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Canyons School DistrictSecondary II

Scope and Sequence

CANYONS SCHOOL DISTRICTSECONDARY II

SCOPE AND SEQUENCE2013 – 2014

★ – Modeling Standards 1


Scope and Sequence

Secondary Strand IIUnit 1: Extending the Number System

N.RN.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 513 to be the cube root of 5 because we want (5

13)3=5(13 )∙3 to hold, so (5 13)

3must equal 5).

N.RN.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.

SimplifyingRadicals

Write radical expressions in equivalent forms.

N.RN.3 Explain why sums and products of rational numbers are rational, why the sum of a rational number and an irrational number is irrational, and why the product of a nonzero rational number and an irrational number is irrational.

A.SSE.3c: Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expressions. ★c. Use the properties of exponents to transform expressions for exponential functions. (For example the expression 1.15t can be rewritten as (1.151/12)12t – 1.01212t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.)

A.APR.1 Understand that polynomials form a system analogous to the integers—namely, they are closed under the operations of addition subtraction, and multiplication; add, subtract, and multiply polynomials.



Scope and Sequence

Unit 1: Extending the Number System

CCSSNecessary Background

KnowledgeVocabulary “I Can…” Statements

N.RN.1 Properties of integer exponents (8.EE.1)

Familiarity with radical notation (8.EE.2)

Rational exponent Radical Radicand Index nth root

I can define the meaning of a rational exponent.

N.RN.2 Properties of integer exponents (8.EE.1)

Familiarity with radical notation (8.EE.2)

Know the relationship between radical notation and rational exponent notation (N.RN.1)

Rational exponent Radical Radicand Index nth root

I can convert radical notation to rational exponent notation, and vice-versa.

I can extend the properties of integer exponents to rational exponents and use them to simplify expressions.

Simplifying Radicals

Use square root and cube symbols (8.EE.2)

Radical expressions Radical form Index Radicand Root

I can write radical expressions in equivalent forms using radical symbols.

I can perform operations on radical expressions.

N.RN.3 Know that √2 is irrational (8.EE.2)

Understand that rational numbers can be written as a

Rational Irrational

I can simplify radical expressions. I can add, subtract, and multiply real numbers. I can explain why adding and multiplying two

rational numbers results in a rational number. I can explain why adding a rational number to an



Scope and Sequence

termination or repeating decimal and that irrational numbers are non-termination, non-repeating decimals .(8.NS.1)

irrational number results in an irrational number. I can explain why multiplying a nonzero number to

an irrational number results in an irrational number.

A.SEE.3c ★ Understand the distributive property in simplifying and expanding expressions

Various types of factoring skills

Factors Coefficients Terms Exponent Base Constant Variable Binomial Monomial Polynomial

I can, given a context, determine the best form of an expression.

A.APR.1 Understand operations and properties of integers, including closure

Add and subtract like terms

Understand the distributive property

Like terms Binomial Trinomial Polynomial Closure

I can add and subtract polynomials. I can multiply polynomials using the distributive

property, and then simplify. I can understand closure of polynomials for

addition, subtraction, and multiplication.



Scope and Sequence

ACT PLAN College and Career Readiness StandardsUnit 1: Extending the Number System

Numbers: Concepts and Properties Work with squares and square roots of numbers Work problems involving positive integer exponents Work with cubes and cube roots of numbers Apply rules of exponents Draw conclusions based on number concepts, algebraic properties, and/or relationships between expressions and

numbersExpressions, Equations, and Inequalities

Add, subtract, and multiply polynomials



Scope and Sequence

Secondary Strand IIUnit 2: Quadratic Functions and Modeling 1

A.SSE.1b: Interpret expressions that represent a quantity in terms of its context.b. Interpret complicated expressions by viewing one or more of their parts as a single entity. (For example, interpret P(1 + r)n as the product of P and a factor not depending on P.)

F.IF.4: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. (Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries, end behavior; and periodicity.)★

F.IF.5: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. (For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.)★

F.IF.7a,b: Graph function 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.F.IF.9: Compare properties of two functions each represented in a different way (algebraically, graphically,

numerically in tables, or by verbal descriptions). (For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.)

F.IF.6: Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.★

F.LE.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.



Scope and Sequence

Unit 2: Quadratic Functions and Modeling 1



A.SSE.1b Understand the meaning of symbols indicating mathematical operations, implied operations, the meaning of exponents, and grouping symbols

Understand the meaning of a rational exponent (II.N.RN.2)

Factors Coefficients Terms Exponents Base Constant Variable Binomial Monomial Polynomial

I can explain the meaning of the part in relationship to the entire expression and to the context of the problem.

I can understand that the product of two binomials is the sum of monomial terms. For example the product of (3x + 2) and (x – 5) is the sum of 3x2, -13x, and -10.

F.IF.4 ★ Graph linear and exponential functions from a table or equation

Increasing Decreasing Interval Intercept Maximum Minimum Symmetry End behavior Quadratic Vertex

I can distinguish linear, quadratic, and exponential relationships based on equations, tables, and verbal descriptions.

Given a function in a table or in algebraic or graphical form, I can identify key features such as x- and y-intercepts, intervals where the function is increasing, decreasing, positive, or negative, relative maximums and minimums, symmetries and end behavior.

