School District of South Orange and Maplewood
Mathematics Department
Appendix C: Level Distinctions in Algebra 2
Leveling is not a function of intelligence or mathematical talent or the ability to learn.
Leveling in math begins with a consideration of the mathematical content that has to be
developed, takes a measure of students’ prior learning, and enacts a plan to maximize
learning across a spectrum of student achievement.
This appendix described the parameters of content by level for each learning objective as delineated in the content
outline. At each level, students will develop every learning objective prescribed by State Standards (adopted in 2008).
Variations and modifications in the content outline are based on evidence of foundational knowledge, the instructional
time required for students to obtain mastery of essential aspects of content, and the opportunity to make mathematical
decisions as the content is developed.
The mathematics identified here establishes the expectations for mastery in each level.
Mathematics Curriculum: Algebra 2
Objective
Level 2
1. Graph real world
phenomena and solve problems
that involve variation.
These objectives identify
deficiencies and gaps in 11-2
students' prerequisite knowledge
and skills that typically lead to
problems over the course of the
school year. Most notably, these are
fractions, negative numbers, graph
interpretation, and algebraic
computation. In obj 1-3, these
crucial topics are developed/clarified
at the beginning of the course to
avoid sticking points during the
year.
2. Apply and explain methods
for solving problems involving
integers and rational numbers.
3. Evaluate and simplify
polynomial expressions
4. Use the properties of real
numbers to evaluate
expressions and formulas, and
solve equations
May 16 2012
-Overt discussion and activities to
distinguish understanding between
expressions and equations.
-Extended practice on the types of
solutions linear equations can have
(no solution, infinite solutions, and
1 solution) with most examples
being teacher-generated.
-Remediate approach to making
mathematical decisions to solve
equations
-Integer coefficients and common
fractions in linear equations
-Stress on students describing their
problem-solving approach
(minimum formalization)
Level 3
Level 4
Level 5
The connections regarding this
objective are made among
students in relatively shorter
times than for level 2.
Prior mastery of topic
expected.
Prior mastery of topic
expected.
The connections regarding this
objective are made among
students in relatively shorter
times than for level 2.
Prior mastery of topic
expected.
Prior mastery of topic
expected.
The connections regarding this
objective are made among
students in relatively shorter
times than for level 2.
Prior mastery of topic
expected.
Prior mastery of topic
expected.
Prior mastery of topic
expected.
Prior mastery of topic
expected.
-Brief practice to review the types
of solutions linear equations can
have (no solution, infinite solutions,
and 1 solution)
-More formal use of vocabulary is
required for student-generated
examples.
-Rational coefficients in linear
equations are now required.
-Student are required to generate
examples on assessment.
Page 2
Mathematics Curriculum: Algebra 2
Objective
5. Solve absolute value
equations
6. Solve and graph
inequalities
Level 2
-Frame absolute value as the
distance away from zero.
-Beginning example:
x =constant.
-Students will be responsible for
integer coefficient equations
embedded in absolute value
equations.
-Primary focus on linear inequalities
with integer coefficients and
checking validity of solutions
-Expressing solutions to inequalities
as intervals on a number line.
-Open and closed interval notations.
7. Recognize, represent,
and use linear functions to
represent real world
phenomena and solve
problems.
-Discuss differences between
discrete and continuous function
-Responsible for scenarios
involving continuous data sets only
-Distinguishing between and
finding x and y intercepts (limited to
integer values)
-Describing domain and range with
inequality notation or by a verbal
expression
8. Analyze and determine
the rate of change using
appropriate graphing
technologies.
-Re-teach graphical and algebraic
representation of slope
-Using computer software (green
globs, equation grapher) to
reinforce understanding
-Scatterplot would only include
integer coordinates and a limited
number of points. (n <10)
May 16 2012
Level 3
-Students will be responsible for
integer coefficient equations
embedded in absolute value
equations.
-Build upon prior knowledge of
absolute value as a scenario
usually requiring two cases.
-Solving absolute value
inequalities with integer and
simple rational solutions.
-Write the absolute value
inequality from a number line.
-Solve absolute value inequalities
using defs of intersection/union.
-Discuss differences between
discrete and continuous function
-Responsible for scenarios
involving continuous and
discrete data sets.
-Distinguishing between and
finding x and y intercepts
(rational values possible)
-Describing domain and range
with inequality notation or by a
verbal expression
-Student-driven discussion to
recall slope definition.
-Student exposure to best-fit lines
found via graphing utility vs.
finding by hand(student choice on
assessment)
-Scatterplot would only include
integer coordinates and a larger
set of points (n >10)
Level 4
Prior familiarity expected
-Solving multistep absolute value
equations
-Extend to more rigorous
algebraic processes such as
3|2x – 5|= 15and |3x – 7| = x+ 2
Prior familiarity expected
Solving absolute value
inequalities with non standard
solutions
-Write the absolute value
inequality from a number line
Prior familiarity expected
-Describing domain and range
with inequality notation
-Introduction to interval notation
-No limit to types of values used
in or found in problems
-Discussion of what a correlation
coefficient is and interpreting
correlation coefficient as a
measure of goodness of fit.
Finding the value numerically on
the calculator.
-Assessment includes finding the
line of best fit on a calculator
Level 5
Prior mastery of topic expected.
Note: Prior familiarity expected.
Diagnostic exam administered
to determine need for review.
Topic extensions:
-Multiple representations both on
a number line and a coordinate
plane
Note: Prior familiarity expected.
Diagnostic exam administered
to determine need for review.
Topic extensions:
-Describing domain and range
with interval notation
-Writing equation from abstract
point sets
Note: Prior familiarity expected.
Diagnostic exam administered
to determine need for review.
Topic extensions:
-Interpreting the meaning of slope
in real life situations
-Interpreting correlation coefficient
as a measure of goodness of fit
-Finding the line of best fit on a
calculator and its use in
extrapolation/interpolation.
Page 3
Mathematics Curriculum: Algebra 2
Objective
9. Select and use
appropriate methods for
solving linear equations.
Level 2
Level 4
-Students required to know linear
formats for use on assessments
-Information embedded in
problems are easily identifiable
-Solutions may be rational.
-Heavier use of information
embedded in verbal passages
(not very easily identified)
-Generate parallel/perpendicular
line sets to satisfy a given
condition.
-Extend discussion to include
piecewise functions where
students generate a graph from a
given piecewise function.
-Comparisons between noncommon dilations (rational
coefficients)
-Calculating dilation constant from
a given point on a function.
-Piecewise functions will also be
written given a graphical
representation of the function only
-Generate piecewise functions
from given scenarios involving
real world examples(i.e. shipping
fees)
11. Perform transformations -Limited to identity function, absolute
on commonly-used
value function and quadratic
functions.
function.
-Describing changes in domain and
range as verbal expressions only
-Function list extended to include
square root functions
-Describing changes in domain
and range as verbal expressions
or inequality notation
12. Graph linear and
absolute value inequalities
-Student-generated ideas of
definition of a half-plane
-Extend to graphs of absolute
value inequalities
Function list extended to include
square root functions and cubic
functions
Topic extensions:
-Describing changes in domain
and range in interval notation
-Linear Inequalities can be
represented in standard form.
-Extend topic to 3 or more graphs
on a coordinate plane.
-Can include absolute value and
linear inequalities on a single
coordinate plane
10. Analyze and explain
the general properties and
behavior of functions of
one variable, using
appropriate graphing
technologies.
-Students are given a list of linear
formats from which to choose.
-Solutions limited to integers and
common fractions
Level 3
-Limited to identity function,
absolute value function and
quadratic function.
-Use of computer software and
graphing calculators to reinforce
parent function identification
-Basic dilations of graphs
Identify and compare the
properties of classes of
functions.
May 16 2012
-Teacher-led discussion to explore
the idea of a half-plane
-Reinforcement of open/closed sets
and their respective symbols
Level 5
Note: Prior familiarity expected.
Diagnostic exam administered
to determine need for review.
Topic extensions:
-Multi-variable representations of
linear equations and the concept
of “partial slope”.
Note: Prior familiarity expected.
Diagnostic exam administered
to determine need for review.
Topic extensions:
-Representing absolute value
functions in piecewise form
-Real world examples of
piecewise functions (i.e. shipping
fees, tax brackets)
Topic extensions:
-Describing changes in domain
and range in interval notation
Brief diagnostic review of topic
-Extend topic to 3 or more graphs
on a coordinate plane.
