Download PreAP Algebra II

PreAP Algebra II
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Weeks 1-3
TEKS
• identify the mathematical domains and ranges of functions and determine reasonable domain and range values
for continuous and discrete situations[1A]
• collect and organize data, make and interpret scatterplots, fit the graph of a function to the data, interpret the
results, and proceed to model, predict, and make decisions and critical judgments.[1B]
• identify and sketch graphs of parent functions, including linear (f(x) = x), quadratic (f(x) = x), exponential (f(x) =
a^x), and logarithmic (f(x) = log_a x) functions, absolute value of x (f(x) = |x|), square root of x (f(x) = x), and
reciprocal of x (f(x) = 1/x);[4A]\
Essential Questions
• How do students interpret number sets; determine their members; and evaluate open and closed sets under
various mathematical operations?
• How do students solve linear equations in various complex forms; isolate specific variables in equations; and
simplify various complex expressions?
• How do students write various algebraic proofs and deal with differing orders in steps? Can they defend proof
using cause and effect?
• How do students solve and graph inequalities in one variable and explain the differences between inequalities
and linear equations?
• How do students determine the parent function of absolute value and compare and contrast the differences
between the forms: ?
• How do students interpret equations and graph them using the slope-intercept form?
• How do students interpret various kinds of evidence and transform that evidence into equations of lines in
standard form using the point-slope form?
• How do students determine the equations for absolute value by interpreting their graphs?
• How do students evaluate and graph systems of inequalities in two variables?
• How do students determine the forms of direct and indirect variation and interpret their meanings from graphs?
• How do students solve systems of equations in two and three variables?
• How do students interpret and solve various kinds of word problems using systems?
• How can students use linear programming to solve problems involving maximizing profit, production, and other
variable relationships.
• How do students evaluate matrix operations and use them to solve systems?
• How do students evaluate matrix operations and solve systems using matrices on a graphing calculator?
Weeks 4-6
TEKS
• collect and organize data, make and interpret scatterplots, fit the graph of a function to the data, interpret the
results, and proceed to model, predict, and make decisions and critical judgments.[1B]
• analyze situations and formulate systems of equations in two or more unknowns or inequalities in two unknowns
to solve problems;[3A]
• use algebraic methods, graphs, tables, or matrices, to solve systems of equations or inequalities[3B]
• interpret and determine the reasonableness of solutions to systems of equations or inequalities for given
contexts.[3C]
Weeks 7-9
TEKS
• analyze situations and formulate systems of equations in two or more unknowns or inequalities in two unknowns
to solve problems;[3A]
• use algebraic methods, graphs, tables, or matrices, to solve systems of equations or inequalities[3B]
• interpret and determine the reasonableness of solutions to systems of equations or inequalities for given
contexts.[3C]
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Weeks 10-12
TEKS
• use tools including factoring and properties of exponents to simplify expressions and to transform and solve
equations[2A]
• identify and sketch graphs of parent functions, including linear (f(x) = x), quadratic (f(x) = x), exponential (f(x) =
a^x), and logarithmic (f(x) = log_a x) functions, absolute value of x (f(x) = |x|), square root of x (f(x) = x), and
reciprocal of x (f(x) = 1/x);[4A]
• extend parent functions with parameters such as a in f(x) = a/x and describe the effects of the parameter
changes on the graph of parent functions[4B]
• determine the reasonable domain and range values of quadratic functions, as well as interpret and determine
the reasonableness of solutions to quadratic equations and inequalities;[6A]
• relate representations of quadratic functions, such as algebraic, tabular, graphical, and verbal descriptions[6B]
• use characteristics of the quadratic parent function to sketch the related graphs and connect between the y =
ax2 + bx + c and the y = a(x - h)2 + k symbolic representations of quadratic functions[7A]
• use the parent function to investigate, describe, and predict the effects of changes in a, h, and k on the graphs
of y = a(x - h)2 + k form of a function in applied and purely mathematical situations.[7B]
Essential Questions
• How do students apply the rules for addition and multiplication of polynomials?
• How do students simplify expressions containing various forms of exponents?
• How can students use formal factoring patterns to factor various polynomials?
