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Unformatted text preview: ttention to the subtle differences. 6.1 Introduction to Exponential and Logarithmic Functions 343 (a) Earthquakes are complicated events and it is not our intent to provide a complete discussion of the science involved in them. Instead, we refer the interested reader to a
solid course in Geology10 or the U.S. Geological Survey's Earthquake Hazards Program
found here and present only a simplified version of the Richter scale. The Richter scale
measures the magnitude of an earthquake by comparing the amplitude of the seismic
waves of the given earthquake to those of a "magnitude 0 event", which was chosen to
be a seismograph reading of 0.001 millimeters recorded on a seismometer 100 kilometers
from the earthquake's epicenter. Specifically, the magnitude of an earthquake is given
by
x
M (x) = log
0.001
where x is the seismograph reading in millimeters of the earthquake recorded 100 kilometers from the epicenter.
i. Show that M (0.001) = 0.
ii. Compute M (80, 000).
iii. Show that a...
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This note was uploaded on 05/03/2013 for the course MATH Algebra taught by Professor Wong during the Fall '13 term at Chicago Academy High School. | 677.169 | 1 |
Calculus I Derivatives Introduction In this chapter we will start looking at the next major topic in a calculus class. We will be looking at derivatives in this chapter (as well as the next chapter). This chapter is devoted almost exclusively to finding derivatives. We will be looking at one application of them in this chapter. We will be leaving most of the applications of derivatives to the next chapter.
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This note was uploaded on 02/28/2008 for the course MATH 75 - 76 taught by Professor Yukich during the Spring '06 term at Lehigh University . | 677.169 | 1 |
Catalog Details
MG5740 - Topics in Geometry for Elementary/Middle School Teachers
Credits: 2 - 4 This course is repeatable.
Topics for this course can vary, but may focus on one or more of the following: analytic and transformational geometry, properties of plane and solid figures, similarity, tessellations, fractals, projective geometry and geometry connections with the physical world. Exploration of geometric concepts may be done via hands-on activities, computer software or calculators. Students may repeat the course with a different topic as its focus with permission of the department chair. | 677.169 | 1 |
Showing 1 to 30 of 174
Mathematics for Elementary Educators II: Support Guide Notes
Competencies
This course provides guidance to help you demonstrate the following 4 competencies:
Competency 121.2.1: Rational Numbers and Proportional Reasoning
The graduate applies the properti
EFP Task 1 (0516)
Not Evident
Articulation of
Response
(clarity,
organization,
mechanics)
Competent
Responses are
organized and focus on
Responses are
the main ideas
Responses are
poorly organized
presented in the
unstructured or
or difficult to
assessmen
Task 2 Diversity Awareness
Sean Freeland
Western Governors University
A. Two Groups of Diversity
Intellectual Giftedness, intelligence, and talents are essential concepts that look different
throughout different cultures. Throughout different school syste
EFP Task 2 (0516)
Not Evident
Approaching
Competence
Competent
Responses are
Responses are
poorly organized or
unstructured or
difficult to follow.
disjointed. Vocabulary
Articulation of
Terminology is
and tone are
Response
misused or
unprofessional or
(c
SRT Task 2 (0516)
(A) State the three conditions of the integral test
To use the integral test, the following 3 statements must be true:
1. Its terms are positive
2. Its terms are decreasing
3. The associated function is continuous
(B) Justify that each c
SRT Task 1 (0516)
(A) Perform the nth term test justifying all work
We do the nth term test to see if the series diverges. We take the limit of the given series as it
approaches infinity to get the information we need to determine the convergence or diver
(GR, SRT2-0516) Task 3
(A) Perform the root test, justifying all work.
If we figure out the limit of the nth root of the series, we will come up with a value we label
(rho). The value is then interpreted to determine the convergence or divergence of the
(GR, SRT2-0516) Task 4
(A) State the three conditions of the alternating series testwithout the alternating part:
For a given series:
(1 )n+1 an
n=1
1. All terms in the series are greater than zero:
2. All terms decrease from the first term:
3. The limit
(A1)
(A2)
(A3)
(B)
Provide a logical argument that demonstrates that when applying any two reflections, the outcome will always either be equivalent to a
rotation or a translation.
A reflection is where a figure is reflected across a line so that a mirror
Crystal Stidman (revision 12/22/2015)
EXP Task 1 (0514)
Requirements:
(A1) Create 2 additional axioms
Axiom 1: Each game is played by two distinct teams
Axiom 2: There are at least four teams.
Axiom 3: Each team has at least three distinct players.
Axiom
(1) Find the legs of triangle AFJ and triangle JFK give the hypotenuse is 1 since it is a unit square:
a.
2
2 a =c
2
2 a 2 12
=
2
2
a2=
a=
1
2
1
2
or
2
2
*Label each leg of large isosceles triangle
2
2
(2) Given that BEFD is a square (meaning all sides a
Introduction:
Geometry is based on clearly stated axioms, so it is a subject that lends itself to the practice
of doing proofs about geometric objects and solving problems by applying those axioms.
In this task you will prove a statement regarding a funda
Definitions
for Task 4
The Definition of a Vector Space
A vector space is a set V of elements (called vectors) and vector addition and scalar
multiplication operations that satisfy the following 10 laws (for all vectors X, Y, and Z in V
and all [real] sca
EFP1: Cultural Studies and Diversity
Task 1
Daniel Burns
7/28/2016
Part A: Culture and Diversity
Culture is defined as a set of common beliefs, customs, traditions, morals, and knowledge that
are shared by a particular group or society. It can be expresse
Running head DIVERSITY IN EDUCATION: FROM ADHD TO GIFTED AND TALENTED
Task 2: Diversity in Education: From ADHD to Gifted and Talented
Daniel Burns
Western Governors University
000508729
1
DIVERSITY IN EDUCATION: FROM ADHD TO GIFTED AND TALENTED
2
As our | 677.169 | 1 |
Mathematics and Statistics
The Department of Mathematics and Statistics provides a robust mathematical experience where students gain valuable skills in problem solving, critical thinking, and effective communication of mathematical concepts and models. Our goal is to provide the highest quality education to prepare you to become our colleagues and peers. Our learning and research environment is designed purposefully to welcome students.
Preparing you for a successful career
Our degree programs prepare students for a variety of future endeavors and careers in business, industry, government, research, and academia. Recent graduates of our degree programs have gone on to successful careers as actuaries, statisticians, financial analysts, college professors, mathematicians, operations research analysts, and educators and many of our graduates pursue doctoral degrees in mathematics or statistics.
Mathematical professions regularly rank near the top in surveys of job satisfaction of all professions. In fact, the Wall Street Journal recently ranked Mathematician, Actuary, and Statistician as the top three professions in the United States.
2017 Regional Mathematics Contest
The Mathematics and Statistics Department at Arkansas State University will host the Northeast Arkansas Regional Mathematics Contest, sponsored by ACTM, on Saturday, March 4, 2017. Secondary school students enrolled in any one of the following courses are invited to participate: Algebra I; Geometry; Algebra II; Trig/Pre-Calculus; Calculus; and Statistics. | 677.169 | 1 |
About this product
Description
Description
Linear algebra is the study of vector spaces and the linear maps between them. It underlies much of modern mathematics and is widely used in applications. Written for students with some mathematical maturity and an interest in abstraction and formal reasoning, this book is suitable for an advanced undergraduate course in linear algebra. | 677.169 | 1 |
Guided Tour of Mathematical Methods For Mathematical methods are essential tools for all physical scientists. This second edition provides a comprehensive tour of the mathematical knowledge and techniques that are needed by students in this area. In contrast to more traditional textbooks,More...
Mathematical methods are essential tools for all physical scientists. This second edition provides a comprehensive tour of the mathematical knowledge and techniques that are needed by students in this area. In contrast to more traditional textbooks, all the material is presented in the form of problems. Within these problems the basic mathematical theory and its physical applications are well integrated. The mathematical insights that the student acquires are therefore driven by their physical insight. Topics that are covered include vector calculus, linear algebra, Fourier analysis, scale analysis, complex integration, Greens functions, normal modes, tensor calculus, and perturbation theory. The second edition contains new chapters on dimensional analysis, variational calculus, and the asymptotic evaluation of integrals. This book can be used by undergraduates, and lower-level graduate students in the physical sciences. It can serve as a stand-alone text, or as a source of problems and examples to complement other textbooks.
Roel Snieder holds the Keck Foundation Endowed Chair of Basic Exploration Science at the Colorado School of Mines. From 1997 to 2000, he served as Dean of the Faculty of Earth Sciences at the University of Utrecht. Snieder has served on the editorial boards of Geophysical Journal International, Inverse Problems, Reviews of Geophysics, and the European Journal of Physics. In 2000, he was elected Fellow of the American Geophysical Union. He is co-author of the textbook The Art of Being a Scientist: A Guide for Graduate Students and their Mentors (Cambridge University Press, 2009). From 2003 to 2011, he was a member of the Earth Science Council of the US Department of Energy. In 2008, Snieder worked for the Global Climate and Energy Project at Stanford University on outreach and education on global energy. That same year, he was a founding member of the humanitarian organization Geoscientists Without Borders, where he served as chair until 2013. In 2011, he was elected Honorary Member of the Society of Exploration Geophysicists | 677.169 | 1 |
Algebra 1 Entire Course Bundle with Pre-Algebra Material
Compressed Zip File
Be sure that you have an application to open this file type before downloading and/or purchasing. How to unzip files.
145.05 MB | N/A pages
PRODUCT DESCRIPTION
Take advantage of the cost savings and teacher helps for the classroom. Over 150 classroom activities, scaffold notes, graphic organizers, games, matching activities, task cards, formative assessments, summative assessments, and more for a sale price of $49.99. This is over $300 worth of savings. New products continually added increasing the value of your purchase. All files are PDF format for ease of use or PowerPoint interactive games.
For this item, the cost for one user (you) is $49.99.
If you plan to share this product with other teachers in your school, please add the number of additional users licenses that you need to purchase.
Each additional license costs only $25.00. | 677.169 | 1 |
SMath Studio - Traditionally, since the appearance of the first programs, their speed and available features are ever increasing, while user interfaces are getting more and more complicated. As a result, the non-professional users are unable to deal with these overcomplicated features. As intuitive user interfaces of modern operating systems evolve and computers become more and more powerful, this non-trivial problem is gradually receiving outstanding solutions in terms of simplicity of use. However, most of the mathematical programs are still utilizing user interfaces inherited from very first pocket calculators. It is no secret that a "paper"-style of recording of mathematical expressions is the most convenient to users.
ArithmeTick The objective in ArithmeTick is to solve as many addition, subtration, multiplication, and division problems as possible before time runs out
Algebra Quick Study Guide Boost Your grades with this illustrated quick-study guide. You will use it from high school all the way to graduate school and beyond. FREE first 3 chapters in the trial version | 677.169 | 1 |
Differential Forms
Book Description
Differential forms are utilized as a mathematical technique to help students, researchers, and engineers analyze and interpret problems where abstract spaces and structures are concerned, and when questions of shape, size, and relative positions are involved. Differential Forms has gained high recognition in the mathematical and scientific community as a powerful computational tool in solving research problems and simplifying very abstract problems through mathematical analysis on a computer. Differential Forms, 2nd Edition, is a solid resource for students and professionals needing a solid general understanding of the mathematical theory and be able to apply that theory into practice. Useful applications are offered to investigate a wide range of problems such as engineers doing risk analysis, measuring computer output flow or testing complex systems. They can also be used to determine the physics in mechanical and/or structural design to ensure stability and structural integrity. The book offers many recent examples of computations and research applications across the fields of applied mathematics, engineering, and physics.
The only reference that provides a solid theoretical basis of how to develop and apply differential forms to real research problems
Includes computational methods for graphical results essential for math modeling
Presents common industry techniques in detail for a deeper understanding of mathematical applications | 677.169 | 1 |
Today's Developmental Math students enter college needing more than just the math, and this has directly impacted the instructor's role in the classroom. Instructors have to teach to different learning styles, within multiple teaching environments, and to a student population that is mostly unfamiliar with how to be a successful college student. Authors Andrea Hendricks and Pauline Chow have noticed this growing trend in their combined 30+ years of teaching at their respective community colleges, both in their face-to-face and online courses. As a result, they set out to create course materials that help today's students not only learn the mathematical concepts but also build life skills for future success. Understanding the time constraints for instructors, these authors have worked to integrate success strategies into both the print and digital materials, so that there is no sacrifice of time spent on the math. Furthermore, Andrea and Pauline have taken the time to write purposeful examples and exercises that are student-centered, relevant to today's students, and guide students to practice critical thinking skills. Beginning and Intermediate Algebra and its supplemental materials, coupled with ALEKS or Connect Math Hosted by ALEKS, allow for both full-time and part-time instructors to teach more than just the math in any teaching environment without an overwhelming amount of preparation time or even classroom time.
Book Description McGraw-Hill Science/Engineering/Math. Book Condition: Good. 0077928008 May have signs of use, may be ex library copy. Book Only. Used items do not include access codes, cd's or other accessories, regardless of what is stated in item title. Bookseller Inventory # Z0077928008Z3 | 677.169 | 1 |
In sum, the volume under review is the first quarter of an important work that surveys an active branch of modern mathematics. Some of the individual articles are reminiscent in style of the early volumes of the first Ergebnisse series and will probably prove to be equally useful as a reference;
This book provides a comprehensive introduction to complex analysis in several variables. One major focus of the book is extension phenomena alien to the one dimensional theory (Hartog's Kugelsatz, theorem of Cartan Thullen, Bochner's theorem).
This classic book gives an excellent presentation of topics usually treated in a complex analysis course, starting with basic notions (rational functions, linear transformations, analytic function), and culminating in the discussion of conformal mappings, including the Riemann mapping theorem and the Picard theorem. | 677.169 | 1 |
History of Analytic Geometry (Dover Books on Mathematics)
Delivery: 10-20 Working Days
(2 reviews)
Specifically designed as an integrated survey of the development of analytic geometry, this classic study takes a unique approach to the history of ideas. The author, a distinguished historian of mathematics, presents a detailed view of not only the concepts themselves, but also the ways in which they extended the work of each generation, from before the Alexandrian Age through the eras of the great French mathematicians Fermat and Descartes, and on through Newton and Euler to the "Golden Age," from 1789 to 1850. Appropriate as an undergraduate text, this history is accessible to any mathematically inclined reader. 1956 edition. Analytical bibliography. Index.
Similar Products
Specifications
Country
USA
Author
Carl B. Boyer
Binding
Paperback
EAN
9780486438320
Edition
Unabridged
Format
Unabridged
ISBN
0486438325
Label
Dover Publications
Manufacturer
Dover Publications
MPN
Illustrations
NumberOfItems
1
NumberOfPages
304
PartNumber
Illustrations
PublicationDate
2004-11-29
Publisher
Dover Publications
ReleaseDate
2004-11-29
Studio
Dover Publications
Most Helpful Customer Reviews
Analytic geometry is where the maths student first encounters the combining of traditional Euclidean geometry with algebra. A profound mix, though perhaps most students won't appreciate it as such. Boyer shows how, slowly, the necessary ideas in analytic geometry came together. He traces the first stirrings back to the classical era of ancient Greece and Rome. But the greatest step may well have been due to Rene Decartes and his laying down of the x and y grid in two dimensions. Plus, of course, analytic geometry was necessary for the development of calculus, with the concept of a slope.
You probably are already familiar with all of the maths that the book covers. What Boyer offers is an appreciation of the great minds that preceded up and made these achievements.
Covers the subject neatly and rigorously. Good writer, has the rare skill of getting mathematics concepts across cleanly. Everyone I show this book to wants their own copy. So I give them mine and get | 677.169 | 1 |
WeBWorK sample practive online
Text
Calculators, Computers, PEDs (Personal Electronic Devices)
Get a cheap, simple calculator and bring it to class. You can use these for
exams and quizzes. TI-30 models cost somewhere in $6-20.
Do not play games, email, text, chat, surf, talk, etc. etc. on your PEDs. Turn these
things off during class and put them away. At times a PED, laptop, or other computer
device may be useful in class. Still, turn them off and put them away until you are
told when they will be useful and allowed. PEDs and computers are not allowed for
exams or quizzes.
This is software like Mathematica and Maple which can do symbolic algebra,
graphing, and many other mathematical things, but SageMath is free. You can
download it yourself from sagemath.org
then install on a Gnu/Linux machine, but it is probably easier to use the
FDLTCC server above.
You can also use URL because this is a secure
connection which is redirected from the "s"
after "http" means secure. You may be told that there is no security certificate,
and there is not one yet for this web site. You can safely add an exception for
this sagemath website to just do math. (You should never do this for online banking
or any other online service which handles anything personal which must be secure.)
Note that these are class accounts, so everyone will be able to see work done
by everyone else. There is no privacy. Put nothing private or even identifiable
to yourself online--which is excellent advice for everything on any
Internet service. Make up worksheet names which you can remember yet which
are not identifiable to yourself, and use these just for math. | 677.169 | 1 |
Check Your Delivery Options
By now Algebra (based on set theoretic notions) and Topology have established their dominance over almost all the disciplines in pure mathematics. Both of these subjects have become so vast that they need their detailed discussion separately. However, an attempt has been made here to present the basic and core topics of these subjects together. The book comprises of three main parts:(i) Algebraic systems: Sets and Functions, Groups, Rings, Fields, Integral domains and Linear (or Vector) spaces; (ii) Metric spaces, and (iii) Topological spaces. The first chapter starts with Sets and Functions. It includes the main features of the Set Theory needed in our subsequent discussions. The next three chapters dwell upon different kinds of algebraic structures as detailed above and cover almost all the necessary information needed by a beginner. Metric spaces have been dealt in detail in Chapter 5 including topics on 'Sequences and their convergence''. Bounded and unbounded sets in the metric spaces are also given. The last chapter deals with the Topological spaces. It gives a detailed account of various types of these spaces and covers almost all important topics in the subject needed for a first course. | 677.169 | 1 |
Section 3.2
Continuity
Math 2321
Name:
Continuity
Objectives:
Dene continuity at a point
Dene continuity on an open interval
Dene continuity on a closed interval
Determine the continuity of various functions
Continuity at a point
A function f is conti
Section 3.3
Rates of Change
Math 2321
Name:
Rates of Change
Objectives:
Find average rate of change based on table of values, graphically, and by function
Dene instantaneous rate of change
Recognize instantaneous rate of change as a limit of average ra
Section 3.4
Denition of Derivative
Math 2321
Name:
Denition of Derivative
Objectives:
Dene the derivative of a function as the limit of the dierence quotient
Find the derivative and use it to nd the slope of the tangent line
Find the equation of the ta
Showing 1 to 1 of 1
Although it is supposed to be a high paced class, he takes his time to explain things so that we understand. The example he gives us are from the book so that we can actually apply them to our homework. He gives us review days before any exam, and the questions on the exam are based off everything he went over from the book. It is easy to understand and to keep up with.
Course highlights:
We learned how to find derivatives, and antiderivatives. We went over more subjects but they revolved most around these two main topics. He gave us sets of rules on how to find each of the above and many examples to gain confidence in our work.
Hours per week:
9-11 hours
Advice for students:
Go over his examples the most, they are direct to the point and they help you solve other problems. Do the homework he assigns because they help you tremendously on the quizzes and exams. Study over your homework and quizzes for exams. On the finals, study over quizzes and all exams. | 677.169 | 1 |
Introduction
Now In this section we are going to discuss along with the application of some other operations of arithmetic like Ratio, Proportion, Rates, Percentage, which are very helpful to solve everyday problems and business related problems. | 677.169 | 1 |
Level 300
This course is designed to offer students the opportunity to discuss the Senior High School curriculum in Mathematics, including the basic principles of curriculum development. Students will survey the development of mathematics education in Ghana and examine issues related to the Senior High School curriculum in mathematics. The course delivery will take the form of lectures, discussions, individual and group work, assignment and presentations | 677.169 | 1 |
Fun with Prime Numbers: The Mysterious World of Mathematics - Kyoto University
Wichtige informationen
Kurs
Online
Wann: Freie Auswahl
Beschreibung
Learn about prime numbers, solve math problems and explore recent discoveries and theoriesWichtige informationen
Voraussetzungen: None. Knowledge of high school level algebra is recommended.
Veranstaltungsort(e)
Wo und wann
Beginn
Lage
Freie Auswahl
Online
Meinungen
Zu diesem Kurs gibt es noch keine Meinungen
Was lernen Sie in diesem Kurs?
Mathematics
GCSE Mathematics
Math
Prime Numbers
Themenkreis
Prime numbers are one of the most important subjects in mathematics. Many mathematicians from ancient times to the 21st century have studied prime numbers. In this math course, you will learn the definition and basic properties of prime numbers, and how they obey mysterious laws. Some prime numbers were discovered several hundred years ago whereas others have only been proven recently. Even today, many mathematicians are trying to discover new laws of prime numbers. Calculating by a pen and paper, you will explore the mysterious world of prime numbers. Join us as we tackle math problems, and work together to discover new laws on prime numbers. Let's study and have fun!
Zusätzliche Informationen
Tetsushi Ito Tetsushi is an associate professor of Department of Mathematics at the Faculty of Science at Kyoto University. He received Ph.D in Mathematical Sciences from University of Tokyo in 2003.
Fun with Prime Numbers: The Mysterious World of Mathematics - Kyoto UniversityedX | 677.169 | 1 |
Math Rules!
PreCalc
Why PreCalculus? Hopefully, because you love math! But also to unlock the door to calculus and higher mathematics. PreCalc helps you in laying ground work and achieving skills needed to do well in Calculus. | 677.169 | 1 |
Intermediate Algebra for College Students
The goal of this series is to provide readers with a strong foundation in Algebra. Each book is designed to develop readers' critical thinking and problem-solving capabilities and prepare readers for subsequent Algebra courses as well as "service" math courses. Topics are presented in an interesting and inviting format, incorporating real world sourced data and encouraging modeling and problem-solving. Algebra and Problem Solving. Functions, Linear Functions, and Inequalities. Systems of Linear Equations and Inequalities. Polynomials, Polynomial Functions, and Factoring. Rational Expressions, Functions, and Equations. Radicals, Radical Functions, and Rational Exponents. Quadratic Equations and Functions. Exponential and Logarithmic Functions. Conic Sections and Nonlinear Systems of Equations. Polynomial and Rational Functions. Sequences, Probability, and Mathematical Induction. For anyone interested in introductory and intermediate algebra and for the combined introductory and intermediate algebra.
"synopsis" may belong to another edition of this title.
From the Publisher:
This text provides a comprehensive coverage of intermediate algebra to help students prepare for precalculus as well as other advanced math. The material will also be useful in developing problem solving, critical thinking, and practical application skills. It contains a vast collection of historical references, multidisciplinary applications, enrichment essays, Critical Thinking.
From the Back Cover:
This book provides a comprehensive coverage of intermediate algebra to help students prepare for precalculus as well as other advanced math. The material will also be useful in developing problem solving, critical thinking, and practical application skills. Real World Data and Visualization is integrated. Paying attention to how mathematics influences fine art and vice versa, the book features works from old masters as well as contemporary artists | 677.169 | 1 |
Product Description:
The purpose of this book is to introduce the basic ideas of mathematical proof to students embarking on university mathematics. The emphasis is on helping the reader in understanding and constructing proofs and writing clear mathematics. This is achieved by exploring set theory, combinatorics and number theory, topics which include many fundamental ideas which are part of the tool kit of any mathematician. This material illustrates how familiar ideas can be formulated rigorously, provides examples demonstrating a wide range of basic methods of proof, and includes some of the classic proofs. The book presents mathematics as a continually developing subject. Material meeting the needs of readers from a wide range of backgrounds is included. Over 250 problems include questions to interest and challenge the most able student as well as plenty of routine exercises to help familiarize the reader with the basic ideas.
21 Day Unconditional Guarantee
REVIEWS for An Introduction to Mathematical | 677.169 | 1 |
Find a Hermosa Beach MathIt is the bridge from the concrete to the abstract study of mathematics. Topics include simplifying expressions, evaluating and solving equations and inequalities, and graphing linear and quadratic functions and relations. Real world applications are presented within the course content and a function's approach is emphasized | 677.169 | 1 |
Synopses & Reviews
Publisher Comments
Renowned professor and author Gilbert Strang demonstrates that linear algebra is a fascinating subject by showing both its beauty and value. While the mathematics is there, the effort is not all concentrated on proofs. Strang's emphasis is on understanding. He explains concepts, rather than deduces. This book is written in an informal and personal style and teaches real mathematics. The gears change in Chapter 2 as students reach the introduction of vector spaces. Throughout the book, the theory is motivated and reinforced by genuine applications, allowing pure mathematicians to teach applied mathematics.
About the Author
Gilbert Strang is Professor of Mathematics at the Massachusetts Institute of Technology and an Honorary Fellow of Balliol College. He was an undergraduate at MIT and a Rhodes Scholar at Oxford. His doctorate was from UCLA and since then he has taught at MIT. He has been a Sloan Fellow and a Fairchild Scholar and is a Fellow of the American Academy of Arts and Sciences. Professor Strang has published a monograph with George Fix, "An Analysis of the Finite Element Method", and has authored six widely used textbooks. He served as President of SIAM during 1999 and 2000 and he is Chair of the U.S. National Committee on Mathematics for 2003-2004. | 677.169 | 1 |
Forum for Science, Industry and Business
The Aftermath of Calculator Use in College Classrooms
13.11.2012
Students may rely on calculators to bypass a more holistic understanding of mathematics, says Pitt researcher
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 | 677.169 | 1 |
0133178579
Topic:
General
Author:
Ross L. Finney
Format:
Hardcover
Edition Description:
4th Edition
Age Level:
Eleventh Grade
Publication Year:
2011
Product Type:
Textbook
ISBN:
9780133178579
Detailed item info
Synopsis
The esteemed author team is back with a fourth edition of Calculus: Graphing, Numerical, Algebraic written specifically for high school students and aligned to the guidelines of the AP Calculus exam. The new edition focuses on providing enhanced student and teacher support; for students, the authors added guidance on the appropriate use of graphing calculators and updated exercises to reflect current data. For teachers, the authors provide lesson plans, pacing guides, and point-of-need answers throughout the Teacher's Edition and teaching resources. Learn more | 677.169 | 1 |
PDF (Acrobat) Document File
Be sure that you have an application to open this file type before downloading and/or purchasing.
1.8 MB | 13 pages
PRODUCT DESCRIPTION
Calculus Antiderivatives Indefinite Integration Task Cards.
This activity is designed for Calculus 1 or AP Calculus. Students find the Antiderivatives or Indefinite Integrals of basic functions, with no substitution. This activity is usually in the beginning of Unit 4, Integration, depending upon your text. Included functions are polynomial, radicals, trigonometric, logarithmic, and exponential functions.
Included in the Lesson:
✓ Task Cards: There are 17 cards and three blank cards for you to customize for problems you may have emphasized.
✓ Master List of questions included with answer key.
✓ Student response sheets with room for students to show their work.
✓ Handout with similar problems. The handout can be used as enrichment, homework, group work, or as an assessment | 677.169 | 1 |
3 2.1.1 Constants and Other Stuff The constants listed are amount some of the main ones, other values can be derived through calculation using modern calculators or computers. The values are typically given with more than 15 places of accuracy so that they can be used for double precision calculations. 2.1.2 Basic Operations · These operations are generally universal, and are described in sufficient detail for our use. · Basic properties include,
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4 2.1.2.1 - Factorial · A compact representation of a series of increasing multiples. 2.1.3 Exponents and Logarithms · The basic properties of exponents are so important they demand some sort of mention · Logarithms also have a few basic properties of use, · All logarithms observe a basic set of rules for their application, | 677.169 | 1 |
math basic math and pre algebra teach yourself
2.
Copyright 0 1996by Debra Anne Ross
All rights reserved under the Pan-American and International
Copyright Conventions. This book may not be reproduced, in
whole or in part, in any form or by any means electronic or
mechanical, including photocopying, recording, or by any
information storage and retrieval system now known or
hereafter invented, without written permission from the
publisher, The Career Press.
MASTER m T H : BASICMATH AND PRE-ALGEBRA
Cover design by The Visual Group
Printed in the U.S.A.by Book-martPress
To order this title, please call toll-free 1-800-CAREER-1(NJ and
Canada: 201-848-0310)to order using VISA or Mastercard, or for
further information on books from Career Press.
Library of Congress Cataloging-in-PublicationData
Ross, Debra h e , 1958
Master math :basic math and pre-algebra/ by Debra Anne Ross.
p. cm.
Includes index.
1. Mathematics. I. Title.
ISBN 1-56414-214-0 (pbk.)
QA39.2.R6547 1996
513.2-dc20
CIP
96-22992
11.
Introduction
Basic Math and Pre-Algebra is the first of three
books in the Master Math series. The second and third
books are entitled Algebra and Re-Calculus and Geo-
metry. The Master Math series presents the general
principles of mathematics from grade school through
college including arithmetic, algebra, geometry, trigo-
nometry, pre-calculus and introductory calculus.
Basic Math and Pre-Algebra is a comprehensive
arithmetic book that explains the subject matter in a
way that makes sense to the reader. It begins with the
most basic fundamental principles and progresses
through more advanced topics to prepare a student for
algebra. Basic Math and Be-Algebra explains the prin-
ciples and operations of arithmetic, provides step-by-
step procedures and solutions and presents examples
and applications.
Basic Math and Pre-Algebra is a reference book
for grade school and middle school students that ex-
plains and clarifies the arithmetic principles they are
learningin school.It is also a comprehensivereference
source for more advanced students learning algebra
1
12.
Basic Math and Pre-Algebra
and pre-calculus. Basic Math and Re-Algebra is inval-
uable for students of all ages, parents, tutors and any-
one needing a basic arithmetic reference source.
The information provided in each book and in
the series as a whole is progressive in difficulty and
builds on itself, which allows the reader to gain per-
spective on the connected nature of mathematics. The
skills required to understand every topic presented are
explained in an earlier chapter or book within the
series. Each of the three books contains a complete
table of contents, a comprehensive index, and the ta-
bles of contents of the other two books in the series so
that specific subjects, principles and formulas can be
easily found. The books are written in a simple style
that facilitates understanding and easy referencing of
sought-afterprinciples, definitions and explanations.
Basic Math and Pre-Algebra and the Master
Math series are not replacements for textbooks but
rather reference books providing explanations and per-
spective. The Master Math series would have been in-
valuable to me during my entire education from grade
school through graduateschool.There is no other source
that provides the breadth and depth of the Master
Math series in a single book or series.
Finally, mathematics is a language-the univer-
sal language. A person struggling with mathematics
should approach it in the same fashion he or she would
approach learning any other language. If someone
moves to a foreigncountry, he or she does not expect to
know the language automatically. It takes practice
and contact with a language in order to master it.
2
13.
Introduction
After a short time in the foreign country he or she
would not say, "I do not know this language well yet. I
must not have an aptitude for it." Yet many people
have this attitude toward mathematics. If time is
spent learning and practicing the principles, math-
ematics will become familiar and understandable.
Don't give up.
3
16.
Basic Math and Pre-Algebra
1.1. Digits and the base ten system
The system of numbers used for counting is
based on groups of ten and is called the base ten
system.
Numbers are made up of digits that each correspond
to a value. For example, the number 5,639,248 or five
million, six hundred thirty-nine thousand, two hundred
forty-eightrepresents:
5 millions, 6 hundred thousands, 3 ten thousands, 9 thousands,
2 hundreds, 4 tens, and 8 ones
This number can also be written:
5,000,000 +600,000 +30,000 +9,000 +200 +40 +8
The ones digit indicates the number of ones, (1).
The tens digit indicates the number of tens, (10).
The hundreds digit indicates the number of
The thousands digit indicates the number of
The ten thousands digit indicates the number of tens
The hundred thousands digit indicates the number of
The millions digit indicates the number of millions,
hundreds, (100).
thousands, (1,000).
of thousands, (10,000).
hundreds of thousands, (100,000).
(1,000,000).
6
17.
Numbers and Their Operations
1.2. Whole numbers
Whole numbers include zero and the counting
numbers greater than zero.
Negative numbers and numbers in the form of
fractions, decimals, percents or exponents are not
whole numbers.
Whole numbers are depicted on the number line and
include zero and numbers to the right of zero.
I L I
Whole Numbers = I I 1 ; >
0 1 2 3 4 5 6 7 8
Whole numbers can be written in the form of a set.
WholeNumbers= (0,1,2, 3,4, 5, 6, 7, 8, 9,10,11,...)
1.3.Addition of whole numbers
This section includes definitions, a detailed
explanation of addition, and a detailed description of
the standard addition technique.
The symbol for addition is +.
Addition is written 6 +2.
The numbers to be added are called addends.
7
18.
Basic Math and Pre-Algebra
The answer obtained in addition is called the sum.
The symbol for what the sum is equal to is the equal
sign =.
Note: The symbol for not equal is #.
For any number n, the following is true: n +0 = n
(Note that letters are often used to represent
numbers.)
A Detailed Explanation of Addition
Consider the followingexamples:
5 + 3 = 8
34 + 76 = 110
439 + 278 = 717
5 + 3 + 2 = 1 0
33 + 28 + 56 = 117
456 + 235 + 649 = 1,340
To add numbers that have two or more digits, it is
easier to write the numbers in a column format with
ones, tens, hundreds, etc., aligned, then add each
column.
Example: Add 5 +3.
5 ones
+3 ones
8 ones
8
22.
Basic Math and Pre-Algebra
A Detailed Description of the StandardAddition
Technique
An alternativeto separating the ones, tens,
hundreds, etc., is to align the proper digits in columns
(onesover ones, tens over tens, etc.).Then add each
column,beginning with the ones, and carry-overdigits
to the left. (Carry over the tens generated in the ones'
column into the tens'column, carry over the hundreds
generated in the tens' column into the hundreds'
column,carry over the thousands generated in the
hundreds'column into the thousands' columns,and so
on.)
Example: Add 35 +46.
35
+46
?
First add 5 +6 = 11,and carry over the 10 into the
tens' column.
1
35
+ 46
1
Add the tens'column.
12
25.
Numbers and Their Operations
12
928
387
49
+ 5
69
Add the hundreds, 1+9 +3 = 13hundreds.
12
928
387
49
+ 5
1369
1.4. Subtraction of whole numbers
This section includes definitions, a detailed
explanation of subtraction, and a detailed description
of the standard subtraction technique.
Subtraction is the process of finding the difference
between two numbers.
The symbol for subtraction is -.
Subtractionis written 6 - 2.
15
26.
Basic Math and Pre-Algebra
The number to be subtracted from is called the
minuend.
The number to be subtracted is the subtrahend.
The answer obtained in subtraction is called the
difference.
Subtraction is the reverse of addition.
If 2 +3 =5, then 5 - 2 = 3 or 5 - 3=2.
For any number n, the following is true: n - 0 = n
(Note that letters are often used to represent
numbers.)
To subtract more than two numbers, subtract the
first two, then subtract the third number from the
difference of the first two, and so on. Subtraction must
be performed in the order that the numbers are listed.
A Detailed Explanation of Subtraction
Consider the examples:
6 - 2 = 4
26 - 8 18
10,322 - 899 = 9,423
To subtract numbers with two or more digits, it is
easier to write the numbers in a column format with
ones, tens, hundreds, etc., aligned, then subtract each
column beginning with the ones. To subtract a digit in
16
32.
Basic Math and Pre-Algebra
To check the subtraction results, add the difference to
the number that was subtracted.
9423
+ 899
10,322
Note that if a larger number is subtracted from a
smaller number, a negative number results. For
example, 5 - 8 = -3 and 40 - 50 = -10. See Section 1.11
"Addition and subtraction of negative and positive
integers."
1.5. Multiplicationofwhole numbers
This section includes definitions, an explanation
of multiplication, and a detailed description of the
standard multiplication technique.
Multiplication is a shortcut for addition.
The symbols for multiplication are x, *, 0 , ( >().
The numbers to be multiplied are called the
multiplicand (thefirst number) and the multiplier (the
second number).
The answer obtained in multiplication is called the
product.
22
33.
Numbers and Their Operations
Multiplication is written using the following
symbols:
6 x 2 , 6 * 2,6 2, (6)(2)
If you add 12 twelves, the answer is 144.
12+12+12+12+12+12+12+12+12+12+12+12 = 144
Adding 12 twelves is the same as multiplying twelve
by twelve
(12)(12)= 144
What is 3 times 2, (3)(2)?It is the value of 3 two
times, or 3 +3 = 6.
Equivalently, what is 2 times 3, (2)(3)?It is the
value of 2 three times or, 2+2+2 = 6.
What is 2 times 4 times 3?
(2)(4)(3)= (2)(4 three times)
= (2)(4+4+4)= (2)(12)= 24
Also, (2)(12)= 2 twelve times
=2+2+2+2+2+2+2+2+2+2+2+2=24
23
34.
Basic Math and Pre-Algebra
Equivalently, 2 times 4 times 3 is:
(2)(4)(3)= (4 two times)(3)= (4+4)(3)
= (8)(3)= 8 three times = 8 +8 +8 = 24
The order in which numbers are multiplied does not
afl'ect the result,just as the order in which numbers
are added does not affect the result.
If a number is multiplied by zero, that number is
multiplied zero times and equals zero.
For any number n, the followingis true: n x 0 = 0
(Note that letters are often used to represent
numbers.)
An Explanationof Multiplication
Consider the followingexamples:
2 8 * 7 = ?
6,846 * 412 = ?
10 * 10 =?
10 * loo=?
10 * 1,000= ?
To multiply numbers with two or more digits, it is
easier to write the numbers in a column format. Then
multiply each digit in the multiplicand (top number),
beginning with the ones'digit, by each digit in the
multiplier (bottom number), beginning with the ones'
digit.
24
37.
Numbers and Their Operations
Note that alignment is important! Each partial
product must be aligned with the right end of the
multiplier digit.
First, multiply the ones'digit in the multiplier with
each digit in the multiplicand beginning with the ones'
digit, 6 * 2 = 12,carry over the 1ten.
1
6846
x 412
2
Multiply the ones'digit in the multiplier with the tens'
digit in the multiplicand, 4 * 2 = 8, then add the 1ten
that was carried over (thereis nothing new to carry).
1
6846
x 412
92
Multiply the ones'digit in the multiplier with the
hundreds'digit in the multiplicand, 8 * 2 = 16, (nothing
carried over to add),carry over the 1thousand.
I 1
6846
x 412
692
27
38.
Basic Math and Pre-Algebra
Multiply the ones'digit in the multiplier with the
thousands'digit in the multiplicand, 6 * 2 = 12, add
the 1thousand that was carried.
1 1
6846
x 412
13692
Resulting in the partial product aligned with the right
end of the ones'digit of the multiplier.
Next, multiply the tens' digit in the multiplier with the
carry).
6846
x 412
ones'digit in the multiplicand, 6 * 1=6, (nothingto
r with the ten
13692
6
Multiply the tens' digit in the mi ltipli
digit in the multiplicand, 4 * 1=4, (nothingto add or
cany).
6846
x 412
?
13692
46
28
39.
Numbers and Their Operations
Multiply the tens' digit in the multiplier with the
hundreds' digit in the multiplicand, 8 * 1= 8,(nothing
to add or carry).
6846
x 412
13692
846
Multiply the tens' digit in the multiplier with the
thousands' digit in the multiplicand, 6* 1=6,(nothing
carried over to add).
6846
x 412
13692
6846
Resulting in the partial product aligned with the right
end of the tens' digit of the multiplier. (Eachpartial
product must be aligned with the right end of the
multiplier digit.)
Next, multiply the hundreds' digit in the multiplier
with the ones'digit in the multiplicand, 6 * 4 = 24, and
carry over the 2 to the tens'column.
29
40.
Basic Math and Pre-Algebra
2
6846
x 412
13692
6846
4
Multiply the hundreds'digit in the multiplier with the
tens' digit in the multiplicand, 4 * 4 = 16, add the 2
over the tens'column,and carry over the 1 to the
hundreds'column.
12
6846
x 412
13692
6846
84
Multiply the hundreds'digit in the multiplier with the
hundreds'digit in the multiplicand, 8 * 4 = 32, add the
1 over the hundreds'column,and carry over the 3 to the
thousands'column.
312
6846
x 412
13692
6846
384
30
41.
Numbers and Their Operations
Multiply the hundreds' digit in the multiplier with the
thousands' digit in the multiplicand, 6 * 4 = 24, and
add the 3 over the thousands' column.
312
6846
x 412
13692
6846
27384
Resulting in the partial product aligned with the right
end of the hundreds' digit of the multiplier. (Each
partial product is aligned with the right end of the
multiplier digit.)
Next, add the three partial products for a total
product.
6846
x 412
13692
6846
27384
2820552
Therefore, 6,846 * 412 = 2,820,552.
31
45.
Numbers and Their Operations
1.6. Division of whole numbers
This section provides definitions and describes
in detail the long division format.
Division evaluates how many times one number is
present in another number.
The symbolsfor division are +, /, r.
The number that gets divided is called the dividend.
The number that does the dividing (or divides into
the dividend)is called the divisor.
The answer obtained after division is called the
quotient.
Division is written 6 + 2, 6 ,(6)/(2),2 p .
2
Division is the inverse of multiplication.
2 * 3 = 6 , 6 + 2 = 3 , 6 + 3 = 2
Six divides by two three times or by three two times.
6 + 2 = 3 , 6 + 3 = 2
6 = 3 + 3 = 2 + 2 + 2
35
46.
Basic Math and Pre-Algebra
Because 6 +6 +6 +6 = 24, there are four sixes in 24.
6 * 4 =24 or 24 i 6 = 4
For any number n, the following is true: n + 0 =
Undefined. (Note that letters are often used to
represent numbers.).
For any number n, the following is true: 0 + n =0
A Description of the Long Division Technique
Example: 4,628 i5 =?.
This division problem is easier to solve by writing it in
a long division format.
Divide the divisor (5) into the left digit(s)of the
dividend. To do this, choose the smallestpart of the
dividend the divisor will divide in to. Because 5 does
not divide into 4, the next smallestpart of the
dividend is 46. Estimate how many times 5 will divide
into 46. First try 9. What is 5 * 9?5 * 9 =45, which is 1
less than 46. Write the 9 over the right end of the
number it will divide into (46), and place 45 under the
46. (See Section 1.18 on rounding, truncating and
estimating for assistance on estimating.)
36
47.
Numbers and Their Operations
9??
5)4628
45
Subtract 46 - 45.
9??
5)4628
45
01
Bring down the next digit (2)to obtain the next
number (12)to divide the 5 into.
9??
5)4628
45
012
Estimate the most number of times 5 will divide into
12. The estimate is 2. Because, 5 * 2 = 10,(with a
remainder of 2), 5 will divide into 12two times. Write
the 2 over the right end of the number it will divide
into (121,and place 10under 12.
37
48.
Basic Math and Pre-Algebra
92?
5)4628
45
012
10-
Subtract 12 - 10.
92?
5)4628
45
012
10
2
-
Bring down the next digit (8)to obtain the next
number (28)to divide 5 into.
92?
5)4628
45
012
10-
28
Estimate the most number of times 5 will divide into
28. The estimate is 5. Because 5 * 5 = 25,(with a
remainder of 3),5 will divide into 28five times. Write
the 5 over the right end of the number it will divide
into (28),and place 25 under 28.
38
49.
Numbers and Their Operations
925
5)4628
45
012
10-
28
25
Subtract 28 - 25.
925
5)4628
45
012
10-
28
25
3
There are no more numbers to bring down, therefore
the division is complete, and 3 is the final remainder.
Therefore, 4,628 + 5= 925plus a remainder of 3.
To check division, multiply the quotient by the divisor
and add the remainder to obtain the dividend.
39
50.
Basic Math and Pre-Algebra
925
x 5
4625
Add the remainder.
4625
+ 3
4628
Example: 160,476+ 364 = ?.
Arrange in the long division format.
3641160476
Divide the divisor (364)into the left digits of the
dividend. To do this, choose the smallest part of the
dividend the divisor will divide in to. 364 will divide
into 1,604.Estimate the most number of times 364
will divide into 1,604.The estimate is 4. 364 * 4 =
1,456with a remainder of 148. Write the 4 over the
right end of the number it will divide into (1,604),and
place 1,456under the 1,604.
4??
364)160476
1456
Subtract 1,604 - 1,456.
40
51.
Numbers and Their Operations
4??
364)160476
1456
0148
Bring down the next digit (7)to obtain the next
number (1,487)to divide 364 into.
4??
364)160476
1456
01487
Estimate the most number oftimes 364 will divide
into 1,487.The estimate is 4. 364 * 4 = 1,456with a
remainder of 31. Write the 4 over the end of the
number it will divide in to (1,487),andplace 1,456
under the 1,487.
44?
3641160476
1456
01487
1456
Subtract 1,487 - 1,456.
41
52.
Basic Math and Pre-Algebra
44?
364)160476
1456
01487
1456
31
Bring down the next digit (6)to obtain the next
number (316)to divide 364 into.
44?
3641160476
1456
01487
1456
316
Because 364 will not divide into 316, place a zero over
the right end of 316.
440
364)160476
1456
01487
1456
316
316 becomes the remainder. Therefore, 160,476~364=
440 plusaremainderof316.
42
53.
Numbers and Their Operations
To check division, multiply the quotient by the divisor
and add the remainder to obtain the dividend.
364
x 440
160160
Add the remainder. + 316
14560
1456
160160
160476
1.7. Divisibility,remainders,factors
and multiples
This section defines and gives examples of
divisibility, remainders, factors and multiples.
The divisibility of a number is determined by how
many times that number can be divided evenly by
another number. For example, 6is divisible by 2
because 2 divides into 6 a total of 3 times with no
remainder. An example of a number that is not
divisible by 2 is 5, because 2 divides into 5 a total of 2
times with 1 remaining.
The remainder is the number left over when a
number cannot be divided evenly by another number.
43
54.
Basic Math and Pre-Algebra
A smaller number is a factor of a larger number if
the smaller number can be divided into the larger
number without producing a remainder. Numbers that
are multiplied together to produce a product are
factors of that product.
The factors of 6 .are6 and 1,or 2 and 3.
6 x 1 = 6 , 2 x 3 = 6
5 is not a factor of 6 because 6 + 5 = 1with a
remainder of 1.
The factors of 10are 2 and 5, or 10and 1,where:
10= (2)(5), 10= (10)(1)
If a and b represent numbers,
Where 2,3, a and b are factors of 6ab.
A multiple of a number is any number that results
after that number is multiplied with any number.
Multiples of zero do not exist because any number
multiplied by zero, including zero, results in zero. It is
possible to create infinite multiples of numbers by
simply multiplying them with other numbers.
44
55.
Numbers and Their Operations
The following are examples of multiples of the
number 3:
3*5=15
3 * 2 = 6
3*4=12
3*20=60
Where 15,6,12and 60 are some of the multiples of 3.
1.8. Integers
Integers include positive numbers and zero
(whole numbers) and also negative numbers.
The set of all integers is represented as follows:
Integers = {...-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,...}
Numbers that are not integers include numbers in
the form of fractions, decimals,percents or exponents.
Consecutiveintegers are integers that are arranged
in an increasing order accordingto their size from the
smallest to the largest without any integers missing in
between.
The following are examples of consecutive integers:
{-10,-9,-8,-7)
{0,1,2,3,4}
{-2,-1,0,1,2,3,4,5)
(99,100,101,102,103,104,105)
45
56.
Basic Math and Pre-Algebra
1.9. Even and odd integers
This section defines and provides examples of
even integers and odd integers.
Even integers are integers that can be divided evenly
by 2.
Even Integers = {...-6,-4,-2,0,2,4,6,8,...}
Zero is an even integer.
Odd integers are integers that cannot be divided
evenly by 2, and therefore are not even.
Odd Integers = {...-7,-5, -3, -1,1,3, 5,7, ...}
Fractions are neither even nor odd. If division of two
numbers yields a fraction the result is not odd or even.
It is possible to immediately determine if a large
number is even or odd by observing whether the digit
in the ones position is even or odd.
Consecutiveeven integers are even integers that are
arranged in an increasing order accordingto their size
without any integers missing in between.
The following is an example of consecutive even
integers: {-10,-8,-6,-4,-2,0, 2,4,6}
Consecutiveodd integers are odd integers that are
arranged in an increasing order accordingto their size
without any integers missing in between.
46
58.
Basic Math and Pre-Algebra
If zero is multiplied by any number the result is zero:
n * O = O
Dividingby zero is "undefined":
n + 0 =undefined
Zero divided by a number (representedby letter n) is
zero:
1.11. Addition and subtractionof
negative and positive integers
This section describes addition and subtraction
of negative integers and positive integers.
Remember the number line.
-5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5
When numbers are added and subtracted, think of
moving alongthe number line.
Begin with the first number and move to the right or
left depending on the sign of the second number and
whether it is being added or subtracted to the first
number.
48
59.
Numbers and Their Operations
To add a positive number, begin at the first number
and move to the right the value of the second number.
2 + 1 = 3
T
-5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5
-2+1 = -1
-5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5
To add a negative number (subtract),begin at the
first number and move to the left the value of the
second number.
2+-1=1
~ ~~~~
w
-5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5
-2+-1= -3
-5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5
To subtract a positive number, begin at the first
number and move to the left the value of the second
number.
2 - 1 = 1
-5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5
49
62.
Basic Math and Pre-Algebra
A summary of the division rules are:
(+) f (+) =+
(-)f (-)=+
(+)i(-)= -
(-)i (+) = -
1.13. The real number line
Real numbers include whole numbers, integers,
fractions, decimals, rational numbers and irrational
numbers.
Real numbers can be expressed as the sum of a
decimal and an integer.
All real numbers except zero are either positive or
negative.
All real numbers correspond to points on the real
number line and all points on the number line
correspond to real numbers.
The real number line reaches from negative infinity
(-m) to positive infinity (v).
< >
-4 -3 -2 -1-.5 0 142 2 5/23n 4
Real numbers include -0.5,a,512 and 7r. (~3.14)
All numbers to the left of zero are negative.
All numbers to the right of zero are positive.
52
63.
Numbers and Their Operations
The distance between zero and a number on the
number line is called the absolute value or the
magnitude of the number.
1.14.Absolute value
The absolute value is the distance between zero
and the number on the number line.
c >
-4 -3 -2 -1-5 0 142 2 5/23 x 4
The absolute value is always positive or zero, never
negative.
The symbol for absolute value of a number
represented by n is In I .
Positive 4 and negative 4 have the same absolute
value.
141 = 4 and 1-41 = 4
Properties of absolute value are (x and y represent
numbers):
1x1 2 0
I x - y l = ly-XI
1x1 l y l = lxyl
Ix+yl 1x1 + lyl
(See Section 1.19for description of and 2.)
53
64.
Basic Math and Pre-Algebra
1.15. Prime numbers
A prime number is a number that can only be
divided evenly (not producing a remainder)by itself
and by 1.
Forexample,7canonlybe dividedevenlyby 7andby 1.
Examples of prime numbers are:
{2,3,5,7,11,13,17,19,23}
Zero and 1are not prime numbers.
The only even prime number is 2.
1.16. Rational vs. irrational numbers
In this section, rational numbers and irrational
numbers are defined.
A number is a rationaZ number if it can be expressed
in the form of a fraction, d y , and the denominator is
not zero.
For example, 2 is a rational number because it can
be expressed as the fraction Wl.
Every integer can be expressed as a fraction and is a
rational number.
(Integer)/l or n/l Where n =any integer.
54
65.
Numbers and Their Operations
Whole numbers are included in the set of integers,
and whole numbers are rational numbers.
