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97805343734Elementary Algebra
Jerome E. Kaufmann and Karen Scwhitters built this text's reputation on clear and concise exposition, numerous examples, and plentiful problem sets. This no-frills text consistently reinforces the following common thread: learn a skill; use the skill to help solve equations; and then apply what they have learned to solve application problems. This simple, straightforward approach has helped many students grasp and apply fundamental problem solving skills necessary for future mathematics
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Tom Clark
Thomas Clark, president of VideoText Interactive, is a life-long teacher of Mathematics and Science with 49 years of experience at all levels. As a result, he is convinced that everyone has the ability to understand mathematics. In the last 20 years, he has focused on the development of multimedia programs that challenge traditional methods of instruction by emphasizing the "why" of Mathematics, and has further directed his attention toward helping homeschooling parents become more effective instructors. He is the author of "Algebra: A Complete Course", and "Geometry: A Complete Course (with Trigonometry)", both of which are available in hard copy and online.
Did you know that mathematics has parts of speech and sentence structure, just as any spoken language? Did you know that understanding the "grammar" of mathematics greatly helps student understanding of problem solving and applications?... More
Join Tom as he offers an entertaining, educational session designed to help you discover the reasons behind several of the traditional trouble spots in math. Topics discussed will be determined by the audience, and may include division of fractions... More
This workshop is designed to help parent-educators understand the scope, sequence and logic of mathematics instruction from preschool through adult. Join Tom as he takes you on a sometimes humorous journey, describing all levels of arithmetic and... More
Are your students learning passively, or are they involved in concept development? Are they figuring things out for themselves, or are they just learning tricks and shortcuts? Come brainstorm with Tom as he humorously explores the reasons... More
Algebra is the study of relations (equations and inequalities). It is therefore essential that students completely and conceptually understand the basic concepts necessary to solve them. In this workshop, Tom will help you develop an inquiry... More
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books.google.fr - Introducing the reader to classical integrable systems and their applications, this book synthesizes the different approaches to the subject, providing a set of interconnected methods for solving problems in mathematical physics. The authors introduce and explain each method, and demonstrate how it can... to Classical Integrable Systems
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ImproveyourThinking SkillsinMaths is written based on the latest Mathematics syllabus issued by the Ministry of Education. The questions in this book are designed to stimulate creative thinking skills solving problem sums.
ImproveyourThinkingSkillsinMaths is written based on the latest Mathematics syllabus issued by the Ministry of Education. The questions in this book are designed to stimulate creative thinking skills in solving problem sums.
ImproveyourThinking SkillsinMaths is written based on the latest Mathematics syllabus issued by the Ministry of Education. The questions in this book are designed to stimulate creative thinking skills problem sums.
Topical Maths Normal (Academic) is a series of books written for students to build a strong foundation in Maths. Each book provides a comprehensive coverage of the Secondary Maths syllabus, and serves as a good revision for students before their examinations.
Topical Maths Normal (Academic) is a series of book written for students to build a strong foundation in Maths. Each book provides a comprehensive coverage of the Secondary Maths syllabus, and serves as a good revision for students before their examinations.
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Intended for students who plan to continue in the calculus sequence, this course involves the study of basic functions: polynomial, rational, exponential, logarithmic, and trigonometric. Topics include a review of the real number system, equations and inequalities, graphing techniques, and applications of functions. Includes problem-solving laboratory sessions. Permission of instructor is required. This course does not count toward the major or minor in mathematics. (Offered annually)
A study of selected topics dealing with the nature of
mathematics, this course has an emphasis on its origins and a focus on mathematics as a creative
endeavor. This course does not normally count toward the major or minor in mathematics. (Offered each semester)
This course offers a standard introduction to the concepts and techniques of the differential calculus of functions of one variable. A problem-solving lab is included as an integral part of the course. This course does not count towards the major in mathematics. (Offered each semester)
This course is a continuation of the topics covered in MATH 130 with an emphasis on integral calculus, sequences, and series. A problem-solving lab is an integral part of the course. Prerequisite: MATH 130 or permission of the instructor. (Offered each semester)
This course emphasizes the process of mathematical reasoning, discovery, and argument. It aims to acquaint students with the nature of mathematics as a creative endeavor, demonstrates the methods and structure of mathematical proof, and focuses on the development of problem-solving skills. Specific topics covered vary from year to year. MATH 135 is required for the major and minor in mathematics. Prerequisite: MATH 131 or permission of the instructor. (Offered each semester)
This course is an introduction to the concepts and methods of linear algebra. Among the most important topics are general vector spaces and their subspaces, linear independence, spanning and basis sets, solution space for systems of linear equations, linear transformations and their matrix representations, and inner products. It is designed to develop an appreciation for the process of mathematical abstraction and the creation of a mathematical theory. Prerequisite: MATH 131, and MATH 135 strongly suggested, or permission of the instructor. Required for the major in mathematics. (Offered
annually)
A continuation of linear algebra with an emphasis on applications. Among the important topics are eigenvalues and eigenvectors, diagonalization, and linear programming theory. The course explores how the concepts of linear algebra are applied in various areas, such as, graph theory, game theory, differential equations, Markov chains, and least squares approximation. Prerequisite: MATH 204. (Offered every third year)
This course offers an introduction to the theory, solution techniques, and applications of ordinary differential equations. Models illustrating applications in the physical and social sciences are investigated. The mathematical theory of linear differential equations is explored in depth. Prerequisites: MATH 232 and MATH 204 or permission of the instructor. (Offered annually)
This course couples reason and imagination to consider a number of theoretic problems, some solved and some unsolved. Topics include divisibility, primes, congruences, number theoretic functions, primitive roots, quadratic residues, and quadratic reciprocity, with additional topics selected from perfect numbers, Fermat's Theorem, sums of squares, and Fibonacci numbers. Prerequisites: MATH 131 and MATH 204 or permission of the instructor. (Offered every third year)
A graph is an ordered pair (V, E) where V is a set of elements called
vertices and E is a set of unordered pairs of elements of V called
edges. This simple definition can be used to model many ideas and
applications. While many of the earliest records of graph theory
relate to the studies of strategies of games such as chess,
mathematicians realized that graph theory is powerful well beyond the
realm of recreational activity. In class, we will begin by exploring
the basic structures of graphs including connectivity, subgraphs,
isomorphisms and trees. Then we will investigate some of the major
results in areas of graph theory such as traversability, coloring and
planarity. Course projects may also research other areas such as
independence and domination. (Offered occasionally)
This course offers a careful treatment of the definitions and major theorems regarding limits, continuity, differentiability, integrability, sequences, and series for functions of a single variable. Prerequisites: MATH 135 and MATH 204. (Offered annually)
This course begins with a generalization of the notions of
limit, continuity, and differentiability (developed in MATH 331), and extends them to the
two-dimensional setting. Next, the Fundamental Theorem of Calculus is extended to line integrals
and then to Green's Theorem. The course culminates with a brief introduction to analysis in the
complex plane. Prerequisites: MATH 232 and MATH 331. (Offered occasionally)
This is an introductory course in probability with an emphasis on the development of the student's ability to solve problems and build models. Topics include discrete and continuous probability, random variables, density functions, distributions, the Law of Large Numbers, and the Central Limit Theorem. Prerequisite: MATH 232 or permission of instructor. (Offered alternate years)
This is a course in the basic mathematical theory of statistics.
It includes the theory of estimation, hypothesis testing, and linear models, and, if time permits, a
brief introduction to one or more further topics in statistics (e.g., nonparametric statistics,
decision theory, experimental design). In conjunction with an investigation of the mathematical
theory, attention is paid to the intuitive understanding of the use and limitations of statistical
procedures in applied problems. Students are encouraged to investigate a topic of their own
choosing in statistics. Prerequisite: MATH 350. (Offered alternate years)
Typical reading: Larsen and Marx, Mathematical Statistics and Its Applications
Drawing on linear algebra and differential equations, this course investigates a variety of mathematical models from the biological and social sciences. In the course of studying these models, such mathematical topics as difference equations, eigenvalues, dynamic systems, and stability are developed. This course emphasizes the involvement of students through the construction and investigation of models on their own. Prerequisites: MATH 204 and MATH 237 or permission of the instructor. (Offered every third year)
An introduction to the axiomatic method as illustrated by
neutral, Euclidean, and non-Euclidean geometries. Careful attention is given to proofs and
definitions. The historical aspects of the rise of non-Euclidean geometry are explored. This course
is highly recommended for students interested in secondary-school teaching. Prerequisite: MATH
331 or MATH 375. (Offered every third year)
Typical reading: Greenberg, Euclidean and Non-Euclidean Geometries: History and
Development
Each time this course is offered, it covers a topic in mathematics that is not usually offered as a regular course. This course may be repeated for grade or credit. Recent topics include combinatorics, graph theory, and wavelets. Prerequisite: MATH 135 and MATH 204 or permission of instructor.
(Offered alternate years)
This course studies abstract algebraic systems such as groups, examples of which are abundant throughout mathematics. It attempts to understand the process of mathematical abstraction, the formulation of algebraic axiom systems, and the development of an abstract theory from these axiom systems. An important objective of the course is mastery of the reasoning characteristic of abstract mathematics. Prerequisites: MATH 135 and MATH 204 or permission of the instructor. (Offered annually)
First order logic is developed as a basis for understanding the nature
of mathematical proofs and constructions and to gain skills in dealing with formal languages.
Topics covered include propositional and sentential logic, logical proofs, and models of theories.
Examples are drawn mainly from mathematics, but the ability to deal with abstract concepts and
their formalizations is beneficial. Prerequisite: MATH 204, PHIL 240, or permission of
instructor. (Offered every third year)
This course covers the fundamentals of point set topology, starting from axioms
that define a topological space. Topics typically include: topological equivalence, continuity,
connectedness, compactness, metric spaces, product spaces, and separation axioms. Some topics
from algebraic topology, such as the fundamental group, might also be introduced. Prerequisite
MATH 331 or permission of the instructor. (Offered occasionally)
This course presents a careful study of various concepts of analysis. Such
topics as convergence and continuity are briefly examined, first on the real line and then in more
general metric spaces. Other topological properties of metric spaces are studied. An examination of
different types of integrals concludes the course. Prerequisite: MATH 331 or permission of
instructor. (Offered occasionally)
An introduction to the theory of functions of a
complex variable. Topics include the geometry of the complex plane, analytic functions, series
expansions, complex integration, and residue theory. When time allows, harmonic fuctions and
boundary-value problems are discussed. Prerequisite: MATH 331 or permission of instructor.
(Offered every third year)
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North Holland - ELSEVIER
H
ISBN: 0444884467
PGS: 268
List: 114.00YOUR PRICE: 108.30
Combinatorial Problems and Exercises
Lovász
1993
The aim of this book is to introduce a range of combinatorial methods for those who want to apply these methods in the solution of practical and theoretical problems. Various tricks and techniques are taught by means of exercises. Hints are given in a separate section and a third section contains all solutions in detail. A dictionary section gives definitions of the combinatorial notions occurring in the book.
Combinatorial Problems and Exercises was first published in 1979. This revised edition has the same basic structure but has been brought up to date with a series of exercises on random walks on graphs and their relations to eigenvalues, expansion properties and electrical resistance. In various chapters the author found lines of thought that have been extended in a natural and significant way in recent years. About 60 new exercises (more counting sub-problems) have been added and several solutions have been simplified.
North Holland - ELSEVIER
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ISBN: 044481504X
PGS: 636
List: 310.00YOUR PRICE: 294.50
Continued Fractions with Applications
N/A
1992
This book is aimed at two kinds of readers: firstly, people working in or near mathematics, who are curious about continued fractions; and secondly, senior or graduate students who would like an extensive introduction to the analytic theory of continued fractions. The book contains several recent results and new angles of approach and thus should be of interest to researchers throughout the field. The first five chapters contain an introduction to the basic theory, while the last seven chapters present a variety of applications. Finally, an appendix presents a large number of special continued fraction expansions. This very readable book also contains many valuable examples and problems.
North Holland - ELSEVIER
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ISBN: 0444892656
PGS: N/A
List: 210.00YOUR PRICE: 199.50
Covering Codes
Cohen
2005
The problems of constructing covering codes and of estimating their parameters are the main concern of this book. It provides a unified account of the most recent theory of covering codes and shows how a number of mathematical and engineering issues are related to covering problems.
Scientists involved in discrete mathematics, combinatorics, computer science, information theory, geometry, algebra or number theory will find the book of particular significance. It is designed both as an introductory textbook for the beginner and as a reference book for the expert mathematician and engineer.
A number of unsolved problems suitable for research projects are also discussed.
North Holland - ELSEVIER
H
ISBN: 0444825118
PGS: N/A
List: 179.00YOUR PRICE: 170.05
Handbook of Coding Theory
N/A
1998
North Holland - ELSEVIER
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ISBN: 0444500871
PGS: N/A
List: 305.00YOUR PRICE: 289.75
HANDBOOK OF COMBINATORICS VOLUME 1
N/A
1995
North Holland - ELSEVIER
H
ISBN: 0444823468
PGS: 1120
List: 275.00YOUR PRICE: 261.25
HANDBOOK OF COMBINATORICS VOLUME 2
N/A
1995
North Holland - ELSEVIER
H
ISBN: 0444823514
PGS: N/A
List: 290.00YOUR PRICE: 275.50
Hypergraphs
Berge
1989 from this standpoint took shape around 1960. In regarding each set as a ``generalised edge'' and in calling the family itself a ``hypergraph'', the initial idea was to try to extend certain classical results of Graph Theory such as the theorems of Turán and König. It was noticed that this generalisation often led to simplification; moreover, one single statement, sometimes remarkably simple, could unify several theorems on graphs. This book presents what seems to be the most significant work on hypergraphs.
North Holland - ELSEVIER
H
ISBN: 0444874895
PGS: N/A
List: 147.00YOUR PRICE: 139.65
Inverse Spectra
Chigogidze
1996
This is a comprehensive introduction into the method of inverse spectra - a powerful method successfully employed in various branches of topology.
The notion of an inverse sequence and its limits, first appeared in the well-known memoir by Alexandrov where a special case of inverse spectra - the so-called projective spectra - were considered. The concept of an inverse spectrum in its present form was first introduced by Lefschetz. Meanwhile, Freudental, had introduced the notion of a morphism of inverse spectra. The foundations of the entire method of inverse spectra were laid down in these basic works.
Subsequently, inverse spectra began to be widely studied and applied, not only in the various major branches of topology, but also in functional analysis and algebra. This is not surprising considering the categorical nature of inverse spectra and the extraordinary power of the related techniques.
Updated surveys (including proofs of several statements) of the Hilbert cube and Hilbert space manifold theories are included in the book. Recent developments of the Menger and Nöbeling manifold theories are also presented.
This work significantly extends and updates the author's previously published book and has been completely rewritten in order to incorporate new developments in the field.
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For courses in secondary or middle school math. This text focuses on all the complex aspects of teaching mathematics in today's classroom and the most current NCTM standards. It demonstrates how to creatively incorporate the standards into teaching along with inquiry-based instructional strateg...
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Contact Information
Hours
Math Center
The Math Center is an open work area where students can work on math assignments with
assistance available. Instructors, paraprofessionals, and student tutors will answer
your questions and provide guidance for study. You can work independently or with
a small group. We are located on the second floor (parking lot level) of the Library
Building, next to the Writing Center and across from Peer Tutoring. You can just drop
by with questions or call 651.450.3895 for more information.
Homework help
Anytime you come to the Math Center for help, we ask that you sign in on the Math
Center's main computer (to help us keep track of the number of students who use the
center). If you're not sure what to do, just ask any staff member wearing the orange
"May I Help You" badge. Your math class instructor also may request that you sign
in on his or her clipboard to verify time spent in the Math Center for course credit.
Students may study and work on math homework while in the Math Center. Students can
also work with math software in the Math Center's Computer Lab. Several staff members
are available. Just raise your hand or ask any staff member with the orange badge
for assistance.
Some math instructors have offices adjacent to the Math Center. You can check posted
schedules to see when your instructor will be available either in his or her office
or in the Math Center.
Resources for students
The Math Center has a wide variety of mathematics books that can be used for reference
by students in the Math Center. There are also solutions manuals, study guides, tests
and answer keys, and a variety of math equipment (protractors, compasses, and rulers)
available for sign-out or on-site use.
Math videos and CD-ROMs are available for most math courses and can be checked out for
two hours or overnight from the Library.
Several mathematics computer software programs, including Derive and Minitab, are
available for use in the Math Center Computer Lab. These can be accessed via the Start
Menu. A staff member can help you and explain the use of the programs.
Gateway Testing
Some instructors require students to complete a Gateway or Competency Test. If your
instructor assigns one of these tests, see any Math Center staff member for details.
Peer Tutoring
Peer Tutoring is located adjacent to the Math Center. Students may make weekly or biweekly appointments
for one-on-one tutoring sessions with a Peer Tutor.
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Beginning Algebra - With CD - 4th edition
Summary: For college-level courses in beginning or elementary algebra.
El ...show morelevel of instructor and adjunct support.
Martin-Gay's series is well known and widely praised for an unparalleled ability to:
Relate to students through real-life applications that are interesting, relevant, and practical.
Martin-Gay believes that every student can:
Test better: The new Chapter Test Prep Video shows Martin-Gay working step-by-step video solutions to every problem in each Chapter Test to enhance mastery of key chapter content.
Study better: New, integrated Study Skills Reminders reinforce the skills introduced in section 1.1, "Tips for Success in Mathematics" to promote an increased focus on the development of all-important study skills.
Learn better: The enhanced exercise sets and new pedagogy, like the Concept Checks, mean that students have the tools they need to learn successfully.
Martin-Gay believes that every student can succeed, and with each successive edition enhances her pedagogy and learning resources to provide evermore relevant and useful tools to help students and instructors achieve success
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OnTRACK lessons, funded by the Texas Education Agency, align with the Texas Essential Knowledge and Skills in ELAR, Mathematics, Science, and Social Studies. Each lesson includes engaging content, interactive experiences, assessment and feedback, and links to additional resources.
Available in TEA's Project Share, OnTRACK lessons supplement classroom instruction and intervention with dynamic learning experiences that use video, graphics, and online activities.
While these lessons are organized into Project Share courses, they do not cover every student expectation in the TEKS for the corresponding SBOE-approved course. Students cannot earn course credit by completing OnTRACK lessons.
Self-paced, credit-bearing courses using OnTRACK materials are available in Algebra I, Algebra II, and Geometry. Contact your ESC to request copies of Teacher-Facilitated OnTRACK Courses for local use.
The OnTRACK Algebra I course consists of six modules (62 total lessons) which may be accessed through the Lessons button in the left menu. The table below provides descriptions of the modules and lessons, along with the TEKS that are addressed in each lesson. (Note, you must be enrolled in the course to access the lessons.)
We recommend that you use Firefox to view these lessons, and that you update your browser plugins before getting started. OnTRACK courses may also be accessed using small devices which operate on Android and IOS operating systems.
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Algebra
Average rating
4.2 out of 5
Based on 8 Ratings and 8 Reviews
Book Description lea... More learning styles by addressing students' unique strengths. The authors talk to students in their own language and walk them through the concepts, showing students both how to do the math and the reasoning behind it. Tying it all together, the use of the Algebra Pyramid as an overarching theme relates specific chapter topics to the `big picture' of algebra.
About Ellyn Gillespie (Author) : Ellyn Gillespie is a published author of young adult books. Some of the published credits of Ellyn Gillespie include Elementary Algebra, Elementary Algebra: Early Systems Equations. View Ellyn Gillespie's profile
About Tom Carson (Author) : Tom Carson is a published author of young adult books. Some of the published credits of Tom Carson include Elementary Algebra, Elementary Algebra: Early Systems Equations. View Tom Carson's profile
Videos
You must be a member of JacketFlap to add a video to this page. Please Log In or Register.
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Mathematics for the Million: How to Master the Magic of Numbers
by Lancelot Hogben Publisher Comments
Taking only the most elementary knowledge for granted, Lancelot Hogben leads readers of this famous book through the whole course from simple arithmetic to calculus. His illuminating explanation is addressed to the person who wants to understand the... (read more)
How To Solve It 2ND Edition a New Aspect of Math
by George Polya Publisher Comments
This perennial best seller was written by an eminent mathematician, but it is a book for the general reader on how to think straight in any field. In lucid and appealing prose, it shows how the mathematical method of demonstrating a proof or finding an... (read more)
Mathematics: From the Birth of Numbers
by Jan Gullberg Publisher Comments
This extraordinary work takes the reader on a long and fascinating journey--from the dual invention of numbers and language, through the major realms of arithmetic, algebra, geometry, trigonometry, and calculus, to the final destination of differential... (read more)
Mathematics : Applications and Concepts - Course 3 (04 Edition)
by Bailey And Pelfrey / Hustchens / Day / Howard Publisher Comments
Mathematics: Applications and Concepts is a three-text Middle School series intended to bridge the gap from Elementary Mathematics to High School Mathematics. The program is designed to motivate middle school students, enable them to see the usefulness... (read more)
Old Dogs, New Math: Homework Help for Puzzled Parents
by Rob Eastaway Publisher Comments students simply memorized their times tables and... (read more)
Mathematical Analysis and Proof: Second Edition
by David S. G. Stirling Publisher Comments
This fundamental and straightforward text addresses a weakness observed among present-day students, namely a lack of familiarity with formal proof. Beginning with the idea of mathematical proof and the need for it, associated technical and logical skills... (read more)
Rapid Math Tricks & Tips: 30 Days to Number Power
by Edward H Julius Publisher Comments... (read more)
Teaching Mathematics Foundations to Middle Years
by Dianne Siemon Publisher Comments
Teaching Mathematics: Foundations to Middle Years connects teacher education students to the bigger picture of mathematics. It shows them how to communicate mathematically, feel positive about mathematics and their role in teaching it and to enter the... (read more)
The Stanford Mathematics Problem Book: With Hints and Solutions
by George Polya Publisher Comments
This volume features a complete set of problems, hints, and solutions based on Stanford University's well-known competitive examination in mathematics. It offers students at both high school and college levels an excellent mathematics workbook. Filled... (read more)
Math and Literature
by Marilyn Burns Publisher Comments
From Quack and Count to Harry Potter, the imaginative ideas in childrens books come to life in math lessons through this unique series. Each resource provides more than 20 classroom-tested lessons that engage children in mathematical problem solving
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usually focus on teaching students on how to develop advanced algebra skills such as systems of equations,
advanced polynomials, imaginary and complex numbers, quadratics, and concepts and includes
the study of trigonometric functions. It also introduces matrices and their properties. The con...
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My Athens
Annual Reviews
How to use this Guide for Mathematics Resources
Welcome to the Mathematics Study Guide. This is written to support the Mathematics Department and its related majors.
The tabs for each language will take you to various resources in print & electronic media - books, journals, databases, blogs, etc. Feel free to email me at fdecker@eastern.edu or use either of the Instant Message links in the box to the right.
Mathematics Department
Credo Reference
The study of numbers, shapes, and other entities by logical means. It is divided
into pure mathematics and applied mathematics, although the
division is not a sharp one and the two branches are interdependent. Applied
mathematics is the use of mathematics in studying natural phenomena. It includes such
topics as statistics, probability, mechanics, relativity, and quantum
mechanics. Pure mathematics is the study of
relationships between abstract entities according to certain rules. It has
various branches, including arithmetic, algebra, geometry, trigonometry, calculus, and topology.
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Hello all, I just began my solve coupled first order differential class. Boy! This thing is really difficult! I just never seem to understand the point behind any topic. The result? My grades go down. Is there any guru who can lend me a helping hand?
Sounds like your bases are not clear. Excelling in solve coupled first order differential requires that your concepts be concrete. I know students who actually start teaching juniors in their first year. Why don't you try Algebrator? I am quite sure, this program will aid you.
I remember having problems with adding fractions, converting fractions and fractional exponents. Algebrator is a truly great piece of math software. I have used it through several algebra classes - Basic Math, Basic Math and Algebra 2. I would simply type in the problem and by clicking on Solve, step by step solution would appear. The program is highly recommended.
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Update now! TI‑Nspire™ OS 3.2 and Software 3.2
Get the free updates
Get the best performance and newest functionality for your TI‑Nspire technology. Update your handheld operating system and computer software to the latest version 3.2 now.
What's cool in TI‑Nspire™ 3.2?
Cool features of the Handheld OS and Software 3.2
Explore math and science concepts using real-world images. In addition to function and 3D graphing, now graph equations and conics.
What's cool in TI‑Nspire™ 3.2?
Cool features of Teacher Software 3.2 and Student Software 3.2
TI‑Nspire software 3.2 lets you and your students create dynamic reports with the PublishView™ feature.
What's cool in TI‑Nspire™ 3.2?
Cool features of Teacher Software 3.2
TI‑Nspire Teacher Software 3.2 lets you demonstrate in color to your entire class. Use your classroom projector to display a virtual image of the TI Nspire CX handheld keypad with the built-in TI-SmartView™ emulator. And now you can create Expression, Image and Chemistry questions.
Release Notes Download
Update to TI‑Nspire™ Operating System (OS) and Software v.3.2
What's new in Handheld OS 3.2 and Software 3.2?
NEW! Perform Conic Graphing
Use this menu option in the Graphing application to access
templates for standard formats of conic equations.
NEW! Use Chemical Notation
Take advantage of the Chem Box feature that lets you
easily write chemical notation.
View pictures
Explore concepts using real-world images. View, graph and analyze the outline
of the Gateway Arch or the path of a basketball. Import your own images and let
your imagination soar.
Explore a new world of data collection
Conduct exciting experiments. Create a hypothesis and test it graphically by collecting
and analyzing data with the built-in Vernier DataQuest™ App and TI‑Nspire data
collection tools. Put the power of scientific discovery in your students' hands.
Graph differential equations
Model real-world events like population growth over a period of time. Visualize
and explore natural phenomena. You can graph and explore slope fields, direction
fields and more.
Energize Statistics class
Study data like a baseball pitcher's earned-run-average over several years. You'll have more
ways to view and analyze data in bar charts and histograms. Plus create summary-level frequency
plots, probability distributions and clustered bar charts.
Graph in 3D
Graph and explore functions in a 3D space with the powerful 3D graphing functionality,
ideal for higher mathematics subjects such as Precalculus and Calculus.
What's new in Teacher Software 3.2?
NEW! Question Types ‑ Expression, Image, Chemistry
Question Capability ‑ Expression
This type facilitates a question where the answer is a number or expression.
Question Capability ‑ Image
This capability lets you create questions where students need to label or
select the correct answer from an image. There are two kinds of Image questions: Label type lets you place labels on an image for a student to fill in. Point On type enables you to place multiple radio buttons or check boxes on an image.
Question Capability ‑ Chemistry
This type allows you to create questions where the answers must be in chemical notation.
These could be: chemical formulas, balanced equations, predicted reaction products, or
predicted yields. It enables consistent formatting of student answers.
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MTH230 Calculus
Course Description
This course demonstrates and exams various concepts of differential calculus. It assists in understanding the basic concepts of differential calculus. These concepts are used to apply differential calculus in business, economics, and science coursework. Applications to real-world problems are emphasized throughout the course. Specific applications to disciplines such as statistics, accounting, finance, and economics are included in this course. A variety of other applications, such as geometry, personal finance, science, and engineering are also presented.
Topics and Objectives
Calculus as an Object Oriented Math
Develop an understanding of a mathematical function as an object
Differentiate the form of various functions and their graphs.
Explain basic concepts of slope, tangent, asymptote, and limit.
Use binomial expansion to compute derivative of a function.
Limit of a Function as an ObjectDerivative as an object
Solve various applied problems by using derivatives.
Graph the first derivative of various functions and find their root.
Compute the derivatives of rational, exponential, logarithmic, and radical functions.
Computing Derivatives of Mixed Functions
Find the derivative of the sum and difference of two functions.
Find the derivative of the ratio and the product of two functions.
Find the derivative of the composition of two functions.
Summarize the Basic Concepts of Differential Calculus
Summarize the Concepts of Continuity, Rate, Limit, Derivative, Tangent of a Function.
Summarize the Product and Quotient Rules of Differentiations.
Review All Topics From Weeks One through Five.
Integration as an area under a curve
Define an area under a curve as an integral of a function.
Evaluate definite integrals.
Apply product, quotient, and chain rules to compute integrals of various functions.
Use integral calculus to solve applied problems.
Use definite integrals to compute the area bounded by two curves.Application of integral calculus in business and economics
Apply integral calculus in business and economics.
Define probability density functions.
Compute the expected value for a given density function.
Apply the integral of density functions to solve applied problems.
Numerical Integration
Compute the probabilities by using the uniform, exponential, normal density functions.
Apply numerical integration to evaluate definite integrals.
Use numerical integration built in Excel function to compute integrals of various density functions, and answer applied probability questions.
Summarize the Basic Concepts of Integral Calculus
Summarize the Concepts of infinite sum, Integral, rules of integration, and Tangent of a Function.
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Algebra 2 Algebra 2 - Probability and Statistics Learn how to break down problems so that you can figure the probability of events. Follow along as Mrs Jenkins explains the Fundamental Counting Principle step by step. Mar 15, 2010, 20:15 PST
Algebra 2 Algebra 2 - Natural Logarithms If lagarithms seem like Greek to you, this interactive algebra 2 lesson will help you sort things out. Play it once, or as many times as you need. Mar 15, 2010, 19:59 PST
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Algebra 2 Algebra 2 - Inverse Matrices Inverse matrices can seem tricky, but as you go through this interactive lesson you will learn step by step how to work through these types of problems. Mar 15, 2010, 19:02 PST
Algebra 2 Algebra 2 - Augmented Matrices Our first lesson in Algebra 2 discusses augmented matrices. Mr. Green will lead you through the lesson step by step, and you will be on your way to breezing through Algebra 2. Mar 15, 2010, 18:28 PST
Algebra 1 Solving Systems of Equations Learn how to solve systems of equations with your own interactive math tutor. You can take the lesson at your own pace, and repeat it as often as you need. Mar 15, 2010, 17:45 PST
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Each of these areas is huge in both academic research and industry application. For many positions a masters or phd is necessary, so you may not want to specialize until late undergrad or graduate school, should you decide to do that.
I heard real analysis is a hardcore math class. But I'm looking into getting a book and self-studying while I'm still not in college.
Meanwhile, "numerical" linear algebra? What's the difference between regular linear algebra and numerical linear algebra? I already took the linear algebra course offered in the community college so I'm just wondering...
EDIT: Nevermind about the numerical linear alg. question. I did a search on google and found the difference.
What are good institutions/universities for applied mathematics? both undergrad and grad.
Check out Gilbert Strang's linear algebra video lectures (first lecture here) at the MIT OCW site online. It's not numerical linear algebra, but it is probably a more advanced level than you have already seen, and probably a prequisite for understanding numerical linear algebra. Plus the lectures are great. If you already know linear algebra at that level, then I recomment picking up Numerical Linear Algebra by Trefethen and Bau,
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DN2266Mathematical Models, Analysis and Simulation Part 17.5 credits
Matematiska modeller, analys och simulering del 1
Course Syllabus
A second course in numerical methods with a mathematical emphasis offering applied and numerical mathematics useful for scientific and engineering modeling. Emphasis on close connection between the properties of mathematical models and their successful numerical treatment.
Number of exercises
Tutoring time
Form of study
Number of places
Schedule
Course responsible
Teacher
Learning outcomes
The overall goal of the course is to give basic knowledge of applied and numerical mathematics useful for scientific and engineering modeling. Especially, the close connection between the properties of mathematical models and their successful numerical treatment is emphasized.
This understanding means that after the course you should be able to
identify and describe discrete equilibrium models using a network approach;
formulate variational problems starting from simple physical principles and derive the corresponding Euler-Lagrange equation such that you can derive basic equations for continuous equilibrium problems in one, two, and three space dimensions;
understand the relation between convergence, consistency, and stability of numerical methods;
understand essential properties of, and proof error estimations for, numerical methods for solving stationary and instationary problems such that you can compare different methods and select suitable algorithms for given problems;
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Introductory Algebra for College Students - 6th edition
Summary: TheBlitzer Algebra Seriescombines mathematical accuracy with an engaging, friendly, and often fun presentation for maximum appeal. Blitzer's personality shows in his writing, as he draws readers into the material through relevant and thought-provoking applications. Every Blitzer page is interesting and relevant, ensuring that students will actually use their textbook to achieve success!106107.97 +$3.99 s/h
Good
Penntext Downingtown, PA
May have minimal notes/highlighting, minimal wear/tear. Ships same or next business day.
$122.31
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A History of Mathematics 1st Edition
0130190748
9780130190741
A History of Mathematics: Blending relevant mathematics and history, this book immerses readers in the full, rich detail of mathematics. It provides a description of mathematics and shows how mathematics was actually practiced throughout the millennia by past civilizations and great mathematicians alike. As a result, readers gain a better understanding of why mathematics developed the way it did. Chapter topics include Egyptian Mathematics, Babylonian Mathematics, Greek Arithmetic, Pre-Euclidean Geometry, Euclid, Archimedes and Apollonius, Roman Era, China and India, The Arab World, Medieval Europe, Renaissance, The Era of Descartes and Fermat, The Era of Newton and Leibniz, Probability and Statistics, Analysis, Algebra, Number Theory, the Revolutionary Era, The Age of Gauss, Analysis to Mid-Century, Geometry, Analysis After Mid-Century, Algebras, and the Twentieth Century. For teachers of mathematics. «Show less
A History of Mathematics: Blending relevant mathematics and history, this book immerses readers in the full, rich detail of mathematics. It provides a description of mathematics and shows how mathematics was actually practiced throughout the millennia by past civilizations... Show more»
Rent A History of Mathematics 1st Edition today, or search our site for other Suzuki
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Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more.
Course Hero has millions of course specific materials providing students with the best way to expand
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Logical Line Fitting: One Step in the EDA Process by Shannon Guerrero, Northern Arizona University Shannon.Guerrero@nau.edu For this presentation, we will be using the following programs: GATOR shows: length (in inches) vs weight (in lbs) of alligato
MAT 155 QUIZ #4 13.1 13.3 This is a take-home quiz. It is due Tuesday April 8. You are allowed to work with others and/or use your notes/book for reference. However, this quiz should represent your work & understanding of the material covered. 1. Co
Probability Review Find the following. 1. The probability of rolling a 6 and an even in two rolls using a standard die.2. The probability of rolling a 6 or an even in one roll using a standard die.3. Given a bag of marbles with 2R, 2B, 2Y, 1G, a.
MAT 155 Activity Cross-Sections of Polyhedra (Section 8.1) Materials: You will the power solids in the back of the room, play-doh, and dental floss Grouping: Work in pairs *Look on pg. 385 & 386 to review types of 2-dimensional figures before starti
Mini-Golf Hole Activity On a clean sheet of paper, design a miniature golf hole that has at least one dog leg. Make a mark where the ball will start and mark the location of the hole. Using what we know about the reflection of light, waves, angles, e
PROPERTIES OF QUADRILATERALSMaterials: Geo-board, rubber bands, protractor, and string. Directions: For each of the following boxes, decide which characteristics always hold for each type of quadrilateral and record your decisions by placing an X i
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MATH120-13S2 (C)Semester Two 2013 Discrete Mathematics
Description
Discrete mathematics is that part of mathematics not involving limit processes. It includes logic, the integers, finite structures, sets and networks.
Discrete mathematics underpins many areas of modern-day science including theoretical computer science, cryptography, coding theory, operations research and computational biology. This course is an introduction to discrete mathematics, and is designed for students interested in mathematics or computer science. Topics covered in the course include: logic, number theory, cryptography, set theory, functions, relations, probability and graph theory.
