ALEX Lesson Plan

     

Systems of Equations: What Method Do You Prefer?

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  This lesson provided by:  
Author:Cathy Bennett
System: Geneva City
School: Geneva High School
The event this resource created for:GEMS
  General Lesson Information  
Lesson Plan ID: 24046

Title:

Systems of Equations: What Method Do You Prefer?

Overview/Annotation:

The purpose of this lesson is to help students apply math concepts of solving systems of equations to real life situations. The students will use the three methods of graphing, substitution, and elimination to solve the system of equations.
This lesson plan was created as a result of the Girls Engaged in Math and Science, GEMS Project funded by the Malone Family Foundation.

 Associated Standards and Objectives 
Content Standard(s):
Mathematics
MA2015 (2016)
Grade: 9-12
Algebraic Connections
2 ) Solve application-based problems by developing and solving systems of linear equations and inequalities. (Alabama)

Mathematics
MA2019 (2019)
Grade: 8
12. Solve systems of two linear equations in two variables by graphing and substitution.

a. Explain that the solution(s) of systems of two linear equations in two variables corresponds to points of intersection on their graphs because points of intersection satisfy both equations simultaneously.

b. Interpret and justify the results of systems of two linear equations in two variables (one solution, no solution, or infinitely many solutions) when applied to real-world and mathematical problems.
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Graph a system of two linear equations, recognizing that the ordered pair for the point of intersection is the x-value that will generate the given y-value for both equations.
  • Recognize that graphed lines with one point of intersection (different slopes) will have one solution, parallel lines (same slope, different y-intercepts) have no solutions, and lines that are the same (same slope, same y-intercept) will have infinitely many solutions.
  • Use substitution to solve a system, given two linear equations in slope-intercept form or one equation in standard form and one in slope-intercept form.
  • Make sense of their solutions by making connections between algebraic and graphical solutions and the context of the system of linear equations.
Teacher Vocabulary:
  • System of linear equations
  • Point of intersection
  • One solution
  • No solution
  • Infinitely many solutions
  • Parallel lines
  • Slope-intercept form of a linear equation
  • Standard form of a linear equation
Knowledge:
Students know:
  • The properties of operations and equality and their appropriate application.
  • Graphing techniques for linear equations (using points, using slope-intercept form, using technology).
  • Substitution techniques for algebraically finding the solution to a system of linear equations.
Skills:
Students are able to:
  • generate a table from an equation.
  • Graph linear equations.
  • Identify the ordered pair for the point of intersection.
  • Explain the meaning of the point of intersection (or lack of intersection point) in context.
  • Solve a system algebraically using substitution when both equations are written in slope-intercept form or one is written in standard form and the other in slope-intercept form.
Understanding:
Students understand that:
  • any point on a line when substituted into the equation of the line, makes the equation true and therefore, the intersection point of two lines must make both equations true.
  • Graphs and equations of linear relationships are different representations of the same relationships, but reveal different information useful in solving problems, and allow different solution strategies leading to the same solutions.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
M.8.12.1: Define variables.
M.8.12.2: Recall how to estimate.
M.8.12.3: Recall how to solve linear equations.
M.8.12.4: Demonstrate how to graph solutions to linear equations.
M.8.12.5: Recall how to graph ordered pairs on a Cartesian plane.
M.8.12.6: Recall that linear equations can have one solution (intersecting), no solution (parallel lines), or infinitely many solutions (graph is simultaneous).
M.8.12.7: Define simultaneous.
M.8.12.8: Recall how to solve linear equations.
M.8.12.9: Recall properties of operations for addition and multiplication.
M.8.12.10: Discover that the intersection of two lines on a coordinate plane is the solution to both equations.
M.8.12.11: Define point of intersection.
M.8.12.12: Recall how to solve linear equations.
M.8.12.13: Demonstrate how to graph on the Cartesian plane.
M.8.12.14: Identify ordered pairs.
M.8.12.15: Recall how to solve linear equations in two variables by using substitution.
M.8.12.16: Create a word problem from given information.
M.8.12.17: Recall how to solve linear equations.
M.8.12.18: Explain how to write an equation to solve real-world mathematical problems.

