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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.

Standard 2 : Use functions to model relationships between quantities. (Major Cluster)Archived
Cluster Standards

This cluster includes the following benchmarks.

Visit the specific benchmark webpage to find related instructional resources.

  • MAFS.8.F.2.4 : Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
  • MAFS.8.F.2.5 : Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.
Cluster Information
Number:
MAFS.8.F.2
Title:
Use functions to model relationships between quantities. (Major Cluster)
Type:
Cluster
Subject:
Mathematics - Archived
Grade:
8
Domain-Subdomain
Functions
Cluster Access Points

This cluster includes the following Access Points.

Cluster Resources

Vetted resources educators can use to teach the concepts and skills in this topic.

Original Student Tutorials
3D Modeling
  • Wind Farm Design Challenge: In this engineering design challenge, students are asked to create the most efficient wind turbine while balancing cost constraints. Students will apply their knowledge of surface area and graphing while testing 3D-printed wind farm blades. In the end, students are challenged to design and test their own wind farm blades, using Tinkercad to model a 3D-printable blade.

Educational Software / Tool
  • Free Graph Paper: A variety of graph paper types for printing, including Cartesian, polar, engineering, isometric, logarithmic, hexagonal, probability, and Smith chart.

Formative Assessments
  • Population Trend: Students are asked to describe the relationship between two quantities in a nonlinear function.

  • Profitable Functions: Students are asked to write a function to model a linear relationship given its graph.

  • Trekking Functions: Students are asked to construct a function to model a linear relationship between two quantities given a table of values.

  • Smart TV: Students are asked to determine the rate of change and initial value of a linear function given a table of values, and interpret the rate of change and initial value in terms of the situation it models.

  • Jet Fuel: Students are asked to analyze and describe the relationship between two linearly related quantities.

  • Graph the Ride: Students are given a verbal description of the relationship between two quantities and are asked to sketch a graph to model the relationship.

  • Construction Function: Students are asked to construct a function to model a linear relationship between two quantities given two ordered pairs in context.

  • Bacterial Growth Graph: Students are given a verbal description of the relationship between two quantities and are asked to sketch a graph to model the relationship.

  • Drain the Pool: Students are asked to determine the rate of change and initial value of a linear function when given a graph, and to interpret the rate of change and initial value in terms of the situation it models.

Lesson Plans
  • Stars: HR Diagram & Classification: In this lesson students will categorize a list of stars based on absolute brightness, size, and temperature. Students will analyze astronomical data presented in charts and plot their data on a special graph called a Hertzsprung-Russell Diagram (H-R Diagram). Using this diagram, they must determine the proper classification of individual stars. Using their data analysis, students completing this lesson will develop two short essay responses to a professional client indicating which stars are Main Sequence Stars and which ones are White Dwarfs, Giants, or Supergiants.

  • Rockets To Pluto: Students will explore current space technology and explore the possibilities of traveling to Pluto. Students will participate in an Engineering Design Challenge in which they will construct and test their rockets to see how far they can go!

    An Engineering Design Challenge is a combination of project-based learning, design thinking, and the engineering design process that develops the innovator's mindset through iteration. This lets students use their own imaginations to design projects according to science and engineering processes.

  • Fast Food Frenzy: In this activity, students will engage critically with nutritional information and macronutrient content of several fast food meals. This is an MEA that requires students to build on prior knowledge of nutrition and working with percentages.

    Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. Click here to learn more about MEAs and how they can transform your classroom.

  • What's My Function?: Students will determine function rules that have been written on cards taped to their backs. They will suggest input values and peers will provide output values to help them determine their function. They will then graph their functions for additional practice.

  • Interpreting Distance-Time Graphs: This lesson helps teachers assess how well students are able to interpret distance-time graphs. It reveals students who interpret distance-time graphs as if they are pictures of situations rather than abstract representations of them as well as those who have difficulty relating speeds to slopes of these graphs.

  • Lines and Linear Equations: This lesson is intended to help teachers 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.

  • Modeling Situations With Linear Equations: This lesson unit is intended to help you 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.
    • Find general relationships between several variables, and express these in different ways by rearranging formulae.
  • Lines and Linear Equations: This lesson unit is intended to help you 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.
  • Interpreting Distance-Time Graphs: This lesson takes a formative assessment approach to assessing whether students can interpret distance-time graphs. Whole-class discussion, group work, and individual activities help students interpret distance-time graphs related to real-world scenarios.
  • Discovering Kepler's Law for the Periods of Planets: Students listen to a video that describes Kepler's determination that planetary orbits are elliptical and then will use data for the solar distance and periods of several of the planets in the solar system, then investigate several hypotheses to determine which is supported by the data.

  • Getting Graphic with Linear Functions: Students will determine whether a function defined by a graph or an equation is a linear function, determine the rates of change and initial value from a table and graph, as well as be able to interpret what the rate of change means as it relates to a situation.

  • In the Real World: This resource provides a Lesson Plan for teaching students how to analyze real-world problems to look for clearly identified values and determine which of them is a constant value and which of them is subject to change (will increase or decrease per unit of time, weight, length, etc.). The students will also be taught how to determine the correct units for each value in an equation written in slope-intercept form.

  • Constructing and Calibrating a Hydrometer: Students construct and calibrate a simple hydrometer using different salt solutions. They then graph their data and determine the density and salinity of an unknown solution using their hydrometer and graphical analysis.

Perspectives Video: Expert
Perspectives Video: Professional/Enthusiasts
Problem-Solving Tasks
  • High School Graduation: This task provides a unique application of modeling with mathematics. Also, students often think that time must always be the independent variable and so may need some help understanding that one chooses the independent and dependent variable based on the way one wants to view a situation.

  • 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. Though students could work independently on the task, collaboration with peers is more likely to result in the exploration of a range of interpretations.

