MAFS.912.F-IF.2.4Archived Standard

For a function that models a relationship between two quantities, 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. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.
General Information
Subject Area: Mathematics
Grade: 912
Domain-Subdomain: Functions: Interpreting Functions
Cluster: Level 2: Basic Application of Skills & Concepts
Cluster: Interpret functions that arise in applications in terms of the context. (Algebra 1 - Major Cluster) (Algebra 2 - 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 Adopted or Revised: 02/14
Date of Last Rating: 02/14
Status: State Board Approved - Archived
Assessed: Yes
Test Item Specifications
    Also assesses:
    MAFS.912.F-IF.3.9

  • Assessment Limits :
    Functions represented algebraically are limited to linear, quadratic, or
    exponential.

    Functions may be represented using tables, graphs or verbally.

    Functions represented using these representations are not limited to
    linear, quadratic or exponential.

    Functions may have closed domains.

    Functions may be discontinuous.

    Items may not require the student to use or know interval notation.

    Key features include x-intercepts, y-intercepts; intervals where the
    function is increasing, decreasing, positive, or negative; relative
    maximums and minimums; symmetries; and end behavior.

  • Calculator :

    Neutral

  • Clarification :
    Students will determine and relate the key features of a function
    within a real-world context by examining the function’s table.

    Students will determine and relate the key features of a function
    within a real-world context by examining the function’s graph.

    Students will use a given verbal description of the relationship
    between two quantities to label key features of a graph of a function
    that model the relationship.

    Students will differentiate between different types of functions using
    a variety of descriptors (e.g., graphically, verbally, numerically, and
    algebraically).

    Students will compare and contrast properties of two functions using
    a variety of function representations (e.g., algebraic, graphic, numeric
    in tables, or verbal descriptions).

  • Stimulus Attributes :
    For F-IF.2.4, items should be set in a real-world context.

    For F-IF.3.9, items may be set in a real-world or mathematical
    context.

    Items may use verbal descriptions of functions.

    Items must use function notation.

  • Response Attributes :
    For F-IF.2.4, items may require the student to apply the basic
    modeling cycle.

    Items may require the student to write intervals using inequalities.

    Items may require the student to choose an appropriate level of
    accuracy.

    Items may require the student to choose and interpret the scale in a
    graph.

    Items may require the student to choose and interpret units.

Sample Test Items (1)
  • Test Item #: Sample Item 1
  • Question:

    Kim is driving from Miami to Key West. The graph shows her distance from Key West.

     

    During what interval is Kim driving the fastest? Drag numbers to the boxes to complete the inequality.

