# Sequences Series Binomial Theorem Worksheet - DOC

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```					        Course Title: Functions, Statistics, and Trigonometry                                                Course: # 440
Credit: 1                                                                                            A/L: 1
Scheduling: Open to Grades 10 - 12                                                                   Semester Block: Meets Daily

Prerequisite: A minimum grade of “C+” in Advanced Algebra level 1 (426), or “A” in Advanced Algebra level 2 (424)

Course Description: A faster paced presentation of the material in F.S.T. level 2, this course will integrate statistical and algebraic concepts, preview calculus,
functions, and intuitive notions of limits. There will be a heavy reliance of the graphic calculator for plotting functions, analyzing data, and simulating
experiments. Graphic calculators are required.
Last Revised: May, 2003
State Standards/Goals                                               Benchmarks                                                     Assessments
Students will:                                                The following forms and types of assessments will be
used in this course:

Formative:
Mission Statement Academic Goals                                                                                               Periodic check of assigned homework.
 Socratic evaluation during instruction time.
Primary:                                                                                                                       Observation of work by individuals and within
groups.
AG.2 Develop computational skills and problem solving                display computational skills by meeting the              Demonstrations by students on board.
strategies.                                                           listed standards for this course.
    use problem solving strategies in solving            Summative:
mathematics problems within this course.                 Quizzes and tests to include short answers,
multiple choice, open response and justification
 Cumulative Final Exam
AG.1 Demonstrate effective skills in reading, writing,               read assignments in textbook.
speaking and listening.                                              do problems on board and explain how they            Available Alternate Assessments (projects) will be
AG.3 Demonstrate proficient use of technology. (See                   arrived at answer.                                   scored by departmental rubric.
Instructional Technology Standards)
AG.5 Demonstrate analytical and creative skills.
    write answers to open responses using the
standard school-wide rubric.                         Formative: will be assessed on assigned homework
readings/problems by observation of their notebooks and
assignment sheets.
State Standards/Goals                                        Benchmarks                                                 Assessments

Students engage in problem solving, communicating,
reasoning, and connection to:

Number Sense
   define a complex numbers and identify the reap   Summative: Test/Quizzes- using short answers &
12.N.1 Define complex numbers (e.g., a + bi) and                 part and the imaginary part in: a + bi           calculations – on the ability to identify the various forms
operations on them, in particular, addition, subtraction,       simplify expressions with complex numbers.       of complex numbers and to perform operations, including
multiplication, and division. Relate the system of              demonstrate knowledge of the Fundamental         using the conjugate to simplify fractions, on them. If
complex numbers to the systems of real and rational              Theorem of Algebra.                              solutions to quadratic equations are complex numbers
numbers.                                                        demonstrate knowledge of the conjugate Zeros     they will test them for extraneous roots.
Theorem.

Summative: Test/quizzes - calculations, graphing &
   extend the meaning of exponents to include
12.N.2 Simplify numerical expressions with powers and                                                             short answers on simplifying numerical expressions
fractional exponents.
roots, including fractional and negative exponents.                                                               containing fractional powers and roots.
   simplify radical and exponential expressions.
   use the graph of a polynomial function to find   Formative: Completing the Pattern; Pascal’s Triangle
the real roots of P(x) = 0 , by sketching and    Revisited; Properties of Cubics.
using a graphing calculator.
Patterns, Relations, and Algebra

Summative: Define: Isosceles triangle form of Pascal’s
   solve polynomial equations and inequalities      Triangle. List properties of Pascal’s triangle and use
12.P.1 Describe, complete, extend, analyze, generalize,
using set theory and graphing on the number      definition to find terms in a given row.
and create a wide variety of patterns, including iterative
line.                                            Refers to standard (AII.P.1)
and recursive patterns such as Pascal’s Triangle.
   define Pascal’s Triangle: Let n and r be                     1
nonnegative integers with r <= n. The (r+1)st              1 1
term in a row n of Pascal’s Triangle is nCr.             1 2 1
1 3 3 1
1. Construct the first 10 rows.
2. Identify different families or sets of numbers in the
diagonals.
3. Relate the numbers in the triangle to the row
numbers.
4. Examine sums of rows. Relate row sums to the row
numbers.
State Standards/Goals                                          Benchmarks                                                    Assessments

12.P.2 Identify arithmetic and geometric sequences and          define a sequence.                                   Summative: Test/Quizzes - short answers and fill-in to
finite arithmetic and geometric series. Use the properties                                                            extend a given sequence.
of such sequences and series to solve problems, including
finding the general term and sum recursively and                                                                      Formative: Infinity in Art; A Prime Number Sieve;
explicitly.                                                                                                           Recursively Defined Curves.

