In implementing the Algebra 2 and Trigonometry process and content
performance indicators, it is expected that students will identify and justify
mathematical relationships, formally and informally. The intent of both the
process and content performance indicators is to provide a variety of ways for
students to acquire and demonstrate mathematical reasoning ability when solving
problems. Local curriculum and local/state assessments must support and allow
students to use any mathematically correct method when solving a problem.
Throughout this document the performance indicators use the words
investigate, explore, discover, conjecture, reasoning, argument, justify,
explain, proof, and apply. Each of these terms is an important
component in developing a student’s mathematical reasoning ability. It is
therefore important that a clear and common definition of these terms be
understood. The order of these terms reflects different stages of the reasoning
process.
Investigate/Explore
- Students will be given situations in which they will be asked to look for
patterns or relationships between elements within the setting.
Discover
- Students will make note of possible patterns and generalizations that result
from investigation/exploration.
Conjecture
- Students will make an overall statement, thought to be true, about the new
discovery.
Reasoning
- Students will engage in a process that leads to knowing something to be true
or false.
Argument
- Students will communicate, in verbal or written form, the reasoning process
that leads to a conclusion. A valid argument is the end result of the
conjecture/reasoning process.
Justify/Explain
- Students will provide an argument for a mathematical conjecture. It may be an
intuitive argument or a set of examples that support the conjecture. The
argument may include, but is not limited to, a written paragraph, measurement
using appropriate tools, the use of dynamic software, or a written proof.
Proof
- Students will present a valid argument, expressed in written form, justified
by axioms, definitions, and theorems.
Apply
- Students will use a theorem or concept to solve an algebraic or numerical
problem.
Students will build new mathematical knowledge through problem solving.
|
A2.PS.1 |
Use a variety of problem solving
strategies to understand new
mathematical content |
|
A2.PS.2 |
Recognize and understand equivalent
representations of a problem situation or a mathematical concept |
Students will solve problems that arise in mathematics and in other contexts.
|
A2.PS.3 |
Observe and explain patterns to
formulate generalizations and
conjectures |
|
A2.PS.4 |
Use multiple representations to
represent and explain problem situations (e.g., verbally, numerically,
algebraically, graphically) |
Students will apply and adapt a variety of appropriate strategies to solve problems.
|
A2.PS.5 |
Choose an effective approach to
solve a problem from a variety of
strategies (numeric, graphic,
algebraic) |
|
A2.PS.6 |
Use a variety of strategies to extend
solution methods to other problems |
|
A2.PS.7 |
Work in collaboration with others to
propose, critique, evaluate, and value alternative approaches to problem solving |
Students will monitor and reflect on the process of mathematical problem solving.
|
A2.PS.8 |
Determine information required to
solve a problem, choose methods for
obtaining the information, and
define parameters for acceptable
solutions |
|
A2.PS.9 |
Interpret solutions within the given
constraints of a problem |
|
A2.PS.10 |
Evaluate the relative efficiency of
different representations and solution methods of a problem |
Students will recognize reasoning and proof as fundamental aspects of mathematics.
|
A2.RP.1 |
Support mathematical ideas using a
variety of strategies |
Students will make and investigate mathematical conjectures.
|
A2.RP.2 |
Investigate and
evaluate conjectures in mathematical terms, using
mathematical strategies to reach a conclusion |
|
A2.RP.3 |
Evaluate conjectures and recognize
when an estimate or approximation is more appropriate than an exact answer |
|
A2.RP.4 |
Recognize when an approximation is
more appropriate than an exact answer |
Students will develop and evaluate mathematical arguments and proofs.
|
A2.RP.5 |
Develop, verify, and explain an
argument, using appropriate
mathematical ideas and language |
|
A2.RP.6 |
Construct logical arguments that
verify claims or counterexamples that refute them |
|
A2.RP.7 |
Present correct mathematical
arguments in a variety of forms |
|
A2.RP.8 |
Evaluate written arguments for
validity |
Students will select and use various types of reasoning and methods of proof.
|
A2.RP.9 |
Support an argument by using a
systematic approach to test more
than one case |
|
A2.RP.10 |
Devise ways to
verify results, using counterexamples and informal indirect
proof |
|
A2.RP.11 |
Extend specific
results to more general cases |
|
A2.RP.12 |
Apply inductive
reasoning in making and supporting mathematical conjectures |
Students will organize and consolidate their mathematical thinking through
communication.
|
A2.CM.1 |
Communicate
verbally and in writing a correct, complete, coherent,and clear
design (outline) and explanation for the steps used in solving a
problem |
|
A2.CM.2 |
Use mathematical representations
to communicate with appropriate accuracy, including numerical tables, formulas,
functions, equations, charts, graphs, and diagrams |
Students will communicate their mathematical thinking coherently and
clearly to peers, teachers, and others.