I can use key features of an algebraic function to graph the function.

F.IF.5 ★ Understand the concept of a function and use function notation (I.FIF.2)

Understand domain

Domain Function Independent variable Dependent variable Discrete

I can identify domains of functions given a graph. I can identify a domain in a particular context.



Scope and Sequence

ContinuousF.IF.7a,b ★

Graph linear and exponential functions showing key features (I.FIF.7)

Interpret key features of a graph

Identify and use transformation of functions

Piecewise Step function Axis of symmetry Absolute value

I can graph quadratic functions expressed in various forms by hand.

I can use technology to model quadratic functions, when appropriate.

I can graph and find key features of piecewise-defined functions, including step functions and absolute value functions.

F.IF.9 Find intercepts, rates of change, end behavior, extreme values, and symmetry of quadratic functions

Intercepts Rates of change End behavior Extreme values Symmetry

I can compare intercepts, maxima and minima, rates of change, and end behavior of two quadratic functions, where one is represented algebraically, graphically, numerically in tables, or by verbal descriptions, and the other is modeled using a different representation.

F.IF.6 Calculate and interpret the rate of change of a linear or exponential function

Average rate of change

Interval Δ Secant line

I can calculate the rate of change in a quadratic function over a given interval from a table or equation.

I can compare rates of change in quadratic functions with those in linear or exponential functions.

F.LE.3 Graph quadratic and exponential functions

Exponential Quadratic Rate of change

I can use a table to observe that exponential functions grow more quickly than quadratic functions.

I can use a graph to observe that exponential functions grow more quickly than quadratic functions.



Scope and Sequence

ACT PLAN College and Career Readiness Standards


Graphical Representations Determine the slope of a line from points of equations Interpret and use information from graphs in the coordinate plane Identify characteristics of graphs based on a set of conditions or on a general equation such as y = ax2 + c



Scope and Sequence


F.BF.1 Write a function that describes a relationship between two quantities. ★a. Determine an explicit expression, a recursive process, or steps for calculation from a context.b. Combine standard function types using arithmetic operations. (For example, build a function that

models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these to the model.)

F.BF.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.)

F.BF.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.



Scope and Sequence




F.BF.1 ★ Write a function describing a linear or exponential relationship (I.F.BF.1)

Explicit Expression Function

I can, given a linear, exponential, or quadratic context, find an explicit algebraic expression or series of steps to model the context with mathematical representations.

I can combine linear, exponential, or quadratic functions using addition, subtraction, or multiplication.

F.BF.3 Identify the effect of vertical translations of graphs of linear and exponential functions on their equations (I.F.BF.3)

Graph parent functions for quadratic and absolute value functions (II.F.IF.7)

Even function Odd function Rigid transformation Dilation Symmetry

I can perform transformation on quadratic and absolute value functions with and without technology.

I can describe the effect of each transformation on functions (e.g., If (f(x) is replaced with f(x + k)).

I can, given the graph of a function, describe the transformations using a specific value of k.

I can recognize which transformations take away the even nature of a quadratic or absolute value of a function.

F.BF.4 Definition of a function Domain and range

f−1 (x ) Inverse Restricted domain

I can determine whether or not a function has an inverse, and find the inverse if it exists.

I can understand that creating an inverse of a quadratic function requires a restricted domain.



Scope and Sequence

ACT PLAN College and Career Readiness StandardsUnit 3: Quadratic Functions and Modeling 2

Graphical Representations Math linear graphs with their equations Identify characteristics of graphs based on a set of conditions or on a general equation such as y = ax2 + c

Numbers: Concepts and Properties Exhibit knowledge of logarithms and geometric sequences

Expressions, Equations, and Inequalities Manipulate expressions and equations Write expressions that require planning and/or manipulating to accurately model a situation

Functions Write an expression for the composite of two simple functions



Scope and Sequence


F.IF.8a: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.

A.SSE.1a: Interpret expressions that represent a quantity in terms of its context.a. Interpret parts of an expression, such as terms, factors, and coefficients.

A.SSE.3 a,b:

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.Application Unit

A.CED.1 Create equations and inequalities in one variable and use them to solve problems. (Include equations arising from linear and quadratic function, and simple rational and exponential functions.)

A.CED.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. (For example, rearrange Ohm’s law V = IR to highlight resistance R).

A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.



Scope and Sequence




F.IF.8 a Identify key features of a quadratic function (II.F.IF.7)

Multiply binomials

Binomial Trinomial Perfect square

trinomial Completing the

square Zero Extreme values

(minimum and maximum)

I can factor quadratics and complete the square to find intercepts, extreme values, and symmetry of the graph.

I can transition between different forms of quadratic functions and identify the advantages of each

A.SSE.1 a Understand the meaning of symbols indicating mathematical operations, implied operations, the meaning of exponents, and grouping symbols

Understand the meaning of a rational exponent


I can identify the parts of an expression, such as terms, factors, and coefficients, bases, exponents, and constant.

A.SSE.3 a, b ★

Understand the distributive property in simplifying and expanding expressions


Factors Coefficients Terms Exponent Base Constant

I can rewrite expressions in different forms using mathematical properties.

