Download Algebra Strand

New York Mathematics Standards Alignment to:
 the Common Core State Standards (CCSS)
 the National Essential Skills Study (NESS)
Please note that the National Essential Skills Study (NESS) is only aligned to the New York Integrated Algebra, Geometry, and Algebra II/Trigonometry Performance Indicators.
The NESS descriptors are not intentionally aligned to the Common Core State Standards (CCSS) or their subparts. Any alignment between NESS descriptors and CCSS is
coincidental.
New York Mathematics
Strands/Bands/
Performance Indicators
Integrated Algebra
Mathematics
Domains/Clusters/
Common Core State Standards
High School
National Essential Skills Study (NESS)
National Rankings
Rank
Problem Solving Strand
Students will new mathematical
knowledge through problem solving.
A.PS.1 Use a variety of problem
solving strategies to understand new
mathematical content
Number & Quantity: Quantities
Reason quantitatively and use units to solve problems.
1. Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and
interpret the scale and the origin in graphs and data displays.
Algebra: Reasoning with Equations & Inequalities
Solve equations and inequalities in one variable.
3. Solve linear equations and inequalities in one variable, including equations
with coefficients represented by letters.
4. Solve quadratic equations in one variable.
a. Use the method of completing the square to transform any quadratic
equation in x into an equation of the form (x – p)2 = q that has the same
solutions. Derive the quadratic formula from this form.
b. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square
roots, completing the square, the quadratic formula and factoring, as
appropriate to the initial form of the equation. Recognize when the
quadratic formula gives complex solutions and write them as a ± bi for real
numbers a and b.
Functions: Interpreting Functions
Interpret functions that arise in applications in terms of the context.
6. Calculate and interpret the average rate of change of a function (presented
symbolically or as a table) over a specified interval. Estimate the rate of
change from a graph.
 2011 International Center for Leadership in Education
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Mathematics – Page 1
New York Mathematics
Strands/Bands/
Performance Indicators
Integrated Algebra
Mathematics
Domains/Clusters/
Common Core State Standards
High School
A.PS.2 Recognize and understand
equivalent representations of a problem
situation or a mathematical concept
Number & Quantity: Quantities
Reason quantitatively and use units to solve problems.
1. Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and
interpret the scale and the origin in graphs and data displays.
Algebra: Seeing Structure in Expressions
Interpret the structure of expressions.
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).
Functions: Interpreting Functions
Analyze functions using different representations.
7. Graph functions expressed symbolically and show key features of the graph,
by hand in simple cases and using technology for more complicated cases.
a. Graph linear and quadratic functions and show intercepts, maxima, and
minima.
b. Graph square root, cube root, and piecewise-defined functions, including
step functions and absolute value functions.
9. Compare properties of two functions each represented in a different way
(algebraically, graphically, numerically in tables, or by verbal descriptions).
For example, given a graph of one quadratic function and an algebraic
expression for another, say which has the larger maximum.
Statistics & Probability: Interpreting Categorical & Quantitative Data
Summarize, represent and interpret data on a single count or measurement
variable.
1. Represent data with plots on the real number line (dot plots, histograms, and
box plots).
Summarize, represent and interpret data on two categorical and
quantitative variables.
6. Represent data on two quantitative variables on a scatter plot, and describe
how the variables are related.
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Mathematics – Page 2
New York Mathematics
Strands/Bands/
Performance Indicators
Integrated Algebra
Students will solve problems that arise
in mathematics and in other contexts.
A.PS.3 Observe and explain patterns to
formulate generalizations and
conjectures
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Algebra: Seeing Structure in Expressions
Write expressions in equivalent forms to solve problems.
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.
Functions: Building Functions
Build a function that models a relationship between two quantities.
1. Write a function that describes a relationship between two quantities.
a. Determine an explicit expression, a recursive process, or steps for
calculation from a context.
b. Combine standard function types using arithmetic operations. For example,
build a function that models the temperature of a cooling body by adding a
constant function to a decaying exponential, and relate these functions to
the model.
2. Write arithmetic and geometric sequences both recursively and with an
explicit formula, use them to model situations, and translate between the two
forms.
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
Apply pattern recognition in data sets and
series to reason or solve problems involving
arithmetic, geometry, exponents, etc.
M16
Mathematics – Page 3
New York Mathematics
Strands/Bands/
Performance Indicators
Integrated Algebra
Mathematics
Domains/Clusters/
Common Core State Standards
High School
A.PS.4 Use multiple representations to
represent and explain problem
situations (e.g., verbally, numerically,
algebraically, graphically)
Number & Quantity: Quantities
Reason quantitatively and use units to solve problems.
1. Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and
interpret the scale and the origin in graphs and data displays.
Algebra: Creating Equations
Create equations that describe numbers or relationships.
1. Create equations and inequalities in one variable and use them to solve
problems. Include equations arising from linear and quadratic functions, and
simple rational and exponential functions.
2. Create equations in two or more variables to represent relationships between
quantities; graph equations on coordinate axes with labels and scales.
4. Rearrange formulas to highlight a quantity of interest, using the same
reasoning as in solving equations. For example, rearrange Ohm’s law V = IR
to highlight resistance R.
Functions: Interpreting Functions
Analyze functions using different representations.
7. Graph functions expressed symbolically and show key features of the graph,
by hand in simple cases and using technology for more complicated cases.
a. Graph linear and quadratic functions and show intercepts, maxima, and
minima.
b. Graph square root, cube root, and piecewise-defined functions, including
step functions and absolute value functions.
8. Write a function defined by an expression in different but equivalent forms to
reveal and explain different properties of the function.
a. Use the process of factoring and completing the square in a quadratic
function to show zeros, extreme values, and symmetry of the graph, and
interpret these in terms of a context.
9. Compare properties of two functions each represented in a different way
(algebraically, graphically, numerically in tables, or by verbal descriptions).
For example, given a graph of one quadratic function and an algebraic
expression for another, say which has the larger maximum.
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
M10
M11
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Apply variables in expressions and
equations to solve problems (i.e., write
mathematical equations for given situation,
create a mathematical model to understand
the relationships between variables, or make
connections between the structures of
mathematically abstract concepts and the
real world).
Evaluate and employ accurate and
appropriate procedures for statistical data
collection, organization, analysis, and
display including making estimates and
predictions, critiquing data, and drawing
inferences (e.g., using the normal curve and
z-scores, line of best fit).
M21
Mathematics – Page 4
New York Mathematics
Strands/Bands/
Performance Indicators
Integrated Algebra
Students will apply and adapt a variety
of appropriate strategies to solve
problems.
A.PS.5 Choose an effective approach
to solve a problem from a variety of
strategies (numeric, graphic, algebraic)
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Number & Quantity: Quantities
Reason quantitatively and use units to solve problems.
1. Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and
interpret the scale and the origin in graphs and data displays.
2. Define appropriate quantities for the purpose of descriptive modeling.
3. Choose a level of accuracy appropriate to limitations on measurement when
reporting quantities.
Algebra: Seeing Structure in Expressions
Write expressions in equivalent forms to solve problems.
3. Choose and produce an equivalent form of an expression to reveal and
explain properties of the quantity represented by the expression.
a. Factor a quadratic expression to reveal the zeros of the function it defines.
b. Complete the square in a quadratic expression to reveal the maximum or
minimum value of the function it defines.
Functions: Interpreting Functions
Analyze functions using different representations.
7. Graph functions expressed symbolically and show key features of the graph,
by hand in simple cases and using technology for more complicated cases.
a. Graph linear and quadratic functions and show intercepts, maxima, and
minima.
b. Graph square root, cube root, and piecewise-defined functions, including
step functions and absolute value functions.
8. Write a function defined by an expression in different but equivalent forms to
reveal and explain different properties of the function.
a. Use the process of factoring and completing the square in a quadratic
function to show zeros, extreme values, and symmetry of the graph, and
interpret these in terms of a context.
9. Compare properties of two functions each represented in a different way
(algebraically, graphically, numerically in tables, or by verbal descriptions).
For example, given a graph of one quadratic function and an algebraic
expression for another, say which has the larger maximum.
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Mathematics – Page 5
New York Mathematics
Strands/Bands/
Performance Indicators
Integrated Algebra
Mathematics
Domains/Clusters/
Common Core State Standards
High School
A.PS.5 (Continued from previous page)
(Continued from previous page)
Statistics & Probability: Interpreting Categorical & Quantitative Data
Summarize, represent and interpret data on a single count or measurement
variable.
1. Represent data with plots on the real number line (dot plots, histograms, and
box plots).
Summarize, represent and interpret data on two categorical and
quantitative variables.
6. Represent data on two quantitative variables on a scatter plot, and describe
how the variables are related.
Functions: Interpreting Functions
Analyze functions using different representations.
8. Write a function defined by an expression in different but equivalent forms to
reveal and explain different properties of the function.
a. Use the process of factoring and completing the square in a quadratic
function to show zeros, extreme values, and symmetry of the graph, and
interpret these in terms of a context.
Functions: Building Functions
Build a function that models a relationship between two quantities.
2. Write arithmetic and geometric sequences both recursively and with an
explicit formula, use them to model situations, and translate between the two
forms.
There is no New York Mathematics Performance Indicator–Common Core
alignment.
A.PS.6 Use a variety of strategies to
extend solution methods to other
problems
A.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
A.PS.8 Determine information required
to solve a problem, choose methods for
obtaining the information, and define
parameters for acceptable solutions
National Essential Skills Study (NESS)
National Rankings
Rank
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
M10
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
There is no New York Mathematics Performance Indicator–Common Core
alignment.
 2011 International Center for Leadership in Education
Mathematics – Page 6
New York Mathematics
Strands/Bands/
Performance Indicators
Integrated Algebra
A.PS.9 Interpret solutions within the
given constraints of a problem
A.PS.10 Evaluate the relative
efficiency of different representations
and solution methods of a problem
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Algebra: Creating Equations
Create equations that describe numbers or relationships.
3. Represent constraints by equations or inequalities, and by systems of
equations and/or inequalities, and interpret solutions as viable or nonviable
options in a modeling context. For example, represent inequalities describing
nutritional and cost constraints on combinations of different foods.
Algebra: Reasoning with Equations & Inequalities
Understand solving equations as a process of reasoning and explain the
reasoning.
2. Solve simple rational and radical equations in one variable, and give
examples showing how extraneous solutions may arise.
There is no New York Mathematics Performance Indicator–Common Core
alignment.
National Essential Skills Study (NESS)
National Rankings
Rank
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
M10
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Reasoning and Proof Strand
Students will recognize reasoning and
proof as fundamental aspects of
mathematics.
A.RP.1 Recognize that mathematical
ideas can be supported by a variety of
strategies
Number & Quantity: Quantities
Reason quantitatively and use units to solve problems.
1. Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and
interpret the scale and the origin in graphs and data displays.
Algebra: Reasoning with Equations & Inequalities
Understand solving equations as a process of reasoning and explain the
reasoning.
1. Explain each step in solving a simple equation as following from the equality
of numbers asserted at the previous step, starting from the assumption that the
original equation has a solution. Construct a viable argument to justify a
solution method.
2. Solve simple rational and radical equations in one variable, and give
examples showing how extraneous solutions may arise.
 2011 International Center for Leadership in Education
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Mathematics – Page 7
New York Mathematics
Strands/Bands/
Performance Indicators
Integrated Algebra
Mathematics
Domains/Clusters/
Common Core State Standards
High School
A.RP.1 (Continued from previous page)
(Continued from previous page)
Statistics & Probability: Interpreting Categorical & Quantitative Data
Summarize, represent and interpret data on a single count or measurement
variable.
2. Use statistics appropriate to the shape of the data distribution to compare
center (median, mean) and spread (interquartile range, standard deviation) of
two or more different data sets.
3. Interpret differences in shape, center, and spread in the context of the data
sets, accounting for possible effects of extreme data points (outliers).
4. Use the mean and standard deviation of a data set to fit it to a normal
distribution and to estimate population percentages. Recognize that there are
data sets for which such a procedure is not appropriate. Use calculators,
spreadsheets, and tables to estimate areas under the normal curve.
Students will make and investigate
mathematical conjectures.
A.RP.2 Use mathematical strategies to
reach a conclusion and provide
supportive arguments for a conjecture
Number & Quantity: Quantities
Reason quantitatively and use units to solve problems.
1. Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and
interpret the scale and the origin in graphs and data displays.
Algebra: Reasoning with Equations & Inequalities
Understand solving equations as a process of reasoning and explain the
reasoning.
1. Explain each step in solving a simple equation as following from the equality
of numbers asserted at the previous step, starting from the assumption that the
original equation has a solution. Construct a viable argument to justify a
solution method.
2. Solve simple rational and radical equations in one variable, and give
examples showing how extraneous solutions may arise.
Statistics & Probability: Interpreting Categorical & Quantitative Data
Summarize, represent and interpret data on a single count or measurement
variable.
2. Use statistics appropriate to the shape of the data distribution to compare
center (median, mean) and spread (interquartile range, standard deviation) of
two or more different data sets.
3. Interpret differences in shape, center, and spread in the context of the data
sets, accounting for possible effects of extreme data points (outliers).
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Mathematics – Page 8
New York Mathematics
Strands/Bands/
Performance Indicators
Integrated Algebra
Mathematics
Domains/Clusters/
Common Core State Standards
High School
A.RP.2 (Continued from previous page)
(Continued from previous page)
4. Use the mean and standard deviation of a data set to fit it to a normal
distribution and to estimate population percentages. Recognize that there are
data sets for which such a procedure is not appropriate. Use calculators,
spreadsheets, and tables to estimate areas under the normal curve.
Summarize, represent and interpret data on two categorical and
quantitative variables.
6. Represent data on two quantitative variables on a scatter plot, and describe
how the variables are related.
a. Fit a function to the data; use functions fitted to data to solve problems in
the context of the data. Use given functions or choose a function suggested
by the context. Emphasize linear, quadratic, and exponential models.
Number & Quantity: Quantities
Reason quantitatively and use units to solve problems.
1. Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and
interpret the scale and the origin in graphs and data displays.
3. Choose a level of accuracy appropriate to limitations on measurement when
reporting quantities.
A.RP.3 Recognize when an
approximation is more appropriate than
an exact answer
Students will develop and evaluate
mathematical arguments and proofs.
A.RP.4 Develop, verify, and explain an
argument, using appropriate
mathematical ideas and language
Algebra: Seeing Structure in Expressions
Write expressions in equivalent forms to solve problems.
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.
Algebra: Reasoning with Equations & Inequalities
Understand solving equations as a process of reasoning and explain the
reasoning.
1. Explain each step in solving a simple equation as following from the equality
of numbers asserted at the previous step, starting from the assumption that the
original equation has a solution. Construct a viable argument to justify a
solution method.
2. Solve simple rational and radical equations in one variable, and give
examples showing how extraneous solutions may arise.
Functions: Building Functions
Build a function that models a relationship between two quantities.
2. Write arithmetic and geometric sequences both recursively and with an
explicit formula, use them to model situations, and translate between the two
forms.
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
M10
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Mathematics – Page 9
New York Mathematics
Strands/Bands/
Performance Indicators
Integrated Algebra
A.RP.5 Construct logical arguments
that verify claims or counterexamples
that refute them
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Algebra: Seeing Structure in Expressions
Write expressions in equivalent forms to solve problems.
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.
Algebra: Reasoning with Equations & Inequalities
Understand solving equations as a process of reasoning and explain the
reasoning.
1. Explain each step in solving a simple equation as following from the equality
of numbers asserted at the previous step, starting from the assumption that the
original equation has a solution. Construct a viable argument to justify a
solution method.
2. Solve simple rational and radical equations in one variable, and give
examples showing how extraneous solutions may arise.
Functions: Building Functions
Build a function that models a relationship between two quantities.
2. Write arithmetic and geometric sequences both recursively and with an
explicit formula, use them to model situations, and translate between the two
forms.
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Mathematics – Page 10
New York Mathematics
Strands/Bands/
Performance Indicators
Integrated Algebra
A.RP.6 Present correct mathematical
arguments in a variety of forms
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Algebra: Seeing Structure in Expressions
Write expressions in equivalent forms to solve problems.
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.
Algebra: Reasoning with Equations & Inequalities
Understand solving equations as a process of reasoning and explain the
reasoning.
1. Explain each step in solving a simple equation as following from the equality
of numbers asserted at the previous step, starting from the assumption that the
original equation has a solution. Construct a viable argument to justify a
solution method.
2. Solve simple rational and radical equations in one variable, and give
examples showing how extraneous solutions may arise.
Functions: Building Functions
Build a function that models a relationship between two quantities.
2. Write arithmetic and geometric sequences both recursively and with an
explicit formula, use them to model situations, and translate between the two
forms.
Statistics & Probability: Interpreting Categorical & Quantitative Data
Summarize, represent and interpret data on a single count or measurement
variable.
3. Interpret differences in shape, center, and spread in the context of the data
sets, accounting for possible effects of extreme data points (outliers).
4. Use the mean and standard deviation of a data set to fit it to a normal
distribution and to estimate population percentages. Recognize that there are
data sets for which such a procedure is not appropriate. Use calculators,
spreadsheets, and tables to estimate areas under the normal curve.
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Mathematics – Page 11
New York Mathematics
Strands/Bands/
Performance Indicators
Integrated Algebra
Mathematics
Domains/Clusters/
Common Core State Standards
High School
A.RP.7 Evaluate written arguments for
validity
Algebra: Seeing Structure in Expressions
Write expressions in equivalent forms to solve problems.
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.
Algebra: Reasoning with Equations & Inequalities
Understand solving equations as a process of reasoning and explain the
reasoning.
1. Explain each step in solving a simple equation as following from the equality
of numbers asserted at the previous step, starting from the assumption that the
original equation has a solution. Construct a viable argument to justify a
solution method.
2. Solve simple rational and radical equations in one variable, and give
examples showing how extraneous solutions may arise.
Functions: Building Functions
Build a function that models a relationship between two quantities.
2. Write arithmetic and geometric sequences both recursively and with an
explicit formula, use them to model situations, and translate between the two
forms.
Statistics & Probability: Interpreting Categorical & Quantitative Data
Summarize, represent and interpret data on a single count or measurement
variable.
3. Interpret differences in shape, center, and spread in the context of the data
sets, accounting for possible effects of extreme data points (outliers).
4. Use the mean and standard deviation of a data set to fit it to a normal
distribution and to estimate population percentages. Recognize that there are
data sets for which such a procedure is not appropriate. Use calculators,
spreadsheets, and tables to estimate areas under the normal curve.
Students will select and use various
types of reasoning and methods of
proof.
A.RP.8 Support an argument by using
a systematic approach to test more than
one case
National Essential Skills Study (NESS)
National Rankings
Rank
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
There is no New York Mathematics Performance Indicator–Common Core
alignment.
 2011 International Center for Leadership in Education
M10
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Mathematics – Page 12
New York Mathematics
Strands/Bands/
Performance Indicators
Integrated Algebra
Mathematics
Domains/Clusters/
Common Core State Standards
High School
A.RP.9 Devise ways to verify results
or use counterexamples to refute
incorrect statements
There is no New York Mathematics Performance Indicator–Common Core
alignment.
A.RP.10 Extend specific results to
more general cases
Algebra: Reasoning with Equations & Inequalities
Solve equations and inequalities in one variable.
3. Solve linear equations and inequalities in one variable, including equations
with coefficients represented by letters.
4. Solve quadratic equations in one variable.
a. Use the method of completing the square to transform any quadratic
equation in x into an equation of the form (x – p)2 = q that has the same
solutions. Derive the quadratic formula from this form.
There is no New York Mathematics Performance Indicator–Common Core
alignment.
A.RP.11 Use a Venn diagram to
support a logical argument
National Essential Skills Study (NESS)
National Rankings
Rank
M10
M10
M10
M21
A.RP.12 Apply inductive reasoning in
making and supporting mathematical
conjectures
There is no New York Mathematics Performance Indicator–Common Core
alignment.
 2011 International Center for Leadership in Education
M10
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Evaluate and employ accurate and
appropriate procedures for statistical data
collection, organization, analysis, and
display including making estimates and
predictions, critiquing data, and drawing
inferences (e.g., using the normal curve and
z-scores, line of best fit).
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Mathematics – Page 13
New York Mathematics
Strands/Bands/
Performance Indicators
Integrated Algebra
Mathematics
Domains/Clusters/
Common Core State Standards
High School
National Essential Skills Study (NESS)
National Rankings
Rank
Communication Strand
Students will organize and consolidate
their mathematical thinking through
communication
A.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
A.CM.2 Use mathematical
representations to communicate with
appropriate accuracy, including
numerical tables, formulas, functions,
equations, charts, graphs, Venn
diagrams, and other diagrams
Algebra: Reasoning with Equations & Inequalities
Understand solving equations as a process of reasoning and explain the
reasoning.
1. Explain each step in solving a simple equation as following from the equality
of numbers asserted at the previous step, starting from the assumption that the
original equation has a solution. Construct a viable argument to justify a
solution method.
Number & Quantity: Quantities
Reason quantitatively and use units to solve problems.
2. Define appropriate quantities for the purpose of descriptive modeling.
3. Choose a level of accuracy appropriate to limitations on measurement when
reporting quantities.
Algebra: Creating Equations
Create equations that describe numbers or relationships.
1. Create equations and inequalities in one variable and use them to solve
problems. Include equations arising from linear and quadratic functions, and
simple rational and exponential functions.
2. Create equations in two or more variables to represent relationships between
quantities; graph equations on coordinate axes with labels and scales.
Algebra: Reasoning with Equations & Inequalities
Represent and solve equations and inequalities graphically.
10. Understand that the graph of an equation in two variables is the set of all its
solutions plotted in the coordinate plane, often forming a curve (which could
be a line).
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.
 2011 International Center for Leadership in Education
M10
M10
M11
M21
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Apply variables in expressions and
equations to solve problems (i.e., write
mathematical equations for given situation,
create a mathematical model to understand
the relationships between variables, or make
connections between the structures of
mathematically abstract concepts and the
real world).
Evaluate and employ accurate and
appropriate procedures for statistical data
collection, organization, analysis, and
display including making estimates and
predictions, critiquing data, and drawing
inferences (e.g., using the normal curve and
z-scores, line of best fit).
Mathematics – Page 14
New York Mathematics
Strands/Bands/
Performance Indicators
Integrated Algebra
A.CM.2 (Continued from previous
page)
Mathematics
Domains/Clusters/
Common Core State Standards
High School
National Essential Skills Study (NESS)
National Rankings
Rank
(Continued from previous page)
12. Graph the solutions to a linear inequality in two variables as a half-plane
(excluding the boundary in the case of a strict inequality), and graph the
solution set to a system of linear inequalities in two variables as the
intersection of the corresponding half-planes.
Functions: Interpreting Functions
Analyze functions using different representations.
77. Graph functions expressed symbolically and show key features of the graph,
by hand in simple cases and using technology for more complicated cases.
a. Graph linear and quadratic functions and show intercepts, maxima, and
minima.
b. Graph square root, cube root, and piecewise-defined functions, including
step functions and absolute value functions.
8. Write a function defined by an expression in different but equivalent forms to
reveal and explain different properties of the function.
a. Use the process of factoring and completing the square in a quadratic
function to show zeros, extreme values, and symmetry of the graph, and
interpret these in terms of a context.
9. Compare properties of two functions each represented in a different way
(algebraically, graphically, numerically in tables, or by verbal descriptions).
For example, given a graph of one quadratic function and an algebraic
expression for another, say which has the larger maximum.
Functions: Linear, Quadratic & Exponential
Construct and compare linear, quadratic, and exponential models and
solve problems.
1. Distinguish between situations that can be modeled with linear functions and
with exponential functions.
a. Prove that linear functions grow by equal differences over equal intervals,
and that exponential functions grow by equal factors over equal intervals.
b. Recognize situations in which one quantity changes at a constant rate per
unit interval relative to another.
Statistics & Probability: Interpreting Categorical & Quantitative Data
Summarize, represent and interpret data on a single count or measurement
variable.
1. Represent data with plots on the real number line (dot plots, histograms, and
box plots).
 2011 International Center for Leadership in Education
Mathematics – Page 15
New York Mathematics
Strands/Bands/
Performance Indicators
Integrated Algebra
A.CM.2 (Continued from previous
page)
Students will communicate their
mathematical thinking coherently and
clearly to peers, teachers, and others.
A.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
Mathematics
Domains/Clusters/
Common Core State Standards
High School
National Essential Skills Study (NESS)
National Rankings
Rank
(Continued from previous page)
Summarize, represent and interpret data on two categorical and
quantitative variables.
6. Represent data on two quantitative variables on a scatter plot, and describe
how the variables are related.
Number & Quantity: Quantities
Reason quantitatively and use units to solve problems.
1. Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and
interpret the scale and the origin in graphs and data displays.
2. Define appropriate quantities for the purpose of descriptive modeling.
3. Choose a level of accuracy appropriate to limitations on measurement when
reporting quantities.
Functions: Interpreting Functions
Understand the concept of a function and use function notation.
2. Use function notation, evaluate functions for inputs in their domains, and
interpret statements that use function notation in terms of a context.
3. Recognize that sequences are functions, sometimes defined recursively,
whose domain is a subset of the integers. For example, the Fibonacci
sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n
≥ 1.
Functions: Building Functions
Build a function that models a relationship between two quantities.
2. Write arithmetic and geometric sequences both recursively and with an
explicit formula, use them to model situations, and translate between the two
forms.
 2011 International Center for Leadership in Education
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Mathematics – Page 16
New York Mathematics
Strands/Bands/
Performance Indicators
Integrated Algebra
A.CM.4 Explain relationships among
different representations of a problem
A.CM.5 Communicate logical
arguments clearly, showing why a
result makes sense and why the
reasoning is valid
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Functions: Interpreting Functions
Interpret functions that arise in applications in terms of the context.
4. For a function that models a relationship between two quantities, interpret
key features of graphs and tables in terms of the quantities, and sketch graphs
showing key features given a verbal description of the relationship. Key
features include: intercepts; intervals where the function is increasing,
decreasing, positive, or negative; relative maximums and minimums;
symmetries; end behavior; and periodicity.
Analyze functions using different representations.
7. Graph functions expressed symbolically and show key features of the graph,
by hand in simple cases and using technology for more complicated cases.
a. Graph linear and quadratic functions and show intercepts, maxima, and
minima.
b. Graph square root, cube root, and piecewise-defined functions, including
step functions and absolute value functions.
8. Write a function defined by an expression in different but equivalent forms to
reveal and explain different properties of the function.
a. Use the process of factoring and completing the square in a quadratic
function to show zeros, extreme values, and symmetry of the graph, and
interpret these in terms of a context.
Algebra: Creating Equations
Create equations that describe numbers or relationships.
4. Rearrange formulas to highlight a quantity of interest, using the same
reasoning as in solving equations. For example, rearrange Ohm’s law V = IR
to highlight resistance R.
Algebra: Reasoning with Equations & Inequalities
Understand solving equations as a process of reasoning and explain the
reasoning.
1. Explain each step in solving a simple equation as following from the equality
of numbers asserted at the previous step, starting from the assumption that the
original equation has a solution. Construct a viable argument to justify a
solution method.
Solve equations and inequalities in one variable.
4. Solve quadratic equations in one variable.
a. Use the method of completing the square to transform any quadratic
equation in x into an equation of the form (x – p)2 = q that has the same
solutions. Derive the quadratic formula from this form.
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Mathematics – Page 17
New York Mathematics
Strands/Bands/
Performance Indicators
Integrated Algebra
A.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.
A.CM.7 Read and listen for logical
understanding of mathematical thinking
shared by other students
A.CM.8 Reflect on strategies of others
in relation to one’s own strategy
Mathematics
Domains/Clusters/
Common Core State Standards
High School
National Essential Skills Study (NESS)
National Rankings
Rank
There is no New York Mathematics Performance Indicator–Common Core
alignment.
M10
There is no New York Mathematics Performance Indicator–Common Core
alignment.
M10
There is no New York Mathematics Performance Indicator–Common Core
alignment.
M10
A.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
A.CM.10 Use correct mathematical
language in developing mathematical
questions that elicit, extend, or
challenge other students’ conjectures
There is no New York Mathematics Performance Indicator–Common Core
alignment.
M10
There is no New York Mathematics Performance Indicator–Common Core
alignment.
 2011 International Center for Leadership in Education
M10
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Mathematics – Page 18
New York Mathematics
Strands/Bands/
Performance Indicators
Integrated Algebra
A.CM.11 Represent word problems
using standard mathematical notation
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Number & Quantity: Quantities
Reason quantitatively and use units to solve problems.
1. Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and
interpret the scale and the origin in graphs and data displays.
Algebra: Creating Equations
Create equations that describe numbers or relationships.
1. Create equations and inequalities in one variable and use them to solve
problems. Include equations arising from linear and quadratic functions, and
simple rational and exponential functions.
2. Create equations in two or more variables to represent relationships between
quantities; graph equations on coordinate axes with labels and scales.
Functions: Interpreting Functions
Interpret functions that arise in applications in terms of the context.
5. Relate the domain of a function to its graph and, where applicable, to the
quantitative relationship it describes. For example, if the function h(n) gives
the number of person-hours it takes to assemble n engines in a factory, then
the positive integers would be an appropriate domain for the function.
Functions: Building Functions
Build a function that models a relationship between two quantities.
1. Write a function that describes a relationship between two quantities.
a. Determine an explicit expression, a recursive process, or steps for
calculation from a context.
b. Combine standard function types using arithmetic operations. For example,
build a function that models the temperature of a cooling body by adding a
constant function to a decaying exponential, and relate these functions to
the model.
2. Write arithmetic and geometric sequences both recursively and with an
explicit formula, use them to model situations, and translate between the two
forms.
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
M10
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Apply variables in expressions and
equations to solve problems (i.e., write
mathematical equations for given situation,
create a mathematical model to understand
the relationships between variables, or make
connections between the structures of
mathematically abstract concepts and the
real world).
M11
Mathematics – Page 19
New York Mathematics
Strands/Bands/
Performance Indicators
Integrated Algebra
A.CM.12 Understand and use
appropriate language, representations,
and terminology when describing
objects, relationships, mathematical
solutions, and rationale
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Number & Quantity: Quantities
Reason quantitatively and use units to solve problems.
1. Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and
interpret the scale and the origin in graphs and data displays.
Algebra: Creating Equations
Create equations that describe numbers or relationships.
1. Create equations and inequalities in one variable and use them to solve
problems. Include equations arising from linear and quadratic functions, and
simple rational and exponential functions.
2. Create equations in two or more variables to represent relationships between
quantities; graph equations on coordinate axes with labels and scales.
Algebra: Reasoning with Equations & Inequalities
Understand solving equations as a process of reasoning and explain the
reasoning.
1. Explain each step in solving a simple equation as following from the equality
of numbers asserted at the previous step, starting from the assumption that the
original equation has a solution. Construct a viable argument to justify a
solution method.
Functions: Interpreting Functions
Interpret functions that arise in applications in terms of the context.
5. Relate the domain of a function to its graph and, where applicable, to the
quantitative relationship it describes. For example, if the function h(n) gives
the number of person-hours it takes to assemble n engines in a factory, then
the positive integers would be an appropriate domain for the function.
Functions: Building Functions
Build a function that models a relationship between two quantities.
1. Write a function that describes a relationship between two quantities.
a. Determine an explicit expression, a recursive process, or steps for
calculation from a context.
b. Combine standard function types using arithmetic operations. For example,
build a function that models the temperature of a cooling body by adding a
constant function to a decaying exponential, and relate these functions to
the model.
2. Write arithmetic and geometric sequences both recursively and with an
explicit formula, use them to model situations, and translate between the two
forms.
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Mathematics – Page 20
New York Mathematics
Strands/Bands/
Performance Indicators
Integrated Algebra
A.CM.13 Draw conclusions about
mathematical ideas through decoding,
comprehension, and interpretation of
mathematical visuals, symbols, and
technical writing
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Number & Quantity: Quantities
Reason quantitatively and use units to solve problems.
3. Choose a level of accuracy appropriate to limitations on measurement when
reporting quantities.
Algebra: Seeing Structure in Expressions
Interpret the structure of expressions.
1. Interpret expressions that represent a quantity in terms of its context.
a. Interpret parts of an expression, such as terms, factors, and coefficients.
Algebra: Creating Equations
Create equations that describe numbers or relationships.
4. Rearrange formulas to highlight a quantity of interest, using the same
reasoning as in solving equations. For example, rearrange Ohm’s law V = IR
to highlight resistance R.
National Essential Skills Study (NESS)
National Rankings
Rank
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Connections Strand
Students will recognize and use
connections among mathematical ideas.
A.CN.1 Understand and make
connections among multiple
representations of the same
mathematical idea
Functions: Interpreting Functions
Interpret functions that arise in applications in terms of the context.
4. For a function that models a relationship between two quantities, interpret
key features of graphs and tables in terms of the quantities, and sketch graphs
showing key features given a verbal description of the relationship. Key
features include: intercepts; intervals where the function is increasing,
decreasing, positive, or negative; relative maximums and minimums;
symmetries; end behavior; and periodicity.
Analyze functions using different representations.
7. Graph functions expressed symbolically and show key features of the graph,
by hand in simple cases and using technology for more complicated cases.
a. Graph linear and quadratic functions and show intercepts, maxima, and
minima.
b. Graph square root, cube root, and piecewise-defined functions, including
step functions and absolute value functions.
8. Write a function defined by an expression in different but equivalent forms to
reveal and explain different properties of the function.
a. Use the process of factoring and completing the square in a quadratic
function to show zeros, extreme values, and symmetry of the graph, and
interpret these in terms of a context.
 2011 International Center for Leadership in Education
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Mathematics – Page 21
New York Mathematics
Strands/Bands/
Performance Indicators
Integrated Algebra
Mathematics
Domains/Clusters/
Common Core State Standards
High School
A.CN.2 Understand the corresponding
procedures for similar problems or
mathematical concepts
Functions: Interpreting Functions
Interpret functions that arise in applications in terms of the context.
4. For a function that models a relationship between two quantities, interpret
key features of graphs and tables in terms of the quantities, and sketch graphs
showing key features given a verbal description of the relationship. Key
features include: intercepts; intervals where the function is increasing,
decreasing, positive, or negative; relative maximums and minimums;
symmetries; end behavior; and periodicity.
Analyze functions using different representations.
7. Graph functions expressed symbolically and show key features of the graph,
by hand in simple cases and using technology for more complicated cases.
a. Graph linear and quadratic functions and show intercepts, maxima, and
minima.
b. Graph square root, cube root, and piecewise-defined functions, including
step functions and absolute value functions.
8. Write a function defined by an expression in different but equivalent forms to
reveal and explain different properties of the function.
a. Use the process of factoring and completing the square in a quadratic
function to show zeros, extreme values, and symmetry of the graph, and
interpret these in terms of a context.
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Mathematics – Page 22
New York Mathematics
Strands/Bands/
Performance Indicators
Integrated Algebra
Students will understand how
mathematical ideas interconnect and
build on one another to produce a
coherent whole.
A.CN.3 Model situations
mathematically, using representations
to draw conclusions and formulate new
situations
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Algebra: Creating Equations
Create equations that describe numbers or relationships.
1. Create equations and inequalities in one variable and use them to solve
problems. Include equations arising from linear and quadratic functions, and
simple rational and exponential functions.
2. Create equations in two or more variables to represent relationships between
quantities; graph equations on coordinate axes with labels and scales.
Functions: Interpreting Functions
Interpret functions that arise in applications in terms of the context.
5. Relate the domain of a function to its graph and, where applicable, to the
quantitative relationship it describes. For example, if the function h(n) gives
the number of person-hours it takes to assemble n engines in a factory, then
the positive integers would be an appropriate domain for the function.
6. Calculate and interpret the average rate of change of a function (presented
symbolically or as a table) over a specified interval. Estimate the rate of
change from a graph.
Functions: Linear, Quadratic & Exponential
Construct and compare linear, quadratic, and exponential models and
solve problems.
1. Distinguish between situations that can be modeled with linear functions and
with exponential functions.
a. Prove that linear functions grow by equal differences over equal intervals,
and that exponential functions grow by equal factors over equal intervals.
b. Recognize situations in which one quantity changes at a constant rate per
unit interval relative to another.
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Mathematics – Page 23
New York Mathematics
Strands/Bands/
Performance Indicators
Integrated Algebra
Mathematics
Domains/Clusters/
Common Core State Standards
High School
A.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
Algebra: Reasoning with Equations & Inequalities
Solve equations and inequalities in one variable.
4. Solve quadratic equations in one variable.
a. Use the method of completing the square to transform any quadratic
equation in x into an equation of the form (x – p)2 = q that has the same
solutions. Derive the quadratic formula from this form.
Functions: Interpreting Functions
Analyze functions using different representations.
8. Write a function defined by an expression in different but equivalent forms to
reveal and explain different properties of the function.
a. Use the process of factoring and completing the square in a quadratic
function to show zeros, extreme values, and symmetry of the graph, and
interpret these in terms of a context.
Number & Quantity: Quantities
Reason quantitatively and use units to solve problems.
2. Define appropriate quantities for the purpose of descriptive modeling.
Algebra: Creating Equations
Create equations that describe numbers or relationships.
4. Rearrange formulas to highlight a quantity of interest, using the same
reasoning as in solving equations. For example, rearrange Ohm’s law V = IR
to highlight resistance R.
A.CN.5 Understand how quantitative
models connect to various physical
models and representations
Students will recognize and apply
mathematics in contexts outside of
mathematics.
A.CN.6 Recognize and apply
mathematics to situations in the outside
world
Algebra: Seeing Structure in Expressions
Write expressions in equivalent forms to solve problems.
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.
Functions: Building Functions
Build a function that models a relationship between two quantities.
1. Write a function that describes a relationship between two quantities.
a. Determine an explicit expression, a recursive process, or steps for
calculation from a context.
b. Combine standard function types using arithmetic operations. For example,
build a function that models the temperature of a cooling body by adding a
constant function to a decaying exponential, and relate these functions to
the model.
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
M10
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Mathematics – Page 24
New York Mathematics
Strands/Bands/
Performance Indicators
Integrated Algebra
A.CN.7 Recognize and apply
mathematical ideas to problem
situations that develop outside of
mathematics
A.CN.8 Develop an appreciation for
the historical development of
mathematics
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Algebra: Creating Equations
Create equations that describe numbers or relationships.
1. Create equations and inequalities in one variable and use them to solve
problems. Include equations arising from linear and quadratic functions, and
simple rational and exponential functions.
2. Create equations in two or more variables to represent relationships between
quantities; graph equations on coordinate axes with labels and scales.
Functions: Building Functions
Build a function that models a relationship between two quantities.
1. Write a function that describes a relationship between two quantities.
a. Determine an explicit expression, a recursive process, or steps for
calculation from a context.
b. Combine standard function types using arithmetic operations. For example,
build a function that models the temperature of a cooling body by adding a
constant function to a decaying exponential, and relate these functions to
the model.
There is no New York Mathematics Performance Indicator–Common Core
alignment.
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
M10
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Mathematics – Page 25
New York Mathematics
Strands/Bands/
Performance Indicators
Integrated Algebra
Mathematics
Domains/Clusters/
Common Core State Standards
High School
National Essential Skills Study (NESS)
National Rankings
Rank
Representation Strand
Students will create and use
representations to organize, record,
and communicate mathematical ideas.
A.R.1 Use physical objects, diagrams,
charts, tables, graphs, symbols,
equations, or objects created using
technology as representations of
mathematical concepts
Algebra: Creating Equations
Create equations that describe numbers or relationships.
1. Create equations and inequalities in one variable and use them to solve
problems. Include equations arising from linear and quadratic functions, and
simple rational and exponential functions.
2. Create equations in two or more variables to represent relationships between
quantities; graph equations on coordinate axes with labels and scales.
Algebra: Reasoning with Equations & Inequalities
Represent and solve equations and inequalities graphically.
10. Understand that the graph of an equation in two variables is the set of all its
solutions plotted in the coordinate plane, often forming a curve (which could
be a line).
12. Graph the solutions to a linear inequality in two variables as a half-plane
(excluding the boundary in the case of a strict inequality), and graph the
solution set to a system of linear inequalities in two variables as the
intersection of the corresponding half-planes.
Functions: Interpreting Functions
Analyze functions using different representations.
7. Graph functions expressed symbolically and show key features of the graph,
by hand in simple cases and using technology for more complicated cases.
a. Graph linear and quadratic functions and show intercepts, maxima, and
minima.
b. Graph square root, cube root, and piecewise-defined functions, including
step functions and absolute value functions.
c. Graph polynomial functions, identifying zeros when suitable factorizations
are available, and showing end behavior.
8. Write a function defined by an expression in different but equivalent forms to
reveal and explain different properties of the function.
a. Use the process of factoring and completing the square in a quadratic
function to show zeros, extreme values, and symmetry of the graph, and
interpret these in terms of a context.
 2011 International Center for Leadership in Education
M10
M11
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Apply variables in expressions and
equations to solve problems (i.e., write
mathematical equations for given situation,
create a mathematical model to understand
the relationships between variables, or make
connections between the structures of
mathematically abstract concepts and the
real world).
Evaluate and employ accurate and
appropriate procedures for statistical data
collection, organization, analysis, and
display including making estimates and
predictions, critiquing data, and drawing
inferences (e.g., using the normal curve and
z-scores, line of best fit).
M21
Mathematics – Page 26
New York Mathematics
Strands/Bands/
Performance Indicators
Integrated Algebra
Mathematics
Domains/Clusters/
Common Core State Standards
High School
A.R.1 (Continued from previous page)
(Continued from previous page)
9. Compare properties of two functions each represented in a different way
(algebraically, graphically, numerically in tables, or by verbal descriptions).
For example, given a graph of one quadratic function and an algebraic
expression for another, say which has the larger maximum.
Statistics & Probability: Interpreting Categorical & Quantitative Data
Summarize, represent and interpret data on a single count or measurement
variable.
1. Represent data with plots on the real number line (dot plots, histograms, and
box plots).
Summarize, represent and interpret data on two categorical and
quantitative variables.
6. Represent data on two quantitative variables on a scatter plot, and describe
how the variables are related.
Algebra: Creating Equations
Create equations that describe numbers or relationships.
1. Create equations and inequalities in one variable and use them to solve
problems. Include equations arising from linear and quadratic functions, and
simple rational and exponential functions.
2. Create equations in two or more variables to represent relationships between
quantities; graph equations on coordinate axes with labels and scales.
Algebra: Reasoning with Equations & Inequalities
Represent and solve equations and inequalities graphically.
10. Understand that the graph of an equation in two variables is the set of all its
solutions plotted in the coordinate plane, often forming a curve (which could
be a line).
12. Graph the solutions to a linear inequality in two variables as a half-plane
(excluding the boundary in the case of a strict inequality), and graph the
solution set to a system of linear inequalities in two variables as the
intersection of the corresponding half-planes.
A.R.2 Recognize, compare, and use an
array of representational forms
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Mathematics – Page 27
New York Mathematics
Strands/Bands/
Performance Indicators
Integrated Algebra
Mathematics
Domains/Clusters/
Common Core State Standards
High School
A.R.2 (Continued from previous page)
(Continued from previous page)
Functions: Interpreting Functions
Analyze functions using different representations.
7. Graph functions expressed symbolically and show key features of the graph,
by hand in simple cases and using technology for more complicated cases.
a. Graph linear and quadratic functions and show intercepts, maxima, and
minima.
b. Graph square root, cube root, and piecewise-defined functions, including
step functions and absolute value functions.
c. Graph polynomial functions, identifying zeros when suitable factorizations
are available, and showing end behavior.
8. Write a function defined by an expression in different but equivalent forms to
reveal and explain different properties of the function.
a. Use the process of factoring and completing the square in a quadratic
function to show zeros, extreme values, and symmetry of the graph, and
interpret these in terms of a context.
9. Compare properties of two functions each represented in a different way
(algebraically, graphically, numerically in tables, or by verbal descriptions).
For example, given a graph of one quadratic function and an algebraic
expression for another, say which has the larger maximum.
Statistics & Probability: Interpreting Categorical & Quantitative Data
Summarize, represent and interpret data on a single count or measurement
variable.
1. Represent data with plots on the real number line (dot plots, histograms, and
box plots).
Summarize, represent and interpret data on two categorical and
quantitative variables.
6. Represent data on two quantitative variables on a scatter plot, and describe
how the variables are related.
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
Mathematics – Page 28
New York Mathematics
Strands/Bands/
Performance Indicators
Integrated Algebra
Mathematics
Domains/Clusters/
Common Core State Standards
High School
A.R.3 Use representation as a tool for
exploring and understanding
mathematical ideas
Algebra: Creating Equations
Create equations that describe numbers or relationships.
1. Create equations and inequalities in one variable and use them to solve
problems. Include equations arising from linear and quadratic functions, and
simple rational and exponential functions.
2. Create equations in two or more variables to represent relationships between
quantities; graph equations on coordinate axes with labels and scales.
Algebra: Reasoning with Equations & Inequalities
Represent and solve equations and inequalities graphically.
10. Understand that the graph of an equation in two variables is the set of all its
solutions plotted in the coordinate plane, often forming a curve (which could
be a line).
12. Graph the solutions to a linear inequality in two variables as a half-plane
(excluding the boundary in the case of a strict inequality), and graph the
solution set to a system of linear inequalities in two variables as the
intersection of the corresponding half-planes.
Functions: Interpreting Functions
Analyze functions using different representations.
7. Graph functions expressed symbolically and show key features of the graph,
by hand in simple cases and using technology for more complicated cases.
a. Graph linear and quadratic functions and show intercepts, maxima, and
minima.
b. Graph square root, cube root, and piecewise-defined functions, including
step functions and absolute value functions.
c. Graph polynomial functions, identifying zeros when suitable factorizations
are available, and showing end behavior.
8. Write a function defined by an expression in different but equivalent forms to
reveal and explain different properties of the function.
a. Use the process of factoring and completing the square in a quadratic
function to show zeros, extreme values, and symmetry of the graph, and
interpret these in terms of a context.
9. Compare properties of two functions each represented in a different way
(algebraically, graphically, numerically in tables, or by verbal descriptions).
For example, given a graph of one quadratic function and an algebraic
expression for another, say which has the larger maximum.
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Mathematics – Page 29
New York Mathematics
Strands/Bands/
Performance Indicators
Integrated Algebra
Mathematics
Domains/Clusters/
Common Core State Standards
High School
A.R.3 (Continued from previous page)
(Continued from previous page)
Statistics & Probability: Interpreting Categorical & Quantitative Data
Summarize, represent and interpret data on a single count or measurement
variable.
1. Represent data with plots on the real number line (dot plots, histograms, and
box plots).
Summarize, represent and interpret data on two categorical and
quantitative variables.
6. Represent data on two quantitative variables on a scatter plot, and describe
how the variables are related.
Students will select, apply, and
translate among mathematical
representations to solve problems
A.R.4 Select appropriate
representations to solve problem
situations
Algebra: Creating Equations
Create equations that describe numbers or relationships.
1. Create equations and inequalities in one variable and use them to solve
problems. Include equations arising from linear and quadratic functions, and
simple rational and exponential functions.
2. Create equations in two or more variables to represent relationships between
quantities; graph equations on coordinate axes with labels and scales.
Algebra: Reasoning with Equations & Inequalities
Represent and solve equations and inequalities graphically.
10. Understand that the graph of an equation in two variables is the set of all its
solutions plotted in the coordinate plane, often forming a curve (which could
be a line).
12. Graph the solutions to a linear inequality in two variables as a half-plane
(excluding the boundary in the case of a strict inequality), and graph the
solution set to a system of linear inequalities in two variables as the
intersection of the corresponding half-planes.
Functions: Interpreting Functions
Analyze functions using different representations.
7. Graph functions expressed symbolically and show key features of the graph,
by hand in simple cases and using technology for more complicated cases.
a. Graph linear and quadratic functions and show intercepts, maxima, and
minima.
b. Graph square root, cube root, and piecewise-defined functions, including
step functions and absolute value functions.
c. Graph polynomial functions, identifying zeros when suitable factorizations
are available, and showing end behavior.
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Mathematics – Page 30
New York Mathematics
Strands/Bands/
Performance Indicators
Integrated Algebra
Mathematics
Domains/Clusters/
Common Core State Standards
High School
A.R.4 (Continued from previous page)
(Continued from previous page)
8. Write a function defined by an expression in different but equivalent forms to
reveal and explain different properties of the function.
a. Use the process of factoring and completing the square in a quadratic
function to show zeros, extreme values, and symmetry of the graph, and
interpret these in terms of a context.
9. Compare properties of two functions each represented in a different way
(algebraically, graphically, numerically in tables, or by verbal descriptions).
For example, given a graph of one quadratic function and an algebraic
expression for another, say which has the larger maximum.
Algebra: Creating Equations
Create equations that describe numbers or relationships.
1. Create equations and inequalities in one variable and use them to solve
problems. Include equations arising from linear and quadratic functions, and
simple rational and exponential functions.
2. Create equations in two or more variables to represent relationships between
quantities; graph equations on coordinate axes with labels and scales.
Algebra: Reasoning with Equations & Inequalities
Represent and solve equations and inequalities graphically.
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.
Functions: Interpreting Functions
Analyze functions using different representations.
7. Graph functions expressed symbolically and show key features of the graph,
by hand in simple cases and using technology for more complicated cases.
a. Graph linear and quadratic functions and show intercepts, maxima, and
minima.
b. Graph square root, cube root, and piecewise-defined functions, including
step functions and absolute value functions.
c. Graph polynomial functions, identifying zeros when suitable factorizations
are available, and showing end behavior.
8. Write a function defined by an expression in different but equivalent forms to
reveal and explain different properties of the function.
A.R.5 Investigate relationships
between different representations and
their impact on a given problem
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Mathematics – Page 31
New York Mathematics
Strands/Bands/
Performance Indicators
Integrated Algebra
A.R.5 (Continued from previous page)
Students will use representations to
model and interpret physical, social,
and mathematical phenomena.
A.R.6 Use mathematics to show and
understand physical phenomena (e.g.,
find the height of a building if a ladder
of a given length forms a given angle of
elevation with the ground)
A.R.7 Use mathematics to show and
understand social phenomena (e.g.,
determine profit from student and adult
ticket sales)
A.R.8 Use mathematics to show and
understand mathematical phenomena
(e.g., compare the graphs of the
functions represented by the equations
y  x 2 and y   x 2 )
Mathematics
Domains/Clusters/
Common Core State Standards
High School
National Essential Skills Study (NESS)
National Rankings
Rank
(Continued from previous page)
a. Use the process of factoring and completing the square in a quadratic
function to show zeros, extreme values, and symmetry of the graph, and
interpret these in terms of a context.
9. Compare properties of two functions each represented in a different way
(algebraically, graphically, numerically in tables, or by verbal descriptions).
For example, given a graph of one quadratic function and an algebraic
expression for another, say which has the larger maximum.
Geometry: Similarity, Right Triangles, & Trigonometry
Define trigonometric ratios and solve problems involving right triangles.
8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles
in applied problems.
Algebra: Creating Equations
Create equations that describe numbers or relationships.
1. Create equations and inequalities in one variable and use them to solve
problems. Include equations arising from linear and quadratic functions, and
simple rational and exponential functions.
Algebra: Reasoning with Equations & Inequalities
Solve equations and inequalities in one variable.
3. Solve linear equations and inequalities in one variable, including equations
with coefficients represented by letters.
Functions: Building Functions
Build new functions from existing functions.
3. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and
f(x + k) for specific values of k (both positive and negative); find the value of
k given the graphs. Experiment with cases and illustrate an explanation of the
effects on the graph using technology. Include recognizing even and odd
functions from their graphs and algebraic expressions for them.
 2011 International Center for Leadership in Education
M10
M10
M11
M10
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Apply variables in expressions and
equations to solve problems (i.e., write
mathematical equations for given situation,
create a mathematical model to understand
the relationships between variables, or make
connections between the structures of
mathematically abstract concepts and the
real world).
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Mathematics – Page 32
New York Mathematics
Strands/Bands/
Performance Indicators
Integrated Algebra
Mathematics
Domains/Clusters/
Common Core State Standards
High School
National Essential Skills Study (NESS)
National Rankings
Rank
Number Sense and Operations
Strand
Students will understand numbers,
multiple ways of representing numbers,
relationships among numbers, and
number systems.
Number Theory
A.N.1 Identify and apply the properties
of real numbers (closure, commutative,
associative, distributive, identity,
inverse) Note: Students do not need to
identify groups and fields, but students
should be engaged in the ideas
Students will understand meanings of
operations and procedures, and how
they relate to one another.
Operations
Number & Quantity: The Real Number System
Use properties of rational and irrational numbers.
3. Explain why the sum or product of two rational numbers is rational; that the
sum of a rational number and an irrational number is irrational; and that the
product of a nonzero rational number and an irrational number is irrational.
A.N.2 Simplify radical terms (no
variable in the radicand)
Number & Quantity: The Real Number System
Extend the properties of exponents to rational exponents.
2. Rewrite expressions involving radicals and rational exponents using the
properties of exponents.
M2
M20
M29
M33
A.N.3 Perform the four arithmetic
operations using like and unlike radical
terms and express the result in simplest
form
Algebra: Reasoning with Equations & Inequalities
Understand solving equations as a process of reasoning and explain the
reasoning.
2. Solve simple rational and radical equations in one variable, and give
examples showing how extraneous solutions may arise.
 2011 International Center for Leadership in Education
Understand and apply basic algebraic
properties (commutative and associative
laws of addition and multiplication,
distributive law of multiplication over
addition, and identities and inverses).
Understand and apply the basic properties
and laws of exponents and scientific
notation to solve problems, including those
with fractional, negative, and zero
exponents.
Factor a composite number into its prime
components and use least common
denominators or least common multiples to
solve equations.
Perform operations with radicals, such as
addition, subtraction, and multiplication.
Perform operations with radicals, such as
addition, subtraction, and multiplication.
M33
Mathematics – Page 33
New York Mathematics
Strands/Bands/
Performance Indicators
Integrated Algebra
Mathematics
Domains/Clusters/
Common Core State Standards
High School
A.N.4 Understand and use scientific
notation to compute products and
quotients of numbers
There is no New York Mathematics Performance Indicator–Common Core
alignment.
A.N.5 Solve algebraic problems arising
from situations that involve fractions,
decimals, percents (decrease/increase
and discount), and
proportionality/direct variation
Number & Quantity: Quantities
Reason quantitatively and use units to solve problems.
1. Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and
interpret the scale and the origin in graphs and data displays.
Algebra: Arithmetic with Polynomials & Rational Expressions
Rewrite rational expressions.
7. (+) Understand that rational expressions form a system analogous to the
rational numbers, closed under addition, subtraction, multiplication, and
division by a nonzero rational expression; add, subtract, multiply, and divide
rational expressions.
Algebra: Reasoning with Equations & Inequalities
Understand solving equations as a process of reasoning and explain the
reasoning.
2. Solve simple rational and radical equations in one variable, and give
examples showing how extraneous solutions may arise.
Number & Quantity: The Real Number System
Extend the properties of exponents to rational exponents.
1. Explain how the definition of the meaning of rational exponents follows from
extending the properties of integer exponents to those values, allowing for a
notation for radicals in terms of rational exponents. For example, we define
51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3
must equal 5.
2. Rewrite expressions involving radicals and rational exponents using the
properties of exponents.
Algebra: Arithmetic with Polynomials & Rational Expressions
Use polynomial identities to solve problems.
5. (+) Know and apply the Binomial Theorem for the expansion of (x + y)n in
powers of x and y for a positive integer n, where x and y are any numbers,
with coefficients determined for example by Pascal’s Triangle. [The Binomial
Theorem can be proved by mathematical induction or by a combinatorial
argument.]
A.N.6 Evaluate expressions involving
factorial(s), absolute value(s), and
exponential expression(s)
National Essential Skills Study (NESS)
National Rankings
Rank
M20
 2011 International Center for Leadership in Education
M1
M3
Understand and apply the basic properties
and laws of exponents and scientific
notation to solve problems, including those
with fractional, negative, and zero
exponents.
Perform operations fluently with positive
and negative numbers, including decimals,
ratios, percents, and fractions, and show
reasoning to justify results.
Use proportional reasoning to solve realworld problems.
Solve and graphically sketch problems
involving two variables that exhibit direct
and indirect variation.
M57
M20
Understand and apply the basic properties
and laws of exponents and scientific
notation to solve problems, including those
with fractional, negative, and zero
exponents.
Use the properties of real (rational and
irrational) numbers and demonstrate
understanding of ordering and absolute
value.
M35
Mathematics – Page 34
New York Mathematics
Strands/Bands/
Performance Indicators
Integrated Algebra
A.N.7 Determine the number of
possible events, using counting
techniques or the Fundamental
Principle of Counting
A.N.8 Determine the number of
possible arrangements (permutations)
of a list of items
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Statistics & Probability: Conditional Probability & the Rules of
Probability
Use the rules of probability to compute probabilities of compound events in
a uniform probability model.
9. (+) Use permutations and combinations to compute probabilities of
compound events and solve problems.
Statistics & Probability: Conditional Probability & the Rules of
Probability
Use the rules of probability to compute probabilities of compound events in
a uniform probability model.
9. (+) Use permutations and combinations to compute probabilities of
compound events and solve problems.
National Essential Skills Study (NESS)
National Rankings
Rank
M32
M51
Determine the probability of single and
compound events and use the Counting
Principle to determine the probability of
independent events occurring jointly.
Determine combinations (the various
groupings a set may be arranged in without
regard to order) and permutations
(arrangements of a set where order matters).
Algebra Strand
Students will represent and analyze
algebraically a wide variety of problem
solving situations.
Variables and Expressions
A.A.1 Translate a quantitative verbal
phrase into an algebraic expression
Algebra: Creating Equations
Create equations that describe numbers or relationships.
1. Create equations and inequalities in one variable and use them to solve
problems. Include equations arising from linear and quadratic functions, and
simple rational and exponential functions.
2. Create equations in two or more variables to represent relationships between
quantities; graph equations on coordinate axes with labels and scales.
Functions: Building Functions
Build a function that models a relationship between two quantities.
1. Write a function that describes a relationship between two quantities.
a. Determine an explicit expression, a recursive process, or steps for
calculation from a context.
b. Combine standard function types using arithmetic operations. For example,
build a function that models the temperature of a cooling body by adding a
constant function to a decaying exponential, and relate these functions to
the model.
 2011 International Center for Leadership in Education
M11
Apply variables in expressions and
equations to solve problems (i.e., write
mathematical equations for given situation,
create a mathematical model to understand
the relationships between variables, or make
connections between the structures of
mathematically abstract concepts and the
real world).
Mathematics – Page 35
New York Mathematics
Strands/Bands/
Performance Indicators
Integrated Algebra
A.A.1 (Continued from previous page)
A.A.2 Write a verbal expression that
matches a given mathematical
expression
Mathematics
Domains/Clusters/
Common Core State Standards
High School
National Essential Skills Study (NESS)
National Rankings
Rank
(Continued from previous page)
2. Write arithmetic and geometric sequences both recursively and with an
explicit formula, use them to model situations, and translate between the two
forms.
There is no New York Mathematics Performance Indicator–Common Core
alignment.
M11
Apply variables in expressions and
equations to solve problems (i.e., write
mathematical equations for given situation,
create a mathematical model to understand
the relationships between variables, or make
connections between the structures of
mathematically abstract concepts and the
real world).
Equations and Inequalities
A.A.3 Distinguish the difference
between an algebraic expression and an
algebraic equation
There is no New York Mathematics Performance Indicator–Common Core
alignment.
M7
M11
A.A.4 Translate verbal sentences into
mathematical equations or inequalities
Algebra: Creating Equations
Create equations that describe numbers or relationships.
1. Create equations and inequalities in one variable and use them to solve
problems. Include equations arising from linear and quadratic functions, and
simple rational and exponential functions.
M7
M27
 2011 International Center for Leadership in Education
Simplify and solve algebraic equations by
identifying and using the correct order of
operations and techniques necessary to carry
out the solution.
Apply variables in expressions and
equations to solve problems (i.e., write
mathematical equations for given situation,
create a mathematical model to understand
the relationships between variables, or make
connections between the structures of
mathematically abstract concepts and the
real world).
Simplify and solve algebraic equations by
identifying and using the correct order of
operations and techniques necessary to carry
out the solution.
Find the solution of linear equations and
inequalities where the variable appears on
either or both sides and in which one or both
sides must be simplified before solving the
equation (e.g., solve x + 2(x - 3) = -4x + 5
for x).
Mathematics – Page 36
New York Mathematics
Strands/Bands/
Performance Indicators
Integrated Algebra
A.A.5 Write algebraic equations or
inequalities that represent a situation
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Algebra: Creating Equations
Create equations that describe numbers or relationships.
1. Create equations and inequalities in one variable and use them to solve
problems. Include equations arising from linear and quadratic functions, and
simple rational and exponential functions.
National Essential Skills Study (NESS)
National Rankings
Rank
M7
M11
M27
A.A.6 Analyze and solve verbal
problems whose solution requires
solving a linear equation in one variable
or linear inequality in one variable
A.A.7 Analyze and solve verbal
problems whose solution requires
solving systems of linear equations in
two variables
Algebra: Creating Equations
Create equations that describe numbers or relationships.
1. Create equations and inequalities in one variable and use them to solve
problems. Include equations arising from linear and quadratic functions, and
simple rational and exponential functions.
Algebra: Reasoning with Equations & Inequalities
Solve equations and inequalities in one variable.
3. Solve linear equations and inequalities in one variable, including equations
with coefficients represented by letters.
Algebra: Reasoning with Equations & Inequalities
Solve systems of equations.
5. Prove that, given a system of two equations in two variables, replacing one
equation by the sum of that equation and a multiple of the other produces a
system with the same solutions.
6. Solve systems of linear equations exactly and approximately (e.g., with
graphs), focusing on pairs of linear equations in two variables.
 2011 International Center for Leadership in Education
M7
M27
Simplify and solve algebraic equations by
identifying and using the correct order of
operations and techniques necessary to carry
out the solution.
Apply variables in expressions and
equations to solve problems (i.e., write
mathematical equations for given situation,
create a mathematical model to understand
the relationships between variables, or make
connections between the structures of
mathematically abstract concepts and the
real world).
Find the solution of linear equations and
inequalities where the variable appears on
either or both sides and in which one or both
sides must be simplified before solving the
equation (e.g., solve x + 2(x - 3) = -4x + 5
for x).
Simplify and solve algebraic equations by
identifying and using the correct order of
operations and techniques necessary to carry
out the solution.
Find the solution of linear equations and
inequalities where the variable appears on
either or both sides and in which one or both
sides must be simplified before solving the
equation (e.g., solve x + 2(x - 3) = -4x + 5
for x).
Solve systems of linear equations
algebraically or graphically.
M40
Mathematics – Page 37
New York Mathematics
Strands/Bands/
Performance Indicators
Integrated Algebra
A.A.8 Analyze and solve verbal
problems that involve quadratic
equations
A.A.9 Analyze and solve verbal
problems that involve exponential
growth and decay
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Algebra: Creating Equations
Create equations that describe numbers or relationships.
1. Create equations and inequalities in one variable and use them to solve
problems. Include equations arising from linear and quadratic functions, and
simple rational and exponential functions.
Algebra: Reasoning with Equations & Inequalities
Solve equations and inequalities in one variable.
4. Solve quadratic equations in one variable.
a. Use the method of completing the square to transform any quadratic
equation in x into an equation of the form (x – p)2 = q that has the same
solutions. Derive the quadratic formula from this form.
b. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square
roots, completing the square, the quadratic formula and factoring, as
appropriate to the initial form of the equation. Recognize when the
quadratic formula gives complex solutions and write them as a ± bi for real
numbers a and b.
Algebra: Seeing Structure in Expressions
Write expressions in equivalent forms to solve problems.
3. Choose and produce an equivalent form of an expression to reveal and
explain properties of the quantity represented by the expression.
c. Use the properties of exponents to transform expressions for exponential
functions. For example the expression 1.15t can be rewritten as (1.151/12)12t
≈ 1.01212t to reveal the approximate equivalent monthly interest rate if the
annual rate is 15%.
Functions: Interpreting Functions
Analyze functions using different representations.
7. Graph functions expressed symbolically and show key features of the graph,
by hand in simple cases and using technology for more complicated cases.
e. Graph exponential and logarithmic functions, showing intercepts and end
behavior, and trigonometric functions, showing period, midline, and
amplitude.
Functions: Linear, Quadratic & Exponential
Construct and compare linear, quadratic, and exponential models and
solve problems.
1. Distinguish between situations that can be modeled with linear functions and
with exponential functions.
a. Prove that linear functions grow by equal differences over equal intervals,
and that exponential functions grow by equal factors over equal intervals.
c. Recognize situations in which a quantity grows or decays by a constant
percent rate per unit interval relative to another.
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
Solve quadratic equations by applying
various tools or techniques.
M47
Express, graph, and interpret exponential
and logarithmic functions.
M48
Mathematics – Page 38
New York Mathematics
Strands/Bands/
Performance Indicators
Integrated Algebra
A.A.10 Solve systems of two linear
equations in two variables
algebraically (See A.G.7)
A.A.11 Solve a system of one linear
and one quadratic equation in two
variables, where only factoring is
required Note: The quadratic equation
should represent a parabola and the
solution(s) should be integers.
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Algebra: Reasoning with Equations & Inequalities
Solve systems of equations.
5. Prove that, given a system of two equations in two variables, replacing one
equation by the sum of that equation and a multiple of the other produces a
system with the same solutions.
6. Solve systems of linear equations exactly and approximately (e.g., with
graphs), focusing on pairs of linear equations in two variables.
Algebra: Reasoning with Equations & Inequalities
Solve systems of equations.
7. Solve a simple system consisting of a linear equation and a quadratic
equation in two variables algebraically and graphically. For example, find the
points of intersection between the line y = –3x and the circle x2 + y2 = 3.
National Essential Skills Study (NESS)
National Rankings
Rank
Solve systems of linear equations
algebraically or graphically.
M40
M36
M47
Simplify polynomials by performing
operations (addition, subtraction,
multiplication, and division) to simplify
expressions (e.g., (2a + 2) + (3a - 1) = 5a +
1).
Solve quadratic equations by applying
various tools or techniques.
Students will perform algebraic
procedures accurately.
Variables and Expressions
A.A.12 Multiply and divide monomial
expressions with a common base, using
the properties of exponents Note: Use
integral exponents only.
A.A.13 Add, subtract, and multiply
monomials and polynomials
Number & Quantity: The Real Number System
Extend the properties of exponents to rational exponents.
1. Explain how the definition of the meaning of rational exponents follows from
extending the properties of integer exponents to those values, allowing for a
notation for radicals in terms of rational exponents. For example, we define
51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3
must equal 5.
Algebra: Arithmetic with Polynomials & Rational Expressions
Perform arithmetic operations on polynomials.
1. Understand that polynomials form a system analogous to the integers,
namely, they are closed under the operations of addition, subtraction, and
multiplication; add, subtract, and multiply polynomials.
 2011 International Center for Leadership in Education
M20
M36
Understand and apply the basic properties
and laws of exponents and scientific
notation to solve problems, including those
with fractional, negative, and zero
exponents.
Simplify polynomials by performing
operations (addition, subtraction,
multiplication, and division) to simplify
expressions (e.g., (2a + 2) + (3a - 1) = 5a +
1).
Mathematics – Page 39
New York Mathematics
Strands/Bands/
Performance Indicators
Integrated Algebra
A.A.14 Divide a polynomial by a
monomial or binomial, where the
quotient has no remainder
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Algebra: Arithmetic with Polynomials & Rational Expressions
Rewrite rational expressions.
6. Rewrite simple rational expressions in different forms; write a(x)/b(x) in the
form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the
degree of r(x) less than the degree of b(x), using inspection, long division, or,
for the more complicated examples, a computer algebra system.
National Essential Skills Study (NESS)
National Rankings
Rank
M20
M62
A.A.15 Find values of a variable for
which an algebraic fraction is undefined
Functions: Interpreting Functions
Understand the concept of a function and use function notation.
2. Use function notation, evaluate functions for inputs in their domains, and
interpret statements that use function notation in terms of a context.
M7
M37
A.A.16 Simplify fractions with
polynomials in the numerator and
denominator by factoring both and
renaming them to lowest terms
Algebra: Arithmetic with Polynomials & Rational Expressions
Rewrite rational expressions.
6. Rewrite simple rational expressions in different forms; write a(x)/b(x) in the
form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the
degree of r(x) less than the degree of b(x), using inspection, long division, or,
for the more complicated examples, a computer algebra system.
7. (+) Understand that rational expressions form a system analogous to the
rational numbers, closed under addition, subtraction, multiplication, and
division by a nonzero rational expression; add, subtract, multiply, and divide
rational expressions.
M29
M36
M62
 2011 International Center for Leadership in Education
Understand and apply the basic properties
and laws of exponents and scientific
notation to solve problems, including those
with fractional, negative, and zero
exponents.
Perform division of a polynomial by a
monomial by dividing powers with like
bases, using the rules for the division of
powers with like bases to simplify fractions
with monomial denominators and reducing
fractions to lowest terms.
Simplify and solve algebraic equations by
identifying and using the correct order of
operations and techniques necessary to carry
out the solution.
Define and apply the properties of relations
and functions (domain, range, function
composition, and inverses) and use algebraic
and graphic methods to determine if a
relation is a function.
Factor a composite number into its prime
components and use least common
denominators or least common multiples to
solve equations.
Simplify polynomials by performing
operations (addition, subtraction,
multiplication, and division) to simplify
expressions (e.g., (2a + 2) + (3a - 1) = 5a +
1).
Perform division of a polynomial by a
monomial by dividing powers with like
bases, using the rules for the division of
powers with like bases to simplify fractions
with monomial denominators and reducing
fractions to lowest terms.
Mathematics – Page 40
New York Mathematics
Strands/Bands/
Performance Indicators
Integrated Algebra
A.A.17 Add or subtract fractional
expressions with monomial or like
binomial denominators
A.A.18 Multiply and divide algebraic
fractions and express the product or
quotient in simplest form
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Algebra: Arithmetic with Polynomials & Rational Expressions
Rewrite rational expressions.
7. (+) Understand that rational expressions form a system analogous to the
rational numbers, closed under addition, subtraction, multiplication, and
division by a nonzero rational expression; add, subtract, multiply, and divide
rational expressions.
Algebra: Arithmetic with Polynomials & Rational Expressions
Rewrite rational expressions.
6. Rewrite simple rational expressions in different forms; write a(x)/b(x) in the
form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the
degree of r(x) less than the degree of b(x), using inspection, long division, or,
for the more complicated examples, a computer algebra system.
National Essential Skills Study (NESS)
National Rankings
Rank
M36
M36
M62
A.A.19 Identify and factor the
difference of two perfect squares
Algebra: Seeing Structure in Expressions
Interpret the structure of expressions.
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).
M36
M47
 2011 International Center for Leadership in Education
Simplify polynomials by performing
operations (addition, subtraction,
multiplication, and division) to simplify
expressions (e.g., (2a + 2) + (3a - 1) = 5a +
1).
Simplify polynomials by performing
operations (addition, subtraction,
multiplication, and division) to simplify
expressions (e.g., (2a + 2) + (3a - 1) = 5a +
1).
Perform division of a polynomial by a
monomial by dividing powers with like
bases, using the rules for the division of
powers with like bases to simplify fractions
with monomial denominators and reducing
fractions to lowest terms.
Simplify polynomials by performing
operations (addition, subtraction,
multiplication, and division) to simplify
expressions (e.g., (2a + 2) + (3a - 1) = 5a +
1).
Solve quadratic equations by applying
various tools or techniques.
Mathematics – Page 41
New York Mathematics
Strands/Bands/
Performance Indicators
Integrated Algebra
A.A.20 Factor algebraic expressions
completely, including trinomials with a
lead coefficient of one (after factoring a
GCF)
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Algebra: Seeing Structure in Expressions
Interpret the structure of expressions.
Write expressions in equivalent forms to solve problems.
3. Choose and produce an equivalent form of an expression to reveal and
explain properties of the quantity represented by the expression.
a. Factor a quadratic expression to reveal the zeros of the function it defines.
National Essential Skills Study (NESS)
National Rankings
Rank
M29
M36
M62
Factor a composite number into its prime
components and use least common
denominators or least common multiples to
solve equations.
Simplify polynomials by performing
operations (addition, subtraction,
multiplication, and division) to simplify
expressions (e.g., (2a + 2) + (3a - 1) = 5a +
1).
Perform division of a polynomial by a
monomial by dividing powers with like
bases, using the rules for the division of
powers with like bases to simplify fractions
with monomial denominators and reducing
fractions to lowest terms.
Equations and Inequalities
A.A.21 Determine whether a given
value is a solution to a given linear
equation in one variable or linear
inequality in one variable
Algebra: Reasoning with Equations & Inequalities
Solve equations and inequalities in one variable.
3. Solve linear equations and inequalities in one variable, including equations
with coefficients represented by letters.
M7
M27
M45
 2011 International Center for Leadership in Education
Simplify and solve algebraic equations by
identifying and using the correct order of
operations and techniques necessary to carry
out the solution.
Find the solution of linear equations and
inequalities where the variable appears on
either or both sides and in which one or both
sides must be simplified before solving the
equation (e.g., solve x + 2(x - 3) = -4x + 5
for x).
Solve linear inequalities and graph the
solution set on a number line.
Mathematics – Page 42
New York Mathematics
Strands/Bands/
Performance Indicators
Integrated Algebra
A.A.22 Solve all types of linear
equations in one variable
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Algebra: Seeing Structure in Expressions
Write expressions in equivalent forms to solve problems.
3. Choose and produce an equivalent form of an expression to reveal and
explain properties of the quantity represented by the expression.
National Essential Skills Study (NESS)
National Rankings
Rank
M7
M27
A.A.23 Solve literal equations for a
given variable
A.A.24 Solve linear inequalities in one
variable
A.A.25 Solve equations involving
fractional expressions Note:
Expressions which result in linear
equations in one variable.
Algebra: Creating Equations
Create equations that describe numbers or relationships.
4. Rearrange formulas to highlight a quantity of interest, using the same
reasoning as in solving equations. For example, rearrange Ohm’s law V = IR
to highlight resistance R.
Algebra: Seeing Structure in Expressions
Write expressions in equivalent forms to solve problems.
3. Choose and produce an equivalent form of an expression to reveal and
explain properties of the quantity represented by the expression.
Algebra: Reasoning with Equations & Inequalities
Understand solving equations as a process of reasoning and explain the
reasoning.
2. Solve simple rational and radical equations in one variable, and give
examples showing how extraneous solutions may arise.
M7
M27
M7
M27
M29
 2011 International Center for Leadership in Education
Simplify and solve algebraic equations by
identifying and using the correct order of
operations and techniques necessary to carry
out the solution.
Find the solution of linear equations and
inequalities where the variable appears on
either or both sides and in which one or both
sides must be simplified before solving the
equation (e.g., solve x + 2(x - 3) = -4x + 5
for x).
Simplify and solve algebraic equations by
identifying and using the correct order of
operations and techniques necessary to carry
out the solution.
Find the solution of linear equations and
inequalities where the variable appears on
either or both sides and in which one or both
sides must be simplified before solving the
equation (e.g., solve x + 2(x - 3) = -4x + 5
for x).
Simplify and solve algebraic equations by
identifying and using the correct order of
operations and techniques necessary to carry
out the solution.
Find the solution of linear equations and
inequalities where the variable appears on
either or both sides and in which one or both
sides must be simplified before solving the
equation (e.g., solve x + 2(x - 3) = -4x + 5
for x).
Factor a composite number into its prime
components and use least common
denominators or least common multiples to
solve equations.
Mathematics – Page 43
New York Mathematics
Strands/Bands/
Performance Indicators
Integrated Algebra
Mathematics
Domains/Clusters/
Common Core State Standards
High School
A.A.26 Solve algebraic proportions in
one variable which result in linear or
quadratic equations
Algebra: Reasoning with Equations & Inequalities
Solve equations and inequalities in one variable.
3. Solve linear equations and inequalities in one variable, including equations
with coefficients represented by letters.
4. Solve quadratic equations in one variable.
a. Use the method of completing the square to transform any quadratic
equation in x into an equation of the form (x – p)2 = q that has the same
solutions. Derive the quadratic formula from this form.
b. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square
roots, completing the square, the quadratic formula and factoring, as
appropriate to the initial form of the equation. Recognize when the
quadratic formula gives complex solutions and write them as a ± bi for real
numbers a and b.
Algebra: Reasoning with Equations & Inequalities
Solve equations and inequalities in one variable.
4. Solve quadratic equations in one variable.
a. Use the method of completing the square to transform any quadratic
equation in x into an equation of the form (x – p)2 = q that has the same
solutions. Derive the quadratic formula from this form.
b. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square
roots, completing the square, the quadratic formula and factoring, as
appropriate to the initial form of the equation. Recognize when the
quadratic formula gives complex solutions and write them as a ± bi for real
numbers a and b.
Algebra: Seeing Structure in Expressions
Write expressions in equivalent forms to solve problems.
3. Choose and produce an equivalent form of an expression to reveal and
explain properties of the quantity represented by the expression.
a. Factor a quadratic expression to reveal the zeros of the function it defines.
Algebra: Reasoning with Equations & Inequalities
Solve equations and inequalities in one variable.
4. Solve quadratic equations in one variable.
b. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square
roots, completing the square, the quadratic formula and factoring, as
appropriate to the initial form of the equation. Recognize when the
quadratic formula gives complex solutions and write them as a ± bi for real
numbers a and b.
A.A.27 Understand and apply the
multiplication property of zero to solve
quadratic equations with integral
coefficients and integral roots
A.A.28 Understand the difference and
connection between roots of a quadratic
equation and factors of a quadratic
expression
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
M3
M7
Use proportional reasoning to solve realworld problems.
Simplify and solve algebraic equations by
identifying and using the correct order of
operations and techniques necessary to carry
out the solution.
Solve quadratic equations by applying
various tools or techniques.
M47
Solve quadratic equations by applying
various tools or techniques.
M47
M36
Simplify polynomials by performing
operations (addition, subtraction,
multiplication, and division) to simplify
expressions (e.g., (2a + 2) + (3a - 1) = 5a +
1).
Solve quadratic equations by applying
various tools or techniques.
M47
Mathematics – Page 44
New York Mathematics
Strands/Bands/
Performance Indicators
Integrated Algebra
Mathematics
Domains/Clusters/
Common Core State Standards
High School
National Essential Skills Study (NESS)
National Rankings
Rank
Students will recognize, use, and
represent algebraically patterns,
relations, and functions.
Patterns, Relations, and Functions
A.A.29 Use set-builder notation and/or
interval notation to illustrate the
elements of a set, given the elements in
roster form
A.A.30 Find the complement of a
subset of a given set, within a given
universe
Functions: Interpreting Functions
Understand the concept of a function and use function notation.
1. Understand that a function from one set (called the domain) to another set
(called the range) assigns to each element of the domain exactly one element
of the range. If f is a function and x is an element of its domain, then f(x)
denotes the output of f corresponding to the input x. The graph of f is the
graph of the equation y = f(x).
Statistics & Probability: Conditional Probability & the Rules of
Probability
Understand independence and conditional probability and use them to
interpret data.
1. Describe events as subsets of a sample space (the set of outcomes) using
characteristics (or categories) of the outcomes, or as unions, intersections, or
complements of other events (“or,” “and,” “not”).
M37
M21
M37
A.A.31 Find the intersection of sets (no
more than three sets) and/or union of
sets (no more than three sets)
Statistics & Probability: Conditional Probability & the Rules of
Probability
Understand independence and conditional probability and use them to
interpret data.
1. Describe events as subsets of a sample space (the set of outcomes) using
characteristics (or categories) of the outcomes, or as unions, intersections, or
complements of other events (“or,” “and,” “not”).
M21
M49
 2011 International Center for Leadership in Education
Define and apply the properties of relations
and functions (domain, range, function
composition, and inverses) and use algebraic
and graphic methods to determine if a
relation is a function.
Evaluate and employ accurate and
appropriate procedures for statistical data
collection, organization, analysis, and
display including making estimates and
predictions, critiquing data, and drawing
inferences (e.g., using the normal curve and
z-scores, line of best fit).
Define and apply the properties of relations
and functions (domain, range, function
composition, and inverses) and use algebraic
and graphic methods to determine if a
relation is a function.
Evaluate and employ accurate and
appropriate procedures for statistical data
collection, organization, analysis, and
display including making estimates and
predictions, critiquing data, and drawing
inferences (e.g., using the normal curve and
z-scores, line of best fit).
Analyze the truth value of compound
sentences by creating truth tables.
Mathematics – Page 45
New York Mathematics
Strands/Bands/
Performance Indicators
Integrated Algebra
Mathematics
Domains/Clusters/
Common Core State Standards
High School
National Essential Skills Study (NESS)
National Rankings
Rank
Coordinate Geometry
A.A.32 Explain slope as a rate of
change between dependent and
independent variables
A.A.33 Determine the slope of a line,
given the coordinates of two points on
the line
A.A.34 Write the equation of a line,
given its slope and the coordinates of a
point on the line
A.A.35 Write the equation of a line,
given the coordinates of two points on
the line
Functions: Interpreting Functions
Interpret functions that arise in applications in terms of the context.
6. Calculate and interpret the average rate of change of a function (presented
symbolically or as a table) over a specified interval. Estimate the rate of
change from a graph.
Functions: Linear, Quadratic & Exponential
Construct and compare linear, quadratic, and exponential models and
solve problems.
1. Distinguish between situations that can be modeled with linear functions and
with exponential functions.
a. Prove that linear functions grow by equal differences over equal intervals,
and that exponential functions grow by equal factors over equal intervals.
b. Recognize situations in which one quantity changes at a constant rate per
unit interval relative to another.
Functions: Linear, Quadratic & Exponential
Construct and compare linear, quadratic, and exponential models and
solve problems.
1. Distinguish between situations that can be modeled with linear functions and
with exponential functions.
b. Recognize situations in which one quantity changes at a constant rate per
unit interval relative to another.
Functions: Linear, Quadratic & Exponential
Construct and compare linear, quadratic, and exponential models and
solve problems.
2. Construct linear and exponential functions, including arithmetic and
geometric sequences, given a graph, a description of a relationship, or two
input-output pairs (include reading these from a table).
Functions: Linear, Quadratic & Exponential
Construct and compare linear, quadratic, and exponential models and
solve problems.
2. Construct linear and exponential functions, including arithmetic and
geometric sequences, given a graph, a description of a relationship, or two
input-output pairs (include reading these from a table).
 2011 International Center for Leadership in Education
Know the equation for the slope of a line
and compute slope given the coordinates of
two points.
M46
Know the equation for the slope of a line
and compute slope given the coordinates of
two points.
M46
M44
M44
M46
Know the equation of a line and interpret
graphically using the slope-intercept form (y
= mx+b) and the point-slope form (y-b =
m(x-a)).
Know the equation of a line and interpret
graphically using the slope-intercept form (y
= mx+b) and the point-slope form (y-b =
m(x-a)).
Know the equation for the slope of a line
and compute slope given the coordinates of
two points.
Mathematics – Page 46
New York Mathematics
Strands/Bands/
Performance Indicators
Integrated Algebra
A.A.36 Write the equation of a line
parallel to the x- or y-axis
A.A.37 Determine the slope of a line,
given its equation in any form
A.A.38 Determine if two lines are
parallel, given their equations in any
form
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Geometry: Expressing Geometric Properties with Equations
Use coordinates to prove simple geometric theorems algebraically.
5. Prove the slope criteria for parallel and perpendicular lines and use them to
solve geometric problems (e.g., find the equation of a line parallel or
perpendicular to a given line that passes through a given point).
Functions: Linear, Quadratic & Exponential
Construct and compare linear, quadratic, and exponential models and
solve problems.
1. Distinguish between situations that can be modeled with linear functions and
with exponential functions.
b. Recognize situations in which one quantity changes at a constant rate per
unit interval relative to another.
Geometry: Expressing Geometric Properties with Equations
Use coordinates to prove simple geometric theorems algebraically.
5. Prove the slope criteria for parallel and perpendicular lines and use them to
solve geometric problems (e.g., find the equation of a line parallel or
perpendicular to a given line that passes through a given point).
National Essential Skills Study (NESS)
National Rankings
Rank
M4
M44
M44
M46
M4
M44
M46
A.A.39 Determine whether a given
point is on a line, given the equation of
the line
Functions: Interpreting Functions
Understand the concept of a function and use function notation.
1. Understand that a function from one set (called the domain) to another set
(called the range) assigns to each element of the domain exactly one element
of the range. If f is a function and x is an element of its domain, then f(x)
denotes the output of f corresponding to the input x. The graph of f is the
graph of the equation y = f(x).
 2011 International Center for Leadership in Education
M44
Understand the properties of and apply
parallel, perpendicular, and intersecting
lines in problem-solving situations.
Know the equation of a line and interpret
graphically using the slope-intercept form (y
= mx+b) and the point-slope form (y-b =
m(x-a)).
Know the equation of a line and interpret
graphically using the slope-intercept form (y
= mx+b) and the point-slope form (y-b =
m(x-a)).
Know the equation for the slope of a line
and compute slope given the coordinates of
two points.
Understand the properties of and apply
parallel, perpendicular, and intersecting
lines in problem-solving situations.
Know the equation of a line and interpret
graphically using the slope-intercept form (y
= mx+b) and the point-slope form (y-b =
m(x-a)).
Know the equation for the slope of a line
and compute slope given the coordinates of
two points.
Know the equation of a line and interpret
graphically using the slope-intercept form (y
= mx+b) and the point-slope form (y-b =
m(x-a)).
Mathematics – Page 47
New York Mathematics
Strands/Bands/
Performance Indicators
Integrated Algebra
A.A.40 Determine whether a given
point is in the solution set of a system
of linear inequalities
A.A.41 Determine the vertex and axis
of symmetry of a parabola, given its
equation (See A.G.10 )
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Algebra: Reasoning with Equations & Inequalities
Represent and solve equations and inequalities graphically.
12. Graph the solutions to a linear inequality in two variables as a half-plane
(excluding the boundary in the case of a strict inequality), and graph the
solution set to a system of linear inequalities in two variables as the
intersection of the corresponding half-planes.
Functions: Interpreting Functions
Analyze functions using different representations.
8. Write a function defined by an expression in different but equivalent forms to
reveal and explain different properties of the function.
a. Use the process of factoring and completing the square in a quadratic
function to show zeros, extreme values, and symmetry of the graph, and
interpret these in terms of a context.
National Essential Skills Study (NESS)
National Rankings
Rank
M65
M47
M66
Find the graphic solution of systems of
linear inequalities (i.e., graph the solution
set or region of the coordinate plane
common to both inequalities).
Solve quadratic equations by applying
various tools or techniques.
Know how to sketch basic conic sections
(e.g., circles, parabolas) by using their
equations and solve systems of non-linear
equations graphically.
Trigonometric Functions
A.A.42 Find the sine, cosine, and
tangent ratios of an angle of a right
triangle, given the lengths of the sides
A.A.43 Determine the measure of an
angle of a right triangle, given the
length of any two sides of the triangle
A.A.44 Find the measure of a side of a
right triangle, given an acute angle and
the length of another side
A.A.45 Determine the measure of a
third side of a right triangle using the
Pythagorean theorem, given the lengths
of any two sides
Geometry: Similarity, Right Triangles, & Trigonometry
Define trigonometric ratios and solve problems involving right triangles.
6. Understand that by similarity, side ratios in right triangles are properties of
the angles in the triangle, leading to definitions of trigonometric ratios for
acute angles.
Geometry: Similarity, Right Triangles, & Trigonometry
Define trigonometric ratios and solve problems involving right triangles.
8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles
in applied problems.
Geometry: Similarity, Right Triangles, & Trigonometry
Define trigonometric ratios and solve problems involving right triangles.
8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles
in applied problems.
Geometry: Similarity, Right Triangles, & Trigonometry
Define trigonometric ratios and solve problems involving right triangles.
8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles
in applied problems.
 2011 International Center for Leadership in Education
M28
M28
M28
M23
Know and apply the six basic trigonometric
functions and ratios and solve right triangles
using basic trigonometric ratios (sine,
cosine, tangent).
Know and apply the six basic trigonometric
functions and ratios and solve right triangles
using basic trigonometric ratios (sine,
cosine, tangent).
Know and apply the six basic trigonometric
functions and ratios and solve right triangles
using basic trigonometric ratios (sine,
cosine, tangent).
Apply the Pythagorean Theorem to right
triangles.
Mathematics – Page 48
New York Mathematics
Strands/Bands/
Performance Indicators
Integrated Algebra
Mathematics
Domains/Clusters/
Common Core State Standards
High School
National Essential Skills Study (NESS)
National Rankings
Rank
Geometry Strand
Students will use visualization and
spatial reasoning to analyze
characteristics and properties of
geometric shapes.
Shapes
A.G.1 Find the area and/or perimeter
of figures composed of polygons and
circles or sectors of a circle Note:
Figures may include triangles,
rectangles, squares, parallelograms,
rhombuses, trapezoids, circles, semicircles, quarter-circles, and regular
polygons (perimeter only).
A.G.2 Use formulas to calculate
volume and surface area of rectangular
solids and cylinders
Geometry: Circles
Find arc length and areas of sectors of circles.
5. Derive using similarity the fact that the length of the arc intercepted by an
angle is proportional to the radius, and define the radian measure of the angle
as the constant of proportionality; derive the formula for the area of a sector.
Geometry: Geometric Measurement & Dimensions
Explain volume formulas and use them to solve problems.
1. Give an informal argument for the formulas for the circumference of a circle,
area of a circle, volume of a cylinder, pyramid, and cone. Use dissection
arguments, Cavalieri’s principle, and informal limit arguments.
Geometry: Geometric Measurement & Dimensions
Explain volume formulas and use them to solve problems.
1. Give an informal argument for the formulas for the circumference of a circle,
area of a circle, volume of a cylinder, pyramid, and cone. Use dissection
arguments, Cavalieri’s principle, and informal limit arguments.
3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve
problems.
 2011 International Center for Leadership in Education
M9
M18
M26
Compute the perimeter and area of common
two-dimensional figures.
Understand the properties of circles (radius,
arc, diameter, chord, secant, and tangent)
and apply circle quantities (lengths of line
segments, angle measure within a circle,
circumference, and area) in problem-solving
situations.
Know the classification and properties of
three-dimensional figures (prisms,
rectangular solids, pyramids, right circular
cylinders, cones, and spheres) and be able to
compute the volume and surface area of
common solids.
Mathematics – Page 49
New York Mathematics
Strands/Bands/
Performance Indicators
Integrated Algebra
Mathematics
Domains/Clusters/
Common Core State Standards
High School
National Essential Skills Study (NESS)
National Rankings
Rank
Students will apply coordinate
geometry to analyze problem solving
situations.
Coordinate Geometry
A.G.3 Determine when a relation is a
function, by examining ordered pairs
and inspecting graphs of relations
A.G.4 Identify and graph linear,
quadratic (parabolic), absolute value,
and exponential functions
Functions: Interpreting Functions
Understand the concept of a function and use function notation.
1. Understand that a function from one set (called the domain) to another set
(called the range) assigns to each element of the domain exactly one element
of the range. If f is a function and x is an element of its domain, then f(x)
denotes the output of f corresponding to the input x. The graph of f is the
graph of the equation y = f(x).
Functions: Interpreting Functions
Analyze functions using different representations.
7. Graph functions expressed symbolically and show key features of the graph,
by hand in simple cases and using technology for more complicated cases.
a. Graph linear and quadratic functions and show intercepts, maxima, and
minima.
b. Graph square root, cube root, and piecewise-defined functions, including
step functions and absolute value functions.
e. Graph exponential and logarithmic functions, showing intercepts and end
behavior, and trigonometric functions, showing period, midline, and
amplitude.
Functions: Linear, Quadratic & Exponential
Construct and compare linear, quadratic, and exponential models and
solve problems.
1. Distinguish between situations that can be modeled with linear functions and
with exponential functions.
a. Prove that linear functions grow by equal differences over equal intervals,
and that exponential functions grow by equal factors over equal intervals.
b. Recognize situations in which one quantity changes at a constant rate per
unit interval relative to another.
c. Recognize situations in which a quantity grows or decays by a constant
percent rate per unit interval relative to another.
2. Construct linear and exponential functions, including arithmetic and
geometric sequences, given a graph, a description of a relationship, or two
input-output pairs (include reading these from a table).
 2011 International Center for Leadership in Education
M30
M37
M48
Know and apply the components and
properties of the rectangular coordinate
system: x–y axis, origin, quadrants, abscissa
(x-coordinate) and ordinate (y-coordinate),
and general representation of a point (x,y).
Define and apply the properties of relations
and functions (domain, range, function
composition, and inverses) and use algebraic
and graphic methods to determine if a
relation is a function.
Express, graph, and interpret exponential
and logarithmic functions.
Express, graph, and interpret polynomial
functions (linear, quadratic, cubic, etc.).
M53
Mathematics – Page 50
New York Mathematics
Strands/Bands/
Performance Indicators
Integrated Algebra
Mathematics
Domains/Clusters/
Common Core State Standards
High School
A.G.5 Investigate and generalize how
changing the coefficients of a function
affects its graph
Functions: Building Functions
Build new functions from existing functions.
3. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and
f(x + k) for specific values of k (both positive and negative); find the value of
k given the graphs. Experiment with cases and illustrate an explanation of the
effects on the graph using technology. Include recognizing even and odd
functions from their graphs and algebraic expressions for them.
Algebra: Reasoning with Equations & Inequalities
Represent and solve equations and inequalities graphically.
12. Graph the solutions to a linear inequality in two variables as a half-plane
(excluding the boundary in the case of a strict inequality), and graph the
solution set to a system of linear inequalities in two variables as the
intersection of the corresponding half-planes.
A.G.6 Graph linear inequalities
A.G.7 Graph and solve systems of
linear equations and inequalities with
rational coefficients in two variables
(See A.A.10)
A.G.8 Find the roots of a parabolic
function graphically Note: Only
quadratic equations with integral
solutions.
Algebra: Reasoning with Equations & Inequalities
Solve systems of equations.
5. Prove that, given a system of two equations in two variables, replacing one
equation by the sum of that equation and a multiple of the other produces a
system with the same solutions.
6. Solve systems of linear equations exactly and approximately (e.g., with
graphs), focusing on pairs of linear equations in two variables.
Represent and solve equations and inequalities graphically.
12. Graph the solutions to a linear inequality in two variables as a half-plane
(excluding the boundary in the case of a strict inequality), and graph the
solution set to a system of linear inequalities in two variables as the
intersection of the corresponding half-planes.
Functions: Interpreting Functions
Analyze functions using different representations.
7. Graph functions expressed symbolically and show key features of the graph,
by hand in simple cases and using technology for more complicated cases.
a. Graph linear and quadratic functions and show intercepts, maxima, and
minima.
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
M48
Express, graph, and interpret exponential
and logarithmic functions.
Express, graph, and interpret polynomial
functions (linear, quadratic, cubic, etc.).
M53
M44
M53
M40
Know the equation of a line and interpret
graphically using the slope-intercept form (y
= mx+b) and the point-slope form (y-b =
m(x-a)).
Express, graph, and interpret polynomial
functions (linear, quadratic, cubic, etc.).
Solve systems of linear equations
algebraically or graphically.
Find the graphic solution of systems of
linear inequalities (i.e., graph the solution
set or region of the coordinate plane
common to both inequalities).
M65
M66
Know how to sketch basic conic sections
(e.g., circles, parabolas) by using their
equations and solve systems of non-linear
equations graphically.
Mathematics – Page 51
New York Mathematics
Strands/Bands/
Performance Indicators
Integrated Algebra
A.G.9 Solve systems of linear and
quadratic equations graphically Note:
Only use systems of linear and
quadratic equations that lead to
solutions whose coordinates are
integers.
A.G.10 Determine the vertex and axis
of symmetry of a parabola, given its
graph (See A.A.41) Note: The vertex
will have an ordered pair of integers
and the axis of symmetry will have an
integral value.
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Algebra: Reasoning with Equations & Inequalities
Solve systems of equations.
6. Solve systems of linear equations exactly and approximately (e.g., with
graphs), focusing on pairs of linear equations in two variables.
7. Solve a simple system consisting of a linear equation and a quadratic
equation in two variables algebraically and graphically. For example, find the
points of intersection between the line y = –3x and the circle x2 + y2 = 3.
Functions: Interpreting Functions
Analyze functions using different representations.
7. Graph functions expressed symbolically and show key features of the graph,
by hand in simple cases and using technology for more complicated cases.
a. Graph linear and quadratic functions and show intercepts, maxima, and
minima.
National Essential Skills Study (NESS)
National Rankings
Rank
M40
M66
M30
M54
M66
Solve systems of linear equations
algebraically or graphically.
Know how to sketch basic conic sections
(e.g., circles, parabolas) by using their
equations and solve systems of non-linear
equations graphically.
Know and apply the components and
properties of the rectangular coordinate
system: x–y axis, origin, quadrants, abscissa
(x-coordinate) and ordinate (y-coordinate),
and general representation of a point (x,y).
Apply transformations (reflection, rotation,
translation, and dilation) of 2-dimensional
figures graphically to interpret, analyze, and
illustrate the concepts of congruency,
similarity, and symmetry.
Know how to sketch basic conic sections
(e.g., circles, parabolas) by using their
equations and solve systems of non-linear
equations graphically.
Measurement Strand
Students will determine what can be
measured and how, using appropriate
methods and formulas.
Units of Measurement
A.M.1 Calculate rates using
appropriate units (e.g., rate of a space
ship versus the rate of a snail)
Number & Quantity: Quantities
Reason quantitatively and use units to solve problems.
1. Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and
interpret the scale and the origin in graphs and data displays.
2. Define appropriate quantities for the purpose of descriptive modeling.
 2011 International Center for Leadership in Education
M13
Use the technique of dimensional analysis to
convert units of measure (e.g.,
kilometers/hour to meters/minute) and apply
ratios in real-world situations (e.g., scale
drawings).
Mathematics – Page 52
New York Mathematics
Strands/Bands/
Performance Indicators
Integrated Algebra
Mathematics
Domains/Clusters/
Common Core State Standards
High School
A.M.2 Solve problems involving
conversions within measurement
systems, given the relationship between
the units
Number & Quantity: Quantities
Reason quantitatively and use units to solve problems.
1. Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and
interpret the scale and the origin in graphs and data displays.
2. Define appropriate quantities for the purpose of descriptive modeling.
National Essential Skills Study (NESS)
National Rankings
Rank
M8
M13
Solve problems using units of metric
measure and convert between metric and
English/customary units.
Use the technique of dimensional analysis to
convert units of measure (e.g.,
kilometers/hour to meters/minute) and apply
ratios in real-world situations (e.g., scale
drawings).
Students will understand that all
measurement contains error and be
able to determine its significance.
Error and Magnitude
A.M.3 Calculate the relative error in
measuring square and cubic units, when
there is an error in the linear measure
Number & Quantity: Quantities
Reason quantitatively and use units to solve problems.
3. Choose a level of accuracy appropriate to limitations on measurement when
reporting quantities.
M12
Understand accuracy and precision of
measurement, round off numbers according
to the correct number of significant figures,
and determine percent error.
Statistics and Probability Strand
Students will collect, organize, display,
and analyze data
Organization and Display of Data
A.S.1 Categorize data as qualitative or
quantitative
Statistics & Probability: Interpreting Categorical & Quantitative Data
Summarize, represent and interpret data on a single count or measurement
variable.
1. Represent data with plots on the real number line (dot plots, histograms, and
box plots).
Summarize, represent and interpret data on two categorical and
quantitative variables.
6. Represent data on two quantitative variables on a scatter plot, and describe
how the variables are related.
 2011 International Center for Leadership in Education
M21
Evaluate and employ accurate and
appropriate procedures for statistical data
collection, organization, analysis, and
display including making estimates and
predictions, critiquing data, and drawing
inferences (e.g., using the normal curve and
z-scores, line of best fit).
Mathematics – Page 53
New York Mathematics
Strands/Bands/
Performance Indicators
Integrated Algebra
Mathematics
Domains/Clusters/
Common Core State Standards
High School
A.S.2 Determine whether the data to be
analyzed is univariate or bivariate
Statistics & Probability: Interpreting Categorical & Quantitative Data
Summarize, represent and interpret data on a single count or measurement
variable.
1. Represent data with plots on the real number line (dot plots, histograms, and
box plots).
Summarize, represent and interpret data on two categorical and
quantitative variables.
6. Represent data on two quantitative variables on a scatter plot, and describe
how the variables are related.
A.S.3 Determine when collected data
or display of data may be biased
A.S.4 Compare and contrast the
appropriateness of different measures of
central tendency for a given data set
Statistics & Probability: Making Inferences & Justifying Conclusions
Understand and evaluate random processes underlying statistical
experiments.
1. Understand statistics as a process for making inferences about population
parameters based on a random sample from that population.
Make inferences and justify conclusions from sample surveys, experiments
and observational studies.
3. Recognize the purposes of and differences among sample surveys,
experiments, and observational studies; explain how randomization relates to
each.
4. Use data from a sample survey to estimate a population mean or proportion;
develop a margin of error through the use of simulation models for random
sampling.
5. Use data from a randomized experiment to compare two treatments; use
simulations to decide if differences between parameters are significant.
6. Evaluate reports based on data.
Statistics & Probability: Interpreting Categorical & Quantitative Data
Summarize, represent and interpret data on a single count or measurement
variable.
2. Use statistics appropriate to the shape of the data distribution to compare
center (median, mean) and spread (interquartile range, standard deviation) of
two or more different data sets.
3. Interpret differences in shape, center, and spread in the context of the data
sets, accounting for possible effects of extreme data points (outliers).
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
M21
M31
M14
M17
M21
M31
Evaluate and employ accurate and
appropriate procedures for statistical data
collection, organization, analysis, and
display including making estimates and
predictions, critiquing data, and drawing
inferences (e.g., using the normal curve and
z-scores, line of best fit).
Understand and apply measures of
dispersion (range, mean deviation, variance,
and standard deviation).
Understand and apply measures of central
tendency (mean, median, and mode, and
representative sampling of a population).
Understand the importance of random
sampling and sample size in generating
representative data.
Evaluate and employ accurate and
appropriate procedures for statistical data
collection, organization, analysis, and
display including making estimates and
predictions, critiquing data, and drawing
inferences (e.g., using the normal curve and
z-scores, line of best fit).
Understand and apply measures of
dispersion (range, mean deviation, variance,
and standard deviation).
Understand and apply measures of central
tendency (mean, median, and mode, and
representative sampling of a population).
M14
Mathematics – Page 54
New York Mathematics
Strands/Bands/
Performance Indicators
Integrated Algebra
Mathematics
Domains/Clusters/
Common Core State Standards
High School
A.S.5 Construct a histogram,
cumulative frequency histogram, and a
box-and-whisker plot, given a set of
data
Statistics & Probability: Interpreting Categorical & Quantitative Data
Summarize, represent and interpret data on a single count or measurement
variable.
1. Represent data with plots on the real number line (dot plots, histograms, and
box plots).
A.S.6 Understand how the five
statistical summary quartiles) is used to
construct a box-and-whisker plot
Statistics & Probability: Interpreting Categorical & Quantitative Data
Summarize, represent and interpret data on a single count or measurement
variable.
1. Represent data with plots on the real number line (dot plots, histograms, and
box plots).
National Essential Skills Study (NESS)
National Rankings
Rank
M21
M14
M21
M43
A.S.7 Create a scatter plot of bivariate
data
Statistics & Probability: Interpreting Categorical & Quantitative Data
Summarize, represent and interpret data on two categorical and
quantitative variables.
6. Represent data on two quantitative variables on a scatter plot, and describe
how the variables are related.
M21
M31
 2011 International Center for Leadership in Education
Evaluate and employ accurate and
appropriate procedures for statistical data
collection, organization, analysis, and
display including making estimates and
predictions, critiquing data, and drawing
inferences (e.g., using the normal curve and
z-scores, line of best fit).
Understand and apply measures of central
tendency (mean, median, and mode, and
representative sampling of a population).
Evaluate and employ accurate and
appropriate procedures for statistical data
collection, organization, analysis, and
display including making estimates and
predictions, critiquing data, and drawing
inferences (e.g., using the normal curve and
z-scores, line of best fit).
Understand and apply the concepts and
applications of quartiles (distributing groups
into four equal sizes), percentiles
(distributing individuals into 100 groups of
equal size), and random distribution to
understand and interpret data.
Evaluate and employ accurate and
appropriate procedures for statistical data
collection, organization, analysis, and
display including making estimates and
predictions, critiquing data, and drawing
inferences (e.g., using the normal curve and
z-scores, line of best fit).
Understand and apply measures of
dispersion (range, mean deviation, variance,
and standard deviation).
Mathematics – Page 55
New York Mathematics
Strands/Bands/
Performance Indicators
Integrated Algebra
Mathematics
Domains/Clusters/
Common Core State Standards
High School
A.S.8 Construct manually a reasonable
line of best fit for a scatter plot and
determine the equation of that line
Statistics & Probability: Interpreting Categorical & Quantitative Data
Summarize, represent and interpret data on two categorical and
quantitative variables.
6. Represent data on two quantitative variables on a scatter plot, and describe
how the variables are related.
a. Fit a function to the data; use functions fitted to data to solve problems in
the context of the data. Use given functions or choose a function suggested
by the context. Emphasize linear, quadratic, and exponential models.
c. Fit a linear function for a scatter plot that suggests a linear association.
National Essential Skills Study (NESS)
National Rankings
Rank
M21
Evaluate and employ accurate and
appropriate procedures for statistical data
collection, organization, analysis, and
display including making estimates and
predictions, critiquing data, and drawing
inferences (e.g., using the normal curve and
z-scores, line of best fit).
Analysis of Data
A.S.9 Analyze and interpret a
frequency distribution table or
histogram, a cumulative frequency
distribution table or histogram, or a
box-and-whisker plot
A.S.10 Evaluate published reports and
graphs that are based on data by
considering: experimental design,
appropriateness of the data analysis,
and the soundness of the conclusions
Statistics & Probability: Interpreting Categorical & Quantitative Data
Summarize, represent and interpret data on a single count or measurement
variable.
1. Represent data with plots on the real number line (dot plots, histograms, and
box plots).
Summarize, represent and interpret data on two categorical and
quantitative variables.
5. Summarize categorical data for two categories in two-way frequency tables.
Interpret relative frequencies in the context of the data (including joint,
marginal, and conditional relative frequencies). Recognize possible
associations and trends in the data.
Statistics & Probability: Making Inferences & Justifying Conclusions
Make inferences and justify conclusions from sample surveys, experiments
and observational studies.
6. Evaluate reports based on data.
M21
M17
M21
 2011 International Center for Leadership in Education
Evaluate and employ accurate and
appropriate procedures for statistical data
collection, organization, analysis, and
display including making estimates and
predictions, critiquing data, and drawing
inferences (e.g., using the normal curve and
z-scores, line of best fit).
Understand the importance of random
sampling and sample size in generating
representative data.
Evaluate and employ accurate and
appropriate procedures for statistical data
collection, organization, analysis, and
display including making estimates and
predictions, critiquing data, and drawing
inferences (e.g., using the normal curve and
z-scores, line of best fit).
Mathematics – Page 56
New York Mathematics
Strands/Bands/
Performance Indicators
Integrated Algebra
Mathematics
Domains/Clusters/
Common Core State Standards
High School
A.S.11 Find the percentile rank of an
item in a data set and identify the point
values for first, second, and third
quartiles
Statistics & Probability: Interpreting Categorical & Quantitative Data
Summarize, represent and interpret data on a single count or measurement
variable.
2. Use statistics appropriate to the shape of the data distribution to compare
center (median, mean) and spread (interquartile range, standard deviation) of
two or more different data sets.
4. Use the mean and standard deviation of a data set to fit it to a normal
distribution and to estimate population percentages. Recognize that there are
data sets for which such a procedure is not appropriate. Use calculators,
spreadsheets, and tables to estimate areas under the normal curve.
Statistics & Probability: Interpreting Categorical & Quantitative Data
Summarize, represent and interpret data on two categorical and
quantitative variables.
6. Represent data on two quantitative variables on a scatter plot, and describe
how the variables are related.
b. Informally assess the fit of a function by plotting and analyzing residuals.
Interpret linear models.
8. Compute (using technology) and interpret the correlation coefficient of a
linear fit.
Statistics & Probability: Interpreting Categorical & Quantitative Data
Interpret linear models.
9. Distinguish between correlation and causation.
Statistics & Probability: Interpreting Categorical & Quantitative Data
Interpret linear models.
9. Distinguish between correlation and causation.
A.S.12 Identify the relationship
between the independent and dependent
variables from a scatter plot (positive,
negative, or none)
A.S.13 Understand the difference
between correlation and causation
A.S.14 Identify variables that might
have a correlation but not a causal
relationship
National Essential Skills Study (NESS)
National Rankings
Rank
M43
M21
M22
M21
M22
 2011 International Center for Leadership in Education
Understand and apply the concepts and
applications of quartiles (distributing groups
into four equal sizes), percentiles
(distributing individuals into 100 groups of
equal size), and random distribution to
understand and interpret data.
Evaluate and employ accurate and
appropriate procedures for statistical data
collection, organization, analysis, and
display including making estimates and
predictions, critiquing data, and drawing
inferences (e.g., using the normal curve and
z-scores, line of best fit).
Interpret data to determine correlation and
distinguish between correlation and cause
and effect.
Evaluate and employ accurate and
appropriate procedures for statistical data
collection, organization, analysis, and
display including making estimates and
predictions, critiquing data, and drawing
inferences (e.g., using the normal curve and
z-scores, line of best fit).
Interpret data to determine correlation and
distinguish between correlation and cause
and effect.
Mathematics – Page 57
New York Mathematics
Strands/Bands/
Performance Indicators
Integrated Algebra
Mathematics
Domains/Clusters/
Common Core State Standards
High School
National Essential Skills Study (NESS)
National Rankings
Rank
Students will make predictions that are
based upon data analysis.
Predictions from Data
A.S.15 Identify and describe sources of
bias and its effect, drawing conclusions
from data
A.S.16 Recognize how linear
transformations of one-variable data
affect the data’s mean, median, mode,
and range
A.S.17 Use a reasonable line of best fit
to make a prediction involving
interpolation or extrapolation
Statistics & Probability: Making Inferences & Justifying Conclusions
Make inferences and justify conclusions from sample surveys, experiments
and observational studies.
3. Recognize the purposes of and differences among sample surveys,
experiments, and observational studies; explain how randomization relates to
each.
6. Evaluate reports based on data.
Statistics & Probability: Interpreting Categorical & Quantitative Data
Summarize, represent and interpret data on a single count or measurement
variable.
2. Use statistics appropriate to the shape of the data distribution to compare
center (median, mean) and spread (interquartile range, standard deviation) of
two or more different data sets.
Statistics & Probability: Interpreting Categorical & Quantitative Data
Interpret linear models.
7. Interpret the slope (rate of change) and the intercept (constant term) of a
linear model in the context of the data.
 2011 International Center for Leadership in Education
M17
M21
M14
M31
M21
Understand the importance of random
sampling and sample size in generating
representative data.
Evaluate and employ accurate and
appropriate procedures for statistical data
collection, organization, analysis, and
display including making estimates and
predictions, critiquing data, and drawing
inferences (e.g., using the normal curve and
z-scores, line of best fit).
Understand and apply measures of central
tendency (mean, median, and mode, and
representative sampling of a population).
Understand and apply measures of
dispersion (range, mean deviation, variance,
and standard deviation).
Evaluate and employ accurate and
appropriate procedures for statistical data
collection, organization, analysis, and
display including making estimates and
predictions, critiquing data, and drawing
inferences (e.g., using the normal curve and
z-scores, line of best fit).
Mathematics – Page 58
New York Mathematics
Strands/Bands/
Performance Indicators
Integrated Algebra
Mathematics
Domains/Clusters/
Common Core State Standards
High School
National Essential Skills Study (NESS)
National Rankings
Rank
Students will understand and apply
concepts of probability.
Probability
A.S.18 Know the definition of
conditional probability and use it to
solve for probabilities in finite sample
spaces
Statistics & Probability: Conditional Probability & the Rules of
Probability
Understand independence and conditional probability and use them to
interpret data.
1. Describe events as subsets of a sample space (the set of outcomes) using
characteristics (or categories) of the outcomes, or as unions, intersections, or
complements of other events (“or,” “and,” “not”).
2. Understand that two events A and B are independent if the probability of A
and B occurring together is the product of their probabilities, and use this
characterization to determine if they are independent.
3. Understand the conditional probability of A given B as P(A and B)/P(B), and
interpret independence of A and B as saying that the conditional probability of
A given B is the same as the probability of A, and the conditional probability
of B given A is the same as the probability of B.
4. Construct and interpret two-way frequency tables of data when two
categories are associated with each object being classified. Use the two-way
table as a sample space to decide if events are independent and to
approximate conditional probabilities. For example, collect data from a
random sample of students in your school on their favorite subject among
math, science, and English. Estimate the probability that a randomly selected
student from your school will favor science given that the student is in tenth
grade. Do the same for other subjects and compare the results.
5. Recognize and explain the concepts of conditional probability and
independence in everyday language and everyday situations. For example,
compare the chance of having lung cancer if you are a smoker with the
chance of being a smoker if you have lung cancer.
Use the rules of probability to compute probabilities of compound events in
a uniform probability model.
6. Find the conditional probability of A given B as the fraction of B’s outcomes
that also belong to A, and interpret the answer in terms of the model.
7. Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and
interpret the answer in terms of the model.
8. (+) Apply the general Multiplication Rule in a uniform probability model,
P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of
the model.
 2011 International Center for Leadership in Education
M5
Examine problem-solving situations
involving simple probability and use
probabilistic reasoning to compare and
communicate the theoretical or empirical
likelihood of events.
Determine the probability of single and
compound events and use the Counting
Principle to determine the probability of
independent events occurring jointly.
M32
Mathematics – Page 59
New York Mathematics
Strands/Bands/
Performance Indicators
Integrated Algebra
A.S.19 Determine the number of
elements in a sample space and the
number of favorable events
A.S.20 Calculate the probability of an
event and its complement
A.S.21 Determine empirical
probabilities based on specific sample
data
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Statistics & Probability: Conditional Probability & the Rules of
Probability
Use the rules of probability to compute probabilities of compound events in
a uniform probability model.
9. (+) Use permutations and combinations to compute probabilities of
compound events and solve problems.
Statistics & Probability: Conditional Probability & the Rules of
Probability
Understand independence and conditional probability and use them to
interpret data.
1. Describe events as subsets of a sample space (the set of outcomes) using
characteristics (or categories) of the outcomes, or as unions, intersections, or
complements of other events (“or,” “and,” “not”).
Use the rules of probability to compute probabilities of compound events in
a uniform probability model.
6. Find the conditional probability of A given B as the fraction of B’s outcomes
that also belong to A, and interpret the answer in terms of the model.
7. Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and
interpret the answer in terms of the model.
8. (+) Apply the general Multiplication Rule in a uniform probability model,
P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of
the model.
Statistics & Probability: Conditional Probability & the Rules of
Probability
Use the rules of probability to compute probabilities of compound events in
a uniform probability model.
9. (+) Use permutations and combinations to compute probabilities of
compound events and solve problems.
National Essential Skills Study (NESS)
National Rankings
Rank
M5
Examine problem-solving situations
involving simple probability and use
probabilistic reasoning to compare and
communicate the theoretical or empirical
likelihood of events.
M5
M5
M32
 2011 International Center for Leadership in Education
Examine problem-solving situations
involving simple probability and use
probabilistic reasoning to compare and
communicate the theoretical or empirical
likelihood of events.
Examine problem-solving situations
involving simple probability and use
probabilistic reasoning to compare and
communicate the theoretical or empirical
likelihood of events.
Determine the probability of single and
compound events and use the Counting
Principle to determine the probability of
independent events occurring jointly.
Mathematics – Page 60
New York Mathematics
Strands/Bands/
Performance Indicators
Integrated Algebra
Mathematics
Domains/Clusters/
Common Core State Standards
High School
A.S.22 Determine, based on calculated
probability of a set of events, if:
 some or all are equally likely to
occur
 one is more likely to occur than
another
 whether or not an event is certain
to happen or not to happen
A.S.23 Calculate the probability of:
 a series of independent events
 a series of dependent events
 two mutually exclusive events
 two events that are not mutually
exclusive
Statistics & Probability: Conditional Probability & the Rules of
Probability
Use the rules of probability to compute probabilities of compound events in
a uniform probability model.
9. (+) Use permutations and combinations to compute probabilities of
compound events and solve problems.
Statistics & Probability: Conditional Probability & the Rules of
Probability
Understand independence and conditional probability and use them to
interpret data.
2. Understand that two events A and B are independent if the probability of A
and B occurring together is the product of their probabilities, and use this
characterization to determine if they are independent.
3. Understand the conditional probability of A given B as P(A and B)/P(B), and
interpret independence of A and B as saying that the conditional probability of
A given B is the same as the probability of A, and the conditional probability
of B given A is the same as the probability of B.
4. Construct and interpret two-way frequency tables of data when two
categories are associated with each object being classified. Use the two-way
table as a sample space to decide if events are independent and to
approximate conditional probabilities. For example, collect data from a
random sample of students in your school on their favorite subject among
math, science, and English. Estimate the probability that a randomly selected
student from your school will favor science given that the student is in tenth
grade. Do the same for other subjects and compare the results.
Use the rules of probability to compute probabilities of compound events in
a uniform probability model.
6. Find the conditional probability of A given B as the fraction of B’s outcomes
that also belong to A, and interpret the answer in terms of the model.
7. Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and
interpret the answer in terms of the model.
8. (+) Apply the general Multiplication Rule in a uniform probability model,
P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of
the model.
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
M5
M5
Examine problem-solving situations
involving simple probability and use
probabilistic reasoning to compare and
communicate the theoretical or empirical
likelihood of events.
Examine problem-solving situations
involving simple probability and use
probabilistic reasoning to compare and
communicate the theoretical or empirical
likelihood of events.
Determine the probability of single and
compound events and use the Counting
Principle to determine the probability of
independent events occurring jointly.
M32
Mathematics – Page 61
New York Mathematics Standards Alignments
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
Mathematics
Domains/Clusters/
Common Core State Standards
High School
National Essential Skills Study (NESS)
National Rankings
Rank
Problem Solving Strand
Students will new mathematical
knowledge through problem solving.
G.PS.1 Use a variety of problem
solving strategies to understand new
mathematical content
Geometry: Similarity, Right Triangles, & Trigonometry
Define trigonometric ratios and solve problems involving right triangles.
6. Understand that by similarity, side ratios in right triangles are properties of
the angles in the triangle, leading to definitions of trigonometric ratios for
acute angles.
7. Explain and use the relationship between the sine and cosine of
complementary angles.
8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles
in applied problems.
Geometry: Expressing Geometric Properties with Equations
Use coordinates to prove simple geometric theorems algebraically.
4. Use coordinates to prove simple geometric theorems algebraically. For
example, prove or disprove that a figure defined by four given points in the
coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies
on the circle centered at the origin and containing the point (0, 2).
5. Prove the slope criteria for parallel and perpendicular lines and use them to
solve geometric problems (e.g., find the equation of a line parallel or
perpendicular to a given line that passes through a given point).
6. Find the point on a directed line segment between two given points that
partitions the segment in a given ratio.
7. Use coordinates to compute perimeters of polygons and areas of triangles and
rectangles, e.g., using the distance formula.
 2011 International Center for Leadership in Education
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Mathematics – Page 62
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
Students will solve problems that arise
in mathematics and in other contexts.
G.PS.2 Observe and explain patterns to
formulate generalizations and
conjectures
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Geometry: Congruence
Understand congruence in terms of rigid motions.
6. Use geometric descriptions of rigid motions to transform figures and to
predict the effect of a given rigid motion on a given figure; given two figures,
use the definition of congruence in terms of rigid motions to decide if they are
congruent.
7. Use the definition of congruence in terms of rigid motions to show that two
triangles are congruent if and only if corresponding pairs of sides and
corresponding pairs of angles are congruent.
8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow
from the definition of congruence in terms of rigid motions.
Geometry: Similarity, Right Triangles, & Trigonometry
Understand similarity in terms of similarity transformations.
2. Given two figures, use the definition of similarity in terms of similarity
transformations to decide if they are similar; explain using similarity
transformations the meaning of similarity for triangles as the equality of all
corresponding pairs of angles and the proportionality of all corresponding
pairs of sides.
3. Use the properties of similarity transformations to establish the AA criterion
for two triangles to be similar.
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
Apply pattern recognition in data sets and
series to reason or solve problems involving
arithmetic, geometry, exponents, etc.
M16
Mathematics – Page 63
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
Mathematics
Domains/Clusters/
Common Core State Standards
High School
G.PS.3 Use multiple representations to
represent and explain problem
situations (e.g., spatial, geometric,
verbal, numeric, algebraic, and
graphical representations)
Geometry: Congruence
Experiment with transformations in the plane.
2. Represent transformations in the plane using, e.g., transparencies and
geometry software; describe transformations as functions that take points in
the plane as inputs and give other points as outputs. Compare transformations
that preserve distance and angle to those that do not (e.g., translation versus
horizontal stretch).
5. Given a geometric figure and a rotation, reflection, or translation, draw the
transformed figure using, e.g., graph paper, tracing paper, or geometry
software. Specify a sequence of transformations that will carry a given figure
onto another.
Understand congruence in terms of rigid motions.
8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow
from the definition of congruence in terms of rigid motions.
Make geometric constructions.
12. Make formal geometric constructions with a variety of tools and methods
(compass and straightedge, string, reflective devices, paper folding, dynamic
geometric software, etc.). Copying a segment; copying an angle; bisecting a
segment; bisecting an angle; constructing perpendicular lines, including the
perpendicular bisector of a line segment; and constructing a line parallel to a
given line through a point not on the line.
13. Construct an equilateral triangle, a square, and a regular hexagon inscribed
in a circle.
Geometry: Similarity, Right Triangles, & Trigonometry
Understand similarity in terms of similarity transformations.
1. Verify experimentally the properties of dilations given by a center and a scale
factor:
a. A dilation takes a line not passing through the center of the dilation to a
parallel line, and leaves a line passing through the center unchanged.
b. The dilation of a line segment is longer or shorter in the ratio given by the
scale factor.
3. Use the properties of similarity transformations to establish the AA criterion
for two triangles to be similar.
Define trigonometric ratios and solve problems involving right triangles.
6. Understand that by similarity, side ratios in right triangles are properties of
the angles in the triangle, leading to definitions of trigonometric ratios for
acute angles.
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
M10
M11
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Apply variables in expressions and
equations to solve problems (i.e., write
mathematical equations for given situation,
create a mathematical model to understand
the relationships between variables, or make
connections between the structures of
mathematically abstract concepts and the
real world).
Evaluate and employ accurate and
appropriate procedures for statistical data
collection, organization, analysis, and
display including making estimates and
predictions, critiquing data, and drawing
inferences (e.g., using the normal curve and
z-scores, line of best fit).
M21
Mathematics – Page 64
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
Mathematics
Domains/Clusters/
Common Core State Standards
High School
G.PS.3 (Continued from previous page)
(Continued from previous page)
7. Explain and use the relationship between the sine and cosine of
complementary angles.
Geometry: Expressing Geometric Properties with Equations
Translate between the geometric description and the equation for a conic
section.
1. Derive the equation of a circle of given center and radius using the
Pythagorean Theorem; complete the square to find the center and radius of a
circle given by an equation.
Use coordinates to prove simple geometric theorems algebraically.
4. Use coordinates to prove simple geometric theorems algebraically. For
example, prove or disprove that a figure defined by four given points in the
coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies
on the circle centered at the origin and containing the point (0, 2).
5. Prove the slope criteria for parallel and perpendicular lines and use them to
solve geometric problems (e.g., find the equation of a line parallel or
perpendicular to a given line that passes through a given point).
6. Find the point on a directed line segment between two given points that
partitions the segment in a given ratio.
7. Use coordinates to compute perimeters of polygons and areas of triangles and
rectangles, e.g., using the distance formula.
Students will apply and adapt a variety
of appropriate strategies to solve
problems.
G.PS.4 Construct various types of
reasoning, arguments, justifications and
methods of proof for problems
Geometry: Congruence
Understand congruence in terms of rigid motions.
8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow
from the definition of congruence in terms of rigid motions.
Prove geometric theorems.
9. Prove theorems about lines and angles. Theorems include: vertical angles are
congruent; when a transversal crosses parallel lines, alternate interior angles
are congruent and corresponding angles are congruent; points on a
perpendicular bisector of a line segment are exactly those equidistant from
the segment’s endpoints.
10. Prove theorems about triangles. Theorems include: measures of interior
angles of a triangle sum to 180°; base angles of isosceles triangles are
congruent; the segment joining midpoints of two sides of a triangle is parallel
to the third side and half the length; the medians of a triangle meet at a point.
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Mathematics – Page 65
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
Mathematics
Domains/Clusters/
Common Core State Standards
High School
G.PS.4 (Continued from previous page)
(Continued from previous page)
11. Prove theorems about parallelograms. Theorems include: opposite sides are
congruent, opposite angles are congruent, the diagonals of a parallelogram
bisect each other, and conversely, rectangles are parallelograms with
congruent diagonals.
Geometry: Similarity, Right Triangles, & Trigonometry
Understand similarity in terms of similarity transformations.
3. Use the properties of similarity transformations to establish the AA criterion
for two triangles to be similar.
Prove theorems involving similarity.
4. Prove theorems about triangles. Theorems include: a line parallel to one side
of a triangle divides the other two proportionally, and conversely; the
Pythagorean Theorem proved using triangle similarity.
5. Use congruence and similarity criteria for triangles to solve problems and to
prove relationships in geometric figures.
Geometry: Circles
Understand and apply theorems about circles.
1. Prove that all circles are similar.
2. Identify and describe relationships among inscribed angles, radii, and chords.
Include the relationship between central, inscribed, and circumscribed
angles; inscribed angles on a diameter are right angles; the radius of a circle
is perpendicular to the tangent where the radius intersects the circle.
Geometry: Congruence
Experiment with transformations in the plane.
3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the
rotations and reflections that carry it onto itself.
4. Develop definitions of rotations, reflections, and translations in terms of
angles, circles, perpendicular lines, parallel lines, and line segments.
5. Given a geometric figure and a rotation, reflection, or translation, draw the
transformed figure using, e.g., graph paper, tracing paper, or geometry
software. Specify a sequence of transformations that will carry a given figure
onto another.
Understand congruence in terms of rigid motions.
6. Use geometric descriptions of rigid motions to transform figures and to
predict the effect of a given rigid motion on a given figure; given two figures,
use the definition of congruence in terms of rigid motions to decide if they are
congruent.
G.PS.5 Choose an effective approach
to solve a problem from a variety of
strategies (numeric, graphic, algebraic)
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Mathematics – Page 66
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
Mathematics
Domains/Clusters/
Common Core State Standards
High School
G.PS.5 (Continued from previous page)
(Continued from previous page)
Geometry: Similarity, Right Triangles, & Trigonometry
Understand similarity in terms of similarity transformations.
1. Verify experimentally the properties of dilations given by a center and a scale
factor:
a. A dilation takes a line not passing through the center of the dilation to a
parallel line, and leaves a line passing through the center unchanged.
b. The dilation of a line segment is longer or shorter in the ratio given by the
scale factor.3. Use the properties of similarity transformations to establish the
AA criterion for two triangles to be similar.
Define trigonometric ratios and solve problems involving right triangles.
6. Understand that by similarity, side ratios in right triangles are properties of
the angles in the triangle, leading to definitions of trigonometric ratios for
acute angles.
7. Explain and use the relationship between the sine and cosine of
complementary angles.
8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles
in applied problems.
Geometry: Geometric Measurement & Dimensions
Explain volume formulas and use them to solve problems.
1. Give an informal argument for the formulas for the circumference of a circle,
area of a circle, volume of a cylinder, pyramid, and cone. Use dissection
arguments, Cavalieri’s principle, and informal limit arguments.
3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve
problems.
Geometry: Modeling with Geometry
Apply Geometric Concepts in modeling situations
1. Use geometric shapes, their measures, and their properties to describe objects
(e.g., modeling a tree trunk or a human torso as a cylinder).
2. Apply concepts of density based on area and volume in modeling situations
(e.g., persons per square mile, BTUs per cubic foot).
There is no New York Mathematics Performance Indicator–Common Core
alignment.
G.PS.6 Use a variety of strategies to
extend solution methods to other
problems
G.PS.7 Work in collaboration with
others to propose, critique, evaluate,
and value alternative approaches to
problem solving
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
M10
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Mathematics – Page 67
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
Students will monitor and reflect on the
process of mathematical problem
solving.
G.PS.8 Determine information required
to solve a problem, choose methods for
obtaining the information, and define
parameters for acceptable solutions
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Geometry: Congruence
Experiment with transformations in the plane.
3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the
rotations and reflections that carry it onto itself.
4. Develop definitions of rotations, reflections, and translations in terms of
angles, circles, perpendicular lines, parallel lines, and line segments.
5. Given a geometric figure and a rotation, reflection, or translation, draw the
transformed figure using, e.g., graph paper, tracing paper, or geometry
software. Specify a sequence of transformations that will carry a given figure
onto another.
Understand congruence in terms of rigid motions.
6. Use geometric descriptions of rigid motions to transform figures and to
predict the effect of a given rigid motion on a given figure; given two figures,
use the definition of congruence in terms of rigid motions to decide if they are
congruent.
Geometry: Similarity, Right Triangles, & Trigonometry
Understand similarity in terms of similarity transformations.
1. Verify experimentally the properties of dilations given by a center and a scale
factor:
a. A dilation takes a line not passing through the center of the dilation to a
parallel line, and leaves a line passing through the center unchanged.
b. The dilation of a line segment is longer or shorter in the ratio given by the
scale factor.
3. Use the properties of similarity transformations to establish the AA criterion
for two triangles to be similar.
Define trigonometric ratios and solve problems involving right triangles.
6. Understand that by similarity, side ratios in right triangles are properties of
the angles in the triangle, leading to definitions of trigonometric ratios for
acute angles.
7. Explain and use the relationship between the sine and cosine of
complementary angles.
8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles
in applied problems.
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Mathematics – Page 68
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
Mathematics
Domains/Clusters/
Common Core State Standards
High School
G.PS.8 (Continued from previous page)
(Continued from previous page)
Geometry: Geometric Measurement & Dimensions
Explain volume formulas and use them to solve problems.
1. Give an informal argument for the formulas for the circumference of a circle,
area of a circle, volume of a cylinder, pyramid, and cone. Use dissection
arguments, Cavalieri’s principle, and informal limit arguments.
3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve
problems.
There is no New York Mathematics Performance Indicator–Common Core
alignment.
G.PS.9 Interpret solutions within the
given constraints of a problem
National Essential Skills Study (NESS)
National Rankings
Rank
M10
G.PS.10 Evaluate the relative
efficiency of different representations
and solution methods of a problem
There is no New York Mathematics Performance Indicator–Common Core
alignment.
M10
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Reasoning and Proof Strand
Students will recognize reasoning and
proof as fundamental aspects of
mathematics.
G.RP.1 Recognize that mathematical
ideas can be supported by a variety of
strategies
Geometry: Congruence
Understand congruence in terms of rigid motions.
8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow
from the definition of congruence in terms of rigid motions.
Prove geometric theorems.
9. Prove theorems about lines and angles. Theorems include: vertical angles are
congruent; when a transversal crosses parallel lines, alternate interior angles
are congruent and corresponding angles are congruent; points on a
perpendicular bisector of a line segment are exactly those equidistant from
the segment’s endpoints.
10. Prove theorems about triangles. Theorems include: measures of interior
angles of a triangle sum to 180°; base angles of isosceles triangles are
congruent; the segment joining midpoints of two sides of a triangle is parallel
to the third side and half the length; the medians of a triangle meet at a point.
 2011 International Center for Leadership in Education
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Mathematics – Page 69
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
Mathematics
Domains/Clusters/
Common Core State Standards
High School
G.RP.1 (Continued from previous page)
(Continued from previous page)
11. Prove theorems about parallelograms. Theorems include: opposite sides are
congruent, opposite angles are congruent, the diagonals of a parallelogram
bisect each other, and conversely, rectangles are parallelograms with
congruent diagonals.
Geometry: Similarity, Right Triangles, & Trigonometry
Understand similarity in terms of similarity transformations.
3. Use the properties of similarity transformations to establish the AA criterion
for two triangles to be similar.
Prove theorems involving similarity.
4. Prove theorems about triangles. Theorems include: a line parallel to one side
of a triangle divides the other two proportionally, and conversely; the
Pythagorean Theorem proved using triangle similarity.
5. Use congruence and similarity criteria for triangles to solve problems and to
prove relationships in geometric figures.
Geometry: Circles
Understand and apply theorems about circles.
1. Prove that all circles are similar.
2. Identify and describe relationships among inscribed angles, radii, and chords.
Include the relationship between central, inscribed, and circumscribed
angles; inscribed angles on a diameter are right angles; the radius of a circle
is perpendicular to the tangent where the radius intersects the circle.
Geometry: Expressing Geometric Properties with Equations
Use coordinates to prove simple geometric theorems algebraically.
5. Prove the slope criteria for parallel and perpendicular lines and use them to
solve geometric problems (e.g., find the equation of a line parallel or
perpendicular to a given line that passes through a given point).
G.RP.2 Recognize and verify, where
appropriate, geometric relationships of
perpendicularity, parallelism,
congruence, and similarity, using
algebraic strategies
National Essential Skills Study (NESS)
National Rankings
Rank
M4
M10
M54
 2011 International Center for Leadership in Education
Understand the properties of and apply
parallel, perpendicular, and intersecting
lines in problem-solving situations.
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Apply transformations (reflection, rotation,
translation, and dilation) of 2-dimensional
figures graphically to interpret, analyze, and
illustrate the concepts of congruency,
similarity, and symmetry.
Mathematics – Page 70
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
Students will make and investigate
mathematical conjectures.
G.RP.3 Investigate and evaluate
conjectures in mathematical terms,
using mathematical strategies to reach a
conclusion
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Geometry: Congruence
Experiment with transformations in the plane.
2. Represent transformations in the plane using, e.g., transparencies and
geometry software; describe transformations as functions that take points in
the plane as inputs and give other points as outputs. Compare transformations
that preserve distance and angle to those that do not (e.g., translation versus
horizontal stretch).
3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the
rotations and reflections that carry it onto itself.
4. Develop definitions of rotations, reflections, and translations in terms of
angles, circles, perpendicular lines, parallel lines, and line segments.
5. Given a geometric figure and a rotation, reflection, or translation, draw the
transformed figure using, e.g., graph paper, tracing paper, or geometry
software. Specify a sequence of transformations that will carry a given figure
onto another.
Understand congruence in terms of rigid motions.
6. Use geometric descriptions of rigid motions to transform figures and to
predict the effect of a given rigid motion on a given figure; given two figures,
use the definition of congruence in terms of rigid motions to decide if they are
congruent.
7. Use the definition of congruence in terms of rigid motions to show that two
triangles are congruent if and only if corresponding pairs of sides and
corresponding pairs of angles are congruent.
8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow
from the definition of congruence in terms of rigid motions.
Prove geometric theorems.
9. Prove theorems about lines and angles. Theorems include: vertical angles are
congruent; when a transversal crosses parallel lines, alternate interior angles
are congruent and corresponding angles are congruent; points on a
perpendicular bisector of a line segment are exactly those equidistant from
the segment’s endpoints.
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Mathematics – Page 71
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
Mathematics
Domains/Clusters/
Common Core State Standards
High School
G.RP.3 (Continued from previous page)
(Continued from previous page)
10. Prove theorems about triangles. Theorems include: measures of interior
angles of a triangle sum to 180°; base angles of isosceles triangles are
congruent; the segment joining midpoints of two sides of a triangle is parallel
to the third side and half the length; the medians of a triangle meet at a point.
11. Prove theorems about parallelograms. Theorems include: opposite sides are
congruent, opposite angles are congruent, the diagonals of a parallelogram
bisect each other, and conversely, rectangles are parallelograms with
congruent diagonals.
Geometry: Similarity, Right Triangles, & Trigonometry
Understand similarity in terms of similarity transformations.
1. Verify experimentally the properties of dilations given by a center and a scale
factor:
a. A dilation takes a line not passing through the center of the dilation to a
parallel line, and leaves a line passing through the center unchanged.
b. The dilation of a line segment is longer or shorter in the ratio given by the
scale factor.
2. Given two figures, use the definition of similarity in terms of similarity
transformations to decide if they are similar; explain using similarity
transformations the meaning of similarity for triangles as the equality of all
corresponding pairs of angles and the proportionality of all corresponding
pairs of sides.
3. Use the properties of similarity transformations to establish the AA criterion
for two triangles to be similar.
Prove theorems involving similarity.
4. Prove theorems about triangles. Theorems include: a line parallel to one side
of a triangle divides the other two proportionally, and conversely; the
Pythagorean Theorem proved using triangle similarity.
5. Use congruence and similarity criteria for triangles to solve problems and to
prove relationships in geometric figures.
Students will develop and evaluate
mathematical arguments and proofs.
G.RP.4 Provide correct mathematical
arguments in response to other
students’ conjectures, reasoning, and
arguments
National Essential Skills Study (NESS)
National Rankings
Rank
There is no New York Mathematics Performance Indicator–Common Core
alignment.
 2011 International Center for Leadership in Education
M10
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Mathematics – Page 72
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
G.RP.5 Present correct mathematical
arguments in a variety of forms
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Geometry: Congruence
Experiment with transformations in the plane.
2. Represent transformations in the plane using, e.g., transparencies and
geometry software; describe transformations as functions that take points in
the plane as inputs and give other points as outputs. Compare transformations
that preserve distance and angle to those that do not (e.g., translation versus
horizontal stretch).
3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the
rotations and reflections that carry it onto itself.
4. Develop definitions of rotations, reflections, and translations in terms of
angles, circles, perpendicular lines, parallel lines, and line segments.
5. Given a geometric figure and a rotation, reflection, or translation, draw the
transformed figure using, e.g., graph paper, tracing paper, or geometry
software. Specify a sequence of transformations that will carry a given figure
onto another.
Understand congruence in terms of rigid motions.
6. Use geometric descriptions of rigid motions to transform figures and to
predict the effect of a given rigid motion on a given figure; given two figures,
use the definition of congruence in terms of rigid motions to decide if they are
congruent.
7. Use the definition of congruence in terms of rigid motions to show that two
triangles are congruent if and only if corresponding pairs of sides and
corresponding pairs of angles are congruent.
8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow
from the definition of congruence in terms of rigid motions.
Prove geometric theorems.
9. Prove theorems about lines and angles. Theorems include: vertical angles are
congruent; when a transversal crosses parallel lines, alternate interior angles
are congruent and corresponding angles are congruent; points on a
perpendicular bisector of a line segment are exactly those equidistant from
the segment’s endpoints.
10. Prove theorems about triangles. Theorems include: measures of interior
angles of a triangle sum to 180°; base angles of isosceles triangles are
congruent; the segment joining midpoints of two sides of a triangle is parallel
to the third side and half the length; the medians of a triangle meet at a point.
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Mathematics – Page 73
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
Mathematics
Domains/Clusters/
Common Core State Standards
High School
G.RP.5 (Continued from previous page)
(Continued from previous page)
11. Prove theorems about parallelograms. Theorems include: opposite sides are
congruent, opposite angles are congruent, the diagonals of a parallelogram
bisect each other, and conversely, rectangles are parallelograms with
congruent diagonals.
Geometry: Similarity, Right Triangles, & Trigonometry
Understand similarity in terms of similarity transformations.
1. Verify experimentally the properties of dilations given by a center and a scale
factor:
a. A dilation takes a line not passing through the center of the dilation to a
parallel line, and leaves a line passing through the center unchanged.
b. The dilation of a line segment is longer or shorter in the ratio given by the
scale factor.
2. Given two figures, use the definition of similarity in terms of similarity
transformations to decide if they are similar; explain using similarity
transformations the meaning of similarity for triangles as the equality of all
corresponding pairs of angles and the proportionality of all corresponding
pairs of sides.
3. Use the properties of similarity transformations to establish the AA criterion
for two triangles to be similar.
Prove theorems involving similarity.
4. Prove theorems about triangles. Theorems include: a line parallel to one side
of a triangle divides the other two proportionally, and conversely; the
Pythagorean Theorem proved using triangle similarity.
5. Use congruence and similarity criteria for triangles to solve problems and to
prove relationships in geometric figures.
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
Mathematics – Page 74
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
Mathematics
Domains/Clusters/
Common Core State Standards
High School
G.RP.6 Evaluate written arguments for
validity
Geometry: Congruence
Understand congruence in terms of rigid motions.
8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow
from the definition of congruence in terms of rigid motions.
Prove geometric theorems.
9. Prove theorems about lines and angles. Theorems include: vertical angles are
congruent; when a transversal crosses parallel lines, alternate interior angles
are congruent and corresponding angles are congruent; points on a
perpendicular bisector of a line segment are exactly those equidistant from
the segment’s endpoints.
10. Prove theorems about triangles. Theorems include: measures of interior
angles of a triangle sum to 180°; base angles of isosceles triangles are
congruent; the segment joining midpoints of two sides of a triangle is parallel
to the third side and half the length; the medians of a triangle meet at a point.
11. Prove theorems about parallelograms. Theorems include: opposite sides are
congruent, opposite angles are congruent, the diagonals of a parallelogram
bisect each other, and conversely, rectangles are parallelograms with
congruent diagonals.
Geometry: Similarity, Right Triangles, & Trigonometry
Understand similarity in terms of similarity transformations.
2. Given two figures, use the definition of similarity in terms of similarity
transformations to decide if they are similar; explain using similarity
transformations the meaning of similarity for triangles as the equality of all
corresponding pairs of angles and the proportionality of all corresponding
pairs of sides.
3. Use the properties of similarity transformations to establish the AA criterion
for two triangles to be similar.
Prove theorems involving similarity.
4. Prove theorems about triangles. Theorems include: a line parallel to one side
of a triangle divides the other two proportionally, and conversely; the
Pythagorean Theorem proved using triangle similarity.
5. Use congruence and similarity criteria for triangles to solve problems and to
prove relationships in geometric figures.
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Mathematics – Page 75
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
Students will select and use various
types of reasoning and methods of
proof.
G.RP.7 Construct a proof using a
variety of methods (e.g., deductive,
analytic, transformational)
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Geometry: Congruence
Experiment with transformations in the plane.
2. Represent transformations in the plane using, e.g., transparencies and
geometry software; describe transformations as functions that take points in
the plane as inputs and give other points as outputs. Compare transformations
that preserve distance and angle to those that do not (e.g., translation versus
horizontal stretch).
3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the
rotations and reflections that carry it onto itself.
4. Develop definitions of rotations, reflections, and translations in terms of
angles, circles, perpendicular lines, parallel lines, and line segments.
5. Given a geometric figure and a rotation, reflection, or translation, draw the
transformed figure using, e.g., graph paper, tracing paper, or geometry
software. Specify a sequence of transformations that will carry a given figure
onto another.
Understand congruence in terms of rigid motions.
6. Use geometric descriptions of rigid motions to transform figures and to
predict the effect of a given rigid motion on a given figure; given two figures,
use the definition of congruence in terms of rigid motions to decide if they are
congruent.
7. Use the definition of congruence in terms of rigid motions to show that two
triangles are congruent if and only if corresponding pairs of sides and
corresponding pairs of angles are congruent.
8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow
from the definition of congruence in terms of rigid motions.
Prove geometric theorems.
9. Prove theorems about lines and angles. Theorems include: vertical angles are
congruent; when a transversal crosses parallel lines, alternate interior angles
are congruent and corresponding angles are congruent; points on a
perpendicular bisector of a line segment are exactly those equidistant from
the segment’s endpoints.
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Mathematics – Page 76
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
Mathematics
Domains/Clusters/
Common Core State Standards
High School
G.RP.7 (Continued from previous page)
(Continued from previous page)
10. Prove theorems about triangles. Theorems include: measures of interior
angles of a triangle sum to 180°; base angles of isosceles triangles are
congruent; the segment joining midpoints of two sides of a triangle is parallel
to the third side and half the length; the medians of a triangle meet at a point.
11. Prove theorems about parallelograms. Theorems include: opposite sides are
congruent, opposite angles are congruent, the diagonals of a parallelogram
bisect each other, and conversely, rectangles are parallelograms with
congruent diagonals.
Geometry: Similarity, Right Triangles, & Trigonometry
Understand similarity in terms of similarity transformations.
1. Verify experimentally the properties of dilations given by a center and a scale
factor:
a. A dilation takes a line not passing through the center of the dilation to a
parallel line, and leaves a line passing through the center unchanged.
b. The dilation of a line segment is longer or shorter in the ratio given by the
scale factor.
2. Given two figures, use the definition of similarity in terms of similarity
transformations to decide if they are similar; explain using similarity
transformations the meaning of similarity for triangles as the equality of all
corresponding pairs of angles and the proportionality of all corresponding
pairs of sides.
3. Use the properties of similarity transformations to establish the AA criterion
for two triangles to be similar.
Prove theorems involving similarity.
4. Prove theorems about triangles. Theorems include: a line parallel to one side
of a triangle divides the other two proportionally, and conversely; the
Pythagorean Theorem proved using triangle similarity.
5. Use congruence and similarity criteria for triangles to solve problems and to
prove relationships in geometric figures.
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
Mathematics – Page 77
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
G.RP.8 Devise ways to verify results
or use counterexamples to refute
incorrect statements
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Geometry: Congruence
Experiment with transformations in the plane.
2. Represent transformations in the plane using, e.g., transparencies and
geometry software; describe transformations as functions that take points in
the plane as inputs and give other points as outputs. Compare transformations
that preserve distance and angle to those that do not (e.g., translation versus
horizontal stretch).
3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the
rotations and reflections that carry it onto itself.
4. Develop definitions of rotations, reflections, and translations in terms of
angles, circles, perpendicular lines, parallel lines, and line segments.
5. Given a geometric figure and a rotation, reflection, or translation, draw the
transformed figure using, e.g., graph paper, tracing paper, or geometry
software. Specify a sequence of transformations that will carry a given figure
onto another.
Understand congruence in terms of rigid motions.
6. Use geometric descriptions of rigid motions to transform figures and to
predict the effect of a given rigid motion on a given figure; given two figures,
use the definition of congruence in terms of rigid motions to decide if they are
congruent.
7. Use the definition of congruence in terms of rigid motions to show that two
triangles are congruent if and only if corresponding pairs of sides and
corresponding pairs of angles are congruent.
8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow
from the definition of congruence in terms of rigid motions.
Prove geometric theorems.
9. Prove theorems about lines and angles. Theorems include: vertical angles are
congruent; when a transversal crosses parallel lines, alternate interior angles
are congruent and corresponding angles are congruent; points on a
perpendicular bisector of a line segment are exactly those equidistant from
the segment’s endpoints.
10. Prove theorems about triangles. Theorems include: measures of interior
angles of a triangle sum to 180°; base angles of isosceles triangles are
congruent; the segment joining midpoints of two sides of a triangle is parallel
to the third side and half the length; the medians of a triangle meet at a point.
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Mathematics – Page 78
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
Mathematics
Domains/Clusters/
Common Core State Standards
High School
G.RP.8 (Continued from previous page)
(Continued from previous page)
11. Prove theorems about parallelograms. Theorems include: opposite sides are
congruent, opposite angles are congruent, the diagonals of a parallelogram
bisect each other, and conversely, rectangles are parallelograms with
congruent diagonals.
Geometry: Similarity, Right Triangles, & Trigonometry
Understand similarity in terms of similarity transformations.
1. Verify experimentally the properties of dilations given by a center and a scale
factor:
a. A dilation takes a line not passing through the center of the dilation to a
parallel line, and leaves a line passing through the center unchanged.
b. The dilation of a line segment is longer or shorter in the ratio given by the
scale factor.
2. Given two figures, use the definition of similarity in terms of similarity
transformations to decide if they are similar; explain using similarity
transformations the meaning of similarity for triangles as the equality of all
corresponding pairs of angles and the proportionality of all corresponding
pairs of sides.
3. Use the properties of similarity transformations to establish the AA criterion
for two triangles to be similar.
Prove theorems involving similarity.
4. Prove theorems about triangles. Theorems include: a line parallel to one side
of a triangle divides the other two proportionally, and conversely; the
Pythagorean Theorem proved using triangle similarity.
5. Use congruence and similarity criteria for triangles to solve problems and
to prove relationships in geometric figures.
There is no New York Mathematics Performance Indicator–Common Core
alignment.
G.RP.9 Apply inductive reasoning in
making and supporting mathematical
conjectures
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
M10
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Mathematics – Page 79
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
Mathematics
Domains/Clusters/
Common Core State Standards
High School
National Essential Skills Study (NESS)
National Rankings
Rank
Communication Strand
Students will organize and consolidate
their mathematical thinking through
communication.
G.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
Geometry: Congruence
Experiment with transformations in the plane.
2. Represent transformations in the plane using, e.g., transparencies and
geometry software; describe transformations as functions that take points in
the plane as inputs and give other points as outputs. Compare transformations
that preserve distance and angle to those that do not (e.g., translation versus
horizontal stretch).
3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the
rotations and reflections that carry it onto itself.
4. Develop definitions of rotations, reflections, and translations in terms of
angles, circles, perpendicular lines, parallel lines, and line segments.
5. Given a geometric figure and a rotation, reflection, or translation, draw the
transformed figure using, e.g., graph paper, tracing paper, or geometry
software. Specify a sequence of transformations that will carry a given figure
onto another.
Understand congruence in terms of rigid motions.
6. Use geometric descriptions of rigid motions to transform figures and to
predict the effect of a given rigid motion on a given figure; given two figures,
use the definition of congruence in terms of rigid motions to decide if they are
congruent.
7. Use the definition of congruence in terms of rigid motions to show that two
triangles are congruent if and only if corresponding pairs of sides and
corresponding pairs of angles are congruent.
8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow
from the definition of congruence in terms of rigid motions.
Prove geometric theorems.
9. Prove theorems about lines and angles. Theorems include: vertical angles are
congruent; when a transversal crosses parallel lines, alternate interior angles
are congruent and corresponding angles are congruent; points on a
perpendicular bisector of a line segment are exactly those equidistant from
the segment’s endpoints.
 2011 International Center for Leadership in Education
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Mathematics – Page 80
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
G.CM.1 (Continued from previous
page)
Mathematics
Domains/Clusters/
Common Core State Standards
High School
National Essential Skills Study (NESS)
National Rankings
Rank
(Continued from previous page)
10. Prove theorems about triangles. Theorems include: measures of interior
angles of a triangle sum to 180°; base angles of isosceles triangles are
congruent; the segment joining midpoints of two sides of a triangle is parallel
to the third side and half the length; the medians of a triangle meet at a point.
11. Prove theorems about parallelograms. Theorems include: opposite sides are
congruent, opposite angles are congruent, the diagonals of a parallelogram
bisect each other, and conversely, rectangles are parallelograms with
congruent diagonals.
Geometry: Similarity, Right Triangles, & Trigonometry
Understand similarity in terms of similarity transformations.
1. Verify experimentally the properties of dilations given by a center and a scale
factor:
a. A dilation takes a line not passing through the center of the dilation to a
parallel line, and leaves a line passing through the center unchanged.
b. The dilation of a line segment is longer or shorter in the ratio given by the
scale factor.
2. Given two figures, use the definition of similarity in terms of similarity
transformations to decide if they are similar; explain using similarity
transformations the meaning of similarity for triangles as the equality of all
corresponding pairs of angles and the proportionality of all corresponding
pairs of sides.
3. Use the properties of similarity transformations to establish the AA criterion
for two triangles to be similar.
Prove theorems involving similarity.
4. Prove theorems about triangles. Theorems include: a line parallel to one side
of a triangle divides the other two proportionally, and conversely; the
Pythagorean Theorem proved using triangle similarity.
5. Use congruence and similarity criteria for triangles to solve problems and to
prove relationships in geometric figures.
 2011 International Center for Leadership in Education
Mathematics – Page 81
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
Mathematics
Domains/Clusters/
Common Core State Standards
High School
G.CM.2 Use mathematical
representations to communicate with
appropriate accuracy, including
numerical tables, formulas, functions,
equations, charts, graphs, and diagrams
Number & Quantity: Quantities
Reason quantitatively and use units to solve problems.
1. Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and
interpret the scale and the origin in graphs and data displays.
2. Define appropriate quantities for the purpose of descriptive modeling.
3. Choose a level of accuracy appropriate to limitations on measurement when
reporting quantities.
Geometry: Expressing Geometric Properties with Equations
Translate between the geometric description and the equation for a conic
section.
1. Derive the equation of a circle of given center and radius using the
Pythagorean Theorem; complete the square to find the center and radius of a
circle given by an equation.
Use coordinates to prove simple geometric theorems algebraically.
4. Use coordinates to prove simple geometric theorems algebraically. For
example, prove or disprove that a figure defined by four given points in the
coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies
on the circle centered at the origin and containing the point (0, 2).
5. Prove the slope criteria for parallel and perpendicular lines and use them to
solve geometric problems (e.g., find the equation of a line parallel or
perpendicular to a given line that passes through a given point).
6. Find the point on a directed line segment between two given points that
partitions the segment in a given ratio.
7. Use coordinates to compute perimeters of polygons and areas of triangles and
rectangles, e.g., using the distance formula.
Geometry: Geometric Measurement & Dimensions
Explain volume formulas and use them to solve problems.
1. Give an informal argument for the formulas for the circumference of a circle,
area of a circle, volume of a cylinder, pyramid, and cone. Use dissection
arguments, Cavalieri’s principle, and informal limit arguments.
2. (+) Give an informal argument using Cavalieri’s principle for the formulas
for the volume of a sphere and other solid figures.
3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve
problems.
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
M10
M11
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Apply variables in expressions and
equations to solve problems (i.e., write
mathematical equations for given situation,
create a mathematical model to understand
the relationships between variables, or make
connections between the structures of
mathematically abstract concepts and the
real world).
Evaluate and employ accurate and
appropriate procedures for statistical data
collection, organization, analysis, and
display including making estimates and
predictions, critiquing data, and drawing
inferences (e.g., using the normal curve and
z-scores, line of best fit).
M21
Mathematics – Page 82
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
Students will communicate their
mathematical thinking coherently and
clearly to peers, teachers, and others.
G.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
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Geometry: Congruence
Experiment with transformations in the plane.
2. Represent transformations in the plane using, e.g., transparencies and
geometry software; describe transformations as functions that take points in
the plane as inputs and give other points as outputs. Compare transformations
that preserve distance and angle to those that do not (e.g., translation versus
horizontal stretch).
3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the
rotations and reflections that carry it onto itself.
4. Develop definitions of rotations, reflections, and translations in terms of
angles, circles, perpendicular lines, parallel lines, and line segments.
5. Given a geometric figure and a rotation, reflection, or translation, draw the
transformed figure using, e.g., graph paper, tracing paper, or geometry
software. Specify a sequence of transformations that will carry a given figure
onto another.
Understand congruence in terms of rigid motions
6. Use geometric descriptions of rigid motions to transform figures and to
predict the effect of a given rigid motion on a given figure; given two figures,
use the definition of congruence in terms of rigid motions to decide if they are
congruent.
7. Use the definition of congruence in terms of rigid motions to show that two
triangles are congruent if and only if corresponding pairs of sides and
corresponding pairs of angles are congruent.
8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow
from the definition of congruence in terms of rigid motions.
Prove geometric theorems
9. Prove theorems about lines and angles. Theorems include: vertical angles are
congruent; when a transversal crosses parallel lines, alternate interior angles
are congruent and corresponding angles are congruent; points on a
perpendicular bisector of a line segment are exactly those equidistant from
the segment’s endpoints.
10. Prove theorems about triangles. Theorems include: measures of interior
angles of a triangle sum to 180°; base angles of isosceles triangles are
congruent; the segment joining midpoints of two sides of a triangle is parallel
to the third side and half the length; the medians of a triangle meet at a point.
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Mathematics – Page 83
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
G.CM.3 (Continued from previous
page)
Mathematics
Domains/Clusters/
Common Core State Standards
High School
National Essential Skills Study (NESS)
National Rankings
Rank
(Continued from previous page)
11. Prove theorems about parallelograms. Theorems include: opposite sides are
congruent, opposite angles are congruent, the diagonals of a parallelogram
bisect each other, and conversely, rectangles are parallelograms with
congruent diagonals.
Geometry: Similarity, Right Triangles, & Trigonometry
Understand similarity in terms of similarity transformations.
1. Verify experimentally the properties of dilations given by a center and a scale
factor:
a. A dilation takes a line not passing through the center of the dilation to a
parallel line, and leaves a line passing through the center unchanged.
b. The dilation of a line segment is longer or shorter in the ratio given by the
scale factor.
2. Given two figures, use the definition of similarity in terms of similarity
transformations to decide if they are similar; explain using similarity
transformations the meaning of similarity for triangles as the equality of all
corresponding pairs of angles and the proportionality of all corresponding
pairs of sides.
3. Use the properties of similarity transformations to establish the AA criterion
for two triangles to be similar.
Prove theorems involving similarity
4. Prove theorems about triangles. Theorems include: a line parallel to one side
of a triangle divides the other two proportionally, and conversely; the
Pythagorean Theorem proved using triangle similarity.
5. Use congruence and similarity criteria for triangles to solve problems and to
prove relationships in geometric figures.
 2011 International Center for Leadership in Education
Mathematics – Page 84
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
G.CM.4 Explain relationships among
different representations of a problem
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Geometry: Congruence
Experiment with transformations in the plane.
2. Represent transformations in the plane using, e.g., transparencies and
geometry software; describe transformations as functions that take points in
the plane as inputs and give other points as outputs. Compare transformations
that preserve distance and angle to those that do not (e.g., translation versus
horizontal stretch).
3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the
rotations and reflections that carry it onto itself.
4. Develop definitions of rotations, reflections, and translations in terms of
angles, circles, perpendicular lines, parallel lines, and line segments.
5. Given a geometric figure and a rotation, reflection, or translation, draw the
transformed figure using, e.g., graph paper, tracing paper, or geometry
software. Specify a sequence of transformations that will carry a given figure
onto another.
Understand congruence in terms of rigid motions
6. Use geometric descriptions of rigid motions to transform figures and to
predict the effect of a given rigid motion on a given figure; given two figures,
use the definition of congruence in terms of rigid motions to decide if they are
congruent.
7. Use the definition of congruence in terms of rigid motions to show that two
triangles are congruent if and only if corresponding pairs of sides and
corresponding pairs of angles are congruent.
8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow
from the definition of congruence in terms of rigid motions.
Geometry: Similarity, Right Triangles, & Trigonometry
Understand similarity in terms of similarity transformations.
1. Verify experimentally the properties of dilations given by a center and a scale
factor:
a. A dilation takes a line not passing through the center of the dilation to a
parallel line, and leaves a line passing through the center unchanged.
b. The dilation of a line segment is longer or shorter in the ratio given by the
scale factor.
2. Given two figures, use the definition of similarity in terms of similarity
transformations to decide if they are similar; explain using similarity
transformations the meaning of similarity for triangles as the equality of all
corresponding pairs of angles and the proportionality of all corresponding
pairs of sides.
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Mathematics – Page 85
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
G.CM.4 (Continued from previous
page)
G.CM.5 Communicate logical
arguments clearly, showing why a
result makes sense and why the
reasoning is valid
Mathematics
Domains/Clusters/
Common Core State Standards
High School
(Continued from previous page)
3. Use the properties of similarity transformations to establish the AA criterion
for two triangles to be similar.
Geometry: Congruence
Experiment with transformations in the plane.
2. Represent transformations in the plane using, e.g., transparencies and
geometry software; describe transformations as functions that take points in
the plane as inputs and give other points as outputs. Compare transformations
that preserve distance and angle to those that do not (e.g., translation versus
horizontal stretch).
3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the
rotations and reflections that carry it onto itself.
4. Develop definitions of rotations, reflections, and translations in terms of
angles, circles, perpendicular lines, parallel lines, and line segments.
5. Given a geometric figure and a rotation, reflection, or translation, draw the
transformed figure using, e.g., graph paper, tracing paper, or geometry
software. Specify a sequence of transformations that will carry a given figure
onto another.
Understand congruence in terms of rigid motions
6. Use geometric descriptions of rigid motions to transform figures and to
predict the effect of a given rigid motion on a given figure; given two figures,
use the definition of congruence in terms of rigid motions to decide if they are
congruent.
7. Use the definition of congruence in terms of rigid motions to show that two
triangles are congruent if and only if corresponding pairs of sides and
corresponding pairs of angles are congruent.
8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow
from the definition of congruence in terms of rigid motions.
Prove geometric theorems
9. Prove theorems about lines and angles. Theorems include: vertical angles are
congruent; when a transversal crosses parallel lines, alternate interior angles
are congruent and corresponding angles are congruent; points on a
perpendicular bisector of a line segment are exactly those equidistant from
the segment’s endpoints.
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Mathematics – Page 86
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
G.CM.5 (Continued from previous
page)
G.CM.6 Support or reject arguments or
questions raised by others about the
correctness of mathematical work
Mathematics
Domains/Clusters/
Common Core State Standards
High School
National Essential Skills Study (NESS)
National Rankings
Rank
(Continued from previous page)
10. Prove theorems about triangles. Theorems include: measures of interior
angles of a triangle sum to 180°; base angles of isosceles triangles are
congruent; the segment joining midpoints of two sides of a triangle is parallel
to the third side and half the length; the medians of a triangle meet at a point.
11. Prove theorems about parallelograms. Theorems include: opposite sides are
congruent, opposite angles are congruent, the diagonals of a parallelogram
bisect each other, and conversely, rectangles are parallelograms with
congruent diagonals.
Geometry: Similarity, Right Triangles, & Trigonometry
Understand similarity in terms of similarity transformations.
1. Verify experimentally the properties of dilations given by a center and a scale
factor:
a. A dilation takes a line not passing through the center of the dilation to a
parallel line, and leaves a line passing through the center unchanged.
b. The dilation of a line segment is longer or shorter in the ratio given by the
scale factor.
2. Given two figures, use the definition of similarity in terms of similarity
transformations to decide if they are similar; explain using similarity
transformations the meaning of similarity for triangles as the equality of all
corresponding pairs of angles and the proportionality of all corresponding
pairs of sides.
3. Use the properties of similarity transformations to establish the AA criterion
for two triangles to be similar.
Prove theorems involving similarity
4. Prove theorems about triangles. Theorems include: a line parallel to one side
of a triangle divides the other two proportionally, and conversely; the
Pythagorean Theorem proved using triangle similarity.
5. Use congruence and similarity criteria for triangles to solve problems and to
prove relationships in geometric figures.
There is no New York Mathematics Performance Indicator–Common Core
alignment.
 2011 International Center for Leadership in Education
M10
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Mathematics – Page 87
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
Students will analyze and evaluate the
mathematical thinking and strategies of
others
G.CM.7 Read and listen for logical
understanding of mathematical thinking
shared by other students
G.CM.8 Reflect on strategies of others
in relation to one’s own strategy
Mathematics
Domains/Clusters/
Common Core State Standards
High School
National Essential Skills Study (NESS)
National Rankings
Rank
There is no New York Mathematics Performance Indicator–Common Core
alignment.
M10
There is no New York Mathematics Performance Indicator–Common Core
alignment.
M10
G.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
G.CM.10 Use correct mathematical
language in developing mathematical
questions that elicit, extend, or
challenge other students’ conjectures
There is no New York Mathematics Performance Indicator–Common Core
alignment.
M10
There is no New York Mathematics Performance Indicator–Common Core
alignment.
 2011 International Center for Leadership in Education
M10
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Mathematics – Page 88
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
G.CM.11 Understand and use
appropriate language, representations,
and terminology when describing
objects, relationships, mathematical
solutions, and geometric diagrams
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Geometry: Congruence
Experiment with transformations in the plane.
1. Know precise definitions of angle, circle, perpendicular line, parallel line, and
line segment, based on the undefined notions of point, line, distance along a
line, and distance around a circular arc.
2. Represent transformations in the plane using, e.g., transparencies and
geometry software; describe transformations as functions that take points in
the plane as inputs and give other points as outputs. Compare transformations
that preserve distance and angle to those that do not (e.g., translation versus
horizontal stretch).
3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the
rotations and reflections that carry it onto itself.
4. Develop definitions of rotations, reflections, and translations in terms of
angles, circles, perpendicular lines, parallel lines, and line segments.
5. Given a geometric figure and a rotation, reflection, or translation, draw the
transformed figure using, e.g., graph paper, tracing paper, or geometry
software. Specify a sequence of transformations that will carry a given figure
onto another.
Understand congruence in terms of rigid motions.
6. Use geometric descriptions of rigid motions to transform figures and to
predict the effect of a given rigid motion on a given figure; given two figures,
use the definition of congruence in terms of rigid motions to decide if they are
congruent.
7. Use the definition of congruence in terms of rigid motions to show that two
triangles are congruent if and only if corresponding pairs of sides and
corresponding pairs of angles are congruent.
8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow
from the definition of congruence in terms of rigid motions.
Geometry: Similarity, Right Triangles, & Trigonometry
Understand similarity in terms of similarity transformations.
1. Verify experimentally the properties of dilations given by a center and a scale
factor:
a. A dilation takes a line not passing through the center of the dilation to a
parallel line, and leaves a line passing through the center unchanged.
b. The dilation of a line segment is longer or shorter in the ratio given by the
scale factor.
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Mathematics – Page 89
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
G.CM.11 (Continued from previous
page)
Mathematics
Domains/Clusters/
Common Core State Standards
High School
National Essential Skills Study (NESS)
National Rankings
Rank
(Continued from previous page)
2. Given two figures, use the definition of similarity in terms of similarity
transformations to decide if they are similar; explain using similarity
transformations the meaning of similarity for triangles as the equality of all
corresponding pairs of angles and the proportionality of all corresponding
pairs of sides.
3. Use the properties of similarity transformations to establish the AA criterion
for two triangles to be similar.
Define trigonometric ratios and solve problems involving right triangles.
6. Understand that by similarity, side ratios in right triangles are properties of
the angles in the triangle, leading to definitions of trigonometric ratios for
acute angles.
7. Explain and use the relationship between the sine and cosine of
complementary angles.
8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles
in applied problems.
Geometry: Expressing Geometric Properties with Equations
Translate between the geometric description and the equation for a conic
section.
1. Derive the equation of a circle of given center and radius using the
Pythagorean Theorem; complete the square to find the center and radius of a
circle given by an equation.
Use coordinates to prove simple geometric theorems algebraically.
4. Use coordinates to prove simple geometric theorems algebraically. For
example, prove or disprove that a figure defined by four given points in the
coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies
on the circle centered at the origin and containing the point (0, 2).
5. Prove the slope criteria for parallel and perpendicular lines and use them to
solve geometric problems (e.g., find the equation of a line parallel or
perpendicular to a given line that passes through a given point).
6. Find the point on a directed line segment between two given points that
partitions the segment in a given ratio.
7. Use coordinates to compute perimeters of polygons and areas of triangles and
rectangles, e.g., using the distance formula.
 2011 International Center for Leadership in Education
Mathematics – Page 90
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
G.CM.12 Draw conclusions about
mathematical ideas through decoding,
comprehension, and interpretation of
mathematical visuals, symbols, and
technical writing
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Geometry: Expressing Geometric Properties with Equations
Translate between the geometric description and the equation for a conic
section.
1. Derive the equation of a circle of given center and radius using the
Pythagorean Theorem; complete the square to find the center and radius of a
circle given by an equation.
Use coordinates to prove simple geometric theorems algebraically.
4. Use coordinates to prove simple geometric theorems algebraically. For
example, prove or disprove that a figure defined by four given points in the
coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies
on the circle centered at the origin and containing the point (0, 2).
5. Prove the slope criteria for parallel and perpendicular lines and use them to
solve geometric problems (e.g., find the equation of a line parallel or
perpendicular to a given line that passes through a given point).
6. Find the point on a directed line segment between two given points that
partitions the segment in a given ratio.
7. Use coordinates to compute perimeters of polygons and areas of triangles and
rectangles, e.g., using the distance formula.
National Essential Skills Study (NESS)
National Rankings
Rank
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Connections Strand
Students will recognize and use
connections among mathematical ideas.
G.CN.1 Understand and make
connections among multiple
representations of the same
mathematical idea
Geometry: Congruence
Experiment with transformations in the plane.
2. Represent transformations in the plane using, e.g., transparencies and
geometry software; describe transformations as functions that take points in
the plane as inputs and give other points as outputs. Compare transformations
that preserve distance and angle to those that do not (e.g., translation versus
horizontal stretch).
3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the
rotations and reflections that carry it onto itself.
4. Develop definitions of rotations, reflections, and translations in terms of
angles, circles, perpendicular lines, parallel lines, and line segments.
5. Given a geometric figure and a rotation, reflection, or translation, draw the
transformed figure using, e.g., graph paper, tracing paper, or geometry
software. Specify a sequence of transformations that will carry a given figure
onto another.
 2011 International Center for Leadership in Education
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Mathematics – Page 91
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
Mathematics
Domains/Clusters/
Common Core State Standards
High School
G.CN.1 (Continued from previouspage)
(Continued from previous page)
Understand congruence in terms of rigid motions.
6. Use geometric descriptions of rigid motions to transform figures and to
predict the effect of a given rigid motion on a given figure; given two figures,
use the definition of congruence in terms of rigid motions to decide if they are
congruent.
7. Use the definition of congruence in terms of rigid motions to show that two
triangles are congruent if and only if corresponding pairs of sides and
corresponding pairs of angles are congruent.
8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow
from the definition of congruence in terms of rigid motions.
Geometry: Circles
Understand and apply theorems about circles.
1. Prove that all circles are similar.
Find arc length and areas of sectors of circles.
5. Derive using similarity the fact that the length of the arc intercepted by an
angle is proportional to the radius, and define the radian measure of the angle
as the constant of proportionality; derive the formula for the area of a sector.
Geometry: Expressing Geometric Properties with Equations
Translate between the geometric description and the equation for a conic
section.
1. Derive the equation of a circle of given center and radius using the
Pythagorean Theorem; complete the square to find the center and radius of a
circle given by an equation.
Use coordinates to prove simple geometric theorems algebraically.
5. Prove the slope criteria for parallel and perpendicular lines and use them to
solve geometric problems (e.g., find the equation of a line parallel or
perpendicular to a given line that passes through a given point).
6. Find the point on a directed line segment between two given points that
partitions the segment in a given ratio.
7. Use coordinates to compute perimeters of polygons and areas of triangles and
rectangles, e.g., using the distance formula.
Geometry: Geometric Measurement & Dimensions
Explain volume formulas and use them to solve problems.
1. Give an informal argument for the formulas for the circumference of a circle,
area of a circle, volume of a cylinder, pyramid, and cone. Use dissection
arguments, Cavalieri’s principle, and informal limit arguments.
3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve
problems.
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
Mathematics – Page 92
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
G.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.
G.CN.3 Model situations
mathematically, using representations
to draw conclusions and formulate new
situations
G.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
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Geometry: Expressing Geometric Properties with Equations
Translate between the geometric description and the equation for a conic
section.
1. Derive the equation of a circle of given center and radius using the
Pythagorean Theorem; complete the square to find the center and radius of a
circle given by an equation.
Use coordinates to prove simple geometric theorems algebraically.
4. Use coordinates to prove simple geometric theorems algebraically. For
example, prove or disprove that a figure defined by four given points in the
coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies
on the circle centered at the origin and containing the point (0, 2).
Geometry: Modeling with Geometry
Apply Geometric Concepts in modeling situations
1. Use geometric shapes, their measures, and their properties to describe objects
(e.g., modeling a tree trunk or a human torso as a cylinder).
2. Apply concepts of density based on area and volume in modeling situations
(e.g., persons per square mile, BTUs per cubic foot).
Geometry: Expressing Geometric Properties with Equations
Translate between the geometric description and the equation for a conic
section.
1. Derive the equation of a circle of given center and radius using the
Pythagorean Theorem; complete the square to find the center and radius of a
circle given by an equation.
Use coordinates to prove simple geometric theorems algebraically.
4. Use coordinates to prove simple geometric theorems algebraically. For
example, prove or disprove that a figure defined by four given points in the
coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies
on the circle centered at the origin and containing the point (0, 2).
5. Prove the slope criteria for parallel and perpendicular lines and use them to
solve geometric problems (e.g., find the equation of a line parallel or
perpendicular to a given line that passes through a given point).
6. Find the point on a directed line segment between two given points that
partitions the segment in a given ratio.
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
M10
M10
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Mathematics – Page 93
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
G.CN.4 (Continued from previous
page)
G.CN.5 Understand how quantitative
models connect to various physical
models and representations
Students will recognize and apply
mathematics in contexts outside of
mathematics.
G.CN.6 Recognize and apply
mathematics to situations in the outside
world
Mathematics
Domains/Clusters/
Common Core State Standards
High School
(Continued from previous page)
7. Use coordinates to compute perimeters of polygons and areas of triangles and
rectangles, e.g., using the distance formula.
Geometry: Similarity, Right Triangles, & Trigonometry
Understand similarity in terms of similarity transformations.
1. Verify experimentally the properties of dilations given by a center and a scale
factor:
a. A dilation takes a line not passing through the center of the dilation to a
parallel line, and leaves a line passing through the center unchanged.
b. The dilation of a line segment is longer or shorter in the ratio given by the
scale factor.
Define trigonometric ratios and solve problems involving right triangles.
6. Understand that by similarity, side ratios in right triangles are properties of
the angles in the triangle, leading to definitions of trigonometric ratios for
acute angles.
7. Explain and use the relationship between the sine and cosine of
complementary angles.
8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles
in applied problems.
Geometry: Modeling with Geometry
Apply Geometric Concepts in modeling situations
1. Use geometric shapes, their measures, and their properties to describe objects
(e.g., modeling a tree trunk or a human torso as a cylinder).
2. Apply concepts of density based on area and volume in modeling situations
(e.g., persons per square mile, BTUs per cubic foot).
Geometry: Modeling with Geometry
Apply Geometric Concepts in modeling situations
1. Use geometric shapes, their measures, and their properties to describe objects
(e.g., modeling a tree trunk or a human torso as a cylinder).
2. Apply concepts of density based on area and volume in modeling situations
(e.g., persons per square mile, BTUs per cubic foot).
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
M10
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Mathematics – Page 94
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
G.CN.7 Recognize and apply
mathematical ideas to problem
situations that develop outside of
mathematics
G.CN.8 Develop an appreciation for
the historical development of
mathematics
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Geometry: Modeling with Geometry
Apply Geometric Concepts in modeling situations
1. Use geometric shapes, their measures, and their properties to describe objects
(e.g., modeling a tree trunk or a human torso as a cylinder).
2. Apply concepts of density based on area and volume in modeling situations
(e.g., persons per square mile, BTUs per cubic foot).
National Essential Skills Study (NESS)
National Rankings
Rank
M10
There is no New York Mathematics Performance Indicator–Common Core
alignment.
M10
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Representation Strand
Students will create and use
representations to organize, record,
and communicate mathematical ideas.
G.R.1 Use physical objects, diagrams,
charts, tables, graphs, symbols,
equations, or objects created using
technology as representations of
mathematical concepts
Geometry: Congruence
Experiment with transformations in the plane.
2. Represent transformations in the plane using, e.g., transparencies and
geometry software; describe transformations as functions that take points in
the plane as inputs and give other points as outputs. Compare transformations
that preserve distance and angle to those that do not (e.g., translation versus
horizontal stretch).
3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the
rotations and reflections that carry it onto itself.
4. Develop definitions of rotations, reflections, and translations in terms of
angles, circles, perpendicular lines, parallel lines, and line segments.
5. Given a geometric figure and a rotation, reflection, or translation, draw the
transformed figure using, e.g., graph paper, tracing paper, or geometry
software. Specify a sequence of transformations that will carry a given figure
onto another.
 2011 International Center for Leadership in Education
M7
M10
M21
Simplify and solve algebraic equations by
identifying and using the correct order of
operations and techniques necessary to carry
out the solution.
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Evaluate and employ accurate and
appropriate procedures for statistical data
collection, organization, analysis, and
display including making estimates and
predictions, critiquing data, and drawing
inferences (e.g., using the normal curve and
z-scores, line of best fit).
Mathematics – Page 95
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
Mathematics
Domains/Clusters/
Common Core State Standards
High School
G.R.1 (Continued from previous page)
(Continued from previous page)
Understand congruence in terms of rigid motions.
8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow
from the definition of congruence in terms of rigid motions.
Make geometric constructions.
12. Make formal geometric constructions with a variety of tools and methods
(compass and straightedge, string, reflective devices, paper folding, dynamic
geometric software, etc.). Copying a segment; copying an angle; bisecting a
segment; bisecting an angle; constructing perpendicular lines, including the
perpendicular bisector of a line segment; and constructing a line parallel to a
given line through a point not on the line.
13. Construct an equilateral triangle, a square, and a regular hexagon inscribed
in a circle.
Geometry: Similarity, Right Triangles, & Trigonometry
Understand similarity in terms of similarity transformations.
1. Verify experimentally the properties of dilations given by a center and a scale
factor:
a. A dilation takes a line not passing through the center of the dilation to a
parallel line, and leaves a line passing through the center unchanged.
b. The dilation of a line segment is longer or shorter in the ratio given by the
scale factor.
3. Use the properties of similarity transformations to establish the AA criterion
for two triangles to be similar.
Geometry: Expressing Geometric Properties with Equations
Translate between the geometric description and the equation for a conic
section.
1. Derive the equation of a circle of given center and radius using the
Pythagorean Theorem; complete the square to find the center and radius of a
circle given by an equation.
Use coordinates to prove simple geometric theorems algebraically.
5. Prove the slope criteria for parallel and perpendicular lines and use them to
solve geometric problems (e.g., find the equation of a line parallel or
perpendicular to a given line that passes through a given point).
6. Find the point on a directed line segment between two given points that
partitions the segment in a given ratio.
7. Use coordinates to compute perimeters of polygons and areas of triangles and
rectangles, e.g., using the distance formula.
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
Mathematics – Page 96
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
Mathematics
Domains/Clusters/
Common Core State Standards
High School
G.R.2 Recognize, compare, and use an
array of representational forms
Geometry: Congruence
Experiment with transformations in the plane.
2. Represent transformations in the plane using, e.g., transparencies and
geometry software; describe transformations as functions that take points in
the plane as inputs and give other points as outputs. Compare transformations
that preserve distance and angle to those that do not (e.g., translation versus
horizontal stretch).
3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the
rotations and reflections that carry it onto itself.
4. Develop definitions of rotations, reflections, and translations in terms of
angles, circles, perpendicular lines, parallel lines, and line segments.
5. Given a geometric figure and a rotation, reflection, or translation, draw the
transformed figure using, e.g., graph paper, tracing paper, or geometry
software. Specify a sequence of transformations that will carry a given figure
onto another.
Understand congruence in terms of rigid motions.
8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow
from the definition of congruence in terms of rigid motions.
Make geometric constructions.
12. Make formal geometric constructions with a variety of tools and methods
(compass and straightedge, string, reflective devices, paper folding, dynamic
geometric software, etc.). Copying a segment; copying an angle; bisecting a
segment; bisecting an angle; constructing perpendicular lines, including the
perpendicular bisector of a line segment; and constructing a line parallel to a
given line through a point not on the line.
13. Construct an equilateral triangle, a square, and a regular hexagon inscribed
in a circle.
Geometry: Similarity, Right Triangles, & Trigonometry
Understand similarity in terms of similarity transformations.
1. Verify experimentally the properties of dilations given by a center and a scale
factor:
a. A dilation takes a line not passing through the center of the dilation to a
parallel line, and leaves a line passing through the center unchanged.
b. The dilation of a line segment is longer or shorter in the ratio given by the
scale factor.
3. Use the properties of similarity transformations to establish the AA criterion
for two triangles to be similar.
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Mathematics – Page 97
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
Mathematics
Domains/Clusters/
Common Core State Standards
High School
G.R.2 (Continued from previous page)
(Continued from previous page)
Geometry: Expressing Geometric Properties with Equations
Translate between the geometric description and the equation for a conic
section.
1. Derive the equation of a circle of given center and radius using the
Pythagorean Theorem; complete the square to find the center and radius of a
circle given by an equation.
Use coordinates to prove simple geometric theorems algebraically.
5. Prove the slope criteria for parallel and perpendicular lines and use them to
solve geometric problems (e.g., find the equation of a line parallel or
perpendicular to a given line that passes through a given point).
6. Find the point on a directed line segment between two given points that
partitions the segment in a given ratio.
7. Use coordinates to compute perimeters of polygons and areas of triangles and
rectangles, e.g., using the distance formula.
Geometry: Congruence
Experiment with transformations in the plane.
2. Represent transformations in the plane using, e.g., transparencies and
geometry software; describe transformations as functions that take points in
the plane as inputs and give other points as outputs. Compare transformations
that preserve distance and angle to those that do not (e.g., translation versus
horizontal stretch).
3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the
rotations and reflections that carry it onto itself.
4. Develop definitions of rotations, reflections, and translations in terms of
angles, circles, perpendicular lines, parallel lines, and line segments.
5. Given a geometric figure and a rotation, reflection, or translation, draw the
transformed figure using, e.g., graph paper, tracing paper, or geometry
software. Specify a sequence of transformations that will carry a given figure
onto another.
Understand congruence in terms of rigid motions.
8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow
from the definition of congruence in terms of rigid motions.
G.R.3 Use representation as a tool for
exploring and understanding
mathematical ideas
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Mathematics – Page 98
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
Mathematics
Domains/Clusters/
Common Core State Standards
High School
G.R.3 (Continued from previous page)
(Continued from previous page)
Make geometric constructions.
12. Make formal geometric constructions with a variety of tools and methods
(compass and straightedge, string, reflective devices, paper folding, dynamic
geometric software, etc.). Copying a segment; copying an angle; bisecting a
segment; bisecting an angle; constructing perpendicular lines, including the
perpendicular bisector of a line segment; and constructing a line parallel to a
given line through a point not on the line.
13. Construct an equilateral triangle, a square, and a regular hexagon inscribed
in a circle.
Geometry: Similarity, Right Triangles, & Trigonometry
Understand similarity in terms of similarity transformations.
1. Verify experimentally the properties of dilations given by a center and a scale
factor:
a. A dilation takes a line not passing through the center of the dilation to a
parallel line, and leaves a line passing through the center unchanged.
b. The dilation of a line segment is longer or shorter in the ratio given by the
scale factor.
3. Use the properties of similarity transformations to establish the AA criterion
for two triangles to be similar.
Geometry: Expressing Geometric Properties with Equations
Translate between the geometric description and the equation for a conic
section.
1. Derive the equation of a circle of given center and radius using the
Pythagorean Theorem; complete the square to find the center and radius of a
circle given by an equation.
Use coordinates to prove simple geometric theorems algebraically.
5. Prove the slope criteria for parallel and perpendicular lines and use them to
solve geometric problems (e.g., find the equation of a line parallel or
perpendicular to a given line that passes through a given point).
6. Find the point on a directed line segment between two given points that
partitions the segment in a given ratio.
7. Use coordinates to compute perimeters of polygons and areas of triangles and
rectangles, e.g., using the distance formula.
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
Mathematics – Page 99
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
Students will select, apply, and
translate among mathematical
representations to solve problems.
G.R.4 Select appropriate
representations to solve problem
situations
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Geometry: Congruence
Experiment with transformations in the plane.
2. Represent transformations in the plane using, e.g., transparencies and
geometry software; describe transformations as functions that take points in
the plane as inputs and give other points as outputs. Compare transformations
that preserve distance and angle to those that do not (e.g., translation versus
horizontal stretch).
3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the
rotations and reflections that carry it onto itself.
4. Develop definitions of rotations, reflections, and translations in terms of
angles, circles, perpendicular lines, parallel lines, and line segments.
5. Given a geometric figure and a rotation, reflection, or translation, draw the
transformed figure using, e.g., graph paper, tracing paper, or geometry
software. Specify a sequence of transformations that will carry a given figure
onto another.
Understand congruence in terms of rigid motions.
8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow
from the definition of congruence in terms of rigid motions.
Make geometric constructions.
12. Make formal geometric constructions with a variety of tools and methods
(compass and straightedge, string, reflective devices, paper folding, dynamic
geometric software, etc.). Copying a segment; copying an angle; bisecting a
segment; bisecting an angle; constructing perpendicular lines, including the
perpendicular bisector of a line segment; and constructing a line parallel to a
given line through a point not on the line.
13. Construct an equilateral triangle, a square, and a regular hexagon inscribed
in a circle.
Geometry: Similarity, Right Triangles, & Trigonometry
Understand similarity in terms of similarity transformations.
1. Verify experimentally the properties of dilations given by a center and a scale
factor:
a. A dilation takes a line not passing through the center of the dilation to a
parallel line, and leaves a line passing through the center unchanged.
b. The dilation of a line segment is longer or shorter in the ratio given by the
scale factor.
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Mathematics – Page 100
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
Mathematics
Domains/Clusters/
Common Core State Standards
High School
G.R.4 (Continued from previous page)
(Continued from previous page)
3. Use the properties of similarity transformations to establish the AA criterion
for two triangles to be similar.
Geometry: Expressing Geometric Properties with Equations
Translate between the geometric description and the equation for a conic
section.
1. Derive the equation of a circle of given center and radius using the
Pythagorean Theorem; complete the square to find the center and radius of a
circle given by an equation.
Use coordinates to prove simple geometric theorems algebraically.
5. Prove the slope criteria for parallel and perpendicular lines and use them to
solve geometric problems (e.g., find the equation of a line parallel or
perpendicular to a given line that passes through a given point).
6. Find the point on a directed line segment between two given points that
partitions the segment in a given ratio.
7. Use coordinates to compute perimeters of polygons and areas of triangles and
rectangles, e.g., using the distance formula.
Geometry: Congruence
Experiment with transformations in the plane.
2. Represent transformations in the plane using, e.g., transparencies and
geometry software; describe transformations as functions that take points in
the plane as inputs and give other points as outputs. Compare transformations
that preserve distance and angle to those that do not (e.g., translation versus
horizontal stretch).
3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the
rotations and reflections that carry it onto itself.
4. Develop definitions of rotations, reflections, and translations in terms of
angles, circles, perpendicular lines, parallel lines, and line segments.
5. Given a geometric figure and a rotation, reflection, or translation, draw the
transformed figure using, e.g., graph paper, tracing paper, or geometry
software. Specify a sequence of transformations that will carry a given figure
onto another.
Geometry: Expressing Geometric Properties with Equations
Use coordinates to prove simple geometric theorems algebraically.
5. Prove the slope criteria for parallel and perpendicular lines and use them to
solve geometric problems (e.g., find the equation of a line parallel or
perpendicular to a given line that passes through a given point).
G.R.5 Investigate relationships
between different representations and
their impact on a given problem
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Mathematics – Page 101
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
Students will use representations to
model and interpret physical, social,
and mathematical phenomena.
G.R.6 Use mathematics to show and
understand physical phenomena (e.g.,
determine the number of gallons of
water in a fish tank)
G.R.7 Use mathematics to show and
understand social phenomena (e.g.,
determine if conclusions from another
person’s argument have a logical
foundation)
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Number & Quantity: Quantities
Reason quantitatively and use units to solve problems.
1. Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and
interpret the scale and the origin in graphs and data displays.
3. Choose a level of accuracy appropriate to limitations on measurement when
reporting quantities.
Geometry: Geometric Measurement & Dimensions
Explain volume formulas and use them to solve problems.
1. Give an informal argument for the formulas for the circumference of a circle,
area of a circle, volume of a cylinder, pyramid, and cone. Use dissection
arguments, Cavalieri’s principle, and informal limit arguments.
3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve
problems.
There is no New York Mathematics Performance Indicator–Common Core
alignment.
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
M10
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Mathematics – Page 102
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
Mathematics
Domains/Clusters/
Common Core State Standards
High School
G.R.8 Use mathematics to show and
understand mathematical phenomena
(e.g., use investigation, discovery,
conjecture, reasoning, arguments,
justification and proofs to validate that
the two base angles of an isosceles
triangle are congruent)
Geometry: Congruence
Experiment with transformations in the plane.
2. Represent transformations in the plane using, e.g., transparencies and
geometry software; describe transformations as functions that take points in
the plane as inputs and give other points as outputs. Compare transformations
that preserve distance and angle to those that do not (e.g., translation versus
horizontal stretch).
3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the
rotations and reflections that carry it onto itself.
4. Develop definitions of rotations, reflections, and translations in terms of
angles, circles, perpendicular lines, parallel lines, and line segments.
5. Given a geometric figure and a rotation, reflection, or translation, draw the
transformed figure using, e.g., graph paper, tracing paper, or geometry
software. Specify a sequence of transformations that will carry a given figure
onto another.
Understand congruence in terms of rigid motions.
6. Use geometric descriptions of rigid motions to transform figures and to
predict the effect of a given rigid motion on a given figure; given two figures,
use the definition of congruence in terms of rigid motions to decide if they are
congruent.
7. Use the definition of congruence in terms of rigid motions to show that two
triangles are congruent if and only if corresponding pairs of sides and
corresponding pairs of angles are congruent.
8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow
from the definition of congruence in terms of rigid motions.
Prove geometric theorems
9. Prove theorems about lines and angles. Theorems include: vertical angles are
congruent; when a transversal crosses parallel lines, alternate interior angles
are congruent and corresponding angles are congruent; points on a
perpendicular bisector of a line segment are exactly those equidistant from
the segment’s endpoints.
10. Prove theorems about triangles. Theorems include: measures of interior
angles of a triangle sum to 180°; base angles of isosceles triangles are
congruent; the segment joining midpoints of two sides of a triangle is parallel
to the third side and half the length; the medians of a triangle meet at a point.
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Mathematics – Page 103
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
Mathematics
Domains/Clusters/
Common Core State Standards
High School
G.R.8 (Continued from previous page)
(Continued from previous page)
11. Prove theorems about parallelograms. Theorems include: opposite sides are
congruent, opposite angles are congruent, the diagonals of a parallelogram
bisect each other, and conversely, rectangles are parallelograms with
congruent diagonals.
Geometry: Similarity, Right Triangles, & Trigonometry
Understand similarity in terms of similarity transformations.
1. Verify experimentally the properties of dilations given by a center and a scale
factor:
a. A dilation takes a line not passing through the center of the dilation to a
parallel line, and leaves a line passing through the center unchanged.
b. The dilation of a line segment is longer or shorter in the ratio given by the
scale factor.
2. Given two figures, use the definition of similarity in terms of similarity
transformations to decide if they are similar; explain using similarity
transformations the meaning of similarity for triangles as the equality of all
corresponding pairs of angles and the proportionality of all corresponding
pairs of sides.
3. Use the properties of similarity transformations to establish the AA criterion
for two triangles to be similar.
Prove theorems involving similarity.
4. Prove theorems about triangles. Theorems include: a line parallel to one side
of a triangle divides the other two proportionally, and conversely; the
Pythagorean Theorem proved using triangle similarity.
5. Use congruence and similarity criteria for triangles to solve problems and to
prove relationships in geometric figures.
Geometry: Expressing Geometric Properties with Equations
Use coordinates to prove simple geometric theorems algebraically.
4. Use coordinates to prove simple geometric theorems algebraically. For
example, prove or disprove that a figure defined by four given points in the
coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies
on the circle centered at the origin and containing the point (0, 2).
5. Prove the slope criteria for parallel and perpendicular lines and use them to
solve geometric problems (e.g., find the equation of a line parallel or
perpendicular to a given line that passes through a given point).
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
Mathematics – Page 104
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
Mathematics
Domains/Clusters/
Common Core State Standards
High School
G.R.8 (Continued from previous page)
(Continued from previous page)
6. Find the point on a directed line segment between two given points that
partitions the segment in a given ratio.
7. Use coordinates to compute perimeters of polygons and areas of triangles and
rectangles, e.g., using the distance formula.
National Essential Skills Study (NESS)
National Rankings
Rank
Algebra Strand
Note: The algebraic skills and concepts
within the Algebra process and content
performance indicators must be
maintained and applied as students are
asked to investigate, make conjectures,
give rationale, and justify or prove
geometric concepts.
Geometry Strand
Students will use visualization and
spatial reasoning to analyze
characteristics and properties of
geometric shapes.
Geometric Relationships
Note: Two-dimensional geometric
relationships are addressed in the
Informal and Formal Proofs band.
G.G.1 Know and apply that if a line is
perpendicular to each of two
intersecting lines at their point of
intersection, then the line is
perpendicular to the plane determined
by them
There is no New York Mathematics Performance Indicator–Common Core
alignment.
 2011 International Center for Leadership in Education
M4
M41
Understand the properties of and apply
parallel, perpendicular, and intersecting
lines in problem-solving situations.
Understand the properties and applications
of the undefined terms of geometry (point,
line, and plane) and their relationship with
intuitive concepts (i.e., collinear points,
coplanar points, opposite rays, and parallel
lines).
Mathematics – Page 105
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
G.G.2 Know and apply that through a
given point there passes one and only
one plane perpendicular to a given line
Mathematics
Domains/Clusters/
Common Core State Standards
High School
There is no New York Mathematics Performance Indicator–Common Core
alignment.
National Essential Skills Study (NESS)
National Rankings
Rank
M4
M41
G.G.3 Know and apply that through a
given point there passes one and only
one line perpendicular to a given plane
There is no New York Mathematics Performance Indicator–Common Core
alignment.
M4
M41
G.G.4 Know and apply that two lines
perpendicular to the same plane are
coplanar
G.G.5 Know and apply that two planes
are perpendicular to each other if and
only if one plane contains a line
perpendicular to the second plane
G.G.6 Know and apply that if a line is
perpendicular to a plane, then any line
perpendicular to the given line at its
point of intersection with the given
plane is in the given plane
There is no New York Mathematics Performance Indicator–Common Core
alignment.
There is no New York Mathematics Performance Indicator–Common Core
alignment.
There is no New York Mathematics Performance Indicator–Common Core
alignment.
M4
M4
M4
M41
 2011 International Center for Leadership in Education
Understand the properties of and apply
parallel, perpendicular, and intersecting
lines in problem-solving situations.
Understand the properties and applications
of the undefined terms of geometry (point,
line, and plane) and their relationship with
intuitive concepts (i.e., collinear points,
coplanar points, opposite rays, and parallel
lines).
Understand the properties of and apply
parallel, perpendicular, and intersecting
lines in problem-solving situations.
Understand the properties and applications
of the undefined terms of geometry (point,
line, and plane) and their relationship with
intuitive concepts (i.e., collinear points,
coplanar points, opposite rays, and parallel
lines).
Understand the properties of and apply
parallel, perpendicular, and intersecting
lines in problem-solving situations.
Understand the properties of and apply
parallel, perpendicular, and intersecting
lines in problem-solving situations.
Understand the properties of and apply
parallel, perpendicular, and intersecting
lines in problem-solving situations.
Understand the properties and applications
of the undefined terms of geometry (point,
line, and plane) and their relationship with
intuitive concepts (i.e., collinear points,
coplanar points, opposite rays, and parallel
lines).
Mathematics – Page 106
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
G.G.7 Know and apply that if a line is
perpendicular to a plane, then every
plane containing the line is
perpendicular to the given plane
G.G.8 Know and apply that if a plane
intersects two parallel planes, then the
intersection is two parallel lines
Mathematics
Domains/Clusters/
Common Core State Standards
High School
There is no New York Mathematics Performance Indicator–Common Core
alignment.
There is no New York Mathematics Performance Indicator–Common Core
alignment.
National Essential Skills Study (NESS)
National Rankings
Rank
M4
M4
M41
G.G.9 Know and apply that if two
planes are perpendicular to the same
line, they are parallel
G.G.10 Know and apply that the lateral
edges of a prism are congruent and
parallel
G.G.11 Know and apply that two
prisms have equal volumes if their
bases have equal areas and their
altitudes are equal
There is no New York Mathematics Performance Indicator–Common Core
alignment.
Geometry: Geometric Measurement & Dimensions
Visualize relationships between two-dimensional and three-dimensional
objects.
4. Identify the shapes of two-dimensional cross-sections of three-dimensional
objects, and identify three-dimensional objects generated by rotations of twodimensional objects.
Geometry: Geometric Measurement & Dimensions
Explain volume formulas and use them to solve problems.
2. (+) Give an informal argument using Cavalieri’s principle for the formulas
for the volume of a sphere and other solid figures.
 2011 International Center for Leadership in Education
M4
M4
M26
M26
Understand the properties of and apply
parallel, perpendicular, and intersecting
lines in problem-solving situations.
Understand the properties of and apply
parallel, perpendicular, and intersecting
lines in problem-solving situations.
Understand the properties and applications
of the undefined terms of geometry (point,
line, and plane) and their relationship with
intuitive concepts (i.e., collinear points,
coplanar points, opposite rays, and parallel
lines).
Understand the properties of and apply
parallel, perpendicular, and intersecting
lines in problem-solving situations.
Understand the properties of and apply
parallel, perpendicular, and intersecting
lines in problem-solving situations.
Know the classification and properties of
three-dimensional figures (prisms,
rectangular solids, pyramids, right circular
cylinders, cones, and spheres) and be able to
compute the volume and surface area of
common solids.
Know the classification and properties of
three-dimensional figures (prisms,
rectangular solids, pyramids, right circular
cylinders, cones, and spheres) and be able to
compute the volume and surface area of
common solids.
Mathematics – Page 107
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
Mathematics
Domains/Clusters/
Common Core State Standards
High School
G.G.12 Know and apply that the
volume of a prism is the product of the
area of the base and the altitude
Geometry: Geometric Measurement & Dimensions
Explain volume formulas and use them to solve problems.
1. Give an informal argument for the formulas for the circumference of a circle,
area of a circle, volume of a cylinder, pyramid, and cone. Use dissection
arguments, Cavalieri’s principle, and informal limit arguments.
G.G.13 Apply the properties of a
regular pyramid, including:
 lateral edges are congruent
 lateral faces are congruent isosceles
triangles
 volume of a pyramid equals onethird the product of the area of the
base and the altitude
G.G.14 Apply the properties of a
cylinder, including:
 bases are congruent
 volume equals the product of the
area of the base and the altitude
 lateral area of a right circular
cylinder equals the product of an
altitude and the circumference of
the base
G.G.15 Apply the properties of a right
circular cone, including:
 lateral area equals one-half the
product of the slant height and the
circumference of its base
 volume is one-third the product of
the area of its base and its altitude
Geometry: Geometric Measurement & Dimensions
Explain volume formulas and use them to solve problems.
1. Give an informal argument for the formulas for the circumference of a circle,
area of a circle, volume of a cylinder, pyramid, and cone. Use dissection
arguments, Cavalieri’s principle, and informal limit arguments.
3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve
problems.
Geometry: Geometric Measurement & Dimensions
Explain volume formulas and use them to solve problems.
1. Give an informal argument for the formulas for the circumference of a circle,
area of a circle, volume of a cylinder, pyramid, and cone. Use dissection
arguments, Cavalieri’s principle, and informal limit arguments.
3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve
problems.
Geometry: Geometric Measurement & Dimensions
Explain volume formulas and use them to solve problems.
1. Give an informal argument for the formulas for the circumference of a circle,
area of a circle, volume of a cylinder, pyramid, and cone. Use dissection
arguments, Cavalieri’s principle, and informal limit arguments.
3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve
problems.
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
M26
M26
M26
M26
Know the classification and properties of
three-dimensional figures (prisms,
rectangular solids, pyramids, right circular
cylinders, cones, and spheres) and be able to
compute the volume and surface area of
common solids.
Know the classification and properties of
three-dimensional figures (prisms,
rectangular solids, pyramids, right circular
cylinders, cones, and spheres) and be able to
compute the volume and surface area of
common solids.
Know the classification and properties of
three-dimensional figures (prisms,
rectangular solids, pyramids, right circular
cylinders, cones, and spheres) and be able to
compute the volume and surface area of
common solids.
Know the classification and properties of
three-dimensional figures (prisms,
rectangular solids, pyramids, right circular
cylinders, cones, and spheres) and be able to
compute the volume and surface area of
common solids.
Mathematics – Page 108
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
G.G.16 Apply the properties of a
sphere, including:
 the intersection of a plane and a
sphere is a circle
 a great circle is the largest circle
that can be drawn on a sphere
 two planes equidistant from the
center of the sphere and
intersecting the sphere do so in
congruent circles

surface area is 4 r

volume is
Mathematics
Domains/Clusters/
Common Core State Standards
High School
National Essential Skills Study (NESS)
National Rankings
Rank
Know the classification and properties of
three-dimensional figures (prisms,
rectangular solids, pyramids, right circular
cylinders, cones, and spheres) and be able to
compute the volume and surface area of
common solids.
Geometry: Geometric Measurement & Dimensions
Explain volume formulas and use them to solve problems.
2. (+) Give an informal argument using Cavalieri’s principle for the formulas
for the volume of a sphere and other solid figures.
M26
2
4 3
r
3
Constructions
G.G.17 Construct a bisector of a given
angle, using a straightedge and
compass, and justify the construction
G.G.18 Construct the perpendicular
bisector of a given segment, using a
straightedge and compass, and justify
the construction
Geometry: Congruence
Make geometric constructions.
12. Make formal geometric constructions with a variety of tools and methods
(compass and straightedge, string, reflective devices, paper folding, dynamic
geometric software, etc.). Copying a segment; copying an angle; bisecting a
segment; bisecting an angle; constructing perpendicular lines, including the
perpendicular bisector of a line segment; and constructing a line parallel to a
given line through a point not on the line.
Geometry: Congruence
Make geometric constructions.
12. Make formal geometric constructions with a variety of tools and methods
(compass and straightedge, string, reflective devices, paper folding, dynamic
geometric software, etc.). Copying a segment; copying an angle; bisecting a
segment; bisecting an angle; constructing perpendicular lines, including the
perpendicular bisector of a line segment; and constructing a line parallel to a
given line through a point not on the line.
 2011 International Center for Leadership in Education
M15
M42
M4
M42
Classify angles by measure (acute, right,
obtuse, and straight) and understand angle
relationships (supplementary,
complementary, and vertical).
Use geometric methods, such as using an
unmarked straightedge and compass, to
complete basic geometric constructions.
Understand the properties of and apply
parallel, perpendicular, and intersecting
lines in problem-solving situations.
Use geometric methods, such as using an
unmarked straightedge and compass, to
complete basic geometric constructions.
Mathematics – Page 109
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
Mathematics
Domains/Clusters/
Common Core State Standards
High School
G.G.19 Construct lines parallel (or
perpendicular) to a given line through a
given point, using a straightedge and
compass, and justify the construction
Geometry: Congruence
Make geometric constructions.
12. Make formal geometric constructions with a variety of tools and methods
(compass and straightedge, string, reflective devices, paper folding, dynamic
geometric software, etc.). Copying a segment; copying an angle; bisecting a
segment; bisecting an angle; constructing perpendicular lines, including the
perpendicular bisector of a line segment; and constructing a line parallel to a
given line through a point not on the line.
Geometry: Congruence
Make geometric constructions.
13. Construct an equilateral triangle, a square, and a regular hexagon inscribed
in a circle.
G.G.20 Construct an equilateral
triangle, using a straightedge and
compass, and justify the construction
National Essential Skills Study (NESS)
National Rankings
Rank
M4
M42
M34
M42
Understand the properties of and apply
parallel, perpendicular, and intersecting
lines in problem-solving situations.
Use geometric methods, such as using an
unmarked straightedge and compass, to
complete basic geometric constructions.
Understand the properties and classification
of polygons (triangles, the family of
quadrilaterals, pentagon, hexagon, etc.) and
apply knowledge of angle and side
relationships of geometric shapes in
problem-solving situations.
Use geometric methods, such as using an
unmarked straightedge and compass, to
complete basic geometric constructions.
Locus
G.G.21 Investigate and apply the
concurrence of medians, altitudes, angle
bisectors, and perpendicular bisectors
of triangles
Geometry: Congruence
Prove geometric theorems.
10. Prove theorems about triangles. Theorems include: measures of interior
angles of a triangle sum to 180°; base angles of isosceles triangles are
congruent; the segment joining midpoints of two sides of a triangle is parallel
to the third side and half the length; the medians of a triangle meet at a point.
M15
M34
G.G.22 Solve problems using
compound loci
There is no New York Mathematics Performance Indicator–Common Core
alignment.
 2011 International Center for Leadership in Education
M66
Classify angles by measure (acute, right,
obtuse, and straight) and understand angle
relationships (supplementary,
complementary, and vertical).
Understand the properties and classification
of polygons (triangles, the family of
quadrilaterals, pentagon, hexagon, etc.) and
apply knowledge of angle and side
relationships of geometric shapes in
problem-solving situations.
Know how to sketch basic conic sections
(e.g., circles, parabolas) by using their
equations and solve systems of non-linear
equations graphically.
Mathematics – Page 110
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
G.G.23 Graph and solve compound
loci in the coordinate plane
Mathematics
Domains/Clusters/
Common Core State Standards
High School
There is no New York Mathematics Performance Indicator–Common Core
alignment.
National Essential Skills Study (NESS)
National Rankings
Rank
M66
Know how to sketch basic conic sections
(e.g., circles, parabolas) by using their
equations and solve systems of non-linear
equations graphically.
Students will identify and justify
geometric relationships formally and
informally.
Informal and Formal Proofs
G.G.24 Determine the negation of a
statement and establish its truth value
G.G.25 Know and apply the conditions
under which a compound statement
(conjunction, disjunction, conditional,
biconditional) is true
There is no New York Mathematics Performance Indicator–Common Core
alignment.
There is no New York Mathematics Performance Indicator–Common Core
alignment.
M49
M10
M49
G.G.26 Identify and write the inverse,
converse, and contrapositive of a given
conditional statement and note the
logical equivalences
G.G.27 Write a proof arguing from a
given hypothesis to a given conclusion
There is no New York Mathematics Performance Indicator–Common Core
alignment.
Geometry: Congruence
Prove geometric theorems.
9. Prove theorems about lines and angles. Theorems include: vertical angles are
congruent; when a transversal crosses parallel lines, alternate interior angles
are congruent and corresponding angles are congruent; points on a
perpendicular bisector of a line segment are exactly those equidistant from
the segment’s endpoints.
10. Prove theorems about triangles. Theorems include: measures of interior
angles of a triangle sum to 180°; base angles of isosceles triangles are
congruent; the segment joining midpoints of two sides of a triangle is parallel
to the third side and half the length; the medians of a triangle meet at a point.
11. Prove theorems about parallelograms. Theorems include: opposite sides are
congruent, opposite angles are congruent, the diagonals of a parallelogram
bisect each other, and conversely, rectangles are parallelograms with
congruent diagonals.
 2011 International Center for Leadership in Education
M49
Analyze the truth value of compound
sentences by creating truth tables.
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Analyze the truth value of compound
sentences by creating truth tables.
Analyze the truth value of compound
sentences by creating truth tables.
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Mathematics – Page 111
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
Mathematics
Domains/Clusters/
Common Core State Standards
High School
G.G.27 (Continued from previous page)
(Continued from previous page)
Geometry: Similarity, Right Triangles, & Trigonometry
Prove theorems involving similarity.
4. Prove theorems about triangles. Theorems include: a line parallel to one side
of a triangle divides the other two proportionally, and conversely; the
Pythagorean Theorem proved using triangle similarity.
5. Use congruence and similarity criteria for triangles to solve problems and to
prove relationships in geometric figures.
Geometry: Congruence
Understand congruence in terms of rigid motions.
8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow
from the definition of congruence in terms of rigid motions.
G.G.28 Determine the congruence of
two triangles by using one of the five
congruence techniques (SSS, SAS,
ASA, AAS, HL), given sufficient
information about the sides and/or
angles of two congruent triangles
National Essential Skills Study (NESS)
National Rankings
Rank
M10
M54
G.G.29 Identify corresponding parts of
congruent triangles
Geometry: Congruence
Understand congruence in terms of rigid motions.
8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow
from the definition of congruence in terms of rigid motions.
M34
M54
 2011 International Center for Leadership in Education
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Apply transformations (reflection, rotation,
translation, and dilation) of 2-dimensional
figures graphically to interpret, analyze, and
illustrate the concepts of congruency,
similarity, and symmetry.
Understand the properties and classification
of polygons (triangles, the family of
quadrilaterals, pentagon, hexagon, etc.) and
apply knowledge of angle and side
relationships of geometric shapes in
problem-solving situations.
Apply transformations (reflection, rotation,
translation, and dilation) of 2-dimensional
figures graphically to interpret, analyze, and
illustrate the concepts of congruency,
similarity, and symmetry.
Mathematics – Page 112
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
Mathematics
Domains/Clusters/
Common Core State Standards
High School
G.G.30 Investigate, justify, and apply
theorems about the sum of the measures
of the angles of a triangle
Geometry: Congruence
Prove geometric theorems.
10. Prove theorems about triangles. Theorems include: measures of interior
angles of a triangle sum to 180°; base angles of isosceles triangles are
congruent; the segment joining midpoints of two sides of a triangle is parallel
to the third side and half the length; the medians of a triangle meet at a point.
National Essential Skills Study (NESS)
National Rankings
Rank
M10
M34
G.G.31 Investigate, justify, and apply
the isosceles triangle theorem and its
converse
Geometry: Congruence
Prove geometric theorems.
10. Prove theorems about triangles. Theorems include: measures of interior
angles of a triangle sum to 180°; base angles of isosceles triangles are
congruent; the segment joining midpoints of two sides of a triangle is parallel
to the third side and half the length; the medians of a triangle meet at a point.
M10
M34
G.G.32 Investigate, justify, and apply
theorems about geometric inequalities,
using the exterior angle theorem
Geometry: Congruence
Prove geometric theorems.
10. Prove theorems about triangles. Theorems include: measures of interior
angles of a triangle sum to 180°; base angles of isosceles triangles are
congruent; the segment joining midpoints of two sides of a triangle is parallel
to the third side and half the length; the medians of a triangle meet at a point.
M10
M34
 2011 International Center for Leadership in Education
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Understand the properties and classification
of polygons (triangles, the family of
quadrilaterals, pentagon, hexagon, etc.) and
apply knowledge of angle and side
relationships of geometric shapes in
problem-solving situations.
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Understand the properties and classification
of polygons (triangles, the family of
quadrilaterals, pentagon, hexagon, etc.) and
apply knowledge of angle and side
relationships of geometric shapes in
problem-solving situations.
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Understand the properties and classification
of polygons (triangles, the family of
quadrilaterals, pentagon, hexagon, etc.) and
apply knowledge of angle and side
relationships of geometric shapes in
problem-solving situations.
Mathematics – Page 113
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
G.G.33 Investigate, justify, and apply
the triangle inequality theorem
Mathematics
Domains/Clusters/
Common Core State Standards
High School
National Essential Skills Study (NESS)
National Rankings
Rank
There is no New York Mathematics Performance Indicator–Common Core
alignment.
M10
M34
G.G.34 Determine either the longest
side of a triangle given the three angle
measures or the largest angle given the
lengths of three sides of a triangle
There is no New York Mathematics Performance Indicator–Common Core
alignment.
M10
M34
G.G.35 Determine if two lines cut by a
transversal are parallel, based on the
measure of given pairs of angles formed
by the transversal and the lines
Geometry: Congruence
Prove geometric theorems.
9. Prove theorems about lines and angles. Theorems include: vertical angles are
congruent; when a transversal crosses parallel lines, alternate interior angles
are congruent and corresponding angles are congruent; points on a
perpendicular bisector of a line segment are exactly those equidistant from
the segment’s endpoints.
 2011 International Center for Leadership in Education
M4
M15
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Understand the properties and classification
of polygons (triangles, the family of
quadrilaterals, pentagon, hexagon, etc.) and
apply knowledge of angle and side
relationships of geometric shapes in
problem-solving situations.
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Understand the properties and classification
of polygons (triangles, the family of
quadrilaterals, pentagon, hexagon, etc.) and
apply knowledge of angle and side
relationships of geometric shapes in
problem-solving situations.
Understand the properties of and apply
parallel, perpendicular, and intersecting
lines in problem-solving situations.
Classify angles by measure (acute, right,
obtuse, and straight) and understand angle
relationships (supplementary,
complementary, and vertical).
Mathematics – Page 114
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
Mathematics
Domains/Clusters/
Common Core State Standards
High School
G.G.36 Investigate, justify, and apply
theorems about the sum of the measures
of the interior and exterior angles of
polygons
Geometry: Congruence
Prove geometric theorems.
10. Prove theorems about triangles. Theorems include: measures of interior
angles of a triangle sum to 180°; base angles of isosceles triangles are
congruent; the segment joining midpoints of two sides of a triangle is parallel
to the third side and half the length; the medians of a triangle meet at a point.
11. Prove theorems about parallelograms. Theorems include: opposite sides are
congruent, opposite angles are congruent, the diagonals of a parallelogram
bisect each other, and conversely, rectangles are parallelograms with
congruent diagonals.
G.G.37 Investigate, justify, and apply
theorems about each interior and
exterior angle measure of regular
polygons
G.G.38 Investigate, justify, and apply
theorems about parallelograms
involving their angles, sides, and
diagonals
Geometry: Congruence
Prove geometric theorems.
10. Prove theorems about triangles. Theorems include: measures of interior
angles of a triangle sum to 180°; base angles of isosceles triangles are
congruent; the segment joining midpoints of two sides of a triangle is parallel
to the third side and half the length; the medians of a triangle meet at a point.
11. Prove theorems about parallelograms. Theorems include: opposite sides are
congruent, opposite angles are congruent, the diagonals of a parallelogram
bisect each other, and conversely, rectangles are parallelograms with
congruent diagonals.
Geometry: Congruence
Prove geometric theorems.
11. Prove theorems about parallelograms. Theorems include: opposite sides are
congruent, opposite angles are congruent, the diagonals of a parallelogram
bisect each other, and conversely, rectangles are parallelograms with
congruent diagonals.
National Essential Skills Study (NESS)
National Rankings
Rank
M10
M34
M10
M34
M10
M34
 2011 International Center for Leadership in Education
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Understand the properties and classification
of polygons (triangles, the family of
quadrilaterals, pentagon, hexagon, etc.) and
apply knowledge of angle and side
relationships of geometric shapes in
problem-solving situations.
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Understand the properties and classification
of polygons (triangles, the family of
quadrilaterals, pentagon, hexagon, etc.) and
apply knowledge of angle and side
relationships of geometric shapes in
problem-solving situations.
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Understand the properties and classification
of polygons (triangles, the family of
quadrilaterals, pentagon, hexagon, etc.) and
apply knowledge of angle and side
relationships of geometric shapes in
problem-solving situations.
Mathematics – Page 115
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
Mathematics
Domains/Clusters/
Common Core State Standards
High School
G.G.39 Investigate, justify, and apply
theorems about special parallelograms
(rectangles, rhombuses, squares)
involving their angles, sides, and
diagonals
Geometry: Congruence
Prove geometric theorems.
11. Prove theorems about parallelograms. Theorems include: opposite sides are
congruent, opposite angles are congruent, the diagonals of a parallelogram
bisect each other, and conversely, rectangles are parallelograms with
congruent diagonals.
National Essential Skills Study (NESS)
National Rankings
Rank
M10
M34
G.G.40 Investigate, justify, and apply
theorems about trapezoids (including
isosceles trapezoids) involving their
angles, sides, medians, and diagonals
There is no New York Mathematics Performance Indicator–Common Core
alignment.
M10
M34
G.G.41 Justify that some quadrilaterals
are parallelograms, rhombuses,
rectangles, squares, or trapezoids
Geometry: Congruence
Prove geometric theorems.
11. Prove theorems about parallelograms. Theorems include: opposite sides are
congruent, opposite angles are congruent, the diagonals of a parallelogram
bisect each other, and conversely, rectangles are parallelograms with
congruent diagonals.
 2011 International Center for Leadership in Education
M34
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Understand the properties and classification
of polygons (triangles, the family of
quadrilaterals, pentagon, hexagon, etc.) and
apply knowledge of angle and side
relationships of geometric shapes in
problem-solving situations.
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Understand the properties and classification
of polygons (triangles, the family of
quadrilaterals, pentagon, hexagon, etc.) and
apply knowledge of angle and side
relationships of geometric shapes in
problem-solving situations.
Understand the properties and classification
of polygons (triangles, the family of
quadrilaterals, pentagon, hexagon, etc.) and
apply knowledge of angle and side
relationships of geometric shapes in
problem-solving situations.
Mathematics – Page 116
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
Mathematics
Domains/Clusters/
Common Core State Standards
High School
G.G.42 Investigate, justify, and apply
theorems about geometric relationships,
based on the properties of the line
segment joining the midpoints of two
sides of the triangle
Geometry: Congruence
Prove geometric theorems.
10. Prove theorems about triangles. Theorems include: measures of interior
angles of a triangle sum to 180°; base angles of isosceles triangles are
congruent; the segment joining midpoints of two sides of a triangle is parallel
to the third side and half the length; the medians of a triangle meet at a point.
National Essential Skills Study (NESS)
National Rankings
Rank
M10
M34
G.G.43 Investigate, justify, and apply
theorems about the centroid of a
triangle, dividing each median into
segments whose lengths are in the ratio
2:1
Geometry: Congruence
Prove geometric theorems.
10. Prove theorems about triangles. Theorems include: measures of interior
angles of a triangle sum to 180°; base angles of isosceles triangles are
congruent; the segment joining midpoints of two sides of a triangle is parallel
to the third side and half the length; the medians of a triangle meet at a point.
M3
M10
M34
G.G.44 Establish similarity of
triangles, using the following theorems:
AA, SAS, and SSS
Geometry: Similarity, Right Triangles, & Trigonometry
Understand similarity in terms of similarity transformations.
3. Use the properties of similarity transformations to establish the AA criterion
for two triangles to be similar.
Prove theorems involving similarity.
5. Use congruence and similarity criteria for triangles to solve problems and to
prove relationships in geometric figures.
M10
M54
 2011 International Center for Leadership in Education
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Understand the properties and classification
of polygons (triangles, the family of
quadrilaterals, pentagon, hexagon, etc.) and
apply knowledge of angle and side
relationships of geometric shapes in
problem-solving situations.
Use proportional reasoning to solve realworld problems.
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Understand the properties and classification
of polygons (triangles, the family of
quadrilaterals, pentagon, hexagon, etc.) and
apply knowledge of angle and side
relationships of geometric shapes in
problem-solving situations.
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Apply transformations (reflection, rotation,
translation, and dilation) of 2-dimensional
figures graphically to interpret, analyze, and
illustrate the concepts of congruency,
similarity, and symmetry.
Mathematics – Page 117
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
Mathematics
Domains/Clusters/
Common Core State Standards
High School
G.G.45 Investigate, justify, and apply
theorems about similar triangles
Geometry: Similarity, Right Triangles, & Trigonometry
Prove theorems involving similarity.
4. Prove theorems about triangles. Theorems include: a line parallel to one side
of a triangle divides the other two proportionally, and conversely; the
Pythagorean Theorem proved using triangle similarity.
5. Use congruence and similarity criteria for triangles to solve problems and to
prove relationships in geometric figures.
National Essential Skills Study (NESS)
National Rankings
Rank
M10
M54
G.G.46 Investigate, justify, and apply
theorems about proportional
relationships among the segments of the
sides of the triangle, given one or more
lines parallel to one side of a triangle
and intersecting the other two sides of
the triangle
Geometry: Similarity, Right Triangles, & Trigonometry
Prove theorems involving similarity.
4. Prove theorems about triangles. Theorems include: a line parallel to one side
of a triangle divides the other two proportionally, and conversely; the
Pythagorean Theorem proved using triangle similarity.
M3
M4
M34
G.G.47 Investigate, justify, and apply
theorems about mean proportionality:
 the altitude to the hypotenuse of a
right triangle is the mean
proportional between the two
segments along the hypotenuse
 the altitude to the hypotenuse of a
right triangle divides the
hypotenuse so that either leg of the
right triangle is the mean
proportional between the
hypotenuse and segment of the
hypotenuse adjacent to that leg
Geometry: Similarity, Right Triangles, & Trigonometry
Prove theorems involving similarity.
4. Prove theorems about triangles. Theorems include: a line parallel to one side
of a triangle divides the other two proportionally, and conversely; the
Pythagorean Theorem proved using triangle similarity.
 2011 International Center for Leadership in Education
M3
M10
M34
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Apply transformations (reflection, rotation,
translation, and dilation) of 2-dimensional
figures graphically to interpret, analyze, and
illustrate the concepts of congruency,
similarity, and symmetry.
Use proportional reasoning to solve realworld problems.
Understand the properties of and apply
parallel, perpendicular, and intersecting
lines in problem-solving situations.
Understand the properties and classification
of polygons (triangles, the family of
quadrilaterals, pentagon, hexagon, etc.) and
apply knowledge of angle and side
relationships of geometric shapes in
problem-solving situations.
Use proportional reasoning to solve realworld problems.
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Understand the properties and classification
of polygons (triangles, the family of
quadrilaterals, pentagon, hexagon, etc.) and
apply knowledge of angle and side
relationships of geometric shapes in
problem-solving situations.
Mathematics – Page 118
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
Mathematics
Domains/Clusters/
Common Core State Standards
High School
G.G.48 Investigate, justify, and apply
the Pythagorean theorem and its
converse
Geometry: Similarity, Right Triangles, & Trigonometry
Define trigonometric ratios and solve problems involving right triangles.
8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles
in applied problems.
National Essential Skills Study (NESS)
National Rankings
Rank
M10
M23
G.G.49 Investigate, justify, and apply
theorems regarding chords of a circle:
 perpendicular bisectors of chords
 the relative lengths of chords as
compared to their distance from the
center of the circle
Geometry: Circles
Understand and apply theorems about circles.
2. Identify and describe relationships among inscribed angles, radii, and chords.
Include the relationship between central, inscribed, and circumscribed
angles; inscribed angles on a diameter are right angles; the radius of a circle
is perpendicular to the tangent where the radius intersects the circle.
M10
M18
G.G.50 Investigate, justify, and apply
theorems about tangent lines to a circle:
 a perpendicular to the tangent at the
point of tangency
 two tangents to a circle from the
same external point
 common tangents of two nonintersecting or tangent circles
Geometry: Circles
Understand and apply theorems about circles.
2. Identify and describe relationships among inscribed angles, radii, and chords.
Include the relationship between central, inscribed, and circumscribed
angles; inscribed angles on a diameter are right angles; the radius of a circle
is perpendicular to the tangent where the radius intersects the circle.
M10
M18
M61
 2011 International Center for Leadership in Education
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Apply the Pythagorean Theorem to right
triangles.
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Understand the properties of circles (radius,
arc, diameter, chord, secant, and tangent)
and apply circle quantities (lengths of line
segments, angle measure within a circle,
circumference, and area) in problem-solving
situations.
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Understand the properties of circles (radius,
arc, diameter, chord, secant, and tangent)
and apply circle quantities (lengths of line
segments, angle measure within a circle,
circumference, and area) in problem-solving
situations.
Use derivatives and the process of
differentiation to determine slopes of
tangent lines, maxima and minima, velocity,
and acceleration.
Mathematics – Page 119
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
Mathematics
Domains/Clusters/
Common Core State Standards
High School
G.G.51 Investigate, justify, and apply
theorems about the arcs determined by
the rays of angles formed by two lines
intersecting a circle when the vertex is:
 inside the circle (two chords)
 on the circle (tangent and chord)
 outside the circle (two tangents,
two secants, or tangent and secant)
There is no New York Mathematics Performance Indicator–Common Core
alignment.
G.G.52 Investigate, justify, and apply
theorems about arcs of a circle cut by
two parallel lines
There is no New York Mathematics Performance Indicator–Common Core
alignment.
National Essential Skills Study (NESS)
National Rankings
Rank
M10
M18
M4
M10
M18
G.G.53 Investigate, justify, and apply
theorems regarding segments
intersected by a circle:
 along two tangents from the same
external point
 along two secants from the same
external point
 along a tangent and a secant from
the same external point
 along two intersecting chords of a
given circle
Geometry: Circles
Understand and apply theorems about circles.
2. Identify and describe relationships among inscribed angles, radii, and chords.
Include the relationship between central, inscribed, and circumscribed
angles; inscribed angles on a diameter are right angles; the radius of a circle
is perpendicular to the tangent where the radius intersects the circle.
 2011 International Center for Leadership in Education
M10
M18
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Understand the properties of circles (radius,
arc, diameter, chord, secant, and tangent)
and apply circle quantities (lengths of line
segments, angle measure within a circle,
circumference, and area) in problem-solving
situations.
Understand the properties of and apply
parallel, perpendicular, and intersecting
lines in problem-solving situations.
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Understand the properties of circles (radius,
arc, diameter, chord, secant, and tangent)
and apply circle quantities (lengths of line
segments, angle measure within a circle,
circumference, and area) in problem-solving
situations.
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Understand the properties of circles (radius,
arc, diameter, chord, secant, and tangent)
and apply circle quantities (lengths of line
segments, angle measure within a circle,
circumference, and area) in problem-solving
situations.
Mathematics – Page 120
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
Mathematics
Domains/Clusters/
Common Core State Standards
High School
National Essential Skills Study (NESS)
National Rankings
Rank
Students will apply transformations and
symmetry to analyze problem solving
situations
Transformational Geometry
G.G.54 Define, investigate, justify, and
apply isometries in the plane (rotations,
reflections, translations, glide
reflections)
Note: Use proper function notation.
G.G.55 Investigate, justify, and apply
the properties that remain invariant
under translations, rotations, reflections,
and glide reflections
Geometry: Congruence
Experiment with transformations in the plane.
1. Know precise definitions of angle, circle, perpendicular line, parallel line, and
line segment, based on the undefined notions of point, line, distance along a
line, and distance around a circular arc.
2. Represent transformations in the plane using, e.g., transparencies and
geometry software; describe transformations as functions that take points in
the plane as inputs and give other points as outputs. Compare transformations
that preserve distance and angle to those that do not (e.g., translation versus
horizontal stretch).
3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the
rotations and reflections that carry it onto itself.
4. Develop definitions of rotations, reflections, and translations in terms of
angles, circles, perpendicular lines, parallel lines, and line segments.
5. Given a geometric figure and a rotation, reflection, or translation, draw the
transformed figure using, e.g., graph paper, tracing paper, or geometry
software. Specify a sequence of transformations that will carry a given figure
onto another.
Geometry: Congruence
Experiment with transformations in the plane.
1. Know precise definitions of angle, circle, perpendicular line, parallel line, and
line segment, based on the undefined notions of point, line, distance along a
line, and distance around a circular arc.
2. Represent transformations in the plane using, e.g., transparencies and
geometry software; describe transformations as functions that take points in
the plane as inputs and give other points as outputs. Compare transformations
that preserve distance and angle to those that do not (e.g., translation versus
horizontal stretch).
3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the
rotations and reflections that carry it onto itself.
4. Develop definitions of rotations, reflections, and translations in terms of
angles, circles, perpendicular lines, parallel lines, and line segments.
 2011 International Center for Leadership in Education
M10
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Apply transformations (reflection, rotation,
translation, and dilation) of 2-dimensional
figures graphically to interpret, analyze, and
illustrate the concepts of congruency,
similarity, and symmetry.
M54
M10
M54
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Apply transformations (reflection, rotation,
translation, and dilation) of 2-dimensional
figures graphically to interpret, analyze, and
illustrate the concepts of congruency,
similarity, and symmetry.
Mathematics – Page 121
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
Mathematics
Domains/Clusters/
Common Core State Standards
High School
G.G.55 (Continued from previous page)
(Continued from previous page)
5. Given a geometric figure and a rotation, reflection, or translation, draw the
transformed figure using, e.g., graph paper, tracing paper, or geometry
software. Specify a sequence of transformations that will carry a given figure
onto another.
Understand congruence in terms of rigid motions.
6. Use geometric descriptions of rigid motions to transform figures and to
predict the effect of a given rigid motion on a given figure; given two figures,
use the definition of congruence in terms of rigid motions to decide if they are
congruent.
Geometry: Congruence
Experiment with transformations in the plane.
2. Represent transformations in the plane using, e.g., transparencies and
geometry software; describe transformations as functions that take points in
the plane as inputs and give other points as outputs. Compare transformations
that preserve distance and angle to those that do not (e.g., translation versus
horizontal stretch).
3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the
rotations and reflections that carry it onto itself.
4. Develop definitions of rotations, reflections, and translations in terms of
angles, circles, perpendicular lines, parallel lines, and line segments.
5. Given a geometric figure and a rotation, reflection, or translation, draw the
transformed figure using, e.g., graph paper, tracing paper, or geometry
software. Specify a sequence of transformations that will carry a given figure
onto another.
Geometry: Congruence
Experiment with transformations in the plane.
4. Develop definitions of rotations, reflections, and translations in terms of
angles, circles, perpendicular lines, parallel lines, and line segments.
G.G.56 Identify specific isometries by
observing orientation, numbers of
invariant points, and/or parallelism
G.G.57 Justify geometric relationships
(perpendicularity, parallelism,
congruence) using transformational
techniques (translations, rotations,
reflections)
National Essential Skills Study (NESS)
National Rankings
Rank
Apply transformations (reflection, rotation,
translation, and dilation) of 2-dimensional
figures graphically to interpret, analyze, and
illustrate the concepts of congruency,
similarity, and symmetry.
M54
M4
M10
M54
 2011 International Center for Leadership in Education
Understand the properties of and apply
parallel, perpendicular, and intersecting
lines in problem-solving situations.
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Apply transformations (reflection, rotation,
translation, and dilation) of 2-dimensional
figures graphically to interpret, analyze, and
illustrate the concepts of congruency,
similarity, and symmetry.
Mathematics – Page 122
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
Mathematics
Domains/Clusters/
Common Core State Standards
High School
G.G.58 Define, investigate, justify, and
apply similarities (dilations and the
composition of dilations and isometries)
Geometry: Similarity, Right Triangles, & Trigonometry
Understand similarity in terms of similarity transformations.
1. Verify experimentally the properties of dilations given by a center and a scale
factor:
a. A dilation takes a line not passing through the center of the dilation to a
parallel line, and leaves a line passing through the center unchanged.
b. The dilation of a line segment is longer or shorter in the ratio given by the
scale factor.
2. Given two figures, use the definition of similarity in terms of similarity
transformations to decide if they are similar; explain using similarity
transformations the meaning of similarity for triangles as the equality of all
corresponding pairs of angles and the proportionality of all corresponding
pairs of sides.
Geometry: Similarity, Right Triangles, & Trigonometry
Understand similarity in terms of similarity transformations.
2. Given two figures, use the definition of similarity in terms of similarity
transformations to decide if they are similar; explain using similarity
transformations the meaning of similarity for triangles as the equality of all
corresponding pairs of angles and the proportionality of all corresponding
pairs of sides.
G.G.59 Investigate, justify, and apply
the properties that remain invariant
under similarities
National Essential Skills Study (NESS)
National Rankings
Rank
M10
M54
M10
M54
G.G.60 Identify specific similarities by
observing orientation, numbers of
invariant points, and/or parallelism
Geometry: Similarity, Right Triangles, & Trigonometry
Prove theorems involving similarity.
4. Prove theorems about triangles. Theorems include: a line parallel to one side
of a triangle divides the other two proportionally, and conversely; the
Pythagorean Theorem proved using triangle similarity.
M10
M54
 2011 International Center for Leadership in Education
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Apply transformations (reflection, rotation,
translation, and dilation) of 2-dimensional
figures graphically to interpret, analyze, and
illustrate the concepts of congruency,
similarity, and symmetry.
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Apply transformations (reflection, rotation,
translation, and dilation) of 2-dimensional
figures graphically to interpret, analyze, and
illustrate the concepts of congruency,
similarity, and symmetry.
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Apply transformations (reflection, rotation,
translation, and dilation) of 2-dimensional
figures graphically to interpret, analyze, and
illustrate the concepts of congruency,
similarity, and symmetry.
Mathematics – Page 123
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
Mathematics
Domains/Clusters/
Common Core State Standards
High School
G.G.61 Investigate, justify, and apply
the analytical representations for
translations, rotations about the origin
of 90º and 180º, reflections over the
lines x  0 , y  0 , and y  x , and
dilations centered at the origin
Geometry: Congruence
Experiment with transformations in the plane.
5. Given a geometric figure and a rotation, reflection, or translation, draw the
transformed figure using, e.g., graph paper, tracing paper, or geometry
software. Specify a sequence of transformations that will carry a given figure
onto another.
National Essential Skills Study (NESS)
National Rankings
Rank
M10
M54
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Apply transformations (reflection, rotation,
translation, and dilation) of 2-dimensional
figures graphically to interpret, analyze, and
illustrate the concepts of congruency,
similarity, and symmetry.
Students will apply coordinate
geometry to analyze problem solving
situations.
Coordinate Geometry
G.G.62 Find the slope of a
perpendicular line, given the equation
of a line
G.G.63 Determine whether two lines
are parallel, perpendicular, or neither,
given their equations
Geometry: Expressing Geometric Properties with Equations
Use coordinates to prove simple geometric theorems algebraically.
5. Prove the slope criteria for parallel and perpendicular lines and use them to
solve geometric problems (e.g., find the equation of a line parallel or
perpendicular to a given line that passes through a given point).
Geometry: Expressing Geometric Properties with Equations
Use coordinates to prove simple geometric theorems algebraically.
5. Prove the slope criteria for parallel and perpendicular lines and use them to
solve geometric problems (e.g., find the equation of a line parallel or
perpendicular to a given line that passes through a given point).
 2011 International Center for Leadership in Education
M4
M44
M4
M44
Understand the properties of and apply
parallel, perpendicular, and intersecting
lines in problem-solving situations.
Know the equation of a line and interpret
graphically using the slope-intercept form (y
= mx+b) and the point-slope form (y-b =
m(x-a)).
Understand the properties of and apply
parallel, perpendicular, and intersecting
lines in problem-solving situations.
Know the equation of a line and interpret
graphically using the slope-intercept form (y
= mx+b) and the point-slope form (y-b =
m(x-a)).
Mathematics – Page 124
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
G.G.64 Find the equation of a line,
given a point on the line and the
equation of a line perpendicular to the
given line
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Geometry: Expressing Geometric Properties with Equations
Use coordinates to prove simple geometric theorems algebraically.
5. Prove the slope criteria for parallel and perpendicular lines and use them to
solve geometric problems (e.g., find the equation of a line parallel or
perpendicular to a given line that passes through a given point).
National Essential Skills Study (NESS)
National Rankings
Rank
M4
M44
M46
G.G.65 Find the equation of a line,
given a point on the line and the
equation of a line parallel to the desired
line
Geometry: Expressing Geometric Properties with Equations
Use coordinates to prove simple geometric theorems algebraically.
5. Prove the slope criteria for parallel and perpendicular lines and use them to
solve geometric problems (e.g., find the equation of a line parallel or
perpendicular to a given line that passes through a given point).
M4
M44
M46
G.G.66 Find the midpoint of a line
segment, given its endpoints
G.G.67 Find the length of a line
segment, given its endpoints
Geometry: Expressing Geometric Properties with Equations
Use coordinates to prove simple geometric theorems algebraically.
6. Find the point on a directed line segment between two given points that
partitions the segment in a given ratio.
Geometry: Expressing Geometric Properties with Equations
Use coordinates to prove simple geometric theorems algebraically.
7. Use coordinates to compute perimeters of polygons and areas of triangles and
rectangles, e.g., using the distance formula.
 2011 International Center for Leadership in Education
M19
M19
Understand the properties of and apply
parallel, perpendicular, and intersecting
lines in problem-solving situations.
Know the equation of a line and interpret
graphically using the slope-intercept form (y
= mx+b) and the point-slope form (y-b =
m(x-a)).
Know the equation for the slope of a line
and compute slope given the coordinates of
two points.
Understand the properties of and apply
parallel, perpendicular, and intersecting
lines in problem-solving situations.
Know the equation of a line and interpret
graphically using the slope-intercept form (y
= mx+b) and the point-slope form (y-b =
m(x-a)).
Know the equation for the slope of a line
and compute slope given the coordinates of
two points.
Compute the distance between two points on
a coordinate plane (length of a line segment)
and determine the midpoint of a line
segment between two points.
Compute the distance between two points on
a coordinate plane (length of a line segment)
and determine the midpoint of a line
segment between two points.
Mathematics – Page 125
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
G.G.68 Find the equation of a line that
is the perpendicular bisector of a line
segment, given the endpoints of the line
segment
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Geometry: Expressing Geometric Properties with Equations
Use coordinates to prove simple geometric theorems algebraically.
5. Prove the slope criteria for parallel and perpendicular lines and use them to
solve geometric problems (e.g., find the equation of a line parallel or
perpendicular to a given line that passes through a given point).
National Essential Skills Study (NESS)
National Rankings
Rank
M4
M44
M46
G.G.69 Investigate, justify, and apply
the properties of triangles and
quadrilaterals in the coordinate plane,
using the distance, midpoint, and slope
formulas
Geometry: Expressing Geometric Properties with Equations
Use coordinates to prove simple geometric theorems algebraically.
7. Use coordinates to compute perimeters of polygons and areas of triangles and
rectangles, e.g., using the distance formula.
M19
M34
M46
G.G.70 Solve systems of equations
involving one linear equation and one
quadratic equation graphically
G.G.71 Write the equation of a circle,
given its center and radius or given the
endpoints of a diameter
Algebra: Reasoning with Equations & Inequalities
Solve systems of equations.
7. Solve a simple system consisting of a linear equation and a quadratic
equation in two variables algebraically and graphically. For example, find the
points of intersection between the line y = –3x and the circle x2 + y2 = 3.
Geometry: Expressing Geometric Properties with Equations
Translate between the geometric description and the equation for a conic
section.
1. Derive the equation of a circle of given center and radius using the
Pythagorean Theorem; complete the square to find the center and radius of a
circle given by an equation.
 2011 International Center for Leadership in Education
M40
M66
M66
Understand the properties of and apply
parallel, perpendicular, and intersecting
lines in problem-solving situations.
Know the equation of a line and interpret
graphically using the slope-intercept form (y
= mx+b) and the point-slope form (y-b =
m(x-a)).
Know the equation for the slope of a line
and compute slope given the coordinates of
two points.
Compute the distance between two points on
a coordinate plane (length of a line segment)
and determine the midpoint of a line
segment between two points.
Understand the properties and classification
of polygons (triangles, the family of
quadrilaterals, pentagon, hexagon, etc.) and
apply knowledge of angle and side
relationships of geometric shapes in
problem-solving situations.
Know the equation for the slope of a line
and compute slope given the coordinates of
two points.
Solve systems of linear equations
algebraically or graphically.
Know how to sketch basic conic sections
(e.g., circles, parabolas) by using their
equations and solve systems of non-linear
equations graphically.
Know how to sketch basic conic sections
(e.g., circles, parabolas) by using their
equations and solve systems of non-linear
equations graphically.
Mathematics – Page 126
New York Mathematics
Strands/Bands/
Performance Indicators
Geometry
G.G.72 Write the equation of a circle,
given its graph. Note: The center is an
ordered pair of integers and the radius
is an integer.
G.G.73 Find the center and radius of a
circle, given the equation of the circle
in center-radius form
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Geometry: Expressing Geometric Properties with Equations
Translate between the geometric description and the equation for a conic
section.
1. Derive the equation of a circle of given center and radius using the
Pythagorean Theorem; complete the square to find the center and radius of a
circle given by an equation.
Geometry: Expressing Geometric Properties with Equations
Translate between the geometric description and the equation for a conic
section.
1. Derive the equation of a circle of given center and radius using the
Pythagorean Theorem; complete the square to find the center and radius of a
circle given by an equation.
National Essential Skills Study (NESS)
National Rankings
Rank
M66
M18
M66
G.G.74 Graph circles of the form
( x  h) 2  ( j  k ) 2  r 2
Geometry: Expressing Geometric Properties with Equations
Translate between the geometric description and the equation for a conic
section.
1. Derive the equation of a circle of given center and radius using the
Pythagorean Theorem; complete the square to find the center and radius of a
circle given by an equation.
 2011 International Center for Leadership in Education
M66
Know how to sketch basic conic sections
(e.g., circles, parabolas) by using their
equations and solve systems of non-linear
equations graphically.
Understand the properties of circles (radius,
arc, diameter, chord, secant, and tangent)
and apply circle quantities (lengths of line
segments, angle measure within a circle,
circumference, and area) in problem-solving
situations.
Know how to sketch basic conic sections
(e.g., circles, parabolas) by using their
equations and solve systems of non-linear
equations graphically.
Know how to sketch basic conic sections
(e.g., circles, parabolas) by using their
equations and solve systems of non-linear
equations graphically.
Mathematics – Page 127
New York Mathematics Standards Alignments
New York Mathematics
Strands/Bands/
Performance Indicators
Algebra 2/Trigonometry
Mathematics
Domains/Clusters/
Common Core State Standards
High School
National Essential Skills Study (NESS)
National Rankings
Rank
Problem Solving Strand
Students will new mathematical
knowledge through problem solving.
A2.PS.1 Use a variety of problem
solving strategies to understand new
mathematical content
Number & Quantity: Quantities
Reason quantitatively and use units to solve problems.
1. Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and
interpret the scale and the origin in graphs and data displays.
2. Define appropriate quantities for the purpose of descriptive modeling.
3. Choose a level of accuracy appropriate to limitations on measurement when
reporting quantities.
Algebra: Creating Equations
Create equations that describe numbers or relationships.
1. Create equations and inequalities in one variable and use them to solve
problems. Include equations arising from linear and quadratic functions, and
simple rational and exponential functions.
Algebra: Reasoning with Equations & Inequalities
Understand solving equations as a process of reasoning and explain the
reasoning.
1. Explain each step in solving a simple equation as following from the equality
of numbers asserted at the previous step, starting from the assumption that the
original equation has a solution. Construct a viable argument to justify a
solution method.
2. Solve simple rational and radical equations in one variable, and give
examples showing how extraneous solutions may arise.
Solve equations and inequalities in one variable.
3. Solve linear equations and inequalities in one variable, including equations
with coefficients represented by letters.
4. Solve quadratic equations in one variable.
a. Use the method of completing the square to transform any quadratic
equation in x into an equation of the form (x – p)2 = q that has the same
solutions. Derive the quadratic formula from this form.
 2011 International Center for Leadership in Education
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Mathematics – Page 128
New York Mathematics
Strands/Bands/
Performance Indicators
Algebra 2/Trigonometry
A2.PS.1 (Continued from previous
page)
A2.PS.2 Recognize and understand
equivalent representations of a problem
situation or a mathematical concept
Mathematics
Domains/Clusters/
Common Core State Standards
High School
(Continued from previous page)
b. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square
roots, completing the square, the quadratic formula and factoring, as
appropriate to the initial form of the equation. Recognize when the
quadratic formula gives complex solutions and write them as a ± bi for real
numbers a and b.
Functions: Interpreting Functions
Interpret functions that arise in applications in terms of the context.
5. Relate the domain of a function to its graph and, where applicable, to the
quantitative relationship it describes. For example, if the function h(n) gives
the number of person-hours it takes to assemble n engines in a factory, then
the positive integers would be an appropriate domain for the function.
Analyze functions using different representations.
7. Graph functions expressed symbolically and show key features of the graph,
by hand in simple cases and using technology for more complicated cases.
a. Graph linear and quadratic functions and show intercepts, maxima, and
minima.
b. Graph square root, cube root, and piecewise-defined functions, including
step functions and absolute value functions.
c. Graph polynomial functions, identifying zeros when suitable factorizations
are available, and showing end behavior.
e. Graph exponential and logarithmic functions, showing intercepts and end
behavior, and trigonometric functions, showing period, midline, and
amplitude.
8. Write a function defined by an expression in different but equivalent forms to
reveal and explain different properties of the function.
a. Use the process of factoring and completing the square in a quadratic
function to show zeros, extreme values, and symmetry of the graph, and
interpret these in terms of a context.
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Mathematics – Page 129
New York Mathematics
Strands/Bands/
Performance Indicators
Algebra 2/Trigonometry
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)
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Algebra: Seeing Structure in Expressions
Write expressions in equivalent forms to solve problems.
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.
Algebra: Creating Equations
Create equations that describe numbers or relationships.
1. Create equations and inequalities in one variable and use them to solve
problems. Include equations arising from linear and quadratic functions, and
simple rational and exponential functions.
2. Create equations in two or more variables to represent relationships between
quantities; graph equations on coordinate axes with labels and scales.
Functions: Building Functions
Build a function that models a relationship between two quantities.
1. Write a function that describes a relationship between two quantities.
a. Determine an explicit expression, a recursive process, or steps for
calculation from a context.
2. Write arithmetic and geometric sequences both recursively and with an
explicit formula, use them to model situations, and translate between the two
forms.
Functions: Interpreting Functions
Interpret functions that arise in applications in terms of the context.
5. Relate the domain of a function to its graph and, where applicable, to the
quantitative relationship it describes. For example, if the function h(n) gives
the number of person-hours it takes to assemble n engines in a factory, then
the positive integers would be an appropriate domain for the function.
Analyze functions using different representations.
7. Graph functions expressed symbolically and show key features of the graph,
by hand in simple cases and using technology for more complicated cases.
a. Graph linear and quadratic functions and show intercepts, maxima, and
minima.
b. Graph square root, cube root, and piecewise-defined functions, including
step functions and absolute value functions.
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
Apply pattern recognition in data sets and
series to reason or solve problems involving
arithmetic, geometry, exponents, etc.
M16
M10
M21
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Evaluate and employ accurate and
appropriate procedures for statistical data
collection, organization, analysis, and
display including making estimates and
predictions, critiquing data, and drawing
inferences (e.g., using the normal curve and
z-scores, line of best fit).
Mathematics – Page 130
New York Mathematics
Strands/Bands/
Performance Indicators
Algebra 2/Trigonometry
A2.PS.4 (Continued from previous
page)
Mathematics
Domains/Clusters/
Common Core State Standards
High School
National Essential Skills Study (NESS)
National Rankings
Rank
(Continued from previous page)
c. Graph polynomial functions, identifying zeros when suitable factorizations
are available, and showing end behavior.
e. Graph exponential and logarithmic functions, showing intercepts and end
behavior, and trigonometric functions, showing period, midline, and
amplitude.
8. Write a function defined by an expression in different but equivalent forms to
reveal and explain different properties of the function.
a. Use the process of factoring and completing the square in a quadratic
function to show zeros, extreme values, and symmetry of the graph, and
interpret these in terms of a context.
Statistics & Probability: Interpreting Categorical & Quantitative Data
Summarize, represent and interpret data on a single count or measurement
variable.
1. Represent data with plots on the real number line (dot plots, histograms, and
box plots).
Summarize, represent and interpret data on two categorical and
quantitative variables.
6. Represent data on two quantitative variables on a scatter plot, and describe
how the variables are related.
a. Fit a function to the data; use functions fitted to data to solve problems in
the context of the data. Use given functions or choose a function suggested
by the context. Emphasize linear, quadratic, and exponential models.
 2011 International Center for Leadership in Education
Mathematics – Page 131
New York Mathematics
Strands/Bands/
Performance Indicators
Algebra 2/Trigonometry
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)
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Functions: Interpreting Functions
Interpret functions that arise in applications in terms of the context.
5. Relate the domain of a function to its graph and, where applicable, to the
quantitative relationship it describes. For example, if the function h(n) gives
the number of person-hours it takes to assemble n engines in a factory, then
the positive integers would be an appropriate domain for the function.
Analyze functions using different representations.
7. Graph functions expressed symbolically and show key features of the graph,
by hand in simple cases and using technology for more complicated cases.
a. Graph linear and quadratic functions and show intercepts, maxima, and
minima.
b. Graph square root, cube root, and piecewise-defined functions, including
step functions and absolute value functions.
c. Graph polynomial functions, identifying zeros when suitable factorizations
are available, and showing end behavior.
e. Graph exponential and logarithmic functions, showing intercepts and end
behavior, and trigonometric functions, showing period, midline, and
amplitude.
8. Write a function defined by an expression in different but equivalent forms to
reveal and explain different properties of the function.
a. Use the process of factoring and completing the square in a quadratic
function to show zeros, extreme values, and symmetry of the graph, and
interpret these in terms of a context.
Statistics & Probability: Interpreting Categorical & Quantitative Data
Summarize, represent and interpret data on a single count or measurement
variable.
1. Represent data with plots on the real number line (dot plots, histograms, and
box plots).
Summarize, represent and interpret data on two categorical and
quantitative variables.
6. Represent data on two quantitative variables on a scatter plot, and describe
how the variables are related.
a. Fit a function to the data; use functions fitted to data to solve problems in
the context of the data. Use given functions or choose a function suggested
by the context. Emphasize linear, quadratic, and exponential models.
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Mathematics – Page 132
New York Mathematics
Strands/Bands/
Performance Indicators
Algebra 2/Trigonometry
A2.PS.6 Use a variety of strategies to
extend solution methods to other
problems
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Functions: Interpreting Functions
Interpret functions that arise in applications in terms of the context.
5. Relate the domain of a function to its graph and, where applicable, to the
quantitative relationship it describes. For example, if the function h(n) gives
the number of person-hours it takes to assemble n engines in a factory, then
the positive integers would be an appropriate domain for the function.
Analyze functions using different representations.
7. Graph functions expressed symbolically and show key features of the graph,
by hand in simple cases and using technology for more complicated cases.
a. Graph linear and quadratic functions and show intercepts, maxima, and
minima.
b. Graph square root, cube root, and piecewise-defined functions, including
step functions and absolute value functions.
c. Graph polynomial functions, identifying zeros when suitable factorizations
are available, and showing end behavior.
e. Graph exponential and logarithmic functions, showing intercepts and end
behavior, and trigonometric functions, showing period, midline, and
amplitude.
8. Write a function defined by an expression in different but equivalent forms to
reveal and explain different properties of the function.
a. Use the process of factoring and completing the square in a quadratic
function to show zeros, extreme values, and symmetry of the graph, and
interpret these in terms of a context.
Statistics & Probability: Interpreting Categorical & Quantitative Data
Summarize, represent and interpret data on a single count or measurement
variable.
1. Represent data with plots on the real number line (dot plots, histograms, and
box plots).
Summarize, represent and interpret data on two categorical and
quantitative variables.
6. Represent data on two quantitative variables on a scatter plot, and describe
how the variables are related.
a. Fit a function to the data; use functions fitted to data to solve problems in
the context of the data. Use given functions or choose a function suggested
by the context. Emphasize linear, quadratic, and exponential models.
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Mathematics – Page 133
New York Mathematics
Strands/Bands/
Performance Indicators
Algebra 2/Trigonometry
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 the 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
Mathematics
Domains/Clusters/
Common Core State Standards
High School
National Essential Skills Study (NESS)
National Rankings
Rank
There is no New York Mathematics Performance Indicator–Common Core
alignment.
M10
There is no New York Mathematics Performance Indicator–Common Core
alignment.
M10
Algebra: Reasoning with Equations & Inequalities
Understand solving equations as a process of reasoning and explain the
reasoning.
2. Solve simple rational and radical equations in one variable, and give
examples showing how extraneous solutions may arise.
Functions: Interpreting Functions
Interpret functions that arise in applications in terms of the context.
5. Relate the domain of a function to its graph and, where applicable, to the
quantitative relationship it describes. For example, if the function h(n) gives
the number of person-hours it takes to assemble n engines in a factory, then
the positive integers would be an appropriate domain for the function.
Functions: Interpreting Functions
Interpret functions that arise in applications in terms of the context.
5. Relate the domain of a function to its graph and, where applicable, to the
quantitative relationship it describes. For example, if the function h(n) gives
the number of person-hours it takes to assemble n engines in a factory, then
the positive integers would be an appropriate domain for the function.
Analyze functions using different representations.
8. Write a function defined by an expression in different but equivalent forms to
reveal and explain different properties of the function.
a. Use the process of factoring and completing the square in a quadratic
function to show zeros, extreme values, and symmetry of the graph, and
interpret these in terms of a context.
 2011 International Center for Leadership in Education
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Mathematics – Page 134
New York Mathematics
Strands/Bands/
Performance Indicators
Algebra 2/Trigonometry
Mathematics
Domains/Clusters/
Common Core State Standards
High School
National Essential Skills Study (NESS)
National Rankings
Rank
Reasoning and Proof Strand
Students will recognize reasoning and
proof as fundamental aspects of
mathematics.
A2.RP.1 Support mathematical ideas
using a variety of strategies
Algebra: Reasoning with Equations & Inequalities
Understand solving equations as a process of reasoning and explain the
reasoning.
1. Explain each step in solving a simple equation as following from the equality
of numbers asserted at the previous step, starting from the assumption that the
original equation has a solution. Construct a viable argument to justify a
solution method.
2. Solve simple rational and radical equations in one variable, and give
examples showing how extraneous solutions may arise.
Functions: Trigonometric Functions
Prove and apply trigonometric identities.
8. Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to calculate
trigonometric ratios.
9. (+) Prove the addition and subtraction formulas for sine, cosine, and tangent
and use them to solve problems.
Geometry: Similarity, Right Triangles, & Trigonometry
Apply trigonometry to general triangles.
9. (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing
an auxiliary line from a vertex perpendicular to the opposite side.
10. (+) Prove the Laws of Sines and Cosines and use them to solve problems.
 2011 International Center for Leadership in Education
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Mathematics – Page 135
New York Mathematics
Strands/Bands/
Performance Indicators
Algebra 2/Trigonometry
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
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Algebra: Reasoning with Equations & Inequalities
Understand solving equations as a process of reasoning and explain the
reasoning.
1. Explain each step in solving a simple equation as following from the equality
of numbers asserted at the previous step, starting from the assumption that the
original equation has a solution. Construct a viable argument to justify a
solution method.
2. Solve simple rational and radical equations in one variable, and give
examples showing how extraneous solutions may arise.
Functions: Trigonometric Functions
Prove and apply trigonometric identities.
8. Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to calculate
trigonometric ratios.
9. (+) Prove the addition and subtraction formulas for sine, cosine, and tangent
and use them to solve problems.
Geometry: Similarity, Right Triangles, & Trigonometry
Apply trigonometry to general triangles.
9. (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing
an auxiliary line from a vertex perpendicular to the opposite side.
10. (+) Prove the Laws of Sines and Cosines and use them to solve problems.
Number & Quantity: Quantities
Reason quantitatively and use units to solve problems.
1. Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and
interpret the scale and the origin in graphs and data displays.
2. Define appropriate quantities for the purpose of descriptive modeling.
3. Choose a level of accuracy appropriate to limitations on measurement when
reporting quantities.
Statistics & Probability: Interpreting Categorical & Quantitative Data
Summarize, represent and interpret data on a single count or measurement
variable.
3. Interpret differences in shape, center, and spread in the context of the data
sets, accounting for possible effects of extreme data points (outliers).
4. Use the mean and standard deviation of a data set to fit it to a normal
distribution and to estimate population percentages. Recognize that there are
data sets for which such a procedure is not appropriate. Use calculators,
spreadsheets, and tables to estimate areas under the normal curve.
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
M10
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Use the technique of dimensional analysis to
convert units of measure (e.g.,
kilometers/hour to meters/minute) and apply
ratios in real-world situations (e.g., scale
drawings).
M13
Mathematics – Page 136
New York Mathematics
Strands/Bands/
Performance Indicators
Algebra 2/Trigonometry
Mathematics
Domains/Clusters/
Common Core State Standards
High School
A2.RP.4 Recognize when an
approximation is more appropriate than
an exact answer
Number & Quantity: Quantities
Reason quantitatively and use units to solve problems.
1. Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and
interpret the scale and the origin in graphs and data displays.
2. Define appropriate quantities for the purpose of descriptive modeling.
3. Choose a level of accuracy appropriate to limitations on measurement when
reporting quantities.
Statistics & Probability: Interpreting Categorical & Quantitative Data
Summarize, represent and interpret data on a single count or measurement
variable.
3. Interpret differences in shape, center, and spread in the context of the data
sets, accounting for possible effects of extreme data points (outliers).
4. Use the mean and standard deviation of a data set to fit it to a normal
distribution and to estimate population percentages. Recognize that there are
data sets for which such a procedure is not appropriate. Use calculators,
spreadsheets, and tables to estimate areas under the normal curve.
Students will develop and evaluate
mathematical arguments and proofs.
A2.RP.5 Develop, verify, and explain
an argument, using appropriate
mathematical ideas and language
Algebra: Seeing Structure in Expressions
Write expressions in equivalent forms to solve problems.
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.
Functions: Trigonometric Functions
Prove and apply trigonometric identities.
8. Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to calculate
trigonometric ratios.
9. (+) Prove the addition and subtraction formulas for sine, cosine, and tangent
and use them to solve problems.
Geometry: Similarity, Right Triangles, & Trigonometry
Apply trigonometry to general triangles.
9. (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing
an auxiliary line from a vertex perpendicular to the opposite side.
10. (+) Prove the Laws of Sines and Cosines and use them to solve problems.
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
M10
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Use the technique of dimensional analysis to
convert units of measure (e.g.,
kilometers/hour to meters/minute) and apply
ratios in real-world situations (e.g., scale
drawings).
M13
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Mathematics – Page 137
New York Mathematics
Strands/Bands/
Performance Indicators
Algebra 2/Trigonometry
A2.RP.5 (Continued from previous
page)
A2.RP.6 Construct logical arguments
that verify claims or counterexamples
that refute claims
Mathematics
Domains/Clusters/
Common Core State Standards
High School
(Continued from previous page)
Geometry: Expressing Geometric Properties with Equations
Translate between the geometric description and the equation for a conic
section.
1. Derive the equation of a circle of given center and radius using the
Pythagorean Theorem; complete the square to find the center and radius of a
circle given by an equation.
Algebra: Seeing Structure in Expressions
Write expressions in equivalent forms to solve problems.
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.
Functions: Trigonometric Functions
Prove and apply trigonometric identities.
8. Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to calculate
trigonometric ratios.
9. (+) Prove the addition and subtraction formulas for sine, cosine, and tangent
and use them to solve problems.
Geometry: Similarity, Right Triangles, & Trigonometry
Apply trigonometry to general triangles.
9. (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing
an auxiliary line from a vertex perpendicular to the opposite side.
10. (+) Prove the Laws of Sines and Cosines and use them to solve problems.
Geometry: Expressing Geometric Properties with Equations
Translate between the geometric description and the equation for a conic
section.
1. Derive the equation of a circle of given center and radius using the
Pythagorean Theorem; complete the square to find the center and radius of a
circle given by an equation.
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Mathematics – Page 138
New York Mathematics
Strands/Bands/
Performance Indicators
Algebra 2/Trigonometry
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
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Algebra: Seeing Structure in Expressions
Write expressions in equivalent forms to solve problems.
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.
Functions: Trigonometric Functions
Prove and apply trigonometric identities.
8. Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to calculate
trigonometric ratios.
9. (+) Prove the addition and subtraction formulas for sine, cosine, and tangent
and use them to solve problems.
Geometry: Similarity, Right Triangles, & Trigonometry
Apply trigonometry to general triangles.
9. (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing
an auxiliary line from a vertex perpendicular to the opposite side.
10. (+) Prove the Laws of Sines and Cosines and use them to solve problems.
Geometry: Expressing Geometric Properties with Equations
Translate between the geometric description and the equation for a conic
section.
1. Derive the equation of a circle of given center and radius using the
Pythagorean Theorem; complete the square to find the center and radius of a
circle given by an equation.
There is no New York Mathematics Performance Indicator–Common Core
alignment.
National Essential Skills Study (NESS)
National Rankings
Rank
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
M10
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
There is no New York Mathematics Performance Indicator–Common Core
alignment.
 2011 International Center for Leadership in Education
Mathematics – Page 139
New York Mathematics
Strands/Bands/
Performance Indicators
Algebra 2/Trigonometry
Mathematics
Domains/Clusters/
Common Core State Standards
High School
A2.RP.10 Devise ways to verify
results, using counterexamples and
informal indirect proof
There is no New York Mathematics Performance Indicator–Common Core
alignment.
A2.RP.11 Extend specific results to
more general cases
There is no New York Mathematics Performance Indicator–Common Core
alignment.
National Essential Skills Study (NESS)
National Rankings
Rank
M10
M10
A2.RP.12 Apply inductive reasoning in
making and supporting mathematical
conjectures
Algebra: Arithmetic with Polynomials & Rational Expressions
Use polynomial identities to solve problems.
5. (+) Know and apply the Binomial Theorem for the expansion of (x + y)n in
powers of x and y for a positive integer n, where x and y are any numbers,
with coefficients determined for example by Pascal’s Triangle. [The Binomial
Theorem can be proved by mathematical induction or by a combinatorial
argument.]
M10
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Communication Strand
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
Algebra: Seeing Structure in Expressions
Interpret the structure of expressions.
1. Interpret expressions that represent a quantity in terms of its context.
a. Interpret parts of an expression, such as terms, factors, and coefficients.
Algebra: Reasoning with Equations & Inequalities
Understand solving equations as a process of reasoning and explain the
reasoning.
1. Explain each step in solving a simple equation as following from the equality
of numbers asserted at the previous step, starting from the assumption that the
original equation has a solution. Construct a viable argument to justify a
solution method.
 2011 International Center for Leadership in Education
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Mathematics – Page 140
New York Mathematics
Strands/Bands/
Performance Indicators
Algebra 2/Trigonometry
Mathematics
Domains/Clusters/
Common Core State Standards
High School
A2.CM.2 Use mathematical
representations to communicate with
appropriate accuracy, including
numerical tables, formulas, functions,
equations, charts, graphs, and diagrams
Number & Quantity: Quantities
Reason quantitatively and use units to solve problems.
1. Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and
interpret the scale and the origin in graphs and data displays.
2. Define appropriate quantities for the purpose of descriptive modeling.
3. Choose a level of accuracy appropriate to limitations on measurement when
reporting quantities.
Algebra: Seeing Structure in Expressions
Write expressions in equivalent forms to solve problems.
3. Choose and produce an equivalent form of an expression to reveal and
explain properties of the quantity represented by the expression.
c. Use the properties of exponents to transform expressions for exponential
functions. For example the expression 1.15t can be rewritten as (1.151/12)12t
≈ 1.01212t to reveal the approximate equivalent monthly interest rate if the
annual rate is 15%.
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.
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
Algebra: Arithmetic with Polynomials & Rational Expressions
Use polynomial identities to solve problems.
5. (+) Know and apply the Binomial Theorem for the expansion of (x + y)n in
powers of x and y for a positive integer n, where x and y are any numbers,
with coefficients determined for example by Pascal’s Triangle. [The Binomial
Theorem can be proved by mathematical induction or by a combinatorial
argument.]
Statistics & Probability: Interpreting Categorical & Quantitative Data
Summarize, represent and interpret data on a single count or measurement
variable.
2. Use statistics appropriate to the shape of the data distribution to compare
center (median, mean) and spread (interquartile range, standard deviation) of
two or more different data sets.
Interpret linear models.
8. Compute (using technology) and interpret the correlation coefficient of a
linear fit.
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
M7
M10
M21
Simplify and solve algebraic equations by
identifying and using the correct order of
operations and techniques necessary to carry
out the solution.
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Evaluate and employ accurate and
appropriate procedures for statistical data
collection, organization, analysis, and
display including making estimates and
predictions, critiquing data, and drawing
inferences (e.g., using the normal curve and
z-scores, line of best fit).
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Mathematics – Page 141
New York Mathematics
Strands/Bands/
Performance Indicators
Algebra 2/Trigonometry
Mathematics
Domains/Clusters/
Common Core State Standards
High School
A2.CM.4 Explain relationships among
different representations of a problem
Functions: Trigonometric Functions
Extend the domain of trigonometric functions using the unit circle.
2. Explain how the unit circle in the coordinate plane enables the extension of
trigonometric functions to all real numbers, interpreted as radian measures of
angles traversed counterclockwise around the unit circle.
Algebra: Reasoning with Equations & Inequalities
Understand solving equations as a process of reasoning and explain the
reasoning.
1. Explain each step in solving a simple equation as following from the equality
of numbers asserted at the previous step, starting from the assumption that the
original equation has a solution. Construct a viable argument to justify a
solution method.
Functions: Trigonometric Functions
Extend the domain of trigonometric functions using the unit circle.
2. Explain how the unit circle in the coordinate plane enables the extension of
trigonometric functions to all real numbers, interpreted as radian measures of
angles traversed counterclockwise around the unit circle.
There is no New York Mathematics Performance Indicator–Common Core
alignment.
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
National Essential Skills Study (NESS)
National Rankings
Rank
M10
M10
M10
There is no New York Mathematics Performance Indicator–Common Core
alignment.
M10
There is no New York Mathematics Performance Indicator–Common Core
alignment.
M10
 2011 International Center for Leadership in Education
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Mathematics – Page 142
New York Mathematics
Strands/Bands/
Performance Indicators
Algebra 2/Trigonometry
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
Mathematics
Domains/Clusters/
Common Core State Standards
High School
National Essential Skills Study (NESS)
National Rankings
Rank
There is no New York Mathematics Performance Indicator–Common Core
alignment.
M10
There is no New York Mathematics Performance Indicator–Common Core
alignment.
M10
Algebra: Creating Equations
Create equations that describe numbers or relationships.
1. Create equations and inequalities in one variable and use them to solve
problems. Include equations arising from linear and quadratic functions, and
simple rational and exponential functions.
Algebra: Creating Equations
Create equations that describe numbers or relationships.
1. Create equations and inequalities in one variable and use them to solve
problems. Include equations arising from linear and quadratic functions, and
simple rational and exponential functions.
Algebra: Reasoning with Equations & Inequalities
Understand solving equations as a process of reasoning and explain the
reasoning.
1. Explain each step in solving a simple equation as following from the equality
of numbers asserted at the previous step, starting from the assumption that the
original equation has a solution. Construct a viable argument to justify a
solution method.
 2011 International Center for Leadership in Education
M10
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Mathematics – Page 143
New York Mathematics
Strands/Bands/
Performance Indicators
Algebra 2/Trigonometry
A2.CM.13 Draw conclusions about
mathematical ideas through decoding,
comprehension, and interpretation of
mathematical visuals, symbols, and
technical writing
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Functions: Linear, Quadratic & Exponential
Construct and compare linear, quadratic, and exponential models and
solve problems.
1. Distinguish between situations that can be modeled with linear functions and
with exponential functions.
a. Prove that linear functions grow by equal differences over equal intervals,
and that exponential functions grow by equal factors over equal intervals.
b. Recognize situations in which one quantity changes at a constant rate per
unit interval relative to another.
c. Recognize situations in which a quantity grows or decays by a constant
percent rate per unit interval relative to another.
2. Construct linear and exponential functions, including arithmetic and
geometric sequences, given a graph, a description of a relationship, or two
input-output pairs (include reading these from a table).
3. Observe using graphs and tables that a quantity increasing exponentially
eventually exceeds a quantity increasing linearly, quadratically, or (more
generally) as a polynomial function.
National Essential Skills Study (NESS)
National Rankings
Rank
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Connections Strand
Students will recognize and use
connections among mathematical ideas.
A2.CN.1 Understand and make
connections among multiple
representations of the same
mathematical idea
Algebra: Reasoning with Equations & Inequalities
Solve equations and inequalities in one variable.
4. Solve quadratic equations in one variable.
a. Use the method of completing the square to transform any quadratic
equation in x into an equation of the form (x – p)2 = q that has the same
solutions. Derive the quadratic formula from this form.
Functions: Interpreting Functions
Analyze functions using different representations.
8. Write a function defined by an expression in different but equivalent forms to
reveal and explain different properties of the function.
a. Use the process of factoring and completing the square in a quadratic
function to show zeros, extreme values, and symmetry of the graph, and
interpret these in terms of a context.
 2011 International Center for Leadership in Education
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Mathematics – Page 144
New York Mathematics
Strands/Bands/
Performance Indicators
Algebra 2/Trigonometry
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
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Algebra: Reasoning with Equations & Inequalities
Solve equations and inequalities in one variable.
4. Solve quadratic equations in one variable.
a. Use the method of completing the square to transform any quadratic
equation in x into an equation of the form (x – p)2 = q that has the same
solutions. Derive the quadratic formula from this form.
Functions: Interpreting Functions
Analyze functions using different representations.
8. Write a function defined by an expression in different but equivalent forms to
reveal and explain different properties of the function.
a. Use the process of factoring and completing the square in a quadratic
function to show zeros, extreme values, and symmetry of the graph, and
interpret these in terms of a context.
Functions: Building Functions
Build a function that models a relationship between two quantities.
1. Write a function that describes a relationship between two quantities.
a. Determine an explicit expression, a recursive process, or steps for
calculation from a context.
b. Combine standard function types using arithmetic operations. For example,
build a function that models the temperature of a cooling body by adding a
constant function to a decaying exponential, and relate these functions to
the model.
c. (+) Compose functions. For example, if T(y) is the temperature in the
atmosphere as a function of height, and h(t) is the height of a weather
balloon as a function of time, then T(h(t)) is the temperature at the location
of the weather balloon as a function of time.
2. Write arithmetic and geometric sequences both recursively and with an
explicit formula, use them to model situations, and translate between the two
forms.
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Mathematics – Page 145
New York Mathematics
Strands/Bands/
Performance Indicators
Algebra 2/Trigonometry
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
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Algebra: Arithmetic with Polynomials & Rational Expressions
Understand the relationship between zeros and factors of polynomials.
3. Identify zeros of polynomials when suitable factorizations are available, and
use the zeros to construct a rough graph of the function defined by the
polynomial.
Geometry: Modeling with Geometry
Apply Geometric Concepts in modeling situations
1. Use geometric shapes, their measures, and their properties to describe objects
(e.g., modeling a tree trunk or a human torso as a cylinder).
Geometry: Modeling with Geometry
Apply Geometric Concepts in modeling situations
2. Apply concepts of density based on area and volume in modeling situations
(e.g., persons per square mile, BTUs per cubic foot).
A2.CN.7 Recognize and apply
mathematical ideas to problem
situations that develop outside of
mathematics
Geometry: Modeling with Geometry
Apply Geometric Concepts in modeling situations
1. Use geometric shapes, their measures, and their properties to describe objects
(e.g., modeling a tree trunk or a human torso as a cylinder).
A2.CN.8 Develop an appreciation for
the historical development of
mathematics
There is no New York Mathematics Performance Indicator–Common Core
alignment.
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
M10
M10
M10
M10
M10
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Mathematics – Page 146
New York Mathematics
Strands/Bands/
Performance Indicators
Algebra 2/Trigonometry
Mathematics
Domains/Clusters/
Common Core State Standards
High School
National Essential Skills Study (NESS)
National Rankings
Rank
Representation Strand
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
Algebra: Seeing Structure in Expressions
Write expressions in equivalent forms to solve problems.
3. Choose and produce an equivalent form of an expression to reveal and
explain properties of the quantity represented by the expression.
a. Factor a quadratic expression to reveal the zeros of the function it defines.
b. Complete the square in a quadratic expression to reveal the maximum or
minimum value of the function it defines.
Algebra: Creating Equations
Create equations that describe numbers or relationships.
2. Create equations in two or more variables to represent relationships between
quantities; graph equations on coordinate axes with labels and scales.
Algebra: Reasoning with Equations & Inequalities
Represent and solve equations and inequalities graphically.
10. Understand that the graph of an equation in two variables is the set of all its
solutions plotted in the coordinate plane, often forming a curve (which could
be a line).
Functions: Interpreting Functions
Understand the concept of a function and use function notation.
3. Recognize that sequences are functions, sometimes defined recursively,
whose domain is a subset of the integers. For example, the Fibonacci
sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n
≥ 1.
Analyze functions using different representations.
7. Graph functions expressed symbolically and show key features of the graph,
by hand in simple cases and using technology for more complicated cases.
a. Graph linear and quadratic functions and show intercepts, maxima, and
minima.
b. Graph square root, cube root, and piecewise-defined functions, including
step functions and absolute value functions.
c. Graph polynomial functions, identifying zeros when suitable factorizations
are available, and showing end behavior.
 2011 International Center for Leadership in Education
M7
M10
Simplify and solve algebraic equations by
identifying and using the correct order of
operations and techniques necessary to carry
out the solution.
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Evaluate and employ accurate and
appropriate procedures for statistical data
collection, organization, analysis, and
display including making estimates and
predictions, critiquing data, and drawing
inferences (e.g., using the normal curve and
z-scores, line of best fit).
M21
Mathematics – Page 147
New York Mathematics
Strands/Bands/
Performance Indicators
Algebra 2/Trigonometry
Mathematics
Domains/Clusters/
Common Core State Standards
High School
A2.R.1 (Continued from previous page)
(Continued from previous page)
e. Graph exponential and logarithmic functions, showing intercepts and end
behavior, and trigonometric functions, showing period, midline, and
amplitude.
Statistics & Probability: Interpreting Categorical & Quantitative Data
Summarize, represent and interpret data on a single count or measurement
variable.
1. Represent data with plots on the real number line (dot plots, histograms, and
box plots).
Summarize, represent and interpret data on two categorical and
quantitative variables.
6. Represent data on two quantitative variables on a scatter plot, and describe
how the variables are related.
Algebra: Seeing Structure in Expressions
Write expressions in equivalent forms to solve problems.
3. Choose and produce an equivalent form of an expression to reveal and
explain properties of the quantity represented by the expression.
a. Factor a quadratic expression to reveal the zeros of the function it defines.
b. Complete the square in a quadratic expression to reveal the maximum or
minimum value of the function it defines.
Algebra: Creating Equations
Create equations that describe numbers or relationships.
2. Create equations in two or more variables to represent relationships between
quantities; graph equations on coordinate axes with labels and scales.
Algebra: Reasoning with Equations & Inequalities
Represent and solve equations and inequalities graphically.
10. Understand that the graph of an equation in two variables is the set of all its
solutions plotted in the coordinate plane, often forming a curve (which could
be a line).
Functions: Interpreting Functions
Understand the concept of a function and use function notation.
3. Recognize that sequences are functions, sometimes defined recursively,
whose domain is a subset of the integers. For example, the Fibonacci
sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n
≥ 1.
A2.R.2 Recognize, compare, and use
an array of representational forms
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Mathematics – Page 148
New York Mathematics
Strands/Bands/
Performance Indicators
Algebra 2/Trigonometry
Mathematics
Domains/Clusters/
Common Core State Standards
High School
A2.R.2 (Continued from previous page)
(Continued from previous page)
Analyze functions using different representations.
7. Graph functions expressed symbolically and show key features of the graph,
by hand in simple cases and using technology for more complicated cases.
a. Graph linear and quadratic functions and show intercepts, maxima, and
minima.
b. Graph square root, cube root, and piecewise-defined functions, including
step functions and absolute value functions.
c. Graph polynomial functions, identifying zeros when suitable factorizations
are available, and showing end behavior.
e. Graph exponential and logarithmic functions, showing intercepts and end
behavior, and trigonometric functions, showing period, midline, and
amplitude.
Statistics & Probability: Interpreting Categorical & Quantitative Data
Summarize, represent and interpret data on a single count or measurement
variable.
1. Represent data with plots on the real number line (dot plots, histograms, and
box plots).
Summarize, represent and interpret data on two categorical and
quantitative variables.
6. Represent data on two quantitative variables on a scatter plot, and describe
how the variables are related.
Functions: Interpreting Functions
Understand the concept of a function and use function notation.
3. Recognize that sequences are functions, sometimes defined recursively,
whose domain is a subset of the integers. For example, the Fibonacci
sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n
≥ 1.
Analyze functions using different representations.
7. Graph functions expressed symbolically and show key features of the graph,
by hand in simple cases and using technology for more complicated cases.
a. Graph linear and quadratic functions and show intercepts, maxima, and
minima.
b. Graph square root, cube root, and piecewise-defined functions, including
step functions and absolute value functions.
c. Graph polynomial functions, identifying zeros when suitable factorizations
are available, and showing end behavior.
A2.R.3 Use representation as a tool for
exploring and understanding
mathematical ideas
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Mathematics – Page 149
New York Mathematics
Strands/Bands/
Performance Indicators
Algebra 2/Trigonometry
Mathematics
Domains/Clusters/
Common Core State Standards
High School
A2.R.3 (Continued from previous page)
(Continued from previous page)
e. Graph exponential and logarithmic functions, showing intercepts and end
behavior, and trigonometric functions, showing period, midline, and
amplitude.
Statistics & Probability: Interpreting Categorical & Quantitative Data
Summarize, represent and interpret data on a single count or measurement
variable.
1. Represent data with plots on the real number line (dot plots, histograms, and
box plots).
Summarize, represent and interpret data on two categorical and
quantitative variables.
6. Represent data on two quantitative variables on a scatter plot, and describe
how the variables are related.
Students will select, apply, and
translate among mathematical
representations to solve problems.
A2.R.4 Select appropriate
representations to solve problem
situations
Algebra: Seeing Structure in Expressions
Write expressions in equivalent forms to solve problems.
3. Choose and produce an equivalent form of an expression to reveal and
explain properties of the quantity represented by the expression.
a. Factor a quadratic expression to reveal the zeros of the function it defines.
b. Complete the square in a quadratic expression to reveal the maximum or
minimum value of the function it defines.
Statistics & Probability: Interpreting Categorical & Quantitative Data
Summarize, represent and interpret data on a single count or measurement
variable.
1. Represent data with plots on the real number line (dot plots, histograms, and
box plots).
Summarize, represent and interpret data on two categorical and
quantitative variables.
6. Represent data on two quantitative variables on a scatter plot, and describe
how the variables are related.
a. Fit a function to the data; use functions fitted to data to solve problems in
the context of the data. Use given functions or choose a function suggested
by the context. Emphasize linear, quadratic, and exponential models.
c. Fit a linear function for a scatter plot that suggests a linear association.
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
Mathematics – Page 150
New York Mathematics
Strands/Bands/
Performance Indicators
Algebra 2/Trigonometry
Mathematics
Domains/Clusters/
Common Core State Standards
High School
A2.R.5 Investigate relationships among
different representations and their
impact on a given problem
Algebra: Seeing Structure in Expressions
Write expressions in equivalent forms to solve problems.
3. Choose and produce an equivalent form of an expression to reveal and
explain properties of the quantity represented by the expression.
a. Factor a quadratic expression to reveal the zeros of the function it defines.
b. Complete the square in a quadratic expression to reveal the maximum or
minimum value of the function it defines.
Statistics & Probability: Interpreting Categorical & Quantitative Data
Summarize, represent and interpret data on a single count or measurement
variable.
1. Represent data with plots on the real number line (dot plots, histograms, and
box plots).
Summarize, represent and interpret data on two categorical and
quantitative variables.
6. Represent data on two quantitative variables on a scatter plot, and describe
how the variables are related.
a. Fit a function to the data; use functions fitted to data to solve problems in
the context of the data. Use given functions or choose a function suggested
by the context. Emphasize linear, quadratic, and exponential models.
c. Fit a linear function for a scatter plot that suggests a linear association.
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)
Functions: Trigonometric Functions
Model periodic phenomena with trigonometric functions.
5. Choose trigonometric functions to model periodic phenomena with specified
amplitude, frequency, and midline.
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
M10
M10
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Mathematics – Page 151
New York Mathematics
Strands/Bands/
Performance Indicators
Algebra 2/Trigonometry
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)
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Statistics & Probability: Making Inferences & Justifying Conclusions
Understand and evaluate random processes underlying statistical
experiments.
1. Understand statistics as a process for making inferences about population
parameters based on a random sample from that population.
Make inferences and justify conclusions from sample surveys, experiments
and observational studies.
4. Use data from a sample survey to estimate a population mean or proportion;
develop a margin of error through the use of simulation models for random
sampling.
Statistics & Probability: Making Inferences & Justifying Conclusions
Make inferences and justify conclusions from sample surveys, experiments
and observational studies.
3. Recognize the purposes of and differences among sample surveys,
experiments, and observational studies; explain how randomization relates to
each.
5. Use data from a randomized experiment to compare two treatments; use
simulations to decide if differences between parameters are significant.
National Essential Skills Study (NESS)
National Rankings
Rank
M10
M10
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Number Sense and Operations
Strand
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)
Number & Quantity: The Real Number System
Extend the properties of exponents to rational exponents.
1. Explain how the definition of the meaning of rational exponents follows from
extending the properties of integer exponents to those values, allowing for a
notation for radicals in terms of rational exponents. For example, we define
51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3
must equal 5.
2. Rewrite expressions involving radicals and rational exponents using the
properties of exponents.
 2011 International Center for Leadership in Education
M1
M20
Perform operations fluently with positive
and negative numbers, including decimals,
ratios, percents, and fractions, and show
reasoning to justify results.
Understand and apply the basic properties
and laws of exponents and scientific
notation to solve problems, including those
with fractional, negative, and zero
exponents.
Mathematics – Page 152
New York Mathematics
Strands/Bands/
Performance Indicators
Algebra 2/Trigonometry
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
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Number & Quantity: The Real Number System
Use properties of rational and irrational numbers.
3. Explain why the sum or product of two rational numbers is rational; that the
sum of a rational number and an irrational number is irrational; and that the
product of a nonzero rational number and an irrational number is irrational.
Algebra: Arithmetic with Polynomials & Rational Expressions
Perform arithmetic operations on polynomials.
1. Understand that polynomials form a system analogous to the integers,
namely, they are closed under the operations of addition, subtraction, and
multiplication; add, subtract, and multiply polynomials.
National Essential Skills Study (NESS)
National Rankings
Rank
Perform operations with radicals, such as
addition, subtraction, and multiplication.
M33
M36
M62
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
Number & Quantity: The Real Number System
Use properties of rational and irrational numbers.
3. Explain why the sum or product of two rational numbers is rational; that the
sum of a rational number and an irrational number is irrational; and that the
product of a nonzero rational number and an irrational number is irrational.
Number & Quantity: The Real Number System
Extend the properties of exponents to rational exponents.
2. Rewrite expressions involving radicals and rational exponents using the
properties of exponents.
Number & Quantity: The Complex Number System
Perform arithmetic operations with complex numbers.
1. Know there is a complex number i such that i2 = –1, and every complex
number has the form a + bi with a and b real.
Number & Quantity: The Complex Number System
Perform arithmetic operations with complex numbers.
1. Know there is a complex number i such that i2 = –1, and every complex
number has the form a + bi with a and b real.
 2011 International Center for Leadership in Education
M33
M39
M33
M25
M25
Simplify polynomials by performing
operations (addition, subtraction,
multiplication, and division) to simplify
expressions (e.g., (2a + 2) + (3a - 1) = 5a +
1).
Perform division of a polynomial by a
monomial by dividing powers with like
bases, using the rules for the division of
powers with like bases to simplify fractions
with monomial denominators and reducing
fractions to lowest terms.
Perform operations with radicals, such as
addition, subtraction, and multiplication.
Apply techniques to obtain a rational
approximation or estimate of a quantity or
number (including irrational numbers such
as radicals).
Perform operations with radicals, such as
addition, subtraction, and multiplication.
Perform operations and solve equations
containing complex numbers.
Perform operations and solve equations
containing complex numbers.
Mathematics – Page 153
New York Mathematics
Strands/Bands/
Performance Indicators
Algebra 2/Trigonometry
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 a  bi Note: This
includes simplifying expressions with
complex denominators.
A2.N.10 Know and apply sigma
notation
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Number & Quantity: The Complex Number System
Perform arithmetic operations with complex numbers.
3. (+) Find the conjugate of a complex number; use conjugates to find moduli
and quotients of complex numbers.
Number & Quantity: The Complex Number System
Perform arithmetic operations with complex numbers.
2. Use the relation i2 = –1 and the commutative, associative, and distributive
properties to add, subtract, and multiply complex numbers.
Functions: Building Functions
Build new functions from existing functions.
3. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and
f(x + k) for specific values of k (both positive and negative); find the value of
k given the graphs. Experiment with cases and illustrate an explanation of the
effects on the graph using technology. Include recognizing even and odd
functions from their graphs and algebraic expressions for them.
National Essential Skills Study (NESS)
National Rankings
Rank
M25
Perform operations and solve equations
containing complex numbers.
Perform operations and solve equations
containing complex numbers.
M25
Apply summation notation to take the sum
of an expression using limits (e.g., take the
sum of 3i + 1 from i = 1 to 5).
M69
Algebra Strand
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
Algebra: Reasoning with Equations & Inequalities
Solve equations and inequalities in one variable.
3. Solve linear equations and inequalities in one variable, including equations
with coefficients represented by letters.
Functions: Interpreting Functions
Analyze functions using different representations.
7. Graph functions expressed symbolically and show key features of the graph,
by hand in simple cases and using technology for more complicated cases.
b. Graph square root, cube root, and piecewise-defined functions, including
step functions and absolute value functions.
 2011 International Center for Leadership in Education
M7
M27
Simplify and solve algebraic equations by
identifying and using the correct order of
operations and techniques necessary to carry
out the solution.
Find the solution of linear equations and
inequalities where the variable appears on
either or both sides and in which one or both
sides must be simplified before solving the
equation (e.g., solve x + 2(x - 3) = -4x + 5
for x).
Mathematics – Page 154
New York Mathematics
Strands/Bands/
Performance Indicators
Algebra 2/Trigonometry
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
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Algebra: Reasoning with Equations & Inequalities
Solve equations and inequalities in one variable.
4. Solve quadratic equations in one variable.
b. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square
roots, completing the square, the quadratic formula and factoring, as
appropriate to the initial form of the equation. Recognize when the
quadratic formula gives complex solutions and write them as a ± bi for real
numbers a and b.
Algebra: Reasoning with Equations & Inequalities
Solve systems of equations.
7. Solve a simple system consisting of a linear equation and a quadratic
equation in two variables algebraically and graphically. For example, find the
points of intersection between the line y = –3x and the circle x2 + y2 = 3.
National Essential Skills Study (NESS)
National Rankings
Rank
Solve quadratic equations by applying
various tools or techniques.
M47
M40
M47
M66
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
Algebra: Reasoning with Equations & Inequalities
Solve equations and inequalities in one variable.
3. Solve linear equations and inequalities in one variable, including equations
with coefficients represented by letters.
Represent and solve equations and inequalities graphically.
12. Graph the solutions to a linear inequality in two variables as a half-plane
(excluding the boundary in the case of a strict inequality), and graph the
solution set to a system of linear inequalities in two variables as the
intersection of the corresponding half-planes.
There is no New York Mathematics Performance Indicator–Common Core
alignment.
 2011 International Center for Leadership in Education
M47
Solve systems of linear equations
algebraically or graphically.
Solve quadratic equations by applying
various tools or techniques.
Know how to sketch basic conic sections
(e.g., circles, parabolas) by using their
equations and solve systems of non-linear
equations graphically.
Solve quadratic equations by applying
various tools or techniques.
Express, graph, and interpret polynomial
functions (linear, quadratic, cubic, etc.).
M53
M57
Solve and graphically sketch problems
involving two variables that exhibit direct
and indirect variation.
Mathematics – Page 155
New York Mathematics
Strands/Bands/
Performance Indicators
Algebra 2/Trigonometry
A2.A.6 Solve an application which
results in an exponential function
A2.A.6 (Continued from previous page)
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Algebra: Seeing Structure in Expressions
Write expressions in equivalent forms to solve problems.
3. Choose and produce an equivalent form of an expression to reveal and
explain properties of the quantity represented by the expression.
c. Use the properties of exponents to transform expressions for exponential
functions. For example the expression 1.15t can be rewritten as (1.151/12)12t
≈ 1.01212t to reveal the approximate equivalent monthly interest rate if the
annual rate is 15%.
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.
Functions: Interpreting Functions
Analyze functions using different representations.
8. Write a function defined by an expression in different but equivalent forms to
reveal and explain different properties of the function.
b. Use the properties of exponents to interpret expressions for exponential
functions. For example, identify percent rate of change in functions such as
y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as
representing exponential growth or decay.
(Continued from previous page)
Functions: Linear, Quadratic & Exponential
Construct and compare linear, quadratic, and exponential models and
solve problems.
2. Construct linear and exponential functions, including arithmetic and
geometric sequences, given a graph, a description of a relationship, or two
input-output pairs (include reading these from a table).
4. For exponential models, express as a logarithm the solution to abct = d where
a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm
using technology.
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
Express, graph, and interpret exponential
and logarithmic functions.
M48
Mathematics – Page 156
New York Mathematics
Strands/Bands/
Performance Indicators
Algebra 2/Trigonometry
Mathematics
Domains/Clusters/
Common Core State Standards
High School
National Essential Skills Study (NESS)
National Rankings
Rank
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
Algebra: Seeing Structure in Expressions
Interpret the structure of expressions.
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).
Write expressions in equivalent forms to solve problems.
3. Choose and produce an equivalent form of an expression to reveal and
explain properties of the quantity represented by the expression.
a. Factor a quadratic expression to reveal the zeros of the function it defines.
Number & Quantity: The Real Number System
Extend the properties of exponents to rational exponents.
1. Explain how the definition of the meaning of rational exponents follows from
extending the properties of integer exponents to those values, allowing for a
notation for radicals in terms of rational exponents. For example, we define
51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3
must equal 5.
2. Rewrite expressions involving radicals and rational exponents using the
properties of exponents.
Number & Quantity: The Real Number System
Extend the properties of exponents to rational exponents.
2. Rewrite expressions involving radicals and rational exponents using the
properties of exponents.
Number & Quantity: The Real Number System
Extend the properties of exponents to rational exponents.
1. Explain how the definition of the meaning of rational exponents follows from
extending the properties of integer exponents to those values, allowing for a
notation for radicals in terms of rational exponents. For example, we define
51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3
must equal 5.
2. Rewrite expressions involving radicals and rational exponents using the
properties of exponents.
 2011 International Center for Leadership in Education
M47
Simplify polynomials by performing
operations (addition, subtraction,
multiplication, and division) to simplify
expressions (e.g., (2a + 2) + (3a - 1) = 5a +
1).
Solve quadratic equations by applying
various tools or techniques.
M20
Understand and apply the basic properties
and laws of exponents and scientific
notation to solve problems, including those
with fractional, negative, and zero
exponents.
M36
M20
M20
M33
Understand and apply the basic properties
and laws of exponents and scientific
notation to solve problems, including those
with fractional, negative, and zero
exponents.
Understand and apply the basic properties
and laws of exponents and scientific
notation to solve problems, including those
with fractional, negative, and zero
exponents.
Perform operations with radicals, such as
addition, subtraction, and multiplication.
Mathematics – Page 157
New York Mathematics
Strands/Bands/
Performance Indicators
Algebra 2/Trigonometry
Mathematics
Domains/Clusters/
Common Core State Standards
High School
A2.A.11 Rewrite algebraic expressions
in radical form as expressions with
fractional exponents
Number & Quantity: The Real Number System
Extend the properties of exponents to rational exponents.
1. Explain how the definition of the meaning of rational exponents follows from
extending the properties of integer exponents to those values, allowing for a
notation for radicals in terms of rational exponents. For example, we define
51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3
must equal 5.
2. Rewrite expressions involving radicals and rational exponents using the
properties of exponents.
Functions: Linear, Quadratic & Exponential
Construct and compare linear, quadratic, and exponential models and
solve problems.
2. Construct linear and exponential functions, including arithmetic and
geometric sequences, given a graph, a description of a relationship, or two
input-output pairs (include reading these from a table).
A2.A.12 Evaluate exponential
expressions, including those with base e
National Essential Skills Study (NESS)
National Rankings
Rank
M20
M33
M11
M48
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
Number & Quantity: The Real Number System
Extend the properties of exponents to rational exponents.
2. Rewrite expressions involving radicals and rational exponents using the
properties of exponents.
Number & Quantity: The Real Number System
Use properties of rational and irrational numbers.
3. Explain why the sum or product of two rational numbers is rational; that the
sum of a rational number and an irrational number is irrational; and that the
product of a nonzero rational number and an irrational number is irrational.
Number & Quantity: The Real Number System
Extend the properties of exponents to rational exponents.
2. Rewrite expressions involving radicals and rational exponents using the
properties of exponents.
 2011 International Center for Leadership in Education
M33
Understand and apply the basic properties
and laws of exponents and scientific
notation to solve problems, including those
with fractional, negative, and zero
exponents.
Perform operations with radicals, such as
addition, subtraction, and multiplication.
Apply variables in expressions and
equations to solve problems (i.e., write
mathematical equations for given situation,
create a mathematical model to understand
the relationships between variables, or make
connections between the structures of
mathematically abstract concepts and the
real world).
Express, graph, and interpret exponential
and logarithmic functions.
Perform operations with radicals, such as
addition, subtraction, and multiplication.
Perform operations with radicals, such as
addition, subtraction, and multiplication.
M33
M33
Perform operations with radicals, such as
addition, subtraction, and multiplication.
Mathematics – Page 158
New York Mathematics
Strands/Bands/
Performance Indicators
Algebra 2/Trigonometry
A2.A.16 Perform arithmetic operations
with rational expressions and rename to
lowest terms
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Algebra: Arithmetic with Polynomials & Rational Expressions
Rewrite rational expressions.
7. (+) Understand that rational expressions form a system analogous to the
rational numbers, closed under addition, subtraction, multiplication, and
division by a nonzero rational expression; add, subtract, multiply, and divide
rational expressions.
National Essential Skills Study (NESS)
National Rankings
Rank
M1
M36
A2.A.17 Simplify complex fractional
expressions
Algebra: Arithmetic with Polynomials & Rational Expressions
Rewrite rational expressions.
7. (+) Understand that rational expressions form a system analogous to the
rational numbers, closed under addition, subtraction, multiplication, and
division by a nonzero rational expression; add, subtract, multiply, and divide
rational expressions.
M1
M33
M36
A2.A.18 Evaluate logarithmic
expressions in any base
Functions: Linear, Quadratic & Exponential
Construct and compare linear, quadratic, and exponential models and
solve problems.
4. For exponential models, express as a logarithm the solution to abct = d where
a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm
using technology.
M11
M48
 2011 International Center for Leadership in Education
Perform operations fluently with positive
and negative numbers, including decimals,
ratios, percents, and fractions, and show
reasoning to justify results.
Simplify polynomials by performing
operations (addition, subtraction,
multiplication, and division) to simplify
expressions (e.g., (2a + 2) + (3a - 1) = 5a +
1).
Perform operations fluently with positive
and negative numbers, including decimals,
ratios, percents, and fractions, and show
reasoning to justify results.
Perform operations with radicals, such as
addition, subtraction, and multiplication.
Simplify polynomials by performing
operations (addition, subtraction,
multiplication, and division) to simplify
expressions (e.g., (2a + 2) + (3a - 1) = 5a +
1).
Apply variables in expressions and
equations to solve problems (i.e., write
mathematical equations for given situation,
create a mathematical model to understand
the relationships between variables, or make
connections between the structures of
mathematically abstract concepts and the
real world).
Express, graph, and interpret exponential
and logarithmic functions.
Mathematics – Page 159
New York Mathematics
Strands/Bands/
Performance Indicators
Algebra 2/Trigonometry
A2.A.19 Apply the properties of
logarithms to rewrite logarithmic
expressions in equivalent forms
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Functions: Linear, Quadratic & Exponential
Construct and compare linear, quadratic, and exponential models and
solve problems.
4. For exponential models, express as a logarithm the solution to abct = d where
a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm
using technology.
National Essential Skills Study (NESS)
National Rankings
Rank
M11
M48
Apply variables in expressions and
equations to solve problems (i.e., write
mathematical equations for given situation,
create a mathematical model to understand
the relationships between variables, or make
connections between the structures of
mathematically abstract concepts and the
real world).
Express, graph, and interpret exponential
and logarithmic functions.
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
There is no New York Mathematics Performance Indicator–Common Core
alignment.
M47
Solve quadratic equations by applying
various tools or techniques.
There is no New York Mathematics Performance Indicator–Common Core
alignment.
M47
Solve quadratic equations by applying
various tools or techniques.
Algebra: Reasoning with Equations & Inequalities
Understand solving equations as a process of reasoning and explain the
reasoning.
2. Solve simple rational and radical equations in one variable, and give
examples showing how extraneous solutions may arise.
Algebra: Reasoning with Equations & Inequalities
Understand solving equations as a process of reasoning and explain the
reasoning.
2. Solve simple rational and radical equations in one variable, and give
examples showing how extraneous solutions may arise.
