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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).
• Algae Blooms
The problem statement describes a changing algae population as reported by the Maryland Department of Natural Resources. In part (a), students are expected to build an exponential function modeling algae concentration from the description given of the relationship between concentrations in cells/ml and days of rapid growth (F-LE.2). The intent of part (b) is for students to gain an appreciation for the exponential growth exhibited despite an apparently modest growth rate of 1 cell division per day.
• 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 I
In the task ''Carbon 14 Dating'' the amount of Carbon 14 in a preserved plant is studied as time passes after the plant has died. In practice, however, scientists wish to determine when the plant died and, as this task shows, this is not possible with a simple measurement of the amount of Carbon 14 remaining in the preserved plant.
• 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.
• Carbon 14 Dating, Variation 2
This exploratory task requires the student to use this property of exponential functions in order to estimate how much Carbon 14 remains in a preserved plant after different amounts of time.
• 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.
• 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.
• Comparing Graphs of Functions
The goal of this task is to use appropriate tools to compare graphs of several functions. In addition, students are asked to study the structure of the different expressions to explain why these functions grow as they do.
• DDT-cay
The purpose of this task is for students to encounter negative exponents in a natural way in the course of learning about exponential functions.
• 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.
• Dido and the Foundation of Carthage
The goal of this task is to interpret the mathematics behind a famous story from ancient mythology, giving rise to linear and quadratic expressions which model the story.
• 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).
• Equal Differences over Equal Intervals 1
Students prove that linear functions grow by equal differences over equal intervals. They will 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.
• Equal Differences over Equal Intervals 2
Linear functions grow by equal differences over equal intervals. In this task students prove the property in general (for equal intervals of any length).
• Equal Factors over Equal Intervals
Examples in this task is designed to help students become familiar with this language "successive quotient". Depending on the students's prior exposure to exponential functions and their growth rates, instructors may wish to encourage students to repeat part (b) for a variety of exponential functions and step sizes before proceeding to the most general algebraic setting in part (c).
• 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. 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. This task is preparatory for standard F.LE.1a.
• Exponential growth versus linear growth I
The purpose of this task it to have students discover how (and how quickly) an exponentially increasing quantity eventually surpasses a linearly increasing quantity.
• Exponential growth versus linear growth II
The purpose of this task it to have students discover how (and how quickly) an exponentially increasing quantity eventually surpasses a linearly increasing quantity.
• Exponential growth versus polynomial growth
This problem shows that an exponential function takes larger values than a cubic polynomial function provided the input is sufficiently large.
• 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.
• Extending the Definitions of Exponents, Variation 2
The goal of this task is to develop an understanding of why rational exponents are defined as they are (N-RN.1), however it also raises important issues about distinguishing between linear and exponential behavior (F-LE.1c) and it requires students to create an equation to model a context (A-CED.2)
• Finding Linear and Exponential Models
The goal of this task is to present students with real world and mathematical situations which can be modeled with linear, exponential, or other familiar functions. In each case, the scenario is presented and students must decide which model is appropriate.
• 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.
• Identifying Exponential Functions
The task is an introduction to the graphing of exponential functions.
• Identifying Functions
This task 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.
• Illegal Fish
This task asks students to interpret the relevant parameters in terms of the real-world context and describe exponential growth.
• In the Billions and Exponential Modeling
This problem provides an opportunity to experiment with modeling real data.
• In The Billions and Linear Modeling
This problem assumes students have completed several preliminary tasks about the fact that linear functions change by equal differences over equal intervals.
• Interesting Interest Rates
Given two bank interest rate scenarios, students will compare returns, write an expression for a balance, and create a table of values for the balances.
• 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.
• 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.
• Linear or exponential?
This task gives a variet 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.
• Mixing Candies
This task assumes students are familiar with mixing problems. This approach brings out different issues than simply asking students to solve a mixing problem, which they can often set up using patterns rather than thinking about the meaning of each part of the equations.
• 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.
• Newton's Law of Cooling
The coffee cooling experiment in this task is a popular example of an exponential model with immediate appeal. The model is realistic and provides a good context for students to practice work with exponential equations.
• 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.
• Possible or Not
Students can look at graphed functions from real-life examples and determine whether the graph makes sense or not in this activity.
• 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.
• Profit of a company, assessment variation
The primary purpose of this task is to assess students' knowledge of certain aspects of the mathematics described in the High School domain A-SSE: Seeing Structure in Expressions.
• 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.
• Rumors
This problem is an exponential function example.
• 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.
• Saturating Exponential
The context here is a familiar one: a cold beverage warms once it is taken out of the refrigerator. Rather than giving the explicit function governing this warmth, a graph is presented along with the general form of the function. Students must then interpret the graph in order to understand more specific details regarding the function.
• 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.
• Solving Problems with Linear and Exponential Models
The goal of this task is to provide examples of exponential and linear functions modeling different real world phenomena. Students will create the appropriate model and then use it to solve linear and exponential equations.
• Stairway - Student Task
This task students to design a stairway for a custom home. They will need to gather information regarding design, safety, and the utility of staircases.
• 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.
• Taxi!
This simple conceptual problem does not require algebraic manipulation, but requires students to articulate the reasoning behind each statement.
• 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.
• Triangular Numbers
The goal of this task is to work on producing a quadratic equation from an arithmetic context.
• 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.
• US Population 1790-1860
This task focuses on the fact that exponential functions are characterized by equal successive quotients over equal intervals. This task can be used alongside F-LE Equal Factors over Equal Intervals.
• US Population 1982-1988
This task provides a preliminary investigation of mathematical modeling using linear functions. In particular, students are asked to make predictions using a linear model without ever writing down an equation for a line. As such, the task could be used to motivate or introduce the observation that linear functions are precisely those that change by constant differences over equal intervals.
• 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.
• What functions do two graph points determine?
Given two points on a plane, students will demonstrate an understanding of unique linear function, unique exponential function, and quadratic function.

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.

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