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Learning Resources

Integers

Interactive curriculum lessons, worked examples, and geometric problem-solving techniques designed to help Kenyan students master CBC, KPSEA, KCSE, and IGCSE mathematics.

Grade 09 Pathway: N/A

First Principles

Objective: Master Multi-Step Operations on Directed Numbers (Integers)

In daily life across Kenya, numbers go above and below zero: from the cool peaks of Mount Kenya at \(-4^\circ\text{C}\) to sea level at Mombasa (\(0\text{ m}\)), and from an M-Pesa account balance of \(+1,500\text{ Ksh}\) to an overdraft (Fuliza) debt of \(-450\text{ Ksh}\). Integers are the set of whole numbers, their negatives, and zero: \[\mathbb{Z} = \{ \dots, -3, -2, -1, 0, 1, 2, 3, \dots \}\]

Interactive Integer Number Line Hopper 🐸

Enter an expression or pick a preset to visualise directional hops on the integer line.

1. Vector & Directional Meaning

Every integer contains two pieces of information: magnitude (distance from 0) and direction (positive to the right, negative to the left). Subtracting a negative is equivalent to reversing direction twice—which results in moving forward: \[a - (-b) = a + b\]

2. Multiplication & Division of Signed Quantities

When multiplying or dividing signed numbers:

  • Like Signs produce a Positive outcome: \((-) \times (-) = (+)\) and \((+) \times (+) = (+)\).
  • Unlike Signs produce a Negative outcome: \((+) \times (-) = (-)\) and \((-) \times (+) = (-)\).

3. Order of Operations (BODMAS / PEMDAS)

In multi-step problems, operations must be strictly evaluated in this order:
Brackets \(\to\) Orders/Powers \(\to\) Division & Multiplication (left to right) \(\to\) Addition & Subtraction (left to right).

Key Formulas

\[ a + b \] Addition of Integers:
• If both signs are identical: Add magnitudes, preserve common sign.
• If signs differ: Subtract the smaller magnitude from the larger magnitude, keep sign of the number with larger magnitude.
\[ a - b = a + (-b) \] Subtraction Rule:
Subtracting a number is identical to adding its additive inverse (opposite). For example, \(5 - (-3) = 5 + 3 = 8\).
\[ (+a) \times (+b) = +(ab) \quad \text{and} \quad (-a) \times (-b) = +(ab) \] Multiplication (Like Signs):
Multiplying two integers with the same sign always produces a positive integer.
\[ (+a) \times (-b) = -(ab) \quad \text{and} \quad (-a) \times (+b) = -(ab) \] Multiplication (Unlike Signs):
Multiplying two integers with opposite signs always yields a negative integer.
\[ \frac{a}{b} \quad (b \neq 0) \] Division Rules:
• \(\frac{-a}{-b} = +\left(\frac{a}{b}\right)\) (Same sign: positive)
• \(\frac{-a}{b} = -\left(\frac{a}{b}\right)\) (Opposite signs: negative)
\[ \text{BODMAS Hierarchy: } \text{Brackets} \to \text{Orders} \to \{\text{Division / Multiplication}\} \to \{\text{Addition / Subtraction}\} \] For equal precedence operations (such as \(\times\) and \(\div\) or \(+\) and \(-\)), work strictly from left to right.

Worked Examples

Level 1 (Easy): Basic Addition and Subtraction
Evaluate: \(-7 + 12 - 9\)
  1. Step 1: Work from left to right. Evaluate \(-7 + 12\). Since signs are opposite, subtract magnitudes: \(12 - 7 = 5\). The larger magnitude is positive, so the intermediate value is \(+5\).
  2. Step 2: Evaluate \(5 - 9\). Subtracting \(9\) from \(5\) gives \(5 + (-9) = -4\).

Final Answer: \(-4\)

Level 2 (Medium): Mixed Operations with Order of Precedence
Evaluate: \(18 - 4 \times (-3) + (-20) \div 5\)
  1. Step 1: Identify high-priority operations (multiplication and division) before addition/subtraction.
  2. Step 2: Perform multiplication: \(4 \times (-3) = -12\). So \(18 - (-12) = 18 + 12 = 30\).
  3. Step 3: Perform division: \((-20) \div 5 = -4\).
  4. Step 4: Combine results: \(30 + (-4) = 30 - 4 = 26\).

