2016-2017 math nation algebra 2 scope & …polynomials. section 1 - topic 2: mafs.912.a...

2016-2017 Math Nation Algebra 2 Scope & Sequence: MAFS Topics and Standards Alignment

www.MathNation.com 1

TABLE OF CONTENTS

SECTIONS PAGES IN THIS DOC

Section 1: Function Overview 2-4

Section 2: Linear Functions 5-9

Section 3: Piecewise-Defined Functions 10-12

Section 4: Quadratics Functions – Part 1 13-16

Section 5: Quadratics Functions – Part 2 17-23

Section 6: Polynomials Functions 24-27

Section 7: Rational Expressions and Equations 28-29

Section 8: Expressions and Equations with Radicals and Rational Exponents 30-32

Section 9: Exponential and Logarithmic Functions 33-37

Section 10: Sequences and Series 38-39

Section 11: Probability 40-44

Section 12: Statistics 45-46

Section 13: Trigonometry – Part 1 47

Section 14: Trigonometry – Part 2 48



SECTION 1: FUNCTIONS

Section 1 - Topic 1:

Adding Functions

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.

In this topic, students will add and subtract

polynomial functions.


Multiplying Functions






polynomials.

In this topic, students will multiply

polynomial functions.


Dividing 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.

In this topic, students will rewrite a rational

expression as the quotient in the form of a

polynomial added to the remainder

divided by the divisor. Students will use

polynomial long division to divide a

polynomial by a polynomial.


Using Synthetic Division to Divide Functions

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.

In this topic, students will use synthetic

division as a method of rewriting rational

expressions when the divisor is in the form

𝑥 − 𝑐.


Composition of Functions

MAFS.912.F-BF.1.1c Write a function that

describes a relationship between two

quantities.

c. Compose functions. For example, if 𝑇(𝑦) is the temperature in the atmosphere as a

function of height and ℎ(𝑡) is the height of a

weather balloon as a function of time, then

𝑇(ℎ(𝑡)) is the temperature at the location of

the weather balloon as a function of time.

In this topic, students will write a function to

model a real-world context by composing

functions and the information within the

context.




Inverse Functions - Part 1

MAFS.912.F-BF.2.4a,c Find inverse functions.

a. Solve an equation of the form 𝑓(𝑥) = 𝑐 for a

simple function, 𝑓, that has an inverse and

write an expression for the inverse.

For example, 𝑓(𝑥) = 2 × 3 or 𝑓(𝑥) = (𝑥 +

1)/(𝑥– 1) for 𝑥 ≠ 1.

c. Read values of an inverse function from a

graph or a table, given that the function has

an inverse.

In this topic, students will investigate inverse

functions. will use a graph or a table of a

function to determine values of the

function’s inverse. Students will find the

inverse of a function.


Inverse Functions - Part 2

MAFS.912.F-BF.2.4a,b,c,d Find inverse

functions.


simple function, 𝑓, that has an inverse and

write an expression for the inverse.

For example, 𝑓(𝑥) = 2 × 3 or 𝑓(𝑥) = (𝑥 +1)/(𝑥– 1) for 𝑥 ≠ 1.

b. Verify by composition that one function is

the inverse of another.



an inverse.

d. Produce an invertible function from a non-

invertible function by restricting the domain.

In this topic, students will continue to work

with inverses. Students will use compositions

to determine if two functions are inverses.

Students will restrict domains to create

invertible functions.


Recognizing Even and Odd Functions

MAFS.912.F-BF.2.3 Identify the effect on the

graph of replacing 𝑓(𝑥) by 𝑓(𝑥) + 𝑘, 𝑘𝑓(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.

In this topic, students will investigate

features of even and odd functions.

Students will determine if functions are

even or odd by examining equations,

tables, and graphs.




Key Features of Graphs of Functions

MAFS.912.F-IF.3.7a Graph functions expressed

symbolically and show key features of the

graph, by hand in simple cases and using

technology in more complicated cases. A.

Graph linear and quadratic functions and

show intercepts, maxima, and minima. This

section focuses on linear functions.

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.

In this topic, students will review key

features of graphs of functions. (solutions,

y-intercepts, positive/negative,

increasing/decreasing, maximum,

minimum,).