Learning Outcomes
• to develop the necessary mathematical skills to recognize and solve a range of problems in discrete mathematics • to understand important ideas from classical number theory, abstract algebra and graph theory • to develop the necessary mathematical skills to understand, analyse and decipher some of the old and modern cryptographic schemes • to develop rigorous thinking based on an axiomatic approach
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Do We Need a New Way to Teach Math?
Students not planning to become engineers or mathematicians are known to complain that they don't "need" certain math classes. Are you among them? Have your math teachers included practical, everyday applications of the principles and theorems you learn in class? What are some examples?
.
For instance, how often do most adults encounter a situation in which they need to solve a quadratic equation? Do they need to know what constitutes a "group of transformations" or a "complex number"? Of course professional mathematicians, physicists and engineers need to know all this, but most citizens would be better served by studying how mortgages are priced, how computers are programmed and how the statistical results of a medical trial are to be understood.
A math curriculum that focused on real-life problems would still expose students to the abstract tools of mathematics, especially the manipulation of unknown quantities. But there is a world of difference between teaching "pure" math, with no context, and teaching relevant problems that will lead students to appreciate how a mathematical formula models and clarifies real-world situations. The former is how algebra courses currently proceed — introducing the mysterious variable x, which many students struggle to understand. By contrast, a contextual approach, in the style of all working scientists, would introduce formulas using abbreviations for simple quantities — for instance, Einstein's famous equation E=mc2, where E stands for energy, m for mass and c for the speed of light.
.
Students: Tell us what you think about these ideas. Would you like to see the math curriculum change? What do you think might be gained — and lost — if math classes focused more on real-life situations? Or do you find that teachers are already doing what Mr. Garfunkel and Mr. Mumford suggest?
Students 13 and older are invited to comment below. Please use only your first name. For privacy policy reasons, we will not publish student comments that include a last name.
From 26 to 27 of 27 Comments
As a student that didn't realize his love of mathematics and consequent application to a degree (and now completion) in the field until the very end of high school, I think I can offer some insight. Although I think an application-based approach might help some, it would certainly dissuade others, it think offering combined and intertwined courses would be better. I know many people who ran from mathematics because they didn't enjoy geometry and assumed it a pre-req for calculus. In reality, we could teach group-theory and pre-algebra together, finance and algebra, computer science and combinatorics. There are many things (abstract algebra for instance) that I believe would be fairly easy from an intuition perspective that would prevent the assumed linear progression towards becoming a mathematician that greatly repels any that fall from the line. I know of a number of people who moved to the arts in spite of clear abstract mathematical minds.
The beauty of mathematics is not in it's applicability but its abstraction, in thinking of what infinite is, of the universe.
Not only what is applied. I ask you: what is the practical use of a painting? You are likely to answer "none, but its beautiful" Well in the same way that people can identify the beauty of a painting, they should identify the beauty of math.
The mathematics of finance, probability etc, is a very small percentage of the science. To diminish the science in order to teach it easier, is not only an insult to the science itself and all the people who spent lifetimes "pushing the envelope", but an insult to students as well, as we are basically admitting that they are are not clever enough to understand the deeper, abstract mathematics.
You propose a dangerous idea: A society of specialists. The accountant to know only accounting, the mathematician math, the poet only poetry. It does not matter if I will never use a quadratic equation. I will never use poetry either, but I want to know it. It is the magic of a truly liberal education.
This idea was called communism and it apparently failed as it does not promote differences it does not praise talent.
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0547016794
9780547016795
111180821X
9781111808211 review material, assignment tracking and time management resources, and practice exercises and online homework to enhance student learning and instruction. With its interactive, objective-based approach, Introductory Algebra provides comprehensive, mathematically sound coverage of topics essential to the beginning algebra course. The Seventh Edition features chapter-opening Prep Tests, real-world applications, and a fresh design--all of which engage students and help them succeed in the course. The Aufmann Interactive Method (AIM) is incorporated throughout the text, ensuring that students interact with and master concepts as they are presented. «Show less... Show more»
Rent Introductory Algebra 7th Edition today, or search our site for other Auf
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LERN 49 - Math Skills Review
(3 units)
This course improves knowledge of basic math. Includes operations and applied problems in
whole numbers, fractions, decimals, percentages, and proportions.
Covers math study strategies such as overcoming math anxiety.
Students who repeat this course will improve skills through further instruction and practice.
(54 hours lecture; 24 hours lab; Pre-Collegiate)
(May be taken three times for credit. May be taken for Credit / No Credit only.)
Prerequisite: LERN 48 or a passing score on the current placement test.
Course Measurable Objectives:
Calculate the additon, subtraction, multiplication and division of integers, decimals and fractions, and percentage and proportions.
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Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more.
Course Hero has millions of course specific materials providing students with the best way to expand
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MATH 343 Fall 2007 Test 2Name:Instructions. Answer carefully and completely. Be sure that your work gives a clear indication of reasoning. Use notation and terminology correctly. If you get stuck on a problem, or get results that don't seem rig
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CALICO JournalVolume 24+ Number 3 + 20071 CALICO IA JOURNAL DEVOTED TO RESEARCH AND DISCUSSION ON TECHNOLOGY AND LANGUAGE LEARNINGDevoted to research and discussion on technology and language learningCALICO Journal.Volume 24, Number 3,2
An oval is used to indicate the beginning or end of an algorithm.A parallelogram indicates the input or output of information.A rectangle indicates the assignment of values to variables; the assigned value may be the result of some computation. S
Technology 21 A Course on Technology for Non-Technologists Abstract There is a need to prepare non-technologists to assume senior management, political and other leadership roles in a highly technological world. Many non-technical college students h
Official course description for Math 256.The course begins with a brief treatment of logic and the logical forms of mathematical statements and their proofs. Emphasis is on the use of logic as a reasoning aid rather than on the theory of logical sys
Mathematics 256 September 15 & 16, 20081. Rules of Inference (Rosen, 1.5) The rules of logic that may (should) be used in arguments are listed in Table 1 on page 66. Study the table carefully. Make sure you understand what each rule says and that it
Mathematics 256 October 10 & 14, 2008Mathematical Induction (Rosen 4.1) Mathematical induction is a form of proof that is used to prove statements of the following structure: n Z+ P (n), where P (n) is some statement about the positive integer n. A
Mathematics 256, Final Exam 9:00 a.m.12:00 noon, December 18, 2008The nal exam will be comprehensive. Approximately 40% of the exam will cover discrete mathematics and 60% will cover linear algebra. Your nal exam score may replace one of the two tes
Euclids Algorithm and Solving Congruences Mathematics 100 A September 22, 2006Denition. The greatest common divisor of two natural numbers a and b, written gcd(a, b), is the largest natural number that divides both a and b. Middle School Algorithm.
36243_1_p1-2912/8/97 8:39 AM Page 23MORE ABOUT FUNCTION PARAMETERSDefault Values for Parameters in FunctionsProblem. We wish to construct a function that will evaluate any real-valued polynomial function of degree 4 or less for a given real val
36243_2_p31-3412/8/97 8:42 AMPage 3136243AdamsPRECEAPPENDIX 2 JA ACS11/17/97pg 31CODES OF ETHICSThe PART OF THE PICTURE: Ethics and Computing section in Chapter 1 noted that professional societies have adopted and instituted codes
14.4 The STL list<T> Class Template114.4 The STL list<T> Class TemplateIn our description of the C+ Standard Template Library in Section 10.6 of the text, we saw that it provides a variety of other storage containers besides vector<T> and that o
15.4 An Introduction to Trees1TREES IN STLThe Standard Template Library does not provide any templates with Tree in their name. However, some of its containers - the set<T>, map<T1, T2> , multiset<T>, and multmap<T1, T2> templates - are generall
10.2 C-Style Arrays1VALARRAYSAn important use of arrays is in vector processing and other numeric computation in science and engineering. In mathematics the term vector refers to a sequence (one-dimensional array) of real values on which various
15.3 Recursion Revisited1EXAMPLE: DRY BONES!The Old Testament book of Ezekiel is a book of vivid images that chronicle the siege of Jerusalem by the Babylonians and the subsequent forced relocation (known as the exile) of the Israelites followin
10.7 An Overview of the Standard Template Library1STL Iterators. The Standard Template Library provides a rich variety of containers:vector list deque stack queue priority_gueue map and multimapset and multiset The elements of a vector<T> can
5.5 Case Study: Decoding Phone Numbers15.5 Case Study: Decoding Phone NumbersPROBLEMTo dial a telephone number, we use the telephones keypad to enter a sequence of digits. For a long-distance call, the telephone system must divide this number i
1.3 Case Study: Revenue Calculation11.3 Case Study: Revenue CalculationPROBLEMSam Splicer installs coaxial cable for the Metro Cable Company. For each installation, there is a basic service charge of $25.00 and an additional charge of $2.00 for
7.7 Case Study: Calculating Depreciation17.7 Case Study: Calculating DepreciationPROBLEMDepreciation is a decrease in the value over time of some asset due to wear and tear, decay, declining price, and so on. For example, suppose that a company
1From Paraconsistent Logic to Universal LogicJean-Yves Bziau"The undetermined is the structure of everything" AnaximanderAbstract During these last years I have been developed a general theory of logics that I have called Universal Logic. In t
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Having the right answer doesn't guarantee understanding. This book helps physics students learn to take an informed and intuitive approach to solving problems. It assists undergraduates in developing their skills and provides them with grounding in important mathematical methods. Starting with a review of basic mathematics, the author presents a thorough analysis of infinite series, complex algebra, differential equations, and Fourier series. Succeeding chapters explore vector spaces, operators and matrices, multivariable and vector calculus, partial differential equations, numerical and complex analysis, and tensors. Additional topics include complex variables, Fourier analysis, the calculus of variations, and densities and distributions. An excellent math reference guide, this volume is also a helpful companion for physics students as they work through their assignments. Dover Original$26$19Numerical Methods by Germund Dahlquist
Åke Björck Practical text strikes balance between students' requirements for theoretical treatment and the needs of practitioners, with best methods for both large- and small-scale computing. Many worked examples and problems. 1974 edition. read more
Methods of Applied Mathematics by Francis B. Hildebrand Offering a number of mathematical facts and techniques not commonly treated in courses in advanced calculus, this book explores linear algebraic equations, quadratic and Hermitian forms, the calculus of variations, more
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MATH LESSONS: ALGEBRA, GEOMETRY, ALGEBRA 2, BASIC MATH
Description: Algebra 1, Algebra 2, Geometry and Basic Math lessons, that work great as lesson plans and for the students to learn in a step by step mode solution of problems and introduction to problems. Keywords:
math, polygon, linear, rational zero theorem solver
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The Undergraduate Program
The Mathematics Department serves a very large number of undergraduate students. We aim for excellence in our teaching of algebra, precalculus, and calculus, for students in every major that requires mathematics. In addition to our teaching staff, we work with Learning Support Services to provide instructional assistance (tutoring), we provide the Mathematics Placement Exam to place students in an appropriate course, and mathematics advising at Undergraduate Affairs. In addition to serving the campus with mathematical resources, we support a large number of undergraduate mathematics majors.
The Pure Mathematics track is designed for students who value the study of mathematics, not only for application, but also for its own sake. Pure mathematicians focus on the big how and why questions of mathematics, and attempt to find new formulae and methods while utilizing insights from a tradition of thousands of years. The Pure Mathematics track is recommended for those interested in graduate study in pure mathematics, and those who seek a rigorous education that involves not only rote computational skills but also rigorous explanations of how mathematics works. To give a well-rounded education in mathematics, the Pure Mathematics track requires an introduction to proof class, and a balance of advanced coursework in algebra, analysis, and geometry. Majors who seek graduate study at top institutions often go beyond the required courses to enroll in graduate courses as well.
Our Math Education track is specially designed for prepare students for a career in K-12 mathematics education. It shares a rigorous approach to advanced mathematics, but requires coursework that is particularly relevant to the K-12 classroom: number theory, classical geometry, and the history of mathematics. In addition, the Math Education track requires experience in Supervised Teaching. Many Math Education majors also participate in CalTeach [LINK TO CALTEACH], to enhance their experience and directly connect with local schools.
Our Computational Mathematics track offers flexibility to students who are interested in mathematics together with its applications – students in Computational Mathematics pick up skills in statistics, computer science, and mathematical modeling, among other topics. Much of the coursework for the Computational Mathematics track is offered outside of the Mathematics Department, offering an interdisciplinary experience.
The Mathematics Minor provides an excellent foundation in mathematics, which can serve a student well in careers that require quantitative analysis. The Mathematics Minor can also provide an enjoyable supplement for students who love mathematics but have already decided to pursue another major. The Mathematics Minor fits particularly well for students pursuing a quantitative major such as physics. We encourage students to consider a combination of major and minor involving mathematics with physics, economics, computer science, and environmental science, for example.
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Math 8/7 is the prealgebra program for students who have completed Math 7/6. It emphasizes the concepts and skills necessary for your child's success in upper-level mathematics courses, including scientific notation; statistics and probability; ratios and proportions; simplifying and balancing equations; factoring algebraic expressions; slope-intercept form; graphing linear inequalities; arcs and sectors; and the Pythagorean theorem.
The solutions manual includes full step-by-step solutions for all lesson and investigation problems and for all 23 cumulative tests. Also includes answers to Supplemental Practice Problems and Facts Practice problems Math 87, Third Edition, Solutions Manual
Review 1 for Math 87, Third Edition, Solutions Manual
Overall Rating:
5out of5
Excellent book!
Date:November 7, 2012
PCWhisperer
Location:Atlanta, GA
Age:55-65
Gender:male
Quality:
5out of5
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5out of5
Meets Expectations:
5out of5
This book brought tears to my eyes! It had me spellbound! I highly recommend it to Tutors and Home-School Instructors. Seriously though, it has made my job teaching Pre-Algebra a snap. I am so much smarter now. Oh yeah, and the price was reasonable as well. nuff said.
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Description
This course is designed to introduce the preservice K-9 teacher with ideas, techniques and approaches to teaching mathematics. Manipulatives, children's literature, problem solving, diagnosis and remediation, assessment, equity issues, and the uses of the calculator are interwoven throughout the topics presented. The math content areas are rational numbers and geometry.
The Viterbo College Teacher Education Program has adopted a Teacher As Reflective Decision Maker Model. Each course is designed to contribute to the development of one or more of the knowledge bases in professional education.
This course contributes to the development of the knowledge bases: Knowledge of the Learner, Curriculum Design, Planning and Evaluation, and Instructional and
Classroom Management.
Goals
To help students:
1. learn to value mathematics;
2. learn to reason mathematically;
3. learn to communicate mathematically;
4. become confident in their mathematical ability; and
5. become problem solvers and posers.
Objectives
Upon successful completion of this course, the student will be able to:
explore, conjecture, reason logically and use a variety of mathematics methods effectively to solve nonroutine problems.
establish classroom environments so that his or her students can explore, conjecture,
reason logically and use a variety of mathematics methods effectively to solve nonroutine problems and develop a lifelong appreciation of math in their lives;
. design and use several forms of assessment, such as portfolios, journals, open-ended problems, tests, and projects
become familiar with educational research on effective teaching of mathematics.
Student Responsibilities
One cannot benefit from or contribute to a class discussion or activity unless one is physically present (this a necessary condition, not a sufficient one). Attendance is
required. Call me (796-3658) if you will not be in class. A valid excuse is necessary to miss class. Unexcused absences may lower your grade for the course.
Assigned readings of the texts and handouts need to be done if meaningful discussion can occur.
As teachers you should appreciate the importance of class participation. Your active participation makes the course go.
Math is not a spectator sport. Assigned problems and textbook exercises are ways for you to develop problem solving skills and reflect on your learning.
Requirements
Six summaries of articles in professonal journals on the following topics (include a copy of the article in your summary; article must be at least two
pages long.)
Geometry <due September 14>
Assessment <due September 28>
Technology <due October 12>
Measurement <due November 2>
Fractions <due November 30>
Equity and mathematics <due December 14>
The purpose of this assignment is to acquaint you with some resources outside of the textbook and to introduce you to some ideas or activities that
you may want to share with the class when we are investigating the appropriate topic.
Please follow these guidelines:
Include a copy of the article with your summary.
Use the reporting form included in your packet.
Articles must be at least two pages long in the original citation.
Articles taken from the internet must be complete (No missing pictures, diagrams, or equations.)
A problem notebook with assigned problems from the text and class.
You must work out the solutions. Merely copying answers from the solutions manual is not appropriate.
Completion of a minimum of 12 hours of field experience working with an elementary student on mathematics
A journal of your sessions with an elementary student. [NOTE: you MUST MEET WITH YOUR STUDENT AND FULFILL THIS REQUIREMENT IN ORDER TO PASS THIS COURSE.]
Two math activities, one on geometry and one on fractions
Four investigations
Two in-class exams
Learning journal
Oral interview
A Note
Some of you may have had mathematics courses that were based on the transmission, or absorption, view of teaching and learning. In this view, students
passively
Consequently, I have three goals when I teach. The first is to help you develop mathematical structures that are more complex, abstract, and powerful than the ones
you currently possess so that you will be capable of solving a wide variety of meaningful problems. The second is to help you become autonomous and self-
motivatedHopefully, I want to help you learn to do something different from and better than what you have experienced as pupils in previous mathematics classes A mathematics methods class is about mathematics, about children as learners of mathematics, about how mathematics can be learned and taught, and about how classrooms can be environments for learning mathematics. It's a class where the students learn about learning mathematics while they themselves are learning mathematics. As SoYou may find this experience frustrating at times. Persevere! Eventually I hope you will own personally the mathematical ideas you once knew unthinkingly or only
peripherally (and sometimes anxiously). I want you to become competent and confident using mathematical ideas and techniques. I want you to be ready to learn
how to get other persons actively involved in problem solving. To nurture a mathematical idea in the mind of a child might be easier if it first thrived in the mind of
the child's teacher.
Americans with Disabilities Act
If you are a person with a disability and require any auxiliary or other accommodations for this class, please see me and Wayne Wojciechowski, the Americans With
Disabilities Act Coordinator (MC 320 - 796- 3085 ) within ten days to discuss your accommodation needs.
It is somewhat surprising and discouraging how little attention has been paid to
the intimate nature of teaching and school learning in the debates on education
that have raged over the past decade. These debates have been so focused on
performance and standards that they have mostly overlooked the means by
which teachers and pupils alike go about their business in real-life classrooms _
how teachers teach and how pupils learn.
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Mathematics, Standard D. MEASUREMENT
Content Standard:
Students in Wisconsin will discover, describe, and generalize
simple and complex patterns and relationships. In the context
of real-world problem situations, the student will use algebraic
techniques to define and describe the problem to determine and
Content Standard:
Students in Wisconsin will use media and technology critically
and creatively to obtain, organize, prepare and share information;
to influence and persuade; and to entertain and be entertained.
B.8.1 Interpret the past using a variety of sources, such as biographies, diaries, journals, artifacts, eyewitness interviews, and other primary source materials, and evaluate the credibility of sources used
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Measure theory is a classical area of mathematics born more than two thousand years ago. Nowadays it continues intensive development and has fruitful connections with most other fields of mathematics as well as important applications in physics. This book gives an exposition of the foundations of modern measure theory and offers three levels of presentation: a standard university graduate course, an advanced study containing some complements to the basic course (the material of this level corresponds to a variety of special courses), and, finally, more specialized topics partly covered by more than 850 exercises. Volume 1 (Chapters 1-5) is devoted to the classical theory of measure and integral. Whereas the first volume presents the ideas that go back mainly to Lebesgue, the second volume (Chapters 6-10) is to a large extent the result of the later development up to the recent years. The central subjects of Volume 2 are: transformations of measures, conditional measures, and weak convergence of measures. These three topics are closely interwoven and form the heart of modern measure theory. less
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Logarithms
This is a site used by the Math Department at the
University of North Carolina - Chapel Hill, to help students taking
lower level math courses.
When you get there, select: Unit 4,
Logarithms. Get out your notes on logarithms.
Read through the Introduction, the
Explanation, and each of the objectives. Match your notes with the
information at the site. Add to your notes when you feel it will
be helpful.
Write an example of the "log
loop."
List the logarithms you can
compute using your calculator.
Write the value of e and write
a short explanation of how it is found and who first studied
it.
Another source of
information on e can be found at the MacTutor
History of Mathematics
Archive. Click on
this link, then click on: Search the Archive > History
Topics > keyword exponential > The number e.
List the basic properties of
logarithms.
Write solutions for the
problems in the four examples.
Now select Logarithm
Equations.
Read the Introduction, the
Explanation, and each of the objectives.
Write solutions to the 4
example problems.
Read the lesson on Exponential
Growth and Decay.
Do the interactive worksheet,
Exponential Growth and Decay, listed in the right column, toward
the top. Write your solutions.
Hand in your answers and
solutions to everything above that was written in
red.
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Degrees and Certificates
Course Details
GREAT IDEAS IN MATHEMATICS
Course Number:
MATH-110
Course Description:
The beauty and significance
of mathematics in the history of human thought. Topics
include primes, the pigeonhole principle, the Fibonacci sequence,
infinity, chaos and fractals. Prerequisites: High school algebra I and
geometry, or equivalent. Offered spring of even-numbered years.
3 credits. Not for General Science majors. (QR)
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It may be true that most books start with commutative algebra, but I think that it's possible to teach alggeom with simple pedagogical examples. My favorite one is about the definition of a point for rings like K x L where K and L are fields - there's no distinction between prime and maximal ideals and that simplifies things a lot.
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Algebra 2 is a continuation of topics first discussed in Algebra 1. Not only is success dependent upon a good understanding of basic calculations (multiplication, division, subtraction and addition), but students will be introduced to more theoretical ideas such as imaginary numbers, matrices an...
...Thanks. This is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures. Together with geometry, analysis, topology, combinatorics, and number theory, algebra is one of the main branches of pure mathematics
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Algebra Problems – Effective Teaching
Algebra problems are one among the significant aspects of mathematics. But, most of the time, people fail to be good at them due to their apparently confusing nature. And the difficulty is that one cannot even avoid them. If you are one among those who fall into this group, there is a good news for you, subliminal messages.
Subliminal messages, a surefire way to make Algebra problems enjoyable
Generally, people turn away from algebra problems because they consider them to complicated to be handled by themselves. But the fact is just the opposite. Often, it is your own negative feelings that make algebra problems tough. If you manage to come out of your negative frame of mind and program your brain about positive thoughts about algebra, you are sure to be good at them. Subliminal messages are messages sent to your subconscious mind so as to get rid of any unwanted habit or feeling. And the same can be applied to assist your brain to get rid of the preconceived idea that algebra problems are beyond your capabilities.
A few more tips to make algebra problems enjoyable
Refrain from pressurizing your brain. In the instance of your failure to solve an algebra problem, do not blame yourself for your incapability. Continue with your positive thoughts about the subject and keep on practicing. Practice helps you get rid of your innate fear for algebra problems. Do not ever adopt the attitude of procrastination. Always feel confident about your potentials and Keep on trying to convince your brain of its capabilities and continue with your attempts. A time is sure to come when you start enjoying algebra problems and treat them as fun games.
Do not simply turn away from algebra problems, they would be there to help you in life with wise logic in times of trouble.
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MATH11160 Technology Mathematics
Course details
In this course students apply essential mathematical concepts, processes and techniques to support the development of mathematical descriptions and models for engineering problems. They investigate and apply the properties of linear, quadratic, exponential and logarithmic functions in appropriate settings, use trigonometric functions to solve triangles and describe periodic phenomena and use vector and matrix algrebra to solve problems in an engineering context. Concepts of elementary statistics to organise and analyse data are covered. Students select appropriate mathematical methods appreciating the importance of underlying assumptions and then use them to investigate and solve problems, and interpret the results. Other important elements of this course are the communication of results, concepts and ideas using mathematics as a language, being able to document the solution to problems in a way that demonstrates a clear, logical and precise approach and communicating, working and learning in peer learning teams where appropriate. Distance education (FLEX) students are required to have significant access to a computer and make frequent use of the internet
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How Much Work is Required: Intuition vs. Mathematical Calculationpart of Pedagogy in Action:Library:Interactive Lectures:Examples This classroom activity presents Calculus II students with some Flash tutorials involving work and pumping liquids and a simple question concerning the amount of work involved in pumping water out of two full containers having the same shape and size but different spatial orientations.
Partial Derivatives: Geometric Visualizationpart of Pedagogy in Action:Library:Interactive Lectures:Examples This write-pair-share activity presents Calculus III students with a worksheet containing several exercises that require them to find partial derivatives of functions of two variables. Afterwards, a series of Web-based animations are used to illustrate the surface of each function, the path of the indicated partial derivative for a specified value of the variable and the value of the derivative at each point along the path.
Mathematical Curve Conjecturespart of Pedagogy in Action:Library:Interactive Lectures:Examples In this activity, a six-foot length of nylon rope is suspended at both ends to model a mathematical curve known as the hyperbolic cosine. In a write-pair-share activity, students are asked to make a conjecture concerning the nature of the curve and then embark on a guided discovery in which they attempt to determine a precise mathematical description of the curve using function notation.
Riemann Sums and Area Approximationspart of Pedagogy in Action:Library:Interactive Lectures:Examples After covering the standard course material on area under a curve, Riemann sums and numerical integration, Calculus I students are given a write-pair-share activity that directs them to predict the best area approximation methods for each of several different functions. Afterwards, the instructor employs a Web-based applet that visually displays each method and provides the corresponding numerical approximations.
U.S. Population Growth: What Does the Future Hold?part of Pedagogy in Action:Library:Interactive Lectures:Examples College Algebra or Liberal Arts math students are presented with a ConcepTest, a Question of the Day and a write-pair-share activity involving U.S. population growth. The results are quite revealing and show that while students may have learned how to perform the necessary calculations, their conceptual understanding concerning exponential growth may remain faulty. Student knowledge (or lack thereof) of the size of our population and its annual growth rate may also be surprising.
Using Satellite Data and Google Earth to Explore the Shape of Ocean Basins and Bathymetry of the Sea Floorpart of Pedagogy in Action:Library:Teaching with Data:Examples This activity is for an introductory oceanography course. It is designed to allow students to use various tools (satellite images, Google Earth) to explore the shape of the sea floor and ocean basins in order to gain a better understanding of both the processes that form ocean basins, as well as how the shape of ocean basins influences physical and biological processes.
Determining the Geologic History of Rocks from a Gravel Depositpart of Examples Gravels deposited as a result of continental glaciation are used to teach introductory-level earth-science students the application of the scientific method in a cooperative learning mode which utilizes hands-on, minds-on analyses. Processes that involve erosion, transportation, and deposition of pebble- and cobble-sized clasts are considered by students in formulating and testing hypotheses.
Limiting Reactants: Industrial Case Studypart of Examples An exercise in which students apply limiting reactants, mass ratios and percent yields to suggest an optimum industrial process. Cost figures are provided but students are told to come up with, and defend, their own criteria for their recommendation.
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Computing Curriculum 2001
Report of Group 2: Supporting Courses
Draft 5.2: July 7, 2000
Supporting Courses
The discipline of computing interacts in essential ways with many other
disciplines at both practical and theoretical levels. Thus, computing
students must gain an understanding of subjects outside of computing, as
well as a mastery of core computing topics. In particular, students
focusing their studies on computing also must have an adequate background
in each of the following areas:
mathematics,
natural science,
application areas,
ethical and social issues,
communication skills, and
understanding of world cultures.
Such areas may be addressed by various combinations of supporting courses
and other experiences. Note that a summary of recommendations appears at
the end of this report.
* * * * * * * * * * *
A. Mathematics
Mathematical methodologies and topics pervade many areas of computing. For
example, Computing Curricula 1991 identified theory as one of the three
primary working methodologies or processes within computer science. The
theory process contains the major elements of definitions and axioms,
theorems, proofs, and interpretation of results. In addition, mathematics
provides a language for working with ideas relevant to computing, specific
tools for analysis and verification, and a theoretical framework for
understanding important computing ideas. For example, functional
programming and problem solving draw directly upon the mathematical
concepts and notations for functions; algorithmic analysis depends heavily
on the mathematical topics of counting, permutations and combinations, and
probability; discussions of concurrency and deadlock draw heavily from
graph theory; and both program verification and computability build upon
formal logic and deduction.
Goals:
Students of computing should:
attain sufficient mathematical maturity and insight to work
comfortably at an abstract, logical level with computing concepts,
attain knowledge of specific topics which support fundamental
areas of computer science,
understand topics in mathematics, not just by themselves, but as
integral parts of numerous areas within computing, and
utilize mathematical methods and insights to support and model
algorithmic thinking as a means to solve various types of problems.
With the pervasive role of mathematics within computing, the computing
curriculum should discuss mathematical concepts early and often. For
example, basic concepts of discrete mathematics should be introduced early
within a student's course work, and later courses should use these concepts
regularly. While different colleges and universities will identify
alternative prerequisite structures according to local needs and
opportunities, a variety of upper-level computing courses should depend
upon prior knowledge of specific mathematical content, and this content
should be reflected in the formal pre-requisites of upper-level courses.
To achieve this level of proficiency, course work should include the
equivalent of the following:
2 to 4 semesters of mathematics supporting specific areas of interest
to the student -- likely including 1 to 2 semesters of calculus.
As a common alternative, the more technically-oriented curriculum might
expand both the amount of discrete mathematics and the coverage of
probability and statistics to yield the following:
2 semesters, emphasizing discrete structures, proof,
basic counting,
1 semester of probability and statistics,
1 to 3 semesters of mathematics supporting specific areas of interest
to the student -- likely including 1 to 2 semesters of calculus.
* * * * * * * * * * *
Coursework Involving Discrete Structures
Basic material on Discrete Structures should be covered early in the
curriculum, and this material should be used in several later course (e.g.,
algorithms, operating systems, programming languages). Such material may
be covered in several ways:
Some schools may offer separate courses on discrete mathematics
within the first two years of a student's program.
Some schools may elect to integrate the material on Discrete
Structures with other introductory computing courses.
The next part of this report illustrates how either of these approaches
might be followed. In each case, topics in discrete mathematics should be
combined into courses with unifying themes and with an emphasis on
applications of various mathematical concepts.
When offered as separate courses, Discrete Structures courses would provide
a solid introduction to discrete mathematics. Since courses in discrete
mathematics sometimes have earned the reputation of being a potpourri of
disjoint and irrelevant details, considerable care should be taken to shape
a coherent and theme-oriented course, where applications are shown for most
or all topics.
Discrete Structures I:
The following commentary provides a base for such a course, drawing heavily
from topics listed in the Discrete Structures Knowledge Units.
Themes:
functions and their properties
proof techniques and their applications
counting technique.
Applications:
RSA encryption
recurrence relations
combinations and permutations
elementary probability
tree properties
Course Outline
Introduction to Logic and Proofs (3 hours)
direct proofs
proof by contradiction
mathematical induction
[these techniques should be used regularly
in what follows]
Boolean Algebra (2 hours)
the algebra of Boolean values and operations
possible application:
  digital circuitry design
Functions (4 hours)
properties, composition
possible applications:
symbolic manipulation
in a symbolic algebra package
reduction, associativity
of composition
the combinatorial
explosion and its dangers
Relations (3 hours)
reflexivity, symmetry, transitivity,
equivalence relations
possible applications:
connection with
relational databases
common operations
(joins, selection)
Sets (2 hours)
Venn diagrams, complements, Cartesian
products, power sets
possible applications:
basics of types and
type inference
Number Theory (6 hours)
factorability, Chinese remainder theorem
properties of primes
modular arithmetic and different bases
possible application:
RSA encryption
hash functions
algorithms for arithmetic
operations within a computer (e.g., multiplication)
Cardinality and counting(12 hours)
Pigeonhole principle
basic finite probability
binomial coefficients
possible applications --
often using induction proofs:
size of power sets
combinations
permutations
Recurrence relations (3 hours)
basic formulae and solutions
possible applications
tree properties (e.g.,
height, number of leaves)
This course covers much of the knowledge units DS1, DS2, DS3, and DS4,
although few of those topics are covered together as a separate unit.
Rather, this course highlights applications, for which solutions require
proof, logic, and counting. In particular, while the above outline
indicates that logic and proofs are introduced early in a 3-hour unit, the
expectation is that these topics will be applied in class and homework
consistently throughout the rest of the semester. From this perspective,
one might consider that logic and proofs take up 30+ hours of this course.
Such a course might have strong appeal to those interested in applied
mathematics or other disciplines, and this discrete math course might be
taught by mathematics faculty. Alternatively, these topics could be
introduced with explicit examples from computing, in which case the course
might be listed in either the mathematics or computing departments.
At some schools, these topics from mathematics might be integrated into
beginning computing courses. For example, an early introduction to
functional programming might be tied to a treatment of the mathematical
topics of functions, sets, relations, and recursion.
Prerequisites: This first mathematics course should have similar
prerequisites to traditional calculus course(s), so that work may proceed
at a reasonable level of rigor -- assuming proficiency with basic algebra,
geometry, trigonometry, logarithms, and exponential functions. In
particular, the above 2 semester requirement for discrete mathematics
assumes that students will have completed successfully a precalculus course
either in high school or college. Students without this background may be
unable to take a discrete math sequence concurrently with typical CS1/CS2
courses; such students will need to work with their advisors to catch up.
Then, later courses in computing should build explicitly on the
mathematical foundation begun early.
* * * * * * * * * * *
Discrete Structures II:
With the importance of discrete mathematics, formal methods, and theory in
many areas of computer science, students should take an intermediate-level
course on discrete mathematics -- building on Discrete Structures I and
serving as a prerequisite for some required upper-level computing courses.
As with Discrete Structures I, a follow-up course must be coherent and
clearly relevant, explicitly stating unifying themes and using mathematical
theory and rigor to obtain interesting results. The following sample
course illustrates how the topics of graphs, complexity, and probability
could provide such integrating themes. While many schools may choose to
cover these topics differently -- perhaps in courses on algorithms,
automata, and statistics, this outline illustrates a mechanism to combine
core topics efficiently and at an appropriate level of rigor.
Themes:
graphs
use of careful proof techniques in applications
algorithmic analysis and complexity
basic probability and statistics
Applications:
Resource allocation graphs
Traveling Salesperson Problem
Course Outline
Graphs (9 hours)
directed and undirected, weighted and unweighted
cycles, connectivity, other properties
basic algorithms and proof techniques
traversals (depth-first and breadth-first)
possible applications:
resource allocation graphs
and deadlock detection
Matrices (6 hours)
basic properties
possible applications:
adjacency matrices and
their powers (include careful proofs)
Basic order analysis (6 hours)
definition of Big O, little-o and theta
(if not covered previously in CS2)
Additional attention to proof techniques is found throughout this course.
Substitution of other topics or applications might support specific,
upper-division computing courses.
For many schools, some of these topics might be covered in a lower-division
course which emphasizes probability and statistics, while other topics
might be covered in upper-level courses in algorithms or automata. When
the topics are combined and integrated with unifying themes, however, the
course can cover core topics efficiently and use the solving of specific
problems as unifying themes.
The design and level of Discrete Structures II are vital for at least two
reasons:
Students should learn the topics listed above relatively early
in their careers, so these topics can provide a foundation for
upper-division courses in computing.
Students must gain proficiency and technical mastery of theory,
formal methods, proof, and analysis; and a one-semester foundation
cannot provide adequate mathematical background and skill.
Rather than separate discrete structures from introductory programming at
the beginning level, the two areas might be merged with both the
mathematics and computer science motivating and supporting the other.
As in the description of Discrete Structures 1 and 2, applications are
stressed for each mathematical topic, although here those applications
largely relate to concepts of programming and algorithms.
CS-M1: Functional Programming, Proof, and Counting
Themes:
functional programming
functions, sets, relations, and their uses
recursion and mathematical induction
proof techniques
Applications:
List Processing
Construction and counting of set constructs
Web interfaces and CGI programming
Course Outline
Introduction to Functional Programming and mechanics (3 hours)
read-evaluate-print cycle
programming environment
Lists and User-defined Procedures (3 hours)
functions and parameters
functional properties and composition
application:
list functions
Boolean Expressions and Conditionals (3 hours)
Boolean algebra: values and logical operations
applications:
conditional execution
simplification of
statements using Boolean expressions
Recursion (6 hours)
on strings, lists, numbers
flat and deep recursion
possible applications:
sets as lists,
with set operations
relations as n-tuples
within lists
Properties of Functions and Function Usage (2 hours)
pre-conditions, post-conditions, and invariants
properties of relations and functions
(symmetry/antisymmetry,
reflexivity, irreflexivity, transitivity)
applications:
checking of input parameter
values
checking of intermediate values
Cardinality and Counting (12 hours)
Pigeonhole principle
basic finite probability
set theory (Venn diagrams, complements,
products, power sets)
connection between recursion and
mathematical induction
possible applications
(combining computation and inductive proof):
Venn diagrams, complements
generating
and counting cardinality of Cartesian products
and power sets
random numbers and simulation
combinations and permutations
Local Bindings and Problem-Solving (4 hours)
tail recursion
evaluation
introduction to efficiency analysis and Big-O
Functional Concepts (4 hours)
map, filter, currying
procedures as values
variable arity
higher-order functions
anonymous functions
Input/Output and Files (3 hours)
simple I/O
streams
possible applications:
sequential file processing
filters
Web interfaces and CGI (4 hours)
HTML forms
elementary HCI principles
I/O with query strings
possible application:
Web-based directory
look-up of file data
While many languages might be used to support the programming component of
this course, some common choices might be either dynamically-typed
languages like Scheme or LISP or statically-typed ones like ML or Haskell.
CS-M2: An Introduction to Object-Oriented Programming and Program Analysis
Themes:
Object-oriented principles
Efficiency and program analysis
Applications:
Searching and Sorting
Course Outline
Classes and objects as packages (records) of state and
methods (procedures and functions) (10 hours)
Calculus, Probability, Statistics, and/or Additional
Mathematics
In addition to these first two courses, students should take one or more
advanced course in mathematics to support electives beyond the computing
core and to be determined though the advising process. Traditionally, this
work centers on one or two semesters of calculus, and this sequence still
is appropriate for many computing students. For example, such an
introduction might include the following topics:
While calculus is appropriate for many computing students, others may find
different mathematics courses to provide better support for the students'
specific interests. Specific choices of additional mathematics courses
should support electives beyond the computing core and be determined though
the advising process. Possible choices might include:
multivariable calculus,
differential equations,
numerical analysis,
logic,
linear algebra,
probability and statistics,
number theory,
abstract algebra, or
statistics.
It is expected that specific upper-level computer science courses will have
one or more of these topics as prerequisites. For example, a computer
graphics course may require as a prerequisite linear algebra, multivariable
calculus, and/or differential equations.
Regardless of the choice of an additional course work in mathematics, all
computing students must develop reasonable facility in mathematical
thinking and a good mathematical skills.
In addition to the mathematics topics and courses described here, other
mathematically-based topics are described elsewhere in these
recommendations. For example, automata is described under the Foundations
of Computing heading, and Algorithms include many concepts and techniques
from graph theory.
* * * * * * * * * * *
B. Science
As noted in Computing Curricula 1991, the process of abstraction (data
collection, hypothesis formation and testing, experimentation, analysis)
represents a vital component of logical thought within the field of
computing. The scientific method represents a basis methodology for much
of the discipline of computer science, and students should have a solid
exposure to this methodology.
Goals:
Computing Students should:
develop firm understanding of the scientific method, and
experience this mode of inquiry in a science course
which incorporates labs as a part of the study of science.
To achieve these goals, students must have direct hands-on experience with
hypothesis formulation, experimental design, hypothesis testing, and data
analysis. While a curriculum may provide this experience in various ways,
it is vital that students must "do science", not just "read about science".
Thus, all computing students should take:
1 semester of a lab-based course which applies the
scientific method
in substantive and extensive ways.
While such exposure might be achieved in lab-based, computer science
courses, most schools (at least in the United States) may require one or
more semesters of a natural science (e.g., in biology, chemistry, or
physics). Although this requirement allows substantial flexibility in terms
of subject matter, any science option must include a lab component to
provide actual experience with the scientific method.
* * * * * * * * * * *
C. Application Areas
With the broad range of applications of computing in today's society,
computing scientists must to be able to work effectively with people from
other disciplines.
Goal:
Many students of computing should:
engage in the in-depth study of some subject, which uses or discusses
computing in a substantive way.
Computing students have a wide range of interests and professional goals.
In some cases, for some students in some programs, diversity of curricular
program may be achieved through novel or innovative approaches beyond the
scope of a standard computing curriculum. (For example, for a double major
in computer science and Asian studies, the Asian studies may not use
computing in a substantive way, but may be very useful to them in
their chosen careers which have involved work with Japanese and other
companies. Similarly, for a double major in computer science and religious
studies, philosophical issues from the second major may be extremely
helpful in a technical career emphasizing public policy.)
For many students, study of computing together with an application area
will be extremely useful. Such work might be accomplished in several ways:
some approaches might include an extended internship experience or the
equivalent of a full semester's work (e.g., 4-5 semester courses or a
minor) that would count toward a major in that discipline. For example,
computing students are encouraged to study such fields as psychology,
sociology, economics, biology, business, or one of the science or
engineering disciplines.
* * * * * * * * * * *
D. Ethical and Social Issues
Computing Curricula 1991 stated, "Undergraduates ... need to understand the
basic cultural, social, legal and ethical issues inherent in the discipline
of computing. ... Students also need to develop the ability to ask serious
questions about the social impact of computing and to evaluate proposed
answers to those questions. ... Finally, students need to be aware of the
basic legal rights of software and hardware vendors and users, and they
also need to appreciate the ethical values that are the basis for those
rights. [p. 11]" Such comments are at least as relevant today, with the
integration of computing into an ever increasing range of areas within
society.
Goals:
Students of computing should:
understand the cultural, social, legal, and ethical context and
consequences of computing,
anticipate possible issues that can arise from technology in general
and computing in particular, and
be able to frame alternative responses to possible ethical and
social issues.
Computing curricula should consider including a course on ethical and
social issues -- possibly taught by a faculty member in philosophy or in a
social science discipline. However, students also should come to
appreciate that these issues are not tangential to computing, and computing
faculty should at least touch upon these issues periodically in regular and
core computing courses.
Possible approaches for including these topics within a curriculum may be
found in the [forthcoming] report of Pedagogy Focus Group 4: Professional
Practices.
* * * * * * * * * * *
E. Communication Skills
A widely-heard contemporary theme is that computer scientists must be able
to communicate effectively with colleagues and clients; no longer can a
computer professional expect to work in isolation most of the time. Thus,
computing students must sharpen their oral and writing skills in a variety
of contexts -- both inside and outside of computing courses.
Goals:
Computing students should be able to:
work effectively in team-based projects,
communicate ideas effectively in written form, and
make both formal and informal presentations orally.
While institutions may work to accomplish these goals in many ways, the
program of each student of computing must include numerous occasions for
improving writing, practicing oral communication (using both speaking and
active listening skills), and working in small groups.
At the least, a computing curriculum should require:
1 semester of a course emphasizing the mechanics and
process of writing.
English composition (grammar, punctuation, and other
mechanics)
writing (production and editing of at least 3 3-4 page
papers),
a significant experience involving small-group projects and
large-group progjects, and
at least one formal oral presentation to a group.
Furthermore, the computing curriculum should integrate writing
and verbal discussion consistently in substantive ways ways. Communication
skills should not separate from computing, but rather should be fully
incorporated into the computing curriculum and its requirements. Some
mechanisms to achieve this could include the following:
requirement of a course in oral communications and/or a course
in technical writing,
attention paid to lab writeups in any computing course with a
formal lab,
for a project-based experience:
* requirement of a formal written proposal with an oral presentation
and defense,
* requirement of a formal written final report with an oral
presentation and defense,
for a capstone course or experience,
* requirement of a formal written final report and oral presentation,
encouragement or requirement for upper-level students to produce
and present posters of their work publicly (e.g., at departmental
seminars, open houses, parents' weekends, or programs for
prospective students),
encouragement or requirement for upper-level students to serve
(under supervision) as tutors or lab assistants, or graders to
help introductory-level students,
inclusion of short- or long-essay problems on tests in computing
courses, with matters of exposition included in the grading scale
(e.g., grading on the basis of what is said, rather than on what
might have been meant),
inclusion of justifications and explanations in lab writeups and
assignments,
student preparation and presentation of selected portions of course
materials, instead of the instructor always lecturing,
inclusion of design documentation as part of a programming
assignment (with the design work submitted and graded before coding
begins),
as part of projects, inclusion of a paper commenting on related
ethical issues, contributions of team-members, and possibility of
purchasing or downloading project components,
assignment of a paper which relates codes of ethics (e.g., from
ACM, DPMA, or IFIP) to practical issues currently under study,
encouragement or requirement of students to contribute to
computer science seminars or colloquia,
participation of students in computing clubs,
presentation of student experiences in internships, coops, or
practica.
While suggestions indicate just a few of the ways that students may gain
proficiency in speaking, writing, and group interactions, every curricular
program should ensure that students have several such experiences.
Overall, students should develop their skills both in non-technical and in
technical courses, so they come to understand that the same communication
principles from other disciplines also apply within the discipline of
computing.
* * * * * * * * * * *
F. Exposure to World Cultures.
Technology has been a primary factor in the interdependency of world
cultures and societies; information technology plays a vital role as
societies interact within the global village. Also, Section 3.2 of
Computing Curriculum 2001 begins, "Computing education is ... affected by
changes in the cultural and sociological context in which it occurs." With
the multinational dimensions of the information technology field, computing
professionals increasingly are called upon to function within an
environment involving different cultures and backgrounds.
Goal:
In order to function within this increasingly multi-cultural world,
students of computing should have a basic understanding of multiple
cultures.
To accomplish this goal, students are strongly encouraged to engage in the
formal study of a foreign language and to take courses that discuss world
cultures and international issues and perspectives. For many students,
this might involve the following course work:
3 semesters of courses discussing world cultures and
international issues and perspectives,
possibly including 2 or more semesters
of a foreign language at the college level.
While these suggestions fall outside the scope of course work required
for a computer science degree, some institutions may choose to incorporate the
above work into general distribution requirements. Further, within a
liberal arts context, a curriculum might emphasize the study of world
cultures further through an additional requirement of 1 to 3 courses in
this area. While such requirements would support the breadth valued within
liberal arts, this further work also would add valuable international
perspectives to benefit future computing professionals.
* * * * * * * * * * *
Summary of Support Course Recommendations:
The following summarizes the recommendations made above.
Required:
Mathematics
2 semesters of discrete mathematics
1-2 semesters of mathematics to support
advanced work in computing, often including
calculus
of one variable (through integral calculus)
an additional 1 or 2 semesters for students in
technically-oriented programs
(e.g., offered by
technically-oriented programs, but not liberal arts programs)
Science
1 semester of a lab-based course which applies
the scientific method
in substantive and extensive ways.
Ethical and Social Issues
considerable exposure to ethical and
social issues of computing, including substantive
work within
computing courses
Communication Skills
1 semester course emphasizing the
mechanics and process of writing.
a
significant experience involving small-group projects,
at least one formal oral presentation to a group
Additional Recommendations:
Application Area
in-depth study of some subject, which uses
or discusses computing in a substantive way,
|
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General
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Mathematics
Chapters
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Honors Calculus :
Students who are interested in math or
science might also want to consider a more challenging Honors version
of Calculus II and III, Math 116 and Math 260 (the analogues of math
114 and math 240, respectively). These courses will cover essentially
the same material as 114 and 240, but more in depth and involve
discussion of the underlying theory as well as computations.
Introduction to concepts
and methods of calculus for students with little or no previous
calculus experience. Polynomial and elementary transcendental
functions and their applications, derivatives, extremum problems,
curve-sketching, approximations; integrals and the fundamental theorem
of calculus.
Brief review of High
School calculus, applications of integrals, transcendental functions,
methods of integration, infinite series, Taylor's theorem. Use of
symbolic manipulation and graphics software in calculus.
Functions of several
variables, vector-valued functions, partial derivatives and
applications, double and triple integrals, conic sections, polar
coordinates, vectors and analytic geometry, first and second order
ordinary differential equations. Applications to physical
sciences. Use of symbolic manipulation and graphics software in
calculus.
Functions of several
variables, partial derivatives, multiple integrals, differential
equations; introduction to linear algebra and matrices with
applications to linear programming and Markov processes. Elements
of probability and statistics. Applications to social and
biological sciences. Use of symbolic manipulation and graphics software
in calculus.
This is an Honors level
version of Math 114 which explores the mathematics more deeply.
L/L 122-123. Community
Math
Teaching Project. (M) Staff.
This course allows Penn
students to teach a series of hands-on activities to students in math
classes at University City High School. The semester starts with
an introduction to successful approaches for teaching math in urban
high schools. The rest of the semester will be devoted to a
series of weekly hands-on activities designed to teach fundamental
aspects of geometry. The first class meeting of each week, Penn
faculty teach Penn students the relevant mathematical background and
techniques for a hands-on activity. During the second session of each
week, Penn students will teach the hands-on activity to a small group
of UCHS students. The Penn students will also have an opportunity
to develop their own activity and to implement it with the UCHS
students.
L/R 170. Ideas in
Mathematics. (C) Natural Science &
Mathematics Sector. Class of 2010 and beyond. Staff. May also be
counted toward the General Requirement in Natural Science &
Mathematics.
Topics from among the
following: logic, sets, calculus, probability, history and philosophy
of mathematics, game theory, geometry, and their relevance to
contemporary science and society.
This course focuses on
the creative side of mathematics, with an emphasis on discovery,
reasoning, proofs and effective communication, while at the same time
studying real and complex numbers, sequences, series, continuity,
differentiability and integrability. Small class sizes permit an
informal, discussion-type atmosphere, and often the entire class works
together on a given problem. Homework is intended to be
thought-provoking, rather than skill-sharpening.
This course focuses on
the creative side of mathematics, with an emphasis on discovery,
reasoning, proofs and effective communication, while at the same time
studying arithmetic, algebra, linear algebra, groups, rings and fields.
Small class sizes permit an informal, discussion-type atmosphere, and
often the entire class works together on a given problem.
Homework is intended to be thought-provoking, rather than
skill-sharpening.
This is an honors version
of Math 240 which explores the same topics but with greater
mathematical rigor.
299. Undergraduate
Research in Mathematics Staff. Prequisite(s): Knowledge
of elementary arithmetic and calculus will be helpful.
Introduction to adeles and their application in Analysis, Geometry and
Number theory.
312. Linear Algebra.
(M)
Staff. Prerequisite(s): MATH 240. Students who have already received
credit for either Math 370, 371, 502 or 503 cannot receive further
credit for Math 312 or Math 313/513. Students can receive credit
for at most one of Math 312 and Math 313/513.
Linear transformations,
Gauss Jordan elimination, eigenvalues and eigenvectors, theory and
applications. Mathematics majors are advised that MATH 312 cannot
be taken to satisfy the major requirements.
313. (CIS 313,
MATH513) Computational Linear Algebra.
Staff. Prerequisite(s):
Math 114 or 115, and some programming experienceMany important problems
in a wide range of disciplines within computer science and throughout
science are solved using techniques from linear algebra. This
course will introduce students to some of the most widely used
algorithms and illustrate how they are actually used.
Some
specific
topics:
the
solution
of systems of linear equations by
Gaussian elimination, dimension of a linear space, inner product, cross
product, change of basis, affine and rigid motions, eigenvalues and
eigenvectors, diagonalization of both symmetric and non-symmetric
matrices, quadratic polynomials, and least squares optimization.
Applications
will
include
the
use
of matrix computations to computer
graphics, use of the discrete Fourier transform and related techniques
in digital signal processing, the analysis of systems of linear
differential equations, and singular value decompositions with
application to a principal component analysis.
The
ideas
and
tools
provided
by this course will be useful to students
who intend to tackle higher level courses in digital signal processing,
computer vision, robotics, and computer graphics.
320. Computer Methods
in Mathematical Science I. (A) Staff. Prerequisite(s):
MATH 240 or concurrent and ability to program a computer, or permission
of instructor.
Students will use
symbolic manipulation software and write programs to solve problems in
numerical quadrature, equation-solving, linear algebra and differential
equations. Theoretical and computational aspects of the methods
will be discussed along with error analysis and a critical comparison
of methods.
Topics will be drawn from
some subjects in combinatorial analysis with applications to many other
branches of math and science: graphs and networks, generating
functions, permutations, posets, asymptotics.
Syllabus for MATH
360-361: a study of the foundations of the differential and integral
calculus, including the real numbers and elementary topology,
continuous and differentiable functions, uniform convergence of series
of functions, and inverse and implicit function theorems. MATH
508-509 is a masters level version of this course.
L/L 361. Advanced
Calculus. (C) Staff. Prerequisite(s):
MATH 360.
Continuation of MATH 360.
L/L 370. Algebra. (C) Staff. Prerequisite(s):
MATHSyllabus for MATH
370-371: an introduction to the basic concepts of modern algebra.
Linear algebra, eigenvalues and eigenvectors of matrices, groups, rings
and fields. MATH 502-503 is a masters level version of this
course.
L/L 371. Algebra. (C) Staff. Prerequisite(s):
MATH 370After a rapid review of
the basic techniques for solving equations, the course will discuss one
or more of the following topics: stability of linear and nonlinear
systems, boundary value problems and orthogonal functions, numerical
techniques, Laplace transform methods.
Method of separation of
variables will be applied to solve the wave, heat, and Laplace
equations. In addition, one or more of the following topics will
be covered: qualitative properties of solutions of various equations
(characteristics, maximum principles, uniqueness theorems), Laplace and
Fourier transform methods, and approximation techniques.
Random variables, events,
special distributions, expectations, independence, law of large
numbers, introduction to the central limit theorem, and applications.
432. Game Theory. (C) Staff.
A mathematical approach
to game theory, with an emphasis on examples of actual games.
Topics will include mathematical models of games, combinatorial games,
two person (zero sum and general sum) games, non-cooperating games and
equilibria.
475. Statistics of
Law. (M)
Staff. Prerequisite(s): Permission of instructor; no formal
mathematical prerequisite, but one year of college calculus would be
helpful.
Introduction to
probability and statistics with illustrative material drawn from
cases. Statistical inference. Basic concepts of information
theory. This course may not be taken to satisfy the requirements of the
major.
480. (MATH550) Elementary
Topics in Advanced Real Analysis. (M) Staff.
Prerequisites: A year of analysis at the 300 level or above (for
example, Mathematics 360-361, or 508-509); a semester of linear algebra
at the 300 level or above (for
example, Mathematics 370) or Permission of Instructor.
499. Supervised
Study. (C)
Staff. Prerequisite(s): Permission of major adviser. Hours and credit
to be arranged.
Study under the direction
of a faculty member. Intended for a limited number of mathematics
majors.
L/L 502. Abstract
Algebra. (A) Staff. Prerequisite(s):
MathAn introduction to
groups, rings, fields and other abstract algebraic systems, elementary
Galois Theory, and linear algebra -- a more theoretical course than
Math 370.
L/L 503. Abstract
Algebra. (B) Staff. Prerequisite(s):
Math 502 or with the permission of the instructorContinuation of Math 502.
504. Graduate
Proseminar in Mathematics. (A505. Graduate
Proseminar in Mathematics. (BConstruction of real
numbers, the topology of the real line and the foundations of single
variable calculus. Notions of convergence for sequences of
functions. Basic approximation theorems for continuous functions
and rigorous treatment of elementary transcendental functions.
The course is intended to teach students how to read and construct
rigorous formal proofs. A more theoretical course than Math 360.
L/L 509. Advanced
Analysis. (B) Staff. Prerequisite(s):
Math 508 or with the permission of the instructor. Linear algebra
is also helpful.
512. Advanced Linear
Algebra.
Staff. Prerequisite(s): Math 114 or 115. Math 512 covers Linear Algebra
at the advanced level with a theoretical approach. Students can
receive credit for at most one of Math 312 and Math 512.
This course is open only to graduate students from departments other
than mathematics. The course as presently taught is merged with
Math 313, with additional work required from students enrolled
in Math 513. For a description of the topics and pre-requisites,
please refer to the description of Math 313.
Introduction to calculus
of variations. The topics will include the variation of a
functional, the Euler-Lagrange equations, parametric forms, end points,
canonical transformations, the principle of least action and
conservation laws, the Hamilton-Jacobi equation, the second variation.
Topics may vary but
typically will include an introduction to topological linear spaces and
Banach spaces, and to Hilbert space and the spectral theorem.
More advanced topics may include Banach algebras, Fourier analysis,
differential equations and nonlinear functional analysis.
549. Topics in
Analysis. (M) Staff. Prerequisite(s):
Math 548 or with the permission of the instructor.
Continuation of Math 548.
560. Selections from
Geometry and Topology. (M) Staff. Corequisite(s):
Math 500 or permission of the instructor.
In the last 25 years
there has been a revolution in image reconstruction techniques in
fields from astrophysics to electron microscopy and most notably in
medical imaging. In each of these fields one would like to have a
precise picture of a 2 or 3 dimensional object which cannot be obtained
directly. The data which is accessible is typically some collection of
averages. The problem of image reconstruction is to build an
object out of the averaged data and then estimate how close the
reconstruction is to the actual object. In this course we
introduce the mathematical techniques used to model measurements and
reconstruct images. As a simple representative case we study
transmission X-ray tomography (CT).In this context we cover the basic
principles of mathematical analysis, the Fourier transform,
interpolation and approximation of functions, sampling theory, digital
filtering and noise analysis.
585. The Mathematics
of Medical Imaging and Measurement. (M) Staff. Prerequisite(s):
Math 584 or with the permission of the instructor.
This course offers
first-hand experience of coupling mathematics with applications.
Topics will vary from year to year. Among them are: Random walks
and Markov chains, permutation networks and routing, graph expanders
and randomized algorithms, communication and computational complexity,
applied number theory and cryptography.
591. Advanced Applied
Mathematics. (M) Staff. Prerequisite(s):
Math 590 or with the permission of the instructor.
Introduction to
mathematics used in physics and engineering, with the goal of
developing facility in classical techniques. Vector spaces,
linear algebra, computation of eigenvalues and eigenvectors, boundary
value problems, spectral theory of second order equations, asymptotic
expansions, partial differential equations, differential operators and
Green's functions, orthogonal functions, generating functions, contour
integration, Fourier and Laplace transforms and an introduction to
representation theory of SU(2) and SO(3). The course will draw on
examples in continuum mechanics, electrostatics and transport problems.
|
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In this book the author has presented the elements of Coordinate Geometry in a manner suitable for beginners and junior students. Only Cartesian and Polar Coordinates are used, and the book forms Part I of the complete work. The Straight Line and Circle have been treated in detail supplemented by... more
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The ninth edition of Calculus and Analytical Geometry has been thoroughly updated to say only what is true and mathematically sound. As in earlier editions, there are Practice Exercises, Questions to Guide Your Review, Additional ExercisesTheory, Examples and Applications , at the end of each section. Preliminaries Limits and Continuity... more
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About the Book : This book is intended as a textbook for a first-year graduate course on algebraic topology, with as strong flavoring in smooth manifold theory. Starting with general topology, it discusses... This book is intended as a textbook for a first-year graduate course on algebraic topology, with as... more
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This title presents a demonstration of how the true mathematician learns to draw unexpected analogies, tackle problems from unusual angles and extract a little more information from the data. It is a collection of truly practical lessons. In lucid and appealing prose, Polya reveals how the mathematical method of demonstrating... more
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This book will serve as suitable text-cum-reference book for both in undergraduate and postgraduate students. The book is divided into two sections-A and B. Section-A consists of six chapters and section-B consists of seven chapters. Section-A deals with Introduction followed by Theory of Curves, Envelopes and Developables. Curves on surfaces... more
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Having trouble with geometry? Do Pi, The Pythagorean Theorem, and angle calculations just make your head spin? Relax. With Head First 2D Geometry, you'll master everything from triangles, quads and polygons to the time-saving secrets of similar and congruent angles -- and it'll be quick, painless, and fun.... more
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From the Foreword by Professor Leonidas J. Guibas Geometry, graphics, and vision all deal in some form with the shape of objects, their motions, as well as the transport of light and its interactions with objects. This book clearly shows how much they have in common and the kinds of... more
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READERSHIP: Students of Arts, Science & Engineering This book presents in an elegant way, the essentials of two and three dimensions of analytical geometry with plenty of examples to illustrate the basic ideas and to bequeath to the students numerous techniques of problem-solving. The exercises provide ample problems to supplement2 new & used from sellers starting at 3,276 In Stock.Ships Free to India in
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The paleantologist Zahoor is trying to do his research while General Zia is launching a campaign to Islamise knowledge. Science is being rewritten and called Islamic Science. The teaching of evolution is banned. Nothing is natural or accidental; everything is revealed and ordained. On a fossil dig in the Salt... more
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This ACM volume deals with tackling problems that can be represented by data structures which are essentially matrices with polynomial entries, mediated by the disciplines of commutative algebra and algebraic geometry. The discoveries stem from an interdisciplinary branch of research which has been growing steadily over the past decade. The... more
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Analysis is the branch of mathematics concerned with issues that have their roots in calculus. The purpose of this course is to take the student on a leisurely and scenic stroll through parts of the world of analysis as to allow for a better understanding of some of the ideas of calculus. The student will meet certain methods and ways of thinking which will be useful for significant mathematical ingsight. The Mathematical Analysis course provides a glimpse of a few of the beauties, curiosities, and even pathologies that, for some, make mathematics more an addiction than an occupation.
Here is a sampling of the questions that will be answered during the course:
+Why are the real numbers so much more important in calculus than the rational
numbers?
+How far does the analogy between the hyperbolic functions and the trigonometric
functions extend?
+Are there more rational numbers than there are integers? (NO!)
+Are there more real numbers than there are rationals? (YES!)
+In the sequence of integers, how often do the prime numbers occur?
+I understand why the square root of two is irrational, but why are e and pi
irrational?
+What does the Heisenberg Uncertainty Principle have to do with integration
by parts?
+Students will be assessed on the basis of their in-class participation,
on homework assignments and, possibly, on a project.
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Fundamentals of Statistics
0131464493
9780131464490
Fundamentals of Statistics: For a one semester or one-term algebra-based Introductory Statistics Courses. Drawing on the author's extensive teaching experience and background in statistics and mathematics, this text promotes student success in introductory statistics while maintaining the integrity of the course. Four basic principles characterize the approach of this text: generating and maintaining student interest; promoting student success and confidence; providing extensive and effective opportunity for student practice; and allowing for flexibly of teaching styles. «Show less
Fundamentals of Statistics: For a one semester or one-term algebra-based Introductory Statistics Courses. Drawing on the author's extensive teaching experience and background in statistics and mathematics, this text promotes student success in introductory statistics... Show more»
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Helping Undergraduates Learn to Read Mathematics
Although most students "learn to read" during their first
year of primary school, or even before, reading is a skill which continues
to develop through primary, secondary and post-secondary school, as the
reading material becomes more sophisticated and as the expectations for
level of understanding increase. However, most of the time spent deliberately
helping students learn to read focuses on literary and historical texts.
Mathematical reading (and for that matter, mathematical writing) is rarely
expected, much less considered to be an important skill, or one which can
be increased by practice and training.
Even as an undergraduate mathematics major, I viewed mathematical reading
as a supplementary way of learning--inferior to learning by lecture or
discussion, but necessary as a way of "filling in the gaps."
Not until graduate school was I responsible for reading new material at
a high level of comprehension. And, as I began to study primarily written
mathematics (texts and articles) rather than spoken mathematics (lectures),
I discovered that the activities and habits needed to learn from written
mathematics are quite different from those involved in learning from a
mathematics lecture or from those used in reading other types of text.
As I consciously considered how to read mathematics more effectively and
to develop good reading habits, I observed in my undergraduate students
an uneasiness and lack of proficiency in reading mathematics.
In response to this situation, I wrote for my students (mostly math
majors in Introductory Abstract Algebra at the University of Chicago) two
handouts, one on reading theorems and the other
on reading definitions. These describe some
of the mental activities which help me to read mathematics more effectively.
I also gave a more specific written assignment, applying some of these
questions to a particular section of assigned reading. My hope was that,
as they were forced to actively engage in reading, they would discover
that reading mathematics could be a profitable pursuit, and that that they
would develop habits which they would continue to use.
More than one such written exercise is needed to significantly affect
the way that students view reading. While the students seemed to understand
the types of questions that are helpful, they needed some practice in carrying
these out, and even more practice using these activities in the absence
of a written assignment.
A Few Mathematical Study Skills... Reading Theorems
In almost any advanced mathematics text, theorems, their proofs, and
motivation for them make up a significant portion of the text. The question
then arises, how does one read and understand a theorem properly? What
is important to know and remember about a theorem?
A few questions to consider are:
What kind of theorem is this? Some possibilties are:
A classification of some type of object (e.g., the classification of
finitely generated abelian groups)
An equivalence of definitions (e.g., a subgroup is normal if, equivalently,
it is the kernel of a group homomorphism or its left and right cosets coincide)
An implication between definitions (e.g., any PID is a UFD)
A proof of when a technique is justified (e.g., the Euclidean algorithm
may be used when we are in a Euclidean domain)
Can you think of others?
What's the content of this theorem? E.g., are there some cases in which
it is trivial, or in which we've already proven it?
Why are each of the hypotheses needed? Can you find a counterexample
to the theorem in the absence of each of the hypotheses? Are any of the
hypotheses unneccesary? Is there a simpler proof if we add extra hypotheses?
How does this theorem relate to other theorems? Does it strengthen
a theorem we've already proven? Is it an important step in the proof of
some other theorem? Is it surprising?
What's the motivation for this theorem? What question does it answer?
We might ask more questions about the proof of theorem. Note that, in
some ways, the easiest way to read a proof is to check that each step follows
from the previous ones. This is a bit like following a game of chess by
checking to see that each move was legal, or like running the spell checker
on an essay. It's important, and necessary, but it's not really the point.
It's tempting to read only in this step-by-step manner, and never put together
what actually happened. The problem with this is that you are unlikely
to remember anything about how to prove the theorem. Once you've read a
theorem and its proof, you can go back and ask some questions to help synthesize
your understanding. For example:
Can you write a brief outline (maybe 1/10 as long as the theorem) giving
the logic of the argument -- proof by contradiction, induction on n, etc.?
(This is KEY.)
What mathematical raw materials are used in the proof? (Do we need
a lemma? Do we need a new definition? A powerful theorem? and do you recall
how to prove it? Is the full generality of that theorem needed, or just
a weak version?)
A Few Mathematical Study Skills... Reading Definitions
Nearly everyone knows (or think they know) how to read a novel, but
reading a mathematics book is quite a different thing. To begin with, there
are all these definitions! And it's not always clear why one would care
to know about these things being defined. So what should you do when you
read a definition?
Ask yourself (or the book) a few questions:
What kind of creature does the definition apply to? integers? matrices?
sets? functions? some pair of these together?
How do we check to see if it's satisfied? (How would we prove that
something satisfied it?)
Are there necessary or sufficient conditions for it? That is, is there
some set of objects which I already understand which is a subset or a superset
of this set?
Does anything satisfy this definition? Is there a whole class
of things which I know satisfy this definition?
Does anything not satisfy this definition? For example?
What special properties do these objects have, that would motivate
us to make this definition?
Is there a nice classification of these things?
Let's apply this to an example, abelian groups:
What kind of creature does it apply to? Well, to groups... in particular,
to a set together with a binary operation.
How do we check to see if it's satisfied? The startling thing is that
we have to compare every single pair of elements! This would be a big job,
so:
Are there necessary or sufficient conditions for it? Well, it's sufficient
that the group be cyclic, as we saw in the homework. Do you know of any
neccesary conditions?
Does anything satisfy this definition? Well, yes... the group of rational
numbers under addition, for example. We have a whole class of things which
satisfy the definition, too -- cyclic groups.
Does anything not satisfy this definition? Yes, matrix groups come
to mind first. There are finite non-abelian groups, but this is
harder to see... do you know of one yet?
What special properties do these objects have, that would motivate
us to make this definition? Some of these properties are obvious, others
are things which we had to prove. One example: If H and K are subgroups
of an abelian group, then HK is also a subgroup.
Is there a nice classification of these things? Why, yes, at least
for a large subcategory of them. We'll get to it later... it says, basically,
that a finite abelian group is always built in a simple way from cyclic
groups (Zn's).
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Pre-Calculus, a one-semester course, covers a variety of topics to prepare students for more advanced calculus courses. The course starts with functions and graphs and moves on to polynomial and rational functions. The course also examines exponential and logarithmic functions, along with trigonometric functions and applications. Students receive introduction to analytic geometry and discrete algebra. The course ends with an introduction to calculus, including lessons on limits, derivatives and integrals.
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Math
Math
Mathematics
Students are required to earn three math credits as part of their graduation requirements. The usual sequence of math courses for college-bound students, particularly those students with interests in science, mathematics, engineering, law, and medicine, is Algebra I, Geometry, Algebra II, Introductory Analysis, and Calculus. Exceptions to this order must be cleared by the administration. The minimal high school math sequence is Algebra I, Basic Geometry, and two of the Integrated Math courses. Algebra I, Basic Geometry, and Algebra II will fulfill the college entrance requirements for those students who are more interested in the humanities. Graphing or scientific calculators are required for some math courses. Calculators with QWERTY keyboard are not permitted.
Algebra I is a course in the language and methods of algebraic expressions and sentences and the first college-preparatory course in mathematics. Students will gain skill and precision in simplifying and solving equations and inequalities in one variable. Analysis of familiar situations and translation of their components to mathematical language is emphasized along with signed numbers, factoring, simplifying radical expressions, graphing on coordinate planes and solving systems of equations.
This course covers all of the topics in Algebra I – Modules A and B, but the course is taught over two semesters rather than one. There is special emphasis on hands-on learning and organizational skills.
This course covers all of the topics in Algebra I – Modules C and D, but the course is taught over two semesters rather than one. There is special emphasis on hands-on learning and organizational skills.
This course is designed for college-bound students. This course improves upon and extends the skills and concepts from Algebra I and stresses verbal precision and applications. The concept of function is introduced with emphasis on linear, quadratic, and logarithmic functions.
This course covers all of the topics taught in Algebra II, plus a thorough discussion and analysis of trigonometry. Graphing calculators are used to enhance the curriculum. This course moves more rapidly and is more challenging than regular Algebra II; it is designed for students planning a career in mathematics, engineering, physics, etc. Computer projects may be included.
Basic Geometry is a course for students who have a need or desire for more mathematics, but who do not require a rigorous approach as presented in regular geometry. This course presents all of the theorems and properties of plane geometry, but does not go into detailed presentation of each theorem or each proof.
This course is open to seniors only. Calculus is designed for students interested in mathematics, science and engineering. It covers a major part of the differential calculus. Applications will be stressed. Graphing calculators are used to enhance the curriculum. Computer projects may be included.
This course covers the topics of regular calculus, but also includes integral calculus. This course prepares students to take the Advanced Placement Exam for Calculus AB for which college credit may be awarded. Computer projects may be included.