12.P.3 Demonstrate an understanding of the binomial             define the Binomial Theorem.4                        Formative: – practice exercise as a review so students
theorem and use it in the solution of problems.4                use the binomial formula to calculate                will know if they should go back over material on their
n                        own: the definition of the Binomial Theorem, calculate a
coefficients: ( a  b) 
n
 ch
a
r 0
n
r
nr
br 4     specific coefficient using the binomial formula by pencil-
and-paper and using the calculator and use the formula to
   relate the binomial formula to probability           find probability distributions.4
distributions.4
Formative: The Binomial Theorem for Rational
Exponents; A Test for Convergence.
4
These topics were covered in Advanced Algebra
and will only be reviewed if time allows – not part of
final exam.

12.P.4 Demonstrate an understanding of the                      identify trigonometric, exponential, and             Summative: Test/Quizzes – short answer and open
trigonometric, exponential, and logarithmic functions.           logarithmic functions.                               response - ability to identify each type of function and to
   solve problems involving exponential,                solve problems using each type. Use the basic properties
trigonometric and logarithmic functions.             of logs to solve problems.
   apply the basic properties of logarithms and
exponents when problem solving.

   apply functional notation and the composition of
12.P.5 Perform operations on functions, including                                                                     Summative: Test/Quizzes - define relation and function,
functions, product of functions, quotient of
composition. Find inverses of functions.                                                                              use functional notation and evaluate functions (linear,
functions and inverse functions.
   distinguish a relation from a function and           logarithmic), graph the functions. Pencil-and-paper
demonstrate the ways of writing and reading a        knowledge of the composition of functions; finding the
composite knowing the Composite g  f: is the        inverses; the ability to determine the domain and range of
function that maps x onto g(f(x)), and whose         a composite function.
domain is the set of all values in the domain of f
for which f(x) is in the domain of g.
State Standards/Goals                                       Benchmarks                                                   Assessments

   define the domain and range of a composite
function.
   find the inverse of a function.
   determine if an inverse is a function using the
Inverse Relation Theorem and Horizontal-line
Test for inverses.

   graph linear, quadratic, cubic, polynomial,
12.P.6 Given algebraic, numeric and/or graphical               rational, logarithmic, exponential , and            Summative: Test/Quizzes – ability to graph by paper-
representations, recognize functions as polynomial,            trigonometric functions and identify the graph of   and-pencil and using a graphic calculator.
rational, logarithmic, exponential, or trigonometric.          each

   place quadratic equation in standard form:
Summative: Test - demonstrate knowledge using
12.P.7 Find solutions to quadratic equations (with real        ax2 + bx + c = 0, identifying the a, b and c
coefficients and real or complex roots) and apply to the                                                           matching, short answers, and open responses : of
values.
solutions of problems.                                        find the quadratic equation given the roots as
rational or complex numbers.
   identify the discriminant and use the               complex roots. Demonstrate ability to graph using
Discriminant Theorem.                               pencil-and-paper and calculator methods.
   solve quadratic equations by factoring,
completing the square and with the quadratic
formula.                                            Formative: Algebraic and Transcendental Numbers;
   use the quadratic formula to find the zeros of a
complex roots.
   solve problems such as projectile paths with
polynomial equations.
   use quadratic regressions and relate it to
Legendre’s method of least squares.
   construct a model for a quadratic trend,
including impressionistic models.
State Standards/Goals                                          Benchmarks                                                      Assessments

