Content Standard(s):
Mathematics 2 ) Solve application-based problems by developing and solving systems of linear equations and inequalities. (Alabama)
Mathematics 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:
Mathematics 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:
Mathematics 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:
Mathematics 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:
