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Choose a topic below to begin reading, revise worked examples and practise examination-style questions.

Number & Numeration Algebra Geometry & Mensuration Trigonometry Statistics Probability Coordinate Geometry & Graphs Functions & Graphs Vectors & Transformations Calculus

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Number & Numeration

MATHEMATICS HIP-HOP

NUMBER & NUMERATION

Mathematics Notes • Learn it. Practice it. Master it.

Learning Objectives

• Identify and classify different types of numbers.
• Perform operations with integers, fractions and decimals.
• Use factors and multiples.
• Find HCF and LCM.

Convert between fractions, decimals and percentages.

Solve problems involving ratio and proportion.

1. Number System

Natural numbers: 1, 2, 3, 4, 5, …

Whole numbers: 0, 1, 2, 3, 4, …

Integers: …, −3, −2, −1, 0, 1, 2, 3, …

Rational numbers can be written as p/q, where q ≠ 0.

Irrational numbers cannot be expressed as a ratio of two integers, e.g. √2 and π.

Real numbers consist of rational and irrational numbers.

2. Place Value

In 47,583: 4 is ten-thousands, 7 thousands, 5 hundreds, 8 tens and 3 units.

47,583 = 40,000 + 7,000 + 500 + 80 + 3.
Remember: the position of a digit determines its value.

3. Factors and Prime Numbers

A factor divides a number exactly with no remainder.

Factors of 12 are 1, 2, 3, 4, 6 and 12.

A prime number has exactly two positive factors: 1 and itself.

2 is the only even prime number.

4. Prime Factorisation

Prime factorisation expresses a number as a product of prime numbers.

Example: 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5.

5. HCF

HCF means Highest Common Factor.

Example: HCF(18, 24) = 6.

6. LCM

LCM means Lowest Common Multiple.

Example: LCM(6, 8) = 24.
Exam Alert: HCF → factors; LCM → multiples.

7. Fractions

A fraction has a numerator and denominator.

Example: 3/5 has numerator 3 and denominator 5.

Equivalent fractions have the same value, e.g. 1/2 = 2/4 = 3/6.

8. Adding Fractions

Example: 1/3 + 1/6 = 2/6 + 1/6 = 3/6 = 1/2.

9. Multiplying Fractions

Example: 2/3 × 4/5 = 8/15.

10. Decimals

Decimals represent parts of a whole using place values based on powers of ten.

Example: 0.375 = 375/1000 = 3/8.

11. Percentages

A percentage means per hundred: x% = x/100.

Example: 20% of 150 = (20/100) × 150 = 30.

12. Ratio

A ratio compares quantities.

Example: 20:30 simplifies to 2:3.

13. Proportion

Two ratios are proportional when they represent the same relationship.

Example: 2:3 = 4:6.
Example: if 3 books cost UGX 15,000, then 5 books cost UGX 25,000 at the same rate.

📝 Practice Zone

1. Find the prime factorisation of 84.

2. Find the HCF of 36 and 48.

3. Find the LCM of 12 and 18.

4. Calculate 3/4 + 1/8.

5. Calculate 5/6 × 9/10.

6. Convert 0.625 into a fraction in simplest form.

7. Find 15% of 240.

8. Simplify 24:36.

9. A school has 240 students. If 35% are girls, calculate the number of girls.

10. An item is bought for UGX 80,000 and sold for UGX 100,000. Find the profit and profit percentage.

⚡ Quick Revision

• HCF = Highest Common Factor.
• LCM = Lowest Common Multiple.
• Prime number = exactly two positive factors.
• Percentage = per hundred.

Ratio = comparison of quantities.

Rational number = can be written as p/q.

Integer = positive, negative or zero whole number.

🎤 Maths Hip-Hop Tip

“Don't just memorize the formula — understand what the formula is doing”

Study pattern: simple explanation → worked example → practice → examination application.

MATHEMATICS HIP-HOP
We Educate the World • Learn it. Practice it. Master it.

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Algebra

MATHEMATICS HIP-HOP

ALGEBRA

Mathematics Notes • Learn it. Practice it. Master it.

Learning Objectives

• Understand algebraic terms, expressions and coefficients.
• Simplify algebraic expressions.
• Expand and factorise algebraic expressions.
• Solve linear equations.
• Solve simultaneous equations.
• Solve and interpret quadratic equations.
• Work with algebraic inequalities.
• Apply algebra to examination and real-life problems.

1. Introduction to Algebra

Algebra uses letters and symbols to represent numbers and quantities. A letter whose value can change is called a variable.

Example: In 3x + 5, x is the variable, 3 is the coefficient of x, and 5 is the constant.

2. Algebraic Terms and Expressions

A term is a part of an expression separated by a plus or minus sign.

Example: In 4x + 3y − 7, the terms are 4x, 3y and −7.

Like terms have the same variables raised to the same powers. Only like terms can be combined by addition or subtraction.

3. Simplifying Algebraic Expressions

Collect like terms and perform the required operations.

Example:
3x + 5x − 2 = 8x − 2

4a + 3b − a + 2b = 3a + 5b

4. Removing Brackets

Use the distributive law to multiply the term outside a bracket by every term inside.

Examples:
3(x + 4) = 3x + 12
2(3x − 5) = 6x − 10

5. Expansion

Expansion means removing brackets by multiplication.

Example:
(x + 3)(x + 2)
= x² + 2x + 3x + 6
= x² + 5x + 6

6. Factorisation

Factorisation is the reverse of expansion. It expresses an algebraic expression as a product of factors.

Example:
6x + 12 = 6(x + 2)

Remember: Look for the highest common factor before factorising.

7. Linear Equations

A linear equation contains a variable whose highest power is 1. The aim is to find the value of the unknown.

Example:
2x + 5 = 17
2x = 12
x = 6

8. Equations with Brackets

Example:
3(x + 2) = 18
3x + 6 = 18
3x = 12
x = 4

9. Simultaneous Equations

Simultaneous equations are equations that are solved together to find two or more unknowns.

Example:
x + y = 10
x − y = 2

Adding the equations gives:
2x = 12
x = 6
Substitute into x + y = 10:
6 + y = 10
y = 4

10. Quadratic Equations

A quadratic equation has the general form:

ax² + bx + c = 0, where a ≠ 0

Quadratic equations may be solved by factorisation, completing the square, or the quadratic formula.

Example by factorisation:
x² + 5x + 6 = 0
(x + 2)(x + 3) = 0
Therefore, x = −2 or x = −3.

11. Quadratic Formula

x = (−b ± √(b² − 4ac)) / 2a

For an equation in the form ax² + bx + c = 0, identify a, b and c carefully before substituting into the formula.

12. Algebraic Inequalities

Inequalities compare quantities using symbols such as <, >, ≤ and ≥.

Example:
2x + 3 > 9
2x > 6
x > 3

Exam Alert: When multiplying or dividing an inequality by a negative number, reverse the inequality sign.

📝 Practice Zone

1. Simplify: 5x + 3x − 4.
2. Simplify: 7a + 2b − 3a + 5b.
3. Expand: 4(x + 6).
4. Expand: (x + 4)(x + 2).
5. Factorise: 8x + 16.
6. Solve: 3x + 7 = 22.
7. Solve: 5(x − 2) = 20.
8. Solve simultaneously: x + y = 12 and x − y = 4.
9. Solve: x² + 7x + 12 = 0.
10. Solve the inequality: 3x − 5 ≥ 10.

⚡ Quick Revision

• Variable: a letter representing an unknown or changing value.
• Coefficient: the numerical factor multiplying a variable.
• Like terms: terms with the same variables and powers.
• Expansion: removing brackets by multiplication.
• Factorisation: expressing an expression as a product of factors.
• Linear equation: an equation with highest variable power 1.
• Quadratic equation: an equation with highest variable power 2.

🎤 Maths Hip-Hop Tip

“In algebra, keep your steps clear. One correct step leads to another.”

Study pattern: Understand → Work an example → Practise → Check your answer → Apply in an exam

MATHEMATICS HIP-HOP
We Educate the World • Learn it. Practice it. Master it.

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Geometry & Mensuration

MATHEMATICS HIP-HOP

GEOMETRY & MENSURATION

Mathematics Notes • Learn it. Practice it. Master it.

Learning Objectives

• Identify common geometric shapes and their properties.
• Understand angles, triangles, quadrilaterals and circles.
• Calculate perimeter and area.
• Calculate surface area and volume.
• Apply geometry and mensuration to examination and real-life problems.

1. Basic Geometric Terms

Point: an exact position. Line: a straight path extending indefinitely. Line segment: part of a line with two endpoints. Angle: the amount of rotation between two rays.

Types of angles:

• Acute: less than 90°
• Right: 90°
• Obtuse: greater than 90° but less than 180°
• Straight: 180°
• Reflex: greater than 180° but less than 360°

2. Angles on a Straight Line

Angles on a straight line add up to 180°.

Example: 180° − 65° = 115°.

3. Angles Around a Point

Angles around a point add up to 360°.

Example: 360° − (80° + 120° + 70°) = 90°.

4. Triangles

A triangle is a three-sided polygon. Its interior angles add up to 180°.

• Equilateral: three equal sides and angles of 60°.
• Isosceles: two equal sides and two equal angles.
• Scalene: all sides and angles different.
• Right-angled: one angle is 90°.

5. Pythagoras' Theorem

In a right-angled triangle:

a² + b² = c²

Here c is the hypotenuse. Example: if a=3 cm and b=4 cm, c²=9+16=25, so c=5 cm.

6. Quadrilaterals

A quadrilateral has four sides and its interior angles add up to 360°.

• Square: four equal sides and four right angles.
• Rectangle: opposite sides equal and four right angles.
• Parallelogram: opposite sides parallel and equal.
• Rhombus: four equal sides.
• Trapezium: one pair of parallel sides.

7. Perimeter

Perimeter is the total distance around a two-dimensional shape.

Rectangle: P = 2(l+w). Example: l=8 cm, w=5 cm → P=26 cm.

8. Area

• Rectangle: A=lw
• Square: A=s²
• Triangle: A=½bh
• Parallelogram: A=bh

Example: triangle with base 10 cm and height 6 cm: A=½×10×6=30 cm².

9. Circles

Important parts include centre, radius, diameter, circumference and chord.

• Diameter = 2r
• Circumference = 2πr or πd
• Area = πr²

Example: r=7 cm, π=22/7: Area=(22/7)×7²=154 cm².

10. Surface Area

Surface area is the total area of the outer surfaces of a solid.

• Cuboid: 2(lw+lh+wh)
• Cube: 6a²

11. Volume

Volume measures the space occupied by a three-dimensional object.

• Cuboid: V=lwh
• Cube: V=a³
• Prism: cross-sectional area × length
• Cylinder: V=πr²h

Example: 8 cm × 5 cm × 3 cm = 120 cm³.

12. Units in Mensuration

Use square units for area and cubic units for volume.

• Length: cm, m, km
• Area: cm², m², km²
• Volume: cm³, m³

📝 Practice Zone

1. Find the missing angle on a straight line if one angle is 72°.
2. Angles around a point are 90°, 110° and 80°. Find the fourth.
3. A triangle has angles 55° and 65°. Find the third.
4. Use Pythagoras to find the hypotenuse when the shorter sides are 6 cm and 8 cm.
5. Find the perimeter of a rectangle 12 cm by 7 cm.
6. Find the area of a triangle with base 14 cm and height 9 cm.
7. Find the area of a circle with radius 7 cm using π=22/7.
8. Find the surface area of a cube with side 5 cm.
9. Find the volume of a cuboid 10 cm × 4 cm × 3 cm.
10. Write the formula for the volume of a cylinder with radius 3 m and height 5 m.

⚡ Quick Revision

• Straight line = 180°
• Around a point = 360°
• Triangle = 180°
• Quadrilateral = 360°
• Pythagoras = a²+b²=c²
• Rectangle area = lw
• Triangle area = ½bh
• Circle area = πr²
• Cuboid volume = lwh

🎤 Maths Hip-Hop Tip

“Draw it, label it, choose the formula, then calculate.”

Understand the shape → Write the formula → Substitute → Calculate → Check the units.

MATHEMATICS HIP-HOP
We Educate the World • Learn it. Practice it. Master it.

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Trigonometry

MATHEMATICS HIP-HOP

TRIGONOMETRY

Angles • Sides • Bearings • Graphs • Exam Skills

MATHS + MUSIC = SUCCESS 🎧📐

🎯 Learning Objectives

By the end of this lesson, you should be able to:

• Identify the sides of a right-angled triangle.
• Use sine, cosine and tangent to solve problems.
• Find unknown sides and angles.
• Solve problems involving angles of elevation and depression.
• Use bearings in practical problems.
• Apply the sine rule and cosine rule.
• Find the area of a triangle using trigonometry.
• Interpret basic trigonometric graphs.
• Solve simple trigonometric equations and use identities.

1. Introduction to Trigonometry

Trigonometry is the study of the relationship between the angles and sides of triangles.

In many school examinations, trigonometry is especially useful when working with right-angled triangles.

RIGHT-ANGLED TRIANGLE

Hypotenuse = side opposite the right angle

Opposite = side opposite the chosen angle

Adjacent = side next to the chosen angle, excluding the hypotenuse

2. SOH–CAH–TOA

SOH

sin θ = Opposite / Hypotenuse

CAH

cos θ = Adjacent / Hypotenuse

TOA

tan θ = Opposite / Adjacent

🧠 Remember

SOH → CAH → TOA

3. Finding an Unknown Side

Example: A right-angled triangle has an angle of 30° and a hypotenuse of 10 cm. Find the opposite side.

Using sine:

sin 30° = Opposite / 10

Since sin 30° = 0.5:

0.5 = Opposite / 10

Opposite = 5 cm

Exam Tip: First identify the known sides and the unknown side. Then choose SOH, CAH or TOA.

4. Finding an Unknown Angle

If the three sides or two relevant sides are known, use the inverse trigonometric functions:

• θ = sin⁻¹(Opposite / Hypotenuse)
• θ = cos⁻¹(Adjacent / Hypotenuse)
• θ = tan⁻¹(Opposite / Adjacent)

Example: If the opposite side is 6 cm and the adjacent side is 8 cm:

tan θ = 6/8

θ = tan⁻¹(0.75) ≈ 36.9°

5. Angles of Elevation and Depression

An angle of elevation is measured upward from a horizontal line of sight.

An angle of depression is measured downward from a horizontal line of sight.

Real-Life Application

Trigonometry can be used to calculate the height of a building, tree, tower or other object when the distance and angle are known.

6. Bearings

A bearing is a direction measured clockwise from North.

• Bearings are normally written using three figures.
• North = 000°
• East = 090°
• South = 180°
• West = 270°

Example: A bearing of 045° means 45° clockwise from North.

Exam Alert ⚠️

Always draw a North line when solving a bearing problem. Label the angle clearly.

7. The Sine Rule

For a triangle with sides and opposite angles labelled in the usual way:

a / sin A = b / sin B = c / sin C

The sine rule is useful when you know a side and its opposite angle, together with another side or angle.

8. The Cosine Rule

The cosine rule can be used to find a missing side or angle in a non-right-angled triangle.

a² = b² + c² − 2bc cos A

It is particularly useful when two sides and the included angle are known, or when all three sides are known and an angle is required.

9. Area of a Triangle Using Trigonometry

When two sides and the included angle are known:

Area = ½ab sin C

Example: If a = 8 cm, b = 10 cm and C = 30°:

Area = ½ × 8 × 10 × sin 30° = 20 cm²

10. Trigonometric Graphs

The three basic trigonometric functions are:

• y = sin x
• y = cos x
• y = tan x

Important features to understand include the shape of the graph, period, maximum and minimum values, and where the graph crosses the axes.

Graph Reminder

For sine and cosine, one complete cycle is 360° (or 2π radians). The tangent graph repeats every 180° (or π radians).

11. Basic Trigonometric Identities

Some important identities include:

sin²θ + cos²θ = 1

tan θ = sin θ / cos θ

Learn these identities carefully because they can simplify more advanced trigonometric problems.

12. Trigonometric Equations

When solving trigonometric equations, always pay attention to the required range of angles.

Example: Solve sin θ = 0.5 for 0° ≤ θ ≤ 360°.

The solutions are:

θ = 30° or 150°

🎤 EXAM METHOD

1. Read the question carefully.
2. Draw and label a diagram.
3. Identify the known information.
4. Choose the correct trigonometric rule.
5. Substitute the values.
6. Use the correct calculator mode.
7. Give the answer with appropriate units and accuracy.

🔥 Practice Zone

1. A right-angled triangle has a hypotenuse of 12 cm and an angle of 40°. Find the side opposite the angle.
2. A triangle has opposite side 7 cm and adjacent side 9 cm. Find the angle.
3. A ladder 5 m long makes an angle of 65° with the ground. Find its vertical height.
4. Convert the bearing 125° into a description of its direction.
5. Use the sine rule to find an unknown side when the relevant opposite angle and another side-angle pair are given.
6. Use the cosine rule to find a missing side when two sides and the included angle are known.
7. Find the area of a triangle with sides 10 cm and 14 cm and included angle 35°.
8. State the period of y = sin x.
9. State the period of y = tan x.
10. Solve sin θ = 0.5 for 0° ≤ θ ≤ 360°.

⚡ Quick Revision

• SOH: sin θ = Opposite/Hypotenuse
• CAH: cos θ = Adjacent/Hypotenuse
• TOA: tan θ = Opposite/Adjacent
• Sine Rule: a/sin A = b/sin B = c/sin C
• Cosine Rule: a² = b² + c² − 2bc cos A
• Triangle Area: ½ab sin C
• Bearing: measured clockwise from North
• Always: draw a diagram and label it.

🎧 MATHS HIP-HOP TIP

“Don’t just memorize SOH–CAH–TOA. Understand which sides and angles you are working with.”

Simple explanation → Worked example → Practice → Examination application.

MATHEMATICS HIP-HOP — WE EDUCATE THE WORLD 🌍

Keep learning. Keep practising. Keep mastering mathematics.

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Statistics

MATHEMATICS HIP-HOP

STATISTICS

Understand the data. Read the pattern. Explain the evidence.

1. Introduction to Statistics

Statistics is the branch of mathematics concerned with collecting, organising, presenting, analysing and interpreting data.

Statistics helps us answer questions such as: How many students passed? What is the average mark? Which result occurs most often? How are the values distributed?

Learning Objectives

• Identify and classify different types of data.
• Construct and interpret frequency tables.
• Draw and interpret statistical diagrams.
• Calculate mean, median, mode and range.
• Use quartiles and cumulative frequency.
• Interpret statistical information in examination questions.

2. Data Collection

Data means information collected for a particular purpose.

Examples include students' marks, ages, heights, number of books owned and daily temperatures.

Primary data is collected directly by the investigator, for example through questionnaires, interviews or measurements.

Secondary data is obtained from an existing source such as a report, database or published table.

3. Discrete and Continuous Data

Discrete data consists of countable values, usually whole numbers. Example: number of children in a family.

Continuous data can take any value within a range and is usually obtained by measurement. Example: height, mass, time or temperature.

Remember This
Counting usually gives discrete data; measuring usually gives continuous data.

4. Frequency Tables

A frequency tells us how many times a value occurs.

Example data: 2, 3, 3, 4, 4, 4, 5, 5

Value	Frequency
2	1
3	2
4	3
5	2

5. Statistical Diagrams

Bar Charts

A bar chart represents categories using separate rectangular bars. The bars normally have gaps between them.

Histograms

A histogram represents continuous grouped data. The bars touch because the intervals are continuous.

When class widths are unequal, frequency density is used:

Frequency density = Frequency ÷ Class width

Frequency Polygons

A frequency polygon is formed by plotting frequencies against class midpoints and joining the points with straight lines.

Pie Charts

A pie chart represents a whole quantity as a circle divided into sectors.

Sector angle = (Frequency ÷ Total frequency) × 360°

6. Mean

The mean is found by adding all the values and dividing by the number of values.

Mean = Sum of values ÷ Number of values

Example: Mean of 4, 6, 8, 10 and 12 = (4 + 6 + 8 + 10 + 12) ÷ 5 = 8.

Mean from a Frequency Table

Mean = Σfx ÷ Σf

Here, x is the value and f is its frequency.

7. Median

The median is the middle value when the data is arranged in order.

Example: 3, 5, 7, 8, 11 has median 7.

