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MAFS.912.A-
APR.1.1
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.
Perform arithmetic
operations on
polynomials. (Algebra
1 - Major Cluster)
Mathematics 912 Algebra:
Arithmetic
with
Polynomials
& Rational
Expressions
Click Here
MAFS.912.A-
APR.2.2
Know and apply the Remainder Theorem: For a
polynomial p(x) and a number a, the remainder
on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).
Understand the
relationship between
zeros and factors of
polynomials. (Algebra
1 - Supporting Cluster)
(Algebra 2 - Major
Cluster)
Mathematics 912 Algebra:
Arithmetic
with
Polynomials
& Rational
Expressions
Click Here
MAFS.912.A-
APR.2.3
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.
Understand the
relationship between
zeros and factors of
polynomials. (Algebra
1 - Supporting Cluster)
(Algebra 2 - Major
Cluster)
Mathematics 912 Algebra:
Arithmetic
with
Polynomials
& Rational
Expressions
Click Here
MAFS.912.A-
APR.3.4
Prove polynomial identities and use them to
describe numerical relationships. For example,
the polynomial identity (x² + y²)² = (x² – y²)² + (2xy)² can be used to generate Pythagorean
triples.
Use polynomial
identities to solve
problems. (Algebra 2 -
Additional Cluster)
Mathematics 912 Algebra:
Arithmetic
with
Polynomials
& Rational
Expressions
Click Here
MAFS.912.A-
APR.3.5
Know and apply the Binomial Theorem for the
expansion of
(xError! Cannot read or display file. in
powers of x and y for a positive integer n, where
Use polynomial
identities to solve
problems. (Algebra 2 -
Additional Cluster)
Mathematics 912 Algebra:
Arithmetic
with
Polynomials
Click Here
x and y are any numbers, with coefficients
determined for example by pascal’s triangle. & Rational
Expressions
MAFS.912.A-
APR.4.6
Rewrite simple rational expressions in different
forms; write a(x)/b(x) in the form q(x) +
r(x)/b(x), where a(x), b(x), q(x), and r(x) are
polynomials with the degree of r(x) less than the
degree of b(x), using inspection, long division,
or, for the more complicated examples, a
computer algebra system.
Rewrite rational
expressions. (Algebra 2
- Supporting Cluster)
Mathematics 912 Algebra:
Arithmetic
with
Polynomials
& Rational
Expressions
Click Here
MAFS.912.A-
APR.4.7
Understand that rational expressions form a
system analogous to the rational numbers, closed
under addition, subtraction, multiplication, and
division by a nonzero rational expression; add,
subtract, multiply, and divide rational
expressions.
Rewrite rational
expressions. (Algebra 2
- Supporting Cluster)
Mathematics 912 Algebra:
Arithmetic
with
Polynomials
& Rational
Expressions
Click Here
MAFS.912.A-
CED.1.1
Create equations and inequalities in one variable
and use them to solve problems. Include
equations arising from linear and quadratic
functions, and simple rational, absolute, and
exponential functions. ★
Create equations that
describe numbers or
relationships. (Algebra
1 - Major Cluster)
(Algebra 2 - Supporting
Cluster)
Mathematics 912 Algebra:
Creating
Equations
Click Here
MAFS.912.A-
CED.1.2
Create equations in two or more variables to
represent relationships between quantities; graph
equations on coordinate axes with labels and
scales. ★
Create equations that
describe numbers or
relationships. (Algebra
1 - Major Cluster)
(Algebra 2 - Supporting
Cluster)
Mathematics 912 Algebra:
Creating
Equations
Click Here
MAFS.912.A-
CED.1.3
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. ★
Create equations that
describe numbers or
relationships. (Algebra
1 - Major Cluster)
(Algebra 2 - Supporting
Cluster)
Mathematics 912 Algebra:
Creating
Equations
Click Here
MAFS.912.A-
CED.1.4
Rearrange formulas to highlight a quantity of
interest, using the same reasoning as in solving
Create equations that
describe numbers or
relationships. (Algebra
Mathematics 912 Algebra:
Creating
Equations
Click Here
equations. For example, rearrange Ohm’s law V = IR to highlight resistance R. ★
1 - Major Cluster)
(Algebra 2 - Supporting
Cluster)
MAFS.912.A-
REI.1.1
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.
Understand solving
equations as a process
of reasoning and
explain the reasoning.
(Algebra 1 - Major
Cluster) (Algebra 2 -
Major Cluster)
Mathematics 912 Algebra:
Reasoning
with
Equations &
Inequalities
Click Here
MAFS.912.A-
REI.1.2
Solve simple rational and radical equations in
one variable, and give examples showing how
extraneous solutions may arise.
Understand solving
equations as a process
of reasoning and
explain the reasoning.
(Algebra 1 - Major
Cluster) (Algebra 2 -
Major Cluster)
Mathematics 912 Algebra:
Reasoning
with
Equations &
Inequalities
Click Here
MAFS.912.A-
REI.2.3
Solve linear equations and inequalities in one
variable, including equations with coefficients
represented by letters.
Solve equations and
inequalities in one
variable. (Algebra 1 -
Major Cluster)
(Algebra 2 - Supporting
Cluster)
Mathematics 912 Algebra:
Reasoning
with
Equations &
Inequalities
Click Here
MAFS.912.A-
REI.2.4
Solve quadratic equations in one variable.
a. Use the method of completing the square
to transform any quadratic equation in x
into an equation of the form (x – p)² = q that has the same solutions. Derive the
quadratic formula from this form.
b. Solve quadratic equations by inspection
(e.g., for x² = 49), taking square roots,
completing the square, the quadratic
formula and factoring, as appropriate to
the initial form of the equation.
Recognize when the quadratic formula
Solve equations and
inequalities in one
variable. (Algebra 1 -
Major Cluster)
(Algebra 2 - Supporting
Cluster)
Mathematics 912 Algebra:
Reasoning
with
Equations &
Inequalities
Click Here
gives complex solutions and write them
as a ± bi for real numbers a and b.
MAFS.912.A-
REI.3.5
Prove that, given a system of two equations in
two variables, replacing one equation by the sum
of that equation and a multiple of the other
produces a system with the same solutions.
Solve systems of
equations. (Algebra 1 -
Additional Cluster)
(Algebra 2 - Additional
Cluster)
Mathematics 912 Algebra:
Reasoning
with
Equations &
Inequalities
Click Here
MAFS.912.A-
REI.3.6
Solve systems of linear equations exactly and
approximately (e.g., with graphs), focusing on
pairs of linear equations in two variables.
Solve systems of
equations. (Algebra 1 -
Additional Cluster)
(Algebra 2 - Additional
Cluster)
Mathematics 912 Algebra:
Reasoning
with
Equations &
Inequalities
Click Here
MAFS.912.A-
REI.3.7
Solve a simple system consisting of a linear
equation and a quadratic equation in two
variables algebraically and graphically. For
example, find the points of intersection between
the line y = –3x and the circle x² + y² = 3.
Solve systems of
equations. (Algebra 1 -
Additional Cluster)
(Algebra 2 - Additional
Cluster)
Mathematics 912 Algebra:
Reasoning
with
Equations &
Inequalities
Click Here
MAFS.912.A-
REI.3.8
Represent a system of linear equations as a
single matrix equation in a vector variable.
Solve systems of
equations. (Algebra 1 -
Additional Cluster)
(Algebra 2 - Additional
Cluster)
Mathematics 912 Algebra:
Reasoning
with
Equations &
Inequalities
Click Here
MAFS.912.A-
REI.3.9
Find the inverse of a matrix if it exists and use it
to solve systems of linear equations (using
technology for matrices of dimension 3 × 3 or
greater).
Solve systems of
equations. (Algebra 1 -
Additional Cluster)
(Algebra 2 - Additional
Cluster)
Mathematics 912 Algebra:
Reasoning
with
Equations &
Inequalities
Click Here
MAFS.912.A-
REI.4.10
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).
Represent and solve
equations and
inequalities graphically.
(Algebra 1 - Major
Cluster) (Algebra 2 -
Major Cluster)
Mathematics 912 Algebra:
Reasoning
with
Equations &
Inequalities
Click Here
MAFS.912.A-
REI.4.11
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
Represent and solve
equations and
inequalities graphically.
Mathematics 912 Algebra:
Reasoning
with
Click Here
f(x) = g(x); find the solutions approximately,
e.g., using technology to graph the functions,
make tables of values, or find successive
approximations. Include cases where f(x) and/or
g(x) are linear, polynomial, rational, absolute
value, exponential, and logarithmic functions. ★
(Algebra 1 - Major
Cluster) (Algebra 2 -
Major Cluster)
Equations &
Inequalities
MAFS.912.A-
REI.4.12
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.
Represent and solve
equations and
inequalities graphically.
(Algebra 1 - Major
Cluster) (Algebra 2 -
Major Cluster)
Mathematics 912 Algebra:
Reasoning
with
Equations &
Inequalities
Click Here
MAFS.912.A-
SSE.1.1
Interpret expressions that represent a quantity in
terms of its context. ★ a. Interpret parts of an expression, such as
terms, factors, and coefficients.
b. Interpret complicated expressions by
viewing one or more of their parts as a
single entity. For example, interpret
Error! Cannot read or display file.as the
product of P and a factor not depending
on P.
Interpret the structure
of expressions.
(Algebra 1 - Major
Cluster) (Algebra 2 -
Major Cluster)
Mathematics 912 Algebra:
Seeing
Structure in
Expressions
Click Here
MAFS.912.A-
SSE.1.2
Use the structure of an expression to identify
ways to rewrite it. For example, see x4- y4 as
(x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²).
Interpret the structure
of expressions.
