# Cluster 4: Represent and solve equations and inequalities graphically. (Algebra 1 - Major Cluster) (Algebra 2 - Major Cluster)Archived Export Print

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.

General Information
Number: MAFS.912.A-REI.4
Title: Represent and solve equations and inequalities graphically. (Algebra 1 - Major Cluster) (Algebra 2 - Major Cluster)
Type: Cluster
Subject: Mathematics - Archived
Domain-Subdomain: Algebra: Reasoning with Equations & Inequalities

## Related Standards

This cluster includes the following benchmarks.

## Related Access Points

This cluster includes the following access points.

## Independent

MAFS.912.A-REI.4.In.10a
Identify and graph the solutions (ordered pairs) on a graph of an equation in two variables.
MAFS.912.A-REI.4.In.11a
Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically.
MAFS.912.A-REI.4.In.12a
Graph a linear inequality in two variables using at least two coordinate pairs that are solutions.
MAFS.912.A-REI.4.In.12b
Graph a system of linear inequalities in two variables using at least two coordinate pairs for each inequality.

## Access Points

MAFS.912.A-REI.4.AP.10a
Identify and graph the solutions (ordered pairs) on a graph of an equation in two variables.
MAFS.912.A-REI.4.AP.11a
Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically
MAFS.912.A-REI.4.AP.12a
Graph a linear inequality in two variables using at least two coordinate pairs that are solutions.
MAFS.912.A-REI.4.AP.12b
Graph a system of linear inequalities in two variables using at least two coordinate pairs for each inequality.

## Related Resources

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

## Educational Game

Solving Inequalities: Inequalities and Graphs of Inequalities:

In this challenge game, you will be solving inequalities and working with graphs of inequalities. Use the "Teach Me" button to review content before the challenge. During the challenge you get one free solve and two hints! After the challenge, review the problems as needed. Try again to get all challenge questions right! Question sets vary with each game, so feel free to play the game multiple times as needed! Good luck!

Type: Educational Game

## 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

Using Technology:

Students are asked to use technology (e.g., spreadsheet, graphing calculator, or dynamic geometry software) to estimate the solutions of the equation f(x) = g(x) for given functions f and g.

Type: Formative Assessment

Case In Point:

Students are asked to explain the relationship between the set of solutions and the graph of an exponential equation.

Type: Formative Assessment

What Is the Point?:

Students are asked to explain the relationship between a given linear equation and both a point on its graph and a point not on its graph.

Type: Formative Assessment

Finding Solutions:

Students are asked to explain the relationship between a given linear equation and both a point on its graph and a point not on its graph.

Type: Formative Assessment

Graphs and Solutions - 2:

Students are asked to find the solution(s) of the equation f(x) = g(x) given the graphs of f and g and explain their reasoning.

Type: Formative Assessment

Using Tables:

Students are asked to find solutions of the equation f(x) = g(x) for two given functions, f and g, by constructing a table of values.

Type: Formative Assessment

Graph a System of Inequalities:

Students are asked to graph a system of two linear inequalities.

Type: Formative Assessment

Which Graph?:

Students are asked to select the correct graph of the solution region of a given system of two linear inequalities.

Type: Formative Assessment

Graphs and Solutions -1:

Students are asked to explain why the x-coordinate of the intersection of two functions, f and g, is a solution of the equation f(x) = g(x).

Type: Formative Assessment

Linear Inequalities in the Half-Plane:

Students are asked to graph all solutions for a non-strict <= or >= linear inequality in the coordinate plane.

Type: Formative Assessment

Graphing Linear Inequalities:

Students are asked to graph a strict < or > linear inequality in the coordinate plane.

Type: Formative Assessment

## Lesson Plans

This lesson allows students to graph systems of inequalities with two variables and view three types of solutions.

Type: Lesson Plan

Solving Systems of Inequalities:

This lesson aligns to the Mathematics Formative Assessment System (MFAS) Task Graph a System of Inequalities (). In this lesson, students with similar instructional needs are grouped according to MFAS rubric levels: Getting Started, Moving Forward, Almost There, and Got It. Students in each group complete an exercise designed to move them toward a better understanding of solutions of systems of inequalities and their graphs.

Type: Lesson Plan

Steel vs. Wooden Roller Coaster Lab:

This lesson is a Follow Up Activity to the Algebra Institute and allows students to apply their skills on analyzing bivariate data. This STEM lesson allows students the opportunity to investigate if there is a linear relationship between a coaster's height and speed. Using technology the students can determine the line of best fit, correlation coefficient and use the line for interpolation. This lesson also uses prior knowledge and has students solve systems of equations graphically to determine which type of coaster is faster.

