Introduction Chapter Notes | Mathematics for Grade 10 PDF Download

Algebraic Language

Key Terms

• Product: Result of multiplication (e.g., 5 × 3 = 15).
• Quotient: Result of division (e.g., 15 ÷ 3 = 5).
• Sum: Result of addition (e.g., 5 + 3 = 8).
• Difference: Result of subtraction (e.g., 5 - 3 = 2).
• Factor: Numbers multiplied to get a product (e.g., 3 and 5 are factors of 15).
• Additive inverse: Number that, when added, gives zero (e.g., -5 for 5, since 5 + (-5) = 0).
• Multiplicative inverse (Reciprocal): Number that, when multiplied, gives 1 (e.g., 1/5 for 5, since 5 × 1/5 = 1).
• Identity for addition: 0 (e.g., 5 + 0 = 5).
• Identity for multiplication: 1 (e.g., 5 × 1 = 5).
• Coefficient: Number multiplying a variable (e.g., in 3x, 3 is the coefficient).
• Solution: Value that makes an equation true (e.g., x = 2 for 2x = 4).
• Input variable: Independent variable (e.g., x in y = 2x).
• Output variable: Dependent variable (e.g., y in y = 2x).

Some Words We Use in Algebra

Operations

• Addition: Combining numbers (e.g., 5x + 3x = 8x).
• Subtraction: Deducting one number from another (e.g., 13y - 12y = y).
• Multiplication: Repeated addition (e.g., 2 × 3y × 6x = 36xy).
• Division: Splitting into parts (e.g., 12x ÷ 4 = 3x).

Types of Expressions

• Monomial: One term (e.g., 6x²).
• Binomial: Two terms (e.g., -3x + 7).
• Trinomial: Three terms (e.g., 100x³ + 45x² - 50x).
• Variable: Symbol for an unknown (e.g., x, s, t).
• Coefficient: Number before a variable (e.g., -15 in -15x³).
• Constant: Number without a variable (e.g., 14 in 33x² + 14).

Some Mathematical Conventions and Expressions

Conventions

• Multiplication sign: Often omitted (e.g., 5 × p = 5p, -3 × (a + 4) = -3(a + 4)).
• Order in products: Write constants first (e.g., 11a, not a11).
• Coefficient of 1: Omitted (e.g., 1a = a, -1a = -a).

Quantities

This section introduces quantities and their units, focusing on the International System of Units (SI).

Understanding Quantities

Quantity: Anything measurable or countable (e.g., height = 1.8 m, where 1.8 is the number, m is the unit).

SI Units: Seven base quantities:

• Length: metre (m).
• Mass: kilogram (kg).
• Time: second (s).
• Electric current: ampere (A).
• Thermodynamic temperature: kelvin (K).
• Amount of substance: mole (mol).
• Luminous intensity: candela (cd).

Practical Units

• Examples: Cement (kg), sand (m³), building height (m), distance (km), petrol (litres).
• Estimation: Centimetre (finger width), metre (arm length), litre (bottle), kilometre (long walk), kilogram (bag of sugar).

Relationship Between Quantities

Relationships

Examples:

• Driving time vs. petrol: More time decreases petrol.
• Sweets eaten vs. calories: More sweets increase calories.
• Square perimeter vs. area: Larger perimeter increases area.
• Triangle base vs. area: Larger base increases area.
• Circle radius vs. circumference: Larger radius increases circumference.

Properties of Operations

Distributive Property

• Definition: a(b + c) = ab + ac (e.g., 5 × 27 = 5 × 20 + 5 × 7).
• Multiplication distributes over addition (e.g., 3(x + 4) = 3x + 12).

Commutative Property

• Addition: x + y = y + x (e.g., 15 + 11 = 11 + 15).
• Multiplication: xy = yx (e.g., 15 × 11 = 11 × 15).

Associative Property

• Addition: x + (y + z) = (x + y) + z (e.g., 1 + (2 + 3) = (1 + 2) + 3).
• Multiplication: x(yz) = (xy)z (e.g., 2 × (3 × 4) = (2 × 3) × 4).

Factorisation and Expansion

Factorisation

• Definition: Writing as a product of factors (e.g., 35 = 7 × 5, xy + xz = x(y + z)).
• Examples: ax + bx = x(a + b), 6x - 3 = 3(2x - 1).

Expansion

• Definition: Writing a product as a sum (e.g., (x + 1)(x + 3) = x² + 4x + 3).

The Hidden Pattern of the Distributive Property

Factorising Quadratics

• Example: x² + 5x + 6 = (x + 2)(x + 3).
  Decompose: x² + 2x + 3x + 6 = x(x + 2) + 3(x + 2).
• Example: x² + 2xy + y² = (x + y)(x + y).

Another Hidden Pattern of the Distributive Property

This section focuses on factorising differences of squares.

Difference of Squares
Pattern: x² - a² = (x + a)(x - a).
Example: x² - 9 = (x + 3)(x - 3).
Decompose: x² + 3x - 3x - 9 = x(x + 3) - 3(x + 3).

Applications

• Compute differences: 11² - 9² = (11 + 9)(11 - 9) = 20 × 2.

Computational Properties of Integers

This section revises rules for integer operations.

