ALEX Classroom Resource

Students:

• Make sense of algebraic expressions by identifying structures within the expression which allow them to rewrite it in useful ways to assist in the solution of given problems.
• Produce the useful equivalent forms of expressions,
• Factor a quadratic expression with leading coefficient of one to reveal the zeros of the function it defines and complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
• Use the vertex form of a quadratic expression to reveal the maximum or minimum value and the axis of symmetry of the function it defines.
• Justify their selection of a form for an expression by explaining which features of the expression are revealed by the particular form and how these features aid in resolving a problem situation.

Students know:

• Techniques for generating equivalent forms of an algebraic expression, including factoring and completing the square for quadratic expressions and using properties of exponents.
• When one form of an algebraic expression is more useful than an equivalent form of that same expression to solve a given problem.

Students are able to:

• Use algebraic properties including properties of exponents to produce equivalent forms of the same expression by recognizing underlying mathematical structures.
• Factor quadratic expressions.
• Complete the square in quadratic expressions.
• Use the vertex form of a quadratic expression to identify the maximum or minimum and the axis of symmetry.

Students understand that:

• Making connections among equivalent expressions reveals the roles of important mathematical features of a problem.

Learning Objectives:
ALGI.6.1: Convert an expression to an alternative format.
ALGI.6.2: Recognize the best format for a specific application.
ALGI.6.3: Match equivalent expressions written in different forms.

a.
ALGI.6.4: Define factor, quadratic expression and zero product property.
ALGI.6.5: Factor a quadratic expression.
ALGI.6.6: Use the zero product property to reveal the zeros in the function.
ALGI.6.7: Solve a one-step equation.
ALGI.6.8: Solve a two-step equation.
ALGI.6.9: Determine the Greatest Common Factor (GCF).

b.
ALGI.6.10: Define maximum and minimum value.
ALGI.6.11: Explain the steps for completing the square.
ALGI.6.12: Given a quadratic expression in which the square has already been completed, determine the maximum or minimum values.

c.
ALGI.6.13: Define roots.
ALGI.6.14: Find the equation using the distributive property.
ALGI.6.15: Locate and identify the roots on a graph using the x-intercepts.
ALGI.6.16: Take given roots and convert into a one-step equation set equal to zero.

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:

• Given a contextual situation that may include linear, quadratic, exponential, or rational functional relationships in one variable.
• Model the relationship with equations or inequalities and solve the problem presented in the contextual situation for the given variable.

Students know:

• When the situation presented in a contextual problem is most accurately modeled by a linear, quadratic, exponential, or rational functional relationship.

Students are able to:

• Write equations in one variable that accurately model contextual situations.

Students understand that:

• Features of a contextual problem can be used to create a mathematical model for that problem.

Learning Objectives:
ALGI.11.1: Solve the equation represented by the real-world situation.
ALGI.11.2: Set up an equation to represent the given situation, using correct mathematical operations and variables.
ALGI.11.3: Given a contextual situation, interpret and defend the solution in the context of the original problem.
ALGI.11.4: Define equation, expression, variable, equality and inequality.
ALGI.11.5: Create inequalities with one variable (Exponential, Quadratic, Linear).
ALGI.11.6: Create equalities with one variable (Exponential, Quadratic, Linear).
ALGI.11.7: Solve two-step equations and inequalities.
ALGI.11.8: Solve one-step equations and inequalities using the four basic operations.
ALGI.11.9: Compare and contrast equations and inequalities.
ALGI.11.10: Recognize inequality symbols including greater than, less than, greater than equal to and less than equal to.