Mississippi Math C.A.20

MathGrades 9–12Algebra (A) Demonstrate Basic Knowledge of Functions, Including their Behavior and Characteristics

The Standard

State and apply the First and Second Fundamental Theorem of Calculus.

2025 Mississippi College- and Career-Readiness Standards for Mathematics · Employ Various Integration Properties and Techniques to Elevate Integrals

Teacher's field guide

What Students Need to Do: Students explain how a continuous function's accumulated area changes at a rate given by the function. They differentiate an integral with a variable upper limit and evaluate a definite integral using an antiderivative.
What Mastery Looks Like: Given A(x)=∫₀ˣ 2t dt, students find A'(x)=2x and explain why. They find A(3)=9 by subtracting the antiderivative's values at 3 and 0.
Common Misconceptions: Students may give x² for A'(x), confusing accumulated area with its rate of change. They may subtract endpoint values in the wrong order or use only the upper endpoint.
How to Assess It: Exit ticket: For A(x)=∫₀ˣ 2t dt, find A'(x) and A(3). Write one sentence explaining the rule you used for each answer.

Lesson moves

On graph paper, shade under y=2t from 0 to x, then add a thin strip and compare its height with 2x.
Ask students to write why A'(x)=2x for A(x)=∫₀ˣ 2t dt, using the height of one added strip.
Give pairs cards labeled 2t, t², ∫₀³ 2t dt=9, 3t², t³, and ∫₀² 3t² dt=8 to match and verify.
Give a cyclist's velocity v(t)=2t meters per second, then ask students to find the distance traveled during the first three seconds.

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