12.P.8 Solve a variety of equations and inequalities using      define the Root of a Power Theorem: For all          Refers to standards 12.P.8, 12.P.11, and 12.P.12
algebraic, graphical, and numerical methods, including           positive integers m > 1 and n  2, and all           A stone is thrown straight up into the air with initial
the quadratic formula; use technology where appropriate.         nonnegative real numbers x,                          velocity v0 = 10 feet per second. If one neglects the
Include polynomial, exponential, logarithmic, and                                         m                           effects of air resistance, after t seconds the height of the
x m  ( n x )m  x n
n
trigonometric functions; expressions involving absolute                                                               stone is h  v 0 t  1 gt 2 (until the stone hits the ground),
2
values; trigonometric relations; and simple rational            define: radical notation, geometric mean             where g  32 feet per second squared (the gravitational
expressions.                                                     rationalizing the denominator and the conjugate.     acceleration at the Earth’s surface). What is the greatest
   simplify products with radicals.                     height that the stone reaches, and when does it reach that
   find quotients with radicals.                        height?
   rationalize the denominators of fractions
   solve equations and inequalities involving           notation for nth roots, find the product with radicals, find
exponential, logarithmic and trigonometric           the quotients with radicals, using powers and roots of
functions.                                           negative numbers, and solving equations with radicals.
Summative: Test/Quizzes - pencil-and paper test on
problems using polynomial, exponential, logarithmic, and
trigonometric functions.

   perform matrix multiplication4                       Formative: Topics are reviewed only to help students
12.P.9 Use matrices to solve systems of linear equations.
Apply to the solution of everyday problems.4                    perform matrix transformations4                      know if they must be revisited.
   use matrices to find the image of a figure under
a composite of transformations4.
   find the matrix for RØ4

4
These topics were covered in Advanced Algebra
and will only be reviewed if time allows – not part of
final exam.
State Standards/Goals                                        Benchmarks                                                   Assessments

12.P.10 Use symbolic, numeric, and graphical methods            factor and multiply polynomials including cube     Summative: Test/Quizzes - calculations and graphing
to solve systems of equations and/or inequalities                plus cube and cube minus cube.                     (by hand and with graphic calculator) to include
involving algebraic, exponential, and logarithmic               solve systems of linear and quadratic equations.   problems involving factoring and multiplying
expressions. Also use technology where appropriate.             solve systems of linear and quadratic              polynomials, simplifying rational expressions with the
Describe the relationships among the methods.                    inequalities.                                      conjugate.
   simplify rational expressions and solve
fractional equations.
   use a graphic calculator to draw linear,
trigonometric functions.

12.P.11 Solve everyday problems that can be modeled             solve combined inequalities and inequalities       Summative: Test/Quizzes - Pencil-and-paper to identify
using polynomial, rational, exponential, logarithmic,            involving absolute values.                         the vertex form of an equation for a parabola, find axis of
trigonometric, and step functions, absolute values, and         solve systems of equations and inequalities in     symmetry and locate the minimum and maximum value
square roots. Apply appropriate graphical, tabular, or           two or more variables.                             of y. (with calculator)
symbolic methods to the solution. Include growth and            model real-word phenomena with a variety of
decay; joint (e.g., I = Prt, y = k(w1 + w2)) and combined        functions including polynomial, rational,
(F = G(m1m2)/d2) variation, and periodic processes.              exponential, logarithmic and trigonometric.
   model problems involving joint and combined
variations.

12.P.12 Relate the slope of a tangent line at a specific        apply max/min quadratic functions to problems      Summative: Test/Quizzes - Pencil-and-paper test to
point on a curve to the instantaneous rate of change.1           such as max height of a projectile, max/min        identify the vertex form of an equation for a parabola,
Identify maximum and minimum values of functions in              areas, and others.                                 find axis of symmetry and locate the minimum and
simple situations. Apply these concepts to the solution of      understand form: y – k = a(x - h)2 and if a > 0    maximum value of y. (with calculator)
problems.                                                        the parabola opens up and the y-coordinate of
1
the vertex is the minimum, if (a < 0) it opens
Topic covered in Pre-Calculus                                   down and the y-coordinate is a maximum.
State Standards/Goals                                                Benchmarks                                                    Assessments