Computer Science ½ credit Prerequisite: Algebra I
This course provides an introduction to fundamental computer concepts and elementary programming techniques. It involves the study of the programming language C++. The course is particularly valuable to students interested in engineering, science, research, and computer programming. The NCAA Clearinghouse will not count this course as a mathematics credit.
This is a course in traditional Euclidean geometry enhanced with some work with three dimensions, as well as integrating the uses of algebra in problem solving. The primary focus is on practicing and understanding deductive logic through proofs of theorems and exercises.
This course is designed and recommended for students who desire to reinforce and expand their knowledge of algebra fundamentals while working toward their graduation requirements in mathematics. Together with Integrated Math 211 reinforce and expand their knowledge of geometry fundamentals while working toward their graduation requirements in mathematics. Together with Integrated Math 111 work toward their graduation requirement in mathematics and already have mastered the competencies necessary for passing the Ohio Graduation Mathematics Test. It may serve as a stand-alone course or as a stepping-stone toward Algebra II.
This course is specifically designed for college-bound students. The course concepts include but are not limited to trigonometry, complex numbers, relation, functions, vectors, polar equations, conic sections and detailed use of graphing calculators to explore these concepts with deeper understanding.
This course is designed for students who have completed the study of trigonometry. It is a thorough course in the detailed study of all types of functions. Trigonometric ideas are used throughout the course. Computer projects may be included.
Probability and Statistics ½ Credit Prerequisite: Algebra I
This semester course emphasizes introductory statistics more than probability. Content includes proper sampling methods, methods for organizing data, calculation of measures of central tendency and measures of dispersion (variability), and finally how predictions can be made. Many areas of study in college require an introductory statistics course; this course would give students a strong base for entering a collegiate level statistics course. It is of particular interest to students who wish to learn to read and interpret statistical data.
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For students desiring to study mathematics, engineering, physics, or chemistry at the university level, linear algebra/matrix theory is an absolute requirement. Linear Algebra is the mathematical study of vectors and vector spaces (also called linear spaces). Matrices are widely used to represent the linear transformations that input one vector and output another. Currently, you have received a solid foundation in Calculus. The only REAL prerequisite for this material is a good knowledge of high school algebra. We have designated AB Calculus as a requirement because everyone will need strong algebra skills and good analytical ability. Linear Algebra has many practical applications, not limited to math/physics/engineering. It shows up in economics, logistics, finance, computer science, operations research, and lots of other disciplines.
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A course that just attempts to define the current research areas of maths. If the landscape is so complex, why can't undergraduates be provided with a map, so to speak, in order to begin to decipher this subject?
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Welcome to Mathematics 10C
In Mathematics 10C you will be encouraged to develop positive attitudes and to gain knowledge and skills through your own exploration of mathematical ideas—often with the help of study partners. As you progress through this course, you will also be encouraged to make connections to what you already know from your personal experiences. Building on your own experiences will give you a solid base for your understanding of mathematics.
There are seven mathematical processes that are critical aspects of learning, doing, and understanding mathematics. The Alberta Program of Studies incorporates the following interrelated mathematical processes. You will undergo these processes on a regular basis to help you achieve the goals of this course and future mathematics courses.
Process
Rationale
Application
Communication
Students must be able to communicate mathematical ideas in a variety of ways and contexts.
You will write, read, and discuss mathematical ideas with your peers and teacher.
Connections
Through connections, students begin to view mathematics as useful and relevant.
You will connect the math that you learn to meaningful contexts.
Mental Mathematics and Estimation
Mental mathematics and estimation are fundamental components of number sense.
You will make predictions about the outcome of events. You will also determine whether mathematical results are reasonable.
Problem Solving
Learning through problem solving should be the focus of mathematics at all grade levels.
You will learn multiple strategies for approaching problem solving.
Reasoning
Mathematical reasoning helps students think logically and make sense of mathematics.
You will use interactive multimedia, calculators, or computers to explore mathematical concepts.
Visualization
The use of visualization in the study of mathematics provides students with opportunities to understand and make connections among concepts.
You will use concrete materials, technology, and a variety of visual representations.
The Alberta Program of Studies outlines what students are expected to learn in mathematics courses. These expectations are written in statements called general outcomes.
There are three general outcomes in Mathematics 10C. They are:
Develop spatial sense and proportional reasoning.
Develop algebraic reasoning and number sense.
Develop algebraic and graphical reasoning through the study of relations.
The general outcomes are further divided into specific learning outcomes related to the topics you will be studying in this course. Specific learning outcomes are subdivided into achievement indicators. These achievement indicators form the basis for the outcomes for each lesson.
Mathematics 10C Textbook and Website Support
There are two approved textbooks for this course. They are Math 10 (McGraw-Hill Ryerson) and Foundations and Pre-calculus Mathematics 10 (Pearson). You will be using one of these textbooks throughout this course. You will find additional support at each textbook's online website— for McGraw-Hill Ryerson and for Pearson. You can find tips for success in mathematics, master sheets, general web links, a digital version of the textbook, web interactive, and other useful learning tools. By choosing a chapter from the pull-down menu, you can access interactive quizzes and web resources for individual chapters.
Mathematics 10C Partners
Mathematics 10C Co-Developers
Learning in an Online Environment
This course is delivered to you in an online environment. You can look forward to using resources, such as interactive multimedia and the Internet, for various activities. You will also have access to computer simulations, computer multimedia, computer graphics, and electronic information to support your learning.
Remember that exploring the Internet can be educational and entertaining. However, be aware that these computer networks may not be censored. You may unintentionally come across offensive or inappropriate articles on the Internet. With that in mind, be aware that perspectives presented on the Internet are there for you to analyze critically and to accept or reject based on that analysis. Since information sources are not always cited, you should always confirm facts with a second reliable source.
Some school jurisdictions may limit access to social networking sites. In such circumstances, you should consult with your teacher, as your teacher may need to adapt lessons to accommodate co-operative learning.
LearnAlberta.ca
LearnAlberta.ca is a protected digital learning environment for Albertans. This Alberta Education portal, found at is a place where you can access resources for projects, homework, help, review, or study.
For example, LearnAlberta.ca contains a large Online Reference Centre that includes multimedia encyclopedias, journals, newspapers, transcripts, images, maps, and more. The National Geographic site contains many current video clips that have been indexed for Alberta Programs of Study. The content is organized by grade level, subject, and curriculum objective. Use the search engine to quickly find key concepts. Check this site often, as new interactive multimedia segments are being added all the time.
LearnAlberta.ca now contains all of the available distributed learning online materials in the "T4T Courses" tab. If you are experiencing technical difficulties with the materials for this course, you can find the materials on LearnAlberta.ca.
If you find a password is required, contact your teacher or school to get one. No fee is required.
Alternative Learning Environments and Distributed Learning
Distributed Learning is a model through which learning is distributed in a variety of delivery formats and mediums—print, digital (online), and traditional delivery methods—allowing teachers, students, and content to be located in different, non-centralized locations. Mathematics 10C students will be completing this course in a variety of learning environments, including traditional classrooms, online/virtual schools, home education, outreach programs, and alternative programs.
Instructional Design
Explanation
The learning model used in Mathematics 10C is designed to be engaging and to have you participate in inquiry and problem solving. You will actively interpret and critically reflect on your learning process. Learning begins within a community setting at the centre of a larger process of teaching and learning. You will be encouraged to share your knowledge and experiences by interaction, feedback, debate, and negotiation.
Components
This course uses the following structure and instructional design to connect you to the relevant curriculum and scientific concepts in Mathematics 10C. These components are used consistently throughout the course and will help you in seeing the context and overall content of the program. The components of the course are described below in the order that you would see them in a typical lesson.
Component
Description
Focus
In the Focus section, the lesson theme is introduced. A real-world context and link to the unit or module theme is established, objectives are identified, and lesson questions are posed.
Assessment
The Assessment section provides a list of activities you are expected to submit as a record of achievement. These items may include a posting to a discussion board, assigned questions from the textbook, a portion of the unit project, or some other work assigned by your teacher.
A Lesson Assignment document will help you to know what is to be submitted as part of your assessment.
Launch
This area helps students to prepare for the lesson ahead. Included in this section are the Are You Ready?, Refresher, and Materials headings.
Are You Ready?
This section provides a short pre-test to help you assess your mastery of prior skills and knowledge. If you are successful with these questions, you can move on to the Discover or Explore sections. If you encounter difficulty with these questions, you can move on to the Refresher to relearn prior skills.
Refresher
The Refresher addresses skills and knowledge gained previously, which will help you tackle the concepts of the lesson. This section may include an overview of a formula (e.g., Pythagorean theorem) or procedure (e.g., how to factor) or a short set of Self-Check questions for practice. This section may also include links or references to previous lessons or multimedia elements that will allow you to review prior skills.
Discover
Discover establishes the inquiry for the lesson. Activities in this section expose you to relationships and concepts to be addressed in the Explore section. Activities within the section will lead you to identify and analyze patterns or trends. Discussion with peers or with teachers will occur to further support inquiry-based learning.
Watch and Listen
Watch and Listen includes both passive and interactive multimedia content (e.g., podcasts, videos, interactive Flash activities). This section also includes a description of what you are supposed to focus on while using the multimedia in order to be active learners.
Try This
Try This includes opportunities to practise and to apply learned concepts outside of a lab environment. These can be simulation activities, questions, webquests, or other activities that provide you with a space to explore different ways of applying new concepts.
You will find Try This questions placed in the Lesson Assignment document.
Math Lab
Math Lab is an activity where you complete an investigation that allows for data collection and analysis. The Math Lab often involves a hands-on component.
Math Lab activities are also found in the Lesson Assignment document.
Share
Share allows you to use the discussion board to gather information from your peers or your teacher and to compare such information to your own results. This component provides opportunities for you to reflect, communicate, and build consensus about work completed during the Discover activities.
Explore
Explore supports the lesson inquiry initiated in the Discover section by formally introducing and developing concepts. You will be introduced to theorems, formulas, and concepts that will enable you to build on prior knowledge.
Read
Read is used to introduce sections of the textbook used for content or skill development. The relevance of the passage to context and lesson inquiry is defined.
Self-Check
Self-Check provides opportunities for you to check your understanding of new concepts learned in the lessons and to make connections to prior learning. These will be in auto-marked form. You can judge by your results in these sections whether you need to seek further clarification from your teacher on certain concepts.
Did You Know?
This section includes information that enriches your learning. Here, you could find historical information or math trivia that may be of interest to you.
Tips
This section includes alternate strategies, algorithms, or shortcuts for calculating values or implementing procedures.
Caution
This component alerts you to common misconceptions or procedural errors that would lead to incorrect work.
Connect
This heading comprises all of those activities which invite you to reflect on the knowledge and skills gained in the lesson and to connect that to the Big Picture. You will also have the opportunity to extend and enrich your learning with the Going Beyond section.
Project Connection
Aspects of the lesson related to the unit project are identified in this section. An activity related to completion of the unit project is described.
Reflect and Connect
Opportunity for you to consider what knowledge and skills have been gained or expanded during a lesson. You are asked to use a variety of reflective techniques (e.g., concept maps, summaries, answering questions). This may involve reflection on specific lesson elements or connecting lesson topic to the unit theme.
Going Beyond
Going Beyond entails the investigation of a sub-topic or examples related to the lesson theme. These sub-topics typically extend beyond the curriculum but may be of interest to you.
Lesson Summary
The Lesson Summary provides information about what has been accomplished in the lesson. The Lesson Summary also addresses the lesson questions posed at the beginning of the lesson and answers those questions based on the material covered in the lesson. All lesson summaries build toward the unit and course summaries and make connections to the Big Picture introduced at the module level.
Glossary
You will create your own glossary in this course. In Module 1: Lesson 1 you will discover the Glossary Terms document. You will add new terms to this handout and save it in a secure location, such as a course folder. (The course folder may be a document folder on your computer, or it may be a physical location such as a binder for storing print outs and pages.) As you encounter new terms, you can add them to your Glossary Terms document. You can update the Glossary Terms document each time you encounter new terms.
You will find the definitions to these terms in the lessons themselves, as well as in your textbook. You can further enrich your understanding of these terms by doing further research on the Internet or by sharing ideas with other students. The glossary is intended to be personal. You should define terms in a way that makes sense to you. You can add examples of how those terms are used. These examples can be in the form of diagrams, illustrations, or worked problems.
Toolkit
The Toolkit is a collection of resources that provide you with further explanations or guidance for completing activities and assignments. The Toolkit includes grid paper templates, Math Lab templates, virtual manipulatives, computer video, and other objects to assist you as you work through each lesson.
Assessment
Your work will be assessed in a number of different ways. Assessment items can be either formative or summative. In a typical lesson, you may be asked to share or discuss the concepts learned with a peer or with your teacher. The results of these discussion items should be recorded for future reference or for your teacher to examine. Other assessment items can include selected textbook questions, Project Connection pieces, and personal reflections. You may also have the opportunity to select your best work for grading.
As a way to help you to recognize the assessment items, you will have a Lesson Assignment document for each lesson. This document contains all of the assessment items for each particular lesson. Save this document to your folder at the beginning of every lesson and add to it as you proceed through the lesson. At the end of the lesson, you can submit the Lesson Assignment document to your teacher. Some of these items will be contribute to your mark, and other items may only serve to help your teacher know how to support your learning.
In this course there will also be opportunities for self-assessment. When you come to the end of a learning section, you can test your knowledge and skills by answering questions related to the section. The solutions are provided as a way for you to compare your own work.
Using the Mathematics 10C Folder
The Mathematics 10C folder serves as the organized collection of samples of your work in Mathematics 10C. It gives you an ongoing record of your efforts, achievements, self-reflection, and progress throughout the course. When you want to show your friends or family what you've been learning, your work is all there for them.
In addition to being able to show others what you have done, the course folder lets you see your progress. It lets you see how your knowledge and skills are growing. It also lets you review and annotate work you have already completed. You may find your course folder useful in preparing for tests, quizzes, and your unit project.
Throughout the course, you will be asked to place your work in the Mathematics 10C folder. This folder may be an electronic folder on a server or a physical folder such as a binder. If you are unsure of the process, your teacher will help you.
Periodically, you will be asked to share items from your course folder with your teacher. This is not always for grading, as often your teacher may use these items to learn more about you and your interests or as a way of tailoring other work assigned to you.
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THIS COURSEWARE IS NOT SUBJECT TO THE TERMS OF A LICENCE FROM A COLLECTIVE OR LICENSING BODY, SUCH AS ACCESS COPYRIGHT.
Terms of Use
NOTICE TO USER: This Terms of Use Agreement is a complete and legal Agreement between You and Her Majesty the Queen in Right of Alberta as represented by the Minister of Education (the "Minister") regarding the use of Mathematics 10C Learn EveryWare (this "Resource"). By using this Resource, You accept all the conditions of this Agreement. If You do not agree with the following terms and conditions, do NOT use this Resource and remove all associated files from your system.
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Unit 1 Introduction
Art Gallery of Alberta, 2010. Photo: Robert Lemermeyer
The Art Gallery of Alberta, located in downtown Edmonton, opened its doors to the public in 2010. Designed by Randall Stout Architects, Inc., the building houses both national and international exhibitions.
With its many curves and jutting shapes suggesting prisms and cones, the architecture of the building truly is capable of capturing the observer's attention and imagination. The design of the building complements the design of other buildings in the vicinity including Edmonton's City Hall and the Winspear Centre.
What are the shapes that you have observed in your community? Are there buildings with interesting designs? Are there monuments or works of art that feature prisms, cones, or spheres?
While there is no doubt that the design of the Art Gallery of Alberta and other such buildings are intended to appeal to the observer, the design of most other objects follow function rather than form.
Nowadays, you can do "one-stop shopping" at the local home improvement centre. A few pieces of lumber to finish building your deck, a bookcase for your room, or a plant for your kitchen are some of the many things you can buy at these supercentres.
The next time you stop at one of these stores, pay attention to the different objects you can find there and how the shape of an object serves its function. For example, books are placed in rectangle-shaped spaces. The drinking glasses in your kitchen cupboard are cylindrical. Why do these objects have these particular shapes?
Can you imagine placing your favourite books on a triangular shelf or drinking from a glass in the shape of a sphere? All of these objects are designed to fulfill their functions. Even the medicine pills sold in pharmacies are designed to be more easily swallowed, chewed, and packaged.
There was a time long ago when the furniture you bought at a store was handmade. No two pieces of furniture were made in the same way. The bookcases and TV stands that you buy from today's stores are all factory-made or prefabricated. In fact, the object is often packaged in pieces, ready for assembly by the buyer. Since the pieces are prefabricated, you can replace the pieces.
When you bring that entertainment unit or bookcase home from the store, you often have to assemble it yourself. It is rare to get a piece that is too wide or a screw that is too short. Each piece in a do-it-yourself kit is made to exact measurements. Each piece is made to exact specifications so that if you needed to order another part, you can get one that is identical in shape and size to the original.
When you plan where to place a new bookcase, you may need to know the dimensions of the bookcase in imperial units. Knowing how SI units convert into imperial units will help you to find the best fit possible.
In this unit you will investigate the following questions:
Lesson
Title
Lesson Questions
1
Basic Measurement Systems and Referents
How can referents be used to estimate measurements?
Why are there two systems of measurement?
2
Using Measurement Instruments
How do you choose the appropriate techniques, tools, and formulae to describe the dimensions of an object?
How can you measure the dimensions of objects of irregular shape or size?
3
Measurement Systems and Conversions
How do the strategies for converting units in the SI compare with those used in the imperial system?
When can proportions be used to solve problems?
4
Surface Area of 3-D Objects
How is the concept of surface area applied to understanding the design of structures?
How do you determine the surface area of a 3-D shape?
5
The Volume of 3-D Objects
How is the concept of volume applied to understanding the design of structures?
How are the formulas for the volumes of solids related to each other?
6
Surface Area and Volume Problem Solving
Why is visualization important to the study of the surface area and volume of 3-D objects?
How does changing the dimensions of an object affect its surface area and volume?
7
Introduction to Trigonometry
In what situations can the concepts of trigonometry be used to solve problems?
How are the sine, cosine, and tangent ratios used to determine information about a right triangle?
8
Solving Right Triangle Problems
How do you approach problems whose solutions are based on trigonometry and its principles?
How is trigonometry used to determine heights and distances that cannot be directly measured?
In this unit you will be working on a project as you learn new concepts in each lesson. This project will be based on a place that is special to you, whether it is a place in your home, in the neighborhood, or in your imagination.
You will start by examining measurement systems by using interactive multimedia and Math Labs. By using referents, which approximate SI and imperial units, you will be able to obtain good measurement estimates. You will learn how to convert between SI and imperial units and determine which units are appropriate to use for a given measurement task and a given measurement instrument. These skills will help you as you develop your project.
You will also investigate the surface area and volume of solids and learn how the properties of 3-D objects are used in design. In this part of the unit, you will conduct hands-on math labs using objects you can find around the house to help you discover the properties of spheres, cones, and pyramids. This knowledge will be transferred to your project where you will describe or design the objects that are found in your special place.
The last two lessons of the unit will focus on the concepts of trigonometry and how these can be used to solve problems where direct measurements are difficult to obtain. You will use these concepts in the analysis of the objects in your special place.
Module 1: Measurement and Its Applications
Lesson 1: Systems of Measurement and Personal Referents
Focus
Most people have a favourite place. That spot might be your bedroom at home, a cabin at the lake, or a beach in a tropical location. Perhaps your favourite place only exists in your imagination; a place where you can go to create, meditate, or take refuge. How would you describe your favourite place to someone who has never been there? One way you would likely describe your place is to explain how big it is—in other words, you could describe its dimensions.
In Canada, two measurement systems are commonly used—the SI (International System of Units) or metric system, and the imperial system. The SI is the measurement system officially adopted by Canada, but the imperial system is used frequently in the trades and in day-to-day conversations. For example, many people only know their height and weight in imperial units of measure—feet and inches and pounds, for example. In this lesson you will take a look at both systems of measurement.
Since most people don't usually carry tape measures around with them, you will also relate these measures to common objects, which will allow you to quickly estimate a measurement. Such objects are called referents.
estimate a linear measure using a referent, and explain the process used
Lesson Questions
How can referents be used to estimate measurements?
Why are there two systems of measurement?
Assessment
Glossary Terms
Project Connection
Lesson 1 Assignment
In this lesson you will complete the Lesson 1 Assignment. Save a copy of the Lesson 1 1Did You Know?
Around 1100 AD, a yard was defined as the length of a man's arm.
The formal system of measurement used in Canada is the SI, but imperial units are still used in ordinary conversation. Imperial unit usage is found, for example, in recipes, construction, house renovation, and gardening. The imperial system of measurement actually has a longer history than the SI. Imperial units of length were initially based on human dimensions. In this lesson you will create a body ruler to help you understand these units.
As you start making measurements in the imperial system, you will need to be familiar with fractions. Use the multimedia piece titled "Exploring Fractions" to practise basic fraction operations. (Make sure you maximize the screen by clicking on the button in the top-right corner of the video.)
Please note that clicking on the link takes you to the LearnAlberta website. This page includes both a video and an interactive component. Go to the left side of the web page, and choose "Exploring Fractions (Object Interactive)."
Every person's body is different. The length of your foot is most likely different from that of a classmate. In this lab you will determine the measurements of your body parts described in a chart in order to use the body parts as referents for measurement.
Math Lab: Body Referents
Go to the Lesson 1 Assignment that you saved to your course folder. Then complete Math Lab: Body Referents.
Share
You now have an opportunity to share, with other students, the answers to Math Lab: Body Referents questions you have just completed.
To make the most of this sharing opportunity you need to do the following:
Ensure you have completed all the questions to the best of your ability and place them in a form that is convenient for sharing.
Post your answers to your class discussion area, or share them using another method as instructed by your teacher.
Review the results you recorded for steps 1 to 3. Are the results of other students similar to yours? Can you explain reasons for any differences?
Review the answers provided to questions 1 and 2. If possible, discuss the answers to these questions with the students who posted them. Your discussion might focus on clarifying meaning, or developing a clearer understanding of other students' strategies and ideas.
Finally, if necessary, revise your answers for questions 3 and 4 to incorporate what you have learned from the sharing you have done. Save a copy of your revised work in your course folder, along with a record of your discussion.
Module 1: Measurement and Its Applications
Explore
Glossary Terms
In this course you will often come across math-related words that may be unfamiliar to you. These words will likely be used over and over again, so it is important that you understand the meaning of these words. You will also need to record the words and their meanings so that you can refer to them when necessary.
In this course you will create your own glossary. Use the document titled Glossary Terms to keep a record of the math terms that you come across in Mathematics 10C.
In this lesson the suggested glossary terms you should add to Glossary Terms include the following:
imperial measurement
referent
SI measurement
When you have finished adding definitions to Glossary Terms, you should save the Word document in your course folder. You will refer to it again in other lessons to add new terms.
Read
Read the textbook that you are using for this course.
Math 10 (McGraw-Hill Ryerson)
Read "Imperial Measurement" on page 22. Try to determine what system of measurement Canada and the United States use.
Read "Link the Ideas" on page 23 to find out about common units used in the imperial system. Pay particular attention to the following:
Foundations and Pre-calculus Mathematics 10 (Pearson)
when the SI system of measures gained popularity among the countries of the world, including Canada
what common measurement is still often reported in imperial units
Read page 6 to find out how imperial units are related to each other. Try to find out from your reading what the differences are between the precision of an imperial measurement device compared to an SI measurement device. You will investigate measurement tools and conversions in Lessons 2 and 3.
A non-standard measurement unit is a unit that you would not normally use to report a measurement. Non-standard units are not found on measuring devices. For example, at one time the height of a horse was measured in hands. Personal referents, such as the ones you developed in the Math Lab, are examples of non-standard measurement units. Referents help you to estimate lengths in standard units. For example, you may know that the length of your foot is 25 cm. If you determine that the width of the hallway is as long as eight of your feet, then you can estimate that the hallway is 8 × 25 cm = 200 cm, or 2-m wide.
Try This
Go to the Lesson 1 Assignment that you saved to your course folder, and complete TT 1, TT 2, and TT 3. Once you have completed these questions, make sure to save your updated Lesson 1 Assignment to your course folder.
Share
Post your results to TT 3 on the discussion board, and consider the responses that others have posted there. Use examples from those posts to support or revise your answers to the questions in this Share section.
Self-Check
So far, you have learned about referents and the two systems of measurement. Test yourself now to see how much you remember. Go to Lesson 1 Self-Check.
Module 1: Measurement and Its Applications
Connect
Project Connection
Think some more about the special place that will be the basis of your Unit 1 Project. Do you see places where you will use metric measures and other places where you will use imperial measures? How will you decide?
Go to the navigation tree and view the Unit 1 Project to review the initial requirements of the project. Please make a few notes and store them in your course folder.
In Lesson 2 you will be ready to represent your place visually using simple shapes such as cylinders, cones, rectangular solids, and spheres. You will also calculate the volumes and surface areas of these solids using both imperial and metric units.
Reflect and Connect
Return to the Lesson 1 Assignment that you saved to your course folder. Then complete RC 1, RC 2, RC 3, and RC 4. Make sure to save your updated Lesson 1 Assignment to your course folder when you have completed the questions. Then submit the Lesson 1 Assignment to your teacher for marks.
Going Beyond
Did You Know?
The Romans used milestones to mark every 1000 steps.
There are other units of measure based on referents. Such units have their origins in agriculture and navigation, for example. Use your favourite Internet search engine to extend your learning by researching the origins of such units of measure as bolt, furlong, league, milestone, and chain. In your search, identify the imperial and metric equivalents of these measures, as well as the referents that are associated with these measures, and explain why these particular referents were chosen.
Module 1: Measurement and Its Applications
Lesson Summary
In Lesson 1 you investigated the following questions:
How can referents be used to estimate measurements?
Why are there two systems of measurement?
In this lesson you used referents to approximate both SI units and imperial units. You examined referents for linear measure including millimetre, centimetre, metre, kilometre, inch, foot, yard, and mile. You used referents to estimate linear measurements, and then you compared those estimates to the actual measurements. You also learned about the origins of the SI and imperial systems of measurement. In your discussions with your peers and with tradespeople in your community, you learned that some trades have adopted the SI, whereas others continue to use the imperial system.
In the next lesson you will use your knowledge of referents to choose appropriate units for measuring, and you will also learn strategies for solving measurement problems.
Module 1: Measurement and Its Applications
Lesson 2: Using Measurement Instruments
Focus
In Lesson 1 you learned how to estimate SI (metric) and imperial measurements using referents. Estimation is a very important skill that helps you to plan ahead and also to check that your results are reasonable. You will continue to build on this skill in Lesson 2.
Perhaps as part of your unit project, you need to create a cylindrical shape, a very small opening, or a quarter pipe. How can you figure out the required measurements? In what units will you measure your object, and with what instrument?
In this lesson you will expand on your ability to measure while you think about how you can measure large, small, or curved objects.
Outcomes
At the end of this lesson, you will be able to
justify the choice of units used for determining a measurement in a problem-solving context
solve problems that involve linear measure, using instruments such as rulers, calipers, or tape measures
describe and explain a personal strategy used to determine a linear measurement; e.g., the circumference of a bottle, the length of a curve, or the perimeter of the base of an irregular 3-D object
Lesson Questions
How do you choose the appropriate techniques, instruments, and formulae to describe the dimensions of an object?
How can you measure the dimensions of objects of irregular shape or size?
Assessment
Your assessment for this lesson includes the following:
Glossary Terms
Project Connection
Lesson 2 Assignment
In this lesson you will complete the Lesson 2 Assignment. Save a copy of the Lesson 2 the 2: Are You Ready? and answer the questions in this section. If you are experiencing difficulty, you may want to use the information and the multimedia in the Refresher section to clarify concepts before completingBelow you will find links to some video clips that may help refresh your memory. These interactive pieces will help you to answer the following questions:
How do I determine the circumference of a circle?
How do I calculate the perimeter of a polygon or an enclosed shape with straight sides?
The multimedia lesson titled "Parts of a Circle and Circumference" reviews the parts of a circle and explores the relationships between the diameter, radius, and circumference of a circle. The value of pi is discussed, and the lesson includes a game and practical math problems that require using the formula for the circumference of a circle.
The LearnAlberta resource from the Mathematics Glossary defines the term perimeter. Go to "Perimeter". It contains an animation to illustrate the definition. Try "Example" at the bottom of the web page.
Materials
Module 1: Measurement and Its Applications
Discover
When you are measuring shapes, sometimes you have to get creative. For example, if you want to measure the circumference of a circle but you only have a tape measure and a piece of string, what can you do?
Try This
Go to the Lesson 2 Assignment that you saved to your course folder. Then complete TT 1. Save your result to your course folder so that you can use your answer for a later comparison.
Watch and Listen
Now that you have tried to measure the diameter of a circle that you have drawn, watch the video clip titled "Measuring a Non-Linear Path."
Module 1: Measurement and Its Applications
Explore
Glossary Terms
In this course you will often come across math-related words that may be unfamiliar to you. In Lesson 1 you started a list of glossary terms and saved the document to your course folder.
Add the following terms to your "Glossary Terms" document:
circumference
diameter
dimensions
irregular shape
trundle wheel
vernier caliper
Then save the updated document in your course folder.
You have the SI (metric) units and the imperial units estimated as body referents, your ruler, an online caliper, and a tape measure, so where do you start? Well, it depends on what you are trying to measure and how accurate a measurement you require.
In construction, sometimes you need a sledge hammer and sometimes you need a ball-peen hammer. The same idea applies in measurement: different instruments provide different scales. You will investigate measurement instruments and their usefulness.
Watch and Listen
Have you ever had to measure a distance that is too long for a tape measure or a metre-stick? Maybe you want to know how far it is to the end of your street. Instead of using a stretched-out tape measure over and over, you can use a trundle wheel. The video titled "Trundle Wheel" will show you how.
A plumber may have to measure the diameter of a PVC (polyvinyl chloride) pipe in order to know if the pipe will be suitable for a repair job. One way to obtain the proper measurement is by using a vernier caliper. Use the multimedia piece titled "Vernier Calipers" to practise using a vernier caliper.
Try This
Return to the Lesson 2 Assignment that you saved to your course folder. Then complete TT 7, TT 8, TT 9, and TT 10. Return your Lesson 2 Assignment to your course folder when you have completed these questions.
Whenever you have been required to measure something in this lesson, you have also been told which measurement instrument to use. However, as you work on projects in your home, you will need to decide for yourself what are the most appropriate instruments and units to use.
When deciding which measurement instrument to use, you will need to consider the following.
Question
Considerations
What measurement instruments do I have available?
You may have limited options. You may have to borrow or purchase the right instrument.
Am I measuring something that is short or long?
You likely won't want to measure the length of a hallway with a ruler or the thickness of a dime with a trundle wheel.
Is the scale on the measurement instrument big enough?
You can't measure the diameter of a beach ball using a vernier caliper!
Are the divisions on the scale small enough to give a precise measurement?
How precise you want the measurement to be may determine which instrument you need to use.
You will also need to decide what unit of measurement to use when measuring an object. Some questions you will need to answer include these ones.
Question
Considerations
Do I need the measurement to be in SI units or imperial units?
You may need to report measurements in SI units if previous measurements were also in SI units.
Am I measuring something that is short or long?
If something is long, like the length of a building, you would likely avoid using inches or millimetres as the units of measure.
What measurement instrument am I using?
The instrument that you use will have a certain scale; e.g., inch or cm or m.
Which units will give me a reasonable answer?
The length of a swimming pool could be reported as 50 000 mm, 5000 cm, 50 m, or 0.05 km. The most appropriate example is 50 m.
Self-Check
SC 1. Match the following scenarios with the correct measurement instruments.
measuring the height of your wall
trundle wheel
measuring the diameter of a table-tennis ball
caliper
measuring the width of a sheet of paper
ruler
measuring the distance across the grocery store parking lot
tape measure
SC 2. You have used your caliper to determine that a Canadian dime has a thickness of 1.22 mm and a diameter of 18.03 mm.
Determine the circumference of this dime.
How many dimes can fit into a container that has the following dimensions?Discover
When you convert measurements, you need to know how units of measurement relate to each other. For example, you should know from your previous math studies that 1 cm = 10 mm or 1 ft = 12 in (see questions 5.b. and 5.d. in Are You Ready?). But do you know how many millimetres there are in a kilometre or how many kilometres there are in a mile?
Open up the document SI and Imperial Conversions Sheet. Use the Internet and your textbook to find the correct relationships between the SI units and the imperial units stated on the sheet. Complete the sheet by writing the correct numbers in the blanks.
Save your completed sheet to your folder so that your teacher can check it for accuracy.
Explore
Glossary Terms
Find the "Glossary Terms" document that you saved to your course folder. Add the following words to the document:
unit analysis
unit conversion
You may also choose to add other terms to help you understand the math you are studying.
You learned in a previous lesson that both the SI and the imperial systems are used in the trades. In order for two tradespeople who use different measurement systems to understand each other, it may be necessary to use measurement conversions.
You can begin your study of measurement systems by examining conversions within the SI.
Here are some examples of problems you might encounter when it comes to unit conversions.
Example: Converting Between SI Units
Problem
Convert 450 cm to millimetres and kilometres.
Solution
One way that you can solve a conversion problem is to set up a proportion, and then use cross-multiplication to find the answer.
Since 1 cm = 10 mm,
The variable x will be in units of centimetres.
Then,
There are 4500 mm in 450 cm.
Another way that you can solve a conversion problem is to use the technique of unit analysis. Unit analysis helps you to keep track of the measurement units to ensure that your result will be expressed in the correct units.
Recall that 1 m = 100 cm and 1 km = 1000 m. If you want the final result to be expressed km, you can show your work in the following way:
There is 0.0045 km in 450 cm.
Example: Converting Between Imperial Units
Problem
On a particular Canadian Football League team, the average height of the players is 6 ft 3 in.
First, determine how tall the average football player is in inches only on this team.
Second, figure out, on average, how many of these football players, lined up head to toe, it would take to stretch across a regulation football field with length 110 yards.
Solution
You already have part of the height in inches, so you just need to convert 6 ft into inches, before adding the extra 3 in.
Since 1 ft = 12 in,
The variable x will be in inches.
Then,
There are 72 inches in 6 ft.
Therefore, the average height is 72 + 3 = 75 in.
Next, you can use the unit analysis method to convert 110 yd to inches. Then you can determine how many times 75 in goes into the result.
Recall that 1 ft = 12 in and 3 ft = 1 yd.
So,
The football field is 3960 in.
Now, 3960 in ÷ 75 in = 52.8.
Therefore, it would take about 53 football players, lying head-to-toe, to line the length of a CFL football field.
Module 1: Measurement and Its Applications
Try This
Practice converting units within a system by completing TT 1 in the Lesson 3 Assignment that you saved to your course folder.
The next step is to learn how to convert measurements between the SI and the imperial system. In the Math Lab: Body Referents in Lesson 1, you established referents for both measurement systems. You can use these referents to make sure each of your calculations is reasonable.
To do so, you would estimate the answer using an appropriate referent; then compare your estimate with your calculation. If the numbers are close, then your calculation is reasonable. If the numbers are different, stop to think about why the numbers are different and where you might have gone wrong in your calculations. Keep this in mind as you read the next section. You will have an opportunity to use referents to estimate in a subsequent Self-Check section.
Read
How do you convert between SI and imperial units? The strategies to do this are the same as those used in the previous conversions. The first thing you have to do is find out the relationship between the units. Once you know this, you can either set up a ratio or prepare to convert using unit analysis.
Read the textbook that you are using for this course to see how these strategies are put into place.