• How can students use grouping to factor various polynomials?
• How can students use various methods (such as the “M” method to factor complex quadratic polynomials?
• How can students use various factoring methods to solve word problems involving polynomial equations?
• How do students solve polynomial inequalities?
• How do students simplify expressions containing zero and negative exponents?
• How can students use factoring methods to simplify rational algebraic expressions?
• How can students find products and quotients of rational algebraic expressions?
• How can students find sums and differences of rational algebraic expressions?
• How do students simplify expressions containing zero and negative exponents?
• How can students use factoring methods to simplify rational algebraic expressions?
• How can students find products and quotients of rational algebraic expressions?
• How can students find sums and differences of rational algebraic expressions?
• How can students solve rational algebraic equations?
Weeks 13-15
TEKS
• use tools including factoring and properties of exponents to simplify expressions and to transform and solve
equations[2A]
• determine the reasonable domain and range values of quadratic functions, as well as interpret and determine
the reasonableness of solutions to quadratic equations and inequalities;[6A]
• relate representations of quadratic functions, such as algebraic, tabular, graphical, and verbal descriptions[6B]
• determine a quadratic function from its roots (real and complex) or a graph.[6C]
• analyze situations involving quadratic functions and formulate quadratic equations or inequalities to solve
problems;[8A]
• compare and translate between algebraic and graphical solutions of quadratic equations[8C]
• solve quadratic equations and inequalities using graphs, tables, and algebraic methods.[8D]
Weeks 16-18
TEKS
• use complex numbers to describe the solutions of quadratic equations.[2B]
• determine the reasonable domain and range values of quadratic functions, as well as interpret and determine
the reasonableness of solutions to quadratic equations and inequalities;[6A]
• determine a quadratic function from its roots (real and complex) or a graph.[6C]
• analyze situations involving quadratic functions and formulate quadratic equations or inequalities to solve
problems;[8A]
• analyze and interpret the solutions of quadratic equations using discriminants and solve quadratic equations
using the quadratic formula;[8B]
• compare and translate between algebraic and graphical solutions of quadratic equations[8C]
• solve quadratic equations and inequalities using graphs, tables, and algebraic methods.[8D]
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Weeks 19-21
TEKS
• use tools including factoring and properties of exponents to simplify expressions and to transform and solve
equations[2A]
• identify and sketch graphs of parent functions, including linear (f(x) = x), quadratic (f(x) = x), exponential (f(x) =
a^x), and logarithmic (f(x) = log_a x) functions, absolute value of x (f(x) = |x|), square root of x (f(x) = x), and
reciprocal of x (f(x) = 1/x);[4A]
• extend parent functions with parameters such as a in f(x) = a/x and describe the effects of the parameter
changes on the graph of parent functions[4B]
• describe and analyze the relationship between a function and its inverse.[4C]
Essential Questions
• How can students understand the meanings of roots and radicals?
• How can students understand that even roots do not have solutions in the real numbers?
• How can students find products and quotients of roots and radicals?
• How can students find sums and differences of roots and radicals?
• How can students solve equations involving roots and radicals?
• How can students find extraneous solutions to equations involving roots and radicals?
• How can students understand the imaginary number “i”, and deal with it in complex numbers?
• How can students identify quadratic equations and their basic characteristics?
• How can students use completing the square to solve for the roots of a quadratic equation?
• How can students discover the vertex form while completing the square?
• How can students discover that the quadratic equation is really just a derivation of completing the square?
• How can students discover that the discriminant determines the natures of the roots of a quadratic equation?
• How can students determine the nature of graphs of quadratic equations from their vertex form?
• How can students graph quadratic equations finding all significant points and the axis of symmetry?
• How can students identify quadratic equations and write their formulas from various evidence?
• How can students use long division to find factors of a polynomial equation?
• How can students use synthetic division to find the roots of a polynomial?
• How can students use the conjugate root theorem to find roots of a polynomial?
• How can students use Descartes’ rule of signs to find the possible signs of roots?
• How can students determine a list of all possible rational roots then use synthetic division to find them?
• How can students graph polynomial equations with many roots?
