Grade 8 Mathematics Study Notes

Strand 1.0 — Numbers

In Grade 8, Strand 1.0 covers five types of numbers: integers, fractions, decimals, squares and square roots, and rates, ratios, proportions and percentages.

Identification of integers

Integers are positive whole numbers, negative whole numbers, and zero.

Examples of integers: −3, +4, +6, −13, +20, −50.

If a number has a decimal part or a fraction part, it is not an integer. For example: 3.1, 0.32, 15½ and 7¼ are not integers.

Note: If a number has zero as its decimal part — such as 4.0, 6.0, 3.00 — then it is an integer. For example, 14.0 can also be written as 14.

Positive integers can be written without the + sign: +200 = 200.

Example 1 — Classify as integers or not integers

a) 8   b) 5.5   c) −20   d) 0.6   e) −43   f) 15¼   g) −⅓   h) 14.0

Solution

Integers: 8, −20, 14.0 — they are whole numbers (14.0 = 14).

Not integers: 5.5, 0.6, 15¼, −⅓ — they have a decimal or fraction part.

Integers in real-life situations

Integers describe situations where things go in two opposite directions from a fixed starting point. That starting point is called the reference point and is given the value zero.

Example 2 — Identify the integers for these situations

a) Five metres below the ground.   b) Two hundred metres above the sea level.

Solution

a) Since the ground is the reference point, below the ground is taken to be negative and above the ground is positive. Therefore five metres below the ground = −5.

b) Sea level is the reference point. Two hundred metres above sea level = +200. (+200 can also be written as 200.)

More examples

• Seven floors above the ground → +7
• Three floors below the ground → −3
• Twenty degrees below zero → −20°C
• Seventy-five degrees above zero → +75°C

Key Points — Identification of integers

• Integers are positive whole numbers, negative whole numbers and zero.
• A number with a decimal or fraction part is not an integer.
• If the decimal part is zero (e.g. 14.0), it is an integer.
• Positive integers can be written with or without the + sign: +200 = 200.
• The reference point is always zero. Above/in front = positive. Below/behind = negative.

Representation of integers on a number line

A number line is a straight line with numbers placed at equal intervals along its length.

• Zero is the starting point.
• All numbers to the right of zero are positive.
• All numbers to the left of zero are negative.

How to represent integers on a number line

1. Draw a straight line with arrows at both ends.
2. Mark zero at the centre.
3. Mark equal intervals to the right and write positive integers.
4. Mark equal intervals to the left and write negative integers.

Example 3 — Represent the integers from −5 to 5

Example 4 — Represent the integers from −40 to 40 (scale: 1 division = 10 units)

Example 5 — Represent numbers from 41 to 50

This range does not include zero, so zero is not shown.

Key Points — Representation on a number line

• A number line is a straight line with numbers at equal intervals.
• Zero is the starting point; positive integers go right, negative integers go left.
• For a large range, use a scale (e.g. 1 division = 10 units).
• A number line does not have to include zero — draw only the section you need.

Addition of integers on a number line

When carrying out addition of integers using a number line, we start from the first number given. We then move to the right a number of steps equal to the number to be added.

Worked examples — addition of integers

Example 6 — Use a number line to evaluate 4 + 6

Start at 4. Move 6 steps to the right. You land at 10.

∴ 4 + 6 = 10

Example 7 — Use a number line to solve −6 + 4

Start at −6. Move 4 steps to the right. You land at −2.

∴ −6 + 4 = −2

Real-life examples

Tyre pressure: Gauge reads 25 units. Pressure increased by 10 units. Start at 25, move 10 right. Final reading = 35 units.

Journey time: Mwende starts at 7:00 a.m. and walks 4 hours. Start at 7, move 4 right. She arrives at 11:00 a.m.

Key Points — Addition of integers on a number line

• Start from the first number given.
• Move to the right by the number of steps equal to the number to be added.
• The number where you land is the answer.

Subtraction of integers on a number line

When subtracting integers using the number line, we start from the first number given then move steps equal to the number to be subtracted to the left.

Worked examples — subtraction of integers

Example 8 — Use a number line to work out 3 − 6

Start at 3. Move 6 steps to the left. The jump crosses zero. You land at −3.

∴ 3 − 6 = −3

Example 9 — Use a number line to solve −2 − 5

Start at −2. Move 5 steps to the left. You land at −7.

∴ −2 − 5 = −7

Real-life examples

Temperature: Water placed in a fridge. Temperature dropped by 25°C from 20°C. Start at 20, move 25 left. Final temperature = −5°C.

Rice store: Store had 150 bags. 5 bags used each week for 10 weeks. 5 × 10 = 50 bags used. Start at 150, move 50 left. 100 bags remain.

Key Points — Subtraction of integers on a number line

• Start from the first number given.
• Move to the left the same number of steps as the number to be subtracted.
• The number where you land is the answer.
• If the jump passes zero, the answer becomes negative.

Combined operations on a number line

A combined operation has both addition and subtraction in the same problem. We carry out each operation in order from left to right, starting each new step from where we stopped in the previous one.

Worked examples — combined operations

Example A — Evaluate −6 + 4 − 3

Step 1: Start at −6. Move 4 steps right (+4). Land at −2.

Step 2: From −2, move 3 steps left (−3). Land at −5.

∴ −6 + 4 − 3 = −5

Example B — Temperature change

Temperature is −3°C. It rises 8°C, then falls 5°C.

Step 1: Start at −3. Move 8 right. Land at +5.

Step 2: From +5, move 5 left. Land at 0.

∴ Final temperature = 0°C

Example C — Floors in a building

A lift is at floor −3. It goes up 7 floors, then comes down 2 floors.

Step 1: Start at −3. Move 7 right. Land at +4.

Step 2: From +4, move 2 left. Land at +2.

∴ Lift stops at floor +2.

Key Points — Combined operations on a number line

• Work from left to right, one operation at a time.
• Start each new step from where the previous step ended.
• + means move right; − means move left.
• The number where you land after the last step is the answer.