For an even number of values, take the mean of the two middle values. Example: 2, 4, 6, 10 → Median = (4 + 6) ÷ 2 = 5.

8. Mode

The mode is the value that occurs most frequently.

Example: 2, 4, 4, 5, 6, 4, 7 → Mode = 4.

9. Range

The range measures the spread from the smallest to the largest value.

Range = Largest value − Smallest value

For 5, 8, 10, 12 and 17, range = 17 − 5 = 12.

10. Measures of Dispersion

Measures of dispersion describe how spread out the data is.

• Range: largest value − smallest value.
• Interquartile range: upper quartile − lower quartile.

11. Quartiles

Quartiles divide ordered data into four parts.

• Lower quartile (Q₁): approximately the 25th percentile.
• Median (Q₂): the 50th percentile.
• Upper quartile (Q₃): approximately the 75th percentile.

Interquartile Range = Q₃ − Q₁

12. Cumulative Frequency

Cumulative frequency is the running total of frequencies.

It is useful when finding the median, quartiles and other percentiles from grouped data.

Example: frequencies 4, 6, 5 and 3 give cumulative frequencies 4, 10, 15 and 18.

13. Interpreting Statistical Diagrams

1. Check the title and what the data represents.
2. Read the scale on each axis.
3. Check the units used.
4. Identify highest and lowest values.
5. Look for patterns, differences or unusual values.
6. Determine whether data is grouped or ungrouped.

EXAM ALERT

Always read the scale carefully. A graph may use intervals such as 2, 5, 10 or 20 rather than increasing by one unit.

For grouped data, identify the class interval before choosing the correct statistical method.

14. Examination-Style Worked Example

The marks of five learners are 12, 15, 18, 15 and 20.

Mean: (12 + 15 + 18 + 15 + 20) ÷ 5 = 80 ÷ 5 = 16.

Median: 12, 15, 15, 18, 20 → 15.

Mode: 15.

Range: 20 − 12 = 8.

15. Practice Zone

1. Find the mean of 6, 8, 10, 12 and 14.
2. Find the median of 4, 9, 2, 7 and 6.
3. Find the mode of 3, 5, 5, 7, 8, 5 and 9.
4. Find the range of 12, 18, 7, 25 and 14.
5. Explain the difference between discrete and continuous data.
6. Find the sector angle for frequency 15 when total frequency is 60.
7. A class has frequencies 4, 7, 5 and 9. Find the cumulative frequencies.
8. State two features that distinguish a histogram from a bar chart.
9. Find the interquartile range when Q₁ = 12 and Q₃ = 28.
10. For 5, 8, 8, 10, 12, 15, 15, 15, find mean, median, mode and range.

16. Quick Revision

• Mean = total ÷ number of values.
• Median = middle value after arranging the data.
• Mode = most frequent value.
• Range = maximum − minimum.
• Frequency = number of times a value occurs.
• Cumulative frequency = running total of frequencies.
• IQR = Q₃ − Q₁.
• Pie-chart angle = frequency ÷ total × 360°.

🎵 MATHS HIP-HOP TIP

“Statistics is not just about calculating numbers — it is about understanding what the numbers are telling you.”

Mathematics Hip-Hop • Learn it. Practice it. Master it. Share it.

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Probability

MATHEMATICS HIP-HOP

PROBABILITY

Measure the chance. Understand the outcome. Make the mathematics clear.

1. Introduction to Probability

Probability is the branch of mathematics used to measure how likely an event is to happen.

Probability is used in games, weather forecasting, insurance, science, business and everyday decision-making.

Learning Objectives

• Understand the language of probability.
• Calculate probabilities of simple events.
• Use the probability scale.
• Find probabilities using equally likely outcomes.
• Use complementary probability.
• Work with combined events and tree diagrams.
• Apply probability to examination-style problems.

2. The Language of Probability

Important terms include:

• Experiment: a process that produces an outcome.
• Outcome: a possible result of an experiment.
• Sample space: the complete set of possible outcomes.
• Event: one outcome or a collection of outcomes of interest.

For example, when a fair die is rolled, the sample space is {1, 2, 3, 4, 5, 6}.

3. The Probability Scale

Probability lies between 0 and 1.

Impossible   0   ───────── 0.5 ───────── 1   Certain

• Probability 0: impossible event.
• Probability 1: certain event.
• Values between 0 and 1 represent different degrees of likelihood.

4. Probability of an Event

When all outcomes are equally likely:

P(E) = Number of favourable outcomes ÷ Total number of possible outcomes

Example: A fair die is rolled. Find the probability of obtaining a 4.

There is 1 favourable outcome and 6 possible outcomes.

P(4) = 1 ÷ 6 = 1/6.

5. Sample Spaces

A sample space lists every possible outcome.

For a coin tossed once:

S = {Head, Tail}

For a die rolled once:

S = {1, 2, 3, 4, 5, 6}

6. Complementary Probability

The complement of an event means the event does not happen.

P(not E) = 1 − P(E)

Example: If P(rain) = 0.3, then:

P(no rain) = 1 − 0.3 = 0.7.

7. Probability from a Frequency Table

If an experiment is repeated many times, experimental probability can be estimated using:

Experimental probability = Frequency of event ÷ Total number of trials

Example: A coin is tossed 50 times and lands on heads 27 times.

Experimental probability of heads = 27 ÷ 50 = 0.54.

8. Mutually Exclusive Events

Two events are mutually exclusive if they cannot happen at the same time.

For example, on one roll of a die, getting a 2 and getting a 5 are mutually exclusive.

P(A or B) = P(A) + P(B)

This rule applies when A and B are mutually exclusive.

9. Independent Events

Two events are independent when the occurrence of one does not affect the probability of the other.

For independent events:

P(A and B) = P(A) × P(B)

Example: A fair coin is tossed twice. The probability of getting two heads is:

1/2 × 1/2 = 1/4.

10. Tree Diagrams

A tree diagram shows possible sequences of events using branches.

When following a branch, multiply probabilities along the branch. When adding separate successful branches, add their probabilities.

Exam Method

1. Write the possible outcomes.
2. Write the probability on each branch.
3. Multiply along each required branch.
4. Add the relevant branches.

11. Probability Without Replacement

When an item is removed and not returned, the total number of objects changes.

Example: A bag contains 3 red balls and 2 blue balls. One red ball is removed without replacement. The probability of drawing another red ball is:

After removing one red ball, 2 red balls remain out of 4 balls.

Probability = 2/4 = 1/2.

12. Expected Frequency

Expected frequency estimates how many times an event should occur over a number of trials.

Expected frequency = Probability × Number of trials

Example: If the probability of success is 0.2 in 100 trials:

Expected frequency = 0.2 × 100 = 20.

EXAM ALERT

Check whether the question involves with replacement or without replacement. In without-replacement questions, the probabilities usually change after an item is taken out.

13. Examination-Style Worked Example

A bag contains 5 red balls and 3 blue balls. One ball is selected at random.

