Proportional Reasoning, Linear Functions & Probability

MYP Unit Framework

Key Concept: RELATIONSHIPS Related Concepts: Equivalence. Models. Justification. Global Context: Globalisation and Sustainability (How do we model growth, fairness, and risk?) Statement of Inquiry: Proportional and linear RELATIONSHIPS model how quantities CHANGE together — and probability models help us make RATIONAL DECISIONS under UNCERTAINTY, from personal choices to global policies.

Inquiry Questions

1. Ratio, Rate and Proportion

Ratio — The Mathematics of Comparison

A ratio compares two quantities of the SAME kind. 'In a class, the ratio of boys to girls is 3:2. For every 3 boys, there are 2 girls.'

Rate — Comparison of Different Kinds

Speed (km/h). Price (₹/kg). Population density (people/km²). 'A RATE compares two QUANTITIES with DIFFERENT UNITS. It describes how ONE quantity changes with respect to ANOTHER.'

Proportion — When Two Ratios Are Equal

a:b :: c:d ⇔ a/b = c/d. Product of extremes = Product of means: ad = bc.

Direct Proportion: y = kx (y ∝ x). Graph is a STRAIGHT LINE through ORIGIN. 'Double x → double y. The RATIO y/x is CONSTANT = k.'

Inverse Proportion: y = k/x. Graph is a HYPERBOLA. 'Double x → HALVE y. The PRODUCT xy is CONSTANT = k.'

Real-World Applications

• Currency exchange. Map scales. Recipe scaling. 'If 5 kg of rice costs ₹200, how much does 8 kg cost?' (Direct proportion — 8×200/5 = ₹320).
• 'If 6 workers can build a wall in 8 days, how long will 4 workers take?' (Inverse proportion — more workers = less time. k = 6×8 = 48. 4 workers: 48/4 = 12 days).

2. Linear Functions — y = mx + c

The Most Important Function

'A LINEAR FUNCTION describes a RELATIONSHIP where the RATE OF CHANGE is CONSTANT. It is the simplest — and most widely used — mathematical model in science, economics, and everyday life.'

Slope (m) = Rate of Change

m = (y₂−y₁)/(x₂−x₁) = Δy/Δx. m > 0: RISING. m < 0: FALLING. m = 0: HORIZONTAL (constant).

y-intercept (c) — The Starting Value

Where the line crosses the y-axis (when x = 0). 'c is the INITIAL VALUE. The FIXED COST. The baseline.'

Graphing Linear Functions

1. Plot the y-intercept (0, c). 2. Use the slope m = rise/run to find another point. 3. Draw a STRAIGHT line.

From Words to Equations

'A taxi charges a FIXED FARE of ₹30 plus ₹12 per kilometre.' Cost = 30 + 12x, where x = distance. 'The slope (12) is the RATE. The intercept (30) is the FIXED part.'

Intersection of Two Lines — Solving Simultaneous Equations Graphically

'The intersection point is the SOLUTION to BOTH equations — the point where two situations are EQUAL.'

3. Probability — From Simple to Compound

Review — Classical Probability

P(E) = n(E)/n(S). 0 ≤ P(E) ≤ 1.

Sample Spaces and Diagrams

• Tree Diagram: Shows ALL possible outcomes of sequential events.
• 'Probability that BOTH events occur = multiply the probabilities along the branches.'

Compound Events

Expected Value — What to Expect in the Long Run

E = Σ (value × probability). A game: win ₹100 with probability 0.1. Win nothing with 0.9. E = 100×0.1 = ₹10. 'If you play this game MANY times, you will win an AVERAGE of ₹10 per play. The expected value tells you whether a game is "fair."'

Risk and Decision-Making

'Probability is not just about dice and cards. It's about RISK. What is the probability of a flood in your city this year? Should you buy insurance? Should the government invest in a vaccine for a disease that affects 1 in 10,000 people? Probability helps us make RATIONAL decisions — even when we cannot be CERTAIN.'

Your Summative Assessment

Task: 'The Linear Modelling Project'

1. Find or collect DATA that shows a LINEAR relationship. 2. Plot the data. 3. Find the equation of the line of best fit. 4. Use your model to MAKE A PREDICTION. 5. Discuss: How reliable is your prediction? What are the LIMITATIONS of a linear model?

ATL Skills