  • Difficulty: N/A
  • Type: GRID: Graphic Response Item Display

Related Courses

This benchmark is part of these courses.
1200310: Algebra 1 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1200320: Algebra 1 Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1200330: Algebra 2 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1200340: Algebra 2 Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1200370: Algebra 1-A (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1200380: Algebra 1-B (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1200400: Foundational Skills in Mathematics 9-12 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1206330: Analytic Geometry (Specifically in versions: 2014 - 2015 (course terminated))
2000350: Anatomy and Physiology (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
2000360: Anatomy and Physiology Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
2020910: Astronomy Solar/Galactic Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
2000320: Biology 1 Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
2000330: Biology 2 Honors (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2022, 2022 and beyond (current))
2003340: Chemistry 1 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
2003350: Chemistry 1 Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
2003360: Chemistry 2 Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
2001320: Earth/Space Science Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
2001340: Environmental Science (Specifically in versions: 2015 - 2022, 2022 and beyond (current))
2000440: Genetics Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
2002410: Integrated Science 1 Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
2002430: Integrated Science 2 Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
2002440: Integrated Science 3 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
2002450: Integrated Science 3 Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
2002500: Marine Science 1 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
2002510: Marine Science 1 Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
2002520: Marine Science 2 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
2002530: Marine Science 2 Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
2003400: Nuclear Radiation (Specifically in versions: 2014 - 2015, 2015 - 2018 (course terminated))
2020710: Nuclear Radiation Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
2003320: Physical Science Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
2003380: Physics 1 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
2003390: Physics 1 Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
2003410: Physics 2 Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
2002540: Solar Energy Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1200500: Advanced Algebra with Financial Applications (Specifically in versions: 2014 - 2015 (course terminated))
1200410: Mathematics for College Success (Specifically in versions: 2014 - 2015, 2015 - 2022 (course terminated))
1200700: Mathematics for College Algebra (Specifically in versions: 2014 - 2015, 2015 - 2022 (course terminated))
7912070: Access Mathematics for Liberal Arts (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022, 2022 - 2023, 2023 and beyond (current))
7912080: Access Algebra 1A (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022, 2022 and beyond (current))
7912090: Access Algebra 1B (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022, 2022 and beyond (current))
7920011: Access Chemistry 1 (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2023, 2023 and beyond (current))
2000500: Bioscience 1 Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 - 2023, 2023 and beyond (current))
2000510: Bioscience 2 Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 - 2023, 2023 and beyond (current))
2002445: Integrated Science 3 for Credit Recovery (Specifically in versions: 2014 - 2015, 2015 - 2020 (course terminated))
2003345: Chemistry 1 for Credit Recovery (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
2003385: Physics 1 for Credit Recovery (Specifically in versions: 2014 - 2015, 2015 - 2020 (course terminated))
1200315: Algebra 1 for Credit Recovery (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1200335: Algebra 2 for Credit Recovery (Specifically in versions: 2014 - 2015, 2015 - 2019 (course terminated))
1200375: Algebra 1-A for Credit Recovery (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1200385: Algebra 1-B for Credit Recovery (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7912100: Fundamental Algebraic Skills (Specifically in versions: 2013 - 2015, 2015 - 2017 (course terminated))
1207300: Liberal Arts Mathematics 1 (Specifically in versions: 2014 - 2015, 2015 - 2022 (course terminated))
7912075: Access Algebra 1 (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022, 2022 and beyond (current))
2003836: Florida's Preinternational Baccalaureate Physics 1 (Specifically in versions: 2015 - 2022, 2022 and beyond (current))
2003838: Florida's Preinternational Baccalaureate Physics 2 (Specifically in versions: 2015 and beyond (current))
7912095: Access Algebra 2 (Specifically in versions: 2016 - 2018, 2018 - 2019, 2019 - 2022, 2022 and beyond (current))
2001341: Environmental Science Honors (Specifically in versions: 2016 - 2022, 2022 and beyond (current))
1200387: Mathematics for Data and Financial Literacy (Specifically in versions: 2016 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.

Formative Assessments

Uphill and Downhill:

Students are asked to interpret key features of a graph (intercepts and intervals over which the graph is increasing) in the context of a problem situation.

Type: Formative Assessment

Elevation Along a Trail:

Students are asked to interpret key features of a graph (symmetry) in the context of a problem situation.

Type: Formative Assessment

Surf’s Up:

Students are given a table of functional values and asked to describe and interpret key features of the graph in the context of the problem.

Type: Formative Assessment

Taxi Ride:

Students are asked to sketch a graph from a verbal description.

Type: Formative Assessment

Bike Race:

Students are asked to evaluate three verbal descriptions and to state why each does or does not match a given graph.

Type: Formative Assessment

Lesson Plans

Compacting Cardboard:

Students investigate the amount of space that could be saved by flattening cardboard boxes. The analysis includes linear graphs and regression analysis along with discussions of slope and a direct variation phenomenon.

Type: Lesson Plan

To The Limit:

"To The Limit" MEA has students identify several factors that can affect a population’s growth. Students will examine photos to list limiting factors and discuss their impact on populations. As a group they will develop a solution to minimize the impact of pollution on fish population.

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.

Type: Lesson Plan

Ferris Wheel:

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

  • Model a periodic situation, the height of a person on a Ferris wheel, using trigonometric functions.
  • Interpret the constants a, b, c in the formula h = a + b cos ct in terms of the physical situation, where h is the height of the person above the ground and t is the elapsed time.

Type: Lesson Plan

Representing Polynomials:

This lesson unit is intended to help you assess how well students are able to translate between graphs and algebraic representations of polynomials. In particular, this unit aims to help you identify and assist students who have difficulties in recognizing the connection between the zeros of polynomials when suitable factorizations are available, and graphs of the functions defined by polynomials as well as recognizing the connection between transformations of the graphs and transformations of the functions obtained by replacing f(x) by f(x + k), f(x) + k, -f(x), f(-x).