(Algebra 1 - Major
Cluster) (Algebra 2 -
Major Cluster)
Mathematics 912 Algebra:
Seeing
Structure in
Expressions
Click Here
MAFS.912.A-
SSE.2.3
Choose and produce an equivalent form of an
expression to reveal and explain properties of the
quantity represented by the expression.★ a. Factor a quadratic expression to reveal
the zeros of the function it defines.
b. Complete the square in a quadratic
expression to reveal the maximum or
minimum value of the function it defines.
Write expressions in
equivalent forms to
solve problems.
(Algebra 1 - Supporting
Cluster) (Algebra 2 -
Major Cluster)
Mathematics 912 Algebra:
Seeing
Structure in
Expressions
Click Here
c. Use the properties of exponents to
transform expressions for exponential
functions. For example the expression
Error! Cannot read or display file.can
be rewritten as
Error! Cannot read or display file.≈ Error! Cannot read or display file.to
reveal the approximate equivalent
monthly interest rate if the annual rate is
15%.
MAFS.912.A-
SSE.2.4
Derive the formula for the sum of a finite
geometric series (when the common ratio is not
1), and use the formula to solve problems. For
example, calculate mortgage payments. ★
Write expressions in
equivalent forms to
solve problems.
(Algebra 1 - Supporting
Cluster) (Algebra 2 -
Major Cluster)
Mathematics 912 Algebra:
Seeing
Structure in
Expressions
Click Here
MAFS.912.C.1.1 Understand the concept of limit and estimate
limits from graphs and tables of values.
Limits and Continuity Mathematics 912 Calculus Click Here
MAFS.912.C.1.2 Find limits by substitution. Limits and Continuity Mathematics 912 Calculus Click Here
MAFS.912.C.1.3 Find limits of sums, differences, products, and
quotients.
Limits and Continuity Mathematics 912 Calculus Click Here
MAFS.912.C.1.4 Find limits of rational functions that are
undefined at a point.
Limits and Continuity Mathematics 912 Calculus Click Here
MAFS.912.C.1.5 Find one-sided limits. Limits and Continuity Mathematics 912 Calculus Click Here
MAFS.912.C.1.6 Find limits at infinity. Limits and Continuity Mathematics 912 Calculus Click Here
MAFS.912.C.1.7 Decide when a limit is infinite and use limits
involving infinity to describe asymptotic
behavior.
Limits and Continuity Mathematics 912 Calculus Click Here
MAFS.912.C.1.8 Find special limits such as
Error! Cannot read or display file.
Limits and Continuity Mathematics 912 Calculus Click Here
MAFS.912.C.1.9 Understand continuity in terms of limits. Limits and Continuity Mathematics 912 Calculus Click Here
MAFS.912.C.1.10 Decide if a function is continuous at a point. Limits and Continuity Mathematics 912 Calculus Click Here
MAFS.912.C.1.11 Find the types of discontinuities of a function. Limits and Continuity Mathematics 912 Calculus Click Here
MAFS.912.C.1.12 Understand and use the Intermediate Value
Theorem on a function over a closed interval.
Limits and Continuity Mathematics 912 Calculus Click Here
MAFS.912.C.1.13 Understand and apply the Extreme Value
Theorem: If f(x) is continuous over a closed
interval, then f has a maximum and a minimum
on the interval.
Limits and Continuity Mathematics 912 Calculus Click Here
MAFS.912.C.2.1 Understand the concept of derivative
geometrically, numerically, and analytically, and
interpret the derivative as an instantaneous rate
of change or as the slope of the tangent line.
Differential Calculus Mathematics 912 Calculus Click Here
MAFS.912.C.2.2 State, understand, and apply the definition of
derivative.
Differential Calculus Mathematics 912 Calculus Click Here
MAFS.912.C.2.3 Find the derivatives of functions, including
algebraic, trigonometric, logarithmic, and
exponential functions.
Differential Calculus Mathematics 912 Calculus Click Here
MAFS.912.C.2.4 Find the derivatives of sums, products, and
quotients.
Differential Calculus Mathematics 912 Calculus Click Here
MAFS.912.C.2.5 Find the derivatives of composite functions
using the Chain Rule.
Differential Calculus Mathematics 912 Calculus Click Here
MAFS.912.C.2.6 Find the derivatives of implicitly-defined
functions.
Differential Calculus Mathematics 912 Calculus Click Here
MAFS.912.C.2.7 Find derivatives of inverse functions. Differential Calculus Mathematics 912 Calculus Click Here
MAFS.912.C.2.8 Find second derivatives and derivatives of higher
order.
Differential Calculus Mathematics 912 Calculus Click Here
MAFS.912.C.2.9 Find derivatives using logarithmic
differentiation.
Differential Calculus Mathematics 912 Calculus Click Here
MAFS.912.C.2.10 Understand and use the relationship between
differentiability and continuity.
Differential Calculus Mathematics 912 Calculus Click Here
MAFS.912.C.2.11 Understand and apply the Mean Value Theorem. Differential Calculus Mathematics 912 Calculus Click Here
MAFS.912.C.3.1 Find the slope of a curve at a point, including
points at which there are vertical tangent lines
and no tangent lines.
Applications of
Derivatives
Mathematics 912 Calculus Click Here
MAFS.912.C.3.2 Find an equation for the tangent line to a curve at
a point and a local linear approximation.
Applications of
Derivatives
Mathematics 912 Calculus Click Here
MAFS.912.C.3.3 Decide where functions are decreasing and
increasing. Understand the relationship between
the increasing and decreasing behavior of f and
the sign of f'.
Applications of
Derivatives
Mathematics 912 Calculus Click Here
MAFS.912.C.3.4 Find local and absolute maximum and minimum
points.
Applications of
Derivatives
Mathematics 912 Calculus Click Here
MAFS.912.C.3.5 Find points of inflection of functions.
Understand the relationship between the
concavity of f and the sign of f". Understand
points of inflection as places where concavity
changes.
Applications of
Derivatives
Mathematics 912 Calculus Click Here
MAFS.912.C.3.6 Use first and second derivatives to help sketch
graphs. Compare the corresponding
characteristics of the graphs of f, f', and f".
Applications of
Derivatives
Mathematics 912 Calculus Click Here
MAFS.912.C.3.7 Use implicit differentiation to find the derivative
of an inverse function.
Applications of
Derivatives
Mathematics 912 Calculus Click Here
MAFS.912.C.3.8 Solve optimization problems. Applications of
Derivatives
Mathematics 912 Calculus Click Here
MAFS.912.C.3.9 Find average and instantaneous rates of change.
Understand the instantaneous rate of change as
the limit of the average rate of change. Interpret
a derivative as a rate of change in applications,
including velocity, speed, and acceleration.
Applications of
Derivatives
Mathematics 912 Calculus Click Here
MAFS.912.C.3.10 Find the velocity and acceleration of a particle
moving in a straight line.
Applications of
Derivatives
Mathematics 912 Calculus Click Here
MAFS.912.C.3.11 Model rates of change, including related rates
problems.
Applications of
Derivatives
Mathematics 912 Calculus Click Here
MAFS.912.C.3.12 Solve problems using the Newton-Raphson
method.
Applications of
Derivatives
Mathematics 912 Calculus Click Here
MAFS.912.C.4.1 Use rectangle approximations to find
approximate values of integrals.
Integral Calculus Mathematics 912 Calculus Click Here
MAFS.912.C.4.2 Calculate the values of Riemann Sums over
equal subdivisions using left, right, and midpoint
evaluation points.
Integral Calculus Mathematics 912 Calculus Click Here
MAFS.912.C.4.3 Interpret a definite integral as a limit of Riemann
sums.
Integral Calculus Mathematics 912 Calculus Click Here
MAFS.912.C.4.4 Interpret a definite integral of the rate of change
of a quantity over an interval as the change of
the quantity over the interval. That
is, Error! Cannot read or display file. f'(x)dx =
f(b) - f(a) (fundamental theorem of calculus).
Integral Calculus Mathematics 912 Calculus Click Here
MAFS.912.C.4.5 Use the Fundamental Theorem of Calculus to
evaluate definite and indefinite integrals and to
represent particular antiderivatives. Perform
analytical and graphical analysis of functions so
defined.
Integral Calculus Mathematics 912 Calculus Click Here
MAFS.912.C.4.6 Use these properties of definite integrals:
Error! Cannot read or display file.[f(x) + g(x)]dx
= Error! Cannot read or display file.
f(x)dx
+ Error! Cannot read or display file.
g(x)dx
Error! Cannot read or display file.k • f(x)dx =
k Error! Cannot read or display file.
f(x)dx
Error! Cannot read or display file. f(x)dx = 0
Error! Cannot read or display file. f(x)dx = -
Error! Cannot read or display file.
f(x)dx
Error! Cannot read or display file. f(x)dx
+ Error! Cannot read or display file.
f(x)dx
= Error! Cannot read or display file.
f(x)dx
If f(x) ≤ g(x) on [a, b], then Error! Cannot read or display file.
f(x)dx
Integral Calculus Mathematics 912 Calculus Click Here
≤ Error! Cannot read or display file. g(x)dx
MAFS.912.C.4.7 Use integration by substitution (or change of
variable) to find values of integrals.
Integral Calculus Mathematics 912 Calculus Click Here
MAFS.912.C.4.8 Use Riemann Sums, the Trapezoidal Rule, and
technology to approximate definite integrals of
functions represented algebraically,
geometrically, and by tables of values.
Integral Calculus Mathematics 912 Calculus Click Here
MAFS.912.C.5.1 Find specific antiderivatives using initial
conditions, including finding velocity functions
from acceleration functions, finding position
functions from velocity functions, and solving
applications related to motion along a line.
Applications of
Integration
Mathematics 912 Calculus Click Here
MAFS.912.C.5.2 Solve separable differential equations, and use
them in modeling.
Applications of
Integration
Mathematics 912 Calculus Click Here
MAFS.912.C.5.3 Solve differential equations of the form
Error! Cannot read or display file.as applied
to growth and decay problems.
Applications of
Integration
Mathematics 912 Calculus Click Here
MAFS.912.C.5.4 Use slope fields to display a graphic
representation of the solution to a differential
equation, and locate particular solutions to the
equation.
Applications of
Integration
Mathematics 912 Calculus Click Here
MAFS.912.C.5.5 Use definite integrals to find the area between a
curve and the x-axis or between two curves.
Applications of
Integration
Mathematics 912 Calculus Click Here
MAFS.912.C.5.6 Use definite integrals to find the average value
of a function over a closed interval.
Applications of
Integration
Mathematics 912 Calculus Click Here
MAFS.912.C.5.7 Use definite integrals to find the volume of a
solid with known cross-sectional area, including
solids of revolution.
Applications of
Integration
Mathematics 912 Calculus Click Here
MAFS.912.C.5.8 Apply integration to model, and solve problems
in physical, biological, and social sciences.
Applications of
Integration
Mathematics 912 Calculus Click Here
MAFS.912.F-
BF.1.1
Write a function that describes a relationship
between two quantities. ★ Build a function that
models a relationship
between two quantities.
(Algebra 1 - Supporting
Mathematics 912 Functions:
Building
Functions
Click Here
a. Determine an explicit expression, a
recursive process, or steps for calculation
from a context.
b. Combine standard function types using
arithmetic operations. For example, build
a function that models the temperature of
a cooling body by adding a constant
function to a decaying exponential, and
relate these functions to the model.
c. Compose functions. For example, if T(y)
is the temperature in the atmosphere as a
function of height, and h(t) is the height
of a weather balloon as a function of
time, then T(h(t)) is the temperature at
the location of the weather balloon as a
function of time.
Cluster) (Algebra 2 -
Major Cluster)
MAFS.912.F-
BF.1.2
Write arithmetic and geometric sequences both
recursively and with an explicit formula, use
them to model situations, and translate between
the two forms. ★
Build a function that
models a relationship
between two quantities.
(Algebra 1 - Supporting
Cluster) (Algebra 2 -
Major Cluster)
Mathematics 912 Functions:
Building
Functions
Click Here
MAFS.912.F-
BF.2.3
Identify the effect on the graph of replacing f(x)
by f(x) + k, k f(x), f(kx), and f(x + k) for specific
values of k (both positive and negative); find the
value of k given the graphs. Experiment with
cases and illustrate an explanation of the effects
on the graph using technology. Include
recognizing even and odd functions from their
graphs and algebraic expressions for them.
Build new functions
from existing functions.
(Algebra 1 - Additional
Cluster) (Algebra 2 -
Additional Cluster)
Mathematics 912 Functions:
Building
Functions
Click Here
MAFS.912.F-
BF.2.4
Find inverse functions.
a. Solve an equation of the form f(x) = c for
a simple function f that has an inverse
and write an expression for the inverse.
Build new functions
from existing functions.
(Algebra 1 - Additional
Cluster) (Algebra 2 -
Additional Cluster)
Mathematics 912 Functions:
Building
Functions
Click Here
For example, f(x) =2 x³ or f(x) =
(x+1)/(x–1) for x ≠ 1. b. Verify by composition that one function
is the inverse of another.
c. Read values of an inverse function from a
graph or a table, given that the function
has an inverse.
d. Produce an invertible function from a
non-invertible function by restricting the
domain.
MAFS.912.F-
BF.2.5
Understand the inverse relationship between
exponents and logarithms and use this
relationship to solve problems involving
logarithms and exponents.
Build new functions
from existing functions.
(Algebra 1 - Additional
Cluster) (Algebra 2 -
Additional Cluster)
Mathematics 912 Functions:
Building
Functions
Click Here
MAFS.912.F-
BF.2.a
Use the change of base formula. Build new functions
from existing functions.
(Algebra 1 - Additional
Cluster) (Algebra 2 -
Additional Cluster)
Mathematics 912 Functions:
Building
Functions
Click Here
MAFS.912.F-
IF.1.1
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).
Understand the concept
of a function and use
function notation.
(Algebra 1 - Major
Cluster) (Algebra 2 -
Supporting Cluster)
Mathematics 912 Functions:
Interpreting
Functions
Click Here
MAFS.912.F-
IF.1.2
Use function notation, evaluate functions for
inputs in their domains, and interpret statements
that use function notation in terms of a context.
Understand the concept
of a function and use
function notation.
(Algebra 1 - Major
Cluster) (Algebra 2 -
Supporting Cluster)
Mathematics 912 Functions:
Interpreting
Functions
Click Here
MAFS.912.F-
IF.1.3
Recognize that sequences are functions,
sometimes defined recursively, whose domain is
a subset of the integers. For example, the
Understand the concept
of a function and use
function notation.
(Algebra 1 - Major
Mathematics 912 Functions:
Interpreting
Functions
Click Here
Fibonacci sequence is defined recursively by f(0)
= f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1. Cluster) (Algebra 2 -
Supporting Cluster)
MAFS.912.F-
IF.2.4
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. ★
Interpret functions that
arise in applications in
terms of the context.
(Algebra 1 - Major
Cluster) (Algebra 2 -
Major Cluster)
Mathematics 912 Functions:
Interpreting
Functions
Click Here
MAFS.912.F-
IF.2.5
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 engines in a factory, then the positive
integers would be an appropriate domain for the
function. ★
Interpret functions that
arise in applications in
terms of the context.
(Algebra 1 - Major
Cluster) (Algebra 2 -
Major Cluster)
Mathematics 912 Functions:
Interpreting
Functions
Click Here
MAFS.912.F-
IF.2.6
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. ★
Interpret functions that
arise in applications in
terms of the context.
(Algebra 1 - Major
Cluster) (Algebra 2 -
Major Cluster)
Mathematics 912 Functions:
Interpreting
Functions
Click Here
MAFS.912.F-
IF.3.7
Graph functions expressed symbolically and
show key features of the graph, by hand in
simple cases and using technology for more
complicated cases. ★ a. Graph linear and quadratic functions and
show intercepts, maxima, and minima.
b. Graph square root, cube root, and
piecewise-defined functions, including
step functions and absolute value
functions.
Analyze functions
using different
representations.
(Algebra 1 - Supporting
Cluster) (Algebra 2 -
Supporting Cluster)
Mathematics 912 Functions:
Interpreting
Functions
Click Here
c. Graph polynomial functions, identifying
zeros when suitable factorizations are
available, and showing end behavior.
d. Graph rational functions, identifying
zeros and asymptotes when suitable
factorizations are available, and showing
end behavior.
e. Graph exponential and logarithmic
functions, showing intercepts and end
behavior, and trigonometric functions,
showing period, midline, and amplitude,
and using phase shift.
MAFS.912.F-
IF.3.8
Write a function defined by an expression in
different but equivalent forms to reveal and
explain different properties of the function.
a. Use the process of factoring and
completing the square in a quadratic
function to show zeros, extreme values,
and symmetry of the graph, and interpret
these in terms of a context.
b. Use the properties of exponents to
interpret expressions for exponential
functions. For example, identify percent
rate of change in functions such as y =
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Error! Cannot read or display file., y =
Error! Cannot read or display file., y =
Error! Cannot read or display file., and
classify them as representing exponential
growth or decay.
Analyze functions
using different
representations.
(Algebra 1 - Supporting
Cluster) (Algebra 2 -
Supporting Cluster)
Mathematics 912 Functions:
Interpreting
Functions
Click Here
MAFS.912.F-
IF.3.9
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
Analyze functions
using different
representations.
(Algebra 1 - Supporting
Cluster) (Algebra 2 -
Supporting Cluster)
Mathematics 912 Functions:
Interpreting
Functions
Click Here
quadratic function and an algebraic expression
for another, say which has the larger maximum.
MAFS.912.F-
LE.1.1
Distinguish between situations that can be
modeled with linear functions and with
exponential functions. ★ a. Prove that linear functions grow by equal
differences over equal intervals, and that
exponential functions grow by equal
factors over equal intervals.
b. Recognize situations in which one
quantity changes at a constant rate per
unit interval relative to another.
c. Recognize situations in which a quantity
grows or decays by a constant percent
rate per unit interval relative to another.
Construct and compare
linear, quadratic, and
exponential models and
solve problems.
(Algebra 1 - Supporting
Cluster) (Algebra 2 -
Supporting Cluster)
Mathematics 912 Functions:
Linear,
Quadratic, &
Exponential
Models
Click Here
MAFS.912.F-
LE.1.2
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). ★
Construct and compare
linear, quadratic, and
exponential models and
solve problems.
(Algebra 1 - Supporting
Cluster) (Algebra 2 -
Supporting Cluster)
Mathematics 912 Functions:
Linear,
Quadratic, &
Exponential
Models
Click Here
MAFS.912.F-
LE.1.3
Observe using graphs and tables that a quantity
increasing exponentially eventually exceeds a
quantity increasing linearly, quadratically, or
(more generally) as a polynomial function. ★
Construct and compare
linear, quadratic, and
exponential models and
solve problems.
(Algebra 1 - Supporting
Cluster) (Algebra 2 -
Supporting Cluster)
Mathematics 912 Functions:
Linear,
Quadratic, &
Exponential
Models
Click Here
MAFS.912.F-
LE.1.4
For exponential models, express as a logarithm
the solution to
Error! Cannot read or display file.= d where a,
c, and d are numbers and the base b is 2, 10, or e;
evaluate the logarithm using technology. ★
Construct and compare
linear, quadratic, and
exponential models and
solve problems.
(Algebra 1 - Supporting
Mathematics 912 Functions:
Linear,
Quadratic, &
Exponential
Models
Click Here
Cluster) (Algebra 2 -
Supporting Cluster)
MAFS.912.F-
LE.2.5
Interpret the parameters in a linear or
exponential function in terms of a context. ★ Interpret expressions
for functions in terms
of the situation they
model. (Algebra 1 -
Supporting Cluster)
(Algebra 2 - Additional
Cluster)
Mathematics 912 Functions:
Linear,
Quadratic, &
Exponential
Models
Click Here
MAFS.912.F-
TF.1.1
Understand radian measure of an angle as the
length of the arc on the unit circle subtended by
the angle; Convert between degrees and radians.
Extend the domain of
trigonometric functions
using the unit circle.
(Algebra 2 - Additional
Cluster)
Mathematics 912 Functions:
Trigonometric
Functions
Click Here
MAFS.912.F-
TF.1.2
Explain how the unit circle in the coordinate
plane enables the extension of trigonometric
functions to all real numbers, interpreted as
radian measures of angles traversed
counterclockwise around the unit circle.
Extend the domain of
trigonometric functions
using the unit circle.
(Algebra 2 - Additional
Cluster)
Mathematics 912 Functions:
Trigonometric
Functions
Click Here
MAFS.912.F-
TF.1.3
Use special triangles to determine geometrically
the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π–x, π+x, and 2π–x in terms of their values for x, where x is any real number.
Extend the domain of
trigonometric functions
using the unit circle.
(Algebra 2 - Additional
Cluster)
Mathematics 912 Functions:
Trigonometric
Functions
Click Here
MAFS.912.F-
TF.1.4
Use the unit circle to explain symmetry (odd and
even) and periodicity of trigonometric functions.
Extend the domain of
trigonometric functions
using the unit circle.
(Algebra 2 - Additional
Cluster)
Mathematics 912 Functions:
Trigonometric
Functions
Click Here
MAFS.912.F-
TF.2.5
Choose trigonometric functions to model
periodic phenomena with specified amplitude,
frequency, and midline. ★ Model periodic
phenomena with
trigonometric functions.
(Algebra 2 - Additional
Cluster)
Mathematics 912 Functions:
Trigonometric
Functions
Click Here
MAFS.912.F-
TF.2.6
Understand that restricting a trigonometric
function to a domain on which it is always
increasing or always decreasing allows its
inverse to be constructed.
Model periodic
phenomena with
trigonometric functions.
(Algebra 2 - Additional
Cluster)
Mathematics 912 Functions:
Trigonometric
Functions
Click Here
MAFS.912.F-
TF.2.7
Use inverse functions to solve trigonometric
equations that arise in modeling contexts;
evaluate the solutions using technology, and
interpret them in terms of the context. ★
Model periodic
phenomena with
trigonometric functions.
(Algebra 2 - Additional
Cluster)
Mathematics 912 Functions:
Trigonometric
Functions
Click Here
MAFS.912.F-
TF.3.8
Prove the Pythagorean identity sin²(θ) + cos²(θ) = 1 and use it to calculate trigonometric ratios.
Prove and apply
trigonometric identities.
(Algebra 2 - Additional
Cluster)
Mathematics 912 Functions:
Trigonometric
Functions
Click Here
MAFS.912.F-
TF.3.9
Prove the addition and subtraction, half-angle,
and double-angle formulas for sine, cosine, and
tangent and use these formulas to solve
problems.
Prove and apply
trigonometric identities.
(Algebra 2 - Additional
Cluster)
Mathematics 912 Functions:
Trigonometric
Functions
Click Here
MAFS.912.G-
C.1.1
Prove that all circles are similar. Understand and apply
theorems about circles.
(Geometry - Additional
Cluster)
Mathematics 912 Geometry:
Circles Click Here
MAFS.912.G-
C.1.2
Identify and describe relationships among
inscribed angles, radii, and chords. Include the
relationship between central, inscribed, and
circumscribed angles; inscribed angles on a
diameter are right angles; the radius of a circle
is perpendicular to the tangent where the radius
intersects the circle.
Understand and apply
theorems about circles.
(Geometry - Additional
Cluster)
Mathematics 912 Geometry:
Circles Click Here
MAFS.912.G-
C.1.3
Construct the inscribed and circumscribed circles
of a triangle, and prove properties of angles for a
quadrilateral inscribed in a circle.
Understand and apply
theorems about circles.
(Geometry - Additional
Cluster)
Mathematics 912 Geometry:
Circles Click Here
MAFS.912.G-
C.1.4
Construct a tangent line from a point outside a
given circle to the circle.
Understand and apply
theorems about circles.
(Geometry - Additional
Cluster)
Mathematics 912 Geometry:
Circles Click Here
MAFS.912.G-
C.2.5
Derive using similarity the fact that the length of
the arc intercepted by an angle is proportional to
the radius, and define the radian measure of the
angle as the constant of proportionality; derive
the formula for the area of a sector.
Find arc lengths and
areas of sectors of
circles. (Geometry -
Additional Cluster)
Mathematics 912 Geometry:
Circles Click Here
MAFS.912.G-
CO.1.1
Know precise definitions of angle, circle,
perpendicular line, parallel line, and line
segment, based on the undefined notions of
point, line, distance along a line, and distance
around a circular arc.
Experiment with
transformations in the
plane. (Geometry -
Supporting Cluster)
Mathematics 912 Geometry:
Congruence Click Here
MAFS.912.G-
CO.1.2
Represent transformations in the plane using,
e.g., transparencies and geometry software;
describe transformations as functions that take
points in the plane as inputs and give other
points as outputs. Compare transformations that
preserve distance and angle to those that do not
(e.g., translation versus horizontal stretch).
Experiment with
transformations in the
plane. (Geometry -
Supporting Cluster)
Mathematics 912 Geometry:
Congruence Click Here
MAFS.912.G-
CO.1.3
Given a rectangle, parallelogram, trapezoid, or
regular polygon, describe the rotations and
reflections that carry it onto itself.
Experiment with
transformations in the
plane. (Geometry -
Supporting Cluster)
Mathematics 912 Geometry:
Congruence Click Here
MAFS.912.G-
CO.1.4
Develop definitions of rotations, reflections, and
translations in terms of angles, circles,
perpendicular lines, parallel lines, and line
segments.
Experiment with
transformations in the
plane. (Geometry -
Supporting Cluster)
Mathematics 912 Geometry:
Congruence Click Here
MAFS.912.G-
CO.1.5
Given a geometric figure and a rotation,
reflection, or translation, draw the transformed
figure using, e.g., graph paper, tracing paper, or
geometry software. Specify a sequence of
transformations that will carry a given figure
onto another.
Experiment with
transformations in the
plane. (Geometry -
Supporting Cluster)
Mathematics 912 Geometry:
Congruence Click Here
MAFS.912.G-
CO.2.6
Use geometric descriptions of rigid motions to
transform figures and to predict the effect of a
given rigid motion on a given figure; given two
figures, use the definition of congruence in terms
of rigid motions to decide if they are congruent.
Understand congruence
in terms of rigid
motions. (Geometry -
Major Cluster)
Mathematics 912 Geometry:
Congruence Click Here
MAFS.912.G-
CO.2.7
Use the definition of congruence in terms of
rigid motions to show that two triangles are
congruent if and only if corresponding pairs of
sides and corresponding pairs of angles are
congruent.
Understand congruence
in terms of rigid
motions. (Geometry -
Major Cluster)
Mathematics 912 Geometry:
Congruence Click Here
MAFS.912.G-
CO.2.8
Explain how the criteria for triangle congruence
(ASA, SAS, SSS, and Hypotenuse-Leg) follow
from the definition of congruence in terms of
rigid motions.
Understand congruence
in terms of rigid
motions. (Geometry -
Major Cluster)
Mathematics 912 Geometry:
Congruence Click Here
MAFS.912.G-
CO.3.9
Prove theorems about lines and angles; use
theorems about lines and angles to solve
problems. Theorems include: vertical angles are
congruent; when a transversal crosses parallel
lines, alternate interior angles are congruent
and corresponding angles are congruent; points
on a perpendicular bisector of a line segment
are exactly those equidistant from the segment’s endpoints.
Prove geometric
theorems. (Geometry -
Major Cluster)
Mathematics 912 Geometry:
Congruence Click Here
MAFS.912.G-
CO.3.10
Prove theorems about triangles; use theorems
about triangles to solve problems. Theorems
include: measures of interior angles of a triangle
sum to 180°; triangle inequality theorem; base
angles of isosceles triangles are congruent; the
segment joining midpoints of two sides of a
triangle is parallel to the third side and half the
length; the medians of a triangle meet at a point.
Prove geometric
theorems. (Geometry -
Major Cluster)
Mathematics 912 Geometry:
Congruence Click Here
MAFS.912.G-
CO.3.11
Prove theorems about parallelograms; use
theorems about parallelograms to solve
problems. Theorems include: opposite sides are
congruent, opposite angles are congruent, the
diagonals of a parallelogram bisect each other,
and conversely, rectangles are parallelograms
with congruent diagonals.
Prove geometric
theorems. (Geometry -
Major Cluster)
Mathematics 912 Geometry:
Congruence
Click Here
MAFS.912.G-
CO.4.12
Make formal geometric constructions with a
variety of tools and methods (compass and
straightedge, string, reflective devices, paper
folding, dynamic geometric software, etc.).
Copying a segment; copying an angle; bisecting
Make geometric
constructions.
(Geometry - Supporting
Cluster)
Mathematics 912 Geometry:
Congruence Click Here
a segment; bisecting an angle; constructing
perpendicular lines, including the perpendicular
bisector of a line segment; and constructing a
line parallel to a given line through a point not
on the line.
MAFS.912.G-
CO.4.13
Construct an equilateral triangle, a square, and a
regular hexagon inscribed in a circle.
Make geometric
constructions.
