 Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
 Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
 Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
Clusters should not be sorted from Major to Supporting and then taught in that order. To do so would strip the coherence of the mathematical ideas and miss the opportunity to enhance the major work of the grade with the supporting clusters.

Also assesses:
 Assessment Limits :
Exponential functions should be in the form  Calculator :
Neutral
 Clarification :
Students will determine whether the realworld context may be
represented by a linear function or an exponential function and give
the constant rate or the rate of growth or decay.Students will choose an explanation as to why a context may be
modeled by a linear function or an exponential function.
Students will interpret the rate of change and intercepts of a linear
function when given an equation that models a realworld context.Students will interpret the xintercept, yintercept, and/or rate of
growth or decay of an exponential function given in a realworld
context  Stimulus Attributes :
Items should be set in a realworld context.Items must use function notation.
 Response Attributes :
Items may require the student to apply the basic modeling cycle.Items may require the student to choose a parameter that is
described within the realworld context.Items may require the student to choose an appropriate level of
accuracy.Items may require the student to choose and interpret the scale in a
graph.Items may require the student to choose and interpret units.
MAFS.912.FLE.2.5
 Test Item #: Sample Item 1
 Question:
The graph of function f models the specific humidity in the atmosphere, in grams of water vapor per kilogram of atmospheric gas , versus temperature, in degrees Celsius (ºC), as shown. Four of its points are labeled.
This question has two parts.
Part A.
Felicia wants to model the raltionship between temperature, in ºC, and specific humidity, in . Select words to complete the statement about the type of model Felicia should use.
The relationship is _____________ because the specific humidity increases by equal ______ over equal intervals of temperature.
Part B.
Which relationship must be true to justify the function type that models the relationship?
 Difficulty: N/A
 Type: : Multiple Types
Related Courses
Related Access Points
Related Resources
Formative Assessments
Lesson Plans
Lesson Study Resource Kit
Original Student Tutorial
Perspectives Video: Expert
Perspectives Video: Professional/Enthusiasts
ProblemSolving Tasks
Professional Development
Unit/Lesson Sequence
Video/Audio/Animation
Virtual Manipulatives
MFAS Formative Assessments
Students are given two functions, one represented verbally and the other by a table, and are asked to compare the rate of change in each in the context of the problem.
Students are given a linear and an exponential function, one represented verbally and the other by a table. Then students are asked to compare the rate of change in each in the context of the problem.
Students are given four verbal descriptions of functions and asked to identify each as either linear or exponential and to justify their choices.
Students are asked to prove that an exponential function grows by equal factors over equal intervals.
Students are asked to prove that a linear function grows by equal differences over equal intervals.
Original Student Tutorials Mathematics  Grades 912
Learn about exponential growth in the context of interest earned as money is put in a savings account by examining equations, graphs, and tables in this interactive tutorial.
Student Resources
Original Student Tutorial
Learn about exponential growth in the context of interest earned as money is put in a savings account by examining equations, graphs, and tables in this interactive tutorial.
Type: Original Student Tutorial
ProblemSolving Tasks
In this example, students are asked to write a function describing the population growth of algae. It is implied that this is exponential growth.
Type: ProblemSolving Task
This problem solving task challenges students to find the linear, exponential and quadratic functions based on two points.
Type: ProblemSolving Task
This problem solving task asks students to predict and model US population based on a chart of US population data from 1982 to 1988.
Type: ProblemSolving Task
This problem solving task asks students to solve five exponential and linear function problems based on a US population chart for the years 17901860.
Type: ProblemSolving Task
The purpose of this task is to give students an opportunity to explore various aspects of exponential models (e.g., distinguishing between constant absolute growth and constant relative growth, solving equations using logarithms, applying compound interest formulas) in the context of a real world problem with ties to developing financial literacy skills. In particular, students are introduced to the idea of inflation of prices of a single commodity, and are given a very brief introduction to the notion of the Consumer Price Index for measuring inflation of a body of goods.
Type: ProblemSolving Task
This task gives a variation of reallife contexts which could be modeled by a linear or exponential function. The key distinguishing feature between the two is whether the change by equal factors over equal intervals (exponential functions), or by a constant increase per unit interval (linear functions). The task could either be used as an assessment problem on this distinction, or used as an introduction to the differences between these very important classes of functions.
Type: ProblemSolving Task
This task requires students to use the fact that on the graph of the linear function h(x) = ax + b, the ycoordinate increases by a when x increases by one. Specific values for a and b were left out intentionally to encourage students to use the above fact as opposed to computing the point of intersection, (p,q), and then computing respective function values to answer the question.
Type: ProblemSolving Task
This task gives students an opportunity to work with exponential functions in a real world context involving continuously compounded interest. They will study how the base of the exponential function impacts its growth rate and use logarithms to solve exponential equations.
Type: ProblemSolving Task
This exploratory task requires the student to use properties of exponential functions in order to estimate how much Carbon 14 remains in a preserved plant after different amounts of time.
Type: ProblemSolving Task
This task involves a fairly straightforward decaying exponential. Filling out the table and developing the general formula is complicated only by the need to work with a fraction that requires decisions about rounding and precision.
Type: ProblemSolving Task
This task requires students to use the fact that the value of an exponential function f(x) = a · b^x increases by a multiplicative factor of b when x increases by one. It intentionally omits specific values for c and d in order to encourage students to use this fact instead of computing the point of intersection, (p,q), and then computing function values to answer the question.
Type: ProblemSolving Task
This problem assumes that students are familiar with the notation x_{0} and ?x. However, the language "successive quotient" may be new.
