Unit 1: Numbers
1.1 Integers
Definition
Integers are whole numbers that can be positive, negative, or zero.
Examples: …, −3, −2, −1, 0, 1, 2, 3, …
Operations with Integers
Addition
- Same sign → add, keep the sign
- Different sign → subtract smaller from larger, keep the larger sign
−3 + (−5) = −8
7 + (−4) = 3
Subtraction
Subtracting an integer means adding its opposite.
5 − 8 = 5 + (−8) = −3
−2 − (−6) = −2 + 6 = 4
Multiplication & Division
- Same signs → positive
- Different signs → negative
−3 × −4 = 12
−12 ÷ 3 = −4
Practice Questions
- −7 + 12 = ?
- 15 − (−9) = ?
- −6 × 7 = ?
- −24 ÷ −6 = ?
1.2 Fractions
Definition
A fraction represents part of a whole and is written as a/b where b ≠ 0.
Operations with Fractions
Addition/Subtraction: Make denominators the same.
2/3 + 1/6 = 5/6
Multiplication: Multiply numerators and denominators.
2/3 × 4/5 = 8/15
Division: Multiply by the reciprocal.
2/3 ÷ 5/6 = 4/5
Practice Questions
- 3/4 + 5/8 = ?
- 7/10 − 1/5 = ?
- 2/3 × 3/7 = ?
- 4/5 ÷ 2/3 = ?
1.3 Decimals
Definition
Decimals are numbers with digits after a decimal point.
Examples: 0.5, 2.75, −0.03
- Add/Subtract: Line up decimal points
- Multiply: Ignore decimal first
- Divide: Make divisor a whole number
Practice Questions
- 3.45 + 2.7 = ?
- 5.6 − 3.25 = ?
- 0.4 × 0.3 = ?
- 2.1 ÷ 0.7 = ?
1.4 Percentages
Definition
Percentage means “per hundred”.
3/4 = 75%
0.2 = 20%
Practice Questions
- Convert 7/8 to a percentage
- Find 20% of 150
- Increase 200 by 15%
- Decrease 80 by 25%
1.5 Powers and Roots
Definition
A power is repeated multiplication. A root reverses a power.
am × an = am+n
am ÷ an = am−n
(am)n = amn
Practice Questions
- Simplify 2³ × 2⁴
- Simplify (3²)³
- Evaluate √81 and ∛64
- Solve 5ˣ = 125
1.6 Ratio, Proportion, and Rates
- Ratio: Comparison of quantities (e.g. 3:4)
- Proportion: Two ratios equal
- Rate: Quantity per unit
Practice Questions
- Simplify 12 : 16
- Solve 2/5 = x/25
- A car travels 180 km in 3 hours. Find speed.
- If 5 pencils cost $3, find the cost of 12 pencils.
Unit 2: Algebra
2.1 Expressions
Definition
An algebraic expression contains numbers, variables, and operations.
3x + 5
2a² − 7b + 4
Simplifying Expressions
3x + 5x − 2 = 8x − 2
Practice Questions
- 5x + 3 + 2x − 7
- 4a² − 2a + 7 + 3a² + a − 5
- 2m − 5 + 7m + 3
- 6x + 4y − 2x + y
2.2 Equations
Definition
An equation shows two expressions are equal.
Practice Questions
- 5x − 7 = 13
- 3(x + 4) = 18
- 2x + 5 = x + 9
- 4(x − 2) + 3 = 19
Inequalities
- x + 5 > 12
- 3x − 7 ≤ 11
- −2x + 3 < 9
- 4 − 3x ≥ 10
2.3 Sequences
IB Mathematics Notes
Unit 1: Numbers
1.1 Integers
Definition
Integers are whole numbers that can be positive, negative, or zero.
Examples
…, −3, −2, −1, 0, 1, 2, 3, …
Rules
Same sign → add, keep the sign
Different sign → subtract, keep the sign of the larger number
−3 + (−5) = −8
7 + (−4) = 3
Subtraction Rule
Subtracting an integer is the same as adding its opposite
5 − 8 = 5 + (−8) = −3
−2 − (−6) = −2 + 6 = 4
Multiplication & Division
Same signs → positive
Different signs → negative
Practice Questions
- −7 + 12 = ?
- 15 − (−9) = ?
- −6 × 7 = ?
- −24 ÷ −6 = ?
1.2 Fractions
Definition
A fraction represents part of a whole and is written as a / b, where b ≠ 0.
Addition & Subtraction
Make denominators the same, then add or subtract numerators
2/3 + 1/6 = 4/6 + 1/6 = 5/6
Multiplication
Multiply numerators and denominators
2/3 × 4/5 = 8/15
Division
Multiply by the reciprocal
2/3 ÷ 5/6 = 2/3 × 6/5 = 4/5
Practice Questions
- 3/4 + 5/8 = ?
- 7/10 − 1/5 = ?
- 2/3 × 3/7 = ?
- 4/5 ÷ 2/3 = ?
1.3 Decimals
Definition
Decimals are numbers with digits after the decimal point.
Operations
Addition/Subtraction: Line up decimal points
Multiplication: Multiply first, then place decimals
Division: Make divisor a whole number
Practice Questions
- 3.45 + 2.7 = ?
