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MAA Review
[Reviewed by William J. Satzer, on 12/29/2008]
Lectures on Surfaces: (Almost) Everything You Wanted to Know about Them does a masterful job of introducing the study of surfaces to advanced undergraduates. Although there are many attractive volumes in the AMS Student Mathematical Library series, this is the first one I've seen that would really have captured my interest as an undergraduate.
One of the reasons why this text works so well (I think) is that the authors are not experts in the area. They take extra care to elaborate or explain things that an expert would not, and then anticipate those places where a newcomer might get stuck. Nonetheless, their scope is ambitious and ranges all the way from triangulation and classification of surfaces to Riemann surfaces, Riemannian geometry on surfaces, and the Gauss-Bonnet theorem. The Euler characteristic in its many guises is ubiquitous.
The book is divided into five chapters consisting of thirty-six lectures. Since this work developed from the MASS (Mathematics Advanced Study Semesters) program at Penn State, the lecture subdivision is a natural consequence of how material was divided for presentation in the classroom. The authors assume that students' background includes the usual calculus sequence, basic linear algebra, rudimentary differential equations, and a bit of real analysis. As important as the prerequisites is an appetite for learning new mathematics at a pretty brisk pace.
The authors begin with basic examples of surfaces and describe various ways of representing surfaces: by an equation, by planar model and quotient space, by local coordinates, or parametrically. The second chapter focuses on the combinatorial structure and topological characterization of surfaces. This includes classification of compact surfaces, with a proof for the orientable case. The authors introduce triangulations and the Euler characteristic of a triangulation; then they go on to define homology groups and Betti numbers and so provide a second interpretation of the Euler characteristic. The third and fourth chapters take up differentiable structures and vector fields on surfaces, Riemann surfaces, and then Riemann metrics. This brings us to geodesics, curvature, the hyperbolic plane, and the Gauss-Bonnet theorem. The final chapter comes back to topology and smooth structures for a discussion of degree and index of vector fields.
It is amazing how much mathematics is naturally associated with the study of surfaces, and how the pieces fit together so remarkably. The authors succeed in pulling in many topics while keeping their story coherent and compelling. This book would work well as the text for a capstone course or independent reading. However, there are relatively few exercises, so an instructor would probably need to develop supplementary problem sets for classroom use
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11C Scientific RPN Calculator 1. 11C Scientific RPN Calculator 1.0 is a product including all the features of the real one. Over 120 built-in functions including :Hyperbolic and Inverse hyperbolic Trig functions.Probability permutations and combinations.Factorial, absolute,... Minimum system requirements: Windows...
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Description
Mathematics for Physical Chemistry, Third Edition , is the ideal text for students and physical chemists who want to sharpen their mathematics skills. It can help prepare the reader for an undergraduate course, serve as a supplementary text for use during a course, or serve as a reference for graduate students and practicing chemists. The text concentrates on applications instead of theory, and, although the emphasis is on physical chemistry, it can also be useful in general chemistry courses.
The Third Edition includes new exercises in each chapter that provide practice in a technique immediately after discussion or example and encourage self-study. The first ten chapters are constructed around a sequence of mathematical topics, with a gradual progression into more advanced material. The final chapter discusses mathematical topics needed in the analysis of experimental data.
Numerous examples and problems interspersed throughout the presentations
Each extensive chapter contains a preview, objectives, and summary
Includes topics not found in similar books, such as a review of general algebra and an introduction to group theory
Provides chemistry specific instruction without the distraction of abstract concepts or theoretical issues in pure mathematics
Recommendations:
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In order to succeed in this system you must first understand it. We
begin by examining some of the components of a mathematics course, and
then looking at how they all fit together.
The most prominent feature of large mathematics courses is the
lecture. Lectures typically meet for an hour (actually, fifty minutes)
on Mondays, Tuesdays and Wednesdays. In a thirteen week semester this
adds up to a maximum of thirty-nine hours. University courses
generally require that students learn in much greater depth and
breadth than high school courses. Considering the volume of material
to be covered in a semester, it is clear that these thirty-nine hours
must be used extremely efficiently by both the instructor and the
student. An instructor may sometimes use the lecture to point out
interesting things not contained in the book, to give alternate
explanations to those presented in the text or to unify the concepts
as presented in the text.
The next feature is the recitation section. These are one-hour
meetings, generally on Thursday or Friday mornings, with a TA. The
main purpose of these section meetings is to reinforce the material
covered that week by focusing on additional examples, especially of
the type assigned for homework. Mathematics is not a spectator sport;
rather, it is a contact sport. The section meetings provide an
interactive setting in smaller groups in which applications of
mathematical concepts may be explored.
The final major component of a course is the help available outside of
class. The many options for assistance are discussed below.
The student should take care to view all of the components of a course,
including lectures, sections, examinations, homework, textbook and
help availability as a collection of elements designed to help them in
their continuing study of mathematics. All of these components are
designed to fit together as a package to help the student understand
the material from as many points of view as possible.
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MATH 220.921:
Mathematical Proof
The main aim of the course is to
learn how write clear and correct mathematical proofs. It provides the
gateway to more advanced mathematics. A little more precisely (though
this is provisional) we cover subjects from:
07/29/2020- There will be extra
office hours on Friday 12--1 and 3--5 and Saturday 12--2.
07/29/2010- Solutions to Homework 8 posted.
07/23/2010- Solutions to Homework 7 posted.
07/22/2010- Homework 8 posted due on Thursday July 29th.
07/13/2010- Homework 7 posted due on Tuesday July 20th.
06/29/2010- Second Midterm will be on Thursday July 8th, with questions
on Chapters 5, 6 and 9.
06/29/2020- Our next lecture on July 6th will be a problem solving
session.
06/29/2010- Solutions to HW6 has been posted and there will be no
homework for next week.
06/22/2010- Homework 6 and solutions to midterm1 and HW 5 have been
posted.
06/02/2010- The first midterm is on chapters 1--5.
06/01/2010- There will be a problem solving session on Monday June 7th
10--12 in MATH100.
06/01/2010- There will be no lecture on Tuesday June 8th and first
Midterm will be on Thursday June 10th.
05/28/2010- Homework 3 and solutions to HW 2 posted.
05/20/2010- Homework 2 posted due on Thursday May 27th.
05/13/2010- Homework 1 posted due on Thursday May 20th.
05/04/2010- Website
Created.
You must have either a score of 64% or higher in one of MATH 101, MATH
103, MATH 105,
SCIE 001, or one of MATH 121, MATH 200, MATH 217, MATH 253, MATH 263.
If you do not have these prerequisites then you must see your lecturer
as soon as possible.
Exams:
There will be two Midterm Exams totally worth 40% of the term grade,
tentatively on June 10th, and one on July 8th.
There will be a final examination held either
the evening of Friday July 30 or Saturday July 31. This exam will
account for
50% of a student's
final grade. The final exam will not generally be weighted higher for
students who perform better on the final exam than they did during the
term, although some allowance may be made for students who perform much
better on the final exam than they did during the term.
Homework
Assignments:
On this web-page
you will find the sections from the text that you should be reading
before to come to class. The instructor will try to observe this
pre-determined schedule.
It is important that you check regularly this
course webpage.
Homework assignments will be posted weekly on this
course website, best 8 grades (out of 9 or 10 homworks in total) is
counted
as 10% of the term grade. Homework is the
essential
educational part of this course. You cannot expect to work problems on
the exams if you have not worked lots of homework problems.
Therefore, it
is important that you spend an adequate time on homework regularly,
each week. Late homework will not be accepted. You can work together on
the homework, but you should always write up your own homework
solutions in your own words.
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Buy print & eBook together and save 40%
Description
Theory and application of a variety of mathematical techniques in economics are presented in this volume. Topics discussed include: martingale methods, stochastic processes, optimal stopping, the modeling of uncertainty using a Wiener process, Itô's Lemma as a tool of stochastic calculus, and basic facts about stochastic differential equations. The notion of stochastic ability and the methods of stochastic control are discussed, and their use in economic theory and finance is illustrated with numerous applications.
Quotes and reviews
@from:R. Kihlstrom @qu:This book will almost certainly become a basic reference for academic researchers in finance. It will also find wide use as a textbook for Ph.D. students in finance and economics. @source:Mathematical Reviews
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Course Detail
Registration
Curriculum & Instruction: Math as a Second Language
EDCI 200 Z3 (CRN: 60994)
3 Credit Hours—Seats Available!
Jump Navigation
About EDCI 200 Z3
This course lays the groundwork for all the Vermont Mathematics Initiative courses that follow. A major theme is understanding algebra and arithmetic through language. The objective is to provide a solid conceptual understanding of the operations of arithmetic, as well as the interrelationships among arithmetic, algebra, and geometry. Topics include arithmetic vs. algebra; solving equations; place value and the history of counting; inverse processes; the geometry of multiplication; the many faces of division; rational vs. irrational numbers and the one-dimensional geometry of numbers. All of the topics in this course are taught in the context of the mathematics curriculum in grades K-6.
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Not only helpful with math, but that's what I use it for. You can type in pretty much any kind of problem and it will solve it, or at least give you helpful information that will lead to the solution. I'm taking an online college algebra class at the moment and I could probably pass the class using nothing but this.
EDIT: This is also really helpful, often even more helpful than WolframAlpha:
Wolfram alpha is REALLY overkill for all high school math. It really should only be used for calculus, and even with calculus it has trouble with large integrations and derivatives.
Personally I don't suggest you use Wolfram Alpha unless you already know how to do the math and just need it to reduce a ridiculously complex problem. Otherwise you're gimping yourself in the long run by relying on a piece of technology to do your work instead of knowing how to do it yourself.
Besides, for simple algebra it only takes a few seconds to solve in your head. It takes 7 times as long to type that into wolfram alphaA generic graphing calculator can do everything you need to know in math class up until calculus. Until that point, using Wolfram Alpha is overkill. The only upside to it I can think of at this moment is the additional notation that you learn from using Wolfram Alpha.
etc. Their full, pay-to-download program (Mathematica) is great as well, but the notation is a bitch to learn.
The notation is just LaTeX. If you're going to do anything in the sciences or maths, you'll need to learn LaTeX anyway.
Mathematica scripting is also pretty strait forward and very user friendlyEssentially, I'd liken Mathematica to photoshop - it's friendly enough for simple and familiar computations, but there's a certain amount of complexity inherent in its versatility. I probably know about 5% of the commands, as a generous estimateNot really sure if on topic or not, but:
For anyone wanting to use those Wolfram addons, but with difficulties with the notation - you could always mess around with the online equation editor: It shows what the formula gives, and the source as well.
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Video Summary: This learning video introduces students to the world of Fractal Geometry through the use of difference equations. As a prerequisite to this lesson, students would need two years of high school algebra (comfort with single variable equations) and motivation to learn basic complex arithmetic. Ms. Zager has included a complete introductory tutorial on complex arithmetic with homework assignments downloadable here. Also downloadable are some supplemental challenge problems. Time required to complete the core lesson is approximately one hour, and materials needed include a blackboard/whiteboard as well as space for students to work in small groups. During the in-class portions of this interactive lesson, students will brainstorm on the outcome of the chaos game and practice calculating trajectories of difference equations
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Short description
This dictionary includes explanations of over 200 mathematical words and phrases. Other features include: multiplication tables; table of squares and cubes; frequently-used fractions, decimals and percentages; metric and imperial units; simple coordinate graphs; angle and circle rules.
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Mathematics is the language and tool of the sciences, a cultural phenomenon with a rich historical traditions, and a
model of abstract reasoning. Historically, mathematical methods
and thinking have proved extraordinarily successful in physics, and engineering. Nowadays, it is used successfully in many new
areas, from computer science to biology and finance. A Mathematics
concentration provides a broad education in various areas of mathematics
in a program flexible enough to accommodate many ranges of interest.
The study of mathematics is an excellent preparation
for many careers; the patterns of careful logical reasoning and
analytical problem solving essential to mathematics are also applicable
in contexts where quantity and measurement play only minor roles.
Thus students of mathematics may go on to excel in medicine, law, politics, or business as well as any of a vast range of scientific
careers. Special programs are offered for those interested in
teaching mathematics at the elementary or high school level or
in actuarial mathematics, the mathematics of insurance. The other
programs split between those which emphasize mathematics as an
independent discipline and those which favor the application of
mathematical tools to problems in other fields. There is considerable
overlap here, and any of these programs may serve as preparation
for either further study in a variety of academic disciplines, including mathematics itself, or intellectually challenging careers
in a wide variety of corporate and governmental settings.
Elementary Mathematics Courses. In order
to accommodate diverse backgrounds and interests, several course
options are available to beginning mathematics students. All courses
require three years of high school mathematics; four years are
strongly recommended and more information is given for some individual
courses below. Students with College Board Advanced Placement
credit and anyone planning to enroll in an upper-level class should
consider one of the Honors sequences and discuss the options with
a mathematics advisor.
Students who need additional preparation for calculus
are tentatively identified by a combination of the math placement
test (given during orientation), college admission test scores
(SAT or ACT), and high school grade point average. Academic advisors
will discuss this placement information with each student and
refer students to a special mathematics advisor when necessary.
Two courses preparatory to the calculus, MATH 105
and 110, are offered. MATH 105 is a course on data analysis, functions
and graphs with an emphasis on problem solving. MATH 110 is a
condensed half-term version of the same material offered as a
self-study course taught through the Math Lab and is only open
to students in MATH 115 who find that they need additional preparation
to successfully complete the course. A maximum total of 4 credits
may be earned in courses numbered 103, 105, and 110. MATH 103
is offered exclusively in the Summer half-term for students in
the Summer Bridge Program.
MATH 127 and 128 are courses containing selected
topics from geometry and number theory, respectively. They are
intended for students who want exposure to mathematical culture
and thinking through a single course. They are neither prerequisite
nor preparation for any further course. No credit will be received
for the election of MATH 127 or 128 if a student already has credit
for a 200-(or higher) level mathematics course.
Each of MATH 115, 185, and 295 is a first course
in calculus and generally credit can be received for only one
course from this list. The Sequence 115-116-215 is appropriate
for most students who want a complete introduction to calculus.
One of MATH 215, 285, or 395 is prerequisite to most more advanced
courses in Mathematics.
The sequences 156-255-256, 175-176-285-286, 185-186-285-286, and 295-296-395-396 are Honors sequences. Students need not be
enrolled in the LS&A Honors Program to enroll in any of these
courses but must have the permission of an Honors advisor. Students
with strong preparation and interest in mathematics are encouraged
to consider these courses.
MATH 185-285 covers much of the material of MATH
115-215 with more attention to the theory in addition to applications.
Most students who take MATH 185 have taken a high school calculus
course, but it is not required. MATH 175-176 assumes a knowledge
of calculus roughly equivalent to MATH 115 and covers a substantial
amount of so-called combinatorial mathematics as well as calculus-related
topics not usually part of the calculus sequence. MATH 175 and
176 are taught by the discovery method: students are presented
with a great variety of problem and encouraged to experiment in
groups using computers. The sequence MATH 295-396 provides a rigorous
introduction to theoretical mathematics. Proofs are stressed over
applications and these courses require a high level of interest
and commitment. Most students electing MATH 295 have completed
a thorough high school calculus. MATH 295-396 is excellent preparation
for mathematics at the advanced undergraduate and graduate level.
Students with strong scores on either the AB or
BC version of the College Board Advanced Placement exam may be
granted credit and advanced placement in one of the sequences
described above; a table explaining the possibilities is available
from advisors and the Department. In addition, there is one course
expressly designed and recommended for students with one or two
semesters of AP credit, MATH 156. Math 156 is an Honors course
intended primarily for science and engineering concentrators and
will emphasize both applications and theory. Interested students
should consult a mathematics advisor for more details.
In rare circumstances and with permission of a Mathematics
advisor, reduced credit may be granted for MATH 185 or 295 after
MATH 115. A list of these and other cases of reduced credit for
courses with overlapping material is available from the Department.
To avoid unexpected reduction in credit, student should always
consult an advisor before switching from one sequence to another.
In all cases a maximum total of 16 credits may be earned for calculus
courses MATH 115 through 396, and no credit can be earned for
a prerequisite to a course taken after the course itself.
Students completing MATH 116 who are principally
interested in the application of mathematics to other fields may
continue either to MATH 215 (Analytic Geometry and Calculus III)
or to MATH 216 (Introduction to Differential Equation -- these
two courses may be taken in either order. Students who have greater
interest in theory or who intend to take more advanced courses
in mathematics should continue with MATH 215 followed by the sequence
MATH 217-316 (Linear Algebra-Differential Equations). MATH 217
(or the Honors version, MATH 513) is required for a concentration
in Mathematics; it both serves as a transition to the more theoretical
material of advanced courses and provides the background required
to optimal treatment of differential equations in MATH 316. MATH
216 is not intended for mathematics concentrators.
Special Departmental Policies. All prerequisite
courses must be satisfied with a grade of C- or above. Students
with lower grades in prerequisite courses must receive special
permission of the instructor to enroll in subsequent courses.
MATH 105. Data, Functions, and Graphs.
Instructor(s):
Prerequisites & Distribution: (4). (MSA). (QR/1). May not be repeated for credit. Students with credit for MATH 103 can elect MATH 105 for only 2 credits. No credit granted to those who have completed any Mathematics course numbered 110 or higher. A maximum of four credits may be earned in MATH 101, 103, 105, and 110.
MATH 105 serves both as a preparatory course to the calculus sequences and as a terminal course for students who need only this level of mathematics. Students who complete MATH 105 are fully prepared for MATH 115. ThisMATH 107. Mathematics for the Information Age.
Section 001.
Instructor(s):
Karen Rhea
Prerequisites & Distribution: Three to four years high school mathematics. (3). (MSA). (QR/1). May not be repeated for credit.
Credits: (3).
Course Homepage: No homepage submitted.
From computers and the Internet to playing a CD or running an election, great progress in modern technology and science has come from understanding how information is exchanged, processed and perceived.
MATH 110. Pre-Calculus (Self-Study).
Instructor(s):
Prerequisites & Distribution: See Elementary Courses above. Enrollment in MATH 110 is by recommendation of MATH 115 instructor and override only. (2). (Excl). May not be repeated for credit. No credit granted to those who already have 4 credits for pre-calculus mathematics courses. A maximum of four credits may be earned in MATH 101, 103, 105, and 110.
The course covers data analysis by means of functions and graphs. MATH 110 serves both as a preparatory class to the calculus sequences and as a terminal course for students who need only this level of mathematics. The course is a condensed, half-term version of MATH 105 (MATH 105 covers the same material in a traditional classroom setting) designed for students who appear to be prepared to handle calculus but are not able to successfully complete MATH 115. Students who complete MATH 110 are fully prepared for MATH 115. Students may enroll in MATH 110 only on the recommendation of a mathematics instructor after the third week of classes.
ENROLLMENT IN MATH 110 IS BY PERMISSION OF MATH 115 INSTRUCTOR ONLY. COURSE MEETS SECOND HALF OF THE TERM. STUDENTS WORK INDEPENDENTLY WITH GUIDANCE FROM MATH LAB STAFF.
MATH 115. Calculus I.
Instructor(s):
Prerequisites & Distribution: Four years of high school mathematics. See Elementary Courses above. (4). (MSA). (BS). (QR/1). May not be repeated for credit. Credit usually is granted for only one course from among 115, 185, and 295. No credit granted to those who have completed MATH 175.
The sequence MATH 115-116-215 is the standard complete introduction to the concepts and methods of calculus. It is taken by the majority of students intending to concentrate in mathematics, science, or engineering as well as students heading for many other fields. The emphasis is on concepts and solving problems rather than theory and proof. All sections are given a uniform midterm and final exam. The course presents the concepts of calculus from three points of view: geometric (graphs); numerical (tables); and algebraic (formulas). Students will develop their reading, writing, and questioning skills.
Topics include functions and graphs, derivatives and their applications to real-life problems in various fields, and definite integrals. MATH 185 is a somewhat more theoretical course which covers some of the same material. MATH 175 includes some of the material of MATH 115 together with some combinatorial mathematics. A student whose preparation is insufficient for MATH 115 should take MATH 105 (Data, Functions, and Graphs). MATH 116 is the natural sequel. A student who has done very well in this course could enter the honors sequence at this point by taking MATH 186. The cost for this course is over $100 since the student will need a text (to be used for MATH 115 and 116) and a graphing calculator (the Texas Instruments TI-83 is recommended).
See MATH 115 for a general description of the sequence MATH 115-116-215.
Topics include the indefinite integral, techniques of integration, introduction to differential equations, and infinite series. MATH 186 is a somewhat more theoretical course which covers much of the same material. MATH 215 is the natural sequel. A student who has done very well in this course could enter the Honors sequence at this point by taking MATH 285.
MATH 127. Geometry and the Imagination.
Section 001.
Instructor(s):
Emina Alibegovic
Prerequisites & Distribution: Three years of high school mathematics including a geometry course. Only first-year students, including those with sophomore standing, may pre-register for First-Year Seminars. All others need permission of instructor. (4). (MSA). (BS). (QR/1). May not be repeated for credit. No credit granted to those who have completed a 200- (or higher) level mathematics course (except for MATH 385 and 485).
First-Year Seminar
Credits: (4).
Course Homepage: No homepage submitted.
This course introduces students to the ideas and some of the basic results in Euclidean and non-Euclidean geometry. Beginning with geometry in ancient Greece, the course includes the construction of new geometric objects from old ones by projecting and by taking slices. The next topic is non-Euclidean geometry. This section begins with the independence of Euclid's Fifth Postulate and with the construction of spherical and hyperbolic geometries in which the Fifth Postulate fails; how spherical and hyperbolic geometry differs from Euclidean geometry. The last topic is geometry of higher dimensions: coordinatization — the mathematician's tool for studying higher dimensions; construction of higher-dimensional analogues of some familiar objects like spheres and cubes; discussion of the proper higher-dimensional analogues of some geometric notions (length, angle, orthogonality, etc. ) This course is intended for students who want an introduction to mathematical ideas and culture. Emphasis on conceptual thinking — students will do hands-on experimentation with geometric shapes, patterns, and ideas.
This course is designed for students who seek an introduction to the mathematical concepts and techniques employed by financial institutions such as banks, insurance companies, and pension funds. Actuarial students, and other mathematics concentrators should elect MATH 424, which covers the same topics but on a more rigorous basis requiring considerable use of calculus. Topics covered include: various rates of simple and compound interest, present and accumulated values based on these; annuity functions and their application to amortization, sinking funds, and bond values; depreciation methods; introduction to life tables, life annuity, and life insurance values. This course is not part of a sequence. Students should possess financial calculators.
MATH 186. Honors Calculus II.
Instructor(s):
Prerequisites & Distribution: Permission of the Honors advisor. (4). (MSA). (BS). (QR/1). May not be repeated for credit. Credit is granted for only one course from among MATH 116, 156, 176, 186, and 296.
Waitlist Code: 5: Students in LSA College Honors may request overrides from the Honors Office; other students may request them from the Math Dept Office, 2084 East Hall.
MATH 214. Linear Algebra and Differential Equations.
Instructor(s):
Prerequisites & Distribution: MATH 115 and 116This course is intended for second-year students who might otherwise take MATH 216 (Introduction to Differential Equations) but who have a greater need or desire to study Linear Algebra. This may include some Engineering students, particularly from Industrial and Operations engineering (IOE), as well as students of Economics and other quantitative social sciences. Students intending to concentrate in Mathematics must continue to elect MATH 217.
While MATH 216 includes 3-4 weeks of Linear Algebra as a tool in the study of Differential Equations, MATH 214 will include roughly 3 weeks of Differential Equations as an application of Linear Algebra.
The sequence MATH 115-116-215 is the standard complete introduction to the concepts and methods of calculus. It is taken by the majority of students intending to concentrate in mathematics, science, or engineering as well as students heading for many other fields. The emphasis is on concepts and solving problems rather than theory and proof. All sections are given a midterm and final exam. Topics Maple software. MATH 285 is a somewhat more theoretical course which covers the same material. For students intending to concentrate in mathematics or who have some interest in the theory of mathematics as well as its applications, the appropriate sequel is MATH 217. Students who intend to take only one further mathematics course and need differential equations should take MATH 216.
MATH 216. Introduction to Differential Equations.
