Content Standard(s):
Mathematics 4. Interpret linear, quadratic, and exponential expressions in terms of a context by viewing one or more of their parts as a single entity.
Example: Interpret the accrued amount of investment P (1 + r)t , where P is the principal and r is the interest rate, as the product of P and a factor depending on time t.
Unpacked Content
Evidence Of Student Attainment:
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.
Interpret the meaning of the parts of an expression. For example, see that 3 + (x-2)2 is a sum of a constant and a square, that the square contains the expression x-2, and that the value of the expression is always greater than 3.
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. Teacher Vocabulary:
Linear expression
Quadratic expression
Exponential expression
Equivalent expressions Knowledge:
Students know:
How to recognize the parts of linear, quadratic and exponential expressions and what each part represents. When one form of an algebraic expression is more useful than an equivalent form of that same expression to solve a given problem. That one or more parts of an expression can be viewed as a single entity. Skills:
Students are able to:
Use algebraic properties to produce equivalent forms of the same expression by recognizing underlying mathematical structures. Interpret expressions in terms of a context. View one or more parts of an expression as a single entity and determine the impact parts of the expression have in terms of the context. Understanding:
Students understand that:
Making connections among the parts of an expression reveals the roles of important mathematical features of a problem. Diverse Learning Needs:
Essential Skills:
Learning Objectives: ALGI.4.1: Define linear, quadratic and exponential functions.
ALGI.4.2: Classify an expression as linear, quadratic or exponential from a table.
ALGI.4.3: Classify an expression as linear, quadratic or exponential from an equation.
ALGI.4.4: Classify an expression as linear, quadratic or exponential from a graph.
ALGI.4.5: Define terms, factors, and coefficients.
ALGI.4.6: Identify factors in linear, exponential and quadratic expressions.
ALGI.4.7: Identify coefficients in linear, exponential and quadratic expressions.
ALGI.4.8: Identify terms in linear, exponential and quadratic expressions.
ALGI.4.9: Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient).
ALGI.4.10: Recognize one or more parts of an exponential expression as a single entity.
ALGI.4.11: Recognize one or more parts of a quadratic expression as a single entity.
ALGI.4.12: Recognize one or more parts of a linear expression as a single entity.
Prior Knowledge Skills:
Recognize ordered pairs.
Identify ordered pairs.
Recognize linear equations.
Recall how to solve problems using the distributive property.
Define linear functions, nonlinear functions, slope, and y-intercept.
Alabama Alternate Achievement Standards
AAS Standard:
Mathematics 6. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
a. Factor quadratic expressions with leading coefficients of one, and use the factored form to reveal the zeros of the function it defines.
b. 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; complete the square to find the vertex form of quadratics with a leading coefficient of one.
c. Use the properties of exponents to transform expressions for exponential functions.
Example: Identify percent rate of change in functions such as y = (1.02)t , y = (0.97)t , y = (1.01)12t , y = (1.2)t/10 , and classify them as representing exponential growth or decay.
Unpacked Content
Evidence Of Student Attainment:
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. Teacher Vocabulary:
Quadratic expression
Zeros
Complete the square
Roots
Zeros
Solutions
x-intercepts
Maximum value
Minimum value
Factor
Roots
Exponents
Equivalent form Vertex form of a quadratic expression Knowledge:
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. Skills:
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. Understanding:
Students understand that:
Making connections among equivalent expressions reveals the roles of important mathematical features of a problem. Diverse Learning Needs:
Essential Skills:
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.
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).
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.
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.
Prior Knowledge Skills:
Identify how many solutions the linear equation may or may not have.
Recall how to solve problems using the distributive property
Explain the distributive property.
Recall solving one-step equations.
Alabama Alternate Achievement Standards
AAS Standard:
Mathematics 9. Select an appropriate method to solve a quadratic equation in one variable.
a. 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. Explain how the quadratic formula is derived from this form.
b. Solve quadratic equations by inspection (such as x 2 = 49), taking square roots, completing the square, the quadratic formula, and factoring, as appropriate to the initial form of the equation, and recognize that some solutions may not be real.
Unpacked Content
Evidence Of Student Attainment:
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. Teacher Vocabulary:
Completing the square
Quadratic equations
Quadratic formula
Inspection Imaginary numbers
Binomials
Trinomials Knowledge:
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. Skills:
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. Understanding:
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). Diverse Learning Needs:
Essential Skills:
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.
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.
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.
Prior Knowledge Skills:
Identify perfect squares and square roots.
Define square root, expressions, and approximations.
Explain the distributive property.
Calculate an expression in the correct order (Ex. exponents, mult./div. from left to right, and add/sub. from left to right).
Recalving one-step equations.
List given information from the problem.
Identify the unknown, in a given situation, as the variable.
Test the found number for accuracy by substitution. Calculate a solution to an equation by combining like terms, isolating the variable, and/or using inverse operations.
Define equation and variable.
Set up an equation to represent the given situation, using correct mathematical operations and variables.
Recognize the correct order to solve expressions with more than one operation.
Calculate a numerical expression (Ex. V=4x4x4).
Choose the correct value to replace each variable in the algebraic expression (Substitution).
Alabama Alternate Achievement Standards
AAS Standard:
