C&I

Curriculum and Instruction

Mathematics

Crosswalk: Comparison of the 1999 Core Curriculum and 2005 Core Curriculum for High School Mathematics
September 2005

Algebra 2 and Trigonometry is the capstone course of the three units of credit required for a Regents diploma.  This course is a continuation and extension of the two courses that preceded it.  While developing the algebraic techniques that will be required of those students that continue their study of mathematics, this course is also intended to continue developing alternative solution strategies and algorithms.  For example, technology can provide to many students the means to address a problem situation to which they might not otherwise have access. Within this course, the number system will be extended to include imaginary and complex numbers.  The families of functions to be studied will include polynomial, absolute value, radical, trigonometric, exponential, and logarithmic functions.  Problem situations involving direct and indirect variation will be solved. Problems resulting in systems of equations will be solved graphically and algebraically. Algebraic techniques will be developed to facilitate rewriting mathematical expressions into multiple equivalent forms.  Data analysis will be extended to include measures of dispersion and the analysis of regression that model functions studied throughout this course.  Associated correlation coefficients will be determined, using technology tools and interpreted as a measure of strength of the relationship.  Arithmetic and geometric sequences will be expressed in multiple forms, and arithmetic and geometric series will be evaluated.  Binomial experiments will provide the basis for the study of probability theory and the normal probability distribution will be analyzed and used as an approximation for these binomial experiments.  Right triangle trigonometry will be expanded to include the investigation of circular functions.  Problem situations requiring the use of trigonometric equations and identities will also be investigated.

The following chart lists the concepts and skills in Geometry (2005 Core) and indicates where it was included in the 1999 Core.

Number Sense and Operations Strand / Algebra / Measurement / Statistics and Probability

Number Sense and Operations Strand

2005 Core Curriculum

1999 Core Curriculum

Performance
Indicator

Concept/Skill

Key Idea

Concept/Skill

A2.N.1

Evaluate numerical expressions with negative and/or fractional exponents, without the aid of a calculator (when the answers are rational numbers)

Math B – 3D

Evaluate expressions with fractional exponents

A2.N.2

Perform arithmetic operations with expressions containing irrational numbers in radical form

Math A – 3A

Operate with radicals:  simplification, multiplication and division, and addition and subtraction

A2.N.3

Perform arithmetic operations with polynomial expressions containing rational coefficients

Math A – 3A

Addition and subtraction of polynomials: combining like terms

Multiplication of polynomials

Division of polynomials by monomials

A2.N.4

Perform arithmetic operations on irrational expressions

Math A – 3A

Operations with radicals

A2.N.5

Rationalize a denominator containing a radical expression

Math B – 2A

Rationalize denominators

 

2005 Core Curriculum

1999 Core Curriculum

Performance

Indicator

Concept/Skill

Key Idea

Concept/Skill

A2.N.6

Write square root of negative numbers in terms of i

Math B – 3D

Simplify square toots with negative radicands

A2.N.7

Simplify powers of i

Math B – 3D

Cyclic nature of the powers of i

A2.N.8

Determine the conjugate of a complex number

 

Implied by not directly stated

A2.N.9

Perform arithmetic operations on complex numbers

Math B – 3D

Basic arithmetic operations with complex numbers

A2.N.10

Know and apply sigma notation

Math B- 6F

Use of ∑-notation

 

Algebra Strand

2005 Core Curriculum

1999 Core Curriculum

Performance

Indicator

Concept/Skill

Key Idea

Concept/Skill

A2.A.1

Solve absolute value equations and inequalities involving linear expressions in one variable

Math B- 7K

Solve equations using absolute values

  • Absolute value inequalities

A2.A.2

Use the discriminate to determine the nature of the roots of a quadratic equation

Math B – 7I

Determine from the discriminant of a quadratic equation whether the roots are imaginary, rational, or irrational

A2.A.3

Solve systems of equations involving one linear equation and one quadratic equation algebraically Note:  This includes rational equations that result in linear equations with extraneous roots.

