Mississippi College and Career Readiness
Standards for Mathematics Scaffolding
Document
Algebra I
September 2016 Page 1 of 97
College- and Career-Readiness Standards for Mathematics
ALGEBRA I
Number and Quantity
The Real Number System (N-RN)
Use properties of rational and irrational numbers
N-RN.3
Explain why:
●
the sum or product
of two rational
numbers is rational;
●
the sum of a rational
and irrational
number is irrational;
and
●
the product of a
nonzero rational
number and an
irrational number is
irrational.
Desired Student Performance
A student should know
● How to identify numbers on
a real number line.
● The significance of rational
and irrational numbers as
subsets of real numbers,
distinguishes between the
two, and provides examples
of each type when prompted.
● How to simplify expressions
including rational terms.
● How to use the properties of
exponents to evaluate
expressions with exponents,
including expressions
containing negative and zero
exponents.
A student should understand
● The meaning of rational
exponents follow the properties
of integer exponents. For
example, 5
1
3
is defined as the
cube root of 5 because (5
1
3
)
3
=
5
1
3
× 5
1
3
× 5
1
3
= 5.
● How to simplify and solve
expressions involving radicals
and rational exponents.
● The sum of rational numbers is
always rational, and the
product of rational numbers is
always rational.
● The sum of a rational number
and an irrational number is
A student should be able to do
● Classify numbers with the real
number system.
● Simplify and solve expressions
involving radicals, and rational
exponents.
● Extend the properties of integer
exponents to rational
exponents.
● Attend to precision
(Mathematical Practice 6),
using clear definitions and
stating the meaning of the
mathematical symbols they
include in their expressions.
● Explain why rational numbers
are closed under addition and
multiplication.
September 2016 Page 2 of 97
College- and Career-Readiness Standards for Mathematics
● How to write repeating
decimals as fractions.
● How to interpret and
compare representations of
square root functions.
● How to use the laws of
exponents to find products
and quotients of monomials.
● How to identify the
properties of the real number
system.
always irrational, and the
product of a rational number
and an irrational number is
always irrational.
September 2016 Page 3 of 97
College- and Career-Readiness Standards for Mathematics
ALGEBRA I
Number and Quantity
Quantities (N-Q)*
Reason quantitatively and use units to solve problems
N.Q.1
Use units as a way to
understand problems
and to guide the
solution of multi-step
problems; choose and
interpret units
consistently in
formulas; choose and
interpret the scale and
the origin in graphs and
data displays. *
Desired Student Performance
A student should know
● How to select appropriate
scales for a graph using
estimation.
● How to plot points on a
coordinate plane.
● The possible x- and y-
values of coordinates in
each quadrant of a
coordinate plane.
● How to plot points on a
graph given a table,
equation, or situation.
● How to interpret bar graphs,
line graphs, and
histograms.
A student should understand
● The meaning of slope and y-
intercept conceptually.
● How to interpret the slope and
y-intercept in statistical
situations.
● How to interpret data
displayed in graphs and make
predictions in real-world
context.
● Relationship between tabular
and graphic representations of
data.
A student should be able to do
● Justify answers to problems
using tables, graphs, formulas,
and equations.
● Measure and collect data,
selecting appropriate units and
degrees of precision for a given
situation.
● Describe the form, direction,
strength, and outliers of an
association using mathematical
terms. For example,
“predicted,” “expected” or
“approximate.”
September 2016 Page 4 of 97
College- and Career-Readiness Standards for Mathematics
● Make predictions based on
linear models and interpret
slope and y-intercept in context.
● Make connections between
solving equations, graphing,
and manipulating expressions.
September 2016 Page 5 of 97
College- and Career-Readiness Standards for Mathematics
ALGEBRA I
Number and Quantity
Quantities (N-Q)*
Reason quantitatively and use units to solve problems
N-Q.2
Define appropriate
quantities for the
purpose of descriptive
modeling. * [Refer to
the Quantities section
of the High School
Number and Quantity
Conceptual Category in
the previous pages of
this document.]
Desired Student Performance
A student should know
● How to convert rates and
units of measurement.
● The appropriate unit for
expressing different
quantities (e.g., length,
area, or volume).
● How to create bar graphs,
line graphs, and
histograms.
A student should understand
● How to choose appropriate units
by defining quantities needed to
model a situation.
● How to express information in
appropriate units and with
understandable scales on
graphs in modeling real-world
situations.
● How to determine what quantity
and unit to express in a final
solution.
● How to determine which
numeric form of their solution is
appropriate (e.g., mixed
A student should be able to do
● Describe data and
relationships from various
representations.
● Determine if a solution is
appropriate for the situation.
● Derive units to represent real-
world situations.
● Recognize whether given
quantities are discrete or
continuous.
● Define inputs and outputs in
specific mathematical models.
● Quantify real-world data based
on relevant attributes and
September 2016 Page 6 of 97
College- and Career-Readiness Standards for Mathematics
fractions, improper fraction,
decimals, or negative/positive
values).
create or choose suitable
measures for the situation.
September 2016 Page 7 of 97
College- and Career-Readiness Standards for Mathematics
ALGEBRA I
Number and Quantity
Quantities (N-Q)*
Reason quantitatively and use units to solve problems
N-Q.3
Choose a level of
accuracy appropriate
to limitations on
measurement when
reporting quantities. *
Desired Student Performance
A student should know
● The rules of significant
digits.
● How to select appropriate
scales for graph using
estimation.
● The relationship between
dependent and
independent variables in a
given data set.
● How to explain and
illustrate how a change in
one variable results in a
change in another
variable.
A student should understand
● How to determine the level of
accuracy needed by reading a
problem.
● How accurately answers can be
reported by recognizing which
quantity most restricts the
solution.
● The tools used to collect and
display data limits the accuracy
of a measurement.
● The analogy to univariate data is
how little a median or mean
really tells us about a set of
data.
● Why significant digits and units
are important in calculations and
measurement context.
A student should be able to do
● Describe the association with
form, direction, strength, and
outliers.
● Describe the precision of a
measurement tool.
● Recognize trends in data and
make predictions in relation to
context with an understanding
of accuracy and limitations.
● Recognize variability in data
and the need to address its
presence in data.
September 2016 Page 8 of 97
College- and Career-Readiness Standards for Mathematics
ALGEBRA I
Algebra
Seeing Structure in Expressions (A-SSE)
Interpret the structure of expressions
A-SSE.1a
Interpret expressions
that represent a
quantity in terms of its
context.*
a. Interpret parts of an
expression, such as
terms, factors, and
coefficients.
Desired Student Performance
A student should know
● How to use substitution to
make new identities.
● How to use the distributive
property to expand
polynomials.
● How to evaluate numerical
expressions involving
parentheses, powers, and
rational numbers.
● How to translate verbal
phrases into mathematical
expressions.
A student should understand
● The mathematical meaning of
the following words: factors,
coefficients, terms, exponent,
base, constant, and variable.
● How to represent and identify
factors, coefficients, terms,
exponents, bases, constants,
and variables components
when given a mathematical
expression.
● How to explain the effect of
changing one part of an
expression by analyzing its
component parts.
● How to write and interpret
complex expressions by
A student should be able to do
● Explain the meaning of the
parts of an expression as they
relate to the entire expression
and to the context of the
problem.
● Extend understanding of the
structure of linear, exponential,
and quadratic functions to
radical, rational, and polynomial
functions.
● Identify the parts of any
expression as terms, factors,
coefficients, exponents,
quotients, divisors, dividends,
remainders, and constants.
September 2016 Page 9 of 97
College- and Career-Readiness Standards for Mathematics
analyzing their component
parts.
● Determining the real-world
context of the variables, factors,
or terms in an expression.
September 2016 Page 10 of 97
College- and Career-Readiness Standards for Mathematics
ALGEBRA I
Algebra
Seeing Structure in Expressions (A-SSE)
Interpret the structure of expressions
A-SSE.1b
Interpret expressions
that represent a
quantity in terms of its
context.*
b. Interpret complicated
expressions by viewing
one or more of their
parts as a single entity.
For example, interpret
P(1+r)
n
as the product
of P and a factor not
depending on P.
Desired Student Performance
A student should know
● How to use substitution to
make new identities.
● How to evaluate numerical
expressions involving
parentheses, powers, and
rational numbers.
● How to use the distributive
property to expand
polynomials.
● How to translate verbal
phrases into mathematical
expressions.
● How to recognize and use
the properties of identity
and equality.
A student should understand
● The mathematical meaning of
the following words: factors,
coefficients, terms, exponent,
base, constant, and variable.
● How to represent and identify
factors, coefficients, terms,
exponents, bases, constants,
and variables components
when given a mathematical
expression.
● How to explain the effect of
changing one part of an
expression by analyzing its
component parts.
● How to write and interpret
complex expressions by
A student should be able to do
● Write an expression containing
identical factors as an
expression using exponents.
● Evaluate open sentences by
performing operations.
● Write formulas using two or
more variables.
● Explain the meaning of the parts
of an expression as they relate
to the entire expression and to
the context of the problem.
● Extend understanding of the
structure of linear, exponential,
and quadratic functions to
radical, rational, and polynomial
functions.
September 2016 Page 11 of 97
College- and Career-Readiness Standards for Mathematics
analyzing their component
parts.
● Determine the real-world
context of the variables, factors,
or terms in an expression.
September 2016 Page 12 of 97
College- and Career-Readiness Standards for Mathematics
ALGEBRA I
Algebra
Seeing Structure in Expressions (A-SSE)
Interpret the structure of expressions
A-SSE.2
Use the structure of an
expression to identify
ways to rewrite it. For
example, see x
4
–y
4
as
(x
2
)2 –(y
2
)
2
, thus
recognizing it as a
difference of squares
that can be factored as
(x
2
–y
2
)(x
2
+ y
2
).
Desired Student Performance
A student should know
● How to simplify expressions
involving rational numbers
and coefficients.
● How to apply properties of
exponents to simplify
expressions.
● How to add, subtract,
multiply, and divide
polynomials.
A student should understand
● Polynomial or rational
expressions can sometimes be
simplified to binomials or
quadratic factors.
● How to find patterns in
repeated calculations, and
make conjectures based on
these patterns.
● How to expand powers and
products of expressions.
● How to factor expressions
completely.
● How to compare the
equivalence relationship
between the original form of an
A student should be able to do
● Use algebraic methods and
mathematical properties to
transform expressions to
determine whether expressions
are equivalent.
● Rearrange terms to rewrite an
equivalent expression.
● Write expressions in equivalent
forms by factoring.
● Apply the difference of squares
theorem to polynomial
expressions and numerical
examples.
● Factor polynomials completely.
● Rewrite algebraic expressions
in different equivalent forms by
September 2016 Page 13 of 97
College- and Career-Readiness Standards for Mathematics
expression and its expanded
form.
using methods such as factoring
or combining like terms.
September 2016 Page 14 of 97
College- and Career-Readiness Standards for Mathematics
ALGEBRA I
Algebra
Seeing Structure in Expressions (A-SSE)
Write expressions in equivalent forms to solve problems
A-SSE.3a
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.
Desired Student Performance
A student should know
● How to simplify expressions
involving rational numbers
and coefficients.
● How to solve multiple-step
equations including
variations of the distributive
property.
● How to apply properties of
exponents to simplify and
rewrite expressions.
● How to add, subtract,
multiply, and divide
polynomial expressions.
A student should understand
● How to expand the product of
linear factors into polynomials
and compare the two
expressions and look for
patterns.
● How to rewrite expressions in
different forms using
mathematical properties.
● The best form to write an
expression given the context of
an expression.
● The relationship between the
factorization of a quadratic
expression and the solutions of
a quadratic equation.
A student should be able to do
● Factor expressions completely
using various factoring methods.
● Apply the Zero-Product Property
to factored expressions.
● Use algebra to simplify long
computations, such as computing
large sums of consecutive
numbers.
● Factor expressions by identifying
a common factor.
● Use difference of squares
factoring to solve equations.
● Explain and justify the
relationship between the
factorization of a quadratic
expression and the solutions of a
quadratic equation.
September 2016 Page 15 of 97
College- and Career-Readiness Standards for Mathematics
ALGEBRA I
Algebra
Seeing Structure in Expressions (A-SSE)
Write expressions in equivalent forms to solve problems
A-SSE.3b
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.
Desired Student Performance
A student should know
● How to simplify expressions
involving rational numbers
and coefficients.
● How to solve multiple-step
equations including
variations of the distributive
property.
● How to apply properties of
exponents to simplify and
rewrite expressions.
● How to add, subtract,
multiply, and divide
polynomial expressions.
A student should understand
● Completing the square is a part
of a process that transforms a
quadratic polynomial into a
difference of squares.
● How to graph quadratic
functions and examine the
graph to find the vertex.
● How to use their knowledge of
quadratics to optimize
quadratic functions.
● How to expand the product of
linear factors into polynomials
and compare the two
expressions and look for
patterns.
A student should be able to do
● Factor expressions completely.
● Apply the Zero-Product
Property to factored
expressions.
● Convert the equation of a
parabola into graphing form by
completing the square.
● Write expressions in equivalent
forms by completing the square
to convey the vertex form, to
find the maximum or minimum
value of a quadratic function,
and to identify and explain the
meaning of the vertex.
● Use difference of squares and
factoring to solve equations.
September 2016 Page 16 of 97
College- and Career-Readiness Standards for Mathematics
● How to rewrite expressions in
different forms using
mathematical properties.
● The optimal form to write an
expression given the context.
● Explain and justify the
relationship between the
factorization of a quadratic
expression and the solutions of
a quadratic equation.
September 2016 Page 17 of 97
College- and Career-Readiness Standards for Mathematics
ALGEBRA I
Algebra
Seeing Structure in Expressions (A-SSE)
Write expressions in equivalent forms to solve problems
A-SSE.3c
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 1.15
t
can be
written as
Desired Student Performance
A student should know
● How to simplify
expressions involving
rational numbers and
coefficients.
● How to solve multiple-step
equations including
variations of the distributive
property.
● How to apply properties of
exponents to simplify and
rewrite expressions.
● How to add, subtract,
multiply, and divide
polynomial expressions.
A student should understand
● How to use properties of
exponents to create equivalent
expressions.
● How to expand the product of
linear factors into polynomials
and compare the two
expressions and look for
patterns.
● How to represent exponential
decay in multiple ways and how
to investigate the effect when
the exponent is 0 or negative.
● How to rewrite expressions in
different forms using
mathematical properties.
A student should be able to do
● Solve complicated equations
and simple exponential
equations by rewriting and
solving an equivalent equation.
● Factor expressions completely
using various factoring skills.
● Apply the zero-property to
factored expressions and
explain meaning of the zeros.
● Use algebra to simplify long
computations, such as
computing large sums of
consecutive numbers.
September 2016 Page 18 of 97
College- and Career-Readiness Standards for Mathematics
[
1
.
15
1
12
]
12
≈
1
.
012
12
to
reveal the approximate
equivalent monthly
interest rate if the
annual rate is 15%.
● The most useful form to write an
expression given the context of
an expression.
● The relationship between the
factorization of a quadratic
expression and the solutions of
a quadratic equation.
● Use factoring strategies,
including difference of squares,
to solve equations.
● Explain and justify the
relationship between the
factorization of a quadratic
expression and the solutions of
a quadratic equation.
September 2016 Page 19 of 97
College- and Career-Readiness Standards for Mathematics
ALGEBRA I
Algebra
Arithmetic with Polynomials and Rational Expressions (A-APR)
Perform arithmetic operations on polynomials
A-APR.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.
Desired Student Performance
A student should know
● How to identify polynomials
and their characteristics.
● How to identify like terms.
● How to use the distributive
property.
● How to find the degree of a
polynomial.
● Rules for adding,
subtracting, and multiplying
integers.
● How to define terms related
to the characteristics of
polynomials. (e.g., terms,
degree, coefficient, leading
coefficient, monomial,
binomial, and trinomials).
A student should understand
● How to add and subtract
polynomials.
● How to simplify the product of
a polynomial by a monomial.
● Polynomials, like integers, are
“closed” under addition,
subtraction, and multiplication.
● How to combine linear and
quadratic polynomials with
addition and subtraction.
● How to multiply a constant by
a linear or quadratic
polynomial.
A student should be able to do
● Write polynomials in standard
form.
● Multiply polynomials using
multiple methods.
● Find squares of binomials
involving sums and
differences.
● Look closely to discern a
pattern or structure when
finding the square of a sum
and difference.
September 2016 Page 20 of 97
College- and Career-Readiness Standards for Mathematics
● The concept of a zero pair.
● The concept of closure.
September 2016 Page 21 of 97
College- and Career-Readiness Standards for Mathematics
ALGEBRA I
Algebra
Arithmetic with Polynomials and Rational Expressions (A-APR)
Understand the relationship between zeros and factors of polynomials
A-APR.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 (limit to 1st-
and 2nd- degree
polynomials).
Desired Student Performance
A student should know
● How to recognize
equivalent expressions.
● How to solve multi-step
equations in one variable.
● How to identify and graph
linear functions.
● How to factor a polynomial
completely.
● How to recognize perfect-
square polynomials.
● How to graph quadratic
functions by hand, showing
intercepts, and maxima or
minima.
● The relationship of the
degree of a polynomial to
A student should understand
● How to factor expressions by
identifying a common factor.
● How to apply the Zero-Product
Property to factored
expressions.
● How factors, zeros, and x-
intercepts, of a polynomial
function are related.
● How factors and roots of a
polynomial function are
related.
● Key features of a parabola by
looking at how the coefficients
affect the graph.
● If the product of two quantities
equals zero, at least one of the
quantities equals zero.
A student should be able to do
● Find zeros by factoring
polynomials of 1st- and 2nd-
degrees and use the Zero-
Product Property.
● Determine the maximum
number of zeros of a
polynomial.
● Recognize that repeated factors
indicate multiplicity of roots and
graph polynomials with repeated
factors.
● Identify the zeros of a
polynomial.
● Find the zeros of a polynomial
from its graph.
September 2016 Page 22 of 97
College- and Career-Readiness Standards for Mathematics
the graph of the polynomial
function.
● Why each factor is set to equal
zero.
● Use the zeros to construct a
rough graph of the function
defined by the polynomial.
September 2016 Page 23 of 97
College- and Career-Readiness Standards for Mathematics
ALGEBRA I
Algebra
Creating Equations (A-CED)*
Create equations that describe numbers or relationships
A-CED.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.*
Desired Student Performance
A student should know
● How to define variables.
● How to translate algebraic
and verbal expressions.
● How to solve multi-step
equations and inequalities
in one variable.
● How to solve equations
and inequalities with
variables on both sides.
● How to rewrite equations
and formulas.
A student should understand
● The relationships between
quantities (e.g., how the
quantities are changing or
growing with respect to each
other); express these
relationships using
mathematical operations to
create an appropriate equation
or inequality to solve.
● Build an equation or inequality
from a mathematical situation.
● Determine when equations
and inequalities are true
sometimes, always, or never.
● Discern when to represent an
equation and inequality using
A student should be able to do
● Construct and solve linear,
exponential, and quadratic
equations in one variable given
real-world situations.
● Construct and solve simple
exponential functions by
examining exponential growth
and decay problems.
● Construct equations that models
geometric change by visualizing
and extending a pattern.
● Extend their understanding of
exponential functions by
examining the multiplier and
starting point in different
representations.
September 2016 Page 24 of 97
College- and Career-Readiness Standards for Mathematics
one variable versus two
variables.
● How to identify linear,
quadratic, and exponential
functions from multiple
representations.
● Understand the relationship of
the zeros of a quadratic function
and the x-intercepts of its graph.
● Model real-world problems
using linear, quadratic, and
simple rational and exponential
functions.
September 2016 Page 25 of 97
College- and Career-Readiness Standards for Mathematics
ALGEBRA I
Algebra
Creating Equations (A-CED)*
Create equations that describe numbers or relationships
A-CED.2
Create equations in two
variables to represent
relationships between
quantities; graph
equations on
coordinate axes with
labels and scales.*
[Note the standard
appears in future
courses with a slight
variation in the
standard language.]
Desired Student Performance
A student should know
● How to translate algebraic
and verbal expressions.
● How to solve equations in
one variable.
● How to solve one-variable
equations with variables on
both sides.
● How to solve literal
equations for specific
variable.
● How to graph linear
equations on a coordinate
axis with labels and scales.
A student should understand
● How to build equations from
mathematical situations.
● How to solve a two-variable
equation.
● When equations are true
sometimes, always, or never.
● The slope and y-intercept can
be used to write and graph an
equation of the line.
● How to explain and illustrate
how a change in one variable
results in a change in another
variable and apply to the
relationships between
A student should be able to do
● Identify the quantities in a
mathematical problem or real-
world situation that should be
represented by distinct variables
and describe what quantities the
variable represents.
● Write and graph an equation of a
direct variation (proportional
relationship).
● Determine appropriate units for
the labels and scale of a graph
depicting the relationship
between equations created in
two variables.
September 2016 Page 26 of 97
College- and Career-Readiness Standards for Mathematics
● How to apply contextual
meaning to slope and
y-intercept.
● How to interpret graphs and
write equations for linear
relations.
● How to justify the
relationship between graph,
table, equation, and
situation.
independent and dependent
variables.
● How to graph and analyze
linear and exponential
functions.
● How to use algebraic and
graphical methods to solve
systems of linear equations in
mathematical and real-world
situations.
● Graph created equations and
inequalities in two variables on a
coordinate plane with
appropriate labels and scales.
● Identify and evaluate linear and
exponential functions.
● Write linear and exponential
equations from a given graph,
table, or situation that describes
the distinct function.
September 2016 Page 27 of 97
College- and Career-Readiness Standards for Mathematics
ALGEBRA I
Algebra
Creating Equations (A-CED)*
Create equations that describe numbers or relationships
A-CED.3
Represent constraints
by equations or
inequalities, and by
systems of equations
and/or inequalities, and
interpret solutions as
viable or non-viable
options in a modeling
context. For example,
represent inequalities
describing nutritional
and cost constraints on
combinations of
different foods.*
Desired Student Performance
A student should know
● How to build equations from
a mathematical or real-
world situation.
● How to solve multiple-step
equations and inequalities
with one variable.
● How to solve and graph
equations and inequalities
of two variables.
● How to solve and graph
systems of equations and
inequalities.
● How to determine when
equations are true
sometimes, always, or
never.
A student should understand
● How to define constraints and
determine their necessity in
modeling real-world
situations.
● Constraints are necessary to
balance a mathematical
model with real-world context.
● When a modeling context
involves constraints.
● How to interpret solutions as
viable or nonviable options in
a modeling context.
● When a problem should be
represented by an equation,
inequality, systems of
equations, and/or inequalities.
● How to represent constraints
by equations or inequalities
A student should be able to do
● Use the graphing method to
solve or estimate the solutions
of complex equations and
inequalities.
● Explain the meaning of
solutions to equations and
inequalities using the context of
the problem.
● Eliminate algebraic solutions
that do not make sense in the
context of the problem.
● Recognize how certain input
and output values may or may
not be reasonable.
● Select an appropriate domain
for a single-variable in a
modeling context.
September 2016 Page 28 of 97
College- and Career-Readiness Standards for Mathematics
and by systems of equations
and/or inequalities.
● Develop necessary constraints
using linear equations and
linear inequalities.
September 2016 Page 29 of 97
College- and Career-Readiness Standards for Mathematics
ALGEBRA I
Algebra
Creating Equations (A-CED)*
Create equations that describe numbers or relationships
A-CED.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 .*
Desired Student Performance
A student should know
● How to simplify expressions
by combining like terms.
● How to solve multiple-step
equations including rational
coefficients and involving
the distributive property.
● How to solve equations with
variables on both sides.
A student should understand
● Formulas are equations with
specific meaning that show the
relationship between two or
more quantities.
● Why rewriting formulas can be
useful.
● How to manipulate an
equation algebraically without
changing its value.
● Two equations that appear to
be very different can describe
the same equation.
● How to solve an equation for a
specific variable.
A student should be able to do
● Solve literal equations using the
same processes used in solving
numerical equations.
● Solve formulas that arise from
real-world situations and are
limited to linear and quadratic
variables.
● Translate a linear equation in
standard form to slope intercept
form.
● Translate a linear equation in
slope intercept form to standard
form.
September 2016 Page 30 of 97
College- and Career-Readiness Standards for Mathematics
ALGEBRA I
Algebra
Reasoning with Equations and Inequalities (A-REI)
Understand solving equations as a process of reasoning and explain the reasoning
A-REI.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.
Desired Student Performance
A student should know
● The order of operations
and how to apply it.
● Zero pairs can be used to
simply addition and
subtraction equations.
● How to simplify expressions
using properties of algebra.
● How to add, subtract,
multiply, and divide rational
numbers.
A student should understand
● How to construct a
mathematically viable
argument justifying a given, or
self-generated, solution
method.
● Equations can have multiple
solutions or no solutions.
● How to work backward to
justify solutions to equations.
A student should be able to do
● Apply and explain the results of
using inverse operations.
● Justify the steps in solving
equations by applying and
explaining the properties of
equality, inverse, and identity.
● Use the names of the properties
to aid in justifying the steps
performed when solving an
equation.
● Find and analyze mistakes in
work samples.
● Choose an appropriate method for
solving an equation.
● Show steps to justify
mathematical methods.
September 2016 Page 31 of 97
College- and Career-Readiness Standards for Mathematics
● Share different ways of solving
equations that lead to the same
solution.
September 2016 Page 32 of 97
College- and Career-Readiness Standards for Mathematics
ALGEBRA I
Algebra
Reasoning with Equations and Inequalities (A-REI)
Solve equations and inequalities in one variable
A-REI.3
Solve linear equations
and inequalities in one
variable, including
equations with
coefficients
represented by letters.
Desired Student Performance
A student should know
● How to simplify expressions
involving rational numbers
and coefficients.
● The order of operations and
how to apply it.
A student should understand
● How to solve equations and
inequalities with variables on
both sides.
● How to solve equations and
inequalities using inverse
operations.
● How to solve equations and
inequalities involving many
variations of the distributive
property.
● How to solve equations and
inequalities involving rational
coefficients.
● Equations can have multiple
solutions or no solutions.
A student should be able to do
● Interpret a situation and
represent it mathematically.
● Deepen understanding of
equations as statements about
numbers that can be true
always, sometimes, or never.
● Extend earlier work with solving
linear equations/inequalities in
one variable to solving literal
equations that are linear in the
variable being solved for.
Include simple exponential
equations that rely only on
application of the laws of
exponents.
September 2016 Page 33 of 97
College- and Career-Readiness Standards for Mathematics
● How solving literal equations
relates to solving numeric
equations.
● Build an equation from a
mathematical situation.
● Rewrite mathematical formulas
in equivalent forms.
● Solve literal equations for a
specified variable.
September 2016 Page 34 of 97
College- and Career-Readiness Standards for Mathematics
ALGEBRA I
Algebra
Reasoning with Equations and Inequalities (A-REI)
Solve equations and inequalities in one variable
A-REI.4a
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.
Desired Student Performance
A student should know
● How to factor quadratic
polynomials.
● How to use factoring to
solve equations.
● How to apply the Difference
of Squares Theorem to
polynomial expressions and
numerical examples.
● How to use difference of
squares factoring to solve
equations.
A student should understand
● Equations can be written in
more than one form.
● Write quadratic equations in
both standard form and vertex
form.
● What different forms for writing
quadratics reveal about the
function.
● How to solve quadratic
equations by completing the
square.
● The connection between the
quadratic formula and the
process of completing the
square.
● The connection between the
roots of a quadratic equation
and the coefficients of a
quadratic equation.
A student should be able to do
● Write quadratic equations in one
variable in both standard form
as well as vertex form and
understand what the
parameters of each form
reveals about the function.
● Derive the quadratic formula by
completing the square of a
general quadratic equation.
● Construct a quadratic equation
given the equation’s two roots.
● Factor non-monic quadratics.
● Identify which process is best to
solve a quadratic equation.
● Identify the y-intercept, zeros,
and vertex of a quadratic
function and use that to create a
rough sketch of the function,
September 2016 Page 35 of 97
College- and Career-Readiness Standards for Mathematics
ALGEBRA I
Algebra
Reasoning with Equations and Inequalities (A-REI)
Solve equations and inequalities in one variable
A-REI.4b
Solve quadratic
equations in one
variable.
b. Solve quadratic
equations by
inspection (e.g., for
2
= 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.
Desired Student Performance
A student should know
● How to identify linear,
exponential, and quadratic
functions by inspection.
● How to extend the property
of exponents to rational
exponents.
● How to factor polynomials by
using the greatest common
factor.
● How to factor general
quadratic polynomials.
● How to use factoring to solve
quadratic equations.
A student should understand
● How to factor a quadratic
expression to reveal the zeros
of the function.
● When solving by inspection,
be able to identify the number
of real roots, their value, and if
there is no real root.
● The similarities and differences
between quadratic functions
and linear functions.
● How to determine the best
method to solve a quadratic
equation.
● How to transform quadratic
equations to and from
standard form, graphing form,
and factored form.
A student should be able to do
● Solve quadratic equations by
taking the square root.
● Solve quadratic equations by
factoring.
● Solve quadratic equations by
inspection.
● Recognize non-real solutions.
● Create a quadratic equation
that describes a given
situation.
● Solve quadratic equations by
inspection, factoring,
completing the square, and the
quadratic formula.
● Complete the square in a
quadratic expression to reveal
September 2016 Page 36 of 97
College- and Career-Readiness Standards for Mathematics
the minimum or maximum
value of the function.
● Transform quadratic equations
to and from standard form,
graphing form, and factored
form.
● Recognize when the
solution(s) to a quadratic
equation is not real, i.e.
complex (when the value of
the expression under the
radical in the quadratic formula
is negative).
September 2016 Page 37 of 97
College- and Career-Readiness Standards for Mathematics
ALGEBRA I
Algebra
Reasoning with Equations and Inequalities (A-REI)
Solve systems of equations
A-REI.5
Given a system of two
equations in two
variables, show and
explain why the sum of
equivalent forms of the
equations produces the
same solution as the
original system.
Desired Student Performance
A student should know
● How to solve multiple-step
equations involving rational
numbers and coefficients.
● How to solve literal
equations for specific
variables.
● How to rewrite equations in
equivalent forms.
● How to evaluate numerical
expressions involving
parentheses, powers, and
rational numbers.
● How to express word
problems using variables
and mathematical notation.
● How to write formulas using
two or more variables.
A student should understand
● Solving system of equations
by using elimination with
addition requires adding the
two equations together to
eliminate one of the variables.
● An equivalent system is
formed whenever one of the
equations is multiplied by a
nonzero number and/or when
one of the equations is
replaced by the sum of a
constant multiple of another
equation and that equation.
● Equations do not have to be
written in standard form to use
elimination.
A student should be able to do
● Provide mathematical
justification for the addition and
subtraction methods of solving
systems of equations
(elimination method).
● Solve system of equations by
using elimination with
multiplication.
● Write and solve a linear system
of equation to represent a
contextual problem.
● Substitute the value from
solving for one of the variable
into either equation, and solve
for the other variable.
● Write the solution to the system
of equation as an ordered pair.
September 2016 Page 38 of 97
College- and Career-Readiness Standards for Mathematics
● How to write linear
equations in standard form.
● A system of intersecting lines
has exactly one solution and is
consistent and independent.
● A system whose graphs
coincide has infinitely many
solutions and is consistent and
dependent.
● A system of parallel lines has
no solution and is consistent.
● Determine if a system of
equation has exactly one
solution, no solution, or
infinitely many solutions.
● Recognize constraints of
systems of equations when
modeling real-world situations.
September 2016 Page 39 of 97
College- and Career-Readiness Standards for Mathematics
ALGEBRA I
Algebra
Reasoning with Equations and Inequalities (A-REI)
Solve systems of equations
A-REI.6
Solve systems of
linear equations
algebraically,
exactly, and
graphically while
focusing on pairs of
linear equations in
two variables.
Desired Student Performance
A student should know
● How to solve multiple step
equations involving rational
numbers and coefficients.
● How to solve literal equations
for specific variables.
● How to rewrite equations in
equivalent forms.
● How to evaluate numerical
expressions involving
parentheses, powers, and
rational numbers.
● How to express word problems
using variables and
mathematical notation.
● How to write formulas using
two or more variables.
● How to write linear equations
in standard form.
● How to graph linear equations
in two variables.
● How to find and interpret slope
and y-intercept.
A student should understand
● Systems of equations can be
solved both graphically and
algebraically.
● A system of intersecting lines
has exactly one solution and is
consistent and independent.
● A system whose graphs
coincide has infinitely many
solutions and is consistent and
dependent.
● A system of parallel lines has no
solution and is consistent.
● How recognizing and comparing
the slopes of a lines can help
solve many problems and reveal
many characteristics of lines.
A student should be able to do
● Graph systems of linear equations.
● Solve systems of linear equations
by graphing.
● Write and solve real-world and
mathematical situation problems
for systems of equations.
● Determine the best method for
solving systems of equations.
● Solve a linear system of equations
using the linear combinations
(elimination method).
● Apply systems of linear equations.
● Determine whether a system of
linear equations has no, one, or
infinitely many solutions.
● Recognize constraints of systems
of equations when modeling real-
world situation.
September 2016 Page 40 of 97
College- and Career-Readiness Standards for Mathematics
ALGEBRA I
Algebra
Reasoning with Equations and Inequalities (A-REI)
Represent and solve equations and inequalities graphically
A-REI.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).
Desired Student Performance
A student should know
● How to plot points on a
coordinate plane.
● How to substitute values
for variables.
● How to find the slope of a
line given two points.
● How to recognize the slope
of a line given an equation
in both standard form and
y-intercept form.
● How to graph points given
a table of values.
A student should understand
● How to find the solutions to an
equation and how they relate to
the graph of the equation.
● A graph/curve is a visual
representation of an equation
or data.
● An ordered pair is a solution to
the equation if it represents a
point on the graph.
● How to graph an equation given
in both standard form and slope
intercept form.
● How to identify characteristics
of a graph given its equation.
● How equations, graphs, and
tables are related.
● How to create a table of values
that satisfy an equation.
A student should be able to do
● Test a point to determine
whether it is on the graph of an
equation.
● Graph an equation by plotting
points.
● Write the equation of a vertical
or horizontal line given its graph
or a point on its graph.
● Write equations of a line given
slope and y-intercept, two points,
or slope and a point.
● Read a graph to identify points
that are solutions to an equation.
● Find the intersection points of
two graphs and understand its
meaning.
● Identify different graphs as
belonging to the same family of
graphs.
September 2016 Page 41 of 97
College- and Career-Readiness Standards for Mathematics
● A continuous curve or a line
contains an infinite number of
solutions.
● Identify solutions and non-
solutions of linear and
exponential equations.
September 2016 Page 42 of 97
College- and Career-Readiness Standards for Mathematics
ALGEBRA I
Algebra
Reasoning with Equations and Inequalities (A-REI)
Represent and solve equations and inequalities graphically
A-REI.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, quadratic,
absolute value, and
exponential functions
Desired Student Performance
A student should know
● How to evaluate
expressions.
● How to construct a table
of values for a given
function.
● How to graph functions
using graphing
technology.
● How to recognize
proportionality in direct
and inverse variation.
● How to graph equations
given in both standard
form and slope intercept
form.
A student should understand
● How technology can be used
to find the domain, range,
points of intersection, and
other attributes use to
characterize families of
graphs.
● How to recognize the
distinguishing features of basic
graphs, such as their general
shape, and the points and
quadrants that they pass
through.
● How to describe the rules for
translating graphs of equations
vertically or horizontally.
A student should be able to do
● Write and graph linear,
quadratic, absolute value, and
exponential functions (by hand
as well as with technology).
● Approximate solutions to
systems of two equations using
graphing technology.
● Approximate solutions to
systems of two equations using
tables of values.
● 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)
.
September 2016 Page 43 of 97
College- and Career-Readiness Standards for Mathematics
● How to decide whether a
situation represents direct or
inverse variation.
● How manipulating parameters
of the symbolic rule will result
in a predictable transformation
of the graph.
● Be able to express that when
f(x) = g(x), the two equations
have the same solution(s).
● Adjust the window setting on
specific graphing technology
devices to approximate
solutions to systems of
equations.
● Compare graphs of linear,
quadratic, absolute value, and
exponential functions.
● Use the graphing method to
solve or estimate the solutions
of complex equations.
● Solve system of equations when
one or both equations is/are not
linear.
● Use intersections of functions to
find solutions to the related
single-variable equations.
● Discuss misconceptions and
assumptions associated with the
standard screen view when
using graphing technology to
graph systems of equations and
approximate intersection points.
September 2016 Page 44 of 97
College- and Career-Readiness Standards for Mathematics
ALGEBRA I
Algebra
Reasoning with Equations and Inequalities (A-REI)
Represent and solve equations and inequalities graphically
A-REI.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.
Desired Student Performance
A student should know
● How to write and graph
linear equations with two
variables.
● How to simplify inequalities
to represent them in a
format that is easy to graph.
● How to find and interpret
the slope of a line and
recognize its relationship in
graphs.
A student should understand
● All points on a half-plane are
solutions to a linear inequality.
● The solutions to a system of
inequalities in two-variables are
the points that lie in the
intersection of the
corresponding half-planes.
● How to graph the solution set of
linear and non-linear
inequalities with two variables.
● How to graph a system of linear
equation and inequality on a
coordinate plane.
● How to explain that the solution
set for a system of linear
inequalities is the intersection
A student should be able to do
● Determine whether the
boundary line should be
included as part of the solution
set.
● Graph the solutions to a linear
inequality in two variables as a
half-plane, excluding the
boundary for non-inclusive
inequalities.
● Graph the solution set to a
system of linear inequalities in
two variables as the intersection
of their corresponding half-
planes.
● Graph constraints using
systems of inequalities.
September 2016 Page 45 of 97
College- and Career-Readiness Standards for Mathematics
of the shaded regions (half-
planes) of both inequalities.
● How to check points in the
intersection of the half-planes
to verify that they represent a
solution to the system of
inequality.
● Use the graph of a two-variable,
linear inequality to solve real-
world mathematical situations.
● Use a system of inequalities to
create a graph of a feasible
region and then analyze
different scenarios based on the
feasible region.
September 2016 Page 46 of 97
College- and Career-Readiness Standards for Mathematics
ALGEBRA I
Functions
Interpreting Functions (F-IF)
Understand the concept of a function and use function notation
F-IF.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).
Desired Student Performance
A student should know
● How to simplify expressions
involving rational numbers
and coefficients.
● How to generate data by
evaluating expressions for
different values of a variable
and organize the data.
● How to justify conjectures
and patterns using
numerical expressions.
● How to translate verbal
phrases into mathematical
expressions.
● How to generalize patterns
using words and algebraic
methods.
A student should understand
● A function is a rule that assigns
each element from a set of
inputs to exactly one element
from a set of outputs.
● The difference between a
relation and a function.
● Domain can also be referred to
as “input” and “x-values”.
● Range can also be referred to
as “output” and y-values”.
● The graph of the function, f, is
the graph of the equation =
()
● The relationship between a
function, a table, and/or graph.
● How to look for and analyze
patterns in input-output tables.
A student should be able to do
● Use the definition of a function
to determine whether a
relationship is a function given a
table, graph, mapping, or words.
● Given the function, f(x), identify
x as an element of the domain,
the input, and f(x) is an element
in the range, the output.
● Find a rule to describe a set of
input and output values.
● Build a function from a real-
world mathematical situation or
word problem.
● Determine whether a
relationship is a function based
on its description or graph.
September 2016 Page 47 of 97
College- and Career-Readiness Standards for Mathematics
● How to recognize different
ways to express a function.
● Provide applications of
mathematical functions and
non-functions.
● Make input-output tables.
September 2016 Page 48 of 97
College- and Career-Readiness Standards for Mathematics
ALGEBRA I
Functions
Interpreting Functions (F-IF)
Understand the concept of a function and use function notation
F-IF.2
Use function notation,
evaluate functions for
inputs in their
domains, and interpret
statements that use
function notation in
terms of a context.
Desired Student Performance
A student should know
● How to simplify expressions
involving rational numbers
and coefficients.
● How to generate data by
evaluating expressions for
different values of a variable
and organize the data.
● How to justify conjectures
and patterns using numerical
expressions.
● How to translate verbal
phrases into mathematical
expressions.
● How to generalize patterns
using words and algebraic
methods.
A student should understand
● A function is a rule that assigns
each element from a set of
inputs to exactly one element
from a set of outputs.
● The graph of the function, f, is
the graph of the equation y=
f(x).
● How to recognize different
ways to define and express a
function.
● How to work with functions
expressed in various form (e.g.,
f(x) notation, tables, and
graphs.
● How to use function notation to
evaluate functions for given
inputs in the domain, including
combinations and compositions
of functions.
A student should be able to do
● Use function notation to express
relationships between contextual
variables.
● Input a value from the domain of
a function and evaluate.
● Create contextual examples that
can be modeled by linear or
exponential functions.
● Use the definition of a function to
determine whether a relationship
is a function given a table,
graph, mapping, or words.
● Given the function, f(x), identify x
as an element of the domain, the
input, and (f) x is an element in
the range, the output.
● Write a relation in function
notation.
● Find a rule to describe a set of
input and output values.
September 2016 Page 49 of 97
College- and Career-Readiness Standards for Mathematics
● The relationship between a
function, a table, mapping
and/or graph.
● How to look for and analyze
patterns in input-output tables.
● Build a function from a real-world
mathematical situation or word
problem.
● Determine whether a
relationship is a function based
on its description or graph.
● Provide applications of
mathematical functions and non-
functions.
● Make input-output tables.
● Identify functions, including
functions represented in
equations, tables, graphs, or
context.
September 2016 Page 50 of 97
College- and Career-Readiness Standards for Mathematics
ALGEBRA I
Functions
Interpreting Functions (F-IF)
Understand the concept of a function and use function notation
F-IF.3
Recognize that
sequences are
functions whose
domain is a subset of
the integers.
Desired Student Performance
A student should know
● How to simplify expressions
involving rational numbers
and coefficients.
● How to generate data by
evaluating expressions for
different values of a variable
and organize the data.
● How to justify conjectures
and patterns using numerical
expressions.
● How to translate verbal
phrases into mathematical
expressions.
● How to generalize patterns
using words and algebraic
methods.
● How to recognize linear
functions.
A student should understand
● How to recognize and explain
that an explicit formula allows
them to find any element of a
sequence without knowing the
previous term.
● The connection between
tables with constant
differences and linear
functions.
● How to look for and analyze
patterns in input-output tables.
● A function is a rule that
assigns each element from a
set of inputs to exactly one
element from a set of outputs.
● A sequence is a function with
a restricted domain.
A student should be able to do
● Identify and generate explicit
formula for arithmetic and
geometric sequences.
● Find an explicit function/rule
that models a real-world
mathematical situation.
● Represent data from a table,
graph, or situation as a
sequence and predict terms in
the sequence.
● Determine if a sequence is
arithmetic, geometric, or
neither.
● Determine whether a
relationship is a linear or
exponential function based on
its description, graph, equation
or table of values.
September 2016 Page 51 of 97
College- and Career-Readiness Standards for Mathematics
ALGEBRA I
Functions
Interpreting Functions (F-IF)
Interpret functions that arise in applications in terms of the context
F-IF.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.*
Desired Student Performance
A student should know
● How to find and interpret
slope as it relates to a graph.
● How to graph the linear
equations and inequalities in
two variables given in both
standard form and slope
intercept form.
● How to generate data by
evaluating expressions for
different values of a variable
and organize the data.
● How to justify conjectures
and patterns using numerical
expressions.
● How to translate verbal
phrases into mathematical
expressions.
A student should understand
● The graph of a function is
often a useful way of
visualizing the relationship of
the function models, and
manipulating a mathematical
expression for a function can
reveal important
characteristics of the function’s
properties.
● How to determine what a
graph looks like.
● How to describe what happens
when x increases/decreases.
● How to identify the x- and y-
intercepts of a graph
● How to determine any
limitations on the
inputs/outputs of the equation.
A student should be able to do
● Describe a parabola, using its
intercepts, minima, maxima,
vertex, symmetry, and whether
it is positively or negatively
oriented.
● Distinguish linear, quadratic
and exponential equations
based on equations, tables,
graphs and verbal
descriptions.
● Given a function, identify key
features such as intercepts;
intervals where the function is
increasing, decreasing,
positive, or negative; relative
maximums and minimums;
symmetries; and end behavior.
● Use key features of a function
to sketch a graph.
September 2016 Page 52 of 97
College- and Career-Readiness Standards for Mathematics
● How to identify a maximum or
minimum y-value (if it exists).
● How to determine whether the
graph has symmetry and
describe the symmetry.
● How to determine the direction
of the graph.
● How to compare the relative
steepness of lines and to build
intuition about positive,
negative, and zero slopes.
● Interpret key features in terms
of context.
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ALGEBRA I
Functions
Interpreting Functions (F-IF)
Interpret functions that arise in applications in terms of the context
F-IF.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.*
Desired Student Performance
A student should know
● How to determine whether
relations are functions using
tables, graphs, mapping, and
context.
● How to graph the linear
equations and inequalities in
two variables given in both
standard form and slope
intercept form.
● How to generate data by
evaluating expressions for
different values of a variable
and organize the data.
● How to justify conjectures
and patterns using numerical
expressions.
A student should understand
● How to interpret key features
of functions.
● How to sketch the graph of
functions showing key
features, with and without
technology.
● How to apply strategies for
finding exponential equations
given the y-intercept and
another point.
● How to relate the domain of a
function to its graph within
context of a given relationship.
● How to determine whether the
domain of a function is
reasonable given the context.
A student should be able to do
● Identify appropriate values for
the domain of a function based
on context.
● Identify the domain of a
function from the graph.
● Use set and interval notation to
represent domain.
● Describe the domain of a
relation by examining an
equation or graph.
● Solidify connections between
tables, equations, graphs and
mathematical situations
representations of functions.
● Find equations of linear,
quadratic and exponential
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● How to translate verbal
phrases into mathematical
expressions.
● How to sketch the graph of a
function that models a
relationship between two
quantities.
functions by using known
quantities to solve for a
missing parameter.
● Interpret fractional exponents.
September 2016 Page 55 of 97
College- and Career-Readiness Standards for Mathematics
ALGEBRA I
Functions
Interpreting Functions (F-IF)
Interpret functions that arise in applications in terms of the context
F-IF.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.*
Desired Student Performance
A student should know
● How to find and interpret
slope as it relates to a
graph.
● How to graph the linear
equations and inequalities
in two variables given in
both standard form and
slope intercept form.
● How to justify conjectures
and patterns using
numerical expressions.
● How to translate verbal
phrases into mathematical
expressions.
● How to generate data by
evaluating expressions for
A student should understand
● How the slope of a graph
relates to a rate of change.
● How to interpret the rate of
change and initial value of a
linear function in terms of the
situation it models and in
terms of its graph or a table of
values.
● The rate of change between
any two points, for non-linear
functions, might not be the
same as the rate of change of
the overall function.
● How to compare the relative
steepness of lines and to build
A student should be able to do
● Calculate the slope between
two points.
● Calculate the rate of change
over a given interval for
rational, square root, cube root,
and polynomial functions with a
context.
● Calculate the rate of change
when presented as an equation
or table.
● Estimate the rate of change
from a graph.
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different values of a variable
and organize the data.
intuition about positive,
negative, and zero slopes.
September 2016 Page 57 of 97
College- and Career-Readiness Standards for Mathematics
ALGEBRA I
Functions
Interpreting Functions (F-IF)
Analyze functions using different representations
F-IF.7a
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 functions
(linear and quadratic)
and show intercepts,
maxima, and minima.
Desired Student Performance
A student should know
● How to find and interpret
slope as it relates to a
graph.
● How to graph the linear
equations and inequalities
in two variables given in
both standard form and
slope intercept form.
● How to generate data by
evaluating expressions for
different values of a variable
and organize the data.
● How to justify conjectures
and patterns using
numerical expressions.
A student should understand
● The graph of a function is
often a useful way of
visualizing the relationship of
the function models, and
manipulating a mathematical
expression for a function can
reveal important
characteristics of the
function’s properties.
● How to interpret key features
of functions.
● How to sketch the graph of
functions showing key
features, with and without
technology.
A student should be able to do
● Graph and identify key features
in linear and quadratic functions
by hand and with technology.
● Distinguish between linear and
quadratic equations based on
equations, tables, graphs and
verbal descriptions.
● Given a linear or quadratic
function, identify key features
such as intercepts—intervals
where the function is
increasing, decreasing,
positive, or negative; relative
maximums and minimums;
symmetries; and end behavior.
● Use key features of a function
to sketch a graph.
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● How to translate verbal
phrases into mathematical
expressions.
● How to relate the domain of a
function to its graph within
context of a given relationship.
● How to sketch the graph of a
function that models a
relationship between two
quantities.
● How to describe what the
graph of a given function looks
like.
● How to describe what
happens when x
increases/decreases.
● How to identify x- and y-
intercepts.
● How to determine any
limitations on the
inputs/outputs of the equation.
● How to determine if there is a
maximum or minimum y-
value.
● How to determine whether the
graph has symmetry.
● How to identify the direction of
the graph.
● Interpret key features in terms
of context.
● Use the graphing method to
solve or estimate the solutions.
equations and inequalities.
● Graph quadratic equations
using vertex form.
September 2016 Page 59 of 97
College- and Career-Readiness Standards for Mathematics
ALGEBRA I
Functions
Interpreting Functions (F-IF)
Analyze functions using different representations
F-IF.7b
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
and piecewise-defined
functions, including
absolute value
functions.
Desired Student Performance
A student should know
● How to find and interpret
slope as it relates to a
graph.
● How to graph the linear
equations and inequalities
in two variables given in
both standard form and
slope intercept form.
● How to generate data by
evaluating expressions for
different values of a variable
and organize the data.
● How justify conjectures and
patterns using numerical
expressions.
A student should understand
● The graph of a function is
often a useful way of
visualizing the relationship of
the function models, and
manipulating a mathematical
expression for a function can
reveal important
characteristics of the function’s
properties.
● How to interpret key features
of functions.
● How to sketch the graph of
functions showing key
features, with and without
technology.
A student should be able to do
● Graph linear, exponential,
quadratic, absolute value, and
piecewise-defined functions by
hand as well as with
technology.
● Distinguish between linear,
exponential, quadratic, square
root, and piecewise-defined
functions in context.
● Interpret functions given in a
different representations. (i.e.,
equations, tables, graphs, and
verbal descriptions.)
● Given a linear or quadratic
function, identify key features
such as intercepts—intervals
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● How to translate verbal
phrases into mathematical
expressions.
● How to relate the domain of a
function to its graph within
context of a given relationship.
● How to sketch the graph of a
function that models a
relationship between two
quantities.
● How to describe what the
graph of a given function looks
like.
● How to describe what happens
when x increases/decreases.
● How to identify x- and y-
intercepts.
● How to determine any
limitations on the
inputs/outputs of the equation.
● How to determine if there is a
maximum or minimum y-value.
● How to determine whether the
graph has symmetry.
● How to identify the direction of
the graph.
where the function is
increasing, decreasing,
positive, or negative; relative
maximums and minimums;
symmetries; and end behavior.
● Use key features of a function
to sketch a graph.
● Interpret key features in terms
of context.
● Use the graphing method to
solve or estimate the solutions
of complex equations and
inequalities.
September 2016 Page 61 of 97
College- and Career-Readiness Standards for Mathematics
ALGEBRA I
Functions
Interpreting Functions (F-IF)
Analyze functions using different representations
F-IF.8a
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.
Desired Student Performance
A student should know
● How to recognize
equivalent expressions.
● How to solve two-step
equations with one variable.
● How to factor a polynomial
completely.
● How to recognize perfect-
square polynomials.
● How to graph quadratic and
linear functions by hand and
using technology.
● How the degree of a
polynomial relates to its
polynomial function.
A student should understand
● Where on a graph you can find
the solutions, zeros, roots, or x-
intercepts of a quadratic
function.
● How to 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 them.
● Which representation is best
when comparing the properties
of quadratic functions.
● How factors and roots of a
polynomial function are related.
A student should be able to do
● Use factoring and completing
the square to find key features
of quadratics.
● Write an equivalent form of a
function defined by an
expression for functions given.
● Apply the zero property to
factored expressions.
● Identify zeros, transformations,
points of discontinuity, and
asymptotes, when suitable
factorizations are available.
● Compare properties of
quadratic functions from
multiple representations.
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● How to identify key features of
a parabola by looking at how
the coefficients affect the
graph.
● If the product of two quantities
equals zero, at least one of the
quantities equals zero.
● Determine the maximum
number of zeros of a
polynomial.
● Model real-world problems
using quadratic functions.
September 2016 Page 63 of 97
College- and Career-Readiness Standards for Mathematics
ALGEBRA I
Functions
Interpreting Functions (F-IF)
Analyze functions using different representations
F-IF.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.
Desired Student Performance
A student should know
● How to recognize
equivalent expressions.
● How to solve multiple-step
equations involving one
variable and rational
numbers.
● How to factor a polynomial
completely.
● How to graph quadratic and
linear functions by hand and
using technology.
● How the degree of a
polynomial relates to its
polynomial function.
A student should understand
● Which representation is best
when comparing the properties
of quadratic functions.
● How factors and roots of a
polynomial function are related.
● Identify key features of a
parabola by looking at how the
coefficients affect the graph.
● Use transformations to simplify
calculations and show that two
expressions are equivalent.
● When it useful to write an
expression as a product of
expressions vs. the standard
form.
● Represent functions
algebraically, graphically,
A student should be able to do
● Express functions using
multiple representations and
compare the properties for
quadratic functions (e.g.,
equation, table of values,
graph, or mathematical
situation).
● Model quadratic functions in
real-world context.
● Determine which
representation is best
when comparing the
properties of quadratic
functions.
● Expand powers and products of
expressions.
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College- and Career-Readiness Standards for Mathematics
numerically in tables, and/or by
verbal description.
● Factor polynomials completely
using various factoring
techniques.
September 2016 Page 65 of 97
College- and Career-Readiness Standards for Mathematics
ALGEBRA I
Functions
Building Functions (F-BF)
Build a function that models a relationship between two quantities
F-BF.1a
Write a function that
describes a
relationship between
two quantities.*
a. Determine an
explicit expression or
steps for calculation
from a context.
Desired Student Performance
A student should know
● How to simplify expressions
including rational polynomial
terms.
● How to evaluate expressions
with exponents.
● How to relate representations
of square root functions
● How to use the laws of
exponents to find products
and quotients of monomials.
● How to use the properties of
exponents to simplify
expressions containing
negative and zero exponents.
● How to make input-output
tables and look for and
analyze patterns.
● How to graph linear equations
and inequalities in two
variables.
A student should understand
● How to write an explicit
formula for a sequence and
use the formula to identify
terms in the sequence.
● How to relate arithmetic
sequence to linear function
and geometric sequence to a
exponential function.
● How to build a function from a
real-world mathematical
situation.
● How to recognize and
describe patterns.
● How to determine the
common difference and
common ratio.
● How manipulating parameters
of the symbolic rule will result
in a predictable transformation
of the graph.
A student should be able to do
● Write an explicit formula
for arithmetic and
geometric functions.
● Represent data from a table,
graph, or situation as a
sequence and predict terms in
the sequence.
● Identify a sequence as
arithmetic, geometric, or neither.
● Determine whether a
relationship is a linear or
exponential function based on
its description, graph, equation
or table of values.
● Build functions and generate
graphs both by hand and using
graphing technology.
● Write a function that describes a
relationship between two
quantities by determining an
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College- and Career-Readiness Standards for Mathematics
● How to solve equations with
elements from a replacement
set.
● How to find function values.
explicit expression or steps for
calculation from a context.
September 2016 Page 67 of 97
College- and Career-Readiness Standards for Mathematics
ALGEBRA I
Functions
Building Functions (F-BF)
Build new functions from existing functions
F-BF.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
Desired Student Performance
A student should know
● How to solve quadratic
equations by inspection,
factoring, completing the
square and the quadratic
formula.
● How to complete the square in
a quadratic expression to
reveal the minimum or
maximum value of the function.
● How to solve quadratic
equations by inspection,
factoring, completing the
square and the quadratic
formula.
● How to complete the square in
a quadratic expression to
A student should understand
● How to describe the rules for
translating graphs of equations.
● How to recognize the
distinguishing features of basic
graphs, such as their general
shape, and the points and
quadrants that they pass
through.
● How to use graphing
technology to explore
transformations of functions.
● How to explore transformations
that preserve characteristics of
graphs of functions and which
do not.
A student should be able to do
● Sketch the graphs of the
equations = , = 1, =
2
, =
3
, =
√
, =
|
|
, and
variations of these equations.
● Perform transformation on
quadratic and absolute value
functions with and without
technology.
● Describe the effects of each
transformation of functions
(e.g., if () is replaced with
( + )).
● Given the Given the graph of a
function, describe all
September 2016 Page 68 of 97
College- and Career-Readiness Standards for Mathematics
algebraic expressions
for them.
reveal the minimum or
maximum value of the function.
● How to identify the effects of
vertical translations of graphs
of linear and exponential
functions on their equations.
● How to graph parent functions
for quadratic and absolute
value functions.
● The meaning and effects that
the coefficients, factors,
exponents, and/or intercepts in
a linear and exponential
function have when describing
the attributes of graphs.
transformations using specific
values of k.
● Recognize which
transformations take away the
even nature of a quadratic or
absolute value function.
September 2016 Page 69 of 97
College- and Career-Readiness Standards for Mathematics
ALGEBRA I
Functions
Linear, Quadratic, and Exponential Models (F-LE)*
Construct and compare linear, quadratic, and exponential models and solve problems
F-LE.1a
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.
Desired Student Performance
A student should know
● How to find and interpret
slope as a rate of change.
● How to apply properties of
exponents to generate
equivalent numerical
expressions.
● How to evaluate square
roots of perfect squares and
cube roots of perfect cubes.
● How to graph a variety of
functions, including
exponential using a table of
values.
A student should understand
● Two families of functions
characterized by laws of growth
are linear functions, which grow
at a constant rate, and
exponential functions, which
grow at a constant percent rate.
● How to distinguish between
constant differences (linear
functions) and constant ratios
(exponential functions) by
recognizing constant growth
patterns versus exponential
growth patterns (e.g.,
compound interest versus
simple interest).
A student should be able to do
● Make conjectures about the
equations, tables, and graphs of
linear and exponential functions.
● Recognize situations in which a
quantity grows or decays by a
constant percent rate per unit
interval relative to another.
● Create and graph linear,
quadratic, and exponential
functions.
● Write and use arithmetic and
geometric sequences recursively
and explicitly to model situations.
● Distinguish between situations
that model linear and exponential
functions.
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College- and Career-Readiness Standards for Mathematics
● How to recognize the
relationship between rises and
runs on a graph and differences
of inputs and outputs in a
symbolic form of the proof.
● The ratio of the rise and run for
any two distinct points on a line
is the same.
● Linear functions with a constant
term of zero describe
proportional relationships.
● Characteristics of graphs,
tables, and equations for linear,
exponential, and quadratic
functions.
● Construct linear and exponential
functions give a graph, table, or
mathematical situation.
● Use exponential functions to
calculate compound interest.
September 2016 Page 71 of 97
College- and Career-Readiness Standards for Mathematics
ALGEBRA I
Functions
Linear, Quadratic, and Exponential Models (F-LE)*
Construct and compare linear, quadratic, and exponential models and solve problems
F-LE.1b
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.
Desired Student Performance
A student should know
● How to find and interpret
slope as a rate of change.
● How to apply properties of
exponents to generate
equivalent numerical
expressions.
● How to evaluate square
roots of perfect squares and
cube roots of perfect cubes.
● How to graph a variety of
functions, including
exponential use of a table of
values.
A student should understand
● How real-world and
mathematical situations can be
modeled by linear functions
when the rate of change of a
quantity is constant.
● When the rate of change is not
constant, the function cannot be
linear.
● How to analyze tables and
graphs to identify exponential or
linear functions.
● The ratio of the rise and run for
any two distinct points on a line
is the same.
● Linear functions with a constant
term of zero describe
proportional relationships.
A student should be able to do
● Recognize situations in which
one quantity changes at a
constant rate per unit interval
relative to another.
● Recognize a linear function
when analyzing a table, graph,
or equation.
● Determine the rate of change of
a linear function in context.
● Make conjectures about
equations, tables, and graphs
of linear and exponential
functions.
● Combine linear and exponential
functions using arithmetic
operations.
September 2016 Page 72 of 97
College- and Career-Readiness Standards for Mathematics
ALGEBRA I
Functions
Linear, Quadratic, and Exponential Models (F-LE)*
Construct and compare linear, quadratic, and exponential models and solve problems
F-LE.1c
Distinguish between
situations that can be
modeled with linear
functions and with
exponential functions.*
c. Recognize situations
in which a quantity
grows or decays by a
constant percent rate
per unit interval relative
to another.
Desired Student Performance
A student should know
● How to find and interpret
slope as a rate of change.
● How to apply properties of
exponents to generate
equivalent numerical
expressions.
● How to evaluate square
roots of perfect squares and
cube roots of perfect cubes.