Type: ProblemSolving Task
This task assumes that students are familiar with the ?x and ?y notations. Students most likely developed this familiarity in their work with slope.
Type: ProblemSolving Task
An important property of linear functions is that they grow by equal differences over equal intervals. In F.LE Equal Differences over Equal Intervals 1, students prove this for equal intervals of length one unit, and note that in this case the equal differences have the same value as the slope.
Type: ProblemSolving Task
This problemsolving task asks students to examine if linear modeling would be appropriate to describe and predict population growth from select years.
Type: ProblemSolving Task
This problemsolving task provides students an opportunity to experiment with modeling real data by using population growth rates from the past two centuries.
Type: ProblemSolving Task
This problemsolving task challenges students to write expressions and create a table to calculate how much money can be gained after investing at different banks with different interest rates.
Type: ProblemSolving Task
This problemsolving task asks students to describe exponential growth through a realworld problem involving the illegal introduction of fish into a lake.
Type: ProblemSolving Task
This problemsolving emphasizes the expectation that students know linear functions grow by constant differences over equal intervals and exponential functions grow by constant factors over equal intervals.
Type: ProblemSolving Task
Virtual Manipulatives
Using this virtual manipulative, students are able to graph a function and a set of ordered pairs on the same coordinate plane. The constants, coefficients, and exponents can be adjusted using slider bars, so the student can explore the affect on the graph as the function parameters are changed. Students can also examine the deviation of the data from the function. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.
Type: Virtual Manipulative
In this online tool, students input a function to create a graph where the constants, coefficients, and exponents can be adjusted by slider bars. This tool allows students to explore graphs of functions and how adjusting the numbers in the function affect the graph. Using tabs at the top of the page you can also access supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.
Type: Virtual Manipulative
Parent Resources
ProblemSolving Tasks
In this example, students are asked to write a function describing the population growth of algae. It is implied that this is exponential growth.
Type: ProblemSolving Task
This problem solving task challenges students to find the linear, exponential and quadratic functions based on two points.
Type: ProblemSolving Task
This problem solving task asks students to predict and model US population based on a chart of US population data from 1982 to 1988.
Type: ProblemSolving Task
This problem solving task asks students to solve five exponential and linear function problems based on a US population chart for the years 17901860.
Type: ProblemSolving Task
The purpose of this task is to give students an opportunity to explore various aspects of exponential models (e.g., distinguishing between constant absolute growth and constant relative growth, solving equations using logarithms, applying compound interest formulas) in the context of a real world problem with ties to developing financial literacy skills. In particular, students are introduced to the idea of inflation of prices of a single commodity, and are given a very brief introduction to the notion of the Consumer Price Index for measuring inflation of a body of goods.
Type: ProblemSolving Task
This task gives a variation of reallife contexts which could be modeled by a linear or exponential function. The key distinguishing feature between the two is whether the change by equal factors over equal intervals (exponential functions), or by a constant increase per unit interval (linear functions). The task could either be used as an assessment problem on this distinction, or used as an introduction to the differences between these very important classes of functions.
Type: ProblemSolving Task
This task requires students to use the fact that on the graph of the linear function h(x) = ax + b, the ycoordinate increases by a when x increases by one. Specific values for a and b were left out intentionally to encourage students to use the above fact as opposed to computing the point of intersection, (p,q), and then computing respective function values to answer the question.
Type: ProblemSolving Task
This task gives students an opportunity to work with exponential functions in a real world context involving continuously compounded interest. They will study how the base of the exponential function impacts its growth rate and use logarithms to solve exponential equations.
Type: ProblemSolving Task
This exploratory task requires the student to use properties of exponential functions in order to estimate how much Carbon 14 remains in a preserved plant after different amounts of time.
Type: ProblemSolving Task
This task involves a fairly straightforward decaying exponential. Filling out the table and developing the general formula is complicated only by the need to work with a fraction that requires decisions about rounding and precision.
Type: ProblemSolving Task
This task requires students to use the fact that the value of an exponential function f(x) = a · b^x increases by a multiplicative factor of b when x increases by one. It intentionally omits specific values for c and d in order to encourage students to use this fact instead of computing the point of intersection, (p,q), and then computing function values to answer the question.
Type: ProblemSolving Task
This problem assumes that students are familiar with the notation x_{0} and ?x. However, the language "successive quotient" may be new.
Type: ProblemSolving Task
This task assumes that students are familiar with the ?x and ?y notations. Students most likely developed this familiarity in their work with slope.
Type: ProblemSolving Task
An important property of linear functions is that they grow by equal differences over equal intervals. In F.LE Equal Differences over Equal Intervals 1, students prove this for equal intervals of length one unit, and note that in this case the equal differences have the same value as the slope.
Type: ProblemSolving Task
This problemsolving task asks students to examine if linear modeling would be appropriate to describe and predict population growth from select years.
Type: ProblemSolving Task
This problemsolving task provides students an opportunity to experiment with modeling real data by using population growth rates from the past two centuries.
Type: ProblemSolving Task
This problemsolving task challenges students to write expressions and create a table to calculate how much money can be gained after investing at different banks with different interest rates.
Type: ProblemSolving Task
This problemsolving task asks students to describe exponential growth through a realworld problem involving the illegal introduction of fish into a lake.
Type: ProblemSolving Task
This problemsolving emphasizes the expectation that students know linear functions grow by constant differences over equal intervals and exponential functions grow by constant factors over equal intervals.
Type: ProblemSolving Task