Transformations of Functions – Part 1


graph of replacing 𝑓(𝑥) by 𝑓(𝑥) + 𝑘, 𝑘𝑓 (𝑥),

𝑓(𝑘𝑥), and 𝑓(𝑥 + 𝑘) for specific values of k


k given the graphs. Experiment with cases and





In this topic, students will review

transformations of functions. Students will

investigate horizontal shifts of functions.

Students will also consider multiple

transformations on a function.


Transformations of Functions – Part 2


graph of replacing 𝑓(𝑥) by 𝑓(𝑥) + 𝑘, 𝑘𝑓 (𝑥),

𝑓(𝑘𝑥), and 𝑓(𝑥 + 𝑘) for specific values of k


k given the graphs. Experiment with cases and





In this topic, students will review

transformations of functions. Students will

investigate horizontal shifts of functions.

Students will also consider multiple

transformations on a function.



SECTION 2: LINEAR FUNCTIONS, EQUATIONS, AND INEQUALITIES


Linear Equations in One Variable - Part 1

MAFS.912.A-CED.1.1 Create equations and

inequalities in one variable and use them to

solve problems.

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.

MAFS.912.A-REI.1.2 Solve simple rational and

radical equations in one variable, and give

examples showing how extraneous solutions

may arise.

MAFS.912.A-SSE.1.1a Interpret expressions that

represent a quantity in terms of its context.

a. Interpret parts of an expression, such as

terms, factors, and coefficients.

In this topic, students will justify the steps to

solve equations. Students will create and

solve equations representing real-world

situations. Additionally, students will

interpret expressions and what the terms

represent.


Linear Equations in One Variable - Part 2

MAFS.912.A-CED.1.4 Rearrange formulas to

highlight a quantity of interest using the same

reasoning as in solving equations.



solve problems.

In this topic, students will solve equations

with multiple variables for a specific

variable.




Linear Equations and Inequalities in Two

Variables

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.

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 nonviable options in a

modeling context.

MAFS.912.F-LE.2.5 Interpret the parameters in a

linear or an exponential function in terms of a

context.





In this topic, students will represent real-

world situations with linear functions.

Students will graph the functions and

interpret key features of the graph.


Key Features of Linear Functions




the quantities and sketch graphs showing key







In this topic, students will review the key

features of linear functions.




Classifying Linear Functions and Finding

Inverses

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. For

example, 𝑓(𝑥) = 2𝑥³ or 𝑓(𝑥) = (𝑥 + 1)/(𝑥– 1) for

𝑥 ≠ 1.





an inverse.



In this topic, students will classify linear

functions as even, odd, or neither.

Additionally, students will find the inverse of

a linear function, if it exists.


Solving Linear Systems - Investigating

Graphing, Substitution, and Elimination

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.

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 𝑓(𝑥) = 𝑔(𝑥); find the solutions approximately (e.g., using

technology to graph the functions, make

tables of values, or find successive

approximations).

In this topic, students investigate solutions

to systems of linear equations. Students will

solve systems by graphing and substitution.

Additionally, students will explore

equivalent systems of equations.




Solving Linear Systems Using Elimination













In this topic, students will solve systems

using the elimination method. Additionally,

student will interpret different terms in a

system of equations.


Solving Linear Systems Using Substitution













modeling context.

In the topic, students will solve systems of

equations by substitution. They will explore

why the x-coordinates of the points where

the graphs of the equations 𝑦 = 𝑓(𝑥) and

𝑦 = 𝑔(𝑥) intersect are the solutions of the

equation 𝑓(𝑥) = 𝑔(𝑥).




Systems of Linear Equations in Three Variables -

Part 1









In this topic, students will write and solve

systems of linear equations in three

variables that represent real-world

situations.


Systems of Linear Equations in Three Variables -

Part 2









In this topic, students will write and solve

systems of linear equations in three

variables that represent real-world

situations.


Systems of Linear Inequalities





modeling context.

In this topic students will create systems of

linear inequalities from real-world situations.



SECTION 3: PIECEWISE-DEFINED FUNCTIONS


Introduction to Piecewise-Defined Functions -

Part 1


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.

MAFS.912.F-IF.3.7b Graph functions expressed

symbolically and show key features of the graph

by hand in simple cases and using technology for

more complicated cases.

b. Graph square root, cube root, and piecewise-

defined functions, including step functions and

absolute value functions.

In this topics, students will explore and

evaluate piecewise-defined functions.

Additionally, students will define key

features for graphs of piecewise-defined

functions.


Introduction to Piecewise-Defined Functions -

Part 2













In this topics, students will explore and

evaluate piecewise-defined functions.

Additionally, students will define key

features for graphs of piecewise-defined

functions.




Graphing and Writing Piecewise-Defined

Functions - Part 1

MAFS.912.A.F-IF.2.4 For a function that models a












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.

In this topic, students will graph piece-

wise defined functions. Additionally,

students will write piece-wise defined

functions and describe key features of

the graphs.


Graphing and Writing Piecewise-Defined

Functions - Part 2

















In this topic, students will graph piece-

wise defined functions. Additionally,

students will write piece-wise defined

functions and describe key features of

the graphs.




Real World Examples of Piecewise-Defined

Functions












In this topic, students will look at real

world examples of piecewise-defined

functions. Students will write and graph

the function that represents the situation.


Absolute Value Functions

















In this topic, students will explore

absolute value functions. Students will

make the connection that absolute

value functions can be written as

piecewise-defined function. Students will

write and graph absolute value

functions.


Transformations of Piecewise–Defined Functions

MAFS.912.F-BF.2.3 Identify the effect on the graph

of replacing 𝑓(𝑥) by 𝑓(𝑥) + 𝑘, 𝑘𝑓(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.

In this topic, students will apply their

knowledge of transformations of

functions to piecewise-defined functions.



SECTION 4: QUADRATICS PART 1


Real-Life Examples of Quadratic Functions

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.

MAFS.912.CED.1.1 Create equations and inequalities in

one variable and use them to solve problems.

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.

In this topic, students will determine

and relate the key features of a

function within a real-world context

by examining the function’s graph.

Students will also consider using the

gravitational constant to write a

quadratic function to represent a

real-life situation.


Solving Quadratic Equations by Factoring

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.

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.

In this topic, students will factor a

quadratic expression to find the

solutions.




Solving Quadratic Equations by Factoring -

Special Cases - Part 1




a. Factor a quadratic expression to reveal the zeros of

the function it defines.

MAFS.912.A-REI.2.4.b Solve quadratic equations in one

variable.

b. Solve quadratic equations by inspection (e.g., for 𝑥2 =49), taking square roots, completing the square, the

quadratic formula, and factoring, as appropriate to the

initial form of the equation. Recognize when the

quadratic formula gives complex solutions and write

them as 𝑎 ± 𝑏i for real numbers 𝑎 and 𝑏.

In this topic, students will look at

special cases of factoring. This topic

focuses on perfect square trinomials.


Solving Quadratic Equations by Factoring -

Special Cases - Part 2




a. Factor a quadratic expression to reveal the zeros of the

function it defines.


variable.





them as 𝑎 ± 𝑏i for real numbers 𝑎 and 𝑏.

In this topic, students will look at

special cases of factoring. This topic

focuses on difference of two

squares.


Complex Numbers - Part 1

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.

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.

In this topic, students will use I to

represent imaginary numbers.

Students will add, subtract, and

multiply complex numbers and use

𝑖2 = −1 to write the answer as a

complex number.




Complex Numbers - Part 2

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.

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.

In this topic, students will use I to

represent imaginary numbers.

Students will add, subtract, and

multiply complex numbers and use

𝑖2 = −1 to write the answer as a

complex number.


Solving Quadratic Equations by Completing

the Square

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.



initial form

MAFS.912.N-CN.3.7 Solve quadratic equations with real

coefficients that have complex solutions.

In this topic, students will transform a

quadratic equations by completing

the square and then solve the

equation by taking the square root.


Solving Quadratics Using the Quadratic

Formula - Part 1




variable.

b. Solve quadratic equations by inspection (e.g., for 𝑥2 =

49), taking square roots, completing the square, the




them as 𝑎 ± 𝑏𝑖 for real numbers 𝑎 and 𝑏.

In this topic, students will use the

quadratic formula to solve

quadratics.




Solving Quadratics Using the Quadratic

Formula - Part 2



MAFS.912.A-REI.2.4 Solve quadratic equations in one

variable.





them as 𝑎 ± 𝑏𝑖 for real numbers 𝑎 and 𝑏.

MAFS.912.A-CED.1.1 Create equations and inequalities in

one variable and use them to solve problems.

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.

In this topic, students will use the

quadratic formula to solve

quadratics.



SECTION 5: QUADRATICS – PART 2

Section 5 – Topic 1:

Graphing Quadratics in Standard Form



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.

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 quadratic function and

an algebraic expression for another, say which

has the larger maximum.

MAFS.912.A-REI.2.4.b Solve quadratic

equations in one variable.

b. Solve quadratic equations by inspection

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

completing the square, the quadratic formula,

and factoring, as appropriate to the initial form

of the equation. Recognize when the

quadratic formula gives complex solutions and

write them as 𝑎 ± 𝑏i for real numbers 𝑎 and 𝑏.

In this topic, students will review the key

features of a quadratic function. Additionally,

they will use key features to sketch the graph

of the quadratic.


Writing Quadratic Equations in Standard Form

from a Graph





In this topic, students will identify key features

from a graph and use those to write the

equation represented by the graph.




Graphing Quadratics in Vertex Form - Part 1
















In this topic, students will identify key features

from the vertex form. Students will use the

features to graph the function.




Graphing Quadratics in Vertex Form - Part 2







MAFS.912.F-IF.3.8.a 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.

















In this topic, students write functions in vertex

form and identify key features. Students will use

the features to graph the function.



Section 5, Topic 5:

Writing Quadratic Equations in Vertex Form

from a Graph

MAFS.912.F-IF.3.8.a 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.





In this topic, students will use the vertex and

other features to write the equation of the

quadratic in vertex form.

Section 5, Topic 6:

Converting Quadratic Equations

MAFS.912.A-SSE.2.3b Choose and produce an

equivalent form of an expression to reveal and

explain properties of the quantity represented

by the expression b. Complete the square in a

quadratic expression to reveal the maximum or

minimum value of the function it defines.

In this topic, students will write quadratic

equations in different forms.

Section 5, Topic 7:

Writing Quadratic Equations When Given a

Focus and Directrix

MAFS.912.G-GPE.1.2 Derive the equation of a

parabola given a focus and directrix.

In this topic, students will use the understand

the relationship between the directrix and

focus of a parabola and use those features to

write the equation of the parabola.




Systems of Equations with Quadratics - Part 1

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.







approximations). Include cases where 𝑓(𝑥)

and/or 𝑔(𝑥) are linear, polynomial, rational,

absolute value, exponential, and logarithmic

functions.

In this topic, students will solve systems of

equations that contain linear and quadratic

equations, as well as systems of two

quadratics.


Systems of Equations with Quadratics - Part 2

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.







approximations). Include cases where 𝑓(𝑥)

and/or 𝑔(𝑥) are linear, polynomial, rational,


functions.

In this topic, students will solve systems of

equations that contain linear and quadratic

equations, as well as systems of two

quadratics.




Transformations with Quadratic Functions


graph of replacing 𝑓(𝑥) by 𝑓(𝑥) 𝑘, 𝑘𝑓(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





In this topic, students will apply their knowledge

of transformations of functions specifically to

quadratic functions.


Key Features of Quadratic Functions











In this topic, students will review all the key of

quadratic functions.




Classifying Quadratic Functions and Finding

Inverses



simple function, f, that has an inverse and write

an expression for the inverse. For example,

𝑓(𝑥) = 2𝑥³ or 𝑓(𝑥) = (𝑥 + 1)/(𝑥– 1) for 𝑥 ≠ 1.





an inverse.




graph of replacing 𝑓(𝑥) by 𝑓(𝑥) + 𝑘, 𝑘𝑓(𝑥),

𝑓(𝑘𝑥), and 𝑓(𝑥 + 𝑘) for specific values of 𝑘


𝑘 given the graphs. Experiment with cases and





In this topic, students will classify quadratic

functions as even, odd, or neither.

Additionally, they will find inverses of quadratic

functions and restrict domains to produce and

invertible function.



SECTION 6: POLYNOMIALS FUNCTIONS


Classifying Polynomials and Closure Property






polynomials.

MAFS.912.A-APR.3.4 Prove polynomial identities

and use them to describe numerical

relationships.

In this topic, students will classify polynomials

which leads into a review of the closure

property as applied to polynomials.


Polynomial Identities - Part 1

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).



relationships.

In this topic, students will prove polynomial

identities. Students will use those identities to

write equivalent expressions and describe

numerical relationships.


Polynomial Identities - Part 2








relationships.

In this topic, students will prove polynomial

identities. Students will use those identities to

write equivalent expressions and describe

numerical relationships.


Recognizing End Behavior of Graphs of

Polynomials






relationship.

In this topic, students will review graphs and

make generalities about end behavior of

polynomial functions. Students will use those

generalities to determine the end behavior

when given a polynomial function.




Using Successive Differences






relationship.

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.

In this topic, students will explore rate of

change and successive differences of different

polynomial functions. They will use their findings

to classify polynomial functions.


Understanding Zeroes of Polynomials











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.

In this topic, students will explore zeroes of

polynomial functions and how this relates to

the degree of the function.




Factoring Polynomials

MAFS.912.A-SSE.2.3 Choose and produce an



by the expression.










polynomial.

In this topic, students will apply their prior

knowledge of factoring and polynomial

identities to factor polynomials of higher

degrees.




Sketching Graphs of Polynomials





polynomial.

MAFS.912.F-IF.3.7c Graph functions expressed




c. Graph polynomial functions, identifying zeros

when suitable factorizations are available and

showing end behavior.

MAFS.912.A-SSE.2.3 Choose and produce an



by the expression.






In this topic, students will apply their knowledge

of zeroes and end behavior of polynomials to

sketch the graph of polynomial functions of

higher degrees.



SECTION 7: RATIONAL EXPRESSIONS AND EQUATIONS

Section 7, Topic 1:

The Remainder Theorem

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).

In this topic, students will understand and apply

the remainder theorem to determine if an

expression is a factor of a polynomial function.

Section 7, Topic 2:

Solving Rational Equations




may arise.





modeling context. For example, represent

inequalities describing nutritional and cost

constraints on combinations of different foods.

In this topic, students will solve a rational

equation in one variable.

Section 7, Topic 3:

Solving Systems of Rational Equations




are the solutions of the equation 𝑓(𝑥) = 𝑔(𝑥);

find the solutions approximately (e.g., using



approximations). Include cases where 𝑓(𝑥) and/or 𝑔(𝑥) are linear, polynomial, rational,


functions.




may arise.

In this topic, students will solve a system of

rational equations.



Section 7, Topic 4:

Using Rational Equations to Solve Real World

Problems



solve problems. Include equations arising from

linear and quadratic functions and simple

rational, absolute, and exponential functions.





modeling context. For example, represent

inequalities describing nutritional and cost

constraints on combinations of different foods.

In this topic, students will use rational equations

solve real-world situations.

Section 7, Topic 5:

Graphing Rational Functions

MAFS.912.F-IF.3.7d Graph functions expressed




d. Graph rational functions, identifying zeros

and asymptotes when suitable factorizations

are available and showing end behavior.








In this topic, students will explore the key

features of rational functions and use those to

graph the function.



SECTION 8: EXPRESSIONS AND EQUATIONS WITH RADICALS AND RATIONAL EXPONENTS


Expressions with Radicals and

Radical Exponents – Part 1

MAFS.912.N-RN.1.1 Explain how the definition of the meaning of rational

exponents follows from extending the properties of integer exponents to

those values, allowing for a notation for radicals in terms of rational

exponents. For example, we define 5

1

3 to be the cube root of 5 because

we want (5

1

3 ) 3 =

5 (

1

3)3, to hold, so

(51

3 ) 3 must equal 5.

MAFS.912.N-RN.1.2 Rewrite expressions involving radicals and rational

exponents using the properties of exponents.

In this topic, students will

understand rational exponents

using the properties of integer

exponents. Students will also

convert between expressions with

radicals and rational exponents.


Expressions with Radicals and

Radical Exponents – Part 2

MAFS.912.N-RN.1.1 Explain how the definition of the meaning of rational

exponents follows from extending the properties of integer exponents to

those values, allowing for a notation for radicals in terms of rational

exponents. For example, we define 5

1

3 to be the cube root of 5 because

we want (5

1

3 ) 3 =

5 (

1

3)3, to hold, so

(51

3 ) 3 must equal 5.

MAFS.912.N-RN.1.2 Rewrite expressions involving radicals and rational

exponents using the properties of exponents.

