Strand: FUNCTIONS (8.F)

Define, evaluate, and compare functions (Standards 8.F.1-3). Use functions to model relationships between quantities (Standards 8.F.4-5).
• Baseball Cards
This task could be put to good use in an instructional sequence designed to develop knowledge related to students' understanding of linear functions in contexts.
• Battery Charging
This task has students engaging in a simple modeling exercise, taking verbal and numerical descriptions of battery life as a function of time and writing down linear models for these quantities. To draw conclusions about the quantities, students have to find a common way of describing them.
• Bike Race
The purpose of this task is for students to interpret two distance-time graphs in terms of the context of a bicycle race. There are two major mathematical aspects to this: interpreting what a particular point on the graph means in terms of the context, and understanding that the "steepness" of the graph tells us something about how fast the bicyclists are moving.
• Chapter 2 - Mathematical Foundation (UMSMP)
This is Chapter 2 of the Utah Middle School Math: Grade 8 textbook. It provides a Mathematical Foundation for Proportional and Linear Relationships.
• Chapter 2 - Student Workbook (UMSMP)
This is Chapter 2 of the Utah Middle School Math: Grade 8 student workbook. It covers Proportional and Linear Relationships.
• Chapter 3 - Mathematical Foundation (UMSMP)
This is Chapter 3 of the Utah Middle School Math: Grade 8 textbook. It provides a Mathematical Foundation for Representations of a Line.
• Chapter 3 - Student Workbook (UMSMP)
This is Chapter 3 of the Utah Middle School Math: Grade 8 student workbook. It covers Representations of a Line.
• Chapter 4 - Mathematical Foundation (UMSMP)
This is Chapter 4 of the Utah Middle School Math: Grade 8 textbook. It provides a Mathematical Foundation for Functions.
• Chapter 4 - Student Workbook (UMSMP)
This is Chapter 4 of the Utah Middle School Math: Grade 8 student workbook. It focuses on Functions.
• Chapter 5 - Mathematical Foundation (UMSMP)
This is Chapter 5 of the Utah Middle School Math: Grade 8 textbook. It provides a Mathematical Foundation for Simultaneous Linear Equations.
• Chapter 5 - Student Workbook (UMSMP)
This is Chapter 5 of the Utah Middle School Math: Grade 8 student workbook. It focuses on Simultaneous Linear Equations.
• Chicken and Steak, variation 1
This task presents a real world situation that can be modeled with a linear function best suited for an instructional context.
• Chicken and Steak, variation 2
This task is intended strictly for instructional purposes with the goal of building understandings of linear relationships within a meaningful and, hopefully, somewhat familiar context.
• Comparing Algorithms: Guess My Rule
Use a function machine to play a game where you guess three mystery algorithms, then check to see if you're correct. In the activity you test function rules and find relationships between those rules and graphs. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Comparing Exponential, Quadratic, and Linear Functions
This interactive requires the student to examine functional relationships to determine whether they are quadratic, exponential, or linear. The classroom activity for the lesson shows the student 3 graphs and has them determine what sort of function they reflect. They also solve word problems using the interactive activity. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
This Teaching Channel video demonstrates a lesson where students analyze patterns and represent functions. (9 minutes)
• Coordinates and the Cartesian Plane
This lesson helps students understand functions and the domain and range of a set of data points.
• Delivering the Mail, Assessment Variation
This task involves constructing a linear function and interpreting its parameters in a context. Thus, this task has a medium level of complexity
• Distance
In this task students interpret two graphs that look the same but show very different quantities. The first graph gives information about how fast a car is moving while the second graph gives information about the position of the car. This problem works well to generate a class or small group discussion. Students learn that graphs tell stories and have to be interpreted by carefully thinking about the quantities shown.
• Distance Across the Channel
This task asks students to find a linear function that models something in the real world. After finding the equation of the linear relationship between the depth of the water and the distance across the channel, students have to verbalize the meaning of the slope and intercept of the line in the context of this situation
• Downhill
This task provides an opportunity to compare and contrast the graph of a function and what it represents with a drawing of the hill and the vertical and horizontal distances traversed with each mile down the slope.
• Exploring Linear Functions: Representational Relationships
This lesson plan helps students better understand linear functions by allowing them to manipulate values and get a visual representation of the result.