A number is an irrationd number if it is not a
rational number and therefore cannot be expressed in
the form of a fraction.
Examples of irrational numbers are numbers that
possess endless non-repeating digits to the right of the
decimal point, such as,
n = 3.1415...,and ./z= 1.414...
1.17. Complex numbers
In this section, complex numbers, real numbers
and imaginary numbers are defined, and addition,
subtraction, multiplication and division of complex
numbers are described.
Complex numbers are numbers involving G,(see
chapter on roots and radicals for a description of $>.
There is no number that when squared equals -1.By
definition ( a ) 2 should equal -1,but it does not,
therefore the symbol i is introduced, such that:
./-x =i& Where x is a positive number and (i)2= -1.
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66.
Basic Math and Pre-Algebra
For example, evaluate ( f i ) 2 .
(&O=(i&)(i&) = (i)2(&)(&)
= (i)2 &5) (5) = (-1)(5)= -5
Complex numbers are numbers involving i and are
generally in the form:
x +iy Where x and y are real numbers.
In the expression,x +iy, x is called the real part and
iy is called the imaginary part.
A real number multiplied by i forms an imaginary
number.
(realnumber&) =imaginary number
A real number added to an imaginary number forms
a complex number.
(real number) +(real number)(i)=complex number
To add or subtract complex numbers, add or subtract
the real parts and the imaginary parts separately.
Example: Add (5+4i)+(3+2i).
(5+4i)+(3+2i)= (5+3)+(4i+2i)=8+6i
Example: Subtract (5+4i) - (3+2i).
(5+4i) - (3+2i)=(5- 3) +(4i - 2i)= 2 +2i
56
67.
Numbers and Their Operations
Complex numbers are multiplied as ordinary
binomials, and (i)2 is replaced by -1.(See Chapter 5,
"Polynomials,"Section 5.4 for multiplication of
polynomials in the second book of the Master Math
series entitled Algebra.)To multiply binomials, each
term in the first binomial is multiplied by each term
in the second binomial, and like terms are combined
(added).
Example: Multiply (5+4i) x (3+2i).
(5+4i)(3+2i)
= (5)(3)+(5)(2i)+(4i)(3)+(4i)(2i)
= 15+10i+12i+8(i)2= 15+22i +8(-1)
= 15+22i - 8= (15 - 8)+22i = 7 +22i
To divide complex numbers, first multiply the
numerator and denominator by what is called the
complex conjugate of the denominator. The complex
conjugate of (3+2i) is (3- 2i), and the complex
conjugate of (3- 2i) is (3+2i). The product of a complex
number and its conjugate is a real number. Remember
to replace (i)2by -1.(See chapters 5 entitled
"Polynomials"and 6 entitled "AlgebraicFractions
with Polynomial Fractions" for division and
multiplication of polynomials and polynomial
fractions in the second book of the Master Math series
entitled Algebra.)
57
69.
Numbers and Their Operations
1.18. Rounding,truncating and
estimating numbers
Quickly estimating the answer to a problem can
be achieved by rounding the numbers that are to be
added, subtracted,multiplied or divided,then adding,
subtracting, multiplying or dividing the rounded
numbers. Numbers can be rounded to the nearest ten,
hundred, thousand, million, etc., dependingon their
size.
For example, estimate the sum of (38+43)by
rounding.
Round 38to the nearest ten. 40
Round 43 to the nearest ten. 40
The estimated sum is 40+40= 80
Compare the estimate with the actual sum.
38+43= 81
80is a good estimate of the actual value, 81.
To round a number to the nearest ten, hundred,
thousand, etc., the last retained digit should either be
increased by one or left unchanged accordingto the
followingrules:
If the left most digit to be dropped is less than 5,
leave the last retained digit unchanged.
If the left most digit to be dropped is greater than 5,
increase the last retained digit by one.
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Basic Math and Pre-Algebra
If the left most digit to be dropped is equal to 5,
leave the last retained digit unchanged if it is even
or increase the last retained digit by one if it is odd.
Example: Round the following numbers to the
nearest ten.
11 roundsto 10
65 roundsto 60
538 roundsto 540
65,236 rounds to 65,240
Example: Round the following numbers to the
nearest hundred.
238 roundsto 200
650 roundsto 600
9,436 rounds to 9,400
750 roundsto 800
740 roundsto 700
Example: Round the following numbers to the
nearest tenth. (See Chapter 3,"Decimals.")
5.01 rounds to 5.0
8.38 rounds to 8.4
9.55 rounds to 9.6
9.65 rounds to 9.6
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72.
Basic Math and Pre-Algebra
Example: Estimate the sum of (56+68 +43) by
rounding.
Round each number and add the estimate.
60 +70 +40 = 170
Compare with the actual numbers.
56+68+43=167
Truncating rounds down to the smaller ten, hundred,
thousand, etc. For example,
43 truncates to 40
768 truncatesto 760
1.19. Inequalities,>, <, 2, S
Inequalities are represented by the symbols for
greater than and less than, and describe expressions
in which the value on one side of the symbol is greater
than the value on the other side of the symbol.
The symbol for greater than is >
The symbol for less than is <
The symbol for greater than or equal to is 2
The symbol for less than or equal to is I
If a and b are numbers,
a < b represents a is less than b, or a is to the left of b
on the number line.
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Numbers and Their Operations
a > b represents a is greater than b, or a is to the right
of b on the number line.
a 1b represents a is less than or equal to b.
a 2 b represents a is greater than or equal to b.
a < c < b represents a is less than c and c is less than b.
a > c > b represents a is greater than c and c is greater
than b.
Examples:
2 < 4
4 > 2
5 < 8
8 > 5
2 < 5 < 8
5 < 8
5 1 5
The number line below describesx > 1.
If c is a positive number, then c > 0. For example, if
c=5,then5>0.
If d is a negative number, then d < 0. For example, if
d = -5,then -5< 0.
Example: If a,b and c represent numbers, and
If a < b
Then a + c < b + c
Also a - c < b - c
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75.
Numbers and Their Operations
Example: Ifa, b and d represent numbers, and
If d<O
Andif a < b
Then ad>bd
(If a negative number is multiplied, the inequality
reverses.)
Example: If a =2, b =3 and d = -4, and
Because -4 < O
And 2 < 3
Then (2)(-4) > (3)(-4)
Equivalently -8> -12
(Remember, -12is to the left of -8on the number line.)
Example: If a, b and d represent numbers, and
If d<O
Andif a > b
Then a + d < b + d
(If a negative number is divided, the inequality
reverses.)
Example: If a =4, b =6 and d = -2,and
Because -2 < 0
And 4 < 6
Then 4 + -2 > 6 + -2
Equivalently -2 > -3
(Remember,-2 is to the right of -3on the number line.)
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Basic Math and Pre-Algebra
In summary, for an inequality:
If a number is added to both sides, the inequality
remains unchanged.
If a number is subtracted from both sides, the
inequality remains unchanged.
If a positive number is multiplied, the inequality sign
remains unchanged.
If a positive number is divided, the inequality sign
remains unchanged.
If a negative number is multiplied, the inequality sign
reverses.
If a negative number is divided, the inequality sign
reverses.
1.20. Factorial
The factorial of a positive integer is the product
of that integer and each consecutive positive integer
from one to that number.
The symbol for factorial is "!"(the explanation
point).
Examples of the factorial of numbers are:
5!= 1 * 2 * 3 * 4 * 5 = 120
7!= 1 * 2 * 3 * 4 * 5 * 6 * 7 =5,040
3!= 1 * 2 * 3 = 6
n! = 1* 2 * 3 * ...n
O! = 1 by definition.
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77.
ChaPter 2
Fractions
2.1 Definitions
2.2 Multiplying fractions
2.3 Adding and subtracting fractions with common
denominators
2.4 Adding and subtracting fractions with different
denominators
2.5 Dividing fractions
2.6 Reducing fractions
2.7 Complex fractions, mixed numbers and improper
2.8 Adding and subtracting mixed numbers
2.9 Comparingfractions: Which is larger or smaller?
fractions
2.1. Definitions
A fraction is a number that is expressed in the
a
form a/bor -
b'
A fraction can also be defined as:
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78.
Basic Math and Pre-Algebra
part section some
or -
whole' whole all
If a pie is cut into 8pieces, then 2pieces are a
fraction of the whole pie.
- 2 pieces2 pieces
whole pie
-
8 pieces in whole pie
The top number is the numerator, and the bottom
number is the denominator.
Numerator
Denominator
=Numerator + Denominator
5
Fractions express division. -= 5 + 8
8
Equivalent fractions are fractions that have equal
value. The following are equivalent fractions:
112= 214= 418= 8/16= 16/32= 32/64
213 =416= 8/12= 16/24= 32/48
315 =6/10= 12/20=24/40= 48/80
2.2. Multiplying fractions
To multiply fractions, multiply the numerators,
multiply the denominators and place the product of
the numerators over the product of the denominators.
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79.
Fractions
For example,
9 5 3 9 x 5 ~ 3 135
2 8 4 2 x 8 ~ 4 64
- -- x - x - = -
2 x 3 6 12 3
3 4 3 x 4 12 2
- = - = -- x - =
If the numerators are less than the denominators,
then after multiplying,the value of the product will be
less than the values of the original fractions.This is
true because multiplying a number or fraction by a
fraction is equivalentto taking a fraction of the first
number or fraction.
For example: If you have a piece of a pie (which
means you have a fraction of a pie),then you eat a
fractionof the piece, you have eaten less than a piece.
In other words, you have eaten a fraction of a fraction.
2.3. Adding and subtracting
fractions with common
denominators
To add or subtract two or more fractionsthat
have the same denominators,simply add or subtract
the numerators and place the sum or difference over
the common denominator.
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Basic Math and Pre-Algebra
Examples:
9 5 3 9 + 5 + 3 - 17
8 8 8 8 8
- + - + - = - -
2.4. Adding and subtracting
fractions with different
denominators
To add or subtract two or more fractions that
have different denominators, a common denominator
must be found. Then, each original fraction must be
multiplied by multiplying-fractions that have their
numerator equal to their denominator, so that new
equivalent fractions are created that have the same
common denominator. These fractions can then be
added or subtracted as shown above.The procedure is:
1.Find a common denominator for two or more
fractions by calculating multiples of each
denominator until one number is obtained that is a
multiple of each denominator. This is called a
common multiple.For example, if there are two
denominators, 4 and 6, the multiples of 4 are: 4,8,
12, 16, .., and the multiples of 6 are: 6, 12, 18,24,...
The smallest number that is a multiple of both 4
and 6 is 12.Therefore, 12 is called the lowest
common multiple and it is also the Lowest common
denominator for 4 and 6.
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81.
Fractions
2. After the lowest common denominator is found,
transform the original fractions into fractions with
common denominators so that they can be added or
subtracted. To do this, multiply each original
fraction by a multiplying-fiaction that has its
numerator equal to its denominator, so as to create
new equivalent fractions that have the same common
denominators. By having the numerator equal to the
denominator in these mu1tiplying-fractions, the
value of each of the original fractions remains
unchanged. For example, (1/2)x (2/2) = (2/4),where
1/2 is the original fraction, 2/2 is the multiplying-
fraction and 2/4 is the resulting equivalent fraction
with its value equal to 1/2.
To determine each multiplying-fraction, compare the
new common denominator with the original
denominators of each fraction, then create
multiplying-fractions for each original fraction such
that when the original denominator is multiplied
with the multiplying-fraction'sdenominator, the
resulting denominator is the common denominator.
3. After the equivalent fractions with their common
denominators have been obtained, add or subtract
the numerators and place the sum or difference over
the common denominator.
4. Reduce the resulting fraction if it is possible, (see
section below on reducing fractions).
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Basic Math and Pre-Algebra
For example, add 1/4+116=?.
First, find the lowest common denominator.
Multiples of4: 4,8, 12, 16, ...
Multiples of 6: 6, 12, 18, ...
The lowest common multiple and lowest common
denominator is 12.
Next, multiply each original fraction by a multiplying-
fraction (with its numerator equal to its denominator)
such that each equivalent fraction will have a
denominator of 12.
For the first fraction, because 4 x 3 = 12,
1/4+?/? =?/12
1/4 x 3/3 = 3/12
3/12 is equivalent to 1/4.
For the second fraction,because 6 x 2 = 12,
1/6+?/? =?I12
1/6 x 2/2 = 2/12
2/12 is equivalent to 1/6.
After the equivalent fractions with their common
denominators have been obtained, add the numerators
and place the sum over the common denominator.
3/12 +2/12 = 5/12
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83.
Fractions
Example: Add
?
2 5 1 0 ?
- + - + -9 5 3 -- -
First, find the lowest common denominator.
Multiples of 2: 2,4, 6, 8, 10, ...
Multiples of 5: 5, 10, 15,20, ...
Multiples of 10: 10,20, ...
The lowest common multiple is 10,therefore the
lowest common denominator is 10. (Notice that the
smallest number that 2, 5, and 10 will all divide into
is 10.)(Also, the factors of 10 are: (2)(5) and (1)(lO).)
Next, multiply each original fraction by a multiplying-
fraction (with its numerator equal to its denominator)
such that each equivalent fraction will have a
denominator of 10.
For the first fraction, because 2 x 5 = 10:
9 5 45
2 5 10
- x - = -
For the second fraction, because 5 x 2 = 10:
5 2 10
5 2 10
- x - = -
For the third fraction, because 10x 1= 10:
85.
Fractions
Therefore, the equation is:
9 x 4 - 5 x 1 - - -3 x 2 -
2 x 4 8 x 1 4 x 2
3 6 5 6 3 6 - 5 - 6 25
8 8 8 8 8
----- _ - _ - -
An alternative method for adding and subtracting
two fractions is as follows:
1.Multiply the two denominators to find a common
denominator. The result will be a common multiple
of each, but not necessarily the lowest common
multiple.
Find the new numerator of the first fraction by
multiplying the numerator of the first fraction with
the original denominator of the second fraction.
Find the new numerator of the second fraction by
multiplying the numerator of the second fraction
with the original denominator of the first fraction.
2. Then, add or subtract the new numerators and place
the result over the common denominator.
3.Reduce the resulting fraction if it is possible (see
section below on reducing fractions).
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86.
Basic Math and Pre-Algebra
For example, add 1/2+314.
Multiply denominators. 2 x 4 =8
The new common denominator is 8.
The first new numerator is 1x 4=4.
The second new numerator is 3 x 2 =6.
The new equivalent fractions are:
4 6 10
8 8 8
- + - = -
To reduce 10/8, divide 2 into both the numerator and
the denominator.
1 0 / 2 5
8 / 2 4
- --
2.5. Dividing fractions
To divide fractions, turn the second fraction
upside down to invert the numerator and denominator.
(An inverted fraction is called a reciprocal.) Then,
multiply the first fraction with the reciprocal fraction.
3 . 5 - 3 8 24 - 6
4 8 4 5 20 5
---. ---x--=---
Where 815 is the reciprocal of 5/8.
Multiplying the reciprocal is equivalent to dividing
because:
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88.
Basic Math and Pre-Algebra
Example: Reduce 30/40.
Write the fraction in factored form and cancel the
factors common to both the numerator and
denominator.
3- 3 x 2 ~ 530
40 2 X 2 X 2 X 5 4
-----
Note that by inspection a 10 could have been canceled
from both the numerator and the denominator.
2.7. Complex fractions,mixed
numbers and improper fractions
In this section, improper fractions, mixed
numbers and complex fractions are defined, and
methods for converting improper fractions to mixed
numbers and converting mixed numbers to improper
fractions are described.
Every integer can be expressed as a fraction. For
example, 6 = 6/1, 23 = 23/1.
Improper fractions are fractions with their
numerators larger than their denominators. Examples
of improper fractions are:
13 6 12- - -
2 ' 1 ' 2
78
89.
Fractions
If a value is represented by an integer and a fraction,
it is called a mixed number. A mixed number always
has an integer and a fraction. The following are
examples of mixed numbers:
1 3
6 - , 25-
2 4
An improper fraction can be expressed as a mixed
number by dividing the numerator with the
denominator. For example:
4 1
- -- 1-
3 3
4 1
3 3
Where - is the improper fraction and 1- is the mixed
number.
When performing calculations, it is generally easier
to work with improper fractions than with mixed
numbers.
Mixed numbers are easily converted to improper
fractions. Following are two methods used to convert
mixed numbers into improper fractions.
Method 1
1.Multiply the integer and the denominator.
2. Add the numerator to the result, which results in a
new numerator.
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90.
Basic Math and Pre-Algebra
3. Place the new numerator over the original
denominator.
For example, convert 6 112to an improper fraction.
Multiply the integer and the denominator. (6)(2)= 12
Add the numerator to the result to obtain the new
numerator. 1+12= 13
Place the new numerator over the denominator to
obtain the improper fraction. 13/2
Method 2
1.Find the common denominator.
2. Add the fractions.
For example, convert 6 112to an improper fraction.
1 1 6 1
2 2 1 2
6 - = 6 + - = - + -
The lowest common denominator is 2. Create
equivalent fractions with common denominators of 2
and add. (112 does not need to be multiplied.)
Therefore, 6 112 = 1312.
Improper fractions are easily converted to mixed
numbers by dividing the numerator with the
denominator.
80
91.
Fractions
For example, convert 25/4 to a mixed fraction.
Divide the numerator by the denominator. 25 + 4 = 6
plus a remainder of 1.
Place the remainder over the divisor resulting in 6 1/4.
To check the result, multiply the integer and the
denominator. 6 x 4 =24
Add the numerator and place the result over the
denominator. 24 +1 =25
25/4 is the original improper fraction.
A complex:fraction has a fraction in the numerator or
in the denominator or in both so that there is one or
more fractions within a fraction.
The following are examples of complexfractions
(x and y represent numbers):
1 / 3 3 x + 2 / y 3- -
4 ' 4 / 5 ' 5 ' X / Y
Complexfractions can be simplifiedby performing
the indicated division of the fractions or sub-fractions.
For example, simplifjlthe following (remember when
dividing fractions, multiply the reciprocal).
1 / 3 1 1 4 1 1 1- = - + 4 4 - + - = - x - = -
4 3 3 1 3 4 1 2
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93.
Fractions
Find a common denominator for the two improper
fractions, 13/2 and 10314.
4 is a multiple of both 2 and 4.
Multiply each fraction by a fraction with its numerator
equal to its denominator such that the result is a
common denominator of 4 in each fraction.
Add the numerators and place the result over the
common denominator.
1 3 x 2 103x1
2 x 2 4 x 1
+
- 26 103 26+103 - 129
4 4 4 4
----- -+-
Therefore, 6 1/2 + 25 314 = 129/4.
2.9. Comparing fractions: Which is
larger or smaller?
In this section, methods for comparing the value
of fractions are described.
Two fractions can be directly compared if they have
the same denominator. For example, which of the
followingfractions is larger?
3/4 or 1/2?
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Basic Math and Pre-Algebra
Because 4 is a common multiple of both fractions, it is
also a common denominator. To obtain a common
denominator, multiply 3/4by 1/1and 1/2by 2/2.
1/2x 2/2=2/4
314x l/l= 3/4
The two fractions become:
3/4and 2/4
Because 3 > 2,
3/4> 2/4
To compare several fractions, rather than converting
all of the fractions to the same denominator, it may be
easier to compare two at a time. Which of the following
are larger?
1/8or 3/4or 5/6?
First, compare:
1/8and 3/4
The common denominator is 8.
To obtain a common denominator, multiply 3/4by 2/2.
3/4x 2/2=6/8
The two fractions become:
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Basic Math and Pre-Algebra
To quickly assess whether one fraction is larger or
smaller than another fraction, it may be useful to
determine whether each fraction is larger or smaller
than 112.
The smaller the numerator is in comparison to the
denominator, the smaller will be the value of fraction.
5/10 > 5/100 > 5/1000 > 5/10,000 > 5/100,000 >
5/1,000,000
58/59 is almost 1.
99/985,236is approximately one-tenthousandth.
86
97.
ChaPter 3
Decimals
3.1 Definitions
3.2 Adding and subtracting decimals
3.3 Multiplying decimals
3.4 Dividing decimals
3.5 Rounding decimals
3.6 Comparing the size of decimals
3.7 Decimals and money
3.1. Definitions
The decimal system and decimals are based on
tenths or the number 10.
The digits to the right of the decimal point are called
decimal fi-actions.
A decimal can be expressed in the form of a fraction,
and a fraction can be expressed in the form of a
decimal.
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98.
Basic Math and Pre-Algebra
Decimals that do not have a digit to the left of the
decimal point are written 0.95or .95.Inserting the
zero before the decimal point prevents the viewer from
mistaking .95for 95.
The digits to the right of the decimal point
correspond to tenths, hundredths, thousandths, ten
thousandths,hundred thousandths, etc. For example,
the digits in the number 48.35679correspond to:
4tens, 8ones. 3 tenths, 5 hundredths, 6thousandths,
7ten thousandths, 9hundred thousandths
The decimal point is always present after the ones
digit even though it may not be written. For example:
7 = 7.0 = 7.00= 7.000= 7.0000000
29 = 29.00= 29.0= 29.000000000
The following are examples of the equivalent forms
of decimals and fractions:
0.1 = 1/10
0.01 = 1/100
0.001 = 1/1,000
0.0001 = 1/10,000
0.00001 = 1/100,000
Decimals have equivalent forms as fractions. For
example, the decimal 0.5is equal to the fraction 5/10.
This canbe provenby dividingthe fraction 5/10or
540.
88
99.
Decimals
Using the long division format:
Insert the decimal point and a zero in the tenth
position.
10 divides into 5.0,0.5times, then multiply 10 x 0.5 =
5.0.
a 5
524
00
10&
Therefore, 0.5 is equal to 5/10.
In general, a fraction can be transformed into its
decimal equivalent by dividing the numeratorby the
denominator. (SeeSection 3.4,"Dividingdecimals.")
a 5
110
0 0
1/2=2 L 2)1,0- 2)1.0
Therefore, 1/2=0.5.
89
100.
Basic Math and Pre-Algebra
Proper names that correspond to decimal fractions
are used to describe measurements or quantities in
various units such as grams, moles, seconds,liters,
meters, etc. For example, if a scientist is measuring
extremely small amounts of a chemical in grams,
proper names for the following decimal quantities are:
0.1 = 10-1 = 100milligrams
0.01 = 10-2 = 10milligrams
0.001 = 10-3 = 1milligram
0.0001 = 10-4 = 100micrograms
0.00001 = 10-5 = 10micrograms
0.000001 = 10-6 = 1microgram
0.000000001 = 10-9 = lnanogram
0.000000000001 = 10-12 = 1picogram
0.000000000000001= 10-15 = 1femptogram
3.2. Adding and subtracting
decimals
To add or subtract decimals, align the decimal
points, then proceed with the addition or subtraction
(as if the decimal points are not there).
To add decimals, arrange in column format, add each
column beginning with the right column, and carry over
digits to the next larger column as necessary.
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Basic Math and Pre-Algebra
11
389.32
65.20
2.00
+ .05
56.57
Add the hundreds' column.
I1
389.32
65.20
2.00
+ -05
456.57
Therefore, 389.32+65.2+2+0.05=456.57.
To subtract decimals, arrange in column format,
subtract each columnbeginning with the right column
and borrow when necessary from the next larger
column. If there are more than two numbers, subtract
the first two, then subtract the third from the
difference, and so on.
Example: Subtract 23 - 3.89.
Arrange in column format and subtract the
hundredths'column.
92
103.
Decimals
23.00
- 3.89
?
To borrow from the tenths' column, first borrow from
the ones'column. Subtract the hundredths and tenths.
2 2 . {9}{10}
- 3 . 8 9
. 1 1
To subtract the ones'column,first borrow from the
tens'. Subtract the ones and tens.
1 9 . 1 1
Therefore, 23 - 3.89 = 19.11.
3.3. Multiplying decimals
To multiply decimals, ignore the decimal points
and multiply the numbers, then place the decimal
point in the product so that there are the same number
of digits to the right of the decimalpoint in the product
as there are in the all of the numbers that were
multiplied combined.
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Basic Math and Pre-Algebra
For example, if 2.2 and 0.3 are multiplied, because
there are two digits to the right of the decimalpoints
in 2.2 and 0.3(onefor each number),there must be two
digits to the right of the decimal point in the product.
Therefore, 2.2 x 0.3=0.66.
Example: Multiply 35.268and 2.5.
There are a total number of four digits to the right of
the decimal points in these two numbers, therefore the
product will have four digits to the right of the decimal
point.
Multiply the 5 in the multiplier with each number in
the multiplicand beginning at the right.
35.268
x 2.5
176340
Multiply the 2 in the multiplier with each number in
the multiplicand beginning at the right.
35.268
x 2.5
176340
70536
Add the partial products.
94
105.
Decimals
35.268
X 2.5
176340
70536
Place the decimalpoint four digits from the right.
Therefore, 35.268 x 2.5 = 88.1700.
3.4. Dividing decimals
To divide decimals, arrange the numbers into
the long division format, move the decimal point in the
divisor to the right until there are no digits to the right
of the decimal point, move the decimal point in the
dividend the same number of places to the right so
that the overall value of the division is unchanged
(zeros may be inserted in the dividend as required),
then divide using long division. The decimal point will
be placed in the quotient above where it moved to in
the dividend.
Example: 10 + 0.5 =?
Arrange the numbers into the long division format.
95
106.
Basic Math and Pre-Algebra
?
0.5&
Move the decimal point in the divisor and the dividend
to the right until there are no digits to the right of the
decimal point in the divisor (insert zeros as required).
?
5.)100
Divide 5 into 10.
2?.
5. )100
Subtract 10 - 10= 0, and bring down the last 0.
2?.
5.)100
U!
000
There is nothing for 5 to divide in to, so place a 0 above
the last 0 in the dividend.
20.
5.)100
U!
000
Therefore, 10+ 0.5 = 20.
96
107.
Decimals
3.5. Rounding decimals
To round decimals, the last retained digit
should either be increasedby one or left unchanged
according to the followingrules:
If the left most digit to be dropped is less than 5, leave
the last retained digit unchanged.
If the left most digit to be dropped is greater than 5,
increase the last retained digit by one.
If the left most digit to be dropped is equal to 5, leave
the last retained digit unchanged if it is even or
increase the last retained digit by one if it is odd.
For example, round the followingto the nearest
integer.
3.4 rounds to 3.
2.5 rounds to 2.
45.64 rounds to 46.
Decimals may also be rounded to the nearest tenth,
hundredth, thousandth, etc., depending on how many
decimalplaces there are and the accuracyrequired.
For example,round the following as specified.
45.689 rounded to the nearest tenth is 45.7.
1.9654rounded to the nearest hundredth is 1.96.
1.545454 rounded to the nearest thousandth is 1.545.
97
108.
Basic Math and Pre-Algebra
When solvingcomplexmathematical or engineering
problems, it is important to retain the same number of
"significant digits"in the intermediate and final
results. The number of decimal places in the resulting
numbers should not exceed the number of decimal
places in the initial numbers because the resulting
numbers cannot be known with greater accuracy than
the original numbers. Therefore,depending on the
least number of significant digits in the initial
numbers, rounding intermediate and final results will
be required to maintain the decimalplaces to ones,
tenths, hundredths,thousandths, etc.
For example, if 45.689 and 1.9654are added, the
accuracy of the result cannot be greater than three
decimal places.
45.689 +1.9654 = 47.6544= 47.654
3.6. Comparingthe size of decimals
To comparethe value of two or more decimals to
determine which decimal is a larger or smaller, the
followingprocedure can be applied:
1.Place the decimals in a column.
2. Align the decimal points.
3. Fill in zeros to the right so that both decimals have
the same number of digits to the right of the decimal
point.
98
109.
Decimals
4. The larger decimal will have the largest digit in the
greatest column (thefarthest to the left).
Example: Compare 0.00025 and 0.000098.
Place the decimals in a column, align the decimal
points, and fill in zeros.
0.000250
0.000098
The 2 in the ten-thousandths place is greater than the
9 in the hundred-thousandths place. Therefore:
0.000250 is larger than 0.000098.
3.7. Decimals and money
There is a relationship between money and
decimals because the money system is based on
decimals. Dollars are represented by whole numbers
and cents are represented by tenths and hundredths of
a dollar.
For example, $25.99 = 25 dollars and 99 cents.
The following are equivalent: Ninety-nine
hundredths of a dollar, 99 cents, 99/100 of a dollar,
and 0.99 dollars.
99
110.
Basic Math and Pre-Algebra
Examples:
10cents is one tenth or 1/10of a dollar or 0.1 dollars.
70 cents is seven tenths or 7/10 of a dollar or 0.7
dollars.
1 cent is one hundredth or M O O of a dollar or 0.01
dollars.
6 cents is six hundredths or 61100of a dollar or 0.06
dollars.
The amount, $0.50 is 1/2of a dollar. Similarly, the
decimal 0.5 equals the fraction 1/2.
The word cent refers to one hundred, (remember a
century has 100years), and in money, cents refers to
the number of hundredths.
100
111.
Chapter 4
Percentages
4.1 Definitions
4.2 Figuring out the percents of numbers
4.3 Adding, subtracting,multiplying and dividing
4.4 Percent increase and decrease (percent change)
4.5 Simple and compound interest
percents
4.1. Definitions
Percent is defined as a rate or proportion per
hundred, one one-hundredth,or M O O .
The symbol for a percent is %.
Apercent is a form of a fraction with its denominator
equal to 100.To remember that percent is per one-
hundred, think of a century, which has 100years.
A percent can be convertedto a fraction or decimal.A
percent can be reduced and manipulated like a fraction
or decimal.
101
114.
Basic Math and Pre-Algebra
Percents of a number can also be determined by
moving the decimal point according to the percent
amount. The following examples represent percents of
the number nine:
1%of9.Move the decimal to the left two digits. = 0.09
10%of 9.Move the decimalto the left one digit. = 0.9
100%of 9.The decimal does not move. = 9
1000%of 9. Move the decimalto the right one digit.
= 90
10,000%of 9.Move the decimal to the right two digits.
= 900
100,000% of 9. Move the decimal to the right three
digits. = 9,000
The following are examples of percents of numbers:
What is 6 percent of 30?
6 x U100 of 30 =6 x U100 x 30
= 6/100x 30= 180/100= 1.80
What is 5 percent of 20?
5 x 1/100of 20 = 5 x 1/100x 20 = 100/100= 1.00
What is 20 percent of 80?
20 x M O O of 80= 20 x 1/100x 80= 1600/100= 16
The examples above can also be solved using an
alternative method.
104
115.
Percentages
What is 6 percent of 30?
Because 1%of 30 is 0.30, and 6%is 6 times 1%,
then 0.30x 6 = 1.80.
What is 5 percent of 20?
Because 10%of 20 is 2, and 5%is half of 10%,
then half of 2 is 1.
What is 20 percent of 80?
Because 10%of 80 is 8, and 20%is 2 times 10%,
then 8x 2 = 16.
4.3. Adding, subtracting,multiplying
and dividing percents
To add or subtract percents, simply add or
subtract as with integers but keep track of the percent.
For example, add or subtract the following:
26% +4%= 30%
158%- 6%= 152%
3%+16%+4%=23%
For example, if you have 50%of your assets in stocks
and 20%of your assets in bonds, then 70%of your
assets are in stocks and bonds.
50%in stocks+20%in bonds= 70%in stocks and bonds
105
117.
Percentages
4.4. Percent increase and decrease
(percent change)
In this section,percent change and discount are
discussed.
To determine the percent that a number has
increased or decreased, the following equation is used:
?- -
amount of increase or decrease (change)
-
original amount 100
This equation can also be written:
amount of increase or decrease
(change) (100) = percent change
original amount
Example: If the price of a house is discounted from
$250,000 to,$200,000, what is the percent decrease or
discount in the price?
$50/000 (100)(100)=
$250,000 - $200,000
$250,000 $250,000
= (0.2)(100) =20%
107
119.
Percentages
4.5. Simple and compound interest
Simple interest is generally computed annually,
therefore the time it will take to earn a certain percent
on money invested in a simple interest account is a
year.
Example: How much simple interest will $5,000
earn in a year and in 6 months at a rate of 4%?
$5,000 x 4% = ($5,000)(0.04)= $200 will be earned in a
year.
In the first 6 months, $5,000will earn:
$200 x (6 months)/(l2 months) = $200 x 1/22= $100
Therefore, in 6 months the $5,000becomes $5,100
and in one year the $5,000 becomes $5,200.
Compound interest is compounded periodically during
the year. To determine the compound interest, divide
the interest rate by the number of times it is
compounded in the year and apply it during each
period.
For example, if the interest rate is 4% and it is
compounded semiannually, then 2%is applied to the
principle every 6 months. If4%is compounded
quarterly then 1%is applied to the principle every 3
months or four times a year.
109
120.
Basic Math and Pre-Algebra
Example: How much interest will $5,000 earn in six
months and in a year at 4% interest compounded
semi-annually?
Divide the interest by the number of times it is
compounded. 4% + 2 = 2%
Apply 2% interest for the first 6-monthperiod.
$5,000x 2% = ($5,000)(0.02)= $100 earned in 6
months.
The second 6-monthperiod begins with $5,100.
Apply 2% interest again for the second 6-monthperiod.
$5,100 x 2%= ($5,100)(0.02)= $102 earned in the
second 6-monthperiod.
In 6 months the $5,000 becomes $5,100
And in one year the $5,000 becomes $5,202
Note that compound interest earns slightly more
than simple interest for the same rate.
110
121.
ChaPter 5
Converting Percentages,
Fractions and Decimals
5.1 Converting fractions to percents
5.2 Convertingpercents to fractions
5.3 Converting fractions to decimals
5.4 Converting decimals to fractions
5.5 Converting percents to decimals
5.6 Converting decimals to percents
5.1. Convertingfractions to percents
To convert a fraction to a percent, divide the
numerator by the denominator, then multiply by 100.
For example,convert 315 into a percent.
Divide the numerator by the denominator using long
division.
111
123.
ConvertingPercentages,Fractions and Decimals
5.3. Convertingfractionsto decimals
Toconvert a fraction to a decimal,divide the
numerator by the denominator.
For example,convert 3/5 into a decimal.
Divide the numerator by the denominator using long
division.
? l6
-3.0
0.0
5p = 5)3.0
Therefore,3/5=0.6.
5.4. Convertingdecimals to fractions
Toconvert a decimal to a fraction,place the
decimal-fractionover the tenth, hundredth,
thousandth, ten-thousandth, hundred-thousandth,etc.,
that it corresponds to, then reduce the fraction.
