Essentials to Algebra 2 Curriculum Map Summer 2012
Pacing / Unit/Essential Questions / Essential Knowledge- Content/Performance Indicators(What students must learn) / Essential Skills
(What students will be able to do) / Vocabulary / Resources
Pearson NYS Algebra 2
5 Days / Unit 1: Equations and Inequalities
How do you solve absolute value equation/inequality and plot on the number line? / A2.A.1 Solve absolute value equations and inequalities involving linear expressions in one variable / Review of Algebra Topics
Student will be able to
- simplify expressions
- write and evaluate algebraic expressions
- represent mathematical phrases and real world quantities using algebraic expressions
- solve multi step equations and check
- distinguish between solution, no solution and identity
- solve literal equations
- solve multi step inequalities and graph them
- write inequality from a sentence using key word at least, at most, fewer, less, more …
Algebra 2 and Trig. Topics
Students will be able to
- solve absolute value equations and check
- solve absolute value inequalities and check for extraneous solution
- distinguish between an “and” problem and an “or” problem and accordingly write the solution / ü Term
ü Constant term
ü Like terms
ü Coefficient
ü Expression
ü Equation
ü Literal Equation
ü Inequality
ü Absolute Value
ü Extraneous solution / 1-3: Algebraic Expressions
(1 day)
1-4: Solving equations.
(4 day)
Supplement with additional worksheets on equations with fractional coefficients
1-5: Solving Inequalities
(1 day)
1-6 Absolute Value Equations
(3 - 4 days)
3 Days / Unit 2: Linear Equations and Functions
How do you distinguish between Direct and Inverse variation?
How do you distinguish between a relation and a function?
How do you find the domain and range of a function?
How do you transformation with functions? / A2.A.5 Use direct and inverse variation to solve for unknown values
A2.A.37 Define a relation and function
A2.A.38 Determine when a relation is a function
A2.A.39 Determine the domain and range of a function from its equation
A2.A.40 Write functions in functional notation
A2.A.41 Use functional notation to evaluate functions for given values in the domain
A2.A.46 Perform transformations with functions and relations:
f(x + a) , f(x) + a, f(−x), − f(x), af(x) / Review of Algebra Topics
Student will be able to
- Determine if a function is linear
- Graph a linear function with/without a calculator.
- Find the Slope of a linear function given an equation, graph or 2 points
- Find the equation for a linear function given two points or a point and a graph.
-
Algebra 2 and Trig. Topics
Student will be able to
- Distinguish between a relation and a function.
- Determine if a relation is a function given a set of ordered pair, mapping diagram, graph or table of values
- Distinguish between direct and indirect variation
- Determine of a given function is direct given a function rule, graph or table of values
- Solve word problems related to direct and indirect variation (ref. to regents questions from jmap.org)
- Distinguish between parallel and perpendicular lines.
- Do linear regression using a graphing calculator
- Determine the correlation between the data sets by viewing or plotting a scatter-plot.
- Perform vertical and horizontal translations
- Graph absolute value equations and perform related translations / ü Relation
ü Function
ü Vertical line test
ü Function Rule
ü Function notation
ü Domain
ü Range
ü Direct Variation
ü Constant of Variation
ü Linear function
ü Linear equation
ü x-intercept
ü y-intercept
ü Slope
ü Standard form of linear function
ü Slope intercept form of linear function
ü Point slope form of linear function
ü Line of best fit
ü Scatter plot
ü Correlation
ü Correlation coefficient
ü Regression
ü Absolute value / 2.1 Relations and functions
Emphasis on domain and Range
(2 days)
2.2 Direct Variation
(2 days)
2.3 Linear Functions and slope-intercept Form
(3 days)
2.4 More about Linear Equations
(1 day)
2.5 Using Linear Model
(1 day)
2.6 Families of functions
(2 – 3 days)
2.7 Absolute value Functions and Graphs
(1 - 2 days)
5 Days / Unit 4: Quadratic Equations and Functions
How do you perform transformations of functions?
How do you factor completely all types of quadratic expressions?
How do you use the calculator to find appropriate regression formulas?
How do you use imaginary numbers to find square roots of negative numbers?
How do you solve quadratic equations using a variety of techniques?
How do you determine the kinds of roots a quadratic will have from its equation?
How do you find the solution set for quadratic inequalities?
How do you solve systems of linear and quadratic equations graphically and algebraically? / A2.A.46 Perform transformations with functions and relations:
f (x + a) , f(x)+ a, f (−x), − f (x), af (x)
A2.A.40 Write functions in functional notation
A2.A.39 Determine the domain and range of a function from its
equation
A2.A.7 Factor polynomial expressions completely, using any combination of the following techniques: common factor extraction, difference of two perfect squares, quadratic trinomials
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
A2.A.20 Determine the sum and
product of the roots of a quadratic
equation by examining its coefficients
A2.A.21 Determine the quadratic
equation, given the sum and product of
its roots
A2.A.13 Simplify radical expressions
A2.A.24 Know and apply the
technique of completing the square
A2.A.25 Solve quadratic equations,
using the quadratic formula
A2.A.2 Use the discriminant to
determine the nature of the roots of a
quadratic equation
A2.A.4 Solve quadratic inequalities in one and two variables, algebraically and graphically
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.
A2.N6 Write square roots of negative numbers in terms of i
A2.N9 Perform arithmetic operations on complex numbers and write the answer in the form a+bi / Review of Algebra Topics
Students will be able to
- use definitions of domain and range to sketch a quadratic
- factor the difference of two squares
- factor completely
- solve quadratic equations by factoring
- use a quadratic equation to model a real situation
- determine a quadratic equation, given integer roots
- graph linear and quadratic functions
Algebra 2 and Trig Topics
Students will be able to
- perform horizontal and vertical translations of the graph of y = x2
- graph a quadratic in vertex form: f(x) =a(x - h)2 + k
- identify and label the vertex as ( h , k )
- identify and label the axis of symmetry of a parabola
- graph parabolas in the form of y = a x2 with various values of a
- graph a quadratic in vertex form:
- f(x) = ax2+bx+c
- find the axis of symmetry algebraically using the standard form of the equation
- identify the y-intercept as ( 0, c )
- find the vertex of a parabola algebraically using the standard form of the equation
- identify the range of parabolas
- sketch a graph of a parabola after finding the axis of symmetry, the vertex, and the y-intercept
- use the calculator to find a quadratic regression equation
- factor using “FOIL”
- finding a GCF
- perfect square trinomials
- difference of two squares
- zero product property
- finding the sum and product of roots
- writing equations knowing the roots or knowing the sum and product of the roots
- solve by taking square roots
- solve by completing the square
- solve by using the quadratic formula
- use the discriminant to find the nature of the roots
- simplify expressions containing complex numbers (include rationalizing the denominator)
- solve quadratic inequalities
- solve systems of quadratics algebraically / ü Parabola
ü Quadratic function
ü Vertex form
ü Axis of symmetry
ü Vertex of the parabola
ü Maximum
ü Minimum
ü Standard form
ü Domain and Range
ü Regressions
ü Factoring
ü Greatest Common Factor
ü Perfect square trinomial
ü Difference of two squares
ü Zero of a function (root)
ü Discriminant
ü Imaginary numbers
ü Complex numbers
ü Conjugates
ü / 4-1 Quadratic functions and transformations
(2 – 3 days)
4-2 Standard form of a quadratic function
(2 days)
4-3 Modeling with quadratic functions
(1 - 2 days)
4-4 Factoring quadratic expressions (4 days)
4-5 Quadratic equations (1-2 days)
4-6 Completing the square (2 -3 days)
4-7 Quadratic Formula (2 days)
4-8 Complex Numbers
(4 - 5 days)
Additional resource at
www.emathinstruction.com
Quadratic Inequalities Page 256-257 (1 day)
4-9 Quadratic Systems
(2 days)
6 Days / Unit 5: Polynomials
How do you perform arithmetic operations with polynomial expressions?
How do you factor polynomials?
How do you solve polynomial equation?
How do you expand a polynomial to the nth
Order?
How do you find the nth term of a binomial expansion? / A2.N.3 Perform arithmetic operations with polynomial expressions containing rational coefficients
A2.A.7 Factor polynomial expressions completely, using any combination of the following techniques: common factor extraction, difference of two perfect squares, quadratic trinomials
A2.A.26 Find the solution to polynomial equations of higher degree that can be solved using factoring and/or the quadratic formula
A2.A.50 Approximate the solution to polynomial equations of higher degree by inspecting the graph
A2.A.36 Apply the binomial theorem to expand a binomial and determine a specific term of a binomial expansion / Review of Algebra Topics
Student will be able to
- combine like terms
- subtract polynomial expressions
- multiply monomials, binomials and trinomials
Algebra 2 and Trig Topics
Students will be able to
- recognize and classify polynomials
- factor polynomials using common factor extraction, difference of two perfect squares and or trinomial factoring.
