# MAFS.8.F.2.5Archived Standard Export Print
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.
General Information
Subject Area: Mathematics
Domain-Subdomain: Functions
Cluster: Level 2: Basic Application of Skills & Concepts
Cluster: Use functions to model relationships between quantities. (Major Cluster) -

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.

Date of Last Rating: 02/14
Status: State Board Approved - Archived
Assessed: Yes
Test Item Specifications

• Assessment Limits :
Linear or nonlinear relationships may use any of the four quadrants. Graph descriptions move from left to right. Functional relationships must be continuous.
• Calculator :

Neutral

• Context :

Allowable

Sample Test Items (5)

• Test Item #: Sample Item 2
• Question: Kim rides a stationary bike for fifteen minutes of exercise.

Kim starts her ride slowly, stops for 2 minutes, and then continues her ride faster than she started.

Use the Connect Line tool to create a possible graph of Kim’s ride. • Difficulty: N/A
• Type: GRID: Graphic Response Item Display

• Test Item #: Sample Item 3
• Question:

Mary and Kim go bike riding on some trails. Graphs of the functions representing one of their rides are shown, where x is the time, in minutes, and y is the distance, in miles.

Select all statements that are true based on the graphs shown. • Difficulty: N/A
• Type: MS: Multiselect

• Test Item #: Sample Item 4
• Question:

Mary and Kim ride their bikes to school each day. Graphs of the functions representing one of their rides are shown, where x is the time, in minutes, and y is the distance, in miles. Write a short story about Mary’s bike ride describing the relationship between time and distance. Include information about changes in the behavior of the function and rate of speed using terms such as increasing, constant, and decreasing.

• Difficulty: N/A
• Type: OR: Open Response

• Test Item #: Sample Item 5
• Question:

Mary and Kim each take 15 mintues to ride their bikes to school. The graphs of the functions that model their rides are shown, where x is the time, in minutes, and y is the distance, in miles. The graphs are divided into time intervals A, B, and C.

Use the graphs to match each statement with the appropriate person or people.

• Difficulty: N/A
• Type: MI: Matching Item

## Related Courses

This benchmark is part of these courses.
1205050: M/J Accelerated Mathematics Grade 7 (Specifically in versions: 2014 - 2015, 2015 - 2020, 2020 - 2022, 2022 and beyond (current))
1205070: M/J Grade 8 Pre-Algebra (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
2002100: M/J Comprehensive Science 3 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
2002110: M/J Comprehensive Science 3, Advanced (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
2003010: M/J Physical Science (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
2003020: M/J Physical Science, Advanced (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1204000: M/J Foundational Skills in Mathematics 6-8 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7812030: Access M/J Grade 8 Pre-Algebra (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022, 2022 and beyond (current))
2002085: M/J Comprehensive Science 2 Accelerated Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
2002200: M/J STEM Environmental Science (Specifically in versions: 2015 - 2022, 2022 and beyond (current))
2001025: M/J STEM Astronomy and Space Science (Specifically in versions: 2015 - 2022, 2022 and beyond (current))

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.

## Related Resources

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

## 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.

Type: Educational Software / Tool

## Formative Assessments

Population Trend:

Students are asked to describe the relationship between two quantities in a nonlinear function.

Type: Formative Assessment

Jet Fuel:

Students are asked to analyze and describe the relationship between two linearly related quantities.

Type: Formative Assessment

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.

Type: Formative Assessment

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.

Type: Formative Assessment

Interpreting Distance-Time Graphs:

This lesson unit is intended to help you assess how well students are able to interpret distance-time graphs and, in particular, to help you identify students who:

• Interpret distance-time graphs as if they are pictures of situations rather than abstract representations of them.
• Have difficulty relating speeds to slopes of these graphs.

Type: Formative Assessment

Lines and Linear Equations:

This lesson is intended to help you assess how well students are able to:

• Interpret speed as the slope of a linear graph.
• Translate between the equation of a line and its graphical representation.

Type: Formative Assessment

## Lesson Plans

Stars: HR Diagram & Classification:

In this Model Eliciting Activity (MEA), 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 MEA 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.

Type: Lesson Plan

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.

Type: Lesson Plan

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.

Type: Lesson Plan

Are We There Yet?:

Students will learn to write and interpret linear functions that represent real world situations, noting the importance of slope and y-intercept.

Type: Lesson Plan

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.

Type: Lesson Plan

## Original Student Tutorials

Math Models and Social Distancing:

Learn how math models can show why social distancing during a epidemic or pandemic is important in this interactive tutorial.

Type: Original Student Tutorial

Cruising Through Functions:

Cruise along as you discover how to qualitatively describe functions in this interactive tutorial.

