Mississippi Math C.G.24

MathGrades 9–12Geometry (G) Geometry

The Standard

Apply Rolle’s Theorem and the Mean Value Theorem and their geometric consequences.

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 check whether a function is continuous on the interval and differentiable between its endpoints. They find where a tangent slope matches the secant slope. When the endpoints have the same height, they look for a horizontal tangent.
What Mastery Looks Like: Given a function and interval, a student checks the conditions before applying either theorem. They calculate the secant slope and solve for a point inside the interval with that tangent slope. They recognize that equal endpoint values make the tangent horizontal.
Common Misconceptions: Students may think equal endpoint heights guarantee a horizontal tangent, even for |x| on [−1,1], which has a sharp corner. They may also use an endpoint as the matching point or assume there can be only one.
How to Assess It: Exit ticket: For f(x)=x²−2x on [0,2], check the conditions, calculate the secant slope, and find a point where the tangent matches it. Why is that tangent horizontal?

Lesson moves

Plot y=x² on graph paper from 0 to 2; draw the endpoint secant and mark where a tangent has the same slope.
Ask: Does a graph that starts and ends at the same height always have a horizontal tangent? Use |x| to defend your answer.
Sort cards for x² on [−1,1], x² on [0,2], and |x| on [−1,1] by which theorem applies; find each valid point.
Give students a smooth distance graph for a 120-mile, two-hour trip; have them mark where the speed is 60 mph.

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