12.P.13 Describe the translations and scale changes of a               describe the difference between scale changes        Summative: Test/Quizzes - pencil-and-paper and
given function f(x) resulting from substitutions for the                and translations.                                    calculator based to demonstrate and identify the results of
various parameters a, b, c, and d in y = af (b(x + c/b)) + d.          describe and identify the results of scale changes   scale changes and translations on all types of functions.
In particular, describe the effect of such changes on                   and translations on polynomial functions.
polynomial, rational, exponential, logarithmic, and                    describe and identify the results of scale changes
trigonometric functions.                                                and translations on polynomial, rational,
exponential, logarithmic and trigonometric
functions.

Geometry

12.G.1 Define the sine, cosine, and tangent of an acute                define the basic (Sin, Cos, Tan) trigonometric       Summative: Test/Quiz – write the formula for the Sine,
angle. Apply to the solution of problems.                               functions of an acute angle.                         Cosine, and Tangent functions. Using short answers
   find the missing sides and angles of a right         apply each of the functions to solve for a missing angle
triangle.                                            and side in a right triangle.

12.G.2 Derive and apply basic trigonometric identities                 derive the basic trigonometric identity: sin2 +
(e.g., sin2 + cos2 = 1, tan2 + 1 = sec2) and the laws               cos2 = 1.2                                          Summative: Test/Quiz – short answer and open
of sines and cosines.                                                  derive the basic trigonometric identity: tan2 +     responses defining the 6 trigonometric functions;
1 = sec22                                           applying the Law of Sines and Cosines and using them in
   define the Sec, Csc, Cot functions.                  problem solving.
   use the Law of Sines and the Law of Cosines in
Formative: Applications of the Law of Sines; Bond
problem solving situations.
Angles in Molecules; The Gregory Series; Landmarks
and Surveying.
2
Topics from advanced trig and will only be covered a
time allows – not part of final exam.
State Standards/Goals                                       Benchmarks                                                     Assessments

   identify the equation of an ellipse and              Review topics so if students are unfamiliar they must
12.G.4 Relate geometric and algebraic representations of       demonstrate how the equation was developed.4         review on their own - create a graphs of conic sections.
4
lines, simple curves, and conic sections.                     write equation for an ellipse in standard form,      Identify formula to conic, identify parts of formula and
identify the length of the horizontal axis, the      tell their relationship to the graph. Write the equations in
length of the vertical axis and the distance         standard form from non-standard form.4
between the foci.4
   identify the equation of the hyperbola and           Formative: Orbits of Celestial Bodies; Using Paper-
demonstrate how the equation was developed.4         folding to Make Conics; Using Drawing to Make Conic
   write equation for an hyperbola in standard          Sections; Eccentricity; Quadric Sections.
form, identify the length of the horizontal axis,
the length of the vertical axis and the distance
between the foci.4
   identify the graph of the general quadratic
equation.4
4
These topics were covered in Advanced Algebra
and will only be reviewed if time allows – not part of
final exam.

Measurement

12.M.1 Describe the relationship between degree and           describe relationship between degrees and            Refers to standard 12.M.1
radian measures, and use radian measure in the solution        radian measurements.                                 In one hour, the minute hand on a clock moves through a
complete circle, and the hour hand moves through 1/12 of
of problems, in particular, problems involving angular        convert from degrees to radians.
velocity and acceleration. 2                                                                                        a circle. Through how many radians do the minute and
   convert from radians to degrees.
the hour hand move between 1:00 p.m. and 6:45 p.m. on
   solve circular arc length and circular sector area
the same day?
2
Topics from Advanced Trig and will only be
covered as time allows – not part of final exam.
Data Analysis, Statistics, and Probability
Summative: Conduct a survey – students will conduct a
survey and includes random sampling techniques.
   design a survey and apply random sampling
12.D.1 Design surveys and apply random sampling tech-          techniques.
niques to avoid bias in the data collection.                                                                        Formative: Graphing and Interpreting Statistical Data;
   define population and sample.                        Statistical Analysis of Tests,; Local Land Use Survey;
Automobile Survey; Statistical Experiments; Class
Survey.
State Standards/Goals                                                Benchmarks                                                   Assessments