Type: Lesson Plan

Pendulum Conundrum Inquiry Lab:

In this exploration, students will answer the following essential questions:

1. How does the length of a pendulum impact how long it takes to swing back and forth?
2. How does the amount of mass hanging from a pendulum impact the amount of time to swing back and forth?
3. How can we calculate the value of acceleration due to gravity (g) from the behavior of a moving pendulum (optional activity for math reinforcement)?

Type: Lesson Plan

Whose Line Is It Anyway?:

In this lesson, students will use graphing calculators to explore linear equations in the form y = mx + b. They will observe the graphs of equations with different values of slope and y-intercept. They will draw conclusions about how the value of slope and y-intercept are visible in the appearance of the graph.

Type: Lesson Plan

Defining Regions Using Inequalities:

This lesson unit is intended to help you assess how well students are able to use linear inequalities to create a set of solutions. In particular, the lesson will help you identify and assist students who have difficulties in: representing a constraint by shading the correct side of the inequality line and understanding how combining inequalities affects a solution space.

Type: Lesson Plan

## Original Student Tutorials

Graphing Linear Inequalities:

Learn to graph linear inequalities in two variables to display their solutions as you complete this interactive tutorial.

Type: Original Student Tutorial

Finding Solutions on a Graph:

Learn to determine the number of possible solutions for a linear equation with this interactive tutorial.

Type: Original Student Tutorial

Solving an Equation Using a Graph:

Follow as we learn why the x-coordinate of the point of intersection of two functions is the solution of the equation f(x) = g(x) in this interactive tutorial.

Type: Original Student Tutorial

Taxi!:

This simple conceptual problem does not require algebraic manipulation, but requires students to articulate the reasoning behind each statement.

Solution Sets:

The typical system of equations or inequalities problem gives the system and asks for the graph of the solution. This task turns the problem around. It gives a solution set and asks for the system that corresponds to it. The purpose of this task is to give students a chance to go beyond the typical problem and make the connections between points in the coordinate plane and solutions to inequalities and equations. Students have to focus on what the graph is showing. When you are describing a region, why does the inequality have to go one way or another? When you pick a point that clearly lies in a region, what has to be true about its coordinates so that it satisfies the associated system of inequalities?

Population and Food Supply:

In this task students use verbal descriptions to construct and compare linear and exponential functions and to find where the two functions intersect (F-LE.2, F-LE.3, A-REI.11).

This task is the last in a series of three tasks that use inequalities in the same context at increasing complexity in 6th grade, 7th grade and in HS algebra. Students write and solve inequalities, and represent the solutions graphically.

Two Squares are Equal:

This classroom task is meant to elicit a variety of different methods of solving a quadratic equation (A-REI.4). Some are straightforward (for example, expanding the square on the right and rearranging the equation so that we can use the quadratic formula); some are simple but clever (reasoning from the fact that x and (2x - 9) have the same square); some use tools (using a graphing calculator to graph the functions f(x) = x^2 and g(x) = (2x-90)^2 and looking for values of x at which the two functions intersect). Some solution methods will work on an arbitrary quadratic equation, while others (such as the last three) may have difficulty or fail if the quadratic equation is not given in a particular form, or if the solutions are not rational numbers.

## Tutorials

Graph the solution to a system of inequalities.:

This video will demonstrate how to graph the solution to a system of inequalities.

Type: Tutorial

Introduction to the Coordinate Plane:

In this video, you will learn about Rene Descartes, and how he bridged the gap between algebra and geometry.

Type: Tutorial

Dependent and independent variables exercise: graphing the equation:

It's helpful to represent an equation on a graph where we plot at least 2 points to show the relationship between the dependent and independent variables. Watch and we'll show you.

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

## Video/Audio/Animation

Graphing Lines 1:

Khan Academy video tutorial on graphing linear equations: "Algebra: Graphing Lines 1"

Type: Video/Audio/Animation

## Virtual Manipulatives

Graph a Line Using Y-Intercept and Slope:

This tutorial will help you to graph a line using its slope and y-intercept, or to identify the slope and y-intercept from a linear equation written in slope-intercept form.

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

## Student Resources

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

## Original Student Tutorials

Graphing Linear Inequalities:

Learn to graph linear inequalities in two variables to display their solutions as you complete this interactive tutorial.

Type: Original Student Tutorial

Finding Solutions on a Graph:

Learn to determine the number of possible solutions for a linear equation with this interactive tutorial.