- Write a polynomial function given its roots.
- Solve polynomial equations /find the roots graphically.
- Divide polynomials by factoring, long division or synthetic division
- Apply the Binomial Theorem to expand a binomial expression
- Find a specific term of a binomial expansion. / ü Polynomial
ü Monomial
ü Binomial
ü Trinomial
ü Degree
ü Root
ü Solution
ü Zero Property / 5-1 Polynomial Functions
(1 day)
5-2 Polynomials, Linear Factors and Zeros
(1 day)
5-3 Solving Polynomial Equations
(2 days)
5-4 Dividing Polynomials
(2 days)
5-7 The Binomial Theorem
(2 days)
5 Days / Unit 6: Rational Expressions and Functions
How do we perform arithmetic operations on rational expressions?
How do we simplify a complex fraction?
How do we solve a rational equation? / A2.A.16 Perform arithmetic operations with rational expressions and rename to lowest terms
A2.A.17 Simplify complex fractional expressions
A2.A.23 Solve rational equations and inequalities / Review of Algebra Topics
All topics in this unit except complex fractions are taught in Integrated Algebra. In Algebra most problems involve monomials and simple polynomials. In Algebra 2 factoring becomes more complex and may require more than one step to factor completely.
Algebra 2 Topics
Students will be able to
- Simplify a rational expression to lowest terms by factoring and reducing
- State any restrictions on the variable
- Multiply and divide rational expressions
- Add and subtract rational expressions
- Simplify a complex fraction
- Solve rational equations (inequalities will be saved for the Alg 2 course) / Simplest form
ü Rational Expression
ü Common factors
ü Reciprocal
ü Least Common Multiple
ü Lowest Common Denominator
ü Common factors
ü Complex Fraction
ü Rational equation / 8-4 Rational Expressions (3-4 days)
8-5 Adding and Subtracting Rational Expressions- includes simplifying complex fractions (4-5 days)
8-6 Solving Rational Equations (2 -3 days)
5 Days / Unit 7:Exponential and Logarithmic Functions
How do you model a quantity that changes regularly over time by the same percentage?
How are exponents and logarithms related?
How are exponential functions and logarithmic functions related?
Which type of function models the data best? / A2.A.6 Solve an application with results in an exponential function.
A2.A.12 Evaluate exponential expressions, including those with base e.
A2.A.53 Graph exponential functions of the form. for positive values of b, including b = e.
A2.A.18 Evaluate logarithmic expressions in any base
A2.A.54 Graph logarithmic functions, using the inverse of the related exponential function.
A2.A.51 Determine the domain and range of a function from its graph.
A2.A.19 Apply the properties of logarithms to rewrite logarithmic expressions in equivalent forms.
A2.A. 27 Solve exponential equations with and without common bases.
A2.A. 28 Solve a logarithmic equations by rewriting as an exponential equation.
A2.S. 6 Determine from a scatter plot whether a linear, logarithmic, exponential, or power regression model is most appropriate. / Students will be able to:
- model exponential growth and decay
- explore the properties of functions of the form
- graph exponential functions that have base e
- write and evaluate logarithmic expressions
- graph logarithmic functions
- derive and use the properties of logarithms to simplify and expand logarithms.
- solve exponential and logarithmic equations
- evaluate and simplify natural logarithmic expressions
- solve equations using natural logarithms / ü asymptote
ü change of base formula
ü common logarithm
ü exponential equation
ü exponential function
ü exponential decay
ü exponential growth
ü logarithm
ü logarithmic equation
ü logarithmic function
ü natural logarithmic function / 7 -1 Exploring Exponential Models
(1 day)
7 - 2 Properties of Exponential functions
(2 days)
7 – 3 Logarithmic Functions as Inverses
(2 days)
- Fitting Curves to Data
Page 459 (1 day)
7 - 4 Properties of Logarithms
(2 – 3 days)
7 - 5 Exponential and Logarithmic Equations
(3 days)
7 - 6 Natural Logarithms pg 478
(2 days)
Summer School 2011 – Essentials of Algebra 2