Prior Knowledge Skills:
  • Define quadrant, coordinate plane, coordinate axes (x-axis and y-axis), horizontal, vertical, and reflection.
  • Demonstrate an understanding of an extended coordinate plane.
  • Draw a four-quadrant coordinate plane.
  • Draw and extend vertical and horizontal number lines.
  • Interpret graphing points in all four quadrants of the coordinate plane in real-world situations.
  • Recall how to graph points in all four quadrants of the coordinate plane.

Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.8.12 Solve two-step linear equations where coefficients are less than 10 and answers are integers.


Mathematics
MA2019 (2019)
Grade: 9-12
Algebra I with Probability
10. Select an appropriate method to solve a system of two linear equations in two variables.

a. Solve a system of two equations in two variables by using linear combinations; contrast situations in which use of linear combinations is more efficient with those in which substitution is more efficient.

b. Contrast solutions to a system of two linear equations in two variables produced by algebraic methods with graphical and tabular methods.
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Choose an appropriate method for solving a system of two linear equations (e.g., substitution, addition, tables, graphing).
  • Solve and justify solutions.
  • Contrast solutions to a system of two linear equations to determine which method is more efficient.
  • Understand that tables and graphs of systems of equations my produce estimates rather than exact solutions.
  • Provide reasonable approximations when appropriate in a graph or table.
Teacher Vocabulary:
  • Solution of a system of linear equations
  • Substitution method
  • Elimination method
  • Graphically solve
  • System of linear equations
  • Solving systems by addition
  • Tabular methods
Knowledge:
Students know:
  • Appropriate use of properties of addition, multiplication and equality.
  • Techniques for producing and interpreting graphs of linear equations.
  • Techniques for producing and interpreting tables of linear equations.
  • The conditions under which a system of linear equations has 0, 1, or infinitely many solutions.
Skills:
Students are able to:
  • Accurately perform the operations of multiplication and addition, and techniques for manipulating equations.
  • Graph linear equations precisely.
  • Create tables and locate solutions from the tables for systems of linear equations.
  • Use estimation to find approximate solutions on a graph.
  • Contrast solution methods and determine efficiency of a method for a given problem situation.
Understanding:
Students understand that:
  • The solution of a linear system is the set of all ordered pairs that satisfy both equations.
  • Solving a system by graphing or with tables can sometimes lead to approximate solutions.
  • A system of linear equations will have 0, 1, or infinitely many solutions.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
ALGI.10.1: Solve a system of equations using three methods (Substitution, Elimination, and Graphing.
ALGI.10.2: Distinguish the similarities and differences between the three methods of solving systems of equations.

Prior Knowledge Skills:
  • Solve a system of equation by graphing.
  • Solve a system of equation by elimination.
  • Solve a system of equation by substitution.
  • Understand the meaning of the solution to a system of equations.
  • Graph a linear equation.

Alabama Alternate Achievement Standards
AAS Standard:
M.A.AAS.11.9 Identify equivalent expressions given a linear expression using arithmetic operations.


