Florida B.E.S.T. MA.912.C.5.6

MathGrades 9–12Apply integrals to solve problems.

The Standard

Find the average value of a function over a closed interval by using definite integrals.

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

Teacher's field guide

What Students Need to Do: Students find total signed accumulation over an interval, then divide by the interval’s length. They connect the result to the height of a rectangle with the same signed area.
What Mastery Looks Like: Students correctly use 1/(b − a) times the integral from a to b of f(x). They evaluate it accurately and explain that the result represents a constant height with the same signed area.
Common Misconceptions: Students often forget to divide by the interval length, so they report total accumulation instead of the average height. They may average endpoint values or use b instead of b minus a in the denominator.
How to Assess It: Exit ticket: Find the average value of f(x) = x² on [0, 3]. Show the integral, the interval length, and the final value.

Lesson moves

Draw f(x) = x + 1 on grid paper over [0, 4], cut out the region, and rearrange it into an equal-base rectangle.
Compare the average value and the average of the endpoints for f(x) = x² on [0, 2], then explain why they differ.
Use a card sort matching functions and intervals with their definite integrals, interval lengths, and average values.
Find the average temperature from T(t) = 70 + 10 sin(πt/12) over a 24-hour period and interpret the result.

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