14


Scope and Sequence

Variable Binomial Monomial Polynomial

Application UnitA.CED.1 Solve linear equations

(I.A.REI.3). Solve exponential

equations that can be solved using laws of exponents (I. A.REI.3)

Solve quadratic equations and inequalities (II.A.REI.4).

Write recursive and explicit equations

Recursive Explicit Rational

I can create one-variable linear, exponential, quadratic, and inequalities from contextual situations (stories).

I can solve and interpret the solution to linear, exponential, quadratic, and inequalities in context.

I can solve compound inequalities. I can use interval notation to represent

inequalities.

A.CED.4 Recognize variables as representing quantities in context

Solve quadratic equations (II.A.REI.4)

Constant Variable Formula Literal equation

I can solve a quadratic formula for a variable of interest.

A.CED.2 Graph a linear equation (I.F.IF.7)

Graph an exponential equation (I.F.IF.7)

Understand the meaning of dependent versus independent variables.

Dependent variable Independent variable Rate of change

I can write and graph an equation to represent a quadratic relationship between two quantities.

I can model a data set using an equation including quadratic relationships.

I can choose appropriate scale for the variables.

15


Scope and Sequence

Understand rate of change

16


Scope and Sequence

ACT PLAN College and Career Readiness StandardsUnit 4: Quadratic Functions and Modeling 3

Expressions, Equations, and Inequalities Multiply two binomials Identify solutions to simple quadratic equations Factor simple quadratics (e.g., the difference of squares and perfect square trinomials) Solve quadratic equations

17


Scope and Sequence

Secondary Strand IIUnit 5: Expressions and Equations 1

A.SSE.2Use the structure of an expression to identify ways to rewrite it. For example, see x

4− y 4as ( x2)2−( y2 )2 , thus

recognizing it as a difference of squares that can be factored as ( x2− y2)( x2+ y2 )

.N.CN.1 Know that there is a complex number i such that i

2=−1 , and every complex number has the form a+bi with a and b real.

N.CN.2 Use the relation i2=−1and the commutative, associative, and distributive properties to add, subtract, and

multiply complex numbers.HONORSN.CN.3

Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.

A.REI.4 Solve quadratic equations in one variablea. Use the method of completing the square to transform any quadratic equation in x into an equation of the

form ( x− p )2=q that has the same solutions. Derive the quadratic formula from this form.

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

quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a±bi for real numbers a and b.

N.CN.7 Solve quadratic equations with real coefficients that have complex solutions.HONORSN.CN.8

Extend polynomial identities to the complex numbers. (For example, rewrite x2+4 as (x+2i)(x-2i).)

HONORSN.CN.9

Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

18


Scope and Sequence

Unit 5: Expressions and Equations 1



A.SSE.2 Understand the distributive property in simplifying and expanding expressions



I can understand that an expression has different forms.

I can justify the different forms based on mathematical properties.

I can interpret different symbolic notation.

N.CN.1 Real number system and its subsets

Real numbers Complex numbers Imaginary numbers i a+bi

I can understand that the set of complex numbers includes the set of all real numbers and the set of imaginary numbers.

I can express numbers in the form a+bi.

N.CN.2 Definition of i Combining like terms

in polynomials (II.1.A.APR.1)

Complex numbers i

I can add, subtract, and multiply complex numbers.

HONORSN.CN.3

Complex numbers Complex plane Rationalizing Denominators

Conjugate Modulus Magnitude Complex plane

I can determine the conjugate of a complex number.

I can define the modulus of a complex number as the positive square root of the sum of the squares of the real and imaginary parts of the complex



Scope and Sequence

i2=−1 number. I can use conjugates to express quotients of

complex numbers in standard form.A.REI.4 Factor

Simplify radicals Understanding of

complex numbers (II.N.CN.1)

Understand the real number and complex number systems (II.N.CN.1)

Radicals Complex numbers Solve Factor Discriminant

I can complete the square. I can solve quadratic equations, including complex

solutions, using completing the square, quadratic formula, factoring, and by taking the square root.

I can derive the quadratic formula from completing the square.

I can recognize when one method is more efficient than the other.

I can interpret the discriminant. I can understand the zero product property and use

it to solve a factorable quadratic equation.N.CN.7 Understand the

meaning of a complex number

Solve a quadratic equation

Complex number Imaginary number Roots Solutions Zeros

I can understand the meaning of a complex number.

I can solve a quadratic equation and understand the nature of the roots

HONORSN.CN.8

Factor quadratics Understand that some

quadratic functions have complex solutions

Know the definition of i

Perform operations on complex numbers

Standard form of a complex number

Conjugates Complex numbers i Factor

I can express a quadratic as a product of two complex factors.



Scope and Sequence

HONORSN.CN.9

Understand number systems

Solve quadratic equations

Know the definition of a complex number (II.N.CN.1)

Know the meaning of algebraically closed (See intro to Extending the Number System)

Fundamental Theorem of Algebra

Solutions Complex Roots Real number system Complex number

system Algebraically closed Multiplicity

I can know that the Fundamental Theorem of Algebra guarantees that any quadratic function will have a solution in the complex number system.