● How to graph a variety of
functions, including
exponential use of a table of
values.
● The relationship between
variables in a function.
A student should understand
● Explicit forms of functions will
show that linear models grow
by a constant rate over equal
intervals.
● Exponential models grow by
equal factors over equal
intervals.
● If the percent rate of change is
not constant for a given
function, the function is not
exponential.
● Constant ratios are like
constant differences, except
you calculate the ratio between
consecutive outputs.
A student should be able to do
● Recognize situations in which a
quantity grows or decays by a
constant percent rate per unit
interval relative to another.
● Write exponential functions
from graphs, tables, and
mathematical and real-world
situations with an explicit
formula.
● Compare graphs, tables,
equations, and situations of
linear and exponential
functions.
● Describe how quantities
increase or decrease
exponentially over intervals.
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College- and Career-Readiness Standards for Mathematics
● When the rate of change is not
constant, the function cannot
be linear.
● Match tables with constant
ratios to exponential functions
and graphs.
● Make conjectures about
equations, tables, and graphs
of linear and exponential
functions.
September 2016 Page 74 of 97
College- and Career-Readiness Standards for Mathematics
ALGEBRA I
Functions
Linear, Quadratic, and Exponential Models (F-LE)*
Construct and compare linear, quadratic, and exponential models and solve problems
F-LE.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).*
Desired Student Performance
A student should know
● How to write the equation of
a line given two points, a
graph, or table.
● How to simplify expressions
involving rational numbers
and coefficients.
● How to generate data by
evaluating expressions for
different values of a variable
and organize the data.
● How to justify conjectures
and patterns using
numerical expressions.
● How to translate verbal
phrases into mathematical
expressions.
● How to generalize patterns
using words and algebraic
methods.
A student should understand
● How to identify sequences
generated by adding a constant
as arithmetic and those
generated by multiplying by a
constant as geometric.
● The vocabulary and notation for
arithmetic sequences as they
develop formulas for the n
th
term.
● How to write sequences from
recursive equations and vice
versa.
● How to convert between explicit
and recursive equations for
arithmetic sequences.
● How to find equations for
geometric sequences and see
relationships between
A student should be able to do
●
Construct linear and exponential
functions given a graph.
●
Construct linear and exponential
function given a description of a
relationship.
●
Construct linear and exponential
functions given two input-output
pairs.
●
Construct arithmetic and
geometric sequences given a
description of a relationship.
●
Construct arithmetic and
geometric sequences given two
input-output pairs.
●
Sort sequences based on their
patterns in their representation.
●
Write rules for arithmetic and
geometric sequences that
September 2016 Page 75 of 97
College- and Career-Readiness Standards for Mathematics
● How to recognize linear
functions.
geometric sequences and
exponential functions.
● How to look for and analyze
patterns in input-output tables.
model real-world problems and
mathematical situations.
● Use technology to model and
compare linear and exponential
functions.
September 2016 Page 76 of 97
College- and Career-Readiness Standards for Mathematics
ALGEBRA I
Functions
Linear, Quadratic, and Exponential Models (F-LE)*
Interpret expressions for functions in terms of the situation they model
F-LE.5
Interpret the
parameters in a linear
or exponential function
in terms of a context.*
Desired Student Performance
A student should know
● How to recognize equivalent
expressions.
● How to solve multiple-step
equations involving one
variable and rational numbers.
● How to factor a polynomial
completely.
● How to graph quadratic and
linear functions by hand and
using technology.
● How to generate data by
evaluating expressions for
different values of a variable
and organize the data.
A student should understand
● How to apply knowledge of
linear and exponential
functions to investigate the
relationship between simple
and compound interest.
● How to represent exponential
decay in multiple
representations.
● How to solidify connections
between a table, equation,
graph, and situational
representations of an
exponential function.
● How to interpret the meaning
of slope and y-intercept of a
A student should be able to do
● Based on the context of a
situation, explain the meaning
of the coefficients, factors,
exponents, and/or intercepts in
a linear or exponential function.
● Apply exponential functions to
real-life situations involving
growth and decay.
● Calculate simple interest.
● Use exponential functions to
calculate compound interest.
● Determine which
representation is best when
comparing the properties of
quadratics.
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College- and Career-Readiness Standards for Mathematics
● How to justify conjectures and
patterns using numerical
expressions.
● How to expand powers and
products of expressions.
linear equation in terms of
context.
● Explain and illustrate how a
change in one variable may
result in a change in another
variable and apply to the
relationships between
independent and dependent
variables.
September 2016 Page 78 of 97
College- and Career-Readiness Standards for Mathematics
ALGEBRA I
Statistics and Probability*
Interpreting Categorical and Quantitative Data (S-ID)
Summarize, represent, and interpret data on a single count or measurement variable
S-ID.1
Represent and analyze
data with plots on the
real number line (dot
plots, histograms, and
box plots). *
Desired Student Performance
A student should know
● How to determine the
mean, median, mode, and
range for a set of data, and
decide how meaningful they
are in specific situations.
● How to identify trends in
data.
● How to perform basic
operations involving rational
numbers.
● How to identify limitations,
or misuses, of visual
representations of data.
A student should understand
● A dot plot includes values from
the range of the data and plots
a point for each occurrence of
an observed value on a
number line.
● A histogram subdivides the
data into class intervals and
uses a rectangle to show the
frequency of observations in
those intervals.
● A box-and-whisker plot shows
the five-number summary of a
distribution. (Five-number
summary includes the
minimum, lower quartile (25
A student should be able to do
● Construct dot plots, histograms,
and box-and-whisker plots for
data on real number lines.
● Analyze data and compare data
in different data sets. (e.g., dot
plots, histograms and box-and-
whisker plots.)
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percentile), median (50
percentile), upper quartile (75
percentile), and the maximum.
● Quartiles are just medians for
the upper and lower halves of
the data set.
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ALGEBRA I
Statistics and Probability*
Interpreting Categorical and Quantitative Data (S-ID)
Summarize, represent, and interpret data on a single count or measurement variable
S-ID.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. *
Desired Student Performance
A student should know
● How to determine the mean,
median, mode, and range
for a set of data, and decide
how meaningful they are in
specific situations.
● How to identify trends in
data.
● How to perform basic
operations involving rational
numbers.
● How to identify limitations, or
misuses, of visual
representations of data.
A student should understand
● A spread describes how the
data lies.
● The shape of a data
distribution might be described
as symmetrical, skewed, flat, or
bell shaped, and it might be
summarized by a statistic-
measuring center (such as
standard deviation or
interquartile range).
● Different distributions can be
compared numerically using
statistics or compared visually
using plots.
● Which statistics to compare,
which plots to use, and what
the results of a comparison
might mean, depending on the
A student should be able to do
● Describe a distribution using
center and spread.
● Use the correct measure of
center and spread to describe
a distribution that is symmetric
or skewed.
● Identify outliers and their
effects on data sets.
● Compare two or more different
data set using the center and
spread of each.
● Analyze data and compare
data in different data sets.
● Compute the mean, median,
interquartile range, and
standard deviation by hand in
simple cases and using
technology with larger data
sets.
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question to be investigated and
the real-life actions to be taken.
September 2016 Page 82 of 97
College- and Career-Readiness Standards for Mathematics
ALGEBRA I
Statistics and Probability*
Interpreting Categorical and Quantitative Data (S-ID)
Summarize, represent, and interpret data on a single count or measurement variable
S-ID.3
Interpret differences in
shape, center, and
spread in the context
of the data sets,
accounting for
possible effects of
extreme data points
(outliers). *
Desired Student Performance
A student should know
● How to determine the mean,
median, mode, and range
for a set of data, and decide
how meaningful they are in
specific situations.
● How to identify trends in
data.
● How to perform basic
operations involving rational
numbers.
● How to identify limitations, or
misuses, of visual
representations of data.
A student should understand
● What shape distributions a
data set can have and how
statistics can affect the shape
and outliers.
● How shapes of graphically
displayed data can describe
data distributions.
● The shape and presence of
extreme values may affect
center and spread.
● The shape of a data
distribution might be described
as symmetrical, skewed, flat,
or bell-shaped, and it might be
summarized by a statistic-
measuring center (such as
A student should be able to do
● Identify a data set by its
shape and describe the data
set as symmetric, skewed,
flat, or bell-shaped.
● Use the outlier rule to identify
outliers in a data set.
● Explain how adding or
removing an outlier affects
measures of center and
spread in real-world and
mathematical situations.
● Compare two or more data
sets using shape, center, and
spread.
● Determine which statistics to
compare, which plots to use,
and what the results of a
September 2016 Page 83 of 97
College- and Career-Readiness Standards for Mathematics
standard deviation or
interquartile range).
● Different distributions can be
compared numerically using
statistics or compared visually
using plots.
● How to explain a decision
based on a graphical display of
data and the corresponding
descriptive statistics.
● How changes in data affect
visual representation of data.
comparison might mean,
depending on the question to
be investigated and the real-
life actions to be taken.
● Discuss the effects of outliers
on the measures of center
and what that would look like
on a graph of the data.
● Discuss the effects of extreme
values on the decision-
making process in the context
of a problem.
● Explain how measures of
spread might affect the
decision-making process
within the context of a set of
data.
● Organize multiple sets of data
for comparison and
articulates similarities and
differences.
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College- and Career-Readiness Standards for Mathematics
ALGEBRA I
Statistics and Probability*
Interpreting Categorical and Quantitative Data (S-ID)
Summarize, represent, and interpret data on two categorical and quantitative variables
S-ID.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.*
Desired Student Performance
A student should know
● How to determine the mean,
median, mode, and range for
a set of data, and decide how
meaningful they are in
specific situations.
● How to identify trends in
data.
● How to perform basic
operations involving rational
numbers.
● How identify limitations, or
misuses, of visual
representations of data.
● How to make and interpret
visual and tabular
representations of data.
A student should understand
● Entries in the “Total” row and
column are called marginal
frequencies.
● Entries in the body of the table
are called joint frequencies.
● The relative frequencies in the
body of the table are called
conditional frequencies.
● How to use two-way tables to
organize and display
categorical data.
● The difference between
quantitative data versus
categorical data.
● What it means for two
categorical data sets to be
independent.
A student should be able to do
● Recognize the differences
between joint, marginal, and
conditional relative frequencies.
● Calculate relative frequencies
including joint, marginal, and
conditional relative frequencies.
● Create and summarize a two-
way frequency table for a set of
categorical data.
