Standard 7: Algebra -- The student accurately describes and applies
concepts and procedures from algebra.
(14) 7.1. Recognize and use appropriate concepts,
procedures, definitions, and properties to simplify expressions and
solve equations.
7.1.a. Explain the distinction between factor and term.
7.1.b. Explain the distinction between expression and equation.
7.1.c. Explain the distinction between simplify and solve.
7.1.d. Know what it means to have a solution to an equation.
7.1.e. Use properties of equality to solve an equation through a series of
equivalent equations.
7.1.f. Use appropriate properties to simplify an expression, resulting in an
equivalent expression.
7.1.g. Recognize the equivalence between expressions with rational exponents and
radicals .
(15) 7.2. Combine and simplify algebraic expressions that
contain polynomials, rational expressions , radicals, positive or
negative rational exponents, and logarithmic expressions .
7.2.a. Find the sum, difference, or product of two
polynomials , then simplify the result.
7.2.b. Factor out the greatest common factor from polynomials of any degree and
from expressions involving
rational exponents.
7.2.c. Factor quadratic polynomials with integer coefficients into a product of
linear terms .
7.2.d. Simplify quotients of polynomials given in factored form, or in a form
which can be factored or determine if
irreducible over the real numbers.
7.2.e. Add, subtract, multiply, and divide two rational expressions of the form,
a/bx + c where a, b, and c are real
numbers such that bx+c ≠ 0 and of the form p(x) / q(x), where p(x) and q(x) are
polynomials.
7.2.f. Simplify products and quotients of single-term expressions with rational
exponents (rationalizing
denominators not necessary ).
7.2.g. Simplify products and quotients of expressions with rational exponents
and rationalize denominator when
necessary.
7.2.h. Simplify rational expressions that involve complex fractions.
7.2.i. Simplify logarithmic expressions.
7.2.j. Factor polynomials over the complex numbers, if possible, and relate to
the Fundamental Theorem of
Algebra.
(16) 7.3. Solve various types of equations and
inequalities numerically , graphically, and algebraically; interpret solutions
algebraically and in the context of the problem; distinguish between exact and
approximate answers.
7.3.a. Solve linear equations in one variable.
7.3.b. Solve linear inequalities in one variable, including those involving
“and” and “or.”
7.3.c. Solve systems of linear and nonlinear equations in two variables.
7.3.d. Solve linear inequalities in two variables (graphically only) and
nonlinear inequalities (numerically,
graphically and algebraically).
7.3.e. Solve absolute value equations of the form |ax + b| = c.
7.3.f. Use a variety of strategies to solve quadratic equations including those
with irrational solutions and
recognize when solutions are non-real. Simplify complex solutions and check
algebraically. Solve
quadratic equations by completing the square and by taking roots.
7.3.g. Solve equations in one variable containing a single radical or two
radicals.
7.3.h. Solve exponential equations in one variable (numerically, graphically and
algebraically).
7.3.i. Solve rational equations in one variable that can be transformed into an
equivalent linear or quadratic
equation (limited to monomial or binomial denominators).
7.3.j. Solve literal equations (formulas) for a particular variable.
7.3.k. Solve logarithmic equations.
7.3.l. Solve rational equations and inequalities with polynomial denominators.
7.3.m. Solve absolute value and polynomial inequalities and justify solution.
(17) 7.4. Demonstrate an understanding of matrices and
their applications.
7.4.a. Add, subtract and multiply 2×2 matrices.
7.4.b. Find the inverse of a 2×2 matrix.
7.4.c. Evaluate the determinant of 2×2 and 3×3 matrices.
7.4.d. Solve 2×2 and 3×3 systems of linear equations using matrices or the
determinant.
(18) 7.5. Demonstrate an understanding of sequences and
series.
7.5.a. Identify a sequence as arithmetic or geometric,
write an expression for the general term, and evaluate the
sum of a series based upon the sequence.
7.5.b. Construct terms of a series from the general formula and use summation
notation.
7.5.c. Use induction to prove theorems about sums of series.
Standard 8: Functions -- The student accurately describes and applies
function concepts and procedures to
understand mathematical relationships .
(19) 8.1. Recognize functional relationships presented in
words , tables, graphs, and symbols.
8.1.a. Recognize whether a relationship given in a
symbolic, graphical, or tabular form is a function.
