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Distinguish between situations that can be modeled with linear functions and with exponential functions. ★
  1. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
  2. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
  3. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
Standard #: MAFS.912.F-LE.1.1Archived Standard
Standard Information
General Information
Subject Area: Mathematics
Grade: 912
Domain-Subdomain: Functions: Linear, Quadratic, & Exponential Models
Cluster: Level 3: Strategic Thinking & Complex Reasoning
Cluster: Construct and compare linear, quadratic, and exponential models and solve problems. (Algebra 1 - Supporting Cluster) (Algebra 2 - Supporting Cluster) -

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.
Date Adopted or Revised: 02/14
Content Complexity Rating: Level 3: Strategic Thinking & Complex Reasoning - More Information
Date of Last Rating: 02/14
Status: State Board Approved - Archived
Assessed: Yes
Related Courses
Related Resources
Formative Assessments
  • How Does Your Garden Grow? 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.
  • Exponential Growth 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.
  • Prove Linear Students are asked to prove that a linear function grows by equal differences over equal intervals.
  • Prove Exponential Students are asked to prove that an exponential function grows by equal factors over equal intervals.
  • Linear or Exponential? Students are given four verbal descriptions of functions and asked to identify each as either linear or exponential and to justify their choices.
Lesson Plans
  • The Towers of Hanoi: Experiential Recursive Thinking
    This lesson is about the Towers of Hanoi problem, a classic famous problem involving recursive thinking to reduce what appears to be a very large and difficult problem into a series of simpler ones.  The learning objective is for students to begin to understand recursive logic and thinking, relevant to computer scientists, mathematicians and engineers.   The lesson is experiential, in that each student will be working with her/his own Towers of Hanoi manipulative, inexpensively obtained.  There is no formal prerequisite, although some familiarity with set theory and functions is helpful.  The last three sections of the lesson involve some more formal concepts with recursive equations and proof by induction, so the students who work on those sections should probably be level 11 or 12 in a K-12 educational system.  The lesson has a Stop Point for 50-minute classes, followed by three more segments that may require a half to full additional class time.  So the teacher may use only those segments up to the Stop Point, or if two class sessions are to be devoted to the lesson, the entire set of segments.  Supplies are modest, and may be a set of coins or some washers from a hardware store to assemble small piles of disks in front of each student, each set of disks representing a Towers of Hanoi manipulative.  Or the students may assemble before the class a more complete Towers of Hanoi at home, as demonstrated in the video.  The classroom activities involve attempting to solve with hand and mind the Towers of Hanoi problem and discussing with fellow students patterns in the process and strategies for solution.
  • You’re Pulling My Leg – or Candy! Students will explore how exponential growth and decay equations can model real-world problems. Students will also discover how manipulating the variables in an exponential equation changes the graph. Students will watch a Perspectives Video to see how exponential growth is modeled in the real world.
  • Modeling Exponential Growth: Having Kittens This lesson assesses how well students can interpret a situation and represent the constraints and variables mathematically, select appropriate mathematical methods to use, make sensible estimates and assumptions, investigate an exponentially increasing sequence, and communicate their reasoning clearly.
  • Functions of Everyday Situations This lesson first assesses how well students can graph relationships between variables from a description of an everyday situation. It also guides student interpretation of algebraic functions in terms of the contexts in which they arise, and reflects on their domains, as well as determines whether they should be discrete or continuous.
  • Appreciation for Car Depreciation Students will use information from the internet or a car dealership's advertisement to identify a car and determine the future value of the car using different depreciation rates over different intervals of time. Students will graph their data to show exponential decay and compare to a linear decrease on the same graph.
  • BIOSCOPES Summer Institute 2013 - Motion This lesson is the first in a sequence of grade 9-12 physical science lessons that are organized around the big ideas that frame motion, forces, and energy. It directly precedes resource # 52648 "BIOSCOPES Summer Institute 2013 - Forces." This lesson is designed along the lines of an iterative 5-E learning cycle and employs a predict, observe, and explain (POE) activity at the beginning of the "Engage" phase in order to elicit student prior knowledge. The POE is followed by a sequence of inquiry-based activities and class discussions that are geared toward leading the students systematically through the exploration of 1-dimensional motion concepts. Included in this resource is a summative assessment as well as a teacher guide for each activity.