• Exploring Reasoning and Proof
Questions requiring geometric reasoning are applied to the icing of a cake in this Annenberg interactive. Students must estimate the amount of frosting needed to cover a whole cake. The classroom activity focuses on pattern recognition and geometric reasoning. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Finding Patterns to Make Predictions
This activity asks students to identify and contemplate mathematical patterns that we see around us. They are asked to represent them in a table and predict the pattern to the 7th, 9th, and nth terms. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Foxes and Rabbits
This task emphasizes the importance of the "every input has exactly one output" clause in the definition of a function by using the example of fox and rabbit populations.
• Function Machine
Students learn how a function machine works in this interactive from Annenberg and then use them to answer questions. Students must then apply function rules to problems, plot points on graphs, and solve rules problems. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Function Rules
The purpose of this task is to connect the a function described by a verbal rule with corresponding values in a table (one of six connections to be made between the four ways to represent a function, the other two being through its graph and through an expression). It also encourages students to think more broadly about functions as relating objects other than numbers, although this broad application is not intended to be assessed. Because of its ambiguity, this task would be more suitable for use in a classroom than for assessment.
• Functional Relationships Between Quantities: Calculating Fuel Consumption
This lesson consists of interactive visualizations to help students examine the relationship between a car's mpg to gallons per mile. They can use the interactive slider to see how the relationship changes as a car's efficiency is changed. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Functions (8.F) - 8th Grade Core Guide
The Utah State Board of Education (USBE) and educators around the state of Utah developed these guides for Mathematics Grade 8 Functions.
• Grade 8 Math Module 5: Examples of Functions from Geometry (EngageNY)
In the first topic of this 15 day 8th grade module, students learn the concept of a function and why functions are necessary for describing geometric concepts and occurrences in everyday life. Once a formal definition of a function is provided, students then consider functions of discrete and continuous rates and understand the difference between the two. Students apply their knowledge of linear equations and their graphs from Module 4 to graphs of linear functions. Students inspect the rate of change of linear functions and conclude that the rate of change is the slope of the graph of a line. They learn to interpret the equation y=mx+b as defining a linear function whose graph is a line. Students compare linear functions and their graphs and gain experience with non-linear functions as well. In the second and final topic of this module, students extend what they learned in Grade 7 about how to solve real-world and mathematical problems related to volume from simple solids to include problems that require the formulas for cones, cylinders, and spheres.
• Grade 8 Math Module 6: Linear Functions (EngageNY)
In Grades 6 and 7, students worked with data involving a single variable. Module 6 introduces students to bivariate data. Students are introduced to a function as a rule that assigns exactly one value to each input. In this module, students use their understanding of functions to model the possible relationships of bivariate data. This module is important in setting a foundation for students work in algebra in Grade 9.
• Grade 8 Unit 4: Functions (Georgia Standards)
In this unit students will recognize a relationship as a function when each input is assigned to exactly one unique output; reason from a context, a graph, or a table, after first being clear which quantity is considered the input and which is the output; produce a counterexample: an input value with at least two output values when a relationship is not a function; explain how to verify that for each input there is exactly one output; and translate functions numerically, graphically, verbally, and algebraically.
• Grade 8 Unit 5: Linear Functions (Georgia Standards)
In this unit students will graph proportional relationships; interpret unit rate as the slope; compare two different proportional relationships represented in different ways; use similar triangles to explain why the slope is the same between any two points on a non-vertical line; derive the equation y = mx for a line through the origin; derive the equation y = mx + b for a line intercepting the vertical axis at b; and interpret equations in y = mx + b form as linear functions.
• Grade 8 Unit 6: Linear Models and Tables (Georgia Standards)
In this unit students will identify the rate of change and the initial value from tables, graphs, equations, or verbal descriptions; write a model for a linear function; sketch a graph when given a verbal description of a situation; analyze scatter plots; informally develop a line of best fit; use bivariate data to create graphs and linear models; and recognize patterns and interpret bivariate data.
• Graphing Equations in Slope Intercept Form
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.
• Graphing Equations in Slope Intercept Form Video
This is a video introduction to the topic.
• Graphing Linear Equations: T-Charts
This teaching module takes the student step-by-step through graphing linear equations. They are shown how to graph by making a T chart, plotting points, and drawing the line.
• Graphit
With this interactive applet students are able to create graphs of functions and sets of ordered pairs on the same coordinate plane.
• 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.