- 5.6 − 3.25 = ?
- 0.4 × 0.3 = ?
- 2.1 ÷ 0.7 = ?
1.4 Percentages
Definition
Percentage means “per hundred” (%).
3/4 = 75%
0.2 = 20%
Practice Questions
- Convert 7/8 to a percentage
- Find 20% of 150
- Increase 200 by 15%
- Decrease 80 by 25%
Unit 2: Algebra
2.1 Expressions
Definition
An algebraic expression contains numbers, variables, and operations.
3x + 5
2a² − 7b + 4
Practice Questions
- Simplify: 5x + 3 + 2x − 7
- Simplify: 4a² − 2a + 7 + 3a² + a − 5
- Expand: 2(x + 3)
- Factorise: x² + 7x + 12
Unit 3: Geometry & Trigonometry
3.1 Angles
Definition
Angles are formed when two rays meet at a point and are measured in degrees.
Practice Questions
- Find the missing angle in a triangle with angles 50° and 60°
- Two angles are supplementary. One is 110°. Find the other.
- Find x if vertically opposite angles are 3x + 10 and 2x + 20
3.4 Trigonometry
SOHCAHTOA
sin θ = opposite / hypotenuse
cos θ = adjacent / hypotenuse
tan θ = opposite / adjacent
Practice Questions
- Find hypotenuse if opposite = 6 and θ = 30°
- Find θ if opposite = 4 and adjacent = 4
- Find missing side if cos A = 0.6 and hypotenuse = 10
Unit 4: Statistics & Probability
4.2 Measures of Central Tendency
Key Measures
Mean = Sum ÷ Number of values
Median = Middle value
Mode = Most frequent value
Range = Highest − Lowest
Practice Questions
- Find mean, median, mode, range: 3, 5, 7, 5, 8
- Find median: 9, 4, 6, 8, 7
4.3 Probability
Definition
Probability measures the likelihood of an event occurring.
P(E) = favourable outcomes ÷ total outcomes
Practice Questions
- Probability of rolling an even number on a die
- Probability of drawing a King from a deck
- Bag has 3 red, 5 blue, 2 green balls. Find P(blue)
Unit 5: Functions
5.1 What is a Function?
Definition
A function is a rule that assigns exactly one output for each input.
Each input has one and only one output.
A function can be represented using:
- Table
- Equation
- Graph
- Word description
Example (Table)
x: 1, 2, 3, 4
f(x): 3, 5, 7, 9
Rule: f(x) = 2x + 1
Example (Equation)
f(x) = 2x + 3
f(0) = 3
f(1) = 5
f(2) = 7
Practice Questions
- For f(x) = 3x − 1, find f(0), f(2), f(5)
- For g(x) = x² + 2, find g(1), g(3), g(4)
- Complete the table for h(x) = 5 − x when x = 0, 1, 2, 3
5.2 Linear Functions
Definition
A linear function is a function whose graph is a straight line.
General Form
y = mx + c
m = gradient (slope)
c = y-intercept
Gradient (Slope)
m = (y₂ − y₁) / (x₂ − x₁)
- m > 0 → line goes up
- m < 0 → line goes down
- m = 0 → horizontal line
Example
Points: (1, 2) and (3, 6)
m = (6 − 2) / (3 − 1) = 2
Finding the Equation
Using y = mx + c
2 = 2(1) + c → c = 0
Equation: y = 2x
Practice Questions
- Find the slope of the line through (2,3) and (5,9)
- Write the equation of the line passing through (1,4) with slope 3
- Find the y-intercept of y = 2x + 5
- Are the points (0,1), (1,3), (2,5) collinear?
Unit 5: Practice Questions
A. Evaluating Functions
- f(x) = 2x + 3 → find f(0), f(1), f(5), f(−2)
- g(x) = x² − 4x + 5 → find g(2), g(3), g(0), g(−1)
- h(x) = 5 − 3x → find h(0), h(1), h(4), h(−2)
- k(x) = 2x² + 3x − 1 → find k(2), k(−1), k(0)
- f(x) = 4x + 7, find x if f(x) = 23
- g(x) = x² + 6x + 8, solve g(x) = 0
B. Linear Functions – Gradient & Equation
Practice Questions
- Find the slope of the line through the points (1, 2) and (4, 8).
- Find the equation of the line with slope 2 passing through the point (2, 3).
- Find the equation of the line with slope −3 passing through the point (−1, 4).
- Determine whether the points (0, 1), (2, 5), and (4, 9) are collinear.
- Graph the line: y = 2x + 1.
- Graph the line: y = −x + 4.
C. Solving Graphically
Practice Questions
- Solve graphically: y = 2x + 1 and y = −x + 4.
- Solve graphically: y = x − 2 and y = 3x + 1.
- Solve graphically: y = 0.5x + 1 and y = −x + 4.
- Solve graphically: y = 2x and y = x + 3.
- Find the intersection point of y = x + 2 and y = −x + 6.
-
Two taxi companies charge different fares:
- Taxi A charges a base fare of $5 plus $2 per km.
- Taxi B charges a base fare of $3 plus $3 per km.
Solve graphically to find the distance at which both taxis cost the same.