  • Falling for Gravity Students will investigate the motion of three objects of different masses undergoing free fall. Additionally, students will:
    • Use spark timers to collect displacement and time data.
    • Use this data to calculate the average velocity for the object during each interval.
    • Graph this data on a velocity versus time graph, V-t. They find the slope of this graph to calculate acceleration.
    • Calculate the falling object's acceleration from their data table and graph this data on an acceleration versus time graph, a-t.
    • Use their Spark timer data paper, cut it into intervals, and paste these intervals into their displacement versus time graph.
  • Which Function? This activity has students apply their knowledge to distinguish between numerical data that can be modeled in linear or exponential forms. Students will create mathematical models (graph, equation) that represent the data and compare these models in terms of the information they show and their limitations. Students will use the models to compute additional information to predict future outcomes and make conjectures based on these predictions.
  • Piles of Paper This hands-on student activity engages learners in exploring the differences between linear and exponential growth by examining the heights of flat versus folded paper. Students collect data, organize it into tables, and then develop algebraic models to represent each type of growth. To deepen understanding, the activity also introduces real-world examples that illustrate both linear and exponential patterns in everyday life.
Original Student Tutorial
  • Exponential Functions Part 2: Growth 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.
Perspectives Video: Expert
Perspectives Video: Professional/Enthusiasts
Problem-Solving Tasks
  • Algae Blooms In this example, students are asked to write a function describing the population growth of algae. It is implied that this is exponential growth.
  • What functions do two graph points determine? This problem solving task challenges students to find the linear, exponential and quadratic functions based on two points.
  • US Population 1982-1988 This problem solving task asks students to predict and model US population based on a chart of US population data from 1982 to 1988.
  • US Population 1790-1860 This problem solving task asks students to solve five exponential and linear function problems based on a US population chart for the years 1790-1860.
  • Rising Gas Prices - Compounding and Inflation 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.
  • Linear or exponential? This task gives a variation of real-life 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.
  • Linear Functions This task requires students to use the fact that on the graph of the linear function h(x) = ax + b, the y-coordinate 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.
  • Comparing Exponentials 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.
  • Carbon 14 Dating, Variation 2 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.
  • Basketball Rebounds 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.
  • Basketball Bounces, Assessment Variation 2 This task asks students to analyze data from a physical context, choose a model that reasonably fits the data, and use the model to answer questions about the physical context. This variant of the task is not scaffolded; for a more scaffolded version, see Basketball Bounces, Assessment Variation 1.
  • Basketball Bounces, Assessment Variation 1 Students are asked to select the best model for a given context and use the model to make predictions. This task assesses students’ modeling skills. Students are tasked to distinguish between situations that can be modeled with linear and exponential functions and recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
  • Exponential Functions This task requires students to use the fact that the value of an exponential function f(x) = a * bX increases by a multiplicative factor of b when x increases by one. Students will use their knowledge about the coordinate plane and graphing to relate two exponential functions. It intentionally omits specific values for c and d to encourage students to apply this knowledge rather than computing the point of intersection, and instead, compute function values to answer the question.
  • Equal Factors over Equal Intervals This problem assumes that students are familiar with the notation x0 and Δx. However, the language "successive quotient" may be new. Examples in the third column of the charts are designed to help students become familiar with this language. Two other tasks, Equal Differences over Equal Intervals 1 (CPALMS resource #43645) and Equal Differences over Equal Intervals 2 (CPALMS resource #43647), illustrate linear functions.
  • Equal Differences over Equal Intervals 2 This task assumes that students are familiar with the Δx and Δy notations. Students most likely developed this familiarity in their work with slope. In “Equal Differences over Equal Intervals 2”, students prove the property in general (for equal intervals of any length). This task should follow “Equal Differences over Equal Intervals 1” (resource #43645), which has students prove that linear equations represent equal differences over equal lengths of one.
  • Equal Differences over Equal Intervals 1

    An important property of linear functions is that they grow by equal differences over equal intervals. In 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.

    In Equal Differences over Equal Intervals 2 (resource #43647), students prove the property in general (for equal intervals of any length).

  • Linear Modeling in the Billions This problem-solving task asks students to examine if linear modeling would be appropriate to describe and predict population growth from select years.
  • Exponential Modeling in the Billions In this problem-solving task, students analyze real data using population growth rates from the past two centuries using exponential functions.
  • Interesting Interest Rates This problem-solving 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.
  • Illegal Fish This problem-solving task asks students to describe exponential growth through a real-world problem involving the illegal introduction of fish into a lake.
  • Identifying Functions This open-ended task presents a table of values for four different types of functions: linear, quadratic, and exponential. Students are asked to indicate which function type corresponds to each list of values.
Professional Development
  • Mathematical Modeling: Insights into Algebra, Teaching for Learning This professional development resource provides a rich collection of information to help teachers engage students more effectively in mathematical modeling. It features videos of two complete lessons with commentary, background information on effective teaching, modeling, and lesson study, full lesson plans to teach both example lessons, examples of student work from the lessons, tips for effective teaching strategies, and list of helpful resources.
    • In Lesson 1 students use mathematical models (tables and equations) to represent the relationship between the number of revolutions made by a "driver" and a "follower" (two connected gears in a system), and they will explain the significance of the radii of the gears in regard to this relationship.
    • In Lesson 2 students mathematically model the growth of populations and use exponential functions to represent that growth.
Video/Audio/Animation
  • MIT BLOSSOMS - Flu Math Games This video lesson shows students that math can play a role in understanding how an infectious disease spreads and how it can be controlled. During this lesson, students will see and use both deterministic and probabilistic models and will learn by doing through role-playing exercises. There are no formal prerequisites, as students in any high school or even middle school math class could enjoy this learning video. But more advanced classes can go into the optional applied probability modeling that accompanies the module in a downloadable pdf file. The primary exercises between video segments of this lesson are class-intensive simulation games in which members of the class 'infect' each other under alternative math modeling assumptions about disease progression. Also there is an occasional class discussion and local discussion with nearby classmates.
MFAS Formative Assessments
  • Exponential Growth 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.
  • How Does Your Garden Grow? 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.
  • Linear or Exponential? Students are given four verbal descriptions of functions and asked to identify each as either linear or exponential and to justify their choices.
  • Prove Exponential Students are asked to prove that an exponential function grows by equal factors over equal intervals.
  • Prove Linear Students are asked to prove that a linear function grows by equal differences over equal intervals.
Original Student Tutorials Mathematics - Grades 9-12
  • Exponential Functions Part 2: Growth 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.
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