Scope and Sequence

ACT PLAN College and Career Readiness StandardsUnit 5: Expressions and Equations 1

Expressions, Equations, and Inequalities Manipulate expressions and equations Identify solutions to simple quadratic equations Solve quadratic equations Factor simple quadratics (e.g., the difference of squares and perfect square trinomials)

Numbers: Concepts and Properties Exhibit some knowledge of the complex numbers Multiply two complex numbers Apply properties of complex numbers



Scope and Sequence

Secondary Strand IIUnit 6: Expressions and Equations 2

A.CED.1 Create equations and inequalities in one variable and use them to solve problems. (Include equations arising from linear and quadratic function, and simple rational and exponential functions.)

A.CED.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. (For example, rearrange Ohm’s law V = IR to highlight resistance R).

F.IF.8b: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.b. Use the properties of exponents to interpret expressions for exponential functions. (For example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay.)

A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

A.REI.7 Solve a simple system consisting of a linear equations 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).

HONORSA.REI.8

Represent a system of linear equations as a single matrix equation in a vector variable

HONORSA.REI.9

Find the inverse of a matrix if it exists and use it to solve systems of linear equations (use technology for matrices of dimension 3x3 or greater).



Scope and Sequence

Unit 6: Expressions and Equations 2



A.CED.1 Solve linear equations (I.A.REI.3).

Solve exponential equations that can be solved using laws of exponents (I. A.REI.3)

Solve quadratic equations and inequalities (II.A.REI.4).

Write recursive and explicit equations

Recursive Explicit Rational

I can create one-variable linear, exponential, quadratic, and inequalities from contextual situations (stories).

I can solve and interpret the solution to linear, exponential, quadratic, and inequalities in context.

I can solve compound inequalities. I can use interval notation to represent inequalities.

A.CED.4 Recognize variables as representing quantities in context

Solve quadratic equations (II.A.REI.4)

Constant Variable Formula Literal equation

I can solve a quadratic formula for a variable of interest.

F.IF.8b Identify key features of exponential functions

I can use the properties of exponents to interpret expressions for exponential functions.

A.CED.2 Graph a linear equation (I.F.IF.7)

Graph an exponential equation (I.F.IF.7)

Understand the meaning

Dependent variable Independent variable Rate of change

I can write and graph an equation to represent a quadratic relationship between two quantities.

I can model a data set using an equation including quadratic relationships.

I can choose appropriate scale for the variables.



Scope and Sequence

of dependent versus independent variables.

Understand rate of changeA.REI.7 Know that a quadratic

function is a vertical parabola and a quadratic equation can be a parabola of any conic section

Understand what a system is and the nature of the solutions

Solve systems (I.A.REI.6)

Quadratic function Unit circle System of equations

I can solve a simple system consisting of a linear equation and a quadratic equation (i.e., parabolas and circles) in two variables graphically.

I can solve a simple system consisting of a linear equation and a quadratic equation (i.e., parabolas and circles) in two variables algebraically.

I can recognize that the solutions of a system that includes a unit circle centered at the origin and a line with a y-intercept of 0 are points on a unit circle.

HONORSA.REI.8

Know what is meant by an element of the matrix

Order of the matrix

Matrix Equation Vector

I can rewrite a system of linear equations in matrix form as AX=B, where X is the vector of variables.

I can solve a system of linear equations using matrices.

HONORSA.REI.9

Represent systems of equations as matrices

Find the determinant of a matrix

Inverse of a matrix Identity matrix Invertible Nonsingular Determinant A-1

I Augmented Matrix

I can use the determinant to determine whether an inverse exists.

For 2 x 2 matrices, apply the following to find the inverse: For

A=[a bc d ] , A−1= 1

det( A ) [ d −b−c a ]= 1

ad−bc [ d −b−c a ] .

I can apply the augmented matrix method, by hand for 2 x 2 matrices and using technology for 3 x 3 or greater, to find A-1.

I can use matrix algebra to solve AX = B as the unique solution X = A-1B.



Scope and Sequence



Scope and Sequence

ACT PLAN College and Career Readiness StandardsUnit 6: Expressions and Equations 2

Expressions, Equations, and Inequalities Solve linear inequalities that require reversing the inequality sign Write expressions, equations, and inequalities for common algebra settings Find solutions to systems of linear equations



Scope and Sequence

Secondary Strand II

Unit 7: Probability

S.CP.1 Describe events as the 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”).

S.CP.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.

S.CP.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.

S.CP.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 of other subjects and compare the results.)

S.CP.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.)

S.CP.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.

S.CP.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.HONORS S.CP.8

Apply the general Multiplication Rule in a uniform probability model, x and interpret the answer in terms of the model.

HONORS S.CP.9

Use permutations and combinations to compute probabilities of compound events and solve problems.

HONORS S.MD.6

Use probability to make fair decisions (e.g., drawing by lots, using a random number generator).



Scope and Sequence

HONORS S.MD.7

Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).



Scope and Sequence

Unit 7: Applications of Probability



S.CP.1 Represent sample spaces (7.SP.8)

Sample space Subset Outcome Union Intersection Complement

I can use correct set notation, with appropriate symbols and words, to identify sets and subsets within a sample space.