Math A – 7A

 

 

 

Math A – 7E

Algebraic solution of systems of linear equations, inequalities, and quadratic-linear pair by substitution method and addition-subtraction method

Solve systems of linear equations, inequalities, and quadratic-linear pair

A2.A.4

Solve quadratic inequalities in one and two variables, algebraically and graphically

Math B – 7K

Solve equations, using fractions, absolute values, and radicals

  • Linear inequalities
  • Quadratic inequalities

A2.A.5

Use direct and inverse variation to solve for unknown values

 

Not addressed

A2.A.6

Solve an application which results in an exponential function

Math B – 7E

Apply exponential functions in the solution of problems

Solve real-world problems by using linear, exponential, and quadratic functions

A2.A.7

Factor polynomial expressions completely

 

Not specifically addressed

 

2005 Core Curriculum

1999 Core Curriculum

Performance

Indicator

Concept/Skill

Key Idea

Concept/Skill

A2.A.8

Apply the rules of exponents to simplify expressions involving negative and/or fractional exponents

Math B – 4B

Manipulate symbolic representations to explore concepts at an abstract level

  • Use positive, negative, and zero exponents and be familiar with the laws used in working with expressions containing exponents

A2.A.9

Rewrite algebraic expressions that contain negative exponents using only positive exponents

Math B – 4B

Manipulate symbolic representations to explore concepts at an abstract level

  • Use positive, negative, and zero exponents and be familiar with the laws used in working with expressions containing exponents

A2.A.10

Rewrite algebraic expressions with fractional exponents as radical expressions

 

Not specifically addressed

A2.A.11

Rewrite algebraic expressions in radical form as expressions with fractional exponents

 

Not specifically addressed

 

A2.A.12

Evaluate exponential expressions, including those with base e

 

Not addressed

A2.A.13

Simplify radical expressions

Math A – 3A

Simplification of fractions

A2.A.14

Perform addition, subtraction, multiplication, and division of radical expressions

Math A – 3A

 

Operations with radicals:  multiplication and division and addition and subtraction

A2.A.15

Rationalize denominators involving algebraic radical expressions

Math B – 2A

Rationalize denominators

 

2005 Core Curriculum

1999 Core Curriculum

Performance

Indicator

Concept/Skill

Key Idea

Concept/Skill

A2.A.16

Perform arithmetic operations with rational expressions and rename to lowest terms

Math B – 3A

Operations with fractions with polynomial denominators

Add and subtract rational fractions with monomial and binomial denominators

A2.A.17

Simplify complex fractional expressions

Math A – 3A

Simplification of fractions – complex fractions not specifically named

A2.A.18

Evaluate logarithmic expressions in any base

Math B – 4B

Manipulate symbolic representations to explore concepts at an abstract level

  • Rewrite the equality logba = c as a = bc
  • Rewrite expressions involving exponents and logarithms

A2.A.19

Apply the properties of logarithms to rewrite logarithmic expressions in equivalent forms

Math B – 4B

 

 

 

 

Math B – 7A

Manipulate symbolic representations to explore concepts at an abstract level

  • Rewrite the equality logba = c as a = bc
  • Rewrite expressions involving exponents and logarithms

Express exponential functions as logs

A2.A.20

Determine the sum and product of the roots of a quadratic equation by examining its coefficients

 

Not addressed

A2.A.21

Determine the quadratic equation, given the sum and product of its roots

 

Not addressed

A2.A.22

Solve radical equations

Math B – 7K

Solve equations using radicals

  • Equations with radicals

A2.A.23

Solve rational equations and inequalities

Math B – 7K

Solve fractional equations – no mention of inequalities

 

2005 Core Curriculum

1999 Core Curriculum

Performance

Indicator

Concept/Skill

Key Idea

Concept/Skill

A2.A.24

Know and apply the technique of completing the square

 

Not addressed

A2.A.25

Solve quadratic equations, using the quadratic formula

 

Not addressed

A2.A.26

Find the solution to polynomial equations of higher degree that can be solved using factoring and/or the quadratic formula

 

Not addressed

A2.A.27

Solve  exponential equations with and without common bases

 

Not addressed

A2.A.28

Solve a logarithmic equation by rewriting as and exponential equation

 

Not addressed

A2.A.29

 