|
A2.CM.3 |
Present organized mathematical
ideas with the use of
appropriate standard notations,
including the use of symbols and
other representations when sharing
an idea in verbal and written form |
|
A2.CM.4 |
Explain relationships among
different representations of a problem |
|
A2.CM.5 |
Communicate logical arguments
clearly, showing why a result makes sense and why the reasoning is valid |
|
A2.CM.6 |
Support or reject arguments or
questions raised by others about the correctness of mathematical work |
Students will analyze and evaluate the mathematical thinking and
strategies of others.
|
A2.CM.7 |
Read and listen for logical
understanding of mathematical
thinking shared by other students |
|
A2.CM.8 |
Reflect on strategies of others in
relation to one’s own strategy |
|
A2.CM.9 |
Formulate
mathematical questions that elicit, extend, or challenge
strategies, solutions, and/or conjectures of others |
Students will use the language of mathematics to express mathematical ideas precisely.
|
A2.CM.10 |
Use correct mathematical language
in developing mathematical
questions that elicit, extend, or
challenge other students’
conjectures |
|
A2.CM.11 |
Represent word
problems using standard mathematical notation |
|
A2.CM.12 |
Understand and
use appropriate language, representations, and terminology
when describing objects, relationships, mathematical
solutions, and rationale |
|
A2.CM.13 |
Draw conclusions
about mathematical ideas through decoding, comprehension,
and interpretation of mathematical visuals, symbols, and
technical writing |
Students will recognize and use connections among mathematical ideas.
|
A2.CN.1 |
Understand and
make connections among multiple representations of the same
mathematical idea |
|
A2.CN.2 |
Understand the
corresponding procedures for similar problems or
mathematical concepts |
Students will understand how mathematical ideas interconnect and build
on one another to produce a coherent whole.
|
A2.CN.3 |
Model situations mathematically,
using representations to draw
conclusions and formulate new
situations |
|
A2.CN.4 |
Understand how
concepts, procedures, and mathematical results in one area
of mathematics can be used to solve problems in other areas
of mathematics |
|
A2.CN.5 |
Understand how
quantitative models connect to various physical models and
representations |
Students will recognize and apply mathematics in contexts outside of mathematics.
|
A2.CN.6 |
Recognize and apply mathematics to
situations in the outside world |
|
A2.CN.7 |
Recognize and
apply mathematical ideas to problem situations that develop
outside of mathematics |
|
A2.CN.8 |
Develop an
appreciation for the historical development of mathematics |
Students will create and use representations to organize, record, and
communicate mathematical ideas.
|
A2.R.1 |
Use physical objects, diagrams,
charts, tables, graphs, symbols,
equations, or objects created
using technology as representations
of mathematical concepts |
|
A2.R.2 |
Recognize,
compare, and use an array of representational forms |
|
A2.R.3 |
Use
representation as a tool for exploring and understanding
mathematical ideas |
Students will select, apply, and translate among mathematical representations
to solve problems.
|
A2.R.4 |
Select appropriate representations
to solve problem situations |
|
A2.R.5 |
Investigate
relationships between different representations and their
impact on a given problem |
Students will use representations to model and interpret physical, social,
and mathematical phenomena.
|
A2.R.6 |
Use mathematics to show and
understand physical phenomena
(e.g., investigate sound waves
using the sine and cosine
functions) |
|
A2.R.7 |
Use mathematics
to show and understand social phenomena (e.g., interpret the
results of an opinion poll) |
|
A2.R.8 |
Use mathematics
to show and understand mathematical phenomena (e.g., use
random number generator to simulate a coin toss) |
Students will understand meanings of operations and procedures, and how they
relate to one another.
Operations
|
A2.N.1 |
Evaluate numerical
expressions with negative and/or
fractional exponents, without the
aid of a calculator (when the
answers are rational numbers) |
|
A2.N.2 |
Perform
arithmetic operations (addition, subtraction,
multiplication, division) with expressions containing
irrational numbers in radical form |
|
A2.N.3 |
Perform arithmetic operations with polynomial expressions
containing rational coefficients |
|
A2.N.4 |
Perform
arithmetic operations on irrational expressions |
|
A2.N.5 |
Rationalize a
denominator containing a radical expression |
|
A2.N.6 |
Write square
roots of negative numbers in terms of i |
|
A2.N.7 |
Simplify powers
of i |
|
A2.N.8 |
Determine the
conjugate of a complex number |
|
A2.N.9 |
Perform
arithmetic operations on complex numbers and write the
answer in the form
Note: This includes simplifying expressions with complex
denominators. |
|
A2.N.10 |
Know and apply
sigma notation |
Students will represent and analyze algebraically a wide variety of problem
solving situations.
