Precalculus McCann Technical School 70 Hodges Cross … · ... finding limits graphically and algebraically, ... successful progress towards completing class work, ... o Limits •

Euchler 284-01

Precalculus 284-01

Northern Berkshire Vocational Regional School District McCann Technical School

70 Hodges Cross Road North Adams, MA 01247

John Euchler

Mathematics Department

March 1, 2011

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Mar 2011

Course Philosophy

In order to fulfill our mission of graduating “individuals who are technically skilled,

academically prepared, and socially responsible individuals” (McCann mission

statement), it is important that our graduates have achieved mathematical competence in

many areas. The Precalculus course is the highest level course aligned to the Massachusetts Frameworks, and provides the opportunity to delve beyond the Trigonometry curriculum into emerging mathematical topics. The goals for grade 12 students enrolled in Precalculus include:

• To introduce concepts and applications of Calculus • To express mathematical ideas coherently both verbally and in writing • To explore the connections that exist within mathematics and with other

disciplines • To develop critical thinking and problem solving skills • To demonstrate understanding of more advanced math concepts • To identify and dispel common math misconceptions

Course Description

The Precalculus course expands on the Trigonometry topics toward inclusion of concepts

which introduce the foundation of a college Calculus course. This year-long, ninety-

minute course that meets on alternating academic weeks addresses topics include

Pythagorean relationships, functions and their graphs, trigonometric functions, right

triangle trigonometry, graphs of six trigonometric functions, inverses, analytic

trigonometry, complex numbers and polar coordinates, continuity, finding limits

graphically and algebraically, and an introduction to the derivative. Projects typically

require students to research topics beyond the scope of the textbook as well as across the

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curriculum. In order to promote mathematical literacy,

there are supplemental reading assignments projects intended to expand the students’ abstraction, reflect upon their understanding, and practice technical writing in anticipation of senior project presentations in the spring of senior year.


Course Syllabus

Instructional Philosophy

This course will allow students to explore and experience mathematics through a variety

of activities and real world applications. Emphasis will be placed on students’

understanding of key concepts and the ability of students to demonstrate their learned

knowledge through exams, projects, discussions and written work. Students will be

encouraged to inquire, discuss, analyze, and question the various topics presented

throughout the course in order to promote complete mastery of topics.

Major Course Projects and Activities

• Assignments o A variety of assignments will be given to students throughout the course to

help reinforce learning objectives, are graded on completeness; solutions being reviewed upon student request. Assignments are valued at 10% of the student’s grade. • Notebook/Portfolio

o Students are required to maintain a course notebook which will include all class notes, homework assignments, writing assignments, and handouts.

o A classroom reference notebook is kept in the classroom. Students produce notes on a daily rotating basis for the class notebook as a reference for absentees. Credit is provided to each student who contributes to the class notebook.

o Quizzes and projects are kept on file in the classroom and are accessible to students in order to review for final examination,

maintain a general student portfolio, and to help prepare for make-up quizzes. o The notebook grade is based on completeness, and carries a 10% grade value. o Weekly “puzzler” questions are offered to students to help maintain a

mathematical attitude during the technical week, as well as providing additional practice solving open-ended as well as unsolved questions.

• Projects

o There are several projects that will be assigned.


o These will serve as extensions to the material learned in class. o Students may be asked to work individually, with partners, or in groups and

may complete such assignments as webpages, PowerPoint presentations, research reports, posters, models, diagrams, etc.

• Attendance/Participation o Daily attendance, preparation, and participation are expected, will be recorded, and are worth 10% of the student grade. This is in accordance with McCann’s Attendance Policy as detailed in the Student/Parent Handbook. o Attendance/participation grades are based on student being present and prepared for class, cooperation, successful progress towards completing class work, participation in daily activities.

COURSE ASSESSMENT PLAN

The following assessment plan is applied for the mathematics students at McCann

Technical:

GRADING SYSTEM: “Report cards are issued quarterly and serve as a guideline for students and their parents to measure achievement. Parents are encouraged to contact teachers and counselors to ensure a continuing participation in student progress. Courses are graded numerically in accordance with the following values.” (2005-2006 McCann Student Handbook)

90-100 80-90 70-80 65-70 0-64 I X


• Other o A variety of other projects and activities may be incorporated as

deemed appropriate by the individual course instructor(s).

