Essential Question(s) A question is essential when it stimulates multi-layered inquiry, provokes deep thought and lively discussion, requires students to consider alternatives and justify their reasoning, encourages re-thinking of big ideas, makes meaningful connections with prior learning, and provides students with opportunities to apply problem-solving skills to authentic situations. Bolded statements represent Essential Questions that span many units and courses. The statements shown in italics represent Essential Questions that are applicable specifically to the content in Unit 1 of Algebra II. When and how is mathematics used in solving real world problems? What characteristics of problems would determine how to model the situation and develop a problem solving strategy? What is the role of complex numbers in the equation solving process? When and why is it necessary to follow set when manipulating numeric or algebraic expressions? How do the ordered pairs on the graph of an equation relate to the equation itself and then to a system which contains the given equation? What are some similarities and differences between the algorithms used for performing operations on rational numbers and the algorithms used for performing operations on rational expressions? Why does the equation solving process sometimes produce extraneous solutions?
Ability to use appropriate vocabulary to categorize polynomials and rational expressions b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)n as the product of P and a factor not depending on P. Note: This is an overarching standard that has applications in multiple units and multiple courses. See the skills and knowledge that are stated in the Standard. Example: When determining the of a student could identify the range as by observing that the denominator is always greater than or equal to 1 and that the numerator is always equal to one and therefore the quotients are always . Example: When analyzing a student would realize that the range of would be . This realization would come from observing that the product of a radical and a negative number would be less than or equal to zero. A.SSE.2 Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2). Note: This is an overarching standard that has applications in multiple units and multiple courses. Ability to use properties of mathematics to alter the structure of an expression Ability to select and then use an appropriate factoring technique Ability to factor expressions completely over complex numbers Recognize higher degree polynomials that are quadratic in nature and can therefore be solved using methods associated with solving quadratic equations. Recognize that rational expressions can be written in different forms and that each form is useful for different reasons. Example: Considering. The second expression would be more useful when determining the zeros of the expression. 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. ★ 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 by Equation (1)
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. function and extraneous solutions Ability to identify extraneous solutions A.REI.11 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); find the solutions approximately, e.g., using 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: Include combinations of linear, polynomial, rational, radical, absolute and exponential 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 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. F.IF.7 functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. ★ Ability to connect experience with graphing linear, exponential and quadratic functions from Algebra I to graphing polynomial functions Ability to identify key features of a function: max, min,
Remainder Theorem 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). When you divide a polynomial by the remainder will be Successive Approximations A.REI.11 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); find the solutions approximately, e.g., using technology to graph the functions, As it appears in this standard, successive approximations refer to the process of using guess and check to determine when . After substituting a value into each expression one would use clues from the results as to what would be a better guess. This process would continue over and over again until the output values from the two expressions come closer to being equal to one another until a specified degree of precision is reached. Example: To find where by using successive approximations you might try
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: Include combinations of linear, polynomial, rational, radical, absolute value, and exponential functions. Progressions from the Common Core State Standards in Mathematics For an in-depth discussion of overarching, “big picture” perspective on student learning of the Common Core State Standards please access the documents found at the site below. Vertical Alignment Vertical curriculum alignment provides two pieces of information: A description of prior learning that should support the learning of the concepts in this unit A description of how the concepts studied in this unit will support the learning of other mathematical concepts. Vertical Alignment Previous Math Courses Algebra II Unit 1 Future Mathematics
Students have been exposed to the quadratic formula and solutions that are square roots of negative numbers. (Algebra 1 Unit 2, 4 and 5) equation has one real and two complex roots. Students subtracted, multiplied and factored polynomials. (Algebra 1 Unit 4) Students have recognized types of relationships that are in linear and exponential models. (Algebra 1 Unit 3) Students have used the process of factoring and completing the square in the quadratic function to show extreme values, and symmetry of the graph, and interpret these in terms of a context. (Algebra 1 Unit 5) Students have used exponential functions to model arithmetic and geometric sequences. (Algebra 1 Unit 2) Polynomials Understand and Manipulate Polynomial Expressions Solve Polynomial Equations Using Algebraic and Graphic Techniques Polynomials are the foundation of the higher level mathematical studies. For example in Calculus students will need to be able to determine which values of “x” will make a polynomial expression take on a positive value a negative value or a value of zero. Students have simplified radicals. (8th Grade – Expressions and Equations) Students have translated between radical and exponential notation. (Algebra 1 Unit 2) Radicals Interpret Radical Expressions Radical Equations Students will need to make use of formulas which contain radicals. The use of such formulas will require students to be adept at manipulating and understanding radical expressions.