Identify an arithmetic or geometric sequence and find the formula for its nth term

Related to Math B – 4A

Represent problem situations symbolically by using sequences

  • Use symbolic form to represent an explicit rule for a sequence

A2.A.30

Determine the common difference in an arithmetic sequence

Related to Math B – 4A

Represent problem situations symbolically by using sequences

  • Use symbolic form to represent an explicit rule for a sequence

A2.A.31

Determine the common ration in a geometric sequence

Related to Math B – 4A

Represent problem situations symbolically by using sequences

  • Use symbolic form to represent an explicit rule for a sequence

 

2005 Core Curriculum

1999 Core Curriculum

Performance

Indicator

Concept/Skill

Key Idea

Concept/Skill

A2.A.32

Determine a specified term of an arithmetic or geometric sequence

Related to Math B – 4A

Represent problem situations symbolically by using sequences

  • Use symbolic form to represent an explicit rule for a sequence

A2.A.33

Specify terms of a sequence, given its recursive definition

Related to Math B – 4A

Represent problem situations symbolically by using sequences

  • Use symbolic form to represent an explicit rule for a sequence

A2.A.34

Represent the sum of a series, using sigma notation

Related to Math B – 4A

Represent problem situations symbolically by using sequences

  • Use symbolic form to represent an explicit rule for a sequence

A2.A.35

Determine the sum of the first n terms of an arithmetic or geometric series

Related to Math B – 4A

Represent problem situations symbolically by using sequences

  • Use symbolic form to represent an explicit rule for a sequence

A2.A.36

Apply the binomial theorem to expand a binomial and determine a specific term of a binomial expansion

Math B – 6F

Binomial Theorem

A2.A.37

Define a relation and function

Math B – 7A

Definition of a relation

A2.A.38

Determine when a relation is a function

Math B – 7A

Determining if a relation is a function

A2.A.39

Determine the domain and rage of a function from its equation

 

Not directly addressed

A2.A.40

Write functions in functional notation

 

Not directly addressed

 A2.A.41

Use functional notation to evaluate functions for given values in the domain

Math B – 7A

Notation for absolute value, composite functions

 

2005 Core Curriculum

1999 Core Curriculum

Performance

Indicator

Concept/Skill

Key Idea

Concept/Skill

A2.A.42

Find the composition of functions

Math B – 3B

 

 

Math B – 3E

 

Math B – 7J

3Develop an understanding of and use the composition of functions and transformations

Evaluate and form the composition of functions

Determine the value of compound functions

A2.A.43

Determine if a function is one-to-one, onto, or both

 

Not addressed

A2.A.44

Define the inverse of a function

Math B – 7A

Definition of inverse function

A2.A.45

Determine the inverse of a function and use composition to justify the result

 

Not directly addressed

A2.A.46

Perform transformations with functions and relations: f(x +a) , f(x) +a, f(-x), -f(x), af(x)

Math B – 7F

Apply and interpret transformations to functions

  • Use ideas of transformations to investigate the relationships between functions

A2.A.47

Determine the center-radius form for the equation of a circle in standard form

Math B – 4L

Use algebraic relationships to analyze the conic sections

A2.A.48

Write the equation of a circle, given its center and a point on the circle

Math B- 4L

Use algebraic relationships to analyze the conic sections – not using a point on the circle

A2.A.49

Write the equation of a circle from its graph

 

Not directly addressed

A2.A.50

Approximate the solution to polynomial equations of higher degree by inspecting the graph

 

Note addressed

A2.A.51

Determine the domain and range of a function from its graph

 

Not directly addressed

A2.A.52

Identify relations and functions, using graphs

 

Implied in Math B –7A

Use function vocabulary and notation

Determining if a relation is a function

 

2005 Core Curriculum

1999 Core Curriculum

Performance

Indicator

Concept/Skill

Key Idea

Concept/Skill

A2.A.53

Graph exponential functions of the form y = bx for positive values of b, including b= e

Math B – 4J

Be able to sketch the effects of changing the value of a in the function y = ax

B = e not addressed

A2.A.54

Graph logarithmic functions, using the inverse of the related exponential function