Type: Lesson Plan

Functions and Everyday Situations:

This lesson unit is intended to help you assess how well students are able to articulate verbally the relationships between variables arising in everyday contexts, translate between everyday situations and sketch graphs of relationships between variables, interpret algebraic functions in terms of the contexts in which they arise and reflect on the domains of everyday functions and in particular whether they should be discrete or continuous.

Type: Lesson Plan

Cup-Activity: writing equations from data:

This is a great lab activity that allows students to develop a true understanding of slope as a rate of change. Students are active and involved and must use higher order thinking skills in order to answer questions. Students work through an activity, measuring heights of cups that are stacked. Students them determine a "rate of change - slope". Students are then asked to put this into slope-intercept form. The important part here is in their determining the y-intercept of the equation. Students then take this further and finally attempt to create a linear inequality to determine how many cups, stacked vertically, will fit under a table.

Type: Lesson Plan

Parts and more Parts-- Parabola Fun:

This is an entry lesson into quadratic functions and their shapes. Students see some real-life representations of parabolas. This lesson provides important vocabulary associated with quadratic functions and their graphs in an interactive manner. Students create a foldable and complete a worksheet using their foldable notes.

Type: Lesson Plan

How high is that railing, anyway?:

This is a short activity where students are able to determine the height of an elevated railing by using the equations associated with freefall. This lesson may also be appropriate for analyzing graphs related to position/velocity/acceleration versus time.

Type: Lesson Plan

Exponential Graphing Using Technology:

This lesson is teacher/student directed for discovering and translating exponential functions using a graphing app. The lesson focuses on the translations from a parent graph and how changing the coefficient, base and exponent values relate to the transformation.

Type: Lesson Plan

Original Student Tutorials

Quadratic Function Part 2: Launches:

Learn about different formats of quadratic equations and their graphs with experiments involving launching and shooting of balls in this interactive tutorial.

This is part 2 of a two-part series: Click HERE to open part 1.

Type: Original Student Tutorial

Quadratic Functions Part 1: Ball Games:

Join us as we watch ball games and explore how the height of a ball bounce over time is represented by quadratic functions, which provides opportunities to interpret key features of the function in this interactive tutorial.

This is part 1 of a two-part series: Click HERE to open part 2.

Type: Original Student Tutorial

Exponential Functions Part 3: Decay:

Learn about exponential decay as you calculate the value of used cars by examining equations, graphs, and tables in this interactive tutorial.

Type: Original Student Tutorial

Linear Functions: Jobs:

Learn how to interpret key features of linear functions and translate between representations of linear functions through exploring jobs for teenagers in this interactive tutorial. 

Type: Original Student Tutorial

Exponential Functions Part 2: Growth:

Learn about exponential growth in the context of interest earned as money is put in a savings account by examining equations, graphs, and tables in this interactive tutorial.

Type: Original Student Tutorial

Exponential Functions Part 1:

Learn about exponential functions and how they are different from linear functions by examining real world situations, their graphs and their tables in this interactive tutorial.

Type: Original Student Tutorial

Perspectives Video: Experts

Jumping Robots and Quadratics:

<p>Jump to it and learn more about how quadratic equations are&nbsp;used in robot navigation problem solving!</p>

Type: Perspectives Video: Expert

Birdsong Series: Mathematically Modeling Birdsong:

Richard Bertram discusses his mathematical modeling contribution to the Birdsong project that helps the progress of neuron and ion channel research.

Download the CPALMS Perspectives video student note taking guide.

Type: Perspectives Video: Expert

Perspectives Video: Professional/Enthusiast

Graphing Torque and Horsepower for Dyno-mite Racing:

<p>SCCA race car drivers discuss how using a chassis dyno to graph horsepower and torque curves helps them maximize potential in their race cars.</p>

Type: Perspectives Video: Professional/Enthusiast

Problem-Solving Tasks

As the Wheel Turns:

In this task, students use trigonometric functions to model the movement of a point around a wheel and, through space. Students also interpret features of graphs in terms of the given real-world context.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

Throwing Baseballs:

This task could be used for assessment or for practice. It allows students to compare characteristics of two quadratic functions that are each represented differently, one as the graph of a quadratic function and one written out algebraically. Specifically, students are asked to determine which function has the greatest maximum and the greatest non-negative root.