I can identify an event as a subset of a set of outcomes (a sample space).

I can draw Venn Diagrams that show relationships (unions, intersections, or complements) between sets within a sample space.

S.CP.2 Understand basic properties of probability (7.SP.5)

Approximate probabilities of chance events through experiment (7.SP.6)

Use Venn diagrams (II.S.CP.1) and two-way frequency tables (I.S.ID.5)

P( A∩B ) is the equivalent of the probability of event A and event B occurring together (II.S.CP.1)

Joint probability Intersection Event Independent events P(A)

P( A∩B ) P(A and B)

I can use appropriate probability notation for individual events as well as their intersection (joint probability).

I can calculate probabilities for events, including joint probabilities, using various methods (e.g. Venn diagram, frequency table).

I can compare the product of probabilities for

individual events (P( A )⋅P(B )) with their joint

probability (P( A∩B )). I can understand that independent events satisfy

the relationship (P( A )⋅P(B )=P( A∩B)).

S.CP.3 Use basic probability notation, particularly

Conditional Independence Conditional

I can understand conditional probability and how it applies to real-life events.



Scope and Sequence

P( A∩B ) (II.S.CP.2)

Understand independent events (11.S.CP.2)

probability

P( A|B ) I can use P( A|B )=

P( A∩B )P(B) to calculate conditional

probabilities. I can understand that events A and B are

independent if and only if they satisfy P( A|B )=P (A )or satisfyP(B|A )=P (B ) .

I can apply the definition of independence to a variety of chance events.

S.CP.4 Summarize categorical data in a variety of ways (I.S.ID.5)

Understand what it means for two events to be independent (II.S.CP.2)

Conditional Independence Joint probability Conditional

probability (P( A|B )) Marginal probability

I can model real-life data using two-way frequency tables.

I can recognize that the conditional probability, P(A|B), represents the joint probability for A and B divided by the marginal probability of B.

I can use P( A|B )=

P( A∩B )P(B) to calculate conditional

probabilities from a two-way frequency table. I can apply the definition of independence to a

variety of chance events as represented by a two-way frequency table.

S.CP.5 Summarize categorical data in a variety of ways (I.S.CP.3)

Understand what it means for two events to be independent (II.S.CP.2)

Find probabilities of events using tree diagrams (7.SP.8)

Understand and calculate conditional probabilities

Conditional probability

Independence

I can interpret conditional probabilities and independence in context.



Scope and Sequence

(II.S.CP.3)S.CP.6 Find probabilities of

compound events (7.SP.8)

Summarize categorical data in two-way frequency tables (I.S.ID.5)

Random variable Probability model

I can find and interpret conditional probabilities using a two-way table, Venn diagram, or tree diagram.

I can understand the difference between compound and conditional probabilities.

S.CP.7 Find the probabilities of compound events (7.SP.8)

Or And P(A) ¿

¿

I can define the probability of event (A or B) as the probability of their union.

Understand and use the formula P(A or B)=P(A)+ P(B) -P(A and B)

HONORSS.CP.8

Probabilities of compound events and tree diagrams (7.SP.8)

Sample space, sets, subsets, outcomes, events, union, intersection, “and”, “or” (II.S.CP.1)

Two-way tables (II.S.CP.3)

Conditional Probability (II.S.CP.3)

Uniform probability model

Multiplication rule P(A and B) = P(A)P(B

given A) = P(B)P(A|B) P(A ∩ B) = P(A)P(B|A)

= P(B)P(A|B)

I can define the probability of event (A and B) as the probability of the intersection of events A and B.

I can understand P(B|A) to mean the probability of event B occurring when A has already occurred.

I can use the Multiplication rule, P(A and B) = P(A)P(B|A) = P(B)P(A|B), to determine P(A and B).

I can determine the probability of dependent and independent events in real contexts.

HONORSS.CP.9

Probabilities of compound events (7.SP.8)

Conditional probability (II.S.CP.3)

Multiplication rule of probability (II.S.SP.8(+))

Factorial Permutation Combination P(n,r) nPr

C(n,r) nCr

I can define n! as the product n (n – 1) … 3 2 ⋅ ⋅ ⋅ ⋅ ⋅1

I can understand that a permutation is a rearrangement of the elements of an ordered list and calculate probabilities using the permutation formula P(n,r) = n (n – 1) … (n – (r – 1)) = n!/(n – r).⋅ ⋅ ⋅



Scope and Sequence

(nr )

I can understand that a combination is the number of ways to choose r items from a set of n elements and calculate probabilities using the combination formula C(n,r) = P(n,r)/r! = n!/[(n – r)! r!].

HONORSS.MD.6

Understand probabilities as chance events (7.SP.5)

Approximate probabilities using experiments (7.SP.6)

Random Random number

tables Random number

generator Fair decision

I can simulate random outcomes using various tools.

I can analyze the fairness of games by determining the probabilities of the possible outcomes.

HONORSS.MD.7

Understand probability as a number representing the likelihood of a chance event (7.SP.5)

Approximate probabilities using experiments (7.SP.6)

Summarize, represent and interpret data on a single count or measurement variable (I.S.ID)

Model random processes using probability (I.S.MD.5)

Random Variability Modeling Sample

I can recognize that data based on random processes are subject to variability.