Weeks 22-24
TEKS
• use tools including factoring and properties of exponents to simplify expressions and to transform and solve
equations[2A]
• extend parent functions with parameters such as a in f(x) = a/x and describe the effects of the parameter
changes on the graph of parent functions[4B]
• use the parent function to investigate, describe, and predict the effects of parameter changes on the graphs of
square root functions and describe limitations on the domains and ranges;[9A]
• relate representations of square root functions, such as algebraic, tabular, graphical, and verbal descriptions;
[9B]
• determine the reasonable domain and range values of square root functions, as well as interpret and determine
the reasonableness of solutions to square root equations and inequalities;[9C]
• determine solutions of square root equations using graphs, tables, and algebraic methods;[9D]
• determine solutions of square root inequalities using graphs and tables;[9E]
• analyze situations modeled by square root functions, formulate equations or inequalities, select a method, and
solve problems[9F]
• connect inverses of square root functions with quadratic functions.[9G]
Weeks 25-27
TEKS
• extend parent functions with parameters such as a in f(x) = a/x and describe the effects of the parameter
changes on the graph of parent functions[4B]
• use quotients of polynomials to describe the graphs of rational functions, predict the effects of parameter
changes, describe limitations on the domains and ranges, and examine asymptotic behavior;[10A]
• analyze various representations of rational functions with respect to problem situations;[10B]
• determine the reasonable domain and range values of rational functions, as well as interpret and determine the
reasonableness of solutions to rational equations and inequalities;[10C]
• determine the solutions of rational equations using graphs, tables, and algebraic methods;[10D]
• determine solutions of rational inequalities using graphs and tables;[10E]
• analyze a situation modeled by a rational function, formulate an equation or inequality composed of a linear or
quadratic function, and solve the problem[10F]
• use functions to model and make predictions in problem situations involving direct and inverse variation.[10G]
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Weeks 28-30
TEKS
• extend parent functions with parameters such as a in f(x) = a/x and describe the effects of the parameter
changes on the graph of parent functions[4B]
• develop the definition of logarithms by exploring and describing the relationship between exponential functions
and their inverses;[11A]
• use the parent functions to investigate, describe, and predict the effects of parameter changes on the graphs of
exponential and logarithmic functions, describe limitations on the domains and ranges, and examine asymptotic
behavior;[11B]
• determine the reasonable domain and range values of exponential and logarithmic functions, as well as interpret
and determine the reasonableness of solutions to exponential and logarithmic equations and inequalities;[11C]
• determine solutions of exponential and logarithmic equations using graphs, tables, and algebraic methods;[11D]
• determine solutions of exponential and logarithmic inequalities using graphs and tables[11E]
• analyze a situation modeled by an exponential function, formulate an equation or inequality, and solve the
problem.[11F]
Essential Questions
• How can students identify various conic sections from their formulas?
• How can students identify various conic sections from other kinds of evidence?
• How can students identify various conic sections from the nature of their squared terms?
• How can students use completing the square to derive the formulas for conic sections?
• How can students graph all conic sections and find their significant characteristics and points?
• How can students derive the formulas for the various conic sections by using the conic sections’ definitions?
• How can students identify and graph rational functions with asymptotes?
• How can students identify and graph exponential functions of both growth and decay?
• How can students convert to and use rational exponents?
• How can students use rational exponents to solve equations, both with common bases and with variables in the
base?
• How can students understand their natures and solve composite functions?
• How can students determine if a function has an inverse and how to find and graph that inverse?
• How can students understand the history and purpose of logarithms?
• How can students understand that logs are the inverses of exponential functions?
• How can students convert to and use logs in real world problems involving exponential growth and decay and
compound interest?
• How can students understand and use the laws of logarithms?
• How can students understand the natural log and the transcendental number “e”?
• How can students use the graphing calculator to solve problems involving logs?
