 Secondary Mathematics I

Strand: FUNCTIONS - Interpreting Linear and Exponential Functions (F.IF)

Understand the concept of a linear or exponential function and use function notation. Recognize arithmetic and geometric sequences as examples of linear and exponential functions (Standards F.IF.1-3). Interpret linear or exponential functions that arise in applications in terms of a context (Standards F.IF.4-6). Analyze linear or exponential functions using different representations (Standards F.IF.7, 9).
• 1,000 is half of 2,000
This real-life modeling task could serve as a summative exercise in which many aspects of students' knowledge of functions are put to work.
• Algebra Why and When video
This video explains why and when algebra is needed instead of arithmetic functions.
• Analyzing Graphs
This task could be used as a review problem or as an assessment problem after many different types of functions have been discussed. Since the different parameters of the functions are not given explicitly, the focus is not just on graphing specific functions but rather students have to focus on how values of parameters are reflected in a graph.
• Applications of Quadratic Functions video
This video introduces and explains the topic.
• As the Wheel Turns
In this task, students use trigonometric functions to model the movement of a point around a wheel and, in the case of part (c), through space (F-TF.5). Students also interpret features of graphs in terms of the given real-world context (F-IF.4).
• Average Cost
For a function that models a relationship between two quantities, students will interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
• Bank Account Balance
The purpose of this task is to study an example of a function which varies discretely over time.
• Cell Phones
This simple task assesses whether students can interpret function notation. The four parts of the task provide a logical progression of exercises for advancing understanding of function notation and how to interpret it in terms of a given context.
• Containers
The purpose of the task is to help students think about how two quantities vary together in a context where the rate of change is not given explicitly but is derived from the context.
• Coordinates and the Cartesian Plane
This lesson helps students understand functions and the domain and range of a set of data points.
• Derivate
Students may use the applet in this lesson to graph a function and a tangent line and view its equation.
• 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).
• Domain and Range
This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review.
• Domain and Range video
This video introduces the concepts of domain and range.
• Domains
The purpose of this task to help students think about an expression for a function as built up out of simple operations on the variable, and understand the domain in terms of values for which each operation is invalid (e.g., dividing by zero or taking the square root of a negative number).
• Exponential Kiss
The purpose of this task is twofold: first using technology to study the behavior of some exponential and logarithmic graphs and secondly to manipulate some explicit logarithmic and exponential expressions.
• Finding the domain
The purpose of this task is to introduce the idea of the domain of a function by linking it to the evaluation of an expression defining the function.
• From the flight deck
This task is designed to help students learns how to Interpret functions that arise in applications in terms of the context.
• Function Flyer
The applet on this site allows the students to manipulate the graph of a function by changing the value of exponents, coefficients and constants.
• FUNCTIONS - Interpreting Linear and Exponential Functions (F.IF) - 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 Interpreting Linear and Exponential Functions (F.IF).
• Functions and the Vertical Line Test
The vertical line test for functions is the focus of this lesson plan.
• GeoGebra
GeoGebra is dynamic online geometry software. Constructions can be made with points, vectors, segments, lines, polygons, conic sections, inequalities, implicit polynomials and functions. All of them can be changed dynamically afterwards.
• Graphing Calculator
A free online graphing calculator.
• Graphing Rational Functions
This task starts with an exploration of the graphs of two functions whose expressions look very similar but whose graphs behave completely differently.
• Graphing Stories
The purpose of this task is to have students represent each indicated relationship of a given variable vs. time graphically with special attention to representing key features of increasing and decreasing intervals, maximums and minimums, intercepts, and constant and variable rates of change.
• Graphit
With this interactive applet students are able to create graphs of functions and sets of ordered pairs on the same coordinate plane.
• Graphs of Power Functions
This task requires students to recognize the graphs of different (positive) powers of x.
This exploration can be done in class near the beginning of a unit on graphing parabolas. Students need to be familiar with intercepts, and need to know what the vertex is. It is effective after students have graphed parabolas in vertex form, but have not yet explored graphing other forms.
• Hoisting the Flag 1
In this task, students are given a scenario and different graphs that could describe the relationship of the quantities in the situation, the height of a flag as a function of time.
• Hoisting the Flag 2
In this task, students are given a scenario and different graphs that could describe the relationship of the quantities in the situation, the height of a flag as a function of time.
• How is the Weather?
This task can be used as a quick assessment to see if students can make sense of a graph in the context of a real world situation. The task could also be used to generate a group discussion on interpreting functions given by graphs.
• Identifying Exponential Functions
The task is an introduction to the graphing of exponential functions.
• Identifying graphs of functions
The goal of this task is to get students to focus on the shape of the graph of an equation and how this changes depending on the sign of the exponent and on whether the exponential is in the numerator or denominator.
• Inductive Patterns
This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review.
• Inductive Patterns video
This video explains patterns and how we can use math with patterns.
• Inductive Reasoning
This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review.
• Inductive Reasoning video
This video introduces and explains the topic.