M7
M33
M7
M27
M33
 2011 International Center for Leadership in Education
Simplify and solve algebraic equations by
identifying and using the correct order of
operations and techniques necessary to carry
out the solution.
Perform operations with radicals, such as
addition, subtraction, and multiplication.
Simplify and solve algebraic equations by
identifying and using the correct order of
operations and techniques necessary to carry
out the solution.
Find the solution of linear equations and
inequalities where the variable appears on
either or both sides and in which one or both
sides must be simplified before solving the
equation (e.g., solve x + 2(x - 3) = -4x + 5
for x).
Perform operations with radicals, such as
addition, subtraction, and multiplication.
Mathematics – Page 160
New York Mathematics
Strands/Bands/
Performance Indicators
Algebra 2/Trigonometry
Mathematics
Domains/Clusters/
Common Core State Standards
High School
A2.A.24 Know and apply the technique
of completing the square
Algebra: Seeing Structure in Expressions
Write expressions in equivalent forms to solve problems.
3. Choose and produce an equivalent form of an expression to reveal and
explain properties of the quantity represented by the expression.
b. Complete the square in a quadratic expression to reveal the maximum or
minimum value of the function it defines.
Algebra: Reasoning with Equations & Inequalities
Solve equations and inequalities in one variable.
4. Solve quadratic equations in one variable.
a. Use the method of completing the square to transform any quadratic
equation in x into an equation of the form (x – p)2 = q that has the same
solutions. Derive the quadratic formula from this form.
Functions: Interpreting Functions
Analyze functions using different representations.
8. Write a function defined by an expression in different but equivalent forms to
reveal and explain different properties of the function.
a. Use the process of factoring and completing the square in a quadratic
function to show zeros, extreme values, and symmetry of the graph, and
interpret these in terms of a context.
Algebra: Reasoning with Equations & Inequalities
Solve equations and inequalities in one variable.
4. Solve quadratic equations in one variable.
b. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square
roots, completing the square, the quadratic formula and factoring, as
appropriate to the initial form of the equation. Recognize when the
quadratic formula gives complex solutions and write them as a ± bi for real
numbers a and b.
Functions: Interpreting Functions
Analyze functions using different representations.
8. Write a function defined by an expression in different but equivalent forms to
reveal and explain different properties of the function.
a. Use the process of factoring and completing the square in a quadratic
function to show zeros, extreme values, and symmetry of the graph, and
interpret these in terms of a context.
A2.A.25 Solve quadratic equations,
using the quadratic formula
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
Solve quadratic equations by applying
various tools or techniques.
M47
Solve quadratic equations by applying
various tools or techniques.
M47
Mathematics – Page 161
New York Mathematics
Strands/Bands/
Performance Indicators
Algebra 2/Trigonometry
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
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Algebra: Arithmetic with Polynomials & Rational Expressions
Understand the relationship between zeros and factors of polynomials.
3. Identify zeros of polynomials when suitable factorizations are available, and
use the zeros to construct a rough graph of the function defined by the
polynomial.
Algebra: Reasoning with Equations & Inequalities
Solve equations and inequalities in one variable.
4. Solve quadratic equations in one variable.
b. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square
roots, completing the square, the quadratic formula and factoring, as
appropriate to the initial form of the equation. Recognize when the
quadratic formula gives complex solutions and write them as a ± bi for real
numbers a and b.
Algebra: Creating Equations
Create equations that describe numbers or relationships.
1. Create equations and inequalities in one variable and use them to solve
problems. Include equations arising from linear and quadratic functions, and
simple rational and exponential functions.
Functions: Building Functions
Build new functions from existing functions.
5. (+) Understand the inverse relationship between exponents and logarithms
and use this relationship to solve problems involving logarithms and
exponents.
Functions: Linear, Quadratic & Exponential
Construct and compare linear, quadratic, and exponential models and
solve problems.
4. For exponential models, express as a logarithm the solution to abct = d where
a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm
using technology.
Interpret expressions for functions in terms of the situation they model.
5. Interpret the parameters in a linear or exponential function in terms of a
context.
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
M47
Solve quadratic equations by applying
various tools or techniques.
Express, graph, and interpret polynomial
functions (linear, quadratic, cubic, etc.).
M53
M7
Simplify and solve algebraic equations by
identifying and using the correct order of
operations and techniques necessary to carry
out the solution.
Express, graph, and interpret exponential
and logarithmic functions.
M48
Mathematics – Page 162
New York Mathematics
Strands/Bands/
Performance Indicators
Algebra 2/Trigonometry
Mathematics
Domains/Clusters/
Common Core State Standards
High School
A2.A.28 Solve a logarithmic equation
by rewriting as an exponential equation
Algebra: Creating Equations
Create equations that describe numbers or relationships.
1. Create equations and inequalities in one variable and use them to solve
problems. Include equations arising from linear and quadratic functions, and
simple rational and exponential functions.
Functions: Building Functions
Build new functions from existing functions.
5. (+) Understand the inverse relationship between exponents and logarithms
and use this relationship to solve problems involving logarithms and
exponents.
Functions: Linear, Quadratic & Exponential
Construct and compare linear, quadratic, and exponential models and
solve problems.
4. For exponential models, express as a logarithm the solution to abct = d where
a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm
using technology.
National Essential Skills Study (NESS)
National Rankings
Rank
M7
Simplify and solve algebraic equations by
identifying and using the correct order of
operations and techniques necessary to carry
out the solution.
Express, graph, and interpret exponential
and logarithmic functions.
M48
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
Algebra: Seeing Structure in Expressions
Write expressions in equivalent forms to solve problems.
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.
Functions: Building Functions
Build a function that models a relationship between two quantities.
2. Write arithmetic and geometric sequences both recursively and with an
explicit formula, use them to model situations, and translate between the two
forms.
 2011 International Center for Leadership in Education
M16
M60
M16
Apply pattern recognition in data sets and
series to reason or solve problems involving
arithmetic, geometry, exponents, etc.
Evaluate and use finite sequence and series
as systematic and useful means of
quantifying things.
Apply pattern recognition in data sets and
series to reason or solve problems involving
arithmetic, geometry, exponents, etc.
Mathematics – Page 163
New York Mathematics
Strands/Bands/
Performance Indicators
Algebra 2/Trigonometry
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
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Algebra: Seeing Structure in Expressions
Write expressions in equivalent forms to solve problems.
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.
Functions: Building Functions
Build a function that models a relationship between two quantities.
2. Write arithmetic and geometric sequences both recursively and with an
explicit formula, use them to model situations, and translate between the two
forms.
Algebra: Seeing Structure in Expressions
Write expressions in equivalent forms to solve problems.
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.
Functions: Building Functions
Build a function that models a relationship between two quantities.
2. Write arithmetic and geometric sequences both recursively and with an
explicit formula, use them to model situations, and translate between the two
forms.
Functions: Interpreting Functions
Understand the concept of a function and use function notation.
3. Recognize that sequences are functions, sometimes defined recursively,
whose domain is a subset of the integers. For example, the Fibonacci
sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n
≥ 1.
Functions: Building Functions
Build a function that models a relationship between two quantities.
2. Write arithmetic and geometric sequences both recursively and with an
explicit formula, use them to model situations, and translate between the two
forms.
Functions: Building Functions
Build a function that models a relationship between two quantities.
2. Write arithmetic and geometric sequences both recursively and with an
explicit formula, use them to model situations, and translate between the two
forms.
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
M3
Use proportional reasoning to solve realworld problems.
Apply pattern recognition in data sets and
series to reason or solve problems involving
arithmetic, geometry, exponents, etc.
M16
M16
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Apply pattern recognition in data sets and
series to reason or solve problems involving
arithmetic, geometry, exponents, etc.
M24
Understand the concepts of recurrence
relations and apply them to solve consumer
mathematics problems involving such things
as percentage rates, personal loans, simple
interest, compound interest, installment
buying, mortgage rates, etc.
M69
Apply summation notation to take the sum
of an expression using limits (e.g., take the
sum of 3i + 1 from i = 1 to 5).
M10
Mathematics – Page 164
New York Mathematics
Strands/Bands/
Performance Indicators
Algebra 2/Trigonometry
Mathematics
Domains/Clusters/
Common Core State Standards
High School
A2.A.35 Determine the sum of the first
n terms of an arithmetic or geometric
series
Functions: Building Functions
Build a function that models a relationship between two quantities.
2. Write arithmetic and geometric sequences both recursively and with an
explicit formula, use them to model situations, and translate between the two
forms.
Algebra: Arithmetic with Polynomials & Rational Expressions
Use polynomial identities to solve problems.
5. (+) Know and apply the Binomial Theorem for the expansion of (x + y)n in
powers of x and y for a positive integer n, where x and y are any numbers,
with coefficients determined for example by Pascal’s Triangle. [The Binomial
Theorem can be proved by mathematical induction or by a combinatorial
argument.]
Functions: Interpreting Functions
Understand the concept of a function and use function notation.
1. Understand that a function from one set (called the domain) to another set
(called the range) assigns to each element of the domain exactly one element
of the range. If f is a function and x is an element of its domain, then f(x)
denotes the output of f corresponding to the input x. The graph of f is the
graph of the equation y = f(x).
Functions: Interpreting Functions
Understand the concept of a function and use function notation.
1. Understand that a function from one set (called the domain) to another set
(called the range) assigns to each element of the domain exactly one element
of the range. If f is a function and x is an element of its domain, then f(x)
denotes the output of f corresponding to the input x. The graph of f is the
graph of the equation y = f(x).
Functions: Interpreting Functions
Understand the concept of a function and use function notation.
1. Understand that a function from one set (called the domain) to another set
(called the range) assigns to each element of the domain exactly one element
of the range. If f is a function and x is an element of its domain, then f(x)
denotes the output of f corresponding to the input x. The graph of f is the
graph of the equation y = f(x).
2. Use function notation, evaluate functions for inputs in their domains, and
interpret statements that use function notation in terms of a context.
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
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
M69
Apply summation notation to take the sum
of an expression using limits (e.g., take the
sum of 3i + 1 from i = 1 to 5).
M70
Understand and apply the binomial theorem
(e.g., explore the relationship of the
binomial theorem with Pascal’s triangle and
the Fibonacci sequence).
M37
M37
M37
Define and apply the properties of relations
and functions (domain, range, function
composition, and inverses) and use algebraic
and graphic methods to determine if a
relation is a function.
Define and apply the properties of relations
and functions (domain, range, function
composition, and inverses) and use algebraic
and graphic methods to determine if a
relation is a function.
Define and apply the properties of relations
and functions (domain, range, function
composition, and inverses) and use algebraic
and graphic methods to determine if a
relation is a function.
Mathematics – Page 165
New York Mathematics
Strands/Bands/
Performance Indicators
Algebra 2/Trigonometry
A2.A.40 Write functions in functional
notation
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Functions: Interpreting Functions
Understand the concept of a function and use function notation.
1. Understand that a function from one set (called the domain) to another set
(called the range) assigns to each element of the domain exactly one element
of the range. If f is a function and x is an element of its domain, then f(x)
denotes the output of f corresponding to the input x. The graph of f is the
graph of the equation y = f(x).
National Essential Skills Study (NESS)
National Rankings
Rank
M37
M53
M56
A2.A.41 Use functional notation to
evaluate functions for given values in
the domain
Functions: Interpreting Functions
Understand the concept of a function and use function notation.
2. Use function notation, evaluate functions for inputs in their domains, and
interpret statements that use function notation in terms of a context.
M37
M53
M56
A2.A.42 Find the composition of
functions
A2.A.43 Determine if a function is oneto-one, onto, or both
Functions: Building Functions
Build a function that models a relationship between two quantities.
1. Write a function that describes a relationship between two quantities.
c. (+) Compose functions. For example, if T(y) is the temperature in the
atmosphere as a function of height, and h(t) is the height of a weather
balloon as a function of time, then T(h(t)) is the temperature at the location
of the weather balloon as a function of time.
There is no New York Mathematics Performance Indicator–Common Core
alignment.
M37
M37
 2011 International Center for Leadership in Education
Define and apply the properties of relations
and functions (domain, range, function
composition, and inverses) and use algebraic
and graphic methods to determine if a
relation is a function.
Express, graph, and interpret polynomial
functions (linear, quadratic, cubic, etc.).
Express a linear function (f(x) = mx + b)
with the appropriate notation and determine
the ordered pairs.
Define and apply the properties of relations
and functions (domain, range, function
composition, and inverses) and use algebraic
and graphic methods to determine if a
relation is a function.
Express, graph, and interpret polynomial
functions (linear, quadratic, cubic, etc.).
Express a linear function (f(x) = mx + b)
with the appropriate notation and determine
the ordered pairs.
Define and apply the properties of relations
and functions (domain, range, function
composition, and inverses) and use algebraic
and graphic methods to determine if a
relation is a function.
Define and apply the properties of relations
and functions (domain, range, function
composition, and inverses) and use algebraic
and graphic methods to determine if a
relation is a function.
Mathematics – Page 166
New York Mathematics
Strands/Bands/
Performance Indicators
Algebra 2/Trigonometry
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: f ( x  a) ,
f ( x) a , f ( x) ,  f (x) , af (x)
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Functions: Building Functions
Build new functions from existing functions.
4. Find inverse functions.
a. Solve an equation of the form f(x) = c for a simple function f that has an
inverse and write an expression for the inverse. For example, f(x) =2 x3 or
f(x) = (x+1)/(x–1) for x ≠ 1.
Functions: Building Functions
Build new functions from existing functions.
4. Find inverse functions.
a. Solve an equation of the form f(x) = c for a simple function f that has an
inverse and write an expression for the inverse. For example, f(x) =2 x3 or
f(x) = (x+1)/(x–1) for x ≠ 1.
b. (+) Verify by composition that one function is the inverse of another.
Functions: Building Functions
Build new functions from existing functions.
3. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and
f(x + k) for specific values of k (both positive and negative); find the value of
k given the graphs. Experiment with cases and illustrate an explanation of the
effects on the graph using technology. Include recognizing even and odd
functions from their graphs and algebraic expressions for them.
National Essential Skills Study (NESS)
National Rankings
Rank
M37
M37
M54
Define and apply the properties of relations
and functions (domain, range, function
composition, and inverses) and use algebraic
and graphic methods to determine if a
relation is a function.
Define and apply the properties of relations
and functions (domain, range, function
composition, and inverses) and use algebraic
and graphic methods to determine if a
relation is a function.
Apply transformations (reflection, rotation,
translation, and dilation) of 2-dimensional
figures graphically to interpret, analyze, and
illustrate the concepts of congruency,
similarity, and symmetry.
Coordinate Geometry
A2.A.47 Determine the center-radius
form for the equation of a circle in
standard form
Geometry: Expressing Geometric Properties with Equations
Translate between the geometric description and the equation for a conic
section.
1. Derive the equation of a circle of given center and radius using the
Pythagorean Theorem; complete the square to find the center and radius of a
circle given by an equation.
M18
M66
 2011 International Center for Leadership in Education
Understand the properties of circles (radius,
arc, diameter, chord, secant, and tangent)
and apply circle quantities (lengths of line
segments, angle measure within a circle,
circumference, and area) in problem-solving
situations.
Know how to sketch basic conic sections
(e.g., circles, parabolas) by using their
equations and solve systems of non-linear
equations graphically.
Mathematics – Page 167
New York Mathematics
Strands/Bands/
Performance Indicators
Algebra 2/Trigonometry
A2.A.48 Write the equation of a circle,
given its center and a point on the circle
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Geometry: Expressing Geometric Properties with Equations
Translate between the geometric description and the equation for a conic
section.
1. Derive the equation of a circle of given center and radius using the
Pythagorean Theorem; complete the square to find the center and radius of a
circle given by an equation.
National Essential Skills Study (NESS)
National Rankings
Rank
M18
M66
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
Geometry: Expressing Geometric Properties with Equations
Translate between the geometric description and the equation for a conic
section.
1. Derive the equation of a circle of given center and radius using the
Pythagorean Theorem; complete the square to find the center and radius of a
circle given by an equation.
Functions: Interpreting Functions
Analyze functions using different representations.
7. Graph functions expressed symbolically and show key features of the graph,
by hand in simple cases and using technology for more complicated cases.
c. Graph polynomial functions, identifying zeros when suitable factorizations
are available, and showing end behavior.
Functions: Interpreting Functions
Interpret functions that arise in applications in terms of the context.
5. Relate the domain of a function to its graph and, where applicable, to the
quantitative relationship it describes. For example, if the function h(n) gives
the number of person-hours it takes to assemble n engines in a factory, then
the positive integers would be an appropriate domain for the function.
Functions: Interpreting Functions
Understand the concept of a function and use function notation.
1. Understand that a function from one set (called the domain) to another set
(called the range) assigns to each element of the domain exactly one element
of the range. If f is a function and x is an element of its domain, then f(x)
denotes the output of f corresponding to the input x. The graph of f is the
graph of the equation y = f(x).
 2011 International Center for Leadership in Education
M66
Understand the properties of circles (radius,
arc, diameter, chord, secant, and tangent)
and apply circle quantities (lengths of line
segments, angle measure within a circle,
circumference, and area) in problem-solving
situations.
Know how to sketch basic conic sections
(e.g., circles, parabolas) by using their
equations and solve systems of non-linear
equations graphically.
Know how to sketch basic conic sections
(e.g., circles, parabolas) by using their
equations and solve systems of non-linear
equations graphically.
Express, graph, and interpret polynomial
functions (linear, quadratic, cubic, etc.).
M53
M37
M37
Define and apply the properties of relations
and functions (domain, range, function
composition, and inverses) and use algebraic
and graphic methods to determine if a
relation is a function.
Define and apply the properties of relations
and functions (domain, range, function
composition, and inverses) and use algebraic
and graphic methods to determine if a
relation is a function.
Mathematics – Page 168
New York Mathematics
Strands/Bands/
Performance Indicators
Algebra 2/Trigonometry
A2.A.53 Graph exponential functions
of the form y  b x for positive values
of b, including b  e
A2.A.54 Graph logarithmic functions,
using the inverse of the related
exponential function
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Functions: Interpreting Functions
Analyze functions using different representations.
7. Graph functions expressed symbolically and show key features of the graph,
by hand in simple cases and using technology for more complicated cases.
e. Graph exponential and logarithmic functions, showing intercepts and end
behavior, and trigonometric functions, showing period, midline, and
amplitude.
Functions: Interpreting Functions
Analyze functions using different representations.
7. Graph functions expressed symbolically and show key features of the graph,
by hand in simple cases and using technology for more complicated cases.
e. Graph exponential and logarithmic functions, showing intercepts and end
behavior, and trigonometric functions, showing period, midline, and
amplitude.
National Essential Skills Study (NESS)
National Rankings
Rank
Express, graph, and interpret exponential
and logarithmic functions.
M48
Express, graph, and interpret exponential
and logarithmic functions.
M48
Trigonometric Functions
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 cofunction and reciprocal relationships
between trigonometric ratios
Geometry: Similarity, Right Triangles, & Trigonometry
Define trigonometric ratios and solve problems involving right triangles.
6. Understand that by similarity, side ratios in right triangles are properties of
the angles in the triangle, leading to definitions of trigonometric ratios for
acute angles.
Functions: Trigonometric Functions
Extend the domain of trigonometric functions using the unit circle.
3. (+) Use special triangles to determine geometrically the values of sine,
cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the
values of sine, cosine, and tangent for π–x, π+x, and 2π–x in terms of their
values for x, where x is any real number.
Geometry: Similarity, Right Triangles, & Trigonometry
Define trigonometric ratios and solve problems involving right triangles.
6. Understand that by similarity, side ratios in right triangles are properties of
the angles in the triangle, leading to definitions of trigonometric ratios for
acute angles.
Geometry: Similarity, Right Triangles, & Trigonometry
Define trigonometric ratios and solve problems involving right triangles.
7. Explain and use the relationship between the sine and cosine of
complementary angles.
 2011 International Center for Leadership in Education
M28
M28
M28
M28
Know and apply the six basic trigonometric
functions and ratios and solve right triangles
using basic trigonometric ratios (sine,
cosine, tangent).
Know and apply the six basic trigonometric
functions and ratios and solve right triangles
using basic trigonometric ratios (sine,
cosine, tangent).
Know and apply the six basic trigonometric
functions and ratios and solve right triangles
using basic trigonometric ratios (sine,
cosine, tangent).
Know and apply the six basic trigonometric
functions and ratios and solve right triangles
using basic trigonometric ratios (sine,
cosine, tangent).
Mathematics – Page 169
New York Mathematics
Strands/Bands/
Performance Indicators
Algebra 2/Trigonometry
A2.A.59 Use the reciprocal and cofunction 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
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Functions: Trigonometric Functions
Extend the domain of trigonometric functions using the unit circle.
3. (+) Use special triangles to determine geometrically the values of sine,
cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the
values of sine, cosine, and tangent for π–x, π+x, and 2π–x in terms of their
values for x, where x is any real number.
Functions: Trigonometric Functions
Extend the domain of trigonometric functions using the unit circle.
2. Explain how the unit circle in the coordinate plane enables the extension of
trigonometric functions to all real numbers, interpreted as radian measures of
angles traversed counterclockwise around the unit circle.
Functions: Trigonometric Functions
Extend the domain of trigonometric functions using the unit circle.
1. Understand radian measure of an angle as the length of the arc on the unit
circle subtended by the angle.
National Essential Skills Study (NESS)
National Rankings
Rank
M28
M64
M18
M64
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
Functions: Trigonometric Functions
Extend the domain of trigonometric functions using the unit circle.
2. Explain how the unit circle in the coordinate plane enables the extension of
trigonometric functions to all real numbers, interpreted as radian measures of
angles traversed counterclockwise around the unit circle.
Functions: Trigonometric Functions
Model periodic phenomena with trigonometric functions.
6. (+) Understand that restricting a trigonometric function to a domain on which
it is always increasing or always decreasing allows its inverse to be
constructed.
M64
M28
M37
 2011 International Center for Leadership in Education
Know and apply the six basic trigonometric
functions and ratios and solve right triangles
using basic trigonometric ratios (sine,
cosine, tangent).
Understand the trigonometric properties of
the unit circle and sketch the graphs of basic
circular functions (y = sin x, y = cos x, and y
= tan x, where the measure of the angle x is
expressed in radians).
Understand the properties of circles (radius,
arc, diameter, chord, secant, and tangent)
and apply circle quantities (lengths of line
segments, angle measure within a circle,
circumference, and area) in problem-solving
situations.
Understand the trigonometric properties of
the unit circle and sketch the graphs of basic
circular functions (y = sin x, y = cos x, and y
= tan x, where the measure of the angle x is
expressed in radians).
Understand the trigonometric properties of
the unit circle and sketch the graphs of basic
circular functions (y = sin x, y = cos x, and y
= tan x, where the measure of the angle x is
expressed in radians).
Know and apply the six basic trigonometric
functions and ratios and solve right triangles
using basic trigonometric ratios (sine,
cosine, tangent).
Define and apply the properties of relations
and functions (domain, range, function
composition, and inverses) and use algebraic
and graphic methods to determine if a
relation is a function.
Mathematics – Page 170
New York Mathematics
Strands/Bands/
Performance Indicators
Algebra 2/Trigonometry
A2.A.64 Use inverse functions to find
the measure of an angle, given its sine,
cosine, or tangent
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Functions: Trigonometric Functions
Model periodic phenomena with trigonometric functions.
7. (+) Use inverse functions to solve trigonometric equations that arise in
modeling contexts; evaluate the solutions using technology, and interpret
them in terms of the context.
National Essential Skills Study (NESS)
National Rankings
Rank
M28
M37
A2.A.65 Sketch the graph of the
inverses of the sine, cosine, and tangent
functions
Functions: Building Functions
Build new functions from existing functions.
4. Find inverse functions.
c. (+) Read values of an inverse function from a graph or a table, given that
the function has an inverse.
M28
M37
A2.A.66 Determine the trigonometric
functions of any angle, using
technology
A2.A.67 Justify the Pythagorean
identities
Functions: Trigonometric Functions
Extend the domain of trigonometric functions using the unit circle.
3. (+) Use special triangles to determine geometrically the values of sine,
cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the
values of sine, cosine, and tangent for π–x, π+x, and 2π–x in terms of their
values for x, where x is any real number.
Functions: Trigonometric Functions
Prove and apply trigonometric identities.
8. Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to calculate
trigonometric ratios.
M28
M10
M55
 2011 International Center for Leadership in Education
Know and apply the six basic trigonometric
functions and ratios and solve right triangles
using basic trigonometric ratios (sine,
cosine, tangent).
Define and apply the properties of relations
and functions (domain, range, function
composition, and inverses) and use algebraic
and graphic methods to determine if a
relation is a function.
Know and apply the six basic trigonometric
functions and ratios and solve right triangles
using basic trigonometric ratios (sine,
cosine, tangent).
Define and apply the properties of relations
and functions (domain, range, function
composition, and inverses) and use algebraic
and graphic methods to determine if a
relation is a function.
Know and apply the six basic trigonometric
functions and ratios and solve right triangles
using basic trigonometric ratios (sine,
cosine, tangent).
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Perform the general solution of triangles by
using the Law of Sines and Law of Cosines
to obtain the angle and side length
measurements of any triangle.
Mathematics – Page 171
New York Mathematics
Strands/Bands/
Performance Indicators
Algebra 2/Trigonometry
A2.A.68 Solve trigonometric equations
for all values of the variable from 0º to
360º
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Functions: Trigonometric Functions
Prove and apply trigonometric identities.
9. (+) Prove the addition and subtraction formulas for sine, cosine, and tangent
and use them to solve problems.
National Essential Skills Study (NESS)
National Rankings
Rank
M28
M64
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
y  A sin Bx or y  A cos Bx
A2.A.71 Sketch and recognize the
graphs of the functions y  sec(x) ,
y  csc(x) , y  tan(x) , and y  cot(x)
A2.A.72 Write the trigonometric
function that is represented by a given
periodic graph
Functions: Trigonometric Functions
Extend the domain of trigonometric functions using the unit circle.
4. (+) Use the unit circle to explain symmetry (odd and even) and periodicity of
trigonometric functions.
Model periodic phenomena with trigonometric functions.
5. Choose trigonometric functions to model periodic phenomena with specified
amplitude, frequency, and midline.
Functions: Trigonometric Functions
Model periodic phenomena with trigonometric functions.
5. Choose trigonometric functions to model periodic phenomena with specified
amplitude, frequency, and midline.
Functions: Trigonometric Functions
Model periodic phenomena with trigonometric functions.
5. Choose trigonometric functions to model periodic phenomena with specified
amplitude, frequency, and midline.
Functions: Trigonometric Functions
Extend the domain of trigonometric functions using the unit circle.
4. (+) Use the unit circle to explain symmetry (odd and even) and periodicity of
trigonometric functions.
 2011 International Center for Leadership in Education
M64
M64
M64
M64
Know and apply the six basic trigonometric
functions and ratios and solve right triangles
using basic trigonometric ratios (sine,
cosine, tangent).
Understand the trigonometric properties of
the unit circle and sketch the graphs of basic
circular functions (y = sin x, y = cos x, and y
= tan x, where the measure of the angle x is
expressed in radians).
Understand the trigonometric properties of
the unit circle and sketch the graphs of basic
circular functions (y = sin x, y = cos x, and y
= tan x, where the measure of the angle x is
expressed in radians).
Understand the trigonometric properties of
the unit circle and sketch the graphs of basic
circular functions (y = sin x, y = cos x, and y
= tan x, where the measure of the angle x is
expressed in radians).
Understand the trigonometric properties of
the unit circle and sketch the graphs of basic
circular functions (y = sin x, y = cos x, and y
= tan x, where the measure of the angle x is
expressed in radians).
Understand the trigonometric properties of
the unit circle and sketch the graphs of basic
circular functions (y = sin x, y = cos x, and y
= tan x, where the measure of the angle x is
expressed in radians).
Mathematics – Page 172
New York Mathematics
Strands/Bands/
Performance Indicators
Algebra 2/Trigonometry
Mathematics
Domains/Clusters/
Common Core State Standards
High School
A2.A.73 Solve for an unknown side or
angle, using the Law of Sines or the
Law of Cosines
Functions: Trigonometric Functions
Prove and apply trigonometric identities.
8. Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to calculate
trigonometric ratios.
Geometry: Similarity, Right Triangles, & Trigonometry
Apply trigonometry to general triangles.
10. (+) Prove the Laws of Sines and Cosines and use them to solve problems.
11. (+) Understand and apply the Law of Sines and the Law of Cosines to find
unknown measurements in right and non-right triangles (e.g., surveying
problems, resultant forces).
Geometry: Similarity, Right Triangles, & Trigonometry
Define trigonometric ratios and solve problems involving right triangles.
8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles
in applied problems.
A2.A.74 Determine the area of a
triangle or a parallelogram, given the
measure of two sides and the included
angle
National Essential Skills Study (NESS)
National Rankings
Rank
Perform the general solution of triangles by
using the Law of Sines and Law of Cosines
to obtain the angle and side length
measurements of any triangle.
M55
M9
M28
M55
A2.A.75 Determine the solution(s) from
the SSA situation (ambiguous case)
Geometry: Similarity, Right Triangles, & Trigonometry
Apply trigonometry to general triangles.
11. (+) Understand and apply the Law of Sines and the Law of Cosines to find
unknown measurements in right and non-right triangles (e.g., surveying
problems, resultant forces).
M10
M28
 2011 International Center for Leadership in Education
Compute the perimeter and area of common
two-dimensional figures.
Know and apply the six basic trigonometric
functions and ratios and solve right triangles
using basic trigonometric ratios (sine,
cosine, tangent).
Perform the general solution of triangles by
using the Law of Sines and Law of Cosines
to obtain the angle and side length
measurements of any triangle.
Understand and apply a systematic
methodology or procedure (e.g., direct or
indirect measurement, direct or indirect
proof, inductive or deductive reasoning) to
model and solve problems.
Know and apply the six basic trigonometric
functions and ratios and solve right triangles
using basic trigonometric ratios (sine,
cosine, tangent).
Mathematics – Page 173
New York Mathematics
Strands/Bands/
Performance Indicators
Algebra 2/Trigonometry
A2.A.76 Apply the angle sum and
difference formulas for trigonometric
functions
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Functions: Trigonometric Functions
Prove and apply trigonometric identities.
9. (+) Prove the addition and subtraction formulas for sine, cosine, and tangent
and use them to solve problems.
National Essential Skills Study (NESS)
National Rankings
Rank
M28
M64
A2.A.77 Apply the double-angle and
half angle formulas for trigonometric
functions
Geometry: Similarity, Right Triangles, & Trigonometry
Apply trigonometry to general triangles.
9. (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing
an auxiliary line from a vertex perpendicular to the opposite side.
M28
Know and apply the six basic trigonometric
functions and ratios and solve right triangles
using basic trigonometric ratios (sine,
cosine, tangent).
Understand the trigonometric properties of
the unit circle and sketch the graphs of basic
circular functions (y = sin x, y = cos x, and y
= tan x, where the measure of the angle x is
expressed in radians).
Know and apply the six basic trigonometric
functions and ratios and solve right triangles
using basic trigonometric ratios (sine,
cosine, tangent).
Measurement Strand
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
Functions: Trigonometric Functions
Extend the domain of trigonometric functions using the unit circle.
1. Understand radian measure of an angle as the length of the arc on the unit
circle subtended by the angle.
Functions: Trigonometric Functions
Extend the domain of trigonometric functions using the unit circle.
1. Understand radian measure of an angle as the length of the arc on the unit
circle subtended by the angle.
2. Explain how the unit circle in the coordinate plane enables the extension of
trigonometric functions to all real numbers, interpreted as radian measures of
angles traversed counterclockwise around the unit circle.
 2011 International Center for Leadership in Education
M64
M64
Understand the trigonometric properties of
the unit circle and sketch the graphs of basic
circular functions (y = sin x, y = cos x, and y
= tan x, where the measure of the angle x is
expressed in radians).
Understand the trigonometric properties of
the unit circle and sketch the graphs of basic
circular functions (y = sin x, y = cos x, and y
= tan x, where the measure of the angle x is
expressed in radians).
Mathematics – Page 174
New York Mathematics
Strands/Bands/
Performance Indicators
Algebra 2/Trigonometry
Mathematics
Domains/Clusters/
Common Core State Standards
High School
National Essential Skills Study (NESS)
National Rankings
Rank
Statistics and Probability Strand
Students will collect, organize, display,
and analyze data.
Collection of Data
A2.S.1 Understand the differences
among various kinds of studies (e.g.,
survey, observation, controlled
experiment)
Statistics & Probability: Making Inferences & Justifying Conclusions
Make inferences and justify conclusions from sample surveys, experiments
and observational studies.
3. Recognize the purposes of and differences among sample surveys,
experiments, and observational studies; explain how randomization relates to
each.
M17
M21
A2.S.2 Determine factors which may
affect the outcome of a survey
Statistics & Probability: Making Inferences & Justifying Conclusions
Make inferences and justify conclusions from sample surveys, experiments
and observational studies.
4
M17
Understand the importance of random
sampling and sample size in generating
representative data.
Evaluate and employ accurate and
appropriate procedures for statistical data
collection, organization, analysis, and
display including making estimates and
predictions, critiquing data, and drawing
inferences (e.g., using the normal curve and
z-scores, line of best fit).
Understand the importance of random
sampling and sample size in generating
representative data.
Organization and Display of Data
A2.S.3 Calculate measures of central
tendency with group frequency
distributions
Statistics & Probability: Making Inferences & Justifying Conclusions
Make inferences and justify conclusions from sample surveys, experiments
and observational studies.
4. Use data from a sample survey to estimate a population mean or proportion;
develop a margin of error through the use of simulation models for random
sampling.
 2011 International Center for Leadership in Education
M14
M31
Understand and apply measures of central
tendency (mean, median, and mode, and
representative sampling of a population).
Understand and apply measures of
dispersion (range, mean deviation, variance,
and standard deviation).
Mathematics – Page 175
New York Mathematics
Strands/Bands/
Performance Indicators
Algebra 2/Trigonometry
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
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Statistics & Probability: Interpreting Categorical & Quantitative Data
Summarize, represent and interpret data on a single count or measurement
variable.
2. Use statistics appropriate to the shape of the data distribution to compare
center (median, mean) and spread (interquartile range, standard deviation) of
two or more different data sets.
3. Interpret differences in shape, center, and spread in the context of the data
sets, accounting for possible effects of extreme data points (outliers).
4. Use the mean and standard deviation of a data set to fit it to a normal
distribution and to estimate population percentages. Recognize that there are
data sets for which such a procedure is not appropriate. Use calculators,
spreadsheets, and tables to estimate areas under the normal curve.
Statistics & Probability: Interpreting Categorical & Quantitative Data
Summarize, represent and interpret data on a single count or measurement
variable.
4. Use the mean and standard deviation of a data set to fit it to a normal
distribution and to estimate population percentages. Recognize that there are
data sets for which such a procedure is not appropriate. Use calculators,
spreadsheets, and tables to estimate areas under the normal curve.
National Essential Skills Study (NESS)
National Rankings
Rank
Understand and apply measures of
dispersion (range, mean deviation, variance,
and standard deviation).
M31
M36
M43
Simplify polynomials by performing
operations (addition, subtraction,
multiplication, and division) to simplify
expressions (e.g., (2a + 2) + (3a - 1) = 5a +
1).
Understand and apply the concepts and
applications of quartiles (distributing groups
into four equal sizes), percentiles
(distributing individuals into 100 groups of
equal size), and random distribution to
understand and interpret data.
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
Statistics & Probability: Interpreting Categorical & Quantitative Data
Summarize, represent and interpret data on two categorical and
quantitative variables.
6. Represent data on two quantitative variables on a scatter plot, and describe
how the variables are related.
a. Fit a function to the data; use functions fitted to data to solve problems in
the context of the data. Use given functions or choose a function suggested
by the context. Emphasize linear, quadratic, and exponential models.
 2011 International Center for Leadership in Education
M21
Evaluate and employ accurate and
appropriate procedures for statistical data
collection, organization, analysis, and
display including making estimates and
predictions, critiquing data, and drawing
inferences (e.g., using the normal curve and
z-scores, line of best fit).
Mathematics – Page 176
New York Mathematics
Strands/Bands/
Performance Indicators
Algebra 2/Trigonometry
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
Mathematics
Domains/Clusters/
Common Core State Standards
High School
Statistics & Probability: Interpreting Categorical & Quantitative Data
Summarize, represent and interpret data on two categorical and
quantitative variables.
6. Represent data on two quantitative variables on a scatter plot, and describe
how the variables are related.
b. Informally assess the fit of a function by plotting and analyzing residuals.
Interpret linear models.
8. Compute (using technology) and interpret the correlation coefficient of a
linear fit.
Statistics & Probability: Interpreting Categorical & Quantitative Data
Interpret linear models.
8. Compute (using technology) and interpret the correlation coefficient of a
linear fit.
National Essential Skills Study (NESS)
National Rankings
Rank
M21
M31
M21
Evaluate and employ accurate and
appropriate procedures for statistical data
collection, organization, analysis, and
display including making estimates and
predictions, critiquing data, and drawing
inferences (e.g., using the normal curve and
z-scores, line of best fit).
Understand and apply measures of
dispersion (range, mean deviation, variance,
and standard deviation).
Evaluate and employ accurate and
appropriate procedures for statistical data
collection, organization, analysis, and
display including making estimates and
predictions, critiquing data, and drawing
inferences (e.g., using the normal curve and
z-scores, line of best fit).
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 ( n Pr ) of n items
taken r at a time
Statistics & Probability: Conditional Probability & the Rules of
Probability
Use the rules of probability to compute probabilities of compound events in
a uniform probability model.
9. (+) Use permutations and combinations to compute probabilities of
compound events and solve problems.
Statistics & Probability: Conditional Probability & the Rules of
Probability
Use the rules of probability to compute probabilities of compound events in
a uniform probability model.
9. (+) Use permutations and combinations to compute probabilities of
compound events and solve problems.
 2011 International Center for Leadership in Education
M51
M51
Determine combinations (the various
groupings a set may be arranged in without
regard to order) and permutations
(arrangements of a set where order matters).
Determine combinations (the various
groupings a set may be arranged in without
regard to order) and permutations
(arrangements of a set where order matters).
Mathematics – Page 177
New York Mathematics
Strands/Bands/
Performance Indicators
Algebra 2/Trigonometry
Mathematics
Domains/Clusters/
Common Core State Standards
High School
A2.S.11 Calculate the number of
possible combinations ( n C r ) of n items
taken r at a time
Statistics & Probability: Conditional Probability & the Rules of
Probability
Use the rules of probability to compute probabilities of compound events in
a uniform probability model.
9. (+) Use permutations and combinations to compute probabilities of
compound events and solve problems.
Statistics & Probability: Conditional Probability & the Rules of
Probability
Use the rules of probability to compute probabilities of compound events in
a uniform probability model.
9. (+) Use permutations and combinations to compute probabilities of
compound events and solve problems.
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
Statistics & Probability: Using Probability to Make Decisions
Calculate expected values and use them to solve problems.
3. (+) Develop a probability distribution for a random variable defined for a
sample space in which theoretical probabilities can be calculated; find the
expected value. For example, find the theoretical probability distribution for
the number of correct answers obtained by guessing on all five questions of a
multiple-choice test where each question has four choices, and find the
expected grade under various grading schemes.
Statistics & Probability: Using Probability to Make Decisions
Calculate expected values and use them to solve problems.
4. (+) Develop a probability distribution for a random variable defined for a
sample space in which probabilities are assigned empirically; find the
expected value. For example, find a current data distribution on the number
of TV sets per household in the United States, and calculate the expected
number of sets per household. How many TV sets would you expect to find in
100 randomly selected households?
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
M51
M32
M51
M5
M32
M5
Determine combinations (the various
groupings a set may be arranged in without
regard to order) and permutations
(arrangements of a set where order matters).
Determine the probability of single and
compound events and use the Counting
Principle to determine the probability of
independent events occurring jointly.
Determine combinations (the various
groupings a set may be arranged in without
regard to order) and permutations
(arrangements of a set where order matters).
Examine problem-solving situations
involving simple probability and use
probabilistic reasoning to compare and
communicate the theoretical or empirical
likelihood of events.
Determine the probability of single and
compound events and use the Counting
Principle to determine the probability of
independent events occurring jointly.
Examine problem-solving situations
involving simple probability and use
probabilistic reasoning to compare and
communicate the theoretical or empirical
likelihood of events.
Mathematics – Page 178
New York Mathematics
Strands/Bands/
Performance Indicators
Algebra 2/Trigonometry
Mathematics
Domains/Clusters/
Common Core State Standards
High School
A2.S.15 Know and apply the binomial
probability formula to events involving
the terms exactly, at least, and at most
Algebra: Arithmetic with Polynomials & Rational Expressions
Use polynomial identities to solve problems.
5. (+) Know and apply the Binomial Theorem for the expansion of (x + y)n in
powers of x and y for a positive integer n, where x and y are any numbers,
with coefficients determined for example by Pascal’s Triangle. [The Binomial
Theorem can be proved by mathematical induction or by a combinatorial
argument.]
A2.S.16 Use the normal distribution as
an approximation for binomial
probabilities
Algebra: Arithmetic with Polynomials & Rational Expressions
Use polynomial identities to solve problems.
5. (+) Know and apply the Binomial Theorem for the expansion of (x + y)n in
powers of x and y for a positive integer n, where x and y are any numbers,
with coefficients determined for example by Pascal’s Triangle. [The Binomial
Theorem can be proved by mathematical induction or by a combinatorial
argument.]
Statistics & Probability: Interpreting Categorical & Quantitative Data
Summarize, represent and interpret data on a single count or measurement
variable.
4. Use the mean and standard deviation of a data set to fit it to a normal
distribution and to estimate population percentages. Recognize that there are
data sets for which such a procedure is not appropriate. Use calculators,
spreadsheets, and tables to estimate areas under the normal curve.
 2011 International Center for Leadership in Education
National Essential Skills Study (NESS)
National Rankings
Rank
M5
M70
M31
Examine problem-solving situations
involving simple probability and use
probabilistic reasoning to compare and
communicate the theoretical or empirical
likelihood of events.
Understand and apply the binomial theorem
(e.g., explore the relationship of the
binomial theorem with Pascal’s triangle and
the Fibonacci sequence).
Understand and apply measures of
dispersion (range, mean deviation, variance,
and standard deviation).
Understand and apply the binomial theorem
(e.g., explore the relationship of the
binomial theorem with Pascal’s triangle and
the Fibonacci sequence).
M70
Mathematics – Page 179