Advanced

Proficient Satisfactory Passing Failing Incomplete (make-up required) Excused

MATHEMATICS DEPARTMENT ACADEMIC GRADING POLICY:

70% Tests & Quizzes • Final Exam (quarter 4)

• Projects 20% Homework/Notebook Portfolio

• 10% Assignments • 10% Notebook

o Lecture Notes o Handouts

10% Attendance/Participation • Cooperation • Class Work • Attendance

FINAL EXAMINATIONS: “Final examinations must be taken when scheduled. A grade of “F” will be given for any examination missed unless previously approved by the Principal. Final examinations will be by course title for all students. No exemptions will be given.”(2005-2006 McCann Student Handbook)

• Quizzes


Timeline:

• First Quarter

o Logic

• Six Definitions of True Statements • Open Sentences • Paradoxes

o Coordinate Geometry Review • Cartesian Plane • Pythagorean Theorem • Distance Formula

o Trigonometric Functions • Definitions • Vertical Line Test • Applications of Trigonometric Functions

• Second Quarter

o Trigonometric Identities • Proofs • Law of Sines • Law of Cosines • Hero’s Formula

o Sketching Trigonometric Functions • Graphs of Linear Functions • Amplitude, Frequency, Period, Wavelength • Applications to Sound • Applications to Light

o Inverse Functions • Inverse Sine, Cosine, Tangent graphs • Solving Equations With Inverse Functions

o Law of Sin • Graphs • Solving Systems by Graphing • Addition Method • Multiplication/Addition Method

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• Third Quarter

o Polar Coordinates • Argand Diagrams • Graphs of Quadratic Functions • DeMoivre’s Theorem • Fractional Dimensions • Sierpinski Tetrahedron • Menger Sponge

o Exponential and Logarithmic Functions • Exponential Models • Exponential Functions • Logarithmic Functions as Inverses • Properties of Logarithms • Exponential and Logarithmic Equations • Natural Logarithms

• Fourth Quarter

o Senior Project Mathematics Paper o Number Theory Project o Limits

• Graphical Representation • Algebraic Solutions • Continuity • Zero/zero and infinity/infinity cases • Introduction to the derivative and slope of a curve o Introduction to Hyperbolic Trigonometric Functions

o Senior Project Mathematics Paper o Number Theory Project o Limits

• Graphical Representation • Algebraic Solutions • Continuity • Zero/zero and infinity/infinity cases • Introduction to the derivative and slope of a curve

o Introduction to Hyperbolic Trigonometric Functions

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Standards

Standard

10.P.2 Demonstrate an understanding of the relationship between various representations of a line. Determine a line's slope and x- and y-intercepts from its graph or from a linear equation that represents the line. Find a linear equation describing a line from a graph or a geometric description of the line, e.g., by using the "point-slope" or "slope y- intercept" formulas. Explain the significance of a positive, negative, zero, or undefined slope. (Learning Standards for Grades 9-10 (November 2000))

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Massachusetts Mathematics Curriculum Framework Learning Standards for Grades 11-12 (November 2000) Course Curriculum Topic Linear Functions

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Exponential & Logarithmic Functions

Rational Expressions & Functions

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12.P.4 Demonstrate an understanding of the trigonometric, exponential, and logarithmic functions.

12.P.5 Perform operations on functions, including composition. Find inverses of functions.

12.P.6 Given algebraic, numeric and/or graphical representations, recognize functions as polynomial, rational, logarithmic, exponential, or trigonometric.

12.P.11 Solve everyday problems that can be modeled using polynomial, rational, exponential, logarithmic, trigonometric, and step functions, absolute values, and square roots. Apply appropriate graphical, tabular, or symbolic methods to the solution. Include growth and decay; joint (e.g., I = Prt, y = k(w1 + w2)) and combined (F = G (m1m2)/d2) variation, and periodic processes. 12.P.5 Perform operations on functions, including composition. Find inverses of functions.


12.P.11 Solve everyday problems that can be modeled using polynomial, rational, exponential, logarithmic, trigonometric, and step functions, absolute values, and square roots. Apply appropriate graphical, tabular, or symbolic methods to the solution. Include growth and decay; joint (e.g., I = Prt, y = k(w1 + w2)) and combined (F = G (m1m2)/d2) variation, and periodic processes.