Type: Problem-Solving Task

Average Cost:

This task asks students to find the average, write an equation, find the domain, and create a graph of the cost of producing DVDs.

Type: Problem-Solving Task

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; 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. Features in one graph, like maximum and minimum points, correspond to features in another graph. For example, on a rainy day, the solar radiation is very low, and the cumulative rainfall graph is increasing with a large slope.

Type: Problem-Solving Task

Logistic Growth Model, Explicit Version:

This problem introduces a logistic growth model in the concrete settings of estimating the population of the U.S. The model gives a surprisingly accurate estimate and this should be contrasted with linear and exponential models.

Type: Problem-Solving Task

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. The goal of this task is to have students appreciate how different constants influence the shape of a graph.

Type: Problem-Solving Task

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. Students also have to pay attention to the scale on the vertical axis to find the correct match. The first and third graphs look very similar at first glance, but the function values are very different since the scales on the vertical axes are very different. The task could also be used to generate a group discussion on interpreting functions given by graphs.

Type: Problem-Solving Task

Sum of Even and Odd:

Students explore and manipulate expressions based on the following statement:

A function f defined for -a < x="">< a="" is="" even="" if="" f(-x)="f(x)" and="" is="" odd="" if="" f(-x)="-f(x)" when="" -a="">< x="">< a.="" in="" this="" task="" we="" assume="" f="" is="" defined="" on="" such="" an="" interval,="" which="" might="" be="" the="" full="" real="" line="" (i.e.,="" a="">

Type: Problem-Solving 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.

Type: Problem-Solving Task

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 the context. It can be used as either an assessment or a teaching task.

Type: Problem-Solving Task

Tutorial

Linear Functions:

In this tutorial, "Linear functions of the form f(x) = ax + b and the properties of their graphs are explored interactively using an applet." The applet allows students to manipulate variables to discover the changes in intercepts and slope of the graphed line. There are six questions for students to answer, exploring the applet and observing changes. The questions' answers are included on this site. Additionally, a tutorial for graphing linear functions by hand is included.

Type: Tutorial

Unit/Lesson Sequence

Sample Algebra 1 Curriculum Plan Using CMAP:

This sample Algebra 1 CMAP is a fully customizable resource and curriculum-planning tool that provides a framework for the Algebra 1 Course. The units and standards are customizable and the CMAP allows instructors to add lessons, worksheets, and other resources as needed. This CMAP also includes rows that automatically filter and display Math Formative Assessments System tasks, E-Learning Original Student Tutorials and Perspectives Videos that are aligned to the standards, available on CPALMS.

Learn more about the sample Algebra 1 CMAP, its features and customizability by watching the following video:

Using this CMAP

To view an introduction on the CMAP tool, please .

To view the CMAP, click on the "Open Resource Page" button above; be sure you are logged in to your iCPALMS account.

To use this CMAP, click on the "Clone" button once the CMAP opens in the "Open Resource Page." Once the CMAP is cloned, you will be able to see it as a class inside your iCPALMS My Planner (CMAPs) app.

To access your My Planner App and the cloned CMAP, click on the iCPALMS tab in the top menu.

All CMAP tutorials can be found within the iCPALMS Planner App or at the following URL: http://www.cpalms.org/support/tutorials_and_informational_videos.aspx

Type: Unit/Lesson Sequence

Virtual Manipulatives

Slope Slider:

In this activity, students adjust slider bars which adjust the coefficients and constants of a linear function and examine how their changes affect the graph. The equation of the line can be in slope-intercept form or standard form. This activity allows students to explore linear equations, slopes, and y-intercepts and their visual representation on a graph. 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.

Type: 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

Data Flyer:

Using this virtual manipulative, students are able to graph a function and a set of ordered pairs on the same coordinate plane. The constants, coefficients, and exponents can be adjusted using slider bars, so the student can explore the affect on the graph as the function parameters are changed. Students can also examine the deviation of the data from the function. 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.

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

STEM Lessons - Model Eliciting Activity

To The Limit:

"To The Limit" MEA has students identify several factors that can affect a population’s growth. Students will examine photos to list limiting factors and discuss their impact on populations. As a group they will develop a solution to minimize the impact of pollution on fish population.

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.

MFAS Formative Assessments

Bike Race:

Students are asked to evaluate three verbal descriptions and to state why each does or does not match a given graph.

Elevation Along a Trail:

Students are asked to interpret key features of a graph (symmetry) in the context of a problem situation.

Surf’s Up:

Students are given a table of functional values and asked to describe and interpret key features of the graph in the context of the problem.

Taxi Ride:

Students are asked to sketch a graph from a verbal description.

Uphill and Downhill:

Students are asked to interpret key features of a graph (intercepts and intervals over which the graph is increasing) in the context of a problem situation.

Original Student Tutorials Mathematics - Grades 9-12

Exponential Functions Part 1:

Learn about exponential functions and how they are different from linear functions by examining real world situations, their graphs and their tables in this interactive tutorial.

Exponential Functions Part 2: Growth:

Learn about exponential growth in the context of interest earned as money is put in a savings account by examining equations, graphs, and tables in this interactive tutorial.

Exponential Functions Part 3: Decay:

Learn about exponential decay as you calculate the value of used cars by examining equations, graphs, and tables in this interactive tutorial.

Linear Functions: Jobs:

Learn how to interpret key features of linear functions and translate between representations of linear functions through exploring jobs for teenagers in this interactive tutorial. 

Quadratic Function Part 2: Launches:

Learn about different formats of quadratic equations and their graphs with experiments involving launching and shooting of balls in this interactive tutorial.

This is part 2 of a two-part series: Click HERE to open part 1.

Quadratic Functions Part 1: Ball Games:

Join us as we watch ball games and explore how the height of a ball bounce over time is represented by quadratic functions, which provides opportunities to interpret key features of the function in this interactive tutorial.

This is part 1 of a two-part series: Click HERE to open part 2.

Student Resources

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

Original Student Tutorials

Quadratic Function Part 2: Launches:

Learn about different formats of quadratic equations and their graphs with experiments involving launching and shooting of balls in this interactive tutorial.

This is part 2 of a two-part series: Click HERE to open part 1.

Type: Original Student Tutorial

Quadratic Functions Part 1: Ball Games:

Join us as we watch ball games and explore how the height of a ball bounce over time is represented by quadratic functions, which provides opportunities to interpret key features of the function in this interactive tutorial.

This is part 1 of a two-part series: Click HERE to open part 2.

Type: Original Student Tutorial

Exponential Functions Part 3: Decay:

Learn about exponential decay as you calculate the value of used cars by examining equations, graphs, and tables in this interactive tutorial.

Type: Original Student Tutorial

Linear Functions: Jobs:

Learn how to interpret key features of linear functions and translate between representations of linear functions through exploring jobs for teenagers in this interactive tutorial. 

Type: Original Student Tutorial

Exponential Functions Part 2: Growth:

Learn about exponential growth in the context of interest earned as money is put in a savings account by examining equations, graphs, and tables in this interactive tutorial.

Type: Original Student Tutorial

Exponential Functions Part 1:

Learn about exponential functions and how they are different from linear functions by examining real world situations, their graphs and their tables in this interactive tutorial.

Type: Original Student Tutorial

Perspectives Video: Expert

Jumping Robots and Quadratics:

<p>Jump to it and learn more about how quadratic equations are&nbsp;used in robot navigation problem solving!</p>

Type: Perspectives Video: Expert

Problem-Solving Tasks

As the Wheel Turns:

In this task, students use trigonometric functions to model the movement of a point around a wheel and, through space. Students also interpret features of graphs in terms of the given real-world context.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

Throwing Baseballs:

This task could be used for assessment or for practice. It allows students to compare characteristics of two quadratic functions that are each represented differently, one as the graph of a quadratic function and one written out algebraically. Specifically, students are asked to determine which function has the greatest maximum and the greatest non-negative root.

Type: Problem-Solving Task

Average Cost:

This task asks students to find the average, write an equation, find the domain, and create a graph of the cost of producing DVDs.

