Algebra II Unit 1 Snap Shot Unit Title Cluster Statements Standards in this Unit Unit 1 Polynomial, Rational and Radical Relationships Perform arithmetic operations with complex numbers. Use complex numbers in polynomial identities and equations. Interpret the structure of expressions. Write expressions in equivalent forms to solve problems. Perform arithmetic operations on polynomials. Understand the relationship between zeros and factors of polynomials. Use polynomial identities to solve problems. Rewrite rational expressions. Understand solving radical, rational and polynomial equations as a process of reasoning and explain the reasoning. Represent and solve equations and inequalities graphically including systems which are comprised of a variety of functions. systems of equations. Analyze polynomial and rational functions using different representations. N.CN.1 N.CN.2 N.CN.7 N.CN.8 (+) N.CN.9 (+) A.SSE.1 A.SSE.2 A.SSE.3* A.APR.1 A.APR.2 A.APR.3 A.APR.4 A.APR.5(+) A.APR.6 A.APR.7(+) A.REI.1 A.REI.2 A.REI.4b A.REI.7 A.REI.11 F.IF.7 G.GPE.2
Ability to connect experience with solving quadratic equations from Algebra I to situations analyzing the discriminant will reveal the nature of the solutions which would include complex solutions To Help students to determine if a quadratic equation has complex roots by looking at the graph( the graph does not intersect the x-axis) or the value of the discriminant ( if ) N.CN.8 (+) Extend polynomial identities to the complex numbers. For example, rewrite Knowledge that a negative number can be thought of as the square of an imaginary number N.CN.9 (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. Knowledge of the connection between the number of roots and the degree of the polynomial; considering multiple roots, complex roots and distinct real roots The Fundamental Theorem of Algebra states “Every polynomial equation of degree n with complex coefficients has n roots in the complex numbers. “ Factor algebraically, then use the graphing calculator to show the relationship to the roots of the polynomial. For example, have students make connections between the zeros of the polynomials and the behavior of the graph at these values. y = -3x3 + 24x2 – 45x y = -3x(x – 3)(x – 5) the graph of this polynomial intersect the x-axis at x=0, x=3 and x=5 y = x4 + 6x3 + 9x2 y = x2(x + 3)2 the of this polynomial will be tangent to the x-axis at x=0 and x = -3 because they are double roots Emphasis should be placed on the ability to solve all types of quadratic polynomials, including those with irrational and complex solutions. Cluster Note: Extend to polynomial and rational expressions.
Ability to select and then use an appropriate factoring technique Ability to factor expressions completely over complex numbers A.SSE.4 Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments. ★ Note: Consider extending this standard to infinite geometric series in curricular implementations of this course description. Knowledge of the difference between an infinite and a finite series Ability to apply the formula for the sum of a finite geometric series : Sum of Finite Geometric Progression The sum in geometric progression (also called geometric series) is given by Equation (1) Multiply both sides of Equation (1) by will have Equation (2) Subtract Equation (2) from Equation (1) A.APR.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; Note: Extend to polynomials beyond quadratics Ability to connect experiences from Algebra I with linear and quadratic polynomials to polynomials of higher degree Students should understand that the integers and polynomials are not closed under division. Example is not an integer therefore the set of integers is not closed division. Similarly does not simplify to a polynomial therefore the set of polynomials is not closed under division.
