         (c) 0 ____ +8
    (d) +10 ____ –6       (e) –11 ____ +6        (f) +11 ____ +11
    3. Arrange the integers in ascending order
    (a) –3, +4, –5, –1     (b) +20, –15, +4, –11   (c) –22, –11, –20, +11
    4. Arrange the integers in descending order.

    (a) +1, –2, –8, +9      (b) +10, –10, +24, –5    (c) –3, –5, –1, –8

    2.3 Addition of integers
    (a) Addition of integers using a number line

    Activity 2.5
    • You need the following materials: white powder or dry loose soil,
    tape measure, manila paper.

    • Make a number line from –4 to +6 on the field.

    Steps:

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    Now use it to work out (–3) + (+4). Follow these steps.
    (i) Stand at –3. Move 4 steps to the right. Where do you stop? That
    is the answer to (–3) + (+4).
    (ii) Repeat this for (+4) + (–3). What do you get?
    • On a paper, draw the number line you made. Discuss the steps you

    followed to find your answer.

    Example 2.4
    On a number line, work out (–7) + (+10)

    Solution
    Stand at –7. Move 10 steps to the right.

    Where do you stop?

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    Practice Activity 2.4

    Using a number line, work out:
    1. (–1) + (+3) 2. (–9) + (+4) 3. (–10) + (+5)
    4. (–2) + (–3) 5. (+3) + (–4) 6. (+10) + (–7)
    Use a number line to add the following integers. Explain your answer.
    7. (–6) + (–2) 8. (+8) + (–2) 9. (–15) + (–12)

    10. (–13) + (–1)

    (b) Addition of integers without using a number line

    Activity 2.6

    1. Work out the following without using a number line.
    (a) (–3) + (+4)     (b) (+4) + (–3)     (c) (–3) + (–4)     (d) (+3) + (+4)
    2. Try your addition without using a number line.
    (a) (–3) + (+4)     (b) (+3) + (–4)     (c) (–3) + (–4)      (d) (+3) + (+4)
    3. From your working in number 1 and 2, which method was easier?

    Discuss your steps in each case.

    Tip:
    (i) When adding numbers with the same sign, the answer takes that sign.
    For example,
    (–3) + (–4) = –(3 + 4) = –7
    (+3) + (+4) = +(3 + 4) = +7
    (ii) When adding numbers with different signs, the answer takes the sign
    of the larger number.
    For example, (–3) + (+4) = +(4 – 3) = +1

                          (+3) + (–4) = –(4 – 3) = –1

    Example 2.5
    Work out:
    (–8) + (–6)
    Solution

    (–8) + (–6) = –(8 + 6) = –14

    Practice Activity 2.5
    Work out the following.
    1. (–8) + (+5) 2. (–6) + (+2) 3. (–20) + (+16)
    4. (–2) + (+10) 5. (–3) + (–3) 6. (–12) + (+6)
    Work out the following. Explain your steps.
    7. (–11) + (+10) 8. (+4) + (+4) 9. (–12) + (+1)
    10. (–9) + (+4)

    2.4 Subtraction of integers

    (a) Subtraction of integers using a number line

    Activity 2.7
    You need the following materials: white powder or dry loose soil, tape
    measure, manila paper.

    Make a number line from –5 to +5 on the field.

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    Use your number line to subtract: (a) (–1) – (+3)       (b) (–1) – (–3)

    Follow these steps:
    • For (–1) – (+3); start at –1 move 3 steps backwards (to the left), where
    do you stop?
    • For (–1) – (–3); start at –1, move 3 steps backward of backward.
    Backward of backward results in forward movement (to the right).

    Where do you stop? Explain your answer.

    Example 2.6

    Using a number line, work out (–1) – (+5).

    Solution

    Stand at –1. Move 5 steps to the left. Where do you stop?

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    Practice Activity 2.6
    Work out the following using a number line.
    1. (+8) – (+3)        2. (–6) – (+2)        3. (–8) – (+3)
    4. (+7) – (+9)        5. (+7) – (+3)        6. (+9) – (+9)
    Use number line to subtract the following. Explain the steps followed.
    7. (–4) – (+1)        8. (+2) – (+8)        9. (+4) – (+4)
    10. (+13) – (+10)
    (b) Subtraction of integers without using a number line
    Activity 2.8
    Work out the following without using a number line.
    1. (–1) – (+3)                          2. (–1) – (–3)
    Follow these steps:
    1. (–1) – (+3) = –(3 + 1)          2. (–1) – (–3) = (–1) + (+3) = 3 – 1

    Explain your steps.

    Tip:
    Since backward of backward results in forward movement, then
    (–1) – (–3) = (–1) + (+3) = (+3) – (+1)

    Example 2.7
    Work out:
    (a) 6 – 3     (b) (–6) – (+3)    (c) (+6) – (–3)      (d) (–6) – (–3)

    Solution
    We work out as shown below:
    (a) 6 – 3 = 3
    (b) (–6) – (+3) = –(6 + 3) = –9
    (c) (+6) – (–3) = 6 + 3 = 9
    (d) Recall that – –3 is replaced by +3 (Backward of backward is forward)

    Thus (–6) – (–3) = –6 + 3 = –3

    Practice Activity 2.7
    Work out the following.
    1. (+12) – (+10)        2. (–15) – (+5)        3. (–7) – (–3)
    4. (–8) – (+2)            5. (+16) – (–3)        6. (–11) – (+6)
    Work out the following. Discuss your answers.
    7. (+7) – (+13)         8. (–10) – (–8)         9. (+9) – (–4)

    10. (–6) – (–13)      11. (–4) – (–12)        12. (–2) – (–3)

    2.5 Additive inverses of numbers
    Activity 2.9
    Add the following integers.
    1. (–4) + (+4) =         2. (–5) + (+5) =        3. (–6) + (+6) =
    4. (–7) + (+7) =        5. (–8) + (+8) =         6. (–9) + (+9) =

    What do you notice when you add?
    State five other positive and negative integers. State their additive

    inverse. Write them on flash cards. Present your findings.

    Tip: For every integer, there is another integer such that the sum of
    the two integers is zero. The pair of integers whose sum is zero are

    additive inverses.

    Example 2.8

    (a) Work out the following.
    (i) (–2) + (+2) =            (ii) (–4) + (+4) =
    (b) Find the additive inverse of: (i) –9        (ii) +6

    Solution
    (a) (i) (–2) + (+2) = 0
    (ii) (–4) + (+4) = 0
    (b) Additive inverse of (+a) is (–a), so that (+a) + (–a) = 0. So we have:
    (i) additive inverse of –9 is +9.

    (ii) additive inverse of +6 is –6.

    Practice Activity 2.8
    Write the additive inverse of the following.
    1. –3          2. –10        3. –11
    4. –14        5. –15        6. +4
    Find additive inverses of the following. Explain your answer.
    7. +6          8. +8          9. +10
    10. +12      11. –7        12. –8

    13. +9        14. +8       15. +15

    2.6 Solving problems involving addition and

    subtraction of integers

    Activity 2.10
    Read the instructions and solve the puzzle. Use the distance between
    integers to find the position.
    I am 7 steps from +2. I am greater than 2.

    Draw a number line to show integers.

    7 steps from +2 is either –5 or +9.

    I am greater than +2. Give the answer now.

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    Justify your answer.
    Example 2.9
    I am 8 steps from –1. I am greater than +6. Where am I?
    Solution

    Draw a number line.

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    I am either at –9 or +7. But I am greater than +6. We know –9 < +6 and

    +7 > +6. Therefore, I am +7.

    Practice Activity 2.9
    1. I am a positive number. Exactly 11 steps from 0. What am I?
    2. Mary is 17 steps from +7. She is next to –9. Where is she?
    3. The temperature of town A is +15 °C at noon. In the evening its
    temperature is +6 °C. What is the difference in the temperature?
    4. I am a negative number. Exactly 9 steps from –1. Where am I?
    5. A positive number is 8 steps away from –3. It is greater than 4. What
    is the number?
    6. A number is 15 steps from +10. The number is less than –4. What is the

    number?

    Revision Activity 2

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    Word list
    Integer         Positive integer         Negative integer
    Locate         Steps                         Compare
    Order           Arrange                     Distance
    Additive inverses

    Task
    Do the following.
    (i) Read each word aloud to your friend.
    (ii) Write the meaning of each of the words above. Discuss with your friend.

    (iii) Write sentences using each of the words above. Read with your friend.

    ◄UNIT 1: Reading, writing, comparing and calculating whole numbers up to 1 000 000UNIT 3:Prime factorisation and divisibility tests►