Type: Problem-Solving Task

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; 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. Features in one graph, like maximum and minimum points, correspond to features in another graph. For example, on a rainy day, the solar radiation is very low, and the cumulative rainfall graph is increasing with a large slope.

Type: Problem-Solving Task

Logistic Growth Model, Explicit Version:

This problem introduces a logistic growth model in the concrete settings of estimating the population of the U.S. The model gives a surprisingly accurate estimate and this should be contrasted with linear and exponential models.

Type: Problem-Solving Task

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. The goal of this task is to have students appreciate how different constants influence the shape of a graph.

Type: Problem-Solving Task

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. Students also have to pay attention to the scale on the vertical axis to find the correct match. The first and third graphs look very similar at first glance, but the function values are very different since the scales on the vertical axes are very different. The task could also be used to generate a group discussion on interpreting functions given by graphs.

Type: Problem-Solving Task

Sum of Even and Odd:

Students explore and manipulate expressions based on the following statement:

A function f defined for -a < x="">< a="" is="" even="" if="" f(-x)="f(x)" and="" is="" odd="" if="" f(-x)="-f(x)" when="" -a="">< x="">< a.="" in="" this="" task="" we="" assume="" f="" is="" defined="" on="" such="" an="" interval,="" which="" might="" be="" the="" full="" real="" line="" (i.e.,="" a="">

Type: Problem-Solving 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.

Type: Problem-Solving Task

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 the context. It can be used as either an assessment or a teaching task.

Type: Problem-Solving Task

Virtual Manipulatives

Slope Slider:

In this activity, students adjust slider bars which adjust the coefficients and constants of a linear function and examine how their changes affect the graph. The equation of the line can be in slope-intercept form or standard form. This activity allows students to explore linear equations, slopes, and y-intercepts and their visual representation on a graph. 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.

Type: 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

Data Flyer:

Using this virtual manipulative, students are able to graph a function and a set of ordered pairs on the same coordinate plane. The constants, coefficients, and exponents can be adjusted using slider bars, so the student can explore the affect on the graph as the function parameters are changed. Students can also examine the deviation of the data from the function. 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.

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

Parent Resources

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

Problem-Solving Tasks

As the Wheel Turns:

In this task, students use trigonometric functions to model the movement of a point around a wheel and, through space. Students also interpret features of graphs in terms of the given real-world context.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

Throwing Baseballs:

This task could be used for assessment or for practice. It allows students to compare characteristics of two quadratic functions that are each represented differently, one as the graph of a quadratic function and one written out algebraically. Specifically, students are asked to determine which function has the greatest maximum and the greatest non-negative root.

Type: Problem-Solving Task

Average Cost:

This task asks students to find the average, write an equation, find the domain, and create a graph of the cost of producing DVDs.

Type: Problem-Solving Task

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; 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. Features in one graph, like maximum and minimum points, correspond to features in another graph. For example, on a rainy day, the solar radiation is very low, and the cumulative rainfall graph is increasing with a large slope.

Type: Problem-Solving Task

Logistic Growth Model, Explicit Version:

This problem introduces a logistic growth model in the concrete settings of estimating the population of the U.S. The model gives a surprisingly accurate estimate and this should be contrasted with linear and exponential models.

Type: Problem-Solving Task

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. The goal of this task is to have students appreciate how different constants influence the shape of a graph.

Type: Problem-Solving Task

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. Students also have to pay attention to the scale on the vertical axis to find the correct match. The first and third graphs look very similar at first glance, but the function values are very different since the scales on the vertical axes are very different. The task could also be used to generate a group discussion on interpreting functions given by graphs.

Type: Problem-Solving Task

Sum of Even and Odd:

Students explore and manipulate expressions based on the following statement:

A function f defined for -a < x="">< a="" is="" even="" if="" f(-x)="f(x)" and="" is="" odd="" if="" f(-x)="-f(x)" when="" -a="">< x="">< a.="" in="" this="" task="" we="" assume="" f="" is="" defined="" on="" such="" an="" interval,="" which="" might="" be="" the="" full="" real="" line="" (i.e.,="" a="">

Type: Problem-Solving 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.

Type: Problem-Solving Task

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 the context. It can be used as either an assessment or a teaching task.

Type: Problem-Solving Task

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