Mathematics
MA2019 (2019)
Grade: 9-12
Algebra I with Probability
13. Represent constraints by equations and/or inequalities, and solve systems of equations and/or inequalities, interpreting solutions as viable or nonviable options in a modeling context. Limit to contexts arising from linear, quadratic, exponential, absolute value, and linear piecewise functions.
Unpacked Content
Evidence Of Student Attainment:
Students:
Given a contextual situation involving constraints,
  • Write equations or inequalities or a system of equations or inequalities that model the situation and justify each part of the model in terms of the context.
  • Solve the equation, inequalities or systems and interpret the solution in the original context including discarding solutions to the mathematical model that cannot fit the real-world situation (e.g., distance cannot be negative).
  • Solve a system by graphing the system on the same coordinate grid and determine the point(s) or region that satisfies all members of the system.
  • Determine the point(s) of the region satisfying all members of the system that maximizes or minimizes the variable of interest in the case of a system of inequalities.
Teacher Vocabulary:
  • Constraint
  • System of equations
  • System of inequalities
  • Profit
  • Boundary
  • Closed half plane
  • Open half plane
  • Half plane
  • Consistent
  • Inconsistent
  • Dependent
  • Independent
  • Region
Knowledge:
Students know:
  • When a particular system of two variable equations or inequalities accurately models the situation presented in a contextual problem.
  • Which points in the solution of a system of linear inequalities need to be tested to maximize or minimize the variable of interest.
Skills:
Students are able to:
  • Graph equations and inequalities involving two variables on coordinate axes.
  • Identify the region that satisfies both inequalities in a system.
  • Identify the point(s) that maximizes or minimizes the variable of interest in a system of inequalities.
  • Test a mathematical model using equations, inequalities, or a system against the constraints in the context and interpret the solution in this context.
Understanding:
Students understand that:
  • A symbolic representation of relevant features of a real-world problem can provide for resolution of the problem and interpretation of the situation and solution.
  • Representing a physical situation with a mathematical model requires consideration of the accuracy and limitations of the model.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
ALGI.13.1: Define systems of equations, constraints, viable solution, and nonviable solution.
ALGI.13.2: Create a system of equations or inequalities to represent the given constraints (linear).
ALGI.13.3: Create an equation or inequality to represent the given constraints (linear).
ALGI.13.4: Determine if a solution to a system of equations or inequalities is viable or nonviable.
ALGI.13.5: Determine if there is one solution, infinite solutions, or no solutions to a system of equations or inequalities.

Prior Knowledge Skills:
  • Recall how to draw a number line.
  • Recognize the symbols for =, >, <, < and >.
  • Substitute for the variable to find the value of a given expression.
  • Choose the correct value to replace each variable in the algebraic expression (Substitution).
  • Convert mathematical terms to mathematical symbols and numbers.
  • Recall how to order positive and negative numbers. (Use number line if needed).
  • Locate the position of integers and/or rational numbers on a horizontal or vertical number line.

Alabama Alternate Achievement Standards
AAS Standard:
M.A.AAS.11.11 Select an equation or inequality involving one operation (limit to addition or subtraction) with one variable that represents a real-world problem. Solve the equation.


Mathematics
MA2019 (2019)
Grade: 9-12
Algebra I with Probability
19. Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x).

a. Find the approximate solutions of an equation graphically, using tables of values, or finding successive approximations, using technology where appropriate.
Note: Include cases where f(x) is a linear, quadratic, exponential, or absolute value function and g(x) is constant or linear.
Unpacked Content
Evidence Of Student Attainment:
Students:
Given two functions (where f(x) is linear, quadratic ,absolute value, or exponential and g(x) is constant or linear) that intersect,
  • Graph each function and identify the intersection point(s).
  • Explain solutions for f(x) = g(x) as the x-coordinate of the points of intersection of the graphs, and explain solution paths.
  • Use technology, tables, and successive approximations to produce the graphs, as well as to determine the approximation of solutions.
Teacher Vocabulary:
  • Functions
  • Linear functions
  • Absolute value functions
  • Exponential functions
  • Intersection
Knowledge:
Students know:
  • Defining characteristics of linear, polynomial, absolute value, and exponential graphs.
  • Methods to use technology and tables to produce graphs and tables for two functions.
Skills:
Students are able to:
  • Determine a solution or solutions of a system of two functions.
  • Accurately use technology to produce graphs and tables for linear, quadratic, absolute value, and exponential functions.
  • Accurately use technology to approximate solutions on graphs.
Understanding:
Students understand that:
  • By graphing y=f(x) and y=g(x) on the same coordinate plane, the x-coordinate of the intersections of the two equations is the solution to the equation f(x) = g(x)
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
ALGI.19.1: Define function, function notation, linear, polynomial, rational, absolute value, exponential, and logarithmic functions, and transitive property.
ALGI.19.2: Explain, using the transitive property, why the x-coordinates of the points of the graphs are solutions to the equations.
ALGI.19.3: Find solutions to the equations y = f(x) and y = g(x) using the graphing calculator.
ALGI.19.4: Solve equations for y.
ALGI.19.5: Demonstrate use of a graphing calculator, including using a table, making a graph, and finding successive approximations.