Type: Original Student Tutorial

## Perspectives Video: Expert

Using Mathematics to Optimize Wing Design:

Nick Moore discusses his research behind optimizing wing design using inspiration from animals and how they swim and fly.

Type: Perspectives Video: Expert

## Perspectives Video: Professional/Enthusiasts

Asymptotic Behavior in Shark Growth Research:

Fishery Scientist from Florida State University discusses his new research in deep sea sharks and the unusual behavior that is found when the data is graphed.

Type: Perspectives Video: Professional/Enthusiast

KROS Pacific Ocean Kayak Journey: Kites, Wind, and Speed:

Lofty ideas about kites helped power a kayak from California to Hawaii.

Related Resources:
KROS Pacific Ocean Kayak Journey: GPS Data Set[.XLSX]
KROS Pacific Ocean Kayak Journey: Path Visualization for Google Earth[.KML]

Type: Perspectives Video: Professional/Enthusiast

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.

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.

## Student Center Activity

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.

Type: Student Center Activity

## 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)

Type: Teaching Idea

## Tutorials

Comparing linear functions from a graph:

In this tutorial, students will compare linear functions from a graph. Students should have an understanding of slope and rate of change before reviewing this tutorial.

Type: Tutorial

Compare linear functions from a table and graph:

This tutorial shows how to compare linear functions that are presented in both a table and graph. Students should have an understanding of rate of change before viewing this video.

Type: Tutorial

Comparing linear functions :

Students will compare linear functions presented in a graph and in a table. Students should have a strong understanding of rate of change before viewing this tutorial.

Type: Tutorial

Interpreting Linear Graphs:

In this tutorial, you will look at several real-world examples of linear graphs and interpret the relationship between the two variables.

Type: Tutorial

## Unit/Lesson Sequences

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.

Type: Unit/Lesson Sequence

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.

Type: Unit/Lesson Sequence

## Virtual Manipulatives

Graphing Lines:

Allows students access to a Cartesian Coordinate System where linear equations can be graphed and details of the line and the slope can be observed.

Type: Virtual Manipulative

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.

Type: Virtual Manipulative

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.

Type: Virtual Manipulative

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).

Type: Virtual Manipulative

## STEM Lessons - Model Eliciting Activity

Stars: HR Diagram & Classification:

In this Model Eliciting Activity (MEA), 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 MEA 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.

## MFAS Formative Assessments

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.

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.

Jet Fuel:

Students are asked to analyze and describe the relationship between two linearly related quantities.

Population Trend:

Students are asked to describe the relationship between two quantities in a nonlinear function.

## Original Student Tutorials Mathematics - Grades 6-8

Cruising Through Functions:

Cruise along as you discover how to qualitatively describe functions in this interactive tutorial.

Math Models and Social Distancing:

Learn how math models can show why social distancing during a epidemic or pandemic is important in this interactive tutorial.

## Student Resources

Vetted resources students can use to learn the concepts and skills in this benchmark.

## Original Student Tutorials

Math Models and Social Distancing:

Learn how math models can show why social distancing during a epidemic or pandemic is important in this interactive tutorial.

Type: Original Student Tutorial

Cruising Through Functions:

Cruise along as you discover how to qualitatively describe functions in this interactive tutorial.

Type: Original Student Tutorial

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.

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.

## Student Center Activity

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.

Type: Student Center Activity

## Tutorials

Comparing linear functions from a graph:

In this tutorial, students will compare linear functions from a graph. Students should have an understanding of slope and rate of change before reviewing this tutorial.

Type: Tutorial

Compare linear functions from a table and graph:

This tutorial shows how to compare linear functions that are presented in both a table and graph. Students should have an understanding of rate of change before viewing this video.

Type: Tutorial

Comparing linear functions :

Students will compare linear functions presented in a graph and in a table. Students should have a strong understanding of rate of change before viewing this tutorial.

Type: Tutorial

Interpreting Linear Graphs:

In this tutorial, you will look at several real-world examples of linear graphs and interpret the relationship between the two variables.

Type: Tutorial

## Virtual Manipulatives

Graphing Lines:

Allows students access to a Cartesian Coordinate System where linear equations can be graphed and details of the line and the slope can be observed.

Type: Virtual Manipulative

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.

Type: Virtual Manipulative

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).

Type: Virtual Manipulative

## Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this benchmark.

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.

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.

## 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)

Type: Teaching Idea

## Virtual Manipulative

Graphing Lines:

Allows students access to a Cartesian Coordinate System where linear equations can be graphed and details of the line and the slope can be observed.

Type: Virtual Manipulative