Standard 4 : Develop an understanding of two-variable systems of equations.



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General Information

Number: MA.8.AR.4
Title: Develop an understanding of two-variable systems of equations.
Type: Standard
Subject: Mathematics (B.E.S.T.)
Grade: 8
Strand: Algebraic Reasoning

Related Benchmarks

This cluster includes the following benchmarks
Code Description
MA.8.AR.4.1: Given a system of two linear equations and a specified set of possible solutions, determine which ordered pairs satisfy the system of linear equations.
Clarifications:
Clarification 1: Instruction focuses on the understanding that a solution to a system of equations satisfies both linear equations simultaneously.
MA.8.AR.4.2: Given a system of two linear equations represented graphically on the same coordinate plane, determine whether there is one solution, no solution or infinitely many solutions.
MA.8.AR.4.3: Given a mathematical or real-world context, solve systems of two linear equations by graphing.
Clarifications:
Clarification 1: Instruction includes approximating non-integer solutions.

Clarification 2: Within this benchmark, it is the expectation to represent systems of linear equations in slope-intercept form only.

Clarification 3: Instruction includes recognizing that parallel lines have the same slope.



Related Access Points

This cluster includes the following access points.

Access Points

Access Point Number Access Point Title
MA.8.AR.4.AP.1a: Given a system of two linear equations displayed on a graph, identify the solution of a system as the point where the two lines intersect.
MA.8.AR.4.AP.1b: Identify the coordinates of the point of intersection for two linear equations plotted on a coordinate plane.
MA.8.AR.4.AP.2: Given a system of two linear equations represented graphically on the same coordinate plane, identify whether there is one solution or no solution.
MA.8.AR.4.AP.3: Given two sets of coordinates for two lines, plot the lines on a coordinate plane and describe or select the solution to a system of linear equations.


Related Resources

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

Formative Assessments

Name Description
Solving System of Linear Equations by Graphing:

Students are asked to solve a system of linear equations by graphing.

Identify the Solution:

Students are asked to identify the solutions of systems of equations from their graphs and justify their answers.

Lesson Plans

Name Description
Changes are Coming to System of Equations:

Use as a follow up lesson to solving systems of equations graphically. Students will explore graphs of systems to see how manipulating the equations affects the solutions (if at all).

Battle on the High Seas: Applying Systems of Linear Equations:

This lesson is designed to introduce solving systems of linear equations in two variables by graphing. Students will find the solutions of systems of linear equations in two variables by graphing "paths" of battleships and paths of launched torpedoes targeting them. The solutions of the systems will represent the intersection of the paths of a targeted ship (modeled by a linear equation) and the path of a torpedo from a battleship (modeled by another linear equation).

A Scheme for Solving Systems:

Students will graph systems of linear equations in slope-intercept form to find the solution to the system. Students will practice with systems that have one solution, no solution, and all solutions. Because the lesson builds upon a group activity, the students have an easy flow into the lesson and the progression of the lesson is a smooth transition into solving systems algebraically.

Where does my string cross?:

Students will graph a system of linear equations using pieces of string that intersect and discover what the point of intersection has to do with both equations. It will get tricky when the strings do not intersect, or when they transform into the same line.

Put Your Money Where Your Mouth Is:

Students will learn to find the solutions to a system of linear equations, by graphing the equations.

Exploring Systems of Equations using Graphing Calculators:

This lesson plan introduces the concept of graphing a system of linear equations. Students will use graphing technology to explore the meaning of the solution of a linear system including solutions that correspond to intersecting lines, parallel lines, and coinciding lines.
Students will also do graph linear systems by hand.

My Candles are MELTING!:

In this lesson, students will apply their knowledge to model a real-world linear situation in a variety of ways. They will analyze a situation in which 2 candles burn at different rates. They will create a table of values, determine a linear equation, and graph each to determine if and when the candles will ever be the same height. They will also determine the domain and range of their functions and determine whether there are constraints on their functions.

Human systems of linear equations:

Students will work in cooperative groups to demonstrate solving systems of linear equations. They will form lines as a group and see where the point of intersection is.

Problem-Solving Tasks

Name Description
Kimi and Jordan:

Students are asked to create and graph linear equations to compare the savings of two individuals. The purpose of the table in (a) is to help students complete (b) by noticing regularity in the repeated reasoning required to complete the table.

Fixing the Furnace:

Students are asked to write equations to model the repair costs of three different companies and determine the conditions for which each company would be least expensive. This task can be used to both assess student understanding of systems of linear equations or to promote discussion and student thinking that would allow for a stronger solidification of these concepts. The solution can be determined in multiple ways, including either a graphical or algebraic approach.

How Many Solutions?:

The student is given the equation 5x-2y=3 and asked, if possible, to write a second linear equation creating systems resulting in one, two, infinite, and no solutions.



Student Resources

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

Problem-Solving Tasks

Title Description
Kimi and Jordan:

Students are asked to create and graph linear equations to compare the savings of two individuals. The purpose of the table in (a) is to help students complete (b) by noticing regularity in the repeated reasoning required to complete the table.

Fixing the Furnace:

Students are asked to write equations to model the repair costs of three different companies and determine the conditions for which each company would be least expensive. This task can be used to both assess student understanding of systems of linear equations or to promote discussion and student thinking that would allow for a stronger solidification of these concepts. The solution can be determined in multiple ways, including either a graphical or algebraic approach.

How Many Solutions?:

The student is given the equation 5x-2y=3 and asked, if possible, to write a second linear equation creating systems resulting in one, two, infinite, and no solutions.



Parent Resources

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

Problem-Solving Tasks

Title Description
Kimi and Jordan:

Students are asked to create and graph linear equations to compare the savings of two individuals. The purpose of the table in (a) is to help students complete (b) by noticing regularity in the repeated reasoning required to complete the table.

Fixing the Furnace:

Students are asked to write equations to model the repair costs of three different companies and determine the conditions for which each company would be least expensive. This task can be used to both assess student understanding of systems of linear equations or to promote discussion and student thinking that would allow for a stronger solidification of these concepts. The solution can be determined in multiple ways, including either a graphical or algebraic approach.

How Many Solutions?:

The student is given the equation 5x-2y=3 and asked, if possible, to write a second linear equation creating systems resulting in one, two, infinite, and no solutions.