(Geometry - Supporting
Cluster)
Mathematics 912 Geometry:
Congruence Click Here
MAFS.912.G-
GMD.1.1
Give an informal argument for the formulas for
the circumference of a circle, area of a circle,
volume of a cylinder, pyramid, and cone. Use
dissection arguments, Cavalieri’s principle, and informal limit arguments.
Explain volume
formulas and use them
to solve problems.
(Geometry - Additional
Cluster)
Mathematics 912 Geometry:
Geometric
Measurement
& Dimension
Click Here
MAFS.912.G-
GMD.1.2
Give an informal argument using Cavalieri’s principle for the formulas for the volume of a
sphere and other solid figures.
Explain volume
formulas and use them
to solve problems.
(Geometry - Additional
Cluster)
Mathematics 912 Geometry:
Geometric
Measurement
& Dimension
Click Here
MAFS.912.G-
GMD.1.3
Use volume formulas for cylinders, pyramids,
cones, and spheres to solve problems. ★ Explain volume
formulas and use them
to solve problems.
(Geometry - Additional
Cluster)
Mathematics 912 Geometry:
Geometric
Measurement
& Dimension
Click Here
MAFS.912.G-
GMD.2.4
Identify the shapes of two-dimensional cross-
sections of three-dimensional objects, and
identify three-dimensional objects generated by
rotations of two-dimensional objects.
Visualize relationships
between two-
dimensional and three-
dimensional objects.
(Geometry - Additional
Cluster)
Mathematics 912 Geometry:
Geometric
Measurement
& Dimension
Click Here
MAFS.912.G-
GPE.1.1
Derive the equation of a circle of given center
and radius using the Pythagorean Theorem;
complete the square to find the center and radius
of a circle given by an equation.
Translate between the
geometric description
and the equation for a
conic section.
(Geometry - Additional
Cluster) (Algebra 2 -
Additional Cluster)
Mathematics 912 Geometry:
Expressing
Geometric
Properties
with
Equations
Click Here
MAFS.912.G-
GPE.1.2
Derive the equation of a parabola given a focus
and directrix.
Translate between the
geometric description
and the equation for a
conic section.
(Geometry - Additional
Cluster) (Algebra 2 -
Additional Cluster)
Mathematics 912 Geometry:
Expressing
Geometric
Properties
with
Equations
Click Here
MAFS.912.G-
GPE.1.3
Derive the equations of ellipses and hyperbolas
given the foci and directrices.
Translate between the
geometric description
and the equation for a
conic section.
(Geometry - Additional
Cluster) (Algebra 2 -
Additional Cluster)
Mathematics 912 Geometry:
Expressing
Geometric
Properties
with
Equations
Click Here
MAFS.912.G-
GPE.2.4
Use coordinates to prove simple geometric
theorems algebraically. For example, prove or
disprove that a figure defined by four given
points in the coordinate plane is a rectangle;
prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing
the point (0, 2).
Use coordinates to
prove simple geometric
theorems algebraically.
(Geometry - Major
Cluster)
Mathematics 912 Geometry:
Expressing
Geometric
Properties
with
Equations
Click Here
MAFS.912.G-
GPE.2.5
Prove the slope criteria for parallel and
perpendicular lines and use them to solve
geometric problems (e.g., find the equation of a
line parallel or perpendicular to a given line that
passes through a given point).
Use coordinates to
prove simple geometric
theorems algebraically.
(Geometry - Major
Cluster)
Mathematics 912 Geometry:
Expressing
Geometric
Properties
with
Equations
Click Here
MAFS.912.G-
GPE.2.6
Find the point on a directed line segment
between two given points that partitions the
segment in a given ratio.
Use coordinates to
prove simple geometric
theorems algebraically.
(Geometry - Major
Cluster)
Mathematics 912 Geometry:
Expressing
Geometric
Properties
with
Equations
Click Here
MAFS.912.G-
GPE.2.7
Use coordinates to compute perimeters of
polygons and areas of triangles and rectangles,
e.g., using the distance formula. ★ Use coordinates to
prove simple geometric
theorems algebraically.
Mathematics 912 Geometry:
Expressing
Geometric
Properties
Click Here
(Geometry - Major
Cluster)
with
Equations
MAFS.912.G-
MG.1.1
Use geometric shapes, their measures, and their
properties to describe objects (e.g., modeling a
tree trunk or a human torso as a cylinder). ★ Apply geometric
concepts in modeling
situations. (Geometry -
Major Cluster)
Mathematics 912 Geometry:
Modeling with
Geometry
Click Here
MAFS.912.G-
MG.1.2
Apply concepts of density based on area and
volume in modeling situations (e.g., persons per
square mile, BTUs per cubic foot). ★ Apply geometric
concepts in modeling
situations. (Geometry -
Major Cluster)
Mathematics 912 Geometry:
Modeling with
Geometry
Click Here
MAFS.912.G-
MG.1.3
Apply geometric methods to solve design
problems (e.g., designing an object or structure
to satisfy physical constraints or minimize cost;
working with typographic grid systems based on
ratios). ★
Apply geometric
concepts in modeling
situations. (Geometry -
Major Cluster)
Mathematics 912 Geometry:
Modeling with
Geometry
Click Here
MAFS.912.G-
SRT.1.1
Verify experimentally the properties of dilations
given by a center and a scale factor:
a. A dilation takes a line not passing
through the center of the dilation to a
parallel line, and leaves a line passing
through the center unchanged.
b. The dilation of a line segment is longer
or shorter in the ratio given by the scale
factor.
Understand similarity
in terms of similarity
transformations.
(Geometry - Major
Cluster)
Mathematics 912 Geometry:
Similarity,
Right
Triangles, &
Trigonometry
Click Here
MAFS.912.G-
SRT.1.2
Given two figures, use the definition of
similarity in terms of similarity transformations
to decide if they are similar; explain using
similarity transformations the meaning of
similarity for triangles as the equality of all
corresponding pairs of angles and the
proportionality of all corresponding pairs of
sides.
Understand similarity
in terms of similarity
transformations.
(Geometry - Major
Cluster)
Mathematics 912 Geometry:
Similarity,
Right
Triangles, &
Trigonometry
Click Here
MAFS.912.G-
SRT.1.3
Use the properties of similarity transformations
to establish the AA criterion for two triangles to
be similar.
Understand similarity
in terms of similarity
transformations.
Mathematics 912 Geometry:
Similarity,
Right
Click Here
(Geometry - Major
Cluster)
Triangles, &
Trigonometry
MAFS.912.G-
SRT.2.4
Prove theorems about triangles. Theorems
include: a line parallel to one side of a triangle
divides the other two proportionally, and
conversely; the Pythagorean Theorem proved
using triangle similarity.
Prove theorems
involving similarity.
(Geometry - Major
Cluster)
Mathematics 912 Geometry:
Similarity,
Right
Triangles, &
Trigonometry
Click Here
MAFS.912.G-
SRT.2.5
Use congruence and similarity criteria for
triangles to solve problems and to prove
relationships in geometric figures.
Prove theorems
involving similarity.
(Geometry - Major
Cluster)
Mathematics 912 Geometry:
Similarity,
Right
Triangles, &
Trigonometry
Click Here
MAFS.912.G-
SRT.3.6
Understand that by similarity, side ratios in right
triangles are properties of the angles in the
triangle, leading to definitions of trigonometric
ratios for acute angles.
Define trigonometric
ratios and solve
problems involving
right triangles.
(Geometry - Major
Cluster)
Mathematics 912 Geometry:
Similarity,
Right
Triangles, &
Trigonometry
Click Here
MAFS.912.G-
SRT.3.7
Explain and use the relationship between the sine
and cosine of complementary angles.
Define trigonometric
ratios and solve
problems involving
right triangles.
(Geometry - Major
Cluster)
Mathematics 912 Geometry:
Similarity,
Right
Triangles, &
Trigonometry
Click Here
MAFS.912.G-
SRT.3.8
Use trigonometric ratios and the Pythagorean
Theorem to solve right triangles in applied
problems. ★ Define trigonometric
ratios and solve
problems involving
right triangles.
(Geometry - Major
Cluster)
Mathematics 912 Geometry:
Similarity,
Right
Triangles, &
Trigonometry
Click Here
MAFS.912.G-
SRT.4.9
Derive the formula A = 1/2 ab sin(C) for the area
of a triangle by drawing an auxiliary line from a
vertex perpendicular to the opposite side.
Apply trigonometry to
general triangles
Mathematics 912 Geometry:
Similarity,
Right
Triangles, &
Trigonometry
Click Here
MAFS.912.G-
SRT.4.10
Prove the Laws of Sines and Cosines and use
them to solve problems.
Apply trigonometry to
general triangles
Mathematics 912 Geometry:
Similarity, Click Here
Right
Triangles, &
Trigonometry
MAFS.912.G-
SRT.4.11
Understand and apply the Law of Sines and the
Law of Cosines to find unknown measurements
in right and non-right triangles (e.g., surveying
problems, resultant forces).
Apply trigonometry to
general triangles
Mathematics 912 Geometry:
Similarity,
Right
Triangles, &
Trigonometry
Click Here
MAFS.912.N-
CN.1.1
Know there is a complex number i such that i² =
–1, and every complex number has the form a + bi with a and b real.
Perform arithmetic
operations with
complex numbers.
(Algebra 2 - Additional
Cluster)
Mathematics 912 Number &
Quantity: The
Complex
Number
System
Click Here
MAFS.912.N-
CN.1.2
Use the relation i² = –1 and the commutative, associative, and distributive properties to add,
subtract, and multiply complex numbers.
Perform arithmetic
operations with
complex numbers.
(Algebra 2 - Additional
Cluster)
Mathematics 912 Number &
Quantity: The
Complex
Number
System
Click Here
MAFS.912.N-
CN.1.3
Find the conjugate of a complex number; use
conjugates to find moduli and quotients of
complex numbers.
Perform arithmetic
operations with
complex numbers.
(Algebra 2 - Additional
Cluster)
Mathematics 912 Number &
Quantity: The
Complex
Number
System
Click Here
MAFS.912.N-
CN.2.4
Represent complex numbers on the complex
plane in rectangular and polar form (including
real and imaginary numbers), and explain why
the rectangular and polar forms of a given
complex number represent the same number.
Represent complex
numbers and their
operations on the
complex plane.
Mathematics 912 Number &
Quantity: The
Complex
Number
System
Click Here
MAFS.912.N-
CN.2.5
Represent addition, subtraction, multiplication,
and conjugation of complex numbers
geometrically on the complex plane; use
properties of this representation for computation.
For example, (–1 + √3 i)³ = 8 because (–1 + √3 i) has modulus 2 and argument 120°.
Represent complex
numbers and their
operations on the
complex plane.
Mathematics 912 Number &
Quantity: The
Complex
Number
System
Click Here
MAFS.912.N-
CN.2.6
Calculate the distance between numbers in the
complex plane as the modulus of the difference,
Represent complex
numbers and their
Mathematics 912 Number &
Quantity: The
Complex
Click Here
and the midpoint of a segment as the average of
the numbers at its endpoints.
operations on the
complex plane.
Number
System
MAFS.912.N-
CN.3.7
Solve quadratic equations with real coefficients
that have complex solutions.
Use complex numbers
in polynomial identities
and equations. (Algebra
2 - Additional Cluster)
Mathematics 912 Number &
Quantity: The
Complex
Number
System
Click Here
MAFS.912.N-
CN.3.8
Extend polynomial identities to the complex
numbers. For example, rewrite x² + 4 as (x +
2i)(x – 2i).
Use complex numbers
in polynomial identities
and equations. (Algebra
2 - Additional Cluster)
Mathematics 912 Number &
Quantity: The
Complex
Number
System
Click Here
MAFS.912.N-
CN.3.9
Know the Fundamental Theorem of Algebra;
show that it is true for quadratic polynomials.
Use complex numbers
in polynomial identities
and equations. (Algebra
2 - Additional Cluster)
Mathematics 912 Number &
Quantity: The
Complex
Number
System
Click Here
MAFS.912.N-
Q.1.1
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. ★
Reason quantitatively
and use units to solve
problems. (Algebra 1 -
Supporting Cluster)
(Algebra 2 - Supporting
Cluster)
Mathematics 912 Number &
Quantity:
Quantities
Click Here
MAFS.912.N-
Q.1.2
Define appropriate quantities for the purpose of
descriptive modeling. ★ Reason quantitatively
and use units to solve
problems. (Algebra 1 -
Supporting Cluster)
(Algebra 2 - Supporting
Cluster)
Mathematics 912 Number &
Quantity:
Quantities
Click Here
MAFS.912.N-
Q.1.3
Choose a level of accuracy appropriate to
limitations on measurement when reporting
quantities. ★ Reason quantitatively
and use units to solve
problems. (Algebra 1 -
Supporting Cluster)
(Algebra 2 - Supporting
Cluster)
Mathematics 912 Number &
Quantity:
Quantities
Click Here
MAFS.912.N-
RN.1.1
Explain how the definition of the meaning of
rational exponents follows from extending the
Extend the properties of
exponents to rational
Mathematics 912 Number &
Quantity: The Click Here
properties of integer exponents to those values,
allowing for a notation for radicals in terms of
rational exponents. For example, we
define Error! Cannot read or display file. to be
the cube root of 5 because we
want Error! Cannot read or display file. =
Error! Cannot read or display file.to hold, so
Error! Cannot read or display file.must equal 5.
exponents. (Algebra 1 -
Major Cluster)
(Algebra 2 - Major
Cluster)
Real Number
System
MAFS.912.N-
RN.1.2
Rewrite expressions involving radicals and
rational exponents using the properties of
exponents.
Extend the properties of
exponents to rational
exponents. (Algebra 1 -
Major Cluster)
(Algebra 2 - Major
Cluster)
Mathematics 912 Number &
Quantity: The
Real Number
System
Click Here
MAFS.912.N-
RN.2.3
Explain why the sum or product of two rational
numbers is rational; that the sum of a rational
number and an irrational number is irrational;
and that the product of a nonzero rational
number and an irrational number is irrational.
Use properties of
rational and irrational
numbers. (Algebra 1 -
Additional Cluster)
Mathematics 912 Number &
Quantity: The
Real Number
System
Click Here
MAFS.912.N-
VM.1.1
Recognize vector quantities as having both
magnitude and direction. Represent vector
quantities by directed line segments, and use
appropriate symbols for vectors and their
magnitudes (e.g., v, |v|, ||v||, v).
Represent and model
with vector quantities.
Mathematics 912 Number &
Quantity:
Vector &
Matrix
Quantities
Click Here
MAFS.912.N-
VM.1.2
Find the components of a vector by subtracting
the coordinates of an initial point from the
coordinates of a terminal point.
Represent and model
with vector quantities.
Mathematics 912 Number &
Quantity:
Vector &
Matrix
Quantities
Click Here
MAFS.912.N-
VM.1.3
Solve problems involving velocity and other
quantities that can be represented by vectors.
Represent and model
with vector quantities.
Mathematics 912 Number &
Quantity:
Vector &
Matrix
Quantities
Click Here
MAFS.912.N-
VM.2.4
Add and subtract vectors.
a. Add vectors end-to-end, component-
wise, and by the parallelogram rule.
Perform operations on
vectors.
Mathematics 912 Number &
Quantity:
Vector &
Click Here
Understand that the magnitude of a sum
of two vectors is typically not the sum of
the magnitudes.
b. Given two vectors in magnitude and
direction form, determine the magnitude
and direction of their sum.
c. Understand vector subtraction v – w as v + (–w), where –w is the additive inverse of w, with the same magnitude as w and
pointing in the opposite direction.
Represent vector subtraction graphically
by connecting the tips in the appropriate
order, and perform vector subtraction
component-wise.
Matrix
Quantities
MAFS.912.N-
VM.2.5
Multiply a vector by a scalar.
a. Represent scalar multiplication
graphically by scaling vectors and
possibly reversing their direction;
perform scalar multiplication component-
wise, e.g., as
cError! Cannot read or display file. =
Error! Cannot read or display file..
b. Compute the magnitude of a scalar
multiple cv using ||cv|| = |c|v. Compute
the direction of cv knowing that when
|c|v ≠ 0, the direction of cv is either along v (for c > 0) or against v (for c < 0).
Perform operations on
vectors.
Mathematics 912 Number &
Quantity:
Vector &
Matrix
Quantities
Click Here
MAFS.912.N-
VM.3.6
Use matrices to represent and manipulate data,
e.g., to represent payoffs or incidence
relationships in a network.
Perform operations on
matrices and use
matrices in
applications.
Mathematics 912 Number &
Quantity:
Vector &
Matrix
Quantities
Click Here
MAFS.912.N-
VM.3.7
Multiply matrices by scalars to produce new
matrices, e.g., as when all of the payoffs in a
game are doubled.
Perform operations on
matrices and use
matrices in
applications.
Mathematics 912 Number &
Quantity:
Vector &
Click Here
Matrix
Quantities
MAFS.912.N-
VM.3.8
Add, subtract, and multiply matrices of
appropriate dimensions.
Perform operations on
matrices and use
matrices in
applications.
Mathematics 912 Number &
Quantity:
Vector &
Matrix
Quantities
Click Here
MAFS.912.N-
VM.3.9
Understand that, unlike multiplication of
numbers, matrix multiplication for square
matrices is not a commutative operation, but still
satisfies the associative and distributive
properties.
Perform operations on
matrices and use
matrices in
applications.
Mathematics 912 Number &
Quantity:
Vector &
Matrix
Quantities
Click Here
MAFS.912.N-
VM.3.10
Understand that the zero and identity matrices
play a role in matrix addition and multiplication
similar to the role of 0 and 1 in the real numbers.
The determinant of a square matrix is nonzero if
and only if the matrix has a multiplicative
inverse.
Perform operations on
matrices and use
matrices in
applications.
Mathematics 912 Number &
Quantity:
Vector &
Matrix
Quantities
Click Here
MAFS.912.N-
VM.3.11
Multiply a vector (regarded as a matrix with one
column) by a matrix of suitable dimensions to
produce another vector. Work with matrices as
transformations of vectors.
Perform operations on
matrices and use
matrices in
applications.
Mathematics 912 Number &
Quantity:
Vector &
Matrix
Quantities
Click Here
MAFS.912.N-
VM.3.12
Work with 2 × 2 matrices as transformations of
the plane, and interpret the absolute value of the
determinant in terms of area.
Perform operations on
matrices and use
matrices in
applications.
Mathematics 912 Number &
Quantity:
Vector &
Matrix
Quantities
Click Here
MAFS.912.S-
CP.1.1
Describe events as subsets of a sample space (the
set of outcomes) using characteristics (or
categories) of the outcomes, or as unions,
intersections, or complements of other events
(“or,” “and,” “not”). ★
Understand
independence and
conditional probability
and use them to
interpret data. (Algebra
2 - Additional Cluster)
Mathematics 912 Statistics &
Probability:
Conditional
Probability &
the Rules of
Probability
Click Here
MAFS.912.S-
CP.1.2
Understand that two events A and B are
independent if the probability of A and B
occurring together is the product of their
Understand
independence and
conditional probability
Mathematics 912 Statistics &
Probability:
Conditional
Click Here
probabilities, and use this characterization to
determine if they are independent. ★ and use them to
interpret data. (Algebra
2 - Additional Cluster)
Probability &
the Rules of
Probability
MAFS.912.S-
CP.1.3
Understand the conditional probability of A
given B as P(A and B)/P(B), and interpret
independence of A and B as saying that the
conditional probability of A given B is the same
as the probability of A, and the conditional
probability of B given A is the same as the
probability of B. ★
Understand
independence and
conditional probability
and use them to
interpret data. (Algebra
2 - Additional Cluster)
Mathematics 912 Statistics &
Probability:
Conditional
Probability &
the Rules of
Probability
Click Here
MAFS.912.S-
CP.1.4
Construct and interpret two-way frequency
tables of data when two categories are associated
with each object being classified. Use the two-
way table as a sample space to decide if events
are independent and to approximate conditional
probabilities. For example, collect data from a
random sample of students in your school on
their favorite subject among math, science, and
English. Estimate the probability that a
randomly selected student from your school will
favor science given that the student is in tenth
grade. Do the same for other subjects and
compare the results. ★
Understand
independence and
conditional probability
and use them to
interpret data. (Algebra
2 - Additional Cluster)
Mathematics 912 Statistics &
Probability:
Conditional
Probability &
the Rules of
Probability
Click Here
MAFS.912.S-
CP.1.5
Recognize and explain the concepts of
conditional probability and independence in
everyday language and everyday situations. For
example, compare the chance of having lung
cancer if you are a smoker with the chance of
being a smoker if you have lung cancer. ★
Understand
independence and
conditional probability
and use them to
interpret data. (Algebra
2 - Additional Cluster)
Mathematics 912 Statistics &
Probability:
Conditional
Probability &
the Rules of
Probability
Click Here
MAFS.912.S-
CP.2.6
Find the conditional probability of A given B as
the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the
model. ★
Use the rules of
probability to compute
probabilities of
compound events in a
uniform probability
model. (Algebra 2 -
Additional Cluster)
Mathematics 912 Statistics &
Probability:
Conditional
Probability &
the Rules of
Probability
Click Here
MAFS.912.S-
CP.2.7
Apply the Addition Rule, P(A or B) = P(A) +
P(B) – P(A and B), and interpret the answer in terms of the model. ★
Use the rules of
probability to compute
probabilities of
compound events in a
uniform probability
model. (Algebra 2 -
Additional Cluster)
Mathematics 912 Statistics &
Probability:
Conditional
Probability &
the Rules of
Probability
Click Here
MAFS.912.S-
CP.2.8
Apply the general Multiplication Rule in a
uniform probability model, P(A and B) =
P(A)P(B|A) = P(B)P(A|B), and interpret the
answer in terms of the model. ★
Use the rules of
probability to compute
probabilities of
compound events in a
uniform probability
model. (Algebra 2 -
Additional Cluster)
Mathematics 912 Statistics &
Probability:
Conditional
Probability &
the Rules of
Probability
Click Here
MAFS.912.S-
CP.2.9
Use permutations and combinations to compute
probabilities of compound events and solve
problems. ★ Use the rules of
probability to compute
probabilities of
compound events in a
uniform probability
model. (Algebra 2 -
Additional Cluster)
Mathematics 912 Statistics &
Probability:
Conditional
Probability &
the Rules of
Probability
Click Here
MAFS.912.S-
IC.1.1
Understand statistics as a process for making
inferences about population parameters based on
a random sample from that population. ★ Understand and
evaluate random
processes underlying
statistical experiments.
(Algebra 2 - Supporting
Cluster)
Mathematics 912 Statistics &
Probability:
Making
Inferences &
Justifying
Conclusions
Click Here
MAFS.912.S-
IC.1.2
Decide if a specified model is consistent with
results from a given data-generating process,
e.g., using simulation. For example, a model
says a spinning coin falls heads up with
probability 0.5. Would a result of 5 tails in a row
cause you to question the model ★
Understand and
evaluate random
processes underlying
statistical experiments.
(Algebra 2 - Supporting
Cluster)
Mathematics 912 Statistics &
Probability:
Making
Inferences &
Justifying
Conclusions
Click Here
MAFS.912.S-
IC.2.3
Recognize the purposes of and differences
among sample surveys, experiments, and
observational studies; explain how
randomization relates to each. ★
Make inferences and
justify conclusions
from sample surveys,
experiments, and
observational studies.
Mathematics 912 Statistics &
Probability:
Making
Inferences &
Click Here
(Algebra 2 - Major
Cluster)
Justifying
Conclusions
MAFS.912.S-
IC.2.4
Use data from a sample survey to estimate a
population mean or proportion; develop a margin
of error through the use of simulation models for
random sampling. ★
Make inferences and
justify conclusions
from sample surveys,
experiments, and
observational studies.
(Algebra 2 - Major
Cluster)
Mathematics 912 Statistics &
Probability:
Making
Inferences &
Justifying
Conclusions
Click Here
MAFS.912.S-
IC.2.5
Use data from a randomized experiment to
compare two treatments; use simulations to
decide if differences between parameters are
significant. ★
Make inferences and
justify conclusions
from sample surveys,
experiments, and
observational studies.
(Algebra 2 - Major
Cluster)
Mathematics 912 Statistics &
Probability:
Making
Inferences &
Justifying
Conclusions
Click Here
MAFS.912.S-
IC.2.6 Evaluate reports based on data. ★ Make inferences and
justify conclusions
from sample surveys,
experiments, and
observational studies.
(Algebra 2 - Major
Cluster)
Mathematics 912 Statistics &
Probability:
Making
Inferences &
Justifying
Conclusions
Click Here
MAFS.912.S-
ID.1.1
Represent data with plots on the real number line
(dot plots, histograms, and box plots). ★ Summarize, represent,
and interpret data on a
single count or
measurement variable.
(Algebra 1 - Additional
Cluster) (Algebra 2 -
Additional Cluster)
Mathematics 912 Statistics &
Probability:
Interpreting
Categorical &
Quantitative
Data
Click Here
MAFS.912.S-
ID.1.2
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. ★
Summarize, represent,
and interpret data on a
single count or
measurement variable.
(Algebra 1 - Additional
Cluster) (Algebra 2 -
Additional Cluster)
Mathematics 912 Statistics &
Probability:
Interpreting
Categorical &
Quantitative
Data
Click Here
MAFS.912.S-
ID.1.3
Interpret differences in shape, center, and spread
in the context of the data sets, accounting for
possible effects of extreme data points (outliers).
★
Summarize, represent,
and interpret data on a
single count or
measurement variable.
(Algebra 1 - Additional
Cluster) (Algebra 2 -
Additional Cluster)
Mathematics 912 Statistics &
Probability:
Interpreting
Categorical &
Quantitative
Data
Click Here
MAFS.912.S-
ID.1.4
Use the mean and standard deviation of a data
set to fit it to a normal distribution and to
estimate population percentages. Recognize that
there are data sets for which such a procedure is
not appropriate. Use calculators, spreadsheets,
and tables to estimate areas under the normal
curve. ★
Summarize, represent,
and interpret data on a
single count or
measurement variable.
(Algebra 1 - Additional
Cluster) (Algebra 2 -
Additional Cluster)
Mathematics 912 Statistics &
Probability:
Interpreting
Categorical &
Quantitative
Data
Click Here
MAFS.912.S-
ID.2.5
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. ★
Summarize, represent,
and interpret data on
two categorical and
quantitative variables.
(Algebra 1 - Supporting
Cluster) (Algebra 2 -
Supporting Cluster)
Mathematics 912 Statistics &
Probability:
Interpreting
Categorical &
Quantitative
Data
Click Here
MAFS.912.S-
ID.2.6
Represent data on two quantitative variables on a
scatter plot, and describe how the variables are
related. ★ a. Fit a function to the data; use functions
fitted to data to solve problems in the
context of the data. Use given functions
or choose a function suggested by the
context. Emphasize linear, and
exponential models.
b. Informally assess the fit of a function by
plotting and analyzing residuals.
c. Fit a linear function for a scatter plot that
suggests a linear association.
Summarize, represent,
and interpret data on
two categorical and
quantitative variables.
(Algebra 1 - Supporting
Cluster) (Algebra 2 -
Supporting Cluster)
Mathematics 912 Statistics &
Probability:
Interpreting
Categorical &
Quantitative
Data
Click Here
MAFS.912.S-
ID.3.7
Interpret the slope (rate of change) and the
intercept (constant term) of a linear model in the
context of the data. ★ Interpret linear models.
(Algebra 1 - Major
Cluster)
Mathematics 912 Statistics &
Probability:
Interpreting
Categorical &
Quantitative
Data
Click Here
MAFS.912.S-
ID.3.8
Compute (using technology) and interpret the
correlation coefficient of a linear fit. ★ Interpret linear models.
(Algebra 1 - Major
Cluster)
Mathematics 912 Statistics &
Probability:
Interpreting
Categorical &
Quantitative
Data
Click Here
MAFS.912.S-
ID.3.9
Distinguish between correlation and causation.
★ Interpret linear models.