12.D.2 Select an appropriate graphical representation for              read, interpret, and graph data in circle graphs,    Summative: Test/Quizzes – short answers and open
a set of data and use appropriate statistics (e.g., quartile            bar graphs, line graphs, scatterplots, stem plots,   response question showing ability to read and interpret
or percentile distribution) to communicate information                  dot plots, stem-and-leaf plots, box-and whisker      each type of graph. Calculate each measure of central
about the data.                                                         blots and histograms.                                tendency including the variance and standard deviation.
    calculate measures of central tendency from
displayed data.
    find outliers in a box-and-whiskers plot.            Formative: – The Quincunx; Cumulative Percentile
    calculate variance and standard deviation.           Curves; Is Your Class Typical?; How Common Is the
Letter “e”?; Sums of Random Digits; Design a Study.

12.D.3 Apply regression results and curve fitting to                   find exponential regression from a data set.         Summative: Test/Quiz – calculations do show knowledge
make predictions from data.                                            find quadratic regression from a data set.           of binomial distribution and use it to answer open
response questions.2

    define uniform, normal and binomial
12.D.4 Apply uniform, normal, and binomial                                                                                   Exploratory only topics
distribution.2
distributions to the solutions of problems.2
    find the mean and standard deviation of a
binomial distribution.2
    solve problems using uniform, normal and
binomial distribution.2
    find probabilities using standard normal
distribution.2
    explain the Central Limit Theorem (CLT).2
2
Topics from Advanced Trig or Pre-calculus and will
only be covered as time allows – not part of final exam.
State Standards/Goals                                         Benchmarks                                                    Assessments

Refers to standard 12.D.6
12.D.5 Describe a set of frequency distribution data by        use concepts of variance and standard deviation      There are 9 points on a paper. No three are on the same
spread (i.e., variance and standard deviation), skewness,       in everyday applications.                            line. How many different triangles can be drawn with
symmetry, number of modes, or other characteristics. Use                                                             vertices on these points?
these concepts in everyday applications.                                                                             Refers to standard 12.D.6
There are eight McBride children, three girls and five
boys. How many different ways are there of forming
groups of McBride children containing at least two of the
three girls?
Refers to standard 12.D.6
Some services that involve electronic access require
clients to choose a six-digit password. In an effort to
increase security of the passwords, clients cannot use
combinations that correspond to actual dates, nor can
they use two identical digits in succession, nor passwords
with one digit appearing three or more times. How many

Formative: The binomial Theorem for Rational
Exponents; Infinity in Art; A Prime Number Sieve;
Probabilities and the Lottery.

12.D.6 Use combinatorics (e.g., “fundamental counting          use series to solve counting problems.               Summative: Test/Quizzes - students can select the
principle,” permutations, and combinations) to solve           define the properties of Pascal’s Triangle.          appropriate formula for finding permutations and
problems, in particular, to compute probabilities of           use the Binomial Theorem to calculate                combinations. Students can correctly calculate
compound events. Use technology as appropriate.                 probabilities.4                                      permutations and combinations and use the correct
   demonstrate knowledge of the difference              formulas on a scientific or graphic calculator.
between permutation and combination problems
and use them to solve everyday problems.
4
These topics were covered in Advanced Algebra
and will only be reviewed if time allows – not part of
final exam.
State Standards/Goals                                     Benchmarks                                                  Assessments

Instructional Technology Standards:
Formative: in-class or take home projects that
1.47 Use formulas in spreadsheet application.                write formula to sum a row or column in a         demonstrates knowledge of how to set up a simulation.
   enter data into a graphic calculators list.
   select and use correct functions from “math” on
a graphic calculator.

Formative: practical - demonstrate ability (practical
   transfer data into Excel from another document.
1.48 Import/export data between spreadsheet and other                                                           application) to enter and use data as part of test on
   construct a graph using Excel or similar
applications. (AG.3 )                                                                                           various math topics and selects proper technology to
complete activities and assignments.