I can analyze experimental designs and sampling strategies.

I can use the results of experiments and data samples to evaluate decisions.

I can recognize the limitations of decisions drawn from sample data, based on how the data were produced.

ACT PLAN College and Career Readiness StandardsUnit 7: Applications of Probability

Probability, Statistics, and Data Analysis Manipulate data from tables and graphs



Scope and Sequence

Analyze and draw conclusions based on information from figures, tables, and graphs Exhibit knowledge of conditional and joint probability Exhibit knowledge of simple counting techniques



Scope and Sequence

Secondary Strand IIUnit 8: Similarity, Right Triangle Trigonometry, and Proof 1

G.SRT.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.

G.SRT.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain, using similarity transformations, the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

G.SRT.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

G.CO.9 Prove theorems about lines and angles. (Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.)

G.CO.10

Prove theorems about triangles. (Theorems include: measures of interior angles of a triangle sum to 180°, base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.)

G.SRT.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.)

G.CO.11

Prove theorems about parallelograms. (Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.)



Scope and Sequence

Unit 8: Similarity, Right Triangle Trigonometry, and Proof 1



G.SRT.1 Describe the effect of dilations on two-dimensional figures using coordinates (8.G.3)

Reason using proportions

Dilation Center of dilation Scale factor Similarity Transformation

Given a line segment, a point not on the line segment, and a dilation factor, I can construct a dilation of the original segment.

I can recognize that the length of the resulting image is the length of the original segment multiplied by the scale factor and that the original and dilated images are parallel to each other.

G.SRT.2 Understand similarity as a sequence of transformations (8.G.4)

Similarity Transformation Corresponding parts

I can decide whether two figures are similar using properties of transformations.

I can understand that in similar triangles, corresponding sides are proportional and corresponding angles are congruent.

G.SRT.3 The sum of the measures of the angles in a triangle is 180 degrees (8.G.5)

If two angles of a triangle are congruent to two corresponding angles of a second triangle, then the third pair of corresponding angles must be congruent

Similarity Transformation AA

I can prove that if two angles of one triangle are congruent to two angles of another triangle, the triangles are similar (AA) using the properties of similarity transformations.

G.SRT.5 Identify corresponding parts of triangles

Congruence Similarity Congruent triangles Similar triangles

I can find lengths of measures of sides and angles of congruent and similar triangles.

I can solve problems in context involving sides or angles of congruent of similar triangles.



Scope and Sequence

Corresponding angles Corresponding sides

I can prove conjectures about congruence or similarity in geometric figures using congruence and similarity criteria.

G.CO.9 Know properties of supplementary, complementary, vertical, and adjacent angles (7.G.5)

Proof Vertical angles Parallel lines Transversal Alternate interior

angles Corresponding angles Perpendicular

bisector Equidistant

I can prove and use theorems about lines and angles, including but not limited to:• Vertical angles are congruent.• When parallel lines are cut by a transversal, congruent angle pairs are created.• When parallel lines are cut by a transversal, supplementary angle pairs are created.• Points on the perpendicular bisector of a line segment are equidistant from the segment’s endpoints.

G.CO.10 Prove theorems about lines and angles (II.G.CO.9)

Interior/exterior angles of a triangle

Supplementary angles Linear pairs Isosceles Base Legs Base angles Vertex angles Midpoint Median of a triangle Auxiliary line

I can prove and use theorems about triangles including, but not limited to:• Prove that the sum of the interior angles of a triangles = 180• Prove that the base angles of an isosceles triangle are congruent. Prove that if two angles of a triangle are congruent, the triangle is isosceles.• Prove the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length.• Prove the medians of a triangle meet at a point.

G.SRT.4 Understand the AA criterion for similar triangles (II.G.SRT.3)

Parallel lines Pythagorean Theorem Similarity Similar triangles

I can prove that a line constructed parallel to one side of a triangle intersecting the other two sides of the triangle divides the intersected sides proportionally.

I can prove that a line that divides two sides of a



Scope and Sequence

triangle proportionally is parallel to the third side. I can prove that if three sides of one triangle are

proportional to the corresponding sides of another triangle, the triangles are similar.

I can prove the Pythagorean Theorem using similarity.

G.CO.11 Know the definition and properties of parallelograms

Parallelogram Diagonal Consecutive angles Opposite angles Bisect

I can prove and use theorems about parallelograms including, but not limited to:• Opposite sides of a parallelogram are congruent.• Opposite angles of a parallelogram are congruent.• The diagonals of a parallelogram bisect each other.• Rectangles are parallelograms with congruent diagonals.



Scope and Sequence

ACT PLAN College and Career Readiness StandardsUnit 8: Similarity, Right Triangle Trigonometry, and Proof 1

Properties of Plane Figures Exhibit knowledge of basic angle properties and special sums of angle measures (e.g. 90°, 180°, 360°) Use properties of isosceles triangles Apply properties of 30°-60°-90°, 45°-45°-90°, similar, and congruent triangles Solve multistep geometry problems that involve integrating concepts, planning, visualization, and/or making connections

with other content areas Use relationships among angles, arcs, and distances in a circle

Measurement Use relationships involving area, perimeter, and volume of geometric figures to compute another measure Use scale factors to determine the magnitude of a size change



Scope and Sequence

Secondary Strand IIUnit 9: Similarity, and Right Triangle Trigonometry 2

G.SRT.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.