Instructor(s):
Prerequisites & Distribution: MATH 116, 119, 156, 176, 186, or 296. Not intended for Mathematics concentrators. (4). (MSA). (BS). (QR/1). May not be repeated for credit. Credit can be earned for only one of MATH 216, 256, 286, or 316 engineering and the sciences. Math concentrators and other students who have some interest in the theory of mathematics should elect the sequence MATH 217-316. After There is a weekly computer lab using MATLAB software. This course is not intended for mathematics concentrators, who should elect the sequence MATH 217-316. MATH 286 covers much of the same material in the honors sequence. The sequence MATH 217-316 covers all of this material and substantially more at greater depth and with greater emphasis on the theory. MATH 404 covers further material on differential equations. MATH 217 and 417 cover further material on linear algebra. MATH 371 and 471 cover additional material on numerical methods.
MATH 217. Linear Algebra.
Instructor(s):
Prerequisites & Distribution: MATH 215, 255, or 285 Engineering and the sciences. Math concentrators and other students who have some interest in the theory of mathematics should elect the sequence MATH 217-316. These courses are explicitly designed to introduce the student to both the concepts and applications of their subjects and to the methods by which the results are proved. Therefore the student entering MATH 217 should come with a sincere interest in learning about proofs. The topics covered include: systems of linear equations; matrix algebra; vectors, vector spaces, and subspaces; geometry of Rn; linear dependence, bases, and dimension; linear transformations; eigenvalues and eigenvectors; diagonalization; and inner products. Throughout there will be emphasis on the concepts, logic, and methods of theoretical mathematics. MATH 417 and 419 cover similar material with more emphasis on computation and applications and less emphasis on proofs. MATH 513 covers more in a much more sophisticated way. The intended course to follow MATH 217 is 316. MATH 217 is also prerequisite for MATH 412 and all more advanced courses in mathematics.
Instructor(s):
Hendrikus Gerardus Derksen
One of the best ways to develop mathematical abilities is by solving problems using a variety of methods. Familiarity with numerous methods is a great asset to the developing student of mathematics. Methods learned in attacking a specific problem frequently find application in many other areas of mathematics. In many instances an interest in and appreciation of mathematics is better developed by solving problems than by hearing formal lectures on specific topics. The student has an opportunity to participate more actively in his/her education and development. This course is intended for superior students who have exhibited both ability and interest in doing mathematics, but it is not restricted to honors students. This course is excellent preparation for the Putnam exam. Students and one or more faculty and graduate student assistants will meet in small groups to explore problems in many different areas of mathematics. Problems will be selected according to the interests and background of the students.
MATH 296. Honors Mathematics II.
Section 001.
Instructor(s):
Brian D Conrad
Prerequisites & Distribution: Prior knowledge of first year calculus and permission of the Honors advisor. (4). (Excl). (BS). (QR/1). May not be repeated for credit. Credit is granted for only one course from among MATH 156, 176, 186, and 296.
Credits: (4).
Course Homepage: No homepage submitted.
The sequence MATH 295-296-395-396 is a more intensive honors sequence than MATH 185-186-285-286. The material includes all of that of the lower sequence and substantially more. The approach is theoretical, abstract, and rigorous. Students are expected to learn to understand and construct proofs as well as do calculations and solve problems. The expected background is a thorough understanding of high school algebra and trigonometry. No previous calculus is required, although many students in this course have had some calculus. Students completing this sequence will be ready to take advanced undergraduate and beginning graduate courses. This sequence is not restricted to students enrolled in the LS&A Honors Program. The precise content depends on material covered in MATH 295 but will generally include topics such as infinite series, power series, Taylor expansion, metric spaces. Other topics may include applications of analysis, Weierstrass Approximation theorem, elements of topology, introduction to linear algebra, complex numbers.
Waitlist Code: 5: Students in LSA College Honors may request overrides from the Honors Office; other students may request them from the Math Dept Office, 2084 East Hall.
MATH 310. Elementary Topics in Mathematics.
Section 001 — Math Games & Theory of Games.
Instructor(s):
Morton Brown
Prerequisites & Distribution: Two years of high school mathematics. (3). (Excl). (BS). May not be repeated for credit.
Credits: (3).
Course Homepage: No homepage submitted.
The current offering of the course focuses on game theory. Students study the strategy of several games where mathematical ideas and concepts can play a role. Most of the course will be occupied with the strructure of a variety of two person games of strategy: tic-tac-toe, tic-tac-toe misere, the French military game, hex, nim, the penny dime game, and many others. If there is sufficient interest students can study: dots and boxes, go moku, and some aspects of checkers and chess. There will also be a brief introduction to the classical Von Neuman/Morgenstern theory of mixed strategy games.
One of the main goals of the course (along with every course in the algebra sequence) is to expose students to rigorous, proof-oriented mathematics. Students are required to have taken MATH 217, which should provide a first exposure to this style of mathematics. A distinguishing feature of this course is that the abstract concepts are not studied in isolation. Instead, each topic is studied with the ultimate goal being a real-world application. As currently organized, the course is broken into four parts: the integers "mod n" and linear algebra over the integers mod p, with applications to error correcting codes; some number theory, with applications to public-key cryptography; polynomial algebra, with an emphasis on factoring algorithms over various fields, and permutation groups, with applications to enumeration of discrete structures "up to automorphisms" (a.k.a. Pólya Theory). MATH 412 is a more abstract and proof-oriented course with less emphasis on applications. EECS 303 (Algebraic Foundations of Computer Engineering) covers many of the same topics with a more applied approach. Another good follow-up course is MATH 475 (Number Theory). MATH 312 is one of the alternative prerequisites for MATH 416, and several advanced EECS courses make substantial use of the material of MATH 312. MATH 412 is better preparation for most subsequent mathematics courses.
MATH 316. Differential Equations.
Section 001.
Instructor(s):
Arthur G Wasserman
Prerequisites & Distribution: MATH 215 and 217. (3). (Excl). (BS). May not be repeated for credit. Credit can be earned for only one of MATH 216, 256, 286, or 316.
Credits: (3).
Course Homepage: No homepage submitted.
This is an introduction to differential equations for students who have studied linear algebra (MATH 217). It treats techniques of solution (exact and approximate), existence and uniqueness theorems, some qualitative theory, and many applications. Proofs are given in class; homework problems include both computational and more conceptually oriented problems. First-order equations: solutions, existence and uniqueness, and numerical techniques; linear systems: eigenvector-eigenvalue solutions of constant coefficient systems, fundamental matrix solutions, nonhomogeneous systems; higher-order equations, reduction of order, variation of parameters, series solutions; qualitative behavior of systems, equilibrium points, stability. Applications to physical problems are considered throughout. MATH 216 covers somewhat less material without the use of linear algebra and with less emphasis on theory. MATH 286 is the Honors version of MATH 316. MATH 471 and/or MATH 572 are natural sequels in the area of differential equations, but MATH 316 is also preparation for more theoretical courses such as MATH 451.
MATH 333. Directed Tutoring.
Instructor(s):
Prerequisites & Distribution: Enrollment in the secondary teaching certificate program with concentration in mathematics. Permission of instructor required. (1-3). (Excl). (EXPERIENTIAL). May be repeated for credit for a maximum of 3 credits. Offered mandatory credit/no credit.
Credits: (1-3).
Course Homepage: No homepage submitted.
An experiential mathematics course for exceptional upper-level students in the elementary teacher certification program. Students tutor needy beginners enrolled in the introductory courses (MATH 385 and MATH 489) required of all elementary teachers.
MATH 351. Principles of Analysis.
Section 001.
Instructor(s):
Morton Brown
Prerequisites & Distribution: MATH 215 and 217. (3). (Excl). (BS). May not be repeated for credit. No credit granted to those who have completed or are enrolled in MATH 451.
Credits: (3).
Course Homepage: No homepage submitted.
The content of this course is similar to that of MATH 451 but MATH 351 assumes less background. This course covers topics that might be of greater use to students considering a Mathematical Sciences concentration or a minor in Math. Course content includes: analysis of the real line, rational and irrational numbers, infinity — large and small, limits, convergence, infinite sequences and series, continuous functions, power series and differentiation.
MATH 354. Fourier Analysis and its Applications.
Section 001.
Instructor(s):
Mahdi Asgari
Prerequisites & Distribution: MATH 216, 256, 286, or 316. (3). (Excl). (BS). May not be repeated for credit. No credit granted to those who have completed or are enrolled in MATH 450 or 454.
Credits: (3).
Course Homepage: No homepage submitted.
This course is an introduction to Fourier analysis at an elementary level, emphasizing applications. The main topics are Fourier series, discrete Fourier transforms, and continuous Fourier transforms. A substantial portion of the time is spent on both scientific/technological applications (e.g., signal processing, Fourier optics), and applications in other branches of mathematics (e.g., partial differential equations, probability theory, number theory). Students will do some computer work, using MATLAB, an interactive programming tool that is easy to use, but no previous experience with computers is necessary.
MATH 371 / ENGR 371. Numerical Methods for Engineers and Scientists.
Section 001.
Instructor(s):
David Gammack
Prerequisites & Distribution: ENGR 101; one of MATH 216, 256, 286, or 316. (3). (Excl). (BS). May not be repeated for credit. No credit granted to those who have completed or are enrolled in Math 471. CAEN lab access fee required for non-Engineering students.
Credits: (3).
Lab Fee: CAEN lab access fee required for non-Engineering students.
Course Homepage: No homepage submitted.
This is a survey course of the basic numerical methods which are used to solve practical scientific problems.
Important concepts such as accuracy, stability, and efficiency are discussed. The course provides an introduction
to MATLAB, an interactive program for numerical linear algebra. Convergence theorems are discussed and
applied, but the proofs are not emphasized.
Objectives of the course
Develop numerical methods for approximately solving problems from continuous mathematics on the
computer
Implement these methods in a computer language (MATLAB)
Apply these methods to application problems
Computer language:
In this course, we will make extensive use of Matlab, a technical computing environment for numerical
computation and visualization produced by The MathWorks, Inc. A Matlab manual is available in the MSCC Lab.
Also available is a MATLAB tutorial written by Peter Blossey.
MATH 396. Honors Analysis II.
Section 001.
Instructor(s):
Mario Bonk
This course is a continuation of MATH 395 and has the same theoretical emphasis. Students are expected to understand and construct proofs. Differential and integral calculus of functions on Euclidean spaces. Students who have successfully completed the sequence MATH 295-396 are generally prepared to take a range of advanced undergraduate and graduate courses such as MATH 512, 513, 525, 590, and many others.
MATH 412. Introduction to Modern Algebra.
Instructor(s):
Prerequisites & Distribution: MATH 215, 255, or 285; and 217. (3). (Excl). (BS). May not be repeated for credit. No credit granted to those who have completed or are enrolled in MATH 512. Students with credit for MATH 312 should take MATH 512 rather than 412. One credit granted to those who have completed MATH 312.
Credits: (3).
Course Homepage: No homepage submitted.
This course is designed to serve as an introduction to the methods and concepts of abstract mathematics. A typical student entering this course has substantial experience in using complex mathematical (calculus) calculations to solve physical or geometrical problems, but is unused to analyzing carefully the content of definitions or the logical flow of ideas which underlie and justify these calculations. Although the topics discussed here are quite distinct from those of calculus, an important goal of the course is to introduce the student to this type of analysis. Much of the reading, homework exercises, and exams consists of theorems (propositions, lemmas, etc.) and their proofs. MATH 217 or equivalent required as background. The initial topics include ones common to every branch of mathematics: sets, functions (mappings), relations, and the common number systems (integers, rational numbers, real numbers, and complex numbers). These are then applied to the study of particular types of mathematical structures such as groups, rings, and fields. These structures are presented as abstractions from many examples such as the common number systems together with the operations of addition or multiplication, permutations of finite and infinite sets with function composition, sets of motions of geometric figures, and polynomials. Notions such as generator, subgroup, direct product, isomorphism, and homomorphism are defined and studied.
MATH 312 is a somewhat less abstract course which substitutes material on finite automata and other topics for some of the material on rings and fields of MATH 412. MATH 512 is an Honors version of MATH 412 which treats more material in a deeper way. A student who successfully completes this course will be prepared to take a number of other courses in abstract mathematics: MATH 416, 451, 475, 575, 481, 513, and 582. All of these courses will extend and deepen the student's grasp of modern abstract mathematics.
MATH 417. Matrix Algebra I.
Instructor(s):
Prerequisites & Distribution: Three courses MATH 513.
Credits: (3).
Course Homepage: No homepage submitted.
Many problems in science, engineering, and mathematics are best formulated in terms of matrices — rectangular arrays of numbers. This course is an introduction to the properties of and operations on matrices with a wide variety of applications. The main emphasis is on concepts and problem-solving, but students are responsible for some of the underlying theory. Diversity rather than depth of applications is stressed. This course is not intended for mathematics concentrators, who should elect MATH 217 or 513 (Honors). Topics include matrix operations, echelon form, general solutions of systems of linear equations, vector spaces and subspaces, linear independence and bases, linear transformations, determinants, orthogonality, characteristic polynomials, eigenvalues and eigenvectors, and similarity theory. Applications include linear networks, least squares method (regression), discrete Markov processes, linear programming, and differential equations.
MATH 419 is an enriched version of MATH 417 with a somewhat more theoretical emphasis. MATH 217 (despite its lower number) is also a more theoretical course which covers much of the material of MATH 417 at a deeper level. MATH 513 is an Honors version of this course, which is also taken by some mathematics graduate students. MATH 420 is the natural sequel, but this course serves as prerequisite to several courses: MATH 452, 462, 561, and 571.
MATH 419. Linear Spaces and Matrix Theory.
Instructor(s):
Prerequisites & Distribution: Four terms of college mathematics in MATH 513.
Credits: (3).
Course Homepage: No homepage submitted.
MATH 419 covers much of the same ground as MATH 417 but presents the material in a somewhat more abstract way in terms of vector spaces and linear transformations instead of matrices. There is a mix of proofs, calculations, and applications with the emphasis depending somewhat on the instructor. A previous proof-oriented course is helpful but by no means necessary. Basic notions of vector spaces and linear transformations: spanning, linear independence, bases, dimension, matrix representation of linear transformations; determinants; eigenvalues, eigenvectors, Jordan canonical form, inner-product spaces; unitary, self-adjoint, and orthogonal operators and matrices, and applications to differential and difference equations.
MATH 417 is less rigorous and theoretical and more oriented to applications. MATH 217 is similar to MATH 419 but slightly more proof-oriented. MATH 513 is much more abstract and sophisticated. MATH 420 is the natural sequel, but this course serves as prerequisite to several courses: MATH 452, 462, 561, and 571.
MATH 425 / STATS 425. Introduction to Probability.
Instructor(s):
Mathematics faculty
This course introduces students to useful and interesting ideas of the mathematical theory of probability and to a number of applications of probability to a variety of fields including genetics, economics, geology, business, and engineering. The theory developed together with other mathematical tools such as combinatorics and calculus are applied to everyday problems. Concepts, calculations, and derivations are emphasized. The course will make essential use of the material of MATH 116 and 215. Topics include the basic results and methods of both discrete and continuous probability theory: conditional probability, independent events, random variables, jointly distributed random variables, expectations, variances, covariances. Different instructors will vary the emphasis. STATS 426 is a natural sequel for students interested in statistics. MATH 523 includes many applications of probability theory.
This course introduces students to the theory of probability and to a number of applications. Topics include the basic results and methods of both discrete and continuous probability theory: conditional probability, independent events, random variables, jointly distributed random variables, expectations, variances, covariances.
There will be approximately 10 problem sets. Grade will be based on two 1-hour midterm exams, 20% each; 20% homework; 40% final exam. pText (required): Sheldon Ross, A First Course in Probability, 6th edition, Prentice-Hall, 2002.
MATH 450. Advanced Mathematics for Engineers I.
Instructor(s):
Prerequisites & Distribution: MATH 215, 255, or 285. (4). (Excl). (BS). May not be repeated for credit. No credit granted to those who have completed or are enrolled in MATH 354 or 454.
Credits: (4).
Course Homepage: No homepage submitted.
Although this course is designed principally to develop mathematics for application to problems of science and engineering, it also serves as an important bridge for students between the calculus courses and the more demanding advanced courses. Students are expected to learn to read and write mathematics at a more sophisticated level and to combine several techniques to solve problems. Some proofs are given, and students are responsible for a thorough understanding of definitions and theorems. Students should have a good command of the material from MATH 215, and 216 or 316, which is used throughout the course. A background in linear algebra, e.g. MATH 217, is highly desirable, as is familiarity with Maple software. Topics include a review of curves and surfaces in implicit, parametric, and explicit forms; differentiability and affine approximations; implicit and inverse function theorems; chain rule for 3-space; multiple integrals; scalar and vector fields; line and surface integrals; computations of planetary motion, work, circulation, and flux over surfaces; Gauss' and Stokes' Theorems; and derivation of continuity and heat equation. Some instructors include more material on higher dimensional spaces and an introduction to Fourier series. MATH 450 is an alternative to MATH 451 as a prerequisite for several more advanced courses. MATH 454 and 555 are the natural sequels for students with primary interest in engineering applications.
MATH 451. Advanced Calculus I.
Instructor(s):
Prerequisites & Distribution: MATH 215 and one course beyond MATH 215; or MATH 255 or 285. Intended for concentrators; other students should elect MATH 450. (3). (Excl). (BS). May not be repeated for credit. No credit granted to those who have completed or are enrolled in MATH 351.
Credits: (3).
Course Homepage: No homepage submitted.
This course has two complementary goals: (1) a rigorous development of the fundamental ideas of calculus, and (2) a further development of the student's ability to deal with abstract mathematics and mathematical proofs. The key words here are "rigor" and "proof"; almost all of the material of the course consists in understanding and constructing definitions, theorems (propositions, lemmas, etc.) and proofs. This is considered one of the more difficult among the undergraduate mathematics courses, and students should be prepared to make a strong commitment to the course. In particular, it is strongly recommended that some course which requires proofs (such as MATH 412) be taken before MATH 451. Topics include: logic and techniques of proof; sets, functions, and relations; cardinality; the real number system and its topology; infinite sequences, limits, and continuity; differentiation; integration, and the Fundamental Theorem of Calculus; infinite series; and sequences and series of functions.
There is really no other course which covers the material of MATH 451. Although MATH 450 is an alternative prerequisite for some later courses, the emphasis of the two courses is quite distinct. The natural sequel to MATH 451 is 452, which extends the ideas considered here to functions of several variables. In a sense, MATH 451 treats the theory behind MATH 115-116, while MATH 452 does the same for MATH 215 and a part of MATH 216. MATH 551 is a more advanced version of Math 452. MATH 451 is also a prerequisite for several other courses: MATH 575, 590, 596, and 597.
MATH 452. Advanced Calculus II.
Section 001 — Multivariable Calculus and Elementary Function Theory.
Instructor(s):
Lukas I Geyer
Prerequisites & Distribution: MATH 217, 417, or 419; and MATH 451. (3). (Excl). (BS). May not be repeated for credit.
Credits: (3).
Course Homepage: No homepage submitted.
This course does a rigorous development of multivariable calculus and elementary function theory with some view towards generalizations. Concepts and proofs are stressed. This is a relatively difficult course, but the stated prerequisites provide adequate preparation. Topics include:
MATH 551 is a higher-level course covering much of the same material with greater emphasis on differential geometry. Math 450 covers the same material and a bit more with more emphasis on applications, and no emphasis on proofs. MATH 452 is prerequisite to MATH 572 and is good general background for any of the more advanced courses in analysis (MATH 596, 597) or differential geometry or topology (MATH 537, 635).
MATH 454. Boundary Value Problems for Partial Differential Equations.
Instructor(s):
Prerequisites & Distribution: MATH 216, 256, 286, or 316. (3). (Excl). (BS). May not be repeated for credit. Students with credit for MATH 354 can elect MATH 454 for one credit. No credit granted to those who have completed or are enrolled in MATH 450.
Credits: (3).
Course Homepage: No homepage submitted.
This course is devoted to the use of Fourier series and other orthogonal expansions in the solution of boundary-value problems for second-order linear partial differential equations. Emphasis is on concepts and calculation. The official prerequisite is ample preparation. Classical representation and convergence theorems for Fourier series; method of separation of variables for the solution of the one-dimensional heat and wave equation; the heat and wave equations in higher dimensions; spherical and cylindrical Bessel functions; Legendre polynomials; methods for evaluating asymptotic integrals (Laplace's method, steepest descent); Fourier and Laplace transforms; and applications to linear input-output systems, analysis of data smoothing and filtering, signal processing, time-series analysis, and spectral analysis. Both MATH 455 and 554 cover many of the same topics but are very seldom offered. MATH 454 is prerequisite to MATH 571 and 572, although it is not a formal prerequisite, it is good background for MATH 556.
MATH 462. Mathematical Models.
Section 001.
Instructor(s):
David Bortz
Prerequisites & Distribution: MATH 216, 256, 286, or 316; and MATH 217, 417, or 419. (3). (Excl). (BS). May not be repeated for credit. Students with credit for MATH 362 must have department permission to elect MATH 462.
Credits: (3).
Course Homepage: No homepage submitted.
This course will cover biological models constructed from difference equations and ordinary differential equations. Applications will be drawn from population biology, population genetics, the theory of epidemics, biochemical kinetics, and physiology. Both exact solutions and simple qualitative methods for understanding dynamical systems will be stressed.
MATH 471. Introduction to Numerical Methods.
Instructor(s):
Prerequisites & Distribution: MATH 216, 256, 286, or 316; and 217, 417, or 419; and a working knowledge of one high-level computer language. (3). (Excl). (BS). May not be repeated for credit. No credit granted to those who have completed or are enrolled in MATH 371 or 472.
Credits: (3).
Course Homepage: No homepage submitted.
This is a survey of the basic numerical methods which are used to solve scientific problems. The emphasis is evenly divided between the analysis of the methods and their practical applications. Some convergence theorems and error bounds are proven. The course also provides an introduction to MATLAB, an interactive program for numerical linear algebra, as well as practice in computer programming. One goal of the course is to show how calculus and linear algebra are used in numerical analysis. Topics may include computer arithmetic, Newton's method for non-linear equations, polynomial interpolation, numerical integration, systems of linear equations, initial value problems for ordinary differential equations, quadrature, partial pivoting, spline approximations, partial differential equations, Monte Carlo methods, 2-point boundary value problems, and the Dirichlet problem for the Laplace equation. MATH 371 is a less sophisticated version intended principally for sophomore and junior engineering students; the sequence MATH 571-572 is mainly taken by graduate students, but should be considered by strong undergraduates. MATH 471 is good preparation for MATH 571 and 572, although it is not prerequisite to these courses.
MATH 475. Elementary Number Theory.
Section 001.
Instructor(s):
Muthukrishnan Krishnamurthy
Prerequisites & Distribution: At least three terms of college mathematics are recommended. (3). (Excl). (BS). May not be repeated for credit.
Credits: (3).
Course Homepage: No homepage submitted.
This is an elementary introduction to number theory, especially congruence arithmetic. Number theory is one of the few areas of mathematics in which problems easily describable to a layman (is every even number the sum of two primes?) have remained unsolved for centuries. Recently some of these fascinating but seemingly useless questions have come to be of central importance in the design of codes and cyphers. The methods of number theory are often elementary in requiring little formal background. In addition to strictly number-theoretic questions, concrete examples of structures such as rings and fields from abstract algebra are discussed. Concepts and proofs are emphasized, but there is some discussion of algorithms which permit efficient calculation. Students are expected to do simple proofs and may be asked to perform computer experiments. Although there are no special prerequisites and the course is essentially self-contained, most students have some experience in abstract mathematics and problem solving and are interested in learning proofs. A Computational Laboratory (Math 476, 1 credit) will usually be offered as an optional supplement to this course. Topics usually include the Euclidean algorithm, primes and unique factorization, congruences, Chinese Remainder Theorem, Hensel's Lemma, Diophantine equations, arithmetic in polynomial rings, primitive roots, quadratic reciprocity, and quadratic fields. MATH 575 moves much faster, covers more material, and requires more difficult exercises. There is some overlap with MATH 412 which stresses the algebraic content. MATH 475 may be followed by Math 575 and is good preparation for MATH 412. All of the advanced number theory courses, MATH 675, 676, 677, 678, and 679, presuppose the material of MATH 575, although a good student may get by with MATH 475. Each of these is devoted to a special subarea of number theory.
MATH 476. Computational Laboratory in Number Theory.
Section 001.
Instructor(s):
Muthukrishnan Krishnamurthy
Prerequisites & Distribution: Prior or concurrent enrollment in MATH 475 or 575. (1). (Excl). (BS). May not be repeated for credit.
