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Strand

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,

and multiply polynomials.

Perform arithmetic

operations on

polynomials. (Algebra

1 – Major Cluster)

Mathematics 912 Algebra:

Arithmetic

with

Polynomials

& Rational

Expressions

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

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

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 –

Mathematics 912 Algebra:

Arithmetic

with

Polynomials

& Rational

Expressions

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 –

Mathematics 912 Algebra:

Arithmetic

with

Polynomials

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

MAFS.912.A-

APR.4.7

Understand that rational expressions form a

system analogous to the rational numbers, closed

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

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

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

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

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

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

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

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

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

b. Solve quadratic equations by inspection

(e.g., for x² = 49), taking square roots,

formula and factoring, as appropriate to

the initial form of the equation.

Solve equations and

inequalities in one

variable. (Algebra 1 –

Major Cluster)

(Algebra 2 – Supporting

Cluster)

Mathematics 912 Algebra:

Reasoning

with

Equations &

Inequalities

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 –

Cluster)

Mathematics 912 Algebra:

Reasoning

with

Equations &

Inequalities

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 –

Cluster)

Mathematics 912 Algebra:

Reasoning

with

Equations &

Inequalities

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 –

Cluster)

Mathematics 912 Algebra:

Reasoning

with

Equations &

Inequalities

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 –

Cluster)

Mathematics 912 Algebra:

Reasoning

with

Equations &

Inequalities

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 –

Cluster)

Mathematics 912 Algebra:

Reasoning

with

Equations &

Inequalities

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

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

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

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

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

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

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

MAFS.912.C.1.1 Understand the concept of limit and estimate

limits from graphs and tables of values.

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.

MAFS.912.C.1.4 Find limits of rational functions that are

undefined at a point.

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.

MAFS.912.C.1.8 Find special limits such as

Error! Cannot read or display file.

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.

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.

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.

MAFS.912.C.2.2 State, understand, and apply the definition of

derivative.

MAFS.912.C.2.3 Find the derivatives of functions, including

algebraic, trigonometric, logarithmic, and

exponential functions.

MAFS.912.C.2.4 Find the derivatives of sums, products, and

quotients.

MAFS.912.C.2.5 Find the derivatives of composite functions

using the Chain Rule.

MAFS.912.C.2.6 Find the derivatives of implicitly-defined

functions.

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.

MAFS.912.C.2.9 Find derivatives using logarithmic

differentiation.

MAFS.912.C.2.10 Understand and use the relationship between

differentiability and continuity.

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

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

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

MAFS.912.C.3.4 Find local and absolute maximum and minimum

points.

Applications of

Derivatives

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

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

MAFS.912.C.3.7 Use implicit differentiation to find the derivative

of an inverse function.

Applications of

Derivatives

MAFS.912.C.3.8 Solve optimization problems. Applications of

Derivatives

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

MAFS.912.C.3.10 Find the velocity and acceleration of a particle

moving in a straight line.

Applications of

Derivatives

MAFS.912.C.3.11 Model rates of change, including related rates

problems.

Applications of

Derivatives

MAFS.912.C.3.12 Solve problems using the Newton-Raphson

method.

Applications of

Derivatives

MAFS.912.C.4.1 Use rectangle approximations to find

approximate values of integrals.

MAFS.912.C.4.2 Calculate the values of Riemann Sums over

equal subdivisions using left, right, and midpoint

evaluation points.

MAFS.912.C.4.3 Interpret a definite integral as a limit of Riemann

sums.

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).

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.

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

≤ 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.

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.

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

MAFS.912.C.5.2 Solve separable differential equations, and use

them in modeling.

Applications of

Integration

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

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

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

MAFS.912.C.5.6 Use definite integrals to find the average value

of a function over a closed interval.

Applications of

Integration

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

MAFS.912.C.5.8 Apply integration to model, and solve problems

in physical, biological, and social sciences.

Applications of

Integration

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

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

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.

Cluster) (Algebra 2 –

Mathematics 912 Functions:

Building

Functions

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.

Cluster) (Algebra 2 –

Mathematics 912 Functions:

Building

Functions

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.

Cluster) (Algebra 2 –

Mathematics 912 Functions:

Building

Functions

MAFS.912.F-

BF.2.a

Use the change of base formula. Build new functions

from existing functions.

Cluster) (Algebra 2 –

Mathematics 912 Functions:

Building

Functions

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

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

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

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

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

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

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

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