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KSSM Mathematics Form 5 • DLP

Chapter 1: Variation 1.1 Direct Variation 📖 TB pp. 2–16
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Cadet Mathematician Class 5 Science 1
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🎯 Learning Standards 1.1.1 & 1.1.2

Welcome to Direct Variation: The Secret of Constant Proportions

In everyday life and science, quantities frequently change together. When you travel further, taxi fare increases; when a chef cooks for more guests, the amount of rice multiplies. In this lesson, you will master the exact mathematical rules and notations that define a Direct Variation (Ubahan Langsung).

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Mathematical Notation & Symbol Decoder

Master the official KSSM SPM mathematical symbols before diving into problem solving.

\( \propto \) "is proportional to" or "varies directly as"
\( y \propto x \)
Bahasa Melayu: "y berubah secara langsung dengan x" (William Emerson, 1768)
\( k \) Constant of Proportionality (Non-zero fixed ratio)
\( k = \frac{y}{x^n} \)
Bahasa Melayu: "Pemalar perkadaran" (nisbah malar yang tetap)
\( = \) Equation Form (Transforms variation into solvable algebra)
\( y \propto x^n \iff y = kx^n \)
Replaces the symbol \( \propto \) with \( = k \)
\( x^n \) Powers & Fractional Roots (\( n \in \{1, 2, 3, \frac{1}{2}, \frac{1}{3}\} \))
\( x^2, \quad x^3, \quad \sqrt{x} = x^{\frac{1}{2}}, \quad \sqrt[3]{x} = x^{\frac{1}{3}} \)
Square, Cube, Square Root (\( \sqrt{x} \)), Cube Root (\( \sqrt[3]{x} \))
\( m = k \) Gradient of line through Origin \( (0, 0) \)
\( \text{Gradient } m = \frac{y_1 - 0}{x_1^n - 0} = k \)
The straight line MUST pass through \( (0, 0) \) (no \( y \)-intercept)
\( \frac{y_1}{x_1^n} = \frac{y_2}{x_2^n} \) Proportion Method (Direct comparison between two states)
\( \frac{y_1}{x_1^n} = \frac{y_2}{x_2^n} = k \)
Alternative SPM method without finding \( k \) explicitly first
⏱️ 💼

Dilemma 1: The Promoter's Wage (TB p. 3)

A part-time promoter earns RM8 per hour (\( \text{Wage} = 8t \)). If working hours double from 10 hours to 20 hours, what happens to the total salary? What if they work 0 hours?

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Dilemma 2: The Pizza Value Trap (TB p. 16)

A restaurant offers a \( 15\text{ cm} \) diameter pizza for RM12 and a \( 30\text{ cm} \) pizza (twice as wide!) for RM35. Does price vary directly as the diameter? Is twice the width worth almost triple the price?

📚 Core Textbook Principles (KSSM Form 5, pp. 2–6)

Definition

When variable \(y\) increases, variable \(x\) also increases at the same rate and vice versa.

\( y \propto x \iff y = kx \)
\(k\) is the non-zero constant of proportionality (\( k = \frac{y}{x} \)).
Powers & Roots

For \(y\) varying directly as \(x^n\), where \( n \in \{1, 2, 3, \frac{1}{2}, \frac{1}{3}\} \):

\( y \propto x^n \iff y = kx^n \)
Constant ratio: \( k = \frac{y}{x^n} \).
The Origin Rule

The graph of \(y\) against \(x^n\) is strictly a straight line passing through the origin \( (0, 0) \).

\( \text{Gradient } m = k = \frac{y}{x^n} \)
If a line has a non-zero \(y\)-intercept (\(c \ne 0\)), it is NOT direct variation!

⚖️ Spring Extension & Hooke's Law

Textbook Example 5 (p. 8) states that extension of a spring, \( x\text{ cm} \), varies directly as load mass, \( w\text{ g} \). Drag the slider to hang different weights and observe the extension and the ratio \( \frac{x}{w} \).

100 g
Extension, \( x \) 1.50 cm
Ratio \( \frac{x}{w} = k \) 0.015
💡 Notice that \( \frac{x}{w} = 0.015 \) is constant for all weights! Therefore:
\( x = 0.015w \)
100g
0 cm 1.5 cm 3.0 cm 4.5 cm

Live Graph: Extension \( x \) vs Mass \( w \)

w (g) x (cm) 0

Straight line passing through \( (0, 0) \). Gradient \( m = k = 0.015 \).

Recorded Data Table (Self Practice 1.1b Q2)

Mass of load, \( w\text{ (g)} \) Extension, \( x\text{ (cm)} \) Ratio \( \frac{x}{w} \) Passes Origin?
00.00-Yes (0, 0)
500.750.015Yes
1001.500.015Yes
2003.000.015Yes (TB Ex. 5)

📈 Power & Radical Variation Lab

Textbook page 6: In Form 5 KSSM, \( y \propto x^n \) where \( n \in \{1, 2, 3, \frac{1}{2}, \frac{1}{3}\} \). Select a power \( n \) and toggle between viewing \( y \) against \( x \) and \( y \) against \( x^n \).

2.0
💡 Discovery Rule: When \( n = 1 \), the graph of \( y \) against \( x \) is already a straight line through the origin.
x y O(0,0)
Function: y = 2x Origin: (0, 0) Gradient \( m = k = \) 2.0

🕵️ The Origin Detective: Spot the Imposter!

SPM Exam Trap: Students often believe that any straight line graph represents direct variation. Test your investigative skills on these 3 mystery suspects.

Suspect Alpha
(0,0)

Case A: Points \( (1, 3), (2, 6), (3, 9) \)

Equation: \( y = 3x \). Straight line passing through \( (0, 0) \).

Suspect Beta
(0, 1.0)

Case B: Points \( (0.4, 1.5), (0.8, 2.0) \) (TB Ex. 3a)

Equation: \( y = 1.25x + 1.0 \). Straight line with \( y \)-intercept \( = 1.0 \).

Suspect Gamma

Case C: Points \( (2, 8), (3, 18), (4, 32) \) (TB Ex. 3b)

Plot of \( y \) against \( x \) is curved, but \( y \) against \( x^2 \) is straight!

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Diagnostic Check (3 Quick Questions)