Florida B.E.S.T. MA.912.C.3.3

MathGrades 9–12Apply derivatives to solve problems.

The Standard

Determine where a function is decreasing and increasing using its derivative.

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

Teacher's field guide

What Students Need to Do: Students calculate or interpret f', then locate input values where it is zero or undefined. They use the sign of f' on each interval to describe the graph's behavior in interval notation.
What Mastery Looks Like: A student finds all critical numbers and tests the sign of f' between them. The student gives correct intervals and supports each interval with a sign chart or clear reasoning.
Common Misconceptions: Students may use the sign of f instead of the sign of f'. They may assume every zero of f' creates a turning point. They may also omit points where f' is undefined or include critical numbers inside open intervals.
How to Assess It: Give the exit ticket: For f(x) = x³ − 3x² − 9x + 1, find f', make a sign chart, and state where f rises or falls.

Lesson moves

Give students a graph of f' and arrow tiles, then have them place up arrows where f'>0 and down arrows where f'<0.
Ask: “A classmate says every zero of f' is a turning point.” Students write a counterexample and explain why the claim fails.
Run a card sort matching function formulas, derivatives, sign charts, and interval statements, with students correcting any unmatched set.
Use a cyclist's position function and its velocity to identify when the cyclist moves forward, moves backward, or changes direction.

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