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Assignments
The problems are exercises for your mathematical health; they provide a means for you to check your understanding of the material. They are meant to be neither difficult, nor tricky, nor involved. If you find that you are stuck on any problem, then review the relevant material; if you remain stuck, then discuss the problem with someone. Of course, you must still think through each problem on your own and write it up in your own words.
Normally the problem presented on Tuesday is due two days later. Although some problems have more than one solution, feel free to write up and pass in the solution presented; just think it through for yourself. You will learn more if you think about the problem before it is presented. Similarly, Thursday's presentation normally treats a problem due that day. If you learn that you have made a mistake in your own solution, then you may request an extension so that you can correct your work. Although this practice may seem odd, remember what counts in the end: that you can solve the problem correctly and that you can explain the solution clearly.
Problems are to be handed in at the end of the indicated class session and will be graded in part on the quality of the write-up | 677.169 | 1 |
Algebra 2 and Trigonometry is a new text for a course in intermediate algebra and trigonometry that continues the approach that has made Amsco a leader in presenting ... 2/Algebra 2 and Trigonometry.pdf
u0022Just DoWhatu0027sRightu0022 by Plato Plato , University of St. Andrews, Mathematics Archive About the author .... Plato (427-347 BC), as a young aristocrat, was Socrates ...
Pearson Pilot Guide Thank you for piloting Prentice Hall Science Explorer. Weu0027ve put together this short guide to cover the basics of the program to help you get started. Explorer/SX Pilot Guide.pdf
Economics Support Materials - Grade 2 Page 3 OVERVIEW These supplemental materials include a variety of lessons and activities in the field of Social Studies with a ...
ii The College Board: Connecting Students to College Success The College Board is a not-for-profit membership association whose mission is to connect students to ...
Section Review 40-1 1. A disease is any change, other than injury, that disrupts the normal functions of the body. 2. Diseases are either inherited, caused by materi ... | 677.169 | 1 |
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Read More Reading Graphs and Charts Mathematics Writing Part II: Mathematics Review Symbols, Terminology, Formulas, and General Mathematical Information Arithmetic Algebra Measurement Part III: Four Full-Length Practice Exams Each practice exam includes the same number of questions as the actual exam The practice exams come complete with answers and explanations for all questions Model essay responses include grader comments | 677.169 | 1 |
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Currently unavailable6Student Solutions Manual for Mathematics for Elementary Teachers
Manipulative Kit for Mathematics for Elementary Teachers
Connect Plus Math Access Card for Mathematics for Elementary Teachers
Mathematics for Elementary Teachers: A Conceptual Approach
Mathematics for Elementary Teachers : A Conceptual Approach
Math for Elementary Teachers : A Conceptual Approach
Manipulative Kit Mathematics for Elementary Teachers
Student's Solution Manual Mathematics for Elementary Teachers
Loose Leaf for Mathematics for Elementary Teachers
Summary
CONNECT MATH ACCESS CARD FOR MATHEMATICS FOR ELEMENTARY TEACHERS
McGraw-Hill conducted in-depth research to create a new and improved learning experience that meets the needs of today's students and instructors. The result is a reinvented learning experience rich in information, visually engaging, and easily accessible to both instructors and students. McGraw-Hill's Connect is a Web-based assignment and assessment platform that helps students connect to their coursework and prepares them to succeed in and beyond the course.
Connect Mathematics enables math instructors to create and share courses and assignments with colleagues and adjuncts with only a few clicks of the mouse. | 677.169 | 1 |
Details about PreMBA Analytical Primer:
This book is a review of the analytical methods required in most of the quantitative courses taught at MBA programs. Students with no technical background, or who have not studied mathematics since college or even earlier, may easily feel overwhelmed by the mathematical formalism that is typical of economics and finance courses. These students will benefit from a concise and focused review of the analytical tools that will become a necessary skill in their MBA classes. The objective of this book is to present the essential quantitative concepts and methods in a self-contained, non-technical, and intuitive way. | 677.169 | 1 |
Program components for student and teacher | math 180, Math 180 is a comprehensive system of curriculum, instruction, assessment, and professional development designed to equip older struggling students with the knowledge.
Online student edition, Teacher login / registration : teachers: if your school or district has purchased print student editions, register now to access the full online version of the book..
Aleks -- assessment and learning, k-12, higher education, Increase student performance and retention with individualized assessment and learning. take control of your classroom, and save time with aleks' powerful learning | 677.169 | 1 |
BEGINNING ALGEBRA
9780131444447
ISBN:
0131444441
Edition: 4 Pub Date: 2004 Publisher: Prentice Hall
Summary: Clearly explained concepts, study skills help, and real-life applications will help the reader to succeed in learning algebra.
Martin-Gay, K. Elayn is the author of BEGINNING ALGEBRA, published 2004 under ISBN 9780131444447 and 0131444441. One hundred forty one BEGINNING ALGEBRA textbooks are available for sale on ValoreBooks.com, twenty one used from the cheapest price of $0.97, or buy new starting at $13 | 677.169 | 1 |
"I am very excited to receive these items and I must say this is the best educational product company I've ever had the pleasure of doing business with. Thank you for the great service."
-- Geraldine K.
Geometry Video Series: DVD - Set of 6
Overview
Item # 532604
Average Rating:
Recommended Grade(s):9-12
Web Price
$159.95
Quantity
Available Quantity 2
Description
Teach Geometry through dynamic skits, mnemonics and computer graphics in a format that is so lucid students understand and retain the material. Each video is 26 minutes long and is numbered for progressive placement, basic as #1. DVD titles: Geometry Basics Angles 101 Triangles Special Triangles Figuring Out Area The Pythagorean Theorem | 677.169 | 1 |
More options
Contributors
Contents/Summary
Preface to the Second Edition-- Preface to the First Edition-- 1. Measure theory and probability-- 2. Independence and conditioning-- 3. Gaussian variables-- 4. Distributional computations-- 5. Convergence of random variables-- 6. Random processes-- Where is the notion N discussed?-- Final suggestions: how to go further?-- References-- Index.
(source: Nielsen Book Data)
Publisher's Summary
Derived from extensive teaching experience in Paris, this second edition now includes over 100 exercises in probability. New exercises have been added to reflect important areas of current research in probability theory, including infinite divisibility of stochastic processes, past-future martingales and fluctuation theory. For each exercise the authors provide detailed solutions as well as references for preliminary and further reading. There are also many insightful notes to motivate the student and set the exercises in context. Students will find these exercises extremely useful for easing the transition between simple and complex probabilistic frameworks. Indeed, many of the exercises here will lead the student on to frontier research topics in probability. Along the way, attention is drawn to a number of traps into which students of probability often fall. This book is ideal for independent study or as the companion to a course in advanced probability theory. (source: Nielsen Book Data) | 677.169 | 1 |
Essentials of Trigonometry
9780534494230
ISBN:
0534494234
Pub Date: 2005 Publisher: Thomson Learning
Summary: Easy-to-understand, ESSENTIALS OF TRIGONOMETRY starts with the right-angle definition, and applications involving the solution of right triangles to help you investigate and understand the trigonometric functions, their graphs, their relationships to one another, and ways in which they can be used in a variety of real-world applications. The accompanying CD-ROM and online tutorials give you the practice you need to i...mprove your grade in the course.
Smith, Karl J. is the author of Essentials of Trigonometry, published 2005 under ISBN 9780534494230 and 0534494234. Nine Essentials of Trigonometry textbooks are available for sale on ValoreBooks.com, and four used from the cheapest price of $9 | 677.169 | 1 |
This course provides a mathematical foundation for the number systems used in secondary and post-secondary mathematics courses, with an emphasis on rigorous logical and set-theoretical foundations of the natural numbers, integers, rational numbers, and real numbers. The course also covers the common algebraic extensions of the number system and familiarizes students with the historical development of the number system.
This course will explore a variety of algebra-related topics including modular arithmetic; characteristics of polynomial, exponential, logarithmic, and trigonometric functions; and algebraic structures including semigroups, groups, rings, integral domains, and fields.
This course explores ways to collect, organize, display, and analyze data and make reasoned decisions based on it. Students use statistical methods based on data, develop and evaluate inferences and predictions about data, and apply probability and distribution theory concepts. The course helps prepare teachers to teach statistical concepts and AP statistics and to critically examine and comprehend data analysis in education literature. Graphing calculators and computer software are incorporated. | 677.169 | 1 |
Microsoft Mathematics
editor's review
download
18
specifications
Microsoft Mathematics is a software designed to help you learn equations in a user-friendly environment.
The interface of the program is clean and intuitive. In the "Worksheet" area you can type an expression on the lower part of the screen and press the "Enter" button to process it.
Results are instantly displayed above this area. For each equation you can edit the entry or view keyboard equivalents.
Furthermore, you can plot an equation or function in the "Graphing" tab (2D or 3D, Cartesian or Polar), as well as insert a data set, parametric functions and inequalities.
On the left side of the screen, you can use the calculator pad to input shortcuts for complex numbers, calculus (e.g. partial derivatives, indefinite integrals), statistics (e.g. geometric means, permutations), trigonometry, linear algebra (e.g. matrix insertions, transpositions) and standard elements (e.g. expand, slope, pi). Also, you can create a "favorite buttons" group.
In addition, you can use the "Undo" and "Redo" buttons, as well as initiate a triangle solver, equation solver, unit converter, formulas and equations (e.g. geometry, physics, chemistry, laws of exponents, logarithm properties, constants).
The program uses a pretty high amount of system resources and contains a comprehensive help file (also includes a glossary). We haven't encountered any kind of problems during our tests and we strongly recommend Microsoft Mathematics to teachers or students. | 677.169 | 1 |
You should already have a book or a teacher or an online course. Why will it make any difference if write you a new book? I can do the problem, but how do I know I'm not just doing your homework for you? You must show SOME knowledge of the subject matter or we're both just wasting out time. | 677.169 | 1 |
9814271400
/ 9814271403
An Introduction to the Analysis of Algorithms
by:Soltys Michael...
Show More invariants. The algorithms considered are the basic and traditional algorithms of computer science, such as Greedy, Dynamic and Divide & Conquer. In addition, two classes of algorithms that rarely make it into introductory textbooks are discussed. Randomized algorithms, which are now ubiquitous because of their applications to cryptography; and Online algorithms, which are essential in fields as diverse as operating systems (caching, in particular) and stock-market predictions. This self-contained book is intended for undergraduate students in computer science and mathematics | 677.169 | 1 |
MATH 120 - Intermediate Algebra
at Cañada College
for Spring 2013 (CRN : 42056 second course in a 2-part series covering elementary and intermediate algebra and is a continuation of MATH 110. Topics include a review of equations, absolute value, lines and graphs, functions, rational exponents, radical expressions and equations, quadratic equations and graphs, exponential functions, and logarithmic functions. Additional topics may include conic sections and systems of equations.
Hybrid courses require regular access to a computer with reliable internet connection and basic computer literacy. Web access is available in the Cañada College Learning Center.
Hybrid courses combine face-to-face classroom instruction with computer-based online learning. Hybrid courses move a significant part of course learning online (51% to 99%) including To Be Arranged Hours and, as a result, reduce the amount of classroom seat time. Students meet on campus for a portion of the class without alternative distance education means of student participation and then work independently to complete the online portion of the course. | 677.169 | 1 |
Precise Calculator has arbitrary precision and can calculate with complex numbers, fractions, vectors and matrices. Has more than 150 mathematical functions and statistical functions and is programmable (if, goto, print, return, for). | 677.169 | 1 |
Mathematics A Concise History and Philosophy
9780387942803
ISBN:
0387942807
Pub Date: 1994 Publisher: Springer Verlag
Summary: This is a concise introductory textbook for a one semester course in the history and philosophy of mathematics. It is written for mathematics majors, philosophy students, history of science students and secondary school mathematics teachers. The only prerequisite is a solid command of pre-calculus mathematics. It is shorter than the standard textbooks in that area and thus more accessible to students who have trouble... coping with vast amounts of reading. Furthermore, there are many detailed explanations of the important mathematical procedures actually used by famous mathematicians, giving more mathematically talented students a greater opportunity to learn the history and philosophy by way of problem solving. Several important philosophical topics are pursued throughout the text, giving the student an opportunity to come to a full and consistent knowledge of their development. These topics include infinity, the nature of motion, and Platonism. This book offers, in fewer pages, a deep penetration into the key mathematical and philosophical aspects of the history of mathematics.
Anglin, W. S. is the author of Mathematics A Concise History and Philosophy, published 1994 under ISBN 9780387942803 and 0387942807. Three hundred six Mathematics A Concise History and Philosophy textbooks are available for sale on ValoreBooks.com, fifty one used from the cheapest price of $19.24, or buy new starting at $35 | 677.169 | 1 |
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About the book:
Designed for use at Key Stages 1 and 2, "New Curriculum Mathematics for Schools" offers an integrated approach to mathematics. The series focuses on six main topics: pattern, number, data handling, shape and space, measurement and algebra. These key themes are taught through activities and are constantly reinforced. Emphasis is placed on purposeful, progressive and pleasurable learning. One teacher's book is provided for each level at Key Stage 2. Each is linked to a particular pupil's book, set of activity cards and copymasters pack. The teacher's books link all the other elements provided at each level to the Attainment Targets, providing a detailed match to the National Curriculum.
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Awesomebookscanada via Canada
Softcover, ISBN 0050044222 Publisher: Oliver & Boyd50044222 Publisher: Oliver & Boyd, 1991 0050044222 Publisher: Oliver & Boyd 0050044222 Publisher: Oliver & Boyd, 1991 Used - Very Good, Usually ships in 1-2 business days, In stock and ready for immediate shipment. Shipped from the US to arrive in 9-15 business days. Order inquiries handled promptly.
Softcover, ISBN 0050044222 Publisher: Oliver & Boyd | 677.169 | 1 |
This book is specially designed for students interested in participating in the Mathematics Olympiad, but even those who just have a casual interest in Mathematics will find the questions here... More > intriguing and challenging. The questions in the book are arranged according to topic, and the detailed solutions and workings can be found at the back of the volume. We sincerely hope that by doing the questions in this book, students will understand and grasp the fundamental techniques required for critical Mathematical thinking.< Less
South African CAPs and RNSC Compatible
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Included are the Learning Outcomes & the prescribed Assessment Standards, which include
NUMBERS, OPERATIONS AND RELATIONSHIPS
PATTERN FUNCTIONS AND ALGEBRA
SPACE & SHAPE
MEASUREMENT
DATA HANDLING< Less
South African CAPs and RNSC Compatible The Included are the Learning Outcomes & the prescribed Assessment Standards, which include
NUMBERS,
OPERATIONS AND RELATIONSHIPS
PATTERN FUNCTIONS AND ALGEBRA
SPACE & SHAPE MEASUREMENT
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THERE MUST BE AN EASIER WAY TO DO MATH. Mathematics, as we learned and came to know too well in school, is complicated. To most people, it is even scary. LEARN STREET MATH attempts to demystify Math... More > as we've known it.
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LEARN STREET MATH: FORWARD MATH is the first in a series of e-books that feature what the author calls "Street Math" tricks. Street Math tricks are easy, fast, and fun to learn techniques to solve most basic arithmetic problems.
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Beginning Algebra (5th Edition)
9780136007029
ISBN:
0136007023
Edition: 5 Pub Date: 2007 Publisher: Pearson
Summary: Clearly explained concepts, study skills help, and real-life applications will help the reader to succeed in learning algebra.
Elayn Martin-Gay is the author of Beginning Algebra (5th Edition), published 2007 under ISBN 9780136007029 and 0136007023. One hundred sixty Beginning Algebra (5th Edition) textbooks are available for sale on ValoreBooks.com, forty four used from the cheapest price of $3.59, or buy n...ew starting at $14 | 677.169 | 1 |
What
lies beneath the surface?
A guide to the Boundary
Element Method
and Green's functions
for the students and professionals: history, ideas and applications
Science, technology,
engineering, and mathematics (STEM) have been identified by many
agencies as key areas in research innovations that will help the US
maintain its leadership in the 21st century. The Boundary Element
Method (BEM) is a numerical method whose applications cover nearly all
the STEM fields. This website is designed to serve both as a guide for
the students who want to learn about the simple, yet powerful,
mathematical concepts of the method and as a resource for the
professionals who are already working in the area of BEM. The website
is an outcome of the National
Science Foundation (NSF) sponsored workshop:
Bridging
Education and Industrial Applications
held on the University of Minnesota campus (Minneapolis, Minnesota)
from April 23 to April 26, 2012. | 677.169 | 1 |
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There will be one paper of two and a half hours duration carrying 80 marks and Internal Assessment of 20 marks.
The paper will be divided into two sections, Section I (40 marks), Section II (40 marks).
Section I: will consist of compulsory short answer questions.
Section II: Candidates will be required to answer four out of seven questions.
The solution of a question may require the knowledge of more than one branch of the syllabus.
1. Pure Arithmetic
Irrational Numbers
(a) Rational, irrational numbers as real numbers, their place in the number system. Surds and rationalization of surds.
(b) Irrational numbers as non-repeating, nonterminating decimals.
(c) Classical definition of a rational number p/q, p, q Î Z, q 0.
Hence, define irrational numbers as what cannot be expressed as above.
(d) Simplifying an expression by rationalising the denominator.
2. Commercial Mathematics
(i) Profit and Loss
The meaning of Marked price, selling price and discount, thus giving an idea of profit and loss on day to day dealings. Simple problems related to Profit and Loss and Discount, including inverse working.
(ii) Compound Interest Compound Interest as a repeated Simple Interest computation with a growing Principal. Use of formula –
A = P(1+r/100)n. Finding CI from the
relation CI = A-P. Simple direct problems based on above formulae.
3. Algebra
(i) Expansions
(a ± b)2
(a ± b)3
(x ± a)(x ± b)
(ii) Factorisation
a2 – b2
a3 ± b3
ax2 + bx + c, by splitting the middle term.
(iii) Changing the subject of a formula.
Concept that each formula is a perfect equation with variables.
Concept of expressing one variable in terms, of another various operators on terms transposing the terms squaring or taking square root etc.
(iv) Linear Equations and Simultaneous (linear) Equations
Solving algebraically (by elimination as well as substitution) and graphically.
Solving simple problems based on these by framing appropriate formulae.
(v) Indices/ Exponents
Handling positive, fractional, negative and "zero" indices.
Simplification of expressions involving various exponents
etc use of laws of exponents.
(vi) Logarithms
(a) Logarithmic form vis-à-vis exponential form: interchanging.
(b) Laws of Logarithms and its use Expansion of expression with the help of laws of logarithm
(v) Pythagoras Theorem Proof and Simple applications of Pythagoras Theorem and its converse.
(vi) Rectilinear Figures
Rectilinear figures or polygons, Different kinds of polygons and its names interior and exterior angles and their relations. Types of regular polygons parallelograms, conditions of parallelograms, Rhombus, Rectangles. Proof and use of theorems on parallelogram.
(b) Being given a figure, to draw its lines of symmetry. Being given part of one of the figures listed above to draw the rest of the figure based on the given lines of symmetry (neat recognizable free hand sketches acceptable).
(ii) Similarity
Axioms of similarity of triangles. Basic theorem of proportionality.
(a) Areas of similar triangles are proportional to the squares on corresponding sides.
(b) Direct applications based on the above including applications to maps and models.
(iii) Loci
Loci: Definition, meaning, Theorems based on Loci.
(a) The locus of a point equidistant from a fixed point is a circle with the fixed point as centre.
(b) The locus of a point equidistant from two interacting lines is the bisector of the angles between the lines.
(c) The locus of a point equidistant from two given points is the perpendicular bisector of the line joining the points.
(iv) Circles
(a) Chord Properties:
A straight line drawn from the center of a circle to bisect a chord which is not a diameter is at right angles to the chord.
The perpendicular to a chord from the center bisects the chord (without proof).
Equal chords are equidistant from the center.
Chords equidistant from the center are equal (without proof).
There is one and only one circle that passes through three given points not in a straight line.
(b) Arc and chord properties:
The angle that an arc of a circle subtends at the center is double that which it subtends at any point on the remaining part of the circle.
Angles in the same segment of a circle are equal (without proof).
Angle in a semi-circle is a right angle.
If two arcs subtend equal angles at the center, they are equal, and its converse.
If two chords are equal, they cut off equal arcs, and its converse (without proof).
If two chords intersect internally or externally then the product of the lengths of the segments are equal.
(c) Cyclic Properties:
Opposite angles of a cyclic quadrilateral are supplementary.
The exterior angle of a cyclic quadrilateral is equal to the opposite interior angle (without proof).
(d) Tangent Properties:
The tangent at any point of a circle and the radius through the point are perpendicular to each other.
If two circles touch, the point of contact lies on the straight line joining their centers.
From any point outside a circle two tangents can be drawn and they are equal in length.
If a chord and a tangent intersect externally, then the product of the lengths of segments of the chord is equal to the square of the length of the tangent from the point of contact to the point of intersection.
If a line touches a circle and from the point of contact, a chord is drawn, the angles between the tangent and the chord are respectively equal to the angles in the corresponding alternate segments.
Note: Proofs of the theorems given above are to be taught unless specified otherwise.
(v) Constructions
(a) Construction of tangents to a circle from an external point.
(b) Circumscribing and inscribing a circle on a triangle and a regular hexagon.
4. Mensuration
Area and circumference of circle, Area and volume of solids – cone, sphere.
(a) Circle: Area and Circumference. Direct application problems including Inner and Outer area..
(b) Three-dimensional solids – right circular cone and sphere: Area (total surface and curved surface) and Volume. Direct application problems including cost, Inner and Outer volume and melting and recasting method to find the volume or surface area of a new solid. Combination of two solids included.
Note: Frustum is not included.
Areas of sectors of circles other than quartercircle and semicircle are not included.
(tossing of one or two coins, throwing a die and selecting a student from a group)
Note:SI units, signs, symbols and abbreviations
(1) Agreed conventions
(a) Units may be written in full or using the agreed symbols, but no other abbreviation may be used.
(b) The letter 's' is never added to symbols to indicate the plural form.
(c) A full stop is not written after symbols for units unless it occurs at the end of a sentence.
(d) When unit symbols are combined as a quotient, e.g. metre per second, it is recommended that they be written as m/s, or as m s-1.
(e) Three decimal signs are in common international use: the full point, the mid-point and the comma. Since the full point is sometimes used for multiplication and the comma for spacing digits in large numbers, it is recommended that the mid-point be used for decimals.
(2) Names and symbols INTERNAL ASSESSMENT
The minimum number of assignments: Three assignments as prescribed by the teacher.
Suggested Assignments
Comparative newspaper coverage of different items.
Survey of various types of Bank accounts, rates of interest offered.
Planning a home budget.
Cutting a circle into equal sections of a small central angle to find the area of a circle by using the formula A = pr2.
To use flat cut outs to form cube, cuboids, pyramids and cones and to obtain formulae for volume and total surface area.
To use a newspaper to study and report on shares and dividends.
Draw a circle of radius r on a ½ cm graph paper, and then on a 2 mm graph paper. Estimate the area enclosed in each case by actually counting the squares. Now try out with circles of different radii. Establish the pattern, if any, between the two observed values and the theoretical value (area = pr2). Any modifications?
Set up a dropper with ink in it vertical at a height say 20 cm above a horizontally placed sheet of plain paper. Release one ink drop; observe the pattern, if any, on the paper. Vary the vertical distance and repeat. Discover any pattern of relationship between the vertical height and the ink drop observed.
You are provided (or you construct a model as shown) – three vertical sticks (size of a pencil) stuck to a horizontal board. You should also have discs of varying sizes with holes (like a doughnut). Start with one disc; place it on (in) stick A. Transfer it to another stick (B or C); this is one move (m). Now try with two discs placed in A such that the large disc is below and the smaller disc is above (number of discs = n=2 now). Now transfer them one at a time in B or C to obtain similar situation (larger disc below). How many moves? Try with more discs (n = 1, 2, 3, etc.) and generalise.
The board has some holes to hold marbles, red on one side and blue on the other. Start with one pair. Interchange the positions by making one move at a time. A marble can jump over another to fill the hole behind. The move (m) equal 3. Try with 2 (n=2) and more. Find relationship between n and m.
Take a square sheet of paper of side 10 cm. Four small squares are to be cut from the corners of the square sheet and then the paper folded at the cuts to form an open box. What should be the size of the squares cut so that the volume of the open box is maximum?
Take an open box, four sets of marbles (ensuring that marbles in each set are of the same size) and some water. By placing the marbles and water in the box, attempt to answer the question: do larger marbles or smaller marbles occupy more volume in a given space?
An eccentric artist says that the best paintings have the same area as their perimeter (numerically). Let us not argue whether such sizes increases the viewer's appreciation, but only try and find what sides (in integers only) a rectangle must have if its area and perimeter are to be equal (note: there are only two such rectangles).
Find by construction the centre of a circle, using only a 60-30 setsquare and a pencil.
Various types of "cryptarithm".
EVALUATION
The assignments/project work are to be evaluated by the subject teacher and by an External Examiner. (The External Examiner may be a teacher nominated by the Head of the school, who could be from the faculty, but not teaching the subject in the section/class. For example, a teacher of Mathematics of Class VIII may be deputed to be an External Examiner for Class X, Mathematics projects.) The Internal Examiner and the External Examiner will assess the assignments independently. | 677.169 | 1 |
Synopses & Reviews
Publisher Comments:
You'll be inspired by this low-stress guide to a high-tech math learning tool!
Your TI-Nspire is unlike any mathematical tool you've ever seen, so you'll really appreciate this plain-English guide to what it can do and how to do it. From loading the batteries and creating a document to performing geometric calculations and constructing statistical graphs, you'll see how to use the TI-Nspire alone and with your PC.
Start here — set up your TI-Nspire handheld, get familiar with the keypad, use the function keys, and configure system settings
Synopsis:
Synopsis:
About the Author
C. C. Edwards has a Ph.D. in mathematics from the University of Wisconsin, Milwaukee, and is currently teaching mathematics on the undergraduate and graduate levels. She has been using technology in the classroom since before Texas Instruments came out with its first graphing calculator, and she frequently gives workshops at national and international conferences on using technology in the classroom. She is the author of TI-83 Plus Graphing Calculator for Dummies and TI-84 Plus Graphing Calculator for Dummies and she has written 40 activities for the Texas Instruments Explorations Web site. She was an editor of Eightysomething!, a newsletter that used to be published by Texas Instruments. She still hasn't forgiven TI for canceling that newsletter.
Just barely five feet tall, CC, as her friends call her, has three goals in life: to be six inches taller, to have naturally curly hair, and to be independently wealthy. As yet, she is nowhere close to meeting any of these goals. When she retires, she plans to become an old-lady carpenter.
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The Dozer, January 17, 2011 (view all comments by The Dozer)
This book makes the TI-89 understandable. If you are taking any math classes above 90 you will need it. It really helps with Statistics. It's a pretty complex calculator so if your not good with technical stuff? Get the book!!
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by Wiley,"Synopsis"
by Wiley, | 677.169 | 1 |
The aftermath of calculator use in college classrooms
Date:
November 12, 2012
Source:
University of Pittsburgh
Summary:
Math instructors promoting calculator usage in college classrooms may want to rethink their teaching strategies, experts say. They have proposed the need for further research regarding calculators' role in the classroom after conducting a limited study with undergraduate engineering students.
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Math instructors promoting calculator usage in college classrooms may want to rethink their teaching strategies, says Samuel King, postdoctoral student in the University of Pittsburgh's Learning Research & Development Center. King has proposed the need for further research regarding calculators' role in the classroom after conducting a limited study with undergraduate engineering students published in the British Journal of Educational Technology.
"We really can't assume that calculators are helping students," said King. "The goal is to understand the core concepts during the lecture. What we found is that use of calculators isn't necessarily helping in that regard."
Together with Carol Robinson, coauthor and director of the Mathematics Education Centre at Loughborough University in England, King examined whether the inherent characteristics of the mathematics questions presented to students facilitated a deep or surface approach to learning. Using a limited sample size, they interviewed 10 second-year undergraduate students enrolled in a competitive engineering program. The students were given a number of mathematical questions related to sine waves -- a mathematical function that describes a smooth repetitive oscillation -- and were allowed to use calculators to answer them. More than half of the students adopted the option of using the calculators to solve the problem.
"Instead of being able to accurately represent or visualize a sine wave, these students adopted a trial-and-error method by entering values into a calculator to determine which of the four answers provided was correct," said King. "It was apparent that the students who adopted this approach had limited understanding of the concept, as none of them attempted to sketch the sine wave after they worked out one or two values."
After completing the problems, the students were interviewed about their process. A student who had used a calculator noted that she struggled with the answer because she couldn't remember the "rules" regarding sine and it was "easier" to use a calculator. In contrast, a student who did not use a calculator was asked why someone might have a problem answering this question. The student said he didn't see a reason for a problem. However, he noted that one may have trouble visualizing a sine wave if he/she is told not to use a calculator.
"The limited evidence we collected about the largely procedural use of calculators as a substitute for the mathematical thinking presented indicates that there might be a need to rethink how and when calculators may be used in classes -- especially at the undergraduate level," said King. "Are these tools really helping to prepare students or are the students using the tools as a way to bypass information that is difficult to understand? Our evidence suggests the latter, and we encourage more research be done in this area."
King also suggests that relevant research should be done investigating the correlation between how and why students use calculators to evaluate the types of learning approaches that students adopt toward problem solving in mathematics.
University of Pittsburgh. "The aftermath of calculator use in college classrooms." ScienceDaily. ScienceDaily, 12 November 2012. <
University of Pittsburgh. (2012, November 12). The aftermath of calculator use in college classrooms. ScienceDaily. Retrieved September 5, 2015 from
University of Pittsburgh. "The aftermath of calculator use in college classrooms 11, 2015 — A new study has found that teachers who report having more symptoms of depression had classrooms that were of lesser quality, and that students in these classrooms had fewer performance gains. ... read more
Sep. 2, 2014 — STEM fields are heavily dominated by males, which is of concern to universities as they try to improve student retention and achievement. One exception is in the field of biology. Of undergraduate ... read more
Feb. 5, 2013 — Researchers have developed a classroom design that gives instructors increased flexibility in how to teach their courses and improves accessibility for students, while slashing administrative ... read more | 677.169 | 1 |
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Learning French to follow some mathematics texts @SabyasachiMukherjee, the symbols might actually not be exactly the same either. Contrarily to popular belief, maths notations are not universal. In my experience, English-based maths usually seems less rigorous in terms of presentation than what you'd find in French books (possibly due to bourbakism). This won't be a problem if you're just reading, it will be if you plan to take exams. | 677.169 | 1 |
Thousands of FREE, short, online videos that are focused on explaining and modeling the learning of specific topics in math...
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Thousands of FREE, short, online videos that are focused on explaining and modeling the learning of specific topics in math (basic arithmetic and math to calculus), statistics, biology, physics, chemistry, finance, and other topics. The topics cover K-12 levels and higher education. The simple and clear presented information enables learners to see and review the topics and how to solve the problems at their pace with as much practice as they wish. In particular there are over a thousand videos just for mathematics. The site also contains a handful of interactive mathematics learning objects that are of the drill and practice (Math Portion) to your Bookmark Collection or Course ePortfolio
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This site contains a catalogue of video clips from course lectures outside of mathematics that use mathematics. The following...
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This site contains a catalogue of video clips from course lectures outside of mathematics that use mathematics. The following are given: the math being used in the video, the link to the video clip, the time segment within the lecture that contains that mathematics application, the university where the course was taught, the course title, the name of the professor, and a brief description of the content and how it might be used in the relevant mathematics course. The site is indexed by mathematics application with four main categories: Statistics, Algebra and Pre-Calculus, First Year Calculus, and Second Year Calculus. The site is meant for mathematics instructors who want to show their students how mathematics is used. Students will get an authentic demonstration on where they will be seeing the math in their other Catalog of Mathematics Applications Found in Non-Mathematics Courses to your Bookmark Collection or Course ePortfolio
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Quoted from the site: "IDEA is an interdisciplinary effort to provide students and teachers around the world with...
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Quoted from the site: "IDEA is an interdisciplinary effort to provide students and teachers around the world with computer based activities for differential equations in a wide variety of disciplines." Currently, IDEA contains nearly twenty activities. These are applications of differential equations to areas as diverse as bungee jumping and salmon migration. Some of these applications are presented as text with illustrations, but others include interactive IDEA: Internet Differential Equations Activities to your Bookmark Collection or Course ePortfolio
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Select this link to open drop down to add material IDEA: Internet Differential Equations Activities to your Bookmark Collection or Course ePortfolio | 677.169 | 1 |
This eModule presents real-world problems that motivate students to work with right triangle geometry. Interactive worksheets and diagrams with feedback help students learn to solve for unknown angles... More: lessons, discussions, ratings, reviews,...
The eNLVM enhances the applets available through the National Library of Virtual Manipulatives (NLVM). It allows teachers to embed applets in lessons, make up questions to which students can submit an... More: lessons, discussions, ratings, reviews,...
This tool lets you plot functions, polar plots, and 3D with just a suitable web browser (within the IE, FireFox, or Opera web browsers), and find the roots and intersections of graphs. In addition, yo... More: lessons, discussions, ratings, reviews,...
This Graphing Calculator website contains hundreds of examples of how to best use the graphing calculator with specific classroom examples. Unlike using a manual, teachers and students can see how to... More: lessons, discussions, ratings, reviews,...
A story from the middle school classroom: Ihor describes how he scheduled a "contest" for students to show what they know about slopes and y-intercepts using the Green Globs software. He also provides... More: lessons, discussions, ratings, reviews,...
A very powerful graphing program that is also especially easy to use. You can graph functions in two or more dimensions using different kinds of coordinates. You can make animations and save as movies... More: lessons, discussions, ratings, reviews,...
Students explore the relationship between equations and their graphs in this hands-on learning environment where they investigate, manipulate, and understand linear, quadratic and other graphs. They ... More: lessons, discussions, ratings, reviews,...
The Vector Algebra Tools are a comprehensive set of vector algebra calculators that are specifically designed for the study of vectors and vector algebra applications in high school and first year of ... More: lessons, discussions, ratings, reviews,...
The "Scrambler" is an amusement park ride with a central rotating hub with three central arms, with spokes at the end of each arm that rotate in a different direction. This applet, approved by the El... More: lessons, discussions, ratings, reviews,...
With a scope that spans the mathematics curriculum from middle school to college, The Geometer's Sketchpad brings a powerful dimension to the study of mathematics. Sketchpad is a dynamic geometry cons... More: lessons, discussions, ratings, reviews,...
Students compare rectangular graphs and polar graphs for functions of the form y = a sin(bx) and r = a sin(b?). Students analyze how the period and amplitude of a Cartesian graph correlate to featu... More: lessons, discussions, ratings, reviews,...
All the familiar capabilities of current TI scientific calculators plus a host of powerful enhancements. Designed with unique features that allow you to enter more than one calculation, compare result | 677.169 | 1 |
Consumer Math Elective
Course Length
2 semesters
Available in
Ignitia
Switched-On Schoolhouse
LIFEPAC
This practical math elective trains students in mathematical applications used in everyday situations. Consumer math includes real-world examples and an emphasis on critical thinking skills to solve problems. Topics in the first semester of this course from our online academy include an overview of basic math skills, personal finance skills, statistics and home recordkeeping, taxes, insurance, and banking services.
Building financial literacy, this course from our online academy's second semester includes topics such as credit cards and loan interest, purchasing items, discounts and markups, travel and transportation costs, vacation spending, retirement planning, and job related services. Encouraging solid financial habits, consumer math is essential for success in adulthood no matter students' desired career paths. Each unit of the course contains quizzes and a test to evaluate progress and student mastery.
Additional Details
Ready to Get Started with Our Online Academy?
Alpha Omega Academy has year-long open enrollment, so you can start this course at any time! Visit the tuition page of our online academy to learn more about pricing or click the button below to get started with enrollment today. Have questions first? Call us at 800.682.7396.
Our son just graduated after being part of the Alpha Omega Academy (AOA) family for 7 years. We originally chose AOA because we wanted a sound Bible-based curriculum that would also prepare our son for college and "life after school." As we moved closer to junior high and high school years, we decided it was essential to have a structured set of courses. We used AOA's Switched-on-Schoolhouse (SOS) ...more for 6th through 10th grades, a combination of SOS and LIFEPAC for 11th grade, and LIFEPAC only for 12th grade. The courses are appropriately challenging so that the extent and breadth of what is learned is of high quality. We found the curriculum to be so well rounded our son was well prepared for our annual standardized testing and for the SAT and ACT tests. We chose the college prep courses and our son was accepted into the college he really wanted to attend. A tremendous benefit with AOA is that they maintain all of the transcripts, and this was a VERY big plus for us for the high school years with college applications and our required state reporting. The AOA graduation ceremony was outstanding and we would not have closed out our season of home education any other way! We highly recommend AOA!Nancy W. – VA | 677.169 | 1 |
Search Results
The University of Akron has created these excellent algebra tutorials that review some of the main topics in the discipline. There are ten lessons, which focus on topics like radicals and exponents, basic algebra,...
Created and maintained by John J. O'Connor and Edmund F. Robertson of the University of St. Andrews, this site contains a cornucopia of materials related to the history of mathematics and well-known mathematicians...
Presented by the University of Illinois at Urbana-Champaign, this page allows visitors access to courses designed with Mathematica to "achieve a better conceptual understanding of the material while still gaining a good...
From the Graduate Texts in Mathematics series comes this textbook on graph theory by Reinhard Diestel from the University of Hamburg. Topics covered include flows, planar graphs, infinite graphs, and Hamilton cycles. ...
PBS presents this game in which students are asked to rearrange and rotate a set of geometric shapes to form the image of a rabbit, candle, fox, and others. Tangram "is an ancient game that originated in China" and it... | 677.169 | 1 |
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Texas Instruments' TI InterActive! 1.1
05/01/02
TI InterActive! is an integrated software package that assists teachers in developing lessons and in-class activities that promote the understanding of math and science concepts, and the improvement of students' problem-solving skills. Designed for the Windows platform, TI InterActive! has word processing capabilities that have been enhanced with the graphing functionality of the TI-83 Plus, as well as the symbolical algebraic system of the TI-92 and TI-89.
The program allows students to write lab reports, graph data and perform algebraic computations all within one document. It enables teachers to create interactive worksheets for students to investigate mathematical functions and concepts graphically, analytically and numerically, and then communicate the discoveries they have made. The program also has a built-in Web browser that lets the user extract tabular data quickly into the document to be graphed and modeled.
Most classrooms today have a computer with Internet access, but many math and science classrooms do not have class sets of graphing products like the TI-83 Plus. In a classroom with just one computer, I use TI Interactive! to prompt student discussion when we are learning about a new function and the transformations of this function. I change the parameters using the slider bar, and ask the students to discuss in their small groups what effect changing this parameter has on the parent function.
In the lab, my students extract data from the Internet, find a mathematical model that fits the data, and interpolate and extrapolate information about the event based on their model. Using TI InterActive!, I save an entire class period. Before we had this program, my class would go to the computer lab and spend the entire time copying and inputting data from the Internet into their handheld graphing products. Now inputting data takes about five seconds, letting instructors move onto the important concepts of the lesson, instead of relying on the students to comprehend the lesson on their own.
Strengths and Weaknesses
TI InterActive! is simple to use compared to the learning curve required by other math and science software. The product not only graphs, but also allows students and teachers to label their graphs - coordinates and elements - easily. The draw features in the graphing environment allow us to create tangent lines, asymptotes and other components of the graph in addition to the function. The program lets students quickly extract data from the Internet and data-collecting devices, such as Calculator-Based Ranger (CBR) and CBL2, and write up lab reports using the graphing, algebraic and word processing capabilities. Teachers can create quizzes, in-class activities, tests and homework assignments. These activities can be saved in multiple formats, including HTML, Microsoft Word and Rich Text Format (RTF).
TI InterActive! works with TI's family of graphing products, including the TI-83, TI-83 Plus, TI-89 and TI-92, as well as the CBR and CBL2 data-collecting devices, but these products are not required. Yet, despite the program's benefits, its word processing capabilities need further development. For one thing, the spell checker is not automatic. My students would like the next version to include automatic formatting capabilities, because algebraic and graphing objects are more cumbersome than pictures to format in text documents | 677.169 | 1 |
often a comprehensive review of all basic mathematical concepts and prepares students for further coursework. The arithmetic is presented with an emphasis on problem solving, skills, and concepts, with some introductory algebra integrated throughout | 677.169 | 1 |
The history of mathematics and number theory from ancient times to the present is covered through methods and concepts, including theorems of Format, Euler, divisibility, factorization, primes, congruencies, diophantine problems, and other topics. Prerequisite: upper-division class standing | 677.169 | 1 |
Algebra & Trigonometry
9780132191401
ISBN:
0132191407
Edition: 3 Pub Date: 2006 Publisher: Prentice Hall
Summary: Gets Them Engaged. Keeps Them Engaged Blitzer's philosophy: present the full scope of mathematics, while always (1) engaging the student by opening their minds to learning (2) keeping the student engaged on every page (3) explaining ideas directly, simply, and clearly so they don't get "lost" when studying and reviewing.
Blitzer, Robert is the author of Algebra & Trigonometry, published 2006 under ISBN 97801...32191401 and 0132191407. Twelve Algebra & Trigonometry textbooks are available for sale on ValoreBooks.com, eight used from the cheapest price of $7.00, or buy new starting at $187 | 677.169 | 1 |
This introduction to computational geometry focuses on algorithms. Motivation is provided from the application areas as all techniques are related to particular applications in robotics, graphics, CAD/CAM, and geographic information systems. Modern insights in computational geometry are used to provide solutions that are both efficient and easy to understand and implement.
Product Description
Review
"An excellent introduction to the field is given here, including a general motivation and usage cases beyond simple graphics rendering or interaction." from the ACM Reviews by William Fahle, University of Texas at Dallas, USA
Most helpful customer reviews
Easy reading, excellent text on the topic. I'm coming from a Geomatics (CompSci/Geog) based background and in my 4th year of University.
Every chapter starts with an overview of the problem with real world examples, simple solutions to this that are not optimized nor consider degenerate cases, and then goes into a 'how can we make this better' style of discussion with excellent justifications along the way.
There is an expectation that you are familiar with basic algorithm design, performance analysis, and data structures.
Most Helpful Customer Reviews on Amazon.com (beta)
Amazon.com:
13 reviews
6 of 7 people found the following review helpful
Good overview but lacking detailMarch 17 2011
By
David
- Published on Amazon.com
Format: Hardcover
This is a very well written book and covers a wide range of computational geometry problems. It is a very good introduction/overview to computational geometry. I would have preferred more detail, specifically code examples. This is a good book to gain an understanding of the topic but not so good if you are actually trying to implement the concepts in code or make the best use existing code libraries.
2 of 2 people found the following review helpful
A textbook for undergraduatesNov. 4 2014
By
Clive McCarthy
- Published on Amazon.com
Format: Hardcover
Verified Purchase
This is a computer science textbook for undergraduates. Lots of "real world" uses for computational geometry to egg-on the unmotivated. Not so good for those that ARE already motivated.
The text glosses over basic tasks such as "whether a point lies to the left or right of a directed line" (page 4) with the expectation that some unnamed library function will do this. For me, not so. Moreover I'd would have liked to read a geometric proof of such things. The foundations are left out, yet elsewhere they waste space to give us Pythagoras' Theorem.
2 of 2 people found the following review helpful
Fantastic explanation -- very clear, with great topicsJuly 30 2013
By
EE Codewright
- Published on Amazon.com
Format: Hardcover
Verified Purchase
It is a joy to read and review this book -- the exposition is crystal clear; the writing style is warm and engaging (not too terse and not too verbose), conveying understanding and not just stating facts, theorems, and algorithms; the graphics are great (numerous richly detailed illustrations); the topics hit the heart of computational geometry; the historical remarks help set context; and the book is beautifully typeset and printed on high quality acid free paper.
1 of 1 people found the following review helpful
Good introduction to the topics but is not good at explaningApril 8 2014
By
mackster
- Published on Amazon.com
Format: Hardcover
Verified Purchase
This is the standard text book for CG, and it nicely introduces us to a lot of concepts. But, unfortunately it is not the best book I have read. Most of the examples albeit few, does not make much sense. The algorithms discussed sometimes cannot be grasped. I often went online to read more about the subject to understand the topics. This also could mean that my grasping of the subject is low :), but that should not matter as long as the material is explained clearly.
Solid introductionAug. 16 2012
By
B. Figares
- Published on Amazon.com
Format: Hardcover
Verified Purchase
Beautiful book, solid contents. I learned a lot from it and had a nice time practicing with the exercises. Lots of examples and problems, a lot of interesting algorithms and techniques, every chapter is a progressive refinement of a particular idea to solve a problem expressed as geometry.
Difficulty level: make sure you know some asymptotic analysis and discrete mathematics to get the best out of it, but could be read by anyone who can code i believe (although again, he'll miss a lot of beautiful mathematics) | 677.169 | 1 |
This Mathematics Textbook for Class 10 is designed by the National Council of Educational Research and Training in India and this makes it highly reliable to the student of Mathematics in class 10. The book consists of 14 chapters with all the topics of mathematics that are crucial for a student of class ten to be well versed in. The book is easy to follow and gives the student a good guide to the subject and prepares him well for his examination.
About the National Council of Educational Research and Training
The National Council of Educational Research and Training provides textbooks for all subjects for students from the class 1 to class 12. The council provides textbooks in English, Hindi and Urdu. Some of the other books provided by the NCERT include Source Book on Assessment for Classes 1 to 5, Environmental Education for Class X and The Reflective Teacher to name a few.
this book gives you a complete study of all the basic theorems and proofs needed for a student . its a very helpful textbook helping to develop the mathematical skills of a student. overall it can be kept for future as a hand book for the basic knowledge of mathematics | 677.169 | 1 |
Paperback
Overview
John Bird's approach to mathematics, based on numerous worked examples and interactive problems, is ideal for level 2 and 3 vocational courses including the BTEC National specifications.
Theory is kept to a minimum, with the emphasis firmly placed on problem-solving skills, making this a thoroughly practical introduction to the mathematics that students need to master.
Now in its seventh edition, Engineering Mathematics has helped thousands of students to succeed in their exams. The new edition includes a section at the start of each chapter that explains and how it relates to real life engineering projects.
It is supported by a fully updated companion website with resources for both students and lecturers. It has 1000 worked problems, 238 multiple-choice questions, and full solutions to all 1800 further questions contained in the 237 practice exercises. All 525 illustrations used in the text can also be downloaded for use in the classroom. | 677.169 | 1 |
Plantation, FL ChemistryVy TJon-Erik R.
...Advanced functions such as Ln and Exponential functions are also explained in the subject. The focus on differences become crucial when dealing with advanced mathematics. Calculus branches into two sections, differential and integral calculus. | 677.169 | 1 |
Summary
Introductory Algebra prepares students for Intermediate Algebra by covering fundamental algebra concepts and key concepts needed for further study. Students of all backgrounds will be delighted to find a refreshing book that appeals to every learning style and reaches out to diverse demographics. Through down-to-earth explanations, patient skill-building, and exceptionally interesting and realistic applications, this worktext will empower students to learn and master algebra in the real world.
Table of Contents
R PREALGEBRA REVIEW
R1 Fractions
R2 Operations with Fractions
R3 Decimals and Percents
1 REAL NUMBERS AND THEIR PROPERTIES
1.1 Introduction to Algebra
1.2 The Real Numbers
1.3 Adding and Subtracting Real Numbers
1.4 Multiplying and Dividing Real Numbers
1.5 Order of Operations
1.6 Properties of the Real Numbers
1.7 Simplifying Expressions
2 EQUATIONS, PROBLEM SOLVING, AND INEQUALITIES
2.1 The Addition and Subtraction Properties of Equality
2.2 The Multiplication and Division Properties of Equality: Applications with Percents | 677.169 | 1 |
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About the Book
Computing Curricula 2001 (CC2001), a joint undertaking of the Institute for Electrical and Electronic Engineers/Computer Society (IEEE/CS) and the Association for Computing Machinery (ACM), identifies the essential material for an undergraduate degree in computer science. This Sixth Edition of "Mathematical Structures for Computer Science" covers all the topics in the CC2001 suggested curriculum for a one-semester intensive discrete structures course, and virtually everything suggested for a two-semester version of a discrete structures course. Gersting's text binds together what otherwise appears to be a collection of disjointed topics by emphasizing the following themes: - Importance of logical thinking - Power of mathematical notation - Usefulness of abstractions | 677.169 | 1 |
...
Show More to write clear, correct, and concise proofs. Unlike similar textbooks, this one begins with logic since it is the underlying language of mathematics and the basis of reasoned arguments. The text then discusses deductive mathematical systems and the systems of natural numbers, integers, rational numbers, and real numbers. It also covers elementary topics in set theory, explores various properties of relations and functions, and proves several theorems using induction. The final chapters introduce the concept of cardinalities of sets and the concepts and proofs of real analysis and group theory. In the appendix, the author includes some basic guidelines to follow when writing proofs. Written in a conversational style, yet maintaining the proper level of mathematical rigor, this accessible book teaches students to reason logically, read proofs critically, and write valid mathematical proofs. It will prepare them to succeed in more advanced mathematics courses, such as abstract algebra and geometry.
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Matrices and Linear Transformations
"Comprehensive . . . an excellent introduction to the subject." — Electronic Engineer's Design Magazine. This introductory textbook, aimed at sophomore- and junior-level undergraduates in mathematics, engineering, and the physical sciences, offers a smooth, in-depth treatment of linear algebra and matrix theory. The major objects of study are matrices over an arbitrary field. Contents include Matrices and Linear Systems; Vector Spaces; Determinants; Linear Transformations; Similarity: Part I and Part II; Polynomials and Polynomial Matrices; Matrix Analysis; and Numerical Methods. The first seven chapters, which require only a first course in calculus and analytic geometry, deal with matrices and linear systems, vector spaces, determinants, linear transformations, similarity, polynomials, and polynomial matrices. Chapters 8 and 9, parts of which require the student to have completed the normal course sequence in calculus and differential equations, provide introductions to matrix analysis and numerical linear algebra, respectively. Among the key features are coverage of spectral decomposition, the Jordan canonical form, the solution of the matrix equation AX = XB, and over 375 problems, many with answers.
Unlike Elementary Matrix Theory, by Howard Eves, this book does share a lot of material with the linear algebra texts published in 2010. However, there is still some material in this book, such as canonical forms, that doesn't typically appear in introductory linear algebra books. While the author does write clearly, the writing is at a more mathematically sophisticated level than the books aimed at introducing linear algebra to students in 2010. Considering the very cheap price of this Dover reprint, this is a worthwhile second book to have on linear algebra. See also Elementary Matrix Theory, by Howard Eves, to find a book with much material that is difficult to find in more recently published books.
Review: Matrices and Linear Transformations
User Review - Darin - Goodreads
Unlike Elementary Matrix Theory, by Howard Eves, this book does share a lot of material with the linear algebra texts published in 2010. However, there is still some material in this book, such as ...Read full review | 677.169 | 1 |
Glen Echo ACT MathAzadeh MClark R.
...During this level course, students gain proficiency in solving linear equations, inequalities, and systems of linear equations. New concepts include solving quadratic equations and inequalities, exploring conics, investigating polynomials, and applying/using matrices to organize and interpret data. Students will also investigate exponential and logarithmic functions.
John T.Daniel B.
...Coast Guard and a Naval Architect and Marine Engineer. I apply advanced mathematics, physics, and technical writing on a daily basis in my job, and I love it! I hope to inspire the same passion I have in these subject areas in the students I work with. | 677.169 | 1 |
About this item
Comments: Brand new. We distribute directly for the publisher. In 1963 Oystein Ore wrote this classic volume, which was published in the New Mathematical Library Series. This elegant book has provided students and teachers with an excellent introduction to the field of graph theory for close to thirty years. Robin Wilson's revision adds strength to the book by updating the terminology and notation, bringing them in line with contemporary usage. Wilson has added new material on interval graphs, the traveling salesman
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$25.59883856352
ISBN: 0883856352
Publisher: Mathematical Association of America
AUTHOR
Ore, Oystein, Wilson, Robin J.
SUMMARY
The New Mathematical Library series features fresh approaches and broad coverage of topics especially suitable for high school and the first two years of college.Ore, Oystein is the author of 'Graphs and Their Uses' with ISBN 9780883856352 and ISBN 0883856 | 677.169 | 1 |
Note: Citations are based on reference standards. However, formatting rules can vary widely between applications and fields of interest or study. The specific requirements or preferences of your reviewing publisher, classroom teacher, institution or organization should be applied.
Math and art : an introduction to visual mathematics
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Abstract:
Explores the potential of mathematics to generate visually appealing objects and reveals some of the beauty of mathematics. This title includes numerous illustrations, computer-generated graphics, photographs, and art reproductions to demonstrate how mathematics can inspire art.Read more...
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schema:Review ; schema:itemReviewed < ; # Math and art : an introduction to visual mathematics schema:reviewBody ""." ; . | 677.169 | 1 |
Mathematics for Physical Chemistry
By
Robert Mortimer, Rhodes College, Memphis, TN, USA
Mathematics for Physical Chemistry is the ideal supplementary text for practicing chemists and students who want to sharpen their mathematics skills while enrolled in general through physical chemistry courses. This book specifically emphasizes the use of mathematics in the context of physical chemistry, as opposed to being simply a mathematics text.This 4e includes new exercises in each chapter that provide practice in a technique immediately after discussion or example and encourage self-study. The early chapters are constructed around a sequence of mathematical topics, with a gradual progression into more advanced material. A final chapter discusses mathematical topics needed in the analysis of experimental data.
Audience
New chemistry researchers; freshmen through juniors, seniors and graduates students enrolled in general through physical chemistry courses; especially students in lower- and upper-division honors chemistry courses.
Book information
Published: June 2013
Imprint: ELSEVIER
ISBN: 978-0-12-415809-2
Reviews
"The text is extremely clear and concise delivering exactly what the student needs to know in a pinch - nothing more, nothing less. It is an indispensable resource for any student of physical chemistry."--Gregory S. Engel, Harvard University
"Mathematics for Physical Chemistry is a comprehensive review of many useful mathematical topics...The book would be useful for anyone studying physical chemistry."--Daniel B. Lawson, University of Michigan-Dearborn
"The student will derive benefit from the clarity, and the professional from a concise compilation of techniques stressing application rather than theory.… Recommended."--John A. Wass for SCIENTIFIC COMPUTING AND INSTRUMENTATION | 677.169 | 1 |
Undergraduate Algebraic Geometry
9780521356626
ISBN:
0521356628
Pub Date: 1988 Publisher: Cambridge University Press
Summary: Algebraic geometry is, essentially, the study of the solution of equations and occupies a central position in pure mathematics. This short and readable introduction to algebraic geometry will be ideal for all undergraduate mathematicians coming to the subject for the first time. With the minimum of prerequisites, Dr Reid introduces the reader to the basic concepts of algebraic geometry including: plane conics, cubics... and the group law, affine and projective varieties, and non-singularity and dimension. He is at pains to stress the connections the subject has with commutative algebra as well as its relation to topology, differential geometry, and number theory. The book arises from an undergraduate course given at the University of Warwick and contains numerous examples and exercises illustrating the theory.
Reid, Miles is the author of Undergraduate Algebraic Geometry, published 1988 under ISBN 9780521356626 and 0521356628. Three hundred one Undergraduate Algebraic Geometry textbooks are available for sale on ValoreBooks.com, fifty one used from the cheapest price of $21.45, or buy new starting at $39.14.[read more]
Ships From:Multiple LocationsShipping:Standard, ExpeditedComments:RENTAL: Supplemental materials are not guaranteed (access codes, DVDs, workbooks). Almost new condition. SKU:9780521356626-2-0-3 Orders ship the same or next business day. Expedite... [more]RENTAL: Supplemental materials are not guaranteed (access codes, DVDs, workbooks). Almost9780521356626-2-0-3 Orders ship the same or next business day. Expedite... [more] | 677.169 | 1 |
9812380647Counting
This book is a useful, attractive introduction to basic counting techniques for upper secondary and junior college students, as well as teachers. Younger students and lay people who appreciate mathematics, not to mention avid puzzle solvers, will also find the book interesting. The various problems and applications here are good for building up proficiency in counting. They are also useful for honing basic skills and techniques in general problem solving. Many of the problems avoid routine and the diligent reader will often discover more than one way of solving a particular problem, which is indeed an important awareness in problem solving. The book thus helps to give students an early start to learning problem-solving heuristics and thinking skills | 677.169 | 1 |
9780201437249
ISBN:
0201437244
Edition: 2 Pub Date: 2000 Publisher: Pearson
Summary: Chapter Zero is designed for the sophomore/junior level Introduction to Advanced Mathematics course. Written in a modified R.L. Moore fashion, it offers a unique approach in which students construct their own understandings. However, while students are called upon to write their own proofs, they are also encouraged to work in groups. There are few finished proofs contained in the text, but the author offers proof ske...tches and helpful technique tips to help students as they develop their proof writing skills. This book is most successful in a small, seminar style class.
Carol Schumacher is the author of Chapter Zero: Fundamental Notions of Abstract Mathematics (2nd Edition), published 2000 under ISBN 9780201437249 and 0201437244. Two hundred fifty six Chapter Zero: Fundamental Notions of Abstract Mathematics (2nd Edition) textbooks are available for sale on ValoreBooks.com, sixty used from the cheapest price of $26.25, or buy new starting at $111.37 | 677.169 | 1 |
ReviewsA Google User
Works very well Nice scientific calculator with graphing, conversions, physics constants etc. Can do symbolic derivatives. It has just been published and looks very promising! I for one would welcome more symbolic stuff some time in future.Works very well Nice scientific calculator with graphing, conversions, physics constants etc. Can do symbolic derivatives. It has just been published and looks very promising! I for one would welcome more symbolic stuff some time in future. | 677.169 | 1 |
28Student Resource Guide To Accompany Excursions In Modern Math
Videos on DVD with Optional Subtitles for Excursions in Modern Mathematics
Customer Reviews
Just what I neededMay 6, 2011 by James Admans
This textbook explains the material in an understandable way. It has lots of examples and the homework section covers everything. I teach a university course based on this textbook and I really like it. It is full of mathematics that students can apply readily to everyday situations, without being heavily computational. This rental textbook is valuable and worthy, the book was in the condition it was said and I am extremely satisfied with dealing with ecampus.
Excursions in Modern Mathematics:
5 out of
5
stars based on
1 user reviews.
Summary
Excursions in Modern Mathematics, Seventh Edition, shows readers that math is a lively, interesting, useful, and surprisingly rich subject. With a new chapter on financial math and an improved supplements package, this book helps students appreciate that math is more than just a set of classroom theories: math can enrich the life of anyone who appreciates and knows how to use it.
This collection of "excursions" into modern mathematics consists of four independent learning modules--1) Social Choice, 2) Management Science, 3) Growth and Symmetry, and 4) Statistics. It is written in an informal, very readable style, with pedagogical features that make material both interesting and clear. Coverage centers around an assortment of real-world examples and applications, demonstrating the beauty and relevance of mathematics.
Author Biography
Peter Tannenbaum has bachelor's degrees in Mathematics and Political Science and a Ph. D. in Mathematics, all from the University of California, Santa Barbara. He has held faculty positions at the University of Arizona, Universidad Simon Bolivar (Venezuela), and is currently professor of mathematics at the California State University, Fresno. His current research interests are in the interface between mathematics, politics and behavioral economics. He is also involved in mathematics curriculum reform and teacher preparation. His hobbies are travel, foreign languages and sports. He is married to Sally Tannenbaum, a professor of communication at CSU Fresno, and is the father of three (twin sons and a daughter). | 677.169 | 1 |
Meike Akveld and Andrew Jobbings
In his classic Knot Book, Colin Adams relays the story of a hapless undergraduate who inquires of his calculus professor, "What kind of math do you study". When the professor replies, "Knot Theory," the student responds, "Oh good! I don't like theory either!" The misunderstanding aside, the student's response points to one of the delights of knot theory: one can learn and understand a fair amount of this subject without needing a lot of mathematical knowledge or technique. Indeed, knot theory might be the perfect way of enticing a mathematically bored middle or high school student into deeper and more interesting waters.
Knots Unravelled is a wonderful gift for such a student. It is slim, easy to read, and filled with delightful mathematical ideas. After beginning with a history of knots and knot theory, each chapter addresses some aspect of the mathematical theory of knots. The topics range from knot diagrams and the Reidemeister moves, to crossing number, unknotting number, 3-colorability, and the Jones polynomial. Only the last chapter, on the Jones polynomial, assumes any high-school algebra. Between the chapters are interludes which explore specific examples of knots. These include the Celtic knots and the more prosaic but tractable torus knots.
The organization and writing of the book are superb. I read it in an hour on a plane flight. The discussion is appropriately informal and is interspersed with exercises called "tasks." Many of the tasks are straightforward but non-trivial. A dedicated high school student could enjoyably pass quite a bit of time drawing pictures or playing with string in an effort to solve them. Solutions to the tasks are in the back, so even if the student is not dedicated, they can still learn how to solve the problems.
For readers who want more mathematical rigor and depth, the aforementioned book by Adams is a great place to start. From there, the reader can progress to any number of other undergraduate and graduate texts. | 677.169 | 1 |
With the advent of widely accessible, inexpensive (or even free) computational tools and Computer Algebra Systems (TI-89, Wolfram|Alpha, etc.), much of what traditionally comprises a high school math curriculum can now easily be done by almost everyone. Factoring polynomials, solving inequalities, graphing linear equations, differentiation and integration -- these are the types of skills high school math students spend most of their time learning, and yet all of it can be done for free by anyone with a web browser.
What does this mean for the high school math curriculum? On the one hand, we could leave it more-or-less the same, insisting that today's student learn what we learned decades ago, while banning or carefully regulating the use of these new tools. On the other hand, we could embrace the tools and the opportunities they create to spend more math class time on different topics and skills, perhaps focusing more on analytic and synthetic problem solving and less on mechanical symbolic manipulation -- but at the risk of students never learning some basic foundations.
So how about it? Binomial coefficients? The angle-addition formulas for trig functions? The conditions under which a function has an inverse? Basic computer programming? Keeping in mind that the vast majority of high school students do not go on to become professional mathematicians, what should the high school math curriculum consist of?
Btw, I post this question (inspired by this discussion) here because this is a community of thoughtful mathematicians. I recognize this discussion may belong in a different forum, but I don't what/where that forum is. Any suggestions are welcome.
Please set this to community-wiki if it does not get voted to close... (subjective yada-yada?) I liked Theo Gray's little rant when I first read it, though. :)
–
Guess who it is.Aug 23 '10 at 5:12
4
To a large extent, I think the premise is false--the skills (generally meaning symbolic manipulations) that can be replaced with CAS are not the core of secondary school curricula today. Further, the notion that secondary school curricula today are the same as they have been for the past decade or two or three is just not true. There is a somewhat cyclical nature to curricula on a national level in the U.S., but even within that, there are shifts, and local variance in the U.S. is very high (and then there's international curricula).
–
IsaacAug 23 '10 at 5:22
1
On the other hand, there is the persistent accusation that things are getting "dumbed down" even more, every year. Having been off academia for quite a while, it might be great if answers to this question can address that, or better yet, the ones fresh from or are still in the "affected" levels can chime in (under the presumption that since you hang around here, you have an idea on what is or isn't missing in your education).
–
Guess who it is.Aug 23 '10 at 5:35
1
...and probably the lesson I most want to be hammered home to students: don't (blindly) trust the computer/calculator. Verify, verify, verify!
–
Guess who it is.Aug 23 '10 at 12:12
1
Statistics. it was an optional class in my HS, but if there is anything in math that is used more often and more maliciously to defraud people and persuade the public opinion I can't think of it. People should be aware of the limitations and the meaning of statistical results.
–
crasicOct 30 '10 at 22:58
7 Answers
7
I spent over a decade thinking that "mathematics" is about completing page after page of identical long division problems. Only after I left school did I discover that mathematics actually has more to it than that. (And boy, am I glad I did!)
My opinions on the matter:
I think there will always be people who like mathematics, and people who don't. However, I do think the number of people who "like mathematics" could be drastically increased if it was taught better.
I don't think it's super-critical exactly which particular mathematical topics you teach. I think it is super-critical that you teach people what mathematics actually is. (A surprising number of people don't understand this.)
I do not think that modern technology somehow makes understanding how to do X or how to do Y by hand "obsolete". Nor do I subscribe to the idea of "banning" calculators, computers and so forth. By all means, use the technology. (But don't let us fall into the trap of just blindly pushing symbols around without comprehending their meaning.)
I think it might be useful to teach a wider range of mathematical subjects. Not in exhaustive detail, obviously. In many cases, that requires a whole heap of complicated ground-work. But introduce the interesting ideas and explore the main properties.
Personally, I don't see why you couldn't do something like group theory with school-age children. (Obviously, we're talking older children here.) "You know how to add and subtract, right? OK, so let's just throw the rule book out the window and invent our own brand-new system of adding and subtracting objects. Hey, look, we invented this from scratch, but now all these interesting properties keep appearing..."
My other pet idea is to take some of the computer software out there which does stuff with user-defined functions. Let the kids play with it, and see what neat stuff they can come up with. If you just type random gibberish, nothing interesting happens. But if you understand coordinate systems and functions and have an intuition for how simple mathematical operators affect numbers, you can make the computer draw trippy stuff.
Perhaps the biggest problem with mathematics is that most people don't understand it. Many people think that mathematics "is about numbers". (That's like saying that science "is about test-tubes"!) Many think that mathematics "is one subject". (Again, that's like claiming that science is only "one subject".)
The way that I do mathematics is that it's this fascinating journey of exploration, experimentation and discovery. But the way the textbooks do it is "Here is a type of problem. Here is the exact procedure for solving this type of problem. Here are 300 identical problems of this type. Go solve them. Your teacher will then compare your answers to the published answer book, and give you some ticks." Wow, how thrilling. :-P
Having said all that, we do seem to currently live in an age where being stupid is something to be proud of. This doesn't just affect mathematics. How to solve that larger meta-problem, I have no idea....
Students should learn math. You know, the stuff that most of us do. When we use the word "math" we certainly don't have in our minds adding, multiplying, factoring polynomials, or even calculus. So why do we teach children that this is what math is?
This view has been in several published places like Devlin's The Math Gene. He makes a good argument that the concept of a group isn't any harder than other stuff done in grade school. The problem is that it is introduced later.
The best thing I've read about math education is A Mathematician's Lament by Paul Lockhart.
Google "The New Math" movement of the 1960's and 1970's,Matt. They TRIED that,it was a resounding disaster. The correct way of doing math is not always the best way to learn it-especially when dealing with grade schoolers.
–
Mathemagician1234Dec 2 '13 at 3:52
@Mathemagician1234 I think the key word in this question is "High School." I'm not proposing we get rid of the basics in elementary school for some discovery based method of teaching. Just like a musician still has to practice scales and basics, math students need to drill the basics as well. In order to stay motivated, music students still get to listen to great music and see what will come from their efforts some day. Math students don't ever get a taste of what will come from working hard, so why do should they do it?
–
MattDec 2 '13 at 7:07
I recall the Asimov short story about humanity that has completely lost the knowledge of how to do simple arithmetic without use of a computer, called "The Feeling of Power". The point is, despite computers being able to do these things for us, it is still important and valuable to know how to do simple operations. At times, we are amazingly close to that point in this very day, certainly in the current crop of kids growing up now.
Basic mathematical skills are tremendously valuable in a variety of places. Just because you CAN factor a polynomial by computer does not mean that at least understanding the tools to do so are no longer necessary for 99% of the people in the world.
Whatever you think the high school mathematics curriculum "should" be, in the United States a curriculum has been put in place, known as the Common Core Standards (CCS), which will significantly - I believe - change American mathematics education. (The CCS extend to all K-12 mathematics.)
I strongly believe that even if something can be done by calculator, a high school student should understand some nuts and bolts of how it's being done.
I spent some time doing volunteer tutoring at CRLS (Cambridge Rindge and Latin School). I still remember the girl at 8th or 9th grade who was studying inequalities, yet she couldn't compute $2-1$ without the help of calculator. The most frustrating case was the kid in 9th or 10th grade who didn't know multiplication tables, he couldn't compute, say, 3 times 7 in his head. I was supposed to help him with square roots. Well, how do you explain that $\sqrt{8} = 2\sqrt{2}$? First of, you have to realize that $8 = 2 \times 2 \times 2$, and that's exactly what he couldn't do. I ended up spending some extra time with him, afterward been told that he passed the test or exam and was very happy with my help - but I wasn't! I felt we were wasting lots of time and just because he didn't know multiplication tables, he couldn't break down numbers into factors, and calculator wasn't really help.
The bottom line: I think in school students still should learn basic arithmetic before moving to more advanced stuff. I think it's a pure idiocy to introduce the kids to set theory while having them relying on their calculators to compute $1 + 1$. Honestly, the advanced stuff can wait till college.
In high school in the 1970s I got tremendous benefit out of a one-term elective course called Mathematical Logic, which was nothing fancy, just basic prepositional stuff. Understanding the basic structure of logical arguments empowers one to learn anything more efficiently. If college-bound students could ditch half of their AP Calculus for a course in Logical Reasoning that includes basic material from Daniel Velleman's book How to Prove It and some basic facts about statistics, they would be in a far better position to pursue many subjects, not just math.
Yes! I agree completely. A course in logic or problem solving skills will be much more useful. (Personally I am also of the opinion that calculus [in highschool] should not be taught as a class in itself, but as part of physics.)
–
Willie WongOct 30 '10 at 20:14
I would include more probability, such as how to conduct a simple probabilistic experiment using statistical tests. Once you understand that, you can do a lot by simply looking up various tests. Without understanding probability, people are easily manipulated using statistics or make poor choices.
I would include definitely include more use of technology, including Wolfram Alpha and basic mathematical programming. These tools allow people to achieve much more.
Some basic mathematical philosophy would be good as well. Game theory can be very thought provoking and is quite simple as well. We could look at other areas even though we wouldn't be able to examine them rigorously. I think students should understand that maths is based on axioms, that not all maths problems can be solved (assuming ZFC is consistent) and that many problems can't be solved efficiently (P vs NP).
In my opinion the things you are suggesting in the last two paragraphs are more of the "I wish I learned this in high school" variety. As enrichment topics for especially bright and interested students, they are attractive. But it makes me think you are not aware of the skill level of the average (or even college-bound) high school student....
–
Pete L. ClarkAug 23 '10 at 13:14
4
...For instance, suppose that a 7-inch pizza costs $5. How much should a 14-inch pizza cost? I would like a high school student to be able to quickly recognize that $11, say, is a great price, and that if the price is any more than $20, they are being cheated. I don't think most high school students have internalized the necessary math for this, so let's a bit risible to bring ZFC into curricular discussions.
–
Pete L. ClarkAug 23 '10 at 13:22
@Pete: I don't believe that everything in the syllabus needs to be examinable
–
CasebashAug 24 '10 at 13:01 | 677.169 | 1 |
Textbooks Collection
Grades 7-8 Mean, Median, And Mode, Rich Miller Iii
Math
This lesson is a math lesson for seventh and eighth grade students on mean, medium, and mode. Through this lesson students will be able to understand the measures of central tendency and their definitions, how to calculate them and what steps are involved, and how the theories can be applied on real life. In this lesson, students are tiered by ability and are able to pick a project based off of their interest and the math concept they are working on. Each activity has a tiered task card to guide the students.
The Effects Of The Use Of Technology In Mathematics Instruction On Student Acheivement, Ron Y. Myers
FIU Electronic Theses and Dissertations
The purpose of this study was to examine the effects of the use of technology on students' mathematics achievement, particularly the Florida Comprehensive Assessment Test (FCAT) mathematics results. Eleven schools within the Miami-Dade County Public School System participated in a pilot program on the use of Geometers Sketchpad (GSP). Three of these schools were randomly selected for this study. Each school sent a teacher to a summer in-service training program on how to use GSP to teach geometry. In each school, the GSP class and a traditional geometry class taught by the same teacher were the study participants. Students' mathematics ...
Honors Scholar Theses
Differential equations are equations that involve an unknown function and derivatives. Euler's method are efficient methods to yield fairly accurate approximations of the actual solutions. By manipulating such methods, one can find ways to provide good approximations compared to the exact solution of parabolic partial differential equations and nonlinear parabolic differential equations.
Active Calculus, Matthew Boelkins, David Austin, Steven Schlicker
Open Education Materials
Active Calculus is different from most existing calculus texts in at least the following ways: the text is free for download by students and instructors in .pdf format; in the electronic format, graphics are in full color and there are live html links to java applets; the text is open source, and interested instructors can gain access to the original source files upon request; the style of the text requires students to be active learners — there are very few worked examples in the text, with there instead being 3-4 activities per section that engage students in connecting ideas, solving problems ...
Calculating The Time Constant Of An Rc Circuit, Sean Dunford
Undergraduate Journal of Mathematical Modeling: One + Two
In this experiment, a capacitor was charged to its full capacitance then discharged through a resistor. By timing how long it took the capacitor to fully discharge through the resistor, we can determine the RC time constant using calculus.
A Math 8 Unit In Scientific Notation Aligned To The New York State Common Core And Learning Standards, Jessica K. Griffin
Education and Human Development Master's Theses
In response to the implementation of new Common Core State Standards (CCSS), this curriculum project was designed to help teachers in the transition to the new standards. The curriculum project will be referred to as a unit plan throughout the paper. The unit plan on Scientific Notation, for the eighth grade mathematics curriculum, is aligned to the New York State Common Core and Learning Standards for Mathematics (NYSCCLSM). The unit plan addresses mathematical modeling, Mathematical Practice Standard 4. The unit plan may provide a way in which teachers can work towards the Common Core State Standards Initiative's goal to ...
Length Of A Hanging Cable, Eric Costello
Undergraduate Journal of Mathematical Modeling: One + Two
The shape of a cable hanging under its own weight and uniform horizontal tension between two power poles is a catenary. The catenary is a curve which has an equation defined by a hyperbolic cosine function and a scaling factor. The scaling factor for power cables hanging under their own weight is equal to the horizontal tension on the cable divided by the weight of the cable. Both of these values are unknown for this problem. Newton's method was used to approximate the scaling factor and the arc length function to determine the length of the cable. A script ...
Introduction To Real Analysis, William F. Trench
Faculty Authored Books
Using a clear and informal approach, this book introduces readers to a rigorous understanding of mathematical analysis and presents challenging math concepts as clearly as possible. This book is intended for those who want to gain an understanding of mathematical analysis and challenging mathematical concepts.
Raphael's School Of Athens: A Theorem In A Painting?, Robert Haas
Journal of Humanistic Mathematics
Raphael's famous painting The School of Athens includes a geometer, presumably Euclid himself, demonstrating a construction to his fascinated students. But what theorem are they all studying? This article first introduces the painting, and describes Raphael's lifelong friendship with the eminent mathematician Paulus of Middelburg. It then presents several conjectured explanations, notably a theorem about a hexagram (Fichtner), or alternatively that the construction may be architecturally symbolic (Valtieri). The author finally offers his own "null hypothesis": that the scene does not show any actual mathematics, but simply the fascination, excitement, and joy of mathematicians at their work.bertElectronic Theses and Dissertations
We provide an analytic read-through of Richard Dedekind's 1901 article "Über die Permutationen des Körpers aller algebraischen Zahlen," describing the principal results concerning infinite Galois theory from both Dedekind's point of view and a modern perspective, noting an apparently uncorrected error in the supplement to the article in the Collected Works. As there is no published English-language translation of the article, we provide an annotated original translation.
Primarily, we will explore the combinatorial and geometric nature of reductive monoids with zero. Such monoids have a decomposition in terms of a Borel subgroup, called the Bruhat decomposition, which produces a finite monoid, R, the Renner monoid. We will explore ...
Electronic Journal of Linear Algebra
Extensions of the Schur complement and the principal pivot transform, where the usual inverses are replaced by the Moore-Penrose inverse, are revisited. These are called the pseudo Schur complement and the pseudo principal pivot transform, respectively. First, a generalization of the characterization of a block matrix to be an M-matrix is extended to the nonnegativity of the Moore-Penrose inverse. A comprehensive treatment of the fundamental properties of the extended notion of the principal pivot transform is presented. Inheritance properties with respect to certain matrix classes are derived, thereby generalizing some of the existing results. Finally, a thorough discussion on the ...
Electronic Thesis and Dissertation Repository
Mapping properties of the Fourier transform between weighted Lebesgue and Lorentz spaces are studied. These are generalizations to Hausdorff-Young and Pitt's inequalities. The boundedness of the Fourier transform on $R^n$ as a map between Lorentz spaces leads to weighted Lebesgue inequalities for the Fourier transform on $R^n$ .
A major part of the work is on Fourier coefficients. Several different sufficient conditions and necessary conditions for the boundedness of Fourier transform on unit circle, viewed as a map between Lorentz $\Lambda$ and $\Gamma$ spaces are established. For a large range of Lorentz indices, necessary and sufficient conditions for ...
Electronic Journal of Linear Algebra
The join of two disjoint graphs G and H, denoted by G ∨ H, is the graph obtained by joining each vertex of G to each vertex of H. In this paper, the signless Laplacian characteristic polynomial of the join of two graphs is first formulated. And then, a lower bound for the i-th largest signless Laplacian eigenvalue of a graph is given. Finally, it is proved that G ∨ K_m, where G is an (n − 2)-regular graph on n vertices, and K_n ∨ K_2 except for n = 3, are determined by their signless Laplacian spectra.
Combinatorial Polynomial Identity Theory, Mayada Khalil Shahada
Electronic Thesis and Dissertation Repository
This dissertation consists of two parts. Part I examines certain Burnside-type conditions on the multiplicative semigroup of an (associative unital) algebra $A$.
A semigroup $S$ is called $n$-collapsing if, for every $a_1,\ldots, a_n \in S$, there exist functions $f\neq g$ on the set $\{1,2,\ldots,n\}$ such that \begin{center} $s_{f(1)}\cdots s_{f(n)} = s_{g(1)}\cdots s_{g(n)}$. \end{center} If $f$ and $g$ can be chosen independently of the choice of $s_1,\ldots,s_n$, then $S$ satisfies a semigroup identity. A semigroup $S$ is called $n$-rewritable if $f ...
Quantization Of Two Types Of Multisymplectic Manifolds, Baran Serajelahi
Electronic Thesis and Dissertation Repository
This thesis is concerned with quantization of two types of multisymplectic manifolds that have multisymplectic forms coming from a Kahler form. In chapter 2 we show that in both cases they can be quantized using Berezin-Toeplitz quantization and that the quantizations have reasonable semiclassical properties.
In the last chapter of this work, we obtain two additional results. The first concerns the deformation quantization of the (2n-1)-plectic structure that we examine in chapter 2, we make the first step toward the definition of a star product on the Nambu-Poisson algebra (C^{\infty}(M),{.,...,.}). The second result concerns the algebraic properties ...
Dissertations
Exponential propagation iterative (EPI) methods provide an efficient approach to the solution of large stiff systems of ODE, compared to standard integrators. However, the bulk of the computational effort in these methods is due to products of matrix functions and vectors, which can become very costly at high resolution due to an increase in the number of Krylov projection steps needed to maintain accuracy. In this dissertation, it is proposed to modify EPI methods by using Krylov subspace spectral (KSS) methods, instead of standard Krylov projection methods, to compute products of matrix functions and vectors. This improvement allowed the benefits ...
Idaho Conference on Undergraduate Research
With the large scale proliferation of networked devices ranging from medical implants like pacemakers and insulin pumps, to corporate information assets, secure authentication, data integrity and confidentiality have become some of the central goals for cybersecurity. Cryptographic hash functions have many applications in information security and are commonly used to verify data authenticity. Our research focuses on the study of the properties that dictate the security of a cryptographic hash functions that use Even-Mansour type of ciphers in their underlying structure. In particular, we investigate the algebraic design requirements of the Grøstl hash function and its generalizations. Grøstl is an ...
Dynamic Approach To K-Forcing, Yair Caro, Ryan Pepper
Theory and Applications of Graphs
The k-forcing number of a graph is a generalization of the zero forcing number. In this note, we give a greedy algorithm to approximate the k-forcing number of a graph. Using this dynamic approach, we give corollaries which improve upon two theorems from a recent paper of Amos, Caro, Davila and Pepper [2], while also answering an open problem posed by Meyer [9].
International Journal of Aviation, Aeronautics, and Aerospace
Both, a global isothermal temperature model and a nonlinear quadratic temperature model of the ISA was developed and presented here. Constrained optimization techniques in conjunction with the least-square-root approximations were used to design best-fit isothermal models for ISA pressure and density changes up to 47 geopotential km for NLPAM, and 86 orthometric km for ISOAM respectively. The mass of the dry atmosphere and the relevant fractional-mass scale heights have been computed utilizing the very accurate eight-point Gauss-Legendre numerical quadrature for both ISOAM and NLPAM. Both, the ISOAM and the NLPAM represent viable alternatives to ISA in many practical applications and ...
Smooth Representation Of Functions On Non-Periodic Domains By Means Of The Fourier Continuation Method, Nicholas Rubel, David Bilyeu, Justin Koo
STAR (STEM Teacher and Researcher) Program Posters
This report examines a new methodology in solving Partial Differential Equations (PDEs) numerically. The report also studies the accuracy of this new method as a PDE solver. This new Fourier Continuation (FC) method is one of a few that avoids the well-known Gibbs Phenomenon, which is the overestimation or underestimation of a function. These estimations are oscillations around a "jump" when a non-periodic function is expressed in terms of sines and cosines. Instead, the FC algorithm creates a smooth, periodic extension of a function over a general domain, as demonstrated by the many examples presented here. The FC algorithm was ...
Dissertations, Theses, and Student Research Papers in Mathematics
The center of a graph is the set of vertices whose distance to other vertices is minimal. The centralizingnumber of a graph G is the minimum number of additional vertices in any graph H where V(G) is the center of H. Buckley, Miller, and Slater and He and Liu provided infinite families of graphs with each centralizing number. We show the number of graphs with each nonzero centralizing number grows super-exponentially with the number of vertices. We also provide a method of altering graphs without changing the centralizing number and give results about the centralizing number of dense ... | 677.169 | 1 |
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An easy-to-follow guide to practical, real-world applications of linear algebra Linear algebra is a branch of mathematics that uses matrices to solve systems of linear equations; it has applications in many disciplines, from sociology and game theory to computer programming, engineering, and business. Linear Algebra For Dummies maps to a typical, college-level linear algebra course, in which students first study matrices and matrix operations, then apply those fundamentals to abstract topics such as vector spaces, linear transformations, determinants, and, finally, eigenvalues and eigenvectors. It gives students theoretical and practical ways of approaching various types of problems, presenting the information in a way that allows readers to fully digest not just the how of solving linear algebraic problems, but also the why. Mary Jane Sterling (Peoria, IL) is the author of Algebra For Dummies (978-0-7645-5325-7), Business Math for Dummies (978-0-470-23331-3), and several other books. She has been teaching at Bradley University for almost 30 years.
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Most Helpful Customer Reviews
I really like the approach of this book, as it uses good analogies, examples, graphs, and figures that appeal to your intuition to give deeper meaning to the subtleties of linear algebra. But it is a shame there are typos and mistakes. One in particular blew my mind - on p. 51 she states " A square matrix is singular if it has a multiplicative inverse; a matrix is non-singular if it does not have a multiplicative inverse." This is exactly opposite of how everyone else defines "singular", but she uses this definition consistently throughout the book, so either it's a mistake or she has decided to rewrite the rules of linear algebra. Either way, it's not helpful!!!
I hate having to make corrections on a book I just paid for, but for the price, what can you expect? Most textbooks on the subject are between two and ten times more expensive, and they have mistakes too. But the other books are dense, dry, and boring. At least this book attempts to make linear algebra interesting, if not fun.
Good explanations, but a disappointing number of substantive typos that make the examples confusing.
I would recommend this book, particularly as a review for those of us who had the subject in college but have forgotten most of it. Just be careful of those typos and keep a pencil handy to make corrections while the details are fresh in your mind.
The book gives a general good overview. However, there are several errors in the examples provided. For example in page 196, the matrices format does not match the example given with the coordinates in the form of x, y and z.
I just finished the text. It is a nice easy review. The amount of theory and proofs is kept to a minimum which should please those who are less gifted mathematically. The major issues as stated elsewhere are the number of typos and errors which were not caught in the editing process. Many are obvious and easy to fix. I would encourage the editors of this series to be more careful in the proofing. Perhaps it is too expensive given the low purchase price of the texts. Maybe we should all send in a list of errata and they could print a new edition.
I have just finished this. I bought it as a refresher, as I had basically not done any linear algebra since first year university subjects 35 years ago. For dusting off cobwebs on the major topics of linear algebra, matrices, it was useful, easy to read and clear. I also wanted an intro to vector spaces as I have this hankering, even at my age, to take on some abstract algebra, and needed to get a handle on the basics. So, as I said, useful.
However, the book - as various other reviewers have said - has far too many errors. This is a serious flaw in any work, but is critically harmful in a volume for dummies. Because obviously, we don't know enough often to pick them up so end up learning the wrong things. Fortunately my cobwebs dusted off sufficiently for me to pick up the ones I saw. But being a dummy, how do I know now what other things I have absorbed that are in fact wrong? This is a very worrying situation. It is not good enough.
This is an excellent resource for anyone currently enrolled in a linear algebra course. I often need to have several different explanations before I thoroughly understand a new concept and this book provides clear, easy to understand explanations that clarified many topics for me. Would highly recommend this book! | 677.169 | 1 |
Prerequisite: Prealgebra or beginning algebra in high school or college. The traditional topics of intermediate algebra through quadratic equations and functions.
MATH 101: Fundamental Mathematical Concepts I. 3 hours.
Prerequisite: One year of high school algebra or MATH 100. Development of the number systems — whole numbers through real numbers. Problem solving strategies, functions, elementary logic and set theory are included.
Prerequisite: MATH 100 or one year of high school algebra and one year of high school geometry. A study of functions and graphs, solutions of equations and inequalities and the properties of polynomial, rational, exponential and logarithmic functions.
MATH 110: Trigonometry. 3 hours.
Prerequisite: MATH 109 or two years of high school algebra and one year of high school geometry. The study of trigonometric, logarithmic and exponential functions and their applications.
MATH 141. Applied Logic. 1 hour.
This course is designed to help students learn to apply the tools of logic to concrete situations, such as those posed on LSAT and GMAT tests. The course will include a discussion of propositional logic, propositional equivalences, rules of inference and common fallacies. Students are strongly encouraged to take PHIL 100 Introduction to Logic and Critical Thinking either prior to or concurrently with this course.
MATH 205 Mathematical Connections. 3 hours.
Prerequisite: At least two years of high school algebra. A quantitative reasoning course for students in the liberal arts, focusing on applications of mathematics to social issues in our world. Contains the study of providing urban services, making social choices, constructing fair voting systems, and planning the fair division of resources.
MATH 211 Precalculus. 3 hours.
Prerequisite: High-school level algebra skills and/or successful completion of College Algebra are required. This course is designed to prepare students for Calculus I. It covers a variety of topics from algebra, with emphasis on the development of rational, exponential, logarithmic and trigonometric functions including their essential properties, graphs and basic applications. Additional topics range from linear systems to conic sections.
MATH 227: Introduction to Statistics. 3 hours.
Prerequisite: One year of high school algebra. A course to acquaint the student with the basic ideas and language of statistics including such topics such as descriptive statistics, correlation and regression, basic experimental design, elementary probability, binomial and normal distributions, estimation and test of hypotheses, and analysis of variance.
MATH 230: Business Calculus. 3 hours.
Prerequisite: Two years of high school algebra. Topics from differential and integral calculus with an emphasis on business applications. This class cannot be used as a prerequisite for MATH 232.
MATH 231: Calculus I. 4 hours.
Prerequisite: Two years of high school algebra and one semester of high school trigonometry. A study of the fundamental principles of analytic geometry and calculus with an emphasis on differentiation.
MATH 232: Calculus II. 4 hours.
Prerequisite: MATH 231 or MATH 236. It is recommended that students receive a grade of C or better in MATH 231 or MATH 236 to be successful in this course. Continuation of Calculus I including techniques of integration and infinite series.
MATH 233: Calculus III. 4 hours.
Prerequisite: MATH 232. It is recommended that students receive a grade of C or better in MATH 231 to be successful in this course. Functions of two variables, partial differentiation, applications of multiple integrals to areas and volumes, line and surface integrals, and vectors.
MATH 234: Introduction to Mathematical Proof. 3 hours.
Prerequisite: MATH 231 or MATH 236. Recommended prerequisite: MATH 232. A careful introduction to the process of constructing mathematical arguments, covering the basic ideas of logic, sets, functions and relations. A substantial amount of time will be devoted to looking at important forms of mathematical argument such as direct proof, proof by contradiction, proof by contrapositive and proof by cases. Applications from set theory, abstract algebra or analysis may be covered at the discretion of the instructor.
Prerequisite: Math ACT score of 28 or better and a course in trigonometry with a grade of B or better. This course is an introduction to single variable calculus with an emphasis on differential calculus. We will cover limits, derivatives, and applications, with an emphasis on both calculational techniques and their theoretical underpinnings. The course will conclude with an exploration of the Riemann sum definition of the definite integral.
Prerequisite: MATH 232. It is recommended that students receive a grade of C or better in MATH 232 to be successful in this course. This course includes an introduction to probability theory, discrete and continuous random variables, mathematical expectation and multivariate distributions.
MATH 327: Mathematical Statistics. 3 hours.
Prerequisite: MATH 326. It is recommended that students receive a grade of C or better in MATH 326 to be successful in this course. This course takes the material from MATH 326 into the applications side of statistics including functions of random variables, sampling distributions, estimations and hypothesis testing.
MATH 330: Geometry. 3 hours.
Prerequisite: MATH 234. Foundations of Euclidian geometry from the axioms of Hilbert and an introduction to non-Euclidian geometry.
MATH 366: Differential Equations. 3 hours.
Prerequisite: MATH 232. A first course in ordinary differential equations.
Prerequisite: MATH 234. An introduction to point-set topology. Metric spaces, connectedness, completeness and compactness are some of the topics discussed.
MATH 493: Senior Seminar. 3 hours.
Modern topics in mathematics are discussed in a seminar setting. Students integrate their study of mathematics throughout their undergraduate years and explore the connections among mathematics and other courses they have pursued. Departmental assessment of the major is included. This course is designed to be a capstone experience taken during the final semester of the senior year.
The history and philosophy of mathematics are discussed in a seminar setting. All students in this course must complete a project wherein familiar questions asked by high school math students are examined and answered in depth. Also, students are required to read and make a presentation on an article from an approved mathematics education journal. Department assessment of the major is included. This course is designed to be a capstone experience taken during the fall semester of the senior year. | 677.169 | 1 |
books.google.com - Explore the main algebraic structures and number systems that play a central role across the field of mathematics Algebra and number theory are two powerful branches of modern mathematics at the forefront of current mathematical research, and each plays an increasingly significant role in different branches... and Number Theory | 677.169 | 1 |
How to Study as a Mathematics Major
Lara Alcock
Explores how to engage with the academic content of a mathematics major
Covers aspects of university life separate from mathematics
Designed to help students to recognize and reflect on their own experiences and to anticipate, articulate and overcome the challenges they encounter
Unique subject-specific study guide for mathematics students
How to Study as a Mathematics Major
Lara Alcock
Description
Every year, thousands of students declare mathematics as their major. Many are extremely intelligent and hardworking. However, even the best will encounter challenges, because upper-level mathematics involves not only independent study and learning from lectures, but also a fundamental shift from calculation to proof.
This shift is demanding but it need not be mysterious -- research has revealed many insights into the mathematical thinking required, and this book translates these into practical advice for a student audience. It covers every aspect of studying as a mathematics major, from tackling abstract intellectual challenges to interacting with professors and making good use of study time. Part 1 discusses the nature of upper-level mathematics, and
explains how students can adapt and extend their existing skills in order to develop good understanding. Part 2 covers study skills as these relate to mathematics, and suggests practical approaches to learning effectively while enjoying undergraduate life.
As the first mathematics-specific study guide, this friendly, practical text is essential reading for any mathematics major.
How to Study as a Mathematics Major
Lara Alcock
Author Information
Lara Alcock is Senior Lecturer in the Mathematics Education Centre at Loughborough University, UK. An accomplished undergraduate and graduate mathematician at Warwick, her doctorate was in mathematics education before holding various academic posts including Assistant Professor of Mathematics Education and Mathematics at Rutgers University, New Jersey. Her research focuses on the challenges students encounter as they make the transition from calculation-based to proof-based mathematics. She was awarded the 2012 MAA Seldon Prize for Research in Undergraduate Mathematics Education. She has been awarded National Teaching Fellows of 2015 by The Higher Education Academy.
How to Study as a Mathematics Major
Lara Alcock
Reviews and Awards
"Students can benefit from just 'picking it up' for a short time - now and then - and reading just about any section. The sections are relatively short - sometimes just two or three pages, but are very informative. One could easily recommend [How to Study as a Mathematics Major] to undergraduates." --Mathematical Association of America
How to Study as a Mathematics Major
Lara Alcock
From Our Blog
By Lara Alcock
Two contrasting experiences stick in mind from my first year at university. First, I spent a lot of time in lectures that I did not understand. I don't mean lectures in which I got the general gist but didn't quite follow the technical details. I mean lectures in which I understood not one thing from the beginning to the end. I still went to all the lectures and wrote everything down ' I was a dutiful sort of student ' but this was hardly the ideal learning experience... | 677.169 | 1 |
2015-2016 University Catalog
This course is designed to enhance mathematical literacy and to stimulate interest in appreciation for mathematics and quantitative reasoning as valuable tools for addressing issues in a constantly changing society. Topics may include, at an introductory level, logical reasoning and problem solving through mathematical games and puzzles; sets, relations, and functions; counting and number concepts (number theory and infinity). Prerequisite: MATH 1314 or SAT Mathematics score of 500+ or ACT Mathematics score of 20+. | 677.169 | 1 |
Currently there is substantial exchange and communication between academic communities around the world as researchers endeavour to discover why so many children 'fail' at a subject that society deems crucial for future economic survival. This book charts current thinking and trends in teacher education around the world, and looks critically at the... more...
If learners in the classroom are to be excited by mathematics, teachers need to be both well informed about current initiatives and able to see how what is expected of them can be translated into rich and stimulating classroom strategies.
The book examines current initiatives that affect teaching mathematics and identifies pointers for action in... more...
Explores the psychology of thinking about post-secondary level mathematics, suggesting that the way it is taught does not correspond to the way it is learned. Addressed to mathematicians and educators in mathematics, considers the nature and cognitive theory of advanced mathematical thinking, and re more...
This text focuses attention on mathematics learners in transition and on their practices in different contexts; on the institutional and socio-cultural framing of the transition processes involved; and on the communication and negotiation of mathematical meanings during transition. more...
The 20 chapters in this book all focus on aspects of mathematical beliefs, from a variety of different perspectives. Thus, contemporary knowledge of the field is synthesized and existing boundaries are extended. more...
This title is concerned with communication in mathematics class-rooms. In a series of empirical studies of project work, it follows students' inquiry co-operation as well as students' obstructions to inquiry co-operation. more...
This text describes the state-of-the-art in this branch of science. Starting from a general perspective on the didactics of mathematics, the 30 original contributions to the book go on to identify certain subdisciplines and suggest an overall structure or 'topology' for the field. more...
Part course text for advanced graduate students and beginning engineers and part reference for the more experienced, develops methods for solving practical and mathematical problems arising in the creation of automatic control systems. Covers information from combinatorics, extremal problems on the more...
This title consists of nine survey articles on various advances in algorithmic combinatorics. The articles cover both recent areas of application and exciting new theoretical developments. The book is accessible to Ph.D. students in discrete mathematics or theoretical computer science and is intended for researchers in the field of combinatorics. more... | 677.169 | 1 |
Details about Intermediate Algebra A Graphing Approach:
Elayn Martin-Gay's success as a developmental math author starts with a strong focus on mastering the basics through well-written explanations, innovative pedagogy and a meaningful, integrated program of learning resources. The revisions to this edition provide new pedagogy and resources to build reader confidence and help readers develop basic skills and understand concepts. Features incorporation of AMATYC and NCTM standards-reflected in an increased emphasis on visualization graphing, and data analysis. In addition, Martin-Gay's 4-step problem solving process-Understand, Translate, Solve and Interpret-is integrated throughout. Also includes new features such as Study Skills Reminders, "Integrated Reviews", and "Concept Checks." For those in need of a graphing utility resource in intermediate algebra, and for readers who need to prepare for advanced algebra or finite math. | 677.169 | 1 |
This is an introductory undergraduate textbook in set theory. In mathematics these days, essentially everything is a set. Some knowledge of set theory is necessary part of the background everyone needs for further study of mathematics. It is also possible to study set theory for its own interest--it is a subject with intruiging results anout simple objects. This book starts with material that nobody can do without. There is no end to what can be learned of set theory, but here is a beginning.
Most Helpful Customer Reviews
Elements of Set Theory is by far the best undergraduate text for introductory set theory in publication. It manages to balance the intuitive with the technical so successfully that the reader is more than prepared to tackle more advanced topics like constructability, forcing, descriptive set theory and so on. However this edition by Academic Press is unreadable. The edition I purchased had 'Transferred to Digital Printing 2009' printed at the bottom of the copyright page and the text looks as if it has been printed by a malfunctioning printer. I purchased this to replace an old (and very well used) cheap, international edition which got me through my undergraduate classes and was frequently referred to during my graduate studies. There is no comparison between the editions the international edition is easy to read - all the text and the symbols are easy to read with clear lines. This edition's display of the text and symbols is blurry and in some cases smudged to the point of unreadability. I sent it back.
This is my favourite Enderton book. Contains elegant concise cogent proof writing and a comprehensive introduction to set theory. I read the first 150 pages in one sitting and the author even inserted dry wit and elucidating diagrams to contribute to stopping the reader from putting the book down.
Most Helpful Customer Reviews on Amazon.com (beta)
Amazon.com:
11 reviews
66 of 67 people found the following review helpful
Excellent introduction to set theory4 April 2000
By
Jakub Zielinski
- Published on Amazon.com
Format: Hardcover
The only reason I won't say it's THE BEST introduction to set theory is that I haven't read ALL such introductions. I am (obviously) a student of logic and I worked my way through the whole book a few years ago. It is an insightful development of set theory, both as a foundation for mathematics and a distinctive mathematical discipline in its own right. Set theory can be developed from a "naive" or an "axiomatic" perspective. The naive approach simply asks the reader to accept arguments about sets on the basis of informed intuition, whereas the axiomatic approach relies on showing how mathematical proofs can be formalized as deductions from a precise axiom system. Enderton's book deftly combines both approaches ; axiomatic considerations are isolated from the rest of the text and identified by a stripe running down the side of the page. Those who are not interested in axioms can avoid dealing with them almost entirely, but enthusiasts of formal rigor (like me!) won't be disappointed either. The axioms, which comprise a system known as Zermelo Fraenkel set theory with Choice, are introduced as needed in the overall development (so Replacement Axioms aren't mentioned until page 179). The text develops relations and functions as well as natural and real number systems, and then goes on to cardinals, orderings, and ordinals. I particularly enjoyed Enderton's well-motivated exposition of ordinals, which clearly shows how these numbers measure the lengths of well-orderings. His treatment of cardinals, transfinite induction, and the Axiom of Choice, is enlightening as well. A final chapter, which includes cofinality and inaccessible cardinals, should whet the student's appetite for further study in set theory. I have a hard time thinking of anything negative to say about this book. Perhaps it would be better if its nicely annotated bibliography were a bit more extensive. If you wanna learn set theory, buy this book!
13 of 14 people found the following review helpful
An Excellent Introduction17 Jun. 2004
By
ktrmes
- Published on Amazon.com
Format: Hardcover
Perhaps because it is a Foundations book -- in my mathematics training it always seemed that the people who did the best job of motivating and explaining (or at least making you feel you understood) the material were Foundations people -- but this book has a presentation polished to the point where the closest genre of mathematics text in level of polish would be intro calculus books, where the problems theorems and proofs have been worked over for many many many years. Here, however, the material is in great part relatively recent - probably the closest to contemporary stuff you can see as an undergraduate -- in Real Analysis, by contrast, you may well just be coming out of the 19th century by graduate school. This polish, I have discovered in later years, facilitates use of this book for self-study and it is a wonderful text for providing rapid refreshment of important concepts. I have over the years referred back to it on a number of occassions and have always been pleasantly reminded what a wonderful book it is. This is a very nice book and the best introduction to the material I have seen (although, given the number of intro books I have seen on the topic, this may not be a strong statement).
5 of 5 people found the following review helpful
The Best Intro. Ever!11 Sept. 2010
By
A Customer 2000
- Published on Amazon.com
Format: Hardcover
I might sound biased, but who doesn't? Enderton's book is the BEST introduction out there if you want to learn. There are better books for conciseness or pendanticalness, what a word :(. But, he introduces the material in such a way that the novice can actual absorb most of it in one go through. It's deep enough that a return trip through a difficult chapter is worth your time. After reading this text you can easily segueway to Moschovakis's book, "Notes on Set Theory" or even attempt books like "Set Theory An Introduction To Independence Proofs (Studies in Logic and the Foundations of Mathematics)" by Kunen.
This book covers all of the basics of set theory (greater in scope than Halmos's "Naive Set Theory" (which is a great book by the way)).
18 of 23 people found the following review helpful
Not great but hard to do better12 July 2005
By
Nathan Oakes
- Published on Amazon.com
Format: Hardcover
The style is readable without being wordy. The book starts with a good, intuitive discussion of sets and the axiomatic method, but follows with a sketchy description of truth tables. The rest of the book is similarly uneven. It is best when introducing some topics with extensive motivation. Its main weaknesses are in the completeness of the explanations and the clarity of the proofs. Several of the proofs were the cause of much head-scratching. That shouldn't happen in an elementary text. There were several spots in the text where the train of thought is not clear. Sections that I particularly thought were sloppy and inadequate were the development of cardinals and the Axiom of Choice.
As math textbooks go, I've read better, but for an undergraduate introduction to set theory, the competition is not very impressive. There are 23 errata listed on his web site. It is a simple matter to pencil in the corrections. One book you should consider as an alternative is Hrbacek & Jech. If the high price is an issue, the text by Stoll does a good job with the basics.
4 of 4 people found the following review helpful
Simply the best21 Sept. 2012
By
Abhi
- Published on Amazon.com
Format: Hardcover
Verified Purchase
The book is nice and simple and well explained. The exercises problems are solvable and are not contest problems like in some books. Pretty much most of the other reviews have summed it up well. I can confidently say that among books written on Set Theory (like by Cohen, Halmos, Stoll, Hrbacek & Jech), that I bought and tried to read, this book is SIMPLY THE BEST introduction to Set Theory. Blindly read this book and no other book. It has enough set theory to get you going in pure mathematics.
Addition to the review on 11/21/2012: I am close to being done with the arithmatic section and I must say that a book on set theory cannot possibly be better than this one. I have all the material I need to know in order to get a good start for Real Analysis. Those parts of the proof that he says "is left as an exercise" are truly trivial after following the material that is covered till that point. This book is SIMPLY THE BEST. Don't think twice. Just get this book and read it top to bottom. There is a good reason why Stanford and Berkeley prescribe this book as the text for their Set Theory courses every year. I know that Hrbacek and Jech is one contender but I am very biased towards Enderton's book as he makes this subject unbelievably simple where as other books (and definitely Halmos's book) makes it seem harder than what it is. | 677.169 | 1 |
The program provides detailed, step-by-step solution in a tutorial-like format to the following problem: Given a 2x2 matrix, or a 3x3 matrix, or a 4x4 matrix, or a 5x5 matrix. Find its inverse matrix by using the Gauss-Jordan elimination method. The check of the solution is given. The program is designed for university students and professors | 677.169 | 1 |
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This interdisciplinary lesson uses musical terms and concepts to teach algebra and geometry. Students will analyze musical scales and frequencies generated by a geometric sequence, and relate sine waves to musical...
Using the real world example of security cameras, this lesson has students explore properties of polygons. Using this example, students will be able to discover a formula as related to polygons. An activity sheet and...
This algebra unit from illuminations provides an in depth exploration of exponential models in context. The model of light passing through water is used to demonstrate exponential functions and related mathematical...
