Lesson Notes By Weeks and Term v4 - SHS 1
NUMBER AND ALGEBRAIC PATTERNS
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NUMBER AND ALGEBRAIC PATTERNS
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Subject: Additional Mathematics
Class: SHS 1
Term: 1st Term
Week: 6
Grade code: 1.1.1.LI.3
Strand code: 1
Sub-strand code: 1
Content standard code: 1.1.1.CS.1
Indicator code: 1.1.1.LI.3
Theme: MODELLING WITH ALGEBRA
Subtheme: NUMBER AND ALGEBRAIC PATTERNS
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This lesson revisits the fundamental laws of indices that learners encountered in JHS and builds upon them to establish more advanced rules. Indices, or powers, are a fundamental concept in mathematics, providing a concise way to represent repeated multiplication. Understanding indices is crucial for advanced topics in algebra, calculus, and science. In Ghana, this knowledge is applicable in fields like finance (calculating compound interest on loans from banks like GCB or Ecobank), science (modelling population growth), and technology (understanding data storage in gigabytes and terabytes).
Materials: Whiteboard/Chalkboard, Markers/Chalk, A4 sheets for group work, Calculators (for verification only).
Introduction (5 mins): Start with a quick recap. Ask the class: "If I write 2 x 2 x 2 x 2 x 2, is there a shorter, more powerful way to write this?" Guide them to the answer 2⁵. Base: The number being multiplied (in this case, 2). Index/Exponent/Power: The number of times the base is multiplied by itself (in this case, 5). The plural of index is indices.
Part A: Recollecting the Basic Laws of Indices (15 mins) (Teacher facilitates a group activity where learners use simple examples to "discover" these laws.)
Let's use examples to remember the rules we already know. Multiplication Law: When multiplying expressions with the same base, add the indices. Derivation: Consider `3² × 3⁴`. `3² = 3 × 3` `3⁴ = 3 × 3 × 3 × 3` So, `3² × 3⁴ = (3 × 3) × (3 × 3 × 3 × 3) = 3⁶` We see that `6 = 2 + 4`. General Law: `aᵐ × aⁿ = aᵐ⁺ⁿ` Division Law: When dividing expressions with the same base, subtract the indices. Derivation: Consider `5⁵ ÷ 5²`. `5⁵ ÷ 5² = (5 × 5 × 5 × 5 × 5) / (5 × 5)` We can cancel out two 5s from the top and bottom, leaving `5 × 5 × 5 = 5³`. We see that `3 = 5 - 2`. General Law: `aᵐ ÷ aⁿ = aᵐ⁻ⁿ` (where a ≠ 0) Power of a Power Law: When an exponential expression is raised to another power, multiply the indices. Derivation: Consider `(2³)².` This means `2³ × 2³`. Using the multiplication law, this is `2³⁺³ = 2⁶`. We see that `6 = 3 × 2`. General Law: `(aᵐ)ⁿ = aᵐⁿ` Power of a Product Law: Derivation: Consider `(2 × 5)³`. This means `(2 × 5) × (2 × 5) × (2 × 5)`. Rearranging gives `(2 × 2 × 2) × (5 × 5 × 5) = 2³ × 5³`. General Law: `(ab)ⁿ = aⁿbⁿ` Power of a Quotient Law: Derivation: Similar to the above, `(3/4)² = (3/4) × (3/4) = (3×3)/(4×4) = 3²/4²`. General Law: `(a/b)ⁿ = aⁿ/bⁿ` (where b ≠ 0)