Equations and Inequalities
|
A2.A.1 |
Solve absolute value equations and
inequalities involving linear
expressions in one variable |
|
A2.A.2 |
Use the
discriminant to determine the nature of the roots of a
quadratic equation |
|
A2.A.3 |
Solve systems of
equations involving one linear equation and one quadratic
equation algebraically Note: This includes rational
equations that result in linear equations with extraneous
roots. |
|
A2.A.4 |
Solve quadratic
inequalities in one and two variables, algebraically and
graphically |
|
A2.A.5 |
Use direct and
inverse variation to solve for unknown values |
|
A2.A.6 |
Solve an
application which results in an exponential function |
Students will perform algebraic procedures accurately.
Variables and Expressions
|
A2.A.7 |
Factor polynomial
expressions completely, using
any combination of the following
techniques: common factor
extraction, difference of two
perfect squares, quadratic
trinomials |
|
A2.A.8 |
Apply the rules
of exponents to simplify expressions involving
negative and/or
fractional exponents |
|
A2.A.9 |
Rewrite algebraic
expressions that contain negative exponents using only
positive exponents |
|
A2.A.10 |
Rewrite algebraic
expressions with fractional exponents as radical expressions |
|
A2.A.11 |
Rewrite algebraic
expressions in radical form as expressions with fractional
exponents |
|
A2.A.12 |
Evaluate exponential expressions,
including those with base e |
|
A2.A.13 |
Simplify radical expressions |
|
A2.A.14 |
Perform addition, subtraction,
multiplication, and division of radical
expressions |
|
A2.A.15 |
Rationalize denominators involving
algebraic radical expressions |
|
A2.A.16 |
Perform arithmetic operations with
rational expressions and rename to
lowest terms |
|
A2.A.17 |
Simplify complex fractional expressions |
|
A2.A.18 |
Evaluate logarithmic expressions in any
base |
|
A2.A.19 |
Apply the properties of logarithms to
rewrite logarithmic expressions in
equivalent forms |
Equations and Inequalities
|
A2.A.20 |
Determine the sum and product of
the roots of a quadratic
equation by examining its
coefficients |
|
A2.A.21 |
Determine the
quadratic equation, given the sum and product of its roots |
|
A2.A.22 |
Solve radical
equations |
|
A2.A.23 |
Solve rational
equations and inequalities |
|
A2.A.24 |
Know and apply
the technique of completing the square |
|
A2.A.25 |
Solve quadratic
equations, using the quadratic formula |
|
A2.A.26 |
Find the solution
to polynomial equations of higher degree that can be solved
using factoring and/or the quadratic formula |
|
A2.A.27 |
Solve exponential
equations with and without common bases |
|
A2.A.28 |
Solve a
logarithmic equation by rewriting as an exponential equation |
Students will recognize, use, and represent algebraically patterns,
relations, and functions.
Patterns, Relations, and Functions
| A2.A.29 |
Identify an arithmetic or
geometric sequence and find the formula for its nth term |
| A2.A.30 |
Determine the common
difference in an arithmetic sequence |
| A2.A.31 |
Determine the common ratio
in a geometric sequence |
| A2.A.32 |
Determine a specified term
of an arithmetic or geometric sequence |
| A2.A.33 |
Specify terms of a
sequence, given its recursive definition |
| A2.A.34 |
Represent the sum of a
series, using sigma notation |
| A2.A.35 |
Determine the sum of the
first n terms of an arithmetic or geometric series |
| A2.A.36 |
Apply the binomial theorem
to expand a binomial and determine a specific term of a binomial
expansion |
| A2.A.37 |
Define a relation and
function |
| A2.A.38 |
Determine when a relation
is a function |
| A2.A.39 |
Determine the domain and
range of a function from its equation |
| A2.A.40 |
Write functions in
functional notation |
| A2.A.41 |
Use functional notation to
evaluate functions for given values in the domain |
| A2.A.42 |
Find the composition of
functions |
| A2.A.43 |
Determine if a function is
one-to-one, onto, or both |
| A2.A.44 |
Define the inverse of a
function |
| A2.A.45 |
Determine the inverse of a
function and use composition to justify the result |
| A2.A.46 |
Perform transformations
with functions and relations:
,
,
,
,
|
Coordinate Geometry
| A2.A.47 |
Determine the center-radius
form for the equation of a circle in standard form |
| A2.A.48 |
Write the equation of a
circle, given its center and a point on the circle |
| A2.A.49 |
Write the equation of a
circle from its graph |
| A2.A.50 |
Approximate the solution to
polynomial equations of higher degree by inspecting the graph |
| A2.A.51 |
Determine the domain and
range of a function from its graph |
| A2.A.52 |
Identify relations and
functions, using graphs |
| A2.A.53 |
Graph exponential functions
