Download foundations of algebra 2

Foundations of Algebra 2
FOUNDATIONS OF ALGEBRA 2
COURSE OF STUDY
POMPTON LAKES PUBLIC SCHOOLS
JUNE 2010
Submitted by
The Mathematics Department
Dr. Terrance Brennan, Superintendent
Vincent Przybylinski, Principal
Anthony Mattera, Vice Principal
Frances J. Macdonald Mathematics Supervisor K-12
Mary Curran, Board of Ed President
Mr. Ray Keating, III, Board of Ed Vice President
Board Members
Mr. William Baig, Mr. Joel Bernstock, Mrs. Catherine Brolsma,
Mrs. Joyce Colfax, Mr. Scott Croonquist, Mr. Tom Salus, Mrs. Stephanie Shaw
1
Foundations of Algebra 2
2
I.
RATIONALE
This year long course in Algebra 2 is not only a preparation for college but also a
course that prepares students to attain success on the SAT’s and the HSPA
II.
DESCRIPTION
This is a full year course of Foundations of Algebra Two. The course is designed
for students who are college bound and have completed the two year course of
Foundations of Algebra I. Graphing Calculator concepts and techniques are
integrated and stressed when they have a direct relationship to the material
being covered. The material covered includes the exploration of real numbers,
equations, inequalities, coordinate planes, graphs, functions, factoring, rational
expressions, irrational and complex numbers, analytical geometry, logarithms,
sequences and series.
III.
THE CORE CURRICULUM CONTENT STANDARDS
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
All students will develop the ability to pose and solve mathematical
problems in mathematics, other disciplines, and everyday
experiences.
All students will communicate mathematically through written, oral,
symbolic, and visual forms of expression.
All students will connect mathematics to other learning by
understanding the interrelationships of mathematical ideas and the
roles that mathematics and mathematical modeling play in other
disciplines and in life.
All students will develop reasoning ability and will become selfreliant, independent mathematical thinkers.
All students will regularly and routinely use calculators, computers,
manipulatives, and other tools to enhance mathematical thinking,
understanding and power.
All students will develop number sense and an ability to represent
numbers in a variety of forms and use numbers in diverse
situations.
All students will develop spatial sense and an ability to represent
geometric properties and relationships to solve problems in
mathematics and in everyday life.
All students will understand, select, and apply various methods of
performing numerical operations.
All students will develop an understanding of and will use
measurement to describe and analyze phenomena.
All students will use a variety of estimation strategies and recognize
situations in which estimation is appropriate.
All students will develop an understanding of patterns,
relationships, and functions and will use them to represent and
explain real world phenomena.
Foundations of Algebra 2
4.12
4.13
4.14
4.15
4.16
IV.
3
All students will develop an understanding of statistics and
probability and will use them to describe sets of data, model
situations, and support appropriate inferences and arguments.
All students will develop algebraic concepts and processes and will
use them to represent and analyze relationships among variable
quantities and to solve problems.
All students will apply the concepts and methods of discrete
mathematics to model and explore a variety of practical situations.
All students will develop an understanding of the conceptual
building blocks of calculus and will use them to model and analyze
natural phenomena.
All students will demonstrate high levels of mathematical thought
through experiences which extend beyond traditional computation,
algebra, and geometry.
STANDARD 9.1 (Career and Technical Education)
All students will develop career awareness and planning, employment skills, and
foundational knowledge necessary for success in the workplace.
Strands and Cumulative Progress Indicators
Building knowledge and skills gained in preceding grades, by the end of Grade
12, students will:
A.
Career Awareness/Preparation
1.
Re-evaluate personal interests, ability, and skills through various
measures including self assessments.
2.
Evaluate academic and career skills needed in various career
clusters.
3.
Analyze factors that can impact an individual’s career
4.
Review and update their career plan and include plan in portfolio.
5.
Research current advances in technology that apply to a sector
occupational career cluster.
B.
Employment skills
1.
Assess personal qualities that are needed to obtain and retain a job
related to career clusters.
2.
Communicate and comprehend written and verbal thoughts, ideas,
directions and information relative educational and occupational
settings.
3.