Math 10 (McGraw-Hill Ryerson)
Read "Link the Ideas" on page 37. As you read, consider how you know whether a conversion is exact or approximate.
Read "Example 1: Convert Between SI and Imperial Units for Length" on page 38. Look for the use of unit analysis in solving a conversion problem.
Foundations and Pre-calculus Mathematics 10 (Pearson)
Read "Example 1: Converting from Metres to Feet" on page 18 to see how a measurement in metres is converted to an equivalent measure in feet.
Read "Example 2: Converting between Miles and Kilometres" on page 19 to see two methods for solving a problem involving a conversion between miles and kilometres.
When you are done, you can test yourself in the Self-Check section.
Self-Check
For each of the following, choose the correct answer.
SC 1. A measure of 2 cm is (larger than, smaller than) an inch.
SC 2. A measure of a mile is (larger than, smaller than) a kilometre.
SC 3. A measure of a yard is (larger than, smaller than) a metre.
SC 4. A measure of 25 cm is (larger than, smaller than) a foot.
SC 5. Convert 90 in to yards, demonstrating unit analysis. For this question, please show all your steps to the solution.
Try This
Go to the Lesson 3 Assignment that you saved to your course folder. Complete TT 2.
Share
You now have learned several ways of converting measurements from one unit to another. You can convert measurements by setting up a proportion and using cross-multiplication. Alternatively, you could use unit analysis. You also have an idea of how metric units compare to imperial units.
Can you describe why working with proportions is a good strategy for doing unit conversions? Have you developed other strategies of your own? Post your ideas to the discussion board. 3 Summary
How do the strategies for converting units in the SI compare with those used in the imperial system?
When can proportions be used to solve problems?
In this lesson you learned how proportional reasoning can be used to convert a measurement within or between the SI and the imperial system.
The SI is based on powers of 10. As a result, conversions in the SI system involve multiplication or division by 10, 100, 1000, and so on.
The imperial system, on the other hand, is not based on powers of any specific value. When using the imperial system, it is more advantageous to set up a proportion. Proportions are best suited to solving problems when an equivalent relationship can be established. Once the proportion is established, you can convert first by cross-multiplying and then by dividing.
You also learned to verify using unit analysis. Unit analysis helps you to convert measurements by cancelling unwanted units. In addition, you solved problems that involve the conversion of units and justified, using mental mathematics, the reasonableness of the solution.
Module 1: Measurement and Its Applications
Lesson 4: Surface Area of 3-D Objects
Focus
Now that you are familiar with estimating and converting between SI (metric) units and imperial units, you will expand your knowledge to include 3-D objects that include curved surfaces. These objects include cylinders, spheres, cones, prisms, and pyramids. You will use the skills gained in Lesson 2 where you explored measuring curved surfaces.
Our surroundings are full of various 3-D objects, many of which can be broken into smaller, basic objects whose surface area and volume can be calculated. You will investigate surface area in this lesson, and by the end of this lesson you will be able to apply and use the surface area calculations needed in your project.
Outcomes
At the end of this lesson, you will be able to solve, using SI and imperial units, problems that involve the surface area of objects, including:
right cones
right cylinders
prisms
pyramids
spheres
Lesson Questions
By the end of this lesson you should feel comfortable solving the following questions:
How is the concept of surface area applied to understanding the design of structures?
How do you determine the surface area of a 3-D object?
Assessment
Glossary Terms
Share: Surface Area Formulas
Project Connection
Lesson 4 Assignment
In this lesson you will complete the Lesson 4 Assignment. Save a copy of the Lesson 4 4How did you do? Did you remember the difference between a prism and a pyramid? Did you remember how to find the area of basic shapes? Great work, if you remembered! If you did not remember, please carefully read this Refresher section.
Let's take a look at what area means and how to find areas of basic shapes. This information will not only be helpful as you prepare to find the surface area of objects in this lesson but also when you explore the volume of those objects in the next lesson.
The Mathematics Glossary defines the term area. Go to "Area" to learn more. It contains an interactive Java applet and Flash animations to illustrate the definition.
The mathematics lesson "Finding Area with Unit Squares" explores the measurement of square units. The lesson presents the formulas for the area of a rectangle, parallelogram, and triangle, and it includes math problems that involve the practical application of these formulas.
The mathematics lesson "Estimating Area Using a Grid" uses inscribed polygons, circumscribed polygons, and the concept of limits to explain how the area of a circle can be measured. The lesson includes a game and a math problem that demonstrate the practical application of the formula for the area of a circle.
Materials
paper
prism net and pyramid net
soup can with label
scissors
scotch tape
a rectangular prism, such as a wooden block, a shoe box, or a cereal box
You will also require these materials to complete Math Lab: The Surface Area of an Orange:
Module 1: Measurement and Its Applications
Discover
Math Lab: The Surface Area of an Orange
Go to the Lesson 4 Assignment that you saved to your course folder. Complete Math Lab: The Surface Area of an Orange. In order to complete the Math Lab, you will need to go to 1 cm × 1 cm Grid Paper and print a copy.
Module 1: Measurement and Its Applications
Explore
Glossary Terms
In Lesson 1 you were asked to open a file called Glossary Terms and compile and save a list of math terms that you come across in this course. Go to your course folder and get the Glossary Terms document.
In this lesson the suggested glossary terms are the following:
3-D object
apex
lateral area
net
prism
pyramid
regular polygon
right cone
right cylinder
sector
sphere
surface area
Once you are finished, save Glossary Terms and return the updated document to your course folder.
Imagine taking a 3-D object and submerging it in a tub of water. The area of the object that is in contact with the water is called the surface area. You can also think of surface area as the measure of how much exposed area that a solid object has.
How would you determine the surface area of an object?
One way to do determine the surface area of the object is to peel off the outer layer of the object and then calculate the area of the peel. The peel is called the net. In Math Lab: The Surface Area of an Orange, you obtained a net of the orange (i.e., the orange peel sections) and added the areas of each part of the net (i.e., each peel section) to obtain the surface area of the orange.
As you move through this lesson, you will examine the nets of other 3-D objects. By adding together the sections of a net, you can determine the surface area of those objects.
Watch and Listen
Use "Surface Area of Prisms" to find out how to use the net of a rectangular prism to determine its surface area.
Try This: Prisms, Pyramids, and Cylinders
Work with a partner to examine the nets of prisms, pyramids, and cylinders. Use the following document titled Surface Area and Volume Investigation to summarize the information you will collect during this investigation. You may want to use the materials outlined in the Launch section as part of the investigation.
For each 3-D object, create a net that can be easily folded into the 3-D object. To get started, you can use the following descriptions. However, you are not limited by the descriptions. You are free to use other ways of creating a net for the object.
Remember to add the information you discover to your "Surface Area and Volume Investigation" document. Then save a copy of your completed handout in your course folder.
Use the cereal box to create the net of a rectangular prism.
Carefully unfold the cereal box and flatten it.
Cut off any flaps that are not part of the surface area. (Some flaps will remain because they are a part of the exterior of the box.)
Use the soup can with a label to create a net of a cylinder. For an idea of how you can do this, follow this procedure.
Trace each circular end onto a sheet of paper.
Cut out each of the ends.
Remove the label from the soup can.
Tape the circular ends to the label so that the net can be rolled into a 3-D cylinder.
Try This
Go to the Lesson 4 Assignment that you saved to your course folder. Then complete TT 1 to TT 3.
Tips
When you have a triangular prism and you are calculating the area of the base (the triangle), you will likely use the following formula:
area = 0.5 base * height
Remember that the base and the height of the triangle must be 90° to each other—that is, perpendicular to each other.
For example, in this triangular prism:
You would first need to calculate the height of the triangle. See the red line drawn in the picture below:
You will learn more about right triangles in Lesson 7.
Share
You and your partner have come up with some possible formulas for each of the four 3-D objects and added them to your Surface Area and Volume Investigation Sheet document. You may be certain about some of the formulas, but you may uncertain of other ones. You may be able to use the discussion board to get some ideas from other pairs.
Read
Read the textbook that you are using for this course.
Math 10 (McGraw-Hill Ryerson)
Read "Example 5: Visualize and Find Surface Areas of Composite Objects" on page 72 to see how a formula can be used to determine the surface area of a rectangular pyramid. Can you identify how finding the surface area of a rectangular pyramid is different from that of a square pyramid?
Foundations and Pre-calculus Mathematics 10 (Pearson)
Read "Example 2: Determining the Surface Area of a Right Rectangular Triangle" on page 29 to see how a formula can be used to determine the surface area of a rectangular pyramid. Try to see what needs to be determined before the formula can be applied.
Self-Check
You have seen in the examples how a formula is applied in finding the surface area of a pyramid. Now use the formulas that you created to find the solutions to the following problems.
Note: Make sure the formulas that you created have been checked by your teacher. You should have submitted the formulas to your teacher after you completed Share: Surface Area Formulas.
SC 1. Determine the surface area of the following pyramid, to the nearest in2.
SC 2. Determine the surface area of the cylinder to the nearest cm2.
SC 3. Determine the surface area of the following prism, rounded to the nearest cm2.
Did You Know?
While the pyramids of the ancient Egyptians are likely recognized by most people with their square bases and four smooth triangular sides, other ancient civilizations also constructed pyramids with slightly different designs. The Aztec and Mayan pyramids were built with tiered steps and a flat top instead of smooth sides and a peak.
Try This
This is the formula for the surface area of a sphere, in terms of the radius.
Add this formula to your list of formulas. You should save your list of formulas to your course folder.
Read
Read the textbook that you are using for this course.
Math 10 (McGraw-Hill Ryerson)
Read "Example 3: Calculate the Surface Area of a Sphere" on page 71 to see how the formula A = 4r2 is used to calculate the surface area of a sphere. In the question shown in this example, the radius of the sphere is not directly given. Pay attention to how the radius is determined.
Read "Example 4: Determine a Dimension When the Surface Area Is Known" on page 71 to understand how to use the surface area of a bowling ball to calculate the radius of the ball. Pay attention to how the formula is rearranged to find the desired result.
Foundations and Pre-calculus Mathematics 10 (Pearson)
Read "Example 1: Determining the Surface Area of a Sphere" on page 47 to see how the formula A = 4r2 is used to calculate the surface area of a sphere. In the question shown in this example, the radius of the sphere is not given. Read the solution and think of another formula that could be used to solve the problem.
Read "Example 2: Determining the Diameter of a Sphere" on pages 47 and 48 to understand how to use the surface area of a lacrosse ball to determine the diameter of the ball. Pay attention to how the formula is rearranged to find the desired result.
Self-Check
SC 5. Find the surface area of the following sphere to the nearest square metre.
SC 6. Determine the radius of a sphere with a surface area of 64cm2. Report your answer to the nearest centimetre.
Try This
From the examples, you have learned that you can determine the surface area of a sphere using the formula A = 4r2. You have also seen how the formula can be used to determine the radius of a sphere, if you know the surface area. Demonstrate what you have learned by going to the Lesson 4 Assignment that you saved to your course folder and completing TT 5.
There are other ways of determining the surface area of 3-D objects besides analyzing their nets. Often in mathematics, you can discover properties of unfamiliar objects by examining the properties of familiar ones.
For example, the cone is a 3-D object that is shaped much like a pyramid. Like a pyramid, a cone has only one base and the lateral faces of the cone meet at a point called the apex.
The illustration above shows that as you increase the number of sides on the base, the number of faces also increases. The area of each face also becomes smaller.
Eventually, the polygon base approaches the shape of a circle and the lateral area of the pyramid approaches the lateral area of the cone.
You can figure out the formula for the surface area of a cone with this idea in mind.
Consider the formula for the surface area of a rectangular pyramid, as shown in the illustration. The height of the triangular faces, or slant height, is labelled s. The sides of the base are labelled a, b, c, and d.
In the case of a cone, the perimeter of the base is really the circumference of a circle, so its surface area formula would be
Read
Read the textbook that you are using for this course.
Math 10(McGraw-Hill Ryerson)
Read "Example 1: Calculate the Surface Area of a Right Cone" on page 69 to see how the formula is used to calculate the surface area of a right cone. See how the parts of the formula relate to the net of a cone.
Foundations and Pre-calculus Mathematics 10 (Pearson)
Read "Example 3: Determining the Surface Area of a Right Cone" on page 32 to see how the formula is used to calculate the surface area of a right cone. Pay careful attention to how the slant height of the cone is determined. What theorem is used?
Self-Check
Now that you have watched some videos and had a chance to talk with your classmates, it is your turn to try some Self-Check questions to see if you have figured out surface area.
SC 7. When assembled, the net in the preceding illustration will create a
cube
cylinder
cone
prism
SC 8. Determine the surface area of the following cone to the nearest square foot.
Try This
Go to the Lesson 4 Assignment that you saved to your course folder. Complete TT 6. Be sure to pay attention to the information that is given. You may have to use a given value to find another value before you can apply the formula.
Share
You've had a chance to try the Self-Check questions to see if you can calculate the surface area of the various 3-D objects. Post your thoughts about which surface area was the most difficult to calculate and why you think it was the most difficult for you. Then read and respond to postings from two other students by suggesting tips or strategies that you tried when you were determining the surface area of different shaped objects.
Module 1 Appendix
Suggested Answers
Module 1: Measurement and Its Applications
Connect
Project Connection
Think about the place that is the focus of your project. What 3-D objects are found in your place? Are there prisms, cones, cylinders, or spheres? If your place is fairly empty of objects, think of the 3-D objects that could occupy your place.
Now go to the Unit 1 Project and complete the Lesson 4 portion of your project.
Reflect and Connect
Go to the Lesson 4 Assignment to complete RC 1, RC 2, and RC 3. Then save your updated Lesson 4 Assignment to your course folder. Then you will submit the Lesson 4 Assignment to your teacher for marks.
Going Beyond
Did yu know that taller trees generally have more leaves? The water in a tree needs to get from the roots to where photosynthesis happens, which is in the leaves and green parts. If you have a really tall tree, the plant has to force the water against gravity up to its leaves.
How is this possible? Consider the study of surface area. If a plant has large leaves, or numerous smaller leaves, then there is more surface area for evaporation to take place. In a plant, this is called transpiration. When transpiration occurs, the water leaving the plant is replaced by water coming up from the ground—and this water has dissolved nutrients in it.
For more information, initiate an Internet search using the keyword "transpiration" to see what else you can learn about a plant and the surface area of its leaves.
Module 1: Measurement and Its Applications
Lesson 4 Summary
In Lesson 4 you investigated the following questions:
How is the concept of surface area applied to understanding the design of structures?
How do you determine the surface area of a 3-D object?
In this lesson you examined the nets of various 3-D objects. These objects included the prism, pyramid, cylinder, sphere, and cone. From the nets, you were able to determine the surface area formulas for each object. These formulas can be used to determine the surface area of any of the 3-D objects investigated, as long as the required information is given.
Whether you are figuring out how much paint you need to buy or how much wood is needed to build a shed, solving problems involving area comes in very handy.
In the next lesson you will investigate volume, another important concept in design. In Lesson 6 you will apply what you have learned about surface area and volume to solve problems.
Module 1: Measurement and Its Applications
Lesson 5: The Volume of 3-D Objects
Focus
Have you ever moved from one residence to another? A move can take a great deal of planning, co-ordination, and time.
You may need to rent a large truck to contain all of your belongings, or you may have to temporarily store your belongings in a storage facility like the one pictured. The more furniture, clothing, and other possessions that you need to transport, the larger the truck or storage unit that you need for the move.
The amount of space becomes an important consideration. This is known as volume. In this lesson you will investigate the concept of volume and learn how to determine the volume of 3-D shapes.
Outcomes
At the end of this lesson, you will be able to determine the volume of a right cone, a right cylinder, a right prism, a right pyramid, or a sphere using an object or its labelled diagram.
Lesson Questions
How is the concept of volume applied to understanding the design of structures?
How are the formulas for the volumes of solids related to each other?
Assessment
Your assessment for this lesson includes the following:
Glossary Terms
Project Connection
Lesson 5 Assignment
In this lesson you will complete the Lesson 5 Assignment. Save a copy of the Lesson 5 5: Are You ReadyThis resource from the Mathematics Glossary defines the term volume. Go to "Volume." You will find an animation to illustrate the definition.
In the interactive mathematics lesson titled "Volume and Displacement," you can calculate the volume of rectangular prisms. You will also learn that the volume of an irregular object can be found by measuring the amount of water the object displaces.
Discover
Math Lab: Comparing the Volume of a Cylinder and a Sphere
Go to the Lesson 5 Assignment that you saved to your course folder. Then complete Math Lab: Comparing the Volume of a Cylinder and a Sphere.
Explore
Glossary Terms
Retrieve your personal glossary handout titled "Glossary Terms" from your course folder, and update it with the following terms:
area
base
volume
Return your updated "Glossary Terms" to the course folder.
Recall that area is the amount of square units occupying an enclosed shape or two-dimensional space. To find the area of a shape, you need to multiply two dimensions of the shape together. For example,
1 cm × 1 cm = 1 cm2
On the other hand, volume measures the amount of cubic units occupying a three-dimensional space. To find the volume of an object, you need to multiply three dimensions of the object together. For example,
1 cm × 1 cm × 1 cm = 1 cm3
Prisms
Watch and Listen
Watch the multimedia presentation titled "Volume of a Prism." See if you can remember the three steps that are needed to find the volume of a prism.
Generally speaking, the formula for the volume of a prism is the following:
volume = area of base × height
Self-Check
Use the formula for the volume of prisms to solve the following problems.
SC 1. Find the volume of the triangular prism shown here.
SC 2. Find the volume of a rectangular prism that has a base measuring 6 in by 4 in and a height of 8 in.
Module 1: Measurement and Its Applications
Cylinders
A cylinder is a prism with a circular base. Although you might not call a cylinder a circular prism, that's exactly what a cylinder is.
Using the formula for the volume of a prism, what could be the formula for the volume of a cylinder?
Save your answer to your course folder; then check in your textbook to see if your answer is correct. If it isn't correct, what parts of your formula were correct? What parts of the formula need to be changed?
Review Example 1 and Example 2.
Example 1
Determine the volume of a soup can with a diameter of 3.5 in and a height of 4.5 in. Show your solution to the nearest tenth of a square inch.
Solution
Example 2
The volume of a cylinder is 200 cm3. Determine the radius of the cylinder to the nearest hundredth of a centimetre if the height of the cylinder is 8 cm.
Solution
Try This
Return to the Lesson 5 Assignment that you saved to your course folder. Now complete TT 1.
Cones
Math Lab: Volume of a Cone
Go to the Lesson 5 Assignment and complete Math Lab: Volume of a Cone.
Read
Go to your textbook to find the formulas for a cylinder and a cone and note how they are similar and how they are different. What fraction of the volume of a cylinder is the volume of a cone with the same height and radius?
Math 10(McGraw-Hill Ryerson)
Read "Link the Ideas" on page 81.
Foundations and Pre-calculus Mathematics 10 (Pearson)
Read the top half of page 40. (You do not need to read "Example 3: Determining the Volume of a Cone" at this time. You will read it later in this lesson.)
Compare your experimental result in the cone investigation with the formulas you have just read in your textbook. Is your experimental result confirmed? Do your results support the finding that the volume of a cone is the volume of a cylinder with the same height and radius?
If your results do not support the ratio, give some reasons why you think this might be the case. Incorporate these comments into your Math Lab. Your teacher will not penalize you if you did not obtain the theoretical ratio. However, he or she will be looking at how you intrepret and explain the data you did obtain.
Read
Read the textbook that you are using for this course.
Math 10(McGraw-Hill Ryerson)
Read part b) of "Example 1: Calculate the Volume of a Right Cylinder and a Right Cone" on page 82 to see two methods for determining the volume of a cone. As you read, identify similarities and differences between the two methods. Which method do you prefer?
Foundations and Pre-calculus Mathematics 10 (Pearson)
Read "Example 3: Determining the Volume of a Cone" on page 40 and "Example 4: Determining an Unknown Measurement" on page 41 to see how the formula for the volume of a cone is applied. Use your calculator to verify the calculations. See the Caution bubble for a tip on using the calculator.
Caution
When you use your calculator to evaluate a quotient, applying brackets in the right places can be the difference between getting a correct answer and a wrong one.
Say that you want to rearrange the formula for the volume of a cone to determine its height. Then becomes
Evaluate the expression , where V = 20 cm3 and r = 2.5 cm.
After substituting, the expression would be .
Can you see what's wrong with the following way of evaluating the expression?
(The solution, 119.4 cm, is much too large for a cone with a volume of only 20 cm3.)
By entering the keystrokes in this way, you would actually be evaluating.
To evaluate the expression correctly, it is important to use brackets around the denominator:
The height of the cone is 3.06 cm. This answer is both reasonable and correct.
Rectangular Pyramids
You have learned that the volume of a cone is the volume of a cylinder with the same radius and height. This ratio is the same for pyramids. In other words, the volume of a pyramid is the volume of the prism with the same height and base area.
Module 1: Measurement and Its Applications
Read
Take a look at the following example, which compares the relationship between the volume of a right rectangular pyramid and the volume of a right rectangular prism. Read the textbook that you are using for this course.
Math 10 (McGraw-Hill Ryerson)
Read "Example 2: Calculate the Volume of a Right Pyramid" on page 83 to see how the formula for the volume of a right pyramid is used to solve a problem.
Foundations and Pre-calculus Mathematics 10 (Pearson)
Read "Example 2: Determining the Volume of a Right Rectangular Pyramid" on page 39 to see how the formula for the volume of a right pyramid is used to solve a problem.
Then read "Example 1: Determining the Volume of a Right Square Pyramid Given Its Slant Height" on page 38 to see how the volume of a pyramid is determined if the slant height (as opposed to the height) of the pyramid is given. Pay attention to how the Pythagorean theorem is used in the solution.
Tip
Whenever you come across a formula with a fraction, there are two ways that you can evaluate it. For example, if you want to enter the formula into your calculator, you can enter either of the following:
Self-Check
Spheres
Retrieve your analysis from Math Lab: Comparing the Volume of a Cylinder and a Sphere that you saved to your course folder.
The height of the can is equal to twice the radius of the ball.
To determine the formula for the volume of a sphere, you can do what you did for the cone. First, what was the ratio of the volume of the tennis ball compared to the volume of the juice container? Did you observe that the volume of the tennis ball was about that of the container?
This means that the volume of the sphere would be .
This formula is correct, but there's a way to simplify the formula by finding another way to express the can's height.
Think about the fact that the can and the ball have the same radius and the same height. Does it make sense to you that the height of the can would be equal to twice the radius of the ball?
So you could write the volume of a sphere as or, more simply, .
Read
Read the textbook that you are using for this course.
Math 10 (McGraw-Hill Ryerson)
Read "Example 4: Finding the Volume of Composite Figures" on pages 84 and 85 to see how the formula for the volume of a sphere is used to solve a problem involving a sphere and a cone.
Then read "Example 3: Calculate an Unknown Dimension When Given a Volume" on page 84 to see how to use the formula to calculate an unknown dimension when given the volume of a sphere.
Foundations and Pre-calculus Mathematics 10(Pearson)
Read "Example 3: Determining the Volume of a Sphere" on page 49 to see how the formula for the volume of a sphere is used to solve a problem involving the volume of the sun.
Then read part b) of "Example 4: Determining the Surface Area and Volume of a Hemisphere" on page 50 to see how to modify the formula to determine the volume of a hemisphere.
Try This
Go to the Lesson 5 Assignment that you saved to your course folder. Complete TT 2 to practise applying the formula for the volume of the sphere. You may have to do something more than apply the formula for the context-based questions.
The following table summarizes the different types of 3-D objects you have examined in this lesson. Included are the formulas that have been developed for these objects.
Cylinder
Cone
Rectangular Pyramid
Sphere
Self-Check
SC 4. Sheila is excavating the basement for her house on a small lot in town. The dimensions of the basement excavation need to be 40-ft long by 30-ft wide and 9-ft deep. The excavated soil is placed in a circular area beside the excavation. The radius of this area is 20 ft. As more soil is added, the soil pile forms the shape of a cone. The highest the excavator can lift the soil is 24 ft.
Going Beyond
Thanks to advances in technology, the world is truly changing. You can have a different perspective on the concept of place by using Google Earth.
Initiate an Internet search using the keywords "real world math" and "volume of solids." Using these search terms, you should find a website titled Real World Math. There's an exercise titled "Volume of Solids" that shows you how to determine the surface area and volume of some of the world's more famous places.
Module 1: Measurement and Its Applications
Lesson 5 Summary
In this lesson you examined the following questions:
How is the concept of volume applied to understanding the design of structures?
How are the formulas for the volumes of solids related to each other?
In this lesson you looked in your surroundings for various three-dimensional shapes including right cones, right cylinders, prisms, pyramids, and spheres. You discovered the relationships between the volumes of related 3-D objects through hands-on labs and by using interactive activities.
You developed strategies for determining the volume of a right cone, a right cylinder, a right prism, a right pyramid, or a sphere using an object or its labelled diagram.
In the next lesson you will use what you have learned about surface area and volume to solve problems in real-world situations.
Module 1: Measurement and Its Applications
Lesson 6: Surface Area and Volume Problem Solving
Focus
For the Unit 1 Project, you have been describing your favourite place. That place might be your bedroom at home, a cabin at the lake, or an ancient castle from your imagination.
How would you describe your favourite place to someone who has never been there? You could definitely use photos and drawings; but you also need a way to describe the size or dimensions of your place.
In the unit project you will need to represent your place with basic three-dimensional objects. You will then calculate the volume of those 3-D objects.
As you have been working on your project and spending time around your home and school, maybe you have noticed some basic geometric shapes in your environment. A single structure or space, such as the pictured castle, often includes a variety of geometrical shapes—prisms, pyramids, cones, cylinders, and spheres, for example.
If you have a structure that is made of more than one shape, how could you calculate its surface area and volume? What strategies can you develop to investigate the surface area and volume of complex shapes?
Outcomes
At the end of this lesson, you will be able to do the following:
Solve problems that involve surface area and the volume of 3-D objects.
Determine an unknown dimension of a right cone, cylinder, prism, pyramid, or sphere, given the object's surface area or volume and the remaining dimensions.
Solve problems that involve surface area or volume, given a diagram of the composite 3-D object.
Describe the relationship between the volumes of right cones and cylinders with the same height and base and right pyramids and prisms with the same height and base.
Lesson Questions
Why is visualization important to the study of the surface area and volume of 3-D objects?
How does changing the dimensions of an object affect its surface area and volume?
Assessment
Your assessment for this lesson includes the following:
Glossary Terms
Share (Your contribution to the discussion board.)
Project Connection: The Woodshed
Lesson 6 Assignment
In this lesson you will complete the Lesson 6 Assignment. Save a copy of the Lesson 6 6: Are You Ready? and answer the questions in this section. If you are experiencing difficulty, you may want to use the information andSelect the "Use It" tab at the bottom of the activity to review the nets of all of the 3-D objects that were previously explored. The activity tests your ability to visualize the net of a given 3-D object.
Select the "Explore It" tab to review the surface area and volume formulas for all of the 3-D objects previously introduced. This applet also enables you to do a side-by-side comparison of two formulas. For example, you can use this feature to compare
the surface area formula of a sphere and the volume formula for a sphere
the volume formulas of a cylinder and a cone
You can also access a video that demonstrates how math is used in designing large inflatable shapes. On the left-hand side of the website, choose "Exploring Surface Area and Volume (Video Interactive)" to view the video.
Module 1: Measurement and Its Applications
Explore
As you look around your surroundings, you may find objects that resemble the 3-D objects studied in the previous two lessons. A soup can, a box, and a ball are examples of cylinders, prisms, and spheres, respectively. A pylon used to alert motorists of traffic obstructions resembles a cone, and some games use pyramid-shaped dice.
While there are many examples of prisms, pyramids, cylinders, spheres, and cones, you may notice that many other objects are actually composites of two or more of these basic 3-D objects.
The image of the industrial propane tank is an example of a composite figure. Can you tell what 3-D objects are used in the tank's design?
Glossary Terms
Throughout Module 1, you have been adding and saving math terms to "Glossary Terms" in your course folder.
In this lesson the suggested terms for your glossary are
composite figure
hemispherical
surface area to volume ratio
Return your updated "Glossary Terms" to your course folder.
Tip
In complex problems that have more than one three-dimensional shape, it is a good strategy to break the problem into parts so you are dealing with only one shape at a time.
In the case of the propane storage tank shown in the photo, you may want to determine its volume by first determining the volume of the cylinder and then determining the volume of the hemispheres (or half-spheres) on each end.
The steps below will help you to solve a surface area or volume problem:
Step 1: Decide which 3-D object or objects can be used to model the problem.
Step 2: Draw a rough sketch of this 3-D object, and label its dimensions.
Step 3: Decide which formulas you will use. You may need to select more than one formula in the case of composite figures. You might also find that you will need to modify the formula to fit the problem.
Step 4: Substitute given values into your formulas to solve the problem.
Read
Work through the following textbook examples that show how problems involving composite figures are solved. In the solutions, pay attention to
Math 10(McGraw-Hill Ryerson)
Foundations and Pre-calculus Mathematics 10(Pearson)
Read "Example 1: Determining the Volume of a Composite Object" on pages 56 and 57. Then read "Example 2: Determining the Surface Area of a Composite Object" on page 57. Finally, read "Example 3: Solving a Problem Related to a Composite Object" on page 58.
Self-Check
SC 1. A grain storage bin has a diameter of 4.8 m. The height of the straight side wall is 10 m. The cone top has an additional height of 1.5 m.
What is the total volume of this bin?
Suppose you want to paint the outside of one of the grain storage bins. If the slant height of the conical roof is 2.83 m, then what total surface area needs to be painted?
SC 2. Mr. Vanilla charges 0.5 cents/cm3 for his ice cream. How much would you pay for one spherical-shaped scoop of ice cream if a scoop of ice cream has a radius of 3.5 cm?
$8.99
$1.50
26 cents
90 cents
SC 3. The right cylinder and right cone shown have the same radius and volume. The cylinder has a height of 12 in. What is h, the height of the cone?
18 in
24 in
36 in
42 in
SC 4. A glass containing water is in the shape of a right circular cylinder with a radius of 3 cm. The height of the water in the glass is 10 cm.
What is the volume of the water in the glass? Be sure to include units of measure in your answer. Show or explain how you obtained your answer.
Five spherical marbles of equal size are dropped into the glass. The water in the glass rises to a height of 11 cm. What is the increase in the volume of the glass contents? Be sure to include units of measure in your answer. Show or explain how you obtained your answer.
What is the volume of one marble? Be sure to include units of measure in your answer. Show or explain how you obtained your answer.
What is the radius of one marble? Be sure to include units of measure in your answer. Show or explain how you obtained your answer.
Try This
How did you do on the Self-Check questions? If you did well, go to the Lesson 6 Assignment and answer TT 1 to practise the steps in solving surface area and volume problems. If you need some help with the Self-Check questions, take the time to review the solutions or ask your teacher for help.
Share
Take a look at the results from Math Lab: Surface Area and Volume Analysis that you saved to your course folder.
It seems reasonable to assume that as the dimensions of an object are doubled, the surface area and volume are also doubled. Yet the results of the investigation proved otherwise. In fact, you may have found that as the dimensions are doubled, the surface area is quadrupled, or increases by a factor of four. At the same time, the volume undergoes an increase by a factor of eight! Where do these numbers come from?
Review your lab results and see if you can see any patterns in the ratios.
You may want to extend the investigation by quadrupling each dimension; then recalculate the surface area or volume.
Develop an explanation for how to predict the increase in surface area or volume.
Use your explanation to predict how the surface area and volume will change when the dimensions of an object are increased by a factor of 7.
Post your explanation on the discussion board.
Respond to postings from two other students whose explanations are different from yours.
Test their explanations by checking if they get the same surface area and volume as you did for an increase by a factor of 7.
Offer suggestions for improvement, or provide alternative strategies.
Once you have received feedback on your own explanation, make any necessary revisions.
Caution
You cannot measure the volume of some objects because they do not have "regular" lengths, widths, or heights.
An object's volume is greater in water than in air.
Share
Use the Internet to investigate these myths, and find a counterexample of each myth. A counterexample is an example that proves that a statement is false.
Share your counterexamples on the discussion board. Post a counterexample for each of the myths in the Caution. Be sure to include the reason or reasons why a counter example shows that the myth is false. Copy four other counterexamples from the postings on the discussion board. Save your findings to your course folder.
Module 1 Appendix
Suggested Answers
Lesson 6
This is a two-step problem. You need to calculate the volume of the lower cylinder and then the upper cone part of the grain bin.
For the cylinder, you will calculate the area of the base first. That will be the area of a circle.
For the volume of the cylinder part, you multiply the base by the height.
Now you need to find the volume of the cone part. Do you remember that the volume of a cone is the volume of a cylinder? You need the base of the cone, and it is the same circle area as the base of the cylinder. So you are ready to find the volume.
Before you write the final statement, does the answer make sense? Look at the picture. Is the volume of the cone that you calculated definitely less than the volume of the cylinder?
So the final step is to add the two volumes:
181.0 m3 + 9.05 m3 = 190.0 m3
You need to use the surface area formulas for a cylinder and a cone.
You will not need to include the areas of the bases of these objects, because they will not be painted. The bottom of the grain bin is sitting on the ground, so it will not need to be painted. The top part of the grain bin, where the base of the cone meets the top face of the cylinder, will also not need to be painted. So the modified formula will be:
Therefore, the surface area is as follows:
SC 2. D
SC 3. C
SC 4.
New volume:
Old volume:
The difference is 310.9 cm3 – 282.6 cm3 = 28.3 cm3.
The volume change was all due to the marbles.
Since there are five marbles and they are the same size, you divide the volume by the number 5.
So, the volume of one marble is 5.7 cm3.
Since you know the volume of the marble and you know it is a sphere, you can use the formula for the volume of a sphere.
Cells that are extended (e.g., cylinder) have much more membrane per unit of cytoplasm, which means these cells have more surface area for each bit of goo inside of them. Extending the outer surface of a cell into fingers, like an amoeba, or indentations, like the red blood cells shown above, can greatly increase the total surface area.
Interestingly, scientists are identifying the ratio between the surface area and volume as a crucial factor in work with nanotechnology. Take a look at the nanobot in the second visual of red blood cells. What do you suppose its function is?
To find out more, perform an Internet search using the following keywords: "nanotechnology," "surface area," "volume," "nanobot," and "red blood cells."
Module 1: Measurement and Its Applications
Lesson 6 Summary
Why is visualization important to the study of the surface area and volume of 3-D objects?
How does changing the dimensions of an object affect its surface area and volume?
In this lesson you solved problems involving the surface area and volume of 3-D objects. You learned how to approach problems involving composite figures. You learned that it is important to visualize each problem before putting pencil to paper.
You also learned that doubling the dimensions of a 3-D object, such as a prism or a cone, actually increases its surface area by a factor of 4 and its volume by a factor of 8.
You also looked at common myths regarding surface area and volume. Through research, you were able to find explanations that dispelled these myths.
In Lesson 7 you will continue your study of measurement by looking at the measures of right triangles. This branch of mathematics is known as trigonometry 7Are you able to tell if a triangle is a right triangle? Test your ability to do so by going to the multimedia piece titled "Right Triangle." On the bottom of the website is an interactive definition of a right triangle.
The multimedia piece titled "Exploring the Pythagorean Theorem" allows you to change the side lengths of a right triangle to see the effect on the length of the hypotenuse. At the website, choose the "Interactive" button near the middle of the page. You will be presented with a visual explanation of the Pythagorean theorem.
Materials
ruler
protractor
calculator
graph paper
In addition, specific materials are required for the Going Beyond section where you have the opportunity to build a simple sundial. The materials you will require to build the sundial will depend on the design that you choose. You will find out what materials you need by doing an Internet search.
Module 1: Measurement and Its Applications
Discover
Watch and Listen
Have you ever wondered how pilots calculate a safe angle of descent when they are landing 747 planes? Go to "Exploring Trigonometry" and view the video, which talks about how trigonometry is used at airports. You will find the video on the right-hand side of the website.
Try This
Go to the Lesson 7 Assignment that you saved to your Math 10C course folder and complete TT 1 to TT 4.
Share
Post the results of TT 1 to TT 4 to the discussion board. Specifically, include the
lengths of your triangle
values of your three ratios
Examine the results that were posted by two or more other students.
Compare the size of your triangle (the lengths of a, b, and c) with the other ones that have been posted on the discussion board. What do you notice?
Compare the ratios , , and with others in your class. What do you notice?
Can you make a generalization from your results?
Try This
Go to the Lesson 7 Assignment that you saved to your course folder. Complete TT 5 and TT 6.
Module 1: Measurement and Its Applications
Explore
Glossary Terms
Retrieve your handout titled "Glossary Terms" from your course folder. The glossary terms for this lesson that you should add to your handout are
adjacent side
cosine ratio
hypotenuse
opposite side
proportional
Pythagorean theorem
reference angle
right triangle
sine ratio
solving a triangle
tangent ratio
The three sides of a right triangle are known by three different names. You need to be able to identify the opposite side, the adjacent side, and the hypotenuse.
The term hypotenuse may already be familiar to you. Have a look at the diagrams where the orange arrows each point to the hypotenuse. Can you see a pattern? If you were given a right triangle, how could you identify the hypotenuse?
The great thing about the hypotenuse is that it never changes locations—it is always directly across from the right angle. Did you discover that pattern when you viewed the diagrams?
The two other sides that you need to identify are the opposite side and the adjacent side. The location of these other two sides will change, depending on which angle is being considered in the question.
For example, if you had the triangle to the right, you can see that the angle we need to find is labelled x. So, in this case, x is the reference angle. Can you tell which side is opposite from angle x?
If you said the bottom, you are correct!
See how the bottom side is opposite (or directly across) from angle x? In this case, we would label the bottom as the opposite side.
What about in the case to the left? Can you identify the opposite side if you are using angle y as the reference angle?
In this case, the left side is opposite from the reference angle.
Now that you can label the hypotenuse and the opposite side of a right triangle, all you need to do is label the remaining side as the adjacent side. You can also think of the adjacent side as the side between the reference angle and the 90° angle. Please review the examples below.
In the Discover section, you calculated three ratios which compare the lengths of the sides of a right triangle. Now that you know the names of the sides, you can apply them to the following triangle.
The ratios that you calculated in the Discover section were , , and .
These ratios have names.
The sine ratio is the ratio of the length of the side opposite the reference angle to the length of the hypotenuse.
The cosine ratio is the ratio of the length of the side adjacent to the reference angle to the length of the hypotenuse.
The tangent ratio is the ratio of the length of the side opposite the reference angle to the length of the side adjacent to the reference angle.
The definitions can be summarized by the following:
Tip
Often in mathematics, angles are referred to by the Greek letter . Just think of this as x.
Notice that the ratios are based on the location of the reference angle.
A great way to remember the trigonometric ratios is by using the following:
SOH CAH TOA
If you chant SOH CAH TOA many times, the chant will stay in your head. (It sounds like "soak-a-toe-ah!") Here's what it means:
Learn and Listen
The interactive piece titled "Shape and Space" allows you to practise identifying the sides of a triangle by dragging and dropping the opposite, adjacent, and hypotenuse sides into their correct locations. You can find the interactive piece on the right-hand side of the website.
Now that you know how to label the sides of a right triangle and know what the different ratios are, you are ready to see how these ratios can help you solve basic trigonometry problems.
Tip
For any calculations involving trigonometry, you must make sure that your calculator is in the "Degree" mode. Have a look at the calculator screen. Typically, calculators will show "Deg," "Rad," or "Grad." You need your calculator to show "Deg." If you see either "Rad" or "Grad," you need to press the mode button until you see "Deg."
If you cannot find the mode button, or if your calculator does not show any of the "Deg," "Rad," or "Grad" modes, then you can find out how to change the mode by reading in the calculator's manual. These manuals can be searched on the Internet by typing your calculator's make and model number into a search engine.
It is extremely important that you are in Degree mode. If not, your calculations will not be correct.
Module 1: Measurement and Its Applications
Read
Here is a chance for you to see how the sine, cosine, and tangent ratios are evaluated and used to determine angles. Pay careful attention to the difference between a trigonometric ratio and the angle that it can be used to find.
Read the textbook that you are using for this course.
Math 10(McGraw-Hill Ryerson)
Carefully read through "Example 1: Write a Tangent Ratio" on page 103 and "Example 2: Calculate a Tangent and an Angle" on page 104 Write Trigonometric Ratios" on page 116. Next, turn to "Example 2: Evaluate Trigonometric Ratios" on page 117. The great thing is that if the labelling of the triangles makes sense, you will see that the sine and cosine ratios follow a similar pattern as the tangent ratio did.
Foundations and Pre-calculus Mathematics 10 (Pearson)
Carefully read through "Example 1: Determining the Tangent Ratios for Angles" on page 72 and "Example 2: Using the Tangent Ratio to Determine the Measure of an Angle" on pages 72 and 73 Determining the Sine and Cosine of an Angle" on page 92. Next, turn to "Example 2: Using Sine or Cosine to Determine the Measure of an Angle" on page 93. The great thing is that if the labelling of the triangles makes sense, you will see that the sine and cosine ratios follow a similar pattern as the tangent ratio did.
Did You Know?
A tree farmer uses a clinometer to measure the angle between a horizontal line and the line of sight to the top of a tree. The farmer measures the distance to the base of the tree. Then the farmer uses the tangent ratio to calculate the height of the tree.
Let's look at some detailed examples of the six possible question types. Go to Finding Angles and Lengths Using Sine, Cosine, and Tangent Ratios. Under the Finding Angles box, choose "Sine," "Cosine," and "Tangent" to see examples of each. Then under the Finding Lengths box, choose "Sine," "Cosine," and "Tangent" to see examples of each.
Self-Check
Now that you have some experience with trigonometry, see how well you can answer the following questions.
SC 1. In the following triangle, what side is adjacent to angle MLN?
SC 2. Calculate the cosine of angle MLN.
SC 3. Calculate the angle measurement of angle MLN, to the nearest degree.
You have learned how to determine the measure of an unknown length or unknown angle in a triangle. You can use sine, cosine, or tangent ratios to set up an equation to solve for the unknown measure. With these techniques, you can determine the measures of all of the unknown lengths and unknown angles in a triangle. This is known as solving a triangle.
Of course, you need to know some information before you can solve a triangle. The minimum information you need to know is one of the following:
The measure of one length and one acute angle.
The measure of two lengths.
Read
Find out how to solve a triangle when you are given different measures of a triangle. Read the textbook that you are using for this course.
Pay attention to the numbers that are used in each calculation. Every time a new measure is calculated, one more number can be used to determine the next unknown measure.
What reasons are there for using the original given measures? What reasons are there for using calculated measures?
Math 10(McGraw-Hill Ryerson)
Read "Example 3: Solve a Right Triangle" on page 129 to see how to solve a triangle when given the measure of one length and one acute angle.
Foundations and Pre-calculus Mathematics 10 (Pearson)
Read "Example 1: Solving a Right Triangle Given Two Sides" on pages 106 and 107 to see two methods for solving a right triangle given two sides.
Read "Example 2: Solving a Right Triangle Given One Side and One Acute Angle" on page 108 to see how to solve a right triangle given the measure of one length and one acute angle.
Try This
Use what you have learned in this lesson to solve some triangles. Remember to use a calculated value to solve for another value only if you are sure of the correctness of the calculated value. If the calculated value is incorrect, then the subsequent values will also be incorrect.
Go to the Lesson 7 Assignment that you saved to your course folder. Then complete TT 7.
Module 1: Measurement and Its Applications
Connect
Project Connection
At this time, you should work on your Unit 1 Project. Go to the Unit 1 Project, and complete the Lesson 7 portion of the project.
Reflect and Connect
Go to the Lesson 7 Assignment that you saved to your course folder. Now complete RC 1 and RC 2.
Going Beyond
Janis Christie/Photodisc/Getty Images
There are many uses of trigonometry. A sundial is a device that measures time based on the position of the Sun. A sundial is designed in such a way that the Sun casts a shadow from a sharp, straight edge onto a flat surface marked with lines that indicate the hours of the day.
In theory, a stick stuck in the ground could form the basis of a sundial. In reality, it's not that simple. Earth's axis is tilted, which means that the apparent movement of the Sun through the sky changes every day. If this isn't accounted for, a sundial that tells perfect time today will be slightly wrong next week and very wrong next month.
See if you can find out how to build a sundial by doing a search on the Internet. Use the search terms "how to build a sundial." Then look through a few of the websites to find a simple set of instructions.
(After you construct a real sundial, perhaps you will be interested in adding a virtual sundial into your unit project!)
In order to construct a sundial accurately, you will need to find the direction "due north" and you will also need to know the latitude of the town or city where you reside.
Note: Grande Prairie has a latitude of approximately 55°, Calgary is 51°, and Edmonton is 53°. You can go online to look up the latitude of your community. The multimedia piece Latitude Table shows a table that includes the latitude of many communities in Alberta.
Module 1: Measurement and Its Applications
Lesson 7 Summary
In what situations can the concepts of trigonometry be used to solve problems?
How are the sine, cosine, and tangent ratios used to determine information about a right triangle?
You learned in this lesson that trigonometry can be used to solve right triangles. A triangle is solved when the given information about the triangle is used to determine the unknown lengths and angles. A triangle is solvable when the following minimum information is known:
the measures of two sides
the measure of one side and the measure of an acute angle
In all other situations where the minimum information is not known, a triangle cannot be solved.
Trigonometry is used to solve problems in design, policing, and air traffic control, to name a few instances. In Lesson 8 you will be applying trigonometric techniques you have learned to problems arising from everyday contexts.
Module 1: Measurement and Its Applications
Lesson 8: Solving Right Triangle Problems
Focus
You may enjoy listening to music. The music may have been recorded digitally—a process that uses trigonometry. Perhaps the music is in MP3 format using data compression, which uses an understanding of the human ear's ability to distinguish between sounds, and this format also requires trigonometry.
You may travel over a bridge today. That bridge was built using an understanding of forces acting at different angles. You will notice that bridges involve many triangles—trigonometry was used when designing the lengths and strengths of those triangles.
You will often see a surveyor at work in your community. Trigonometry helps the surveyor determine sides of a triangle that are difficult to access. An angle that cannot be reached may be measured from places that can be reached.
You saw an example of this principle in Lesson 7 when you read about building a tree house. The heights of trees and tall buildings can be determined by knowing the distance to the base of the tree or building. Similarly, distances across rivers or busy roads can be determined by using angle and length measurements from one side of the river or the road.
In the last lesson you used the trigonometric ratios and the Pythagorean theorem to find sides and angles in right triangles. In this lesson you will apply those skills to solve real world problems.
Outcomes
At the end of this lesson, you will be able to
solve a problem that involves right triangles
solve a problem that involves one or more right triangles by applying the primary trigonometric ratios or the Pythagorean theorem
solve a problem that involves indirect measurement using the trigonometric ratios, the Pythagorean theorem, and measurement instruments such as a clinometer or a metre-stick
Lesson Questions
How do you approach problems whose solutions are based on trigonometry and its principles?
How is trigonometry used to determine heights and distances that cannot be directly measured?
Assessment
Your assessment for this lesson includes the following:
Glossary Terms
Project Connection: The Staircase
Lesson 8 Assignment
In this lesson you will complete the Lesson 1 Assignment. Save a copy of the Lesson 8 8In this lesson you will be using the trigonometric ratios and the Pythagorean theorem to solve problems you encounter in your everyday life. These problems will all involve right angle triangles. Do you remember what makes a triangle a right angle triangle?
"Space and Shape: Trigonometry" provides a good review before you begin problem solving. On the right-hand side of the website, click on "Interactive." When you are finished, choose "Video."
The video at "Space and Shape: Similarity and Congruence" shows the difference between similar and congruent shapes. For example, are stop signs congruent or similar to other stop signs? View the video to find out. On the right-hand side of the website, choose "Video."
Once you have viewed the video, choose "Interactive." Viewing the interactive information is a fantastic way to figure out if you are truly comfortable with congruent and similar shapes. The site allows you to move and reshape triangles to determine if they are congruent, similar, or neither when compared to each other.
Please review your work from Lesson 7 for the definition of the trigonometric ratios. You will also need to solve triangles in which two pieces of information are given and you have to find either a length or an angle. Remember the six types of solving triangles from Lesson 7.
Materials
tape measure
clear plastic ruler
clear plastic protractor
clear tape
cotton string
small weight (e.g., a metal washer)
You will also need the following items to complete Math Lab: Clinometer.
1 plastic protractor
1 soda straw
1 paper clip
1 toothpick or another paper clip
1 6–8 in (15–20 cm) length of thread
1 roll of fishing line or relatively inflexible string (The length depends upon use.)
Module 1: Measurement and Its Applications
Explore
Glossary Terms
Find the "Glossary Terms" handout that you have saved to your course folder. Add these terms to your glossary:
angle of depression
angle of elevation
clinometer
congruent triangles
similar triangles
Return your updated "Glossary Terms" to your course folder.
man, house, tree: Image Club ArtRoom/Getty Images
Sometimes you will see either the expression angle of elevation or angle of depression when you are solving a word problem involving angles.
The angle of elevation is the angle you measured with your clinometer. It is the angle from the horizontal to the line of sight as shown in the diagram.
The angle of elevation is useful to know in problems where the observer is looking upwards at something.
man: Image Club ArtRoom/Getty Images
The angle of depression, on the other hand, refers to those instances when the observer is looking downwards at something. Like the angle of elevation, the angle of depression is the angle between the horizontal and the line of sight. The difference is that the line of sight, in this case, is directed downwards.
Try This
Go to the Lesson 8 Assignment that you saved to your course folder. Complete TT 1 to TT 5.
Module 1: Measurement and Its Applications
Share
Post your conclusions on the discussion board. Check the responses posted by other students to see if other students agree with you. If not, discuss the differences with the other students and try to reach a resolution.
Explain how you reached a resolution. Save your work to your course folder.
Did You Know?
Trigonometry is used in the fields of design, music, navigation, cartography, manufacturing, physics, optics, projectile motion, and other disciplines that involve angles, fields, waves, harmonics, and vectors.
Trigonometry problems vary in complexity. Some problems involve only one right triangle and one or two steps. Other problems may involve two triangles and may require several calculations. You can approach these problems by following these guidelines:
Sketch the scenario: Set up the problem with a drawing.
Find the right triangle(s): You will need to identify the triangle or triangles in your sketch. Label the given lengths and/or angles. Also, label the length or the angle that you need to find.
Write a trigonometric equation: Use SOH CAH TOA and the information given in the problem to select an equation—sine, cosine, or tangent—to solve.
Solve the equation: Rearrange the equation, and solve for the unknown length or angle.
Example
Retrieve your data from Math Lab: Creating a Clinometer that you saved to your course folder. This example will take you through the steps that were just outlined to help you determine the height of the structure you measured.
One of the required measurements from the Math Lab is the angle of elevation from the horizontal to the top of the structure that you measured. Read the Caution bubble to see how to make sure you read the information correctly.
This is because the clinometer started at 90° and then rotated through 30° to reach 120°.
If you recorded 120° as the angle of elevation, it means that you started at the horizontal and rotated your line of sight 120° upwards. Since a 90° rotation would mean you would be looking straight up, 120° would mean you are now looking slightly behind you.
For this example, assume that the measurements taken were the following:
The distance between you and the base of the tree is 10 m.
The string on the clinometer passes through 140°.
The distance from the ground to the eye level of the person taking the measurement is 1.5 m.
Follow the steps to solve for the height of the tree.
Sketch the Scenario
The angle of elevation is 140° − 90° or 50°.
A sketch might look like the following.
man, tree: Image Club ArtRoom/Getty Images
Find the Right Triangle(s)
The right triangle is coloured red. The distance between you and the tree is 10 m. The angle of elevation is 50°.
The length that needs to be found is y.
Write a Trigonometric Equation
Using 50° as the reference angle, the known side is the adjacent side, and the required side is the opposite side.
The ratio that contains both of these sides is tangent (TOA). So the equation is
Solve the Equation
Multiplying both sides of the equation by 10 gives the following:
Don't forget to add the height to eye-level!
The height of the tree is 11.9 m + 1.5 m = 13.4 m.
Try This
Go to the Lesson 8 Assignment that you saved to your course folder. Complete TT 6.
Read
You now have an idea of the approach you can take with trigonometry problems. Read the following examples to see how you can apply this same approach given different circumstances. Read the textbook that you are using for this course.
Math 10 (McGraw-Hill Ryerson)
Read "Example 2: Calculate a Distance Using Angle of Depression" on page 128 to see how you can calculate a distance using the angle of depression. Look for similarities between the approach to this problem and to the one you just applied to an angle of elevation problem.
Foundations and Pre-calculus Mathematics 10 (Pearson)
Read "Example 3: Solving an Indirect Measurement Problem" on page 100 to see how to approach a similar problem where a measurement is made indirectly. Why can't you use tangent to solve this problem?
SC 4. A person in a hot air balloon is 150 m above the ground. An object is 285 m away from the balloon on a line directly beneath the balloon. What is the angle of depression of the person's line of sight to the object on the ground rounded to the nearest degree?
28°
32°
62°
58°
Caution
It is important to realize that the same angle can be produced from many different triangles. You should observe that, although the side lengths that make up the triangle may vary, the value of a ratio, such as sin 30°, does not vary.
Some trigonometry problems need to be modelled with two triangles. For this type of problem, you will need to determine the measure of either a length or an angle from each triangle. You may need to calculate the measure of a length or angle on one triangle before you can determine the length on another triangle.
Example
In the diagram, the given distances are as follows:
AB = 8 ft
BC = 11 ft
CD = 9 ft
What is the distance from point A to point D?
Solution
The length AD is labelled y in triangle ACD. There is not enough information to solve triangle ACD. You would need to know the length of another side in triangle ACD in order to solve the triangle. However, there is enough information to solve triangle ABC.
Notice that triangles ACD and ABC have length AC in common. You can solve triangle ABC for length AC. Once you know the length of AC, you will have enough information to solve for all unknown measures of triangle ACD, including length AD.
In triangle ABC,
In triangle ACD, you now know the length of the side opposite 62°, and you are solving for the hypotenuse y.
Read
Read the textbook that you are using for this course. Read through the following additional example to see how a trigonometry problem can be modelled by two right triangles. Pay attention to the extra step at the end. Why is this step necessary?
Math 10 (McGraw-Hill Ryerson)
Read "Example 4: Solve a Problem Using Trigonometry" on page 130, which demonstrates how to solve a two-triangle trigonometry problem involving forest fires.
Foundations and Pre-calculus Mathematics 10 (Pearson)
Read "Example 2: Solving a Problem with Triangles in the Same Plane" on pages 115 and 116, which demonstrates how to solve a two-triangle trigonometry problem involving the height of a building.
Try This
Remember as you model and solve problems involving two triangles that you can only solve a triangle if you have enough information. If you don't have enough information in one triangle, then you need to solve the other triangle to get the values you need for the first triangle.
Module 1: Measurement and Its Applications
Connect
You have seen that trigonometry can be used to model real-world problems. In fact, the concepts of trigonometry are applicable in the design of new homes, in the reconstruction of traffic accidents, in the use of navigation, and in other areas.
In this section of Lesson 8 you will have an opportunity to connect what you have learned to a number of everyday contexts.
Project Connection: The Staircase
It is time for you to do another common problem. Of course, it is still a collaborative approach—you can still work together with a partner. Go to the Unit 1 Project and complete the Lesson 8 portion of the project.
Reflect and Connect
Go to the Lesson 8 Assignment and complete the Reflect and Connect activity.
Going Beyond
Did you know that trigonometry is the main math tool behind the technology of GPS (global positioning system)? Your car or your phone may have built-in GPS, so you never need to be lost! So how does GPS work?
GPS measures the satellite signals closest to your location by receiving signals from a minimum of three satellites, which is referred to as triangulation.
GPS locks onto a position and uses trigonometry to calculate its position. This position is measured in latitude and longitude. From that point, as long as the satellite stays locked onto your location, then GPS can provide the speed, the distance, and, most important of all, a map to your destination. Initiate an Internet search for GPS for a much more detailed explanation.
The trigonometry used in GPS systems does go beyond what you have learned in this course. An Internet search for GPS and spherical trigonometry can help you learn more.
Module 1: Measurement and Its Applications
Lesson 8 Summary
How do you approach problems whose solutions are based on trigonometry and its principles?
How is trigonometry used to determine heights and distances that cannot be directly measured?
At the beginning of this lesson, you learned the steps for approaching problems based on trigonometry. You learned that the key first step is to sketch a diagram of the problem.
Then it is so important to correctly identify the right triangle in the problem and label the opposite, adjacent, and hypotenuse sides. You can then set up a sine, cosine, or tangent equation. This equation can then be solved to yield a solution.
In this lesson you also built a simple clinometer which helps you to measure heights which are otherwise difficult to measure directly. The height of a tree or a flagpole or your house can be measured from the safety of the ground. There is no need to climb the structure and drop down a measuring tape!
You have now completed the final lesson of Module 1. If you have not already done so, you should go to the Unit 1 Project and make sure that all of your activities have been completed and submitted to your teacher for marks.
Unit 1 Conclusion
As you look closely at your home and your community, you will see many objects of interesting shape and size. In some instances, an object's form is designed to be visually inspiring. For example, the design of buildings in modern architecture often emphasizes creativity and beauty.
On the other hand, even in architecture, the buildings need to be designed to fulfill their functions. A library will be designed in a different way than a hardware store. Likewise, a bank will incorporate security into its design while a shopping mall will emphasize accessibility.
On a smaller scale, a drinking glass is designed in the shape of a cylinder for many practical reasons. One reason is that a cylindrical glass is easier to hold. Such a glass is also easily stored with other glasses of the same shape.
Similarly, a bookcase has rectangular openings so that more books can be stored and retrieved than if they were placed on a bookcase with circular openings.
In this unit you examined the 3-D objects around you. You began by reviewing the SI and imperial measurement systems. You used referents to obtain approximate measurements of various objects. You also learned how to convert measurements between the SI and imperial systems.
In your project work, you used a 3-D rendering program to create your own special place. This place contained objects whose dimensions you measured.
In this unit you built upon your knowledge of volume and surface area by extending those concepts to cylinders, cones, pyramids, and spheres. You investigated these properties by conducting math labs and using interactive multimedia.
Later, you calculated the volume and surface area of the 3-D objects found in your special place that you described in the Unit 1 Project.
At the end of this unit, you learned about trigonometry and how its concepts are used to determine measurements that are not easily obtained directly. For example, you learned to use a clinometer to determine the angle of elevation to the top of a tall structure. You then used the tangent ratio to find the height of the structure without actually measuring it.
At the end of the unit, you had the opportunity to apply trigonometry concepts to word problems. You learned that sketching the scenario and choosing the right ratio were important steps in finding the solutions to these problems.
The following table summarizes the learning outcomes and corresponding learning activities in this unit.
Specific Outcome
Major Learning Activities That Address Specific Outcome
Solve problems that involve linear measurement using
SI and imperial units of measure
estimation strategies
measurement strategies
Lesson 1
Math Lab: Body Referents
Lesson 2
Video: Measuring a Non-Linear Path
Apply proportional reasoning to problems that involve conversions between SI and imperial units of measure.
Lesson 3
SI and Imperial Conversions Sheet
Solve problems, using SI and imperial units, that involve the surface area and volume of 3-D objects, including:
Unit 1 Project: Place
Do you have a special place—somewhere you enjoy and feel really good about when you are there? What kind of a place is it? How would you describe the place to someone else? Is it a place created by people or a place in nature? "Place" has different meanings across different cultures.
In various cultures around the world, and close to home, place and location are the same and yet not. Traditionally in a mathematical sense, place and location are the same, describable using grids and coordinates. But that changes as we visualized abstract concepts in three- and four-dimensional space. So how can we address place and location in an understandable manner without throwing up our arms in frustration and disbelief that something so simple is suddenly so hard?
Location can be taken as the physical description of place. The GPS coordinates will not change. It may be a specific point on a Cartesian plane. In the end, location can be pointed to or occupied. Place, on the other hand, is more ephemeral; while a location may convey a specific experience, like a summer at the lake, a rock concert, or cultural festivity, place is an experience that is apart from location.
Place dwells within the memory of experience. Place can be that quiet meditational state that we use to recharge ourselves; it can be that contemplation state we use when searching for the solution to a complex problem in math or in life. Place dwells within us and yet it may need a physical presence or location for the complete experiential memory to be re-experienced. For those who follow a religious philosophy, the spiritual presence of a supreme being is always with them yet it is more palpable in a church or other house of worship. The patriot concept of national identity is not so much founded in where one resides but where one places their loyalties. Nationalism is a state of mind, an inner place of identity not a segregated location.
In the end, place is a state of mind that we occupy for a time before moving on to a location where we need to be.
—Ken Ealey
Watch and Listen
Watch the video titled Interview with an Elder, Part 1 that describes how the concept of measurement is perceived in another culture. Pay careful attention to the description of how measurements were taken by the Nehiyawak (Cree) people.
The video titled Interview with an Elder, Part 2 is one example of a meaningful place—a Nehiyawak (Cree) teepee used to create a home in nature that is friendly to Mother Earth. Listen to Elder Bill Sewepagaham describe the critical steps in constructing a teepee.
Complete each part of the Unit 1 Project. You will find instructions in each lesson under the Project Connection heading about when you should work on your project.
Save your responses and your work to the course folder. Have a discussion with your teacher about how your project will be submitted. Your teacher may want you to submit each component as it is completed, or your teacher may want to wait until the entire project is completed prior to submission.
You should also contact your teacher about scoring criteria for the project.
Introduction
Working with a group or on your own, your task is to create an interpretation of what place means to you. Place may be in the form of a geographical place, a home, a sculpture, something as fanciful as a castle, or anything else that comes to mind.
The whole project will be more fun if this place you describe has real meaning to you. Feel free to use a drawing, a model, a sculpture, or a photo to visually represent your place. This is the first step in your project.
Lesson 1
Think some more about the special place that will be the basis of your Unit 1 Project. Do you see places where you will use metric measures and other places where you will use imperial measures? How will you decide?
Lesson 2
What shapes are found in your special place? Use a 3-D modelling program to add a shape to your project creation, and make sure you record its dimensions. Save your 3-D model to the project folder in your course folder.
Lesson 3
There is no Project Connection for Lesson 3.
Lesson 4
Search the Internet to locate a drawing application. You might want to try an application called Google SketchUp. You may wish to view the tutorials that explain how to use this application.
Use the application to draw at least one of each of the following:
right cone
right cylinder
rectangular prism
pyramid
sphere
Save your 3-D objects.
Select at least three of these objects that occupy or could occupy your place. Add these to your Unit 1 Project. For each of these objects, include the following:
a drawing
the measurements, such as length, radius, slant height, and so on
detailed calculations of the surface area
Lesson 5
Choose two or three of the basic shapes—i.e., prism, pyramid, sphere, cylinder, or cone—that you have studied from your Google SketchUp or model.
Give reasons why you chose the shapes you did for your place.
Decide what dimensions make up the base and the height.
Take the measurements you need using Google SketchUp or physical measurements from your model.
Show the calculation of the volume along with an explanation of your strategies. (This information will look like the solutions to the Self-Check questions that you have completed in Lesson 5.)
Lesson 6: The Woodshed
Let's apply what you have been learning to a very practical situation. Ian wants to build a shed to store the wood he will need to fuel his wood-burning stove over the winter.
He has room for a woodshed 8-ft deep and 16-ft long. He wants to stack the wood 6-ft high inside the shed. Ian has figured out that this wood will last him all winter.
He wants the shed to be covered on three sides and the roof with steel cladding that costs $3.99 per square foot. The front opening is to be 8-ft high and the back wall 6-ft high, so the shed will have a roof slanted to help snow slide off. The roof will have a 2-ft overhang in both the front and the back and no overhang on the sides. Calculate how much cladding steel he will need and its cost.
A cord of wood is 4 ft × 4 ft × 8 ft. How many cords does his woodshed hold?
On average, one piece of firewood is 2-ft long, with a diameter of 8 in. How many pieces of wood will Ian be able to fit into his new shed?
Save your work in your folder.
Lesson 7
You have been introduced to six different question types in this lesson:
finding a missing side length using each of the three trigonometry ratios
finding a missing angle using each of the three trigonometry ratios
For this part of your project, please create three questions using three of the question types. These questions should relate to your place.
As well as submitting your questions and detailed solutions to your teacher, you will also post the three questions to your class discussion board. Note: Do not post your solutions; only post the questions.
Then you will find solutions for the three questions that another student has posted. First, you will place your solutions to the other student's questions in your course folder. Next, you will provide your solutions to the other student.
When a classmate answers your three questions, let the student know if he or she correctly answered your questions. If the student made a mistake, let this person know where the error occurred and provide the correct detailed answer.
You will be assessed on the following:
the three questions you post, along with the diagram(s) that go with your questions
your answers to a classmate's questions
the feedback you provided to a classmate who has answered your questions (includes accuracy and a detailed solution, as well as politeness)
For example, your project in Google SketchUp may resemble the image below. You could then submit a question and solution, such as the following:
Question 1: If the teepee is 14-ft tall and the teepee makes an angle of 60° with the ground, what is the radius of the teepee?
Solution to question 1: The opposite side is 14 ft. The variable x is the adjacent side.
So when you think of SOH CAH TOA, you can see that you will use TOA or tan.
Lesson 8: The Staircase
Anya has just purchased her first house. It is beautifully finished on the outside.
The inside does need some work, the house is not very large, and there are some space challenges.
Anya has found one problem. The staircase to her upper bedroom is very steep—almost like a slanted ladder. She wants a staircase that is easy to climb. Unfortunately, Anya is unsure of how to design this staircase in the space she has. Can you help?
Your first task is to figure out a new location for the staircase and to sketch a diagram. Since Anya wants a staircase that is easy to climb, you need to look up standard rise and run. You could research using the Internet by entering the keywords "staircase standard rise run."
You will not likely have a diagram detailed enough to answer Anya's questions about your design. Your new knowledge of trigonometry and the Pythagorean theorem will help you answer these questions.
What is the length of the new staircase?
What is the height and width of each new stair?
What is the difference between the angle made by the new staircase and the old staircase?
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10.4 Chapter Summary and Concluding Remarks
In this chapter we looked past algebra, opening the door to the idea that although algebra is extremely powerful, there are many problems whose solutions can't be found by algebraic techniques and require t numerical methods of solving equations. We also took a sneak peek into Calculus, otherwise known as the mathematics of change.
For polynomials of degree 5 and higher, there's no "quadratic formula" type solution; in other words, there's no solution that involves roots, rational numbers, addition/subtraction, and multiplication/division!
Not every equation can be solved using algebra!
The Interval Bisection Method is one technique that can be used to solve an equation for which there is no algebraic way of solving it. To find the solution, we go through an iterative process, coming up with better and better estimates of the solution.
Calculus is the mathematics of change. While algebra solves static equations, equations in the world of calculus involve quantities that are in motion. This very important topic of mathematics was first developed to help us understand the motion of planets in the Universe.
We hope that you've enjoyed your journey through algebra and along the way have picked up some general problem solving techniques that you can use in any class that requires analytical reasoning. Your thoughts mean a lot to us and if you have any suggestions regarding the text, please feel free to contact us.
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Academics
Upper School Mathematics
Philosophy
The philosophy of the mathematics program in the Upper School is reflected in its goals, which are to provide the student with the information and skills necessary for advanced work in mathematics and the sciences and for making sensible, responsible decisions in a highly technological society.
Course offerings include geometry in Form III, algebra II in Form IV, precalculus in Form V, and calculus in Form VI, although some students may take a different sequence of courses because of acceleration. A course in statistics may be offered some years. Courses are generally offered at two levels: a standard (B) level and an accelerated (A) level.
Geometry
This is a standard course in Euclidean geometry. Properties such as congruence, similarity, symmetry, and area of plane figures are studied. Two-column proofs are introduced and used extensively. A lab, using Geometer's Sketchpad, meets on a regular basis in the computer room. Basic trigonometry, polygon vocabulary and properties, circle properties, reflections, rotations, and translations are some of the other topics that are investigated. Text: McDougal Littell, Geometry.
Algebra II
The skills and concepts learned in algebra I are refined and expanded in algebra II. Quadratic equations, complex numbers, coordinate geometry, relations and functions, variation, radicals, and exponential and logarithmic functions are a few of the topics studied. Word problems receive considerable attention. Graphing calculators, the use of which is integral to the course work, are required of all students. Text: Pearson Prentice Hall.
Precalculus
This course is a rigorous study of algebraic and transcendental functions, including polynomial, trigonometric, and logarithmic functions with applications. The limit concept is studied, and the operations of differentiation and integration are introduced. Graphing calculators, the use of which is integral to the course work, are required of all students. Text: Pearson Prentice Hall.
Applied Calculus
This course offers a review of exponential, logarithmic, polynomial, and trigonometric functions followed by an introduction to the concepts of calculus: limits, derivatives and integrals. Practical applications are emphasized. Other topics in mathematics, such as probability, statistics, and combinatorics, may be discussed at the instructor's discretion. This course may not be offered every year. Text: Lial, Calculus with Applications, Pearson, Addison-Wesley.
Calculus AB
This is a standard first-term college course in differential and integral calculus that follows the AP curriculum. Limits are investigated, leading to a study of differentiation and integration. Application problems from physics, engineering, business, and economics are an essential part of the course. Graphing calculators, the use of which is integral to the course work, are required of all students. Text: Pearson Prentice Hall.
Statistics
Topics in this course include collecting data, constructing and interpreting graphical displays, counting techniques, probability, the normal distribution, confidence intervals, measures of spread, correlation and regression, and the mathematics of voting. This course may not be offered every year. Text: Elementary Statistics; Pearson Prentice Hall.
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Book Description: For combined differential equations and linear algebra courses teaching students who have successfully completed three semesters of calculus. This complete introduction to both differential equations and linear algebra presents a carefully balanced and sound integration of the two topics. It promotes in-depth understanding rather than rote memorization, enabling students to fully comprehend abstract concepts and leave the course with a solid foundation in linear algebra. Flexible in format, it explains concepts clearly and logically with an abundance of examples and illustrations, without sacrificing level or rigor. A vast array of problems supports the material, with varying levels from which students/instructors can choose.
Buyback (Sell directly to one of these merchants and get cash immediately)
Currently there are no buyers interested in purchasing this book. While the book has no cash or trade value, you may consider donating it
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This 170-page student text and exercise book is designed to give the student clear understanding and proficiency in basic arithmetic, including addition, subtraction, multiplication, and division. This book employs the highly motivational approach of integrating the teaching of math with the historical development of math to show why and how each arithmetic operation was invented. The student is given clear instructions on how to perform each operation and plenty of exercises to develop mastery and proficiency.