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Periodic Functions & Trigonometry

Conic Sections

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12.P.4 Demonstrate an understanding of the trigonometric, exponential, and logarithmic functions.


12.P.8 Solve a variety of equations and inequalities using algebraic, graphical, and numerical methods, including the quadratic formula; use technology where appropriate. Include polynomial, exponential, logarithmic, and trigonometric functions; expressions involving absolute values; trigonometric relations; and simple rational expressions.

12.P.11 Solve everyday problems that can be modeled using polynomial, rational, exponential, logarithmic, trigonometric, and step functions, absolute values, and square roots. Apply appropriate graphical, tabular, or symbolic methods to the solution. Include growth and decay; joint (e.g., I = Prt, y = k(w1 + w2)) and combined (F = G (m1m2)/d2) variation, and periodic processes.

12.G.1 Define the sine, cosine, and tangent of an acute angle. Apply to the solution of problems.

12.G.2 Derive and apply basic trigonometric identities (e.g., sin2θ + cos2θ = 1, tan2θ + 1 = sec2θ) and the laws of sines and cosines.

12.M.1 Describe the relationship between degree and radian measures, and use radian measure in the solution of problems, in particular, problems involving angular velocity and acceleration. 12.G.4 Relate geometric and algebraic representations of lines, simple curves, and conic sections.


Polar Coordinates

Euchler 284-01

PC.N.1 Plot complex numbers using both rectangular and polar coordinates systems. Represent complex numbers using polar coordinates, i.e., a + bi= r(cosθ+ isinθ). Apply DeMoivre’s theorem to multiply, take roots, and raise complex numbers to a power.

PC.P.1 Use mathematical induction to prove theorems and verify summation formulas . PC.P.2 Relate the number of roots of a polynomial to its degree. Solve quadratic equations with complex coefficients. PC.P.3 Demonstrate an understanding of the trigonometric functions (sine, cosine, tangent, cosecant, secant, and cotangent). Relate the functions to their geometric definitions. PC.P.4 Explain the identity sin2q+ cos2q= 1. Relate the identity to the Pythagorean theorem. PC.P.5 Demonstrate an understanding of the formulas for the sine and cosine of the sum or the difference of two angles. Relate the formulas to DeMoivre’s theorem and use them to prove other trigonometric identities. Apply to the solution of problems. PC.P.6 Understand, predict, and interpret the effects of the parameters a, b, and c on the graph of y = asin((x - b)) + c; similarly for the cosine and tangent. Use to model periodic processes. (12.P.13) PC.P.9 Relate the slope of a tangent line at a specific point on a curve to the instantaneous rate of change. Explain the significance of a horizontal tangent line. Apply these concepts to the solution of problems.


Vocational/Technical Education Curriculum Frameworks Strands 1, 4, 5, and 6

Strand 4: Employability

4.b Develop employability skills to secure and keep employment in chosen field 4.B.01a Apply strategies to enhance effectiveness of all types of communications in

the workplace 4.B.03a Locate information from books, journals, magazines, and the Internet 4.B.06a Explain information presented graphically 4.B.07a Use writing/publishing/presentation applications 4.B.08a Apply basic skills for work-related oral communication

4.c Solve problems using critical thinking 4.C.01a Demonstrate skills used to define and analyze a given problem 4.C.04a Explain strategies used to formulate ideas, proposals and solutions to

problems 4.C.05a Select potential solutions based on reasoned criteria

Strand 6: Underlying Use of Technology

6.c Demonstrate ability to use technology for research, problem solving, and communication 6.C.03a Demonstrate the use of appropriate electronic sources to conduct research

(e.g., Web sites, online periodical databases, and online catalogs) 6.C.04a Demonstrate proper style (with correct citations) when integrating electronic

research results into a research project 6.C.05a Collect, organize, analyze, and graphically present data using the most

appropriate tools 6.C.06a Present information, ideas, and results of work using any of a variety of

communications technologies (e.g., multimedia presentations, Web pages, videotapes, desktop-published documents)

Vocational/Technical Education Curriculum Frameworks Strand 3:Embedded Academics

Automotive Technology


3.B.19c

3.B.22c

3.B.24c

3.B.26c

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12.D.6 Use combinatorics (e.g., "fundamental counting principle," permutations, and combinations) to solve problems, in particular, to compute probabilities of compound events. Use technology as appropriate.