G.SRT.7: Explain and use the relationship between the sine and cosine of complementary angles.

G.SRT.8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.F.TF.8:

Prove the Pythagorean identity sin

2

( ) + cosθ2

( ) = 1 and use it to find sin ( ), cos ( ), or tan ( ), givenθ θ θ θ sin ( ), cos ( ), or tan ( ), and the quadrant of the angle.θ θ θ

HONORS N.CN.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.

HONORS N.CN.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 + √3i)3

= 8 because (-1 + √3i) has modulus 2 and argument 120°.)

HONORS N.CN.6:

Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.

HONORS Define trigonometric ratios and write trigonometric expressions in equivalent forms.HONORS Prove trigonometric identities using definitions, the Pythagorean Theorem, or other relationships and use the

relationships to solve problems.HONORS F.TF.9:

Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.



Scope and Sequence

Unit 9: Similarity, and Right Triangle Trigonometry 2



G.SRT.6 Understand that corresponding angles of similar triangles are congruent and ratios of corresponding sides are equal

Similar triangles Ratio Right triangle

I can understand that the ratio of two sides in one triangle is equal to the ratio of the corresponding two sides of all other similar triangles.

I can define sine, cosine, and tangent as the ratio of sides in a right triangle.

G.SRT.7 Understand that the acute angles of a right triangle are complementary

Know the right triangle definitions of sine and cosine (II.G.SRT.6)

Complementary angles

Sine Cosine.

I can demonstrate the relationship between sine and cosine in the acute angles of a right triangle.

I can explain the relationship between the sine and cosine in complementary angles.

G.SRT.8 Apply the Pythagorean Theorem in real-world and mathematical problems in two and three dimensions (8.G.7)

Apply right triangle trigonometric ratios to solve right triangles (II.G.SRT.6)

Pythagorean Theorem Sine Cosine Tangent Angle of elevation Angle of depression

I can use the Pythagorean Theorem and trigonometric ratios to find missing measures in triangles in contextual situations.

F.TF.8 Apply the Pythagorean Theorem to determine unknown side lengths (8.G.7)

Define trigonometric

Sine Cosine Tangent

I can prove sin2

( ) + cosθ 2( ) = 1θ for right triangles

in the first quadrant. I can, if given sin ( ), cos ( ),θ θ or tan ( )θ for 0< <90θ ,

find sin ( ), cos ( ),θ θ or tan ( ).θ



Scope and Sequence

ratios (II.G.SRT.6)HONORS N.CN.4

Complex numbers Graphing polar

coordinates Trigonometric identities

on the unit circle Modulus

Complex plane Rectangular form Polar form Modulus Argument

I can convert between the rectangular form, z = x + yi , and polar form, z = r(cos +i sin )θ θ , of a complex number.

I can graph complex numbers on a complex plane in both rectangular and polar form.

I can justify rectangular and polar forms of a complex number as representing the same number.

HONORSN.CN.5

Complex numbers Complex plane

Complex plane I can represent geometrically the sum, difference, product, and conjugation of complex numbers on the complex plane.

I can show that the conjugate of a complex number in the complex plane is the reflection across the x-axis.

I can evaluate the power of a complex number, in rectangular form, using the polar form of the complex number.

HONORSN.CN.6

Distance formula Midpoint formula Modulus Complex plane

Complex plane Modulus

I can show that the distance between two complex numbers is equivalent to the modulus of the difference by applying the distance formula.

I can find the midpoint of a segment between two complex numbers by taking the average of the numbers at its endpoints using the midpoint formula.

HONORS Sine, cosine, tangent Pythagorean Theorem

Sine Cosine Tangent Secant Cosecant Cotangent

I can show how sine, cosine, and tangent are related using trigonometric identities.

I can define secant, cosecant, and cotangent in terms of sine, cosine and tangent.

I can define the six trigonometric functions using the unit circle.



Scope and Sequence

Unit circleHONORS Pythagorean Theorem

Trigonometric definitions Identity Pythagorean Theorem

I can prove trigonometric identities based on the Pythagorean Theorem.

I can simplify trigonometric expressions and solve trigonometric equations using identities.

I can justify half angle and double angle formulas for trigonometric values.

HONORSF.TF.9

Sine, cosine, and tangent Special right triangles

Sine Cosine Tangent

I can prove the addition and subtraction formulas for sine, cosine, and tangent using trigonometric identities.

I can solve problems using addition and subtraction of trigonometric functions.



Scope and Sequence

ACT PLAN College and Career Readiness StandardsUnit 9: Similarity, and Right Triangle Trigonometry 2

Numbers: Concepts and Properties Exhibit some knowledge of the complex numbers Apply properties of complex numbers

Properties of Plan Figures Use the Pythagorean Theorem

Functions Apply basic trigonometric ratios to solve right-triangle problems Use trigonometric concepts and basic identities to solve problems Exhibit knowledge of unit circle trigonometry



Scope and Sequence

Secondary Strand IIUnit 10: Circles with Coordinates and without Coordinates

G.C.1 Prove that all circles are similar.G.C.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.)