Properties

• Subtraction: Smaller - larger = negative (e.g., 10 - 30 = -20).
• Adding negative: Same as subtracting positive (e.g., 3 + (-10) = 3 - 10 = -7).
• Subtracting negative: Same as adding positive (e.g., 3 - (-10) = 3 + 10 = 13).
• Zero sum: Positive + negative = 0 (e.g., 3 + (-3) = 0).

Multiplication:

• Negative × positive = negative (e.g., -15 × 6 = -90).
• Negative × negative = positive (e.g., -15 × (-6) = 90).

Calculator Use

• Keys: (-) for negative numbers, - for subtraction.

Equivalence

This section explains equivalent expressions.

Numerical Expressions

• Equivalent if same value (e.g., 20 + (30 + 70) = 20 + 30 + 70).

Algebraic Expressions

• Equivalent if same value for all x (e.g., 2x + 3x = 5x).
• Verify by simplifying: (b + 2) - (b - 5) = (b + 2) - b + 5 = 7.

Rules:

• x + (y + z) = x + y + z.
• x + (y - z) = x + y - z.
• x - (y + z) = x - y - z.
• x - (y - z) = x - y + z.

Simplifying Expressions

This section teaches combining like terms to simplify expressions.

Simplification

• Like terms: Same variable and exponent (e.g., 26x + 4x = 30x).
• Unlike terms: Cannot combine (e.g., 2x + 40 ≠ 42x).
• Example: x(x + 2) - 7x = x² + 2x - 7x = x² - 5x.

From Numerical Calculations to Algebraic Calculations

Expanded Form

• Number: 5432 = 5 × 10³ + 4 × 10² + 3 × 10 + 2.
• Polynomial: 5x³ + 4x² + 3x + 2 (x = 10).

Polynomial Operations

• Addition: (7x³ + 6x² + 5x + 4) + (4x³ + 3x² + 2x + 1) = 11x³ + 9x² + 7x + 5.
• Sum and product: (x + 2) + (x - 3) = 2x - 1; (x + 2)(x - 3) = x² - x - 6.

Modelling Some Situations

This section uses algebra to represent real-world relationships.

Formulas

• Days in weeks: d = 7w (d = days, w = weeks).
• Minutes in hours: m = 60h (m = minutes, h = hours).
• Learners vs. teachers: l = 25t (l = learners, t = teachers).

Solution, Expression, Equation, Identity, and Impossibility

This section defines key algebraic concepts.

Definitions

• Expression: Value depends on variable (e.g., 5x + 3).
• Equation: True for specific values (e.g., 5x + 3 = 103, x = 20).
• Identity: True for all values (e.g., 2x + 3x = 5x).
• Impossibility: Never true (e.g., x + 10 = x).
• Solution: Value making equation true (e.g., x = 9 for 2x + 6 = 24).

Examples

• 4x + 12 ≠ 7x + 3 (not equivalent, equation true at x = 3).
• 10x + 40 ≠ 10x + 50 (impossibility).

The Scientific Calculator

This section explains calculator use and order of operations.

Calculator Features

• Algebraic logic: Enter expressions left to right.
• Display: Natural (like paper) or linear (single line).
• Keys: (-) for negatives, - for subtraction, x⁻¹ for reciprocals.

Order of Operations

• BODMAS: Brackets, Orders (exponents), Division/Multiplication, Addition/Subtraction.
• Example: 7 + 3 ÷ 10 × 1 = 7 + 0.3 = 7.3.

Sequences

• Constant operations: 3 + 3 = 6, +3 = 9, etc., generates 3, 6, 9, 12, ….
• Multiplication: 2 × 3 = 6, ×3 = 18, generates 2, 6, 18, 54, ….

Powers and Roots

• Power: y^x (e.g., 5³ = 125).
• Square root: √25 = 5 (principal root).
• Example: √(9 + 4) ≠ √9 + √4.

Fractions

• Chain calculations: (8.74 + 9.48) ÷ (5.6 × 3.4).
• Reciprocal: 2/5 = 2 × 5⁻¹.

Points to Remember

• Algebraic terms: Product (×), quotient (÷), sum (+), difference (-).
• Expression types: Monomial (1 term), binomial (2 terms), trinomial (3 terms).
• Conventions: Omit ×, write constants first, drop 1 in 1a.
• SI units: Metre (m), kilogram (kg), second (s), etc.
• Properties: Distributive (a(b + c) = ab + ac), commutative (xy = yx), associative (x(yz) = (xy)z).
• Factorisation: xy + xz = x(y + z); x² - a² = (x + a)(x - a).
• Equivalence: Same value for all x (e.g., 2x + 3x = 5x).
• Simplification: Combine like terms (e.g., 26x + 4x = 30x).
• BODMAS: Order for calculations.
• Calculator: Use (-) for negatives, x⁻¹ for reciprocals.

Difficult Words

• Coefficient: Number multiplying a variable (e.g., 3 in 3x).
• Reciprocal: Multiplicative inverse (e.g., 1/5 for 5).
• Monomial: Expression with one term (e.g., 6x²).
• Binomial: Expression with two terms (e.g., -3x + 7).
• Trinomial: Expression with three terms (e.g., x² + 2x + 1).
• Identity: Equation true for all values (e.g., 2x + 3x = 5x).
• Impossibility: Equation never true (e.g., x + 10 = x).
• BODMAS: Rule for order of operations.

Summary