8.1.b. Determine the domain and range of a function.
8.1.c. Understand and interpret function notation, particularly as it relates to
graphic displays of data.
8.1.d. Demonstrate an understanding of parametric equations.
(20) 8.2. Represent basic functions (linear, quadratic,
exponential, and reciprocal), piecewise-defined functions (varying
over sub-intervals of the domain), and the following advanced functions (cubic,
quartic, logarithmic, square root,
cube root, absolute value, and rational functions of the type f(x) = 1/ x-a)
using and translating among words,
tables, graphs, and symbols
8.2.a. Evaluate functions to generate a graph.
8.2.b. Describe relationships between the algebraic features of a function and
the features of its graph and/or its
tabular representation.
8.2.c. Use simple transformations (horizontal and vertical shifts, reflections
about axes, shrinks and stretches) to
create the graphs of new functions using linear, quadratic, and/or absolute
value functions, cubic quartic,
exponential, logarithmic, square root, cube root, absolute value, piecewise and
rational functions of the
type f(x) = 1/x-a.
8.2.d. Algebraically construct new functions using addition and subtraction
(e.g., profit function), multiplication,
division, and composition.
8.2.e. Given an algebraic representation of a rational function, find the
intercepts, asymptotes (horizontal,
vertical, and slant), and holes (discontinuities), then sketch the graph.
8.2.f. Given a graph or graphical features, including degrees, intercepts,
asymptotes, and/or holes
discontinuities), generate an algebraic representation of a polynomial or
rational function
8.2.g. Sketch the graph of a polynomial given the degree, zeros, max/min values,
and/or initial conditions.
8.2.h. Graphically/numerically construct new functions using addition,
subtraction and composition.
8.2.i. Identify the components of composite functions (e.g., given f ° g° h ,
find suitable functions f, g, and h) and
determine the domain and range.
(21) 8.3. Analyze and interpret features of a function.
8.3.a. Describe patterns in the function’s rate of change,
identifying intervals of increase, decrease, constancy,
and, if possible, relate them to the function’s description in words or
graphically (using graphic calculator).
8.3.b. Identify y- intercepts and zeros using symbols, graphs, and tables.
8.3.c. Identify extrema and trends using graphs and tables.
8.3.d. Recognize and sketch, without the use of technology, the graphs of the
following families of functions:
linear, quadratic, cubic, quartic, exponential, logarithmic, square root, cube
root, absolute value, and
rational functions of the type f(x) = 1/ x-a, using the symmetry of odd and even
functions when
appropriate.
8.3.e. Understand the relationship between the degree of a polynomial and the
number of roots; interpret the
multiplicity of roots graphically.
(22) 8.4. Model situations and relationships using a variety of basic functions
(linear, quadratic, logarithmic, exponential,
and reciprocal) and piecewise-defined functions.
8.4.a. Choose a function suitable for modeling a real
world situation presented using words or data.
8.4.b. Determine and interpret the meaning of rates of change, intercepts,
zeros, extrema, and trends.
8.4.c. Abstract mathematical models from word problems and interpret solutions
in the context of these source
problems.
8.4.d. Identify and justify whether a result obtained from a function model has
real world significance.
(23) 8.5. Recognize, analyze, and interpret inverse
functions.
8.5.a. Explain the conceptual meaning of inverse functions
using graphs, tables, in words, and arrow diagrams.
8.5.b. Define what it means for a function to be one-to-one, identify examples
and non-examples (algebraic and
graphical), and generate examples (algebraic and graphical).
8.5.c. Find and verify the inverse function algebraically, graphically, and
numerically; restrict the domain of a
function when necessary.
(24) 8.6. Recognize, analyze, interpret, and model with
trigonometric functions.
8.6.a. Represent and interpret trigonometric functions
using the unit circle.
8.6.b. Demonstrate an understanding of radians and degrees by converting between
units, finding areas of
sectors, and determining arc lengths of circles.
8.6.c. Find exact values (without technology) of sine, cosine and tangent for
unit circle and for multiples of π / 6
and π / 4; evaluate trigonometric ratios; and distinguish between exact and
approximate values when
evaluating trigonometric ratios/functions.
8.6.d. Sketch graphs of sine, cosine, and tangent functions, without technology;
identify the domain, range,
intercepts, and asymptotes.