Select and utilize appropriate technology in the design and
implementation of teacher-approved projects relevant to
occupational and /or higher educational settings
4.
Evaluate the following academic and career skills as they relate to
home, school, community, and employment.
Communication
Foundations of Algebra 2
5.
4
Punctuality
Time management
Organization
Decision making
Goal setting
Resources allocation
Fair and equitable competition
Safety
Employment application
Teamwork
Demonstrate teamwork and leadership skills that include student
participation in real world applications of career and technical
educational skills.
All students electing further study in career and technical education will
also: participate in a structural learning experience that demonstrates
interpersonal communication, teamwork, and leadership skills.
V.
UNIT
A.
FROM DATA TO EQUATIONS
TIME LINE 20 days CCCS 4.1, 4.2, 4.3, 4.4, 4.5, 4.6, 4.8, 4.10, 4.11,
4.13, 9.1.B.4
1
Objectives
a.
Identify linear equations and linear relationships between
variable in a table.
b.
Write and graph an equation describing a real world linear
relationship.
c.
Write an equation in slope intercept form, given two points
on the line or the slope and a point on the line
d.
Graph a scatter plot and identify the data correlation.
e.
Use a graphics calculator to find the correlation coefficient
and to make predictions using the line of best fit.
f.
Solve problems involving direct variation.
g.
Solve problems by writing and solving linear equations.
h.
Write an equation in the point slope form.
i.
Identify, solve and graph inequalities.
j.
Identify, solve and graph absolute value.
k.
Find the mean, median, mode and range of data.
l.
Write and read box and whisker plots, histograms, frequency
tables, and stem and leaf plots.
2.
Content
a.
Tables and graphs of linear equations
b.
Slopes and intercepts
c.
Scatter plots and correlations
Foundations of Algebra 2
3.
5
d.
Direct variation and proportion
e.
Solutions of equations
f.
Solutions of inequalities
g.
Applications using TI-83 Plus/TI-84 Plus Graphics Calculator
Assessments
A technician at the local oil change shop kept track of the number
of cars that received an oil change during the day, and the time of
day the cars were serviced. The data is displayed below.
Oil changes
Time of day
3
10
a.m.
3
11
a.m.
5
Noon
4
1
p.m.
6
2
p.m.
7
3
p.m.
6
4 p.m.
Display the data on a scatter plot, and determine whether the
correlation of the data can be represented by a linear model. If
there is a linear correlation, is there a positive or negative
correlation?
B.
OPERATIONS WITH NUMBERS AND FUNCTIONS
TIME LINE 22 days CCCS 4.1, 4.2, 4.3, 4.4, 4.5, 4.6, 4.8, 4.10, 4.11,
4.13
1.
Objectives
a.
Compare and identify number systems.
b.
Identify properties of real numbers, and use these properties
to perform operations with rational numbers.
c.
Perform operations, and evaluate expressions using the
properties of exponents.
d.
Identify and compare relations and functions, and use the
vertical line test to identify functions from their graphs.
e.
Use functions to model real world applications, and give
appropriate domain and range restrictions for the situation.
f.
Use function notation to define, evaluate and operate with
functions.
g.
Use the slope formula to write and identify increasing and
decreasing linear functions.
h.
Identify and use properties of functions to add, subtract,
multiply, and divide functions.
i.
Define the composition of functions and describe the
relationship between dependent and independent variables.
j.
Determine and define the inverse of a function.
2.
Content
a.
Operations with numbers and functions
b.
Definition and properties of real numbers
c.
Properties, operations and evaluation of exponents
d.
Definition of function
Foundations of Algebra 2
3.
6
e.
Domain and range of functions
f.
Function notation
g.
Linear functions
h.
Slope formula
I.
Increasing and decreasing functions
j.
Applications of functions
k.
Applications using TI-82 Graphics Calculator
l.
Composition of functions
m.
Inverse of functions
Assessments
Tariq Aziz plans to decorate dolls to sell at a crafts fair. The
decorations cost $26.50 and the dolls cost $6.75 each.
a.
Write a function expressing the cost, C(x), of the
project in terms of the numbers of dolls decorated, x.
b.