This interdisciplinary lesson ties earth science concepts in with algebra. The forest-fire danger rating index is applied to a mathematical model. Students will learn real-world meaning of the intercepts and slope in...
This lesson will help students understand the law of cosines and the Pythagorean theorem. The material has students find out whether a triangle is acute, obtuse, or right; determine a formula for the law of cosines to... | 677.169 | 1 |
Elayn Martin-Gay firmly believes that every student can succeed, and her developmental math textbooks and video resources are motivated by this belief. Basic College Mathematics with Early Integers, Second Edition was written to help students effectively make the transition from arithmetic to algebra. The new edition offers new resources like the Student Organizer and now includes Student Resources in the back of the book to help students on their quest for success.
Editorial Reviews
About the Author
El Martin-Gay developed an acclaimed series of lecture videos to support developmental mathematics students. These highly successful videos originally served as the foundation materials for her texts. Today, the videos are specific to each book in her series. She has also created Chapter Test Prep Videos to help students during their most "teachable moment"—as they prepare for a test—along with Instructor-to-Instructor videos that provide teaching tips, hints, and suggestions for every developmental mathematics course, including basic mathematics, prealgebra, beginning algebra, and intermediate algebra.
Elayn is the author of 12 published textbooks and numerous multimedia interactive products, all specializing in developmental mathematics courses. She has participated as an author across a broad range of educational materials: textbooks, videos, tutorial software, and courseware. This offers an opportunity for multiple combinations for an integrated teaching and learning package, offering great consistency for the studentImportant Information
More About the Author
An award-winning instructor and best-selling author, El developed an acclaimed series of lecture videos to support developmental mathematics students in their quest for success. These highly successful videos originally served as the foundation material for her texts. Today, the videos are specific to each book in the Martin-Gay series. Elayn also pioneered the Chapter Test Prep Video to help students as they prepare for a test--their most "teachable moment!"
Elayn's experience has made her aware of how busy instructors are and what a difference quality support makes. For this reason, she created the Instructor-to-Instructor video series. These videos provide instructors with suggestions for presenting specific math topics and concepts in basic mathematics, prealgebra, beginning algebra, and intermediate algebra. Seasoned instructors can use them as a source for alternate approaches in the classroom. New or adjunct faculty may find the videos useful for review. Her textbooks and acclaimed video program support Elayn's passion of helping every student to succeed.
Most Helpful Customer Reviews
This textbook is supposed to come with a Student Access Code, but if you buy it used or rent it, chances are you won't get a usable code. The book is not even needed (says my professor) if you have the access code, as all the info is taught by Elaiyne Martin-Gay herself online! Just know what your buying is all I'm saying!
This book is much better than the last book i purchased it has more detail to the things i need to refresh to pass my course. I recomended this book to anyone having problems in freshmen college math.Thanks Amazon!
For those of us who had done the basic math as kids in the 1940s and 50s, this book was Greek. However, with a great teacher to cut through the excess verbage and some wrong answers, one did learn to use the new rules of progression. I learned "Please excuse my dear Aunt Sally" to know what order to solve the problems. After that it was easy.
A good text that is well written with clear examples for principles, and also includes a lot of useful exercises for practice. However, be careful to avoid purchasing the "Annotated Instructor's Edition" if you are purchasing it for paid-, accredited-coursework! Why? Turns out the answers being provided is okay if you are a disciplined student; HOWEVER, if your professor requires the text to be used in class for "GRADED" activities, BEWARE! It will cost you a penalty in grading and/or dismissal from class!
I had to buy this for a course and found it for a great price on Amazon. Not only was it a great price but it was also great condition as well. The book itself was poorly written and very uninformative. | 677.169 | 1 |
Perfect for the one-term course, Essentials of Precalculus with Calculus Previews, Fifth Edition provides a complete, yet concise, introduction to precalculus concepts, focusing on important topics that will be of direct and immediate use in most calculus courses. Consistent with Professor Zill's eloquent writing style , this full-color text offers numerous exercise sets and examples to aid in student comprehension, while graphs and figures throughout serve to illuminate key concepts. The exercise sets include engaging problems that focus on algebra, graphing, and function theory, the sub-text of many calculus problems. The authors are careful to use calculus terminology in an informal and accessible way to facilitate the students successful transition into future calculus courses. With an outstanding collection of student and instructor resources, Essentials of Precalculus with Calculus Previews offers a complete teaching and learning package. | 677.169 | 1 |
My opinion regarding the TI-89: Good, and it's about time, and WHY doesn't it include the geometry package from the TI-92 and ***WHY*** doesn't it include the statistical package from the TI-83???
I can forgive TI for not including the Cabri geometry - their stated market is college engineers. BUT, not to include the statistical package is insane, and I told their reps that at NCTM and at a later workshop given by my school district. I have students who take both the AP stats and the calculus (I teach both). Which machine do they buy? The downloaded stats package from the TI website - also required for the TI-86 - is inferior to the 83 built-in version. However, the CAS and differential equation capabilites of the 89 can't be matched by the 83. <sigh>
HP has already had a CAS calculator approved for the exam. But, as is true with all technology, if you don't *understand* the mathmatics, then all you're doing is punching buttons. One of students two years ago showed me his HP and its CAS. He still got a 1 on the exam. | 677.169 | 1 |
Sample records for quantitative problem solving
Presents four guidelines for teaching quantitativeproblem-solving based on research results: analyze difficulties of students, develop a system of heuristics, select and map key relations, and design instruction with proper orientation, exercise, and feedback. Discusses the four guidelines and uses flow charts and diagrams to show how the…
Women are underrepresented in science, technology, engineering and mathematics (STEM) areas in university settings; however this may be the result of attitude rather than aptitude. There is widespread agreement that quantitativeproblem-solving is essential for graduate competence and preparedness in science and other STEM subjects. The research…
A case is presented for the importance of focusing on (1) average ability students, (2) substantive mathematical content, (3) real problems, and (4) realistic settings and solution procedures for research in problemsolving. It is suggested that effective instructional techniques for teaching applied mathematical problemsolving resembles "mathematical laboratory" activities, done in small group problemsolving settings.
No longer solely the domain of Mathematics, problemsolving permeates every area of today's curricula. Ideally students are applying heuristics strategies in varied contexts and novel situations in every subject taught. The ability to solveproblems is a basic life skill and is essential to understanding technical subjects. Problem-solving is a…
Examined the transition from other-regulation to self-regulation by studying correlational problemsolving among 10 secondary school students individually tutored in problemsolving. Quantitative discourse analysis supported the idea of a transition from other to self-regulation and qualitative analysis supported the idea of assimilation of the…
The purpose of this book is to teach the basic principles of problemsolving in both mathematical and non-mathematical problems. The major components of the book consist of learning to translate verbal discussion into analytical data, learning problemsolving methods for attacking collections of analytical questions or data, and building a…
A history of texture analysis (TA) evolution is shown, beginning from the first experimental and theoretical attempts to find and characterize preferred orientations of crystal lattices of grains in real polycrystalline samples. Stages of formation of TA theoretical apparatus, its basic elements, and also application of its capabilities for quantitatively describing anisotropic properties of textured samples are discussed. Attention is also paid to the limitations and difficulties associated with the analysis. The application of the quantitative TA apparatus is demonstrated by example describing elastic properties of textured materials up to multiphase samples containing pores and cracks. A wide scope of TA includes the analysis based on neutron scattering which has been effectively developed at the Frank Laboratory of Neutron Physics. A practical opportunity to determine the bulk crystallographic textures of single-phase and multiphase materials is offered by the use of modern neutron diffractometers, including the SKAT diffractometer at the IBR-2 pulsed reactor. This is especially important for studying samples of natural rocks. The examples given show how the neutron scattering data for the quantitative TA are used in combination with other physical and petrological methods for solving fundamental problems of geology and geophysics based on the analysis of a structure and properties of the Earth's lithosphere matter. The review includes a detailed list of references of original works concerning the TA elaboration, overview publications and monographs, and also information on the most popular TA-related software.
This collection of nine papers, prepared for a conference held at Northwestern University in 1978, presents varied perspectives on applied problemsolving. Assessing applied problemsolving, planning for interest and motivation, developing a theory, reviewing research findings, considering learning disabilities, analyzing through information…
In a unique school--university partnership, methods students collaborated with fifth graders to use the engineering design process to build their problem-solving skills. By placing the problem in the context of a client having particular needs, the problem
Insight problemsolving was investigated with the matchstick algebra problems developed by Knoblich, Ohlsson, Haider, and Rhenius (1999). These problems are false equations expressed with Roman numerals that can be made true bymoving one matchstick. In a first group participants examined a static two-dimensional representation of the false algebraic expression and told the experimenter which matchstick should be moved. In
Learn some basic math skills while at the same time learning some programming skills This short lesson focuses on solving simple math problem using computer programming. In this case, the examples given will be in Python (click on this link for more information: Official Tutorial for the Python programming language.). Computer programming can and has often been used to solve very complex mathematical problems along the lines of calculating ? ...
This issue of ENC Focus focuses on the topic of inquiry and problemsolving. Featured articles include: (1) "Inquiry in the Everyday World of Schools" (Ronald D. Anderson); (2) "In the Cascade Reservoir Restoration Project Students Tackle Real-World Problems" (Clint Kennedy with Advanced Biology Students from Cascade High School); (3) "Project…
Two studies were conducted to describe how students perform direct current (D-C) circuit problems. It was hypothesized that problemsolving in the electricity domain depends largely on good visual processing of the circuit diagram and that this processing depends on the ability to recognize when two or more electrical components are in series or…
Algebra word problems were analyzed in terms of the information integration tasks that are required to solve the problems. These tasks were classified into three levels: value assignment, value derivation, and equation construction. Novices (35 first year algebra students) and experts (13 analytic geometry students) were compared on the proportion…
Several problem-solving interventions that utilise a "circle" approach have been applied within the field of educational psychology, for example, Circle Time, Circle of Friends, Sharing Circles, Circle of Adults and Solution Circles. This research explored two interventions, Solution Circles and Circle of Adults, and used thematicA teaching unit on genetics and human inheritance using problem-solving methodology was undertaken with fourth-level Spanish Secondary Education students (15 year olds). The goal was to study certain aspects of the students' learning process (concepts, procedures and attitude) when using this methodology in the school environment. The change…
The culminating energy project is introduced and the technical problemsolving process is applied to get students started on the project. By the end of the class, students should have a good perspective on what they have already learned and what they still need to learn to complete the project.
? Self Assessment Exercise: Each of the nine items presents two opposing statements: - If you feel of formulas involve learning how to apply basic concepts and principles. 2. When I am learning a new concept in When I am learning a new concept, a problemsolving course, I do not I focus on learning
This work provides a correlation study of the role of the following cognitive variables on problemsolving in elementary physical chemistry: scientific reasoning (level of intellectual development/developmental level), working-memory capacity, functional mental ("M") capacity, and disembedding ability (i.e., degree of perceptual field…
Exploring our patent system is a great way to engage students in creative problemsolving. As a result, the authors designed a teaching unit that uses the study of patents to explore one avenue in which scientists and engineers do science. Specifically, through the development of an idea, students learn how science and technology are connected. The activities described here promote scientific literacy by helping students appreciate science as a human endeavor and making connections between science, technology, and society.
We describe a modeling approach to help students learn expert problemsolving. Models are used to present and hierarchically organize the syllabus content and apply it to problemsolving, but students do not develop and ...
Based on current models of problemsolving within cognitive psychology, this study focused on the spontaneous problemsolving strategies used by children as they first learned LOGO computer programming, and on strategy transformations that took place during the problemsolving process. The research consisted of a six weeks programming training…
Discusses the use of microcomputers and software as problem-solving tools, including comments on "TK! Solver," automatic problem-solving program (reviewed in detail on pp.84-86 in this same issue). Also discusses problem-solving approaches to bridge the disciplines, such as music/physics, junior high science/mathematics (genetics),…
This 14-page monograph addresses the need to teach problemsolving and other higher order thinking skills. After summarizing research and positions of various organizations, it defines several models and describes cognitive and attitudinal components of problemsolving and the types of knowledge that are required. The authors provide a list of principles for teaching problemsolving and include a list of references.
Helping students to construct robust understanding of physics concepts and develop good solving skills is a central goal in many physics classrooms. This thesis examine students' problemsolving abilities from different perspectives and explores strategies to scaffold students' learning. In studies involving analogical problemsolving…
This learning object from Wisc-Online covers trade discount word problems. The lesson teaches a method of solving these problems which requires students to memorize only one equation. Example problems are included.
This paper describes how human-technology interaction in modern ambient technology environments can be best informed by conceptualizing of such environments as problemsolving systems. Typically, such systems comprise multiple human and technological agents that meet the demands imposed by problem constraints through dynamic collaboration. A key assertion is that the design of expert problemsolving systems can benefit from an
Unit 16 1 Algorithms and ProblemSolving · Introduction · What is an Algorithm? · Algorithm Properties · Example · Exercises #12;Unit 16 2 What is an Algorithm? What is an Algorithm? · An algorithm. · The algorithm must be general, that is, it should solve the problem for all possible input sets to the problem
solving framework and answer sheet you design during TA Orientation. The second tool is the Warm. How do I form cooperative groups? 27 III. What criteria do I use to assign students to groups? 31 IV of the group role sheets are available on the bookshelf in room 146, or you can make copies of the following
Tested the hypothesis that adolescent psychiatric patients would be deficient with respect to normal controls in their interpersonal problem-solving skills by comparing 33 patients and 53 high school student controls on 7 tasks reflecting different aspects of problemsolving. With IQ covaried out, controls obtained significantly higher scores on the tasks evaluating optional thinking, social means-ends thinking, and role taking,
ProblemSolving systems customarily use backtracking to deal with obstacles that they encounter in the course of trying to solve a problem. This paper outlines an approach in which the possible obstacles are investigated prior to the search for a solution. This provides a solution strategy that avoids backtracking.
Few studies have explicitly attended to the personal problem-solving process within the counseling literature, perhaps due in part to the dearth of relevant assessment instruments. To examine the dimensions underlying the applied problem-solving process, an exploratory factor analysis was conducted using data collected from four samples of college…
The booklet considers the nature of creativity in children and examines classroom implications. Among the topics addressed are the following: theories about creativity; research; developments in brain research; the creative process; creative problemsolving; the Structure of Intellect ProblemSolving (SIPS) model; a rationale for creativity in the…
Describes the Future ProblemSolving Program, in which students from the U.S. and around the world are tackling some complex challenges facing society, ranging from acid rain to terrorism. The program uses a creative problemsolving process developed for business and industry. A sixth-grade toxic waste cleanup project illustrates the process.…
In this activity, learners use cooperation and logical thinking to find solutions to network problems on the playground. Learners act both as computer routers, figuring out with each other how to effectively get data to the place it's being sent, and as the actual data, because the learners travel various edges of a network to get to their destination or "home" point. Learners use geometry skills to determine the most efficient routes in the network.
An experiment compared solving of operational and diagnostic problems after different instruction about a fictitious device. Solution of both kinds of problems was facilitated by instruction (1) that focused on functional relations among components of the device or (2) that focused on states of the individual components. For operational problems,…
Shaped charges were first used more than 30 years ago to perforate casing, cement sheath and reservoir to provide selective communication between the reservoir and well bore. Since then, advances have been made in design of shaped charge sand ancillary equipment. Penetration has increased considerably. Miniaturization of equipment allows passing through relatively small restrictions and effectively communicating with reservoirs. Gun debris has been drastically reduced and in some designs eliminated. Deep reservoirs can be effectively penetrated under down-hole environments exceeding 500 F and 20,000 psi. This work covers the problems encountered and reviews successful devices employing shaped charge and explosive technology.
Fourteen fifth-grade students gather at the front of the classroom as their summer school instructor introduces Jonathan Bostic as the mathematics teacher for the week. Before examining any math problems, Bostic sits at eye level with the students and informs them that they will solveproblems over the next four days by working individually as…
The development of a computer problemsolving system is reported that considers physical problems faced by an artificial robot moving around in a complex environment. Fundamental interaction constraints with a real environment are simulated for the robot by visual scan and creation of an internal environmental model. The programming system used in constructing the problemsolving system for the simulated robot and its simulated world environment is outlined together with the task that the system is capable of performing. A very general framework for understanding the relationship between an observed behavior and an adequate description of that behavior is included.
Problemsolving is an essential skill for nuclear engineering graduates entering the workforce. Training in qualitative and quantitative aspects of problemsolving allows students to conceptualise and execute solutions to complex problems. Solutions to problems in high consequence fields of study such as nuclear engineering require rapid and…
TRIZ (pronounced TREES), the Russian acronym for the theory of inventive problemsolving, enables a person to focus his attention on finding genuine, potential solutions in contrast to searching for ideas that "may" work through a happenstance way. It is a patent database-backed methodology that helps to reduce time spent on the problem,…
This paper studies the modes of thought that occur during the act of solvingproblems in mathematics. It examines the two main instantiations of mathematical knowledge, the conceptual and the structural, and their role in the afore said act. It claims that awareness of mathematical structure is the lever that educes mathematical knowledge existing in the mind in response to
PSE - 1 Air PSE (ProblemSolving Environment) MODELLING OF AIR POLLUTION IN THE LOS ANGELES BASIN COMPUTER MODELS An air pollution model is a computer program that computes how the different chemical emissions or decreasing car use. An air pollution model is never exact in its attempt to simulate
The conceptual, experimental, and practical aspects of the development of a robot computer problemsolving system were investigated. The distinctive characteristics were formulated of the approach taken in relation to various studies of cognition and robotics. Vehicle and eye control systems were structured, and the information to be generated by the visual system is defined.
This webpage offers some basic principles for teaching problemsolving that foster critical thinking and decision-making skills. It includes a 5-step implementation model developed by D.R. Woods and a brief list of references. [The Forshay & Kirkley paper is cataloged separately and linked as a related resource.
Often after students solve a problem they believe they have accomplished their mission and stop further exploration. The purpose of this article is to discuss ways to encourage students to "look back" so as to maximise their learning opportunities. According to Polya, by "looking back" at a completed solution, by reconsidering and re-examining the…
This article describes steps of teaching problemsolving to college students and provides examples in the context of a university course. The steps involve (1) identifying the types of problems and types of problemsolving methods to be covered, (2) instructing the students in problem-recognition and problemsolving methods, along with ways of…
We describe a theory of quantitative representations and processes that makes novel predictions about student problem-solving and learning during the transition from arithmetic to algebraic competence or \\Five steps common to different problemsolving models are listed. Next, seven specific abilities related to solvingproblems are discussed and examples given. Sample activities, appropriate to help in developing these specific abilities, are suggested. (LS)
Statistical Education Through ProblemSolving (STEPS) was a collaborative project between seven universities throughout the United Kingdom "to develop problem-based teaching and learning materials for statistics." The materials draw on specific problems arising in Biology, Business, Geography and Psychology to help students learn that statistical issues are "important natural parts of the process of reaching conclusions." The software developed as a result of this project, which utilizes the computer and graphical illustration to support learning, is available to educational institutions free of charge and can be downloaded from this website. (Note that other organizations are expected to purchase the software.) A glossary of statistical terms is provided in the software program as well as on this website. Although the funding for the project ended in 1995 and the website was last updated in January 2004, the material is still current and useful for teaching statistics. The authors note that the STEPS modules are intended to be used to support existing coursework, and "not intended to replace lecturing staff or to provide a self-study course in statistics."
The Physics Education Research Group at the University of Minnesota has developed an interview tool to investigate physics faculty views about the learning and teaching of problemsolving. In the part of the interview dealing with grading, faculty members were asked to evaluate a set of five student solutions and explain their reasons for the grades that they assigned. Preliminary analysis on two of the five student solutions was done on six physics faculty members from a large research university. The results indicate that faculty members hold conflicting beliefs when grading between valuing reasoning in student solutions and wanting to give students the benefit of the doubt. This paper illustrates the hypothesis that physics faculty hold conflicting values when grading, and describes how the research university faculty resolved their conflicts.
A robot computer problemsolving system which represents a robot exploration vehicle in a simulated Mars environment is described. The model exhibits changes and improvements made on a previously designed robot in a city environment. The Martian environment is modeled in Cartesian coordinates; objects are scattered about a plane; arbitrary restrictions on the robot's vision have been removed; and the robot's path contains arbitrary curves. New environmental features, particularly the visual occlusion of objects by other objects, were added to the model. Two different algorithms were developed for computing occlusion. Movement and vision capabilities of the robot were established in the Mars environment, using LISP/FORTRAN interface for computational efficiency. The graphical display program was redesigned to reflect the change to the Mars-like environment.
Discusses the need for library science to examine users' search behaviors in the context of social cognition and creative problemsolving. Topics covered include the psychology of search behavior, problemsolving approaches to library instruction in online searching, and problemsolving protocols. (29 references) (CLB)
Describes a problem-solving-oriented teacher-inservice program designed to provide opportunities that allow elementary teachers to focus on personal experience as a way of achieving self-understanding and a way of reconstructing their personal meanings about problemsolving and problem-solving instruction. Concludes that the program had positive…
A tutorial outline of the polyhedral theory that underlies linear programming (LP)-based combinatorial problemsolving is given. Design aspects of a combinatorial problem solver are discussed in general terms. Three computational studies in combinatorial problemsolving using the polyhedral theory developed in the past fifteen years are surveyed: one addresses the symmetric traveling salesman problem, another the optimal triangulation of
Mathematics teachers often experience difficulties in teaching students to become skilled problem solvers. This paper evaluates the effectiveness of two interactive computer programs for high school mathematics problemsolving. Both programs present students with problems accompanied by instruction on domain-specific knowledge required in different episodes of problemsolving. The first program is based on a direct instructional approach to learning,
This book presents interactive problem-solving situations based on the principles of Total Quality Management. Following the introductory chapter, the second chapter describes the two stages of the problem-solving process: (1) analysis of the problem and its causes, and (2) identification of a solution. Each stage is comprised of three…
Problemsolving lies at the heart of mathematical learning. Children need opportunities to write, discuss, and solveproblems on a regular basis. The problems must incorporate grade-appropriate content and be "accessible and engaging to the students, building on what they know and can do." Teachers also play a key role in establishing a classroom… problem…
An interactive decomposition method is developed for solving the multiple criteria (MC) problem. Based on nonlinear programming duality theory, the MC problem is decomposed into a series of subproblems and relaxed master problems. Each subproblem is a bicriterion problem, and each relaxed master problem is a standard linear program. The prime objective of the decomposition is to simplify and facilitate
problemsolving · Key to development of successful strategies: iterative process or co- development of the necessary information to address the problem. · Analysis is the consideration of the nature of the information: deconstruction - breaking down the information, identifying relationships, determination
This page from the site "Stella's Stunners" presents twenty-five thinking strategies that are useful in solvingproblems. They help students monitor their thought processes and thus help learners become better problem solvers.
Musicians practice to build endurance, flexibility, and dexterity. They practice to maintain good performance, to sight-read better, to memorize, and simply, to enjoy music making. There are other motivations for practice, but one, more than others, is a catalyst for consequential change in musical development--practicing to solve performance…
Experimental electronic neural network solves "traveling-salesman" problem. Plans round trip of minimum distance among N cities, visiting every city once and only once (without backtracking). This problem is paradigm of many problems of global optimization (e.g., routing or allocation of resources) occuring in industry, business, and government. Applied to large number of cities (or resources), circuits of this kind expected to solveproblem faster and more cheaply.
RIDGES is a mnemonic device designed to give upper elementary and high school students a structure to follow when solving word problems. RIDGES stands for Read the problem; I know statement; Draw a picture; Goal statement; Equation development; and Solve the equation. (VW)
The two articles dealing with problemsolving and technology in this publication should be useful to those developing the kinds of materials, experiences, and thinking that elementary school industrial arts offers children. The first article accepts problemsolving as an educational goal and reports a timely and universally acceptable approach.…
In recent years an increasing amount of interest has been generated in the application of dialectical methodology to strategic and operational problem-solving. This paper first examines the existing research evidence and then introduces the Dialectical Problem-Solving Technology (DPST) based on the Dialectical Materialism Inquiry System. The results of an empirical investigation into the effectiveness and impact of High Structure (DPST),
Introduction: The macroscopic perspective is one of the frameworks for research on problemsolving in mathematics education. Coming from this perspective, our study addresses the stages of thought in mathematical problemsolving, offering an innovative approach because we apply sequential relations and global interrelations between the different…
This article addresses two unsolved measurement issues in dynamic problemsolving (DPS) research: (a) unsystematic construction of DPS tests making a comparison of results obtained in different studies difficult and (b) use of time-intensive single tasks leading to severe reliability problems. To solve these issues, the MicroDYN approach is…
Solvingproblems and creating processes and procedures from the ground up has long been part of the IT department's way of operating. IT staffs will continue to encounter new problems to solve and new technologies to be implemented. They also must involve their constituents in the creation of solutions. Nonetheless, for many issues they no longer…
This set of papers was originally developed for a conference on Issues and Directions in Mathematics ProblemSolving Research held at Indiana University in May 1981. The purpose is to contribute to the clear formulation of the key issues in mathematical problem-solving research by presenting the ideas of actively involved researchers. An…
Developed as part of the ABCs of Construction National Workplace Literacy Project, this instructional module is designed to help individuals employed as pipefitters learn to solveproblems with charts and tables. Outlined in the first section is a five-step procedure for solvingproblems involving tables and/or charts: identifying the question to…Two questions are dealt with: (1) Can those strategies or behaviors which enable experts to solveproblems well be characterized, and (2) Can students be trained to use such strategies? A problem-solving course for college students is described and the model on which the course is based is outlined in an attempt to answer these questions. The…
This article describes a semester-long pen-pal project in which preservice teachers composed mathematical problems and the middle school students worked for solutions. The college students assessed the solution and the middle school students provided feedback regarding the problem itself. (Contains 6 figures.)
This paper describes the problem-solving behaviors of 12 mathematicians as they completed four mathematical tasks. The emergent problem-solving framework draws on the large body of research, as grounded by and modified in response to our close observations of these mathematicians. The resulting "Multidimensional Problem-Solving Framework" has four…
Robot problems are examined in the context of semantic networks which are used to represent the state of a problem and the operators useful for solving it. Graph transformation algorithms are discussed as an aid to problemsolving. Although these form only a small subset of the first-order predicate calculus based systems, considerations such as subgoal circularity, partially specified states and multiple manipulators sharing the same environment may warrant this simplification. PMID:21869103
... SOMEONE WITH EMOTIONAL & BEHAVIORAL NEEDS Cognitive Problems (Disorientation, Perception, Attention, Learning & Problem-Solving) Cognition is the process ... What Are Some Other Cognitive Problems? What Is Perception? Remember What Is Attention or Concentration? More Resources ... problemsolving could be improved by means of HPS. Three typical problems in introductory courses of mechanics—the inclined plane, the simple pendulum and the Atwood machine—are taken as the object of the present study. The solving strategies of these problems in the eighteenth and nineteenth century constitute the historical component of the study. Its philosophical component stems from the foundations of mechanics research literature. The use of HPS leads us to see those problems in a different way. These different ways can be tested, for which experiments are proposed. The traditional solving strategies for the incline and pendulum problems are adequate for some situations but not in general. The recourse to apparent weights in the Atwood machine problem leads us to a new insight and a solving strategy for composed Atwood machines. Educational implications also concern the development of logical thinking by means of the variety of lines of thought provided by HPS class of inverse problems in remote sensing can be characterized by Q = F(x), where F is a nonlinear and noninvertible (or hard to invert) operator, and the objective is to infer the unknowns, x, from the observed quantities, Q. Since the number of observations is usually greater than the number of unknowns, these problems are formulated as optimization problems, which can be solved by a variety of techniques. The feasibility of neural networks for solving such problems is presently investigated. As an example, the problem of finding the atmospheric ozone profile from measured ultraviolet radiances is studied.
SOLVING NP SEARCH PROBLEMS WITH MODEL EXPANSION by Faraz Hach B.Sc., Sharif University of Science in the School of Computing Science c Faraz Hach 2007 SIMON FRASER UNIVERSITY Fall 2007 All rights the permission of the author. #12;APPROVAL Name: Faraz Hach Degree: Master of Science Title of thesis: SOLVING NP
Two forms of cooperation in distributed problemsolving are considered: task-sharing and result-sharing. In the former, nodes assist each other by sharing the computational load for the execution of subtasks of the overall problem. In the latter, nodes assist each other by sharing partial results which are based on somewhat different perspectives on the overall problem. Different perspectives arise because
This document deals with the observation of students in a direct translation scheme in the solution of word problems in a university freshman-level Intermediate Algebra class. It is felt that since successful problem solvers of algebraic equations often have as much difficulty in solving word problems as do other students in the classes, the…
The purpose of my research was to produce a problemsolving evaluation tool for physics. To do this it was necessary to gain a thorough understanding of how students solveproblems. Although physics educators highly value problemsolving and have put extensive effort into understanding successful problemsolving, there is currently no efficient way to evaluate problemsolving skill. Attempts have been made in the past; however, knowledge of the principles required to solve the subject problem are so absolutely critical that they completely overshadow any other skills students may use when solving a problem. The work presented here is unique because the evaluation tool removes the requirement that the student already have a grasp of physics concepts. It is also unique because I picked a wide range of people and picked a wide range of tasks for evaluation. This is an important design feature that helps make things emerge more clearly. This dissertation includes an extensive literature review of problemsolving in physics, math, education and cognitive science as well as descriptions of studies involving student use of interactive computer simulations, the design and validation of a beliefs about physics survey and finally the design of the problemsolving evaluation tool. I have successfully developed and validated a problemsolving evaluation tool that identifies 44 separate assets (skills) necessary for solvingproblems. Rigorous validation studies, including work with an independent interviewer, show these assets identified by this content-free evaluation tool are the same assets that students use to solveproblems in mechanics and quantum mechanics. Understanding this set of component assets will help teachers and researchers address problemsolving within the classroom.
The traditional approach to teaching science problemsolving is having the students work individually on a large number of problems. This approach has long been overtaken by research suggesting and testing other methods, which are expected to be more effective. To get an overview of the characteristics of good and innovative problem-solving teaching strategies, we performed an analysis of a number of articles published between 1985 and 1995 in high-standard international journals, describing experimental research into the effectiveness of a wide variety of teaching strategies for science problemsolving. To characterize the teaching strategies found, we used a model of the capacities needed for effective science problemsolving, composed of a knowledge base and a skills base. The relations between the cognitive capacities required by the experimental or control treatments and those of the model were specified and used as independent variables. Other independent variables were learning conditions such as feedback and group work. As a dependent variable we used standardized learning effects. We identified 22 articles describing 40 experiments that met the standards we deemed necessary for a meta-analysis. These experiments were analyzed both with quantitative (correlational) methods and with a systematic qualitative method. A few of the independent variables were found to characterize effective strategies for teaching science problemsolving. Effective treatments all gave attention to the structure and function (the schemata) of the knowledge base, whereas attention to knowledge of strategy and the practice of problemsolving turned out to have little effect. As for learning conditions, both providing the learners with guidelines and criteria they can use in judging their own problem-solving process and products, and providing immediate feedback to them were found to be important prerequisites for the acquisition of problem-solving skills. Group work did not lead to positive effects unless combined with other variables, such as guidelines and feedback.
The activity of posing and solvingproblems can enrich learners' mathematical experiences because it fosters a spirit of inquisitiveness, cultivates their mathematical curiosity, and deepens their views of what it means to do mathematics. To achieve these goals, a mathematical problem needs to be at the appropriate level of difficulty,…
The major task in solving a physics problem is to construct an appropriate model of the problem in terms of physical principles. The functions performed by such a model, the information which needs to be represented, and the knowledge used in selecting and instantiating an appropriate model are discussed. An example of a model for a mechanics…
An attempt to implement problemsolving as a teacher of ninth grade algebra is described. The problems selected were not general ones, they involved combinations and represented various situations and were more complex which lead to the discovery of Steiner triple systems.
This article describes competitions across a range of curricular areas that develop students' problemsolving skills by setting authentic, real-world tasks. As individuals or members of a team, students in these competitions are challenged with finding solutions to problems faced not only in today's scientific and technological world, but also in…
The purpose of the study reported in this paper is to explore some of the common difficulties with mathematical word problems experienced by preservice primary teachers. It examines weaknesses in students' content and procedural knowledge, with a particular focus on how they apply these aspects of knowledge to solving closed word problems\\
In this lesson, students will learn how to use their knowledge of beginning, middle, and end to solve word problems that include result unknown, change unknown, and start unknown. They will learn how to use a modified story map to write an equation to represent the problem.
This article describes a design problem that not only takes students through the technological design process, but it also provides them with real-world problem-solving experience as it relates to the manufacturing and engineering fields. It begins with a scenario placing the student as a custom wheel designer for an automotive manufacturing…
GIS Live is a live, interactive, web problem-solving (WPS) program that partners Geographic Information Systems (GIS) professionals with educators to implement geospatial technologies as curriculum-learning tools. It is a collaborative effort of many government agencies, educational institutions, and professional organizations. Problem-based…
This study investigates the belief that solving a large number of physics problems helps students better learn physics. We investigated the number of problemssolved, student confidence in solving these problems, academic achievement, and the level of conceptual understanding of 49 science high school students enrolled in upper-level physics classes from Spring 2010 to Summer 2011. The participants solved an average of 2200 physics problems before entering high school. Despite having solved so many problems, no statistically significant correlation was found between the number of problemssolved and academic achievement on either a mid-term or physics competition examination. In addition, no significant correlation was found between the number of physics problemssolved and performance on the Force Concept Inventory (FCI). Lastly, four students were selected from the 49 participants with varying levels of experience and FCI scores for a case study. We determined that their problemsolving and learning strategies was more influential in their success than the number of problems they had solved.
Students rarely have the opportunity to delve into the unknown and brainstorm solutions to cutting-edge, unsolved science problems that affect thousands of people. To counter this trend, the following activity was developed to expose students to issues and problems surrounding cancer treatment using an inquiry-based approach. Through this activity, students step into the role of "real" scientists and brainstorm possible treatment options by working collaboratively, utilizing problemsolving strategies, and creativity to explore science and technology.
Some easily graded measures of problem-solving processes are introduced, and the impact of a month-long intensive problem-solving course on a selected group of college freshmen and sophomores is demonstrated. The measures are thought to have shown themselves to be both reliable and informative. (MP)
This paper, presented at the 2001 Physics Education Research Conference, discusses the physics education research group at Rensselaer which is working to develop an assessment tool that will measure the problem-solving ability of introductory physics students. In its final form, the tool will consist of approximately 30-40 multiple-choice questions related to a limited number of classical mechanics topics. There are currently four types of questions included in the exam: attitudinal questions, quantitativeproblems that require students to identify the underlying principles used in solving the problem but not an explicit solution, questions that ask students to compare posed problems in terms of solution method, and quantitativeproblems requiring a solution. Although the assessment is still under development, preliminary validation studies have been performed on questions requiring students to identify underlying principles. Specifically, both an ANOVA and a Fisher LSD test have been performed. These evaluations showed that wrong answers on assessment questions correlate to below average performance on the problemsolving portion of the final course exam.
Bertrand's paradox is a famous problem of probability theory, pointing to a possible inconsistency in Laplace's principle of insufficient reason. In this article we show that Bertrand's paradox contains two different problems: an "easy" problem and a "hard" problem. The easy problem can be solved by formulating Bertrand's question in sufficiently precise terms, so allowing for a non ambiguous 'universal average'.Bertrand's paradox is a famous problem of probability theory, pointing to a possible inconsistency in Laplace's principle of insufficient reason. In this article, we show that Bertrand's paradox contains two different problems: an "easy" problem and a "hard" problem. The easy problem can be solved by formulating Bertrand's question in sufficiently precise terms, so allowing for a non-ambiguous universal average.EUREK A is a problem-solving system that operates through a form of analogical reasoning. The system was designed to study how relatively low-level memory, reasoning, and learn- ing mechanisms can account for high-level learning in human problem solvers. Thus, EUREK A's design has focused on is- sues of memory representation and retrieval of analogies, a t the expense of complex
a foundation for school mathematics programs by considering the broad issues of equity, curriculum, teachingProblemSolving 1 NCTM National Mathematics Standards The following comes from the website standards.nctm.org/document/chapter1/index.htm Introduction We live in a mathematical world. Whenever we
-term pleasure. ii #12;ACKNOWLEDGEMENTS There were many individuals whose contribution had solidified this thesis with solving the problem directly that can be costly or even infeasible. The concept of reduction is not only`ere efficace, comparemment `a essayer de le r´esoudre directement, ce qui pour- rait ^etre co^uteux ou m biology, and apply them to new sets of facts.…
Second and third grade students used the creative problemsolving strategy developed by Sidney Parnes and Alex Osborn in their social studies classes. The second graders, finding few biographies written for students reading on a first or second grade level, interviewed community members, collected photographs of them, and wrote their biographies,…
Many traditional classroom science and technology activities often ask students to complete prepackaged labs that ensure that everyone arrives at the same "scientifically accurate" solution or theory, which ignores the important problem-solving and creative aspects of scientific research and technological design. Students rarely have the…
In a 1981 diagnostic test, the Ministry of Education in Singapore found its country facing a challenge: Only 46 percent of students in grades 2-4 could solve word problems that were presented without such key words as "altogether" or "left." Yet today, according to results from the Trends in International Mathematics and Science Study (TIMSS…
The author introduces and studies a class of vector fields which are defined on a given polyhedron and solve linear programming problems. A Dikin-type algorithm is constructed. Relationships with double-bracket equations and entropy-type barrier functions are established
Performance and data from some cognitive models suggested that emotions, experienced during problemsolving, should be taken into account. Moreover, it is proposed that the cognitive science approach using both theoretical and experi- mental data may lead to a better understanding of the phenomena. A closer investigation of ACT-R cognitive architecture (Anderson 1993) revealed some properties analogous to phenomena known
Arguments for and against the use of computers in mathematics classes have centered on whether students benefit from or are merely hindered by practicing computational skills. This paper claims that the true essence of mathematics lies not in computation, basically a mechanical operation, but in problem-solving. Since no amount of computational…
This study investigates the internal structure and construct validity of Complex ProblemSolving (CPS), which is measured by a "Multiple-Item-Approach." It is tested, if (a) three facets of CPS--"rule identification" (adequateness of strategies), "rule knowledge" (generated knowledge) and "rule application" (ability to control a system)--can be…
This paper investigates the impact of group dynamics on metacognitive behaviours of students (aged 13-14) during group collaborative problemsolving attempts involving a design-based real-world applications project. It was discovered that group dynamics mediated the impact of metacognitive judgments related red flag situations and metacognitive…
This naturalistic inquiry investigated how instructional designers engage in complex and ambiguous problemsolving across organizational boundaries in two corporations. Participants represented a range of instructional design experience, from novices to experts. Research methods included a participant background survey, observations of…
A set of computer implemented models are presented which can assist in developing problemsolving strategies. The three levels of expertise which are covered are beginners (those who have completed at least one university physics course), intermediates (university level physics majors in their third year of study), and professionals (university…
The Research Utilization and ProblemSolving (RUPS) Model--an instructional system designed to provide the needed competencies for an entire staff to engage in systems analysis and systems synthesis procedures prior to assessing educational needs and developing curriculum to meet the needs identified--is intended to facilitate the development of…
This research attempts to examine the collaborative problemsolving methods towards critical thinking based on economy (AE) and non economy (TE) in the SPM level among students in the lower sixth form. The quasi experiment method that uses the modal of 3X2 factorial is applied. 294 lower sixth form students from ten schools are distributed…Today is highly speed progressing the computer-based education, which allowes educators and students to use educational programming language and e-tutors to teach and learn, to interact with one another and share together the results of their work. In this paper we will be concentrated on the use of GeoGebra programme for solvingproblems of physics. We have brought an example from physics of how can be used GeoGebra for finding the center of mass(centroid) of a picture(or system of polygons). After the problem is solved graphically, there is an application of finding the center of a real object(a plate)by firstly, scanning the object and secondly, by inserting its scanned picture into the drawing pad of GeoGebra window and lastly, by finding its centroid. GeoGebra serve as effective tool in problem-solving. There are many other applications of GeoGebra in the problems of physics, and many more in different fields of mathematics.
The present study's main objective is to examine whether problem orientation and problem-solving skills differ according to generalized anxiety disorder (GAD) symptom level or clinical status (seeking help for GAD). Its secondary goal is to examine whether two cognitive variables (intolerance of uncertainty and beliefs about worry) vary according to GAD symptom level or clinical status. Three groups of subjects
Three groups of 50 freshman and 50 seniors each, majoring in technology, engineering, and humanities, completed the Personal Problem-Solving Inventory and the Technological Problem-Solving Inventory. There were few differences in personal problemsolving but significant differences by major in technological problemsolving. Few differences between…
The major perspectives on problemsolving of the twentieth century are reviewed--associationism, Gestalt psychology, and cognitive science. The results of the review on teaching problemsolving and the uses of computers to teach problemsolving are included. Four major issues related to the teaching of problemsolving are discussed: (1)…
The research described here seeks to characterize the "managerial" aspects of expert and novice problem-solving behavior, and to describe the impact of managerial or "executive" actions on success or failure in problemsolving. A framework for analyzing protocols of problem-solving sessions based on "episodes" of problem-solving behavior and…
This document consists of three papers. The first, "A Parallel Model of (Sequential) ProblemSolving," describes a parallel model designed to solve a class of relatively simple problems from elementary physics and discusses implications for models of problem-solving in general. It is shown that one of the most salient features of problemsolving,…
Both science and technology education have a commitment to teaching process; investigations or scientific method in science,\\u000a design in technology, and problemsolving in both areas. The separate debates in science and technology education reveal different\\u000a curricular emphases in processes and content, reflecting different goals, and pedagogic and educational research traditions.\\u000a This paper explores these differences and argues that each
. We show that several discrepancy-like problems can be solved in NC nearly achieving the discrepancies guaranteed by a probabilistic analysis and achievable sequentially. For example, we describe an NC algorithm that given\\u000a a set system (X, S) , where X is a ground set and S?2\\u000a \\u000a X\\u000a , computes a set R?X so that for each S?\\u000a S
The objectives of this study were as follows: (1) Determine the relationship between learning strategies and performance in problemsolving, (2) Explore the role of a student's declared major on performance in problemsolving, (3) Understand the decision making process of high and low achievers during problemsolving. Participants (N = 65) solvedproblems using the Interactive multimedia exercise (IMMEX)
Several artificial-intelligence search techniques have been tested as means of solving the swath segment selection problem (SSSP) -- a real-world problem that is not only of interest in its own right, but is also useful as a test bed for search techniques in general. In simplest terms, the SSSP is the problem of scheduling the observation times of an airborne or spaceborne synthetic-aperture radar (SAR) system to effect the maximum coverage of a specified area (denoted the target), given a schedule of downlinks (opportunities for radio transmission of SAR scan data to a ground station), given the limit on the quantity of SAR scan data that can be stored in an onboard memory between downlink opportunities, and given the limit on the achievable downlink data rate. The SSSP is NP complete (short for "nondeterministic polynomial time complete" -- characteristic of a class of intractable problems that can be solved only by use of computers capable of making guesses and then checking the guesses in polynomial time).
Fifty-nine second-year medical students were asked to solve 12 Piagetian formal operational tasks. The purpose was to describe the formal logical characteristics of this medical student sample (59 of a total 65 possible) in terms of their abilities to solveproblems in four formal logical schemata-combinatorial logic, probabilistic reasoning, propositional logic, and proportional reasoning. These tasks were presented as videotape demonstrations or in written form, depending on whether or not equipment manipulation was required, and were scored using conventional, prespecified scoring criteria. The results of this study show approximately 96% of the sample function at the transitional (Piaget's 3A level) stage of formal operations on all tasks and approximately 4% function at the full formal (Piaget's 3B level) stage of formal operations on all tasks. This sample demonstrates formal level thinking to a much greater degree than other samples reported in the literature to date and suggests these students are adequately prepared and developed to meet the challenge of their training (i.e., medical problemsolving).
We describe the concept of distributed problemsolving and define it as the cooperative solution of problems by a decentralized and loosely coupled collection of problem solvers. This approach to problemsolving offers the promise of increased performance and provides a useful medium for exploring and developing new problem-solving techniques. We present a framework called the contract net that specifies
This guide presents a classroom problemsolving model designed to help teachers conduct their own classroom research. It suggests developing a procedure for identifying the instructional problems influencing reading achievement. The model is presented in steps that can be used independently or in concert with other steps. Practice activities are…
This paper reviews the presentation of problemsolving and process aspects of mathematics in curriculum documents from Australia, UK, USA and Singapore. The place of problemsolving in the documents is reviewed and contrasted, and illustrative problems from teachers' support materials are used to demonstrate how problemsolving is now more often treated as a teaching method, rather than a
The purpose of this study was to describe the problem-solving behaviors of experts and novices engaged in solving seven chemical equilibrium problems. Thirteen novices (five high-school students, five undergraduate majors, and three nonmajors) and ten experts (six doctoral students and four faculty members) were videotaped as they individually solved standard chemical equilibrium problems. The nature of the problems was such
Recent work has shown that captive rooks, like chimpanzees and other primates, develop cooperative alliances with their conspecifics. Furthermore, the pressures hypothesized to have favoured social intelligence in primates also apply to corvids. We tested cooperative problem-solving in rooks to compare their performance and cognition with primates. Without training, eight rooks quickly solved a problem in which two individuals had to pull both ends of a string simultaneously in order to pull in a food platform. Similar to chimpanzees and capuchin monkeys, performance was better when within-dyad tolerance levels were higher. In contrast to chimpanzees, rooks did not delay acting on the apparatus while their partner gained access to the test room. Furthermore, given a choice between an apparatus that could be operated individually over one that required the action of two individuals, four out of six individuals showed no preference. These results may indicate that cooperation in chimpanzees is underpinned by more complex cognitive processes than that in rooks. Such a difference may arise from the fact that while both chimpanzees and rooks form cooperative alliances, chimpanzees, but not rooks, live in a variable social network made up of competitive and cooperative relationships. PMID:18364318
This work is one facet of an integrated approach to diagnostic problemsolving for aircraft and space systems currently under development. The authors are applying a method of modeling and reasoning about deep knowledge based on a functional viewpoint. The approach recognizes a level of device understanding which is intermediate between a compiled level of typical Expert Systems, and a deep level at which large-scale device behavior is derived from known properties of device structure and component behavior. At this intermediate functional level, a device is modeled in three steps. First, a component decomposition of the device is defined. Second, the functionality of each device/subdevice is abstractly identified. Third, the state sequences which implement each function are specified. Given a functional representation and a set of initial conditions, the functional reasoner acts as a consequence finder. The output of the consequence finder can be utilized in diagnostic problemsolving. The paper also discussed ways in which this functional approach may find application in the aerospace field.
Introductory programming courses, also known as CS1, have a specific set of expected outcomes related to the learning of the most basic and essential computational concepts in computer science (CS). However, two of the most often heard complaints in such courses are that (1) they are divorced from the reality of application and (2) they make the learning of the basic concepts tedious. The concepts introduced in CS1 courses are highly abstract and not easily comprehensible. In general, the difficulty is intrinsic to the field of computing, often described as "too mathematical or too abstract." This dissertation presents a small-scale mixed method study conducted during the fall 2009 semester of CS1 courses at Arizona State University. This study explored and assessed students' comprehension of three core computational concepts---abstraction, arrays of objects, and inheritance---in both algorithm design and problemsolving. Through this investigation students' profiles were categorized based on their scores and based on their mistakes categorized into instances of five computational thinking concepts: abstraction, algorithm, scalability, linguistics, and reasoning. It was shown that even though the notion of computational thinking is not explicit in the curriculum, participants possessed and/or developed this skill through the learning and application of the CS1 core concepts. Furthermore, problem-solving experiences had a direct impact on participants' knowledge skills, explanation skills, and confidence. Implications for teaching CS1 and for future research are also considered.
In "How to Solve It", accomplished mathematician and skilled communicator George Polya describes a four-step universal solving technique designed to help students develop mathematical problem-solving skills. By providing a glimpse at the grace with which experts solveproblems, Polya provides definable methods that are not exclusive to…
In this study, we built on previous neuroimaging studies of mathematical cognition and examined whether the same cognitive processes are engaged by two strategies used in algebraic problemsolving. We focused on symbolic algebra, which uses alphanumeric equations to represent problems, and the model method, which uses pictorial representation. Eighteen adults, matched on academic proficiency and competency in the two methods, transformed algebraic word problems into equations or models, and validated presented solutions. Both strategies were associated with activation of areas linked to working memory and quantitative processing. These included the left frontal gyri, and bilateral activation of the intraparietal sulci. Contrasting the two strategies, the symbolic method activated the posterior superior parietal lobules and the precuneus. These findings suggest that the two strategies are effected using similar processes but impose different attentional demands. PMID:17509541
EPR spectroscopy is a very powerful biophysical tool that can provide valuable structural and dynamic information on a wide variety of biological systems. The intent of this review is to provide a general overview for biochemists and biological researchers on the most commonly used EPR methods and how these techniques can be used to answer important biological questions. The topics discussed could easily fill one or more textbooks; thus, we present a brief background on several important biological EPR techniques and an overview of several interesting studies that have successfully used EPR to solve pertinent biological problems. The review consists of the following sections: an introduction to EPR techniques, spin labeling methods, and studies of naturally occurring organic radicals and EPR active transition metal systems which are presented as a series of case studies in which EPR spectroscopy has been used to greatly further our understanding of several important biological systems. PMID:23961941
Problemsolving has been a core theme in education for several decades. Educators and policy makers agree on the importance of the role of problemsolving skills for school and real life success. A primary purpose of this study was to investigate the influence of cognitive abilities on mathematical problemsolving performance of students. The…
Measured problem-solving abilities of narcotics abusers using the modified means-ends problem-solving procedure. Good subjects had more total relevent means (RMs) for solvingproblems, used more introspective and emotional RMs, and were better at RM recognition, but did not have more sufficient narratives than poor subjects. (Author/BEF)
While much research has focused on the processes of marital problemsolving, the content of marital problemsolving has received considerably less attention. This study examined the initial efforts to develop a method for assessing marital problemsolving content. Married individuals (N=36) completed a demographic information sheet, the Dyadic…
Considerable evidence indicates that domain specific knowledge in the form of schemes is the primary factor distinguishing experts from novices in problem- solving skill. Evidence that conventional problem-solving activity is not effective in schema acquisition is also accumulating. It is suggested that a major reason for the ineffectiveness of problemsolving as a learning device, is that the cognitive processes
The study challenged the current practices in cognitive load measurement involving complex problemsolving by manipulating the presence of pictures in multiple rule-based problem-solving situations and examining the cognitive load resulting from both off-line and online measures associated with complex problemsolving. Forty-eight participants…
This article explores how teachers can foster an environment that facilitates social problemsolving when toddlers experience conflict, emotional dysregulation, and aggression. This article examines differences in child development and self-regulation outcomes when teachers engage in problemsolving "for" toddlers and problemsolving "with"…
Purpose: To expose engineering students to using modern technologies, such as multimedia packages, to learn, visualize and solve engineering problems, such as in mechanics dynamics. Design/methodology/approach: A multimedia problem-solving prototype package is developed to help students solve an engineering problem in a step-by-step approach. A…
Methodological issues in the use of protocol analysis for research into human problemsolving processes are examined through a case study in which two students were videotaped as they worked together to solve mathematical problems "out loud." The students' chosen strategic or executive behavior in examining and solving a problem was studied,…
TRIZ, as a problem-solving process, is seldom used or brought into an organization in a vacuum. There is almost always an existing structure of tools and processes in use into which TRIZ enters. TRIZ can be brought into an organization as a replacement, or in collaboration with the most commonly used innovation and creativity tools in use such as Creative
This study examined how toddlers gain insights from source video displays and use the insights to solve analogous problems. The sample of 2- and 2.5-year-olds viewed a source video illustrating a problem-solving strategy and then attempted to solve analogous problems. Older, but not younger, toddlers extracted the problem-solving strategy depicted in the video and spontaneously transferred the strategy to solve isomorphic problems. Transfer by analogy from the video was evident only when the video illustrated the complete problem goal structure, including the character's intention and the action needed to achieve a goal. The same action isolated from the problem-solving context did not serve as an effective source analogue. These results illuminate the development of early representation and processes involved in analogical problemsolving. Theoretical and educational implications are discussed. PMID:24077465
In this study, stochastic computational intelligence techniques are presented for the solution of Troesch's boundary value problem. The proposed stochastic solvers use the competency of a feed-forward artificial neural network for mathematical modeling of the problem in an unsupervised manner, whereas the learning of unknown parameters is made with local and global optimization methods as well as their combinations. Genetic algorithm (GA) and pattern search (PS) techniques are used as the global search methods and the interior point method (IPM) is used for an efficient local search. The combination of techniques like GA hybridized with IPM (GA-IPM) and PS hybridized with IPM (PS-IPM) are also applied to solve different forms of the equation. A comparison of the proposed results obtained from GA, PS, IPM, PS-IPM and GA-IPM has been made with the standard solutions including well known analytic techniques of the Adomian decomposition method, the variational iterational method and the homotopy perturbation method. The reliability and effectiveness of the proposed schemes, in term of accuracy and convergence, are evaluated from the results of statistical analysis based on sufficiently large independent runs.