In this topic, students will

understand rational exponents

using the properties of integer

exponents. Students will also

convert between expressions with

radicals and rational exponents.


Solving Equations with Radicals

and Rational Exponents - Part 1

MAFS.912.A-REI.1.2 Solve simple rational and radical equations in one

variable, and give examples showing how extraneous solutions may

arise.

In this topic, students will write and

solve equations with radicals and

rational exponents. Students will

also understand what extraneous

solutions are.


Solving Equations with Radicals

and Rational Exponents - Part 2

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.

MAFS.912.A-REI.1.2 Solve simple rational and radical equations in one

variable, and give examples showing how extraneous solutions may

arise.

MAFS.912.A-CED.1.4 Rearrange formulas to highlight a quantity of

interest using the same reasoning as in solving equations. For example,

rearrange Ohm’s law, 𝑉 = 𝐼𝑅, to highlight resistance, 𝑅.

In this topic, students will write and

solve equations with radicals and

rational exponents. Students will

also understand what extraneous

solutions are.




Graphing Square Root and Cube

Root Functions- Part One

MAFS.912.F-IF.3.7 b Graph functions expressed symbolically and show

key features of the graph by hand in simple cases and using


b. Graph square root, cube root, and piecewise-defined functions,

including step functions and absolute value functions.

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.

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

quadratic function and an algebraic expression for another, say which


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 n engines in a factory, then the positive integers would be

an appropriate domain for the function.

MAFS.912.F-BF.2.3 Identify the effect on the graph of replacing 𝑓(𝑥) by 𝑓(𝑥) + 𝑘, 𝑘𝑓 (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.

In this topic, students will graph

square root and cube root

functions. Students will use the

graphs to solve real-world

problems. Additionally, students

will apply their knowledge of

transformations of functions.




Graphing Square Root and Cube

Root Functions – Part Two

MAFS.912.F-IF.3.7 b Graph functions expressed symbolically and show

key features of the graph by hand in simple cases and using


b. Graph square root, cube root, and piecewise-defined functions,

including step functions and absolute value functions.

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.

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

quadratic function and an algebraic expression for another, say which


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 n engines in a factory, then the positive integers would be

an appropriate domain for the function.

MAFS.912.F-BF.2.3 Identify the effect on the graph of replacing 𝑓(𝑥) by

𝑓(𝑥) + 𝑘, 𝑘𝑓(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.


square root and cube root

functions. Students will use the

graphs to solve real-world

problems. Additionally, students

will apply their knowledge of

transformations of functions.



SECTION 9: EXPONENTIAL AND LOGARITHMIC FUNCTIONS


Real World Exponential Growth and

Decay - Part 1

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.

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.

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 nonviable options in a modeling

context. For example, represent inequalities describing

nutritional and cost constraints on combinations of different

foods.

MAFS.912.F-IF.3.8.b Write a function defined by an expression in

different but equivalent forms to reveal and explain different

properties of the function.

b. Use the properties of exponents to interpret expressions for

exponential functions.

MAFS.912.F-LE.2.5 Interpret the parameters in a linear or an

exponential function in terms of a context.

In this topic, students will explore and

solve problems involving exponential

growth and decay in the context of real-

world situations.




Real World Exponential Growth and

Decay - Part 2





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.

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 nonviable options in a modeling

context. For example, represent inequalities describing

nutritional and cost constraints on combinations of different

foods.

MAFS.912.F-IF.3.8.b Write a function defined by an expression in





MAFS.912.F-LE.2.5 Interpret the parameters in a linear or an

exponential function in terms of a context.

Students will write an equation in one

variable that represents a real-world

context. Students will write and solve an

equation in one variable that represents

a real-world context. Students will identify

the quantities in a real-world situation

that should be represented by distinct

variables.

In this topic, students will explore and

solve problems involving exponential

growth and decay in the context of real-

world situations.




Interpreting Exponential Equations

MAFS.912.F-IF.3.8b Write a function defined by an expression in









MAFS.912.A-SSE.2.3c Choose and produce an equivalent form of

an expression to reveal and explain properties of the quantity

represented by the expression.

c. Use the properties of exponents to transform expressions for


MAFS.912.A-SSE.1.1b Interpret expressions that represent a

quantity in terms of its context.

b. Interpret complicated expressions by viewing one or more of

their parts as a single entity.

In this topic, students will write

exponential functions in equivalent forms

to make observations about what the

function represents in a real-world

context. Additionally, they will use the

functions to solve problems.


Euler’s Number

In this video, students will investigate how

we derive Euler's Number. Euler's Number

will be used in succeeding videos.




Graphing Exponential Functions


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.

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 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.


exponential functions and find the

solution for a system of exponential

functions.


Transformations of Exponential

Functions

MAFS.912.F-BF.2.3 Identify the effect on the graph of replacing

𝑓(𝑥) by 𝑓(𝑥) + 𝑘, 𝑘𝑓 (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.

In this topic, students will apply their

knowledge of transformations of

functions to exponential functions.


Key Features of Exponential

Functions


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.

Include recognizing even and odd functions from their graphs

and algebraic expressions for them.

In this topic, students will explore the key

features of exponential functions.


Logarithmic Functions - Part 1


a. Solve an equation of the form 𝑓(𝑥) = 𝑐 for a simple function,

𝑓, that has an inverse and write an expression for the inverse.

For example, 𝑓(𝑥) = 2 × 3 or 𝑓(𝑥) = (𝑥 + 1)/(𝑥– 1) for 𝑥 ≠ 1.

In this topic, students will discover that a

logarithmic function is the inverse of an

exponential function.




Logarithmic Functions - Part 2


a. Solve an equation of the form 𝑓(𝑥) = 𝑐 for a simple function,

𝑓, that has an inverse and write an expression for the inverse.

For example, 𝑓(𝑥) = 2 × 3 or 𝑓(𝑥) = (𝑥 + 1)/(𝑥– 1) for 𝑥 ≠ 1.

MAFS.912.F-IF.3.7e. Graph functions expressed symbolically and

show key features of the graph by hand in simple cases and

using technology for more complicated cases.

e. Graph exponential and logarithmic functions, showing

intercepts and end behavior, and trigonometric functions,

showing period, midline, and amplitude and using phase shift.

In this topic, students will continue to

build their understanding of logarithmic

functions, as well as graph the functions.


Common and Natural Logarithms

MAFS.912.F-LE.1.4 For exponential models, express as a logarithm

the solution to 𝑎𝑏𝑐𝑡 = 𝑑, where 𝑎, 𝑐, and 𝑑 are numbers and the

base, 𝑏, is 2, 10, or 𝑒; evaluate the logarithm using technology.

MAFS.912.F-BF.2.a: Use the change of base formula.

In this topic, students will extend their

knowledge of logarithms to bases other

than 10. Students will learn and apply the

Change of Base formula.



SECTION 10: SEQUENCES AND SERIES


Arithmetic Sequences - Part 1

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.

MAFS.912.F-BF.1.1a Write a function that


quantities.

a. Determine an explicit expression, a recursive

process, or steps for calculation from a

context.

In this topic, students will write an explicit and

recursive formula for an arithmetic sequence.

Students will apply the formula to real-world

situations.


Arithmetic Sequences - Part 2





forms.



quantities.



context.


recursive formula for an arithmetic sequence.


situations.


Geometric Sequences - Part 1





forms.



quantities.



context.


recursive formula for a geometric sequence.


situations.




Geometric Sequences - Part 2





forms.



quantities.



context.


recursive formula for a geometric sequence.


situations.


Introduction to Geometric Series – Part 1

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.

In this topic, students will be introduced to the

concept of geometric series.


Introduction to Geometric Series – Part 2





mortgage payments.

In this topic, students will derive the formula for

a sum of a finite geometric series with a

common ratio not equal to 1.


Sum of Geometric Series





mortgage payments.

In this topic, students will apply the formula for

the sum of a finite geometric series.


Calculating Loan Payments





mortgage payments.

In this topic, students will use the formula for

the sum of a finite geometric series to

calculate loan payments.



SECTION 11: PROBABILITY


Sets and Venn Diagrams - Part 1

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”).

In this topic, students will explore and be able

to identify the basic elements of Venn diagrams

including intersection, union, and complement.


Sets and Venn Diagrams - Part 2

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”).

In this topic, students will create and analyze

Venn diagrams using the various components

of intersection, union, and complement.


Probability and the Addition Rule - Part 1

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.

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.