Weeks 31-33
TEKS
• describe a conic section as the intersection of a plane and a cone;[5A]
• sketch graphs of conic sections to relate simple parameter changes in the equation to corresponding changes in
the graph;[5B]
• identify symmetries from graphs of conic sections;[5C]
• identify the conic section from a given equation[5D]
• use the method of completing the square.[5E]
Weeks 34-36
TEKS
• use tools including factoring and properties of exponents to simplify expressions and to transform and solve
equations[2A]
• use complex numbers to describe the solutions of quadratic equations.[2B]
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AP
• Polynomial Expressions, Functions, and Equations. Students extend their understanding of functions from
linear* settings to include polynomial functions, operations on these functions, and the solution of polynomial
equations using complex numbers. Students use polynomials, especially quadratics, to model situations with
graphical and symbolic representations, and they translate among these representations to represent and discuss
the qualitative behavior of the associated functions. (AP AII 1)
• Student operates with monomials, binomials, and polynomials, applies these operations to analyze the graphical
behavior of polynomial functions, and applies the composition of functions to model and solve problems. (AP AII
1.1)
• Adds, subtracts, and multiplies polynomial expressions to solve problems. (AP AII 1.1.1)
• Analyzes and describes graphs of polynomial functions by examining their intercepts, zeros, domain and range,
and local (turning points) and global (end) behavior. (AP AII 1.1.2)
• Uses factoring, properties of exponents, and knowledge of the related contextual needs to transform
expressions and solve problems. (AP AII 1.1.3)
• Applies the composition of functions to model and solve problems, explaining the results. (AP AII 1.1.4)
• Student represents, compares, translates among representations, and graphically, symbolically, and tabularly
represents, interprets, and solves problems involving quadratic functions. (AP AII 1.2)
• Identifies, interprets, and translates among different representations of quadratic functions, realizing that their
graphs are parabolas. (AP AII 1.2.1)
• Determines reasonable domain and range values for quadratic functions within a context, and tests the
reasonableness of solutions to quadratic equations (zeros of quadratic functions). (AP AII 1.2.2)
• Identifies any points of intersection of the graph of a quadratic equation of the form y = ax2 and the graph of a
line of the form y = k, and relates the points of intersection to the solutions of the quadratic equation ax2 = k. (AP
AII 1.2.3)
• Sketches a quadratic function’s graph, and recognizes the relationships between the coefficients of a quadratic
function and characteristics of its graph (e.g., shape, position, intercepts, zeros, maximum, minimum, symmetry,
vertex). (AP AII 1.2.4)
• Formulates equations and inequalities based on quadratic functions, solves them using factoring, completing the
square, and technology, and interprets their solutions in terms of the original problem context. (AP AII 1.2.5)
• Develops the quadratic formula, and applies it to the solution of quadratic equations and the interpretation of the
nature of their roots. (AP AII 1.2.6)
• Constructs and solves quadratic inequalities in one and two variables, and represents their solutions graphically.
(AP AII 1.2.7)
• Student represents, applies, and discusses the properties of complex numbers. (AP AII 1.3)
• Defines, plots, and computes with complex numbers. (AP AII 1.3.1)
• Describes how the associative, commutative, and distributive properties of operations on real numbers extend to
operations on complex numbers. (AP AII 1.3.2)
• Solves quadratic equations with real coefficients over the set of complex numbers. (AP AII 1.3.3)
• Exponential, Logarithmic, and Other Functions. Students develop exponential, logarithmic, and other nonlinear
functions (rational, radical, absolute value, and piecewise-defined) to represent, investigate, and solve problems
in mathematics and real-world contexts. (AP AII 2)
• Student represents geometric or exponential growth with exponential functions and equations, and applies such
functions and equations to solve problems in mathematics and real-world contexts. (AP AII 2.1)
• Extends the properties of rational exponents to real exponents, relating expressions with rational exponents to
the corresponding radical expressions. (AP AII 2.1.1)
• Recognizes exponential functions from their verbal description and tabular, graphical, or symbolic
representations, and translates among these representations. (AP AII 2.1.2)
• Describes the effects of changes in the coefficient, base, and exponent on the growth described by an
exponential function. (AP AII 2.1.3)
• Approximates solutions to an exponential equation, and relates the solutions to the points of intersection of the
graph of the exponential equation and the graph of a horizontal line. (AP AII 2.1.4)
• Analyzes a problem situation modeled by an exponential function, formulates an equation or inequality, and
solves the problem. (AP AII 2.1.5)
• Uses exponential functions to solve problems involving compound interest and exponential growth and decay in
mathematics and real-world contexts. (AP AII 2.1.6)
• Graphs and analyzes the behavior of exponential functions. (AP AII 2.1.7)
• Student defines logarithmic functions and uses them to solve problems in mathematics and real-world contexts.