-Can include absolute value and
linear inequalities on a single
coordinate plane
Page 4
Mathematics Curriculum: Algebra 2
Level 3
Level 4
Level 5
-Limited to 2x2 systems of
equations.
-Integer coefficients for all
equations
-Coefficients are integers and
solutions limited to integers and
common fractions (halves,
quarters, etc)
-Systems given are in same linear
format
14. Solve real world
-feasible regions limited to 3 lines
problems using systems with only one being oblique
-distinguish between points on
of inequalities.
boundary lines and those inside
the feasible region
-No limit to solutions for 2x2
systems.
-Extend topic to include 3x3
systems.
-All solutions to 3x3 systems kept
as integers.
-Systems given are in mixed
linear formats
-No limit to coefficients and
solutions.
-Graphical representation of 2x2
systems
Topic extensions:
-Solving 3 dimensional systems of linear
equations.
-Graphical representation of the case of
no solution and infinite solutions for 3
dimensional systems.
-Specifying the vector solutions of an
infinite solution 3x3 system.
-Feasible regions limited to 3
oblique lines or 4 lines (at most 2
being oblique)
-uniform formats presented
Brief diagnostic review of
topic
-mixed formats presented
Topic extensions:
-Modeling real-world scenarios as linear
inequalities (e.g. animal dietary
requirements).
15. Use linear
programming to solve
real-world problems**
Not Taught
-Constraints given
-No more than 3 constraints
-Only one line as oblique
-All integer solutions
-No constraints given
-Non integer solutions likely
16. Describe and
perform operations on
matrices. Solve
systems through
matrix multiplication,
using inverses.
Not Taught
-Operations limited to 2x2
matrices by hand
-Systems of equations limited to
a 2x2 systems using inverses
-No constraints given
-All integer solutions
-no limit to formats of lines
-compound inequalities often
-feasible regions extended to
uncommon quadrilaterals and
pentagons
-extend to discrete math with
lattice points
- Writing and solving systems of
equations of 3x3 system using
matrices on graphing utility
- Use and application of
Cramer’s Rule
-See the advantage of using
matrices to solve systems
Objective
Level 2
13. Solve systems of
linear equations
May 16 2012
- Writing and solving systems of
equations of 3x3 systems using
matrices
- Use and application of Cramer’s Rule
-See the advantages of using matrices
to solve systems
Topic extensions:
- Solving nxn and nxm systems.
-Introduction to other matrix-oriented
techniques such as Gaussian
elimination
Page 5
Mathematics Curriculum: Algebra 2
Objective
Level 2
Level 3
Level 4
Level 5
-Graphing quadratics using the
axis of symmetry
-The axis of symmetry unlimited
-find standard form of a quadratic
given any type of roots (rational,
irrational, and imaginary)
- Ability to interchangeably
express quadratic functions in
factored form, vertex form, and
standard form
-Leading coefficient can be
composite.
-Unlimited exposure to
operations complex numbers
(and conjugates) with rational
and irrational coefficients
- application of binomial theorem
introduced
-Graphing quadratic functions in
standard form, vertex form and/or
intercept/factored form.
-Ability to interchangeably express
quadratic functions in standard,
vertex or factored form.
-Finding the standard form of a
quadratic given any roots
(rational, irrational, and imaginary)
-Distinguish between methods of
solving quadratics needed when
leading coefficient is not equal to
one, but still an integer
-Solving general quadratics
(when leading coefficient not
necessarily equal to one) using
completing the square.
Topic extensions:
-Discovering the completing the
square algorithm as a natural
extension of how to form a
perfect square trinomial.
-Geometric interpretation of
completing the square.
17. Recognize and use
connections among
significant values of a
quadratic function, points
on the graph of the
function, and its symbolic
representation.
-Graphing quadratics from tables
- The axis of symmetry will only be
an integer value
-Intercepts of parabolas limited to
integers
- Quadratics will have integer
coefficients
-Graphing quadratics from tables
and using the axis of symmetry
- The axis of symmetry will only
be an integer value or simple
fraction
-find standard form of a quadratic
given integer (distinct) roots only.
-Sum and difference of cubes
formulas for factoring given
-Use of zero product property for
quadratics given in factored form
18. Identify properties of
imaginary and complex
numbers and operate on
them.
Solve simple quadratic
equations with imaginary
solutions.
-Limited to complex outcomes with
integer coefficients
-Complex conjugates limited to
monomials
-Manipulation of complex numbers
in a+bi (standard form) via FOIL
and distribution.
-Extend to binomial complex
numbers to powers greater than
or equal to 3.
-Complex conjugates extended
to binomials with integer
coefficients.
19. Solve quadratic
equations by completing
the square.
-Solving limited to simple
quadratics where b=0
-Extend topic to include solving
perfect square binomials
-Extend exposure to quadratic
trinomials where b is not a factor
of the leading coefficient
May 16 2012
-Complex number operations.
-Graphical representation of
imaginary numbers
Page 6
Mathematics Curriculum: Algebra 2
Objective
Level 2
Level 3
Level 4
Level 5
20. Solving quadratic
equations using the Quadratic
Formula
-Reinforce standard form
(quadratics must be set equal
to zero)
-Distinguish between possible
types of solution sets (one real,
two real, or two complex)
-Use discriminant to generalize
outcomes for solutions
-Solutions limited to integers
-Extend types of equations to
those with binomial products
(FOIL) necessary before
standard form can be
achieved
-Solutions limited to integers
and common rational values
21. Write a quadratic function
in vertex form.
Transform graphs of quadratic
functions in vertex form.
-Quadratics given in vertex form
where students need to
generate graphs
-Students cross convert Vertex
and Standard Form of a
quadratic to see the
connections between the two
forms.
Not Taught
-Derivation of the quadratic
formula using completing the
square
-Using the discriminant to
determine the nature of roots.
Topic extensions:
-Use of the discriminant in solving
for an unknown coefficient
-Solving quadratic inequalities
algebratically and graphically.
- Converting standard form
quadratics into vertex form using
completing the square.
-Extend to quadratic modeling for
best fit
-Derivation of the quadratic formula
using completing the square.
-Using the discriminant to determine
the nature of roots.
Topic extensions:
-Use of the discriminant to solve real
world problems (e.g. does a ball
reach a specific height?)
-Solving quadratic inequalities
algebraically and graphically.
- Converting standard form
quadratics into vertex form using
completing the square.
-Students required to know
formulas for sum/difference of
cubes
-Equations may need preliminary
work to be put into standard form
before factoring
-Composite leading coefficients
now presented in factoring
quadratic trinomials.
-Automaticity in factoring of
quadratics and sum/difference of
cubes.
Topic extensions:
-Use of substitution of variables to
factor higher order polynomials in
quadratic form.
-Use of multiplicity of roots and
end behaviors to determine
equations of graph.
-Polynomial and synthetic division
are all assessed with divisors that
are factors of the dividend
-Factoring higher order polynomials
using rational zero theorem, polynomial division & quadratic formula.
Topic extensions:
-Use of Descartes’ Rule of Signs.
-Applying the concept of continuity &
intermediate value to finding zeros.
22. Classify and factor
polynomials
23. Find the zeroes of
polynomials and graph
polynomial functions
May 16 2012
Not Taught
-Extend to reverse the process
where students given graph and
need to write the vertex form
which would represent it.
-Classify polynomials by degree
& number of terms
-Multiply poly expressions
limited to trinomial/trinomial
-Recognize greatest common
monomials factors (integer
coefficients)
-Students given formulas for
sum/difference of cubes
-Factoring quadratic trinomials
that are given in standard form.
Not Taught
Page 7
Mathematics Curriculum: Algebra 2
Objective
Level 2
Level 3
Level 4
Level 5
-Evaluating composite functions.
-Finding the inverse of higher order
functions along w/ its domain &range
-Use of composite functions to prove
an inverse relationship.
Topic extensions:
-Showing the inverse as a sequential
undoing of function operations.
-Finding the domains of complex
composite functions (involving
polynomial and radical functions)
algebraically.
-Graphing general radical functions
using transformations.
-Solving radical equations.
Topic extensions:
Determining the cause of extraneous
solutions when solving radical
equations.
24. Write and evaluate
composite functions and
inverse functions
-Limited to linear and quadratic
functions for function
composition
-Students are taught to verbally
analyze and list what
operations a function is
performing and use that list
to create a definition for the
inverse of that function.