Determine the cost of decorating 25 dolls.
c.
How many dolls can be decorated with a budget of
$262.75?
C.
EQUATIONS AND FUNCTIONS
TIME LINE 10 days CCCS 4.1, 4.2, 4.3, 4.4, 4.5, 4.6, 4.8, 4.10, 4.11,
4.13
1.
Objectives
a.
Identify the image and prim age points and the axis of
symmetry of a set of ordered pairs.
b.
Determine the relationship of coordinate of points reflected
over the y-axis, the x-axis, and the line y=x.
c.
Define and graph translations, reflections, and scaler
transformations of the absolute value function.
d.
Solve equations involving absolute value symbols by using
graphing and algebraic methods.
e.
Use TI-83 Plus/TI-84 Plus Graphics Calculator to graph and
transform equations.
2.
Content
a.
Symmetry with x-axis, y-axis and line y=x
b.
Transformations
c.
Applications using TI-82 Graphics Calculator
3.
Assessment
Given the graph of y = 3(x + 2), what function would be obtained by
moving the graph down 2 and moving it 6 to the right.
D.
SYSTEMS OF EQUATIONS
TIME LINE 15 days CCCS 4.1, 4.2, 4.3, 4.4, 4.5, 4.6, 4.8, 4.11, 4.13,
4.16
1.
Objectives
a.
Identify each type of system of two linear equations.
Foundations of Algebra 2
7
b.
3.
Solve a system of two linear equations using elimination by
adding.
c.
Solve and graph linear inequalities.
d.
Represent real world data in a matrix
e.
Perform operations with matrices
f.
Identify the inverse of a matrix
g.
Solve systems of equations using matrices.
2.
Content
a.
Dependent Systems
b.
Independent Systems
c.
Consistent Systems
d.
Inconsistent Systems
e.
Solution of systems using graphs
f.
Solution of systems using elimination
g.
Applications using TI-83 Plus/TI-84 Plus Graphics Calculator
Assessments
The Art Department and the Homecoming Committee at a local
school are ordering supplies. The supplies they need are listed in
the table.
Paint
Brushe Pape Glue Sticks Tap
s
r
e
Art Department
11
14
4
13
6
Homecoming
Committee
10
17
8
16
8
If a bottle of paint costs $4, a paint brush costs $2, a ream of
colored paper costs $8, a box of glue sticks costs $3, and a roll of
tape costs $2, use matrices to find the total cost of supplies for
each group.
E.
QUADRATIC FUNCTIONS
TIME LINE 20 days CCCS 4.1, 4.2, 4.3, 4.4, 4.5, 4.6, 4.8, 4.11, 4.13,
4.16
1.
Objectives
a.
Write quadratic functions as a product of two linear
functions.
b.
Approximate the minimum or maximum value of a quadratic
function and the x-intercepts of its graph.
c.
Solve quadratic equations.
d.
Use the distance formula.
f.
Analyze graphs of quadratic functions.
g.
Find the vertex, axis of symmetry, and direction of opening
for the graphs of quadratic functions in the form
Foundations of Algebra 2
2.
3.
F.
8
/(x) = a(x-h)2+k.
h.
Use the quadratic formula to solve quadratic equations that
model real-world situations.
i..
Determine the number of real-number solutions using the
discriminant.
j.
Solve quadratic equations with imaginary solutions.
k.
Identify complex numbers.
Content
a.
Products of linear functions
b.
Minimum or maximum value of a quadratic function
c.
X-intercepts / roots of quadratic graphs.
d.
Quadratic formula
e.
Graphs of quadratic equations
f.
Discriminant
g.
Nature of roots
h.
Imaginary numbers
i.
Complex numbers
j.
Applications using TI-83 Plus/TI-84 Plus Graphics Calculator
Assessments
During a tennis game, Venus hits the ball at a height of 3 feet
above the ground, with a horizontal velocity of 65 feet per second
and a vertical velocity of 7 feet per second. If she is 43 feet away
from the net when she hits the ball, use the following parametric
equations to answer the three questions:
{x (t) = 65t
{y (t) = 3 + 7t
t²
a)
How high will the ball be when it reaches the 3-foot high net?
