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Mathematics - Secondary Curriculum Secondary Mathematics I
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Strand: FUNCTIONS - Linear and Exponential (F.LE)

Construct and compare linear and exponential models and solve problems (Standards F.LE.1 3). Interpret expressions for functions in terms of the situation they model. (Standard F.LE.5).

Standard F.LE.2

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).

  • Allergy medication
    The purpose of the task is to help students become accustomed to evaluating exponential functions at non-integer inputs and interpreting the values.
  • Basketball Bounces, Assessment Variation 1
    This task asks students to analyze a set of data from a physical context, choose a model that reasonably fits the data, and use the model to answer questions about the physical context.
  • Basketball Bounces, Assessment Variation 2
    This task asks students to analyze a set of data from a physical context, choose a model that reasonably fits the data, and use the model to answer questions about the physical context.
  • 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.
  • Boiling Water
    This task examines linear models for the boiling point of water as a function of elevation. Two sets of data are provided and each is modeled quite well by a linear function.
  • Boom Town
    The purpose of this task is to give students experience working with simple exponential models in situations where they must evaluate and interpret them at non-integer inputs.
  • 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.
  • Choosing an appropriate growth model
    The goal of this task is to examine some population data from a modeling perspective. Because large urban centers and their growth are governed by many complex factors, we cannot expect a simple model (linear, quadratic, or exponential) to give accurate values or predictions over large stretches of time. Deciding on an appropriate model is a delicate process requiring careful analysis.
  • Decaying Dice
    This task provides concrete experience with exponential decay. It is intended for students who know what exponential functions are, but may not have much experience with them, perhaps in an Algebra 1 course.
  • Do two points always determine a linear function II?
    This task is designed as a follow-up to the task F-LE Do Two Points Always Determine a Linear Function? Linear equations and linear functions are closely related, and there advantages and disadvantages to viewing a given problem through each of these points of view. This task is not intended for assessment purposes: rather it is intended to show the depth of the standard F-LE.2 and its relationship to other important concepts of the middle school and high school curriculum, including ratio, algebra, and geometry.
  • Do two points always determine a linear function?
    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 in order to find the explicit equation of the line through two points (when that line is not vertical).
  • Do two points always determine an exponential function?
    This task asks students to 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).
  • Exponential Parameters
    The task provides a reasonably straight-forward introduction to interpreting the parameters of an exponential function in terms of a modeling context. The task has students both generate an exponential expression from a contextual description, and in reverse, interpret parameters in a context from an algebraic expression.
  • Finding Parabolas through Two Points
    In this task students are asked to find all quadratic functions described by given equations.
  • FUNCTIONS - Linear and Exponential (F.LE) - Sec Math I Core Guide
    The Utah State Board of Education (USBE) and educators around the state of Utah developed these guides for the Secondary Mathematics I - Linear and Exponential (F.LE).
  • Functions and the Vertical Line Test
    The vertical line test for functions is the focus of this lesson plan.
  • Graphs and Functions
    This lesson plan is designed to help the student understand how to plot functions on the Cartesian plane and how the graphing of functions leads to lines and parabolas.
  • Identifying Exponential Functions
    The task is an introduction to the graphing of exponential functions.
  • Introduction to Functions
    This lesson introduces students to functions and how they are represented as rules and data tables. They also learn about dependent and independent variables.
  • Introduction to the Materials (Math 1)
    Introduction to the Materials in the Mathematics One of the The MVP classroom experience begins by confronting students with an engaging task and then invites them to grapple with solving it. As students ideas emerge, take form, and are shared, the teacher orchestrates the student discussions and explorations towards a focused mathematical goal. As conjectures are made and explored, they evolve into mathematical concepts that the community of learners begins to embrace as effective strategies for analyzing and solving problems.
  • Modeling: Having Kittens
    This lesson unit is intended to help you assess how well students are able to interpret a situation and represent the constraints and variables mathematically, select appropriate mathematical methods to use, make sensible estimates and assumptions and investigate an exponentially increasing sequence.
  • Module 1: Sequences - Student Edition (Math 1)