A distributed ProblemSolving Environment (PSE) is proposed to help users solve partial differential equation (PDE) based problems in scientific computing. The system inputs a problem description and outputs a program flow, a C-language source code for the problem and also a document for the program. Each module is distributed on distributed computers. The PSE contains all the information of
Presents results from Singaporean and Australian studies on the relationships between the cognitive variables and problemsolving performance in three electrochemistry problems of different degrees of familiarity for comparisons. Concludes that idea association, problem translating skill, prior problemsolving experience, specific knowledge, and…
Models of the cognitions used by engineering students to solveproblems have always been a part of engineering education. Many, such as the engineering mechanics model (select free-body, draw vector diagram, write equilibrium equations, and solve equilibrium equations), have been part of the introduction of students to engineering topics for a long time. More recently, student problem-solving processes are being
We introduce a greedy local search procedure called GSAT for solving propositional satisfiability problems.Our experiments show that this procedure can be used to solve hard, randomly generated problems that are an order of magnitude larger than those that can be handled by more traditional approaches such as the Davis-Putnam procedure or resolution. We also show that GSAT can solve structured
Cognitive load theory was used to hypothesize that a general problem-solving strategy based on a make-as-many-moves-as-possible heuristic could facilitate problem solutions for transfer problems. In four experiments, school students were required to learn about a topic through practice with a general problem-solving strategy, through a conventional problemsolving strategy or by studying worked examples. In Experiments 1 and 2 using junior high school students learning geometry, low knowledge students in the general problem-solving group scored significantly higher on near or far transfer tests than the conventional problem-solving group. In Experiment 3, an advantage for a general problem-solving group over a group presented worked examples was obtained on far transfer tests using the same curriculum materials, again presented to junior high school students. No differences between conditions were found in Experiments 1, 2, or 3 using test problems similar to the acquisition problems. Experiment 4 used senior high school students studying economics and found the general problem-solving group scored significantly higher than the conventional problem-solving group on both similar and transfer tests. It was concluded that the general problem-solving strategy was helpful for novices, but not for students that had access to domain-specific knowledge. PMID:25000309
A two-step approach to sensitivity analysis of model output in large computational models is proposed. A preliminary screening exercise is suggested in order to identify the subset of the most potentially explanatory factors. Afterwards, a quantitative method is recommended on the subset of preselected inputs. The advantage of the proposed procedure is that, very often, among a large number of
Four well-articulated models that offer structured approaches to problemsolving were identified in the engineering research literature. These models provided a conceptual base for the study reported here. Four undergraduates enrolled in statics and two engineering faculty members provided think-aloud data as they solved two statics problems. The data were used to develop a coding system for characterizing engineering students behavioral and cognitive processes. These codes were used to analyze students problemsolving procedures in a detailed manner, particularly differences between good and not-so-good problem solvers. The analyses provide a picture of how students and faculty solveproblems at a cognitive level, and indicate that published problem-solving models are incomplete in describing actual problem-solving processes.
Problemsolving is recognized as a valuable educational experience in science. Thus genetics, essentially a problem-solving science included in almost all high school biology courses, offers a fruitful area for studying student problem-solving performance. The research reported in this article describes the performance of 30 high school students solving 119 problems generated by the computer program GENETICS CONSTRUCTION KIT (Jungck
This study examines associations between the quality of the interparental relationship and how well 68 family triads (mother, father, preadolescent son) solved salient problems which arose at home. Four aspects of the interparental relationship (marital satisfaction, parental agreement, conflict during family problemsolving, and parental coalitions) were included in a regression analysis which controlled for family structure and child externalizing. A longitudinal design assessed families when mean child age was 9.7 years and 2 years later. Parental agreement consistently facilitated family problemsolving. However, strong parental coalitions inhibited family problemsolving, which may be attributed to frustrated autonomy needs of preadolescent males in response to the parental coalition. Stepfamilies had less effective problemsolving at Time 1. The results confirm the benefits of parental agreement to child outcomes via enhanced family problemsolving but show a reverse effect when agreement occurs in the context of coalitions against a preadolescent son. PMID:8222879
Solvingproblems presented in multiple representations is an important skill for future physicists and engineers. However, such a task is not easy for most students taking introductory physics courses. We conducted teaching/learning interviews with 20 students in a first-semester calculus-based physics course on several topics in introductory mechanics. These interviews helped identify the common difficulties students encountered when solving physics problems posed in multiple representations as well as the hints that help students overcome those difficulties. We found that most representational difficulties arise due to the lack of students' ability to associate physics knowledge with corresponding mathematical knowledge. Based on those findings, we developed, tested and refined a set of problem-solving exercises to help students learn to solveproblems in graphical and equational representations. We present our findings on students' common difficulties with graphical and equational representations, the problem-solving exercises and their impact on students' problemsolving abilities.
We consider a linear Hopfield network for solving quadratic programming problems with equation constraints. The problem is reduced to the solution of ordinary linear differential equations with arbitrary square matrix. Because of some properties of this matrix special methods are required for good convergence of the system. After some comparative studies of neural network models for solving this problem we
Hierarchically organized knowledge about actions has been postulated to explain planning in problemsolving. Perdix, a simulation of problemsolving in geometry with schematic planning knowledge, is described. Perdix' planning knowledge enables it to augment the problem space it is given by constructing auxiliary lines. The planning system also…
An excellent starting point for exercising creativity is the area of problemsolving. With a bag of creative problemsolving tools and techniques, problems will no longer represent setbacks but instead, opportunities to introduce innovations that will support the company's initiative of continuous improvement. PMID:10387779
The primary application of dimensional analysis (DA) is in problemsolving. Typically, the problem description indicates that a physical quantity Y(the unknown) is a function f of other physical quantities A[subscript 1], ..., A[subscript n] (the data). We propose a qualitative problem-solving procedure which consists of a parallel decomposition…
In this article, we discuss how to use a diagrammatic approach to solve the classic sailors and the coconuts problem. It provides us an insight on how to tackle this type of problem in a novel and intuitive way. This problem-solving approach will be found useful to mathematics teachers or lecturers involved in teaching elementary number theory,…
Describes the result of implementing the Problem List Generator, a computer-based tool designed to help clinical pathology veterinary students learn diagnostic problemsolving. Findings suggest that student problemsolving ability improved, because students identified all relevant data before providing a solution. (MES)
The tacit mental models of many research and development institutions dedicated to sustainable rural development is that they exist to solve development problems. This has led to a diagnostic and often reactive problem-solving mode of action, and to a culture of trouble-shooting experts who develop solutions. When practiced exclusively, the problem-solving mode is self-limiting because the energy that could create
the one parameter family of behind states comprising the burned Hugoniot and wave curves. For curved and show how the curved detonation jump conditions can be solved to compute the curved detonation Hugoniot
In this paper, multiple criteria optimization has been investigated. A new decision support system (DSS) has been developed for interactive solving of multiple criteria optimization problems (MOPs). The weighted-sum (WS) approach is implemented to solve the MOPs. The MOPs are solved by selecting different weight coefficient values for the criteria…
Various researchers have associated meaningful problemsolving with methods guided directly by a conceptual knowledge base. By contast, a meaningless solving course, or sequence of operations, is essentially independent of the solver's conceptual understanding of the problem under consideration. This paper is the first to document a meaningless,…Presents a scenario in which two people solve a programming problem by discussing various number sequences and functions. The problem is redefined as one related to number theory and operations research. (DDR)
't be afraid of a little algebra. Sleep on it if need be. Ask. The Problems. 1. Let f(n) be the number N be the set of positive integers. Define f on n by f(1) = 1, f(2n) = f(n) and f(2n + 1) = f(n) + 1
. Abbass, and Charles Newton School of Computer Science, University of New South Wales, ADFA Campus to be useful for solving MOPs (Zitzler and Thiele 1999). EAs have some advantages over traditional results when compared with the Strength Pareto Evolutionary Algorithm (SPEA) (Zitzler and Thiele 1999
in the standard school curriculum. These sessions stimulated interest in mathematics and helped students compete the ideas. The American Institute of Mathematics continues to support the training of new teams to create solving related to the night's theme. Rich mathematical discussions ensue as teachers explore new ideas
Review of descriptions of the 12 problem-solving tasks developed since the last review (Ray, 1955) of this topic, indicating that the newer tasks are more sophisticated in design and provide for better experimental control than those used prior to 1953. Validity, reliability, sensitivity, trainability, problem structure, and problem difficulty are discussed as criteria for the selection of tasks to be used in studies of skilled problem-solving performance.
This paper explores the way in which we define and deal with social problems such as crime and proposes a new way of thinking about them. Criminality, poverty, illiteracy, addiction and child abuse are some of society's most acute and intractable problems. Despite countless attempted remedies, these complex social problems have continued to grow around the world. Although we have
The employee timetabling problem (ETP) is concerned with assigning a number of employees into a given set of shifts over a fixed period of time, e.g. a week, while meeting the employee's preferences and organizational work regulations. The problem also attempts to optimize the performance criteria and distribute the shifts equally among the employees. The problem is considered a classical
This paper is an attempt to paint a picture of problemsolving in Chinese mathematics education, where problemsolving has been viewed both as an instructional goal and as an instructional approach. In discussing prob- lem-solving research from four perspectives, it is found that the research in China has been much more content and experience-based than cognitive and empirical-based. We
The purpose of this paper is to solve the Capacitated Vehicle Routing Problem with Time Windows (CVRPTW) on Grid'5000 using the ParadisEO framework. In this respect, four packages developed in ParadisEO are exploited. First, EO package (Evolving Objects) is used to create an evolutionary algorithm to solve the mono-objective CVRPTW. Then, a related multi-objective problem is solved with MOEO package
Adaptive problemsolving contributes to individual and family health and development. In this article, the effect of the cooperative family learning approach (CFLA) on group family problemsolving and on cooperative parenting communication is described. A pretest or posttest experimental design was used. Participant families were recruited from Head Start programs and exhibited two or more risk factors. Participant preschool
Examined whether use of robotics had a greater effect on elementary school children's achievement in science concepts and problem-solving abilities than use of battery-powered motorized manipulatives or no manipulatives. Found no significant difference in achievement from use of robotics except in programming language problemsolving. Both…
We consider a linear Hopfield network for solving quadratic programming problems with equation constraints. The problem is reduced to the solution of the ordinary linear differential equations with arbitrary square matrix. Because of some properties of this matrix the special methods are required for good convergence of the system. After some comparative study of neural network models for solving this
Understanding students' poor performance on mathematical problemsolving in physics Jonathan introductory, algebra-based physics students perform poorly on mathematical problemsolving tasks in physics. There are at least two possible, distinct reasons for this poor performance: (1) Students lack the mathematical
Intercultural problemsolving and negotiation involves interaction of two or more cultures. These processes may be formally modeled using the Evolutionary Systems Design (ESD) framework implemented by appropriate computer group support systems (GSS). The ESD\\/GSS combination provides an ESD computer culture for intercultural problemsolving and negotiation in a same place\\/same time or telework mode. With this, players in a
Motivated by the growing importance of data quality in data-intensive, global business environments and by burgeoning data quality activities, this study builds a conceptual model of data quality problemsolving. The study analyzes data quality activities at five organizations via a five-year longitudinal study. The study finds that experienced practitioners solve data quality problems by re- flecting on and explicating
This paper presents the IPS-I-model: a model that describes the process of information problemsolving (IPS) in which the Internet (I) is used to search information. The IPS-I-model is based on three studies, in which students in secondary and (post) higher education were asked to solve information problems, while thinking aloud. In-depth analyses…
The present study examined the effects of the Iliad expert system on diagnostic problemsolving of third-year (n = 97) medical students. Students used Iliad to work-up simulated cases to supplement the education they received in their medicine clerkship. The results of the research provided evidence that the Iliad expert system did improve student diagnostic problemsolving and decision making.
Examines the relationship of social problemsolving to health behaviors as reported by 126 undergraduate students. Findings revealed significant relationships between elements of social problemsolving and wellness and accident prevention behaviors, and traffic and substance risk taking. However, correlations revealed differences between men and…
This publication features articles that illustrate how several Northwest teachers are using problemsolving to achieve rigorous and imaginative learning in their classrooms. Articles include: (1) "Open-Ended ProblemSolving: Weaving a Web of Ideas" (Denise Jarrett); (2) "Teenager or Tyke, Students Learn Best by Tackling Challenging Math" (Suzie…
In this study, it was hypothesized that problemsolving success is dependent upon two related but district types of mathematical knowledge, content indicators and connectedness indicators. Results did indeed display that the problemsolving success of 188 undergraduate students was related to these two indicators. The correlations of content…
There has been much publicity the past few years, regarding students' lack of basic skills, their inability to think clearly, and their poor use of problemsolving strategies. To focus on this need, the following program has been designed to help elementary teachers introduce problemsolving in an organized manner adding very little, if any extra material to the curriculum.
This study is an investigation of student understanding of population genetics and how students developed, used and revised conceptual models to solveproblems. The students in this study participated in three rounds of problemsolving. The first round involved the use of a population genetics model to predict the number of carriers in a population. The second round required them
known results on two problem formulations. Keywords: Timetabling, hybrid heuristic, tabu searchSolving the Course Timetabling Problem with a Hybrid Heuristic Algorithm Zhipeng L¨u1,2 and Jin, iterated local search, constraint solving. 1 Introduction In recent decades, timetabling has become an area
Examines the effects of database assistance on clinical problemsolving across three cohorts of medical students and two database interfaces. Discusses the relationship between personal domain knowledge and problemsolving, personal domain knowledge and database searching, and comparisons of different interface styles in information retrieval…
Difficulties in social interaction are a central feature of Asperger syndrome. Effective social interaction involves the ability to solve interpersonal problems as and when they occur. Here we examined social problem-solving in a group of adults with Asperger syndrome and control group matched for age, gender and IQ. We also assessed…
Formalizing the Cooperative ProblemSolving Process Michael J. Wooldridge Dept. of Computing research is to build systems that are capable of cooperative problemsolving. To this end, a number process: no mathematical model of the entire process has yet been de scribed. In this paper, we rectify
Purpose – To test the validity of the presumed characteristics of professional services by studying their manifestation in the problemsolving that occurs in service production. Design\\/methodology\\/approach – The paper uses medical research as secondary data to study the existence of associations between the presumed characteristics of professional services and problemsolving in the medical context. A systematic review of
Suggests a method for solving verbal problems in chemistry using a linguistic algorithm that is partly adapted from two artificial intelligence languages. Provides examples of problemssolved using the mental concepts of translation, rotation, mirror image symmetry, superpositioning, disjoininng, and conjoining. (TW)
This study used a computerized simulation and problem-solving tool along with artificial neural networks (ANN) as pattern recognizers to identify the common types of strategies high school and college undergraduate chemistry students would use to solve qualitative chemistry problems. Participants were 134 high school chemistry students who used…
Human factor plays an important role in ensuring lean process management to be successful and provides good proposition for the success of the organization in the long run. One of the main elements of people is their problemsolving capability in identifying and eliminating wastages. The purpose of this paper was to review problemsolving capabilities in lean process management;
In this study, I interviewed 22 engineering Co-Op students about their workplace problemsolving experiences and reflections and explored: 1) Of Co-Op students who experienced workplace problemsolving, what are the different ways in which students experience workplace problemsolving? 2) How do students perceive a) the differences between workplace problemsolving and classroom problemsolving and b) in what areas are they prepared by their college education to solve workplace problems? To answer my first research question, I analyzed data through the lens of phenomenography and I conducted thematic analysis to answer my second research question. The results of this study have implications for engineering education and engineering practice. Specifically, the results reveal the different ways students experience workplace problemsolving, which provide engineering educators and practicing engineers a better understanding of the nature of workplace engineering. In addition, the results indicate that there is still a gap between classroom engineering and workplace engineering. For engineering educators who aspire to prepare students to be future engineers, it is imperative to design problemsolving experiences that can better prepare students with workplace competency.
Tests to assess problem-solving ability being provided for the Air Force are described, and some details on the development and validation of these computer-administered diagnostic achievement tests are discussed. Three measurement approaches were employed: (1) sequential problemsolving; (2) context-free assessment of fundamental skills and…
Many students feel insecure making their first attempts to solve programming problems. Despite finishing the introductory programming course successfully, these students refrain from pursuing their CS studies. Hence, this aversion towards problemsolving and programming is not fully explained by lack of subject understanding and performance. In order to better understand the components of students' comfort, a first attempt to
We explored the affective states that students experienced during effortful problemsolving activities. We conducted a study where 41 students solved difficult analytical reasoning problems from the Law School Admission Test. Students viewed videos of their faces and screen captures and judged their emotions from a set of 14 states (basic…
Twenty middle grades students were interviewed to gain insights into their reasoning about problem-solving strategies using a ProblemSolving Justification Scheme as our theoretical lens and the basis for our analysis. The scheme was modified from the work of Harel and Sowder (1998) making it more broadly applicable and accounting for research…
This paper describes a method of understanding student problem-solving behavior during computer-assisted instruction using trigonometry as the example domain. Instead of attempting to model the student's process for solvingproblems, techniques which infer the equivalence between two adjacent steps in the student's process are used to determine…
Offers a new approach to teaching problemsolving in technology education that encourages students to apply problem-solving skills to improving the human condition. Suggests that technology teachers incorporate elements of a Taoist approach in teaching by viewing technology as a tool with a goal of living a harmonious life. (JOW)
An observation on teaching introductory programming courses on SLN for a period of two terms led me to believe that online students try various ways to solve a problem. In the beginning, I got the impression that some of their approaches for a solution were wrong; but after a little investigation, I found that some of the problem-solving…
Background: Treatments to help persons with Alzheimer's disease (AD) improve and\\/or compensate for deteriorating functional abilities have largely focused on cognitive rather than executive functions. Problemsolving is an executive function integral to most activities of daily living that is compromised by AD. Successful treatment of problem?solving deficits in persons with AD could potentially increase the amount of time a
This manual is to be used by leaders of RUPS (Research Utilizing ProblemSolving) workshops for school or district administrators. The workshop's goal is for administrators to develop problemsolving skills by using the RUPS simulation situations in a teamwork setting. Although workshop leaders should be familiar with the RUPS materials and…
A major goal of education is to help learners store information in long-term memory and use that information on later occasions to effectively solveproblems (Vockell 2010). Therefore, this author began to use the Rubik's cube to help students learn to problemsolve. There is something special about this colorful three-dimensional puzzle that…
A meta-analytic review of empirical studies that have investigated incubation effects on problemsolving is reported. Although some researchers have reported increased solution rates after an incubation period (i.e., a period of time in which a problem is set aside prior to further attempts to solve), others have failed to find effects. The…
A problem in learning to solve mathematics word problems students have been facing is to transfer the learned problem-solving knowledge from one story context to another story context. Some studies have provided evidence showing that structure facilitates transfer of learning to solve word problems. However, it is still under development for what…
Pat Wagener of Los Medanos College describes an inquiry project with his Developmental Math students: "Through my classroom inquiry into teaching problem-solving, I have shown that students can learn to solveproblems in ways that help them develop "habits of mind" with problemsolving processes with the following features in the instructional plan: Students get lots of problemsolving practice, with an emphasis on long term learning of habits of mind Students are introduced to the idea of multiple representations early, and this approach is reinforced through the curriculum materials in meaningful ways and in all aspects of the course Students have many opportunities to share their problemsolving publicly through board work "
Two experiments tested a total of 509 participants on insight problems (the radiation problem and the nine-dot problem). Half of the participants were first exposed to a 1-min movie that included a subliminal hint. The hint raised the solution rate of people who did not recognize it. In addition, the way they solved the problem was affected by the hint. In Experiment 3, a novel technique was introduced to address some methodological concerns raised by Experiments 1 and 2. A total of 80 participants solved the 10-coin problem, and half of them were exposed to a subliminal hint. The hint facilitated solving the problem, and it shortened the solution time. Some implications of subliminal priming for research on and theorizing about insight problemsolving are discussed. PMID:23392651
Quantum computer algorithms can exploit the structure of random satisfiability problems. This paper extends a previous empirical evaluation of such an algorithm and gives an approximate asymptotic analysis accounting for both the average and variation of amplitudes among search states with the same costs. The analysis predicts good performance, on average, for a variety of problems including those near a phase transition associated with a high concentration of hard cases. Based on empirical evaluation for small problems, modifying the algorithm in light of this analysis improves its performance. The algorithm improves on both GSAT, a commonly used conventional heuristic, and quantum algorithms ignoring problem structure.
This study examines 8th grade students' coordination of quantitative units arising from word problems that can be solved via a set of equations that are reducible to a single equation with a single unknown. Along with Unit-Coordination, Quantitative Unit Conservation also emerges as a necessary construct in dealing with such problems. We base our…
To solve a problem, an ordinary computer system executes an existing program. When no such program is available, an AGI system may still be able to solve a concrete problem instance. This paper introduces a new approach to do so in a reasoning system that adapts to its environment and works with insuffcient knowledge and resources. The related approaches are compared, and several conceptual issues are analyzed. It is concluded that an AGI system can solve a problem with or without a problem-specific program, and therefore can have human-like creativity and exibility.
Genetic algorithms (GA) have been applied to a number of optimisation problems with some success (1). The algorithms mimic the process of natural selection, with the effect of creating a number of potentially optimal solutions to some complex search problem. One of the major disadvantages of genetic algorithms is that they are very slow. In this paper we discuss the
Employee timetabling is the operation of assigning employees to tasks in a set of shifts during a fixed period of time, typically a week. We present a general definition of employee timetabling problems (ETPs) that captures many real-world problem formulations and includes complex constraints. The proposed model of ETPs can be represented in a tabular form that is both intuitiveExamined differences between students who perceived themselves as "successful" and "unsuccessful" problem solvers. Results revealed "successful" and "unsuccessful" problem solvers differed in number of problems acknowledged, on self-report ratings about the personal problemsolving process, and on ratings made by interviewers on several cognitive…
In this study an analysis is made regarding the Theory of Inventive ProblemSolving Technique (TRIZ), which emerged in Russia in 1946 and has been commonly used in the USA and Europe in the past few last decades. TRIZ is a method that is used successfully to solve the problems arising during the process of product development. Within this study
The ability to solveproblems in a variety of contexts is becoming increasingly important in our rapidly changing technological society. Problem-solving is a complex process that is important for everyday life and crucial for learning physics. Although there is a great deal of effort to improve student problemsolving skills throughout the educational system, national studies have shown that the majority of students emerge from such courses having made little progress toward developing good problem-solving skills. The Physics Education Research Group at the University of Minnesota has been developing Internet computer coaches to help students become more expert-like problem solvers. During the Fall 2011 and Spring 2013 semesters, the coaches were introduced into large sections (200+ students) of the calculus based introductory mechanics course at the University of Minnesota. This dissertation, will address the research background of the project, including the pedagogical design of the coaches and the assessment of problemsolving. The methodological framework of conducting experiments will be explained. The data collected from the large-scale experimental studies will be discussed from the following aspects: the usage and usability of these coaches; the usefulness perceived by students; and the usefulness measured by final exam and problemsolving rubric. It will also address the implications drawn from this study, including using this data to direct future coach design and difficulties in conducting authentic assessment of problem-solving.
SUMMARY Objectives Depression, loss, and physical illness are associated with suicide in the elderly. However, the nature of individual vulnerability remains poorly understood. Poor problemsolving has been suggested as a risk factor for suicide in younger adults. Unresolved problems may create an accumulation of stressors. Thus, those with perceived deficits in problem-solving ability may be predisposed to suicidal behavior. To test this hypothesis, we investigated whether elderly suicide attempters perceived their problemsolving as deficient. Methods Sixty-four individuals aged 60 and older participated in the study including depressed suicide attempters, depressed non-attempters, and non-depressed controls. The social problemsolving inventory-revised: short-version was used to measure participants' perceived social problemsolving, assessing both adaptive problem-solving dimensions (positive problem orientation and rational problemsolving) and dysfunctional dimensions (negative problem orientation, impulsivity/carelessness, and avoidance). Results Depressed elderly who had attempted suicide perceived their overall problemsolving as deficient, compared to non-suicidal depressed and non-depressed elderly. Suicide attempters perceived their problems more negatively and approached them in a more impulsive manner. On rational problemsolving and avoidant style sub-scales, suicide attempters did not differ from non-suicidal depressed. However, both depressed groups reported lower rational problemsolving and higher avoidance compared to non-depressed controls. Conclusions A perception of life problems as threatening and unsolvable and an impulsive approach to problemsolving appear to predispose vulnerable elderly to suicide attempts. PMID:19405045
This study compared adolescents with Asperger's syndrome with typically developing adolescents on a novel problem-solving task that presented videotaped scenarios in real-life-type social contexts. The Asperger's group was impaired in several aspects of problem-solving, including recounting the pertinent facts, generating possible high-quality problem solutions, and selecting optimal and preferred solutions. This group's solutions differed most from those of the typically
In this paper, we propose a branch and bound method for solving the job-shop problem. It is based on one-machine scheduling problems and is made more efficient by several propositions which limit the search tree by using immediate selections. It solved for the first time the famous 10 \\\\times 10 job-shop problem proposed by Muth and Thompson in 1963.
When a problem is identified in practice, it is important to clarify exactly what it is and establish the cause before seeking a solution. This solution-seeking process should include input from those directly involved in the problematic situation, to enable individuals to contribute their perspective, appreciate why any change in practice is necessary and what will be achieved by the change. This article describes some approaches to identifying and analysing problems in practice so that effective solutions can be devised. It includes a case study and examples of how the Five Whys analysis, fishbone diagram, problem tree analysis, and Seven-S Model can be used to analyse a problem. PMID:22848969
In a pilot project implemented at the University of Kansas, a team of instructors from the education and chemistry departments modified the introductory chemistry laboratory curriculum to center on problem-based inquiry learning units. The new laboratory
Transportation Problem (TP) is one of the basic operational research problems, which plays an important role in many practical applications. In this paper, a bio-inspired mathematical model is proposed to handle the Linear Transportation Problem (LTP) in directed networks by modifying the original amoeba model Physarum Solver. Several examples are used to prove that the provided model can effectively solve Balanced Transportation Problem (BTP), Unbalanced Transportation Problem (UTP), especially the Generalized Transportation Problem (GTP), in a nondiscrete way.
In this project, a plan for solving word problems based on the students' level of development was developed. A 10-week implementation of a plan for solving word problems at the concrete level of development included the use of a flow chart or plan to map out and solve word problems. Students then used the flow chart and manipulatives to develop…
Some research studies, many of which used quantitative methods, have suggested that graphics calculators can be used to effectively enhance the learning of mathematics. More recently research studies have started to explore students' styles of working as they solveproblems with technology. This paper describes the use of a software application…
Behavioral and neuroimaging findings indicate that distinct cognitive and neural processes underlie solvingproblems with sudden insight. Moreover, people with less focused attention sometimes perform better on tests of insight and creative problemsolving. However, it remains unclear whether different states of attention, within individuals, influence the likelihood of solvingproblems with insight or with analysis. In this experiment, participants (N = 40) performed a baseline block of verbal problems, then performed one of two visual tasks, each emphasizing a distinct aspect of visual attention, followed by a second block of verbal problems to assess change in performance. After participants engaged in a center-focused flanker task requiring relatively focused visual attention, they reported solving more verbal problems with analytic processing. In contrast, after participants engaged in a rapid object identification task requiring attention to broad space and weak associations, they reported solving more verbal problems with insight. These results suggest that general attention mechanisms influence both visual attention task performance and verbal problemsolving. PMID:24459538
Studies indicate that the use of multiple representations in teaching helps students become better problem solvers. We report on a study to investigate students' difficulties with multiple representations. We conducted teaching/learning interviews with 20 students in a first semester calculus-based physics course. Each student was interviewed four times during the semester, each time after they had completed an exam in class. During these interviews students were first asked to solve a problem they had seen on the exam, followed by problems that differed in context and type of representation from the exam problem. Students were provided verbal scaffolding to solve the new problems. We discuss the common difficulties that students encountered when attempting to transfer their problemsolving skills across problems in different representations.
Feed filters were installed in Syncrude hydrotreater units to protect the catalyst beds from plugging by fine solids in the feed. Severe filter fouling occurred after a process flow sheet change. The root cause of fouling was revealed through a step-by-step scientific investigation. It was first confirmed that the fouling problem was related to a process flow sheet change that
This paper studies the resolution of (augmented) weighted matching problems within a constraint programming (CP) framework. The first contribution of the paper is a set of techniques that improves substantially the performance of branch-and-bound algorithms based on constraint propagation and the second contribution is the introduction of weighted matching as a global constraint ( WeightedMatching), that can be propagated using
A b s t r ac t This paper studies the resolution of (augmented) weighted matching problems within a constraint programming framework. The first contribution of the paper is a set of branch-and-bound techniques that improves substantially the performance of algorithms based on constraint propagation and the second contribution is the introduction of weighted matching as a global constraint (
All bodily movements made by 51 undergraduate and 2 graduate students were recorded during the solution of mental problems of the type included in the average adult level of the latest revision of the Stanford-Binet scale in which the task is to specify how a given number of pints of water can be measured by means of 2 containers ofWe consider a two dimensional membrane. The goal is to flnd properties of the membrane or properties of a force on the membrane. The data is natural fre- quencies or mode shape measurements. As a result, the functional relationship between the data and the solution of our inverse problem is both indirect and nonlinear. In this paper we describe three
While there has been recent interest in research on planning and reasoning about actions,nearly all research results have been theoretical. We know of no previous examples of aplanning system that has made a significant impact on a problem of practical importance.One of the primary goals during the development of the SIPE-2 planning system has beenthe balancing of efficiency with expressiveness
Network analysis provides an effective practical system for planning and controlling…
Use of Ontology for Solving Interoperability Problems between Enterprises Hui Liu1,2 , Anne enterprises, the semantic issues are important. To date, they are more and more focused on ontology. This paper presents how to use ontology in the PBMEI method, aimed at solving enterprise interoperability
SProblem solving has a long and successful history in mathematics education and is valued by many teachers as a way to engage and facilitate learning within their classrooms. The potential benefit for using problemsolving in the development of algebraic thinking is that "it may broaden and develop students' mathematical thinking beyond the…
This training manual is for teachers participating in the Research Utilizing ProblemSolving (RUPS) workshops. The workshops last for four and one-half days and are designed to improve the school setting and to increase teamwork skills. The teachers participate in simulation exercises in which they help a fictitious teacher or principal solve a…
Many innovations in organizations result when people discover insightful solutions to problems. Insightful problem-solving was considered by Gestalt psychologists to be associated with productive, as opposed to re-productive, thinking. Productive thinking is characterized by shifts in perspective which allow the problem solver to consider new,…
Presents an information processing view of personal problemsolving which involves how people take in information, process information into plans for solutions to personal problems, and carry out plans. Presents a definition of "problem." Offers suggestions for research and for counseling. (Author/NB)
Working memory is one of the cognitive processes thought to differentiate insight and analytic forms of problemsolving. The present research examined memory involvement in the solution of insight versus analytic problems. Participants completed verbal and spatial working memory and short-term memory measures and a series of analytic and insight problems. Results demonstrated a relationship between working-memory capacity and the
This paper presents a new method for solving a linear programming problem, which is an extended version of the one previously presented by the author. The optimal solution of a linear programming problem is composed of some inequality constraints in their equality form. Then, it is possible to recognize the problem of finding the equality constraints which constitute the optimal…The problem of finding a maximum clique of an undirected graph is formulated and solved as a linearly constrained indefinite quadratic global optimization problem. Theoretical upper and lower bounds on the size k of the maximum clique are derived from the global optimization formulation, and a relationship between the set of distinct global maxima of the optimization problem and the
Bipolar preference problems: framework, properties and solving techniques Stefano Bistarelli1 preferences, that we call bipolar prefer- ence problems. Although seemingly specular notions, these two kinds the notion of arc consis- tency to bipolar problems, and we show how branch and bound (with or without
Asserts that within the context of problem-based learning environments, professors can encourage students to use computers as problem-solving tools. The ten-step Integrating Technology for InQuiry (NteQ) model guides professors through the process of integrating computers into problem-based learning activities. (SWM)
In this media-rich lesson plan, students learn how critical thinking and problemsolving are used in advanced manufacturing fields, then apply what they've learned in activities that are based on real-world scenarios.
Mechanical engineering Mechanical engineering is about solvingproblems, designing processes, and making products to improve the quality of human life and shape the economy. Mechanical engineers apply, from power stations to cars, robots and computers. The professional training mechanical engineersGenetic algorithm is one of the most interesting heuristic search techniques. It depends basically on three operations; selection, crossover and mutation. The outcome of the three operations is a new population for the next generation. Repeating these operations until the termination condition is reached. All the operations in the algorithm are accessible with today's molecular biotechnology. The simulations show that with this new computing algorithm, it is possible to get a solution from a very small initial data pool, avoiding enumerating all candidate solutions. For randomly generated problems, genetic algorithm can give correct solution within a few cycles at high probability.
Reflection is essential in order to learn from problemsolving. This thesis explores issues related to how reflective students are and how we can improve their capacity for reflection on problemsolving. We investigate how students naturally reflect in their physics courses about problemsolving and evaluate strategies that may teach them reflection as an integral component of problem-solving. Problem categorization based upon similarity of solution is a strategy to help them reflect about the deep features of the problems related to the physics principles involved. We find that there is a large overlap between the introductory and graduate students in their ability to categorize. Moreover, introductory students in the calculus-based courses performed better categorization than those in the algebra-based courses even though the categorization task is conceptual. Other investigations involved exploring if reflection could be taught as a skill on individual and group levels. Explicit self-diagnosis in recitation investigated how effectively students could diagnose their own errors on difficult problems, how much scaffolding was necessary for this purpose, and how effective transfer was to other problems employing similar principles. Difficulty in applying physical principles and difference between the self-diagnosed and transfer problems affected performance. We concluded that a sustained intervention is required to learn effective problem-solving strategies. Another study involving reflection on problemsolving with peers suggests that those who reflected with peers drew more diagrams and had a larger gain from the midterm to final exam. Another study in quantum mechanics involved giving common problems in midterm and final exams and suggested that advanced students do not automatically reflect on their mistakes. Interviews revealed that even advanced students often focus mostly on exams rather than learning and building a robust knowledge structure. A survey was developed to further evaluate students' attitudes and approaches towards problemsolving. The survey responses suggest that introductory students and even graduate students have different attitudes and approaches to problemsolving on several important measures compared to physics faculty members. Furthermore, responses to individual survey questions suggest that expert and novice attitudes and approaches to problemsolving may be more complex than naively considered.
The quality movement's original problem-solving model was the Plan-Do-Check-Act model of Shewhart and popularized by Deming. Whether called the problem-solving process (Xerox), the Quality Improvement Cycle (AT&T), or the quality-Improvement story (Florida Power & Light), these more recent models are basically variations of the scientific method and the Plan-Do-Check-Act (PDCA) cycle. They offer a systematic approach, a standardization to the
Recently, Mangasarian [18, 19] has discussed the idea of solving certain classes of linear complementarity problems as linear\\u000a programs. The present paper (1) demonstrates how these complementarity problems are related to the theory of polyhedral sets\\u000a having least elements and (2) discusses the question of whether the linear programming approach can be recommended for solving\\u000a them.
Wepropose a novel framework to solve the state assignment problem arising from the signal transition graph (STG) representation of an asynchronous circw"t. Wefirst establish a relation between STGS ad finite state machines (R3ds). Then we solve the STG state assignment problem by minittdzing the number of states in the corresponding F3vl and by using a critical racefree state assignment technique.
Patterns of problem-solving among 5-to-7 year-olds' were examined on a range of literacy (reading and spelling) and arithmetic-based (addition and subtraction) problem-solving tasks using verbal self-reports to monitor strategy choice. The results showed higher levels of variability in the children's strategy choice across Years I and 2 on the arithmetic (addition and subtraction) than literacy-based tasks (reading and spelling). However, across all four tasks, the children showed a tendency to move from less sophisticated procedural-based strategies, which included phonological strategies for reading and spelling and counting-all and finger modellingfor addition and subtraction, to more efficient retrieval methods from Years I to 2. Distinct patterns in children's problem-solving skill were identified on the literacy and arithmetic tasks using two separate cluster analyses. There was a strong association between these two profiles showing that those children with more advanced problem-solving skills on the arithmetic tasks also showed more advanced profiles on the literacy tasks. The results highlight how different-aged children show flexibility in their use of problem-solving strategies across literacy and arithmetical contexts and reinforce the importance of studying variations in children's problem-solving skill across different educational contexts. PMID:19994481Although the solution, within standard quantum physics, of the problem of outcomes has been published several times, many authors continue to treat measurement as an unsolved fundamental dilemma. The solution lies in the formation of entangled subsystems, the non-local nature of the measurement state, and the resulting distinction between mixed-state local outcomes and the pure-state global outcome. Upon "measurement" (i.e. entanglement), the quantum system and its measurement apparatus both decohere and collapse into local mixed states while the unitarily-evolving global state remains coherent and un-collapsed. The states we observe are the local, collapsed states. Considerable experimental evidence supports this conclusion. Theoretical objections to this conclusion are rebutted, and a new perspective on measurement and entanglement is noted.
Accurately assigning folds for divergent protein sequences is a major obstacle to structural studies and underlies the inverse protein folding problem. Herein, we outline our theories for fold-recognition in the "twilight-zone" of sequence similarity (<25% identity). Our analyses demonstrate that structural sequence profiles built using Position-Specific Scoring Matrices (PSSMs) significantly outperform multiple popular homology-modeling algorithms for relating and predicting structures given only their amino acid sequences. Importantly, structural sequence profiles reconstitute SCOP fold classifications in control and test datasets. Results from our experiments suggest that structural sequence profiles can be used to rapidly annotate protein folds at proteomic scales. We propose that encoding the entire Protein DataBank (~1070 folds) into structural sequence profiles would extract interoperable information capable of improving most if not all methods of structural modeling.
This study solves the Grad-Shafranov equation with a fixed plasma boundary by utilizing a meshless method for the first time. Previous studies have utilized a finite element method (FEM) to solve an equilibrium inside the fixed separatrix. In order to avoid difficulties of FEM (such as mesh problem, difficulty of coding, expensive calculation cost), this study focuses on the meshless methods, especially RBF-MFS and KANSA's method to solve the fixed boundary problem. The results showed that CPU time of the meshless methods was ten to one hundred times shorter than that of FEM to obtain the same accuracy.
This article describes the research project, which is being performed for NMPC by Power Technologies, Inc., involving the use of lightning-activated camera systems to photograph lightning strikes to a rural distribution line. Since photograph lightning strikes to a rural distribution line. Since photographs can allow the precise location of the lightning flash and power system flashovers to be observed, they are extremely valuable to engineers who are trying to make better sense of the lightning damage problem. When electrical measurements, such as fault and surge recordings, are combined with photographic data, an overall understanding of each lightning flash and its impact on the system is attained. This can hopefully lead to improved lightning protection practices and systems. The study is being performed on a 13.2 kV distribution system that is located on an exposed plateau near Little Falls, NY (about 80 miles northwest of Albany, NY). Four automated camera systems and a substation fault recorder are utilized. All camera locations afford excellent views of lines and equipment likely to be struck by lightning. The fault recorder is used to measure the fault currents and voltage sags which occur during line flashovers. Also, the National Lightning Detection Network (NLDN) is used to confirm storm activity and camera triggering efficiency. After each storm, all data is analyzed to determine how lightning affected the power system. Areas being investigated include: What are the relative portions of lightning flashovers caused by induced surges (nearby strikes) and direct lightning hits to the line How often do shielding failures occur What system relaying, construction and overvoltage protection practices afford the best lightning protection What system relaying, construction and overvoltage protection practices afford the best lightning protection What system relaying, construction and overvoltage protection practices are problematic
Students' Difficulties in Transfer of ProblemSolving Across Representations Dong-Hai Nguyen and N-2601 Abstract. Studies indicate that the use of multiple representations in teaching helps students become better problem solvers. We report on a study to investigate students' difficulties with multiple
The College of Family Physicians of Canada has used in its certification examination a new type of structured problem-solving examination called the Formal Oral. A series of preselected problem areas such as the complaint, relevant data base, investigation, and treatment are scored by two examiners. (Editor/PG)
Creative problem-solving has been linked to successful adjustment to the demands of daily life. The ability to recognize problems as opportunities can be an essential skill when dealing with uncertainty and adapting to continuous changes, both in personal and professional lives. Family and consumer sciences (FCS) professionals should strive to…
84 #12;Chapter 6 Diffusion: Diffusive initial value problems and how to solve them Selected Reading of the simplest partial dif- ferential equations for diffusive initial value problems in the absence of advection be written T t = · T (6.0.1) where T is the temperature and = k/(cP ) is the thermal diffusivity (which has
The personal problem-solving process can be functionally analyzed from a cognitive-behavioral perspective into at least four major performance classes: (1) decision making; (2) problem exploration, differentiation, and definition; (3) identification of response alternatives; and (4) performance of an intended solution response. The personal…
Can AI Planners Solve Practical Problems? by David E. Wilkins Arti cial Intelligence Center SRI of a planning system that has made a signi cant impact on a problem of practical importance. One of the primary Institute, and SRI International. Research performed at the Department of Civil Engineering, Stanford
Examines successful/unsuccessful distinctions between novices and experts in problemsolving in terms of genetic knowledge, use of production rules, strategy selection, use of critical cues, use of logic, understanding of probability, and the thinking process itself. Suggests five implications for genetics instruction and provides three problems…
Solving the GPS problem in almost linear complexity Shamgar Gurevich University of Wisconsin. The Global Positioning System (GPS) was built to fulfill this task. It works as follows: Satellites send white noise. The GPS Problem is: Design S, and an effective method of extracting (b, 0) from S and R
In this paper, the Unit Commitment (UC) problem is presented and solved, following an innovative approach based on a metaheuristic procedure. The problem consists on deciding which electric generators must be committed, over a given planning horizon, and on defining the production levels that are required for each generator, so that load and spinning reserve requirements are verified, at minimum
New product development is notoriously difficult, and software new product development particularly so. Although a great deal of research has investigated new product development, projects developing new software products continue to have problems meeting their goals. In fact, one line of research proposes new product development is difficult because it must solve an ongoing stream of complex problems. I integrate
The small sample size problem is often encountered in pattern recognition. It results in the singularity of the within-class scatter matrix Sw in Linear Discriminant Analysis (LDA). Different methods have been proposed to solve this problem in face recognition literature. Some methods reduce the dimension of the original sample space and hence unavoidably remove the null space of Sw, which
This paper revisits an efficient procedure for solving posynomial geometric programming (GP) problems, which was initially developed by Avriel et al. The procedure, which used the concept of condensation, was embedded within an algorithm for the more general (signomial) GP problem. It is shown here that a computationally equivalent dual-based algorithm may be independently derived based on some more recent
This paper compares two linear interior point programming algorithms and an interior point quadratic programming algorithm that are used to solve the optimal power flow problem. The paper focuses on the numerical oscillations that occur because of the sequential linearization of the problem. Two methods to reduce the oscillations are discussed and implemented on a six bus test system
Help your child to health: problem-solving without recourse to drugs or treatment $99 Does your child have problems learning? Is your child able to follow through tasks? Could your child be suffering, health, and learning and shows you how to help your child to achieve balanced activity in each without
It is well documented that when solvingproblems experts first search for underlying concepts while students tend to look for equations and previously worked examples. The overwhelming majority of end-of-chapter (EOC) problems in most introductory physics textbooks contain only material and examples discussed in a single chapter, rarely requiring…
Thomas Kuhn's conceptions of the influence of paradigms on the progress of science form the framework for analyzing how medical educators have approached research on medical problemsolving. A new paradigm emphasizing multiple types of problems with varied solution strategies is proposed. (Author/MLW)
In this letter, the delayed projection neural network for solving convex quadratic programming problems is proposed. The neural network is proved to be globally exponentially stable and can converge to an optimal solution of the optimization problem. Three examples show the effectiveness of the proposed network. PMID:17131675
In this paper, we examine student success on three variants of a test item given in different representational formats (verbal, pictorial, and graphical), with an isomorphic problem statement. We confirm results from recent papers where it is mentioned that physics students' problem-solving competence can vary with representational format and that…
A project has been developed for KS3 maths, funded by the Bowland Trust ( with additional support from the DCSF. It consists of a teaching resource of about 20 case-study problems aimed at developing thinking, reasoning and problem-solving skills and has been distributed to all UK secondary schools. Each case study includes…
A simple level set method for solving Stefan problems is presented. This method can be applied to problems involving dendritic solidification. Our method consists of an implicit finite difference scheme for solving the heat equation and a level set approach for capturing the front between solid and liquid phases of a pure substance. Our method is accurate with respect to some exact solutions of the Stefan problem. Results indicate that this method can handle topology changes and complicated interfacial shapes and that it can numerically simulate many of the physical features of dendritic solidification.
The generalized traveling problem (GTSP) is an extension of the classical traveling salesman problem. The GTSP is known to be an NP-hard problem and has many interesting applications. In this paper we present a local-global approach for the generalized traveling salesman problem. Based on this approach we describe a novel hybrid metaheuristic algorithm for solving the problem using genetic algorithms. Computational results are reported for Euclidean TSPlib instances and compared with the existing ones. The obtained results point out that our hybrid algorithm is an appropriate method to explore the search space of this complex problem and leads to good solutions in a reasonable amount of time.
Background The aim of electroencephalogram (EEG) source localization is to find the brain areas responsible for EEG waves of interest. It consists of solving forward and inverse problems. The forward problem is solved by starting from a given electrical source and calculating the potentials at the electrodes. These evaluations are necessary to solve the inverse problem which is defined as finding brain sources which are responsible for the measured potentials at the EEG electrodes. Methods While other reviews give an extensive summary of the both forward and inverse problem, this review article focuses on different aspects of solving the forward problem and it is intended for newcomers in this research field. Results It starts with focusing on the generators of the EEG: the post-synaptic potentials in the apical dendrites of pyramidal neurons. These cells generate an extracellular current which can be modeled by Poisson's differential equation, and Neumann and Dirichlet boundary conditions. The compartments in which these currents flow can be anisotropic (e.g. skull and white matter). In a three-shell spherical head model an analytical expression exists to solve the forward problem. During the last two decades researchers have tried to solve Poisson's equation in a realistically shaped head model obtained from 3D medical images, which requires numerical methods. The following methods are compared with each other: the boundary element method (BEM), the finite element method (FEM) and the finite difference method (FDM). In the last two methods anisotropic conducting compartments can conveniently be introduced. Then the focus will be set on the use of reciprocity in EEG source localization. It is introduced to speed up the forward calculations which are here performed for each electrode position rather than for each dipole position. Solving Poisson's equation utilizing FEM and FDM corresponds to solving a large sparse linear system. Iterative methods are required to solve these sparse linear systems. The following iterative methods are discussed: successive over-relaxation, conjugate gradients method and algebraic multigrid method. Conclusion Solving the forward problem has been well documented in the past decades. In the past simplified spherical head models are used, whereas nowadays a combination of imaging modalities are used to accurately describe the geometry of the head model. Efforts have been done on realistically describing the shape of the head model, as well as the heterogenity of the tissue types and realistically determining the conductivity. However, the determination and validation of the in vivo conductivity values is still an important topic in this field. In addition, more studies have to be done on the influence of all the parameters of the head model and of the numerical techniques on the solution of the forward problem. PMID:18053144
This study compared three different methods of teaching five basic algebra rules to college students. All methods used the same procedures to teach the rules and included four 50-question review sessions interspersed among the training of the individual rules. The differences among methods involved the kinds of practice provided during the four review sessions. Participants who received cumulative practice answered 50 questions covering a mix of the rules learned prior to each review session. Participants who received a simple review answered 50 questions on one previously trained rule. Participants who received extra practice answered 50 extra questions on the rule they had just learned. Tests administered after each review included new questions for applying each rule (application items) and problems that required novel combinations of the rules (problem-solving items). On the final test, the cumulative group outscored the other groups on application and problem-solving items. In addition, the cumulative group solved the problem-solving items significantly faster than the other groups. These results suggest that cumulative practice of component skills is an effective method of training problemsolving. PMID:12102132
The aim of the present research was to indices and characteristics of scale validation for family problemsolving scale. The sample size of 55 couples (110 people) were selected among married men and women in Tehran and assigned to adjusted/compatible and maladjusted/incompatible groups. ENRICH marital satisfaction scale and the new FPS scale was used as research tools. Analysis of the aspects revealed 2 aspects out of 30: communication and problemsolving. Studying internal correlation of total scores of the scales and subscales showed the association rate between total score and the aspects of communication and problemsolving was 0.95. Reliability index of total score re-test was 0.91 and that of communication and problemsolving was 0.78 and 2.89, respectively. Internal correlation of total score, communication and problemsolving was 0.91, 0.78 and 0.83, respectively. As this scale is significantly associated with ENRICH marital satisfaction scale, is permanent and can distinguish adjusted/compatible and maladjusted/incompatible couples, it can be applied for clinical and research purposes.
This paper presents a new distribution and route planning problem, General Delivery Problem (GDP) which is more general than the well-known Vehicle Routing Problem. To solve a GDP, a three-phase framework heuristic approach based on decomposition techniques is introduced. The decomposition techniques are employed to divide an original problem into a set of sub-problems, which can reduce the problem size. A kind of decomposition technique, Capacity Clustering Algorithm (CCA), is embedded into the framework with Simulated Annealing (SA) to solve a special GDP. The proposed three-phase framework with the above two algorithms is compared with five other decomposition methods in a distribution instance of the Regional Fire and Emergency Center in the north of France.
This paper studies the application of evolutionary algorithms for bi-objective travelling salesman problem. Two evolutionary\\u000a algorithms, including estimation of distribution algorithm (EDA) and genetic algorithm (GA), are considered. The solution\\u000a to this problem is a set of trade-off alternatives. The problem is solved by optimizing the order of the cities so as to simultaneously\\u000a minimize the two objectives of travelling
Combinatorial problems such as scheduling, resource allocation, and configuration may involve many attributes that can be\\u000a subject of user preferences. Traditional optimization approaches compile those preferences into a single utility function\\u000a and use it as the optimization objective when solving the problem, but neither explain why the resulting solution satisfies\\u000a the original preferences, nor indicate the trade-offs made during problem
This document focuses on four children in a small group, solving word problems aloud. Different aspects of how children in small groups approach problems were revealed; certain characteristics of their attempts to solveproblems suggest a variety of questions for further research. These pupils were part of a project that involved six groups of…
Teaching physics to first-year university students (in the USA: junior/senior level) is often hampered by their lack of skills in the underlying mathematics, and that in turn may block their understanding of the physics and their ability to solveproblems. Examples are vector algebra, differential expressions and multi-dimensional integrations, and the Gauss and Ampère laws learnt in electromagnetism courses. To enhance those skills in a quick and efficient way we have developed 'Integrating Mathematics in University Physics', in which students are provided with a selection of problems (exercises) that explicitly deal with the relation between physics and mathematics. The project is based on computer-assisted instruction (CAI), and available via the Internet ( or search or click to: CONECT). Normally, in CAI a predefined student-guiding sequence for problemsolving is used (systematic problemsolving). For self-learning this approach was found to be far too rigid. Therefore, we developed the 'adventurous problemsolving' (APS) method. In this new approach, the student has to find the solution by developing his own problem-solving strategy in an interactive way. The assessment of mathematical answers to physical questions is performed using a background link with an algebraic symbolic language interpreter. This manuscript concentrates on the subject of APS.
Nine freshmen in a ninth-grade accelerated algebra class were asked to solve five nonroutine combinatorial problems. The four mathematically gifted students were successful in discovering and verbalizing the generality that characterized the solutions to the five problems, whereas the five nongifted students were unable to discover the hidden…
This book is designed to provide elementary and middle school teachers with motivating problem-solving activities to use with their students. The text contains interesting and challenging problems from mathematics, language arts, social studies, and natural science which are divided into sections of activities of short, middle, and longer duration…
In this paper we propose new hybrid methods for solving the multidimensional knapsack problem. They can be viewed as matheuristics that combine mathematical programming with the variable neighbourhood decomposition search heuristic. In each iteration a relaxation of the problem is solved to guide the generation of the neighbourhoods. Then the problem is enriched with a pseudo-cut to produce a sequence of not only lower, but also upper bounds of the problem, so that integrality gap is reduced. The results obtained on two sets of the large scale multidimensional knapsack problem instances are comparable with the current state-of-the-art heuristics. Moreover, a few best known results are reported for some large, long-studied instances.
While humans may solveproblems by applying any one of a number of different problemsolving strategies, computerized problemsolving is typically brittle, limited in the number of available strategies and ways of combining ...
Responds to Heppner and Krauskopf's article on an information processing approach to personal problemsolving. Presents a four-point summary model of problemsolving and examines what information processing adds to the area of problemsolving. (NB)
Environmental problems are difficult to solve because their causes and effects are not easily understood. When attempts are made to analyze causes and effects, the principal challenge is organization of information into a framework that is logical, technically defensible, and easy to understand and communicate. When decisionmakers attempt to solve complex problems before an adequate cause and effect analysis is performed there are serious risks. These risks include: greater reliance on subjective reasoning, lessened chance for scoping an effective problemsolving approach, impaired recognition of the need for supplemental information to attain understanding, increased chance for making unsound decisions, and lessened chance for gaining approval and financial support for a program/ Cause and effect relationships can be modeled. This type of modeling has been applied to various environmental problems, including cumulative impact assessment (Dames and Moore 1981; Meehan and Weber 1985; Williamson et al. 1987; Raley et al. 1988) and evaluation of effects of quarrying (Sheate 1986). This guidance for field users was written because of the current interest in documenting cause-effect logic as a part of ecological problemsolving. Principal literature sources relating to the modeling approach are: Riggs and Inouye (1975a, b), Erickson (1981), and United States Office of Personnel Management (1986).
't worry this is a common enough first reaction. But to be come competent at programming you have to get to the problems you see. These smaller tasks can be handled either by separate programs you write methods you tokens by frequency of occurrence 3)Print out in order most frequent first #12;Split the input problemThis paper will discuss the conventional wisdom that limits one`s problemsolving effectiveness and then explore new and unique knowledge and skills that help one break out of the old paradigms. One will discover how there is no such thing as a single right answer; how there is an infinite set of solutions to any problem; and how to find the most creative and innovative solutions such that the problem does not recur. One will see how these new methods can be used by almost anyone on any event-based problem. Several recent examples will be presented to support understanding of this new approach structures of Si layers of varying thicknesses and InAs nanowires of varying radii are computed as test problems.
In this lesson, students explore linear patterns, write a pattern in symbolic form, and solve linear equations using algebra tiles, symbolic manipulation, and the graphing calculator. The lesson starts with the presentation of the yo-yo problem. Students then complete a hands-on activity involving a design created with pennies that allows them to explore a linear pattern and express that pattern in symbolic form. Algebra tiles are introduced as the students practice solving linear equations. Working from the concrete to the abstract is especially important for students who have difficulty with mathematics, and algebra tiles help students make this transition. In addition to using algebra tiles, students also use symbolic manipulation and the graphing calculator. Finally, the students return to solve the yo-yo problem. A feature of this lesson is the effective use of peer tutors in this inclusion classroom. Student worksheets are included to print.
The aim of this study on 42 seventh graders (ages 12–13) was to determine whether and to what extent students' metacognitive\\u000a level is linked to their conceptualization and performance in problemsolving at school, especially science problems. This\\u000a hypothesis is supported by a number of studies showing that metacognition is a factor in learning. Two indexes were devised\\u000a for the
A 4-6 week unit for use with college-bound high school students, combining the introduction of chemistry with a methodical method of problemsolving and a review of the mathematics needed for high school chemistry. It includes the vocabulary used in describing the physical properties of matter, the metric system and decimals, a progression of problems dealing with the derived quantities of density and heat, and the calculation of percentage of error.
Traditionally, the multistage inspection problem has been formulated as consisting of a decision schedule where some manufacturing stages receive full inspection and the rest none. Dynamic programming and heuristic methods (like local search) are the most commonly used solution techniques. A highly constrained multistage inspection problem is presented where all stages must receive partial rectifying inspection and it is solved using a real-valued genetic algorithm. This solution technique can handle multiple objectives and quality constraints effectively.
We examine the possibility of using the standard Newton's method for solving a class of nonlinear eigenvalue problems arising from electronic structure calculation. We show that the Jacobian matrix associated with this nonlinear system has a special structure that can be exploited to reduce the computational complexity of the Newton's method. Preliminary numerical experiments indicate that the Newton's method can be more efficient for small problems in which a few smallest eigenpairs are needed.
The multiple instance problem arises in tasks where the training examples are ambiguous: asingle example object may have many alternative feature vectors (instances) that describe it,and yet only one of those feature vectors may be responsible for the observed classification ofthe object. This paper describes and compares three kinds of algorithms that learn axis-parallelrectangles to solve the multiple-instance problem. Algorithms
A new algorithm is presented, designed to solve tridiagonal matrix problems efficiently withparallel computers (multiple instruction stream, multiple data stream (MIMD) machines withdistributed memory). The algorithm is designed to be extendable to higher order bandeddiagonal systems.I. IntroductionCurrently, there are several popular methods for parallelization of the tridiagonal problem.The "most important" of these have recently been described with a unified approach,through
With the advent of on-line (time-sharing) computer systems and graphic terminals, we have available a new dimension in numerical problemsolving capabilities. Rather than simply use the new power to achieve fast turnaround, we can develop interactive routines which are easy to use and also take advantage of the insight and visual capabilities of the human problem solver. Several on-line
Ant colony optimization algorithm is a novel simulated evolutionary algorithm, which provides a new method for complicated combinatorial optimization problems. In this paper the algorithm is used for solving the knapsack problem. It is improved in selection strategy and information modification, so that it can not easily run into the local optimum and can converge at the global optimum. The experiments show the robustness and the potential power of this kind of meta-heuristic algorithm.
This study—to our knowledge the first to model the dynamics of knowledge creation in an engineering problemsolving context—addresses a gap in the literature by illustrating "engineering epistemology," nurtured by "ba," as a critical knowledge asset that facilitates superior problem resolution. Rich narratives generated by phenomenological interviews with US product engineers were interpreted using Nonaka and Takeuchi's knowledge-creation model and
One of the objectives of Cognitive Robotics is to construct robot systems that can be directed to achieve realworld goals by high-level directions rather than complex, low-level robot programming. Such a system must have the ability to represent, problem-solve and learn about its environment as well as communicate with other agents. In previous work, we have proposed ADAPT, a Cognitive Architecture that views perception as top-down and goaloriented and part of the problemsolving process. Our approach is linked to a SOAR-based problem-solving and learning framework. In this paper, we present an architecture for the perceptive and world modelling components of ADAPT and report on experimental results using this architecture to predict complex object behaviour. A novel aspect of our approach is a 'mirror system' that ensures that the modelled background and foreground objects are synchronized with observations and task-based expectations. This is based on our prior work on comparing real and synthetic images. We show results for a moving object that collides and rebounds from its environment, hence showing that this perception-based problemsolving approach has the potential to be used to predict complex object motions.
Solving the Robots Gathering Problem Mark Cieliebak1 , Paola Flocchini2 , Giuseppe Prencipe3 a set of n > 2 simple autonomous mobile robots (decentralized, asyn- chronous, no common coordinate, deterministic) moving freely in the plane and able to sense the positions of the other robots. We study
Our approach to the study of learning of mathematical problem-solving extends the notion of narrative learning environments to include the dynamics of collaborative dialogs and related emergent narratives. This perspective favours the conception of the dialogical aspects of interaction as shared achievements of co- participants and as central meaning-making procedures, based on our qualitative analysis of transcripts from online collaborative
This study investigated the cognitive benefits of learning how to program by determining the degree of cognitive transfer of programming skills at a construct level to solving analogous problems in other domains. Subjects, who were students enrolled in four sections of the beginning Pascal programming course and two sections of a calculus course,…
The purpose of this study was to investigate how modeling-based instruction combined with an interactive-engagement teaching approach promotes students' problemsolving abilities. I focused on students in a calculus-based introductory physics course, based on the matter and interactions curriculum of Chabay & Sherwood (2002) at a large state…
A three-phase plan was developed to solve any waste water odor problem, e.g., those encountered in the chemical manufacturing industry. Phase 1 consists of an evaluation of the odor emissions from both stack and open (fugitive) sources and their impact on ambient odors. The critical odor sources and their required degree of control are thereby defined. Phase 2 consists of
Outlines the problem-solving team training process used at Harvard University (Massachusetts), including the size and formation of teams, roles, and time commitment. Components of the process are explained, including introduction to Total Quality Management (TQM), customer satisfaction, meeting management, Parker Team Player Survey, interactive…
Some Finance ProblemsSolvedThis cutting-edge volume offers a complete primer on conducting problem-solving based assessments in school or clinical settings. Presented are an effective framework and up-to-date tools for identifying and remediating the many environmental factors that may contribute to a student's academic, emotional, or behavioral difficulties, and for…
Global changes in educational discourse have an impact on educational systems, so teacher education programs need to be transformed to better train teachers and to contribute to their professional development. In this process learning styles and problemsolving skills should be considered as individual differences which have an impact in…
The Proceedings of the 1998 Puerto Rico conference on Solving Forest Insect Problems Through Research (sponsored in part by the International Union of Forestry Research Organizations) are available at this Website. The proceedings include the program, abstracts from presentations and posters, and contact information for presenters.
learning. Our work comes in the context of growing interest in interactive, human-in-the-loop learning that people formulate to refine the behavior of a system. We focus on analyzing and learning within Ensemble describe a study we ran to observe human problemsolving behavior with the system, review insights we
Described are procedures followed in developing, administering, and scoring a set of mathematical problem-solving superitems and examining their construct validity through a recently developed evaluation technique associated with a taxonomy of the structure of learned outcomes. Data strongly support the validity of the underlying theoretical…
This classroom note shows how Fibonacci numbers with negative subscripts emerge from a problem-solving context enhanced by the use of an electronic spreadsheet. It reflects the author's work with prospective K-12 teachers in a number of mathematics content courses. (Contains 4 figures.)
Computer and computational scientists at Pacific Northwest National Laboratory (PNNL) are studying and designing collaborative problemsolving environments (CPSEs) for scientific computing in various domains. Where most scientific computing efforts focus at the level of the scientific codes, file systems, data archives, and networked computers, our analysis and design efforts are aimed at developing enabling technologies that are directly meaningful
The purpose of this exploratory qualitative study was to determine the reasoning processes used by paramedics to solve clinical problems. Existing research documents concern over the accuracy of paramedics' clinical decision-making, but no research was found that examines the cognitive processes by which paramedics make either faulty or accurate…
Social Conflict and Negotiative ProblemSolving is an instructional system currently under development by the Improving Teaching Competencies Program (ITCP) of Northwest Regional Educational Laboratory (NWREL). In accordance with the Resource Allocation Management Plan (RAMP, 1975) of ITCP, this report presents a plan of evaluation activities for…
Describes a project that helps students integrate biological concepts using both creativity and higher-order problem-solving skills. Involves students playing the roles of junior scientists aboard a starship in orbit around a class M planet and using a description of habitats, seasonal details, and a surface map of prominent geographic features to…
In CSCL systems, students who are solvingproblems in group have to negotiate with each other by exchanging proposals and arguments in order to resolve the conflicts and generate a shared solution. In this context, argument construction assistance is necessary to facilitate reaching to a consensus. This assistance is usually provided with isolated arguments by demand, but this does not
Explains an algorithm which details procedures for solving a broad class of genetics problems common to pre-college biology. Several flow charts (developed from the algorithm) are given with sample questions and suggestions for student use. Conclusions are based on the authors' research (which includes student interviews and textbook analyses).…
Hemispheric involvement in reasoning abilities has been debated for some time, and it remains unclear whether the right hemisphere's involvement in problemsolving is modality specific or dependent on the type of spatial reasoning required. In the current study, 2 types of nonverbal reasoning abilities were examined, spatial reasoning and proportional reasoning, in 109 patients with cerebrovascular disease that was
of Human Genetics, University of Pittsburgh This research was sponsored in part by the NIH National, molecular genetics, microsatellite genotyping, pattern matching, FASTMAP. #12; #12; ABSTRACT The HumanAutomating Computational Molecular Genetics: Solving the Microsatellite Genotyping Problem See
of Human Genetics, University of Pittsburgh This research was sponsored in part by the NIH National, molecular genetics, microsatellite genotyping, pattern matching, FAST-MAP. #12;#12;ABSTRACT The Human GenomeAutomating Computational Molecular Genetics: Solving the Microsatellite Genotyping Problem See
The mathematics problemsolving approaches of a group of elementary and secondary ESL students were investigated through a performance assessment accompanied by think-aloud procedures. Students were enrolled in ESL mathematics classes in a Title VII project implementing the Cognitive Academic Learning Approach (CALLA). In this approach, curriculum content is used to develop academic language and learning strategies are taught explicitly
Problemsolving is a critical skill for engineering students and essential to development of creativity and innovativeness. Essential to such learning is an ease of communication and allowing students to address the issues at hand via the terminology, attitudes, humor and empathy, which is inherent to their frame of mind as novices, without the…
As technologies have become an integral part of our lives, the way we read and understand text has changed drastically. In this paper, we discuss how various technologies support learners' reading and writing skills within the context of meaningful learning. Next, using elaborated cases, we argue that situating learners in problemsolving…
This study explored and documented students' responses to opportunities for collective knowledge building and collaboration in a problem-solving process within complex environmental challenges and pressing issues with various dimensions of knowledge and skills. Middle-school students ("n" =?16; age 14) and high-school students…
In this article, we present an integer sequence approach to solve the classic water jugs problem. The solution steps can be obtained easily by additions and subtractions only, which is suitable for manual calculation or programming by computer. This approach can be introduced to secondary and undergraduate students, and also to teachers and…
The equation of motion for a mass that moves under the influence of a central, inverse-square force is formulated and solved as a problem in complex variables. To find the solution, the constancy of angular momentum is first established using complex variables. Next, the complex position coordinate and complex velocity of the particle are assumed…
This study was conducted to investigate the quality and nature of the students' interactions during asynchronous online problemsolving in two sections of College Algebra taught by the author. In a shared-work section, students worked independently for an initial phase and had access to classmates' work during a follow-up phase. Students in the…
In this paper a new method is suggested for solving the problem in which the objective function is a linear fractional function, and where the constraint functions are in the form of linear inequalities. The proposed method is based mainly upon revised primal dual simplex algorithm (RPDSA).The algorithm can be combined with interior-point methods to move from an interior point
This book is part of the "Windows on Literacy: Language, Literacy & Vocabulary" program and shows students ways to solveproblems, including drawing a picture and using a calculator. The suggested grade range is K-3; the guided reading level is N-P; the basal results level is Grade 2-Grade 3; and the Windows on Literacy Stage is Fluent Plus…
Solving the GPS Problem in Almost Linear Complexity Speaker: Shamgar Gurevich, UW Madison. Abstract (GPS) was built to ful...ll this task. It works as follows: Satellites send to earth their location. For simplicity, the Figure 1: Satellites communicate location in GPS. location of a satellite is a bit b 2 f 1g
Butterfield found that internal Ss tended to make more constructive responses to frustration-type situations than did extrenal Ss. Therefore, this study predicted that internal Ss would rate themselves as more confident with regard to problem-solving abilities than would external Ss. (Author)
This paper describes how multiple interacting swarms of adaptive mobile agents can be used to solveproblems in networks. The paper introduces a new architectural description for an agent that is chemically inspired and proposes chemical interaction as the principal mechanism for inter-swarm communication. Agents within a given swarm have behavior that is inspired by the foraging activities of ants,
This study explores difficulties that prospective elementary mathematics teachers have with the concepts of ratio and proportion, mainly when they are engaged in solvingproblems using algorithm procedures. These difficulties can be traced back to earlier experiences when they were students of junior and high school. The reflection on these…
The article describes four step-by-step methods to sharpen intuitive capacities for problem-solving and innovation. Visionary and transpersonal knowledge processes are tapped to gain access to relatively deep levels of intuition. The methods are considered useful for overcoming internal blockages or resistance, developing organizational mission…
In this article, we share our learning experience as a Lesson Study team. The Research Lesson was on Figural Patterns taught in Year 7. In addition to helping students learn the skills of the topic, we wanted them to develop a problem-solving disposition. The management of these two objectives was a challenge to us. From the lesson observation and…
such as the motion of a red blood cell in plasma. Keywords: SPH, Stokes flow, microfluidics, red blood cell 1). If the dynamics is dominated by friction and inertial effects can be neglected, the flow through narrow channelsSolving microscopic flow problems using Stokes equations in SPH P. Van Liedekerkea, , B. Smeetsb
Discusses how to train teams in problem-solving skills. Topics include team training, the use of technology, instructional strategies, simulations and training, theoretical framework, and an event-based approach for training teams to perform in naturalistic environments. Contains 68 references. (Author/LRW)
Examines didactic interventions and claims their impact on pupil behavior aids the study of cognitive processes. Studies problemsolving in teaching mathematics. Examines the functions of tutorial interventions in surface features and instability of representation. Finds tutorial interventions demonstrate a substantial increase in performance…
Liquid\\/liquid extraction (LLE) is a powerful separation technique that is finding wider application in the CPI to solve difficult environmental problems, particularly in the removal of trace organic compounds from wastewater streams. LLE is usually only applied when more conventional techniques such as steam stripping or distillation are not suitable. This is because LLE usually involves the introduction of a
This investigation describes the way in which a case study participant (aged 7) used maps (including large- and small-scale maps, dynamic and static maps) to solveproblems in a technology game-based context. The participant demonstrated the capacity to decipher graphical information when simultaneously moving between maps with different representations, orientations, perspectives and scales as he played a Pokemon Game Boy.
WebQuests have been a popular alternative for collaborative group work that utilizes internet resources, but studies have questioned how effective they are in challenging students to use higher order thinking processes that involve creative problemsolving. This article explains how different levels of inquiry relate to categories of learning…
The eight papers presented in this monograph are a result of the ProblemSolving and Critical Thinking Research Workshop that was held in conjunction with the 1990 National Educational Computing Conference (NECC). The intent of the workshop was to provide a unique forum for researchers to share ideas in a special area of educational computing. The…
These model lessons from the primary grades are on the techniques of advertising drawn from a unit on, "Creating and Producing Tools and Techniques". They include behaviorial objectives, teaching and motivational strategies, evaluation techniques. The model lessons follow the problemsolving inquiry approach in social studies using multimedia…
We organised two experimental teaching designs involving web resources in two different French universities. In this paper, we describe these experiments and analyse the students' behaviours. Our aim is to observe whether the use of specific online resources favours the development of problem-solving activities.This paper introduces an artistic model of planning and problemsolving. The model is based on a case study of processes engaged in by a college art student during the course of producing a senior thesis in batik (a wax-resist fabric dyeing process). Based on the premise that knowledge of the creative process is essential to understanding the…
In this study 34 spontaneous analogies produced by 16 college freshmen while solving qualitative physics problems are analyzed. A number of the analogies were invalid in the sense that they led to an incorrect answer from the physicist's point of view. However, many were valid, and a few were powerful in the sense that they seemed not only to help…
The purpose of this study was to investigate the effects of online (web-based) creative problem-solving (CPS) activities on student technological creativity and to examine the characteristics of student creativity in the context of online CPS. A pretest-posttest quasi-experiment was conducted with 107 fourth-grade students in Taiwan. The…
Collaborative problemsolving (CPS) requires sharing goals/attention and coordinating actions--all deficient in HFASD. Group differences were examined in CPS (HFASD/typical), with a friend versus with a non-friend. Participants included 28 HFASD and 30 typical children aged 3-6 years and their 58 friends and 58 non-friends. Groups were matched on…
Presents a model for introducing inquiry and problem-solving into middle grade history classes. It is based on an educational approach suggested by John Dewey. The author uses the model to explore two seemingly contradictory statements by Abraham Lincoln about slavery. (AV)
The author discusses the need for defining what education is and how it can help change the world. The article specifically looks at sincere but vague answers freshmen often give in social issues courses, such as "To solve this problem, we should educate them," and why statements such as these do not make the case for education sufficiently.
72 InterMath 1 --Professional and Cognitive Development through ProblemSolving with Technology). The development of mathematical understanding occurs when technology is used as a cognitive tool that supports to deliver the curriculum through web-based materials and to explore the mathematics using cognitive tools
These materials are the handouts for school administrators participating in RUPS (Research Utilizing ProblemSolving) workshops. The purposes of the workshops are to develop skills for improving schools and to increase teamwork skills. The handouts correspond to the 16 subsets that make up the five-day workshop: (1) orientation; (2) identifying…When students are learning to develop algorithms, they very often spend more time dealing with issues of syntax rather than solving the problem. Additionally, the textual nature of most programming environments works against the learning style of the majority of students. RAPTOR is a visual programming environment, designed specifically to help students envision their algorithms and avoid syntactic baggage. RAPTOR
frameworks that might be used in a computational agent society, like relativized or oracle computing Abstract We propose a new framework to study problemsolving in a computational society. Such a society experiences of other agents. Our history-based comput- ing framework generalizes existing computational
This book is dedicated to George Polya, who focused on problemsolving as the means for teaching and learning mathematics. The first chapter is a reprint of his article "On Learning, Teaching, and Learning Teaching." Then, G. L. Alexanderson paints a portrait of "George Polya, Teacher," including some anecdotes that exemplify Polya's art of…
Strategies for Solving High-Fidelity Aerodynamic Shape Optimization Problems Zhoujie Lyu Aerodynamic shape optimization based on high-fidelity models is a computational intensive endeavor. The techniques are tested using the Common Research Model wing benchmark defined by the Aerodynamic Design
Solving ShapeAnalysis Problems in Languages with Destructive Updating MOOLY SAGIV Tel This article concerns the static analysis of programs that perform destructive updating on heap allocated destructive updating of the input list and (2) a program that searches a list and splices a new element
Solving ShapeAnalysis Problems in Languages with Destructive Updating Mooly Sagiv 1;2 and Thomas concerns the static analysis of programs that perform destructive updating on heapallocated storage. We programs --- including ones in which a significant amount of destructive updating takes place --- our
This article explores how differences in problem representations change both the performance and underlying cognitive processes of beginning algebra students engaged in quantitative reasoning. Contrary to beliefs held by practitioners and researchers in mathematics education, students were more successful solving simple algebra story problems than…
Prior theories have assumed that human problemsolving involves estimating distances among states and performing search through the problem space. The role of mental representation in those theories was minimal. Results of our recent experiments suggest that humans are able to solve some difficult problems quickly and accurately. Specifically, in solving these problems humans do not seem to rely on distances or on search. It is quite clear that producing good solutions without performing search requires a very effective mental representation. In this paper we concentrate on studying the nature of this representation. Our theory takes the form of a graph pyramid. To verify the psychological plausibility of this theory we tested subjects in a Euclidean Traveling Salesman Problem in the presence of obstacles. The role of the number and size of obstacles was tested for problems with 6-50 cities. We analyzed the effect of experimental conditions on solution time per city and on solution error. The main result is that time per city is systematically affected only by the size of obstacles, but not by their number, or by the number of cities.
This study investigated the differential effects of two problem-solving instructional approaches--schema-based instruction (SBI) and general strategy instruction (GSI)--on the mathematical word problem-solving performance of 22 middle school students who had learning disabilities or were at risk for mathematics failure. Results indicated that the…
This study investigated whether activating elements of prior knowledge can influence how problem solvers encode and solve simple mathematical equivalence problems (e.g., 3 + 4 + 5 = 3 + __). Past work has shown that such problems are difficult for elementary school students (McNeil and Alibali, 2000). One possible reason is that children's experiences in math classes may encourage them to think about equations in ways that are ultimately detrimental. Specifically, children learn a set of patterns that are potentially problematic (McNeil and Alibali, 2005a): the perceptual pattern that all equations follow an "operations = answer" format, the conceptual pattern that the equal sign means "calculate the total", and the procedural pattern that the correct way to solve an equation is to perform all of the given operations on all of the given numbers. Upon viewing an equivalence problem, knowledge of these patterns may be reactivated, leading to incorrect problemsolving. We hypothesized that these patterns may negatively affect problemsolving by influencing what people encode about a problem. To test this hypothesis in children would require strengthening their misconceptions, and this could be detrimental to their mathematical development. Therefore, we tested this hypothesis in undergraduate participants. Participants completed either control tasks or tasks that activated their knowledge of the three patterns, and were then asked to reconstruct and solve a set of equivalence problems. Participants in the knowledge activation condition encoded the problems less well than control participants. They also made more errors in solving the problems, and their errors resembled the errors children make when solving equivalence problems. Moreover, encoding performance mediated the effect of knowledge activation on equivalence problemsolving. Thus, one way in which experience may affect equivalence problemsolving is by influencing what students encode about the equations. PMID:24324454
This paper presents a comparative analysis of prospective elementary teachers' mathematical problemsolving-related beliefs in Cyprus and England. Twenty-four participants, twelve from a well-regarded university in each country, were interviewed qualitatively at the exit point of their undergraduate teacher education studies. Analyses revealed both similarities and differences in the ways in which prospective teachers in each country construe both mathematical problems and mathematical problemsolving, indicating not only that their beliefs are culturally situated but also that the concepts of "mathematical problem" and "problemsolving" have different meanings cross-culturally. Such findings challenge the received view in mathematics education research of definitional convergence with respect to both mathematical problems and problemsolving. Some implications for policy making are discussed.
The Texas Instruments TI-89 calculator is an advanced scientific calculator that has both graphing and programming capabilities, and advanced-mathematics software. In this article the TI-89 has been used for solving a variety of problems in physical chemistry. The applications in this paper include calculations with units, solving higher-order equations that are set up in chemical equilibrium problems, differential- and integral-calculus-based calculations in thermodynamics and quantum mechanics, numerical and analytic solutions of first-order differential equations in chemical kinetics, and regression analysis of data collected in a kinetics experiment. The appropriate calculator keystrokes are included for all the examples in this paper. Many complex and interesting problems can be studied with relative ease, thus allowing teachers to introduce modern scientific techniques in the classroom.
This paper presents a new approach to solve Fractional Programming Problems (FPPs) based on two different Swarm Intelligence (SI) algorithms. The two algorithms are: Particle Swarm Optimization, and Firefly Algorithm. The two algorithms are tested using several FPP benchmark examples and two selected industrial applications. The test aims to prove the capability of the SI algorithms to solve any type of FPPs. The solution results employing the SI algorithms are compared with a number of exact and metaheuristic solution methods used for handling FPPs. Swarm Intelligence can be denoted as an effective technique for solving linear or nonlinear, non-differentiable fractional objective functions. Problems with an optimal solution at a finite point and an unbounded constraint set, can be solved using the proposed approach. Numerical examples are given to show the feasibility, effectiveness, and robustness of the proposed algorithm. The results obtained using the two SI algorithms revealed the superiority of the proposed technique among others in computational time. A better accuracy was remarkably observed in the solution results of the industrial application problems.
This paper tries to solve open Job-Shop Scheduling Problems (JSSP) by translating them into Boolean Satisfiability Testing Problems (SAT). The encoding method is essentially the same as the one proposed by Crawford and Baker. The open problems are ABZ8, ABZ9, YN1, YN2, YN3, and YN4. We proved that the best known upper bounds 678 of ABZ9 and 884 of YN1 are indeed optimal. We also improved the upper bound of YN2 and lower bounds of ABZ8, YN2, YN3 and YN4. of demographic information, and two problem-solving sessions. Ten teacher/experts also completed the relatedness rating task and problem -solving sessions. For each rating by the students and teacher/experts, the data were transformed into a network using the Pathfinder algorithm, where each node in the network represented one of the physics concepts. Two statistical comparisons were made between the students' and teacher/experts' data: Pearson-r comparison of relatedness data and a Pearson -r comparison of the Pathfinder graphs. The results indicated that there was: (1) A structure to the thermodynamics concepts held by both the students and the teacher/experts. (2) A significant statistical difference in the Pathfinder networks among the teacher/experts. The differences were primarily localized to concepts dealing with gas laws. (3) No increase in the statistical similarity (comparing teacher/experts and students) in the networks during the instructional period. (4) A change in the students' conceptual networks indicating: (a) an acceptance by the students of certain "deep structures," (b) a time-delayed acceptance of some organizing ideas, and/or (c) gaps in the students' understanding of key ideas. (5) A "weak" rather than "strong" restructuring of the concepts by students. (6) Statistically significant similarities in local networks involving pairs of physics concepts and the problem-solving strategies used by the students. Overall this study corroborated much of the research dealing with experts and novices including studies that indicated that there are differences among novices regarding conceptual understanding and the problem-solving strategies they used. Finally, this technique using concept relatedness data and the associated Pathfinder graphs to diagnose conceptual understanding and problem-solving strategies holds potential for classroom teachers interested in better matching the learner and the learning.
This study investigated the role of strategy instruction and cognitive abilities on word problemsolving accuracy in children with math difficulties (MD). Elementary school children (N = 120) with and without MD were randomly assigned to 1 of 4 conditions: general-heuristic (e.g., underline question sentence), visual-schematic presentation…
We discuss the effect of administering conceptual and quantitative isomorphic problem pairs (CQIPP) back to back vs. asking students to solve only one of the problems in the CQIPP in introductory physics courses. Students who answered both questions in a CQIPP often performed better on the conceptual questions than those who answered the corresponding conceptual questions only. Although students often took advantage of the quantitative counterpart to answer a conceptual question of a CQIPP correctly, when only given the conceptual question, students seldom tried to convert it into a quantitative question, solve it and then reason about the solution conceptually. Even in individual interviews, when students who were only given conceptual questions had difficulty and the interviewer explicitly encouraged them to convert the conceptual question into the corresponding quantitativeproblem by choosing appropriate variables, a majority of students were reluctant and preferred to guess the answer to the conceptual question based upon their gut feeling.
Problemsolving begins with problem identification. Conventional knowledge suggests that because patrol officers work specific geographical areas (beats) on a fairly constant basis, they come to see where the problems exist; thus, police experience alone can be relied on to identify crime problems. However, few have examined whether officers are…
The purpose of this study was to analyze how school psychologists engaged in problem analysis during problem-solving consultation. Five aspects of the problem analysis process were examined: 1) the types of questions participants asked during problem identification, 2) the types of data participants requested, 3) the frequency of requests for each…
Outside the psychologist's laboratory, thinking proceeds on the basis of a great deal of interaction with artefacts that are recruited to augment problem-solving skills. The role of interactivity in problemsolving was investigated using a river-crossing problem. In Experiment 1A, participants completed the same problem twice, once in a low interactivity condition, and once in a high interactivity condition (with order counterbalanced across participants). Learning, as gauged in terms of latency to completion, was much more pronounced when the high interactivity condition was experienced second. When participants first completed the task in the high interactivity condition, transfer to the low interactivity condition during the second attempt was limited; Experiment 1B replicated this pattern of results. Participants thus showed greater facility to transfer their experience of completing the problem from a low to a high interactivity condition. Experiment 2 was designed to determine the amount of learning in a low and high interactivity condition; in this experiment participants completed the problem twice, but level of interactivity was manipulated between subjects. Learning was evident in both the low and high interactivity groups, but latency per move was significantly faster in the high interactivity group, in both presentations. So-called problem isomorphs instantiated in different task ecologies draw upon different skills and abilities; a distributed cognition analysis may provide a fruitful perspective on learning and transfer. PMID:25616778
Intricate predatory strategies are widespread in the salticid subfamily Spartaeinae. The hypothesis we consider here is that the spartaeine species that are proficient at solving prey-capture problems are also proficient at solving novel problems. We used nine species from this subfamily in our experiments. Eight of these species (two Brettus, one Cocalus, three Cyrba, two Portia) are known for specialized invasion of other spiders' webs and for actively choosing other spiders as preferred prey ('araneophagy'). Except for Cocalus, these species also use trial and error to derive web-based signals with which they gain dynamic fine control of the resident spider's behaviour ('aggressive mimicry').The ninth species, Paracyrba wanlessi, is not araneophagic and instead specializes at preying on mosquitoes. We presented these nine species with a novel confinement problem that could be solved by trial and error. The test spider began each trial on an island in a tray of water, with an atoll surrounding the island. From the island, the spider could choose between two potential escape tactics (leap or swim), but we decided at random before the trial which tactic would fail and which tactic would achieve partial success. Our findings show that the seven aggressive-mimic species are proficient at solving the confinement problem by repeating 'correct' choices and by switching to the alternative tactic after making an 'incorrect' choice. However, as predicted, there was no evidence of C. gibbosus or P. wanlessi, the two non-aggressive-mimic species, solving the confinement problem. We discuss these findings in the context of an often-made distinction between domain-specific and domain-general cognition. PMID:25392261
Only a subset of adults acquires specific advanced mathematical skills, such as integral calculus. The representation of more sophisticated mathematical concepts probably evolved from basic number systems; however its neuroanatomical basis is still unknown. Using fMRI, we investigated the neural basis of integral calculus while healthy participants were engaged in an integration verification task. Solving integrals activated a left-lateralized cortical network including the horizontal intraparietal sulcus, posterior superior parietal lobe, posterior cingulate gyrus, and dorsolateral prefrontal cortex. Our results indicate that solving of more abstract and sophisticated mathematical facts, such as calculus integrals, elicits a pattern of brain activation similar to the cortical network engaged in basic numeric comparison, quantity manipulation, and arithmetic problemsolving. PMID:18596607
When subjects are given the balls-and-boxes problem-solving task (Kotovsky & Simon, 1990), they move rapidly toward the goal after an extended exploratory phase, despite having no awareness of how to solve the task. We investigated possible non-conscious learning mechanisms by giving subjects three runs of the task while recording ERPs. Subjects showed significant differences in their ERP components during the exploratory phase between correct and incorrect moves. Exploratory incorrect moves were associated with a shallower response-locked N1 component and a larger response-locked P3 component compared with exploratory correct moves. Subjects who solved the task more quickly exhibited a trend towards larger N1 and P3 components. These results suggest that the brain processes information about the correctness of a move well before subjects are aware of move correctness. They further suggest that relatively simple attentional and error-monitoring processes play an important role in complex problem-solving. PMID:20600180
These modules view aspects of computer use in the problem-solving process, and introduce techniques and ideas that are applicable to other modes of problemsolving. The first unit looks at algorithms, flowchart language, and problem-solving steps that apply this knowledge. The second unit describes ways in which computer iteration may be used…
There is strong evidence that children show selectivity in their reliance on others as sources of information, but the findings to date have largely been limited to contexts that involve factual information. The present studies were designed to determine whether children might also show selectivity in their choice of sources within a problem-solving context. Children in two age groups (20 to 24 months and 30 to 36 months; total N = 60) were presented with a series of conceptually difficult problemsolving tasks, and were given an opportunity to interact with adult experimenters who were depicted as either good helpers or bad helpers. Participants in both age groups preferred to seek help from the good helpers. The findings suggest that even young children evaluate others with reference to their potential to provide help and use this information to guide their behavioral choices. PMID:23484915A set of testable propositions based on the collaboration and meaning analysis process (C-MAP) are presented. The C-MAP involves the conscious externalisation of knowledge to support knowledge transfer, the development of innovated knowledge and the development of cognitive similarity in intense problemsolving teams (Rentsch, J.R., Delise, L.A., and Hutchison, S., 2008a. Transferring meaning and developing cognitive similarity in decision
Problem-solving in school mathematics has traditionally been considered as belonging only to the concrete symbolic mode of\\u000a thinking, the mode which is concerned with making logical, analytical deductions. Little attention has been given to the place\\u000a of the intuitive processes of the ikonic mode. The present study was designed to explore the interface between logical and\\u000a intuitive processes in the
This study examined the social organization of Guatemalan Mayan fathers' engagement with school-age children in a group problem-solving task. Twenty-nine groups of Mayan fathers varying in extent of Western schooling and 3 related school-age children (ages 6–12 years) constructed a puzzle together. Groups with fathers with 0 to 3 grades more often constructed the puzzle through shared multiparty collaboration involving
The main objective of this study is (a) to explore the relationship among cognitive style (field dependence\\/independence),\\u000a working memory, and mathematics anxiety and (b) to examine their effects on students' mathematics problemsolving. A sample\\u000a of 161 school girls (13–14 years old) were tested on (1) the Witkin's cognitive style (Group Embedded Figure Test) and (2)\\u000a Digit Span Backwards Test, with
Simulation driven design helped Lockheed Martin Aeronautical Systems solve a fatigue-related problem on the cargo door of the C5 transport plane. Cracks in the area of the door's upper hinge had led the Air Force to impose a special visual inspection of the door prior to each ADS mission. Use of dynamics analysis software enabled Lockheed Martin to quickly find
This 86-page practice guide (pdf) provides educators with five specific, evidence-based recommendations for improving students' mathematical problemsolving in grades 4 through 8 by incorporating such activities into regular instruction. The guide contains detailed suggestions and strategies for carrying out each recommendation, including potential roadblocks with possible approaches for overcoming them. It concludes by suggesting a four-step process for incorporating the recommendations into a lesson. The guide includes an extensive list of references.
This paper creates and analyzes a new quantum algorithm called the Amplified Quantum Fourier Transform (QFT) for solving the following problem: The Local Period Problem: Let L = {0,1 . . . N-1} be a set of N labels and let A be a subset of M labels of period P, i.e. a subset of the form A=\\{j:j=s+rP,r=0,1ldots M-1\\} where {P? sqrt{N}} and {M ? N}, and where M is assumed known. Given an oracle f : L? {0,1} which is 1 on A and 0 elsewhere, find the local period P and the offset s.
We solve the elliptic equations associated with the Hamiltonian and momentum constraints, corresponding to a system composed of two black holes with arbitrary linear and angular momentum. These new solutions are based on a Kerr-Schild spacetime slicing which provides more physically realistic solutions than the initial data based on conformally flat metric/maximal slicing methods. The singularity/inner boundary problems are circumvented by a new technique that allows the use of an elliptic solver on a Cartesian grid where no points are excised, simplifying enormously the numerical problem. PMID:11136031
Four approaches for calculating downlink interferences for shaped-beam antennas are described. An investigation of alternative mixed-integer programming models for satellite synthesis is summarized. Plans for coordinating the various programs developed under this grant are outlined. Two procedures for ordering satellites to initialize the k-permutation algorithm are proposed. Results are presented for the k-permutation algorithms. Feasible solutions are found for 5 of the 6 problems considered. Finally, it is demonstrated that the k-permutation algorithm can be used to solve arc allotment problems.
An experiment using a sample of 11th graders compared text editing and worked examples approaches in learning to solve dilution and molarity algebra word problems in a chemistry context. Text editing requires students to assess the structure of a word problem by specifying whether the problem text contains sufficient, missing, or irrelevant…
This research investigated how fourth and fifth grade students spontaneously "unpacked" a word problem when generating a graphic representation to aid in problem solution. Relationships among the type of graphic representation produced, spatial visualization, drawing ability, gender, and problemsolving also were examined and described.…
The adoption of problem-based learning as a teaching method in the advertising and public relations programs offered by the Business TAFE (Technical and Further Education) School at RMIT University is explored in this paper. The effect of problem-based learning on student engagement, student learning and contextualised problem-solving was… Patients with Huntington's disease performed the solvable problems significantly worse than the normal control subjects, but there was no difference in performance between the two groups in inhibiting aberrant problems. These results suggest that early Huntington's disease patients exhibit a precocious impairment in their ability to plan the resolution of complex arithmetic word problems without deficit in their ability to eliminate aberrant problems. This dissociation of performance fits with what we have found in such patients using script-sequencing tasks (Allain et al., 2004) and with neuropsychological data obtained by Watkins et al. (2000). These results are consistent with what is known about the neuropathological progression of Huntington's disease in which neuronal loss progresses in a dorso-to-ventral direction and with what was shown in patients with circumscribed frontal lobe damage. In these patients, impairments in planning solvable word problems were more frequent when lesions were in the lateral prefrontal regions. PMID:15629205AN algorithm which efficiently solves large systems of equations arising from the discretization of a single second-order elliptic partial differential equation is discussed. The global domain is partitioned into not necessarily disjoint subdomains which are traversed using the Schwarz Alternating Procedure. On each subdomain the multigrid method is used to advance the solution. The algorithm has the potential to decrease solution time when data is stored across multiple levels of a memory hierarchy. Results are presented for a virtual memory, vector multiprocessor architecture. A study of choice of inner iteration procedure and subdomain overlap is presented for a model problem, solved with two and four subdomains, sequentially and in parallel. Microtasking multiprocessing results are reported for multigrid on the Alliant FX-8 vector-multiprocessor. A convergence proof for a class of matrix splittings for the two-dimensional Helmholtz equation is given. 70 refs., 3 figs., 20 tabs. the 2nd grade, children (n = 206; on average, 7 years, 6 months) were assessed on general language comprehension, working memory, nonlinguistic reasoning, processing speed (a control variable), and foundational skill (arithmetic for WPs; word reading for text comprehension). In spring, they were assessed on WP-specific language comprehension, WPs, and text comprehension. Path analytic mediation analysis indicated that effects of general language comprehension on text comprehension were entirely direct, whereas effects of general language comprehension on WPs were partially mediated by WP-specific language. By contrast, effects of working memory and reasoning operated in parallel ways for both outcomes. PMID:25866461
This research project investigates students' development of problemsolving schemata while using strategies that facilitate the process of using solved examples to assist with a new problem (case reuse). Focus group learning interviews were used to explore students' perceptions and understanding of several problemsolving strategies. Individual clinical interviews were conducted and quantitative examination data were collected to assess students' conceptual understanding, knowledge organization, and problemsolving performance on a variety of problem tasks. The study began with a short one-time treatment of two independent, research-based strategies chosen to facilitate case reuse. Exploration of students' perceptions and use of the strategies lead investigators to select one of the two strategies to be implemented over a full semester of focus group interviews. The strategy chosen was structure mapping. Structure maps are defined as visual representations of quantities and their associations. They were created by experts to model the appropriate mental organization of knowledge elements for a given physical concept. Students were asked to use these maps as they were comfortable while problemsolving. Data obtained from this phase of our study (Phase I) offered no evidence of improved problemsolving schema. The 11 contact hour study was barely sufficient time for students to become comfortable using the maps. A set of simpler strategies were selected for their more explicit facilitation of analogical reasoning, and were used together during two more semester long focus group treatments (Phase II and Phase III of this study). These strategies included the use of a step-by-step process aimed at reducing cognitive load associated with mathematical procedure, direct reflection of principles involved in a given set of problems, and the direct comparison of problem pairs designed to be void of surface similarities (similar objects or object orientations) and sharing physical principles (conservation of energy problems). Overall, our results from the final two phases of this project indicate that these strategies are helpful in facilitating student ability to identify important information from given problems. The promising results from our study have significant implications for further research, curriculum material development, and instructionIn addition to cognitive research on school leaders' problemsolving, this study focuses on the situated and personal nature of problem framing by combining insights from cognitive research on problemsolving and sense-making theory. The study reports the results of a case study of two school leaders solvingproblems in their daily context by…
To study the buckling and post-buckling behavior of shells and various other structures, one must solve a nonlinear 2-point boundary problem. Since closed-form analytic solutions for such problems are virtually nonexistent, numerical approximations are inevitable. This makes the availability of accurate and reliable software indispensable. In a series of papers Lentini and Pereyra, expanding on the work of Keller, developed PASVART: an adaptive finite difference solver for nonlinear 2-point boundary problems. While the program does produce extremely accurate solutions with great efficiency, it is hindered by a major limitation. PASVART will only locate isolated solutions of the problem. In buckling problems, the solution set is not unique. It will contain singular or bifurcation points, where different branches of the solution set may intersect. Thus, PASVART is useless precisely when the problem becomes interesting. To resolve this deficiency we propose a modification of PASVART that will enable the user to perform a more complete bifurcation analysis. PASVART would be combined with the Thurston bifurcation solution: as adaptation of Newton's method that was motivated by the work of Koiter 3 are reinterpreted in terms of an iterative computational method by Thurston.
Recently a large scale study of points in the MSSM parameter space which are problematic at the Large Hadron Collider (LHC) has been performed. This work was carried out in part to determine whether the proposed International Linear Collider (ILC) could be used to solve the LHC inverse problem. The results suggest that while the ILC will be a valuable tool, an energy upgrade may be crucial to its success, and that, in general, precision studies of the MSSM are more difficult at the ILC than has generally been believed.
The split equality problem (SEP) has extraordinary utility and broad applicability in many areas of applied mathematics. Recently, Byrne and Moudafi (2013) proposed a CQ algorithm for solving it. In this paper, we propose a modification for the CQ algorithm, which computes the stepsize adaptively and performs an additional projection step onto two half-spaces in each iteration. We further propose a relaxation scheme for the self-adaptive projection algorithm by using projections onto half-spaces instead of those onto the original convex sets, which is much more practical. Weak convergence results for both algorithms are analyzed. PMID:24574882
This paper argues for a need to develop methods for examining temporal patterns in computer-supported collaborative learning (CSCL) groups. It advances one such quantitative method--Lag-sequential Analysis (LsA)--and instantiates it in a study of problem-solving interactions of collaborative groups in an online, synchronous environment. LsA…
We hypothesize that typical example problems used in quantitative domains such as algebra and probability can be represented in terms of subgoals and methods that these problems teach learners. The "quality" of these subgoals and methods can vary, depending on the features of the examples. In addition, the likelihood of these subgoal's being recognized in novel problems and the likelihood of learners' being able to modify an old method for a new problem may be functions of the training examples learners study. In Experiment 1, subjects who studied examples predicted to teach certain subgoals were often able to recognize those subgoals in nonisomorphic transfer problems. Subjects who studied examples demonstrating two methods rather than one exhibited no advantages in transfer. Experiment 2 demonstrated that if the conditions for applying a method are highlighted in examples, learners are more likely to appropriately adapt that method in a novel problem, perhaps because they recognize that the conditions do not fully match those required for any of the old methods. Overall, the results indicate that the subgoal/method representational scheme may be useful in predicting transfer performance. PMID:2266861
An improved method of model-based diagnosis of a complex engineering system is embodied in an algorithm that involves considerably less computation than do prior such algorithms. This method and algorithm are based largely on developments reported in several NASA Tech Briefs articles: The Complexity of the Diagnosis Problem (NPO-30315), Vol. 26, No. 4 (April 2002), page 20; Fast Algorithms for Model-Based Diagnosis (NPO-30582), Vol. 29, No. 3 (March 2005), page 69; Two Methods of Efficient Solution of the Hitting-Set Problem (NPO-30584), Vol. 29, No. 3 (March 2005), page 73; and Efficient Model-Based Diagnosis Engine (NPO-40544), on the following page. Some background information from the cited articles is prerequisite to a meaningful summary of the innovative aspects of the present method and algorithm. In model-based diagnosis, the function of each component and the relationships among all the components of the engineering system to be diagnosed are represented as a logical system denoted the system description (SD). Hence, the expected normal behavior of the engineering system is the set of logical consequences of the SD. Faulty components lead to inconsistencies between the observed behaviors of the system and the SD. Diagnosis the task of finding faulty components is reduced to finding those components, the abnormalities of which could explain all the inconsistencies. The solution of the diagnosis problem should be a minimal diagnosis, which is a minimal set of faulty components. The calculation of a minimal diagnosis is inherently a hard problem, the solution of which requires amounts of computation time and memory that increase exponentially with the number of components of the engineering system. Among the developments to reduce the computational burden, as reported in the cited articles, is the mapping of the diagnosis problem onto the integer-programming (IP) problem. This mapping makes it possible to utilize a variety of algorithms developed previously for IP to solve the diagnosis problem. In the IP approach, the diagnosis problem can be formulated as a linear integer optimization problem, which can be solved by use of well-developed integer-programming algorithms. This concludes the background information.
In schema-based theories of cognition, vertical transfer occurs when a learner constructs a new schema to solve a transfer task or chooses between several possible schemas. Vertical transfer is interesting to study, but difficult to measure. Did the student solve the problem using the desired schema or by an alternative method? Perhaps the problem cued the student to use certain resources without knowing why? In this paper, we consider some of the threats to validity in problem design. We provide a theoretical framework to explain the challenges faced in designing vertical transfer problems, and we contrast these challenges with horizontal transfer problem design. We have developed this framework from a set of problems that we tested on introductory mechanics students, and we illustrate the framework using one of the problems.
The purpose of this study was to analyze how school psychologists engage racial/cultural diversity when conceptualizing problems during consultation in a multiracial context. Four school psychologists were recruited to engage in computer-simulated problem-solving consultation. Each school psychologist was presented with three fictional…
The effectiveness of problemsolving as a learning tool is often diminished because students typically use only an algorithmic approach to get to the answer. We discuss a way of encouraging students to reflect on the solution to their problem by requiring them--after they have arrived at their solution--to draw solution maps. A solution map…
Various domains require practitioners to encounter and resolve ill-structured problems using collaborative problem-solving. As such, problem-solving is an essential skill that educators must emphasize to prepare learners for practice. One potential way to support problem-solving is through further investigation of instructional design methods that…
Despite increased interest in real life problemsolving with both children and adults, the question of whether problemsolving is related to psychological adjustment remains unanswered. To examine whether college students' self-appraisal of their problemsolving skills is related to their psychological adjustment, 671 students took the Problem…
A regression design was used to test the unique and interactive effects of self-efficacy beliefs and metacognitive prompting on solving mental multiplication problems while controlling for mathematical background knowledge and problem complexity. Problem-solving accuracy, response time, and efficiency (i.e. the ratio of problemssolved correctly…
Problem-solving instruction facilitates children in becoming successful real-world problem solvers. Research that incorporates problem-solving instruction has been limited for students with mild and moderate intellectual disabilities. However, this population of students needs increased opportunities to learn the skills of problemsolving. Using a…
The diagnosis problem arises when a system's actual behavior contradicts the expected behavior, thereby exhibiting symptoms (a collection of conflict sets). System diagnosis is then the task of identifying faulty components that are responsible for anomalous behavior. To solve the diagnosis problem, the present invention describes a method for finding the minimal set of faulty components (minimal diagnosis set) that explain the conflict sets. The method includes acts of creating a matrix of the collection of conflict sets, and then creating nodes from the matrix such that each node is a node in a search tree. A determination is made as to whether each node is a leaf node or has any children nodes. If any given node has children nodes, then the node is split until all nodes are leaf nodes. Information gathered from the leaf nodes is used to determine the minimal diagnosis set.
Cooperators that refuse to participate in sanctioning defectors create the second-order free-rider problem. Such cooperators will not be punished because they contribute to the public good, but they also eschew the costs associated with punishing defectors. Altruistic punishers—those that cooperate and punish—are at a disadvantage, and it is puzzling how such behaviour has evolved. We show that sharing the responsibility to sanction defectors rather than relying on certain individuals to do so permanently can solve the problem of costly punishment. Inspired by the fact that humans have strong but also emotional tendencies for fair play, we consider probabilistic sanctioning as the simplest way of distributing the duty. In well-mixed populations the public goods game is transformed into a coordination game with full cooperation and defection as the two stable equilibria, while in structured populations pattern formation supports additional counterintuitive solutions that are reminiscent of Parrondo's paradox.
