Course Title: Algebra 2
Course Title:Algebra 2
Levels:2, 3, 4, 5
Grade:11
Length of Course:One Year
Credits:5.0
Prerequisites:Algebra
Description:
The purpose of the Algebra 2 course is to build a strong foundation in preparation for students to ultimately take Calculus. Beginning with a review of Algebra 1 topics to enhance critical skills and concepts, students will move onto more advanced topics. A real-world orientation is emphasized in guiding the approaches to exercises and problems, and technology is integrated, wherever appropriate, in the form of graphing calculators and computer programs.
Building ona more intuitive foundation developed in Algebra 1, the course formalizes the idea of what makes a function and emphasizes various families of functions. In addition to becoming proficient with linear, quadratic, and exponential functions, students will begin to develop an understanding of logarithmic and polynomial functions as well. These topics will be utilized both for their abstract ideas and for modeling real-world phenomena.
Evaluation:
Student performance will be measured using a variety of assessments, including projects, teacher-generated tests and quizzes, and a common departmental Midterm and Final Exam. Assessments will emphasize how well key concepts have been understood as well as the depth to which required skills have been mastered.
Text:Algebra 2 (Glencoe McGraw-Hill)
ColumbiaHigh School
algebra ii curriculum
Learning ObjectivesThe student will … / Content Outline: Unit 1–
Core Content Review / Instructional Materials / Notes
1. Graph real world phenomena and solve problems that involve variation.
NJCCCS4.1 A3
4.3 B1, 2,4, C1, 2, 3,
4.5 ABCDE
2. Apply and explain methods for solving problems involving integers and rational numbers.
NJCCCS4.1 A1, 2, 3,
B1,2, C
4.3D3
3. Evaluate and simplify polynomial expressions
NJCCCS4.1 A3
4.3D1, 3 / Graphing data on a Cartesian plane:
A. Connect aspects of a situation with its graph
- 1.Identify the x axis and y axis
- 2. Graph and describe coordinate pairs
- 3. Place independent and dependent events
- on correct axis
- 4. Analyze and explain the direction and
- shape of the graph
- B. Graph direct variation situations and
- interpret their graphs
variation
2.Properties of the graph (start at origin, is
linear, etc.)
C. Connect situation, recursive formula, & graph
- A. Properties and operations with integers
- 1. Working with negative numbers
B. Properties and operations with rational numbers
1. Recognize fractions as division
2. Simplify rational expressions
Add & subtract polynomials
Multiply monomials and binomials
A. Distributing a monomial over a binomial
B. Multiplying 2 binomials using repeated
distribution (FOIL)
Divide a polynomial by a monomial / Printed Materials:
District constructed supplemental packets (Obj 1, 2, 3)
Investigations:
Technology:
Obj 2BbUse TI-83 FRAC
function to check if a decimal can be written as a fraction
Supplies: graph paper / NJCCC Process Standards are coded for Objective 1, but are also enacted for all other objectives across this curriculum.
Objectives 1-3 enable students to review and reinforce prior learning. These objectives are foundational in the development of new course content.
See pacing chart to determine the approximate time required by level for this review.
