Where Does e Hide in Plain Sight?

Open with three very different real-world images: a carbon-14 dated artifact, a spreading epidemic curve, and a bank statement compounding daily. Pose the driving question without revealing the unifying thread.

• Three unrelated scenes share one mathematical fingerprint
• The driving question: why is base e the natural base?
• Set the expectation: prediction before evidence

Ask the learner to commit to one prediction: which of these scenarios is governed by exponential growth with base e?

• Make one independent choice before evidence is shown

A simulation with sliders for a rate constant and time. Learners watch carbon-14 decay, bacterial culture growth, and a viral-video share count evolve on the same plot, then compare curves with different bases.

• Adjust rate constant k
• Observe how all three curves share the same shape
• Compare base 2, base 10, and base e to see why e fits best

Show the actual measured curves side by side: carbon-14 fraction remaining over thousands of years, a flu outbreak case count over weeks, and a savings account balance under continuous compounding. Point out the same shape repeated three times.

• Carbon-14 halves every 5,730 years — exponential decay
• Epidemic early-phase growth is exponential
• Continuous compounding produces the same form

Explain the proportionality rule: when the rate of change equals k times the current amount, the only function satisfying dy/dt = k·y is y = y₀·e^(kt). Contrast with bases 2 or 10, which require awkward conversion factors.

• Rule: dy/dt = k·y
• Only the exponential with base e solves it cleanly
• Bases 2 and 10 work but introduce a conversion constant

A new scenario the learner has not seen: a cup of cooling coffee where Newton's law of cooling applies. A small widget lets them drag a temperature-difference slider and watch the curve follow e^(-kt), confirming the same pattern.

• Recognize the rate-proportional-to-amount pattern in a new context
• Predict the form of the solution before running it
• Apply the idea to cooling

Show what the rule does not cover: logistic growth that saturates, doubling-time problems where base 2 is more intuitive, and processes with delays. Clarify that base e describes the unbounded-exponential core, not every growth story.

• Logistic curves level off — not pure exponential
• Base 2 is convenient when only the doubling time matters
• The proportionality rule is the boundary condition

Directly answer the opening question, restate the unifying rule, and list the three real-world cases plus the cooling example as evidence that base e is everywhere a rate is proportional to the amount.

• Exponential growth with base e appears whenever rate ∝ amount
• Carbon-14, epidemics, compound interest, cooling all obey this rule
• The base e is natural because it solves dy/dt = k·y directly