(AP AII 2.2)
• Defines a logarithm as a solution to an exponential equation, and recognizes the inverse relationship between
functions defined by logarithms and exponential expressions, showing this relationship graphically. (AP AII 2.2.1)
• Solves problems by applying properties of logarithms [log xy = log x + log y, log (_x) = log x – log y, and log(xa)
= a log(x)] to construct equivalent forms of y a logarithmic expression. (AP AII 2.2.2)
• Applies the inverse relationship between exponential and logarithmic functions to solve problems in
mathematics and real-world contexts. (AP AII 2.2.3)
• Student interprets and represents rational and radical functions and solves rational and radical equations.(AP
AII 2.3)
• Models and solves problems using direct, inverse, joint, and combined variation. (AP AII 2.3.1)
• Models problem situations by constructing equations and inequalities based on rational functions, uses a variety
of methods to solve them, and interprets the solutions in terms of the problem situation. (AP AII 2.3.2)
• Adds, subtracts, multiplies, and evaluates rational functions and simplifies rational expressions with linear and
quadratic denominators. (AP AII 2.3.3)
• Describes the graphs of rational functions, describes limitations on the domains and ranges, and examines
asymptotic behavior. (AP AII 2.3.4)
• Uses properties of radicals to solve equations and identifies extraneous roots when they occur. (AP AII 2.3.5)
• Student interprets and models step and other piecewise- defined functions, including functions involving
absolute value. (AP AII 2.4)
• Analyzes a problem situation to determine or interpret reasonable domain and range values for
piecewise-defined functions representing the situation. (AP AII 2.4.1)
• Interprets, constructs, and applies step functions (e.g., greatest integer/ floor) and other piecewise-defined
functions, including absolute value functions, to model and solve problems. (AP AII 2.4.2)
• Translates among verbal, graphical, tabular, and symbolic representations of step functions and other
piecewise-defined functions, including absolute value functions. (AP AII 2.4.3)
• Systems of Equations and Inequalities and Matrices. Students construct, solve, and interpret solutions to
systems of linear equations in two variables. Students represent cross-categorized data in matrices and perform
operations on matrices to model and interpret problem situations. Students model and solve systems of equations
with technology. (AP AII 3)
• Student constructs, solves, and interprets solutions of systems of linear equations in two variables representing
mathematical and real-world contexts. (AP AII 3.1)
• Constructs a system of linear inequalities in two variables to represent a mathematical or real-world setting. (AP
AII 3.1.1)
• Analyzes and explains the reasoning used to solve systems of linear equations in two variables. (AP AII 3.1.2)
• Solves a system of linear equations in two variables using symbolic methods and graphically, and interprets the
meaning of the solution. (AP AII 3.1.3)
• Student represents and interprets cross-categorized data in matrices, develops properties of matrix addition,
and uses matrix addition and its properties to solve problems. (AP AII 3.2)
• Represents numerical or relational data categorized by two variables in a matrix and labels the rows and
columns. Interprets the meaning of a particular entry in a matrix in terms of the labels of its row and column. (AP
AII 3.2.1)
• Uses matrix row and column sums to analyze data. (AP AII 3.2.2)
• Develops the properties of matrix addition, and adds and subtractsmatrices to solve problems. (AP AII 3.2.3)
• Student multiplies matrices, verifies the properties of matrix multiplication, and uses the matrix form for a system
of linear equations to structure and solve systems consisting of two or three linear equations in two or three
unknowns, respectively, with technology. (AP AII 3.3)
• Verifies the properties of matrix multiplication, and multiplies matrices to solve problems. (AP AII 3.3.1)
• Constructs a system of linear equations modeling a real-world situation, and represents the system as a matrix
equation (Ax = b), that is, (AP AII 3.3.2)
• Solves a system consisting of two or three linear equations in two or three unknowns, respectively, by solving
the related matrix equation Ax = b, using technology to find x = A–1b. (AP AII 3.3.3)
• Experiments, Surveys, and Observational Studies. Students distinguish among experiments, surveys, and
observational studies. They design studies, collect and analyze data using appropriate methods, draw
conclusions, and communicate results. They evaluate studies reported in the media. (AP AII 4)
• Student identifies problems that can be addressed through collection and analysis of experimental data, designs
and implements simple comparative experiments, and draws appropriate conclusions from the collected data.
