ALEX Classroom Resource

Students:

• Select an appropriate method (taking square roots, factoring, completing the square, or quadratic formula) for solving a quadratic equation in one variable based on its original form.
• Use completing the square to transform any quadratic equation into the form (x-p)2=q.
• Derive the quadratic formula from (x-p)2=q.
• Recognize that some solutions may not be real or namely that they may be imaginary or complex numbers.
• Provide reasonable approximations when appropriate in a graph or table.

Students know:

• Any real number has two square roots, that is, if a is the square root of a real number then so is -a.
• The method for completing the square.
• A quadratic equation in standard form (ax²+bx+c=0) has real roots when b²-4ac is greater than or equal to zero and complex roots when b²-4ac is less than zero.

Students are able to:

• Take the square root of both sides of an equation.
• Factor quadratic expressions in the form x2+bx+c where the leading coefficient is one.
• Use the factored form to find zeros of the function.
• Complete the square.
• Use the quadratic formula to find solutions to quadratic equations.
• Manipulate equations to rewrite them into other forms.

Students understand that:

• Solutions to a quadratic equation must make the original equation true and this should be verified.
• When the quadratic equation is derived from a contextual situation, proposed solutions to the quadratic equation should be verified within the context given, as well as mathematically.
• Different procedures for solving quadratic equations are necessary under different conditions.
• If ab=0, then at least one of a or b must be zero (a=0 or b=0) and this is then used to produce the two solutions to the quadratic equation.
• Whether the roots of a quadratic equation are real or complex is determined by the coefficients of the quadratic equation in standard form (ax²+bx+c=0).

Students:

• Use elimination to solve a system of linear equations.
• Select an appropriate method (graphical, tabular, substitution, or elimination) for solving a system of linear equations.
• Decide when substitution is a more efficient method than elimination and vice versa.
• Contrast solutions produced algebraically with those produced using a table or graph.
• Provide reasonable approximations when appropriate in a graph or table.

Students know:

• Algebraic techniques for manipulating and solving equations.

Students are able to:

• graph a system of linear equations.
• Generate a table from an equation.
• Find the solution to a system by graphing, completing a table, substitution, and elimination.
• Justify which method they used.

Students understand that:

• When the properties of operations and equality are applied to systems of equations, the resulting equations have the same solution as the original.

Students:

• Solve quadratic equations where both sides of the equation have evident square roots by inspection.
• Transform quadratic equations to a form where the square root of each side of the equation may be taken, including completing the square.
• Use the method of completing the square on the equation in standard form ax²+bx+c=0 to derive the quadratic formula.
• Identify quadratic equations which may be solved efficiently by factoring, and then use factoring to solve the equation.
• Use the quadratic formula to solve quadratic equations.
• Explain when the roots are real or complex for a given quadratic equation, and when complex write them as a ± bi.
• Demonstrate that a proposed solution to a quadratic equation is truly a solution by making the original true.

Students know:

• Any real number has two square roots, that is, if a is the square root of a real number then so is -a.
• The method for completing the square.
• Notational methods for expressing complex numbers.
• A quadratic equation in standard form (ax²+bx+c=0) has real roots when b²-4ac is greater than or equal to zero and complex roots when b²-4ac is less than zero.

Students are able to:

• Accurately use properties of equality and other algebraic manipulations including taking square roots of both sides of an equation.
• Accurately complete the square on a quadratic polynomial as a strategy for finding solutions to quadratic equations.
• Factor quadratic polynomials as a strategy for finding solutions to quadratic equations.
• Rewrite solutions to quadratic equations in useful forms including a ± bi and simplified radical expressions.
• Make strategic choices about which procedures (inspection, completing the square, factoring, and quadratic formula) to use to reach a solution to a quadratic equation.

Students understand that:

• Solutions to a quadratic equation must make the original equation true and this should be verified.
• When the quadratic equation is derived from a contextual situation, proposed solutions to the quadratic equation should be verified within the context given, as well as mathematically.
• Different procedures for solving quadratic equations are necessary under different conditions.
• If ab=0, then at least one of a or b must be zero (a=0 or b=0) and this is then used to produce the two solutions to the quadratic equation.
• Whether the roots of a quadratic equation are real or complex is determined by the coefficients of the quadratic equation in standard form (ax²+bx+c=0).

Learning Objectives:
ALGI.9.1: Define quadratic equation and zero product property.
ALGI.9.2: Solve one-step equations using addition and subtraction that are set equal to zero.
ALGI.9.3: Solve two-step equations using addition and subtraction that are set equal to zero.

a.
ALGI.9.4: Define completing the square.
ALGI.9.5: Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x - p)²= q that has the same solutions.
ALGI.9.6: Derive the quadratic formula from the form (x - p)= q.

b.
ALGI.9.7: Define quadratic formula, factoring, square root, complex number, and real number.
ALGI.9.8: Solve quadratic equations by completing the square.
ALGI.9.9: Solve quadratic equations by the quadratic formula.
ALGI.9.10: Solve quadratic equations by factoring.
ALGI.9.11: Solve quadratic equations by taking square roots.
ALGI.9.12: Recognize when the quadratic formula gives complex solutions.
ALGI.9.13: Write complex solutions as a ±bi for real numbers a and b.

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.

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.

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.

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.

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.