Herding of sheep by dogs is a powerful example of one individual causing many unwilling individuals to move in the same direction. Similar phenomena are central to crowd control, cleaning the environment and other engineering problems. Despite single dogs solving this 'shepherding problem' every day, it remains unknown which algorithm they employ or whether a general algorithm exists for shepherding. Here, we demonstrate such an algorithm, based on adaptive switching between collecting the agents when they are too dispersed and driving them once they are aggregated. Our algorithm reproduces key features of empirical data collected from sheep–dog interactions and suggests new ways in which robots can be designed to influence movements of living and artificial agents. PMID:25165603This paper presents a mathematical model based on fuzzy logic for a computational problemsolving system. The fuzzy logic uses truth degrees as a mathematical model to represent vague algorithm. The fuzzy logic mathematical model consists of fuzzy solution and fuzzy optimization modules. The algorithm is evaluated based on a software metrics calculation that produces the fuzzy set membership. The fuzzy solution mathematical model is integrated in the fuzzy inference engine that predicts various solutions to computational problems. The solution is extracted from a fuzzy rule base. Then, the solutions are evaluated based on a software metrics calculation that produces the level of fuzzy set membership. The fuzzy optimization mathematical model is integrated in the recommendation generation engine that generate the optimize solution.
Problemsolving is an important goal in almost all physics classes. In this study we explore students' perceptions and understanding of the purpose of two different problemsolving approaches. In Phase I of the study, introductory algebra-based physics students were given an online extra credit problem-solving assignment. They were randomly assigned one of three problem-solving strategies: questioning, structure mapping and traditional problemsolving. In Phase II of the study, eight student volunteers were individually assigned to work problems using one of the strategies in two sessions of semi-structured interviews. The first session investigated students' general problemsolving approaches a few weeks after they had completed the online extra credit assignment. The second session investigated students' perceptions of problemsolving strategies and how they relate to the extra credit assignments. In this article, we describe students' perceptions of the purpose of the activities and their underlying problemsolving techniques.
The paper describes results of a teaching experiment with five high school (Year 10 and 11) students. Four qualitative characteristics were established: the first step of solution, main information extracted from the problem, generalisation from a problem and completion of solution. From these characteristics the corresponding quantitative indices were introduced and analysed. The structure of two of them, specific SFS
This study compared the effect of lecture-based instruction to that of problem-based instruction on learner performance (on near-transfer and far-transfer problems), problemsolving processes (reasoning strategy usage and reasoning efficiency), and attitudes (overall motivation and learner confidence) in a Genetics course. The study also analyzed the effect of self-regulatory skills and prior-academic achievement on performance for both instructional strategies. Sixty 11th grade students at a public math and science academy were assigned to either a lecture-based instructional strategy or a problem-based instructional strategy. Both treatment groups received 18 weeks of Genetics instruction through the assigned instructional strategy. In terms of problemsolving performance, results revealed that the lecture-based group performed significantly better on near-transfer post-test problems. The problem-based group performed significantly better on far-transfer post-test problems. In addition, results indicated the learners in the lecture-based instructional treatment were significantly more likely to employ data-driven reasoning in the solving of problems, whereas learners in the problem-based instructional treatment were significantly more likely to employ hypothesis-driven reasoning in problemsolving. No significant differences in reasoning efficiency were uncovered between treatment groups. Preliminary analysis of the motivation data suggested that there were no significant differences in motivation between treatment groups. However, a post-research exploratory analysis suggests that overall motivation was significantly higher in the lecture-based instructional treatment than in the problem-based instructional treatment. Learner confidence was significantly higher in the lecture-based group than in the problem-based group. A significant positive correlation was detected between self-regulatory skills scores and problemsolving performance scores in the problem-based group, but not in the lecture-based group. The difference between correlation coefficients for the two treatment groups was not statistically significant. Further, a significant positive correlation between prior academic achievement and problemsolving performance scores was detected in both treatment groups. Once more, however, the difference between correlation coefficients for the two treatment groups was not statistically significant. Results from this research study are discussed. Limitations of the research study are identified and discussed. Recommendations for future research are presented. Finally, implications of the research study for educational research and practice are presented.
In this paper, we explore the use of isomorphic problem pairs (IPPs) to assess introductory physics students' ability to solve and successfully transfer problem-solving knowledge from one context to another in mechanics. We call the paired problems "isomorphic" because they require the same physics principle to solve them. We analyze written…
In this study, problemsolving provided deeper meaning and understanding through the implementation of a structured problemsolving strategy with the teaching of rational numbers. This action-research study was designed as a quasi-experimental model...
This exercise uses the Think-Pair-Share technique to initiate the problem-solving process. It focuses on a common first step in economic problemsolving: identifying relevant and irrelevant information.
\\u000a Collaborative problemsolving is defined as problemsolving activities that involve interactions among a group of individuals.\\u000a Collaborative problemsolving is considered a necessary skill for success in today's world and schooling. The purpose of this\\u000a chapter is to further investigate the role of computer-based feedback in collaborative problemsolving, in particular, the\\u000a effect of narration plus on-screen text versus | 677.169 | 1 |
Algebra I
The student will solve quadratic equations
in one variable both algebraically and graphically. Graphing
calculators will be used both as a primary tool in solving
problems and to verify algebraic solutions.
The student will estimate square roots at
least to the nearest tenth and use a calculator to compute
decimal approximations of radicals.
Precalculus
The student will recognize multiple
representations of functions (linear, quadratic, absolute value,
step, and exponential functions) and convert between a graph, a
table, and symbolic form. A transformational approach to graphing
will be employed through the use of graphing calculators.
Standards for Grades 9-12
II. Numerical and Algebraic Concepts and
Operations
B. Use tables and graphs as tools to
interpret expressions, equations, and inequalities, using
technology whenever appropriate.
The student will represent linear
equations or inequalities as tables or graphs.
D. Develop an understanding of and facility
in manipulating algebraic expressions, performing elementary
operations on matrices, and solving equations and inequalities.
The student will solve equations and
inequalities using a variety ofmethods
to include graphing, spreadsheets, and symbol
manipulation, explaining procedures used.
The student will gather and plot data,
fit a graph to plotted points, use the graph to
illustrate the relationship between variables, predict
outcomes, and form a generalized equation in a real-world
context.
The student will recognize multiple
representations of functions (linear, quadratic, absolute
value, step, and exponential functions) and convert
between a graph, a table, and symbolic form. A
transformational approach to graphing will be employed
through the use of graphing calculators.
A Numerical Approach
x2 – x – 3 = 0
x2 – x = 3
x(x-1) = 3
x
x – 1
x(x – 1)
3
Iteration and Interpolation
x2 + 3x – 6 = 0
x2 + 3x + 6 = 0
2x2 - 5x – 8 = 0
Symbolic
Graphic
Numeric
Robust
Easy to Remember
Easy to Use
Bisection Method
x2 – x – 3 = 0
x
x2 – x - 3
Find an interval where there is a sign
change.
Find the midpoint of that interval.
If the midpoint is a root, we're
done.
Of the two intervals .. left point to
midpoint or midpoint to right point
choose the one in which there is a sign change. | 677.169 | 1 |
...I have worked throughout the digital revolution of the past 30 years as a electronic designer, programmer, to discrete electronic components that are required to make CPU's, ADC's, DAC's, memory, etc. The topics that are included in today's discrete math are tools that I've used continuously to ...Simply going through the motions is not learning for the student. Explain, with real life examples, why the knowledge will make a difference for the better for the student. By letting the student know how the knowledge will improve them you create a real desire for the student to learn the subject. | 677.169 | 1 |
Introduction to Mathematical Reasoning Numbers, Sets and Functions
9780521597180
ISBN:
0521597188
Pub Date: 1998 Publisher: Cambridge Univ Pr
Summary: Eccles, Peter J. is the author of Introduction to Mathematical Reasoning Numbers, Sets and Functions, published 1998 under ISBN 9780521597180 and 0521597188. Two hundred thirty five Introduction to Mathematical Reasoning Numbers, Sets and Functions textbooks are available for sale on ValoreBooks.com, fifty used from the cheapest price of $54.92, or buy new starting at $54 | 677.169 | 1 |
Videos tagged "mathematics"
In this video, we learn how to find any term in a geometric sequence if we are just given the first few terms. Our first step is to find the common ratio. Then, we look at some patterns of finding consecutive terms and see that we can represent repeatedly multiplying the common ratio by using an exponent. Going through some examples then, we see that we can find the 100th term by take the common ratio, and taking it to the 99th power, and then multiplying that by the first term.
In this video, we take the first few terms of a geometric sequence, and practice finding the common ratio so that we can find the next consecutive term. To find the common ratio, we divide consecutive terms together. In this example, the common ratio is a decimal, and so we can easily multiply it to get as many terms as we want.
In this video, we look at exponential functions and graphs and see how to shift them horizontally right or left. We play around with a table of values and changing the function around. To really understand, we then use the Desmos graphing calculator to create an exponential function with a value h subtracting from the exponent. There is a slider for the value, h, so we can easily change the values and see how it changes the graph of the function.
In this video, we look at exponential functions and graphs, and see how to shift the graph up and down vertically. We first play around with a function and table of values to see what happens when we add a constant value to the end of a function. We then use the Desmos graphing calculator to create an exponential function with a slider where we can input different values for k, and see how changing the function results in a vertical shift of its graph.
In this video, we look at graphs of exponential functions. We look one function that is increasing, and another function that is decreasing and compare the growth of both. We use the function to create a table of values that we can then graph the points. Since this is an exponential function, we graph a smooth curve through the points to represent all the possible points that are solutions to the function. Exponential functions are different from linear functions in that they are curved and grow or decrease really fast.
In this video, we go a step further with looking at exponential functions and graphs, and talk about the end behavior of the functions. We look at two different graphs, and talk about what is happening as the graph is increasing or decreasing along x. We see that exponential graphs grow increasingly large without bound. They also get really close to zero, as we go the other direction. To help understand what the graphs are doing, we use the function, and think about multiplying and dividing to understand the end behavior.
In this video, we begin looking at number patterns where we repeatedly multiply by the same number. We call these number patterns geometric sequences. A geometric sequence always has a starting value for the first term, and then each term after that is multiplied by a number we call the common ratio. To help understand geometric sequences, we compare them as well to arithmetic sequences and number patterns that grow by adding the same number repeatedly.
Can technology help them prove it's their land?
In the national forests of Gujarat, India, the tribal people have been seen as encroachers, thieves who dare to produce food for their families on land claimed by the government. Rama Bhai and his family have worked land in the Sagai village for generations.
Sagai has no electricity, no running water. No one kept records of who was farming which plots of land. So when the laws changed and they were allowed to claim the land, they faced a challenge – how could they prove they'd been farming specific plots in the past?
The technology we use to find our way to unfamiliar places came to their rescue. Learn how GPS operates, and how it, along with Google Maps, saved the day for the poorest of the poor in India.
As Seen on Public Television!
Recalculating covers numerous educational standards across several subject areas including Science, Mathematics, and Social Studies for grades 5-12. To learn more about this educational program, and which standards it covers specific to your grade, subject area, and which standards your district is using, visit our educational program summary section for this video here:
Subject Areas:
■ Economics
■ Science & Technology
■ Mathematics
■ World History/Geography
Topics:
■ GPS & Satellites
■ India
■ Map Skills
■ Property Rights
■ Area & Perimeter
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Now that we have intuited that the dynamics of the customer base populations will be essentially 1-dimensional at long times, we pursue this intuition to develop an analytic description of the customer base population dynamics at all times. We use eigenvalue-eigenvector analysis, which identifies directions in the state space along which dynamics are 1-dimensional, and the scaling factors associated with these directions. | 677.169 | 1 |
Laurys Station Algebra 2Ray A.
...Algebra is all about how fast something is moving. Calculus is about how fast something is accelerating. Getting from Algebra to Calculus requires a touch to change the way the student sees formulas and equations.
Alex D | 677.169 | 1 |
Synopsis
Vectors and tensors are among the most powerful problem-solving tools available, with applications ranging from mechanics and electromagnetics to general relativity. Understanding the nature and application of vectors and tensors is critically important to students of physics and engineering. Adopting the same approach used in his highly popular A Student's Guide to Maxwell's Equations, Fleisch explains vectors and tensors in plain language. Written for undergraduate and beginning graduate students, the book provides a thorough grounding in vectors and vector calculus before transitioning through contra and covariant components to tensors and their applications. Matrices and their algebra are reviewed on the book's supporting website, which also features interactive solutions to every problem in the text where students can work through a series of hints or choose to see the entire solution at once. Audio podcasts give students the opportunity to hear important concepts in the book explained by the author.
Daniel Fleisch is an Associate Professor in the Department of Physics at Wittenberg University, where he specializes in electromagnetics and space physics. He is the author of A Student's Guide to Maxwell's Equations (Cambridge University Press, 2008). | 677.169 | 1 |
Algebra (Graduate Texts in Mathematics)
Algebra fulfills a definite need to provide a self-contained, one volume, graduate level algebra text that is readable by the average graduate student and flexible enough to accomodate a wide variety of instructors and course contents. The guiding philosophical principle throughout the text is that the material should be presented in the maximum usable generality consistent with good pedagogy. Therefore it is essentially self-contained, stresses clarity rather than brevity and contains an unusually large number of illustrative exercises. The book covers major areas of modern algebra, which is a necessity for most mathematics students in sufficient breadth and depth | 677.169 | 1 |
Algebra 2 Home (52 worksheets available to subscribers)
Algebra 2 Funsheets is included with an AlgebraFunSheets subscription.
Most Algebra 2 worksheets will be standard worksheets, since it is harder to format
Algebra 2 problems into a "funsheet" format. These worksheets are simply meant as an
extra resource for Algebra 2 teachers.
In addition, Algebra 1 topics that are redundant in Algebra 2 will only be found on the
Algebra 1 worksheet site.
For Algebra 2 Funsheets that may be available, it is suggested that teachers require students
to show their work on a separate sheet of paper. | 677.169 | 1 |
The aim of these investigations is not to provide drill (although links to other resources on the web that do have been...
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The aim of these investigations is not to provide drill (although links to other resources on the web that do have been included in places), but to encourage students to think about why things happen the way they do in calculus. Such an understanding can be greatly useful both when rote memorization fails and when studying a new concept. Indeed, many concepts from the single variable calculus studied in APSC 171 form the foundations of later courses. The investigations have been designed to be quick and self-contained and should take at most ten or fifteen minutes eachQ's (applets for calculus) to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material MathQ's (applets for calculus)
Select this link to open drop down to add material MathQ's (applets for calculus)Mathway is a mathematics problem solving tool where students can select their math course - Basic Math, Pre-Algebra, Algebra,...
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Mathway is a mathematics problem solving tool where students can select their math course - Basic Math, Pre-Algebra, Algebra, Trigonometry, PreCalculus, Calculus or Statistics and enter a problem. The computer solves the problem and shows the steps for the solution. It also has a worksheet generatorway to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Mathway
Select this link to open drop down to add material Mathway ableThis first-year calculus book is centered around the use of infinitesimals, an approach largely neglected until recently for...
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'This first-year
This site offers several tutorials on algebra, trigonometry, calculus, differentail equations, complex variables, matrix...
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This site offers several tutorials on algebra, trigonometry, calculus, differentail equations, complex variables, matrix algebra, and tables. Cyber Exam which contains quizzes and tests, and Cyber Board which answers FAQs and more.O.S. MATHematics to your Bookmark Collection or Course ePortfolio
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Select this link to open drop down to add material S.O.S. MATHematics OU Calculus I Problem Archive to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material The OU Calculus I Problem Archive
Select this link to open drop down to add material The OU Calculus I Problem Archives for the Calculus Phobe to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Tutorials for the Calculus Phobe
Select this link to open drop down to add material Tutorials for the Calculus Phobe to your Bookmark Collection or Course ePortfolio | 677.169 | 1 |
Introductory Mathematical Analysis for Business, Economics, and the Life and Social introductory textbook beginning with noncalculus topics such as equations, functions, matrix algebra, linear programming, mathematics of finance, and probability, then progressing through both single and multivariable calculus, including continuous random variables. | 677.169 | 1 |
Written in a step-by-step format, this practical guide begins by reviewing background mathematics, probability basics, and descriptive measures. The book goeMULTIPLY your chances of learning STATISTICS Trying to understand statistics but feeling a bit uncertain? Here's your solution. "Statistics Demystifed," Second Edition helps you master this fundamental topic with ease.
Written in a step-by-step format, this practical guide begins by reviewing background mathematics, probability basics, and descriptive measures. The book goes on to demonstrate statistics in action with coverage of sampling, estimation, hypotheses, prediction, regression, correlation, causation, order, and chaos. Detailed examples, concise explanations, and worked-out problems make it easy to understand the material, and end-of-chapter quizzes and a final exam help reinforce learning.
Community Reviews
I've been tutoring math privately for the last 5 years, and I really enjoy the whole Demystified series. While I do not think they are particularly useful for someone trying to learn these skills for the first time, they are great for anyone trying to get a quick refresher course. They have also been a great resource to me as a tutor; I often pull practice problems that give my students a slightly different approach to some topics. | 677.169 | 1 |
Description:
This course, a continuation of MATH 8A, includes solving expressions and equations;graphing points on a coordinate plane, identifying translations, rotations, reflections, and dilations of a graph; and using ratios, proportions, percents, and decimals. MATH 8B also explores problem-solving strategies and the basic fundamentals of geometry, including angles, triangles, prisms, and cylinders. Note: Due to the nature of the lesson assignments for this course, we are unable to accept assignments submitted via e-mail. View TEKS.
Required Materials: Scanner and PDF Software or Digital Notepad System
You will submit all lessons for this course electronically. Your work for each lesson will need to be saved as a PDF in order to submit the lesson for grading. If you have multiple pages, those pages will need to be saved as one file before uploading.
There are several ways to save your completed lesson work as a PDF. One of the most common ways is to use a scanner to scan your pages into a digital document. Some scanners will allow you to scan your pages and save them as a PDF. If you are not able to scan to a PDF, you will need to convert your scanned pages to PDF files. You can do a Google search for "PDF creator online" to find software to convert your pages to PDF files. One of the most popular sites is PrimoPDF.
If you have several PDF files, you will need to merge those files into one large file before submitting your assignment for grading. Again, do a Google search for "Merging PDF files" to find online software to complete this task. PDFMerge! is a good site for merging files. Another good resource is PDF Binder. You can find any of these tools by doing a Google search for its name.
We have also provided instructions for the ACECAD DigiMemo L2 and the ACECAD DigiMemo 692 for both PC (Windows XP/Vista/7) and Mac (OS X) users. Please note that students are not required to use the ACECAD DigiMemo. This is just one tool that will allow students to create a PDF.
IMPORTANT: TTUISD staff is only able to offer limited technical support for scanners or the digital notepad system. Please refer to the support documentation bundled with your particular device and/or seek direct support from its manufacturer. | 677.169 | 1 |
Contents
CHAPTER 9 Polar Coordinates and Complex Numbers
9.1 Polar Coordinates 348
9.2 Polar Equations and Graphs 351
9.3 Slope, Length, and Area for Polar Curves 356
9.4 Complex Numbers 360
CHAPTER 10 Infinite Series
10.1 The Geometric Series
10.2 Convergence Tests: Positive Series
10.3 Convergence Tests: All Series
10.4 The Taylor Series for ex, sin x, and cos x
10.5 Power Series
CHAPTER 11 Vectors and Matrices
11.1 Vectors and Dot Products
11.2 Planes and Projections
11.3 Cross Products and Determinants
11.4 Matrices and Linear Equations
11.5 Linear Algebra in Three Dimensions
CHAPTER 12 Motion along a Curve
12.1 The Position Vector 446
12.2 Plane Motion: Projectiles and Cycloids 453
12.3 Tangent Vector and Normal Vector 459
12.4 Polar Coordinates and Planetary Motion 464
CHAPTER 13 Partial Derivatives
13.1 Surfaces and Level Curves 472
13.2 Partial Derivatives 475
13.3 Tangent Planes and Linear Approximations 480
13.4 Directional Derivatives and Gradients 490
13.5 The Chain Rule 497
13.6 Maxima, Minima, and Saddle Points 504
13.7 Constraints and Lagrange Multipliers 514
C H A P T E R 11
Vectors and Matrices
This chapter opens up a new part of calculus. It is multidimensional calculus, because
the subject moves into more dimensions. In the first ten chapters, all functions
depended on time t or position x-but not both. We had f(t) or y(x). The graphs
were curves in a plane. There was one independent variable (x or t) and one dependent
variable (y or f). Now we meet functions f(x, t) that depend on both x and t. Their
graphs are surfaces instead of curves. This brings us to the calculus of several variables.
Start with the surface that represents the function f(x, t) or f(x, y) or f(x, y,,t). I
emphasize functions, because that is what calculus is about.
EXAMPLE 1 f(x, t) = cos (x - t) is a traveling wave (cosine curve in motion).
At t = 0 the curve is f = cos x. At a later time, the curve moves to the right
(Figure 11.1). At each t we get a cross-section of the whole x-t surface. For a wave
traveling along a string, the height depends on position as well as time.
A similar function gives a wave going around a stadium. Each person stands up
and sits down. Somehow the wave travels.
EXAMPLE 2 f (x, y) = 3x + y + 1 is a sloping roof (fixed in time).
The surface is two-dimensional-you can walk around on it. It is flat because
+
3x y + 1 is a linear function. In the y direction the surface goes up at 45". If y
increases by 1, so doesf . That slope is 1. In the x direction the roof is steeper (slope 3).
There is a direction in between where the roof is steepest (slope fi).
EXAMPLE 3 f (x, y, t) = cos(x - y - t) is an ocean surface with traveling waves.
This surface moves. At each time t we have a new x-y surface. There are three
variables, x and y for position and t for time. I can't draw the function, it needs four
dimensions! The base coordinates are x, y, t and the height is f. The alternative is a
movie that shows the x-y surface changing with t.
At time t = 0 the ocean surface is given by cos (x - y). The waves are in straight
lines. The line x - y = 0 follows a crest because cos 0 = 1. The top of the next wave
is on the parallel line x - y = 2n, because cos 2n = 1. Figure 11.1 shows the ocean
surface at a fixed time.
The line x - y = t gives the crest at time t. The water goes up and down (like people
in a stadium). The wave goes to shore, but the water stays in the ocean.
11 Vectors and Matrices
Fig. 11.1 Moving cosine with a small optical illusion-the darker Fig. 11.2 Linear functions give planes.
bands seem to go from top to bottom as you turn.
Of course multidimensional calculus is not only for waves. In business, demand is
a function of price and date. In engineering, the velocity and temperature depend on
position x and time t . Biology deals with many variables at once (and statistics is
always looking for linear relations like z = x + 2y). A serious job lies ahead, to carry
derivatives and integrals into more dimensions.
In a plane, every point is described by two numbers. We measure across by x and
up by y. Starting from the origin we reach the point with coordinates (x, y). I want
to describe this movement by a vector-the straight line that starts at (0,O) and ends
at (x, y). This vector v has a direction, which goes from (0,O) to (x, y) and not the
other way.
In a picture, the vector is shown by an arrow. In algebra, v is given by its two
components. For a column vector, write x above y:
v= [,I (x and y are the components of v).
Note that v is printed in boldface; its components x and y are in lightface.? The
vector - v in the opposite direction changes signs. Adding v to - v gives the zero
vector (different from the zero number and also in boldface):
and v v = [
X-X
Y-Y
]=[:I
-0.
Notice how vector addition or subtraction is done separately on the x's and y's:
?Another way to indicate a vector is 2 You will recognize vectors without needing arrows.
11.I Vectors and Dot Products
Fig. 11.3 Parallelogram for v + w, stretching for 2v, signs reversed for -v.
The vector v has components v , = 3 and v, = 1. (I write v, for the first component
and v, for the second component. I also write x and y, which is fine for two com-
ponents.) The vector w has w , = - 1 and w, = 2. To add the vectors, add the com-
ponents. To draw this addition, place the start of w at the end of v. Figure 11.3 shows
how w starts where v ends.
VECTORS WITHOUT COORDINATES
In that head-to-tail addition of v + w, we did something new. The vector w was moved
away from the origin. Its length and direction were not changed! The new arrow is
parallel to the old arrow-only the starting point is different. The vector is the same
as before.
A vector can be defined without an origin and without x and y axes. The purpose
of axes is to give the components-the separate distances x and y. Those numbers
are necessary for calculations. But x and y coordinates are not necessary for head-
to-tail addition v + w, or for stretching to 2v, or for linear combinations 2v + 3w.
Some applications depend on coordinates, others don't.
Generally speaking, physics works without axes-it is "coordinate-free." A velocity
has direction and magnitude, but it is not tied to a point. A force also has direction
and magnitude, but it can act anywhere-not only at the origin. In contrast, a vector
that gives the prices of five stocks is not floating in space. Each component has a
meaning-there are five axes, and we know when prices are zero. After examples
from geometry and physics (no axes), we return to vectors with coordinates.
EXAMPLE 1 (Geometry) Take any four-sided figure in space. Connect the midpoints
of the four straight sides. Remarkable fact: Those four midpoints lie in the same plane.
More than that, they form a parallelogram.
Frankly, this is amazing. Figure 11.4a cannot do justice to the problem, because it
is printed on a flat page. Imagine the vectors A and D coming upward. B and C go
down at different angles. Notice how easily we indicate the four sides as vectors, not
caring about axes or origin.
I will prove that V = W. That shows that the midpoints form a parallelogram.
What is V? It starts halfway along A and ends halfway along B. The small triangle
at the bottom shows V = $A + 3B. This is vector addition-the tail of 3B is at the
head of 4A. Together they equal the shortcut V. For the same reason W = 3C + 3D.
The heart of the proof is to see these relationships.
One step is left. Why is +A + 3B equal to $C + i D ? In other words, why is A + B
equal to C + D? (I multiplied by 2.) When the right question is asked, the answer
jumps out. A head-to-tail addition A + B brings us to the point R. Also C + D brings
us to R. The proof comes down to one line:
A + B = P R = C + D . Then V = + A + $ B e q u a l s W = + C + + D .
11 Vectors and Matrices
Fig. 11.4 Four midpoints form a parallelogram (V = W). Three medians meet at P.
EXAMPLE 2 (Also geometry) In any triangle, draw lines from the corners to the
midpoints of the opposite sides. To prove by vectors: Those three lines meet at a point.
Problem 38 finds the meeting point in Figure 11 . 4 ~Problem 37 says that the three
.
vectors add to zero.
EXAMPLE 3 (Medicine) An electrocardiogram shows the sum of many small vectors,
the voltages in the wall of the heart. What happens to this sum-the heart vector
V-in two cases that a cardiologist is watching for?
Case 1 . Part of the heart is dead (infarction).
Case 2. Part of the heart is abnormally thick (hypertrophy).
A heart attack kills part of the muscle. A defective valve, or hypertension, overworks
it. In case 1the cells die from the cutoff of blood (loss of oxygen). In case 2 the heart
wall can triple in size, from excess pressure. The causes can be chemical or mechanical.
The effect we see is electrical.
The machine is adding small vectors and bbprojecting"them in twelve directions. The
leads on the arms, left leg, and chest give twelve directions in the body. Each graph
shows the component of V in one of those directions. Three of the projections-
two in the vertical plane, plus lead 2 for front-back-produce the "mean QRS vector"
in Figure 11.5. That is the sum V when the ventricles start to contract. The left
ventricle is larger, so the heart vector normally points down and to the left.
Pace
SA
Fig. 11.5 V is a sum of small voltage vectors, at the moment of depolarization.
1I.I Vectors and Dot Products
Fig. 11.6 Changes in V show dead muscle and overworked muscle.
We come soon to projections, but here the question is about V itself. How does
the ECCi identify the problem?
Case 1: Heart attack The dead cells make no contribution to the electri-
cid potential. Some small vectors are missing. Therefore the sum V
turns away from the infarcted part.
Chse 2: Hypertrophy The overwork increases the contribution to the
potential. Some vectors are larger than normal. Therefore V turns
toward the thickened part.
When V points in an abnormal direction, the ECG graphs locate the problem. The
P, Q, R, S, T waves on separate graphs can all indicate hypertrophy, in different
regions of the heart. Infarctions generally occur in the left ventricle, which needs the
greatest blood supply. When the supply of oxygen is cut back, that ventricle feels it
first. The result can be a heart attack (= myocardial infarction = coronary occlusion).
Section 11.2 shows how the projections on the ECG point to the location.
First come the basic facts about vectors-components, lengths, and dot products.
COORDINATE VECTORS AND LENGTH
To compute with vectors we need axes and coordinates. The picture of the heart is
"coordinate-free," but calculations require numbers. A vector is known by its compo-
nents. The unit vectors along the axes are i and j in the plane and i, j, k in space:
Notice h~oweasily we moved into three dimensions! The only change is that vectors
have three components. The combinations of i and j (or i, j, k) produce all vectors v
in the plane (and all vectors V in space):
402 S11 Vectors and Matrices
Those vectors arealso written v = (3, 1)andV = (1, 2, - 2). The components of the vector
are also the coordinates of a point. (The vector goes from the origin to the point.) This
relation between point and vector is so close that we allow them the same notation:
P = (x, y, z)and v = (x, y, z) = xi + yj + zk.
The sum v + V is totally meaningless. Those vectors live in different dimensions.
From the components we find the length. The length of (3, 1) is 32 + 12 = 10.
This comes directly from a right triangle. In three dimensions, V has a third com-
ponent to be squared and added. The length of V = (x, y, z) is IVI = x 2 + Y + z2.
2
Vertical bars indicate length, which takes the place of absolute value. The length
of v = 3i + j is the distance from the point (0, 0) to the point (3, 1):
IvI= v+ v= /10 IVl = 12+ 22 + (-2)2 = 3.
A unit vector is a vector of length one. Dividing v and V by their lengths produces
unit vectors in the same directions:
1/3
L-2/3
S IvA
Each nonzero vector has a positive length vj. The direction of v is given
by a unit vector u = v/lvI. Then length times direction equals v.
A unit vector in the plane is determined by its angle 0 with the x axis:
u= sin ] = (cos 0)i + (sin 0)j is a unit vector: lu12 = cos 20 + sin 20 = 1.
LsinO]
In 3-space the components of a unit vector are its "direction cosines":
U = (cos a)i + (cos l)j + (cos y)k: o, fl, y = angles with x, y, z axes.
Then cos 2a + cos 2f + cos 2 y = 1. We are doing algebra with numbers while we are
doing geometry with vectors. It was the great contribution of Descartes to see how
to study algebra and geometry at the same time.
r-~1
k= 10
f. IlI [_6]2
I= L0
Fig. 11.7 Coordinate vectors i, j, k. Perpendicular vectors v *w = (6)(1) + (- 2)(3) = 0.
THE DOT PRODUCT OF TWO VECTORS
There are two basic operations on vectors. First, vectors are added (v+ w). Second,
a vector is multiplied by a scalar (7v or - 2w). That leaves a natural question-how
do you multiply two vectors? The main part of the answer is-you don't. But there
11.1 Vectors and Dot Products 403
is an extremely important operation that begins with two vectors and produces a
number. It is usually indicated by a dot between the vectors, as in v - w, so it is called
the dot product.
DEFINITION I The dot product multiplies the lengths Ivl times Iwl times a cosine:
v * w = iv IwlI cos 0, 0 = angle between v and w.
EXAMPLE I0has length 3, 2Ihas length /8, the angle is 45 .
The dot product is Ivl |wl cos 0 = (3)( )(1/ ), which simplifies to 6. The square
roots in the lengths are "canceled" by square roots in the cosine. For computing v . w,
a second and much simpler way involves no square roots in the first place.
DEFINITION 2 The dot product v * w multiplies component by component and adds:
V*W = V 1 W 1 + V2 W 2
S )+(0)(2)=6.
=(3)(2
The first form Ivl Iwl cos 0 is coordinate-free. The second form vlw, + v 2W computes
2
with coordinates. Remark 4 explains why these two forms are equal.
I4B The dot product or scalar product or inner product of three-dimensional
vectors is
V *W=VIWI cos =V
1 1 + V2W 2+ V 3W 3 . (4)
[2
If the vectors are perpendicular then 0 = 90 and cos 6 = 0 and V W = 0.
r21
I
3 L6
6
=32 (not perpendicular)
-1
21
L
-1
2 = 0 (perpendicular).
2
These dot products 32 and 0 equal IVI IWi cos 0. In the second one, cos 0 must be
zero. The angle is 7n/2 or - n/2-in either case a right angle. Fortunately the cosine
is the same for 0 and - 0, so we need not decide the sign of 0.
Remark 1 When V = W the angle is zero but not the cosine! In this case cos 0 = 1
2
and V . V = IVl . The dot product of V with itself is the length squared:
V.V= (V, V2,V3 )(VI, V2 , V3) = V+ v+ v2 = IV2
Remark 2 The dot product of i = (1, 0, 0) with j = (0, 1, 0) is i j = 0. The axes are
perpendicular. Similarly i -k = 0 and j . k = 0. Those are unit vectors: i i= j =j
k-k= 1.
Remark 3 The dot product has three properties that keep the algebra simple:
1. V-W= WV 2. (cV). W= c(V. W) 3. (U+V).W= UW+V.W
When V is doubled (c = 2) the dot product is doubled. When V is split into i, j, k
components, the dot product splits in three pieces. The same applies to W, since
11 Vectors and Matrices
Fig. 11.8 Length squared = (V - W) (V - W), from coordinates and the cosine law.
V W = W V. The nine dot products of i, j, k are zeros and ones, and a giant splitting
of both V and W gives back the correct V W:
Remark 4 The two forms of the dot product are equal. This comes from computing
IV - wI2 by coordinates and also by the "law of cosines":
+
with coordinates: IV - WI2 = (Vl - Wl)2 (V2 - W2)2+ (V3 - W3)2
from cosine law: IV - WI2 = IV12 + IWI2 - 21VI IWI cos 8.
Compare those two lines. Line 1 contains V and V and V:. Their sum matches
: :
IV12 in the cosine law. Also W +
: W + W matches IWI2.Therefore the terms contain-
: :
ing - 2 are the same (you can mentally cancel the - 2). The definitions agree:
- 2(V1Wl + V2W2 + V3W3) equals - 21VI IWI cos 8 equals - 2V W.
The cosine law is coordinate-free. It applies to all triangles (even in n dimensions).
Its vector form in Figure 11.8 is IV - WI2 = lV12 - 2V W + IWI2. This application to
V W is its brief moment of glory.
Remark 5 The dot product is the best way to compute the cosine of 8:
cos 8 = -
vow
IVl lWl '
Here are examples of V and W with a range of angles from 0 to n:
i and 3i have the same direction cos 8 = 1 8=0
i (i + j) = 1 is positive cosB=l/& 8=n/4
i and j are perpendicular: i j = 0 cos 8 = 0 12
8 = 71
i.(-i+ j)= - 1 is negative cos 8 = - 1 / f i 8=344
i and - 3i have opposite directions cos 8 = - 1 8=n
Remark 6 The Cauchy-Schwarz inequality IV WI < IVI I WI comes from lcos 8 < 1.
1
The left side is IVI IWI lcos 81. It never exceeds the right side IVI IWI. This is a key
inequality in mathematics, from which so many others follow:
Geometric mean f i < arithmetic mean &x + y) (true for any x 3 0 and y 3 0).
Triangle inequality IV + W I < IVI + IWI ( (VI, IWI, IV + WI are lengths of sides).
These and other examples are in Problems 39 to 44. The Schwarz inequality
IV WI < IVI IW( becomes an equality when lcos 8 = 1 and the vectors are
1 .
11.I Vectors and Dot Products 405
11.1 EXERCISES
Read-through questions 11 True or false in three dimensions:
A vector has length and a . If v has components 6 and 1. If both U and V make a 30" angle with W, so does
-8, its length is Ivl= b and its direction vector is u = U+V.
c . The product of Ivl with u is d . This vector goes 2. If they make a 90" angle with W, so does U + V.
from (0,O) to the point x = e ,y = f . A combination 3. If they make a 90" angle with W they are perpendicular:
of the coordinate vectors i = g and j = h produces u*v=o.
v = i i + i j.
12 From W = (1, 2, 3) subtract a multiple of V = (1, 1, 1) so
To add vectors we add their k . The sum of (6, - 8) and that W - cV is perpendicular to V. Draw V and W and
(1,O) is I . To see v + i geometrically, put the m of i W - cv.
at the n of v. The vectors form a 0 with diagonal
v + i. (The other diagonal is P .) The vectors 2v and -v 13 (a) What is the sum V of the twelve vectors from the center
are q and r . Their lengths are s and t . of a clock to the hours?
(b) If the 4 o'clock vector is removed, find V for the other
In a space without axes and coordinates, the tail of V can eleven vectors.
be placed u . Two vectors with the same v are the
same. If a triangle starts with V and continues with W, the (c) If the vectors to 1, 2, 3 are cut in half, find V for the
twelve vectors.
third side is w . The vector connecting the midpoint of V
to the midpoint of W is x . That vector is v the third 14 (a) By removing one or more of the twelve clock vectors,
-
side. In this coordinate-free form the dot product is V W = make the length IVI as large as possible.
2 .
(b) Suppose the vectors start from the top instead of the
Using components, V *W = A and (1,2, 1)- center (the origin is moved to 12 o'clock, so v12 = 0). What
(2, - 3, 7) = B . The vectors are perpendicular if c . is the new sum V*?
The vectors are parallel if D . V V is the same as E . 15 Find the angle POQ by vector methods if P = (1, 1, O),
The dot product of U + V with W equals F . The angle 0 = (0, 0, O),Q = (1, 2, -2).
between V and W has cos 8 = G . When V W is negative
then 8 is H . The angle between i +j and i + k is I . 16 (a) Draw the unit vectors u1 = (cos 8, sin 8) and u2 =
The Cauchy-Schwarz inequality is J , and for V = i + j (cos 4, sin 4). By dot products find the formula for
and W = i + k it becomes 1 Q K . cos (e - 4).
In 1-4 compute V + W and 2V - 3W and IVI2 and V W and (b) Draw the unit vector u, from a 90" rotation of u2. By
cos 8. dot products find the formula for sin (8 + 4).
1 v = (1, 1, 1)' w = (-1, -1, -1) 17 Describe all points (x, y) such that v = xi + yj satisfies
(a)Ivl=2 (b)Iv-il=2
2 V=i+j, W=j-k
(c)v0i=2 (d)vWi=lvl
3 V=i-2j+k, W=i+j-2k
18 (Important) If A and B are non-parallel vectors from the
4 V = ( l , 1, 1, l), W = ( l , 2, 3,4) origin, describe
5 (a) Find a vector that is perpendicular to (v,, 0,). (a) the endpoints of tB for all numbers t
(b) Find two vectors that are perpendicular to (v,, v,, v,). (b) the endpoints of A + tB for all t
6 Find two vectors that are perpendicular to (1, 1,O) and to (c) the endpoints of sA + tB for all s and t
each other. (d) the vectors v that satisfy v A = v B
7 What vector is perpendicular to all 2-dimensional vectors? 19 ( a ) I f v + 2 w = i a n d 2 v + 3 w = j find vand w.
What vector is parallel to all 3-dimensional vectors? (b)If v = i + j and w = 3 i + 4 j then i = v+
8 In Problems 1-4 construct unit vectors in the same direc- W .
tion as V. 20 If P = (0,O) and R = (0, 1) choose Q so the angle PQR is
9 If v and w are unit vectors, what is the geometrical mean- 90". All possible Q's lie in a
ing of v * w? What is the geometrical meaning of (v * w)v? Draw
a figure with v = i and w = (3/5)i + (4/5)j.
+
21 (a) Choose d so that A = 2i 3j is perpendicular to
B = 9i +dj.
10 Write down all unit vectors that make an angle 8 with the (b) Find a vector C perpendicular to A = i +j + k and
vector (1,O). Write down all vectors at that angle. . B=i-k.
406 11 Vectors and Matrices
22 If a boat has velocity V with respect to the water and the 35 The vector from the earth's center to Seattle is ai+ bj + ck.
water has velocity W with respect to the land, then . (a) Along the circle at the latitude of Seattle, what two
The speed of the boat is not IVI + IWI but functions of a, b, c stay constant? k goes to the North Pole.
23 Find the angle between the diagonal of cube and (a) an (b) On the circle at the longitude of Seattle-the
edge (b) the diagonal of a face (c) another diagonal of the meridian-what two functions of a, b, c stay constant?
cube. Choose lines that meet. (c) Extra credit: Estimate a, b, c in your present position.
24 Draw the triangle PQR in Example 1 (the four-sided figure The O" meridian through Greenwich has b = 0.
in space). By geometry not vectors, show that PR is twice as 36 If (A+ BIZ= (AI2+ (BI2,
prove that A is perpendicular to B.
long as V. Similarly lPRl= 21WI. Also V is parallel to W
because both are parallel to . So V = W as before. 37 In Figure 11.4, the medians go from the corners to the
midpoints of the opposite sides. Express MI, M2, in terms
M3
(a) If A and B are unit vectors, show that they make equal
of A, B, C. Prove that MI + M2 + M3 = 0 What relation
.
angles with A + B.
holds between A, B, C?
(b) If A, B, C are unit vectors with A + B + C = 0, they
form a triangle and the angle between any two 38 The point 3 of the way along is the same for all three
is medians. This means that A + $M3 = 3M, = . Prove
(a) Find perpendicular unit vectors I and J in the plane that those three vectors are equal.
that are different from i and j. 39 (a) Verify the Schwarz inequality I W < I I I 1 for V =
V I V W
(b) Find perpendicular unit vectors I, J , K different from i + 2 j + 2 k and W = 2 i + 2 j + k .
i, j, k. (b) What does the inequality become when V = (A,
&)
If I and J are perpendicular, take their dot products with and W = (&, &)?
A = a1 + bJ to find a and b.
40 By choosing the right vector W in the Schwarz inequality,
28 Suppose I= (i +I)/* and J = (i - j)/& Check I J = 0 + +
show that (V, + V2 + V3)2< 3(V; Vi v:). What is W?
and write A = 2i + 3j as a combination a1 + bJ. (Best method:
use a and b from Problem 27. Alternative: Find i and j from 41 The Schwarz inequality for ai + bj and ci + dj says that
I and J and substitute into A.) +
(a2+ b2)(c2 d2)2 (ac + bd)2. Multiply out to show that the
difference is 2 0.
29 (a) Find the position vector OP and the velocity vector
PQ when the point P moves around the unit circle (see figure) 42 The vectors A, B, C form a triangle if A+ B + C = 0. The
with speed 1. (b) Change to speed 2. triangle inequality I + BJ< IA(+ I I says that any one side
A B
length is less than . The proof comes from Schwarz:
+ +
30 The sum (A i)2 (A j)2 (A k)2 equals .
31 In the semicircle find C and D in terms of A and B. Prove
that C D = 0 (they meet at right angles).
+ -
32 The diagonal PR has (PRI2= (A B) (A + B) = A A + 43 True or false, with reason or example:
A B + B A + B B. Add lQS12 from the other diagonal to
prove the parallelogram law: I P R ~ ~ + IQSI2 = sum of squares (a) I + W12 is never larger than lV12 + IWI2
V
of the four side lengths. (b) In a real triangle I + W never equals IV(+ IWI
V I
33 If (1, 2, 3), (3,4, 7), and (2, 1, 2) are corners of a parallelo- (c) V W equals W V
gram, find all possible fourth corners. (d) The vectors perpendicular to i + j + k lie along a line.
34 The diagonals of the parallelogram are A +B
and
. If they have the same length, prove that A B = 0 44 If V = i + V
2k choose W so that V W = I I IW( and
and the region is a I + Wl = IVl+ IWl.
V
I1.2 Planes and Projections 407
45 A methane molecule h~asa carbon atom at (0, 0,O) and 46 (a)Find a vector V at a 45" angle with i and j.
hydrogen atoms at (1, 1, -I), (1, -1, I), (-1, 1, I), and (b) Find W that makes a 60" angle with i and j.
(-1, -1, -1). Find (c) Explain why no vector makes a 30" angle with i and j.
(a) the distance between hydrogen atoms
(b) the angle between vectors going out from the carbon
atom to the hydrogen ,atoms.
11.2 Planes and Projections
The most important "curves" are straight lines. The most important functions are
linear. Those sentences take us back to the beginning of the book-the graph of
mx + b is a line. The goal now is to move into three dimensions, where graphs are
surfaces. Eventually the surfaces will be curved. But calculus starts with the flat
surfaces that correspond to straight lines:
What are the most important surfaces? Planes.
What are the most important functions? Still linear.
The geometrical idea of a plane is turned into algebra, by finding the equation of a
plane. Not just a general formula, but the particular equation of a particular plane.
A line is determined by one point (x,, yo) and the slope m. The point-slope equation
is y - yo = m(x - x,). That is a linear equation, it is satisfied when y = yo and x = xo,
and dyldx is m. For a plane, we start again with a particular point-which is now
(x, ,yo, I:,). But the slope of a plane is not so simple. Many planes climb at a 45"
angle-with "slope 1"-and more information is needed.
The direction of a plane is described by a vector N. The vector is not in the plane,
but perpmdicular to the plane. In the plane, there are many directions. Perpendicular
to the plane, there is only one direction. A vector in that perpendicular direction is
a normal vector.
The normal vector N can point "up" or "down". The length of N is not crucial (we
often make it a unit vector and call it n). Knowing N and the point Po = (x,, yo, z,),
we know the plane (Figure 11.9). For its equation we switch to algebra and use the
dot product-which is the key to perpendicularity.
N is described by its components (a, b, c). In other words N is ai + bj + ck. This
vector is perpendicular to every direction in the plane. A typical direction goes from
N = ai + bj + ck
t normal vector
Fig. 11.9 The normal vector to a plane. Parallel planes have the same N.
11 Vectors and Matrices
Po to another point P = (x, y, z) in the plane. The vector from Po to P has components
(x - xo, y - yo, z - z,). This vector lies in the plane, so its dot product with N is zero:
116 The plane through Po perpendicular to N = (a, b, c) has the equation
(a, b,c)*(x-xo, y-yo, z-zo) = O or
4x-x0) + qy-yo)+ c(z-zo)= 0. (1)
The point P lies on the plane when its coordinates x, y, z satisfy this equation.
EXAMPLE 1 The plane through Po = (1,2,3) perpendicular to N = (1, 1, 1) has the
+
equation (x - 1) + (y - 2) + (z - 3) = 0. That can be rewritten as x + y z = 6.
Notice three things. First, Po lies on the plane because 1 + 2 + 3 = 6. Second, N =
(1, 1, 1) can be recognized from the x, y, z coefficients in x + y + z = 6. Third, we could
change N to (2,2,2) and we could change Po to (8,2, - 4)-because N is still perpen-
dicular and Po is still in the plane: 8 + 2 - 4 = 6.
The new normal vector N = (2,2,2) produces 2(x - 1) + 2(y - 2) + 2(z - 3) = 0.
That can be rewritten as 2x + 2y + 22 = 12. Same normal direction, same plane.
The new point Po = (8, 2, - 4) produces (x - 8) + (y - 2) + (z + 4) = 0. That is
another form of x + y + z = 6. All we require is a perpendicular N and a point Po in
the plane.
EXAMPLE 2 The plane through (1,2,4) with the same N = (1, 1, 1) has a different
equation: (x - 1) + (y - 2) + (z - 4) = 0. This is x + y + z = 7 (instead of 6). These
planes with 7 and 6 are parallel.
+
Starting from a(x - x,) + b(y - yo) c(z - 2,) = 0, we often move ax, + by, + cz,
to the right hand side-and call this constant d:
1I D With the Poterms on the right side, the equation of the plane is N P = d:
a x + b y + c z = a x o + byo+czo=d. (2)
A different d gives a puraIle1 plane; d = 0 gives a plane throzcgh the origin.
EXAMPLE 3 The plane x - y + 3z = 0 goes through the origin (0, 0, 0). The normal
vector is read directly from the equation: N = (1, - 1, 3). The equation is satisfied by
Po = (1, 1,O) and P = ( l , 4 , 1). Subtraction gives a vector V = (0, 3, 1) that is in the
plane, and N V = 0.
The parallel planes x - y + 32 = d have the same N but different d's. These planes
miss the origin because d is not zero (x = 0, y = 0, z = 0 on the left side needs d = 0
on the right side). Note that 3x - 3y + 9z = - 15 is parallel to both planes. N is
changed to 3N in Figure 11.9, but its direction is not changed.
EXAMPLE 4 The angle between two planes is the angle between their normal vectors.
The planes x - y + 3z = 0 and 3y + z = 0 are perpendicular, because (1, - 1, 3)
(0, 3, 1) = 0. The planes z = 0 and y = 0 are also perpendicular, because (O,0, 1)
(0, 1,O) = 0. (Those are the xy plane and the xz plane.) The planes x + y = 0 and
x + z = O m a k e a 6 0 ° angle, becausecos60°=(l, 1,0)*(1,0,l)/dfi=+.
The cosine of the angle between two planes is IN, N,I/IN,I IN,I. See Figure 11.10.
11.2 Planes and Projections 409
1
1, 1)
-k
I
=mx·-· b
= i rI.h
Fig. 11.10 Angle between planes = angle between normals. Parallel and perpendicular to a
line. A line in space through P0 and Q.
Remark 1 We gave the "point-slope" equation of a line (using m), and the "point-
normal" equation of a plane (using N). What is the normal vector N to a line?
The vector V = (1, m) is parallel to the line y = mx + b. The line goes across by 1
and up by m. The perpendicular vector is N = (- m, 1). The dot product N V is
- m + m = 0. Then the point-normal equation matches the point-slope equation:
- m(x - x0 ) + 1(y - yo) = 0 is the same as y - yo = m(x - xo). (3)
Remark 2 What is the point-slope equation for a plane? The difficulty is that a
plane has different slopes in the x and y directions. The function f(x, y)=
m(x - x0 ) + M(y - yo) has two derivatives m and M.
This remark has to stop. In Chapter 13, "slopes" become "partialderivatives."
A LINE IN SPACE
In three dimensions, a line is not as simple as a plane. A line in space needs two
equations. Each equation gives a plane, and the line is the intersection of two planes.
The equations x + y + z = 3 and 2x + 3y + z = 6 determine a line.
Two points on that line are P0 = (1, 1, 1) and Q = (3, 0, 0). They satisfy both equations
so they lie on both planes. Therefore they are on the line of intersection. The direction
of that line, subtracting coordinates of P0 from Q, is along the vector V = 2i - j - k.
The line goes through P0 = (1, 1, 1) in the direction of V = 2i - j - k.
Starting from (xo, Yo, zo) = (1, 1, 1), add on any multiple tV. Then x = 1 + 2t and
y= 1 - t and z = 1 - t. Those are the components of the vector equation
P = P0 + tV-which produces the line.
Here is the problem. The line needs two equations-or a vector equation with a
parameter t. Neither form is as simple as ax + by + cz = d. Some books push ahead
anyway, to give full details about both forms. After trying this approach, I believe
that those details should wait. Equations with parameters are the subject of
Chapter 12, and a line in space is the first example. Vectors and planes give plenty
to do here-especially when a vector is projected onto another vector or a plane.
PROJECTION OF A VECTOR
What is the projection of a vector B onto another vector A? One part of B goes along
A-that is the projection. The other part of B is perpendicularto A. We now compute
these two parts, which are P and B - P.
S11 Vectors and Matrices
In geometry, projections involve cos 0. In algebra, we use the dot product (which
is closely tied to cos 0). In applications, the vector B might be a velocity V or a force
F:
An airplane flies northeast, and a 100-mile per hour wind blows due
east. What is the projection of V = (100, 0) in the flight direction A?
Gravity makes a ball roll down the surface 2x + 2y + z = 0. What are
the projections of F = (0, 0, - mg) in the plane and perpendicular to
the plane?
The component of V along A is the push from the wind (tail wind). The other
component of V pushes sideways (crosswind). Similarly the force parallel to the
surface makes the ball move. Adding the two components brings back V or F.
B N=2i+2i+k
U N =. .. . _
downhill force:
B- projection force
tailwind =
of F force
projection on plane
of V on A IBI sin 0 A
00i \\
A'B "r
cros
IBI cos 0-
IAI force of gravity
F = -mgk
Fig. 14.14 Projections along A of wind velocity V and force F and vector B.
We now compute the projection of B onto A. Call this projection P. Since its
direction is known-P is along A-we can describe P in two ways:
1) Give the length of P along A
2) Give the vector P as a multiple of A.