For example,convert the following decimals into
their fractional form.
113
127.
Converting Percentages, Fractions and Decimals
5.6. Convertingdecimalsto percents
To convert a decimal to a percent, multiply by
100.
Note that multiplying by 100is equivalent to moving
the decimal point two places to the right.
Example: Convert the decimal 0.25 to its percent
form.
Multiply 0.25 by 100.
0.25 x 100= 25%
Two final notes:
If two decimals are multiplied, the product will have
the same number of digits to the right of its decimal
point as in the multiplicand and multiplier combined.
When convertingbetween fractions, percents and
decimals, think about whether the result seems
reasonable. For example, if the fraction 314 is
converted to a percent and the result is 0.0075%,it
does not seem reasonable that 0.0075%could be
equivalent to three-fourths.Instead, 75%does seem
reasonable.
117
128.
Chanter 6
Ratios, Proportions
and Variation
6.1 Definitions
6.2 Comparing ratios to fractions and percents
6.3 Variation and proportion
6.1. Definitions
In this section,ratios and proportions are
defined.
Ratios depict the relation between two similar
values with respect to the number of times the first
contains the second. A ratio represents a comparison
between two quantities. For example,if the ratio
between apples and oranges in a fruit bowl is 3 to 2,
then for every 3 applesthere are 2 oranges.
Ratios are written 3 to 2, 3:2 and 3/2.
The difference between ratios and fractions is that
fractions represent part-per-whole,and ratios
represent part-per-part.
118
129.
Ratios, Proportions and Variation
Proportions are a comparative relation between the
size, quantity, etc., of objects or values. Mathematical
proportions represent two equal ratios. For example,
the following are equivalent:
1/2 =2/4, 1:2 =2:4, 112is proportional to 2/4, 1:2::2:4
Note that ina proportion,the product ofthe"extreme"
outer numbers equals the product of the "mean"
middle numbers. 1:2::2:4 can be written 1x 4 =2 x 2.
When equivalent fractions, such as 112 = 214, are
cross-multiplied(by multiplying the opposite
numerators with the opposite denominators),the two
products are always equal. This principle is useful
when determining the quantity of one of the items
when the ratio is known.
For example, how many apples are there in a
mixture containing 10oranges, if the ratio is 3 apples
to 2 oranges? To solve this, set up the proportion:
3 apples - ? apples
2 oranges 10 oranges
-
Using cross-multiplicationthe equation becomes:
3 ~ 1 0 = 2 x ?
30=2x?
Divide both sides by 2.
3012 = 1x ?
15 =?
Therefore, there are 15 apples in this mixture.
119
130.
Basic Math and Pre-Algebra
6.2. Comparing ratios to fractions
and percents
The differencebetween ratios and fractions is
that fractions represent part-per-whole,and ratios
represent part-per-part.
Ratios can be expressed in the form of a fraction (or
in the form of division).
The definition of a ratio differs from that of a
fraction but all the rules for manipulating fractions
apply to ratios.
The ratio 1to 2 (or 1:2 or 1/2)can be expressed as
50%or 0.50.
Example: If Tom ate 3 pies for every 5 pies Ted ate,
what is the ratio of pies eaten by Tom and Ted?
Tom : Ted
(3 pies) / (5pies)
Tom ate 3/5the number of pies that Ted ate.
Tom ate 3/5 = 0.60 = 60%of the number of pies that
Ted ate.
If a ratio of two values is known, the percent that one
of the values is of the total can be determined by
adding the parts to get the whole, placing the one value
over the whole, and multiplying by 100.
120
131.
Ratios, Proportions and Variation
For example, what percent of the total number of
pies did Tom and Ted eat?
The ratio is (3pies) to (5pies), which represents
partlpart.
And the percent of the total number is
(part/whole)(100).
If the total number or whole is 3 pies +5 pies = 8 pies.
Therefore, the percent of the total number of pies Tom
ate is 3 pied8 pies = 3/8 = 0.375 and 0.373 x 100=
37.5%
Also, the percent of the total number of pies Ted ate is
5 pies/8 pies = 5/8 = 0.625and 0.625x 100= 62.5%
6.3. Variation and proportion
Variation describes how one quantity varies or
changes as another quantity varies or changes.
If as one number increases then decreases, another
number increases then decreases simultaneously, the
numbers are in direct variation or direct proportion to
each other. The equation for direct variation or direct
proportion is:
121
132.
Basic Math and Pre-Algebra
The k represents some constant number, and x and y
represent numbers that vary directly.As x increases,y
increases. As x decreases,y decreases.
If as one number increasesthen decreases, another
number decreasesthen increases simultaneously,the
numbers are in indirect variation and are said to be
inversely proportional.The equation for indirect
variation representing inverselyproportional
quantities is:
xy =k which can be rearranged as y =Wx
The k represents some constant number, and x and y
represent numbers that vary indirectly. As x increases,
y decreases.As x decreases, y increases.
122
133.
Powers and Exponents
7.1 Definition of the exponent or power
7.2 Negative exponents
7.3 Multiplying exponents with the same base
7.4 Multiplying exponents with different bases
7.5 Dividing exponents with the same base
7.6 Dividing exponents with different bases
7.7 Raising a power of a base to a power
7.8 Distributing exponents into parenthesis
7.9 Addition of exponents
7.10 Subtraction of exponents
7.11 Exponents involvingfractions
7.1. Definition of the exponent or
power
An exponent represents a number that is
multiplied by itself the number of times as the
exponent or power defines.
123
134.
Basic Math and Pre-Algebra
The following exponent represents 2 raised to the 6th
power:
26 =2 x 2 x 2x 2 x 2 x 2 =64
2 is called the base and 6 is the exponent or power.
Examples of exponents are 26,257 and 21965.
If a positive whole number is raised to a power
greater than one, the result is a larger number.
If a number (except zero) is raised to the 1stpower,
the result is the number itself. 31 = 3.
If a number is raised to the zero power, the result is
1.
3 0 = 1
aO= 1
a = any number except zero
00=undefined
7.2. Negative exponents
In this section, negative numbers raised to even
and odd powers and positive and negative numbers
raised to negative powers are presented.
If a negative number is raised to a positive even
power, the result is a positive number. For example:
(-2)' = (-2)(-2)(-2)(-2)(-2)(-2) = 64
124
136.
Basic Math and Pre-Algebra
7.4. Multiplying exponentswith
different bases
To multiply exponents with different bases,
each exponent must be expressed individually, then
multiplied. The powers cannot be added directly.
For example:
32 x 24= (3x 3)x (2x 2 x 2 x 2)=9x 16 = 144
ab x dC = abdc Where a, b, c and d represent numbers.
To multiply exponents with different bases but the
same power, the following law of exponents applies (a,
b and c represent numbers):
aCxbc=(axb)c
Which can also be written:
7.5. Dividing exponents with the
samebase
To divide exponents with the same base,
subtract the powers.
126
140.
Basic Math and Pre-Algebra
Consider the following examples (a,b and c
represent numbers):
(ab)2= (a2x b2)=a2b2
(2b)2=(22x b2)=4b2
If numbers or variables inside parenthesis are added
or subtracted, the exponent outside cannot be
distributed as above. Instead:
(a+b)2= (a+b)*(a+b)
=a2+ ab +ab +b2= a2+ 2ab +b2
(See multiplication of binomials in Chapter 5 of the
second book in the Master Math series,Algebra.)
7.9. Addition of exponents
To add exponents whether the base is the same
or different, express each exponent individually,then
add. The powers cannot be added directly.
130
142.
Basic Math and Pre-Algebra
Example: Express the followingfraction.
abxc abc
(2bP (22xb2) 4b2
= -(abY -- -
A number raised to a negative power is equivalent to
one-over that number.
If a positive fraction is raised to a power, and the
numerator is smaller than the denominator, the value
of the resulting fraction is less.
(1/2)2= (1/2)(1/2)= 114 Where 1/4< 112.
If a positive fraction is raised to a power, and the
numerator is larger than the denominator, the value of
the resulting fraction is greater.
(512)2= (5/2)(512)=2514 Where, 2514 > 512.
Remember, 5/2 = 10/4.
Consider the following examples with fractional
exponents:
3112 = &
32/3=(3113)2 =(32)113
132
143.
ChaPter 8
Logarithms
8.1
8.2
8.3
8.4
8.5
8.6
Definition of the logarithm
Common (base ten) and natural logarithm
Solving equations with logarithms or exponents
Exponential form and logarithmic form
Laws of logarithms: addition, subtraction,
multiplication, division, power and radical
Examples: the Richter scale, pH and radiometric
dating
8.1. Definition of the logarithm
The logarithm is the exponent of the power to
which a base number must be raised to equal a given
number.
A logarithm is the inverse of an exponent.
Each exponent has an inverse logarithm.
133
144.
Basic Math and Pre-Algebra
Consider the followingexponential equation (xand y
represent numbers):
x =a y
Where a is an integer and is the base of the exponent.
The inverse of this equation is:
y =log,x
Where a is the base of the logarithm.
The inverse is obtained by taking the base a logarithm
of both sides of the exponential equation.
x =a y
Because log,(a) cancels, the equation becomes:
Example: Convert the exponential equation 24 = 16
to a logarithmic equation.
Because the exponent has a power of 2, take the base 2
logarithm of both sides.
Because log2(2)cancels, the equation becomes:
4 = log2(16)
134
145.
Logarithms
8.2. Common (baseten) and natural
logarithm
The two most common logarithms are the
common logarithm, also called the base ten logarithm,
and the natural logarithm.
The common or base ten logarithm is written:
loglox or simply logx
The natural logarithm is written:
log& or lnx
Where e = 2.71828182846.
The relationship between the common logarithm and
the natural logarithm can be written:
Or equivalently:
In x = (2.3026)logx
135
146.
Basic Math and Pre-Algebra
8.3. Solving equations with
logarithms or exponents
To solve logarithmic equations, take the inverse
or exponent.
Example: Solve the equation y =log x for x.
Because the base is 10,raise both sides of the
equation by base 10.
Because lO(10gio) cancels itself, lO(10gX) = x
The equation becomes:
1 o y = x
Example: Solve the equation y =In x for x.
Because the base is e, raise both sides of the equation
by e to isolate x.
Because eln cancels itself, e(lnx)= x
The equation becomes:
136
147.
Logarithms
To solve exponential equations, take the inverse or
logarithm.
Example: Solvethe equationy = 10xfor x.
Because the base is 10,take log10 of both sides of the
equation.
logy =log(10.)
Because loglo(10)cancels itself, log(1 0 ~ )=x
The equation becomes:
logy=x
Example: Solve the equationy =e~ for x.
Because the base is e, take In of both sides of the
equation.
In y =ln(ex)
Because ln(e)cancels itself, ln(ex)=x
The equation becomes:
lny=x
137
149.
Logarithms
To approximate or compute numerical values of
logarithms or exponents, refer to tables contained in
mathematical handbooks and selected algebra books,
or use the function keys of technical calculators.
8.5. Laws of logarithms:addition,
subtraction, multiplication,
division,power and radical
The following are laws of logarithms
(a represents any base):
Also, note that: logb a = l/logab
Where a and b represent numbers.
Because a represents any base, these laws apply to
base e (natural logarithms) or base ten (common
logarithms). Proofs of the laws of logarithms can be
found in selected algebra books.
139 | 677.169 | 1 |
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About this product
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Description
The Australian Curriculum Edition Targeting Maths Year 5 Student Book has been specifically written to meet the Australian Curriculum requirements of primary school Year 5. The Targeting Maths Year 5 Student Book features: Australian Curriculum Alignment - Complete coverage of all Year 5 year content descriptions for mathematics; Fully integrated Problem-solving Program - Actively builds students' abilities to think mathematically, problem solve and communicate their answers in a variety of ways; Term Investigations - Students plan and work through problems that extend their mathematical fluency. In-stage topic alignment for composite classes - Each Targeting Maths Student Book in Stage 3 (Student Book Year 5 and Year 6) match topic-by-topic. Regular Revision - Regular revision pages maintain skills.
Key Features
Author(s)
Garda Turner
Publisher
Pascal Press
Date of Publication
30/09/2013
Language
English
Format
Paperback
ISBN-10
1742152244
ISBN-13
9781742152240
Subject
School Textbooks & Study Guides: Maths, Science & Technical
Series Title
Targeting Maths
Publication Data
Place of Publication
Glebe
Country of Publication
Australia
Imprint
Pascal Press
Dimensions
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Fantastic for home schooling, together with our regular curriculum. Would have benefited from knowing that there was no answers with it, and a link to how to get them: as I had to contact several people to find out how to access answers. But, book fantastic! | 677.169 | 1 |
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Best Math Software for Windows
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A standalone Windows program that computes Parallel Analysis criteria (eigenvalues) to determine the number of factors to retain in a factor analysis by performing a Monte Carlo simulation. The user can specify 3-300 variables, 50-2500 subjects, andKonSi Malmquist Index SoftWare is convenient program for your research of efficiency dynamics using Malmquist Index method. Malmquist Index and its components are calculated for many periods. The program creates reports with calculated values of Fixed -...
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The aim of this Reader is not merely to afford the student a certain amount of experience in reading scientific German, but to attack the subject systematically. The selections are not chosen at random. They are arranged progressively and consist of fundamental definitions, descriptions, processes and problems of Arithmetic, Algebra, Geometry, Physics and Chemistry. These are linguistically the most important subjects for scientific and engineering students to read first, because they contain the terms and modes of expression which recur in all subsequent reading, and because they contain these terms in the simplest possible connections | 677.169 | 1 |
Product Description:
MASTER ONE LIFE'S MOST USEFUL SKILLS--EVEN IF YOU'VE NEVER BEEN GOOD AT MATH to master the subject than "Algebra Demystified! Entertaining author and experienced teacher Rhonda Huettenmueller provides all the math background you need and uses practical examples, real data, and a totally different approach to life the "myst" from algebra. With "Algebra Demystified, you master algebra one simple step at a time--at your own speed. Unlike most books on the subject, general concepts are presented first --and the details follow. In order to make the process as clear and simple as possible, long computations are presented in a logical, layered progression with just one execution per step. THIS ONE-OF-A-KIND SELF-TEACHING TEXT OFFERS: * Questions at the end of every chapter and section to reinforce learning and pinpoint weaknesses * A 100-questions final exam for self-assessment * An intensive focus on word problems and fractions--help where it's most often needed * "Detailed examples and solutions Whether you want to learn more about algebra, refresh your skills, or improve your classroom performance, "Algebra Demystified is the perfect shortcut.
REVIEWS for Algebra Demystified | 677.169 | 1 |
PRIOR KNOWLEDGE: Students should know how to evaluate expressions with exponents, have worked with perfect squares, have a knowledge of inverse operations, place numbers on a number line, understand the concept of irrational numbers, and multiply rational numbers.
STANDARDS DRIVEN: Students are primarily expected to understand that squares and square roots are inverse operations, that the square root pf perfect squares are rational and square roots of non-perfect squares are irrational, use rational numbers to approximate the value of irrational roots. Students will be expected to extend this to approximating irrational roots to determine which rational approximation is closer to irrational roots. This lesson follows a lesson on solving multi-step linear equations in one variable and precedes a lesson on number sets.
MATERIALS NOTES:
This is an editable word file that includes a partial key as margin notes / comments (these can be hidden in the "Review" Pane) I create my documents in compatibility mode because it allows users more control over pictures and graphs. To ensure that you are getting the best quality, you should not save the file in any new format as certain design elements may be | 677.169 | 1 |
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Do your students dislike carrying a big textbook around campus? We can provide an unbound, three-hole-punched version of the traditional text so that your students can carry just what they need. This unbound version comes with access to MyMaythLab or MyStatLab at a significant discount from the price of the regular text. These authors have created a book to really help students visualize mathematics for better comprehension. By creating algebraic visual side-by-sides to solve various problems in the examples, the authors show students the relationship of the algebraic solution with the visual, often graphical, solution. In addition, the authors have added a variety of new tools to help students better use the book for maximum effectiveness to not only pass the course, but truly understand the material. | 677.169 | 1 |
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Unformatted text preview: ELEMENTARY MATHEMATICS W W L CHEN and X T DUONG c W W L Chen, X T Duong and Macquarie University, 1999. This work is available free, in the hope that it will be useful. Any part of this work may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, with or without permission from the authors. Chapter 9 COMPLEX NUMBERS 9.1. Introduction It is easy to see that the equation x 2 + 1 = 0 has no solution x R . In order to solve this equation, we have to introduce extra numbers into our number system. Define the number i by writing i 2 + 1 = 0. We then extend the collection of all real numbers by adjoining the number i, which is then combined with the real numbers by the operations addition and multiplication in accordance with the rules of arithmetic for real numbers. The numbers a + b i, where a, b R , of the extended collection are then added and multiplied in accordance with the rules of arithmetic for real numbers, suitably extended, and the restriction i 2 + 1 = 0. Note that the number a + 0i, where a R , behaves like the real number a . Definition. We denote by C the collection of all complex numbers; in other words, the collection of all numbers of the form a + b i, where a, b R ....
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This note was uploaded on 10/19/2010 for the course MATHEMATIC Math123 taught by Professor Goh during the Spring '10 term at UCLA. | 677.169 | 1 |
Solving Systems of Linear Equations Task Cards
PDF (Acrobat) Document File
Be sure that you have an application to open this file type before downloading and/or purchasing.