Mathematics as a science is a vast, integrated body of abstract principles derived inductively from concrete observations. Math is a marvelous human creation that began tens of thousands of years ago to satisfy certain human survival needs. Our unique approach in this book is to present math as it developed historically whereby each new idea or method can be seen to flow logically from previous discoveries and observations. For example, the student learns how addition is merely a rapid form of counting and how multiplication is merely a rapid form of addition. Our inductive approach allows the student to keep abstract concepts in math intimately tied to reality rather than as dry, floating abstractions. Furthermore, we show how each new discovery served to satisfy a human need and consequently benefited people, which makes learning math more interesting and enjoyable.
This student version is divided into two parts. Part I is the main text (80 pages) which gives some historical background for each arithmetic operation and clearly demonstrates to the student how to perform the operation. Part I contains some exercises to allow the student to immediately check if he or she understands the method. The bulk of the exercises are in Part II (84 pages) which, for each chapter of Part I, includes numerical problems, word problems, and short essay-type questions. The exercises are designed to help the student understand the concepts clearly and develop proficiency in the arithmetic operations.
In addition to the exercises in Part II, the teacher�s version of this book contains 38 timed math sheets covering all arithmetic operations. Prior to every math class the teacher will hand out one or two of the sheets for the students to complete as quickly and accurately as possible while being timed. In this manner the teacher can monitor the proficiency of the student.
This book is not intended to cover every math topic that the student is required to master as per his or her particular education district. It�s designed to build a solid, conceptual understanding of basic arithmetic. Fractions and decimals are not covered in this book. They will be covered in the next book of this series.
This book is designed so that the student can perform all the work right in the book if desired. Otherwise the work could be performed in a separate notebook that can be purchased from any local supply store. If you purchase this student's version, then we highly recommended purchasing at least one copy of the teacher�s version because it contains all the answers to the exercises. (See below.)
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Mathematics Department
The goal of the Math Department is to provide a curriculum that enables each student to challenge himself to achieve to his fullest potential – in math and outside it, in high school and beyond. We help prepare students for the highest levels of science courses at St. Thomas, which require mathematical skills, and for their mathematics courses at the university level, as well. For many highly motivated students, this includes the challenge to earn college credit through the Advanced Placement program.
We endeavor to help students learn to appreciate the beauty of mathematics, the richness of its language, the elegance of its logic, the power of its applications, and the joy in discovering and understanding its ideas and results. In the passage above, Galileo is writing of these varied aspects of our subject.
Through our math courses, students hone their skills of careful and creative thought, and they begin to see mathematics everywhere: the abstract equations they learn to solve trace the path of a thrown ball or an orbiting planet; the number of opposing spirals on a pine cone or a pineapple (eight and 13), and the seed head of coneflower (34 and 55), are all related numbers, related in turn to the graceful spiral of a seashell and to ratios of lengths in a perfect pentagon!
Students use cutting-edge technology for their investigations: from CBR devices for their study of the mathematics of motion to the Geometer's Sketchpad to construct animated kaleidoscopic images applying the properties of congruence transformations.
This course provides additional challenges within and enrichment to the Algebra I curriculum. Geared toward students for whom an accelerated pace of study is more appropriate, it emphasizes additional challenges, depth, and creative problem-solving within the topics outlined in Algebra I. TI30XIIS Calculator Required.
ADVANCED ALGEBRA I / GEOMETRY
This course is an accelerated Algebra One course (see Algebra One) and Advanced Plane Geometry (see below). Both Adv Algebra and Adv Geometry courses are covered within the year. TI30XIIS Calculator Required.
PLANE GEOMETRY
This course is a study of lines, planes, constructions, proofs, angles, perpendicular lines,
This course is a study of polynomials, special products, factoring, equations, radicals, quadratics, conics, inequalities, solving systems of equations, exponents, logarithms, trigonometric functions, formulas, identities, law of sines, law of cosines, and solutions of triangles. Pre-requisite: 80 in Adv Algebra I/Geometry or Adv Geometry for sophomores and 85 in Adv Geometry for juniors. TI84 Plus Silver Calculator Required.
TRIGONOMETRY / ALGEBRA III
This course is a study of angles, right triangles, trigonometric functions, identities, equations, law of sines, law of cosines, trig graphs, trig inverses, an introduction to probability and statistics, and continuing the study of polar coordinates, exponents, logarithms, conics, rational expressions, and sequences and series. TI84 Plus Silver Calculator Required.
PRE-CALCULUS
This course is a two semester course covering all of Trigonometry and other topics leading up to Calculus, including relations and functions, logarithmic and exponential functions, conic sections, graphing techniques, sequences and series, probability and statistics, and limits. Pre-requisite: 78-86 in Adv Algebra II/Trig or 78-89 in Algebra II/Trig or 83-94 in Algebra II and recommendation. TI84 Plus Silver Calculator Required.
This course is a two-semester introduction to the fundamental concepts of Calculus covered in a first semester college course. Students will not take the Calculus AP test at the end of the year; rather, a school final exam will be given at the end of each semester. Pre-requisite: 70-74 in Advanced Pre-Calculus or 75-92 in Pre-Calculus. TI89 Calculator Required.
ADVANCED PLACEMENT CALCULUS AB
This course is a two semester AP Calculus course covering the requirements set forth by the College Board for one semester of college calculus. The AP exam is taken in May. Pre-requisite: 93+ in Pre-Calculus; 75-92 in Advanced Pre-Calculus. TI89 Calculator Required.
ADVANCED PLACEMENT CALCULUS BC
This course is a two semester AP Calculus course covering the requirements set forth by the College Board for two semesters of college calculus. The AP exam is taken in May. Pre-requisite: 93 in Adv. Pre-Calculus and teacher recommendation. TI89 Calculator Required.
MATHEMATICAL PROBLEM SOLVING
This course is an elective designed to help interested students develop their mathematical reasoning and problem-solving skills. While the course will introduce some important new ideas, most of the material itself will not be new. However, through a review of fundamental concepts from earlier algebra and geometry courses – including ratios, polynomials, exponentials, logarithms, probability, triangles, circles, and three-dimensional geometry – we will apply those ideas to ever more challenging and intriguing problems. The review itself and the problem-solving skills gained will be of help both in preparing for standardized tests and in future math courses. Pre-requisite: Algebra II. (One semester)
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Virtual Nerd isn't the typical one-size-fits-all approach to learning algebra. Instead, you'll find a library of interactive lessons that you can use on your own time, at your own pace, and in a way that suits your needs.
Whether it's solving inequalities or working through quadratic equations, Virtual Nerd can help you pinpoint where you need help and provide the instruction you need so you can move on to the next level.
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Overview of the Mathematics program
It is the goal of the Mathematics program to develop mathematically proficient and confident students. At all levels our program is aligned with the New York State standards and is designed to help students develop skills in computation and problem-solving. With the proper foundation laid at the elementary level, the secondary curriculum provides alternate programs of study to insure that the needs of each student are addressed.
For the majority of high school students a three-year sequence of study in Regents-endorsed math courses is available. This can be enhanced with a choice of electives in the senior year. For students experiencing difficulty, an extensive support program is available to help ensure achievement at the required level of proficiency. Students seeking an additional challenge may take part in the enriched and accelerated honors courses that culminate with the study of calculus in the high school. All students, regardless of their level of accomplishment, are urged to study four years of mathematics.
"The mathematical sciences particularly exhibit order, symmetry, and limitation, and these are the greatest form of beautiful." Aristotle
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Overview
Main description
Building a Better Path To Success!Table of contents
Chapter 2: Linear Equations and Inequalities in One Variable2.1 Solving Linear Equations in One Variable2.2 Applications of Linear Equations2.3 Geometry Applications and Solving Formulas2.4 More Applications of Linear Equations2.5 Linear Inequalities in One Variable2.6 Compound Inequalities in One Variable2.7 Absolute Value Equations and Inequalities
Chapter 3: Linear Equations in Two Variables and Functions3.1 Introduction to Linear Equations in Two Variables3.2 Slope of a Line and Slope-Intercept Form3.3 Writing an Equation of a Line3.4 Linear and Compound Linear Inequalities in Two Variables3.5 Introduction to Functions
Chapter 4: Solving Systems of Linear Equations4.1 Solving Systems of Linear Equations in Two Variables4.2 Solving Systems of Linear Equations in Three Variables4.3 Applications of Systems of Linear Equations4.4 Solving Systems of Linear Equations Using Matrices
Chapter 5: Polynomials and Polynomial Functions5.1 The Rules of Exponents5.2 More on Exponents and Scientific Notation5.3 Addition and Subtraction of Polynomials and Polynomial Functions5.4 Multiplication of Polynomials and Polynomial Functions5.5 Division of Polynomials and Polynomial Functions
Chapter 6: Factoring Polynomials6.1 The Greatest Common Factor and Factoring by Grouping6.2 Factoring Trinomials6.3 Special Factoring TechniquesPutting It All Together6.4 Solving Quadratic Equations by Factoring6.5 Applications of Quadratic Equations
Chapter 9: Quadratic Equations and Functions9.1 The Square Root Property and Completing the Square 9.2 The Quadratic FormulaPutting It All Together9.3 Equations in Quadratic Form9.4 Formulas and Applications9.5 Quadratic Functions and Their Graphs9.6 Applications of Quadratic Functions and Graphing Other Parabolas9.7 Quadratic and Rational Inequalities
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Browse By Age
GeoGebra: Award-winning Geometry Software (that's also free)
GeoGebra was recently announced the 2009 winner of the "Microsoft Education Award" by The Tech Museum. This award is only the latest feather in GeoGebra's much-decorated cap.
GeoGebra is dynamic mathematics software for all levels of education that joins arithmetic, geometry, algebra and calculus. It offers multiple representations of objects in its graphics, algebra, and spreadsheet views that are all dynamically linked. It is particularly well-suited to middle and high school mathematics.
While other interactive software (e.g. Cabri Geometry, Geometer's Sketchpad) focus on dynamic manipulations of geometrical objects, the idea behind GeoGebra is to connect geometric, algebraic, and numeric representations in an interactive way. You can do constructions with points, vectors, lines, conic sections as well as functions and change them dynamically afterwards. Furthermore, GeoGebra allows you to directly enter and manipulate equations and coordinates. Thus you can easily plot functions, work with sliders to investigate parameters, find symbolic derivatives, and use powerful commands like Root or Sequence.
The best part is that unlike Geometer SketchPad and Cabri Geometry, GeoGebra is absolutely free.
[Note: This post has been authored by guest writer, Sidhanth Venkatasubramaniam].
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Creativity, Giftedness, and Talent Development in Mathematics (HcCreativity, Giftedness, and Talent Development in Mathematics (Hc) Book Description
A Volume in The Montana Mathematics Enthusiast: Monograph Series in Mathematics Education Series Editor Bharath Sriraman, The University of Montana Our innovative spirit and creativity lies beneath the comforts and security of today's technologically evolved society. Scientists, inventors, investors, artists and leaders play a vital role in the advancement and transmission of knowledge. Mathematics, in particular, plays a central role in numerous professions and has historically served as the gatekeeper to numerous other areas of study, particularly the hard sciences, engineering and business. Mathematics is also a major component in standardized tests in the U.S., and in university entrance exams in numerous parts of world. Creativity and imagination is often evident when young children begin to develop numeric and spatial concepts, and explore mathematical tasks that capture their interest. Creativity is also an essential ingredient in the work of professional mathematicians. Yet, the bulk of mathematical thinking encouraged in the institutionalized setting of schools is focused on rote learning, memorization, and the mastery of numerous skills to solve specific problems prescribed by the curricula or aimed at standardized testing. Given the lack of research based perspectives on talent development in mathematics education, this monograph is specifically focused on contributions towards the constructs of creativity and giftedness in mathematics. This monograph presents new perspectives for talent development in the mathematics classroom and gives insights into the psychology of creativity and giftedness. The book is aimed at classroom teachers, coordinators of gifted programs, mathcontest coaches, graduate students and researchers interested in creativity, giftedness, and talent development in mathematics.
Popular Searches
The book Creativity, Giftedness, and Talent Development in Mathematics (Hc) by Bharath Sriraman
(author) is published or distributed by Information Age Publishing [159311978X, 9781593119782].
This particular edition was published on or around 2008-07-31 date.
Creativity, Giftedness, and Talent Development in Mathematics (Hc) has Hardcover binding and this format has 312 number of pages of content for use.
This book by Bharath Sriraman
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Teaching Overview
Human beings have a natural desire to do things as well as possible. And when there are several possibilities of doing something, they try to choose the best alternative they can afford. This desire requires two different skills: one needs to decide on the aspects of the real-world that are relevant for the problem context, to recast them in an abstract optimization model, to find an optimal solution to this model, and to reflect on the effects of the implementation process of this optimal solution in real-life. This procedure is embedded in a larger process that is called Operations Research (or System Design and Optimization). The second skill concerns the design of outstanding algorithms to solve such abstract optimization models arising from the modelling of real-life problems. The development of such algorithms, their complexity analysis, and their simulation is summarized under the name of Mathematical Optimization.
The Institute for Operations Research offers lectures, seminars, and student projects in both fields of Optimization
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Electrical And Electronic Calculations
Course Outline
Upon completion of this course you will be able to perform all calculations within Basic Electricity and Basic Electronics field. You will also be able to perform calculations essential to microcomputers. This course covers essential knowledge which is used daily, on the job by electronics technicians as well as engineers. This is a single stream course consisting of 13 lessons and 13 examinations.
Topics of study covered within this course include the following:
Decimal Numbers and Arithmetic
Negative Numbers
Fractions
Powers and Roots
Powers of 10
Logarithms
The Metric System
Algebra
Methods of Solving Equations
Simultaneous Linear Equations
Trigonometry
Computer Mathematics
Complex Numbers
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More About
This Textbook
Overview
This third edition of Arem's CONQUERING MATH ANXIETY workbook presents a comprehensive, multifaceted treatment approach to reduce math anxiety and math avoidance. The author offers tips on specific strategies, as well as relaxation exercises. The book's major focus is to encourage students to take action. Hands-on activities help readers explore both the underlying causes of their problem and viable solutions. Many activities are followed by illustrated examples completed by other students. The free accompanying CD contains recordings of powerful relaxation and visualization exercises for reducing math anxiety 5, 2002
Great tool!
I think that this book is a great tool for those who suffer math anxiety. It has great exercises for those who want to be able to approach math with out the fear of failure that often comes along with math. People need to know that math anxiety is real, is experienced by many people and can be overcome!
Was this review helpful? YesNoThank you for your feedback.Report this reviewThank you, this review has been flagged.
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The purpose of this course is to continue the development of the Big Ideas and Supported Ideas addressed in the Next Generation Sunshine State Standards across the PreK-12 curriculum. The content should include, but not be limited to, numeration, whole numbers, fractions, decimals, percents, integers, geometry, measurement, estimation, graphing, number theory, ratio and proportion, probability, statistics, data analysis, algebraic thinking.
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Introduction: Mathematics Problem Solving
by
James W. Wilson
University of Georgia
Mathematics problem solving is our topic. There is a danger that we have
become so glib about "problem solving" that we no longer find
sufficient substance in the term. I often use terms like "investigation,"
"exploration," "open-ended," "problem contexts,"
or "constructing mathematics" to warn students that the answer
getting exercises they have been led to believe are problem solving are
only a small part of the process. Perhaps the following quote from Polya
captures some of the spirit of problem solving in mathematics.
Your problem may be modest; but if it challenges your curiosity and
brings into play your inventive faculties, and if you solve it by your
own means, you may experience the tension and enjoy the triumph of discovery.
Such experiences at a susceptible age may create a taste for mental work
and leave their imprint on mind and character for a lifetime. (Polya,
How to Solve It, 1945, p. v.)
We will spend our time in this course solving problems and posing more problems
than we solve.
It is useful to have a framework to think about the processes involved in
mathematical problem solving. Most formulations of a problem solving framework
in U. S. textbooks attribute some relationship to Polya's problem solving
stages (1945). These stages were described by 1) understanding the problem,
2) making a plan, 3) carrying out the plan, and 4) looking back.
Polya also stated that problem solving was a major theme of doing mathematics
and when he wrote about what he expected of students, he used the language
of "teaching students to think" (1965). "How to think"
is a theme that underlies much of genuine inquiry and problem solving in
mathematics. Unfortunately, much of the well-intended efforts of teaching
students "how to think" in mathematics problem solving gets transformed
into teaching "what to think" or "what to do." This
is, in particular, a byproduct of an emphasis on procedural knowledge about
problem solving such as we have in the linear framework of U. S. mathematics
textbooks and the very limited problems/exercises included in lessons. To
quote Polya again:
Thus a teacher of mathematics has a great opportunity. If he fills
his allotted time with drilling his students in routine operations he kills
their interest, hampers their intellectual development, and misuses his
opportunity. But if he challenges the curiosity of his students by setting
them problems proportionate to their knowledge, and helps them to solve
their problems with stimulating questions, he may give them a taste for,
and some means of, independent thinking.(Polya, 1945, p. v.)
There is a dynamic and cyclic nature of genuine problem solving. A student
may begin with a problem and engage in thought and activity to understand
it. The student attempts to make a plan and in the process may discover
a need to understand the problem better. Or when a plan has been formed,
the student may attempt to carry it out and be unable to do so. The next
activity may be attempting to make a new plan, or going back to develop
a new understanding of the problem, or posing a new (possibly related) problem
to work on.
The framework at the right is useful for illustrating the dynamic, cyclic interpretation
of Polya's stages. Any of the arrows could describe student activity (thought) in the process
of solving mathematics problems. Clearly, genuine problem experience in
mathematics can not be captured by the outer, one-directional arrows alone.
It is not a theoretical model. Rather, it is a framework for discussing
various pedagogical, curricular, instructional, and learning issues involved
with the goals of mathematical problem solving in our schools.
One aspect of "Looking Back" is the generation of new problems.
New problems and investigations may spring from any of the following:
1. New problems suggested by the one we have worked on.
2. Generalizations of the results of a problem.
3. Generalizations of the strategies and techniques.
4. Searches for "better" solutions.
5. Searches for alternative solutions.
6. Explorations to understand the problem, its results, or its strategies.
7. Serendipity
8. Curiosity
9. Imagination
10. Stubbornness
There will be many examples in the material from this course.
References
Polya, G. (1945) How to Solve It: A New Aspect of Mathematical Method.
Princeton, NJ: Princeton University Press.
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Lulu Marketplace
GOLDen Mathematics: Algebra – The Basics
Chapter 1 only from GOLDen Mathematics: Elementary & Intermediate Algebra. See that book's description for more information. Topics covered: an introduction to algebra and the real number system including working with a calculator. (2 sections, 17 pages)
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MQR.4.4 Functions of More Than One Variable
The mathematics curriculum in grades 9-12 generally focuses on functions of one variable. Real-world applications, however, often require consideration of more than one variable. This unit provides opportunities for students to work with functions of more than one variable.
Instructional Days (suggested)
10 - 15 days
Click on subtopics below to see resources from the Ohio Resource Center
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Algebra Printables Slideshow (Grades 9-12)
Get a sneak peek of what's inside our printable book "Algebra (9-12)," with this slideshow. Students will practice problem solving, quadratic equations, polynomials, statistics, and more. These worksheets and lessons will be an excellent algebra review for high
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I'm doing some literature review for my research and I came across this article by L.A Steen in Middle Matters. He was arguing about the Algebra for All standard in the US and part of the article includes description of what is algebra. I thought I should share them in this blog because it is something very important teachers should be aware of when they teach algebra or what they conceive what algebra is and for. Oftentimes, when students ask what algebra is and what they are going to need it for, teachers lazy answer is "Algebra is just like your math in the grades only that this time you work with letters instead of numbers!"
Algebra is the language of mathematics, which itself is the language of the information age. The language of algebra is the Rosetta Stone of nature and the passport to advanced mathematics (Usiskin, 1995).
It is the logical structure of algebra, not the solutions of its equations, that made algebra a central component of classical education.
As a language, algebra is better learned earlier and harder, when learned later.
In the Middle Ages, algebra meant calculating by rules (algorithms). During the Renaissance, it came to mean calculation with signs and symbols–using x's and y's instead of numbers. (Even today, lay persons tend to judge algebra books by the symbols they contain: they believe that more symbols mean more algebra, more words, less.) I think that many algebra classes still promote this view.
In subsequent centuries, algebra came to be primarily about solving equations and determining unknowns. School algebra still focuses on these three aspects: employing letters, following procedures, and solving equations. This is still very true. You can tell by the test items and exercises used in classes.
In the twentieth century algebra moved rapidly and powerfully beyond its historical roots. First it became what we might call the science of arithmetic–the abstract study of the operations of arithmetic (addition, subtraction, multiplication, etc.). As the power of this "abstract algebra" became evident in such diverse fields as economics and quantum mechanics, algebra evolved into the study of all operations, not just the four found in arithmetic.
Algebra is said to be the great gatekeeper because knowledge and understanding of which can let people into rewarding careers.
Algebra is the new civil right (Robert Moses). It means access. It means success. It unlocks doors to productive careers and gives everyone access to big ideas.
And I like the education battle cry Algebra for All. Of course not everyone is very happy about this. Steen for example wrote in 1999:
No doubt about it: algebra for all is a wise educational goal. The challenge for educators is to find means of achieving this goal that are equally wise. Algebra for all in eighth grade is clearly not one of them–at least not at this time, in this nation, under these circumstances. The impediments are virtually insurmountable:
Relatively few students finish seventh grade prepared to study algebra. At this age students' readiness for algebra–their maturity, motivation, and preparation–is as varied as their height, weight, and sexual maturity. Premature immersion in the abstraction of algebra is a leading source of math anxiety among adults.
Even fewer eighth grade teachers are prepared to teach algebra. Most eighth grade teachers, having migrated upwards from an elementary license, are barely qualified to teach the mix of advanced arithmetic and pre-algebra topics found in traditional eighth grade mathematics. Practically nothing is worse for students' mathematical growth than instruction by a teacher who is uncomfortable with algebra and insecure about mathematics.
Few algebra courses or textbooks offer sufficient immersion in the kind of concrete, authentic problems that many students require as a bridge from numbers to variables and from arithmetic to algebra. Indeed, despite revolutionary changes in technology and in the practice of mathematics, most algebra courses are still filled with mindless exercises in symbol manipulation that require extraordinary motivation to master.
Most teachers don't believe that all students can learn algebra in eighth grade. Many studies show that teachers' beliefs about children and about mathematics significantly influence student learning. Algebra in eighth grade cannot succeed unless teachers believe that all their students can learn it. (all italics, mine)
I shared these here because in my part of the globe the state of algebra education is very much like what Steen described. You may also want to read about the expressions and equations that makes algebra a little more complicated to students.
Author
This site is my contribution to narrowing the gap between research and practice in mathematics teaching and learning. I share teaching and learning materials and blog about reforms, issues, and teaching practices in mathematics.
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This graphing calculator for your Pocket PC provides 2D, 3D and 4D graphing for on your computer and mobile device. It includes helpful tutorials that show you how to plot graphs, calculate equations and solve formulas.
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Guide for Math Students
Welcome to the Math Club's Guide to Undergraduate Life! If you're here, you are probably a current or prospective undergraduate interested in mathematics. Princeton is a great place to do math. The department is relatively large, including 65 faculty members, 66 graduate students, and 73 undergraduate math majors. Due to associations with students and faculty in other departments, the greater math community is even larger.
The close-knit undergraduate community includes students with academic interests ranging from algebraic number theory to finance economics, and personal interests ranging from rowing to juggling. Whatever your own interests, we're sure you'll find some people who share them! Poke around the Guide, drop by some of our events, and feel free to get in touch!
The Guide contains detailed descriptions of the Club's activities, as well as advice about the issues most important to undergraduates in the department. For ease of browsing, it is broken down into the following sections:
Course Guide. General overviews of each of the major sub-fields of mathematics, as well as detailed descriptions of the courses in that area and advice on which courses to take—and in what order.
Interdisciplinary Study. Articles describing popular interdisciplinary programs combining math and another subject, including computer science, physics, economics, and finance. If you're interested in studying math and one of these other subjects in parallel, check it out!
Applied Math. Guidance on how to focus on applied math at Princeton, including information about the Program in Applied and Computational Mathematics (PACM) and its certificate.
Undergraduate Research. Advice on how to get involved in undergraduate research, both early in your Princeton career and through junior and senior independent work. You'll also find suggestions about the research process and the mechanics of choosing advisers and problems.
Summer Opportunities. Information about the various summer programs of interest to math majors and those interested in mathematics. This includes research opportunities, teaching jobs at camps like PROMYS, and industry internships.
Social Life. All about the Club's social side. Yes, Math Club members are people, too. We enjoy board games, movies, pizza and pie (and Pi Day!), and dinners, among other things. Check out the page, then come hang out!
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◊Business Technology
Pre-Algebra Mathematics
Prepares students who want to strengthen computational and problem-solving skills before proceeding to an algebra course. Reviews arithmetic and measurements (both metric and American). Teaches the concept of variables, operations involving signed numbers, simplifying algebraic expressions, solving equations and inequalities in one variable, solving simple formulas, ratio and proportion, and solving application problem using equations.
Prereq: MATH 1 or placement at MATH 22, and ENG 19 with grade C or better or placement at least ENG 22; or consent.
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The Math Curriculum is standard-based, integrated and aligned with the Washington State Essential Academic Learning Requirements. The Common Core Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students.These practices rest on important "processes and proficiencies" with longstanding importance in mathematics education. The first of these are problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency
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Catalog of MAA Publications 2011 Annual : Page 8
NEW Lie Groups A Problem-Oriented Introduction via Matrix Groups Harriet Pollatsek ■ MAA Textbooks Can be used as supplementary reading in a linear algebra course or as a primary text in a "bridge" course that helps students make the transition to courses that emphasize definition and proofs, as well as for an upper level elective. The work of the Norwegian mathematician So-phus Lie extends ideas of symmetry and leads to many applications in mathematics and physics. Ordinarily, the study of the "objects" in Lie's theory (Lie groups and Lie algebras) requires exten-sive mathematical prerequisites beyond the reach of the typical undergrad-uate. By restricting to the special case of matrix Lie groups and relying on ideas from multivariable calculus and linear algebra, this lovely and im-portant material becomes accessible even to college sophomores. Working with Lie's ideas fosters an appreciation of the unity of mathematics and the sometimes surprising ways in which mathematics provides a language to describe and understand the physical world. This is the only book in the undergraduate curriculum to bring this material to students so early in their mathematical careers. Geometric Transformations IV Circular Transformations I. M. Yaglom Translated by Abe Shenitzer ■ Anneli Lax NML The familiar plane geometry of high school— figures composed of lines and circles—takes on a new life when viewed as the study of properties that are preserved by special groups of transformations. No longer is there a single, universal geometry: different sets of transformations of the plane correspond to intriguing, disparate geometries. This book is the concluding Part IV of Geometric Transformations , but it can be studied independently of Parts I, II, and III, which appeared in this series as Volumes 8, 21, and 24. Part I treats the geometry of rigid motions of the plane (isometries); Part II treats the geometry of shape-preserving transformations of the plane (similarities); Part III treats the geometry of transformations of the plane that map lines to lines (affine and projective transformations) and introduces the Klein model of non-Euclidean geometry. The present Part IV develops the geometry of transformations of the plane that map circles to circles (conformal or anallagmatic geometry). The notion of inversion, or reflection in a circle, is the key tool employed. Applications include ruler-and-compass constructions and the Poincaré model of hyper-bolic geometry. The straightforward, direct presentation assumes only some background in high school geometry and trigonometry. Numerous exercises lead the reader to a mastery of the methods and concepts. The second half of the book contains detailed solutions of all the problems. 164 pp., 2009 List: $63.95 ISBN: 978-0-88385-759-5 MAA Member: $51.95 Hardbound Catalog Code: LIG/YD11 Visual Group Theory Nathan Carter ■ Classroom Resource Materials Could serve as a text in abstract algebra/ group theory at the undergraduate level, or as supplementary reading at the graduate level. 296 pp., 2009 List: $46.95 ISBN: 978-0-88385-648-2 MAA Member: $36.95 Paperbound Catalog Code: NML-44/YD11 Over 300 illustrations printed in full color. In a New York Times article, Steven Strogatz of Cornell University calls Visual Group Theory a "terrific new book." He describes the book as "one of the best introductions to group theory—or to any branch of higher math—I've ever read." The more than 300 illustrations in Visual Group Theory bring groups, sub-groups, homomorphisms, products, and quotients into clear view. Every topic and theorem is accompanied with a visual demonstration of its mean-ing and import, from the basics of groups and subgroups through advanced structural concepts such as semidirect products and Sylow theory. Although the book stands on its own, the free software Group Explorer makes an excellent companion. It enables the reader to interact visually with groups, including asking questions, creating subgroups, defining homomorphisms, and saving visualizations for use in other media. It is open source software available for Windows, Macintosh, and Unix systems from Flatland Edwin Abbott Notes and commentary by William F. Lindgren & Thomas F. Banchoff ■ Spectrum Flatland , Edwin Abbott's story of a two-dimen-sional universe as told by one of its inhabitants who is introduced to the mysteries of three-dimensional space, has enjoyed an enduring popularity from the time of its publication in 1884. This fully annotated edition enables the modern-day reader to under-stand and appreciate the many "dimensions" of this classic satire. Mathe-matical notes and illustrations enhance the usefulness of Flatland as an elementary introduction to higher-dimensional geometry. Historical notes show connections to late-Victorian England and to classical Greece. Citations from Abbott's other writings, as well as the works of Plato and Aristotle, serve to interpret the text. Commentary on language and literary style in-cludes numerous definitions of obscure words. An appendix gives a compre-hensive account of the life and work of Flatland 's remarkable author. 334 pp., 2009 List: $71.95 ISBN: 978-0-88385-757-1 MAA Member: $57.50 Hardbound Catalog Code: VGT/YD11 296 pp., 2010 List: $14.99 ISBN: 978-0-52175-994-6 Paperbound Catalog Code: FTL/YD11 5 8 To Order : Call 1.800.331.1622 or Online at
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what is the different between spec math and math?
please anyone explain..
thanks!
spec math is totally different from math(or the full term : mathematical studies)
spec math is totally much more difficult that math studies. cos in math studies you just learn mostly on some algebra, differentiation and intergration.. and questions are mostly on these stuff..
where as for spec math, the questions are normally much more on proving equations like " show that this = that ", then you'll have to apply some theories or laws to prove that this = that. spec math is normally taken by engineering students. though some medic students does take it as well. say in AUSMAT 17, out of 300 students in our batch, about 70 were medic students. and out of this 70 medic students less than 5 did actually take spec mathahhh... okay. okay. looks like calculus is gonna b a tough subject, huh? how bout the other question? the difference between applicable math and discrete math is??ahhh... okay. okay. looks like calculus is gonna b a tough subject, huh? how bout the other question? the difference between applicable math and discrete math is??
yeah.. its quite true alright... what ever you study for your matriculation.. it might seem related to what you study in form 5.. in fact it is related.. but when you really get into it.. its a whole new level for you and definitely not as easy as it seems in form 5
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0738609277
9780738609270
California Star Mathematics, Grades 8-9: Every eighth and ninth grade student in California must participate in the STAR program Are you ready for the STAR Mathematics Exam? REA's California STAR Grades 8 & 9 Mathematics test prep helps you sharpen your skills and pass the exam! Fully aligned with the learning standards of the California Department of Education, this second edition of our popular test prep provides the up-to-date instruction and practice that eighth and ninth grade students need to improve their math skills and pass this important state-required exam. The comprehensive review features student-friendly, easy-to-follow lessons and examples that reinforce the key concepts tested on the STAR, including: ArithmeticAlgebraGeometryData AnalysisStatisticsWord ProblemsFocused lessons explain math concepts in easy-to-understand language that's suitable for eighth and ninth grade students at any learning level. Our tutorials and targeted drills increase comprehension while enhancing your math skills. Color icons and graphics throughout the book highlight practice problems, charts, and figures. The book contains four diagnostic tests that are perfect for classroom quizzes, homework, or extra study. A full-length practice exam lets you test your knowledge and reinforces what you've learned. The practice test comes complete with detailed explanations of answers, allowing you to focus on areas in need of further study. REA's test-taking tips and strategies give you an added boost of confidence so you can succeed on the exam. Whether used in a classroom, at home for self-study, or as a textbook supplement, teachers, parents, and students will consider this book a "must-have" prep for the STAR. REA test preps have proven to be the extra support students need to pass their challenging state-required tests. Our comprehensive test preps are teacher-recommended and written by experienced educators. «Show less
Rent California Star Mathematics, Grades 8-9 2nd Edition today, or search our site for other Hearne Tests
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Fundamentals of Piecewise Polynomial Interpolation, used in graphics, that takes you from linear interpolation to all the common curve drawing methods used in graphics. It uses only Algebra and avoids more complex mathematic notations as much as possible. It also includes a reference with all the common methods used for graphics curves.
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Precalculus Functions and Graphs
9780072867398
ISBN:
0072867396
Edition: 6 Pub Date: 2007 Publisher: McGraw-Hill College
Summary: The Barnett, Ziegler, Byleen College Algebra series is designed to be user friendly and to maximize student comprehension. The goal of this series is to emphasize computational skills, ideas, and problem solving rather than mathematical theory. Precalculus introduces a unit circle approach to trigonometry and can be used in one or two semester college algebra with trig or precalculus courses. The large number of peda...gogical devices employed in this text will guide a student through the course. Integrated throughout the text, students and instructors will find Explore-Discuss boxes which encourage students to think critically about mathematical concepts. In each section, the worked examples are followed by matched problems that reinforce the concept being taught. In addition, the text contains an abundance of exercises and applications that will convince students that math is useful. A Smart CD is packaged with the seventh edition of the book. This CD reinforces important concepts, and provides students with extra practice problems.[read more]
ALTERNATE EDITION: Lightly used instructor's edition. May contain answers/notes in margins. Ships same day or next business day. Free USPS Tracking Number. Excellent Customer Service. Ships from TN[less]
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Overview - AGS MATHEMATICS: CONCEPTS TEACHER RES LIBR CD-ROM
High-Interest Content with Low Readability.
AGS Mathematics: Concepts & Pathways introduces or remediates critical math concepts. Students will be ready to take the next step in math with text that offers engaging elements to motivate learning. These texts make it easy for you to engage students who struggle with reading, language, or a learning disability.
The students who would benefit from these textbooks are those who:
• divide their time between regular classrooms and sheltered environments.
• read below grade level.
• need dedicated support to make lessons understandable.
• may move directly to work or transition programs.
Mathematics: Concepts
Give students the basic math concepts they will need throughout life. Text includes many features that stimulate learning, such as highlighted vocabulary words with definitions, step-by-step examples, sidebar features, notes, chapter reviews, and test-taking tips. This textbook leads students to success in understanding arithmetic operations, introduces algebraic concepts, and includes problem solving and estimation.
Mathematics: Pathways
Students will be ready to take the next step in math with this text that offers many features that stimulate learning, such as highlighted vocabulary words with definitions, step-by-step examples, sidebar features, notes, chapter reviews, and test-taking tips. In addition, the curriculum includes hands-on manipulative activities and exercises that let students construct models that demonstrate selected lesson concepts. Throughout the text, students apply math skills to real-life situations.
• Teacher's Edition - includes the complete Student Text with teaching suggestions, lesson overviews, tips on learning styles, and a variety of activities. • Teacher's Resource LibraryCD-ROM - offers hundreds of activities, the Student Workbook, a Self-Study Guide for students who want to work at their own pace, two forms of chapter tests, plus midterm and final tests.
• Solutions Key on CD-ROM - presents an easy-to-use electronic format of the Classroom Resource Binder.
• Skill Track Software Site License - includes hundreds of multiple choice items relating to the textbook's content and provides group and individual reports for monitoring student progress.