• Heart Rate Monitoring
In this task, students are asked to draw a graph that represents heart rate as a function of time from a verbal description of that function. Then they use the graph to draw conclusions about the context, for instance they have to understand that a heart rate greater than 100 beats per minute occurs when the graph is above the line y=100.
While not a full-blown modeling problem, this task does address some aspects of modeling as described in Standard for Mathematical Practice 4.
• Intercepts of Linear Equations
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.
• Intercepts of Linear Equations video
This video introduces the topic.
• Interpreting Distance-Time Graphs
This lesson unit is intended to help educators assess how well students are able to interpret distance and time graphs.
• Introducing Functions
The goal of this task is to motivate the definition of a function by carefully analyzing some different relationships. In some of these relationships, one quantity can be determined in terms of the other while in others this is not possible. In this way, students are led to see what is special about a function, namely that to each input there corresponds one and only one output.
• 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 Linear Functions
This task lets students explore the differences between linear and non-linear functions. By contrasting the two, it reinforces properties of linear functions.
• IXL Game: Nonlinear functions
Designed for eighth graders this game will help the student understand linear functions, specifically how to identify linear and nonlinear functions. This is just one of many online games that supports the Utah Math core. Note: The IXL site requires subscription for unlimited use.
• Linear Function Machine
By putting different values into the linear function machine students will explore simple linear functions.
• Linear Functions
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.
• Linear Functions video
This video compares proportional and non-proportional linear functions.
• Lines and Linear Equations
This lesson unit is intended to help educators assess how well students are able to interpret speed as the slope of a linear graph and translate between the equation of a line and its graphical representation.
• Manipulating Graphs
This video demonstrates how to use the slope-intercept of a line to the graph of that line. The classroom activity has them demonstrate their understanding by finding equations for a set of lines through the origin. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Modeling Situations With Linear Equations
This lesson unit is intended to help educators assess how well students use algebra in context, and in particular, how well students explore relationships between variables in everyday situations, find unknown values from known values, find relationships between pairs of unknowns, and express these as tables and graphs, as well as find general relationships between several variables, and express these in different ways by rearranging formulae.
• Modeling with a Linear Function
The primary purpose of this task is to elicit common misconceptions that arise when students try to model situations with linear functions. This task, being multiple choice, could also serve as a quick assessment to gauge a class' understanding of modeling with linear functions.
• 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.
• Non-Linear Functions
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.
• Non-linear Functions video
This video introduces non-linear functions.
• Parallel Lines
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.
• Parallel Lines video
This video explains the concept.
• Pennies to Heaven
The goal of this task is to give students a context to investigate large numbers and measurements. Students need to fluently convert units with very large numbers in order to successfully complete this task.
• Riding by the Library
In this task students draw the graphs of two functions from verbal descriptions. Both functions describe the same situation but changing the viewpoint of the observer changes where the function has output value zero. This small twist forces the students to think carefully about the interpretation of the dependent variable.
• Sequencer
By using this applet students are able to create sequences by changing the values of starting numbers, multipliers, and add-ons.
In this task, students will help Bill to find the best price for buying printed jerseys for the baseball team.
In this task, students are asked to match equations with linear graphs.
In this task, students use equations to solve a problem with a restaurant check.
• Teacher Desmos: LEGO Prices
In this activity, students use sliders to explore the relationship between price and number of pieces for various Star Wars LEGO sets and to make several predictions based on that model. Students will also interpret the parameters of their equation in context.
• Teacher Desmos: Linear Functions Collection
A collection of activities designed for algebra students studying linear functions as tables, graphs, and equations.
• Teacher Desmos: Marbleslides Lines
In this activity, students will transform lines so that the marbles go through the stars. Students will test their ideas by launching the marbles, and have a chance to revise before trying the next challenge.
• 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.
• Tides
This is a simple task about interpreting the graph of a function in terms of the relationship between quantities that it represents.
• US Garbage, Version 1
In this task, the rule of the function is more conceptual: We assign to a year (an input) the total amount of garbage produced in that year (the corresponding output). Even if we didn't know the exact amount for a year, it is clear that there will not be two different amounts of garbage produced in the same year. Thus, this makes sense as a "rule" even though there is no algorithmic way to determine the output for a given input except looking it up in the table.
• Vertical Line Test
This interactive applet asks the student to connect points on a plane in order to build a function and then test it to see if it's valid.
• Video Streaming
Given a scenario of monthly plans for video streaming students must determine what type of functions model this situation.

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.