3.09 MB | 55 pages
PRODUCT DESCRIPTION
Save time and engage students in valuable skills practice with this resource! The Solving Systems of Linear Equations Task Cards features 32 linear systems that use the elimination and substitution methods. Solutions and coefficients are all integers.
This resource correlates to the Common Core State Standards in Mathematics too! High School Algebra standard Reasoning with Equations and Inequalities C.6: Solve systems of linear equations exactly and approximately, focusing on pairs of linear equations in two variables.
This resource includes:
-- 32 unique task cards
-- Differentiation ideas for use
-- Directions for use as a set of task cards, a whole group presentation for use with a projector and/or smart board, and as a walkaround activity.
-- Organized Student Answer Sheets
-- Answer Key
This resource is useful for classrooms of all sizes and time frames. With 32 problems, teachers can modify the number of problems to complete into convenient groups of 4, tailoring to their own classroom needs.
Remember that this is for use by only one teacher per purchase. Additional licenses are available for half of the original purchase price | 677.169 | 1 |
Math for Kids - English Welcome to AL KALAM mobile application. You can using this application for learn 1, 2, 3 in English. Have fun learning with AL KALAM mobile a
Jasymca Jasymca is a symbolic calculator. It solves and manipulates equations, handles basic calculus problems, and provides a few more typical functions of computer algebra systems
TVH-72g Graphing Calculator Graphing calculator for the educational use of schools and students who may have difficulty, are unable or may not wish to acquire a stand-alone feature-laden graphing calculator
Math Training Math Training is a fun way to practice your basic arithmetic | 677.169 | 1 |
The PLACe
Math Help
Math Anxiety Stops Here!
The PLACe is committed to providing tutorial services for the academic achievement of Miramar College students in the areas of mathematics and science. To enhance students' performance, we offer one-on-one or group tutoring in pre-algebra, beginning and intermediate algebra, college algebra, statistics, geometry, trigonometry, pre-calculus, calculus, differential equations, biology and chemistry.
The PLACe also provides supplemental instruction in beginning and intermediate algebra classes, and workshops on selected math topics. In addition, we also have supplementary materials for our math students: interactive computer tutorials, video tapes, audio tapes, most course textbooks and solutions manuals that students can use.
The PLACe also recognizes that many students may have math anxiety, and we are equipped to help students conquer their fears about math.
About Math Anxiety
Our tutors can help with your math anxiety. They've been trained to help you find ways to lower the stress and worry about math. They love math but understand it can be a challenge for students.
Math Anxiety Advice from Our Tutors:
Don't talk negatively to yourself. Act and talk to yourself with confidenceLearners review the fundamental laws of algebra including the commutative law of addition, the commutative law of multiplication, the associative law of addition, the associative law of multiplication, and the distributive law. Examples are givenIn this animated activity, learners listen to instructions for simplifying a complex fraction by finding the least common denominator. This learning object requires a computer that is equipped with speakers.
In this animated object, learners use an algebraic formula to solve the following problem: An airplane travels a certain distance with the wind in the same amount of time that it takes to travel a shorter distance against the wind. Given a constant wind speed, what is the speed of the plane without a wind?
In this animated and interactive object, learners examine the definitions and formulas for radius, diameter, circumference, and area. Students also solve practice problems involving the circumference and area of a circle.
This learning object gives a description of direct, inverse, joint, quadratic, and combined variation, and walks the student through a problem of each type. The students learn to create the correct formula from the given problem as well as to solve it. There are also technical application problems given at the end for more practice. | 677.169 | 1 |
Description
These video tutorials are presented by Roderick V. James PhD. EE, Adjunct Professor at Houston Community College. Dr. James has been involved with education for the past ten years. His past teaching experience at the Keller Graduate School of Management of DeVry University included the duties of Curriculum Manager for Project Management.Â
Currently, Dr. James teaches a range of math courses including Fundamentals of Mathematics I & II, College Algebra, Trigonometry, Pre-Calculus, Calculus, Statistics and Finite Mathematics.
The Business of X
In the study of mathematics, once you get past arithmetic you run into theses pesky variables that gives students fits. Every one seems comfortable with 6+4 -(10÷2)=5. In general students have fewer problems with arithmetic and "order of operations" than they do with Algebra, The principal reason for this is the introduction of variables that are generally called "x".
What is this business of "x"?
In math we encounter two types of quantities, constants and variables. Constants are numbers like 3, 5, 11, 1725. These numbers have a value that NEVER changes. Five will always be 5.
Variables, on the other hand change, can change their value depending on the circumstance. If we have a problem involving several unknown quantities then we can assign each unknown quantity its own variable. For example if there are three unknowns we can call them x, y, and z. To solve our problem we find the values for x, y and z that are meaningful for the particular problem.
Another aspect of Algebra that give students trouble is the concept of a function. In simple terms a function describes the relationship between variables. When we say y is a function of x we are talking about two variables and the relation between them. For example if you live in the United States you are accustomed to hearing the temperature given in degrees Fahrenheit. If you live elsewhere you are are probably accustomed to hearing the temperature given in degrees Celsius! There is a relationship between these two measures so that one can be converted to the other. This is an example of a function. Since I know the relationship I can find one given the other.
Lastly we introduce you to the "vocabulary" of math. There are certain key words that lead to a precise mathematical operation. For example it I have 5 oranges more than you, we can turn that into a mathematical expression. Even if I do not know exactly how many oranges you have, I can say that you have "x" oranges. In that case I will have "x + 5" oranges. Thus "more" translates to the mathematical operation "addition".
What we are attempting to do is give you a brief introduction to the wonderful world of Algebra!!!!!
Do you know how to install the app?
Download MathCast | 677.169 | 1 |
Basic College Math Documents
Showing 1 to 30 of 133
Name
Chapter 2: Linear Functions
MATH 111 Sec. 009
Goals of Chapter 2
Calculate and interpret average rates of change
Understand how representations of data can be biased
Recognize that a constant rate of change denotes a linear relationship
Construct a l
Math 111I Sec 001 Name
Group Quiz 5 Solutions
Directions: Read each question carefully and answer in the space provided. The use of calculators is allowed, but not necessary. There are 10 points possible. To receive partial credit on any problem work must | 677.169 | 1 |
Calculus II Advice
Calculus II Documents
Showing 1 to 11 of 11Introduction to the Equation Editor!
Alrighty Say we want to type in the integral of x^2/5 from x=0 to x=4. How to do this? Look above. On
the Insert tab, youll find a button that says Equation. Itll have the Pi symbol on it. Click that symbol.
Itll bring
Examination 1 - Math 142, Frank Thorne (thorne@math.sc.edu)
Thursday, September 26, 2013
Instructions and Advice:
There are seven questions. Each of them is essentially the same as one of the required or additional problems
from your homework, sometimes
Homework 13 - Math 142, Frank Thorne (thornef@mailbox.sc.edu)
Due Friday, December 5
(a) What is a power series?
(b) Describe an example of a power series which converges for all values of x.
(c) Describe an example of a power series which converges onlyShowing 1 to 1 of 1
A warning. This class is very difficult, and the professor can be tough. It will teach you, however, how to stick with something once you start it, how to problem solve, and how to work with your peers to figure out concepts.
Course highlights:
This class covers the fundamentals of Calculus II. This is important in any math related field. You also learn how to problem solve, which is especially important in engineering.
Hours per week:
9-11 hours
Advice for students:
Be prepared to be challenged, and don't be scared. This class is tough, but once its over, you will feel very accomplished. | 677.169 | 1 |
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Mathematical Analysis: Foundations and Advanced Techniques for Functions of Several Variables builds upon the basic ideas and techniques of differential and integral calculus for functions of several variables, as outlined in an earlier introductory volume. The presentation is largely focused on the foundations of measure and integration theory.The book begins with a discussion of the geometry of Hilbert spaces, convex functions and domains, and differential forms, particularly k-forms. The exposition continues with an introduction to the calculus of variations with applications to geometric optics and mechanics. The authors conclude with the study of measure and integration theory - Borel, Radon, and Hausdorff measures and the derivation of measures. An appendix highlights important mathematicians and other scientists whose contributions have made a great impact on the development of theories in analysis.This work may be used as a supplementary text in the classroom or for self-study by advanced undergraduate and graduate students and as a valuable reference for researchers in mathematics, physics, and engineering. One of the key strengths of this presentation, along with the other four books on analysis published by the authors, is the motivation for understanding the subject through examples, observations, exercises, and illustrations.
Title:
Mathematical Analysis
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Birkhauser Boston
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Foundations and Advanced Techniques for Functions of Several Variables | 677.169 | 1 |
Beschreibung
Beschreibung
Each undergraduate course of algebra begins with basic notions and results concerning groups, rings, modules and linear algebra. That is, it begins with simple notions and simple results. Our intention was to provide a collection of exercises which cover only the easy part of ring theory, what we have named the "Basics of Ring Theory". This seems to be the part each student or beginner in ring theory (or even algebra) should know - but surely trying to solve as many of these exercises as possible independently. As difficult (or impossible) as this may seem, we have made every effort to avoid modules, lattices and field extensions in this collection and to remain in the ring area as much as possible. A brief look at the bibliography obviously shows that we don't claim much originality (one could name this the folklore of ring theory) for the statements of the exercises we have chosen (but this was a difficult task: indeed, the 28 titles contain approximatively 15.000 problems and our collection contains only 346). The real value of our book is the part which contains all the solutions of these exercises. We have tried to draw up these solutions as detailed as possible, so that each beginner can progress without skilled help. The book is divided in two parts each consisting of seventeen chapters, the first part containing the exercises and the second part the solutions. | 677.169 | 1 |
Product Description:
Integration is one of the two cornerstones of analysis. Since the fundamental work of Lebesgue, integration has been presented in terms of measure theory. This introductory text starts with the historical development of the notion of the integral and a review of the Riemann integral. From here, the reader is naturally led to the consideration of the Lebesgue integral, where abstract integration is developed via measure theory. The important basic topics are all covered: the Fundamental Theorem of Calculus, Fubini's Theorem, $L_p$ spaces, the Radon-Nikodym Theorem, change of variables formulas, and so on. The book is written in an informal style to make the subject matter easily accessible. Concepts are developed with the help of motivating examples, probing questions, and many exercises. It would be suitable as a textbook for an introductory course on the topic or for self-study.
REVIEWS for An Introduction to Measure and Integration | 677.169 | 1 |
The Math Barrier: An Unfortunate Reality
When you're a math major in college like I was, you can be guaranteed at least two things. First, many people will assume you're a human calculator because they'll ask you to make outrageous computations on the spot. Second, fellow students will ask you for help with their own math homework. Not coincidentally, I found that in both of these cases, people were not only struggling with math, but also failing to understand what math even is. And in most cases, they didn't view math the way I view math: as something that enables us all to be better thinkers who reason more logically and solve problems more strategically. Instead, they viewed math as an obstacle—a confusing barrier they needed to just get past so they could leave it behind forever.
Math is a very real barrier for many students, as described in the recent New York Times article, Community College Students Face a Very Long Road to Graduation by Ginia Bellafante. According to the article, "over 60 percent of all students entering community colleges must take what are called developmental math courses … [that] are algebra-based and focus on linear and quadratic equations." Bellafante writes that these developmental courses are required for graduation and four-year degree programs, "but more than 70 percent of those students never complete the classes, leaving them unable to obtain their degrees."
While all students have unique personal circumstances that impact their ability to access and complete courses, I'd like to examine the aspects of these math courses that are within the control of schools and educators. After all, linear and quadratic equations are topics that high school students learned in the classes I myself taught, and they're even topics that elementary and middle school students are successfully learning about through our DreamBox lessons—often before they even enter high school. So why are adults at community college having such a difficult time passing the courses? Three explanations were highlighted in the article:
"Many [students] have been taught so poorly before they arrive."
"They have developed a debilitating reliance on calculators."
"The pedagogy* tends to focus on computation rather than the underlying concepts, leaving the practice of math to seem far removed from the students' experiences."
It wasn't entirely clear from the article whether the pedagogy noted in point three referred to students' math classes prior to entering community college or after entering community college, but I was relieved to see pedagogy highlighted as a key factor because it is crucial to understanding why students struggle with math. Whether we're talking about K–12 math classes or college courses, it's important to know what "better" teaching and learning in mathematics looks like. For good descriptions, I suggest this profile of Cornell mathematician Steven Strogatz in The Atlantic. Or this great piece on the AMS teaching blog by Morgan Mattingly, a current undergraduate student at the University of Kentucky. Morgan is a double major in Mathematics and STEM Education, and here is part of how she describes her classroom and learning experience:
I have seen firsthand that it occasionally takes experimentation to figure out which method or tool to use in problem solving. Through discussion and group work with my classmates, I noticed that it is not always blatantly obvious that we should draw a picture or use induction or reformulate a hypothesis to find the crux move in a solution.
Despite the fact that these two articles describe how students can engage in learning and doing mathematics in a way that is enjoyable, empowering, and inspiring, the unfortunate reality is that most developmental math courses are not structured this way, either in K–12 or at the college level. As evidence, consider the community college student in Bellafante's article who complains, "This developmental algebra is a stainless-steel wall and there's no way up it, around it or under it." The Atlantic article profiling Strogatz describes similar problems at other colleges and universities, noting that math classes for liberal arts majors are "typically presented in lecture format … and only serve to further disenfranchise students" (Lahey, 2014).
These developmental math classes likely adhere to a flawed model of cognition that Grant Wiggins and Jay McTighe refer to as the "climb the ladder" approach. This approach requires learners to "master all relevant discrete skills before they can be expected to apply them in more integrated, complex, and authentic ways" (Wiggins & McTighe, p. 45). In Schooling by Design, they describe the critical flaws with this approach and why it would logically lead many students to drop out of a math class:
Because [low-achieving students] are less likely to have acquired the basics on the same schedule as more advanced learners, struggling learners are often confined to an educational regimen of low-level activities, rote memorization of discrete facts, and mind-numbing skill-drill worksheets. The unfortunate reality is that many of these students will never get beyond the first rung of the ladder and, therefore, have minimal opportunities to actually use what they are learning in a meaningful fashion. Who wouldn't be inclined to drop out under such conditions? (p. 45)
I would add a fourth reason to the list of causes for these high dropout and failure rates in community college courses: assessment and grading practices. Bellafante's article describes the following aspects of one community college assessment system for developmental math:
The final exam accounted for 35 percent of the total grade
Students received demerits for poor attendance (presumably impacting the grade)
Homework impacted the final grade substantially enough that students knew uncompleted assignments would prevent the final exam from making a significant difference
Each of these three components of the assessment system is familiar and typical, but each has flaws that negatively impact students because these grading practices don't accurately communicate student learning. For example, Bellafante describes a student who believed he could score well on the final—certainly the most current data of student understanding and presumably the most summative as well—but still fail the course because he didn't complete enough homework. Many other assessment systems enable the opposite—allowing a student who completed all her homework to pass the course even if she miserably failed the final. Connecting behaviors and responsibility to a content-based grade is dangerous; it's misleading at best and unfair and illegitimate at worst. For more insights into why these and other familiar grading practices are detrimental to student learning and achievement, I highly recommend Ken O'Connor's book, A Repair Kit for Grading: Fifteen Fixes for Broken Grades.
Given these unintentionally harmful instructional and grading practices in developmental math courses, it's not surprising that developmental math is an insurmountable barrier for many college students. And as Bellafante's article noted, "more than 40 percent of the students in the class failed, at least one for the third time." These high failure rates at colleges all across the country are prompting many schools, organizations, and companies to find or create solutions for students and educators. Educational technology and blended learning are being embraced as potential supports, not only for college students, but also as intervention alternatives for high school students. In theory, these are promising options because students can learn at their own pace and access coursework and lessons from virtually anywhere at any time (provided they have the necessary technology access).
Unfortunately, many blended learning courses and programs merely digitize the same flawed instructional and assessment practices used in the developmental math courses. As Michael Fullan and Katelyn Donnelly wrote in Alive in the Swamp: Assessing Digital Innovations in Education,
Technology–enabled innovations have a different problem, mainly pedagogy and outcomes. Many of the innovations, particularly those that provide online content and learning materials, use basic pedagogy—most often in the form of introducing concepts by video instruction and following up with a series of progression exercises and tests (p. 25).
Patrick White, a student at Harvard's Graduate School of Education participating in Justin Reich's course on MOOCs, recently described the impact of some of these "innovations" on students. In his blog post, "Thoughtless vs. Thoughtful Blended Learning," Patrick writes about high school students taking night school courses for credit recovery, a popular use of blended learning:
… night school instruction was questionable … I heard over and over again about students who never watched or read through any of the instruction material. They simply clicked through screens until they got to assessments and Googled to find answers. Even in rooms where teachers did not allow that practice, instruction from the computer relied on basic "read this" followed by "now answer these questions" approach no different than many textbook-style education methods. Students never had the chance to engage in any activities, projects, or even class discussions to augment their learning. It was all basic regurgitation aided by a teacher who frequently did not even share a background with the content that a student was "learning."
Students need to do more than simply regurgitate basic facts to earn or recover math credits—they need to be powerful and confident mathematical thinkers. These problems in developmental math classroom practices, online math instruction, and blended learning models are topics I'll be discussing in my presentation at the 2014 iNACOL Blended and Online Learning Symposium. Because students deserve the very best learning opportunities and experiences available—with and without technology—I'll be sharing thoughts about the evolution of blended and competency-based schooling. It's important for all educators to look beyond the current schooling and lesson models and envision better educational experiences for both students and educators.