Credits: (1).
Course Homepage: No homepage submitted.
Students will be provided software with which to conduct numerical explorations. Students will submit reports of their findings weekly. No programming necessary, but students interested in programming will have the opportunity to embark on their own projects. Participation in the laboratory should boost the student's performance in MATH 475 or MATH 575. Students in the lab will see mathematics as an exploratory science (as mathematicians do). Students will gain a knowledge of algorithms which have been developed (some quite recently) for number-theoretic purposes, e.g., for factoring. No exams.
Instructor(s):
This course is designed for students who intend to teach junior high or high school mathematics. It is advised that the course be taken relatively early in the program to help the student decide whether or not this is an appropriate goal. Concepts and proofs are emphasized over calculation. The course is conducted in a discussion format. Class participation is expected and constitutes a significant part of the course grade. Topics covered have included problem solving; sets, relations and functions; the real number system and its subsystems; number theory; probability and statistics; difference sequences and equations; interest and annuities; algebra; and logic. This material is covered in the course pack and scattered points in the text book. There is no real alternative, but the requirement of MATH 486 may be waived for strong students who intend to do graduate work in mathematics. Prior completion of MATH 486 may be of use for some students planning to take MATH 312, 412, or 425.
MATH 489. Mathematics for Elementary and Middle School Teachers.
Instructor(s):
Prerequisites & Distribution: MATH 385 or 485. (3). (Excl). May not be repeated for credit. May not be used in any graduate program in mathematics.
Credits: (3).
Course Homepage: No homepage submitted.
This course, together with its predecessor MATH 385, provides a coherent overview of the mathematics underlying the elementary and middle school curriculum. It is required of all students intending to earn an elementary teaching certificate and is taken almost exclusively by such students. Concepts are heavily emphasized with some attention given to calculation and proof. The course is conducted using a discussion format. Class participation is expected and constitutes a significant part of the course grade. Enrollment is limited to 30 students per section. Although only two years of high school mathematics are required, a more complete background including pre-calculus or calculus is desirable. Topics covered include fractions and rational numbers, decimals and real numbers, probability and statistics, geometric figures, and measurement. Algebraic techniques and problem-solving strategies are used throughout the course.
MATH 490. Introduction to Topology.
Section 001 — An Introduction to Point-Set and Algebraic Topology.
Instructor(s):
Elizabeth A Burslem
Prerequisites & Distribution: MATH 412 or 451 or equivalent experience with abstract mathematics. (3). (Excl). (BS). May not be repeated for credit.
Credits: (3).
Course Homepage: No homepage submitted.
This course in an introduction to both point-set and algebraic topology. Although much of the presentation is theoretical and proof-oriented, the material is well-suited for developing intuition and giving convincing proofs which are pictorial or geometric rather than completely rigorous. There are many interesting examples of topologies and manifolds, some from common experience (combing a hairy ball, the utilities problem). In addition to the stated prerequisites, courses containing some group theory (MATH 412 or 512) and advanced calculus (MATH 451) are desirable although not absolutely necessary. The topics covered are fairly constant but the presentation and emphasis will vary significantly with the instructor. These include point-set topology, examples of topological spaces, orientable and non-orientable surfaces, fundamental groups, homotopy, and covering spaces. Metric and Euclidean spaces are emphasized. Math 590 is a deeper and more difficult presentation of much of the same material which is taken mainly by mathematics graduate students. MATH 433 is a related course at about the same level. MATH 490 is not prerequisite for any later course but provides good background for MATH 590 or any of the other courses in geometry or topology.
MATH 501. Applied & Interdisciplinary Mathematics Student Seminar.
Section 001.
Prerequisites & Distribution: At least two 300 or above level math courses, and graduate standing; Qualified undergraduates with permission of instructor only. (1). (Excl). May be repeated for credit for a maximum of 6 credits. Offered mandatory credit/no credit.
Credits: (1).
Course Homepage: No homepage submitted.
The Applied and Interdisciplinary Mathematics (AIM) student seminar is an introductory and survey course in the methods and applications of modern mathematics in the natural, social, and engineering sciences. Students will attend the weekly AIM Research Seminar where topics of current interest are presented by active researchers (both from U-M and from elsewhere). The other central aspect of the course will be a seminar to prepare students with appropriate introductory background material. The seminar will also focus on effective communication methods for interdisciplinary research. MATH 501 is primarily intended for graduate students in the Applied & Interdisciplinary Mathematics M.S. and Ph.D. programs. It is also intended for mathematically curious graduate students from other areas. Qualified undergraduates are welcome to elect the course with the instructor's permission.
Student attendance and participation at all seminar sessions is required. Students will develop and make a short presentation on some aspect of applied and interdisciplinary mathematics.
MATH 512. Algebraic Structures.
Section 001 — Basic Structures of Modern Abstract Algebra.
Instructor(s):
Robert L Griess Jr
Prerequisites & Distribution: MATH 451 or 513. (3). (Excl). (BS). May not be repeated for credit.
Credits: (3).
Course Homepage: No homepage submitted.
Student Body: mainly undergrad math concentrators with a few grad students from other fields
Background and Goals: This is one of the more abstract and difficult courses in the undergraduate program. It is frequently elected by students who have completed the 295--396 sequence. Its goal is to introduce students to the basic structures of modern abstract algebra (groups, rings, and fields) in a rigorous way. Emphasis is on concepts and proofs; calculations are used to illustrate the general theory. Exercises tend to be quite challenging. Students should have some previous exposure to rigorous proof-oriented mathematics and be prepared to work hard. Students from Math 285 are strongly advised to take some 400-500 level course first, for example, Math 513. Some background in linear algebra is strongly recommended
Content: The course covers basic definitions and properties of groups, rings, and fields, including homomorphisms, isomorphisms, and simplicity. Further topics are selected from (1) Group Theory: Sylow theorems, Structure Theorem for finitely-generated Abelian groups, permutation representations, the symmetric and alternating groups (2) Ring Theory: Euclidean, principal ideal, and unique factorization domains, polynomial rings in one and several variables, algebraic varieties, ideals, and (3) Field Theory: statement of the Fundamental Theorem of Galois Theory, Nullstellensatz, subfields of the complex numbers and the integers mod p.
Alternatives: Math 412 (Introduction to Modern Algebra) is a substantially lower level course covering about half of the material of Math 512. The sequence Math 593--594 covers about twice as much Group and Field Theory as well as several other topics and presupposes that students have had a previous introduction to these concepts at least at the level of Math 412.
Subsequent Courses: Together with Math 513, this course is excellent preparation for the sequence Math 593 — 594.
Text Book: Abstract Algebra, Second Edition by David Dummit and Richard Foote.
MATH 513. Introduction to Linear Algebra.
Instructor(s):
William E Fulton
Prerequisites & Distribution: MATH 412. (3). (Excl). (BS). May not be repeated for credit. Two credits granted to those who have completed MATH 214, 217, 417, or 419.
Credits: (3).
Course Homepage: No homepage submitted.
Prerequisites: Math 412 or Math 451 or permission of the instructor
Background and Goals: This is an introduction to the theory of abstract vector spaces and linear transformations. The emphasis is on concepts and proofs with some calculations to illustrate the theory.
Content: Topics are selected from: vector spaces over arbitrary fields (including finite fields); linear transformations, bases, and matrices; eigenvalues and eigenvectors; applications to linear and linear differential equations; bilinear and quadratic forms; spectral theorem; Jordan Canonical Form. This corresponds to most of the first text with the omission of some starred sections and all but Chapters 8 and 10 of the second text.
Alternatives: Math 419 (Lin. Spaces and Matrix Thy) covers much of the same material using the same text, but there is more stress on computation and applications. Math 217 (Linear Algebra) is similarly proof-oriented but significantly less demanding than Math 513. Math 417 (Matrix Algebra I) is much less abstract and more concerned with applications.
Subsequent Courses: The natural sequel to Math 513 is Math 593 (Algebra I). Math 513 is also prerequisite to several other courses: Math 537, 551, 571, and 575, and may always be substituted for Math 417 or 419.
Section 001.
This course is a continuation of MATH520 (a year-long sequence). It covers the topics of reserving models for life insurance; multiple-life models including joint life and last survivor contingent insurances; multiple-decrement models including disability, retirement and withdrawal; insurance models including expenses; and business and regulatory considerations.
MATH 523. Risk Theory.
Section 001 — Risk Management.
Instructor(s):
Conlon
Required Text: "Loss Models-from Data to Decisions", by Klugman, Panjer and
Willmot, Wiley 1998.
Background and Goals: Risk management is of major concern to all
financial institutions and is an active area of modern finance. This course is
relevant for students with interests in finance, risk management, or insurance.
It provides background for the professional exams in Risk Theory offered by the
Society of Actuaries and the Casualty Actuary Society. Contents: Standard distributions used for claim frequency models and for loss
variables, theory of aggregate claims, compound Poisson claims model, discrete time and continuous time models for the aggregate claims variable, the Chapman-Kolmogorov equation for expectations of aggregate claims variables, the
Poisson process, estimating the probability of ruin, reinsurance schemes
and their implications for profit and risk.
Credibility theory, classical theory for independent events, least
squares theory for correlated events, examples of random variables where the
least squares theory is exact.
Grading: The grade for the course will be determined from
performances on homeworks, a midterm and a final exam.
MATH 525 / STATS 525. Probability Theory.
Section 001.
Instructor(s):
Gautam Bharali
Prerequisites & Distribution: MATH 451 (strongly recommended) or 450. MATH 425 would be helpful. (3). (Excl). (BS). May not be repeated for credit.
Credits: (3).
Course Homepage: No homepage submitted.
Background: This course is a fairly rigorous study of the mathematical basis of probability theory. There is some overlap of topics with Math 425, but in Math 525, there is a greater emphasis on the proofs of major results in probability theory. This course and its sequel - Math 526 - are core
courses for the Applied and Interdisciplinary Mathematics (AIM) program.
Content: The notion of a probability space and a random variable, discrete and continuous random variables, independence and expectation, conditional probability and conditional expectations, generating functions and moment generating functions, the Law of Large Numbers, and the Central Limit Theorem comprise the essential core of this course. Further topics, to be decided later (and, if feasible, selected according to audience interest), will be covered in the last month of the semester.
Alternatives: EECS 501 covers some of the above material at a lower level of mathematical rigor. Math 425 (Introduction to Probability) is recommended for students with substantially less mathematical preparation.
Instructor(s):
Virginia R Young
Risk management is of major concern to all financial institutions and is an active area of modern finance. This course is relevant for students with interests in insurance, risk management, or finance. We will cover the following topics: advanced topics in credibility theory, risk measures and premium principles, optimal (re)insurance, reinsurance products, and reinsurance pricing.
I assume that you have taken MATH 523, Risk Theory. In fact, one can think of this course as a continuation of MATH 523 with emphasis on applying the material learned in Risk Theory to more practical settings.
The official text for the course is a set of notes available at UM.CourseTools. In addition, an excellent book concerning modern reinsurance products is Integrating Corporate Risk Management by Prakash Shimpi, published by Texere. I suggest that you buy this book, but I do not require that you do so.
MATH 531. Transformation Groups in Geometry.
Section 001.
Instructor(s):
Emina Alibegovic
Prerequisites: MATH 412 or 512 would be helpful, but neither is necessary.
Text required: None.
Text recommended: Armstrong, Groups and Symmetry; Lyndon: Groups and Geometry.
textbook comment: Your class notes and my handouts will be sufficient. The books
I listed contain some of the material we will cover, but not all of it.
Course description:
The purpose of this course is to explore the close ties between geometry and
algebra. We will study Euclidean and hyperbolic spaces and groups of their
isometries. Our discussions will include, but will not be limited to, free
groups, triangle groups, and Coxeter groups. We will talk about group actions
on spaces, and in particular group actions on trees.
MATH 555. Introduction to Functions of a Complex Variable with Applications.
Section 001.
Instructor(s):
Divakar Viswanath
Prerequisites & Distribution: MATH 450 or 451. (3). (Excl). (BS). May not be repeated for credit.
Credits: (3).
Course Homepage: No homepage submitted.
Recent Texts: Complex Variables and Applications, 6th ed. (Churchill and Brown);
Student Body: largely engineering and physics graduate students with some math and engineering undergrads, and graduate students in the Applied and Interdisciplinary Mathematics (AIM) program
Background and Goals: This course is an introduction to the theory of complex valued functions of a complex variable with substantial attention to applications in science and engineering. Concepts, calculations, and the ability to apply principles to physical problems are emphasized over proofs, but arguments are rigorous. The prerequisite of a course in advanced calculus is essential. This course is a core course for the Applied and Interdisciplinary Mathematics (AIM) graduate program. Content: Differentiation and integration of complex valued functions of a complex variable, series, mappings, residues, applications. Evaluation of improper real integrals, fluid dynamics. This corresponds to Chapters 1--9 of Churchill. Alternatives: Math 596 (Analysis I (Complex)) covers all of the theoretical material of Math 555 and usually more at a higher level and with emphasis on proofs rather than applications. Subsequent Courses: Math 555 is prerequisite to many advanced courses in science and engineering fields.
MATH 557. Methods of Applied Mathematics II.
Section 001.
Prerequisites & Distribution: MATH 217, 419, or 513; 451 and 555. (3). (Excl). (BS). May not be repeated for credit.
Credits: (3).
Course Homepage: No homepage submitted.
Prerequisites: (1) one of the following: Math 217, 419, or 513 (i.e. a
course in linear algebra); (2) one of the following: Math 216, 256, 286, 316, or 404 (i.e. a course in differential equations); (3) Math 451
(or an equivalent course in
advanced calculus); (4) Math 555 (or an equivalent course in complex
variables).
Text: There is no required text. Lecture notes will be made available
to students from the instructor's website. Recommended texts will be
announced in class.
Audience: Graduate students and advanced undergraduates in applied
mathematics, engineering, or the natural sciences.
Background and Goals: In applied mathematics, we often try to
understand a physical process by formulating and analyzing mathematical
models which in many cases consist of differential equations with
initial and/or boundary conditions. Most of the time, especially if the
equation is nonlinear, an explicit formula for the solution is not
available. Even if we are clever or lucky enough to find an explicit
formula, it may be difficult to extract useful information from it and
in practice, we must settle for a sufficiently accurate approximate
solution obtained by numerical or asymptotic analysis (or a combination
of the two). This course is an introduction to the latter of these two
approximation methods. The material covered in the textbook includes
the nature of asymptotic approximations, asymptotic expansions of
integrals and applications to transform theory (Fourier and Laplace), regular and singular perturbation theory for differential equations
including transition point analysis, the use of matched expansions, and
multiple scale methods. The time remaining after studying these topics
will be devoted to the derivation of several famous canonical model
equations of applied mathematics (e.g. the Korteweg-de Vries equation
and the nonlinear Schroedinger equation) using multiscale asymptotics.
Students will come to understand how these equations arise again and
again from fields of study as diverse as water wave theory, molecular
dynamics, and nonlinear optics.
Grading: Students will be evaluated on the basis of homework
assignments and also participation and lecture attendance.
MATH 558. Ordinary Differential Equations.
Section 001.
Instructor(s):
Andrew J Christlieb
Prerequisites & Distribution: MATH 450 or 451. (3). (Excl). (BS). May not be repeated for credit.
Credits: (3).
Course Homepage: No homepage submitted.
Prerequisites: Basic Linear Algebra, Ordinary Differential Equations (math 216), Multivariable Calculus (215) and Either Advanced Calculus (math 451) or an advanced mathematical methods course (e.g. Math 454); preferably both.
Course Objective:
This course is an introduction to the modern qualitative theory of ordinary differential equations with emphasis on geometric techniques and visualization. Much of the motivation for this approach comes from applications. Examples of applications of differential equations to science and engineering are a significant part of the course. There are relatively few proofs.
Course Description: Nonlinear differential equations and iterative maps arise in the mathematical description of numerous systems throughout science and engineering. Such systems may display complicated and rich dynamical behavior. In this course we will focus on the theory of dynamical systems and how it is used in the study of complex systems. The goal of this course is to provide a broad overview of the subject as well as an in-depth analysis of specific examples. The course is intended for students in mathematics, engineering, and the natural sciences. Topics covered will include aspects of autonomous and driven two variable systems including the geometry of phase plane trajectories, periodic solutions, forced oscillations, stability, bifurcations and chaos. Applications to problems from physics, engineering and the natural sciences will arise in the course by way of examples in lecture ad through the homework problems. We will cover material from Chapters 1-5 and 8-13 of the text.
Textbook
Nonlinear Ordinary Differetial Equations, Oxford Press. by: D.W. Jordan and P. Smith
References
Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer. John Guckenheimer and Philip Holmes
Nonlinear Differential Equations and Dynamical Systems, Springer. Ferdinand Verhulst
Applications of Centre Manifold Theory, Springer. J. Carr
Nonlinear Systems, Chambridge. P.G. Drazin
Course Objectives:
To provide first-year graduate students with basic understanding of linear programming, its importance, and applications. To discuss algorithms for linear programming, available software and how to use it intelligently.
Section 001.
Instructor(s):
John R Stembridge
Prerequisite: MATH 512 or an equivalent level of mathematical maturity.
This course will be an introduction to algebraic combinatorics.
Previous exposure to combinatorics will not be necessary, but
experience with proof-oriented mathematics at the introductory
graduate or advanced undergraduate level, and linear algebra, will be needed.
Most of the topics we cover will be centered around enumeration and
generating functions. But this is not to say that the course is only
about enumeration — questions about counting are a good starting point
for gaining a deeper understanding of combinatorial structure.
Some of the topics to be covered include sieve methods, the matrix-tree
theorem, Lagrange inversion, the permanent-determinant method, the transfer matrix method, and ordinary and exponential generating
functions.
Recommended text: R. Stanley, Enumerative Combinatorics, Vol. I
Cambridge Univ. Press, 1997.
MATH 567. Introduction to Coding Theory.
Section 001.
Instructor(s):
Hendrikus Gerardus Derksen
Prerequisites & Distribution: One of MATH 217, 419, 513. (3). (Excl). (BS). May not be repeated for credit.
Credits: (3).
Course Homepage: No homepage submitted.
Student Body: Undergraduate math majors and EECS graduate students
Background and Goals: This course is designed to introduce math majors to an important area of applications in the communications industry. From a background in linear algebra it will cover the foundations of the theory of error-correcting codes and prepare a student to take further EECS courses or gain employment in this area. For EECS students it will provide a mathematical setting for their study of communications technology.
Content: Introduction to coding theory focusing on the mathematical background for error-correcting codes. Shannon's Theorem and channel capacity. Review of tools from linear algebra and an introduction to abstract algebra and finite fields. Basic examples of codes such and Hamming, BCH, cyclic, Melas, Reed-Muller, and Reed-Solomon. Introduction to decoding starting with syndrome decoding and covering weight enumerator polynomials and the Mac-Williams Sloane identity. Further topics range from asymptotic parameters and bounds to a discussion of algebraic geometric codes in their simplest form.
MATH 571. Numerical Methods for Scientific Computing I.
Section 001.
Instructor(s):
James F EppersonThis course is an introduction to numerical linear algebra, a core subject in scientific computing. Three general problems are considered: (1) solving a system of linear equations, (2) computing the eigenvalues and eigenvectors of a matrix, and (3) least squares problems. These problems often arise in applications in science and engineering, and many algorithms have been developed for their solution. However, standard approaches may fail if the size of the problem becomes large or if the problem is ill-conditioned, e.g. the operation count may be prohibitive or computer roundoff error may ruin the answer. We'll investigate these issues and study some of the accurate, efficient, and stable algorithms that have been devised to overcome these difficulties.
The course grade will be based on homework assignments, a midterm exam, and a final exam. Some homework exercises will require computing, for which Matlab is recommended.
MATH 572. Numerical Methods for Scientific Computing II.
Section 001.
Instructor(s):
Divakar ViswanathMath 572 is an introduction to numerical methods for solving
differential equations. These methods are widely used in science
and engineering. The four main segments of the course will
cover the following topics:
MATH 592. Introduction to Algebraic Topology.
Section 001.
Instructor(s):
Igor Kriz
The purpose of this course is to introduce basic concepts
of algebraic topology, in particular fundamental group, covering spaces and homology. These methods provide the
first tools for proving that two topological spaces are
not topologically equivalent (example: the bowling ball
is topologically different from the teacup).
Other simple applications of the methods will
also be given, for example fixed point theorems for
continuous maps.
Prerequisites: basic knowledge of point set topology, such as
from 590 or 591.
Books: There is no ideal text covering all this material on
exactly the level needed (basic but rigorous). Recommended texts
include
Munkres: Elements of Algebraic topology (for homology)
and
J.P.May: A concise course in algebraic topology (for fundamental
group and covering spaces).
Both texts include topics which will not be covered in 592, and are
also suitable textbooks for the next course in algebraic topology, 695.
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Algebra Homework Help
Ordinary algebra is a topic almost everyone studies to some extent in high school. Even so, it's easy to forget basic skills, and many people find themselves having difficulty in math classes later in life because of those forgotten skills. We offer Algebra homework help to get you caught up and ready to take this subject by storm.
Typical topics in a basic, college-level class in ordinary algebra will include:
Graphs, Functions, and Models
Functions, Equations, and Inequalities
Polynomial and Rational Functions
Exponential and Logarithmic Functions
Systems of Equations and Matrices
Conic Sections
Sequences, Series, and Probability
A truly great website for getting help and extra practice in ordinary algebra at all levels is the Virtual Math Lab of West Texas A&M University.
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In Precalculus students use the following as tools to express generalizations and to analyze and understand a variety of mathematical relationships and real-world phenomena:
Functions
Equations
Sequences
Series
Vectors
Limits
Precalculus topic instructional simulation
Modeling is an overarching theme of this Precalculus course support service. Students build on and expand their experiences with functions from Algebra I, Geometry and Algebra II as they continue to explore the characteristics and behavior of functions (including rate of change and limits), and the most important families of functions that model real world phenomena (especially transcendental functions).
Teachers can guide their students through deeper study of functions, equations, sequences, series, vectors, and limits to enable them to successfully express generalizations and to analyze and understand a variety of mathematical relationships and real-world phenomena.
Though I am an experienced teacher, every section or topic in Precalculus provides me with a new way to introduce and teach the course. I find creative problems, demonstrations, and interactive animations to engage each student. Agile Mind gives me access to resources that make me a better teacher. As a result I have better students.
- Marty Romero, Math Chair, Wallis Annenberg HS, Los Angeles, CA
Sign up for a tour to experience how Agile Mind services can work for you and your students. You will be contacted by one of our representatives.
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MATHEMATICS COURSES - MATH
MATH 1020. Fundamentals of Geometry (3) Su, F, S
An introduction to the definitions, methods, and logic of geometry.
Prerequisite: MATH ND0960 or placement test.
MATH QL1030. Contemporary Mathematics (3) Su, F, S
Topics from mathematics which convey to the student the beauty and utility of
mathematics, and which illustrate its application to modern society. Topics
include geometry, statistics, probability, and growth and form. Prerequisite:
MATH 1010 or ACT Math score 23 or higher or placement test.
MATH QL1040. Introduction to Statistics (3) Su, F, S
Basic concepts of probability and statistics with an emphasis on applications.
Prerequisite: MATH 1010 or Math ACT score 23 or higher or placement test.
MATH SI1220. Calculus II (4) Su, F, S
MATH 1630. Discrete Mathematics Applied to Computing (4)
An overview of the fundamentals of algorithmic, discrete mathematics applied
to computation using a contemporary programming language. Topics include logic,
proofs, sets, functions, counting, relations, graphs, trees, Boolean algebra,
and models of computation. This course includes programming. Prerequisites: MATH
QL1050 or MATH QL1080, and CS SI1400 or ability to program in a contemporary
computer language and the consent of the instructor.
MATH 2010. Mathematics for Elementary Teachers I (3) Su, F, S
Prospective elementary school teachers revisit mathematics topics from the
elementary school curriculum and examine them from an advanced perspective
including arithmetic, number theory, set theory and problem solving.
Prerequisite: MATH QL1050.
MATH 2020. Mathematics for Elementary Teachers II (3) Su, F, S
Prospective elementary school teachers revisit mathematics topics from the
elementary school curriculum and examine them from an advanced perspective
including probability, statistics, geometry and measurement. Prerequisite: MATH QL1050 and
MATH 2010.
MATH 2110. Foundations of Algebra (3)
An introduction to Abstract Algebra, Number Theory and Logic with an emphasis
on problem solving and proof writing. Prerequisite: MATH SI1210.
MATH 3410, 3420. Probability and Statistics (3-3) F, S
MATH 3550. Introduction to Mathematical Modeling (3) F or S
Formulation, solution and interpretation of mathematical models for problems occurring
in areas of physical, biological and social science. Prerequisite: MATH 2210,
MATH 2270 or
2280, or consent from instructor.
MATH 3610. Graph Theory (3) F
Principles of Graph Theory including methods and models, special types of graphs, paths
and circuits, coloring, networks, and other applications. Prerequisite: MATH SI1210.
MATH 3750. Dynamical Systems (3) S (alternate years)
MATH 3810. Complex Variables (3) F or S or Su
Analysis and applications of a function of a single complex variable. Analytic function
theory, path integration, Taylor and Laurent series and elementary conformal mapping are
studied. Prerequisite: MATH 2210.
MATH 4110. Modern Algebra I (3) F
Logic, sets, and the study of algebraic systems including groups, rings, and fields.
Prerequisite: MATH 2270.
Basic topics in secondary mathematics are taught to prospective teachers using a
variety of methods of presentation and up-to-date technology, including the use
of graphing calculators and computers. Prerequisite: MATH SI1220.
Aspects of teaching advanced mathematics in a high school setting, including methods of
presentation, exploration, assessment and classroom management. An emphasis is placed on
the use of computers, graphing calculators, and other technology. Prerequisite:
MTHE
3010.
MTHE SI3060. Probability and Statistics for Elementary Teachers (3) F
Basic Probability and statistics with an emphasis on topics and methods pertinent to
prospective elementary school teachers. Prerequisite: MATH 2010 and MATH 2020.
MTHE SI3070. Geometry for Elementary Teachers (3) F
Basic Geometry with an emphasis on the topics and methods pertinent to prospective
elementary school teachers. Prerequisite: MATH 2010 and MATH 2020.
MTHE SI3080. Number Theory for Elementary Teachers (3) S
MTHE 4010. Capstone Mathematics for High School Teachers I (3) S
Prospective high school teachers revisit mathematics topics from the
secondary school curriculum and examine them from an advanced perspective.
The major emphasis is on topics from algebra. Prerequisites: MATH
2110 and MATH 3120.
MTHE 4020. Capstone Mathematics for High School Teachers II (3) S
Prospective high school teachers revisit mathematics topics from the
secondary school curriculum and examine them from an advanced perspective.
The major emphasis is on topics from geometry. Prerequisite: MTHE
4010.
Topics in secondary mathematics are taught to in-service teachers using a
variety of methods and technology to make them better prepared for teaching
secondary mathematics. Expository presentations about a current mathematics
education research area are expected.
MTHE 6350. Linear Algebra (3)
MTHE 6410, 6420. Probability and Statistics (3-3)
The mathematical content of probability and statistics at the undergraduate post
calculus level. An understanding of the application of probability and statistics is also
stressed. Co-requisite: MTHE 5310 or prerequisite of MTHE 5220 and consent of
instructor. Further prerequisites: MTHE 6410 for 6420.
MTHE 6550. Introduction to Mathematical Modeling (3)
Formulation, solution and interpretation of mathematical models for problems occurring
in areas of physical, biological and social science. Prerequisite: MTHE 5310 and 5350.
MTHE 6610. Graph Theory (3)
Principles of Graph Theory including methods and models, special types of graphs, paths
and circuits, coloring, networks, and other applications. Prerequisite: MTHE 5210.
MTHE 6640. Differential Equations II (3)
MTHE 6650. Complex Variables (3)
Analysis and applications of a function of a single complex variable. Analytic function
theory, path integration, Taylor and Laurent series and elementary conformal mapping are
studied. Prerequisite: MTHE 5310 and 5350.
MTHE 6660. Modern Algebra I (3)
Logic, sets, and the study of algebraic systems including groups, rings, and fields.
Prerequisite: MTHE 5350.
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Algebra I
October 5, 2012
The Algebra
I students have been working on solving equations for particular
variables. This math skill is one
of the most important skills in mathematics. The students are working on understanding what solutions are
and how to find them. The steps
for solving these equations are a balancing act. They have to keep both sides of the equal sign balanced,
like weights on a scale. Another
aspect of solving equations is that students have the ability to check their
solutions that they get. This
ability to check their solutions increases the learning and depth of knowledge
of the content
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Cauchy's Cours d'analyse: An Annotated Translation (Sources and Studies in the History of Mathematics and Physical Sciences)
In 1821, Augustin-Louis Cauchy (1789-1857) published a textbook, the Cours d'analyse, to accompany his course in analysis at the Ecole Polytechnique. It is one of the most influential mathematics books ever written. Not only did Cauchy provide a workable definition of limits and a means to make them the basis of a rigorous theory of calculus, but he also revitalized the idea that all mathematics could be set on such rigorous foundations. Today, the quality of a work of mathematics is judged in part on the quality of its rigor, and this standard is largely due to the transformation brought about by Cauchy and the Cours d'analyse. For this translation, the authors have also added commentary, notes, references, and an index.
The 18th century was a wealth of knowledge, exploration and rapidly growing technology and expanding record-keeping made possible by advances in the printing press. In its determination to preserve ...
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In the Mathematics Area of Foundation Programme (FP) at Sultan Qaboos University (SQU), courses are offered at two levels: basic and advanced. Basic Mathematics is offered to those students who do not pass the Placement Test, while Advanced Mathematics is offered to those who do not pass the Exit Test. Furthermore, Advanced Mathematics has two separate streams: (i) Mathematics for Social Sciences and (ii) Mathematics for Sciences. Of these, "Mathematics for Social Sciences" is offered to the students admitted to the colleges of Arts & Social Sciences, Commerce & Economics, Education, and Law. The course, "Mathematics for Sciences" is offered to the colleges of Agriculture & Marine Sciences, Engineering, Medical & Health Sciences, Nursing, and Science. The medium of instruction of the courses is: Arabic for Arabic medium programmes, and English for English medium and bilingual programmes.
The objectives of the Mathematics courses are to ensure that the students are equipped with the mathematical understanding and skills necessary to meet the cognitive and practical requirements of degree programmes in a variety of disciplines.
The course learning outcomes are designed and categorized into the following three courses (the course ending with an odd integer indicates that the course is offered in English language, while the one ending with an even integer indicates that it is offered in Arabic language) :
E-Learning with Moodle: The site has the online course materials for the FP Mathematics courses. Students are advised to visit the site frequently as all course announcements will be done through Moodle. In addition to important announcements, the site has several useful materials such as practice questions, review material, summaries and plotting utilities.
Getting Help: Students are encouraged to visit the instructors and tutors who teach the course during their office hours. If the office hours are not suitable, they can be seen by appointment. The schedule of the course team office hours will be posted in Moodle.
Important Remarks:
Students are not allowed to use cellular phones in the classroom. They should turn off their phones before entering the class. Also, cellular phones are not allowed to be used as calculators or for any other purpose.
Course material is cumulative and the student may be tested on any material previously covered in the course in any quiz or test.
The final exam is comprehensive and will include all material covered in the course.
The student should write his/her name, ID number and Section number on the front page of the answer paper.
Students should read carefully the instructions given in the question paper.
No books, lecture notes, dictionaries, electronic translators, graphing and programming calculators or mobile phones will be permitted in the exam room.
Sharing of calculators, erasers, pens, pencils, etc. will not be allowed.
Academic Dishonesty
All forms of academic dishonesty is prohibited and penalties are decided based on the University rules regulations. Academic dishonesty included (but not limited to) cheating, plagiarism, copying, collusion, falsification, signing for another person, etc. For more details please see Pages 36 and 37 of SQU Undergraduate Academic Regulations, 2005.
Punctuality
Students are required to attend their classes on time. Late attendance is not acceptable. The instructor has the right to refuse admission to latecomers.
General Study Skills
General Study Skills (GSS) is an area of FP with specific learning outcomes. The relevant learning outcomes of GSS are to be embedded into each of the three FP areas. With regard to the Mathematics courses, the students should be able to satisfy the following GSS learning outcomes (OAC, 2007):
1. Work in pairs or groups and participate accordingly i.e. take turns, initiate a discussion, interrupt
10. Use a contents page and an index to locate information in a book.
11. Extract relevant information from a book or article using a battery of reading strategies (e.g. skimming, scanning, etc.).
12. Locate a book/journal in the library using the catalogue.
13. Find topic¬-related information in a book/journal in the library using the catalogue.
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Welcome to Coordinate Algebra and Coordinate Algebra with Support. 1st and 4th period are year long classes that will meet every day, and will have a state end of course test in May that will count as 20% of each student's grade for the course. 3rd period is a semester long class that will meet every day for 1 semester and take the state end of course test in December, it will also count for 20% of those student's grades. This year we are teaching a brand new curriculum that is aligned with the national Common Core standards. We will start of this week right away. I would highly suggest getting signed up for the Cornerstones for Success tutoring program. That way at if any point in the year the student needs help, they will have the appropriate paperwork filled out to participate.
Recommended Materials:
I would recommend that each student gets their own Texas instruments, TI-30xs calculator to use throughout the semester or year. The school has calculators that can be issued to students like a text book, but it is better if the student has their own. Students will also need to have some sort of notebook/folder/binder to take notes in and keep their work organized. They will also need to be prepared with a pencil every day in class.
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Department of Mathematics
Student Handbook: Learning Outcomes
All students will be appraised of the departmental learning outcomes objectives which are as follows:
Upon the completion of the core curriculum in Mathematics, the student should be able to:
1) Analyze polynomial and transcendental functions of one or more variables with respect to:
a) operations of functions, graphs, existence of inverse functions b) existence of limits c) continuity, differentiability (both explicit and implicit) and partial differentiation. d) integrability with techniques of integration e) representation of functions through infinite series f) interpretation and summary of information from graphs of functions g) be able to prove limit theorems in simple cases by using Epsilon - Delta methods
2) Demonstrate an understanding for the applications of the derivative and the integral in:
Upon the completion of the Mathematics Program, the student should be able to:
6) Demonstrate quantitative literacy:
a) be able to analyze, interpret, and present data in a logical and scientific manner. b) know basic counting methods, and basic knowledge of statistics and probability
7) Demonstrate an understanding of the principles and techniques of applying mathematics to real world problems:
a) use techniques of linear algebra and differential equations to solve various applied problems b) understand the importance and widespread existence of nonlinear problems and the role of the linear theory in developing insight into these problems c) grasp the concept of "dynamical" systems and their importance in comparison to "static" problems
8) Understand the role of the computer in mathematics by implementing and understanding the importance and limitations of algorithms for:
a) numerical methods for approximating integrals, series and numbers b) different methods for graphing continuous and discontinuous functions in two and three dimensions c) numerical methods for approximating solutions of linear systems and differential equations
9) Communicate clearly and effectively in an organized fashion the basic concepts and principles of mathematics, from calculus to modern applications and theory:
a) communicate, in both oral and written fashion, mathematical concepts and methods in a precise manner b) present historical perspectives and implications of mathematical ideas c) understand research in mathematics by actively doing research in a specific area d) analyze some application problems using modeling techniques to observe patterns, interconnections, and underlying structures
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Mathematics is a subject that entails counting, computing and calculating of numbers and at times even variables. Earlier, abacus was used by man for the purpose of mastering the skill of counting but with the passage of time more sophisticated calculators were developed. Thanks to technological advancement now there are various types of electronic calculators which are available for purchase, a percentage calculator being one of them. These calculators can be extremely handy in many situations. For instance, if you had to calculate some percentages then it is advisable to use a percent calculator.
Using the technology of a percentage calculator or any other kind of calculator for that matter has its own advantages and disadvantages. Calculators are considered as a normal tool these days, which can prove to be indispensable at times. There are two kinds of calculators: handheld calculators and online versions and an example of the latter would be the percent calculator. Online calculators are different from handheld calculators in the sense that they are far more superior because they provide a lot more functions. Some of these net calculators can even plot an equation into a graphical form.
Popular math calculators such as the percent calculator or other types of calculators are used by people from different walks of life such as technicians, students, engineers and teachers. Online calculators, including the percentage calculator, equip the user with a superior understanding of mathematical operations. These calculators assist them in the process of verifying their knowledge of mathematical formulae and theory. With the help of such a tool, they will be able to visualize a possible value of an unknown answer. Technicians and engineers rely on online calculators heavily because their line of work calls for the use of such devices.
A lot of people have prejudices against mathematics and they are just scared of what the subject entails. On the contrary, mathematics is a subject that is very logical and unless the individual understands the logic behind it, he/she would always find it hard to figure things out. Online calculators like percentage calculator can remove some of the prejudices against mathematics to a certain extent. If you are wondering how a percent calculator or any calculator can help one understand mathematics, then the answer lies in the tendency of such calculators to provide explanations to its workings.
In order to understand how such calculators can help you understand math, make use of a high quality and ultra efficient percentage calculator. This can easily be located online in various websites and you just have to ensure that the option which you have chosen provides explanation of how the answer or solution was obtained. Now use the percent calculator to solve a sum that you do not understand. Once you verify the accuracy of the answer, you can then access the explanation part and see the step-by-step instructions on how the answer was calculated.
If you combine online calculators with online self-tutor resources then you will get the ultimate "dream team" to help you combat all your math problems. Using an online percent calculator is not at all difficult – you just have to enter some information from the sum that you are looking to solve. After this, you just have to click on a mouse button and the percentage calculator would do all the hard work for you and display the answer on your computer screen. So the next time when you are having difficulty with your mathematics homework or anything related to mathematics then make use of free online calculators as these magnify the beauty of mathematics.
Given the rising popularity of the Internet it is but natural for people to shift to an online percentage calculator to assist them in their work. Amongst the many advantages of a percent calculator one of the foremost is its convenience which adds to the fun of solving
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Student placement in a mathematics course is subject to ACT-MATH scores or the COMPASS placement test scores or Academic Services Center approval. Students with ACT-MATH scores of 20 or above may enroll in any math course with numbers up to and including MATH 146 or MATH 165. Students with an ACT-MATH score below 20, or no ACT-MATH score, are required to take the COMPASS placement test. Placement of students is based on the level of achievement on the test.
ASC 090 Math Prep (2 credits)
This course improves basic math computational skills: addition, subtraction, multiplication, and division of whole numbers, fractions, and decimals. Includes a study of percents and application of percents. This course may be required due to Compass test results and the course placement policy. (F, S, Su)
ASC 091 Algebra Prep I (2)
This course is designed for students with little or no algebra background who need to prepare for further study in mathematics or who need to review basic algebra concepts. It includes topics such as real numbers, fundamental operations, variables, equations, inequalities and applications. (F, S, Su, O)
MATH 120 Basic Mathematics I (2)
A review of whole numbers, fractions and decimal numbers in conjunction with the fundamental application of ratios, rates, unit rates, proportions and percents in solving everyday problems. The application of business and consumer mathematics such as simple and compound interest, purchasing and checkbook reconciliation. (F, S, Su)
MATH 135 Applied Mathematics (2)
A review of mathematics including fractions, decimals, percentages and basic algebra which incorporates algebraic fractions and equations with variables. Emphasis is placed on the strategies of problem-solving using agricultural applications. (F)
MATH 136 Technical Trigonometry (2)
A study of the fundamentals of trigonometry. Right triangle trigonometry, the Law of Sines, the Law of Cosines and Vectors. Emphasis is placed on problem-solving for the technology fields. Prerequisite: MATH 132. (F, S, O)
MATH 138 Applied Trigonometry (3)
A theory/lab course studying the fundamentals and applications of trigonometry, including right and oblique triangles, the Law of Sines, the Law of Cosines, vectors, angular velocity, graphs and complex numbers.
MATH 146 Applied Calculus I (4)
Review of algebra, including linear, quadratic, exponential and logarithmic functions. Calculus topics for this course will be limits, continuity, rates of change, derivatives, extrema, anti-derivatives and integrals. Emphasis is placed on real-data application. Course is intended for those majoring in business, management, economics, or the life or social sciences. Prerequisite: MATH 103 or MATH 104 or placement exam. (F) ND:MATH
MATH 147 Applied Calculus II (4)
Integrals, multivariable calculus, introduction to differential equations, probability and calculus, sequences and series, introduction to trigonometric functions, derivatives and integrals of trigonometric functions. Emphasis is placed on real-data application. Course is intended for those majoring in business, management, economics or the life or social sciences. Prerequisite: MATH 146. (S) ND:MATH
MATH X92 Experimental Course (1-9)
A course designed to meet special departmental needs during new course development. It is used for one year after which time the course is assigned a different number.
MATH 299 Special Topics (1-5)
A special purpose class or activity to be used for a mathematics course in process of development, for classes occasionally scheduled to meet student needs or interests, or offered to utilize particular faculty resources. (F, S, Su)
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MATH 549: Introduction to Number Theory
Course ID
Mathematics 549
Course Title
MATH 549: Introduction to Number Theory
Credits
3
Course Description
Number theory is a branch of mathematics that involves the study of integer properties. Topics covered include factorization, prime numbers, continued fractions and congruences as well as more sophisticated tools, such as quadratic reciprocity, Diophantine equations and number theoretic functions. However, many results and open questions in number theory can be understood by those without an extensive background in mathematics. Additional topics might include Fermat's Last Theorem, twin primes, Fibonacci numbers and perfect numbers. 349/549
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MA 371
MA 371 - Real Analysis
Fundamental Concepts a) Set Theory i) Basic notions ii) Relations and functions b) The Real Number System i) Ordered fields ii) The rational numbers as an ordered field iii) The real numbers as a complete ordered field iv) Sequences of real numbers and their properties v) The Cauchy criterion
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Grade 11 Mathematics
MBF3C1:Foundations for College Mathematics, Grade 11, College Preparation
This course enables students to broaden their understanding of mathematics as a problem-solving tool in the real world. Students will extend their understanding of quadratic relations, as well as of measurement and geometry; investigate situations involving exponential growth; solve problems involving compound interest; solve financial problems connected with vehicle ownership; and develop their ability to reason by collecting, analysing, and evaluating data involving one and two variables. Students will consolidate their mathematical skills as they solve problems and communicate their thinking. Prerequisite: Foundations of Mathematics, Grade 10, Applied
MCR3U1:Functions, Grade 11, University Preparation
This course introduces the mathematical concept of the function by extending students' experiences with linear and quadratic relations. Students will investigate properties of discrete and continuous functions, including trigonometric and exponential functions; represent functions numerically, algebraically, and graphically; solve problems involving applications of functions; and develop facility in simplifying polynomial and rational expressions. Students will reason mathematically and communicate their thinking as they solve multi-step problems. Prerequisite: Principles of Mathematics, Grade 10, Academic
MCF3M1:Functions and Applications, Grade 11, University/College Preparation
This course introduces basic features of the function by extending students' experiences with quadratic relations. It focuses on quadratic, trigonometric, and exponential functions and their use in modelling real-world situations. Students will represent functions numerically, graphically, and algebraically; simplify expressions; solve equations; and solve problems relating to financial and trigonometric applications. Students will reason mathematically and communicate their thinking as they solve multi-step problems. Prerequisite: Principles of Mathematics, Grade 10, Academic, or Foundations of Mathematics, Grade 10, Applied
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Course Description: Real number system, order of operations. Algebraic problem solving, solving linear equations. Cartesian coordinate system, graphs of equations. Exponents and radicals.Factoring polynomials, solving equations by factoring.Credits not applicable toward graduation.Four Credits.
Note: This course serves as a pre-requisite for MATH 110 (College Algebra), MATH 130 (Introductory Statistics), or MATH 155 (Mathematics, A Way of Thinking). You must earn at least a "C" grade to qualify for the next course in your sequence.
(b) Students will work through pre-algebra ALEKS modules indicated as necessary.
2. Students will improve their mastery of algebraic skills.
(a) Students will take ALEKS assessment of algebra knowledge and skills.
(b) Students will work through the ALEKS modules indicated as necessary.
(c) Students will take indicated exams to demonstrate their learning.
3. Students will develop their ability to apply algebraic thinking and procedures to problem solving.
(a) Students will work through the ALEKS modules that focus on problem solving.
Course Procedures and Policies:
MATH 001: Math 001, "Introductory Algebra", is a not-for-graduation-credit course intended to prepare students for the various courses for which 001 is a pre-requisite, namely MATH 110 (College Algebra), MATH 130 (Introductory Statistics), and MATH 155 (Mathematics, A Way of Thinking). The material is essentially the first year of algebra, which would typically be taken initially in high school, which explains why this course is numbered 001, and why the 4 credits you will earn here do not count toward graduation, even though they do count toward full-time status.
Your placement score indicated that you have not mastered the material in this course, whatever the reason. Your goal here is to finally learn this material and master the necessary skills so that you can be successful in the courses you eventually need to take as part of your college program
ALEKS: ALEKS (Assessment and LEarning in Knowledge Spaces) is a web-based program designed to carefully assess what students know and what they are ready to learn, and then to methodically tutor them in the given material, in this case Introductory Algebra.
Probably the best thing about ALEKS is that it allows each student to take a course specifically designed for their needs – each student in the class will be working at their own pace and working on material they are ready to learn. The implication of this is that I will not be "lecturing" on textbook sections the way you might be used to seeing. My role as instructor here is to monitor your learning and to engage in individual tutoring as the need arises.
Another advantage to using ALEKS is that since it is web-based you can work on your course at your convenience. ALEKS will remember where you left off and will always make sure that you have shown readiness before presenting new material.
By the way, even though you will be expected to do a considerable amount of ALEKS work on your own time, it is very important to understand that it is important to DO YOUR OWN WORK! If you get someone else to do the work you will only be frustrated when ALEKS thinks you know more than you do and starts asking questions you are not ready for. Also the exams must be taken on your own so having someone work through the online material for you will not help your performance on those exams, and hence on your grade for the course.
Textbook: The textbook we will be using is published by McGraw-Hill, who also handles ALEKS for institutions of higher education. Our text has been precisely integrated with ALEKS, so that you can use your book for explanations, worked examples and practice problems as we move our way through the course material. In fact I will be using a feature called "textbook integrations" in which the material will be presented in the same order as the book covers it and quizzes will be given as you finish chapters in the text.
Number of
Absences
Points
0
+25
1
+20
2
+15
3
+10
4
+5
5
0
6 or more
-2 points each
Attendance: A major factor in learning mathematics is a regular and focused schedule of practice. Can you imagine learning to play the piano by only practicing a few minutes a week! You need to practice virtually every day, and for considerable time each day. It takes the same sort of discipline to solidly learn algebra.
My attendance policy is given in the table at the right. Because it is so important that you put in the time, I have a system that rewards regular attendance. I think that a student who has missed as many as 6 classes should seriously consider dropping the course, but as far as my grading system is concerned, I will subtract 2 points for each absence beyond 5, so a person who attends every class will earn 25 points, while a person who misses, say, 8 classes will LOSE 6 points.
In general I will not distinguish between "excused" and "unexcused" absences, although I do consider absences due to participation in a school event, such as an athletic trip or a theatrical production, to NOT be "absences". In this case, however, it is still important that you put in the extra time to catch up.
ALEKS hours
this week
Points
6 or more
+5
5 or more
+4
4 or more
+3
3 or more
+2
less than 3
-1
ALEKS Time: ALEKS keeps track of how much time you have put in
as well as how much progress you have made. I will be using your ALEKS time as part of the grading scheme, as summarized in the table at the right. Each week there will be a grade assigned based on the time you have spent working on ALEKS over the previous week.
The times INCLUDE the 3 hours plus spent in class, so that 6 hours for 5 points means you would need to work at least 3 hours outside of class to earn those points. In general college students are expected to work 2 hours outside of class for each hour in class. I have made this number a little smaller because I am trying to build in some time for studying the text book.
Some people will need more time to learn the material that others – life is not fair and some people learn things more quickly than others. I do expect each of you, however, to put in roughly 12 total hours per week working on learning the material. This does mean that some of you who are farther along than others might end up finishing the course at some point during the semester! ALEKS will tell you how far along you are and some of you will have a starting point farther along than others.
By the way, there are several "short weeks" this fall. Labor Day (9/4) week and Mid-semester break (10/20) week have only three class meetings rather than four, so these two weeks the 6-5-4-3 (hours) will become 5-4-3-2. Thanksgiving (11/23) week is very short, only 1 class meeting, on Monday; this week the numbers are 2 hrs = 5 points, 1 hours = 3 points, 1 hour = 1 point.