(Algebra 1 - Major
Cluster)
Mathematics 912 Statistics &
Probability:
Interpreting
Categorical &
Quantitative
Data
Click Here
MAFS.912.S-
MD.1.1
Define a random variable for a quantity of
interest by assigning a numerical value to each
event in a sample space; graph the corresponding
probability distribution using the same graphical
displays as for data distributions. ★
Calculate expected
values and use them to
solve problems
Mathematics 912 Statistics &
Probability:
Using
Probability to
Make
Decisions
Click Here
MAFS.912.S-
MD.1.2
Calculate the expected value of a random
variable; interpret it as the mean of the
probability distribution. ★ Calculate expected
values and use them to
solve problems
Mathematics 912 Statistics &
Probability:
Using
Probability to
Make
Decisions
Click Here
MAFS.912.S-
MD.1.3
Develop a probability distribution for a random
variable defined for a sample space in which
theoretical probabilities can be calculated; find
the expected value. For example, find the
theoretical probability distribution for the
number of correct answers obtained by guessing
on all five questions of a multiple-choice test
where each question has four choices, and find
Calculate expected
values and use them to
solve problems
Mathematics 912 Statistics &
Probability:
Using
Probability to
Make
Decisions
Click Here
the expected grade under various grading
schemes. ★ MAFS.912.S-
MD.1.4
Develop a probability distribution for a random
variable defined for a sample space in which
probabilities are assigned empirically; find the
expected value. For example, find a current data
distribution on the number of TV sets per
household in the United States, and calculate the
expected number of sets per household. How
many TV sets would you expect to find in 100
randomly selected households? ★
Calculate expected
values and use them to
solve problems
Mathematics 912 Statistics &
Probability:
Using
Probability to
Make
Decisions
Click Here
MAFS.912.S-
MD.2.5
Weigh the possible outcomes of a decision by
assigning probabilities to payoff values and
finding expected values. ★ a. Find the expected payoff for a game of
chance. For example, find the expected
winnings from a state lottery ticket or a
game at a fast-food restaurant.
b. Evaluate and compare strategies on the
basis of expected values. For example,
compare a high-deductible versus a low-
deductible automobile insurance policy
using various, but reasonable, chances of
having a minor or a major accident.
Use probability to
evaluate outcomes of
decisions
Mathematics 912 Statistics &
Probability:
Using
Probability to
Make
Decisions
Click Here
MAFS.912.S-
MD.2.6
Use probabilities to make fair decisions (e.g.,
drawing by lots, using a random number
generator). ★ Use probability to
evaluate outcomes of
decisions
Mathematics 912 Statistics &
Probability:
Using
Probability to
Make
Decisions
Click Here
MAFS.912.S-
MD.2.7
Analyze decisions and strategies using
probability concepts (e.g., product testing,
medical testing, pulling a hockey goalie at the
end of a game). ★
Use probability to
evaluate outcomes of
decisions
Mathematics 912 Statistics &
Probability:
Using
Probability to
Make
Decisions
Click Here
MAFS.K12.MP.1.1 Make sense of problems and persevere in
solving them.
Mathematically proficient students start by
explaining to themselves the meaning of a
problem and looking for entry points to its
solution. They analyze givens, constraints,
relationships, and goals. They make conjectures
about the form and meaning of the solution and
plan a solution pathway rather than simply
jumping into a solution attempt. They consider
analogous problems, and try special cases and
simpler forms of the original problem in order to
gain insight into its solution. They monitor and
evaluate their progress and change course if
necessary. Older students might, depending on
the context of the problem, transform algebraic
expressions or change the viewing window on
their graphing calculator to get the information
they need. Mathematically proficient students
can explain correspondences between equations,
verbal descriptions, tables, and graphs or draw
diagrams of important features and relationships,
graph data, and search for regularity or trends.
Younger students might rely on using concrete
objects or pictures to help conceptualize and
solve a problem. Mathematically proficient
students check their answers to problems using a
different method, and they continually ask
themselves, “Does this make sense?” They can understand the approaches of others to solving
complex problems and identify correspondences
between different approaches.
Make sense of
problems and persevere
in solving them.
Mathematics K12 Mathematical
Practice Click Here
MAFS.K12.MP.2.1 Reason abstractly and quantitatively.
Mathematically proficient students make sense
of quantities and their relationships in problem
situations. They bring two complementary
abilities to bear on problems involving
Reason abstractly and
quantitatively.
Mathematics K12 Mathematical
Practice Click Here
quantitative relationships: the ability to
decontextualize—to abstract a given situation and represent it symbolically and manipulate the
representing symbols as if they have a life of
their own, without necessarily attending to their
referents—and the ability to contextualize, to pause as needed during the manipulation process
in order to probe into the referents for the
symbols involved. Quantitative reasoning entails
habits of creating a coherent representation of
the problem at hand; considering the units
involved; attending to the meaning of quantities,
not just how to compute them; and knowing and
flexibly using different properties of operations
and objects.
MAFS.K12.MP.3.1 Construct viable arguments and critique the
reasoning of others.
Mathematically proficient students understand
and use stated assumptions, definitions, and
previously established results in constructing
arguments. They make conjectures and build a
logical progression of statements to explore the
truth of their conjectures. They are able to
analyze situations by breaking them into cases,
and can recognize and use counterexamples.
They justify their conclusions, communicate
them to others, and respond to the arguments of
others. They reason inductively about data,
making plausible arguments that take into
account the context from which the data arose.
Mathematically proficient students are also able
to compare the effectiveness of two plausible
arguments, distinguish correct logic or reasoning
from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete
referents such as objects, drawings, diagrams,
and actions. Such arguments can make sense and
Construct viable
arguments and critique
the reasoning of others.
Mathematics K12 Mathematical
Practice Click Here
be correct, even though they are not generalized
or made formal until later grades. Later, students
learn to determine domains to which an
argument applies. Students at all grades can
listen or read the arguments of others, decide
whether they make sense, and ask useful
questions to clarify or improve the arguments.
MAFS.K12.MP.4.1 Model with mathematics.
Mathematically proficient students can apply the
mathematics they know to solve problems
arising in everyday life, society, and the
workplace. In early grades, this might be as
simple as writing an addition equation to
describe a situation. In middle grades, a student
might apply proportional reasoning to plan a
school event or analyze a problem in the
community. By high school, a student might use
geometry to solve a design problem or use a
function to describe how one quantity of interest
depends on another. Mathematically proficient
students who can apply what they know are
comfortable making assumptions and
approximations to simplify a complicated
situation, realizing that these may need revision
later. They are able to identify important
quantities in a practical situation and map their
relationships using such tools as diagrams, two-
way tables, graphs, flowcharts and formulas.
They can analyze those relationships
mathematically to draw conclusions. They
routinely interpret their mathematical results in
the context of the situation and reflect on
whether the results make sense, possibly
improving the model if it has not served its
purpose.
Model with
mathematics.
Mathematics K12 Mathematical
Practice Click Here
MAFS.K12.MP.5.1 Use appropriate tools strategically.
Mathematically proficient students consider the
Use appropriate tools
strategically.
Mathematics K12 Mathematical
Practice Click Here
available tools when solving a mathematical
problem. These tools might include pencil and
paper, concrete models, a ruler, a protractor, a
calculator, a spreadsheet, a computer algebra
system, a statistical package, or dynamic
geometry software. Proficient students are
sufficiently familiar with tools appropriate for
their grade or course to make sound decisions
about when each of these tools might be helpful,
recognizing both the insight to be gained and
their limitations. For example, mathematically
proficient high school students analyze graphs of
functions and solutions generated using a
graphing calculator. They detect possible errors
by strategically using estimation and other
mathematical knowledge. When making
mathematical models, they know that technology
can enable them to visualize the results of
varying assumptions, explore consequences, and
compare predictions with data. Mathematically
proficient students at various grade levels are
able to identify relevant external mathematical
resources, such as digital content located on a
website, and use them to pose or solve problems.
They are able to use technological tools to
explore and deepen their understanding of
concepts.
MAFS.K12.MP.6.1 Attend to precision.
Mathematically proficient students try to
communicate precisely to others. They try to use
clear definitions in discussion with others and in
their own reasoning. They state the meaning of
the symbols they choose, including using the
equal sign consistently and appropriately. They
are careful about specifying units of measure,
and labeling axes to clarify the correspondence
with quantities in a problem. They calculate
accurately and efficiently, express numerical
Attend to precision. Mathematics K12 Mathematical
Practice Click Here
answers with a degree of precision appropriate
for the problem context. In the elementary
grades, students give carefully formulated
explanations to each other. By the time they
reach high school they have learned to examine
claims and make explicit use of definitions.
MAFS.K12.MP.7.1 Look for and make use of structure.
Mathematically proficient students look closely
to discern a pattern or structure. Young students,
for example, might notice that three and seven
more is the same amount as seven and three
more, or they may sort a collection of shapes
according to how many sides the shapes have.
Later, students will see 7 × 8 equals the well
remembered 7 × 5 + 7 × 3, in preparation for
learning about the distributive property. In the
expression x² + 9x + 14, older students can see
the 14 as 2 × 7 and the 9 as 2 + 7. They
recognize the significance of an existing line in a
geometric figure and can use the strategy of
drawing an auxiliary line for solving problems.
They also can step back for an overview and
shift perspective. They can see complicated
things, such as some algebraic expressions, as
single objects or as being composed of several
objects. For example, they can see 5 – 3(x – y)² as 5 minus a positive number times a square and
use that to realize that its value cannot be more
than 5 for any real numbers x and y.
Look for and make use
of structure.
Mathematics K12 Mathematical
Practice Click Here
MAFS.K12.MP.8.1 Look for and express regularity in repeated
reasoning.
Mathematically proficient students notice if
calculations are repeated, and look both for
general methods and for shortcuts. Upper
elementary students might notice when dividing
25 by 11 that they are repeating the same
calculations over and over again, and conclude
Look for and express
regularity in repeated
reasoning.
Mathematics K12 Mathematical
Practice Click Here |
they have a repeating decimal. By paying
attention to the calculation of slope as they
repeatedly check whether points are on the line
through (1, 2) with slope 3, middle school
students might abstract the equation (y – 2)/(x – 1) = 3. Noticing the regularity in the way terms
cancel when expanding (x – 1)(x + 1), (x – 1)(x² + x + 1), and (x – 1)(x³ + x² + x + 1) might lead them to the general formula for the sum of a
geometric series. As they work to solve a
problem, mathematically proficient students
maintain oversight of the process, while
attending to the details. They continually
evaluate the reasonableness of their intermediate
results.