Will it “clear” the net?
b)
How far beyond the net will it travel before it hits the ground?
c)
Will it land in front of the far court’s 39-foot baseline?
POLYNOMIAL FUNCTIONS
TIME LINE 15 days CCCS 4.1, 4.2, 4.3, 4.4, 4.5, 4.6, 4.8, 4.11,
4.13, 4.16
1.
Objectives
a.
Write the factored form of a polynomial.
2.
Content
a.
Special products - difference between two squares
b.
Special products - factorable trinomials
c.
The Factor Theorem
d.
Applications using TI-83 Plus/TI-84 Plus Graphics Calculator
3. Assessments
Use a graph, synthetic division, substitution, and factoring to solve
x4+ 63x² = 0
Foundations of Algebra 2
9
G.
ANALYTIC GEOMETRY
TIME LINE 20 days CCCS 4.1, 4.2, 4.3, 4.4, 4.5, 4.6, , 4.7, 4.8, 4.11,
4.13, 4.16, 9.1.B.3
1.
Objectives
a.
Write an equation of a parabola when given any two of the
following: focus, directrix, vertex
b.
Identify the vertex, focus, and directrix of a parabola from its
equation; then sketch a graph
c.
Write the equation of a circle when given the coordinates of
the center and the length of the radius
d.
Determine the center and radius of a circle when given an
equation of the circle
e.
Determine the coordinates of the center, vertices, covertices, and foci when given an equation of an ellipse with
either a horizontal or vertical major axis
f.
Determine the coordinates of the center and foci, and find
the lengths of the axes when given an equation of a
hyperbola
2.
Content
a.
Parabola - focus, directrix, vertex
b.
Equation of a parabola
c.
Circle - center, radius
d.
Equation of a circle
e.
Ellipse - center, vertices, co-vertices, foci, horizontal and
vertical major axes
f.
Equation of an ellipse
g.
Hyperbola - center, foci, and lengths of the axes
h.
Equation of a hyperbola
3.
Assessment
Graph the equation (x+2)² + (y+3)² = 9. Label the center and the
radius.
H.
TRIGONOMETRIC FUNCTIONS
TIME LINE 20 days CCCS 4.1, 4.2, 4.3, 4.4, 4.5, 4.6, 4.8, 4.10,
4.11, 4.13, 4.14, 4.16
1.
Objectives
a.
Identify the trigonometric ratios in special right triangles.
b.
Apply the special right triangle relationships to find missing
lengths of sides of special triangles.
c.
Determine sides or angles of right triangles using
trigonometric functions or their inverse.
2.
Content
a.
Special right triangles - 45 45 90 and 30 60 90
b.
Trigonometric ratios -sine, cosine, tangent
c.
Inverse trigonometric functions and applications
Foundations of Algebra 2
10
d.
Principal values
e.
Applications of Trigonometric Functions
f.
Degree measure of angles
g.
Applications using TI-83 Plus/TI-84 Plus Graphics Calculator
Assessments
For ΔABC, find the measure of  A to the nearest degree.
AC=33 CB = 56
3.
A
33
C

56
I.
DISCRETE MATHEMATICS: SERIES AND PATTERNS
TIME LINE 10 days CCCS 4.1, 4.2, 4.3, 4.4, 4.5, 4.6, 4.8, 4.10, 4.11,
4.13, 4.14, 4.16
1.
Objectives
a.
Determine whether a given sequence is an arithmetic
sequence.
b.
Determine whether a given sequence is a geometric
sequence.
c.
Find the nth term of an arithmetic sequence.
d.
Find the sum of the first n terms of an arithmetic series.
e.
Find the nth term of a geometric sequence.
f.
Find the sum of the first n terms of a geometric series.
2.
Content
a.
Arithmetic sequence
b.
Differences between terms
c.
Geometric sequence
d.
Ratio between terms
e.
Arithmetic series
f.
Nth term of an arithmetic sequence
g.
Sum of the first n terms of an arithmetic series
h.
Geometric series
I.
nth term of a geometric series
j.
Sum of a Geometric series
k.
Functions of TI-82 graphics calculator
3.