Exams and Quizzes: ALEKS has the ability to construct quizzes at points indicated by the instructor. Since I have integrated ALEKS with our text book I will ask it to give you a quiz on the material in each chapter. These quizzes will be 10 questions worth 2 points each, for a total of 20 points per quiz. There will be 11 such quizzes, a Review chapter and then 10 chapters covering the course material. ALEKS will keep you updated on the next deadline – I have set up a schedule which will allow you to work through the course material by the end of the semester. These quizzes may be taken at any point prior to the deadline, and may be taken twice – the higher of the two grades will count. I will also go through and look at your work so that I may give some partial credit if it is appropriate to do so.
You will also take a paper and pencil exam of my design at midterm and during finals week. I imagine that some of you will not be on schedule and this will no doubt affect your performance on these two exams, but part of success in a course is learning the material within a designated amount of time.
In fact, the final exam will be worth 200 points and will be a combination of two things – 100 points will be based on a paper-and-pencil test you will take during finals week and 100 points will be based on the percentage of ALEKS topics you show mastery of in a final assessment to be taken the last couple days of the semester.
Grading System: At present, and I want to reserve the right to make adjustments to this system as the semester wears on, I see your grade being determined by these four factors:
(5) Final Exam: 200 points possible (100 for the percent of ALEKS topics, 100 on exam)
This makes for a total of 620 points. Grades will be assigned according to the scale: A = 90% or higher, B = 80% or higher, C = 70% or higher, D = 60% or higher. You need at least a "C" grade to be allowed to advance to the next course in your sequence.
Help outside the classroom:
If you find yourself having trouble with the material PLEASE get some help! There is regular tutoring provided in the learning center, either on a scheduled or a drop-in basis, and I want to encourage you in the strongest terms to come see me if you have questions. I have regular office hours, listed at the top of the syllabus, and it is my role here to help you master the material. I'd much rather work with you and try to get you over the hurdles than have you fail in the course. So take an active role here in monitoring your learning and do something about it if you are having trouble!
Schedule: Because ALEKS allows students to work at their individual pace you will be at a variety of places in the material throughout the semester. Still, in order to pass the course and move into the subsequent course you will need to finish the material within the semester's time constraints.
It is possible that some of you will actually complete the ALEKS course before the calendar indicates the semester is over, and that's fine. I will still have you take the midterm exam on October 13 and the final exam on December 14 with the rest of the class. If you do finish early your ALEKS time and attendance points will be based on the amount of time you were working on the material.
It is also possible that some of you may reach December without completing the material. ALEKS offers a guarantee that if you put in a reasonable amount of time during the semester and do not pass the course your license to use ALEKS can be extended so that you can continue to work on finishing the course during the following semester – in this case you will be given a grade of "I" (Incomplete) so that you can work on completing the course during the next semester. Of course, this is far from ideal since it means you could not yet enroll in the course you need to take for your major, so it should be your goal to see that that does not occur.
Americans with Disability Act Statement
ALEKS
Your textbook should come with a username and password so that you can log onto ALEKS (Assessment and LEarning in Knowledge Spaces). Then to be enrolled in my specific course you need the course code, which is:
GQGG6-3KTQN
The first day of class you will each log in and we will take a look at the basics of using ALEKS. I will ask you to work your way through the tutorial so that you become familiar with how to enter mathematical expressions. Then on the second day of class I will have you take the initial ALEKS assessment to get a baseline rating of your skills and readiness for the material in this course.
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Linear Algebra II
Linear Algebra is both rich in theory and full of interesting applications; in this course the student will try to balance both. This course includes a review of topics learned in Linear Algebra I. Upon successful completion of this course, the student will be able to: Solve systems of linear equations; Define the abstract notions of vector space and inner product space; State examples of vector spaces; Diagonalize a matrix; Formulate what a system of linear equations is in terms of matrices; Give an example of a space that has the Archimedian property; Use the Euclidean algorithm to find the greatest common divisor; Understand polar form and geometric interpretation of the complex numbers; Explain what the fundamental theorem of algebra states; Determine when two matrices are row equivalent; State the Fredholm alternative; Identify matrices that are in row reduced echelon form; Find a LU factorization for a given matrix; Find a PLU factorization for a given matrix; Find a QR factorization for a given matrix; Use the simplex algorithm; Compute eigenvalues and eigenvectors; State Shur's Theorem; Define normal matrices; Explain the composition and the inversion of permutations; Define and compute the determinant; Explain when eigenvalues exist for a given operator; Normal form of a nilpotent operator; Understand the idea of Jordan blocks, Jordan matrices, and the Jordan form of a matrix; Define quadratic forms; State the second derivative test; Define eigenvectors and eigenvalues; Define a vector space and state its properties; State the notions of linear span, linear independence, and the basis of a vector space; Understand the ideas of linear independence, spanning set, basis, and dimension; Define a linear transformation; State the properties of linear transformations; Define the characteristic polynomial of a matrix; Define a Markov matrix; State what it means to have the property of being a stochastic matrix; Define a normed vector space; Apply the Cauchy Schwarz inequality; State the Riesz representation theorem; State what it means for a nxn matrix to be diagonalizable; Define Hermitian operators; Define a Hilbert space; Prove the Cayley Hamilton theorem; Define the adjoint of an operator; Define normal operators; State the spectral theorem; Understand how to find the singular-value decomposition of an operator; Define the notion of length for abstract vectors in abstract vector spaces; Define orthogonal vectors; Define orthogonal and orthonormal subsets of R^n; Use the Gram-Schmidt process; Find the eigenvalues and the eigenvectors of a given matrix numerically; Provide an explicit description of the Power Method. (Mathematics 212)Mathematics and Statistics2011-11-11T11:22:52Course Related MaterialsFoundations of Development Policy, Spring 2009
" This course explores the foundations of policy making in developing countries. The goal is to spell out various policy options and to quantify the trade-offs between them. We will study the different facets of human development: education, health, gender, the family, land relations, risk, informal and formal norms and institutions. This is an empirical class. For each topic, we will study several concrete examples chosen from around the world. While studying each of these topics, we will ask: What determines the decisions of poor households in developing countries? What constraints are they subject to? Is there a scope for policy (by government, international organizations, or non-governmental organizations (NGOs))? What policies have been tried out? Have they been successful?"Duflo, EstherBusinessSocial Sciences2010-10-07T04:39:16Course Related MaterialsBusiness organisations and their environments: culture
We know that culture guides the way people behave in society as a whole. But culture also plays a key role in organisations, which have their own unique set of values, beliefs and ways of doing business. This unit explores the concepts of national and orgBusinessSocial Sciences2009-08-13T00:25:40Course Related MaterialsTopics in Philosophy of Science: Social Science, Fall 2006
This course offers an advanced survey of current debates about the ontology, methodology, and aims of the social sciences.Haslanger, SallyHumanitiesSocial Sciences2008-01-27T10:00:48Course Related MaterialsNorms
This module will define a norm and give examples and properties of it.Justin RombergMichael HaagMathematics and StatisticsScience and Technology2007-10-30T11:42:00Course Related MaterialsNormas
Este modulo definirá una norma y da unos ejemplos y sus propiedades.Justin RombergMichael HaagScience and Technology2007-08-20T05:12:00Course Related MaterialsManagerial Psychology Laboratory, Fall 2004
Core subject for students majoring in management science. Surveys individual and social psychology and organization theory interpreted in the context of the managerial environment. Laboratory involves projects of an applied nature in behavioral science. Emphasizes use of behavioral science research methods to test hypotheses concerning organizational behavior. Instruction and practice in communication include report writing, team decision-making, and oral and visual presentation. Twelve units may be applied to the General Institute Laboratory Requirement.Ariely, DanBusiness2006-03-20T23:57:00Course Related MaterialsManagerial Psychology Laboratory, Spring 2003
Surve laboratories and its effect on communication patterns in the organization. 15.301 is a core subject for students majoring in management science. A laboratory is a required element of the course for these students. It involves projects of an applied nature in behavioral science. Emphasizes use of behavioral science research methods to test hypotheses concerning organizational behavior. Instruction and practice in communication include report writing, team decision-making, and oral and visual presentation.Allen, Thomas JohnBusiness2006-03-20T23:57:00Course Related MaterialsForms of Political Participation: Old and New, Spring 2005
How, stability and change in political regimes, the capacity of states to carry out their objectives, and international politics.Tsai, LilySocial Sciences2006-03-20T23:56:00Course Related MaterialsFoundations of Development Policy, Spring 2004
ExplDuflo, EstherSocial Sciences2006-03-20T23:47:00Course Related Materials
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Mathematics
More than mere computation, the content and methods of mathematics derive from and reside in the core of human thought itself. The discipline's lifeblood flows from problems present in spatial and numerical processes, in difficulties that arise from logic, science, applications, and social phenomena, and from problems that arise internally from the many sub-disciplines of pure and applied mathematics.
At the foundations of the discipline, mathematical researchers puzzle over problems of consistency, the nature of truth, problems of logic, and determining means for ensuring correct derivation of mathematical theorems. The skills that accrue to students engaged in these kinds of analyses, and the intellectual capacity to transfer mathematical methods to myriad applications, afford expanding life and career opportunities to well-schooled practitioners of mathematics.
The mission of the department of mathematics at Wheaton College is to prepare students to be transforming agents of Christ in a needy world beset by difficult problem. Most, if not all, of these problems require careful analysis and the application of insightful problem solving skills. Located on the main floor of Wheaton's science building, students and faculty in the department of mathematics and computer science interact with physical and biological scientists, geologists and social scientists as well as other disciplines at the intellectual cross-roads of Wheaton's curriculum to develop cross-disciplinary approaches to problem solving.
Department graduates enter graduate schools in mathematics, computer science, or related disciplines. Others undertake careers or advanced training in actuarial science, teaching, economics, business, and statistics. No matter their eventual fields of service, while at Wheaton mathematics and computer science students study and work individually and in small groups with department faculty whose professional interests include differential geometry, dynamical systems, fractal geometry and chaos theory, math modeling, computing, applied mathematics, probability and statistics, knot theory, math analysis, and modern algebra. Multiple opportunities exist for funded summer research, faculty-student mentored publishing efforts, and working in the department as a recitation or teaching assistant. In the end, department majors hone their skills for a lifetime of service for "Christ and His kingdom."
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Investigations in Geometry
CAS MA 150
Credits:
4
An immersion experience in mathematical thinking and mathematical habits of mind. Students investigate topics in Euclidean and non-Euclidean geometry starting from basic elementary material and leading to an overview of current research topics.
Note that this information may change at any time. Please visit the Student Link for the most up-to-date course information.
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"Very few books do justice to material that is suitable for both professional software engineers and graduate students. This book does just that, without losing its focus or stressing one audience over the other." Marlin Thomas, Computing Reviews
Book Description
Modern Computer Arithmetic focuses on arbitrary-precision algorithms for efficiently performing arithmetic operations such as addition, multiplication and division, and related topics such as modular arithmetic. The authors present algorithms that are ready to implement in your favourite language, while keeping a high-level description and avoiding too low-level or machine-dependent details.
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Math 101: Calculus with Problem Solving. This course provides an introduction to the two fundamental notions of calculus: the derivative and the integral. The five days a week meeting format allows for review of pre-calculus topics as needed. Admission to Math 101 is by placement via Carleton Placement Exam #1 (CP#1) only. After completion of Math 101 students may enter Math 121.
Math 111: Calculus 1. A first introduction to the calculus that develops the derivative and the integral. Designed for students with little or no previous exposure to calculus. Placement is through Carleton Placement Exam #1 (CP#1) or Carleton Placement Exam #2 (CP#2). Upon successful completion of Math 111 students may enter Math 121.
Math 211: Multivariable Calculus. Develops the calculus in two or more dimensions. Successful completion of Math 121, an AP BC score of 4 or 5, or a placement via Carleton Placement Exam #3 (CP#3) is required. Upon successful completion of Math 211 students may take Math 232.
AP Credit
Calculus AB: Score of 4 or 5: take Math 121. Score of 3: take Carleton Placement Exam #2 (CP#2) to determine which Calculus class you should enroll in. If you successfully complete Math 121 with a grade of C- or better, then you will receive 6 credits which count toward the mathematics major and graduation requirements.
Calculus BC: Score of 4 or 5: take Math 211. After successfully completing Math 211 with a grade of C- or better, you will receive 12 credits which count toward a mathematics major and graduation requirements (for Math 111 and 121).
Other Options into Mathematics Courses
Math 115 or Math 215: Introduction to statistics and probability. See the statistics section below.
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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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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...
| 677.169 | 1 |
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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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
| 677.169 | 1 |
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
| 677.169 | 1 |
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Math for Electricity and Electronics Provides a
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Mathematics For The Trades - 8th
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Practical Math Basic Mathematic Concepts Applied to
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Classic American Tech textbook presents basic mathematic concepts
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Introduction to Mathematical
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| 677.169 | 1 |
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.
| 677.169 | 1 |
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
| 677.169 | 1 |
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
| 677.169 | 1 |
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.
| 677.169 | 1 |
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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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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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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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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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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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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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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◊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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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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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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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:
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Course at a glance...
All assignments listed in one week are due by 6:00 a.m. the Monday
of the following week. Tests must be taken on the date indicated either at 8:00-8:50
a.m. or at 4:00-4:50 p.m. in room Hibbard 101.
All assignments assigned prior to a test are due PRIOR to the time the
test is taken. Any homework practice or Quiz associated with the Unit that
is NOT completed prior to taking the Test for that unit will be recorded as a
"0". Be sure to have everything completed prior to taking the test!
- Note:
If additional help is needed throughout the course,
see the tutors in the Math CARE Center in HHH 218
or check out other related videos at BrightStorm
or Kahn Academy.
Although you may find a
calculator helpful to check your answers, you will need to fine tune your
basic arithmetic skills without the aid of a calculator. Therefore,
calculator-based examples and problems are not included in the
assigned readings. Be
sure you know your Basic Multiplication Facts.
UNIT 1– be sure to list key concepts in this unit
on workbook pages 21-22
For those wanting to complete Math 10 on the "Fast
Track" and complete Math 20 prior to taking a summer class starting June 10, here is the
Combined Detailed Schedule for 10 and 20. NOTE: one must register for and pay for Math 20 as a SUMMER special course for this option. One may also take a slower pace and complete the special SUMMER Math 20 by June 28 before the last half summer classes begin or in time for Fall. Both options of these special SUMMER offerings of Math 20 MUST be completed by 6/ 28.
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MATH 50 - Pre-Algebra
(3 units)
This course covers the Fundamental principles of mathematics designed to ease
the transition from arithmetic to algebra. Concepts, computational skills,
thinking skills and problem-solving skills are balanced to build proficiency
and mastery.
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Book DescriptionMost universities recommend this for 1st year undergraduates, probably rightly. Varian is quite clear with intuitive explanations of the basic concepts. However in terms of the maths he becomes incredibly confusing, as he insists on relegating all calculus (which is vital to even 1st year micro) to appendices and using deltas (aka triangles) in the main text. This has three consequences. Firstly it makes his reasoning less rigorous. Secondly he doesn't really explain how the calculus relates to the intuitive concepts or present the mathematical steps in too much detail, so it is often difficult to follow. Thirdly you have to spend a lot of time piecing together proofs from triangles and actual partial derivatives if you want to make use of a proof for an essay or exercise. Some of my friends used the Perloff text (I think it's called microeconomic theory and applications of calculus or something like that) instead, and claimed it was better. I used Varian in first year and got a first in micro, but had to rely on my maths for economists textbook a lot. Definitely too basic for finalists.
This is a great introductory intermediate text if that makes sense. It's more involved and mathematical than the micro section in a typical introductory economics textbook (nothing a good grasp of calculus won't cope with) but less challenging mathematically than an advanced intermediate student would expect. If you've outgrown the former but not yet the latter then this is perfick! Nothing more likely to discourage studying than struggling with both concepts and calculations at the same time. With Varian, you'll get the theory right which should set you (me) up for the hard work to come.
Hal Varian's Intermediate Microeconomics was the recommended text book for my recent 2nd year micro economics module at university.
The material is presented in a very clear and straightforward manner, using minimal mathematical notation. If you want to use a book to help you understand the actual concepts of intermediate microeconomics this is the book for you. The explanation of concepts is concise and the book covers the vast majority of topics covered in a 2nd year micro course.
One word of warning; for most intermediate courses in economics in the UK this book falls short of the level of analytics likely to be required. It is however still a very useful complement to either a more technical text or lecture notes.
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In maths A Level, you have 4 core modules (C1, C2, C3, C4) and 2 applied modules (S, D, and M).
In further maths A Level, you have 2 or 3 further pure modules (FP1 and FP2 and/or FP3), and then either 3 or 4 applied modules, depending on how many further pure units you took.
There shouldn't be any mechanics in the other modules, so if you just want mechanics, I'd just stick to M1, M2 and M3.
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The videos in this application are designed to teach you what you need to know about Vectors and Vector Functions. This means that we`ll cover basics like vector magnitude, length, notation, and equations. Additionally, we`ll look at more advanced applications and vector concepts like dot products, cross products, torque, domain, limits, and problems that require you to find where lines intersect planes or find equations of planes. To learn about these concepts, we`ll cover the topic through a series of video lessons, each of which will cover pertinent ideas and related problems. The video content in this application will include a lesson on each of the following: * Vectors: Finding Magnitude or Length * Vectors: Finding Equations of Lines * Vectors: The Dot Product * The Cross Product of Two Vectors * Torque: An Application of the Cross Product * Finding Where a Line Intersects a Plane * Domain of a Vector Function * Limit of a Vector Function * Finding the Equation of a Plane Given 3 Points
This is one of several Calculus apps from me, PatrickJMT. I have been putting up math videos for a few years on YouTube and now have the most popular `math only` channel on YouTube!, Affter much encouragement and many requests from my YouTube friends, I`ve finally decided to organize the videos and put them out as an App. I`ve been teaching math for >8 years at the college/university level and tutoring for over 20 years. In the past, I have taught at Vanderbilt University (a top 20 ranked university), the University of Louisville and at Austin Community College.
The "Download" link for Vectors & Vector Functions: PatrickJMT Calculus Videos 1.1 directs you to the iTunes AppStore, where you have to continue the download process.You must have an iTunes account to download the application. This download link may not be available in some countries.
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Ray's Arithmetic Curriculum by Mott Media
Used in the 1800's, Ray's Arithmetic taught math to generations.
This set presents principles and follow up each one with examples which include difficult problems to challenge the best students. Students who do not master a concept the first time can return to it later, work the more difficult problems, and master the concepts. Thus in these compact volumes is a complete arithmetic course to study in school, to help in preparation for ACT and SAT tests, and to use for reference throughout a lifetime.
NOTE: The publisher, Mott Media, made the decision to keep prices down by switching from hardback to paperback. When each of the books in the series
is reprinted, it will be in the paperback version. At present, the Primary Arithmetic and Intellectual Arithmetic are paperbacks.
Ray's Arithmetic 8-Volume Set
Ray's Arithmetic 8 Volume Set
By Joseph Ray, Publisher: Mott Media
Included in the Ray's Arithmetic 8-Volume Set are one of each of the following books:
Key to Ray's New Higher Arithmetic
Key to Ray's Higher Arithmetic
Key to Ray's Higher Arithmetic has answers to problems in the higher book.
This key provides basic answers.
Hardback
ISBN-13: 9780880620567
List $12.99
Sale Price $11.95
Parent Teacher Guide
Parent-Teacher Guide for Ray's New Arithmetics
By Ruth Beechick, Publisher: Mott Media
The Ray's New Arithmetics Parent-Teacher Guide gives unit by unit helps for teaching; suggests grade levels for each book; provides progress chart samples for each grade and tests for each unit.
It is written by Dr. Ruth Beechick who is known for her practical and academic approach to teaching. If you want help with teaching, planning, and structuring your curriculum, then you need this guide.
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In this calculus learning exercise, students find the limit using the Limit Comparison Test and solve problems with series based on the p-series. They tell whether an equation will converge or diverge. There are 7 problems.
In this Calculus worksheet, students assess their understanding of various topics, including the derivatives of trigonometric functions, evaluating integrals, sigma notation, and convergent and divergent series. The one page interactive worksheet contains fifty-two problems. Answers are not provided.
In this college level Calculus learning exercise, students use the ratio test to determine if a series converges or diverges. The one page learning exercise contains six problems. Solutions are not provided.
Students analyze geometric series in detail. They determine convergence and sum of geometric series, identify a series that satisfies the alternating series test and utilize a graphing handheld to approximate the sum of a series.
Students investigate sequences and series numerically, graphically, and symbolically. In this sequences and series lesson, students use their Ti-89 to determine if a series is convergent. Students find the terms in a sequence and series and graph them. Students use summation notation to determine the sum of a sequence.
In this infinite series instructional activity, students use comparisons to determine convergence for improper integrals. They use the integral test for infinite series. Students state the reasons they believe a given integral is converging or diverging.
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This course will interest you if you need to create mathematical models or if you use numerical software in industry, science, commerce or research. It's concerned with the skills needed to represent real optimization problems as mathematical models, and with techniques used in numerical analysis and operational research for solving these models by computer. Explaining how and when modelling and numerical techniques can be applied, the course covers solutions of non-linear equations; systems of linear and non-linear equations and mathematical modelling; linear and integer programming; and non-linear optimization for unconstrained and constrained minimisation problems. Knowledge from Level 2 study of calculus and matrices is assumed.
Modules at Level 3 assume that you are suitably prepared for study at this level. If you want
to take a single module to satisfy your career development needs or pursue particular interests,
you don't need to start at Level 1 but you do need to have adequately prepared yourself for OU study
in some other way. Check with our Student Registration & Enquiry Service to
makeThe course is divided into three blocks of work: solutions of non-linear equations, systems of linear and non-linear equations and mathematical modelling; linear and integer programming; and non-linear optimization for unconstrained and constrained minimization problems. About a quarter of your study time will be devoted to practical work. Computer programming is not part of the course.
In the broad area of operational research, the course will enable you to formulate a real problem in mathematical terms; to recognise whether the problem can be solved numerically; to choose a suitable method; to understand the conditions required for the method to work; to evaluate the results and to estimate their accuracy and their sensitivity to changes in the data.
Optimization is a practical subject, although it is supported by a growing body of mathematical theory. Problems that require the creation of mathematical models and their numerical solutions arise in science, technology, business and economics as well as in many other fields. Creating and solving a mathematical model usually involves the following main stages:
formulation of the problem in mathematical terms: this is the creation of a mathematical model
devising a method of obtaining a numerical solution from the mathematical model
making observations of the numerical quantities relevant to the solution of the problem
calculating the solution, usually with a computer or at least with a scientific calculator
interpreting the solution in relation to the real problem
evaluating the success or failure of the mathematical model.
Many of the problems discussed in the course arise in operational research and optimization: for example, how to get the most revenue from mining china clay when there is a choice of several mines. In this example the mathematical model consists of a set of linear inequalities defining the output from each mine, the number of mines that can be worked, the correct blend of clay and the total amount of clay mined each year. The method of solving the problem uses mixed linear and integer programming; the numerical data that need to be observed include the financial implications of opening a mine, the number of mines that can be worked with the labour force, and the quality of clay from potential mines. These data will be fed into a computer, which will combine them with the chosen method of solving the equations to produce solutions consisting of outputs from each mine in each year of operation.
This course examines all the stages but concentrates on: the first stage, creating the mathematical model; the second stage, devising a method; the fourth stage, calculating numerical solutions; and the fifth stage, interpreting the solution. Each of the three blocks of work takes about ten weeks of study:
Block II Formulation and numerical solution of linear programming problems using the revised simplex method; formulation of integer programming problems and the branch and bound method of solution; sensitivity analysis.
Block III Formulation and numerical solution of unconstrained and constrained non-linear optimization problems using, among others, the DFP and BFGS methods with line searches; illustrative applications.
You will learn
Successful study of this course should enhance your skills in:
mathematical modelling
operational research
linear programming and non-linear optimization methods
the use of iterative methods in problem solving
the use of Computer Algebra Packages for problem solving.
Entry
This is a Level 3 course. Level 3 courses build on study skills and subject knowledge acquired from studies at Levels 1 and 2. They are intended only for students who have recent experience of higher education in a related subject, preferably with The Open University. You are expected to bring to the course some knowledge of:
Calculus Definition of differentiation and integration; ability to differentiate and integrate a variety of functions; Taylor's theorem with remainder; partial derivatives; understanding of continuity and convergence
You could get the necessary background from our Level 2 mathematics courses Pure mathematics (M208), or Mathematical methods and models (MST209), or the equivalent. Students are more likely to successfully complete this course if they have acquired their prerequisite knowledge through passing at least one of these recommended OU courses.