12.M.1 Describe the relationship between degree and radian measures, and use radian measure in the solution of problems, in particular, problems involving angular velocity and acceleration.

12.P.11 Solve everyday problems that can be

modeled using polynomial, rational, exponential, logarithmic, trigonometric, and step functions, absolute values, and square roots. Apply appropriate graphical, tabular, or symbolic methods to the solution. Include growth and decay; joint (e.g., I = Prt, y = k(w1 + w2)) and combined (F = G(m1m2)/d2) variation, and periodic processes.


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11/12 Data Analysis, Probability and Statistics

11/12 Measurement

11/12 Patterns, relations, algebra


Mar 2011

3.B.13

3.B.17

3.B.20

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10.P.2 Demonstrate an understanding of the relationship between various representations of a line. Determine a line's slope and x- and y-intercepts from its graph or from a linear equation that represents the line. Find a linear equation describing a line from a graph or a geometric description of the line, e.g., by using the "point-slope" or "slope y- intercept" formulas. Explain the significance of a positive, negative, zero, or undefined slope.


12.G.4 Relate geometric and algebraic representations of lines, simple curves, and conic sections.

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11/12

11/12


Patterns, relations, algebra

Geometry

Mar 2011

Carpentry/Cabinetry

3.B.13c

3.B.14c

Electricity:

3.B.15c

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12.P.11 Solve everyday problems that can be modeled using polynomial, rational, exponential, logarithmic, trigonometric, and step functions, absolute values, and square roots. Apply appropriate graphical, tabular, or symbolic methods to the solution. Include growth and decay; joint (e.g., I = Prt, y = k(w1 + w2)) and combined (F = G(m1m2)/d2) variation, and periodic processes.

10.P.2 Demonstrate an understanding of the relationship between various representations of a line. Determine a line's slope and x- and y-intercepts from its graph or from a linear equation that represents the line. Find a linear equation describing a line from a graph or a geometric description of the line, e.g., by using the "point-slope" or "slope y-intercept" formulas. Explain the significance of a positive, negative, zero, or undefined slope.

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Mar 2011

Computer Assisted Drafting:

Information Technology:

3.B.27c

Machine Technology: 3.B.13c

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Solve a variety of equations and inequalities using algebraic, graphical, and numerical methods, including the quadratic formula; use technology where appropriate. Include polynomial, exponential, logarithmic, and trigonometric functions; expressions involving absolute values; trigonometric relations; and simple rational expressions.

Solve a variety of equations and inequalities using algebraic, graphical, and numerical methods, including the quadratic formula; use technology where appropriate. Include polynomial, exponential, logarithmic, and trigonometric functions; expressions involving absolute values; trigonometric relations; and simple rational expressions.

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Mar 2011

12.P.8

12.P.8 11/12 Patterns, relations, algebra

3.B.14c

3.B.16

Metal Fabrication:

3.B.13c

3.B.14c

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12.M.1 Describe the relationship between degree and radian measures, and use radian measure in the solution of problems, in particular, problems involving angular velocity and acceleration.



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Mar 2011


11/12 Measurement

Instructional Activities

Several methods of instruction are used throughout the course. Students are exposed to

the material through lecture, practice, cooperative learning groups, projects, and

presentations. Beyond the textbook there are several projects that are completed

throughout the year and described below:

familiar enough with the information to speak fluently and answer questions. Appendix

A contains student handouts. Fractional Dimension Project – Students are introduced to the concept of dimensions

between 1 and 2 and between 2 and 3 such as the Sierpinski Gasket, Sierpinski

Tetrahedron, and Menger Sponge. Student pairs attempt to determine if coastlines of US

states have fractional dimension using box-counting and segment-counting methods along

with logarithm conversions and web-based Internet applets. Number Theory Project – Students research and present the definitions, inventors/

discoverers, history, and applications of various numbers (such as hailstone numbers,

vampire numbers, emordnilaps, etc.) existing in recreational mathematics as well as open

(unsolved) problems in number theory.