● Analyze two-way tables to
determine if two categorical
variables are associated or
independent.
● Interpret relative frequencies in
the context of a given data set.
● Recognize possible
associations and trends in data.
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College- and Career-Readiness Standards for Mathematics
● How changes in data affect
visual representations of
data.
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College- and Career-Readiness Standards for Mathematics
ALGEBRA I
Statistics and Probability*
Interpreting Categorical and Quantitative Data (S-ID)
Summarize, represent, and interpret data on two categorical and quantitative variables
S-ID.6a
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.
Desired Student Performance
A student should know
● How to determine the
mean, median, mode, and
range for a set of data, and
decide how meaningful
they are in specific
situations.
● How to identify trends in
data.
● How to identify limitations,
or misuses, of visual
representations of data.
● How to make and interpret
visual and tabular
representations of data.
● How changes in data affect
visual representations of
data.
A student should understand
● Functions may be used to
describe data.
● How to identify the difference
between association and
causation.
● How to analyze tables and
graphs to identify exponential
or linear functions.
● How to make conjectures
about equations, tables, and
graphs of linear, quadratic,
and exponential functions.
● How to distinguish between
constant differences (linear
functions) and constant ratios
(exponential functions) by
recognizing constant growth
A student should be able to do
● Create a scatter plot from two
quantitative variables and
analyze possible associations
between two variables.
● Describe the form, strength,
and direction of the
relationship.
● Categorize data as linear,
exponential, or quadratic
based on its graphical display,
function, or table of data.
● Use algebraic methods and
technology to fit a function to
the data and use the function
to predict values in the context
of the data.
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patterns versus exponential
growth patterns.
● Explain the meaning of slope,
y-intercept, the constant and
coefficients, in terms of the
context of the data.
● Formulate a line of best fit
given data presented in a table
or in a graph.
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College- and Career-Readiness Standards for Mathematics
ALGEBRA I
Statistics and Probability*
Interpreting Categorical and Quantitative Data (S-ID)
Summarize, represent, and interpret data on two categorical and quantitative variables
S-ID.6b
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.
Desired Student Performance
A student should know
● How to determine the mean,
median, mode, and range for a
set of data, and decide how
meaningful they are in specific
situations.
● How to identify trends in data.
● How to perform basic
operations involving rational
numbers.
● How to identify limitations, or
misuses, of visual
representations of data.
● How to make and interpret
visual and tabular
representations of data.
A student should understand
● The residual in a regression
model is the difference
between the observed y-value
and its predicted y-value.
● Residuals measure how much
the data deviate from the
regression line.
● How to represent the
residuals from a function and
the data set it models
numerically and graphically.
● How to use line of fit and
scatter plots to evaluate
trends and make predictions.
A student should be able to do
● Graph the residuals and
evaluate the fit of the linear
equations.
● Fit functions to data.
● Informally assess the fit of a
function by analyzing residuals
from the residual plot.
● Find residuals with and without
technology and analyze their
meaning.
● Write equations of best-fit lines
using linear regression.
● Find a curve of best-fit in the
form of a polynomial function
for data.
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College- and Career-Readiness Standards for Mathematics
● How changes in data affect
visual representations of data.
● How to write linear equations
given a point and slope, two
points, or graph.
● If the data suggest a linear
relationship, the relationship
can be modeled with a
regression line, and its
strength and direction can be
expressed through a
correlation coefficient.
● Calculate and interpret the
correlation coefficient for linear
regression models.
September 2016 Page 90 of 97
College- and Career-Readiness Standards for Mathematics
ALGEBRA I
Statistics and Probability*
Interpreting Categorical and Quantitative Data (S-ID)
Summarize, represent, and interpret data on two categorical and quantitative variables
S-ID.6c
Represent data on two
quantitative variables
on a scatter plot, and
describe how the
variables are related.*
c. Fit a linear function
for a scatter plot that
suggests a linear
association.
Desired Student Performance
A student should know
● How to interpret the slope and
y-intercept of a linear model in
the context of the data.
● How to write and graph linear
equations given a point and
slope, two points, or graph.
● How to determine the mean,
median, mode, and range for
a set of data, and decide how
meaningful they are in specific
situations.
● How to identify trends in data.
● How to perform basic
operations involving rational
numbers.
● How to make and interpret
visual and tabular
representations of data.
A student should understand
● How to use lines of fit and
scatter plots to evaluate trends
and make predictions.
● How to identify the difference
between association and
causation.
● How to determine whether the
graph of real-world data shows
a positive correlation, negative
correlation, or no correlation.
● How to use the function for the
line of fit to predict values
inside the range of the data for
a real-world situation.
● Some models are better than
others at making predictions.
A student should be able to do
● Fit a linear function for a
scatter plot that suggests a
linear correlation.
● Fit a linear function (trend line)
to a scatter plot with and
without technology.
● Create a scatter plot from two
quantitative variables and
analyze possible associations
between two variables.
● Describe the form, strength,
and direction of the
relationship.
● Determine whether the graph
shows a positive, negative, or
no correlation.
● Interpret the meaning of
positive and negative
September 2016 Page 91 of 97
College- and Career-Readiness Standards for Mathematics
● How changes in data affect
visual representations of data.
correlated graphs in context of
the data.
● Use algebraic methods and
technology to fit a function to
the data and use the function
to predict values.
September 2016 Page 92 of 97
College- and Career-Readiness Standards for Mathematics
ALGEBRA I
Statistics and Probability*
Interpreting Categorical and Quantitative Data (S-ID)
Interpret linear models
S-ID.7
Interpret the slope
(rate of change) and
the intercept (constant
term) of a linear model
in the context of the
data.*
Desired Student Performance
A student should know
● How to generate data by
evaluating expressions for
different values of a variable
and organize the data.
● How to find the slope of a line
given a graph, table, or two
points on a line.
● How to recognize and justify if
a line has a positive, negative,
zero, or undefined slope.
● How to interpret slope by
describing how y is expected
to change when x changes by
one unit.
● How to simplify expressions
involving rational numbers.
A student should understand
● How to explain the meaning of
slope (rate of change) and y-
intercept (constant term) in
context.
● How to explain and illustrate
how a change in one variable
may result in a change in
another variable and apply to
the relationships between
independent and dependent
variables.
● How the slope of a graph
relates to a rate of change.
● How to interpret the rate of
change and initial value of
linear function in terms of the
A student should be able to do
● Write the equation of a line
given a graph, table of values,
or mathematical situation.
● Determine the rate of change
and constant term when given a
graph, table, or mathematical
situation and interpret its
meaning in context.
● Identify the quantities in a
mathematical problem or real-
world situation that should be
represented by distinct variables
and describe what quantities the
variable represents.
● Calculate the slope between two
points.
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College- and Career-Readiness Standards for Mathematics
situation it models and in
terms of its graph or a table of
values.
● The rate of change between
any two points, for non-linear
functions might not be the
same as the rate of change of
the overall function.
● How to compare the relative
steepness of lines and to build
intuition about positive,
negative, and zero slopes.
● Solve problems that involve
interpreting slope as a rate of
change.
● Estimate the rate of change
from a graph.
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College- and Career-Readiness Standards for Mathematics
ALGEBRA I
Statistics and Probability*
Interpreting Categorical and Quantitative Data (S-ID)
Interpret linear models
S-ID.8
Compute (using
technology) and
interpret the correlation
coefficient of a linear
fit.*
Desired Student Performance
A student should know
● How to interpret the slope
and y-intercept of a linear
model in the context of the
data.
● How to write and graph
linear equations given a
point and the slope, two
points, or graph.
● How to determine the
mean, median, mode, and
range for a set of data, and
decide how meaningful they
are in specific situations.
● How to identify trends in
data.
A student should understand
● Correlation coefficients
measure the strength of
association for a data set.
● Correlation coefficients are a
calculation based on the data
that returns a number
between -1 and 1.
● Correlation does not imply
causation.
● Correlation coefficient does
not detect nonlinear
association.
● How to input data using
statistical or graphing
technology and calculate its
correlation coefficient.
A student should be able to do
● Calculate the correlation
coefficient of a linear fit using
technology.
● Interpret the correlation
coefficient of a linear fit as a
measure of how well the data fit
the relationship.
● Investigate relationships
between quantities by using
points on scatter plots.
● Fit a linear function (trend line)
to a scatter plot with and
without technology.
● Create a scatter plot from two
quantitative variables and
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College- and Career-Readiness Standards for Mathematics
● How to identify limitations,
or misuses, of visual
representations of data.
● How changes in data affect
visual representations of
data.
● Some models are better than
others at making predictions.
analyze possible associations
between two variables.
● Describe the form, strength,
and direction of the
relationship.
● Define, explain, and determine
positive, negative, or no
correlation in context.
September 2016 Page 96 of 97
College- and Career-Readiness Standards for Mathematics
ALGEBRA I
Statistics and Probability*
Interpreting Categorical and Quantitative Data (S-ID)
Interpret linear models
S-ID.9
Distinguish between
correlation and
causation.*
Desired Student Performance
A student should know
● How to interpret the slope
and y-intercept of a linear
model in the context of the
data.
● How to write and graph
linear equations given a
point and the slope, two
points, or graph.
● How to determine the mean,
median, mode, and range for
a set of data, and decide
how meaningful they are in
specific situations.
● How to identify trends in
data.
A student should understand
● The difference between
correlation (association) and
causation (cause-and-effect).
● Correlation refers to how
closely two sets of information
or data are related.
● Causal relationship between
two things or events exists if
one occurs because of the
other.
● When two variables have a
correlation, it does not mean
that a change in one causes a
change in the others.
● Correlation does not imply
causation.
A student should be able to
do
● Investigate relationships
between quantities by using
points on scatter plots.
● Fit a linear function (trend
line) to a scatter plot with and
without technology.
● Create a scatter plot from two
quantitative variables and
analyze possible associations
between two variables.
● Describe the form, strength,
and direction of the
relationship.
● Define positive, negative, or
no correlation and explain
September 2016 Page 97 of 97
College- and Career-Readiness Standards for Mathematics
● How to identify limitations, or
misuses, of visual
representations of data.
● How changes in data affect
visual representations of
data.
● How to use lines of fit and
scatter plots to evaluate trends
and make predictions.
● No model is perfect. Some
models are better than others
at making predictions.
why correlation does not
imply causation.
● Interpret the meaning of
positive and negative
correlated graphs in context
of the data.
● Estimate the correlation
coefficient between two
variables.