Vocabulary:
quadrant ordered pair
integer
rational numbers
monomial binomial
Distributive Property
ColumbiaHigh School
algebra ii curriculum
Learning ObjectivesThe student will… / Content Outline: Unit 2 –
Expressions, Equations, and Inequalities / Instructional materials / Notes
4. Use the properties of real numbers to evaluate expressions and formulas, and solve equations
NJCCCS 4.1 A3
4.3 B1, C1, D1,3
5. Solve absolute value equations
NJCCCS4.3 C1, D1, 2, 3
6. Solve and graph inequalities
NJCCCS 4.1 A3
4.3 D2, 3 / A. Expressions and Formulas
1. Expressions vs. Equations: Differences and
commonalities
2. Variables as unknowns, varying quantities,
and in formulas
3.Evaluating Expressions & Using Formulas
4. Order of operations to evaluate expressions
5. Using formulas by substituting for the
independent variable and simplifying
B. Properties of Real Numbers
1. Classifying numbers: Rational, integers,
natural numbers, whole numbers, irrational
2. Properties: Commutative, Associative,
Distributive, Additive & multiplicative inverses
C. Solving Equations
1. Solving linear equations with one variable
2. Translating verbal & algebraic expressions
3. Reverse order of operations to solve equation
4. Properties of Equality: Reflexive, Symmetric,
Transitive, Substitution
5. Solving for a particular variable in a formula
D. Solving Absolute Value Equations
1.Solve for variables inside abs val brackets
2.Separate into 2 equations, find 2 solutions
E. Inequalities
1. Solve single/multi step inequalities
2. Recognize that the solution is a RANGE of
solutions rather than a single value
3. Graph the range of solutions on a number line
4. Solve compound & abs value inequalities / Printed Materials:
Algebra 2 Chapter 1 sections 1-6
Investigations:
Technology:
Supplies: / See pacing chart to determine the level of development and approximate time required by level for these and all subsequent objectives.
Definitions
variable
expression
formula
absolute value
real numbers
inverse
Properties:
Equality: Symmetric,
Reflexive, Transitive, Substitution
ColumbiaHigh School
algebra ii curriculum
Learning ObjectivesThe student will … / Content Outline: Unit 3 –
Linear Relations and Functions / Instructional Materials / Notes
7. Recognize, represent, and use linear functions to represent real world phenomena and solve problems.
NJCCCS 4.1 A3
4.3 B1, 2, C1, 2
8. Analyze and determine the rate of change using appropriate graphing technologies.
NJCCCS 4.1 A3
4.3C1, B1, 2, D3
9. Select and use appropriate methods for solving linear equations.
NJCCCS 4.1 A3
4.3C1, D2, 3 / A. Relations and Functions
1. Definitions and Properties
2. Representations of types of functions
3.Determine whether a relation is a function by
looking at a table, graph, or equation
4.Vertical Line test for functions
5.Identify domain and range, by looking at a
table, graph, or equation
6.Distinguish independent depend.variables
7. Identify & graph discrete real world data sets
8. Distinguish between discrete vs. continuous
domain & range in real data; graph both
C. Evaluating a function
1.Introduction to Euler’s notation
2.Substitute independent quantity into the fx
notation and simplifying
D. Forms of Linear Relations and Functions
1.Slope-intercept form (y=mx + b)
2.Standard form (Ax + By = C)
3. Graph a line by finding both x and y intercepts
4.Roots of equations and zeros of functions
5.Set the dependent var to zero (y=0) & solve
Rate of Change and Slope
1. Recognize properties of different slopes:
positive, negative, zero, or undefined
2.Using the slope formula
3.Finding the slope of a line from a graph
Writing Linear Equations(from given information)
1. Slope intercept. form y = mx + b
2.Point slope form y – y1 = m(x – x1)
3. The relationship between slopes and intercepts
for parallel and perpendicular lines
Scatter Plots and Lines of Regression
1. Correlation-strong/weak,negative/positive/none
2. Find a line of best fit by hand & w/ith a TI-83+ / Printed Materials:
Algebra 2 Chapter 2 sections 1-5
Investigations:
Relations
Determine whether a relation is a function by looking at a table, graph, or equation
Domain and range
Identify domain and range, by looking at a table, graph, or equation
Application
Use function notation to write and solve equations that model real-world situations
Distinguishing discrete vs. continuous domain/range
Students compare their results to explain the significant differences.
Scatterplots
1.Students use a graphing calculator to find correlation coefficient, regression line slope, and y-intercept.
2. Students use data from their graphing calculator to create a y = mx + b prediction equation in order to extrapolate what would happen if the trend continues.
Technology: TI-83+
Supplies: graph paper / For Slope
1.Show students that slope is best expressed as a fraction. Let them discover why slope as a decimal is not as useful.
2. Move a negative to the top of slope fraction. This way Rise is up or down but run is always to the right.
3. Have students use a ruler to draw a random line across an entire sheet of graph paper. Then have them find the slope of that line by finding the two “best” lattice points counting the rise and run.
4. Simplify the rise/run to lowest terms and see that the slope is still valid
5. Parallel perpendicular
Have students graph lines with opposite-sign slopes. Let them come to realize that a negative (opposite) slope does not create a perpendicular line. Rather it must be an opposite and reciprocal
Definitions
Domain & range
ColumbiaHigh School
algebra ii curriculum
Learning ObjectivesThe student will … / Content Outline: Unit 4 –
Parent Functions and Transformations / Instructional Materials / Notes
10. Analyze and explain the general properties and behavior of functions of one variable, using appropriate graphing technologies.
Identify and compare the properties of classes of functions.
NJCCCS 4.1 A3
4.3 B2, 4, C1, D3
11. Perform transformations on commonly-used functions.
NJCCCS 4.1 A3
4.3B2, 3, C1, D3
12. Graph linear and absolute value inequalities
NJCCCS 4.1 A3
4.3 B1, C1, D3 / A. Parent Functions
1.Constant function
2.Identity function (fx = x)
3.Absolute value
4. Piecewise function
5.Simple quadratic function (centered parabola)
6. Compares properties of classes of functions
Transformations (on the functions listed above)
1. Translations
2.Reflections across the x & y intercepts
3. Dilations of graphs
4. Effects of parameter changes in equations
Graphing inequalities
1.Dashed vs solid boundary lines
2.Defining a half plane
3. Absolute value inequalities ad their graphs / Printed Materials:
Algebra 2 Chapter 2 sections 7-8
Investigations:
Translations
Have students graph each of the 4 families at the origin, then have them draw an image of each graph at a translation point of their choice. Below the translation, they must write a modified (translated) version of the function’s equation. Circulate among students to verify their equations are correct, then let volunteers present their creations on the board.
Technology: TI-83+
Supplies: graph paper
ColumbiaHigh School
algebra ii curriculum
Learning ObjectiveThe student will.. / Content Outline Unit 5 –
Systems of Equations and Inequalities / Instructional Materials / Notes
13. Solve systems of linear equations
NJCCCS 4.1 A3
4.3 A3, B2, C1, D3
14. Solve real world problems using systems of inequalities.
NJCCCS 4.1 A3
4.3 B1, C1, D3 / Solving Systems of Equations by Graphing
1.Solve a system using a tablenote its limitations
2.Solve a system by plotting two lines on a graph and estimating the coordinates of their point of intersection (if any)
3.Classification of Systems: consistent, identity
(dependent), inconsistent (no intersection at all)
Solving Systems of Equations Algebraically *
Solving Systems by Substitution
1.Use when equations do not share the same form
2.Choose substitution so that only one variable
remains in the resultant equation
3. Substitute back to find the remaining variable
Solving Systems by Elimination
1.Best used when equations are of the same form
2. Recognize that multiplying both side of an
equation doesn’t effect the equality
3. Recognize that adding two correct equations
results in another correct equation.
4.Show how eliminating variables from a system
results in isolating one variable.
5) Use back substitution to find the remaining
variable (same as substitution)
Systems with No Solution and Infinite Solutions
Identifying identity, inconsistent, and consistent
systems algebraically.