Your regional or national centre will be able to tell you where you can see reference copies, or you can buy selected materials from Open University Worldwide Ltd.
Regulations
As a student of The Open University, you should be aware of the content of the Module Regulations and the Student Regulations which are
available on our Essential documents website.
If you have a disability
YouWe recommend you access the internet at least once a week during the course to download course resources and assignments, and to keep up to date with course news.
Computing requirements
How to register
To register a place on this course return to the top of the page and use the Click to register button.
Student Reviews
"One course has to be the least enjoyable and, for me, I am afraid it was M373. The only letter ..."
Read more
"This was a thoroughly testing course that I strongly recommend as a Level 3 module for anyone doing a maths
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Mathematics
Find Mathematics Schools or Programs Near You:
Program Summary
A general program that focuses on the analysis of quantities, magnitudes, forms, and their relationships, using symbolic logic and language. Includes instruction in algebra, calculus, functional analysis, geometry, number theory, logic, topology and other mathematical specializations.
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Hist. 638: This course examines the
development of mathematics from the inventions of calculus to the supposed foundational
crisis at the turn of the 20th century. Among its topics are the differential equations
devised for mechanics and astronomy by Euler, Lagrange, and Laplace, the metric system
proposed during the French Revolution, the evolution of satisfactory foundations for
mathematical analysis from Cauchy to Weierstrass, the algebras of Galois and Boole, the
creation of non-Euclidean geometries, and Cantor's transfinite sets. Students will
explore internal controversies and the dynamics of mathematics in larger intellectual and
social settings, such as the rise to power of Russia and Prussia as well as the evolution
of two modern research-intensive universities, the Ecole polytechnique and University of
Berlin.
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(Original post by Dreamweaver)
I was in exactly the same position as you a couple of weeks ago. Chapter 2 mixed is painful. Thankfully, the actual exam questions don't seem to be too bad.
Yeah the last 2 chapters are hard to self teach. Livemaths seems really good for this. Matrices aren't as bad as Vectors (IMHO) so it might be worth starting with those although vectors do pop up in one or two of the questions. How are you finding the integration?
(Original post by JohnyTheLad)Yeah it's like WTF at the beginning. Get the edexcel FP3 book. It explains most bits well. for a unit vector, you basically write the vector out and divide it by its modulus.
Scalar dor product -> you get a number
Vector cross product -> you get a vector
Both have useful applications, i.e. finding the area of a triangle etc..
No, not just death. I want it to be locked up in permanent spiked chastity, forced to worship the feet of the many women who were forced to go through D1 and be caned and whipped eternally. And as for D2, we can have it castrated and forced to become a sissy maid.
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Students will use concepts to describe, classify, model, and predict topical phenomena, aided by inductive (experimental) and deductive (rational) methods,
including calculus.
Students will develop analytical skills appropriate to solve both symbolic and numerical problems involving quantities associated with the topical phenomena. When
obtaining a solution from calculator, spreadsheet or simulation software, students will critically evaluate the method (e.g., How valid are my assumptions?) and significance (e.g., how certain are my values and do my units check?) of.
Students will be able to describe the role physical quantities and principles play in existing environmental and technological systems. Students will be able to understand foundational concepts they can use in later classes, such as the waves and electricity & magnetism.
The mission of the Mathematical Sciences General Education component is:
to educate students in excellent problem solving skills and the quantitative analysis of Mathematics, Statistics, Physics, and Computer Science,
to challenge students to live out their faith in their vocation as they become servant leaders in society, church , and the world, and
encourages the development of
knowledge, skills, and attitudes of intellect, character, and faith that Christians use in
lives of service, leadership and reconciliation.
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Math made easier: advice from experts
Many students struggle with various kinds of math, including positive and negative number signs, fractions, factoring, graphing and word problems, instructors in the department of mathematics and statistics said.
In fall 2011, the success rate for college algebra, a core math course, was 59 percent, said Mellisa Hardeman, senior instructor in the department. The success rate dropped anther percentage point the following year, she said.
In fall 2012, 50 to 60 percent of pre-core math students had difficulties solving math problems, said Denise LeGrand, director of the Mac I math lab.
Ike McPhearson, math tutor, explained why students may have trouble comprehending math. One reason is that students may come from a home where education is not valued, he said.
A bad experience with an instructor can also change students' attitudes about math.
"You can't take yourself too seriously as a teacher," said Hardeman. Instructors can never give a student too much help passing math, she said.
Students who took a math course in high school before going to college are less likely to struggle with math, Hardeman said. Some students go to college years after graduating high school, however, and may forget everything they learned in their math classes.
Fortunately, there are a number of strategies that can help students overcome these challenges and develop a better understanding of math.
"In order to make math easy for students, show different ways of how to understand it," said McPherson, who has tutored high school and college students. Another way of making math fun for students is to create different games, he said.
According to LeGrand, the most important way to become better at math is to practice math exercises for 20 to 30 minutes.
"They won't see the results right away," said LeGrand, " but if they go to class and focus on work required, they will be successful and they will build confidence."
In addition, students can get help from tutors at the math lab. Each semester, the lab hires 12 tutors, LeGrand said.
For the math-impaired, there is a new math course called Quantitative and Mathematical Reasoning. The course was designed for students who are not science, technology, engineering or mathematics majors. It focuses on practical math, for example, currency exchange rates. The course fulfills the core math requirement, in place of college algebra.
Pre-core math courses, developmental math courses students take if they do not have the prerequisites for college math classes, are becoming more successful, said Tracy Watson, coordinator for pre-core math. The success rate for those courses rose to 77 percent in fall 2012, she said. Previously, the success rate was 37 percent for a 4-year period, she said.
This semester, there are 80 math majors at the university.
"We all like how math works because it all fits together," Watson said.
"Students who major in math develop a sense of thinking and solving problems," said Thomas McMillan, department chair.
Once students better understand math, they will have the confidence to solve not only math problems, but problems in everyday life as well
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Introductory Introductory Algebra or Beginning Algebra. The engaging Martin-Gay workbook series presents a student-friendly approach to the concepts of basic math and algebra, giving students ample opportunity to practice skills and see how those skills relate to both their lives and the real world. The goals of the worktexts are to build confidence, increase motivation, and encourage mastery of basic skills and concepts. Martin-Gay ensures that students have the most up-to-date, relevant text preparation for their next math course; enhances ... MOREstudents' perception of math by exposing them to real-life situations through graphs and applications; and ensures that students have an organized, integrated learning system at their fingertips. The integrated learning resources program features text-specific supplements including Martin-Gay's acclaimed tutorial videotapes, CD videos, and MathPro 5. Introductory Algebra is typically a 1-semester course that provides a solid foundation in algebraic skills and reasoning for students who have little or no previous experience with the topic.& The goal is to effectively prepare students to transition into Intermediate Algebra.
R. Prealgebra Preview.
Factors and the Least Common Multiple. Fractions. Decimals and Percents.
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Libros de: ECONOMIA CUANTITATIVA
Learn the science of collecting information to make effective decisions Everyday decisions are made without the benefit of accurate information. Optimal Learning develops the needed principles for gathering information to make decisions, especially when collecting ...
Hirsch, Devaney, and Smale's classic "Differential Equations, Dynamical Systems, and an Introduction to Chaos" has been used by professors as the primary text for undergraduate and graduate level courses covering differential equations. It provides a ...
Volume II is devoted to generalized linear mixed models for binary, categorical, count, and survival outcomes. The second volume has seven chapters also organized in four parts. The first three parts in volume II cover ...
Although there are currently a wide variety of software packages suitable for the modern statistician, R has the triple advantage of being comprehensive, widespread, and free. Published in 2008, the second edition of Statistiques avec ...
Packed with more than a hundred color illustrations and a wide variety of puzzles and brainteasers, Taking Sudoku Seriously uses this popular craze as the starting point for a fun-filled introduction to higher mathematics. How ...
Making good decisions under conditions of uncertainty - which is the norm - requires a sound appreciation of the way random chance works. As analysis and modelling of most aspects of the world, and all ...
Designed specifically for business, economics, or life/social sciences majors, "Calculus: An Applied Approach, 9E, International Edition" motivates students while fostering understanding and mastery. The book emphasizes integrated and engaging applications that show students the real-world ...
This book introduces in a systematic manner a general nonparametric theory of statistics on manifolds, with emphasis on manifolds of shapes. The theory has important and varied applications in medical diagnostics, image analysis, and machine ...
Graphical models in their modern form have been around since the late 1970s and appear today in many areas of the sciences. Along with the ongoing developments of graphical models, a number of different graphical ...
This book provides analysis of stochastic processes from a Bayesian perspective with coverage of the main classes of stochastic processing, including modeling, computational, inference, prediction, decision-making and important applied models based on stochastic processes. In ...
Business Statistics: First European Edition provides readers with in-depth information on business, management and economics. It includes robust and algorithmic testbanks, high quality PowerPoint slides and electronic versions of statistical tables. Furthermore, the text features ...
The aim of this book is to facilitate the use of Stokes' Theorem in applications. The text takes a differential geometric point of view and provides for the student a bridge between pure and applied ...
This book provides a comprehensive description of the state-of-the-art in modelling global and national economies. It introduces the long-run structural approach to modelling that can be readily adopted for use in understanding how economies work,
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MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications 4th Edition walks readers through the ins and outs of this powerful software for technical computing. The first chapter describes basic features of the program and shows how to use it in simple arithmetic operations with scalars. The next two chapters focus on the topic of arrays (the basis of MATLAB), while the remaining text covers a wide range of other applications. MATLAB: An Introduction with Applications 4th Edition is presented gradually and in great detail, generously illustrated through computer screen shots and step-by-step tutorials, and applied in problems in mathematics, science, and engineering.
This training guide introduces development practitioners, policy analysts, and students to social accounting matrices (SAMs) and their use in policy analysis. There are already a number of books that ...
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Algebra 2 Polynomials review was created using smart notebook software. This review covers simplifying using rules of exponents, adding polynomials, subtracting polynomials, multiplying polynomials, dividing polynomials by a monomial and polynomials by binomials, and expanding binomials. I use this review with student response boards for my students to work out the problem and show me their answers. This review is a great way for my students as well as for myself to see how ready they are for the assessment. Elizabeth Welch
NOTEBOOK (SMARTboard) File
Be sure that you have an application to open this file type before downloading and/or purchasing.
164.81
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Quick Graph: Your Scientific Graphing Calculator
It is a powerful, high quality, graphic calculator that takes full advantage of the multitouch display and the powerful graphic capabilities of the iPad and iPhone, both in 2D and 3D. A simple, yet intuitive interface that makes it easy to enter and/or edit equations and visualize them in mathematical notation.
It's capable of displaying explicit and implicit (opt) equations as well as inequalities (opt) in both 2D and 3D, in all standard coordinate systems: cartesian, polar, spherical and cylindrical, all with amazing speed and beautiful results, which can be copied, emailed or saved to the photo library.
"It's ok to write yet another graphing app, so long as it is the best one. And this is"
-- Review by RightyC1
Please keep in mind that in this version, you now have to specify y=, x=, z= and so on, whenever you want to plot an equation.
The advanced feature set gives you access to some of the new features, such as implicit graphs and tracing. You need to specify the dependent variable now, since just typing "x^2" without the y=, will assume the expression to be "x^2=0" and will try to plot it as an implicit graph.
Up to 6 equations can be visualized simultaneously, in both 2D and 3D modes, this limitation can be removed by purchasing the advanced feature set. All the features from the original application are present and will remain free.
It also includes an evaluate feature, to evaluate equations at specific points, as well as a library where you can store commonly used equations.
Last year our school district was fortunate to receive a $12,000 grant to fund a handheld computing project. We chose the iPod touch (a.k.a., "iTouch") as our handheld solution for a variety of reasons. The iTouch is fast and portable. The students stay on task because we can control the apps they are using. There are apps available in all subject areas that focus on specific classroom objectives. We purchased 18 iTouch units for our high school and 30 units for our middle school, grades 5-8. They were implemented as "portable labs" (15 to 20 units in a small bag that can be used by any of the classroom teachers). With many different teachers using the iTouch sets, we learned a lot about using them in the classroom.
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The course syllabus is a general plan for the course; deviations announced
to the class by the instructor may be necessary.
Text:Mathematics
for Elementary Teachers , first edition, and the accompanying Class Activities
manual by Sybilla Beckmann, published by Addison-Wesley. These can be purchased
from the UGA bookstore and other bookstores. Please
bring the activity manual to class.
Course topics: Visualization.
Angles. Geometric shapes and their properties. Constructions with straightedge
and compass. Transformation geometry: reflections, translations, rotations.
Symmetry. Congruence. Similarity. Measurement, especially length, area, and
volume. Converting measurements. Principles underlying calculations of areas
and volumes. Why various area and volume formulas are valid. Area versus perimeter.
The behavior of area and volume under scaling.
Course objectives:
To strengthen and deepen knowledge and understanding of measurement and basic
geometry and how they are used to solve a wide variety of problems. In particular,
to strengthen the understanding of and the ability to explain why various procedures
and formulas in mathematics work. To strengthen the ability to communicate clearly
about mathematics, both orally and in writing. To promote the exploration and
explanation of mathematical phenomena. To show that many problems can be solved
in a variety of ways.
Class work:
This class is part of your preparation as a professional. As a professional,
you should engage in collegial discussions about professional practice and you
should constantly seek to enhance and refine your professional knowledge. To
receive a full participation score, your work in class must consistently exhibit
several or all of the following:
interest in mathematical ideas
interest in different ways of approaching mathematical ideas
careful listening to different ways of solving a problem
careful evaluation of proposed methods of solution
attempts to connect the course material to your experiences with children
and teachers at schools
There will be regular homework assignments. I encourage you to work on homework assignments with your classmates. Of course,
you should always write your homework up on your own, using your own words to
express the ideas you have discussed with others. Do not allow anyone to copy
your work. When you discuss assignments with others, all partners should "give
and take" ideas.
Late homework will not be accepted.
Please consult with me as soon as possible if you are unable to hand in an assignment
due to an illness or emergency.
Writing Intensive Program:
This section of MATH 5030 is part of the Writing
Intensive Program. The Writing Intensive Program is designed to
help courses teach the writing process within various disciplines. Although
you have taken English courses on writing, and although these courses will help
you with all your writing, mathematical writing has its own special features.
In mathematics, we seek coherent, logical explanations, in which the
desired conclusion is deduced from starting assumptions. Our graduate assistant,
Peter Petrov, has been trained by the Writing Intensive Program to help you
learn to write good mathematical explanations. By participating in the Writing
Intensive Program we have also learned about ways to use writing to deepen your
understanding of the course concepts.
How your grade will be calculated:
We will grade all your work on a 5.25 point scale, and we will assign points
as follows:
# of points
description
characteristics
5.25 points
exemplary
work that could serve as a model for other students
5 points
very good
correct work that is careful and thorough
4 points
competent
good, solid work that is largely correct
3 points
basic
work that has merit but also has significant shortcomings
2 points
emerging
work that shows effort but is seriously flawed
0 points
no credit
no work submitted, or no serious effort shown
Grading criteria: We will determine your score on assignments
and tests by the extent to which your work meets the following criteria:
The work is factually correct, or nearly so, with only minor, inconsequential
flaws.
The work addresses the specific question or problem that was posed. It is
focused, detailed, and precise. Key points are emphasized. There are no irrelevant
or distracting points.
The work could be used to teach a student: either a child or another college
student, whichever is most appropriate.
The work is clear, convincing, and logical. An explanation should be convincing
to a skeptic and should not require the reader to make a leap of faith.
Clear, complete sentences are used. Mathematical terms and symbols are used
correctly. If applicable, supporting pictures, diagrams, and/or equations
are used appropriately and as needed.
The work is coherent.
Your grade will be based on tests, homework, and a comprehensive final exam. I expect to give 2 tests and 2 announced
quizes during the semester. I will calculate your course score using the following
percentages.
term tests, 20% each
40%
quizzes, 7% each
14%
class participation (please see criteria above under class work)
3%
homework
15%
final exam
28%
Makeup exams or quizzes will not be given. If an exam or quiz is missed due
to an illness or emergency, I will calculate a grade for the exam or quiz using
a relevant portion of the final exam.
I expect to assign letter grades as follows.
for scores from
up to
letter grade
4.6
5.25
A
4
4.6
B
3.5
4
C
2.5
3.5
D
below 2.5
F
Materials needed: Please
have a calculator available. Please bring your activity manual to class. You
may wish to have colored pencils or markers on hand since we will frequently
solve problems with the aid of pictures.
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Concrete Mathematics A Foundation for Computer Science
9780201558029
ISBN:
0201558025
Edition: 2 Pub Date: 1994 Publisher: Addison-Wesley
Summary: This book introduces the mathematics that supports advanced computer programming and the analysis of algorithms. The primary aim of its well-known authors is to provide a solid and relevant base of mathematical skills - the skills needed to solve complex problems, to evaluate horrendous sums, and to discover subtle patterns in data. It is an indispensable text and reference not only for computer scientists - the auth...ors themselves rely heavily on it! - but for serious users of mathematics in virtually every discipline. Concrete Mathematics is a blending of CONtinuous and disCRETE mathematics. "More concretely," the authors explain, "it is the controlled manipulation of mathematical formulas, using a collection of techniques for solving problems." The subject matter is primarily an expansion of the Mathematical Preliminaries section in Knuth's classic Art of Computer Programming, but the style of presentation is more leisurely, and individual topics are covered more deeply. Several new topics have been added, and the most significant ideas have been traced to their historical roots. The book includes more than 500 exercises, divided into six categories. Complete answers are provided for all exercises, except research problems, making the book particularly valuable for self-study. Major topics include: Sums Recurrences Integer functions Elementary number theory Binomial coefficients Generating functions Discrete probability Asymptotic methods This second edition includes important new material about mechanical summation. In response to the widespread use of the first edition as a reference book, the bibliography and index have also been expanded, and additional nontrivial improvements can be found on almost every page. Readers will appreciate the informal style of Concrete Mathematics. Particularly enjoyable are the marginal graffiti contributed by students who have taken courses based on this material. The authors want to convey not only the importance of the techniques presented, but some of the fun in learning and using them. 0201558025B04062001
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In this resource from the DfE Standards Unit, students will learn to: use past examination papers creatively, explore, identify, and use pattern and symmetry in algebraic
express the nth term algebraically. (GCSE Grades A - F
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Instead of memorizing formulas and equations, Videotext Algebra helps students to understand math through mastery learning, encouraging them to solidify each concept before moving on the next. A copy of the print materials needed for this module is included.
Module C in Videotext Algebra, this unit covers:
Solution Sets (Equations, Inequalities, Graphing Terms, Intercepts)
Special Cases (Absolute Value)
Relations from Solutions (Given Slope & intercept, Given Slope & One Solution, Given two solutions, special cases)
This is a wonderful product! My child struggled with Algebra & this was the answer. They explain things very well & the best part is that he can do it on his own. It is very exspensive but worth every penny. I can't wait to use it with the rest of my children.
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COURSE DESCRIPTION:
This course is designed to prepare students to use advanced algebraic skills and concepts in mathematics and other related disciplines. It includes a study of linear, quadratic, polynomial, trigonometric and logarithmic functions. A graphing calculator is required.
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Doing the Homework
So how do you actually solve homework problems?
It may help to know how the mind solves problems.
You may be used to problems that you can solve at once, and you may
imagine that all problems are solved this way: if you can't do a problem
at once, you cannot do it at all.
Problems in standardized exams are like this: a typical standardized exam
problem is a simple test of one thing.
But there are other kinds of problems, problems that are composed of several
different pieces.
To solve such a problem, you have to
analyze
it ("analyze" means literally, to take apart), and that takes time,
energy, and sometimes creativity.
One difficulty is that the text should be a model, and the text contains
such smooth and polished solutions and proofs -- compared to what students
can produce -- that students may wonder if there is something special
about the mathematicians who originally proved the theorems.
But in fact, the mathematicians who originally developed the theorems did
not come up with anything looking like what is in the textbook.
For example ...
First, there are some obscure results that seem to talk about esoteric
things, like the assortment of results about lengths of lines and
centers of mass by Gilles Personne de Roberval, Gregory St. Vincent,
Evangelista Torricelli that we now say, with hindsight, led to the
Fundamental Theorem of the Calculus.
Frequently, it isn't clear at the time where (if anywhere) these results
are going.
Then comes the one or two or more mathematicians who come up with
something that we call the Fundamental Theorem.
James Gregory didn't seem to know the significance of his result,
which is similar to that of
Isaac Barrow, who did.
Of course, this theorem seems very strange to us today, since it is
constructed out of classical geometric notions.
Then comes the one or more mathematicians who the popular books will
say made the great accomplishment.
For example,
Isaac Newton
(who was very conscious of his public image and cultivated a reputation
for "genius") developed a system of mathematical results, including
a version of the Fundamental Theorem of the Calculus; Newton, who was
Barrow's student, is generally regarded as the inventer of the calculus.
Frequently, the great accomplishment doesn't work right, or is
unintelligible, or has errors, or, like Newton's Calculus, all of the
above.
Sometimes we need people, like
Gottfried Liebniz,
simply to make sense out of the invention.
In Liebniz's case, he had to figure out what a function is (this
after Newton was integrating and differentiating functions -- or
at least that is what Newton said he was doing when he was dividing
zero by zero(!?)).
Okay, so people understand what the invention is about, but one is not
supposed to divide zero by zero, so now comes the long train of people
who work out ways to finesse the mess
(
André Marie Ampère,
who came up with one of the major early versions of the Mean Value
Theorem),
to clean up the mess
(
Augustin Louis Cauchy,
and his limits, although some limit-like things were already around),
and even clean up the mess made by people who cleaned up the mess
(
Karl Theodor Wilhelm Weierstrass,
whose epsilons and deltas made the notion of the limit more precise
but --- more work ahead, folks! --- less intelligible), which can
lead some people to think that there must be a better way
(
Abraham Robinson
who used mathematical logic to circumvent all those nasty limits
... but at what cost?).
And of course, then there are the people who write the texts: in the
case of the Calculus, starting with
Jean Le Rond d'Alembert,
and his Cours d'Analyse.
But that is not it, for the first text really ... well, we can and
should do better.
So after two centuries of successive textbook writers figuring out how
to do things better than their competitors, we have the highly staged
productions you can now buy for outrageous prices.
Meanwhile, the research never ends, as mathematicians develop increasingly
powerful versions of the Fundamental Theorem, from Green's Theorem to
Stokes' Theorem to the recent
index theorem
of Michael Atiyah and Isadore Singer, for which they won the 2004
Abel Prize, the highest award in mathematics.
(For more on the fundamental theorem, see
the Thomas Calculus page on the history of the fundamental theorem.)
This is quite typical of mathematics.
For an entire book about how just one formula took a hundred
years to clean up --- assuming it is now cleaned up --- see Proofs
and Refutations by
Imre Lakatos.
So when something first appears, it is long and complicated, and we can see
the huge amount of work involved.
Here are two twentieth century examples.
Albert Einstein
spent seven years developing his theory of General Relativity.
This meant working on several interconnected complicated problems.
Andrew Wiles
spent seven years constructing a proof of Fermat's Last Theorem, a
deceptively simple-looking problem (show that there is no tuple of
four positive integers x, y, z, n,
with n > 2, such that xn + yn
= zn).
Both of these are relatively new results, and no doubt we will find easier
ways to present and prove them in the century ahead.
So it is true that even mathematicians find mathematics to be hard.
Sometimes it takes a lot of time and effort to solve a problem that looks
simple.
Nothing great was ever accomplished without a lot of
work.
To analyze a complicated or otherwise hard problem, one tries to take it
apart, concentrate on it (or parts of it, or problems similar to it),
return to it (because you probably won't get it all at once), explore it
from a variety of perspectives, etc.
That is what one should do consciously, with the goal of getting the
Unconscious, the great machine in the largely unseen depths of your
mind, to do some work.
For it is the Unconscious that solves hard problems.
To get your Unconscious to solve a problem, you must do the following:
Keep badgering the Unconscious by returning to the problem and spending
time on it.
The Unconscious is somewhat lazy, and gauges how important things are by
the amount of time and energy the Conscious invests on it: the way to
get the Unconscious to work on something is to repeatedly work on it
Consciously.
Work on the correct problem.