1.60 Select the appropriate technology tool for a task.      select and use appropriate software to perform
(AG.3 )                                                       investigations.
CRITICAL THINKING AND MATHEMATICAL: FST 440, level 1, is designed to promote critical thinking skills by requiring students to use logical
processes in the solution of a wide range of applicable problems. In many instances, students will be expected to explain their work to others or to
the whole class.

STUDY SKILLS: During the study of FST 440, students will be expected to participate in activities which will promote and reinforce the following
study skills: reading, writing, public speaking, listening, note taking, time management, problem solving, use of a graphics calculator, project
development, and the application of mathematics to current problems.

KEY RESOURCES:

A. Scott Foresman, Functions/Statistics and Trigonometry, 1998, University of Chicago School Mathematics Project (UCSMP)

Assessment Source Book
Study Skills Handbook
User’s Handbook
Visual Aids

B. Triola, Mario F., Elementary Statistics, 1986, The Benjamin/Cummings Publishing Company

C. TI-83 or TI-83 Plus (preferred) Calculator

D. Student’s notes, worksheets, exams, and homework assignments

E. Massachusetts Mathematics Curriculum Framework
OUTLINE/TIMELINE:

I. Chapter 1 – Exploring Data (7-8 days)

A.   Tables and Graphs
B.   Stemplots and dotplots
C.   Measures of centers
D.   Quartiles, percentiles, and box plots
E.   Histograms
F.   Variance and standard deviation

II. Chapter 2 - Functions and Models (review)        (7-9days)

A.   Language and symbolism
B.    Linear functions and models
C.    Line of best fit
D.    Exponential and quadratic functions and models
E.    Correlation
F.    Step functions

III. Chapter 3 -Transformations (graphs and data)       (8-10 days)

A.   Size changes and Scale changes
B.   Graph-Translation Theorem
C.   Inverse functions
D.   Composite functions
E.   Symmetry and similarity

IV. Chapter 4 – Circular Functions     (12-14 days)

A.   Measure of Angles and Rotations
B.   Lengths of Arcs
C.   Area of Sectors
D.   Sines, Cosines, and Tangents
E.   Scale changes of circular functions
F.   Graph-Standardization Theorem

V. Chapter 5 - Trigonometric Functions      (7-9 days)

A.   Trigonometric Ratios in Right Triangles
B.   Right Triangle ratio identities (SIN, COS, TAN)
C.   Properties and graphs of SIN, COS, TAN
D.   Law of Sines and Law of Cosines
E.   Physical applications through modeling
F.   Trig identities - reciprocal identities and their graphs (CSC, SEC, CTN)
VI. Chapter 13 - Trig Identities and Polar Coordinates (13.1 - 13.4)          (6-8 days)

A.   Proving Trig identities
B.   Conversion between Cartesian and Polar Coordinates
C.   Graphing on a Polar scale and Polar functions
D.   Symmetry and Reflections

VII. Chapter 6 – Root, Power, and Logarithm Functions (7-9 days)

A.   nth root functions
B.   Rational Power Function
C.    Logarithm Functions
D.    e and Natural Logarithms
E.    Properties of Logarithms
F.    Solving Exponential Equations

VIII. Chapter 7 - Probability and Simulation     (5-7 days)

A.   Fundamental properties of Probability
B.    Addition principles and multiplication principles of counting
C.    Permutations
D.    Independent events and conditional events
E.    Probability distribution
F.    Random numbers and Monte Carlo methods

IX. Additional Topics (NOT on final Exam)

Chapter 10 - Binomial and Normal Probability Distributions

A.   Binomial probability distribution
B.   Representing probability by areas
C.   Standard normal probability distribution
D.   Other normal distributions (z-score)
E.   Inferential statistics
G.    Central Limit Theorem and sampling

Others (as time allows)
G. Polynomial functions and modeling (Chapter 9, 11.1 and 11.7)
H. Trigonometry of complex numbers (Chapter 13.5, 13.6)
I. Sequences, series and combinations (Chapter 8 - all)
J. Quadratic Relations (Chapter 12 - all)

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