    The Mathematics Vision Project, Secondary Math One Module 1, Sequences is written as two intertwined learning cycles that begin by alternating from arithmetic sequence to geometric sequences, so students can compare and contrast features as they represent both types of sequences with tables, graphs, story contexts, diagrams, and equations.
  • Module 1: Sequences - Teacher Notes (Math 1)
    The Mathematics Vision Project, Secondary Math One Module 1 Teacher Notes, Sequences is written as two intertwined learning cycles that begin by alternating from arithmetic sequence to geometric sequences, so students can compare and contrast features as they represent both types of sequences with tables, graphs, story contexts, diagrams, and equations.
  • Module 2: Linear & Exponential Functions - Student Edition (Math 1)
    The Mathematics Vision Project, Secondary Math One Module 2, Linear and Exponential Functions, begins with a learning cycle that introduces contexts with continuous domains and defining linear functions as having a constant rate of change and exponential functions as having a constant ratio over equal intervals.
  • Module 2: Linear & Exponential Functions - Teacher Notes (Math 1)
    Mathematics Vision Project, Secondary Math One Module 2 Teacher Notes, Linear and Exponential Functions, begins with a learning cycle that introduces contexts with continuous domains and defining linear functions as having a constant rate of change and exponential functions as having a constant ratio over equal intervals.
  • Moore's Law and Computers
    The goal of this task is to construct and use an exponential model to approximate hard disk storage capacity on personal computers.
  • More Complicated Functions: Introduction to Linear Functions
    This lesson is designed to reinforce the concept of linear functions and ask students to write functions using English, tables and algebra.
  • Paper Folding
    This is a very open-ended task designed for students to develop some of the basic ideas surrounding exponential growth.
  • Population and Food Supply
    In this task students construct and compare linear and exponential functions and find where the two functions intersect.
  • Predicting the Past
    The purpose of this instructional task is to provide an opportunity for students to use and interpret the meaning of a negative exponent in a functional relationship.
  • Rumors
    This problem is an exponential function example.
  • Sample Assessment Task: Golf Balls in Water
    This sample task requires that students model approximately linear data with a linear function. To do so, they must use contextual information and construct representations of linear relationships as well as interpret parameters of the linear models in the context. Use the navigation at the upper right of this page to access the task.
  • Sandia Aerial Tram
    Students are asked to write an equation for a function (linear, quadratic, or exponential) that models the relationship between the elevation of the tram and the number of minutes into the ride.
  • Sequencer
    By using this applet students are able to create sequences by changing the values of starting numbers, multipliers, and add-ons.
  • Snail Invasion
    The purpose of this task is to give students experience modeling a real-world example of exponential growth, in a context that provides a vivid illustration of the power of exponential growth, for example the cost of inaction for a year.
  • Student Task: Table Tiling
    In this task, students must work out how many whole, half and quarter tiles tiles are needed to cover the tops of tables of different sizes.
  • Temperatures in degrees Fahrenheit and Celsius
    Temperature conversions provide a rich source of linear functions which are encountered not only in science but also in our every day lives when we travel abroad. The first part of this task provides an opportunity to construct a linear function given two input-output pairs. The second part investigates the inverse of a linear function while the third part requires reasoning about quantities and/or solving a linear equation.
  • Two Points Determine an Exponential Function I
    Given the graph of a function students must find the value of 2 variables.
  • Two Points Determine an Exponential Function II
    Given the graph of a function students must find the value of 2 variables.
  • Valuable Quarter
    Successful work on this task involves modeling a bank account balance with an exponential function and then solving an exponential equation arising from the given information. This can be done either by extracting a root or taking a logarithm: either method will require a calculator in order to evaluate the expressions. Students will also need to be familiar with the context of annual interest and of compounding interest.


UEN logo http://www.uen.org - in partnership with Utah State Board of Education (USBE) and Utah System of Higher Education (USHE).  Send questions or comments to USBE Specialist - Lindsey  Henderson and see the Mathematics - Secondary website. For general questions about Utah's Core Standards contact the Director - Jennifer  Throndsen.

These materials have been produced by and for the teachers of the State of Utah. Copies of these materials may be freely reproduced for teacher and classroom use. When distributing these materials, credit should be given to Utah State Board of Education. These materials may not be published, in whole or part, or in any other format, without the written permission of the Utah State Board of Education, 250 East 500 South, PO Box 144200, Salt Lake City, Utah 84114-4200.