G.C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

HONORSG.C.4

Construct a tangent line from a point outside a given circle to the circle.

G.C.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.

G.GPE.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.

G.GPE.2 Derive the equation of a parabola given a focus and directrix.



Scope and Sequence

Unit 10: Circles with Coordinates and without Coordinates



G.C.1 The length of a dilated line segment is equal to the length of the original segment multiplied by the scale factor (II.G.SRT.1)

Circle Radius Dilation

I can define a circle as the set of points equidistant to a given center point.

I can prove that all circles are similar.

G.C.2 Understand that all circles are similar (II.G.C.1)

Define circle, radius, and diameter (7.G.4)

Inscribed angle Central angle Circumscribed angle Radius Chord Diameter Perpendicular Tangent line

I can use circle relationships to find the measures of central, inscribed, and circumscribed angles of a circle.

I can use circle relationships to show that the measure of the inscribed angle on a diameter is a right angle.

I can use circle relationships to show that the radius of a circle is perpendicular to a tangent line where the radius intersects the circle.

G.C.3 Use a variety of construction methods

Know the relationship between an inscribed angle and its intercepted arc (II.G.C.2)

Inscribed Circumscribed Angle Quadrilateral

I can inscribe a circle in a triangle. I can circumscribe a circle about a triangle. I can prove that opposite angles in a quadrilateral

inscribed in a circle are supplementary.

HONORS G.C.4

Use a variety of construction techniques

Tangent Circle

I can construct a line from a point tangent to a point on the circle.

G.C.5 Understand similarity (II.GRT)

Calculate circumference

Sector Arc length Constant of

I can use the concept of similarity to understand that arc length intercepted by a central angle is proportional to the radius.

46


Scope and Sequence

and area of a circle proportionality Radian Circumference Area

I can develop the definition of radians as a unit of measure by relating to arc length.

I can discover that the measure of the central angle in radians is the constant of proportionality.

I can derive the formula for the area of a sector.G.GPE.1 Use the Pythagorean

Theorem to find the distance between two points

Use the method of completing the square to transform equations into desired forms (II.REI.4)

Circle Center of a circle Radius of a circle Completing the

square

I can use the Pythagorean Theorem to find the distance between two points.

I can find the center of a circle, given its equation.

G.GPE.2 Find the distance between two points

Find the midpoint of a segment

Focus Directrix Midpoint

I can develop the geometric definition of a parabola, including a focus and directrix.

I can use the distance formula to derive the equation of a parabola.

47


Scope and Sequence

ACT PLAN College and Career Readiness StandardsUnit 10: Circle with Coordinates and without Coordinates

Graphical Representations Recognize special characteristics of parabolas and circles (e.g., the vertex of a parabola and the center or radius of a circle)

Properties of Plane Figures Use relationships among angles, arcs, and distances in a circle

48


Scope and Sequence

Secondary Strand IIUnit 11: Volume and Coordinate Proofs

G.GMD.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.)

HONORS G.GMD.2

Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.

G.GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. ★

G.GPE.6 Find the point on a directed line segment between two given points that partitions that segment in a given ratio.

G.GPE.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).)

49


Scope and Sequence

Unit 11: Volume and Coordinate proofs



G.GMD.1 Use mathematical language and a logical progression of ideas to present an argument

Cylinder Right prism Pyramid Cone Dissection argument Limit argument Cavalieri’s principle

I can develop the formulas for the circumference of a circle, area of a circle, and volume of a cylinder, pyramid and cone using a variety of arguments.

G.GMD.3 ★ Know the formulas for the volumes of cones, cylinders, and spheres (Math 8.G.9)

Pyramid Cylinder Cone Sphere Volume Length Width Height Base Radius π

I can find the volume of cylinders, cones, and spheres in contextual problems.

HONORSG.GMD.2

Know the formulas for the volumes of cones, cylinders, and spheres (Math 8.G.9)

Cavalieri’s principle Cross-section Altitude Parallel Sphere Cone Cylinder

I can show understanding of Cavalieri’s Principle. I can use Cavalieri’s Principle to find volumes of

solid figures.

50


Scope and Sequence

G.GPE.6 Know how to find a ratio between two values and how to solve for an unknown value given a ratio or proportion (6.RP)

Directed line segment a:b ratio Coordinates

I can use coordinate geometry to divide a segment into a given ratio.

G.GPE.4 Calculate slopes of lines and the distances between points using the distance formula

Know the relationships between parallel and perpendicular lines and the basic properties of polygons and circles

Radius Center Diameter Inscribed Altitude Diagonal Perpendicular Bisector Median Parallel Midpoint Pythagorean Theorem

I can use coordinates to prove simple geometric theorems algebraically.

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Scope and Sequence

ACT PLAN College and Career Readiness StandardsUnit 11: Volume and Coordinate Proofs

Measurement Use relationships involving area, perimeter, and volume of geometric figures to compute another measure Use scale factors to determine the magnitude of a size change

Graphical Representations Solve problems integrating multiple algebraic and/or geometric concepts

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