Lesson 1: 2-1 Average and Instantaneous Rate of Change
Duration of Days: 1
Lesson Objective
Upon completion of this lesson, students will be able to:
Calculate the average rate of change of a function over a specified interval [a, b] using the formula.
Identify the average rate of change as the slope of the secant line connecting two points on a curve.
Estimate the average rate of change from various representations, including analytical (equations), graphical (visual curves), and numerical (tables of values).
Interpret the meaning of the average rate of change in a real-world context, including the appropriate use of units (e.g., meters per second, gallons per hour).
Distinguish between the "net change" of a function and its "average rate of change" over a given period.
Communicate mathematical reasoning by explaining what a positive, negative, or zero average rate of change signifies regarding the behavior of the function.
How does the average rate of change of a function relate to the slope of a secant line?
In what ways can we interpret "average rate of change" in real-world contexts (e.g., physics, biology, economics)?
How do the units of the independent and dependent variables determine the units of the rate of change?
What happens to the secant line as the interval between two points becomes infinitely small? (Bridging to the Instantaneous Rate of Change).
Average Rate of Change (AROC)
Secant Line
Difference Quotient
Interval
Function Notation
Independent Variable
Dependent Variable
Slope
Net Change
Units of Measure
Displacement (in the context of position functions)
Velocity (Average)
F-IF.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
F-IF.3: Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.
F-IF.4: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
F-IF.5: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.
F-IF.6: Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over an interval. Interpret average rate of change of a function in terms of an application, such as speed, change in temperature, or population growth rate.
This lesson transitions students from Algebra 2/Pre-Calculus "slope" to the Calculus concept of the Average Rate of Change (AROC). Students will use the formula to calculate the slope of the secant line passing through $(a, f(a)) and (b, f(b)). The lesson emphasizes interpreting these values in context—specifically identifying that AROC represents a constant rate that would yield the same net change over a specific interval. Students will work with functions presented as equations, tables of data, and graphs.
Purpose: To establish the foundational "Pre-Calculus" pillar of the derivative. Before students can understand the instantaneous rate of change (the derivative), they must be fluent in calculating and interpreting the average rate of change across varying intervals.
DOK Level 1 (Recall): Calculating the slope between two points using a given function and the AROC formula.
DOK Level 2 (Skill/Concept): Estimating the AROC from a table of values or a graph where the exact coordinates must be identified.
DOK Level 3 (Strategic Thinking): Interpreting the meaning of the AROC in a real-world scenario, including correct units
For Struggling Learners (Scaffolding):The "Slope Bridge": Explicitly label x1, x2, y1, and y2 before moving to the function notation f(a) and f(b) to reduce cognitive load.
For Advanced Learners (Extension):
Mean Value Theorem Preview: Provide a graph and ask students to find a point on the curve where the "tilt" (tangent) looks the same as the secant line they just calculated.
AP College Board Assessments