Math B – 7B

Represent and analyze logarithmic and exponential functions

A2.A.55

Express and apply the six trigonometric functions as ratios of the sides of  right triangle

Math B – 5A

Use trigonometry as a method to measure indirectly

  • Right triangle trigonometry

A2.A.56

Know the exact and approximate values of the sine, cosine, and tangent of 0˚, 30˚ 45˚, 60˚, 90˚ 180˚, and 270˚ angles

Math B – 5E

Determine the trigonometric functions in terms of the unit circle

  • Special angles 30˚, 45˚, 60˚

A2.A.57

Sketch and se the reference angle for angles in standard position

Math B – 5C

 

 

Math B – 5E

Derive and apply formulas relating angle measure and arc degree measure in a circle

  • Reference and conterminal angles

Reference angles

A2.A.58

Know and apply the co-function and reciprocal relationships between trigonometric ratios

 

Not directly addressed

A2.A.59

Use the reciprocal and co-function relationships to find the value of the secant, cosecant, and cotangent of 0˚, 30˚ 45˚, 60˚, 90˚ 180˚, and 270˚ angles

Math B – 5E

Special angles 30˚, 45˚, 60˚

A2.A.60

Sketch the unit circle and represent angles in standard position

Math B – 5E

Define the trigonometric functions in terms of the unit circle

  • Sine, cosine, tangent, and their reciprocal functions on the unit circle

 

2005 Core Curriculum

1999 Core Curriculum

Performance

Indicator

Concept/Skill

Key Idea

Concept/Skill

A2.A.61

Determine the length of an arc of a circle, given its radius and the measure of its central angle

 

Not addressed

A2.A.62

Find the value of trigonometric functions, if given a point on the terminal side of angle θ

Application of 5E

Define the trigonometric functions in terms of the unit circle

A2.A.63

Restrict the domain of the sine, cosine, and tangent functions to ensure the existence of an inverse function

 

Not addressed

A2.A.64

Use inverse functions to find the measure of an angle, given its sine, cosine, or tangent

 

Not directly addressed

A2.A.65

Sketch the graph of the inverses of the sine, cosine, and tangent functions

 

Not addressed

A2.A.66

Determine the trigonometric functions of any angle, using technology

 

Technology not addressed

A2.A.67

Justify the Pythagorean identities

Math B – 7H

Apply axiomatic structure to algebra and geometry

  • Use the quotient identities, reciprocal identities, and the Pythagorean identities

A2.A.68

Solve trigonometric equations for all values of the variable from 0˚ to 360˚

Math B – 7Q

Develop methods to solve trigonometric equations and verify trigonometric functions

A2.A.69

Determine amplitude, period, frequency, and phase shift, given the graph or equation of a periodic function

 

Not directly addressed

A2.A.70

Sketch and recognize one cycle of a function of the form y = AsinBx or y = Acos Bx

 

Not addressed

A2.A.71

Sketch and recognize the graphs of the functions y = sec(x), y = csc(x), y = tan(x), and y = cot(x)

 

Not addressed

 

2005 Core Curriculum

1999 Core Curriculum

Performance

Indicator

Concept/Skill

Key Idea

Concept/Skill

A2.A.72

Write the trigonometric function that is represented by a given periodic graph

 

Not addressed

A2.A.73

Solve for an unknown side of angle,  using the Law of Sines or the Law of Cosines

Math B – 5F

Relate trigonometric relationships to general solutions of triangles

  • Law of Sines

A2.A.74

Determine the area of a triangle or a parallelogram, given the measure of two sides and the included angle

Math B – 5F

Relate trigonometric relationships to the area of a triangle

  • Application of the sine function in the solution of the area of a triangle

A2.A.75

Determine the solution(s) from the SSA situation (ambiguous case)

Math B- 5F

The ambiguous case

A2.A.76

Apply the angle sum and difference formulas for trigonometric functions

Math B – 5C

Derive and apply formulas relating angle measure and arc degree measure in a circle

  • Sum and difference of two angles

A2.A.77

Apply the double-angle and half-angle formulas for trigonometric functions

Math B – 5C

Derive and apply formulas relating angle measure and arc degree measure in a circle