-Students use composition of
functions to verify inverses.
-Extend to include square root
and cube root functions
-No limit to types of functions
used.
-Evaluating composite functions.
-Use of composite functions to
prove an inverse relationship.
-Students explore the properties of
inverse functions including the
graphical representation of inverse
functions
25. Use reciprocals to solve
equations using exponents
and radicals.
-Radicals limited to square roots
and cube roots.
-Graphing of square root
functions only
-Connecting the domain of a
square root function to the
argument its square root as a
way to understand the
behavior of the function’s
graph.
-Square root arguments limited
to domains extending to
positive infinity.
-Radicals limited to square roots
and cube roots. Connection
to rational exponents is
introduced and explored.
-Radicals limited to square roots
and cube roots.
-Include graphs of cube root
functions
-No limit to argument of square
root or direction of graph.
-Graphing general radical
functions using transformations.
In the form: y=a*sqrt(x-h)+k
The transformations do not
include transformations when
there is a coeff of x under radical.
-Solving radical equations.
Topic extensions:
-Graphing square root inequalities
Determining the cause of
extraneous solutions when solving
radical equations.
-Radicals limited to square roots
and cube roots.
-Include binomials with one
radical term
-Performing radical operations
using properties of radicals.
No limit to radical types used
-Performing radical operations using
properties of radicals.
-There are no limits to numerical
types of rational exponents.
-Simplifying radicals using properties
of radicals. Students work with large
numbers to reinforce their number
sense with respect to evaluating
radicals.
Graph radical functions.
26. Perform operations that
contain radical expressions.
27. Evaluate expressions
with exponents that are
negative and/or fractions.
May 16 2012
-Emphasis on cross converting -No limit to numerical types of
rational exponents to radical rational exponents
expressions. Rational
exponents limited to square
and cube roots.
Page 8
Mathematics Curriculum: Algebra 2
Level 2
Level 3
Level 4
Level 5
28. Solve and graph
exponential functions. Solve
exponential equations.
Not Taught
-Exponential bases limited to
integers and common fractions
-Solving exponential equations
limited to those problems where
same base used throughout.
-Exponential bases include
uncommon fractions
-Extend graph translation theorem
to include exponentials
(movement of asymptotes)
-Solving exponential equations not
requireing logarithms whose base
is not the same throughout.
-Graphing exponential functions
using transformations.
-Comparing the rate of growth/decay
of exponential functions to the rate of
growth/decay of power functions.
29. Understand properties of
logarithms and use natural
logs and e
Not Taught
Not Taught
-Graphing log functions of different
bases.
-Solving log equations using log
properties.
-Changing the base of a log
(particularly to base e or 10)
-Deriving log properties and
change of base as simple
extensions of properties of
exponents. (for instructional
purposes but not assessment
purposes)
-Solve equations with different
bases (integers only)
-Graphing log functions of different
bases.
-Finding the inverse of log and
exponential functions.
-Solving log equations using log
properties.
-Changing the base of a log
(particularly to base e or 10)
Topic extensions:
-Graphing log functions using log
properties instead of a calculator.
-Deriving log properties and change
of base as simple extensions of
properties of exponents.
-Graphing and recognizing logistic
growith functions as constrained
exponential growth.
Objective
May 16 2012
Page 9
Mathematics Curriculum: Algebra 2
Objective
Level 2
Level 3
Level 4
Level 5
30. Solve problems involving
rational functions.
Not Taught
Not Taught
-Focus on graphing the parent
function and the translation of the
parent function
Not Taught
Not Taught
Not Taught
-Graphing of rational functions.
-Identifying slant asymptotes.
-Solving rational equations.
Topic extensions:
-Distinguishing between holes
and vertical asymptotes.
-Introduction to the concept of
end behavior as a basis for
defining asymptotes.
-Understanding the differences
between simplified and
unsimplified rational functions.
- Derive the standard form
equations of parabolas, circles,
ellipses and hyperbolas using the
distance formula and their
respective defintions as a
constrained locus of points.
- Graph a given equation of a
parabola, circle, ellipse and
hyperbola and perform
appropriate transformations on its
center/foci.
Curriculum Extension:
31. Graph and write equations
of parabolas, circles, ellipses
and hyperbolas.
May 16 2012
Page 10