In this topic, students will find probability of one

event taking place and apply the addition rule

to find the probability that one event OR a

separate event will take place.




Probability and the Addition Rule - Part 2

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.















In this topic, students will find probability of one

event taking place and apply the addition rule

to find the probability that one event OR a

separate event will take place.


Probability and Independence

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.

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 probabilities, and use this characterization

to determine if they are independent.

In this topic, students will determine whether or

not two events are dependent or independent,

and use that knowledge to calculate

probabilities of those events.




Conditional 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.

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.








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 probabilities, and use this characterization

to determine if they are independent.

In this topic, students will find the conditional

probability of various real-world situations, as

well as determine and justify independence of

two events,



Section 11 - Topic 7

Two-Way Frequency Tables - Part 1

































In this topic, students will find and interpret

probability from a two-way frequency table.




Two-Way Frequency Tables - Part 2

































In this topic, students will create two-way

frequency tables, as well as find and interpret

probability from the tables they create.



SECTION 12: STATISTICS


Statistics and Parameters

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.

In this topic, students will identify the

population, sample, variable of interest,

parameters, and statistics of interest in various

real-world situations.


Statistical Studies – Part 1





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.

In this topic, students will learn the different

ways to gather data, as well as the 3 principles

of experimental design, and use this

knowledge to identify the best method of data

collection different situations.


Statistical Studies – Part 2

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.





In this topic, students will identify bias in various

sampling techniques, and determine which

sampling techniques work for differing

situations.


The Normal Distribution – Part 1

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.

MAFS.912.S-IC.2.6 Evaluate reports based on

data.

In this topic, students will use the Empirical Rule

to determine the percentage of values

between two data points.












curve.


data.

In this topic, students will calculate and

interpret the z-score in various real-world

situations.










curve.


data.

In this topic, students will find the probability

that an event will occur using the mean and

standard deviation to calculate the z-score.

Students will combine their knowledge of z-

core and the Empirical Rule to interpret data.



SECTION 13: TRIGONOMETRY – PART 1


The Unit Circle - Part 1

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.

In this topic, students will use their

knowledge of special right triangles to

find the angles measure of the angles

formed by rays intersecting the unit circle

and coordinates on the unit circle.


The Unit Circle - Part 2





unit circle.

In this topic, students will use their

knowledge of special right triangles to

find the angles measure of the angles

formed by rays intersecting the unit circle

and coordinates on the unit circle.


Radian Measure - Part 1

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.





unit circle.

In this topic, students will find missing

angles and radian measures on a unit

circle using knowledge of converting

between degrees and radians.


Radian Measure - Part 2








unit circle.

In this topic, students will find missing

angles and radian measures on a unit

circle using knowledge of special right

triangle and reference angles.


More Conversions with Radians




In this topic, students will find equivalent

forms of trigonometric functions by

converting between degrees and

radians.


Arc Measure




In this topic, students will find the length of

the arc on the unit circle subtended by

the angle. Students will also use the arc

length to find the measure of the central

angle, as well as apply their knowledge of

arc length to real-world scenarios.



SECTION 14: TRIGONOMETRY – PART 2


Pythagorean Identity

MAFS.912.F-TF.3.8 Prove the Pythagorean

identity

In this topic, students will prove the Pythagorean

Identitiy, and use it to calculate trigonometric

ratios.


Sine and Cosine Graph – Part 1

MAFS.912.F-TF.2.5 Choose trigonometric

functions to model periodic phenomena with

specified amplitude, frequency, and midline.

In this topic, students will explore periodic

functions and identify the period, amplitude,

and frequency; and use special right triangle

ratios to graph trigonometric functions.


Sine and Cosine Graph – Part 2




In this topic, students will explore periodic

functions and identify the period, amplitude,

frequency and midline, and use special right

triangle ratios to graph trigonometric functions.


Transformations of Trigonometric Functions




In this topic, students will use and apply their

knowledge of transformations to graph various

trigonometric functions. Students will identify key

features of trigonometric functions including

period, amplitude and frequency.


Modeling with Trigonometric Graphs




In this topic, students will use and apply their

knowledge of transformations solve real-world

problems using trigonometric functions. Students

will identify key features of trigonometric

functions including period, amplitude and

frequency.

2016-2017 math nation algebra 2 scope & …polynomials. section 1 - topic 2: mafs.912.a...

Documents