There is a difference between
working on a problem and worrying about it.
When working on the problem, one is conscious of the problem, and aspects
and approaches to the problem, and if that is what one concentrates on,
that is what the Unconscious will try to deal with.
But if one spends the time worrying about the problem, obsessing over one's
inability to solve it quickly, wondering what will happen if the problem
doesn't get solved, all that will do is get the Unconscious to think up
dire consequences of not solving the problem --- and thinking up dire
consequences is one of those things that the Unconscious seems to like to
do.
Getting your Unconscious to work for you rather than against you requires
being in the right
state of Consciousness,
and having the right
attitude.
So work on the problem, but avoid worrying about it.
Try approaching it from various angles.
There is not necessarily one right way to do it, so think up examples,
find analogies and related problems, try to find different ways of looking
at it, and in general, find additional material for the Unconscious to use.
And don't be discouraged by ideas that the Unconscious turns up that turn out
to be wrong.
One of the great secrets is that most inspirations are useless: useful
inspirations are remembered because they are so hard won, and come
after a long train of flops.
For more, see the page on
the Unconscious.
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Math Course 3 - Geometric Shapes v2.0
Math Course 3, Geometric Shapes: Step by Step 2.0 addresses the properties of common geometric shapes including triangles, quadrilaterals, circles, cubes, prisms, and cones. Common geometric theorems are explained and proofs are provided. Concepts of symmetry, and transformation of shapes by translations, rotations, reflections, and magnifications, and uses of transformations in map scales and tessellations are also covered. The Geometric Shapes: Step by Step Course has 38 Lessons organized in 8 Units.
Math Course 3 - Geometric Shapes Help Guide
The Help Guide provides guidance on achieving the 36 learning outcomes contained in the 8 Units of the Geometric Shapes: Step by Step Course. The Help Guide supports and augments the course textbook and workbook. Intended as an instructional aid for use by parent, tutor or teacher.
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Clicking the above tags will only show resources created after July 2009. For a full tag cloud click here.
Linear and Algebraic Functions Unit
Unit Overview This is an introductory unit for 7th graders in functions and algebra. Students Students will work in teams to discuss their findings through a classroom wiki and decide where they want to concentrate their efforts. The teams will present their findings to an authentic audience and must choose between Parent Teacher Group, School Board, Select Board, Vermont Legislature or the creation of a website to reach a more global audience. To prepare students to use data collected and analyzed for this unit, students will learn the importance of algebraic functions. Students will collect, discover and display data for linear and non-linear relationships. They will do several concrete experiments involving two variables and record how their values change in relationship to each other. They will also research similar experiments on the internet and compare this additional data to original data.
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Math in the News
Algebraic Modeling in Life Sciences Has Its Proponents
They note that because researchers and scientists often lack enough information to build quantitative models, algebraic models can fill temporary knowledge gaps. The advantage, they say, is that when more becomes known, the math behind algebraic models can be used to construct kinetic models. Another plus is that algebraic models are more intuitive than differential-equation models, which makes them more accessible to life scientists.
Discrete-time algebraic models, created from finite-state variables such as Boolean networks, are now used to model a variety of biochemical networks, including metabolic, gene regulatory, and signal transduction networks, according to the duo.
"The exciting thing about algebraic models from an educational perspective is that they highlight aspects of modern-day biology and can easily fit in both the biology and mathematics curricula," observed Robeva. They offer a quick way for introducing biology students to constructing and using mathematical models in the context of contemporary problems, she said. "As educators, we should actively be looking for the best ways to seize this opportunity for advancing mathematical biology."
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Technology and Online Tools – Cometdocs Blog
Anyone who works in the field of science or is studying science will need to perform various calculations on a daily basis. Just about every self-respecting scientist has a scientific calculator of their choice or some type of applications that they prefer, but these calculating devices are not always handy. For example, sometimes you might have to do calculations when you are out of the office or you simply don't have these tools at your disposal.
In situations such as this, web applications can come in really handy. Here is a list of five fantastic web applications that provide access to great online scientific calculators that all engineers, chemists, mathematicians and statisticians can use on the go.
1. Google
Google is an excellent tool for performing simple calculations. All you need to do is type the word "calculator" in the search box. A calculator will appear, allowing you to perform just about any calculation you need, including more sophisticated functions like working with logarithms, trigonometric functions and more.
If Google's calculator is not advanced enough for you, you should check out Wolphram Alpha, the knowledge search engine. This tool is pretty amazing. Type in an equation and you'll get a solution immediately. Or just paste a function into the search engine and you'll be presented with its graphical representation, properties and alternative definitions, all in a matter of seconds.
Eeweb calcultor is a free tool developed by and for the electrical engineering community. It is pretty simple to use and will probably fulfill all of your calculating needs. In the right-hand box, you'll be presented with a list of all scientific constants, which you can immediately use in your calculations. Also, you can perform on-the-spot conversions between various metrics used for length, speed, force, mass and much more. There is also an equation-solver which is pretty intuitive and fast.
On first look, the Web 2.0 calc does not look like it can do much, but check the bottom menu to see all of its calculating powers. It is an excellent tool that enables you to calculate a matrix, solve equations, and perform calculations with fractions.
Graphcalculator is a pretty simple tool. It allows you to perform an analysis of functions in the coordinate system. Enter functions and they will be visually represented with the ability to calculate plot and intersection points as well.
Microsoft PowerPoint is without a doubt the industry leader when it comes to programs for creating presentations and slideshows on your computer. It remains the industry standard from presentations because it allows you to easily create not only professional looking presentations, but also very versatile ones. PowerPoint definitely excels in giving users many options when creating their presentations.
Of course, most know that you can add text, pictures and videos to presentations, but not too many people are aware of the fact that you can also add sounds and music to your presentations in PowerPoint very easily. Here's how to do it in PowerPoint 2010.
1. Considering that you have opened up an existing presentation that you would like to insert some sounds into, start by clicking on the Insert tab, which is on the top toolbar in PowerPoint. Then go to "Audio" and click on "Audio from File…"
You can also choose "Clip Art Audio," which offer you various sound effects that you might want to use in your presentation, or you can choose to "Record Audio" and put down a script of your own if you have a microphone at your disposal, but for the sake of this tutorial, we will be using an audio file that is already located on your computer.
2. Once you click on "Audio from File," find the audio file that you want to use, and then click on "Insert."
3. You will now notice a sound icon on your presentation. It is a little more advanced than the sound icon that was given in previous versions of PowerPoint, and it gives you more options. Basically, it looks just like the face of your car stereo – there is a play/pause button, rewind, fast-forward, and a mute button if you want to turn the sound off. Feel free to test your sound out.
4. If you want to edit your sounds further and customize the sounds, go to the "Playback" tab that is located under "Audio Tools." With these options you can do a variety of things. You can edit the duration of your sound, fade it in or out and decrease or increase its volume.
You can also decide whether you want your sound to play just while this particular slide is being shown or throughout the entire presentation. You can also set the sound to loop once it's done or just stop once it's ended, and last but not least, you can also hide the sound icon so that it does not appear on the slideshow when you are viewing it in your presentation.
And that is pretty much all there is to it. By adding sounds and music to your PowerPoint presentation, you are giving your presentation a unique character and adding yet another feature that is aimed toward grabbing the attention of your audience and keeping it for the duration of your slideshow.
daily basis in their profession.
Often people who work in design get plans or ideas for new projects in PDF, since it is one of the most versatile and popular formats for sending all types of documents. Thankfully, there are online tool available that enable you to swiftly and effectively covert PDFs to AutoCAD-compatible formats like DWG and DXF. These are the three best tools for performing such a conversion, and best of all, every one of them is completely online-based and free.
1. PDF to AutoCAD was definitely one of the first online services that offered this kind of conversion, and for free no less. In order to turn your PDFs into AutoCAD compatible DWG or DXF formats, all you need to do is follow these two simple steps:
Find and upload your PDF by clicking the Browse button. If you happen to accidentally upload the wrong PDF, this tool offers you the option to choose another PDF without having to refresh the page. Once you upload your PDF, enter your email address and click Send. After a couple of minutes, you will receive an email with a link to the page where you can download your drawing.
2. Cometdocs, as you know, is our very popular online file conversion service. It converts between many different file formats, and of course, it does PDF to DWG and DFX as well. Because the converter performs a lot of different conversions, there is one extra step involved.
First you select the PDF you want to convert. Then you select the conversion type you want (PDF to DWG or DFX in this case). Next, you enter the email where you want your converted file to be sent, and then you click send to begin the conversion process. A link to your converted file should arrive to your email shortly after.
3. PDF to DXF allows users to transform their PDF drawings into an AutoCAD-compatible DFX file format. The service is, like the two above, completely free and boasts great accuracy. When it comes to conversion process, it works just like PDF to AutoCAD.
All three of these great online tool are not only free, but also give you fast conversions and deliver quality and accurate DWG and DFX files right to your personal email in a matter of minutes.
Microsoft Excel is one of the most powerful software solutions for manipulating, analyzing, calculating and presenting data. Its power lies in the fact that you can basically turn raw data into powerful information using one of the many available Excel functions, visualization tools, and options for data manipulation and calculation. In short, we can perform a variety of functions with data in Excel and use it to make important business decisions.
A lot of important decisions in business are made based on information found on Excel charts and tables. We can confidently say that Excel is an industry leader in spreadsheet analysis. It is used for calculating budgets, creating reports, presenting charts and diagrams, managing projects, calculating mortgage and much, much more.
However, one downside is that Excel is pretty complex tool and you will probably need month to learn how to start using it professionally. It applications are so plentiful and complex that no matter how good you are at Excel, there is also room to get better and always something new to learn.
If you don't have the money for Excel tutorials in the form of books, lecture or courses, there are some resources for learning Excel that can be used for free. Here is a fairly extensive list of some of the best Excel blogs and YouTube channels you can follow to learn more about this amazing software.
YouTube Channels
Sometimes the best way to learn and understand something is to see an example being performed in front of you. That is why videos are such a powerful tool for learning something new. That is why we suggest that you head to YouTube and search to find videos demonstrating Excel functions that you might not necessarily understand completely. The chances that you will find what you are looking for are pretty good. You can also use YouTube to discover the underbelly of Excel and uncover tricks and hacks that might not even be taught in official Excel books and tutorials.
You can find a large collection of Excel tricks and hacks at this channel.
Blogs and Websites for learning Excel
If you have a specific problem with Excel, you can post questions on the official Microsoft Excel forum. However, if you want to learn new stuff for free, the best way to do it is to follow blogs.
Blogs provide a great way for learning about Excel because they are updated regularly and allow readers and authors to interact better than static websites. Here is a list of some of the best blogs for learning Excel. Some of them are for advanced users, so don't get discouraged if you don't understand a thing. Dig deeper to find more basic articles and tips. There is useful information for everyone.
The Chandoo website contains lots of tutorials for learning Excel. There are tutorials for both beginners and advanced users. Also, there is a forum where users can ask and answer Excel related questions.
This website is a great resource for everyone learning Excel. In the tips section, you can find tips about almost any aspect of Excel. There is also a section with useful templates and an interesting blog which is regularly updated with new features and tutorials.
When you get to this website, at first you might be confused because of you are seeing many different sections, but once you spend some time here, you will be able to navigate easily through the many tutorials and tips that the blog has to offer.
This website is not actually a tutorial blog, but it offers a collection of the most interesting tweets on the topic of Excel. If you are having a hard time learning complicated formulas and tables, this website will certainly make you feel better, because you'll realize that you are not alone.
Sometimes we want to share a certain song or an mp3 file on our blog with our readers. We could put a simple download link, but that is not always a good choice because of copyright issues. Also, users would have to download the song to their computers in order to listen to it if there is no integrated player for streaming songs directly on the blog.
Some users solve this issue by creating a video for the song, uploading it to YouTube, and then embedding the YouTube video on the blog. However, that is a pretty complicated way to do it, and there are much easier methods out there. Here are several excellent services that you can use to embed songs onto your website or blog.
SoundCloud is a popular community for musicians and music lovers alike. Users can upload their own music creations and listen to the music of others. It is widely used and attracts a lot of traffic daily. One of its many benefits is an option for sharing a song from SoundCloud onto social media sites and blogs.
Here is how you can embed a song from SoundCloud onto your blog:
First, you need to upload your song on SoundCloud, or if you want to share someone else's music, you can do that as well. There are really plenty of songs to choose from, and some of them are public domain. Go to the song, and click on "Share." You will now see options to share it on various social media sites like Tumblr, Facebook, Twitter and more. As you see, at the bottom there is an option to use an embed code and insert a song directly into the HTML code of our blog.
However, that is not all. Click on the edit widget, and there you will see what the embedded player will look like. You can choose your own special colors or enable comments as well.
Grooveshark is another very popular music-related website. You can find almost any artist or song here and listen for free. Additionally, you can upload your own songs as well if you can't find what you are looking for on the site.
When you find a song you want to share, click "Share – Embed" as you see in the picture. You will be given an HTML code which you will be able to insert directly into the HTML section of your website.
Choose from different templates and adjust the widget size as well. Before you copy the code, you can preview the player and see how it will look on your blog.
DivShare is a sharing and storage site where you can keep files for free. It is similar to sites like Mediafire and Hulkshare in its uses. However, one of the great advantages that it has is that DivShare enables you to share music in a variety of ways. You can provide links to a file for people who want to just listen to it streaming instead of downloading it. And you can also embed music files on your blog and site for people to play the songs on. One of the best things about DivShare is that it offers you a variety of different designs and looks to choose from when picking what the embedded player on your site will look like.
All you have to do is create a DivShare account, upload the file, find the track on your dashboard, click on it, and then click "Embed," after which you will be taken to a page that allows you to pick the look of your player and then displays the embed code on the right.
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MT2116 Abstract mathematics
Exclusions
Prerequisites (applies to degree students only)
Syllabus
This course is an introduction to mathematical reasoning. Students are introduced to the fundamental concepts and constructions of mathematics. They are taught how to formulate mathematical statements in precise terms, and how such statements can be proved or disproved.
The course is designed to enable students to:
develop their ability to think in a critical manner
formulate and develop mathematical arguments in a logical manner
improve their skill in acquiring new understanding and expertise
acquire an understanding of basic pure mathematics, and the role of logical argument in mathematics.
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"I learned about polynomials, and playing Sudoku was fun." Carmen Gutierrez said in addition to solving challenges she enjoyed playing games on the iPad and computers. "I like math because it's unviversal," Carmen, a 12-year-old seventh-grader at ...
So then…change the task required for bitcoin mining from polynomials to protein folding and reward scientific advancement instead. Jean-Claude Morin. The task is unique to each miner because it must include it's own address and the previous block ...
"I plug integers into polynomials and see what integers I get out," she explained. "This is a question that's really easy to ask, but it's very hard to get our hands on the solution. Over the last several hundred years, this has been a question people ...
For example, the guidelines say, even someone who is studying to teach English will be expected to "perform arithmetic operations on polynomials" and "demonstrate knowledge of the physiology of multicellular organisms." Asked why teaching candidates in ...
After three weeks of working through the examples, I recaptured my ninth grade ability to factor polynomials. And so I waited for the day in the Phd program when a professor would ask me to advance to the board and demonstrate this invaluable talent.
That sort of practical problem solving does not come from factoring polynomials but working real world problems with practical application. Back to 1895! Dennis Elam is an assistant professor at Texas A&M San Antonio and a 1966 graduate of Andrews High ...
May 7, 1988 — Given the opportunity to solve multi-variable polynomials and algorithms, most Americans would probably not pass. They have enough problems balancing checkbooks. Coert Olmstead thrives on the complex problems. For his efforts, the math ...
I have to admit I don't factor many polynomials or solve many quadratic equations in my day to day life. But I do use math. I definitely use algebra. I solve for X. Math is something you do use at work - at the grocery store - in deciding whether to ...
Limit to books that you can completely read online
Include partial books (book previews)
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The exercise sets given below indicate a minimum level
course and thus all should be assigned. However, the instructor is free to
choose appropriate exercises to supplement these exercise sets.Every other odd problem is expressed as eoo.
The exercise sets given below indicate a minimum level
course and thus all should be assigned. However, the instructor is free to
choose appropriate exercises to supplement these exercise sets.
Chapter 2: Functions
Section
Math 1111 (Review)
Math 1113 (Items to be Tested)
2.1 What is a Function?
13-24 all
29-35 all, 45-57 odd, 59 a,b, 60, 62, a,b
2.2 Graphs of Functions
19-39
37-47 all
2.3 Incr & Decr / Average Rate of Change
1-25 all
2.4 Transformations of Functions
1-9 odd, 19-39 odd,
43-51 odd
29-32 all, 41, 42, 49d, 52d,
53, 54, 61-69 odd
2.5 Maxima and Minima
19-38, 41-44
2.6 Modeling
Skip
2.7 Combining Functions
1-10, 17, 19, 29-32, 35, 36
11,12, 21-27 odd, 33, 34,
37-40 all, 45-50 all, 59, 61, 62
2.8 One to One Functions & Inverses
1-17 odd, 21, 23, 31, 33, 35, 37
19, 20, 25-29 odd, 36,
38-49 all, 51, 52,
53-59 odd, 65, 67, 69
Chapter 3: Polynomial and Rational Functions
Section
Math 1111 (Review)
Math 1113 (Items to be Tested)
3.1 Polynomial Functs
1, 3, 11, 13, 15
5 – 10 all, 17 – 35 odd, 37, 73, 78, 79, 80, 82
3.2 Dividing Polynomials
13, 17, 21, 51 – 60 odd (For 55 & 56, use the
obvious factors and long division to find the real zeros of the
function)
3.5 Complex Zeros
13 – 29 odd, 31 – 39 odd
3.6 Rational Functs
7 – 11 all, 16, 17 – 23 odd, 33 – 63 odd, 65,
75, 77, 79
Chapter 4: Exponential and Logarithmic Functions
Section
Math 1111 (Review)
Math 1113 (Items to be Tested)
4.1 Exponential Functs
1-17 odd, 19-23
all, 25-37odd, 38
4.2 Logarithmic Functs
1-35 (odd)
37-46 all, 47-53
odd, 54, 56, 57-63 odd, 62
4.3 Laws of Logs
13-33 (odd), 39, 41, 49-55 (odd)
11, 29-37odd, 43-47
odd, 57, 66
4.4 Exponential and Logarithmic Equations
1-17 odd, 35-41 odd
19-33, 43-65 odd, 75-81 odd
The domain of logarithmic functions should be
stressed so that students discard extraneous solutions.
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Please Note: Pricing and availability are subject to change without notice.
Mighty Math Astro Algebra
Learn and improve algebra knowledge with this galactic adventure!
Mighty MathHoughton Mifflin Learning TechnologyAstro AlgebraHoughton Mifflin Learning Technologyintroduces concepts and problem-solving skills essential to understanding basic algebra and pre-algebra concepts by taking students on a galactic adventure with over 90 missions. Graph functions, translate word problems into solvable equations, and identify algebraic expressions using Virtual Manipulatives®.
Develops familiarity with whole and rational numbers, decimals, fractions, ratios, proportions, and percents
Introduces expressions, equations, functions, inequalities, and graphs
Covers points, functions, equations, and inequalities with number lines and coordinate planes
FEATURES
Play with the Basic Building Blocks of Algebra - You understand equations better when you can build them with VariaBlox. Manipulate the Blox to combine like terms, substitute or solve for a variable and multiply, and factor simple algebraic expressions. An alert warns you when your equation is out of balance!
Capture Satellites, Meteoroids and Unusual Outer-Space Treasures - Drag objects into the Cargo Bay to sort them and solve a mathematical problem. Use the special sorting bins for translating between fractions, decimals, and percents and identifying equivalent rational numbers or exponential expressions. The Proportion Tool can help you solve scale factor, equivalent ratio, proportion, and percent problems.
Represent Multiple Points and Graphs of Linear Equations and Inequalities - Use the Grapher Station to solve problems and complete missions by identifying slopes and intercepts and solving systems of equations and inequalities. In some missions, a Number Line replaces the coordinate grid and you can graph fractions, decimals, expressions, and inequalities.
Special Features! - The math fun in Astro Algebra takes place in two modes, On-Duty Mode and Off-Duty Mode. You begin the program in On-Duty Mode, where you are directed through missions. You can also go into Off-Duty Mode at any time in order to freely explore the tools and stations at your own pace, view trophies or play Equation of Mystery in the Cargo Bay. Surf the AstroNet, a simulated universe-wide Internet* that contains data on math topics and terms. Learn about important algebra concepts before or during a mission. Use the Calculator to evaluate functions, compute solutions, and create tables of values.
Astro Algebra's Grow Slide automatically records your progress and adjusts to guide your learning. The activities in Astro Algebra offer dozens of math topics and hundreds of problems in 7th, 8th and 9th grade algebra. As you learn and progress, the Grow Slide advances, offering more challenging problems. Students, parents, or teachers can also set the difficulty of each activity or choose a specific topic related to schoolwork.
Note: The AstroNet is contained wholly within the Astro Algebra CD-ROM and is not connected to the Internet or World Wide Web. The Internet cannot be accessed through the AstroNet.
Learning Opportunities
Solving for Variables
Expressions, Equations & Inequalities
Functions
Graphing on Number Line & Coordinate Grid
Ratios & Proportions
Slope & Intercept
Fractions, Decimals & Percents
Exponents
Problem Solving & Reasoning
Algebra Terminology & Notation
Universal Access
This product contains Universal Access features including TouchWindow and Single Switch compatibility to address a variety of learning styles and abilities.
Mighty MathHoughton Mifflin Learning TechnologyAstro AlgebraHoughton Mifflin Learning Technologyfrom Riverdeep
System Requirements
WindowsHoughton Mifflin Learning TechnologyRequires:
Win 95/98
486/33MHz, Pentium or better
8 MB RAM
9 MB available disk space
SVGA, 640x480, 256 colors
2X CD-ROM
Win-compatible sound card
MacintoshHoughton Mifflin Learning TechnologyRequires:
OS 7.5.6-9.x
68030/25MHz, 68040 or PowerPC
8 MB RAM
9 MB available disk space
640x480, 256 colors
2X CD-ROM
Mighty Math Astro Algebra from Riverdeep
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The Apex Maths CD-ROMs and books offer extension and enrichment for all through problem solving and mathematical investigations. The Apex Maths CD-ROMs and books offer extension and enrichment for all through problem solving and mathematical...
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any direct applications. Alternative methods are examined, and explanations are supplied of the fundamental materials and reasoning behind theories and examples.
No other current books deal with this subject, and the author is a leading authority in the field of computer arithmetic. The text introduces the Conventional Radix Number System and the Signed-Digit Number System, as well as Residue Number System and Logarithmic Number System. This book serves as an essential, up-to-date guide for students of electrical engineering and computer and mathematical sciences, as well as practicing engineers and computer scientists involved in the design, application, and development of computer arithmetic units.
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Introductory And Intermediate Algebra For College Students - With 2 Cds - 3rd edition
Summary: TheBlitzer Algebra Seriescombines mathematical accuracy with an engaging, friendly, and often fun presentation for maximum student appeal. Blitzer's personality shows in his writing, as he draws students 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! KEY TOPICS: Variables, Real Numbers, and Mathematical Models; Linear Equations and Inequ...show morealities in One Variable; Linear Equations in Two Variables; Systems of Linear Equations; Exponents and Polynomials; Factoring Polynomials; Rational Expressions; Basics of Functions; Inequalities and Problem Solving; Radicals, Radical Functions, and Rational Exponents; Quadratic Equations and Functions; Exponential and Logarithmic Functions; Conic Sections and Systems of Nonlinear Equations; Sequences, Series, and the Binomial Theorem. MARKET: for all readers interested in algebra some signs of wear, and may have some markings on the inside. 100% Money Back Guarantee. Shipped to over one million happy customers. Your purchase benefits world literacy!
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Odyssey Trigonometry
Purpose
This Internet resource provides introductory information, concept or skill development, practice, and assessment in Mathematics for grade 9, 10, 11, and 12 students who are at grade level, Special Education, reading below grade level, advanced, or English Language Learners, in a single student, small group, whole class, or computer lab situation.
Brief Description
Odyssey Trigonometry combines curricula and assessment to create high-quality content.
Each chapter contains lessons on various subtopics that include direct instruction, guided practice, and independent practice. Skills are taught explicitly and completely in the activity, and then practiced and applied in subsequent activities.
Introduction: Included at the beginning of each lesson telling the student what topic they will be covering.