of the form
for
positive values of b, including
|
| A2.A.54 |
Graph logarithmic
functions, using the inverse of the related exponential function |
| A2.A.55 |
Express and apply the six
trigonometric functions as ratios of the sides of a right triangle |
| A2.A.56 |
Know the exact and
approximate values of the sine, cosine, and tangent of 0º, 30º, 45º,
60º, 90º, 180º, and 270º angles |
| A2.A.57 |
Sketch and use the
reference angle for angles in standard position |
| A2.A.58 |
Know and apply the
co-function and reciprocal relationships between trigonometric ratios |
| A2.A.59 |
Use the reciprocal and
co-function relationships to find the value of the secant, cosecant,
and cotangent of 0º, 30º, 45º, 60º, 90º, 180º, and 270º angles |
| A2.A.60 |
Sketch the unit circle and
represent angles in standard position |
| A2.A.61 |
Determine the length of an
arc of a circle, given its radius and the measure of its central angle |
| A2.A.62 |
Find the value of
trigonometric functions, if given a point on the terminal side of
angle
|
| A2.A.63 |
Restrict the domain of the
sine, cosine, and tangent functions to ensure the existence of an
inverse function |
| A2.A.64 |
Use inverse functions to
find the measure of an angle, given its sine, cosine, or tangent |
| A2.A.65 |
Sketch the graph of the
inverses of the sine, cosine, and tangent functions |
| A2.A.66 |
Determine the trigonometric
functions of any angle, using technology |
| A2.A.67 |
Justify the Pythagorean
identities |
| A2.A.68 |
Solve trigonometric
equations for all values of the variable from 0º to 360º |
| A2.A.69 |
Determine amplitude,
period, frequency, and phase shift, given the graph or equation of a
periodic function |
| A2.A.70 |
Sketch and recognize one
cycle of a function of the form
or
|
| A2.A.71 |
Sketch and recognize the
graphs of the functions
,
,
,
and
|
| A2.A.72 |
Write the trigonometric
function that is represented by a given periodic graph |
| A2.A.73 |
Solve for an unknown side
or angle, using the Law of Sines or the Law of Cosines |
| A2.A.74 |
Determine the area of a
triangle or a parallelogram, given the measure of two sides and the
included angle |
| A2.A.75 |
Determine the solution(s)
from the SSA situation (ambiguous case) |
| A2.A.76 |
Apply the angle sum and
difference formulas for trigonometric functions |
| A2.A.77 |
Apply the double-angle and
half-angle formulas for trigonometric functions |
Students will determine what can be measured and how, using appropriate
methods and formulas.
Units of Measurement
| A2.M.1 |
Define radian measure |
| A2.M.2 |
Convert
between radian and degree measures |
Students will collect,
organize, display, and analyze data.
| A2.S.1 |
Understand the differences
among various kinds of
studies (e.g.,
survey, observation, controlled experiment) |
| A2.S.2 |
Determine factors which may
affect the outcome of a survey |
Students will collect,
organize, display, and analyze data.
| A2.S.3 |
Calculate measures of
central tendency with group frequency distributions |
| A2.S.4 |
Calculate measures of
dispersion (range, quartiles, interquartile range, standard deviation,
variance) for both samples and populations |
| A2.S.5 |
Know and apply the
characteristics of the normal distribution |
Students will make predictions that are based upon data analysis.
Predictions from
Data
| A2.S.6 |
Determine from a scatter
plot whether a linear, logarithmic, exponential, or power regression
model is most appropriate |
| A2.S.7 |
Determine the function for
the regression model, using appropriate technology, and use the
regression function to interpolate and extrapolate from the data |
| A2.S.8 |
Interpret within the linear
regression model the value of the correlation coefficient as a measure
of the strength of the relationship |
Students will understand and apply concepts of probability.
Probability
A2.S.9 |
Differentiate between situations requiring permutations and those requiring combinations |
A2.S.10 |
Calculate the number of possible permutations of n items taken r at a time |
A2.S.11 |
Calculate the number of possible combinations of n items taken r at a time |
| A2.S.12 |
Use permutations, combinations, and the Fundamental Principle of Counting to determine the number of elements in a sample space and a specific subset (event) |
A2.S.13 |
Calculate theoretical probabilities, including geometric applications |
A2.S.14 |
Calculate empirical probabilities |
| A2.S.15 |
Know and apply the binomial probability formula to events involving the terms exactly, at least, and at most |
A2.S.16 |
Use the normal distribution as an approximation for binomial probabilities |
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04/20/2005