(AP AII 4.1)
• Describes how well-designed experiments use random assignment to balance the variation of some factors in
order to isolate the effects of a treatment. (AP AII 4.1.1)
• Designs a simple comparative experiment to answer a question: determines treatments, identifies methods of
measuring variables, randomly assigns units to treatments, and collects data, distinguishing between explanatory
and response variables. (AP AII 4.1.2)
• Organizes and displays data from an experiment; summarizes the data using measures of center and spread,
including the mean and standard deviation; identifies patterns and trends in tables and plots; and communicates
methods used and the results of the experimental study to nontechnical persons. (AP AII 4.1.3)
• Student distinguishes among surveys, observational studies, and designed experiments and relates each type
of investigation to the research questions it is best suited to address. Student recognizes that an observed
association between a response variable and an explanatory variable does not necessarily imply that the two
variables are causally linked. Student recognizes the importance of random selection in surveys and random
assignment in experimental studies. Student communicates the purposes, methods, and results of a statistical
study, and evaluates studies reported in the media. (AP AII 4.2)
• Distinguishes among questions best explored through a sample survey, an observational study, or a designed
experiment. Recognizes that an observed association between a response variable and an explanatory variable
does not necessarily imply that the two variables are causally linked. Illustrates the different types of conclusions
that may be drawn from surveys, observational studies, and designed experiments. (AP AII 4.2.1)
• Evaluates possible factors involved in a given problem and what information they provide relative to the question
of interest. Formulates specific questions and identifies quantitative measures that may be used in providing
answers to the question of interest. (AP AII 4.2.2)
• Describes advantages and disadvantages of using different methods of measuring variables. Explains how
biases can occur in studies and their effects on study outcomes. (AP AII 4.2.3)
• Compares and contrasts the random sampling of units from a population and the random assignment of
treatments to experimental units. (AP AII 4.2.4)
• Explains why most research questions do not have unique answers and why several approaches to answering
the same question may be appropriate; explains why different studies of the same research question, conducted
differently, may yield very different results and why this is to be expected. (AP AII 4.2.5)
• Communicates, both orally and in writing, the purposes, methods, and results of a statistical study using
nontechnical language. (AP AII 4.2.6)
• Evaluates study results reported in the media. (AP AII 4.2.7)
• Permutations, Combinations, and Probability Distributions. Students solve ordering, counting, and related
probability problems. Students recognize a binomial probability setting and compute the probability distribution for
a binomial count. Students recognize settings where the normal model may be used, and they apply the empirical
rule to solve problems. (AP AII 5)
• Student solves ordering, counting, and related probability problems. Student recognizes a binomial probability
setting and computes the probability distribution for a binomial count. (AP AII 5.1)
• Uses permutations, combinations, and the multiplication rule for counting to solve counting and probability
problems. (AP AII 5.1.1)
• Recognizes a binomial probability setting, and develops and graphs the probability distribution for a binomial
count. (AP AII 5.1.2)
• Student identifies data settings in which the normal distribution may be useful. Student describes characteristics
of the normal distribution and uses the empirical rule to solve problems. (AP AII 5.2)
• Identifies settings in which the normal distribution may be useful, and describes characteristics of the normal
distribution. (AP AII 5.2.1)
• Uses graphical displays and the empirical rule to evaluate the appropriateness of the normal model for a given
set of data. Uses the empirical rule to estimate the probability that an event will occur in a specific interval that
can be described in terms of whole numbers of standard deviations about the mean. (AP AII 5.2.2)