System Requirements for CD-ROMs and Software:
Acrobat Reader 4 or 5 - requires 8MB RAM or 64MB RAM, respectively, to install from resource cd. This step may not be necessary if Acrobat Reader 4 is currently installed on your computer.
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With all those Xs and Ys, pre-algebra can be very intimidating. But with the right preparation, pre-algebra is as easy as 1-2-3! Building on your knowledge of basic math, The Standard Deviants eliminate the intimidation factor by presenting the material in an easy-to-understand manner using plenty of examples and computer graphics.Standard Deviants - Pre-Algebra Power Program 2 Standard Deviants - Pre-Algebra Power Program 2 picks up where Part 1 left off - basic linear equations and variables. From there, you'll tackle one of the trickier, yet more interesting applications: graphing!About Standard Deviants: Recommended by teachers and professors across the country, the Standard
Deviants approach to teaching is anything but standard. By simplifying complex
subjects and presenting the material with humorous skits, computer graphics
and a fun, approachable format, the Standard Deviants make even the most
difficult subjects enjoyable!
The Standard Deviants DVDs are the perfect way to learn and review at your
own pace with real-time, immediate feedback – all at the touch of a button!
The Standard Deviants combine cutting-edge technology, interactive quizzes,
award-winning educational material and a troupe of young actors and
comedians. Everything you need to learn is at your fingertips. Recommended
for junior high, high school, college and beyond!
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Trigonometric Functions and Calculus
for
Liberal Arts and Business Majors
A Complete Text Resource on the World Wide Web
by Stefan Waner and Steven R. Costenoble
Table of Contents
1. Modeling with the Sine Function
Exercises
2. The Six Trigonometric Functions
Exercises
3. Derivatives of Trigonometric Functions
Exercises
4. Integrals of Trigonometric Functions
Exercises
Back to Main Page
Introduction
Trigonometric functions are often omitted in basic calculus courses for students not majoring in the mathematical sciences. However, the sine and cosine functions are extremely useful in modeling cyclical trends, such as the seasonal variation of demand for certain items, or the cyclical nature of recession and prosperity. The four sections we present here are designed with this in mind; we are concerned less with the geometry of triangles than on applications of the trigonometric functions in modeling real life situations.
In the first section, we focus on the use of the sine function to model cyclical phenomena, postponing the introduction of the other trigonometric functions to Section 2. Sections 3 and 4 deal with the calculus of these functions. Of particular interest is the tabular approach to integration by parts that we use to deal with integrals of products of trigonometric and exponential functions.
We would welcome comments and suggestions for improving this resource further.
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Mathematics
It is important for all students to achieve their full
potential in the subject. To enable this to happen students are encouraged
to take an active part in the learning processes they experience and to
enjoy their Mathematics lessons. A variety of learning methods is employed,
which includes classwork, projects and investigational work. Further to
this, to broaden students' experience, practical lessons on different topics
such as databases, spreadsheets and the use of computers, probability and
data handling are included.
Equipment
Students are issued with an Exercise book and text book
which they must bring to every lesson. They are expected to supply for themselves
Pen, Pencil, Ruler, Eraser, protractor, compass and Electronic Calculator.
On Entrance
All possible care is taken to ensure students are placed
in appropriate groups. If any doubt exists it is important that concern
is expressed immediately, therefore contact either Mr. R. McGuigan or Mr.
G. Gorman.
Behaviour
Students are expected to behave in a reasonable manner
that does not interfere with the learning environment of anyone else. They
are expected to be punctual, well prepared, with all homework complete,
and with all necessary equipment. This will enable all students to enjoy
their lessons and progress.
Homework
Homework is an essential part of the course and will set
to support and reinforce students' classroom activities. The amount and
timing of homework will vary depending on classroom needs.
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Mathematics...
Quick and reliable knowledge of multiplication tables is an essential cornerstone of a student's mathematical ability. The Dingo Bingo Multiplication Tables game is a pleasant and efficient way of ensuring students systematically develop and...
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'Math Expression Generator' is an application which generates math expressions taking into considerations supplied parameters. It uses custom defined parameters, base elements and operations which you can freely define to generate the mathematical...
Math Stars Plus is an educational application which includes a series of games that will help kids improve their Math skills. The application includes six different games, namely, Number game, Math detective, Math squares, Math challenge, Basket...
- Load your data. The program can read excel, ascii, and matlab files. Column headers can be included in the files. Use "load X=Y" if both X-data (design) and Y-data (responses) are on the same file. - Choose "Columns" for X...
UMS Math Editor is the program used for typing plain text and mathematical formulas. The distinctive feature of UMS Math Editor is that it not only allows you a simple way to enter mathematical formulas in a regular word processing environment,...
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OpenEuclide is an educational mathematical program for Euclidian geometry. This geometry program is for educational purposes and it allows proving and verifying geometrical theorems for Euclidian or axiomatic geometry. It allows drawing basicGraphicMaster is a powerful mathematical function plotter with extensive features. GraphicMaster allows scientists and students to visualize mathematical functions and data. In contrast to other function plotters, GraphicMaster has some...
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Rewards Multiply is an educational program that combines the study of multiplication tables with fun. The method is easy, after completing a round, the program offers a reward. It includes 12 predefined rewards, but in the 'Program settings' you...
Scientific Notebook 5.50.0.2953 creates mathematical documents with text and graphics. The program can build reports, homework, and exams. You can even export these documents to Rich Text Format, allowing you to share them with colleagues...
K9-Teasers is a collection of 6 games providing early stage learning of mathematics, sorting, and spelling. The Games are: Math Flash Cards Sorting numbers higher to lower or lower to higher Math Pyramid Math Tables Spelling (9 categories over 270...
Crocodile Mathematics is user-friendly mathematical modelling software for secondary school geometry and numeracy. Mathematical modelling is made simple - link shapes, numbers, equations and graphs to create your models. Welcome to the world of...
MatCalc stands for 'Materials Calculator' and it is a software package for computer simulation of the kinetics of microstructural processes.The MatCalc precipitation kinetics module is based on a new model for the evolution of precipitates in...
Valgetal is an addictive & educational game especially created to strengthen mathematics skills in the four basic operations: addition, subtraction, multiplication, and division. This is a Tetris-style game with colorful graphics and clear...
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ALNUSET is a dynamic, interactive system for enhancing the teaching and learning of algebra, numerical sets and functions in secondary schools. It's suitable for use with students from 12 to 17 and is easy to integrate in classroom practice....
This calculator calculates derivative functions of high orders in symbolic and numeric forms. Numerical coefficients and values are calculated with precision 27-36 digits. Obtained symbolic formula can be used with calculators Precision series
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EDCU11022 Numeracy in Action
Course details
This course encompasses both the content and pedagogy of mathematical problem-solving and modelling, chance and data, and pre-algebra and algebra aspects of the P-10 Mathematics Syllabus in Queensland Schools. It endeavours to enhance educator competency in developing an awareness of the essential ideas in the teaching and learning of these mathematical topics in schools
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La Matefest-Infofest
If you're interested in maths and computers but you're a little over-awed by the idea of actually studying those things, then just come to our next Matefest-Infofest and find out what it's all about. This hands-on open day is organized each year by students at the UB and it's a perfect way for you to see what we do in the area of mathematics and computer studies. There are also videogames, workshops in origami (the Japanese art of paper folding), a maths gymkhana and a whole range of stands and workshops to visit. A wonderful opportunity to see how much maths is really around you and to let the mathematician in you come out! Don't miss it!
Faculty-related university extension courses
The UB's university extension courses are courses of varying lengths designed to provide in-service training for people who are already working and a level of specialist study for students. Applicants are not required to have a degree qualification to do one of these courses. At the link provided below and listed in the thematic area Experimental Sciences and Mathematics, you'll find courses related to activities at the Faculty of Mathematics.
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Basic Math for Computer science engineer - Books/Videos - MathOverflow [closed]most recent 30 from Math for Computer science engineer - Books/VideosAMBROSE2010-07-16T14:45:51Z2010-07-16T14:45:51Z
<p>Hi All,</p>
<p>I am a Computer scienece engineering graduate working as a Technical Lead in a Software firm .</p>
<p>My day today work deals development of application which is always has limited time.</p>
<p>Now after some years almost don't remember the academics like basic math required for Computer graduate .</p>
<p>I want to develop my own product so wanted to think in terms of algorithm and maths .Before that wanted to refresh or re-learn the academics stuff .</p>
<p>Can anyone suggest the required math books ? (Remember I have already quoted "almost don't remember the academics" ).</p>
<p>If some could understand what I am asking for ., please provide the requested .</p>
<p>Thanks in Advance ,
Ambrose J</p>
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MATH PRIN.F/FOOD SERVICE OCCUPATIONS
by STRIANESE
No options of this product are available.
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Description
Math Principles for Food Service Occupations teaches students that the understanding and application of mathematics is critical for all food service jobs, from entry level to executive chef or food service manager. All the mathematical problems and concepts presented are explained in a simplified, logical, step by step manner. Now out in the 5th edition, this text is unique because it follows a logical step-by-step process to illustrate and demonstrate the importance of understanding and using math concepts to effectively make money in this demanding business. Part 1 trains the student to use the calculator, while Part 2 reviews basic math fundamentals. Subsequent parts address math essentials in food preparation and math essentials in food service record keeping while the last part of the book concentrates on managerial math. Learning objectives and key words have also been expanded and added at the beginning of each chapter to identify key information, and case studies have been added to help students understand why knowledge of math can solve problems in the food service industry. The content meets the required knowledge and competencies for business and math skills as required by the American Culinary Federation.
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Mathematics is a linguistic activity with a clear and precise communication of meaning, in which it always has a certainty of the proof. Mathematics provides a wide variety of skills, along with background and theory of practice that may be used to pursue graduate work, research, teaching in secondary schools, and various types of industries.
Beginning Salary Range
According to the National Association of Colleges and Employers (NACE) Summer 2011 Salary Survey, beginning salaries for graduates with a Bachelor of Arts or a Bachelor of Science degree in Mathematics start at about $53,914.
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Ashley,
I do believe that one should have a reader when dealing with math. It is a
very difficult concept to learn if you cannot physically see the graphs and
charts. I used a reader and believe it is the best way to approach any sort
of math course. I took basic math lab, elementary algebra, intermediate
algebra, and contemporary math and look forward to a statistics class in the
future. A reader was my saving grace in all of those courses, because I
don't know nemmeth code and didn't have the time to learn it when I needed
to. I learned enough to get by, but not enough to actually read a textbook
and be able to completely workout a problem in Braille. I am sure that APH
has materials to make tactile graphics and with a little bit of creativity
they can be made with materials you can get at a hobby shop like pipe
cleaners and such. I wish anyone taking a math class luck, as it is a very
cumbersome subject for me personally.
Alicia
-----Original Message-----
From: nabs-bounces at acb.org [mailto:nabs-bounces at acb.org] On Behalf Of
bookwormahb at earthlink.net
Sent: Monday, August 01, 2011 4:51 PM
To: Discussion list for NABS, National Alliance of Blind Students.
Subject: Re: [nabs] Fw: Accessibility Questions for MML+
Alicia,
It seems to me many in ACB think technology can solve all our issues, and
that isn't the case. Unless you have
tactile graphics or a tiger embosser to produce tactile graphics, math is
not accessible via a computer.
Math books have diagrams, charts and graphs. How would you make a pie chart
accessible or a scatter plot or a function graph accessible on a computer?
Um, to my knowledge, there is no way. Remember Birkir just told us that
rendering math books in electronic
format via Word is not accessible. Yes online math software like MML would
be more accessible with HTML.
But that doesn't solve the accessibility of the graphs and the textbook.
All blind students I know have had to use a reader; some have used RFB but
they still had to copy down the problems. Math is something where you have
to work the problems out; teachers do not care as much about the answer;
they want to see the five, six or seven steps you took to solve the problem.
So a blind student would work the problems in braille if they know it, and
then for a test they dictate the work to a scribe.
Some math can be done on the computer; it depends on what class.
I believe braille will be very valuable in this case for homework.
I think APH has materials to make tactile graphs. I have tunnel vision and
used it to see the graphs and charts; with numbers and word problems, I used
a reader. I also had my first math class in college, preparing for college
math, in audio from RFB.
I had to copy down the stuff from the book the reader was reading. Math is
a weak skill for me too, and I just barely got through it.
Ashley
-----Original Message-----
From: Starner, Alicia M.
Sent: Monday, August 01, 2011 9:59 AM
To: 'Discussion list for NABS,National Alliance of Blind Students.'
Subject: Re: [nabs] Fw: Accessibility Questions for MML+
Amen! I agree with you 100%. There is no reason why in this day and age with
the technological advances that are available to us that classes should be
unaccessible. For me as a totally blind person, math concepts were extremely
difficult and took a lot of time to comprehend, but that should not be
compounded by textbooks that are not accessible and supplemental programs
such as MML that are largely unusable. Good luck with your math course and
hope it goes well.
Alicia
-----Original Message-----
From: nabs-bounces at acb.org [mailto:nabs-bounces at acb.org] On Behalf Of Birkir
R. Gunnarsson
Sent: Monday, August 01, 2011 8:19 AM
To: nabs at acb.org
Subject: Re: [nabs] Fw: Accessibility Questions for MML+
Hi
You can take the class but also file the appropriate complaints. And
if you are refused accessible textbook material and a reader, and you
have no money to pay one, there's no way you can do the class anyway.
Math is not accessible in a Word file unless it is extremely simple
math or if it is created using MathType plug in. Publishers havenot,
thus far, botherred to do that, it is quite a bit of work.
But you can do the course but also file the complaint. If we do not do
that and we find work arounds, sometimes at our own expense, our
problems will never become a priority for anyone, and a long term
solution will never be found.
Unemployment rate amongst blind kids is staggering, I believe over
70%, it's not because we're stupid or unambitiou, or because social
security is so high it's better than working, it's in large part
because the world works visually and there's to little attention paid
to people who can't use that medium, espcially true with books and
text books. As we move into eBooks and online platforms there is 0
reason this should not be different. All that designers and content
authors have to do is to follow guidelines and standards that usually
result in better experience for everyone (remember our problems are
similar to those of people using cell phones to access online sites).
That's why we must speak up at every opportunity, rather than fail
courses, or barely pass, may be at our own expense.
And ther's no reason one couldn't do both, try to take the course, but
also draw attention to the fact the school needs to find a different
platform that is accessible to all, or that the platform provider has
to take steps to include everyone.
Of course it's ultimately up to Netta to decide what she wants to do,
but it's important to know she has every right to demand accessible
education, at whatever cost to the school, and that accessibility to
math can be achieved quite easily with the willingness and the right
tools.
Cheers and good luck
-B
|
id: 06151249
dt: j
an: 2013b.00623
au: Szeredi, Éva
ti: Forming the concept of congruence. II.
so: Teach. Math. Comput. Sci. 10, No. 1, 1-12 (2012).
py: 2012
pu: ,
la: EN
cc: G55 U65 D45 C35
ut: teacher education; concept formation; acquisition of mathematical concepts;
transformation geometry; manipulative materials; teaching methods;
classroom techniques
ci:
li:
ab: Summary: This paper is a continuation of [the author, Teach. Math. Comput.
Sci. 9, No. 2, 181‒192 (2011; ME 2012a.00515)], where I gave
theoretical background to the topic, description of the traditional
method of representing the isometries of the plane with its effect on
the evolution of congruence concept. In this paper, I describe a new
method of representing the isometries of the plane. This method is
closer to the abstract idea of 3-dimensional motion. The planar
isometries are considered as restrictions of 3-dimensional motions and
these are represented with free translocations given by flags. About
the terminology: I use some important concepts connected to teaching of
congruence, which have to be distinguished. My goal is to analyse
different teaching methods of the 2-dimensional congruencies. I use the
term 3-dimensional motion for the orientation preserving (direct)
3-dimensional isometry (which is also called rigid motion or rigid body
move). When referring the concrete manipulative representation of the
planar congruencies I will use the term translocation.
rv:
|
Mathematics and Computer Science Courses
Mathematics Courses
MAT 090 Intermediate Algebra Review (2 cr.)
(Fall and Spring, Course Offered Every Year)
This course is designed as a preparation for college algebra and other 100-level mathematics courses covering the following topics: the real number system, exponents, roots, radicals, polynomials, factoring, rational expressions, equations and inequalities, graphing linear equations and inequalities, graphing quadratic equations, and word problems. Counts as two credit hours toward course load and full-time student status but does not count as college credit.
MAT 130 Exploring With Mathematics (3 cr.)
(Fall, Spring and Summer, Course Offered Every Year)
This course emphasizes reasoning and communicating to clarify and refine thinking in practical areas of life. Students will gain confidence in their ability to apply their mathematical skills to applied problems and decision making. Topics will be chosen from: set theory, probability, visual representation of information, geometry, and graph theory.
MAT 141 College Algebra (3 cr.)
(Fall, Spring and Summer, Course Offered Every Year)
This course is a study of the algebra of functions. Topics covered include polynomial and rational functions, exponential functions and logarithmic functions. Graphing calculators will be used. Credit not allowed for both MAT 141 and MAT 144. Credit in this course is not given to students who already have credit for MAT 211.
MAT 143 Trigonometry (2 cr.)
(Fall, Spring and Summer, Course Offered Every Year)
The course will emphasize the use of analytic trigonometry in a wide variety of applications. Topics covered will include trigonometric relationships in triangles, trigonometric functions and trigonometric identities. Graphing calculators will be used. Credit not allowed for both MAT 143 and MAT 144. Credit in this course is not given to students who already have credit for MAT 211.
MAT 144 Functions and Graphs (3 cr.)
(Fall and Spring, Course Offered Every Year)
This course is a study of the algebra and geometry of functions. Topics covered include polynomial and rational functions, exponential and logarithmic functions, and trigonometric functions. Graphing calculators will be used. After completing this course, a student would have an appropriate background for MAT-211, Calculus I. Credit not allowed for both MAT 141 or MAT 143 and MAT 144. Credit in this course is not given to students who already have credit for MAT 211.
MAT 160 Fundamental Concepts of Mathematics I: Problem Solving, Number, Operation and Measurement (3 cr.)
(Fall, Course Offered Every Year)
For prospective elementary teachers. Introduction to mathematical concepts, their understanding and communication. Topics include an introduction to problem solving, set operation and their application to arithmetic, numeration systems, arithmetic, and measurement. Emphasis is on developing a deep understanding of the fundamental ideas of elementary school mathematics. Does not apply toward the math/science general education requirement for graduation.
MAT 211 Calculus I (4 cr.)
(Fall and Spring, Course Offered Every Year)
A study of functions, limits, continuity, the derivative and the integral. Applications of differentiation and integration include maxima, minima, marginal cost and revenue, rectilinear motion, and areas. Students will use graphing calculators. May be taken without prerequisite courses with department's permission. Prerequisites: MAT 141 and MAT 143 or MAT 144).
MAT 212 Calculus II (4 cr.)
(Fall and Spring, Course Offered Every Year)
A continuation of the calculus of functions of one variable. Topics include volumes of rotation, transcendental functions, integration techniques, polar coordinates, parametric equations and infinite series. Students will use graphing calculators and computer packages. May be taken without prerequisite with department's permission. Prerequisite: MAT 211.
MAT 245 Statistics I (3 cr.)
(Fall, Spring and Summer, Course Offered Every Year)
A general introduction to descriptive and inferential statistics, designed for non-mathematics majors. Topics include elementary probability, distributions, estimation of population parameters, confidence intervals, hypothesis testing, correlation and regression. Students will use statistical analysis technology.
MAT 248 Statistical Concepts and Methods for Mathematicians (3 cr.)
(Spring, Course Offered Every Year)
An introduction to statistics for mathematically inclined students, focusing on the process of statistical investigations. Observational studies, controlled experiments, sampling, randomization, descriptive statistics, probablility distributions, significance tests, confidence intervals, one- and two-sample inference procedures, linear regression. Statistical software will be used throughout the course. Credit in this course is not given to students who already have credit for MAT-245. Prerequisite: MAT 211.
MAT 250 Introduction to Mathematical Reasoning (3 cr.)
(Fall, Course Offered Every Year)
This course is a study of logic and an introduction to various techniques of mathematical proof, including direct proof, indirect proof and proof by induction. Students will be involved actively in the construction and exposition of proofs from multiple representations - visually, numerically, symbolically - and will present their reasoning in both oral and written form. Topics covered include sets and basic properties of the integers, rational numbers and real numbers. Throughout the course, students will explore strategies of problem solving and active mathematical investigation. After completing this course, a student would have an appropriate background for upper level theoretical mathematics courses. Prerequisite: MAT 212 or Corequisite: MAT 212 with permission of the instructor.
MAT 260 Fundamental Concepts of Mathematics II: Geometry, Algebra, Functions, Data Analysis, and Probablility (3 cr.)
(Spring, Course Offered Every Year)
The second course intended for prospective elementary teachers continues an in-depth introduction to mathematical concepts focusing on student understanding and communication. Topics include geometric concepts (shape and space, area and volume, transformations and symmetry), algebraic concepts (patterns, equations, and functions), and statistical concepts (designing investigations, gathering & analyzing data, and basic probability). The course will utilize investigative activities and instructional technology. Emphasis is on developing a deep understanding of the fundamental ideas of elementary school mathematics and transitioning from inductive to deductive reasoning. Does not apply toward the math/science general education requirement for graduation. Prerequisites: MAT 160 and MAT 245 or MAT 211.
MAT 290 Honors Math Lab (1 cr.)
(Fall, Course Offered Every Year)
Students work in teams to explore via computer various mathematical concepts. The experiment-conjecture-proof technique allows students to experience some of the excitement of discovering mathematics. During the lab period, the teams interact in a cooperative setting and discuss the meaning of what they are learning. All of the labs contain dynamical graphical displays which enhance the students' understanding of the topics studied. At the end of each experiment, students submit a written report describing their findings. Pre or Corequisite courses: MAT 211, 212 or 314.
MAT 299 Introduction to Mathematics Research (1-3 cr.)
(Fall and Spring, Course Offered Every Year)
This course will provide opportunities for freshmen and sophomores to participate in original research in mathematicsMAT 314 Calculus III (4 cr.)
(Fall and Spring, Course Offered Every Year)
A study of vectors in two and three dimensions and multivariable calculus. This includes three-dimensional analytic geometry, partial differentiation and multiple integration, and line integrals. Students will use technology for exploration and problem solving. May be taken without prerequisite with department's permission. Prerequisites: MAT 212.
MAT 340 Probability and Mathematical Statistics (3 cr.)
(Fall, Odd-Numbered Years Only)
The study of probability and statistical inference. Emphasis is placed on the theoretical development of probability distributions, discrete, continuous, and multivariate, and the sampling distributions used in statistical inference. Prerequisites: MAT 212 and either MAT 245 or MAT 248.
MAT 345 Statistics II (3 cr.)
(Spring, Odd-Numbered Years Only)
A continuation of MAT 245 which includes one- and two-sample inference, two-way tables, simple and mutiple regression, and analysis of variance. Applications of these topics will be drawn from business, the social and natural sciences and other areas. Students will use statistical analysis technology. Prerequisite: MAT 245, MAT 248 or PSY 200.
MAT 371 Mathematical Modeling (3 cr.)
(Spring, Course Offered Every Year)
A study of mathematical models used in the social and natural sciences and their role in explaining and predicting real world phenomena. The course will emphasize the development of the skills of model building and will address the use of various types of models, such as continuous, discrete, and statistical models. Prerequisites: CS 101, MAT 248, and MAT 314.
MAT 395 Junior Seminar - Research Methods in Mathematics (2 cr.)
(Spring, Course Offered Every Year)
This course is a junior-level seminar and research development course. Students will be exposed to topcs in contemporary mathematics as a basis for investigating and extending problems, making conjectures, and developing mathematical arguments. Students will work collaboratively to solve problems, develop research questions, and make presentations. Students will develop rsearch topics and will review both the literature and the methods of research in those areas of mathematics. Through review of the literature and through problem investigaiton & development, students will improve oral and written communication of mathematical understanding as well as their ability to investigate new mathematics independently. Prerequisites: MAT 250, MAT 314, and Junior Standing or permission of the instructor.
MAT 410 Advanced Calculus (3 cr.)
(Spring, Course Offered Every Year)
A rigorous treatment of the foundations of calculus. A study of the algebraic and topological properties of the real numbers; one-variable calculus, including limits, continuity, differentiation, Riemann integration, and series of functions. Prerequisites: MAT 250, MAT 314.
MAT 495 Senior Seminar (2 cr.)
(Fall, Course Offered Every Year)
A culminating seminar that brings together work done across the major and builds upon MAT 395. Students will work together on a group research project in select areas of mathematics and will read from a variety of sources to broaden their appreciation of mathematical history and literature. The students will improve oral and written communication skills through class discussion, formal presentations and a variety of written assignments. They will also make and implement plans for postgraduate education and careers. Prerequisites: MAT 395 and Senior Standing or permission of the instructor.
MAT 498 Honors Thesis in Mathematics (3 cr.)
(Fall and Spring, Course Offered Every Year)
In conjunction with a faculty mentor, the student will formulate and execute an original research project that will culminate in a paper and a presentation. The research project must meet Honors Program thesis requirements as well as the expectations of the mathematics faculty. Open to seniors in the Honors and/or Teaching Fellows Programs only. Second semester juniors may enroll with permission of the faculty mentor.
MAT 499 Research in Mathematics (1-3 cr.)
(Fall and Spring, Course Offered Every Year)
In conjunction with a faculty mentor, the student will formulate and execute an original research project that will culminate in a paper and a presentation. Open to juniors and seniors majoring in mathematics and to others by permission of the department. May be repeated for credit for a maximum of six credit hours.
MAT 760 Mathematical Knowledge for Teaching (2 cr.)
(Varies, Contact Department Head)
Introduces licensure students to the philosphy and objectives of mathematics education. The course will focus on the content of school mathematics and examine closely both state and national recommended standards of school mathematics curricula. The emphasis of the course will be on developing a deep understanding of school mathematics and pedagogical content knowledge - the mathematical knowledge for teaching. Technologies appropriate for conceptual understanding of mathematics will be introduced. A related field component will be required at a local school site. MAT 250 and the instructor's consent required.
MAT 764 Methods Secondary/Middle Math (3 cr.)
(Fall, Course Offered Every Year)
A study of the philosophy and objectives of mathematics education, emphasizing methods and materials needed for teaching mathematics in the middle and secondary schools. Attention is given to the importance of planning for instruction and evaluating both the instruction and student performance. Students must demonstrate their skills in planning, teaching, and evaluating. Instructor's consent required.
Computer Studies Courses
CS 101 Beginning Programming (3 cr.)
(Fall and Spring, Course Offered Every Year)
Students learn how a computer works and how to make it work as they design, code, debug and document programs to perform a variety of tasks. This course is intended for students who have not programmed a computer before, but may also serve as an introduction to Java (or other language) even if the student DOES know some programming.
CS 120 Spreadsheets (1 cr.)
(Fall and Spring, Course Offered Every Year)
Introduction to and development of skills in the creation and use of spreadsheets. The student will also learn how to set up and create graphs from spreadsheets and to create macros. Extensive use of microcomputer software such as Excel.
CS 121 Spreadsheets II (1 cr.)
(Spring, Course Offered Every Year)
This course is a continuation of CS-120. Students will learn how to use Excel as a practical business tool with in-depth use of formulas and functions and efficient worksheet and workbook design. Some topics in Excel databases and the creation of simple macros will also be covered. Prerequisite: CS 120 or competency in spreadsheets.
CS 140 Databases (1 cr.)
(Fall, Course Offered Every Year)
Creating a database structure, entering and updating data, generating reports based on querying the database. This course includes a project. Hands-on use of software such as MS Access.
CS 156 Web Site Design & Management (3 cr.)
(Fall and Spring, Course Offered Every Year)
This course requires extensive use of an HTML editor and a web design package to create web pages and web sites. Students will also learn site planning management. This will include learning to plan web sites and planning and assessing visitor involvement. Specific topics and techniques include: tables, frames, forms, cascading style sheets, use of animation and sound, and image creation and manipulation. Additional topics will include dynamic content, Javascript, XML, file management, file transfer protocol and web site evaluation.
CS 160 SAS Programming (3 cr.)
(Varies, Contact Department Head)
A course in programming in the high-level programming language of SAS which is used extensively in business, government and education. By the end of the course the student will be able to immediately apply her skills in real-life programming solutions. Applications in data gathering and manipulation, report generation and elementary statistical procedures. No previous programming experience is required. Prerequisite: computer literacy. Prior experience in statistics is recommended.
CS 230 Web Programming with Databases (3 cr.)
(Fall, Course Offered Every Year)
This course focuses on the server side of client-server programming for the Web,
especially database programming. There will be a study of fundamentals of databases including
normalization and security, and students will apply this knowledge to real web-database
applications. Current tools: JavaScript (prerequisite), PHP (a programming language), SQL (Structured Query
Language). Prerequisites: CS 140, CS 156.
CS 240 Visual Basic (3 cr.)
(Varies, Contact Department Head)
An introduction to programming in Visual Basic. Emphasis will be placed on the event-driven, graphical nature of Visual Basic, as opposed to procedure-oriented programming. Topics include form layout, event-driven Windows programming concepts, variables and data types, objects and properties, control structures, file management, accessing databases, linking applications, Web page development from a Visual Basic application, and developing and using ActiveX controls. This course is intended for those with programming experience. May be taken without prerequisite course with instructor's consent. Prerequisite: CS 101.
CS 299 Introduction to Computer Studies Research (1-4 cr.)
(Fall and Spring, Course Offered Every Year)
This course will provide opportunities for freshmen and sophomores to participate in original research in computer scienceCS 301 Data Structures & Algorithms (3 cr.)
(Fall, Odd-Numbered Years Only)
Topics include the sequential and linked allocation of lists, stacks, queues, trees and graphs. Students gain maturity by writing complex algorithms and through studying run time analysis and program integrity. Prerequisite: CS 212.
CS 311 Computer Organization (3 cr.)
(Fall, Even-Numbered Years Only)
The fundamentals of logic design, the organization and structuring of the major hardware components of computers. Prerequisite: CS 203.
CS 312 Information Systems Management (3 cr.)
(Varies, Contact Department Head)
The main theme of the course is solving problems and creating opportunities with technology in an organizational setting. Topics include how information systems affect and are affected by organizational goals and strategies; basic overviews of the components of an information system; hardware, software, data storage and retrieval, and network communications; the Internet; the information systems development process; and systems development as planned organizational change. Prerequisite: Completion of the General Education fundamental computer skills competency requirement.
CS 326 Networking and Operating Systems (3 cr.)
(Spring, Even-Numbered Years Only)
In the ever-shifting and related fields of operating systems and networking, this
course teaches the fundamental aspects of computing systems including security, memory
management, job scheduling, synchronization, client-server programming and distributed
programming. There will also be significant hands-on application of principles in the lab.
Prerequisites: CS 203, CS 212.
CS 355 Computer Graphics and Modeling (3 cr.)
(Spring, Odd-Numbered Years Only)
This course is about visualizing models on the computer screen, including 2D and 3D images, perspective, shading, animation and stereo. The course will use and study numerical models of such interesting phenomena as geometric objects, fractals, trajectories and propogation of waves. Prerequisites: MAT 211 and CS 212.
CS 370 Ethics and Information Technology (1 cr.)
(Fall and Spring, Course Offered Every Year)
Discussion of the ethical and legal issues created by the introduction of information technology into every day life. Codes of ethics for computer users. Topics may include, but are not limited to, information ownership, individual privacy, computer crime, communications and freedom of expression, encryption and security.
CS 407 Software Engineering (3 cr.)
(Spring, Even-Numbered Years Only)
Introduction to the principles of design, coding and testing of software projects; the software development cycle; and managing the implementation of large computer projects. Students undertake a large team project. Prerequisites: CS 212 and CS 230.
CS 420 Computer Science Seminar (1 cr.)
(Fall, Course Offered Every Year)
Current developments and themes in computer science. An introduction to industry as it exists in the Research Triangle area, to journals in the field of computer science, and to societies and associations dedicated to the advancement of computing. Includes field trips, speakers and discussions of selected topics. Course open to juniors and seniors only. Prerequisites: 6 credits from CS.
CS 421 Topics in Computer Science (3 cr.)
(Spring, Course Offered Every Year)
Topics of current interest in computer science not covered in other courses. Prerequisites vary with topic studied.
CS 498 Honors Thesis in Computer Studies (3 cr.)
(Fall and Spring, Course Offered Every Year)
With a faculty mentor, the student will formulate and execute an original research project that will culminate in a paper and a presentation. The research project must meet Honors Program thesis requirements as well as the expectations of the computer science faculty. Enrollment limited to seniors or second semester juniors in the Honors and/or Teaching Fellows Programs.
CS 499 Computer Studies Research (1-4 cr.)
(Fall and Spring, Course Offered Every Year)
With a faculty mentor, the student will formulate and execute an original research project that will culminate in a paper and a presentation. Open to juniors and seniors majoring in CSC or CIS or others with permission of the department. May be repeated for credit for a maximum of six hours.
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Math and Science Workbook
This study guide provides students with review and practice of the math and science questions they're sure to see on the ACT exam. Each section ...Show synopsisThis study guide provides students with review and practice of the math and science questions they're sure to see on the ACT exam. Each section begins with review and is followed by a 17-question practice set with complete answers and explanations to help students improve their scores.Hide synopsis
Description:Good. 1427797706 Some visible wear, and minimal interior marks....Good. 1427797706 Some visible wear, and minimal interior marks. Unbeatable customer service, and we usually ship the same or next day. Over one million satisfied customers!
Description:Very Good. 1609780582 Item in very good condition and at a...Very Good. 1609780582 Item in very good condition and at a great price! Textbooks may not include supplemental items i.e. CDs, access codes etc...
Description:Good. Paperback has been read, but remains in clean condition....Good. Paperback has been read, but remains in clean condition. All pages are intact, and the cover is intact. There is light highlighting or handwriting through out the book
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Module Description
This course revises ideas associated with continuous functions, including the idea of an inverse, differentiation and integration, and sets them in a more fundamental context which permits a better understanding of their properties. Differential equations are introduced, and methods for solving them are studied. The properties of inequalities are reviewed. Complex numbers in Cartesian form are introduced.
Syllabus
Geometry and Trigonometry:
- Pythagoras' theorem; trigonometric functions.
- Basic manipulation of inequalities
Functions of one variable:
- the functions exp, ln, xa, |x|, trigonometric and hyperbolic functions; their domains and their graphs;
Learning & Teaching Methods
This course consists of 30 contact hours given at 3 hours per week commencing in week 2 (the first teaching week). There will be a test at the end of term and five assessed problem sheets throughout the term. Three revision lectures will also be given in the summer term.
Assessment
25 per cent Coursework Mark, 75 per cent Exam Mark
Other details: Information about coursework deadlines can be found in the "Coursework Information" section of the Current Students, Useful Information Maths web pages: Coursework Information
Exam Duration and Period
1:30 hour exam during Summer Examination period.
Other information
'A' level Maths or equivalent normally required. Available independently to Socrates/IP students spending all relevant terms at Essex.
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Short description (Read more) • Number and Algebra • Geometry and Measures • Handling Data In addition, there exists a Publications Guide. Our mathLoci Constructions and 3D Co-ordinates is a module within the Geometry and Measures principle section our Grades 6, 7 & 8 publications. It is one module out of a total of six modules in that principle section, the others being: • 2D Shapes and 3D Solids • Angles, Bearings and Scale Drawings • Transformations • Pythagoras' Theorem, Trigonometry and Similarity • Measures and Measurements (Less)
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