Solving Systems of Inequalities by Graphing
1.Graph intersecting half-planes created by a
system of linear inequalities
2.Find the resultant “Feasible Region” of a
system of linear inequalities
3.Use of constraints such as x 0, etc., to limit
graph to specific quadrants in real-world examples
4. Find the vertices of an enclosed region
5.Find vertices (corners) of the feasible region by
Inspection, look for intercepts lattice points / Printed Materials:
Algebra 2 Chapter 3 sections 1-3
Investigations:
Break-Even point analysis: When does one function “pass up” another?Finding the Best Cellular Phone Plan: Given brochures for two companies’ cellular phone plans and a list of family members with varying usage requirements, students will plot a system of piecewise linear functions that represent the two plans and decide, based on their point(s) of intersection which plan is less expensive for each family member’s usage requirements.
Technology: TI-83+
Supplies: graph paper / Equations containing more than one variable cannot be solved conclusively. There will always be another variable in the solution.
To solve an equation with two variables, we seek a set of values that will simultaneously satisfy both equations. This occurs at the point of intersection of the two functions on a graph.
*Level 4 should alsosolve 3x3 systems with Substitution and Elimination
ColumbiaHigh School
algebra ii curriculum
Learning ObjectivesThe student will … / Content OutlineUnit 6 –
Linear Programming and Matrices
This is a Level 4 & 5 only unit / Instructional Materials / Notes
15. Use linear programming to solve real-world problems**
NJCCCS4.1 A3, B3, C1
4.3 B1, 2, C1, 2
16. Describe and perform operations on matrices. Solve systems through matrix multiplication, using inverses.
NJCCCS 4.1 B3 / Linear programming
1. Find the maximum and minimum values of a
Function, given various constraints
2. Solve real world optimization problems using
linear programming
Matrices
1.Describe a matrix according to its dimensions
2. Find the determinant & inverse of a 2x2 matrix
3. Use matrices to solve 2x2 systems by hand
Use Cramer’s Rule
Use inverse matrices
4. Use matrices to solve 3x3 systems with a TI-83+ / Printed Materials:
Algebra 2 Chapter 3 sections 4 -5
Chapter 4 sections 1, 3, 5, 6
Investigations:
Technology: TI-83+
Supplies: graph paper / **Level 3 and 4 only
ColumbiaHigh School
algebra ii curriculum
Learning ObjectivesThe student will … / Content Outline Unit 7 – Quadratic Functions / Instructional Materials / Notes
17. Recognize and use connections among significant values of a quadratic function, points on the graph of the function, and its symbolic representation.
NJCCCS 4.1 A3
4.3 B1, 2, C1,2, D2
18. Identify properties of imaginary and complex numbers and operate on them.
Solve simple quadratic equations with imaginary solutions.
NJCCCS – This objective exceeds the range of math identified in the Standards for grades 9-12. / A. Graphing a quadratic functionby using a table
1. Identify the quadratic term, linear term, constant
2. Make a table of values and graph the parabola
3. Identify the absolute maximum minimum of
quadratic function (at the vertex)
B. Solving Quadratic Equations by Graphing
1.Examine the graph of a quadratic equation to
find its roots
2.Graph the quadratic function by making a table
of values
3.Estimate the roots by locating any x-intercepts
on the graph.
C. Solving quadratics equations by factoring*
Imaginary number
1. Factor radicals to extract a negative replace w/i
2. Properties of imaginary numbers
3. Operations on imaginary numbers (+ - x ÷ )
4. Solve quadratic equations w/imaginary solutions
Complex Numbers (Level 2)
1. Definition of Complex Number in form.
2. Arithmetic with complex numbers (add /subtract)
3. Multiplying complex numbers (using FOIL)
Complex conjugates
1. Divide complex numbers
2. Divide a complex number by a constant:
3. Separate fractions
4. Divide a complex number by imaginary number
5. Divide2 complex numbers-use complex conjugate
Complex Numbers (Levels 3-4)
1. Perform operations with complex numbers
2. Find complex roots for quadratic equations
3. Rationalize complex fractions by using the
complex conjugate / Printed Materials:
Algebra 2 Chapter 5 sections 1-4
Investigations:
Technology: TI-83+
Supplies: graph paper / Standard form of a
Quadratic Function -()
Axis of symmetry -(), with vertex, y-intercept c
The number of possible roots are:
Two real solutions (roots)
One real solution
No real solutions (any algebraic roots would be imaginary)