  • Double and half angles for sine and cosine

 

Measurement Strand

2005 Core Curriculum

1999 Core Curriculum

Performance

Indicator

Concept/Skill

Key Idea

Concept/Skill

A2.M.1

Define radian measure

Math B - 4M

Math B – 5E

Radian measure

Radian measure

A2.M.2

Convert between radian and degree measures

 

Not specifically addressed

 

Statistics and Probability Strand

2005 Core Curriculum

1999 Core Curriculum

Performance

Indicator

Concept/Skill

Key Idea

Concept/Skill

A2.S.1

Understand the differences among various kinds of studies (e.g., survey, observation, controlled experiment)

 

Not directly addressed

A2.S.2

Determine factors which may affect the outcome of a survey

 

Not directly addressed

A2.S.3

Calculate measures of central tendency with group frequency distributions

Math A – 5D

 

Math B – 6F

Use statistical methods including the measures of central tendency to describe and compare data

Create and interpret applications of discrete and continuous probability distributions

  • Measures of central tendencies

A2.S.4

Calculate measures of dispersion (range, quartiles, interquartile range, standard deviation, variance) for both samples and populations

Math A - 5D

 

Math B – 6F

Use statistical methods including the measures of central tendency to describe and compare data

Create and interpret applications of discrete and continuous probability distributions

  • Quartiles and percentiles
  • Range
  • Measures of dispersion
  • Variance using the calculator
  • Standard deviation using the calculator

 

2005 Core Curriculum

1999 Core Curriculum

Performance

Indicator

Concept/Skill

Key Idea

Concept/Skill

A2.S.5

Know and apply the characteristics of the normal distribution

Math B – 5G

 

Math B – 7P

Apply the normal curve and its properties to familiar contexts

Use the normal curve to answer questions about data

  • Standard deviation for grouped data
  • Measures of central tendency

A2.S.6

Determine from a scatter plot whether a linear, logarithmic, exponential, or power regression model is most appropriate

Math B – 6E

Use curve fitting to fit data

  • Linear, logarithmic, exponential, and power regressions from scatter plots
  • Linear correlation coefficient

A2.S.7

Determine the function for the regression model, using appropriate technology, and use the regression function to interpolate and extrapolate from the data

Math B – 6G

Make predictions based on interpolations and extrapolations of data

A2.S.8

Interpret within the linear regression model the value of the correlation coefficient as a measure of the strength of the relationship

Math B – 6E

Use curve fitting to fit data

  • Linear correlation coefficient

A2.S.9

Differentiate between situations requiring permutations and those requiring combinations

Math A – 6D

Determine probabilities, using permutations and combinations

A2.S.10

Calculate the number of possible permutations (nPr) of n items taken r at a time

Math A – 6D

Determine probabilities, using permutations and combinations

A2.S.11

Calculate the umber of possible combinations (nCr) of n items taken r at a time

Math A – 6D

Determine probabilities, using permutations and combinations

 

2005 Core Curriculum

1999 Core Curriculum

Performance

Indicator

Concept/Skill

Key Idea

Concept/Skill

A2.S.12

Use permutations, combinations, and the Fundamental Principle of Counting to determine the number of elements in a sample space and a specific subset (event)

Math A – 6D

Use the concept of random variable in computing probabilities

  • Counting principle
  • Sample space

A2.S.13

Calculate theoretical probabilities, including geometric applications

Math A – 6A

Theoretical versus empirical probability

A2.S.14

Calculate empirical probabilities

Math 7/8 – 6C

Math A – 6A

Conduct and predict outcomes of experiments with independent events

Theoretical versus empirical probability

A2.S.15

Know and apply the binomial probability formula to events involving he terms exactly, at least, and at most

Math B – 6C

Interpret probabilities in real-world situations

  • Applications of the probability of exactly, at least, or at most r successes in n trials of a Bernoulli experiment
  • Simple applications of the binomial theorem

A2.S.16

Use the normal distribution as an approximation for binomial probabilities

Math B – 6F

Create and interpret applications of discrete and continuous probability distributions

  • Normal approximation for the binomial distribution

Last Updated: June 18, 2010