(a) Probability of red = 5 ÷ 8 = 5/8.

(b) Probability of blue = 3 ÷ 8 = 3/8.

(c) Check: 5/8 + 3/8 = 8/8 = 1.

14. Practice Zone

1. A fair die is rolled. Find the probability of getting an even number.
2. A fair die is rolled. Find the probability of getting a number greater than 4.
3. A coin is tossed. State the sample space.
4. If P(A) = 0.35, find P(not A).
5. A bag contains 4 green and 6 yellow balls. Find the probability of selecting a green ball.
6. A coin is tossed twice. Find the probability of getting two tails.
7. A card is selected from a standard pack of 52 cards. Find the probability of selecting a heart.
8. A machine produces 8 defective items in 200 items. Estimate the experimental probability of a defective item.
9. If the probability of an event is 0.25 and there are 400 trials, find the expected frequency.
10. Explain why probabilities cannot be less than 0 or greater than 1.

15. Quick Revision

• Probability measures how likely an event is.
• 0 = impossible; 1 = certain.
• P(E) = favourable outcomes ÷ total outcomes, when outcomes are equally likely.
• P(not E) = 1 − P(E).
• For mutually exclusive events: add probabilities.
• For independent events: multiply probabilities.
• Without replacement means the probabilities can change after each selection.
• Expected frequency = probability × number of trials.

🎵 MATHS HIP-HOP TIP

“Probability is about measuring chance — slow down, identify the outcomes, then let the mathematics do the work.”

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Coordinate Geometry & Graphs

MATHEMATICS HIP-HOP • MATHEMATICS NOTES

COORDINATE GEOMETRY & GRAPHS

Understand coordinates, gradients, straight-line equations and graphs step by step.

📚 Introduction

Coordinate geometry connects algebra with geometry. It allows us to describe the position of points, calculate distances and gradients, find equations of lines and interpret graphs.

In this chapter, we move from simple coordinates to straight-line graphs and real-life applications.

🎯 Learning Objectives

• Identify and plot points on the Cartesian plane.
• Use the four quadrants correctly.
• Calculate distance and midpoint.
• Find and interpret the gradient of a line.
• Use y = mx + c to describe straight lines.
• Find equations of straight lines.
• Recognise parallel and perpendicular lines.
• Interpret intercepts and real-life graphs.

1. The Cartesian Plane

The Cartesian plane is formed by two perpendicular axes: the x-axis (horizontal) and y-axis (vertical). Their intersection is the origin (0,0).

A point is written as an ordered pair (x,y). The x-coordinate is always written first.

Worked Example 1

For A(4,3), move 4 units right and 3 units up. For B(−2,5), move 2 units left and 5 units up.

2. The Four Quadrants

Quadrant	x	y
I	+	+
II	−	+
III	−	−
IV	+	−

3. Distance Between Two Points

d = √[(x₂ − x₁)² + (y₂ − y₁)²]

Worked Example 2

Distance between A(2,3) and B(8,11):

d = √[(8−2)² + (11−3)²] = √(36+64) = 10 units.

4. Midpoint of a Line Segment

M = ((x₁+x₂)/2, (y₁+y₂)/2)

Worked Example 3

Midpoint of (2,4) and (10,8) = ((2+10)/2,(4+8)/2) = (6,6).

5. Gradient of a Straight Line

Gradient measures the steepness and direction of a line.

m = (y₂ − y₁)/(x₂ − x₁)

• Positive gradient: rises left to right.
• Negative gradient: falls left to right.
• Zero gradient: horizontal line.
• Vertical line: gradient is undefined.

Worked Example 4

Through (2,3) and (6,11): m = (11−3)/(6−2) = 2.

6. Equation of a Straight Line

y = mx + c

m = gradient   |   c = y-intercept

Worked Example 5

For y = 3x + 2, gradient = 3 and y-intercept = 2.

7. Finding an Equation from a Point and Gradient

Start with y = mx + c, then substitute the known point and gradient.

Worked Example 6

A line has gradient 2 and passes through (3,7).

7 = 2(3) + c → 7 = 6 + c → c = 1.

Equation: y = 2x + 1.

8. Parallel Lines

Parallel straight lines have the same gradient. For example, y = 2x + 4 and y = 2x − 7 are parallel.

9. Perpendicular Lines

For non-vertical straight lines, perpendicular gradients satisfy:

m₁ × m₂ = −1

If one line has gradient 2, a perpendicular line has gradient −1/2.

10. Intercepts

The y-intercept occurs when x = 0. The x-intercept occurs when y = 0.

Worked Example 7

For y = 2x − 6: y-intercept = (0,−6). Setting y=0 gives x=3, so x-intercept = (3,0).

11. Real-Life Graphs

Coordinate graphs can represent distance against time, cost against quantity, temperature changes and motion. Always identify the axes, units, scale, gradient and intercepts before interpreting a graph.

🧠 Examination Alert

• Write coordinates as (x,y).
• Be careful with negative signs.
• Use the correct graph scale.
• For perpendicular lines, m₁m₂ = −1.
• Show your substitution and working.

✍️ Practice Zone

1. State the quadrant containing A(4,6), B(−3,5), C(−2,−7), D(8,−4).
2. Find the distance between (1,2) and (7,10).
3. Find the midpoint of (−2,4) and (6,12).
4. Find the gradient through (2,5) and (8,17).
5. State the gradient and y-intercept of y = −3x + 8.
6. Find the equation with gradient 4 through (2,11).
7. Determine whether y = 5x + 2 and y = 5x − 9 are parallel.
8. Find the gradient perpendicular to a line with gradient −3.
9. Find the x- and y-intercepts of y = 2x − 10.
10. A taxi fare is C = 500x + 2,000. Explain the gradient and intercept.

⚡ Quick Revision

Distance: √[(x₂−x₁)²+(y₂−y₁)²]

Midpoint: ((x₁+x₂)/2,(y₁+y₂)/2)

Gradient: (y₂−y₁)/(x₂−x₁)

Straight line: y = mx + c

Parallel: same gradient

Perpendicular: m₁m₂ = −1

🎤 Mathematics Hip-Hop Tip

Don't just memorise the formula. See the movement. A gradient tells the story of how y changes when x changes.

🔥 Challenge Zone

A line passes through A(−2,5) and B(4,−7).

(a) Find its gradient. (b) Find its equation. (c) Find its x-intercept. (d) Find the equation of a perpendicular line through (2,3).

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Functions & Graphs

MATHEMATICS HIP-HOP • MATHEMATICS NOTES

FUNCTIONS & GRAPHS

Understand functions, notation, domain, range and graphs from first principles.

📚 Introduction

A function describes a relationship between an input and an output. It gives a rule that tells us what output is produced for each permitted input.

Functions are important because they help us describe patterns, equations, graphs and real-life relationships.