• Influenza Epidemic
The principal purpose of this task is to probe students' ability to correlate symbolic statements about a function using function notation with a graph of the function, and to interpret their answers in terms of the quantities between which the function describes a relationship.
• Interpreting the Graph
Students will use the graph (for example, by marking specific points) to illustrate the statements in (a) and (d). If possible, label the coordinates of any points you draw.
• 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.
• IXL Game: Linear Functions: Standard Form
This game will help the student understand linear functions, specifically the standard form by finding x- and y-intercepts. This is just one of many online games that supports the Utah Math core. Note: The IXL site requires subscription for unlimited use.
• Lake Sonoma
This task asks students to describe features of a graph. It provides an opportunity to introduce (or use) mathematical terminology that makes communication easier and more precise, such as: periodic behavior, maxima, minima, outliers, increasing, decreasing, slope.
• Laptop Battery Charge 2
This task uses a situation that is familiar to students to solve a problem they probably have all encountered before: How long will it take until an electronic device has a fully charged battery? A linear model can be used to solve this problem. The task combines ideas from statistics, functions and modeling. It is a nice combination of ideas in different domains in the high school curriculum.
• Linear Functions
The applet in this lesson allows students to manipulate variables and see the changes in the graphed line.
• Logistic Growth Model, Abstract Version
This task is for instructional purposes only and students should already be familiar with some specific examples of logistic growth functions such as that given in ''Logistic growth model, concrete case.''
• Logistic Growth Model, Explicit Version
This problem introduces a logistic growth model in the concrete setting of estimating the population of the U.S.
• Mathemafish Population
In this problem, students use given data points to calculate the average rate of change of a function over a specific interval, foreshadowing the idea of limits and derivatives to students.
• Model air plane acrobatics
This task could serve as an introduction to periodic functions and as a lead-in to sinusoidal functions. By visualizing the height of a plane that is moving along the circumference of a circle several times, students get the idea that output values of the height functions will repeat themselves after each complete revolution. They also connect the situation with key features on the graph, for example they interpret the midline and amplitude of the function as the height of the center of the circle and its radius.
• Modeling London's Population
The purpose of this task is to model the population data for London with a variety of different functions. In addition to the linear, quadratic, and exponential models, this task introduces an additional model, namely the logistic model.
• 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.
• Module 3: Features of Functions - Student Edition (Math 1)
The Mathematics Vision Project, Secondary Math One Module 3, Features of Functions, is the culminating functions module in Secondary Math I. In this module, students broaden their thinking about functions to relationships that are not either linear or exponential.
• Module 3: Features of Functions - Teacher Notes (Math 1)
The Mathematics Vision Project, Secondary Math One Module 3 Teacher Notes, Features of Functions, is the culminating functions module in Secondary Math I. In this module, students broaden their thinking about functions to relationships that are not either linear or exponential.
• Module 8: Connecting Algebra & Geometry - Student Edition (Math 1)
The Mathematics Vision Project, Secondary Math One Module 8, Connecting Algebra and Geometry, students use the Pythagorean Theorem to find the distance between two points and to derive the distance formula.
• Module 8: Connecting Algebra & Geometry - Teacher Notes (Math 1)
The Mathematics Vision Project, Secondary Math One Module 8 Teacher Notes, Connecting Algebra and Geometry, students use the Pythagorean Theorem to find the distance between two points and to derive the distance formula.
• Multi-Function Data Flyer
The applet in this lesson allows students to plot ordered pairs and then change the values in order to observe the effects of those changes.
• Oakland Coliseum
This deceptively simple task asks students to find the domain and range of a function from a given context. The function is linear and if simply looked at from a formulaic point of view, students might find the formula for the line and say that the domain and range are all real numbers. However, in the context of this problem, this answer does not make sense, as the context requires that all input and output values are non-negative integers, and imposes additional restrictions.
• Pizza Place Promotion
Students will use a function that models a relationship between two quantities to figure out how a pizza restaurant's promotion that prices pizza based on a function of time causes the cost to fluctuate.
• Playing Catch
This task gives the graph of the height of a ball over time and asks for a story that could be represented by this graph. The graph is the mathematical representation of a situation and features of the graph correspond to specific moments in the story the graph tells. The purpose of the task is to get away from plotting graphs by focusing on coordinate points and instead looking at the bigger picture a qualitative view.
• Points on a Graph
This task is designed to get at a common student confusion between the independent and dependent variables. This confusion often arises in situations like (b), where students are asked to solve an equation involving a function, and confuse that operation with evaluating the function.
• Polynomials video
The video introduces and explains the topic.
• 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.
• Random Walk II
This task follows up on ''The Random Walk,'' looking in closer detail at what outcomes are possible. These problems form a bridge between work on functions and work on probability. The task is better suited for instruction than for assessment as it provides students with a non standard setting in which to interpret the meaning of functions. Students should carry out the process of flipping a coin and modeling this Random Walk in order to develop a sense of the process before analyzing it mathematically.
• Rate of Change and Slope
This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review.
• Rate of Change and Slope video
This video introduces the concepts.
Through this lesson students will understand how to graph functions.
• Representing Functions and Relations
This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review.
• Representing Functions and Relations video
Explains how algebra can be used to describe, represent and predict relations.
• Representing Patterns
This collection of resources to teach graphing equations in slope intercept form includes warm-up exercises, a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review.
• Representing Patterns video
This video introduces tables and graphs as representations of patterns.
• Representing Polynomials
This lesson unit is intended to help educators assess how well students are able to translate between graphs and algebraic representations of polynomials.
• Running Time
This task provides an application of polynomials in computing. This purpose of this task is to serve as an introduction, and motivation, for the study of end behavior of polynomials, content specifically addresses in standard F-IF.C7c.
• Secondary I Textbook
Secondary I Textbook is composed of modules that are aligned with the Utah Core State Standards for Mathematics. Each lesson begins with a worthwhile task that has been designed to develop mathematical understanding, solidify that understanding, or allow for practice of the new concepts, while focusing on the mathematical goals of the chosen learning cycle.
• Sequencer
By using this applet students are able to create sequences by changing the values of starting numbers, multipliers, and add-ons.
• Snake on a Plane
This task has students approach a function via both a recursive and an algebraic definition, in the context of a famous game of antiquity that they may have encountered in a more modern form.
The task is a seemingly straightforward modeling task that can lead to more involved tasks if the instructor expands on it. In this task, students also have to interpret the units of the input and output variables of the solar radiation function.
This video introduces and explains the topic.
This task consists of a set of 2 short questions.
Students are given four graphs, four equations, four tables, and four rules. Their task is to match each graph with an equation, a table and a rule.
• Telling a Story With Graphs
In this task students are given graphs of quantities related to weather. The purpose of the task is to show that graphs are more than a collection of coordinate points, that they can tell a story about the variables that are involved and together they can paint a very complete picture of a situation, in this case the weather.
• Temperature Change
This task gives an easy context to introduce the idea of average rate of change. This problem could be done as a Think-Pair-Share activity. After posing the question, students can decide what they think and why and then discuss their answer with their neighbor.
• The Aquarium
The purpose of this task is to connect graphs with real life situations. Graphs tell a story. Specific features of a graph connect to specific features of a story. A point on a graph captures a specific instant in the story.
• The Canoe Trip, Variation 1
The purpose of this task is to give students practice constructing functions that represent a quantity of interest in a context, and then interpreting features of the function in the light of that context. It can be used as either an assessment or a teaching task.
• The Canoe Trip, Variation 2
The primary purpose of this task is to lead students to a numerical and graphical understanding of the behavior of a rational function near a vertical asymptote, in terms of the expression defining the function. The canoe context focuses attention on the variables as numbers, rather than as abstract symbols.
• The Customers
The purpose of this task is to introduce or reinforce the concept of a function, especially in a context where the function is not given by an explicit algebraic representation. Further, the last part of the task emphasizes the significance of one variable being a function of another variable in an immediately relevant real-life context. Instructors might prepare themselves for variations on the problems that the students might wander into (e.g., whether one person could have two home phone numbers) and how such variants affect the correct responses.
• The High School Gym
In this task, students will calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval.
• The Parking Lot
The purpose of this task is to investigate the meaning of the definition of function in a real-world context where the question of whether there is more than one output for a given input arises naturally. In more advanced courses this task could be used to investigate the question of whether a function has an inverse.
• The Random Walk
This task requires interpreting a function in a non-standard context. While the domain and range of this function are both numbers, the way in which the function is determined is not via a formula but by a (pre-determined) sequence of coin flips. In addition, the task provides an opportunity to compute some probabilities in a discrete situation. The task could be used to segue the discussion from functions to probability, in particular the early standards in the S-CP domain.
• The Restaurant
The purpose of this task is to get students thinking about the domain and range of a function representing a particular context.
• The story of a flight
This task uses data from an actual flight computer.
• Using Function Notation I
This task deals with a student error that may occur while students are completing F-IF Average Cost.
• Using Function Notation II
The purpose of the task is to explicitly identify a common error made by many students, when they make use of the "identity" f(x+h)=f(x)+f(h). The task has students find a single explicit example for which the identity is false, but it is worth emphasizing that in fact the identity fails for the vast majority of functions.
• Warming and Cooling
This task is meant to be a straight-forward assessment task of graph reading and interpreting skills. This task helps reinforce the idea that when a variable represents time, t=0 is chosen as an arbitrary point in time and positive times are interpreted as times that happen after that.
• Words - Tables - Graphs
The purpose of the task is to show that graphs can tell a story about the variables that are involved.
• Yam in the Oven
The purpose of this task is to give students practice interpreting statements using function notation. It can be used as a diagnostic if students seem to be having trouble with function notation, for example interpreting f(x) as the product of f and x. 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.