Prior Knowledge Skills:
  • Test the formula V= lwh and V=Bh with the experimental findings.
  • Apply area formulas to solve real-world mathematical problems.
  • Define algebraic expression and variable.

Alabama Alternate Achievement Standards
AAS Standard:
M.A.AAS.12.18 Interpret the meaning of a point on the graph of a line. (Ex.: On a graph of football ticket purchases, trace the graph to a point and tell the number of tickets purchased and the total cost.)


Local/National Standards:

 

Primary Learning Objective(s):

The students will be able to apply math concepts transferring from concrete thinking to abstract thinking and apply their knowledge of systems of equations to real-world problems.

Additional Learning Objective(s):

The students will develop strategies for solving word problems using systems of equations by letting x and y represent the unknown quantities. They will use charts and translate the verbal statements into a system of linear equations.

 Preparation Information 

Total Duration:

Greater than 120 Minutes

Materials and Resources:

Graphic organizers, paper, ruler

Technology Resources Needed:

Graphing calculators will be used for verifying student graphs.

Background/Preparation:

The students may need to visit the website for strategies in solving real life problems with a system of equations. problem solving strategies

  Procedures/Activities: 
1.)The students will be taught how to solve a system of equations by graphing. Have the students access the video clip: http://www.youtube.com/watch?v=qxHCEwrpMw0&NR=1 The students will then complete the attached worksheets. retrieved at http://www.kutasoftware.com/FreeWorksheets/Systems
(system sof equations)
This video clip will teach students how to solve systems of equations using substitution.

2.)The students will be taught how to solve a system of equations by substitution. Have the students use the interactive website. After the students have completed the interactive, the teacher will provide them with problems from their textbook to solve by substitution. retreived at: http://www.kutasoftware.com/FreeWorksheets/Systems
(systems of equations using substitution)
This website will explain to students how to solve systems of equations by substitution.

3.)The students will be taught how to solve a system of equations by elimination. Have the students watch the video clip in order to see the elimination method. The teacher will provide the students with problems from the attachment or from their textbook to solve by the elimination method. http://www.youtube.com/watch?v=FHSLKmyyxHA&feature=related http://www.kutasoftware.com/FreeWorksheets/SystemsWorksheets retrieved at:
(systems of elimination)
This video clip will teach students how to solve systems of equations by elimination.

4.)The students will be given problems in which they will decide the best method for solving. They will also solve one problem using all three methods to see that the same solution will be obtained. Worksheet attached: http://www.tutor-usa.com/Tutorial/WorksheetsPdf/A_Systems-of-Equations-all-methods.pdf
(Systems using all methods)
The worksheet reinforces all methods for solving systems of equations.

5.)The students will then complete the attached worksheet on modeling real life problems using a chart to interpret the problems. They will solve the problems using the method they prefer in each problem.

6.)As a real world application have the students complete the attached handout: systems of equations activity. This activity is found at mathwarehouse.com. A grading rubric is attached.


Attachments:
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  Assessment  

Assessment Strategies

The students will be assessed informally as the teacher interacts with them after the explanation of each of the methods in the lesson. The teacher may choose to give a quiz after each concept is taught. The students will be evaluated on their charts and equations solved in the real world problems.

Acceleration:

The teacher may see if the students can solve the real-world problems with only one variable and get the same result they did with the two unknowns.

Intervention:

A worksheet on solving equations may be used at the beginning of the lesson to remind students how to solve an equation. Extra examples should be worked to be sure students understand and can interpret the real life problems.


View the Special Education resources for instructional guidance in providing modifications and adaptations for students with significant cognitive disabilities who qualify for the Alabama Alternate Assessment.