Supplemental Reading and Viewing and Posting – Students continue to expand their exploration of mathematical concepts of time, dimension, and various infinities through readings such as Edwin Abbott’s Flatland, Martin Gardner’s The No-Sided Professor and others collected in Fantasia Mathematica, and viewing Charles and Ray Eames’ Powers of Ten and A Rough Sketch and two versions of the film Flatland. After each reading, students respond to guiding questions and post on a class Wiki (https://siximpossible.wikispaces.com/) devoted to the to the topic of dimensions.


Resources

Abbot, Edwin A. Flatland Dover Publications, New York, NY. 1992. Blitzer, Robert. Precalculus, Second Edition. Prentice Hall. Upper Saddle River, NJ: 2004.

Blatner, David. The Joy of Pi. Walker Publishing, New York, NY: 1997.

Eames, Charles and Ray. Powers of Ten. W. H. Freeman, New York, NY: 1994.

Fadiman, Clifton. Fantasia Mathematica. Simon & Shuster, New York, NY: 1958

Pickover, Clifford A. Wonders of Numbers. Oxford University Press, New York, NY: 2003.

Sweltz, Frank and Hartzler, J.S. Mathematical Modeling in the Secondary Curriculum. National Council of Teachers of Mathematics. Reston, Virginia: 1991.


Appendix A – Fractional Dimension

Fractional Dimension Project

Answer the following questions:

1. The US east cost is said to be 1000 miles long. The coast of Maine (a state on the east coast) is said to be 1000 miles long. Based on your conclusions in the map measuring activities, how can both these statements be true?

2. Put these in order from smallest to highest fractional dimension:

The coastline of Texas The coastline of Maine The coastline of Connecticut The coastline of Massachusetts

3. The coastline of Louisiana is shrinking at a rate of three feet per year.

How could this possibly be known?

Why is this relevant to you and I living in Berkshire County/Southern VT?

4a. Complete the <a href="http://www.cbc.yale.edu/courseware/swingboxdim.html"> Map activity</a>for the bent line, semicircle, Connecticut, Norway, and Koch curve.

b. Record the dimension for each in <a href="http://www.mccanntech.org/teachers/ jeuchler/CoastlineFractalActivity.doc">this table.</a>

c. Complete the <a href="http://polymer.bu.edu/java/java/coastline/ coastlineapplet.html">Create a coastline activity</a> using <a href="http:// www.mccanntech.org/teachers/jeuchler/CoastlineFractalActivity.doc">the same sheet as above</a>. Use log vs log, linear scale for graphs.

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Appendix B – Number Theory Project

Number Theory Project

Assignment: Each student will select at random one type of number previously unencountered in mathematics courses (such as emirps, superfactorials, vampire numbers, Hailstone numbers) and create a poster and present briefly on her/his findings.

The poster should include:

• The name of the type of number,

• The definition of the number,

• An example of the type of number,

• The symbol for the number, if one exists,

• The inventor(s) or discoverer(s) of the number,

• Any history associated with the number,

• Any application of the number

McCann Technical School has a 1.6-to-1 student to computer ratio, therefore, no images nor text should be handwritten on the poster! (Learn to love Photoshop!)

An “overall aesthetic” will be part of the grade.

(The class will design a rubric for this assignment which will be posted on the school website.)

Value of poster = One (1) quiz.


Name __________________________ Precalculus: Number Theory Project

List of Numbers

abundant/deficient numbers aleph naught amicable numbers apocalyptic number chess number Drake number Emirps emordnilap numbers Factorions goliath number hailstone numbers happy numbers Kaprekar numbers narcissistic numbers perfect numbers persistent number Primorial repunits and binary Skewes’ number triangular/hexagonal numbers universe number vampire numbers


Number Theory Peer Observation Sheet Presenter Name(s):

Name of Number:

Comments:

(Please make the effort in your critique to demonstrate that you have actually seen the presentation!) Note well: The above comments will not be used in grading the presenter!

Mar 2011 Euchler 284-01 Page 24 of 24

Documents

Precalculus McCann Technical School 70 Hodges Cross … · ... finding limits graphically and algebraically, ... successful progress towards completing class work, ... o Limits •