Type: Original Student Tutorial

Solving an Equation Using a Graph:

Follow as we learn why the x-coordinate of the point of intersection of two functions is the solution of the equation f(x) = g(x) in this interactive tutorial.

Type: Original Student Tutorial

## Educational Game

Solving Inequalities: Inequalities and Graphs of Inequalities:

In this challenge game, you will be solving inequalities and working with graphs of inequalities. Use the "Teach Me" button to review content before the challenge. During the challenge you get one free solve and two hints! After the challenge, review the problems as needed. Try again to get all challenge questions right! Question sets vary with each game, so feel free to play the game multiple times as needed! Good luck!

Type: Educational Game

Taxi!:

This simple conceptual problem does not require algebraic manipulation, but requires students to articulate the reasoning behind each statement.

Solution Sets:

The typical system of equations or inequalities problem gives the system and asks for the graph of the solution. This task turns the problem around. It gives a solution set and asks for the system that corresponds to it. The purpose of this task is to give students a chance to go beyond the typical problem and make the connections between points in the coordinate plane and solutions to inequalities and equations. Students have to focus on what the graph is showing. When you are describing a region, why does the inequality have to go one way or another? When you pick a point that clearly lies in a region, what has to be true about its coordinates so that it satisfies the associated system of inequalities?

Population and Food Supply:

In this task students use verbal descriptions to construct and compare linear and exponential functions and to find where the two functions intersect (F-LE.2, F-LE.3, A-REI.11).

This task is the last in a series of three tasks that use inequalities in the same context at increasing complexity in 6th grade, 7th grade and in HS algebra. Students write and solve inequalities, and represent the solutions graphically.

Two Squares are Equal:

This classroom task is meant to elicit a variety of different methods of solving a quadratic equation (A-REI.4). Some are straightforward (for example, expanding the square on the right and rearranging the equation so that we can use the quadratic formula); some are simple but clever (reasoning from the fact that x and (2x - 9) have the same square); some use tools (using a graphing calculator to graph the functions f(x) = x^2 and g(x) = (2x-90)^2 and looking for values of x at which the two functions intersect). Some solution methods will work on an arbitrary quadratic equation, while others (such as the last three) may have difficulty or fail if the quadratic equation is not given in a particular form, or if the solutions are not rational numbers.

## Tutorials

Graph the solution to a system of inequalities.:

This video will demonstrate how to graph the solution to a system of inequalities.

Type: Tutorial

Introduction to the Coordinate Plane:

In this video, you will learn about Rene Descartes, and how he bridged the gap between algebra and geometry.

Type: Tutorial

Dependent and independent variables exercise: graphing the equation:

It's helpful to represent an equation on a graph where we plot at least 2 points to show the relationship between the dependent and independent variables. Watch and we'll show you.

Type: Tutorial

## Video/Audio/Animation

Graphing Lines 1:

Khan Academy video tutorial on graphing linear equations: "Algebra: Graphing Lines 1"

Type: Video/Audio/Animation

## 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 topic.

Taxi!:

This simple conceptual problem does not require algebraic manipulation, but requires students to articulate the reasoning behind each statement.

Solution Sets:

The typical system of equations or inequalities problem gives the system and asks for the graph of the solution. This task turns the problem around. It gives a solution set and asks for the system that corresponds to it. The purpose of this task is to give students a chance to go beyond the typical problem and make the connections between points in the coordinate plane and solutions to inequalities and equations. Students have to focus on what the graph is showing. When you are describing a region, why does the inequality have to go one way or another? When you pick a point that clearly lies in a region, what has to be true about its coordinates so that it satisfies the associated system of inequalities?

Population and Food Supply:

In this task students use verbal descriptions to construct and compare linear and exponential functions and to find where the two functions intersect (F-LE.2, F-LE.3, A-REI.11).

This task is the last in a series of three tasks that use inequalities in the same context at increasing complexity in 6th grade, 7th grade and in HS algebra. Students write and solve inequalities, and represent the solutions graphically.

Two Squares are Equal:

This classroom task is meant to elicit a variety of different methods of solving a quadratic equation (A-REI.4). Some are straightforward (for example, expanding the square on the right and rearranging the equation so that we can use the quadratic formula); some are simple but clever (reasoning from the fact that x and (2x - 9) have the same square); some use tools (using a graphing calculator to graph the functions f(x) = x^2 and g(x) = (2x-90)^2 and looking for values of x at which the two functions intersect). Some solution methods will work on an arbitrary quadratic equation, while others (such as the last three) may have difficulty or fail if the quadratic equation is not given in a particular form, or if the solutions are not rational numbers.