Figure 11.1lb shows the projection P and its length. The hypotenuse is IBI. The
length is IPI = HBI 0. The perpendicular component B - P has length IBI sin 0. The
cos
cosine is positive for angles less than 900. The cosine (and P!) are zero when A and
B are perpendicular. IBI cos 0 is negative for angles greater than 900, and the pro-
jection points along -A (the length is IBI Icos 01). Unless the angle is 0O 300 or 450
or
or 600 or 900, we don't want to compute cosines-and we don't have to. The dot
product does it automatically:
A'B
IA|I BI cos 0 = A -B so the length of P along A is IBI cos 0 - (4)
Notice that the length of A cancels out at the end of (4). If A is doubled, P is
unchanged. But if B is doubled, the projection is doubled.
What is the vector P? Its length along A is A . B/IAI. If A is a unit vector, then
JAl = 1 and the projection is A . B times A. Generally A is not a unit vector, until we
divide by IAI. Here is the projection P of B along A:
fA*B\/ A \ A-B
P = (length of P)(unit vector) (27 1 -1) ,A. lAI/
/k,
IAI
\
IAI"
JAI | JA AIA
11.2 Planes and Projections 411
EXAMPLE 5 For the wind velocity V = (100, 0) and flying direction A = (1, 1), find P.
Here V points east, A points northeast. The projection of V onto A is P:
A V 100 A V 100
length |PJ vector P= A = - (1, 1)= (50, 50).
Al 2 Al| 2
EXAMPLE 6 Project F = (0, 0, - mg) onto the plane with normal N = (2, 2, 1).
The projection of F along N is not the answer. But compute that first:
S mg P N - (2 , 2, 1).
INI 3 IN12 9
P is the component of F perpendicular to the plane. It does not move the ball. The
in-plane component is the difference F - P. Any vector B has two projections, along
A and perpendicular:
A.B
The projection P = - 2 A is perpendicular to the remaining component B - P.
A1
EXAMPLE 7 Express B = i -j as the sum of a vector P parallel to A = 3i + j and a
vector B - P perpendicular to A. Note A . B = 2.
A*B 2 6 2 4 12.
Solution P= 2
A= A= i+ j. Then B - P = i- j.
|Al 10 10 10 10 10
Check: P (B - P) = (f6)(f0) - (2o)({) = 0. These projections of B are perpendicular.
2
Pythagoras: P1 + B- P12 equals IB1. Check that too: 0.4 + 1.6 = 2.0.
2
Question When is P = 0? Answer When A and B are perpendicular.
EXAMPLE 8 Find the nearest point to the origin on the plane x + 2y + 2z = 5.
The shortest distancefrom the origin is along the normal vector N. The vector P to
the nearest point (Figure 11.12) is t times N, for some unknown number t. We find t
by requiring P = tN to lie on the plane.
The plane x + 2y + 2z = 5 has normal vector N = (1, 2, 2). Therefore P = tN =
(t, 2t, 2t). To lie on the plane, this must satisfy x + 2y + 2z = 5:
t+2(2t) + 2(2t) = 5 or 9t=5 or t= . (6)
Then P = !N = (6, , ). That locates the nearest point. The distance is INI = .
N
This example is important enough to memorize, with letters not numbers:
The steps are the same. N has components a, b, c. The nearest point on the plane is
a multiple (ta, tb, tc). It lies on the plane if a(ta) + b(tb) + c(tc) = d.
Thus t = d/(a2 + b2 + c2). The point (ta, tb, tc) = tN is in equation (7). The distance
to the plane is ItNI = Idl/INI.
412 11 Vectors and Matrices
2j+2k
= i + 3j + 2k
= tN IPI 5 5
ne 2y
4+4 3 5-111 =2
1-
- +4+4
mex + 2y + 2z = 5 Q + tN
Fig. 11.12 Vector to the nearest point P is a multiple tN. The distance is in (7) and (9).
Question How far is the plane from an arbitrary point Q = (xl, yl, z 1 )?
Answer The vectorfrom Q to P is our multiple tN. In vector form P = Q + tN. This
reaches the plane if P -N = d, and again we find t:
(Q + tN) N = d yields t = (d - Q . N)/IN12.
This new term Q N enters the distance from Q to the plane:
distance = ItNI = d - Q NI/INI = Id - ax 1 - by, - cz1 l/ a2 + b 2 c 2.
When the point is on the plane, that distance is zero-because ax, + by, + czx = d.
When Q is i + 3j + 2k, the figure shows Q . N = 11 and distance = 2.
PROJECTIONS OF THE HEART VECTOR
An electrocardiogram has leads to your right arm-left arm-left leg. You produce the
voltage. The machine amplifies and records the readings. There are also six chest
leads, to add a front-back dimension that is monitored across the heart. We will
concentrate on the big "Einthoven triangle," named after the inventor of the ECG.
The graphs show voltage variations plotted against time. The first graph plots the
voltage difference between the arms. Lead II connects the left leg to the right arm.
Lead III completes the triangle, which has roughly equal sides (especially if you are
a little lopsided). So the projections are based on 600 and 1200 angles.
The heart vector V is the sum of many small vectors-all moved to the same
origin. V is the net effect of action potentials from the cells-small dipoles adding to
a single dipole. The pacemaker (S-A node) starts the impulse. The atria depolarize
to give the P wave on the graphs. This is actually a P loop of the heart vector-the
LEAD aVR LtAU III
Fig. A The graphs show the component of the moving heart vector along each lead. These
figures are reproduced with permission from the CIBA Collection of Medical Illus-
trations by Frank H. Netter, M.D. Copyright 1978 CIBA-GEIGY, all rights reserved.
11.2 Planes and Projections
graphs only show its projections. The impulse reaches the A V node, pauses, and
moves quickly through the ventricles. This produces the QRS complex-the large
sharp movement on the graph.
The total QRS interval should not exceed 1/10 second (2i spaces on the printout).
V points first toward the right shoulder. This direction is opposite to the leads, so
the tracings go slightly down. That is the Q wave, small and negative. Then the heart
vector sweeps toward the left leg. In positions 3 and 4, its projection on lead I
(between the arms) is strongly positive. The R wave is this first upward deflection in
each lead. Closing the loop, the S wave is negative (best seen in leads I and aVR).
Question 1 How many graphs from the arms and leg are really independent?
Answer Only two! In a plane, the heart vector V has two components. If we know
two projections, we can compute the others. (The ECG does that for us.) Different
vectors show better in different projections. A mathematician would use 90" angles,
with an electrode at your throat.
Question 2 How are the voltages related? What is the aVR lead?
Answer Project the heart vector V onto the sides of the triangle:
'The lead vectors have L,- L,, + L,,, = O-they form a triangle.
'The projections have V, - V,, + V,,, = V L, - V L,, + V L,,, = 0.
J'
The aVR lead is - i L , - Qh,. is pure algebra (no wire . By vector addition it points
It
toward the electrode on the right arm. Its length is 3 if the other lengths are 2.
Including aVL and aVF to the left arm and foot, there are six leads intersecting at
equal angles. Visualize them going out from a single point (the origin in the chest).
QRS
loop
L
Fig. B Heart vector goes around the QRS loop. Projections are spikes on the ECG.
Question 3 If the heart vector is V = 2i - j, what voltage differences are recorded?
Answer The leads around the triangle have length 2. The machine projects V:
Lead I is the horizontal vector 2i. So V L, = 4.
Lead I1 is the - 60" vector i - f i j . So V L,, = 2 + fi.
Lead I11 is the - 120" vector - i - f i j . So V L,,, = - 2 + f.
i
The first and third add to the second. The largest R waves are in leads I and 11. In
aVR the projection of V will be negative (Problem 46), and will be labeled an S wave.
II Vectors and Matrlces
Question 4 What about the potential (not just its differences).Is it zero at the center?
Answer It is zero ifwe say so. The potential contains an arbitrary constant C. (It is
like an indefinite integral. Its differences are like definite integrals.) Cardiologists
define a "central terminal" where the potential is zero.
The average of V over a loop is the mean heart vector H. This average requires
[ Vdt, by Chapter 5. With no time to integrate, the doctor looks for a lead where the
area under the QRS complex is zero. Then the direction of H (the axis) is perpendicu-
lar to that lead. There is so much to say about calculus in medicine.
11.2 EXERCISES
Read-through questions Find an x - y - z equation for planes 7-10.
A plane in space is determined by a point Po = (xo, yo, zo) 7 The plane through Po = (1,2, -1) perpendicular to N =
and a a vector N with components (a, b, c). The point +
i j
P = (x, y, z) is on the plane if the dot product of N with b
8 The plane through Po = (1,2, -1) perpendicular to N =
is zero. (That answer was not P!) The equation of this plane
is a( c ) + b( d ) + c( ) = 0. The equation is also
i+2j-k
written as ax + by + cz = d, where d equals f . A parallel 9 The plane through (1,0, 1) parallel to x + 2y + z = 0
plane has the same g and a different h . A plane
through the origin has d = i .
10 The plane through (xo,yo, zo) parallel to x + y + z = 1.
11 When is a plane with normal vector N parallel to the
The equation of the plane through Po = (2, 1,O) perpendic-
vector V? When is it perpendicular to V?
ular to N = (3,4, 5) is I . A second point in the plane is
P = (0, 0, k ). The vector from Po to P is I , and it is (a) If two planes are perpendicular (front wall and side
m to N. (Check by dot product.) The plane through Po = wall), is every line in one plane perpendicular to every line
(2, 1,0) perpendicular to the z axis has N = n and equa- in the other?
tion 0 . (b) If a third plane is perpendicular to the first, it might
The component of B in the direction of A is P , where be (parallel) (perpendicular) (at a 45" angle) to the second.
8 is the angle between the vectors. This is A B divided by Explain why a plane cannot
q . The projection vector P is IBI cos 8 times a r (a) contain (1, 2, 3) and (2, 3,4) and be perpendicular to
vector in the direction of A. Then P = ( IBI cos 8)(A/IAI)sim- N=i+j
plifies to 8 . When B is doubled, P is t . When A is
(b) be perpendicular to N = i + j and parallel to V = i + k
doubled, P is u . If B reverses direction then P v . If
A reverses direction then P w . (c) contain (1, 0, O), (0, 1, O), (0, 0, I), and (1, 1, 1)
(d) contain (1, 1, - 1) if it has N = i + j - k (maybe it can)
When B is a velocity vector, P represents the x . When
(e) go through the origin and have the equation
B is a force vector, P is Y . The component of B perpen-
dicular to A equals . The shortest distance from (0, 0,O) a x + b y + c z = 1.
to the plane ax + by + cz = d is along the A vector. The The equation 3x + 4y + 72 - t = 0 yields a hyperplane in
distance is B and the closest point on the plane is P = four dimensions. Find its normal vector N and two points P,
c . The distance from Q = (xl, y,, z,) to the plane is Q on the hyperplane. Check (P - Q) N = 0.
D .
15 The plane through (x, y, z) perpendicular to ai + bj ck +
goes through (0, 0,0) if . The plane goes through
Find two points P and Po on the planes 1-6 and a normal (xo, Yo 20) if- -
vector N. Verify that N (P - Po) = 0.
16 A curve in three dimensions is the intersection of
1 x+2y+3z=O 2 x+2y+3z=6 3 the yzplane surfaces. A line in four dimensions is the intersection of
hyperplanes.
4 the plane through (0, 0,0) perpendicular to i + j - k
17 (angle between planes) Find the cosine of the angle
5 the plane through (1, 1, 1) perpendicular to i + j - k
between x + 2 y + 2 z = 0 and (a) x + 2 z = 0 (b) x + 2 z = 5
6 the plane through (0, 0,O) and (1,0,0) and (0, 1, 1). (c) X = 0.
11.2 Planes and Projections 415
18 N is perpendicular to a plane and V is along a line. Draw 36 The distance between the planes x + y + 5z = 7 and
the angle 8 between the plane and the line, and explain why 3x + 2y + z = 1 is zero because
V N/IVI JNIis sin 8 not cos 0. Find the angle between the xy
plane and v = i j k
+ +d .
In Problems 37-41 all points and vectors are in the xy plane.
In 19-26 find the projection P of B along A. Also find IPI. 37 The h e 3x + 4y = 10 is perpendicular to the vector N =
. On the line, the closest point to the origin is P =
tN. Find t and P and !PI.
21 B = unit vector at 60" angle with A
38 Draw the line x + 2y = 4 and the vector N = i + 2j. The
closest point to Q = (3, 3) is P = Q + tN. Find t. Find P.
22 B = vector of length 2 at 60" angle with A
39 A new way to find P in Problem 37: minimize x2 y2 = +
x2 + (9 3 ~ )By calculus find the best x and y.
- ~ .
25 A is perpendicular to x - y + z = 0, B = i +j. 40 To catch a drug runner going from (0,O) to (4,O) at 8
26 A is perpendicular to x - y + z = 5, B = i + j + 5k. meters per second, you must travel from (0, 3) to (4,O) at
meters per second. The projection of your velocity
27 The force F = 3i - 4k acts at the point (1,2,2). How much vector onto his velocity vector will have length
force pulls toward the origin? How much force pulls vertically
down? Which direction does a mass move under the force F? 41 Show by vectors that the distance from (xl ,y1) to the line
ax + by = d is I - axl - byll/JW.
d
28 The projection of B along A is P = . The projec-
tion of B perpendicular to A is . Check the dot 42 It takes three points to determine a plane. So why does
product of the two projections. +
ax + by cz = d contain four numbers a, b, c, d? When does
ex +fy + gz = 1 represent the same plane?
29 P=(x,y,z) is on the plane a x + b y + c z = 5 if P * N =
IPI IN1 cos 8 = 5. Since the largest value of cos 8 is 1, the small- 43 (projections by calculus) The dot product of B - tA with
est value of IPI is . This is the distance between itself is JBI2 2tA B + t2(AI2.(a) This has a minimum at
-
t= . (b) Then tA is the projection of .A
30 If the air speed of a jet is 500 and the wind speed is 50, figure showing B, tA, and B - tA is worth 1000 words.
what information do you need to compute the jet's speed over 44 From their equations, how can you tell if two planes are
land? What is that speed? (a) parallel (b) perpendicular (c) at a 45" angle?
31 How far is the plane x + y - z = 1 from (0, 0, 0) and also
from (1, 1, -l)? Find the nearest points.
Problems 45-48 are about the ECG and heart vector.
32 Describe all points at a distance 1 from the plane
x+2y+2z=3. 45 The aVR lead is -$L,-iL,,. Find the aVL and aVF
leads toward the left arm and foot. Show that
33 The shortest distance from Q =(2, 1, 1) to the plane aVR + aVL + aVF = 0. They go out from the center at 120"
+ +
x y z = 0 is along the vector . The point P = angles.
+ +
Q tN = (2 + t, 1 t, 1 + t) lies on the plane if t = .
Then P = and the shortest distance is 46 Find the projection on the aVR lead of V = 2i -j in
(This distance is not IPI.) Question 3.
34 The plane through (1, 1, 1) perpendicular to N = 47 If the potentials are rp,, = 1 (right arm) and (PLA = 2 and
i + 2j + 2k is a distance from (0, 0, 0). cpLL = - 3, find the heart vector V. The diflerences in potential
are the projections of V.
35 (Distance between planes) 2x - 2y + z = 1 is parallel
to 2x -2y + z = 3 because . Choose a vector Q on 48 If V is perpendicular to a lead L, the reading on that lead
the first plane and find t so that Q + tN lies on the second is . If J V(t)dt is perpendicular to lead L, why is the
plane. The distance is ltNl= . area under the reading zero?
416 11 Vectors and Matrices
After saying that vectors are not multiplied, we offered the dot product. Now we
contradict ourselves further, by defining the cross product. Where A B was a number,
the cross product A x B is a vector. It has length and direction:
The length is IAl IBI 1 sin 81. The direction is perpendicular to A and B.
The cross product (also called vector product) is defined in three dimensions only.
A and B lie on a plane through the origin. A x B is along the normal vector N,
perpendicular to that plane. We still have to say whether it points "up" or "down"
along N.
The length of A x B depends on sin 8, where A B involved cos 8. The dot product
rewards vectors for being parallel (cos 0 = 1). The cross product is largest when A is
perpendicular to B (sin n/2 = 1). At every angle
That will be a bridge from geometry to algebra. This section goes from definition to
formula to volume to determinant. Equations (6) and (14) are the key formulas for
A x B.
Notice that A x A = 0. (This is the zero vector, not the zero number.) When B is
parallel to A, the angle is zero and the sine is zero. Parallel vectors have A x B = 0.
Perpendicular vectors have sin 8 = 1 and IA x BI = JAlIBI = area of rectangle with
sides A and B.
Here are four examples that lead to the cross product A x B.
EXAMPLE 1 (From geometry) Find the area of a parallelogram and a triangle.
Vectors A and B, going out from the origin, form two sides of a triangle. They produce
'the parallelogram in Figure 11.13, which is twice as large as the triangle.
The area of a parallelogram is base times height (perpendicular height not sloping
height). The base is [A[. The height is IBI [sin 81. We take absolute values because
height and area are not negative. Then the area is the length of the cross product:
area of parallelogram = IAl IB( [sin 8 = IA x BI.
1 (2)
, height turning 4 axis
base 1 A 1
area lAlIBl(sin81=IAxBI n
moment ) ~ l I F I s i 8 ixj
Fig. 11.13 Area ( A x B ( and moment (R x F(. Cross products are perpendicular to the page.
EXAMPLE 2 (From physics) The torque vector T = R x F produces rotation.
The force F acts at the point (x, y, z). When F is parallel to the position vector R =
xi + yj + zk, the force pushes outward (no turning). When F is perpendicular to R,
the force creates rotation. For in-between angles there is an outward force IF1 cos 8
F
and a turning force I 1 sin 8. The turning force times the distance (RI is the moment
JRI(FIsin 8.
11.3 Cross Products and Determinants 417
The moment gives the magnitude and sign of the torque vector T = R x F. The
direction of T is along the axis of rotation, at right angles to R and F.
EXAMPLE 3 Does the cross product go up or down? Use the right-handrule.
Forces and torques are probably just fine for physicists. Those who are not natural
physicists want to see something turn.t We can visualize a record or compact disc
rotating around its axis-which comes up through the center.
At a point on the disc, you give a push. When the push is outward (hard to do),
nothing turns. Rotation comes from force "around" the axis. The disc can turn either
way-depending on the angle between force and position. A sign convention is
necessary, and it is the right-hand rule:
A x B points along your right thumb when the fingers curl from A toward B.
This rule is simplest for the vectors i, j, k in Figure 11.14-which is all we need.
Suppose the fingers curl from i to j. The thumb points along k. The x-y-z axes
form a "right-handed triple." Since li| = 1 and I|j= 1 and sin 7n/2 = 1, the length of i x j
is 1. The cross product is i x j = k. The disc turns counterclockwise-its angular
velocity is up-when the force acts at i in the direction j.
Figure 11.14b reverses i and j. The force acts at j and its direction is i. The disc
turns clockwise (the way records and compact discs actually turn). When the fingers
curl from j to i, the thumb points down. Thus j x i = - k. This is a special case of an
amazing rule:
The cross product is anticommutative: B x A = - (A x B). (3)
That is quite remarkable. Its discovery by Hamilton produced an intellectual revolu-
tion in 19th century algebra, which had been totally accustomed to AB = BA. This
commutative law is old and boring for numbers (it is new and boring for dot pro-
ducts). Here we see its opposite for vector products A x B. Neither law holds for
matrix products.
ixj=k
ixk
turning jx k = i
axis
screw going in screw coming out
Fig. 11.14 ixj=k=-(jxi) ixk=-j=-(kxi) jxk=i=-(kxj).
EXAMPLE 4 A screw goes into a wall or out, following the right-hand rule.
The disc was in the xy plane. So was the force. (We are not breaking records here.)
The axis was up and down. To see the cross product more completely we need to
turn a screw into a wall.
Figure 11.14b shows the xz plane as the wall. The screw is in the y direction. By
turning from x toward z we drive the screw into the wall-which is the negative y
direction. In other words i x k equals minus j. We turn the screw clockwise to make
it go in. To take out the screw, twist from k toward i. Then k x i equals plus j.
tEverybody is a natural mathematician. That is the axiom behind this book.
418 11 Vectors and Matrices
To summarize: k x i = j and j x k = i have plus signs because kij and jki are in the
same "cyclic order" as ijk. (Anticyclic is minus.) The z-x-y and y-z-x axes form right-
handed triples like x-y-z.
THE FORMULA FOR THE CROSS PRODUCT
We begin the algebra of A x B. It is essential for computation, and it comes out
beautifully. The square roots in IAI IBI Isin 01 will disappear in formula (6) for A x B.
(The square roots also disappeared in A *B, which is IAI IBI cos 0. But IAl IBI tan 0
would be terrible.) Since A x B is a vector we need to find three components.
Start with the two-dimensional case. The vectors a, i + a2j and b, i + b2j are in the
xy plane. Their cross product must go in the z direction. Therefore A x B = ? k
and there is only one nonzero component. It must be IAI IBI sin 0 (with the correct
sign), but we want a better formula. There are two clean ways to compute A x B,
either by algebra (a) or by a bridge (b) to the dot product and geometry:
(a) (ai+a x (bi+b2j)=albixi+ab 2 ixj + a2 bjxi+ab2
2 j) 2 jx j. (4)
On the right are 0, ab 2 k, -a 2 b1 k, and 0. The cross product is (ab 2 - a2 b,)k.
1 Its
(b) Rotate B= bli + b2j clockwise through 90o into B* = b2 i- b j. length is
unchanged (and B - B* = 0). Then IAI IBI sin 0 equals IAl IB*I cos 0, which is A " B*:
IAIIBI sin = A B*= a1 b2 2 bl. (5)
SF In the xy plane, A x B equals (ab 2 - a2 b)k. The parallelogram with
sides A and B has area |a1 - a2b 1. The triangle OAB has area -|a, b2 - a2 b 1.
b2
EXAMPLE 5 For A = i + 2j and B = 4i + 5j the cross product is (1 5 - 2 - 4)k = - 3k.
Area of parallelogram = 3, area of triangle = 3/2. The minus sign in A x B = - 3k is
absent in the areas.
Note Splitting A x B into four separate cross products is correct, but it does not
follow easily from IAl IBI sin 0. Method (a) is not justified until Remark 1 below. An
algebraist would change the definition of A x B to start with the distributive law
(splitting rule) and the anticommutative law:
Ax(B+C)=(AxB)+(AxC) and AxB=-(BxA).
THE CROSS PRODUCT FORMULA (3 COMPONENTS)
We move to three dimensions. The goal is to compute all three components of A x B
(not just the length). Method (a) splits each vector into its i, j, k components, making
nine separate cross products:
(ali+ a2j + a 3k) x (bai + b2 j + b3k) = alb 2(i x i) + alb 2(i x j) + seven more terms.
Remember i x i = j x j = k x k = 0. Those three terms disappear. The other six terms
come in pairs, and please notice the cyclic pattern:
FORMULA A x B = (a2b 3 - a 3 b2)i + (a3b, - a1 b3)j + (alb2 - a2 b,)k. (6)
The k component is the 2 x 2 answer, when a3 = b3 = 0. The i component involves
indices 2 and 3, j involves 3 and 1, k involves 1 and 2. The cross product formula is
11.3 Cross Products and Determinants 419
written as a "determinant" in equation (14) below-many people use that form to
compute A x B.
EXAMPLE6 (i+2j+3k) x (4i+5j+6k)= (2*6- 35)i+(3*4- 16)j+(1 5- 24)k.
The i, j, k components give A x B = - 3i + 6j - 3k. Never add the - 3, 6, and - 3.
Remark 1 The three-dimensional formula (6) is still to be matched with A x B from
geometry. One way is to rotate B into B* as before, staying in the plane of A and B.
Fortunately there is an easier test. The vector in equation (6) satisfies all four geo-
metric requirements on A x B: perpendicular to A, perpendicular to B, correct length,
right-hand rule. The length is checked in Problem 16-here is the zero dot product
with A:
A (A x B)= al(a 2b3 - a3b 2)+ a2(a 3b - ab 3 )+ a 3(ab 2 - a2 b)= 0. (7)
Remark 2 (Optional) There is a wonderful extension of the Pythagoras formula
a2 + b 2 = c2 . Instead of sides of a triangle, we go to areasof projections on the yz, xz,
and xy planes. 32 + 62 + 32 is the square of the parallelogram area in Example 6.
For triangles these areas are cut in half. Figure 11.15a shows three projected trian-
gles of area 1. Its Pythagoras formula is (1)2 + (1)2 + (½)2 = (area of PQR) 2.
EXAMPLE 7 P = (1, 0, 0), Q = (0, 1, 0), R = (0, 0, 1)lie in a plane. Find its equation.
Ideafor any P, Q, R: Find vectors A and B in the plane. Compute the normalN = A x B.
Solution The vector from P to Q has components -1, 1, 0. It is A = j - i (subtract
to go from P to Q). Similarly the vector from P to R is B = k - i. Since A and B are
in the plane of Figure 11.15, N = A x B is perpendicular:
(j - i) x (k - i)=(j x k)- (i x k)-(j x i)+(i x i)= i+ j+ k. (8)
The normal vector is N = i + j + k. The equation of the plane is 1x + ly + z = d.
With the right choice d = 1, this plane contains P, Q, R. The equation is x + y + z = 1.
EXAMPLE 8 What is the area of this same triangle PQR?
Solution The area is half of the cross-product length IA x BI = Ii + j + ki = 3.
R = (0, 0, 1), planex+y +z = 1
normal N = i +j + k
B=k - i Q= (0, 1,0) AI cos 0
A=j-i
P
P=(1.. , 0)
,
Fig. 11.15 Area of PQR is /3/2. N is PQ x PR. Volume of box is IA (B x C)I.
DETERMINANTS AND VOLUMES
We are close to good algebra. The two plane vectors ali + a2j and b i+ b2j are the
1
sides of a parallelogram. Its area is a1b2 - a2 bl, possibly with a sign change. There
420 11 Vectors and Matrices
is a special way to write these four numbers-in a "square matrix." There is also a
name for the combination that leads to area. It is the "determinant of the matrix":
The matrix is a , its determinant is = ajb2 - a2bl.
b, b2 b b2
1
This is a 2 by 2 matrix (notice brackets) and a 2 by 2 determinant (notice vertical
bars). The matrix is an array of four numbers and the determinant is one number:
21 21 10
Examples of determinants: = 6 - 4 = 2, = 0, = 1.
4 3 2 1 0 1
The second has no area because A = B. The third is a unit square (A = i, B = j).
Now move to three dimensions, where determinants are most useful. The parallelo-
gram becomes a parallelepiped. The word "box" is much shorter, and we will use it,
but remember that the box is squashed. (Like a rectangle squashed to a parallelogram,
the angles are generally not 900.) The three edges from the origin are A = (a,, a 2, a3 ),
B=(bl, b2 ,b 3), C=(c1 , C2,c 3). Those edges are at right angles only when A B=
A C = B*C= 0.
Question: What is the volume of the box? The right-angle case is easy-it is length
times width times height. The volume is IAI times IBI times ICI, when the angles are
90'. For a squashed box (Figure 11.15) we need the perpendicular height, not the
sloping height.
There is a beautiful formula for volume. B and C give a parallelogram in the base,
and lB x CI is the base area. This cross product points straight up. The third vector
A points up at an angle-its perpendicular height is JAl cos 0. Thus the volume is
area IB x CI times JAI times cos 0. The volume is the dot product of A with B x C.
11G The triple scalar product is A (B x C). Volume of box = IA (B x C)I.
Important: A . (B x C) is a number, not a vector. This volume is zero when A is in
the same plane as B and C (the box is totally flattened). Then B x C is perpendicular
to A and their dot product is zero.
Usefulfacts: A (Bx C)=(Ax B)C=C (Ax B)=B.(C xA).
All those come from the same box, with different sides chosen as base-but no change
in volume. Figure 11.15 has B and C in the base but it can be A and B or A and C.
The triple product A- (C x B) has opposite sign, since C x B = - (B x C). This order
ACB is not cyclic like ABC and CAB and BCA.
To compute this triple product A . (B x C), we take B x C from equation (6):
A (B x C) = al(b 2 c3 - b 3 C2) + a 2 (b 3 c 1 - blC 3 ) + a 3 (blc 2 - b 2 Cl). (9)
The numbers a,, a2, a3 multiply 2 by 2 determinants to give a 3 by 3 determinant!
There are three terms with plus signs (like alb 2 c3). The other three have minus signs
(like -alb 3c2). The plus terms have indices 123, 231, 312 in cyclic order. The minus
terms have anticyclic indices 132, 213, 321. Again there is a special way to write the
nine components of A, B, C-as a "3 by 3 matrix." The combination in (9), which
11.3 Cross Products and Determinants
gives volume, is a "3 by 3 determinant:"
a1 a2 a3
, determinant = A (B x C)= bl b2 b3
C1 c2 c3
A single number is produced out of nine numbers, by formula (9). The nine numbers
are multiplied three at a time, as in a, blc2-except this product is not allowed. Each
row and column must be represented once. This gives the six terms in the determinant:
The trick is in the _+ signs. Products down to the right are "plus":
With practice the six products like 2 2 2 are done in your head. Write down only
8 + 1 + 1 - 2 - 2 - 2 = 4. This is the determinant and the volume.
Note the special case when the vectors are i, j, k. The box is a unit cube:
1 0 0
1+0+0
volume of cube = 0 1 0 = = 1.
-0-0-0
0 0 1
If A, B, C lie in the same plane, the volume is zero. A zero determinant is the test
to see whether three vectors lie in a plane. Here row A = row B - row C:
Zeros in the matrix simplify the calculation. All three products with plus signs-
down to the right-are zero. The only two nonzero products cancel each other.
If the three - 1's are changed to + l's, the determinant is - 2. The determinant can
be negative when all nine entries are positive! A negative determinant only means
that the rows A, B, C form a "left-handed triple." This extra information from the
sign-right-handed vs. left-handed-is free and useful, but the volume is the absolute
value.
The determinant yields the volume also in higher dimensions. In physics, four
dimensions give space-time. Ten dimensions give superstrings. Mathematics uses all
dimensions. The 64 numbers in an 8 by 8 matrix give the volume of an eight-
dimensional box-with 8! = 40,320 terms instead of 3! = 6. Under pressure from my
class I omit the formula.
11 Vectors and Matric@r
Question When is the point (x, y, z) on the plane through the origin containing B
and C? For the vector A = xi + yj + zk to lie in that plane, the volume A (B x C)
must be zero. The equation of the plane is determinant = zero.
Follow this example for B = j - i and C = k - i to find the plane parallel to B and C:
This equation is x + y + z = 0. The normal vector N = B x C has components 1,1,1.
H
T E CROSS PRODUCT AS A DETERMINANT
There is a connection between 3 by 3 and 2 by 2 determinants that you have to see.
The numbers in the top row multiply determinants from the other rows:
The highlighted product al(b2c3- b3c2) gives two of the six terms. AN six products
contain an a and b and c from diflerent columns. There are 3! = 6 different orderings
of columns 1,2, 3. Note how a3 multiplies a determinant from columns 1 and 2.
Equation (13) is identical with equations (9) and (10). We are meeting the same six
terms in different ways. The new feature is the minus sign in front of a,-and the
common mistake is to forget that sign. In a 4 by 4 determinant, a l , - a,, a3, - a,
would multiply 3 by 3 determinants.
Now comes a key step. We write A x B as a determinant. The vectors i, j, k go in
the top row, the components of A and B go in the other rows. The "determinant" is
exactly A x B:
This time we highlighted the j component with its minus sign. There is no great
mathematics in formula (14)-it is probably illegal to mix i, j, k with six numbers but
it works. This is the good way to remember and compute A x B. In the example
( j - i) x (k - i) from equation (8), those two vectors go into the last two rows:
The k component is highlighted, to see a1b2- a2bl again. Note the change from
equation (1I), which had 0,1, - 1 in the top row. That triple product was a number
(zero). This cross product is a vector i + j + k.
11.3 Cross Products and Determinants
Review question 1 With the i, j, k row changed to 3,4,5, what is the determinant?
Answer 3 * 1 + 4 - 1 k 5 1 = 12. That triple product is the volume of a box.
Review question 2 When is A x B = 0 and when is A *(Bx C) = O? Zero vector,
zero number.
Answer When A and B are on the same line. When A, B,C are in the same plane.
Review question 3 Does the parallelogram area IA x B equal a 2 by 2 determinant?
I
Answer If A and B lie in the xy plane, yes. Generally no.
Reviewquestion4 What are the vector triple products (A x B)-x and C
A x (Bx C)?
Answer Not computed yet. These are two new vectors in Problem 47.
Review question 5 Find the plane through the origin containing A = i + j + 2k and
B = i + k. Find the cross product of those same vectors A and B.
Answer The position vector P = x i + yj + zk is perpendicular to N = A x B:
Read-through questions If A, B, C lie in the same plane then A (B x C) is I .
For A = xi + yj + zk the first row contains the letters J .
The cross product A x B is a a whose length is b . So the plane containing B and C has the equation K =
Its direction is c to A and B. That length is the area of 0. When B = i +j and C = k that equation is . B x C is
a d whose base is IA( and whose height is e . When
-
9 M .
A=ali+a2jandB=b,i+b2j,theareais f .Thisequals
a 2 b y 2 s .IngeneralIA*B12+I~xB12= h . A 3 by 3 determinant splits into N 2 by 2 determinants.
They come from rows 2 and 3, and are multiplied by'.:the
The rules for cross products are A x A = i and
entries in row 1. With i, j, k in row 1, this!determinant equals
AxB=-( I ) and A x ( B + C ) = A x B + k . In
the 0 product. Its j component is p , including the
particular A x B needs the I -hand rule to decide its
Q sign which is easy to forget.
direction. If the fingers curl from A towards B (not more than
180°), then m points n . By this rule i x j = 0
andixk= P andjxk= q .
Compute the cross products 1-8 from formula (6) or the ddter-
+ + +
The vectors ali a 2 j+ a3k and bli b2j b3k have cross
minant (14). Do one example both ways.
product r i+ s j+ t k. The vectors A =
i + j + k and B = i + j have A x B = u . (This is also the 1(ixj)xk 2(ixj)xi
3 by 3 determinant v .) Perpendicular to the plane con- 3 (2i + 3j) x (i + k) 4 (2i + 3j + k) x (2i+ 3j - k)
taining (0, 0, O), (1, 1, I), (1, 1,0) is the normal vector N =
w . The area of the triangle with those three vertices is 5 (2i+3j+k)x(i-j-k) 6(i+j-k)x(i-j+k)
x ,which is half the area of the parallelogram with fourth
7 (i + 2j + 3k) x (4i - 9j)
vertex at Y .
Vectors A, B, C from the origin determine a . Its vol- 8 (i cos 8 +j sin 8) x (i sin 8 -j cos 8)
ume (A * ( A ))( comes from a 3 by 3 B . There are six /B(
9 When are (A x B( = (A((B(and IA (B x C)(= /A1 (C(?
terms, c with a plus sign and D with minus. In every
10 True or false:
term each row and E is represented once. The rows
(1,0, O), (0,0, I), and (0, 1, 0) have determinant = F . That (a) A x B never equals A B.
box is a G , but its sides form a H -handed triple in (b)IfA x B=Oand A*B=O, theneither A = O o r B = O .
the order given. (c) I f A x B = A x C and A#O, then B = C .
424 11 Vectors and Matrlces
In 11-16 find IA x BI by equation (1) and then by computing 33 When B = 3i + j is rotated 90" clockwise in the xy plane
A x B and its length. it becomes B* = . When rotated 90" counterclock-
11 A = i + j + k , B = i 12 A = i + j , B = i - j
wise it is . When rotated 180" it is .
34 From formula (6) verify that B (A x B) = 0.
13 A = - B 14 A = i + j , B = j + k
35 Compute
15 A = a l i + a2j, B = b,i + b2j
16 A = ( a l , a2, a3), B = ( b , , b2, b3)
In Problem 16 (the general case), equation (1) proves that the
length from equation (6) is correct. 36 Which of the following are equal
17 True or false, by testing on A = i, B = j, C = k: +
(A B) x B, (- B) x (-A), IAI IBl Isin 01, (A + C) x (B - C),
(a) A x (A x B) = 0 (b) A .(B x C) = (A x B) C +
HA - B) x (A B).
(c) A (B x C) = C .(B x A) 37 Compare the six terms on both sides to prove that
+
(d) (A - B) x (A B) = 2(A x B).
18 (a) From A x B = - (B x A) deduce that A x A = 0.
+
(b) Split (A B) x (A + B) into four terms, to deduce that
(A x B)= -(B x A).
The matrix is "transposed"-same determinant.
What are the normal vectors to the planes 19-22? 38 Compare the six terms to prove that
19 (2, 1, 0) (x, y, z) = 4 20 3x + 4z = 5
This is an "expansion on row 2." Note minus signs.
39 Choose the signs and 2 by 2 determinants in
Find N and the equation of the plane described in 23-29.
23 Contains the points (2, 1, I), (1, 2, I), (1, 1, 2)
24 Contains the points (0, 1, 2), (1, 2, 3), (2, 3, 4)
25 Through (0, 0, O), (1, 1, I), (a, b, c) [What if a = b = c?] +
40 Show that (A x B) (B x C) + (C x A) is perpendicular to
26 Parallel to i + j and k B-A and C - B and A-C.
27 N makes a 45" angle with i and j
28 N makes a 60" angle with i and j Problems 41-44 compute the areas of triangles.
29 N makes a 90" angle with i and j 41 The triangle PQR in Example 7 has squared area
($12)~ = ((t2 + (j)2+ ((f2,from the 3D version of Pythagoras
30 The triangle with sides i and j is as large as the
in Remark 2. Find the area of PQR when P = (a, 0, 0), Q =
parallelogram with those sides. The tetrahedron with edges
(0, b, O), and R = (0, 0, c). Check with #A x BI.
i, j, k is as large as the box with those edges. Extra
credit: In four dimensions the "simplex" with edges i, j, k, 1 42 A triangle in the xy plane has corners at (a,, b,), (a2,b2)
has volume = . and (a,, b,). Its area A is half the area of a parallelogram.
Find two sides of the parallelogram and explain why
31 If the points (x, y, z), (1, 1, O), and (1, 2, 1) lie on a plane
through the origin, what determinant is zero? What equation A = *[(a2 al)(b3- b l ) - (a3 - al)(b2- bl)l.
-
does this give for the plane?
43 By Problem 42 find the area A of the triangle with corners
32 Give an example of a right-hand triple and left-hand triple. (2, 1) and (4, 2) and (1, 2). Where is a fourth corner to make a
Use vectors other than just i, j, k. parallelogram?
11.4 Matrices and Linear Equations 425
44 Lifting the triangle of Problem 42 up to the plane z = 1 47 (a) The triple cross product (A x B) x C is in the plane of
gives corners (a,, bl , I), (a,, b, , I), (a,, b3, 1). The area of the A and B, because it is perpendicular to the cross product
triangle times 3 is the volume of the upside-down pyramid
from (0, 0,0) to these corners. This pyramid volume is 4 the +
(b) Compute (A x B) x C when A = a,i + a2j a3k, B =
box volume, so 3 (area of triangle) = 4 (volume of box): bli+ b2j+ b3k, C = i .
a1 bl 1 (c) Compute (A C)B - (B C)A when C = i. The answers
1 in (b) and (c) should agree. This is also true if C = j or C =
area o triangle = - a, b2 1
f
2
. k or C = c , i + c2j + C, k. That proves the tricky formula
a3 b3 1
Find the area A in Problem 43 from this determinant.
48 Take the dot product of equation (*) with D to prove
45 (1) The projections of A = a,i + a2j+ a,k and B =
bli + b2j + b3k onto the xy plane are
(2) The parallelogram with sides A and B projects to a
parallelogram with area . 49 The plane containing P = (0, 1, 1) and Q = (1, 0, 1) and
(3) General fact: The projection onto the plane normal to R = (1, 1,O) is perpendicular to the cross product N =
the unit vector n has area (A x B) n. Verify for n = k. . Find the equation of the plane and the area of
triangle PQR.
46 ( a ) F o r A = i + j - 4 k a n d B = -i +j,compute(AxB)*i
and (A x B) j and (A x B) k. By Problem 45 those are 50 Let P =(I, 0, -I), Q = (1, 1, I), R = (2, 2, 1). Choose S so
the areas of projections onto the yz and xz and xy planes. that PQRS is a parallelogram and compute its area. Choose
A
(b) Square and add those areas to find I x BI2. This is T, U , V so that OPQRSTUV is a box (parallelepiped) and
the Pythagoras formula in space (Remark 2). compute its volume.
11.4 Matrices and Linear Equations
We are moving from geometry to algebra. Eventually we get back to calculus, where
functions are nonlinear-but linear equations come first. In Chapter 1, y = mx + b
produced a line. Two equations produce two lines. If they cross, the intersection point
solves both equations-and we want to find it.
Three equations in three variables x, y, z produce three planes. Again they go
through one point (usually). Again the problem is to find that intersection point
-which solves the three equations.
The ultimate problem is to solve n equations in n unknowns. There are n hyper-
planes in n-dimensional space, which meet at the solution. We need a test to be sure
they meet. We also want the solution. These are the objectives of linear algebra, which
joins with calculus at the center of pure and applied mathematics.?
Like every subject, linear algebra requires a good notation. To state the equations
and solve them, we introduce a "matrix." The problem will be Au = d. The solution
will be u = A-'d. It remains to understand where the equations come from, where
the answer comes from, and what the matrices A and A - stand for. '
W W
T O EQUATIONS IN T O UNKNOWNS
Linear algebra has no reason to choose one variable as special. The equation y - yo =
m(x - xo) separates y from x. A better equation for a line is ax + by = d. (A vertical
?Linear algebra dominates some applications while calculus governs others. Both are essential.
A fuller treatment is presented in the author's book Linear Algebra and Its Applications
(Harcourt Brace Jovanovich, 3rd edition 1988), and in many other texts.
11 Vectors and Matrices
line like x = 5 appears when b = 0. The first form did not allow slope m = oo .) This
section studies two lines:
By solving both equations at once, we are asking (x, y) to lie on both lines. The
practical question is: Where do the lines cross? The mathematician's question is: Does
a solution exist and is it unique?
To understand everything is not possible. There are parts of life where you never
know what is going on (until too late). But two equations in two unknowns can have
no mysteries. There are three ways to write the system-by rows, by columns, and
by matrices. Please look at all three, since setting up a problem is generally harder
and more important than solving it. After that comes the concession to the real world:
we compute x and y.
EXAMPLE 1 How do you invest $5000 to earn $400 a year interest, if a money market
account pays 5% and a deposit account pays lo%?
Set up equations by rows: With x dollars at 5% the interest is .05x. With y dollars at
10% the interest is .10y. One row for principal, another row for interest:
Same equations by columns: The left side of (2) contains x times one vector plus y
times another vector. The right side is a third vector. The equation by columns is
Same equations by matrices: Look again at the left side. There are two unknowns x
and y, which go into a vector u. They are multiplied by the four numbers 1, .05, 1,
and .lo, which go into a two by two matrix A. The left side becomes a matrix times
a vector:
NOWyou see where the "rows" and "columns" came from. They are the rows and
columns of a matrix. The rows entered the separate equations (2). The columns
entered the vector equation (3). The matrix-vector multiplication Au is defined so
that all these equations are the same:
Au by rows: [: : I[:] [ =
a1x + b1y
a2x + b2y
] (each is
a dot product)
Au by columns: [I [;I
: = x [ ~ l +] y[:l] (combination of
column vectors)
A is the coeficient matrix. The unknown vector is u. The known vector on the right
side, with components 5000 and 400, is d. The matrix equation is Au = d.
1I.4 Matrices and Linear Equations
Fig. 11.16 Each row of Au = d gives a line. Each column gives a vector.
This notation Au = d continues to apply when there are more equations and more
unknowns. The matrix A has a row for each equation (usually m rows). It has a column
for each unknown (usually n columns). For 2 equations in 3 unknowns it is a 2 by 3
matrix (therefore rectangular). For 6 equations in 6 unknowns the matrix is 6 by 6
(therefore square). The best way to get familiar with matrices is to work with them.
Note also the pronunciation: "matrisees" and never "matrixes."
Answer to the practical question The solution is x = 2000, y = 3000. That is the
intersection point in the row picture (Figure 11.16). It is also the correct combination
in the column picture. The matrix equation checks both at once, because matrices
are multiplied by rows or by columns. The product either way is d:
Singular case In the row picture, the lines cross at the solution. But there is a case
that gives trouble. When the lines areparallel, they never cross and there is no solution.
When the lines are the same, there is an infinity of solutions:
2x+y=O 2x+ y = o
parallel lines same line (5)
2x+y=1 4x+2y=O
This trouble also appears in the column picture. The columns are vectors a and b.
The equation Au = d is the same as xa + yb = d. We are asked to find the combination
of a and b (with coefficients x and y) that produces d. In the singular case a and b lie
along the same line (Figure 11.17). No combination can produce d, unless it happens
to lie on this line.
parallel lines cross at solution
lines * s = 1, y = 1
xa+yb
,y=I misses d
Fig. 11.17 Row and column pictures: singular (no solution) and nonsingular ( x = y = 1).
428 11 Vectors and Matrices
The investment problem is nonsingular,and 2000 a + 3000 b equals d. We also drew
Example 2: The matrix A multiplies u = (1, 1) to solve x + 2y = 3 and x - y = 0:
Au=[ 2] -11 1+2[].
1 1 1-1 0 2]=
By columns 1 + -1 0 [
The crossing point is (1, 1) in the row picture. The solution is x = 1, y = 1 in the
column picture (Figure 11.17b). Then 1 times a plus 1 times b equals the right side d.
SOLUTION BY DETERMINANTS
Up to now we just wrote down the answer. The real problem is to find x and y when
they are unknown. We solve two equations with letters not numbers:
a x + b y = d,
a2 x + b2y= d2.
The key is to eliminate x. Multiply the first equation by a2 and the second equation
by a,. Subtract the first from the second and the x's disappear:
(ab 2 - a 2 bl)y = (ad 2 - a2 d,).
To eliminate y, subtract b, times the second equation from b2 times the first:
(b2 al - b a2 )x = (b2 d, - bid 2 ).
What you see in those parentheses are 2 by 2 determinants! Remember from
Section 11.3:
Fa1 b1 a1 b1
The determinant of is the number alb 2 - a2 bl.
a2 b2 a2 b2
This number appears on the left side of (6) and (7). The right side of (7) is also a
determinant-but it has d's in place of a's. The right side of (6) has d's in place of
b's. So x and y are ratios of determinants, given by Cramer's Rule:
dl bl al di
d2 b2 a2 d2
11 H Cramer's Rule The solution is x -,
a2 bi a2 b2
a2 b2 a2 b2
The investment example is solved by three determinants from the three columns:
1 1 .05 5000 1 1 5000
= 100 = 150.
.05 .10 400 .10 .05 400
Cramer's Rule has x = 100/.05 = 2000 and y = 150/.05 = 3000. This is the solution.
The singular case is when the determinant of A is zero-and we can't divide by it.
111 Cramer's Rule breaks down when det A = 0-which is the singular case.
Then the lines in the row picture are parallel, and one column is a multiple of
the other column.
11.4 Matrices and Linear Equations
EXAMPLE 3 The lines 2x + y = 0, 2x + y = 1 are parallel. The determinant is zero:
:
[
The lines in Figure 11.17a don't meet. Notice the columns: I is a multiple of [:I.
One final comment on 2 by 2 systems. They are small enough so that all solution
methods apply. Cramer's Rule uses determinants. Larger systems use elimination
(3 by 3 matrices are on the borderline). A third solution (the same solution!) comes
from the inverse matrix A - ' , to be described next. But the inverse is more a symbol
for the answer than a new way of computing it, because to find A-' we still use
determinants or elimination.
THE INVERSE OF A MATRIX
The symbol A-' is pronounced "A inverse." It stands for a matrix-the one that
solves Au = d . I think of A as a matrix that takes u to d. Then A-' is a matrix that
takes d back to u. If Au = d then u = A - 'd (provided the inverse exists). This is exactly
like functions and inverse functions: g(x) = y and x = gP'(y). Our goal is to find A - '
when we know A.
The first approach will be very direct. Cramer's Rule gave formulas for x and y,
the components of u. From that rule we can read off A - ' , assuming that D =
a , b2 - a2bl is not zero. D is det A and we divide by it:
The matrix on the right (including 1/D in all four entries) is A-'. Notice the sign
pattern and the subscript pattern. The inverse exists if D is not zero-this is impor-
tant. Then the solution comes from a matrix-vector multiplication, A-' times d. We
repeat the rules for that multiplication:
DEFINITION A matrix M times a vector v equals a vector of dot products:
row 2
Equatioin (8) follows this rule with M = A-' and v = d. Look at Example 1:
There st,ands the inverse matrix. It multiplies d to give the solution u:
The formulas work perfectly, but you have to see a direct way to reach A - Id. Multiply
both sides of Au = d by A - ' . The multiplication "cancels" A on the left side, and
leaves u = A-'d. This approach comes next.
430 11 Vectors and Matrices
MATRIX MULTIPLICATION
To understand the power of matrices, we must multiply them. The product of A- 1
with Au is a matrix times a vector. But that multiplication can be done another way.
First A-' multiplies A, a matrix times a matrix. The product A -'A is another matrix
(a very special matrix). Then this new matrix multiplies u.
The matrix-matrix rule comes directly from the matrix-vector rule. Effectively, a
vector v is a matrix V with only one column. When there are more columns, M times
V splits into separate matrix-vector multiplications, side by side:
DEFINITION A matrix M times a matrix V equals a matrix of dot products:
[row F (row 1)'v, (row 1)*v 2]
MV= I v1V V ]
2 (10)
Lrow 2L (row 2) v, (row 2) v2 1
S2][5 6 [1-5+2*7 1-6+2-8 [19 22.
EXAMPLE 4
3 47 8 3.5+4.7 3.6+4.8 43 50
EXAMPLE 5 Multiplying A` times Aproduces the "identity matrix" 1
0
alb2- a2bl
A-'A =
Sb2
A --2
a
-bl
a, a, b, 0 -a2bi +abl1
2
D (11)
D a2 b2 L 1
This identity matrix is denoted by I. It has l's on the diagonal and O's off the diagonal.
It acts like the number 1. Every vector satisfies Iu = u.
(Inverse matrix and identity
:iJ atrix) AA-' =I and A-A =I and Iu= u:
plA-c=
Nt acd
A= t f , , d W [e c
t. a [,
0 1 y (12)
Note the placement of a, b, c, d. With these letters D is ad - bc.
The next section moves to three equations. The algebra gets more complicated (and
4 by 4 is worse). It is not easy to write out A-'. So we stay longer with the 2 by 2
formulas, where each step can be checked. Multiplying Au = d by the inverse matrix
gives A - 1Au = A - 'd-and the left side is Iu = u.
os 01 [cos 0
sin 01
in 01
v= 0l
Fig. 11.18 Rotate v forward into Av. Rotate d backward into A-'d.
11.4 Matrices and Linear Equations
cos 8 -sin 8
[.in 8 cos e ] rotates every v to Av, through the angle 8.
Question 1 Where is the vector v =
Question 2 What is A- '?
1:l
L J
rotated to?
Question 3 Which vector u is rotated into d =
cos 8 -sin 8
[:I ?
Solution 1 v rotates into Av =
Solution .3 If Au = d then u = A - ld =
[ cos 0 sin 81
-sin 8 cos 8
[;I = [['in
cos 8
81
Historical note I was amazed to learn that it was Leibniz (again!) who proposed the
notation we use for matrices. The entry in row i and column j is aij. The identity
matrix has a l l = a,, = 1 and a,, = a,, = 0. This is in a linear algebra book by Charles
Dodgson-better known to the world as Lewis Carroll, the author of Alice in
Wonderland. I regret to say that he preferred his own notation iu
instead of aij.
"I have turned the symbol toward the left, to avoid all chance of confusion with " 5.
It drove his typesetter mad.
PROJECTION ONTO A P A E = L A T S U R S FllllNG BY A LINE
LN ES Q A E
We close with a genuine application. It starts with three-dimensional vectors a, b, d
and leads to a 2 by 2 system. One good feature: a, b, d can be n-dimensional with no
change in the algebra. In practice that happens. Second good feature: There is a
calculus problem in the background. The example is tofit points by a straight line.
