Florida B.E.S.T. MA.912.C.5.3

MathGrades 9–12Apply integrals to solve problems.

The Standard

Solve differential equations of the form dY/dt=kY as applied to growth and decay problems.

Florida B.E.S.T. Standards for Mathematics

Teacher's field guide

What Students Need to Do: Students identify situations where the rate of change is proportional to the current amount. They use an initial value to build Y = Y0e^(kt), then find k, time, or amount and interpret the result.
What Mastery Looks Like: A student can move from Y' = kY and Y(0) = Y0 to Y = Y0e^(kt). Given two values from a context, the student can find k and explain whether the model shows growth or decay.
Common Misconceptions: Students often model constant percent change with a linear equation. They may use a positive value of k for decay or treat k as a discrete multiplier. Some forget to use the initial value when finding the constant.
How to Assess It: Exit ticket: A medicine amount starts at 500 mg and decreases continuously at 12% per hour. Write the initial-value equation, solve it, and find the amount after 6 hours.

Lesson moves

Run a dice decay lab: roll 60 dice, remove sixes each round, graph survivors, and compare the data with an exponential model.
Ask students to explain why losing 20% each hour is exponential rather than subtracting the same amount each hour.
Use card sets that match a context, differential equation, initial condition, solution, and graph, then have pairs justify each match.
Model caffeine decay using a 100 mg dose and a five-hour half-life, then calculate the amount remaining through one school day.

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