Direct Instruction Videos: The video segments are 3-5 minutes in duration. Once a student has viewed the entire video, he can repeat sections again. Printable transcripts are available in the activity.
Guided Practice: Re-teaches, hint buttons, and example problems support the emphasis on repetition and practice. Feedback in the guided practice sections is based on the critical mistake guidance which provides scaffolded support for students as the feedback guides them to understanding of why correct answers are correct and incorrect answers are incorrect.
Hint: Hint buttons link students to specific help with the skill being taught.
Vocabulary: The vocabulary tab leads students to a vocabulary list. Three levels of vocabulary support are provided and words are defined and pronounced for the student.
Quizzes and Chapter Tests: Quizzes and chapter tests provide feedback for students and teachers about student mastery of the skills presented.
Offline Worksheets: Printable worksheets provide additional practice opportunities.
Toolkits: Toolkits include a calculator, charts and documents, algebra tiles, base ten blocks, calculator, coordinate grapher, counters, data representation tool, fraction tool, geoboard, number line, probability tool, solid shaper, and transformation tool.
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A big advantage of numerical
mathematics is that a numerical solution can be obtained for
problems, where an analytical solution does not exist. An
additional advantage is, that a numerical method only uses
evaluation of standard functions and the operations:
addition, subtraction, multiplication and division. Because
these are just the operations a computer can perform,
numerical mathematics and computers form a perfect
combination.
An analytical method gives
the solution as a mathematical formula, which is an
advantage. From this we can gain insight in the behavior and
the properties of the solution, and with a numerical
solution (that gives the function as a table) this is not
the case. On the other hand some form of visualization may
be used to gain insight in the behavior of the solution. To
draw a graph of a function with a numerical method is
usually a more useful tool than to evaluate the analytical
solution at a great number of points.
In this book we discuss
several numerical methods for solving ordinary differential
equations. We emphasize those aspects that play an important
role in practical problems. In this introductory text we
confine ourselves to ordinary differential equations with
the exception of the last chapter in which we discuss the
heat equation, a parabolic partial differential equation.
The techniques discussed in
the introductory chapters,
for e.g. interpolation, numerical quadrature and the
solution of nonlinear equations, may also be used outside
the context of differential equations. They have been
included to make the book self contained as far as the
numerical aspects are concerned.
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Appendices
This textbook is designed to teach the university mathematics student the basics of linear algebra and the techniques of formal mathematics. There are no prerequisites other than ordinary algebra, but it is probably best used by a student who has the "mathematical maturity" of a sophomore or junior. The text has two goals: to teach the fundamental concepts and techniques of matrix algebra and abstract vector spaces, and to teach the techniques associated with understanding the definitions and theorems forming a coherent area of mathematics. So there is an emphasis on worked examples of nontrivial size and on proving theorems carefully.
This book is copyrighted. This means that governments have granted the author a monopoly --- the exclusive right to control the making of copies and derivative works for many years (too many years in some cases). It also gives others limited rights, generally referred to as "fair use," such as the right to quote sections in a review without seeking permission. However, the author licenses this book to anyone under the terms of the GNU Free Documentation License (GFDL), which gives you more rights than most copyrights (see appendix GFDL). Loosely speaking, you may make as many copies as you like at no cost, and you may distribute these unmodified copies if you please. You may modify the book for your own use. The catch is that if you make modifications and you distribute the modified version, or make use of portions in excess of fair use in another work, then you must also license the new work with the GFDL. So the book has lots of inherent freedom, and no one is allowed to distribute a derivative work that restricts these freedoms. (See the license itself in the appendix for the exact details of the additional rights you have been given.)
Notice that initially most people are struck by the notion that this book is free (the French would say gratuit, at no cost). And it is. However, it is more important that the book has freedom (the French would say liberté, liberty). It will never go "out of print" nor will there ever be trivial updates designed only to frustrate the used book market. Those considering teaching a course with this book can examine it thoroughly in advance.
Adding new exercises or new sections has been purposely made very easy, and the hope is that others will contribute these modifications back for incorporation into the book, for the benefit of all.
Depending on how you received your copy, you may want to check for the latest version (and other news) at
Topics.
The first half of this text (through Chapter M:Matrices) is basically a course in matrix algebra, though the foundation of some more advanced ideas is also being formed in these early sections. Vectors are presented exclusively as column vectors (since we also have the typographic freedom to avoid writing a column vector inline as the transpose of a row vector), and linear combinations are presented very early. Spans, null spaces, column spaces and row spaces are also presented early, simply as sets, saving most of their vector space properties for later, so they are familiar objects before being scrutinized carefully.
You cannot do everything early, so in particular matrix multiplication comes later than usual. However, with a definition built on linear combinations of column vectors, it should seem more natural than the more frequent definition using dot products of rows with columns. And this delay emphasizes that linear algebra is built upon vector addition and scalar multiplication. Of course, matrix inverses must wait for matrix multiplication, but this does not prevent nonsingular matrices from occurring sooner. Vector space properties are hinted at when vector and matrix operations are first defined, but the notion of a vector space is saved for a more axiomatic treatment later (Chapter VS:Vector Spaces). Once bases and dimension have been explored in the context of vector spaces, linear transformations and their matrix representation follow. The goal of the book is to go as far as Jordan canonical form in the Core (part C), with less central topics collected in the Topics (part T). A third part contains contributed applications (part A), with notation and theorems integrated with the earlier two parts.
Linear algebra is an ideal subject for the novice mathematics student to learn how to develop a topic precisely, with all the rigor mathematics requires. Unfortunately, much of this rigor seems to have escaped the standard calculus curriculum, so for many university students this is their first exposure to careful definitions and theorems, and the expectation that they fully understand them, to say nothing of the expectation that they become proficient in formulating their own proofs. We have tried to make this text as helpful as possible with this transition. Every definition is stated carefully, set apart from the text. Likewise, every theorem is carefully stated, and almost every one has a complete proof. Theorems usually have just one conclusion, so they can be referenced precisely later. Definitions and theorems are cataloged in order of their appearance in the front of the book (\miscref{definition}{Definitions}, \miscref{theorem}{Theorems}), and alphabetical order in the index at the back. Along the way, there are discussions of some more important ideas relating to formulating proofs (\miscref{technique}{Proof Techniques}), which is part advice and part logic.
Origin and History.
This book is the result of the confluence of several related events and trends.
At the University of Puget Sound we teach a one-semester, post-calculus linear algebra course to students majoring in mathematics, computer science, physics, chemistry and economics. Between January 1986 and June 2002, I taught this course seventeen times. For the Spring 2003 semester, I elected to convert my course notes to an electronic form so that it would be easier to incorporate the inevitable and nearly-constant revisions. Central to my new notes was a collection of stock examples that would be used repeatedly to illustrate new concepts. (These would become the Archetypes, appendix A.) It was only a short leap to then decide to distribute copies of these notes and examples to the students in the two sections of this course. As the semester wore on, the notes began to look less like notes and more like a textbook.
I used the notes again in the Fall 2003 semester for a single section of the course. Simultaneously, the textbook I was using came out in a fifth edition. A new chapter was added toward the start of the book, and a few additional exercises were added in other chapters. This demanded the annoyance of reworking my notes and list of suggested exercises to conform with the changed numbering of the chapters and exercises. I had an almost identical experience with the third course I was teaching that semester. I also learned that in the next academic year I would be teaching a course where my textbook of choice had gone out of print. I felt there had to be a better alternative to having the organization of my courses buffeted by the economics of traditional textbook publishing.
I had used TeX and the Internet for many years, so there was little to stand in the way of typesetting, distributing and "marketing" a free book. With recreational and professional interests in software development, I had long been fascinated by the open-source software movement, as exemplified by the success of GNU and Linux, though public-domain TeX might also deserve mention. Obviously, this book is an attempt to carry over that model of creative endeavor to textbook publishing.
As a sabbatical project during the Spring 2004 semester, I embarked on the current project of creating a freely-distributable linear algebra textbook. (Notice the implied financial support of the University of Puget Sound to this project.) Most of the material was written from scratch since changes in notation and approach made much of my notes of little use. By August 2004 I had written half the material necessary for our Math 232 course. The remaining half was written during the Fall 2004 semester as I taught another two sections of Math 232.
While in early 2005 the book was complete enough to build a course around and Version 1.0 was released. Work has continued since, filling out the narrative, exercises and supplements.
However, much of my motivation for writing this book is captured by the sentiments expressed by H.M. Cundy and A.P. Rollet in their Preface to the First Edition of
Mathematical Models (1952), especially the final sentence,
This book was born in the classroom, and arose from the spontaneous
interest of a Mathematical Sixth in the construction of simple
models. A desire to show that even in mathematics one could have
fun led to an exhibition of the results and attracted considerable
attention throughout the school. Since then the Sherborne collection
has grown, ideas have come from many sources, and widespread interest
has been shown. It seems therefore desirable to give permanent
form to the lessons of experience so that others can benefit by
them and be encouraged to undertake similar work.
How To Use This Book.
Chapters, Theorems, etc. are not numbered in this book, but are instead referenced by acronyms. This means that Theorem XYZ will always be Theorem XYZ, no matter if new sections are added, or if an individual decides to remove certain other sections. Within sections, the subsections are acronyms that begin with the acronym of the section. So Subsection XYZ.AB is the subsection AB in Section XYZ. Acronyms are unique within their type, so for example there is just one Definition B, but there is also a Section B:Bases. At first, all the letters flying around may be confusing, but with time, you will begin to recognize the more important ones on sight. Furthermore, there are lists of theorems, examples, etc. in the front of the book, and an index that contains every acronym. If you are reading this in an electronic version (PDF or XML), you will see that all of the cross-references are hyperlinks, allowing you to click to a definition or example, and then use the back button to return. In printed versions, you must rely on the page numbers. However, note that page numbers are not permanent! Different editions, different margins, or different sized paper will affect what content is on each page. And in time, the addition of new material will affect the page numbering.
Chapter divisions are not critical to the organization of the book, as Sections are the main organizational unit. Sections are designed to be the subject of a single lecture or classroom session, though there is frequently more material than can be discussed and illustrated in a fifty-minute session. Consequently, the instructor will need to be selective about which topics to illustrate with other examples and which topics to leave to the student's reading. Many of the examples are meant to be large, such as using five or six variables in a system of equations, so the instructor may just want to "walk" a class through these examples. The book has been written with the idea that some may work through it independently, so the hope is that students can learn some of the more mechanical ideas on their own.
The highest level division of the book is the three Parts: Core, Topics, Applications (part C, part T, part A). The Core is meant to carefully describe the basic ideas required of a first exposure to linear algebra. In the final sections of the Core, one should ask the question: which previous Sections could be removed without destroying the logical development of the subject? Hopefully, the answer is "none." The goal of the book is to finish the Core with a very general representation of a linear transformation (Jordan canonical form, Section JCF:Jordan Canonical Form). Of course, there will not be universal agreement on what should, or should not, constitute the Core, but the main idea is to limit it to about forty sections. Topics (part T) is meant to contain those subjects that are important in linear algebra, and which would make profitable detours from the Core for those interested in pursuing them. Applications (part A) should illustrate the power and widespread applicability of linear algebra to as many fields as possible.
The Archetypes (appendix A) cover many of the computational aspects of systems of linear equations, matrices and linear transformations. The student should consult them often, and this is encouraged by exercises that simply suggest the right properties to examine at the right time. But what is more important, this a repository that contains enough variety to provide abundant examples of key theorems, while also providing counterexamples to hypotheses or converses of theorems. The summary table at the start of this appendix should be especially useful.
I require my students to read each Section prior to the day's discussion on that section. For some students this is a novel idea, but at the end of the semester a few always report on the benefits, both for this course and other courses where they have adopted the habit. To make good on this requirement, each section contains three Reading Questions. These sometimes only require parroting back a key definition or theorem, or they require performing a small example of a key computation, or they ask for musings on key ideas or new relationships between old ideas. Answers are emailed to me the evening before the lecture. Given the flavor and purpose of these questions, including solutions seems foolish.
Every chapter of part C ends with "Annotated Acronyms", a short list of critical theorems or definitions from that chapter. There are a variety of reasons for any one of these to have been chosen, and reading the short paragraphs after some of these might provide insight into the possibilities. An end-of-chapter review might usefully incorporate a close reading of these lists.
Formulating interesting and effective exercises is as difficult, or more so, than building a narrative. But it is the place where a student really learns the material. As such, for the student's benefit, complete solutions should be given. As the list of exercises expands, the amount with solutions should similarly expand. Exercises and their solutions are referenced with a section name, followed by a dot, then a letter (C,M, or T) and a number. The letter `C' indicates a problem that is mostly computational in nature, while the letter `T' indicates a problem that is more theoretical in nature. A problem with a letter `M' is somewhere in between (middle, mid-level, median, middling), probably a mix of computation and applications of theorems. So solution MO.T13 is a solution to an exercise in Section MO:Matrix Operations that is theoretical in nature. The number `13' has no intrinsic meaning.
More on Freedom.
This book is freely-distributable under the terms of the GFDL, along with the underlying TeX code from which the book is built. This arrangement provides many benefits unavailable with traditional texts.
No cost, or low cost, to students. With no physical vessel (i.e. paper, binding), no transportation costs (Internet bandwidth being a negligible cost) and no marketing costs (evaluation and desk copies are free to all), anyone with an Internet connection can obtain it, and a teacher could make available paper copies in sufficient quantities for a class. The cost to print a copy is not insignificant, but is just a fraction of the cost of a traditional textbook when printing is handled by a print-on-demand service over the Internet. Students will not feel the need to sell back their book (nor should there be much of a market for used copies), and in future years can even pick up a newer edition freely.
Electronic versions of the book contain extensive hyperlinks. Specifically, most logical steps in proofs and examples include links back to the previous definitions or theorems that support that step. With whatever viewer you might be using (web browser, PDF reader) the "back" button can then return you to the middle of the proof you were studying. So even if you are reading a physical copy of this book, you can benefit from also working with an electronic version.
A traditional book, which the publisher is unwilling to distribute in an easily-copied electronic form, cannot offer this very intuitive and flexible approach to learning mathematics.
The book will not go out of print. No matter what, a teacher can maintain their own copy and use the book for as many years as they desire. Further, the naming schemes for chapters, sections, theorems, etc. is designed so that the addition of new material will not break any course syllabi or assignment list.
With many eyes reading the book and with frequent postings of updates, the reliability should become very high. Please report any errors you find that persist into the latest version.
For those with a working installation of the popular typesetting program TeX, the book has been designed so that it can be customized. Page layouts, presence of exercises, solutions, sections or chapters can all be easily controlled. Furthermore, many variants of mathematical notation are achieved via TeX macros. So by changing a single macro, one's favorite notation can be reflected throughout the text. For example, every transpose of a matrix is coded in the source as {\tt\verb!\transpose{A}!}, which when printed will yield $\transpose{A}$. However by changing the definition of {\tt\verb!\transpose{ }!}, any desired alternative notation (superscript t, superscript T, superscript prime) will then appear throughout the text instead.
The book has also been designed to make it easy for others to contribute material. Would you like to see a section on symmetric bilinear forms? Consider writing one and contributing it to one of the Topics chapters. Should there be more exercises about the null space of a matrix? Send me some. Historical Notes? Contact me, and we will see about adding those in also.
You have no legal obligation to pay for this book. It has been licensed with no expectation that you pay for it. You do not even have a moral obligation to pay for the book. Thomas Jefferson (1743 -- 1826), the author of the United States Declaration of Independence, wrote,
If nature has made any one thing less susceptible than all others
of exclusive property, it is the action of the thinking power called
an idea, which an individual may exclusively possess as long as he
keeps it to himself; but the moment it is divulged, it forces itself
into the possession of every one, and the receiver cannot dispossess
himself of it. Its peculiar character, too, is that no one possesses
the less, because every other possesses the whole of it. He who
receives an idea from me, receives instruction himself without
lessening mine; as he who lights his taper at mine, receives light
without darkening me. That ideas should freely spread from one to
another over the globe, for the moral and mutual instruction of
man, and improvement of his condition, seems to have been peculiarly
and benevolently designed by nature, when she made them, like fire,
expansible over all space, without lessening their density in any
point, and like the air in which we breathe, move, and have our
physical being, incapable of confinement or exclusive appropriation.
Letter to Isaac McPherson
August 13, 1813
However, if you feel a royalty is due the author, or if you would like to encourage the author, or if you wish to show others that this approach to textbook publishing can also bring financial compensation, then donations are gratefully received. Moreover, non-financial forms of help can often be even more valuable. A simple note of encouragement, submitting a report of an error, or contributing some exercises or perhaps an entire section for the Topics or Applications are all important ways you can acknowledge the freedoms accorded to this work by the copyright holder and other contributors.
Conclusion.
Foremost, I hope that students find their time spent with this book profitable. I hope that instructors find it flexible enough to fit the needs of their course. And I hope that everyone will send me their comments and suggestions, and also consider the myriad ways they can help (as listed on the book's website at
| 677.169 | 1 |
The calculus of differential forms has significant advantages over traditional methods as a tool for teaching electromagnetic (EM) field theory. First, films clarify the relationship between field intensity and flux density, by providing distinct mathematical and graphical representations for the two types of fields. Second, Ampere's and Faraday's laws obtain graphical representations that are as intuitive as the representation of Gauss's law. Third, the vector Stokes theorem and the divergence theorem become special cases of a single relationship that is easier for the student to remember, apply, and visualize than their vector formulations. Fourth, computational simplifications result from the use of forms: derivatives are easier to employ in curvilinear coordinates, integration becomes more straightforward, and families of vector identities are replaced by algebraic rules. In this paper, EM theory and the calculus of differential forms are developed in parallel, from an elementary, conceptually oriented point of view using simple examples and intuitive motivations. We conclude that because of the power of the calculus of differential forms in conveying the fundamental concepts of EM theory, it provides an attractive and viable alternative to the use of vector analysis in teaching electromagnetic field theory.
| 677.169 | 1 |
Quadratic equations were really giving me a hard time. Then I got Algebra Buster , and it helped me not only with quadratic but also with pretty much any equation or expression I could think of! Hillary Sill, VI.
Algebra homework has always given me sleepless nights but once I started using Algebra Buster it has been fun. Its made my life easy and study enjoyable. Bud Pippin, UT
I am a 9th grade student and always wondered how some students always got good marks in mathematics but could never imagine that Ill be one of them. Hats off to Algebra Buster ! Now I have a firm grasp over algebra and my approach to problem solving is more methodical. Linda Taylor, KY
As a math teacher, Im always looking for new ways to help my students. Algebra Buster not only allows me to make proficient lesson plans, it also allows my students to check their answers when I am not available. S.H., Texas08-21 :
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Crossing The River With Dogs : Problem Solving For College Students - (rev edition
ISBN13:978-1931914147 ISBN10: 1931914141 This edition has also been released as: ISBN13: 978-0470412244 ISBN10: 0470412240
Summary: Students who often complain when faced with challenging word problems will be engaged as they acquire essential problem solving skills that are applicable beyond the math classroom. The authors of Crossing the River with Dogs: Problem Solving for College Students: Use the popular approach of explaining strategies through dialogs from fictitious students. Present all the classic and numerous non-traditional problem solving strategies (from drawing diagrams to matrix l...show moreogic, and finite differences). Provide a text suitable for students in quantitative reasoning, developmental mathematics, mathematics education, and all courses in between. Challenge students with interesting, yet concise problem sets that include classic problems at the end of each chapter. With Crossing the River with Dogs, students will enjoy reading their text and will take with them skills they will use for a lifetime
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\input zb-basic
\input zb-matheduc
\iteman{ZMATH 2010d.00087}
\itemau{Ammann, Claudia; Frauendiener, J\"org; Holton, Derek}
\itemti{German undergraduate mathematics enrolment numbers: background and change.}
\itemso{Int. J. Math. Educ. Sci. Technol. 41, No. 4, 435-449 (2010).}
\itemab
\itemrv{~}
\itemcc{B40 A45}
\itemut{recruitment; retention; enrolment; tertiary education}
\itemli{doi:10.1080/00207390903564629}
\end
| 677.169 | 1 |
Thomas and received an A in the course. Linear Algebra is the study of matrices and their properties. The applications for linear algebra are far reaching whether you want to continue studying advanced algebra or computer science.
| 677.169 | 1 |
for
Concerning the last questionwhydiscretemath,
"Why isn't discretemathematics offered to high school students without calculus :notbackground?"
Not only is that possible, but it wasdonehadbeenthenorm in the past aspartofwithin the "New Math" movementcurriculum, when everyone had to learn about sets and functions inhighschool. This ended in a PR disaster andahugebacklashagainstmathematics, because generations of students were lost and got turned off by mathematics for life(; some of them later became politicians who decide on our funding!funding.Consequently,itwasabandoned.(Apparently,calculusinHSwasintroducedasapartofthesamepackageandsurvived.)
I'd be interested to know if there are any high school -– college partnerships that offer discrete mathematics to H.S. students with strong analytical skills, and how do they handle the pre-requisitesprerequisites question.
for the last question why discrete math isn't offered to high school students without calculus: not only is that possible, but it was done in the past as part of the "New Math" movement, when everyone had to learn about sets and functions. This ended in a PR disaster, because generations of students were lost and got turned off by mathematics for life (some of them became politicians who decide on our funding!) I'd be interested to know if there are any high school - college partnerships that offer discrete mathematics to H.S. students with strong analytical skills, and how do they handle the pre-requisites question.
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TLE Online Companion is a 32 page USER'S GUIDE with online PIN-code access to THE LEARNING EQUATION lessons, bundled with Tussy/Gustafson's INTERMEDIATE ALGEBRA, SECOND EDITION. Delivered entirely over the Internet, students can access 15 lessons per course, hand-picked by Alan Tussy to enhance the presentation of specific concepts in the course. The TLE ONLINE COMPANION is adapted from the full version of THE LEARNING EQUATION line of developmental mathematics courseware products. Designed for learner-focused, computer classroom, lab-based, and distance learning courses, the pedagogical model employs a "Guided Inquiry" approach whereby students construct their own understanding of concepts. Instead of passively being fed information, students are actively involved in tasks requiring them to discover or apply mathematical concepts. Each lesson has seven interactive components: Introduction, Tutorial, Examples, Summary, Practice and Problems, Extra Practice, and Self Check. The interactive learning content is the perfect compliment to the textbooks, designed to engage and enrich the student's learning experience by addressing multiple learning styles. Using the power of the most comprehensive and powerful course management system available, student progress is tracked from whatever location they choose to learn. The auto-enrollment feature via PIN codes, customizable grade book, world-class test generator for printed and on line assessments, and outstanding communication tools makes managing the learning experience fast and easy.
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The book Student Guide for Acerra's The Learning Equation Online Intermediate Algebra Lessons by Acerra
(author) is published or distributed by Brooks Cole [053440717X, 9780534407179].
This particular edition was published on or around 2003-3-24 date.
Student Guide for Acerra's The Learning Equation Online Intermediate Algebra Lessons
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Olympiad Problem Books
Many countries hold an annual mathematical olympiad competition for high school students. These contests are typically the culmination of a series of preliminary examinations. Contests usually consist of about 6 essay questions, requiring the students to write proofs of challenging problems. Although these problems can be quite difficult, they require no advanced mathematics, only a thorough knowledge of high school math topics and the ability to handle non-routine problems. Contestants are typically given an hour or more to solve each problem. Winners of the national olympiad often then represent the country at the prestigious International Mathematical Olympiad (IMO).
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Calculus hasnt changed, but your students have. Many of todays students have seen calculus before at the high school level. However, professors report nationwide that students come into their calculus courses with weak backgrounds in algebra and trigonometry, two areas of knowledge vital to the read more...
Discrete Mathematics provides an introduction to some of the fundamental concepts in modern mathematics. Abundant examples help explain the principles and practices of Discrete Mathematics. The book intends to cover material required by readers for whom mathematics is just a tool, as well as read more...
The seventh edition of this classic text has retained the features that make it popular, while updating its treatment and inclusion of Computer Algebra Systems and Programming Languages. The exercise sets include additional challenging problems and projects which show practical applications of the read more...
Once considered an unimportant branch of topology, graph theory has come into its own through many important contributions to a wide range of fields and is now one of the fastest-growing areas in discrete mathematics and computer science. This new text introduces basic concepts, definitions, read more...
This is a complete revision of a classic, seminal, and authoritative text that has been the model for most books on the topic written since 1970. It explores the building of stochastic (statistical) models for time series and their use in important areas of application forecasting, model read more...
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