* Levels 3, 4, and 5 only
ColumbiaHigh School
algebra ii curriculum
Learning ObjectivesThe student will … / Content Outline Unit 7 –
Quadratic Functions / Instructional Materials / Notes
19. Solve quadratic equations by completing the square.
NJCCCS 4.3 B1,2,C1, D2,3
20. Solving quadratic equations using the Quadratic Formula
NJCCCS 4.3D2, 3
21. Write a quadratic function in vertex form.
Transform graphs of quadratic functions in vertex form.
NJCCCS 4.1 B1, 4.3 B3 / Completing the Square
1) Identify perfect squares of integers monomials
2) Recognize that perfect squares of binomials are
trinomials (prove with FOIL)
3) Completing the square of a perfect-square
trinomial (using c = (b/2)2
4) Solving equations by completing the square
5) Realize this strategy is best when lead a terms
equal to one, when b is even.
6) For lead terms not equal to 1, try to factor out a
The Quadratic Formula and the Discriminant
1) Derivation of the Quadratic Formula (optional)
2) The four stages of the Quadratic Formula:
a) Standardize: Put quadratic equation into
standard form equal to zero. Id.values of a, b,c
b) Substitute c)Simply
d) Split: Separate the plus/minus expression to
create (up to) 2 solutions
The Discriminant
1) Use discriminant to determine the nature of the
solutions to the equation, and number of x-
intercepts on the graph
Transformations with Quadratic Equations
1) Find the vertex of a parabola
2) Use completing the square to change a quadratic equation to vertex form if practical: b is even
3) Use to change a quadratic equation
into vertex form if completing the square is too
tedious (if or b is odd.)
Graphing equations in Vertex Form
1) Use the a, h, k, and the y-intercept to graph
2) Plot symmetrical points on either side of vertex (if needed) to establish the dilation of the parabola.
3) Recognize the necessity if vertex is on the y-axis / Printed Materials:
Algebra 2 Chapter 5 sections 5-7
Investigations:
Technology: TI-83+
Supplies: graph paper / Quadratic Formula
The Quadratic Formula is a last resort when solving a quadratic equation. It can solve any quadratic equation but is more time consuming than other methods such as factoring and completing the square.
Possible outcomes of a QF: 0,1, or 2 real roots
The Discriminant ()
Vertex Form of a Parabola
ColumbiaHigh School
Algebra ii Curriculum
Learning ObjectivesThe student will … / Content Outline Unit 8- Polynomial Functions / Instructional Materials / Notes
22. Classify and factor polynomials
NJCCCS 4.3 B4, C1, D1, 2, 3
23. Find the zeroes of polynomials and graph polynomial functions
NJCCCS 4.3D2, 3 / Classify polynomials by degree number of terms
Multiply polynomial expressions
Factor polynomials
1. Recognize greatest common monomials factors
2. Factor binomial special cases:
a. difference of two squares,
b. difference of two cubes,
c. sum of two cubes
3. Factor quadratic trinomials
1. Use factoring and the Zero-Product Property to
solve for the zeroes of polynomials
2. Recognize two new factoring special cases:
Sum and Difference of Cubes
3. Use Descartes’ Rule of Signs to determine the
number and type of real zeroes
Graph polynomial functions
1. Sketch polynomial graphs using the zeroes of the
function
2. Identify double roots as tangent to the x-axis on
the graph
3. Identify the number of real roots and complex
Roots from a graph / Printed Materials:
Algebra 2
Chapter 6 sections 1, 3, 4, 5, 7
Investigations:
Technology: TI-83+
Supplies: graph paper
ColumbiaHigh School
algebra ii curriculum