There are three ways to state the problem, and they look different:
+
1. Solve xa yb = d as well as possible (three equations, two unknowns x and y).
2. Project the vector d onto the plane of the vectors a and b.
3. Finld the closest straight line ("least squares") to three given points.
Figure 11.19 shows a three-dimensional vector d above the plane of a and b. Its
projection onto the plane is p = xa + yb. The numbers x and y are unknown, and
our goal is to find them. The calculation will use the dot product, which is always
the key to right angles.
The diflerence d - p is the "error." There has to be an error, because no combination
of a and b can produce d exactly. (Otherwise d is in the plane.) The projection p is
the closest point to d, and it is governed by one fundamental law: The error is
perpendicular to the plane. That makes the error perpendicular to both vectors a
and b:
a*(xa+yb-d)=O and b-(xa+yb-d)=O. (13)
11 Vectors and Matrices
Rewrite those as two equations for the two unknown numbers x and y:
(a a)x + (a b)y = a d
(14)
(b a)x + (b b)y = b d.
These are the famous normal equations in statistics, to compute x and y and p.
EXAMPLE 7 For a = (1, 1, 1) and b = (l,2, 3) and d = (0, 5,4), solve equation (14):
3x+ 6y= 9 x = -1
gives SO p = -a + 2b = (1, 3,5) = projection.
6x + 14y = 22 y= 2
Notice the three equations that we are not solving (we can't): xa + yb = d is
x + y=O
x + 2y = 5 with the 3 by 2 matrix A =
x+3y=4
For d = (0, 5,4) there is no solution; d is not in the plane of a and b. For p = (1,3, 5)
there is a solution, x = - 1 and y = 2. The vector p is in the plane. The error d - p
is (- 1,2, - 1). This error is perpendicular to the columns (1, 1, 1) and (l,2, 3), so it is
perpendicular to their plane.
SAME EXAMPLE (written as a line-fitting problem) Fit the points (1,O) and (2, 5) and
(3,4) as closely as possible ("least squares") by a straight line.
+
Two points determine a line. The example asks the linef = x yt to go through three
points. That gives the three equations in (IS), which can't be solved with two un-
knowns. We have to settle for the closest line, drawn in Figure 11.19b. This line is
computed again below, by calculus.
Notice that the closest line has heights 1, 3, 5 where the data points have heights
0,5,4. Those are the numbers in p and d! The heights 1 , 3 , 5 fit onto a line; the heights
0, 5 , 4 do not. In the first figure, p = (1, 3, 5) is in the plane and d = (0, 5,4) is not.
Vectors in the plane lead to heights that lie on a line.
Notice another coincidence. The coefficients x = - 1 and y = 2 give the projection
+
- a 2b. They also give the closest line f = - 1 + 2t. All numbers appear in both
figures.
kl; closest line f = -1
: 2
Fig. 11.19 Projection onto plane is (1, 3, 5) with coefficients -1, 2. Closest line has heights
1, 3, 5 with coefficients -1, 2. Error in both pictures is -1, 2, -1.
11.4 Matrices and Linear Equations 433
Remark Finding the closest line is a calculus problem: Minimize a sum of squares.
The numbers x and y that minimize E give the least squares solution:
E(x, y) = (x + y - 0)2 + (x + 2y - 5)2 + (x + 3y - 4)2 . (16)
Those are the three errors in equation (15), squared and added. They are also the
three errors in the straight line fit, between the line and the data points. The projection
minimizes the error (by geometry), the normal equations (14) minimize the error (by
algebra), and now calculus minimizes the error by setting the derivatives of E to zero.
The new feature is this: E depends on two variables x and y. Therefore E has two
derivatives. They both have to be zero at the minimum. That gives two equations for
x and y:
x derivative of E is zero: 2(x + y) + 2(x + 2y - 5) + 2(x + 3y - 4) =0
y derivative of E is zero: 2(x + y) + 2(x + 2y - 5)(2) + 2(x + 3y - 4)(3) = 0.
When we divide by 2, those are the normal equations 3x + 6y = 9 and 6x + 14y =
22. The minimizing x and y from calculus are the same numbers -1 and 2.
The x derivative treats y as a constant. The y derivative treats x as a constant.
These are partialderivatives. This calculus approach to least squares is in Chapter 13,
as an important application of partial derivatives.
We now summarize the least squares problem-to find the closest line to n data
points. In practice n may be 1000 instead of 3. The points have horizontal coordinates
bl, b2 , ... , b,. The vertical coordinates are dl, d 2 , ... , d.. These vectors b and d,
together with a = (1, 1, ... , 1), determine a projection-the combination p = xa + yb
that is closest to d. This problem is the same in n dimensions-the error d - p is
perpendicular to a and b. That is still tested by dot products, p a = d a and p b =
d - b, which give the normal equations for x and y:
(a . a)x + (a . b)y = a . d (n) x + (Xbi)y =
=di
(17)
(b . a)x + (b " b)y = b . d (Eb,)x + (Ibý)y = Ebid,.
44K The least squares problem projects d onto the plane of a and b. The
projection is p = xa + yb, i n dimensions. The closest line is f = x + t, in two
dimensions. The normal equations (17) give the best x and y.
11.4 EXERCISES
Read-through questions A matrix-vector multiplication produces a vector of dot
m from the rows, and also a combination of the n
The equations 3x + y = 8 and x + y = 6 combine into the vec-
tor equation x a + y b = = d. The left side is
Au, with coefficient matrix A = d and unknown vector ] [uL,
BL L
[a
[ b]L ,1 I
1-
u= e . The determinant of A is f , so this problem
is not g . The row picture shows two intersecting h If the entries are a, b, c, d, the determinant is D = o . A-
The column picture shows xa + yb = d, where a = I and is [ p ] divided by D. Cramer's Rule shows components
b = i . The inverse matrix is A- = k . The solution of u = A- 'd as ratios of determinants: x = q /D and y =
is u= A-'d= I r /D.
434 11 Vectors and Matrices
A matrix-matrix multiplication MV yields a matrix of dot 1.8 Try Cramer's Rule when there is no solution or infinitely
products, from the rows of s and the columns of t : many:
3x+ y = o 3x+ y = l
or
6x+2y=2 6x+2y=2.
19 Au = d is singular when the columns of A are .
A solution exists if the right side d is . In this solvable
case the number of solutions is
20 The equations x - y = dl and 9x - 9y = d2 can be solved
The last line contains the u matrix, denoted by I. It has if
the property that IA = AI = v for every matrix A, and
Iu = w for every vector u. The inverse matrix satisfies 21 Suppose x = $ billion people live in the U.S. and y = 5
A- 'A = x . Then Au = d is solved by multiplying both billion live outside. If 4 per cent of those inside move out and
sides by v , to give u = z . There is no inverse matrix 2 per cent of those outside move in, find the populations dl
when A . inside and d2 outside after the move. Express this as a matrix
multiplication Au = d (and find the matrix).
The combination xa + yb is the projection of d when the
error B is perpendicular to C and D . If a = 22 In Problem 21 what is special about a l + a2 and bl + b2
(1, 1, I), b = (1, 2, 3), and d = (0, 8, 4), the equations for x and (the sums down the columns of A)? Explain why dl d2 equ- +
y are E . Solving them also gives the closest F to the als x + y.
data points (1, O), G , and (3,4). The solution is x = 0, y = 23 With the same percentages moving, suppose dl = 0.58 bil-
2, which means the best line is H . The projection is lion are inside and d2 = 4.92 billion are outside at the end. Set
O + 2b= I . The three error components are J .
a up and solve two equations for the original populations x
Check perpendicularity: K = 0 and = 0..Applying
and y.
calculus to this problem, x and y minimize the sum of squares
E= M . 24 What is the determinant of A in Problems 21-23? What
is A- '? Check that A- 'A = I.
In 1-8 find the point (x, y) where the two lines intersect (if they
do). Also show how the right side is a combination of the 25 The equations ax + y = 0, x + ay = 0 have the solution
columns on the left side (if it is). Also find the determinant D. x = y = 0. For which two values of a are there other solutions
(and what are the other solutions)?
x+y=7 2 2x+y=11
x-y=3 x+y=6 26 The equations ax + by = 0, cx + dy = 0 have the solution
x = y = 0. There are other solutions if the two lines are
3x- y = 8 4 x+2y=3 . This happens if a, b, c, d satisfy .
x-3y=O 2x+4y=7
27 Find the determinant and inverse of A = [i 1
2 . Do the
2x-4y=O 6 lOx+y=l same for 2A, A-', -A, and I.
x-2y=o x+y=l
28 Show that the determinant of A-' is l/det A:
ax + by = 0 8 ax+by=l
2ax + 2by = 2 cx + dy = 1
Solve Problem 3 by Cramer's Rule.
Try to solve Problem 4 by Cramer's Rule.
A-' =
d/(ad - bc)
- c/(ad - bc)
- b/(ad - bc)
a/(ad - bc)
29 Compute AB and BA and also BC and CB:
I
What are the ratios for Cramer's Rule in Problem 5?
If A = I show how Cramer's Rule solves Au = d.
Draw the row picture and column picture for Problem 1.
Verify the associative law: AB times C equals A times BC.
Draw the row and column pictures for Problem 6.
30 (a) Find the determinants of A, B, AB, and BA above.
Find A- ' in Problem 1. .
(b) Propose a law for the determinant of BC and test it.
Find A-' in Problem 8 if ad - bc = 1.
A 2 by 2 system is singular when the two lines in the row
picture . This system is still solvable if one equation
31 For A = [: :] and B =
g h
[ ]e f
write out AB and
is a of the other equation. In that case the two lines factor its determinant into (ad - bc)(eh -fg). Therefore
are and the number of solutions is det(AB) = (det A)(det B).
11.5 Linear Algebra 435
+ +
32 Usually det (A B) does not equal det A det B. Find 39 Plot the three data points (-1, 2), (0, 6), (1,4) in a plane.
examples of inequality and equality. +
Draw the straight line x yt with the same x and y as in
Problem 38. Locate the three errors up or down from the data
'
33 Find the inverses, and check A- 'A = I and BB- = I, for points and compare with Problem 38.
A=[' 0 2 '
1 and B=[O
2 2 '
'1. ' 40 Solve equation (14) to find the combination xa + yb of
a = (1, 1, 1) and b = (-1, 1, 2) that is closest to d = (1, 1, 3).
Draw the corresponding straight line for the data points
34 In Problem 33 compute AB and the inverse of AB. Check (-1, I), (1, I), and (2, 3). What is the vector of three errors and
'
that this inverse equals B- times A- '. what is it perpendicular to?
35 The matrix product ABB- 'A- ' equals the mat- 41 Under what condition on dl, d,, d3 do the three points
(0, dl), (1, d,), (2, d3)lie on a line?
rix. Therefore the inverse of AB is . Important: The
associative law in Problem 29 allows you to multiply BB-' 42 Find the matrices that reverse x and y and project:
first.
'
36 The matrix multiplication C - B - 'A- 'ABC yields the
matrix. Therefore the inverse of ABC is
37 The equations x + 2y + 32 and 4x + 5y + cz = 0 always
have a nonzero solution. The vector u = (x, y, z) is required
43 Multiplying by P = [:: ::] projects u onto the 45' line.
to be to v = (1, 2, 3) and w = (4, 5, c). So choose u =
(a) Find the projection Pu of u = [;I.
(b) Why does P times P equal P?
38 Find the combination p = xa + yb of the vectors a =
(c) Does P - ' exist? What vectors give Pu = O?
(1, 1, 1) and b = (-1, 0, 1) that comes closest to d = (2, 6,4).
(a) Solve the normal equations (14) for x and y. (b) Check that 44 Suppose u is not the zero vector but Au = 0. Then A - '
the error d - p is perpendicular to a and b. can't exist: It would multiply and produce u.
11.5 Linear Algebra
This section moves from two to three dimensions. There are three unknowns x, y, z
and also three equations. This is a t the crossover point between formulas and
algorithms-it is real linear algebra. The formulas give a direct solution using det-
erminants. The algorithms use elimination and the numbers x, y, z appear at the
end. In practice that end result comes quickly. Computers solve linear equations by
elimination.
The situation for a nonlinear equation is similar. Quadratic equations
+ +
ax2 bx c = 0 are solved by a formula. Cubic equations are solved by Newton's
method (even though a formula exists). For equations involving x or x lo, algorithms
take over completely.
Since we are at the crossover point, we look both ways. This section has a lot to
do, in mixing geometry, determinants, and 3 by 3 matrices:
1. The row picture: three planes intersect at the solution
2. The column picture: a vector equation combines the columns
3. The formulas: determinants and Cramer's Rule
4. Matrix multiplication and A - '
5. The algorithm: Gaussian elimination.
Part of our goal is three-dimensional calculus. Another part is n-dimensional algebra.
And a third possibility is that you may not take mathematics next year. If that
436 11 Vectors and Matrices
happens, I hope you will use mathematics. Linear equations are so basic and impor-
tant, in such a variety of applications, that the effort in this section is worth making.
An example is needed. It is convenient and realistic if the matrix contains zeros.
Most equations in practice are fairly simple-a thousand equations each with 990
zeros would be very reasonable. Here are three equations in three unknowns:
x+ y = 1
x + 2z = 0 (1)
- 2y + 2z = -4.
In matrix-vector form, the unknown u has components x, y, z. The right sides 1, 0, - 4
go into d. The nine coefficients, including three zeros, enter the matrix A:
1 1 0 x 1
1 0 2 = 0 or Au=d. (2)
0 -2 2 z -4
The goal is to understand that system geometrically, and then solve it.
THE ROW PICTURE: INTERSECTING PLANES
Start with the first equation x + y = 1. In the xy plane that produces a line. In three
dimensions it is a plane. It has the usual form ax + by + cz = d, except that c happens
to be zero. The plane is easy to visualize (Figure 11.20a), because it cuts straight down
through the line. The equation x + y = 1 allows z to have any value, so the graph
includes all points above and below the line.
The second equation x + 2z = 0 gives a second plane, which goes through the
origin. When the right side is zero, the point (0, 0, 0) satisfies the equation. This time y
is absent from the equation, so the plane contains the whole y axis. All points (0, y, 0)
meet the requirement x + 2z = 0. The normal vector to the plane is N = i + 2k. The
plane cuts across, rather than down, in 11.20b.
Before the third equation we combine the first two. The intersection of two planes
is a line. In three-dimensional space, two equations (not one) describe a line. The
points on the line have to satisfy x + y = 1 and also x + 2z = 0. A convenient point
is P = (0, 1, 0). Another point is Q = (-1, 2, -). The line through P and Q extends out
in both directions.
The solution is on that line. The third plane decides where.
z
x = -2,
+y= 1 intersect ution
line of fi
two plar
P
x
Fig. 11.20
X-- x
First plane, second plane, intersection line meets third plane at solution.
.=-4
11.5 Linear Algebra
The third equation - 2y + 22 = - 4 gives the third plane-which misses the origin
because the right side is not zero. What is important is the point where the three
planes meet. The intersection line of the first two planes crosses the third plane.
We used determinants (but elimination is better) to find x = - 2, y = 3, z = 1. This
solution satisfies the three equations and lies on the three planes.
A brief comment on 4 by 4 systems. The first equation might be x + y + z - t = 0.
It represents a three-dimensional "hyperplane" in four-dimensional space. (In physics
this is space-time.) The second equation gives a second hyperplane, and its intersection
with the first one is two-dimensional. The third equation (third hyperplane) reduces
the intersection to a line. The fourth hyperplane meets that line at a point, which is
the solution. It satisfies the four equations and lies on the four hyperplanes. In this
course three dimensions are enough.
F
COLUMN PICTURE: COMBINATION O COLUMN VECTORS
There is an extremely important way to rewrite our three equations. In (1) they were
separate, in (2) they went into a matrix. Now they become a vector equation:
The columns of the matrix are multiplied by x, y, z. That is a special way to see matrix-
vector multiplication: Au is a combination of the columns of A. We are looking for
the numbers x, y, z so that the combination produces the right side d.
The column vectors a, b, c are shown in Figure 11.21a. The vector equation is
xa + yb + zc = d. The combination that solves this equation must again be x = - 2,
y = 3, z = 1. That agrees with the intersection point of the three planes in the row
picture.
1
ney c,',
{ a, b, c in
same plane
d not in that plane:
no solution
O = lc+2b-2a
Fig. 11.21 Columns combine to give d. Columns combine to give zero (singular case).
H H NES
T E DETERMINANT AND T E I V R E MATRIX
For a 3 by 3 determinant, the section on cross products gave two formulas. One was
the triple product a (b x c). The other wrote out the six terms:
det A = a (b x c) = al(b2c3- b3c2)+ a2(b3c,- blc3) + a3(b,cz - b2cl).
II Vectors and Matrices
Geometrically this is the volume of a box. The columns a, b, c are the edges going out
from the origin. In our example the determinant and volume are 2:
A slight dishonesty is present in that calculation, and will be admitted now. In
Section 11.3 the vectors A, B, C were rows. In this section a, b, c are columns. It doesn't
matter, because the determinant is the same either way. Any matrix can be
"transposedw-exchanging rows for columns-without altering the determinant. The
six terms (alb2c3is the first) may come in a different order, but they are the same six
terms. Here four of those terms are zero, because of the zeros in the matrix. The sum
of all six terms is D = det A = 2.
Since D is not zero, the equations can be solved. The three planes meet at a point.
The column vectors a, b, c produce a genuine box, and are not flattened into the same
plane (with zero volume). The solution involves dividing by D-which is only possible
if D = det A is not zero.
I 14L When the determinant D is not zero, A bas an inverse: AA-' = A-'A =
I. Then the equations Au = d have one and only one solution u = A - 'd. I
The 3 by 3 identity matrix I is at the end of equation (5). Always Iu = u.
We now compute A-', first with letters and then with numbers. The neatest
formula uses cross products of the columns of A-it is special for 3 by 3 matrices.
rbxc i
Every entry is divided by D: The inverse matrix is A- - ' -1
D
a I. (4)
To test this formula, multiply by A. Matrix multiplication produces a matrix of dot
products-from the rows of the first matrix and the columns of the second, A- 'A = I:
a m ( b x c )be(bxc) ca(bxc) 1 0 0
ae(cxa) be(cxa) cg(cxa)
D
axb b) b)
a m ( a x b 0 ( a xb) c m ( a x
On the right side, six of the triple products are zero. They are the off-diagonals like
b (b x c), which contain the same vector twice. Since b x c is perpendicular to b, this
triple product is zero. The same is true of the others, like a (a x b) = 0. That is the
volume of a box with two identical sides. The six off-diagonal zeros are the volumes
of completely flattened boxes.
On the main diagonal the triple products equal D. The order of vectors can be abc
or bca or cab, and the volume of the box stays the same. Dividing by this number D,
which is placed outside for that purpose, gives the 1's in the identity matrix I.
Now we change to numbers. The goal is to find A-' and to test it.
11.5 Llnear Algebra
That comes from the formula, and it absolutely has to be checked. Do not fail to
multiply A-' times A (or A times A- '). Matrix multiplication is much easier than
the formula for A-'. We highlight row 3 times column 1, with dot product zero:
Remark on A- ' Inverting a matrix requires D # 0 We divide by D = det A. The
.
cross products b x c and c x a and a x b give A-' in a neat form, but errors are
easy. We prefer to avoid writing i, j, k. There are nine 2 by 2 determinants to be
calculated, and here is A-' in full-containing the nine "cofcretors~'divided by D:
Important: The first row of A-' does not use the first column of A, except in 1/D.
In other words, b x c does not involve a. Here are the 2 by 2 determinants that
produce 4, - 2, 2-which is divided by D = 2 in the top row of A-':
The second highlighted determinant looks like + 2 not -2. But the sign matrix on
the right assigns a minus to that position in A-'. We reverse the sign of blc3 - b3cl,
to find the cofactor b3c1 - blc3 in the top row of (6).
f f
To repeat: For a row o A-I, cross out the corresponding column o A. Find the three
2 by 2 determinants, use the sign matrix, and divide by D.
The multiplication BB-I = I checks the arithmetic. Notice how : in B leads to a
zero in the top row of B-'. To find row 1, column 3 of B-' we ignore column 1 and
row 3 of B. (Also: the inverse of a triangular matrix is triangular.) The minus signs
come from the sign matrix.
THE SOLUTION u = A- 'd
The purpose of A-' is to solve the equation Au = d. Multiplying by A-' produces
Iu = A-'d. The matrix becomes the identity, Iu equals u, and the solution is
immediate:
11 Vectors and Matrices
By writing those components x, y, z as ratios ofdeterminants, we have Cramer's Rule:
The solution is x = - b cl y=- la d cl z=- la b 4
Id
- (10)
la b cl' }a b cl' la b el'
The right side d replaces, in turn, columns a and b and c. All denominators are D =
a (b x c). The numerator of x is the determinant d (b x c) in (9). The second numera-
tor agrees with the second component d (c x a), because the cyclic order is correct.
The third determinant with columns abd equals the triple product d (a x b) in A - 'u.
Thus (10) is the same as (9).
EXAMPLE A: Multiply by A-' to find the known solution x = - 2, y = 3, z = 1:
EXAMPLE B: Multiply by B-' to solve Bu = d when d is the column (6, 5, 4):
"= B-'d=
EXAMPLE C: Put d = (6, 5,4) in each column of B. Cramer's Rule gives u = (1, 1,4):
This rule fills the page with determinants. Those are good ones to check by eye,
without writing down the six terms (three + and three -).
The formulas for A-' are honored chiefly in their absence. They are not used by
the computer, even though the algebra is in some ways beautiful. In big calculations,
the computer never finds A - '-just the solution.
We now look at the singular case D = 0. Geometry-algebra-algorithm must all
break down. After that is the algorithm: Gaussian elimination.
H
T E SINGULAR CASE
Changing one entry of a matrix can make the determinant zero. The triple product
a *(bx c), which is also the volume, becomes D = 0. The box is flattened and the
matrix is singular. That happens in our example when the lower right entry is changed
from 2 to 4:
S= 11 0 2
I has determinant D = 0.
11.5 Linear Algebra
This does more than change the inverse. It destroys the inverse. We can no longer
divide by D. There is no S - '.
What happens to the row picture and column picture? For 2 by 2 systems, the
singular case had two parallel lines. Now the row picture has three planes, which
need not be parallel. Here the planes are not parallel. Their normal vectors are the
rows of S, which go in different directions. But somehow the planes fail to go through
a common point.
What happens is more subtle. The intersection line from two planes misses the
third plane. The line is parallel to the plane and stays above it (Figure 11.22a). When
all three planes are drawn, they form an open tunnel. The picture tells more than the
numbers, about how three planes can fail to meet. The third figure shows an end
view, where the planes go directly into the page. Each pair meets in a line, but those
lines don't meet in a point.
Fig. 11.22 The row picture in the singular case: no intersection point, no solution.
When two planes are parallel, the determinant is again zero. One row of the matrix
is a multiple of another row. The extreme case has all three planes parallel-as in a
matrix with nine 1's.
The column picture must also break down. In the 2 by 2 failure (previous section),
the columns were on the same line. Now the three columns are in the same plane. The
combinations of those columns produce d only if it happens to lie in that particular
plane. Most vectors d will be outside the plane, so most singular systems have no
solution.
When the determinant is zero, Au = d has no solution or infinitely many.
H
T E ELIMINATION ALGORITHM
Go back to the 3 by 3 example Au = d. If you were given those equations, you would
never think of determinants. You would-quite correctly-start with the first equa-
tion. It gives x = 1 - y, which goes into the next equation to eliminate x:
Stop there for a minute. On the right is a 2 by 2 system for y and z . The first equation
and first unknown are eliminated-exactly what we want. But that step was not
organized in the best way, because a "1" ended up on the left side. Constants should
stay on the right side-the pattern should be preserved. It is better to take the same
11 Vectors and Matrices
step by subtracting the fist equation from the second:
Same equations, better organization. Now look at the corner term -y. Its coefficient
-1 is the secondpivot. (The first pivot was + 1, the coefficient of x in the first corner.)
We are ready for the next elimination step:
Plan: Subtract a multiple of the "pivot equation" from the equation below it.
Goal: To produce a zero below the pivot, so y is eliminated.
Method: Subtract 2 times the pivot equation to cancel - 2y.
The answer comes by back substitution. Equation (12) gives z = 1. Then equation (11)
gives y = 3. Then the first equation gives x = - 2. This is much quicker than determi-
nants. You may ask: Why use Cramer's Rule? Good question.
With numbers elimination is better. It is faster and also safer. (To check against
error, substitute -2, 3, 1 into the original equations.) The algorithm reaches the
answer without the determinant and without the inverse. Calculations with letters use
det A and A - '.
Here are the steps in a definite order (top to bottom):
Subtract a multiple of equation 1 to produce Ox in equation 2
Subtract a multiple of equation 1 to produce Ox in equation 3
y
Subtract a multiple of equation 2 (new) to produce O in equation 3.
EXAMPLE (notice the zeros appearing under the pivots):
Elimination leads to a triangular system. The coefficients below the diagonal are zero.
First z = 2, then y = 1, then x = - 2. Back substitution solves triangular systems (fast).
As a final example, try the singular case Su = d when the corner entry is changed
from 2 to 4. With D = 0, there is no inverse matrix S - l . Elimination also fails, by
reaching an impossible equation 0 = - 2:
The three planes do not meet at a point-a fact that was not obvious at the start.
Algebra discovers this fact from D = 0. Elimination discovers it from 0 = -2. The
chapter is ending at the point where my linear algebra book begins.
11.5 Linear Algebra
One final comment. In actual computing, you will use a code written by profession-
als. The steps will be the same as above. A multiple of equation 1 is subtracted from
each equation below it, to eliminate the first unknown x. With one fewer unknown
and equation, elimination starts again. (A parallel computer executes many steps at
once.) Extra instructions are included to reduce roundoff error. You only see the
result! Hut it is more satisfying to know what the computer is doing.
In the end, solving linear equations is the key step in solving nonlinear equations.
The central idea of differential calculus is to linearize near a point.
11.5 EXERCISES
Read-through questions 2 The planes x + y = 0, x + y + z = 1, and y + z = 0 intersect
at u = (x, y, z).
Three equations in three unknowns can be written as Au =
d. The a u has components x, y, z and A is a b . The 3 The point u = (x, A z) is on the planes x = y, y = z,
row picture has a c for each equation. The first two x-z=l.
planes intersect in a d , and all three planes intersect in
4 A combination of a = (1, 0, 0) and b = (0, 2, 0) and c =
a e , which is f -. The column picture starts with
(0, 0, 3) equals d = (5, 2, 0).
vectors a, b, c from the columns of g and combines them
to produce h . The vector equation is i = d. 5 Show that Problem 3 has no solution in two ways: find
the determinant of A, and combine the equations to produce
The determinant of A i.s the triple product i . This is o = 1.
the volume of a box, whose edges from the origin are k .
If det A = I then tht: system is m . Otherwise there 6 Solve Problem 2 in two ways: by inspiration and Cramer's
is an n matrix such that A-'A = 0 (the P mat- Rule.
rix). In this case the solution to Au = d is u = q . 7 Solve Problem 4 in two ways: by inspection and by com-
puting the determinant and inverse of the diagonal matrix
'
The rows of A- are the cross products b x c, r ,
s , divided by D. The entries of A-' are 2 by 2 t ,
divided by D. The upper left entry equals u . The 2 by 2
determinants needed for a row of A-' do not use the corre-
sponding v of A.
The solution is u = A-'d. Its first component x is a ratio 8 Solve the three equations of Problem 1 by elimination.
of determinants, Id bcl divided by w . Cramer's Rule
breaks down when det A = x . Then the columns a, b, c 9 The vectors b and c lie in a plane which is perpendicular
lie in the same Y . There is no solution to xa + yb + zc = to the vector . In case the vector a also lies in that
d, if d is not on that 2: . In a singular row picture, the plane, it is also perpendicular and a = 0. The
intersection of planes 1 and 2 is A to the third plane. of the matrix with columns in a plane is .
In practice u is computed by B . The algorithm starts 10 The plane a, x + b , y + c1z = d l is perpendicular to its
by subtracting a multiple of row 1 to eliminate x from c . normal vector N, = . The plane a 2 x + b2y + c2z =
If the first two equations are x - y = 1 and 3x + z = 7, this d2 is perpendicular to N 2 = . The planes meet in a
elimination step leaves D . Similarly x is eliminated from line that is perpendicular to both vectors, so the line is parallel
the third equation, and then E is eliminated. The equ- to their product. If this line is also parallel to the
ations are solved by back F . When the system has no third plane and perpendicular to N,, the system is .
solution, we reach an im.possible equation like G . The The matrix has no , which happens when
example x - y = 1,3x + z = 7 has no solution if the third equ- (N1 x N 2 ) * N 3=O.
ation is H . Problems 11-24 use the matrices A, B, C.
Rewrite 1-4 as matrix equations Au = d (do not solve).
1 d = (0, 0, 8) is a combination of a = (1, 2, 0) and b = (2, 3, 2)
and c = (2, 5, 2).
444 11 Vectors and Matrices
11 Find the determinants IAl, IBI, ICI. Since A is triangular, 27 Find the determinants of these four permutation matrices:
its determinant is the product .
12 Compute the cross products of each pair of columns in B
(three cross products).
13 Compute the inverses of A and B above. Check that
A-'A = I and B-'B = I. and QP = . Multiply u = (x, y, z) by each permuta-
tion to find Pu, Qu, PQu, and QPu.
28 Find all six of the 3 by 3 permutation matrices (including
. With this right side d, why I), with a single 1 in each row and column. Which of them
are "even" (determinant 1) and which are "odd" (determinant
- I)?
is u the first column of the inverse?
29 How many 2 by 2 permutation matrices are there, includ-
15 Suppose all three columns of a matrix add to zero, as in ing I ? How many 4 by 4?
C above. The dot product of each column with v = (1, 1, 1) is
. All three columns lie in the same . The 30 Multiply any matrix A by the permutation matrix P and
determinant of C must be . explain how PA is related to A. In the opposite order explain
how AP is related to A.
16 Find a nonzero solution to Cu = 0. Find all solutions to
31 Eliminate x from the last two equations by subtracting
Cu = 0.
the first equation. Then eliminate y from the new third equa-
17 Choose any right side d that is perpendicular to v = tion by using the new second equation:
(1, 1, 1) and solve Cu = d. Then find a second solution. x + y+ z=2 x+y =1
18 Choose any right side d that is not perpendicular to v = (a) x + 3 y + 3 z = 0 (b) x + z=3
(1, 1, 1). Show by elimination (reach an impossible equation) x+3y+7z=2 y+z=5.
that Cu = d has no solution.
After elimination solve for 2, y, x (back substitution).
19 Compute the matrix product AB and then its determinant. 32 By elimination and back substitution solve
How is det AB related to det A and det B?
x+2y+2z=o x-y =1
20 Compute the matrix products BC and CB. All columns of
(a) 2 x + 3 y + 5 z = 0 (b) x -2=4
CB add to , and its determinant is .
2y + 22 = 8 y-z=7.
21 Add A and C by adding each entry of A to the corre-
33 Eliminate x from equation 2 by using equation 1:
sponding entry of C. Check whether the determinant of A + C
equals det A + det C.
22 Compute 2A by multiplying each entry of A by 2. The
determinant of 2A equals times the determinant of
A. Why can't the new second equation eliminate y from the third
23 Which four entries of A give the upper left corner entry p equation? Is there a solution or is the system singular?
of A-', after dividing by D = det A? Which four entries of A Note: If elimination creates a zero in the "pivot position,"
give the entry q in row 1, column 2 of A-'? Find p and q. try to exchange that pivot equation with an equation below
it. Elimination succeeds when there is a full set of pivots.
24 The 2 by 2 determinants from the first two rows of B are
-1 (from columns 2, 3) and -2 (from columns 1, 3) and 34 The pivots in Problem 32a are 1, -1, and 4. Circle those
(from columns 1, 2). These numbers go into the as they appear along the diagonal in elimination. Check that
third of B- ',after dividing by and chang- the product of the pivots equals the determinant. (This is how
ing the sign of . determinants are computed.)
35 Find the pivots and determinants in Problem 31.
25 Why does every inverse matrix A- ' have an inverse?
26 From the multiplication ABB- 'A- ' = I it follows that
the inverse of AB is . The separate inverses come in
order. If you put on socks and then shoes, the
r'
36 Find the inverse of A = 0 1 O l and also of B =
1
inverse begins by taking off .
11.5 Linear Algebra
37 The symbol aij stands for the entry in row i, column j. 39 Compute these determinants. The 2 by 2 matrix is invert-
Find al and a,, in Problem 36. The formula Zaiibikgives ible if . The 3 by 3 matrix (is)@ not) invertible.
the entry in which row and column of the matrix product
.=
AB?
38 Write down a 3 by 3 singular matrix S in which no two
rows are parallel. Find a combination of rows 1 and 2 that is
parallel to row 3. Find a combination of columns 1 and 2 that =- + J
is parallel to column 3. Find a nonzero solution to Su = 0. -
MIT OpenCourseWare
Resource: Calculus Online Textbook
Gilbert Strang
The following may not correspond to a particular course on MIT OpenCourseWare, but has been
provided by the author as an individual learning resource.
For information about citing these materials or our Terms of Use, visit: | 677.169 | 1 |
IMPACT Mathematics: Algebra and More for the Middle, Grades Course 3, Student Edition
IMPACT Mathematics: Algebra and More, Course 3, Student Edition
Summary
Quick Review Math Handbookhelps students refresh their memory of mathematics concepts and skills. Each Handbook consists of 3 parts:HotWords,HotTopics, andHotSolutions. .HotWords are important mathematical terms. TheHotWords section includes a glossary of terms, a collection of common or significant mathematical patterns, and lists of symbols and formulas in alphabetical order. . .HotTopics are key concepts that students need to know. Each chapter inHotTopics has several topics that give students to-the-point explanations of key mathematical concepts. Each section includes Check It Out exercises to assist students to check for understanding, and there is an exercise set at the end of each topic.. .HotSolutions (found in the back of the handbook) give students easy-to-locate answers to the Check It Out and What Do You Know? Problems.. | 677.169 | 1 |
The world's most advanced symbolic and numeric computing engine, Maple 13, comes together with a high-performance multi-domain modeling and simulation tool, MapleSim 2, for more effective and engaging virtual simulations and exercises, and dramatically more efficient models for real time simulations in research.
Maple 13
Maple 13 is a substantial release with many new features across three key areas including completely new 3-D plotting facilities, powerful new learning and problem-solving tools, and additional resources to enable users to find answers to questions quickly. New plotting facilities include extensive annotation tools and fly-through animations, making 3-D plots more meaningful and easier to interpret. New tools, such as tutors for complex variables and numerical analysis, point-and-click access to control systems design tools, and enhanced step-by-step problem solvers for calculus, help students explore, visualize, and understand mathematical concepts. Maple 13's leading-edge solvers include revolutionary techniques for finding solutions to differential equations that are beyond the scope of standard methods.
The Maple Adoption Program offers a simple and effective way to introduce Maple into the classroom. Educators receive a free home use copy of Maple, and students receive a substantial discount on the Student Edition of Maple. If you're not already teaching with Maple, this is the easiest way to get started. If you are already teaching with Maple, this program gets Maple to your students at the lowest available cost.
Clickable Math Examples
Maplesoft now offers a collection of interactive Maple applications that use Clickable Math methods to solve standard problems. A variety of Calculus applications and Engineering applications are available. These applications give detailed explanations of which tools to use and where to find them in Maple.
Twenty Years and Counting... It was twenty years ago in May that I started with Maplesoft (known then as Waterloo Maple Software) in a tiny office at 608 Weber Street North in Waterloo, Ontario. Read More>>
Does Modelica Matter? Modelica is an open language for (lumped parameter) modeling and simulation and is generating a growing following, especially in Europe. Modelica is also at the heart of simulation tools like MapleSim. Read More>>
This one-hour demonstration and Q&A forum introduces MapleSim™ 2, the high-performance, multi-domain modeling and simulation tool, and Maple™ 13, the technical computing software for mathematicians, engineers, and scientists. These products are based on Maplesoft's core technologies, including the world's most advanced symbolic computation engine and revolutionary physical modeling techniques. Together, they provide a platform where students can work confidently with everything from theoretical concepts to the subtleties and art of design.
Maple 13's unique blend of computational power and ease-of-use makes it an essential tool for doing mathematics. Its smart document environment provides revolutionary Clickable Math techniques for solving problems from any technical discipline, ensuring that students are instantly productive and engaged. The results can be incorporated in rich, interactive, live documents that are as professional looking as a textbook. In this one-hour demonstration and Q&A forum, you will learn how Maple 13 is redefining math education and opening new horizons in technical research, with such features as interactive tutors, context-sensitive menus for mathematical operations, sophisticated visualization tools, and extensive mathematical capabilities. See how Maple 13 helps teachers bring complex problems to life, students focus on concepts rather than the mechanics of solutions, and researchers develop more-sophisticated algorithms or models.
MapleSim Training Tuesday, May 12, 2009 2:00pm ET.
This MapleSim Training webinar is a 1 hour session that demonstrates how to get started with MapleSim. It starts with a step-by-step introduction to the MapleSim modeling environment. It also provides details on using Maple 13 for further analysis of MapleSim models, multibody system modeling, and creating custom components.
Hollywood Math
Wednesday, May 13, 2009 2:00 pm ET.
Over its storied and intriguing history, Hollywood has entertained us with many mathematical moments in film. John Nash in "A Beautiful Mind," the brilliant janitor in "Good Will Hunting," the number theory genius in "Pi," and even Abbott and Costello are just a few of the Hollywood "mathematicians" that come to mind. During
This webinar will, by the use of introductory and more complex examples, demonstrate how Maplesoft's products can be used when teaching and learning controls engineering concepts. The concepts will be demonstrated using MapleSim, a new high-performance multi-domain modeling and simulation tool, as well as with Maple and its Dynamic Systems package, which offers a large selection of analytic and graphing tools for linear time-invariant systems, essential in control systems development.
In addition to these articles, check out the Media Center for all the latest coverage on Maplesoft.
Exclusive benefits for instructors teaching with Maple. Sign up today!
Need help with your mathematics courses?
At the Maplesoft Student Help Center you'll find numerous resources explaining the best ways to use Maple, letting you focus on what's really important: understanding math. | 677.169 | 1 |
Product Description
The Prentice-Hall mathematics series is designed to help students develop a deeper understanding of math through an emphasis on thinking, reasoning, and problem-solving. Comprehensive in scope, teachers may incorporate trigonometry, statistics, or pre-calculus readiness as well as more traditional topics. A mix of print and digital materials helps engage students with visual and dynamic activities alongside textbook instruction.
In the "Getting Ready to Learn" portion of the lesson, "Check your readiness" exercises help students see where they might need to review before the lesson. "Check skills you'll need" list out the skills used in the lesson, and new vocabulary is listed before it's introduced.
Sidebar helps tell students where to go for help in the textbook if they need to review, or note when an online tutor video is available.
The lesson itself includes "quick check" problems for students to see if they understand the concept just introduced; "key concepts" boxes that summarize definitions, formulas, & properties, online activities for review and practice; vocabulary sidebars and features that help focus on the language of math; and multiple types of practice activities that feature new material, integrate older material, and provide challenges. A homework video tutor for every lesson is provided online.
Designed to especially help students prepare for high-stakes tests like the SAT and ACT, as well as standardized tests, test-taking strategies are included in each chapter. | 677.169 | 1 |
Ultimate Math Refresher for the Gre, Gmat & Sat
9780967759401
ISBN:
0967759404
Pub Date: 1999 Publisher: Lighthouse Review, Incorporated
Summary: A comprehensive math review for the GRE, GMAT, and SAT. This math refresher workbook is designed to clearly and concisely state the basic math rules and principles of arithmetic, algebra, and geometry which a student needs to master. This is accomplished through a series of carefully sequenced practice sets designed to build a student's math skills step-by-step. The workbook emphasizes basic concepts and problem solv...ing skills. Strategies for specific question types on the GRE, GMAT, and SAT are the focus of the Lighthouse Review self study programs.
Lighthouse Review, Inc. Staff is the author of Ultimate Math Refresher for the Gre, Gmat & Sat, published 1999 under ISBN 9780967759401 and 0967759404. Seven Ultimate Math Refresher for the Gre, Gmat & Sat textbooks are available for sale on ValoreBooks.com, three used from the cheapest price of $9.36, or buy new starting at $13.35.[read more | 677.169 | 1 |
Syllabus
If you're a Algebra 2 Student, I have a class TI-84 graphing calculators. I highly recommend purchasing a graphing calculator to all students that plan on using math in college.
Expectations
Work effectively every day
Be in class on time
Be respectful to Mr. Hill and fellow students.
Try, try, and try again
Grading
Students will be graded on the school wide policy located in the handbook. Students with I and/or M in their grade will not receive credit. Each quarter will calculated from 3 weighted categories: · 50% of your grade will be from Tests. · 30% of your grade will be from Assignment Quizzes. · 20% of your grade will be from Assignments The Semester grade will then be a made up of the following: · 40% of your grade will be from Quarter 1 · 40% of your grade will be from Quarter 2 · 20% of your grade will be from Final Exam Tests If a student receives less than a 60% on a test, they can go back and do test corrections and retake if they'd like. Test corrections are done by redoing the problems they missed and then writing 2 or more sentences on either how they did the problem or what they needed to change. These sentences will be graded on math understanding. Test corrections will give the student 60% of their missed points back if done completely and correctly. Quizzes If a student is absent or they receive a low score on a quiz, they can retake the quiz after/before school. Their retake score, if better, will replace the original score. Quiz retakes must be done on the students own time and not during class. All students must have a passing grade on the test. Homework
All homework is graded by the students. All assignments must receive a passing score in order to earn a credit in my course.
Attendance and Tardy Policy I'll follow all standard procedures at Columbia High School and more. Missed work from an excused absence is to be turned in no later the two class periods from the return of the absence. Since the greatest factor for success is coming to class, unexcused absences and tardies will have to be made up. I'll assign extra work and consequences as need to help my students come to class on time and learn math. Classroom Rules
•Arrive to school and class on time prepared and ready to learn. •Are courteous in interactions with other students and staff. •Resolve differences amicably and with positive intentions. •Seek help from staff in difficult situations. •Dress appropriately for a positive and safe learning environment. •Follow directions from all staff. •Treat our campus and school property with respect. •Technology and food is to be used/eaten at the teacher's discretion. Final Notes With the introduction of the Common Core Standards, more is required from our students here at Columbia. All math assignments, tests, and quizzes students receive will have all the problems written down and all work shown to arrive at the correct solutions. If a student receives a low grade on any assignment, the student may ask to redo or find a way to receive their desired grade. Goals
Students will learn to persevere through math
Students will be able to defend their work
Students will be able use models to solve problems
Students will be attentive to math details and understand patterns
Mr. Hill mhill@nsd131.org Office Hours: Before school and after school by appointment. Mr. Hill reserves the right to edit, change, and/or enhance this document at any time. | 677.169 | 1 |
The aim of this book is to throw light on various facets of geometry through development of four geometrical themes.The first theme is about the ellipse, the shape of the shadow cast by a circle. The next, a natural continuation of the first, is a study of all three types of conic sections, the ellipse, the parabola and the hyperbola.The third theme... more...
The abstract homotopy theory is based on the observation that analogues of much of the topological homotopy theory and simple homotopy theory exist in many other categories (e.g. spaces over a fixed base, groupoids, chain complexes, module categories). Studying categorical versions of homotopy structure, such as cylinders and path space constructions,... more...
Riemannian geometry has today become a vast and important subject. This new book of Marcel Berger sets out to introduce readers to most of the living topics of the field and convey them quickly to the main results known to date. These results are stated without detailed proofs but the main ideas involved are described and motivated. This enables the... more...
An ingenious problem-solving solution for befuddled math students.
A bestselling math book author takes what appears to be a typical geometry workbook, full of solved problems, and makes notes in the margins adding missing steps and simplifying concepts so that otherwise baffling solutions are made perfectly clear. By learning how to interpret... more... | 677.169 | 1 |
First Course in Mathematical Modeling
Offering a solid introduction to the entire modeling process, "A First Course in Mathematical Modeling, 5E, International Edition" delivers an ...Show synopsisOffering a solid introduction to the entire modeling process, "A First Course in Mathematical Modeling, 5E, International Edition" delivers an excellent balance of theory and practice, giving students hands-on experience developing and sharpening their skills in the modeling process. Throughout the book, students practice key facets of modeling, including creative and empirical model construction, model analysis, and model research. The authors apply a proven six-step problem-solving process to enhance students' problem-solving capabilities - whatever their level. Rather than simply emphasizing the calculation step, the authors first ensure that students learn how to identify problems, construct or select models, and figure out what data needs to be collected. By involving students in the mathematical process as early as possible - beginning with short projects - the book facilitates their progressive development and confidence in mathematics and modeling | 677.169 | 1 |
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Making Math Is a User Interface for Computer-Based Math™
Scott Gray
In this talk from the Wolfram Technology Conference, Scott Gray, director of Making Math at O'Reilly Media, shares his views on Computer-Based Math™ and gives an overview of the Mathematica-based online course platform he's developing as a user interface to mathematics.
Channels: Technology Conference
This talk extends the 2013 Wolfram Technology Conference talk, Measuring Discrete Symmetry, from 5 to 16 or more points, comparing it to the work The Reflexive Universe (1976, Delacorte Press), by Arthur M. Young. ...
This talk discusses the theory, implementations, and applications of quantile regression. Quantile regression is more robust than linear regression and can be used to obtain more complete pictures of distributions.
This talk explores how users can learn math by making math with the Making Math platform, which combines Mathematica, courseware content, and learning management. These technologies enable learners to engage with course materials and respond ...
New in Mathematica 10, Dataset introduces a general-purpose container for storing tabular or hierarchical data using lists and associations. This talk covers the query language of Dataset, ascending and ...
GeometricaPlus covers two types of interactive functions. The first one concerns libraries of geometrical objects, and the second one concerns intersections. In this talk interactive applications and animations illustrate how ...
This talk highlights improvements made to the Mathematica-based math courses at the University of Illinois at Urbana-Champaign. Three changes are highlighted: new, instructive homework assignments were created; ... | 677.169 | 1 |
3/27Elementary and Intermediate Algebra Concepts & Applications
Elementary and Intermediate Algebra : Concepts and Applications
MathXL Tutorials on CD for Elementary and Intermediate Algebra : Concepts and Applications
Summary
This workbook provides one worksheet for each section of the text, organized by section objective. Each worksheet lists the associated objectives from the text, provides fill-in-the-blank vocabulary practice, and exercises for each objective. | 677.169 | 1 |
The only thing you need a calculator for is to perform computations, and that's where we draw the line.
There is nothing special about the fx-991ES calculator that any normal scientific calculator cannot perform, but the only thing that distinguishes it from other calculators is its natural display. This is the only reason that I use it as I can easily spot mistakes I might have made.using a calculator or a software program to do calculus just takes the fun away from maths, especially if ure a student .... it undermines ure mathematical development and moreso ure ability to do problem solving.
on the other hand, it may be a necessity (to use a calculator etc) if u do stuff that requires calculations involving calculus, for a living (as in a permanent job) on a daily basis | 677.169 | 1 |
but more advanced forms of math that use different ideas from calculus as well | 677.169 | 1 |
Description: This is a lively textbook for an introductory course in numerical methods, MATLAB, and technical computing, with an emphasis on the informed use of mathematical software. The book makes extensive use of computer graphics, including interactive graphical expositions of numerical algorithms. The topics covered include an introduction to MATLAB; linear equations; interpolation; zeros and roots; least squares; quadrature; ordinary differential equations; Fourier analysis; random numbers; eigenvalues and singular values; and partial differential equations. | 677.169 | 1 |
Lesson study is a professional development process that teachers engage in to systematically examine their practice. This book examines how it effectively works in different contexts and models of teacher learning, while advancing the knowledge base. more...
Build student success in math with the only comprehensive guide for developing math talent among advanced learners. The authors, nationally recognized math education experts, offer a focused look at educating gifted and talented students for success in math. More than just a guidebook for educators, this book offers a comprehensive approach to mathematics... more...
ICM 2010 proceedings comprises a four-volume set containing articles based on plenary lectures and invited section lectures, the Abel and Noether lectures, as well as contributions based on lectures delivered by the recipients of the Fields Medal, the Nevanlinna, and Chern Prizes. The first volume will also contain the speeches at the opening and closing... more...
This Ebook is designed for science and engineering students taking a course in numerical methods of differential equations. Most of the material in this Ebook has its origin based on lecture courses given to advanced and early postgraduate students. This Ebook covers linear difference equations, linear multistep methods, Runge Kutta methods and finite... more... | 677.169 | 1 |
3.
Dynamic Mathematics for Everyone
What does this mean?
• for Everyone
FOSS (free and open source software)
Translatable (and already translated into more
than 50 languages by volunteers)
4.
• FOSS (Free and open source software)
• Can be downloaded for all platforms
• Can be used on school LANs directly
via a web browser (no install)
• Can be used offline
Why GeoGebra?
9.
Dynamic Square
– With active
• Click once to get Point A.
• Click again to get Point B.
• Dialog box will open … 4 is there… click on OK.
9
10.
Square Dynamics
– CLICK on the Move tool
– You can click and drag point A or point B.
– The square is ALWAYS a square.
10
11.
Square Dynamics
– You can click and drag the square itself.
This will only move the square
- the dimensions of the square
depend on A and B!
- Try to click and drag point C
or point D.
– Why can't you click and drag points C or D?
– Answer: They are dependent on A and B.
11
12.
• Many, many applications
– Ready to use applets are available.
– Is easy to use, powerful and dynamic.
– Teachers can use it to better understand what
they are teaching.
– Teachers and students can freely use it anywhere.
– Students can build animations, simulators, …
Why GeoGebra? | 677.169 | 1 |
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Which OCCC general education math class is best for you?
Which OCCC general education math class is best for you?
The OCCC math department currently offers four classes that satisfy the college's general education requirement: College Algebra, Precalculus, Contemporary Math, and Introduction to Statistics. Each of these courses was developed to satisfy particular needs. To decide which one is best for you, compare each class's design to the needs of your degree program.
Important: Whichever course you choose, check that it will transfer to the four-year school of your choice. For the Oklahoma Regents Transfer Grid, go to:
Math 1503 - Contemporary Mathematics
Prerequisite: (R) (W), MATH 0403 or adequate Math Placement Test Score, either within the last year.
3 CREDITS A study of the mathematics needed for critical evaluation of quantitative information and arguments (including logic, critical appraisal of graphs and tables); use of simple mathematical models, and an introduction to elementary statistics.
This course is meant for students who plan to enter degree fields that will require little (if any) formal mathematics. It is a survey of the modern use of mathematics. It is often taught through hands-on projects.
Example degrees: English Performing and Visual Arts History
Math 2013 - Introduction to Statistics
Prerequisite: (R), MATH 0403 or equivalent or adequate Math Placement Test Score, either within the last year.
In this course, you will .learn how to summarize and analyze data collected from surveys and experiments. You will also learn to make and effectively comm communicate conclusions based on data. This information is helpful in analyzing information you receive through newspapers, magazines, TV, radio, and journals.
Example degrees: Psychology Journalism Political Science
Math 1513 - College Algebra
Prerequisite: (R), MATH 0403 or adequate Math Placement Test Score, either within the last year.
3 CREDITS The student will demonstrate an understanding of the general concepts of relation and function and specifically of polynomial, exponential, and logarithmic functions; the ability to solve systems of equations by utilizing matrices and determinants; and the ability to solve practical problems using algebra.
This course was designed to prepare students for a degree in fields that require some mathematical knowledge beyond that usually taught in the developmental math sequence, but does not require any of the engineering calculus sequence (Math 2014, Math 2214, Math 2314). It is a preparatory class for Calculus for Business, Life Sciences, and Social Sciences (Math 1743), as well as many science courses.
Example degrees: Business Biology Chemistry
Math 1533- Precalculus and Analytic Geometry
Prerequisite: (R) (W), MATH 0403, or adequate Math Placement Test Score, either within the last year.
3 CREDITS This course is intended to serve students for whom Calculus and Analytic Geometry I is a requirement. Topics will include conic sections, systems of equations (both linear and nonlinear), and a general discussion of functions with emphasis on polynomial, rational, exponential, and logarithmic functions.
This course was designed to prepare students for success in the engineering calculus sequence (Math 2014, Math 2214, Math 2314). | 677.169 | 1 |
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