8.6.e. Use transformations (horizontal and vertical shifts, reflections about
axes, period and amplitude changes)
to create new trigonometric functions (algebraic, tabular, and graphical).
8.6.f. Know and apply the identity co²s x + sin² x = 1 and generate related
identities; apply sum and half-angle
identities.
8.6.g. Solve linear and quadratic equations involving trigonometric functions.
8.6.h. Generate algebraic and graphical representation of inverse trigonometric
functions (arcsin, arccos,
arctan), and determine domain and range.
8.6.i. Use trigonometric and inverse trigonometric functions to solve
application problems.
Standard 1: Reasoning and Problem Solving -- The
student uses logical reasoning and mathematical
knowledge to define and solve problems.
25) 1.1. Analyze a situation and describe the problem(s)
to be solved.
1.1.a. Extract necessary facts and relationships from the
given information.
1.1.b. Identify and supply additional information needed to solve each problem.
(26) 1.2. Formulate a plan for solving the problem.
1.2.a. Evaluate the advantages and disadvantages of
different strategies , representations, and tools (including
various forms of technology) for solving the problem.
1.2.b. Choose concepts, strategies, representations, models, and tools
well-suited to solving the problem.
(27) 1.3. Use logical reasoning and mathematical knowledge to obtain and justify
correct solutions.
1.3.a. Correctly execute a plan to solve the problem.
1.3.b. Evaluate and revise the solution method when it appears unlikely to
produce a reasonable or suitably
accurate result.
1.3.c. Evaluate potential solutions for appropriateness, accuracy, and
suitability to the context of the original
problem.
1.3.d. Provide oral, written, and/or symbolic explanations of the reasoning used
to obtain a solution.
1.3.e. Make and justify a multi-step mathematical argument providing appropriate
evidence at each step.
1.3.f. Use a variety of approaches (inductive and deductive, estimations,
generalizations, formal, and/or
informal methods of proof) to justify solutions.
Standard 2: Communication -- The student can interpret and communicate
mathematical knowledge and
relationships in both mathematical and everyday language.
(28) 2.1. Summarize and interpret mathematical information
which may be in oral or written formats.
2.1.a. Summarize and interpret many different types of
graphs.
2.1.b. Recognize and explain the meaning of information presented using
mathematical notation.
2.1.c. Create symbolic representations for situations described in everyday
language.
(29) 2.2. Use symbols, diagrams, graphs, and words to
clearly communicate mathematical ideas, reasoning, and their
implications.
2.2.a. Identify the variables and constants used.
2.2.b. Identify units associated with these variables and constants.
2.2.c. Use correct mathematical symbols, terminology, and notation.
(30) 2.3. Produce mathematically valid oral, written,
and/or symbolic arguments to support a position or conclusion, using
both mathematical and everyday language.
2.3.a. Create explanations that are appropriate to the
needs of the audience and the situation.
2.3.b. Use appropriate details or evidence to support the explanation.
Standard 3: Connections -- The student extends mathematical thinking
across mathematical content areas,
and to other disciplines and real life situations.
(31) 3.1. Use mathematical ideas and strategies to analyze
relationships within mathematics and in other disciplines and
real life situations.
3.1.a. Compare and contrast the different mathematical
concepts and procedures that could be used to
complete a particular task.
3.1.b. Recognize patterns and apply mathematical concepts and procedures in
other subject areas and real
world situations.
(32) 3.2. Understand the importance of mathematics as a
language.
3.2.a. Connect mathematical definitions and procedures
with underlying math concepts.
3.2.b. Transfer mathematical vocabulary, concepts, and procedures to other
disciplinary contexts and the real
world.
3.2.c. Construct procedures and concepts from mathematical definitions.
(33) 3.3. Make connections by using multiple
representations, e.g., analytic, numeric, and geometric.
3.3.a. Integrate mathematical content areas by using
multiple representations.
3.3.b. Use multiple representations to demonstrate understanding of links
between math and other disciplines,
and real world situations.
(34) 3.4. Abstract mathematical models from word problems,
geometric problems, and applications.
3.4.a. Recognize and clarify mathematical structures that
are embedded in other contexts.
3.4.b. Describe geometric objects and shapes algebraically.
3.4.c. Compare and contrast different mathematical models.