Final Answer: \(26\)

Level 3 (Hard): Multi-Step Nested Brackets and Mixed Signs
Evaluate: \(-3 \left[ 5 - 2(-4 + 1) \right] + \frac{-36}{-4 + (-2)}\)
  1. Step 1: Resolve inner bracket: \(-4 + 1 = -3\).
  2. Step 2: Resolve brackets: \(5 - 2(-3) = 5 - (-6) = 5 + 6 = 11\).
  3. Step 3: Multiply outside term: \(-3 \times 11 = -33\).
  4. Step 4: Resolve denominator of the division fraction: \(-4 + (-2) = -6\).
  5. Step 5: Compute division: \(\frac{-36}{-6} = +6\).
  6. Step 6: Complete addition: \(-33 + 6 = -27\).

Final Answer: \(-27\)

Common Mistakes

Mistake 1: Left-to-Right Blindness (Ignoring BODMAS)
Incorrect calculation: In \(8 - 2 \times 3\), writing \((8 - 2) \times 3 = 6 \times 3 = 18\).
Correction Multiplication takes precedence over subtraction: \(8 - (2 \times 3) = 8 - 6 = 2\).
Why it feels right Natural language reads strictly left-to-right, but arithmetic expressions require operator hierarchy.
Mistake 2: "Two Negatives Always Make a Positive" Overgeneralization
Incorrect calculation: In \(-4 + (-6)\), writing \(+10\) because "two negatives make a positive".
Correction The rule "two negatives make a positive" applies only to multiplication and division \((-4 \times -6 = 24)\) or double negation \(-(-6) = +6\). Adding two debts simply creates a larger debt: \(-4 + (-6) = -10\).
Mistake 3: Dropping Negative Signs across Brackets
Incorrect calculation: \(-(3 - 7) = -3 - 7 = -10\).
Correction The negative sign distributes over every term in the bracket: \(-(3 - 7) = -( -4 ) = +4\), or \(-3 - (-7) = -3 + 7 = 4\).

Real World

Opening balance: \(+2,400\text{ Ksh}\)
Bulk maize purchase: \(-6,000\text{ Ksh}\) (Current: \(2400 - 6000 = -3600\text{ Ksh}\))
Customer payment via Lipa na M-Pesa: \(+5,100\text{ Ksh}\) (Current: \(-3600 + 5100 = +1500\text{ Ksh}\))
Transport and county council cess: \(-850\text{ Ksh}\)
Closing balance: \(1500 - 850 = +650\text{ Ksh}\).
Match 1 (Win 3 – 1): \(+2\)
Match 2 (Loss 0 – 3): \(-3\)
Match 3 (Loss 1 – 2): \(-1\)
Match 4 (Win 4 – 0): \(+4\)
Net Goal Difference: \((+2) + (-3) + (-1) + (+4) = +2\).

Practice

A shopkeeper owes Ksh 400 to a wholesale supplier and Ksh 150 to a transport driver. He pays Ksh 200 towards the supplier's debt. What is his total remaining net debt in Ksh? (Type only the number, e.g., 350)
Review the concepts above.
A geography student starts an excursion at 250 metres above sea level and descends 430 metres down the escarpment into the Great Rift Valley. What is the student's final elevation in metres relative to sea level? (Type only the integer, e.g., -180)
Review the concepts above.
An agricultural lorry removes crop waste from a market, reducing the waste dump by 3 tonnes per trip (represented as -3). What is the total change in the dump's weight after 4 identical trips? (Type only the integer, e.g., -12)
Review the concepts above.
A farmer has an M-Pesa overdraft loan of -1,500 shillings. If she makes equal repayments of 300 shillings per week, how many weeks will it take her to clear the debt completely to 0 shillings? (Type only the number, e.g., 5)
Review the concepts above.
Evaluate the following multi-step integer expression: \[ -5 \times (4 - 12) + (-24) \div 6 - 15 \] (Type only the integer, e.g., 21)
Review the concepts above.
A research submarine in the Indian Ocean near Mombasa starts at an elevation of -120 metres. It ascends at a rate of 15 metres per minute for 6 minutes, and then descends at a rate of 22 metres per minute for 4 minutes. What is the submarine's final elevation in metres? (Type only the integer, e.g., -118)
Review the concepts above.