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Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). ★
Standard #: MAFS.912.F-LE.1.2Archived Standard
Standard Information
General Information
Subject Area: Mathematics
Grade: 912
Domain-Subdomain: Functions: Linear, Quadratic, & Exponential Models
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 2: Basic Application of Skills & Concepts - More Information
Date of Last Rating: 02/14
Status: State Board Approved - Archived
Assessed: Yes
Related Courses
Related Resources
Formative Assessments
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.
  • Functions and Everyday Situations # This lesson unit is intended to help you assess how well students are able to articulate verbally the relationships between variables arising in everyday contexts, translate between everyday situations and sketch graphs of relationships between variables, interpret algebraic functions in terms of the contexts in which they arise and reflect on the domains of everyday functions and in particular whether they should be discrete or continuous.
  • Piles of Paper # Piles of Paper is a student activity that demonstrates linear and exponential growth using heights of flat and folded paper. Data tables are created and then algebraic models are developed. Real world types of linear and exponential growth are also introduced.
  • Determining the Hubble Constant # Students will graph distance/velocity data of real galaxies to arrive at their own value of the Hubble constant (H). Once they have calculated their own value of H, they will use it to determine distances to real galaxies with known recessional velocities.
Original Student Tutorial
  • Creating Exponential Functions # Follow as we construct an exponential function from a graph, from a table of values, and from a description of a relationship in the real world in this interactive tutorial.
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.
  • Two Points Determine an Exponential Function II # This problem solving tasks asks students to find the values of points on a graph.
  • Two Points Determine an Exponential Function I # This problem solving task asks students to graph a function and find the values of points on a graph.
  • Sandia Aerial Tram #

    The task provides an opportunity for students to engage in detailed analysis of the rate of change of the elevation.

  • Rumors # This problem is an exponential function example that uses the real-world problem of how fast rumors spread.
  • Do two points always determine an exponential function? # This problem complements the problem "Do two points always determine a linear function?'' There are two constraints on a pair of points R1 and R2 if there is an exponential function f(x) = ae^bx whose graph contains R1 and R2.
  • Do two points always determine a linear function? # In a geometric context, two distinct points ??1 and ??2 always determine a unique line in the Cartesian plane (this is one of the basic postulates of Euclidean geometry). Only the non-vertical lines, however, can be described by the graph of a function ??(??)=????+??. This task focuses on producing an explicit function ??(??) as long as the line is not vertical. This problem allows the student to think geometrically about lines and then relate this geometry to linear functions. Or the student can work algebraically with equations to find the explicit equation of the line through two points (when that line is not vertical).
  • Carbon 14 Dating in Practice II # This problem introduces the method used by scientists to date certain organic material. It is based not on the amount of the Carbon 14 isotope remaining in the sample but rather on the ratio of Carbon 14 to Carbon 12. This ratio decreases, hypothetically, at a constant exponential rate as soon as the organic material has ceased to absorb Carbon 14, that is, as soon as it dies.
  • 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.
  • Population and Food Supply # In this task students use verbal descriptions to construct and compare linear and exponential functions and to find where the two functions intersect (F-LE.2, F-LE.3, A-REI.11).
  • Doubling your money # This task asks students to write equations to predict how much money will be in a savings account at the end of each year, based on different factors like interest rates.
Unit/Lesson Sequences
  • Sample Algebra 1 Curriculum Plan Using CMAP #
    This sample Algebra 1 CMAP is a fully customizable resource and curriculum-planning tool that provides a framework for the Algebra 1 Course. The units and standards are customizable and the CMAP allows instructors to add lessons, worksheets, and other resources as needed. This CMAP also includes rows that automatically filter and display Math Formative Assessments System tasks, E-Learning Original Student Tutorials and Perspectives Videos that are aligned to the standards, available on CPALMS. Learn more about the sample Algebra 1 CMAP, its features and customizability by watching the following video:

    Using this CMAP

    To view an introduction on the CMAP tool, please . To view the CMAP, click on the "Open Resource Page" button above; be sure you are logged in to your iCPALMS account.To use this CMAP, click on the "Clone" button once the CMAP opens in the "Open Resource Page." Once the CMAP is cloned, you will be able to see it as a class inside your iCPALMS My Planner (CMAPs) app. To access your My Planner App and the cloned CMAP, click on the iCPALMS tab in the top menu.All CMAP tutorials can be found within the iCPALMS Planner App or at the following URL: http://www.cpalms.org/support/tutorials_and_informational_videos.aspx
  • Direct and Inverse Variation # "Lesson 1 of two lessons teaches students about direct variation by allowing them to explore a simulated oil spill using toilet paper tissues (to represent land) and drops of vegetable oil (to simulate a volume of oil). Lesson 2 teaches students about inverse variation by exploring the relationship between the heights of a fixed amount of water poured into cylindrical containers of different sizes as compared to the area of the containers' bases." from Insights into Algebra 1 - Annenberg Foundation.
Virtual Manipulatives
  • Linear Function Machine # In this activity, students plug values into the independent variable to see what the output is for that function. Then based on that information, they have to determine the coefficient (slope) and constant(y-intercept) for the linear function. This activity allows students to explore linear functions and what input values are useful in determining the linear function rule. 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.
  • Introduction to Functions # This lesson is designed to introduce students to functions as rules and independent and dependent variables. The lesson provides links to discussions and activities that motivate the idea of a function as a machine as well as proper terminology when discussing functions. Finally, the lesson provides links to follow-up lessons designed for use in succession to the introduction of functions.
  • Data Flyer # 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.
MFAS Formative Assessments
Original Student Tutorials Mathematics - Grades 9-12
  • Creating Exponential Functions # Follow as we construct an exponential function from a graph, from a table of values, and from a description of a relationship in the real world in this interactive tutorial.
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