  • 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.

  • 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.

  • Tides: This is a simple task about interpreting the graph of a function in terms of the relationship between quantities that it represents.

  • 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. This task could be used in different ways: To generate a class discussion about graphing. As a quick assessment about graphing, for example during a class warm-up. To engage students in small group discussion.

  • Downhill: This task would be especially well-suited for instructional purposes. Students will benefit from a class discussion about the slope, y-intercept, x-intercept, and implications of the restricted domain for interpreting more precisely what the equation is modeling.

  • Velocity vs. 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.

  • Chicken and Steak, Variation 1: In this problem-solving task students are challenged to apply their understanding of linear relationships to determine the amount of chicken and steak needed for a barbecue, which will include creating an equation, sketching a graph, and interpreting both. This resource also includes annotated solutions.

  • Video Streaming: Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x,y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

  • Chicken and Steak, Variation 2: In this problem-solving task students are challenged to apply their understanding of linear relationships to determine the amount of chicken and steak needed for a barbecue, which will include creating an equation, sketching a graph, and interpreting both. This resource also includes annotated solutions.

  • Delivering the Mail: This problem-solving task involves constructing a linear function and interpreting its parameters in a mail delivery context. It includes annotated solutions.

  • Distance Across the Channel: This problem-solving 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. Commentary and illustrated solutions are included.

  • Drip, Drop, Drip, Drop: Students design an experiment to model a leaky faucet and determine the amount of water wasted due to the leak. Using the data they gather in a table, students graph and write an equation for a line of best fit. Students then use their derived equation to make predictions about the amount of water that would be wasted from one leak over a long period of time or the amount wasted by several leaks during a specific time period.
Professional Developments
  • Proportional Reasoning: In this resource, a special kind of functional relationship is explored: the proportional relationship. Teachers may find the resource useful for professional development, especially the videos. Students develop proportional reasoning skills by comparing quantities, looking at the relative ways numbers change, and thinking about proportional relationships in linear functions.
    This resource has four objectives. Students learn to differentiate between relative and absolute meanings of "more" and determine which of these is a proportional relationship, compare ratios without using common denominator algorithms, differentiate between additive and multiplicative processes and their effects on scale and proportionality, and interpret graphs that represent proportional relationships or direct variation.

  • Mathematical Modeling: Insights into Algebra, Teaching for Learning: This professional development resource provides a rich collection of information to help teachers engage students more effectively in mathematical modeling. It features videos of two complete lessons with commentary, background information on effective teaching, modeling, and lesson study, full lesson plans to teach both example lessons, examples of student work from the lessons, tips for effective teaching strategies, and list of helpful resources.
    • In Lesson 1 students use mathematical models (tables and equations) to represent the relationship between the number of revolutions made by a "driver" and a "follower" (two connected gears in a system), and they will explain the significance of the radii of the gears in regard to this relationship.
    • In Lesson 2 students mathematically model the growth of populations and use exponential functions to represent that growth.
Student Center Activity
  • Edcite: Mathematics Grade 8: Students can practice answering mathematics questions on a variety of topics. With an account, students can save their work and send it to their teacher when complete.

Teaching Idea
  • Learning About Rate of Change in Linear Functions Using Interactive Graphs: This resource features two pairs of interactive graphs to help students explore rate of change and linear relationships. "Users can drag a slider on an interactive graph to modify a rate of change (cost per minute for phone use) and learn how modifications in that rate affect the linear graph displaying accumulation (the total cost of calls). In this first part, Constant Cost per Minute, the cost per minute for phone use remains constant over time. In the second part, Changing Cost per Minute, the cost per minute for phone use changes after the first sixty minutes of calls." (from NCTM's Illuminations)

Tutorials
Unit/Lesson Sequences
  • Linear Functions and Slope: This session on linear function and slope contains five parts, multiple problems and videos, and interactive activities geared to help students recognize and understand linear relationships, explore slope and dependent and independent variables in graphs of linear relationships, and develop an understanding of rates and how they are related to slopes and equations. Throughout the session, students use spreadsheets to complete the work, and are encouraged to think about the ways technology can aid in teaching and understanding. The solutions for all problems are given, and many allow students to have a hint or tip as they solve. There is even a homework assignment with four problems for students after they have finished all five parts of the session.
  • 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 Equations: This resource provides guided practice for writing and graphing linear functions.

  • 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.

  • Graphing Lines: Allows students access to a Cartesian Coordinate System where linear equations can be graphed and details of the slope and various types of line equations can be explored.

  • Function Flyer: In this online tool, students input a function to create a graph where the constants, coefficients, and exponents can be adjusted by slider bars. This tool allows students to explore graphs of functions and how adjusting the numbers in the function affect the graph. Using tabs at the top of the page you can also access 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.

  • Number Cruncher: In this activity, students enter inputs into a function machine. Then, by examining the outputs, they must determine what function the machine is performing. This activity allows students to explore functions and what inputs are most useful for determining the 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.

  • Graphs and Functions: This lesson is designed to introduce students to plotting points and graphing functions in the Cartesian coordinate system. The lesson provides links to discussions and activities that transition from functions as rules to the graphs of those functions. Finally, the lesson provides links to follow-up lessons designed for use in succession to an introduction to graphing.
  • Equation Grapher: This interactive simulation investigates graphing linear and quadratic equations. Users are given the ability to define and change the coefficients and constants in order to observe resulting changes in the graph(s).

Worksheet
  • Mass vs. Weight: This laboratory activity worksheet engages students in an exploration of mass and weight. Students use a balance and spring scale to measure the masses and weights of a several of objects. Students will analyze the data and determine the the relationship between mass and weight using graphing skills. An answer key is attached to assist the use of this resource.