Content Standard(s):
Mathematics 24. Distinguish between situations that can be modeled with linear functions and those that can be modeled with exponential functions.
a. Show that linear functions grow by equal differences over equal intervals, while exponential functions grow by equal factors over equal intervals.
b. Define linear functions to represent situations in which one quantity changes at a constant rate per unit interval relative to another.
c. Define exponential functions to represent situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
Unpacked Content
Evidence Of Student Attainment:
Students:
Given a linear or exponential function,
Create a sequence from the functions and examine the results to demonstrate that linear functions grow by equal differences, and exponential functions grow by equal factors over equal intervals.
Use slope-intercept form of a linear function and the general definition of exponential functions to justify through algebraic rearrangements that linear functions grow by equal differences, and exponential functions grow by equal factors over equal intervals. Given a contextual situation modeled by functions, determine if the change in the output per unit interval is a constant being added or multiplied to a previous output, and appropriately label the function as linear, exponential, or neither. Teacher Vocabulary:
Linear functions
Exponential functions Constant rate of change
Constant percent rate of change
Intervals
Percentage of growth
Percentage of decay Knowledge:
Students know:
Key components of linear and exponential functions.
Properties of operations and equality Skills:
Students are able to:
Accurately determine relationships of data from a contextual situation to determine if the situation is one in which one quantity changes at a constant rate per unit interval relative to another (linear).
Accurately determine relationships of data from a contextual situation to determine if the situation is one in which one quantity grows or decays by a constant percent rate per unit interval relative to another (exponential). Understanding:
Students understand that:
Linear functions have a constant value added per unit interval, and exponential functions have a constant value multiplied per unit interval.
Distinguishing key features of and categorizing functions facilitates mathematical modeling and aids in problem resolution. Diverse Learning Needs:
Essential Skills:
Learning Objectives: ALGI.24.1: Define linear function and exponential function.
ALGI.24.2: Distinguish between graphs of a line and an exponential function.
ALGI.24.3: Identify the graph of an exponential function.
ALGI.24.4: Identify the graph of a line.
ALGI.24.5: Plot points on a coordinate plane from a given table of values.
a.
ALGI.24.6: Divide each y-value in a table of values by its successive y-value to determine if the quotients are the same, to prove an exponential function.
ALGI.24.7: Subtract each y-value in a table of values by its successive y-value to determine if the differences are the same, to prove a linear function.
ALGI.24.8: Apply rules for adding, subtracting, multiplying, and dividing integers.
b.
ALGI.24.9: Define constant rate of change as slope.
ALGI.24.10: Subtract each y-value in a table of values by its successive y-value to determine if the differences are the same, to prove a linear function.
ALGI.24.11: Recognize the calculated difference is the constant rate of change.
ALGI.24.12: Apply rules for adding, subtracting, multiplying, and dividing integers.
c.
ALGI.24.13: Define exponential growth and decay.
ALGI.24.14: Divide each y-value in a table of values by its successive y-value to determine if the quotients are the same, to prove an exponential function.
ALGI.24.15: Apply the rules of multiplication and division of integers.
Prior Knowledge Skills:
Recognize ordered pairs.
Identify ordered pairs.
Recognize linear equations.
Recall how to solve problems using the distributive property.
Define linear and nonlinear functions, slope, and y-intercept.
Analyze the graph to determine the rate of change.
Alabama Alternate Achievement Standards
AAS Standard:
Mathematics 27. Interpret the parameters of functions in terms of a context. Extend from linear functions, written in the form mx + b , to exponential functions, written in the form abx .
Example: If the function V(t) = 19885(0.75)t describes the value of a car after it has been owned for t years, 1985 represents the purchase price of the car when t = 0, and 0.75 represents the annual rate at which its value decreases.
Unpacked Content
Evidence Of Student Attainment:
Students:
Given a contextual situation that may be modeled by a linear or exponential function,
Create a function that models the situation.
Define and justify the parameters (all constants used to define the function) in terms of the original context. Teacher Vocabulary:
Knowledge:
Students know:
Key components of linear and exponential functions. Skills:
Students are able to:
Communicate the meaning of defining values (parameters and variables) in functions used to model contextual situations in terms of the original context. Understanding:
Students understand that:
Sense making in mathematics requires that meaning is attached to every value in a mathematical expression. Diverse Learning Needs:
Essential Skills:
Learning Objectives: ALGI.27.1: Recall the formula of an exponential function.
ALGI.27.2: Recall the slope-intercept form of a linear function.
ALGI.27.3: Define b as growth or decay factor in the context of an exponential problem.
ALGI.27.4: Define k as the initial amount in the context of an exponential problem.
ALGI.27.5: Define m as the rate of change in the context of a linear problem.
ALGI.27.6: Define b as the initial amount in the context of a linear problem.
Prior Knowledge Skills:
Solve problems with exponents.
Discuss strategies for solving real-world and mathematical problems.
Recognize ordered pairs.
Identify parts of the Cartesian plane.
Recall how to plot points on a Cartesian plane.
Distinguish the difference between linear and nonlinear functions.
Define qualitative, increase, and decrease.
Recall how to name points from a graph (ordered pairs).
Recall how to find the rate of change (slope) in a linear equation.
Recall how to complete an input/output function table.
Analyze real-world situations to identify the rate of change and initial value from a table, graph, or description.
Define function, rate of change, and initial value.
Alabama Alternate Achievement Standards
AAS Standard:
