Advanced Functions, Modelling and Statistics
MYP Unit Framework
Key Concept: RELATIONSHIPS Related Concepts: Models. Representation. Validity. Global Context: Scientific and Technical Innovation (How do mathematical models help us understand and predict COMPLEX SYSTEMS?) Statement of Inquiry: Mathematical MODELS represent RELATIONSHIPS between variables — and evaluating the VALIDITY and LIMITATIONS of these models is essential for making sound decisions in science, policy, and everyday life.
Inquiry Questions
| Type | Question |
|---|---|
| Factual | What is an exponential function? How do you find a line of best fit? What is the difference between correlation and causation? |
| Conceptual | Why are exponential models appropriate for SOME growth patterns and NOT others? How do we know if a model is 'good enough' — and what are the ETHICAL implications of acting on imperfect models? |
| Debatable | Can statistics ever 'prove' anything — or only suggest? Should policy decisions be based on mathematical models even when the models involve SIGNIFICANT UNCERTAINTY? |
1. Exponential Functions — Growth and Decay
The Form: f(x) = a · bˣ
a = initial value (y-intercept). b = growth/decay factor. b > 1 = GROWTH. 0 < b < 1 = DECAY.
The Most Powerful Pattern in Mathematics
'Exponential growth is when a quantity increases by a CONSTANT PERCENTAGE per unit time — not a constant AMOUNT. It starts SLOWLY. Then it EXPLODES.'
Compound Interest: A = P(1 + r/100)ⁿ
'Compound interest is exponential growth applied to MONEY. Einstein allegedly called it "the eighth wonder of the world." Borrowing at compound interest builds wealth for lenders — and DEBT TRAPS for borrowers.'
Exponential Decay
Radioactive decay: N(t) = N₀e^(−λt). Half-life: T₁/₂ = ln 2 / λ. 'Carbon-14 dating uses exponential decay to determine the age of ARCHAEOLOGICAL ARTIFACTS up to ~50,000 years old.'
The COVID-19 Lesson
'During the early pandemic (2020), exponential growth was NOT an abstraction. Cases doubled every few days. "Flattening the curve" meant REDUCING the growth rate — so hospitals would not be OVERWHELMED. The mathematics was literally LIFE AND DEATH. Understanding exponential growth is a civic responsibility.'
2. Trigonometric Functions — Modelling the Periodic
Review: y = a sin(bx + c) + d
a = AMPLITUDE (height). b = frequency (period = 2π/b). c = PHASE SHIFT (horizontal). d = VERTICAL SHIFT.
Modelling Real Periodic Phenomena
- Tides: The height of water at a harbour varies SINUSOIDALLY with a period of ~12.4 hours. 'You can PREDICT the tides for any date in the future — because the gravitational relationship between Earth, Moon, and Sun is known with extraordinary precision.'
- Temperature: Daily temperature varies (approximately) sinusoidally over 24 hours. Yearly temperature over 365 days.
- Sound waves: Pure tones are sinusoidal. 'The A above middle C is 440 Hz — 440 complete oscillations per second. The wave is y = sin(2π × 440t).'
Fitting a Sine Curve to Data
Given data, find amplitude (max−min)/2, vertical shift (max+min)/2, period (from graph), and phase shift. Write the equation. Use it to PREDICT.
3. Bivariate Statistics — Relationships Between Two Variables
Scatter Plots — The First Step
Plot (x,y) pairs. Look for: PATTERN. DIRECTION (positive/negative). FORM (linear/non-linear). STRENGTH (tight cluster or loose cloud). OUTLIERS.
Correlation — Measuring the Relationship
Pearson's r: A number between −1 and +1 measuring STRENGTH and DIRECTION of LINEAR relationship. r = +1: perfect positive. r = −1: perfect negative. r = 0: no linear correlation. 'r measures LINEAR correlation ONLY. Two variables can have r ≈ 0 and still have a STRONG non-linear relationship. Always PLOT the data first.'
Line of Best Fit (Linear Regression): y = mx + c
The line that MINIMISES the sum of squared residuals (vertical distances from data points to the line). Used for PREDICTION: 'If x = ___, what do we PREDICT y to be?'
Interpolation vs. Extrapolation
- Interpolation: Predicting WITHIN the range of the data. REASONABLY RELIABLE.
- Extrapolation: Predicting BEYOND the range of the data. DANGEROUS. 'The line of best fit ASSUMES the linear trend continues — which may be completely WRONG outside the observed range.'
CORRELATION ≠ CAUSATION — The Eternal Warning
- Ice cream sales and drowning deaths are POSITIVELY correlated. Does ice cream CAUSE drowning? NO. Both increase in SUMMER. The hidden variable: temperature.
- 'When you see a correlation — ask: Is there a DIRECT causal link? A REVERSE causal link (Y causes X)? A THIRD VARIABLE causing both? Is it COINCIDENCE?'
Your Summative Assessment — The Mathematical Exploration
Task: Choose a REAL-WORLD phenomenon. Collect or find DATA. Fit a MATHEMATICAL MODEL (linear, quadratic, exponential, or trigonometric). Use your model to make a PREDICTION. Discuss: How well does your model fit the data? What are the LIMITATIONS of your model? What are the IMPLICATIONS of using your model to make decisions?
'This is a MINI version of the IB DP Internal Assessment (IA). The skills you develop here — choosing a topic, collecting data, fitting models, evaluating validity, communicating findings — are the EXACT skills the DP Mathematics IA requires.'
ATL Skills
| Skill | Focus |
|---|---|
| Critical Thinking | Distinguishing correlation from causation. Evaluating model validity. |
| Information Literacy | Finding and interpreting data. Using technology to analyse. |
| Communication | Writing a structured mathematical exploration. |