add, subtract, and multiply polynomials. Ability to show that when polynomials are added, subtracted or multiplied that the result is another polynomial A.APR.2 Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x). Ability to make connections between factors, roots and evaluating functions Ability to use both long division and synthetic division Ability to use the graph of a polynomial to assist in the efficiency of the process for complicated cases Students should be able to identify integer roots of a polynomial equation by graphing. Using these integer roots a student should then use division to factor until reaching quadratic form allowing them to identify all roots of the polynomial equation. A.APR.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. Knowledge of the differences in the end behavior of the graphs as dictated by the leading coefficient and whether the function is even or odd Ability to capture the graphical behavior of polynomial functions which have roots with multiplicity greater than one Students should be able to factor higher degree polynomials that have rational and are therefore easier to factor. Students should be able to produce a rough graph of a polynomial that is in factored form. Cluster Note: This cluster has many possibilities for optional enrichment, such as relating the example in A.APR.4 to the solution of the system u2+v2=1, v = t(u+1), relating the Pascal triangle property of binomial coefficients to (x+y)n+1 = (x+y)(x+y)n, deriving explicit formulas for the coefficients, or proving the binomial theorem by induction. A.APR.4 Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity Knowledge of the process for proving identities Ability to see, use and manipulate the structure in an expression as needed to an identity To an identity, logical steps should be used to manipulate one side of the equation to match the other side of the equation. When proving polynomial identities one may not use properties of equality and must restrict their work to using techniques such as factoring, combining like terms, multiplying polynomials etc.
multiply, and divide rational expressions. Note: A.APR.7 requires the general division algorithm for polynomials A.REI.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. Note: Students should focus on and master A.REI.1 for linear equations and be able to extend and apply their reasoning to other types of equations in future courses. Ability to identify the mathematical property (addition property of equality, distributive property, etc.) used at each step in the solution process as a means of justifying a step Example: A student could justify each step of solving equations as shown below. A.REI.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. Note: The limitations on rational functions to the rational expressions in this standard. Limitations: In this course rational functions are limited to those whose numerators are of degree at most one and denominators of degree at most 2. Ability to connect prior experience with solving simple equations in one variable to solving equations which require new strategies and additional steps Ability to make connections between the domain of a function and extraneous solutions Ability to identify extraneous solutions Students should be asked to make a connection between solving linear equations in one variable and solving rational equations in one variable whose numerators are of degree at most one. A.REI.4b Solve quadratic equations by inspection ( e.g. for ), taking square roots, completing the square, the Ability to solve quadratic equations using various methods and recognize the most efficient method
quadratic formula and factoring as appropriate to the initial form of the equations. Recognize when the quadratic formula gives complex solutions and write them as for real numbers a and b. Ability to use the value of the discriminant to determine if a quadratic equation has one double solution, two unique solutions or no real solutions A.REI.7 geometric problems in the coordinate plane leading to a system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example find the points of intersection between the line and the circle . Knowledge of the algebraic and graphic representations of quadratic relations as well as quadratic functions A.REI.11 Explain why the x-coordinates of the points the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. ★ Note: This is an overarching standard that will be revisited as each function is studied. Ability to connect experience with solving systems of equations graphically from Algebra I to solving systems that include polynomial, exponential ,rational, root , absolute value and logarithmic functions Ability to show the equality of two functions using multiple representations This standard is included many times throughout Algebra I and Algebra II. By the end of this unit students should understand that the coordinates of the point(s) of intersection of graphs of two equations represent the solution(s) to the system created by the two equations.
Quadratic Expressions and Equations Graphing Quadratics/ F.IF.7 A.APR.3 Practice This activity is best used as a review or for skill maintenance. Guiding questions will show how this activity can be used to introduce transformation of graphs. This activity is called “Pass the Problem.” Pass the Problem works well as a means of reviewing multiple concepts and can be used to provide distributed practice over time. Quadratic Expressions and Equations This activity provides a means for having students to work collaboratively and provides a legitimate reason for movement while providing practice that should help to strengthen student’s proficiency with graphing quadratics and identifying key characteristics. Quadratic Expressions and Equations Systems of Equations/ A.REI.11 Practice This lesson seed describes a cooperative learning activity where students work collaboratively where students will build their proficiency with solving systems of equations graphically. Systems of Equations The activity in this lesson seed requires students to solve systems of equations which are comprised of one linear equation and one quadratic equation. It could easily be adapted to solve systems comprised of one linear and any other type of function covered in this unit or other units. Students also explore systems which have two, one or no solutions.