🎯 Learning Objectives

• Understand the meaning of a function.
• Use function notation such as f(x).
• Identify inputs, outputs, domain and range.
• Evaluate functions for given values.
• Represent functions using tables, mappings and graphs.
• Work with linear, quadratic and exponential functions.
• Understand composite and inverse functions.
• Interpret transformations and real-life applications.

1. What Is a Function?

A function is a rule that assigns exactly one output to each permitted input.

Think of a function as a mathematical machine:

Input → Function Rule → Output

For example, if the rule is f(x) = 2x + 3, an input of 4 produces:

f(4) = 2(4) + 3 = 11

2. Function Notation

Instead of writing “y = 2x + 3”, we can write:

f(x) = 2x + 3

Here, f is the name of the function and x is the input.

Worked Example 1

Given f(x) = 3x − 2, find f(5).

Substitute x = 5:

f(5) = 3(5) − 2 = 15 − 2 = 13.

3. Input and Output

The input is the value placed into the function. The output is the value produced by the function.

For f(x) = x² + 1:

• Input 2 → output 5.
• Input 3 → output 10.
• Input −2 → output 5.

4. Domain and Range

The domain is the set of permitted input values.

The range is the set of output values produced by the function.

Domain: possible x-values.

Range: resulting y-values.

5. Functions as Tables

A table can show the relationship between input and output.

x	f(x)=2x+1
0	1
1	3
2	5
3	7

6. Functions and Graphs

A function can be represented on a coordinate plane by plotting ordered pairs (x, f(x)).

For example, for f(x)=2x+1, the points include (0,1), (1,3), (2,5) and (3,7). These points lie on a straight line.

7. Linear Functions

A linear function commonly has the form:

f(x)=mx+c

Here, m is the gradient and c is the y-intercept.

Worked Example 2

For f(x)=−2x+5, identify the gradient and y-intercept.

Gradient = −2.

y-intercept = 5.

8. Quadratic Functions

A quadratic function can be written as:

f(x)=ax²+bx+c

Its graph is called a parabola.

If a > 0, the parabola opens upward. If a < 0, it opens downward.

Worked Example 3

For f(x)=x²−4, find f(3).

f(3)=3²−4=9−4=5.

9. Exponential Functions

An exponential function has the variable in the exponent, for example:

f(x)=aˣ

Exponential functions are useful for describing growth and decay, such as population growth, compound interest and radioactive decay.

10. Composite Functions

A composite function applies one function after another.

It is written as:

(f ∘ g)(x)=f(g(x))

Worked Example 4

Let f(x)=2x+1 and g(x)=x+3.

Then f(g(x)) = 2(x+3)+1 = 2x+7.

11. Inverse Functions

An inverse function reverses the effect of the original function.

If f changes x into y, the inverse changes y back into x.

The inverse is written as f⁻¹(x).

Worked Example 5

Find the inverse of f(x)=2x+3.

Write y=2x+3.

Swap x and y: x=2y+3.

x−3=2y.

y=(x−3)/2.

Therefore, f⁻¹(x)=(x−3)/2.

12. Transformations of Graphs

Graphs can be moved, reflected or stretched.

• f(x)+a → moves the graph upward by a units.
• f(x)−a → moves it downward by a units.
• f(x−a) → moves it right by a units.
• f(x+a) → moves it left by a units.
• −f(x) → reflection in the x-axis.

13. Real-Life Applications

Functions are used whenever one quantity depends on another.

• Cost depending on number of items.
• Distance depending on time.
• Profit depending on sales.
• Population depending on time.
• Temperature depending on time.
• Electricity consumption depending on usage.

Worked Example 6 — Real Life

A company charges a fixed fee of 2,000 shillings plus 500 shillings for every item delivered.

If x is the number of items, the total cost is:

C(x)=500x+2000

The gradient 500 represents the cost per item, while 2,000 is the fixed charge.

🧠 Examination Alert

• Remember that a function gives exactly one output for each permitted input.
• When evaluating f(a), substitute a everywhere x occurs.
• Do not confuse domain with range.
• For composite functions, work from the inside function outward.
• Check inverse-function steps carefully when swapping x and y.
• Label graph axes and scales clearly.

✍️ Practice Zone

1. Given f(x)=4x−3, find f(6).
2. Given f(x)=x²+2, find f(−3).
3. State the domain and range of a function from a supplied set of permitted inputs and outputs.
4. Complete a table for f(x)=3x+2 when x=−2, −1, 0, 1 and 2.
5. State the gradient and y-intercept of f(x)=−5x+7.
6. For f(x)=x²−5x+6, calculate f(2).
7. If f(x)=2x+1 and g(x)=x−4, find f(g(x)).
8. Find the inverse of f(x)=3x−6.
9. Describe the transformation from y=f(x) to y=f(x)+4.
10. A taxi fare is C(x)=1,000x+3,000. Explain the meaning of 1,000 and 3,000.

⚡ Quick Revision

Function: one output for each permitted input.

Notation: f(x).

Domain: input values.

Range: output values.

Linear: f(x)=mx+c.

Quadratic: f(x)=ax²+bx+c.

Composite: (f∘g)(x)=f(g(x)).

Inverse: reverses the original function.

🎤 Mathematics Hip-Hop Tip

Think of a function like a beat machine: put in an input, follow the rule, and get one output. Once you understand the rule, the notation and graphs become much easier.

🔥 Challenge Zone

Let f(x)=2x−3 and g(x)=x²+1.

(a) Find f(5).

(b) Find g(−2).

(c) Find f(g(x)).

(d) Find g(f(x)).

(e) Explain why f(g(x)) and g(f(x)) are not generally the same.

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Vectors & Transformations

📐 VECTORS & TRANSFORMATIONS

MATHEMATICS HIP-HOP — Mathematics Notes

Learn the mathematics of movement, direction and position through clear explanations, worked examples and examination practice.

Learning Objectives

By the end of this lesson, you should be able to:

• distinguish between scalar and vector quantities;
• represent vectors using diagrams and column vectors;
• add, subtract and multiply vectors by scalars;
• calculate the magnitude of a vector;
• use position vectors in geometric problems;
• describe translations, reflections, rotations and enlargements;
• use matrices to represent common transformations; and
• solve examination problems involving vectors and transformations.

1. What Is a Vector?

A vector is a quantity that has both magnitude and direction.

Examples include displacement, velocity and force.

A scalar has magnitude only. Examples include mass, time, temperature and distance.

Scalar	Vector
Distance	Displacement
Speed	Velocity
Mass	Force
Time	Acceleration

2. Representing Vectors

A vector can be represented by an arrow. The length represents its magnitude and the arrowhead shows its direction.

Vectors can also be written as column vectors:

ab

Here, a represents horizontal movement and b represents vertical movement.

Worked Example 1
The vector from A to B is

43.

This means 4 units horizontally and 3 units vertically.

3. Equal and Negative Vectors

Two vectors are equal if they have the same magnitude and direction, even if they are drawn in different positions.

The negative of a vector has the same magnitude but the opposite direction.

If

a=xy,

then

-a=-x-y.

4. Adding and Subtracting Vectors

When vectors are written as column vectors, add or subtract corresponding components.

ab + cd = a+cb+d

ab - cd = a-cb-d

Worked Example 2
Let

a=35,    b=2-1.

Then

a+b = 54.

Also,

a-b = 16.

5. Multiplying a Vector by a Scalar

A scalar is an ordinary number. Multiplying a vector by a scalar changes its magnitude and may change its direction if the scalar is negative.

kab = kakb

Worked Example 3

32-4 = 6-12.

6. Magnitude of a Vector

For a vector

a=xy,

its magnitude is found using Pythagoras' theorem.

|a|=√(^x2+y2)

Worked Example 4
Find the magnitude of

68.

|a|=√(⁶²⁺⁸²) =√(³⁶⁺⁶⁴) =√(¹⁰⁰) =10.

7. Position Vectors

A position vector describes the position of a point relative to the origin.

If point A has coordinates (x, y), its position vector is:

→OA=xy

8. Finding a Vector Between Two Points

If A and B have position vectors a and b, then:

→AB=b-a

Worked Example 5
A has position vector

23

and B has position vector

75.

Therefore:

→AB = 75 - 23 = 52.

9. Parallel Vectors

Two vectors are parallel when one is a scalar multiple of the other.

If

b=ka,

then a and b are parallel.

Worked Example 6

a=23,    b=69.

Since

b=3a,

the vectors are parallel.

10. Midpoints Using Vectors

If A and B have position vectors a and b, the position vector of the midpoint M is:

→OM=a+b2

11. What Is a Transformation?

A transformation changes the position, size or orientation of a shape on a coordinate plane.

The main transformations studied are:

• translation;
• reflection;
• rotation; and
• enlargement.

12. Translation

A translation moves every point the same distance in the same direction. The shape does not rotate or change size.

A translation can be represented by a column vector.

Translation by

ab

means:

(x,y)→(x+a,y+b).

Worked Example 7
Translate P(2, 3) by

4-2.

New point:

(2+4,\;3-2)=(6,1).

13. Reflection

A reflection produces a mirror image of a shape in a line called the mirror line.

Important reflections include:

Mirror line	Coordinate rule
x-axis	(x, y) → (x, −y)
y-axis	(x, y) → (−x, y)
y = x	(x, y) → (y, x)
y = −x	(x, y) → (−y, −x)

Worked Example 8
Reflect P(4, 2) in the x-axis.
The x-coordinate stays the same and the y-coordinate changes sign:

(4,2)→(4,-2).

14. Rotation

A rotation turns a shape about a fixed point called the centre of rotation. The angle and direction must be stated.

Common rotations about the origin are:

Rotation	Coordinate rule
90° anticlockwise	(x, y) → (−y, x)
90° clockwise	(x, y) → (y, −x)
180°	(x, y) → (−x, −y)

Worked Example 9
Rotate P(3, 1) through 90° anticlockwise about the origin:

(3,1)→(-1,3).

15. Enlargement

An enlargement changes the size of a shape while preserving its overall shape.

The amount of enlargement is described by the scale factor.

For centre at the origin:

(x,y)→(kx,ky)

where k is the scale factor.

Worked Example 10
Enlarge P(2, 3) by scale factor 2 about the origin:

(2,3)→(4,6).

16. Transformation Matrices

Transformations can also be represented using matrices. For a point written as a column vector, a matrix is multiplied by the coordinate vector.

Reflection in the x-axis:

1   00   -1

Reflection in the y-axis:

-1   00   1

Rotation 90° anticlockwise:

0   -11   0

Rotation 90° clockwise:

0   1-1   0

17. Combined Transformations

More than one transformation can be performed in sequence. The order matters.

EXAM ALERT: If a question asks for two transformations, perform them in the stated order. Applying the transformations in the opposite order can produce a different final image.

18. Worked Examination Example

Question: Point A is (2, 1). It is translated by

34

and then reflected in the x-axis. Find the final coordinates.

Step 1 — Translation:

(2,1)→(5,5)

Step 2 — Reflection in x-axis:

(5,5)→(5,-5)

Answer: The final coordinates are (5, −5).

19. Practice Zone

1. Add 42 and 35.
2. Find 2-34.
3. Find the magnitude of 512.
4. A has position vector 14 and B has position vector 69. Find →AB.
5. Find the midpoint of A(2, 4) and B(8, 10).
6. Translate P(−2, 5) by 4-3.
7. Reflect Q(5, −2) in the y-axis.
8. Rotate R(3, 2) through 90° anticlockwise about the origin.
9. Enlarge S(−2, 3) by scale factor 3 about the origin.
10. Describe the transformation represented by the matrix 1   00   -1.

20. Quick Revision

• Vector: magnitude + direction.
• Vector between two points: final position − initial position.
• Magnitude: √(^x2+y2).
• Translation: move every point by the same vector.
• Reflection: mirror image in a line.
• Rotation: turn about a centre through a specified angle and direction.
• Enlargement: change size using a scale factor and centre.
• Combined transformations: follow the stated order carefully.

🎯 Mathematics Hip-Hop Tip

Direction matters! When working with vectors, keep track of which point you are moving from and which point you are moving to. For transformations, always record the centre, angle, direction, mirror line or scale factor before calculating.

🔥 Challenge Zone

A triangle has vertices A(1, 1), B(4, 1) and C(2, 4).

1. Find the vector AB.
2. Find the length of AB.
3. Translate the triangle by -23.
4. Take the translated point A and rotate it 90° anticlockwise about the origin.
5. State the final coordinates of A.

Try it yourself before checking your working.

MATHEMATICS HIP-HOP

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Calculus

∫ CALCULUS

MATHEMATICS HIP-HOP — Mathematics Notes

From first principles to differentiation and integration — understand change, gradients, rates and accumulated quantities step by step.

Learning Objectives

By the end of this lesson, you should be able to:

• understand the idea of a limit;
• find gradients from first principles;
• differentiate common algebraic functions;
• use differentiation to find gradients and stationary points;
• solve practical maximum and minimum problems;
• understand the meaning of integration;
• integrate common functions;
• use definite integrals to find areas; and
• apply calculus techniques in examination questions.

1. What Is Calculus?

Calculus is the branch of mathematics used to study change and accumulation.

It has two major ideas:

• Differentiation — investigates rates of change and gradients.
• Integration — investigates accumulation and area.

Maths Hip-Hop idea: Differentiation asks, “How fast is it changing?” Integration asks, “How much has accumulated?”

2. Limits

A limit describes the value that a function approaches as the input gets closer and closer to a particular value.

As x approaches a, f(x) approaches L.

We write this as: lim f(x) = L as x → a.

Worked Example 1
Find the limit of x + 3 as x approaches 2.

Substitute x = 2:
2 + 3 = 5

Therefore, the limit is 5.

3. Differentiation from First Principles

The derivative of a function measures its instantaneous rate of change.

For a function f(x), the derivative from first principles is based on:

f′(x) = lim [f(x + h) − f(x)] / h as h → 0

Worked Example 2
Use first principles to differentiate f(x) = x².

f(x + h) = (x + h)²
= x² + 2xh + h²

Therefore:
[f(x + h) − f(x)] / h
= [x² + 2xh + h² − x²] / h
= 2x + h

As h approaches 0:
f′(x) = 2x.

4. Basic Differentiation Rules

Power rule: If y = xⁿ, then dy/dx = n xⁿ⁻¹.

Constant rule: The derivative of a constant is 0.

Constant multiple: The constant can be carried through differentiation.

Sum/difference: Differentiate each term separately.

Worked Example 3
Differentiate:
y = 5x³ − 4x² + 7x − 9

dy/dx = 15x² − 8x + 7.

5. Differentiating Common Functions

Function	Derivative
xⁿ	n xⁿ⁻¹
constant	0
sin x	cos x
cos x	−sin x
eˣ	eˣ
ln x	1/x

6. Gradient of a Curve

For a curve y = f(x), the derivative gives the gradient at a particular point.

Worked Example 4
For y = x³ − 3x, find the gradient when x = 2.

dy/dx = 3x² − 3
At x = 2:
3(2²) − 3 = 12 − 3 = 9.

The gradient is 9.

7. Stationary Points

A stationary point occurs where the gradient is zero.

At a stationary point: dy/dx = 0

Worked Example 5
Find the stationary points of y = x² − 4x + 3.

dy/dx = 2x − 4
Set the derivative equal to zero:
2x − 4 = 0
x = 2

Substitute x = 2 into the original equation:
y = 4 − 8 + 3 = −1

Therefore the stationary point is (2, −1).

8. Maximum and Minimum Points

Stationary points can represent a local maximum or a local minimum.

A useful test is the second derivative:

If d²y/dx² > 0: local minimum.

If d²y/dx² < 0: local maximum.

Worked Example 6
For y = x² − 4x + 3:
dy/dx = 2x − 4
d²y/dx² = 2

Since the second derivative is positive, the stationary point (2, −1) is a minimum.

9. Applications of Differentiation

Differentiation is useful in real situations involving:

• velocity and acceleration;
• growth and decay;
• maximum area;
• minimum cost;
• maximum profit;
• rates of change; and
• optimisation.

Worked Example 7 — Optimisation
A rectangle has perimeter 20 cm. Let its length be x cm. Then its width is 10 − x cm.

Area A = x(10 − x)
A = 10x − x²

dA/dx = 10 − 2x
For maximum area:
10 − 2x = 0
x = 5

Width = 10 − 5 = 5 cm.
The rectangle with maximum area is therefore a square of side 5 cm.

10. Integration

Integration is the reverse process of differentiation in many basic cases. It is also used to calculate accumulated quantities and areas under curves.

∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C, provided n ≠ −1.

C is the constant of integration.

11. Indefinite Integration

Worked Example 8
Integrate y = 6x² + 4x − 5.

∫(6x² + 4x − 5) dx
= 2x³ + 2x² − 5x + C.

Answer: 2x³ + 2x² − 5x + C

12. Integration of Common Functions

Function	Integral
xⁿ	xⁿ⁺¹/(n + 1) + C
constant k	kx + C
1/x	ln|x| + C
eˣ	eˣ + C
sin x	−cos x + C
cos x	sin x + C

13. Definite Integrals

A definite integral has limits. It gives a numerical result rather than a family of functions.

∫ from a to b of f(x) dx = F(b) − F(a)

where F(x) is an antiderivative of f(x).

Worked Example 9
Find the area under y = x² from x = 0 to x = 2.

∫ x² dx = x³/3

Evaluate from 0 to 2:
2³/3 − 0³/3
= 8/3 square units.

14. Area Under a Curve

When a curve lies above the x-axis, the definite integral between two x-values gives the area under the curve.

EXAM ALERT: If a curve crosses the x-axis, do not automatically treat the entire definite integral as a positive area. Determine the regions separately and consider their signs.

15. Area Between Two Curves

When one curve lies above another, the area between them can be found using:

Area = ∫ (upper function − lower function) dx

Worked Example 10
If the upper curve is y = 6 and the lower curve is y = x² between x = −2 and x = 2, then:

Area = ∫ from −2 to 2 of (6 − x²) dx.

The antiderivative is 6x − x³/3.
Evaluating gives:
Area = 64/3 square units.

16. Differentiation and Integration Together

Think of the relationship this way:

• Differentiation moves from a function to its rate of change.
• Integration can move from a rate of change back to an accumulated function.
• The constant of integration is needed for indefinite integration.

17. Examination Method

1. Read exactly what the question asks for.
2. Write the relevant differentiation or integration rule.
3. Show each algebraic step clearly.
4. For stationary points, set the first derivative equal to zero.
5. For classification, use the second derivative where appropriate.
6. For definite integrals, substitute the upper and lower limits carefully.
7. Include correct units for practical quantities such as area, distance or volume.
8. Check whether your final answer is sensible.

18. Practice Zone

1. Differentiate y = 7x⁴ − 3x² + 5x.
2. Differentiate y = 4x³ − 6x + 2 and find the gradient when x = 1.
3. Find the stationary point of y = x² − 6x + 8.
4. State whether the stationary point in Question 3 is a maximum or minimum.
5. Find ∫(8x³ − 4x + 7) dx.
6. Find the area under y = x² from x = 1 to x = 3.
7. Find ∫(3x² + 2x − 1) dx.
8. A rectangle has perimeter 30 cm. Use differentiation to determine the dimensions that give maximum area.
9. Find the derivative of y = sin x + eˣ.
10. Explain in your own words the difference between differentiation and integration.

19. Quick Revision

• Calculus: mathematics of change and accumulation.
• Derivative: rate of change / gradient.
• First principles: derivative based on a limit as h approaches zero.
• Stationary point: first derivative equals zero.
• Second derivative positive: local minimum.
• Second derivative negative: local maximum.
• Integration: reverse process of differentiation in basic cases.
• Definite integral: gives a numerical accumulated value, often an area.
• Indefinite integral: includes the constant C.

🎯 Mathematics Hip-Hop Tip

Don't just memorise the rules — understand the movement. Differentiation tells you how a quantity is changing. Integration tells you how those small changes build into a total. Once you understand that relationship, calculus becomes much easier to remember and apply.

🔥 Challenge Zone

Consider the function:

y = x³ − 6x² + 9x + 2

1. Find dy/dx.
2. Find the x-values of the stationary points.
3. Find the corresponding y-values.
4. Find d²y/dx².
5. Classify each stationary point as a maximum or minimum.

Show all your working — this is examination-level practice.

MATHEMATICS HIP-HOP

Learn it. Practice it. Master it. Share it. 🎵📐