Assessments
Write the first five terms of the geometric sequence using the given
explicit formula.
t n = 10 ∙ (1/5) n
J
CALCULATORS
1.
Objective
Foundations of Algebra 2
a.
b.
c.
2.
3.
11
Use a calculator to perform basic 4 functions
Know how to use the calculator to solve problems on HSPA
Know how to use the TI graphing calculator to solve
problems
Content
a.
Demonstration of calculator use
b.
Demonstration of TI Graphing Calculators
c.
Key strokes
Assessment
All topics must include the use of graphing calculators
VI.
EVALUATIONS
A.
Tests
B.
Quizzes
C.
Assessments
D.
Semester Exams
E.
Homework
F.
Classwork
G.
Research Projects
H.
Writing assignments
I.
Assessments which require students to:
1.
Use calculator functions to perform basic computation
2.
Check manual computation with calculators
3.
Determine appropriate use of calculator and its functions
VII.
BENCHMARKS
A.
(Semester I Exam)
1.
Tables and graphs of linear equations
2.
Slopes and intercepts
3.
Scatter plots and correlations
4.
Direct variation and proportion
5.
Solutions of equations
6.
Solutions of inequalities
7.
Applications using TI-83 Plus Graphics Calculator
8.
Operations with numbers and functions
9.
Definition and properties of real numbers
10.
Properties, operations and evaluation of exponents
11.
Definition of function
12.
Domain and range of functions
13.
Function notation
14.
Linear functions
15.
Slope formula
16.
Increasing and decreasing functions
17.
Applications of functions
18.
Applications using TI-82 Graphics Calculator
Foundations of Algebra 2
19.
Composition of functions
20.
Inverse of functions
21.
Symmetry with x-axis, y-axis and line y=x
22.
Transformations
23.
Dependent Systems
24.
Independent Systems
25.
Consistent Systems
26.
Inconsistent Systems
27.
Solution of systems using graphs
28.
Solution of systems using elimination
29.
Applications using TI-82 Graphics Calculator
BENCHMARKS
A.
(Semester II Exam)
1.
Products of linear functions
2.
Minimum or maximum value of a quadratic function
3.
X-intercepts / roots of quadratic graphs.
4.
Completing square
5.
Quadratic formula
6.
Graphs of quadratic equations
7.
Discriminant
8.
Nature of roots
9.
Imaginary numbers
10.
Complex numbers
11.
Applications using TI-83 Plus/TI-84 Plus Graphics Calculator
12.
Special products - difference between two squares
13.
Special products - factorable trinomials
14.
The Factor Theorem
15.
Synthetic Division
16.
Inverse variation
17.
Constant of variation
18.
Direct variations
19.
Rational equations
20.
Parabola - focus, directrix, vertex
21.
Equation of a parabola
22.
Circle - center, radius
23.
Equation of a circle
24.
Ellipse - center, vertices, co-vertices, foci, horizontal and
vertical major axes
25.
Equation of an ellipse
26.
Hyperbola - center, foci, and lengths of the axes
27.
Equation of a hyperbola
28.
Arithmetic sequence
29.
Differences between terms
30.
Geometric sequence
31.
Ratio between terms
12
Foundations of Algebra 2
32.
33.
34.
35.
36.
37.
38.
13
Arithmetic series
Nth term of an arithmetic sequence
Sum of the first n terms of an arithmetic series
Geometric series
nth term of a geometric series
Sum of a Geometric series
Functions of TI-83 Plus/TI-84 Plus Graphics Calculator
V.
AFFIRMATIVE ACTION
Evidence of:
A-1 minorities and females incorporated into plans
A-2 human relations concepts being taught
A-3 teaching plans to change ethnic and racial stereotypes
VI.
BIBLIOGRAPHY
Schultz, James. Hollowell, Kathleen. Ellis, Jr., Wade. HRW Advanced Algebra
Orlando : Holt, Rinehart and Winston, 1997. (textbook)
Foster, Alan G., Winters, Leslie J., Gordon, Berchie W., Rath, James N. and
Gell,Joan M. Algebra 2 with Trigonometry. Westerville, Ohio,
Glencoe/McGraw-Hill, 1995.