After working in public education for over ten years as a classroom teacher and curriculum leader, I saw the very real challenges students and teachers experience in mathematics. Because I have seen how technology can help students become powerful mathematical thinkers, I joined DreamBox Learning as the Curriculum Director to help imagine and create new learning experiences for students and educators. Every day, our team supports student learning of mathematics starting as early as preschool, and we design our lessons and assessments so that students are confident thinkers and doers of mathematics no matter what their grade level or where they start. One of Strogatz's quotes from the Atlantic article nicely sums up our approach at DreamBox:
As with any game, or playing music, or making a piece of art, it's doing the real thing that's inspiring. My students are actually making mathematics—in many cases, for the first time in their lives. And they're loving it. And why wouldn't they? It's a joyous, glorious experience. At every level. Little kids can make math. It may be the mathematical equivalent of finger painting, but it's still math.
The work of math educators won't be complete until all students are loving to do mathematics, comfortable in making mathematics, and confident enough that they will see mathematics as empowering rather than exasperating. Mathematics should never be a barrier for anyone.
* This reference was the only time the term pedagogy appeared in the entire 3,000-word New York Times article. Understandably, this article was neither intended to be an educational research study nor an analysis of educational institutions and practices. The words used to describe problems and solutions to important educational problems influence our ability to focus on the root causes of these problems. Therefore educators should take note that the terms assessment, learning, and instruction never appeared in the article. And the words curriculum and test only appeared twice. As educators create solutions to the important problems highlighted in this article, the role of these elements is critical because they constitute the overwhelming portion of the students' schooling experience.
VP of Learning for DreamBox Learning, Inc., Hudson is a learning innovator and education leader who frequently writes and speaks about learning, education, and technology. Prior to joining DreamBox, Hudson spent more than 10 years working in public education, first as a high school mathematics teacher and then as the K–12 Math Curriculum Coordinator for the Parkway School District, a K–12 district of over 17,000 students in suburban St. Louis. While at Parkway, Hudson helped facilitate the district's long-range strategic planning efforts and was responsible for new teacher induction, curriculum writing, and the evaluation of both print and digital educational resources. Hudson has spoken at national conferences such as ASCD, SXSWedu, and iNACOL. | 677.169 | 1 |
Math Proofs Demystified
Overview philosophy and law students grappling with proofs. This book is the perfect resource for demystifying the techniques and principles that govern the mathematical proof area, and is done with the standard "Demystified" level, questions and answers, and accessibility.
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Stan Gibilisco is one of McGraw-Hill's most prolific and popular authors. His clear, reader-friendly writing style makes his math and science books accessible to a wide audience, and his background in research makes him an ideal editor for professional handbooks. He is the author of Geometry Demystified; Trigonometry Demystified; Statistics Demystified; Everyday Math Demystified; Physics Demystified; Electronics Demystified; Electricity Demystified; The TAB Encyclopedia of Electronics for Technicians and Hobbyists; Teach Yourself Electricity and Electronics; and The Illustrated Dictionary of Electronics. Booklist named his McGraw-Hill Encyclopedia of Personal Computing a "Best Reference" of 1996.
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This book is different from any other math proofs book I have found. It teaches you how to think logically. So many people can't do that these days. If you are an advanced student and you want to see how the most complicated proofs are done, you probably don't need this book. But if you're an average person, and you have to work with 'or argue with' people logically, this is a great book. I have also spoken with several beginning geometry and algebra students who were helped by this book, especially when they went on to 'Geometry Demystified.'
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More than 1 year ago
I'm currently enrolled into geometry honors and was desperately in need of proof practice. This book provided weak or nearly any knowledge that could have helped me during class. I would not recommend this book to any readers. The algebra book was fine, but the proof and geometry demystified books are horrible. Please consider saving your money and not spending it on worthless books. | 677.169 | 1 |
A Special Application of Absolute Value Techniques in Authentic Problem Solving
Stupel, Moshe
International Journal of Mathematical Education in Science and Technology, v44 n4 p587-595 2013
There are at least five different equivalent definitions of the absolute value concept. In instances where the task is an equation or inequality with only one or two absolute value expressions, it is a worthy educational experience for learners to solve the task using each one of the definitions. On the other hand, if more than two absolute value expressions are involved, the definition that is most helpful is the one involving solving by intervals and evaluating critical points. In point of fact, application of this technique is one reason that the topic of absolute value is important in mathematics in general and in mathematics teaching in particular. We present here an authentic practical problem that is solved using absolute values and the "intervals" method, after which the solution is generalized with surprising results. This authentic problem also lends itself to investigation using educational technological tools such as GeoGebra dynamic geometry software: mathematics teachers can allow their students to initially cope with the problem by working in an inductive environment in which they conduct virtual experiments until a solid conjecture has been reached, after which they should prove the conjecture deductively, using classic theoretical mathematical tools. (Contains 3 figures.) | 677.169 | 1 |
Mathematics and Statistics Colloquiums
The primary mission of the Department of Mathematics and Statistics is to provide high quality mathematics education to mathematics majors, to students in other majors enrolled in mathematical content courses, to students in general education courses, and to students in developmental mathematics courses. As a service to the College and University, it also is part of the department's mission to provide leadership and expertise in questions of mathematics and statistics as necessary across the disciplines.
Mathematics and Statistics Colloquiums
The Role of Mathematics, Statistics, and Computer Science in Bioinformatics
Dr. Todd A. Gibson
As an example to introduce the field of bioinformatics, Dr. Gibson discusses the Human Genome Project, a 13-year, $3 billion project launched in 1990 by the Department of Energy and the National Institute of Health. The project, drawing from several academic disciplines, mapped genetic variation among human genomes – quite a task, considering the variations within human DNA. Currently, scientists are working on a follow-up study, The 1000 Genomes Project. For this project, the genomes of over 2,600 people from 26 populations around the world are being mapped. Dr. Gibson's talk explains the role of mathematics, statistics, and computer science in the history and breadth of the bioinformatics field. | 677.169 | 1 |
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Description
Many colleges and universities require students to take at least
one math course, and Calculus I is often the chosen option.
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focus on the skills students most need to succeed in a subject | 677.169 | 1 |
Key Stage Three (Years 7 and 8)
The Maths department aims to teach the subject so that students not only develop fluency with essential skills but also understand why important methods and techniques 'work'. Real-life contexts are used to impress upon students the immense power of the subject and the fact that without Maths, many things we take for granted (computers, mobile phones, the Internet, etc.) simply would not exist. The Key Stage 3 programme of study is taught in four discrete modules: Number; Shape and Space; Algebra; and Data Handling. At the end of each module, students are taught 'how' to revise for examinations by developing the skills of note-making and condensing information. These revision sessions are followed by four end-of -module tests, which are largely based on past GCSE questions, ensuring that students become used to the various types of problems that they are likely to tackle in real life.
Key Stage Four (Years 9, 10 and 11)
Students complete a three-year study for two separate GCSE qualifications: Mathematics and Numeracy. The structure is the same as Key Stage 3 to ensure continuity and consistency of methodology; four discrete modules are again delivered, followed by the application of revision skills and subsequent end-of- module testing. Again, the questions within these tests are from previous GCSE papers which ensures that students prepare for external examinations in an incremental fashion while at the same time allowing department staff to effectively target areas for improvement with individual pupils. Both qualifications mentioned above are assessed solely by means of external examination; there is no coursework element with- in the Mathematics GCSE specification.
Key Stage Five
Students follow the WJEC A Level specification. Six modules are delivered by staff members between whom there is much specialist knowledge. The content at this level builds on that studied at GCSE level and students are afforded the opportunity to study concepts that underpin almost every aspect of our daily lives, such as derivatives, the motion of objects and the in-depth analysis of quantitative data. Studying Mathematics at A Level makes you see the whole world in a different light.
"I like Mathematics because the teachers make difficult topics easy to understand" – Bryony McNulty
"I look forward to Mathematics because the teachers are always so positive" – Megan Rosser | 677.169 | 1 |
Showing 1 to 3 of 6
Learning calculus is so interesting and exciting! It truly is a perfect mathematical science, and there are so many practical applications, particularly if you are thinking about going into a STEM field.
Course highlights:
The highlights of this course are without a doubt the integral and volume sections. Learning about the area under a curve has so many applications, and learning about volumes of rotated functions is interesting and fun.
Hours per week:
6-8 hours
Advice for students:
Learn how to study math! Do extra problems, work as hard as you can, and never be ashamed if you don't understand something or need to get some extra help. It will be worth it, but you have to put in the effort!
Course Term:Fall 2015
Professor:Mrs Nawrocki
Jan 27, 2017
| Would recommend.
This class was tough.
Course Overview:
I would recommend it if you enjoy a challenge. This course challenges you and teaches you skills that are very helpful if you are planning to go to college. It can also get you credit for that class for college cheaper than it is at a university due to it being an AP class.
Course highlights:
The highlights of this course is that it's at an upper level than other math classes available. It makes you think. You can see how math is related to the real world, and solve problems that you didn't think could be solved before.
Hours per week:
6-8 hours
Advice for students:
My advice for this class, is to study and to make sure you're caught up. This class goes at a faster rate than most classes, and so you don't want to get caught behind. Make sure you understand the content, and if you don't ask for help. Go to tutoring when needed, and make flash cards for the equations you learn about.
Course Term:Fall 2016
Professor:Jesse Ruiz
Course Required?Yes
Course Tags:Math-heavyBackground Knowledge ExpectedGreat Discussions
Jan 09, 2017
| Would highly recommend.
Not too easy. Not too difficult.
Course Overview:
Having a math class that is different than an algebra or geometry one is definitely nice for a change. Most students found the concepts to be either more or less difficult than algebra so if you don't like algebra then this class is probably more up your alley.
Course highlights:
I gained a real life application of math. Students always ask teachers when they're ever going to need to use their math skills in real life. This class is a perfect example of the kinds of information you will use most often. Trends, patterns, probabilities are all relevant.
Hours per week:
3-5 hours
Advice for students:
Do your homework!! Statistics is an introductory class so it's highly likely that everyone going in will have little to no knowledge on the subject. This makes doing your homework extremely important because if you don't do it, you will have no bank of information to use on the tests or final. | 677.169 | 1 |
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Economics Masters Refresher Course in Mathematics Lecturer: Jonathan Wadsworth Teaching Assistant: Tanya Wilson Aim: to refresh your maths and statistical skills, to the level needed for success in the Royal Holloway economics masters. Why do we need maths/statistics? -to improve your understanding of Economics. -As such everything in the course tailored to try to bring out the relevance of the techniques you will (re)learn help the analysis of Economic issues
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Economics Masters Refresher Course in Mathematics How to do it? Lectures 10-1 and classes 2:30-3:30 each day. In the afternoons and evening you will be expected to: –Read the text –Do the daily problem set. –Prepare to present your answers to the class the next day.
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Vectors and matrices Learning objectives. By the end of this lecture you should: –Understand the concept of vectors and matrices –Understand their relationship to economics –Understand vector products and the basics of matrix algebra Introduction Often interested in analysing the economic relationship between several variables Use of vectors and matrices can make the analysis of complex linear economic relationships simpler
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Lecture 1. Vectors and matrices 1. Definitions Vectors are a list of numbers or variables – where the order ultimately matters –E.g. a list of prices, a list of marks in a course test. –(Price labour, Price capital ) –(25, 45, 65, 85) Since this is just a list can store the same information in different ways Can enter the list either horizontally or vertically (25, 45, 65, 85)or
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Dimensions of a vector If a set of n numbers is presented horizontally it is called a row vector with dimensions 1 x n (so the number of rows is always the 1 st number in a dimension and the number of columns is always the 2 nd number) Eg a = (25, 45, 65, 85) is a 1 x 4 row vector If a set of n numbers is presented vertically it is called a column vector with dimensions n x 1 Eg (Note tend to use lower case letters (a b c etc) to name vectors ) Definitions
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Special Types of vector A null vector or zero vector is a vector consisting entirely of zeros e.g. (0 0 0)is a 1 x 3 null row vector A unit vector is a vector consisting entirely of ones denoted by the letter i e.g. is a 4 x 1 unit column vector Definitions
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Adding vectors General rule If a = (a 1, a a n ) and b = (b 1, b b n ) then a+b = (a 1 + b 1, a 2 + b 2,......a n + b n ) e.g. (0 2 3 ) + ( 1 0 4) = (0+1, 2+0, 3+4) = (1 2 7) (same rule for addition of column vectors ) Note that the result is a vector with the same dimension1 x n Note you can only add two vectors if they have the same dimensions e.g. you cannot add (0 2 3) and (0 1)(1 x 3 & 1 x 2) Note also that sometimes adding two vectors may be mathematically ok, but economic nonsense Eg simply adding factor prices together does not give total input price More definitions and some rules
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Multiplying vectors In general you cannot multiply two row vectors or two column vectors together Eg a = (1 2) b = (2 3) ? But there are some special cases where you can And you can often multiply a column vector by a row vector and vice versa. We'll meet them when we do matrices More on multiplication
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Example. Miki buys two apples and three pears from the Spar shop. Apples cost £0.50 each; pears cost £0.40 each. How much does she spend in total? Answer: This can be seen as an example of a vector product –Write the prices as a vector: p=(0.5, 0.4) –Write the quantities as a vector q=(2, 3) –Total Expenditure (=Price*Quantity) is the sum of expenditures on each good –found by multiplying the first element of the first vector by the first element of the second vector and multiplying the second element of the first vector by the second element of the second vector & adding the results –Spending = 2x x0.4 = £ Vector products – a special case of vector multiplication
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1.What are the dimensions of the following? 2. Can you add the following (if you can, provide the answer)? 3. Find the dot product Instant Quiz
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Idea: vectors can also be thought of as co-ordinates in a graph. A n x 1 vector can be a point in n –dimensional space E.g. x = (5 3) – which might be a consumption vector C= (x 1,x 2 ) Geometry 5 3 x1x1 x2x2 A straight line is drawn out from the origin with definite length and definite direction is called a radius vector
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Using this idea can give a geometric interpretation of scalar multiplication of a vector, vector addition or a "linear combination of vectors" Eg If x = (5 3) then 2x = (10 6) and the resulting radius vector will overlap the original but will be twice as long Geometry 5 3 x1x1 x2x2 Similarly multiplication by a negative scalar will extend a radius vector in the opposite quadrant 10 6
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Also can think of a vector addition as generating a new radius vector between the 2 original vectors Eg If u = (1 4) and v = (3 2) then u+v = (4, 6) and the resulting radius vector will look like this Geometry 1 2 x1x1 x2x2 Note that this forms a parallelogram with the 2 vectors as two of its sides u+v u v
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Can also think of a geometric representation of vector inner (dot) product Remember A geometric representation of this is that the dot product measures how much the 2 vectors lie in the same direction Special Cases a)If x.y=constant then the radius vectors overlap Eg x=(1 1) y =(2 2) So x.y= (1*2 + 1*2) = 4
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Can also think of a geometric representation of vector inner (dot) product Remember A geometric representation of this is that the dot product measures how much the 2 vectors lie in the same direction Special Cases If x.y= 0 then the radius vectors are perpendicular (orthogonal) Eg x=(5 0) y =(0 4) So x.y= (5*0 + 0*4) = 0 Y=(0,4) X=(5 0) x ┴ y
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Will return to these issues when deal with the idea of linear programming (optimising subject to an inequality rather than an equality constraint)
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Linear dependence A group of vectors are said to be linearly dependent if (and only if) one of them can be expressed as a linear combination of the other If not the vectors are said to be linearly independent Equally linear dependence means that there is a linear combination of them involving non-zero scalars that produces a null vector E.g. are linearly dependent Proof. Because x=2y or x – 2y = 0 Butare linearly independent Proof. Suppose x – ay = 0 for some a or other. In other words, Then 2 – a = 0 and also 1 – 3a = 0. So a = 2 and a = 1/3 – a contradiction
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Linear dependence Generalising, a group of m vectors are said to be linearly dependent if there is a linear combination of (m-1) of the vectors that yields the m th vector Formally, the second definition: Let x 1, x 2, …x m be a set of m nx1 vectors If for some scalars a 1, a 2, …a m-1 a 1 x 1 + a 2 x 2 + … a m-1 x m-1 = x m then the group of vectors are linearly dependent, But also a 1 x 1 + a 2 x 2 + … a m-1 x m-1 – x m = 0 which is the first definition Summary. To prove linear dependence find a linear combination that produces the null vector. If you try to find such a linear combination but instead find a contradiction then the vectors are linearly independent
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Quiz II A group of vectors are said to be linearly independent if there is no linear combination of them that produces the null vector 1. are linearly independent. Prove it. 2.are linearly independent. Prove it. 3. What about?
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Quiz answers A group of vectors are said to be linearly independent if there is no linear combination of them that produces the null vector 1. are linearly independent. Suppose not, so that 2a + 0 = 0 and a+1 = 0 for some a, then a = 0 and a = -1 – a contradiction. 2. Suppose not then 2a+0 = 0 and 2a+3 = 0 for some a, but then a = 0 and a = -1.5 – a contradiction 3. No… x = y + 2z so x – y – 2z = 0
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Rank The rank of a group of vectors is the maximum number of them that are linearly independent are linearly independent. So the rank of this group of vectors is 2 are linearly dependent, (2x=y) So the rank is 1 Any two vectors in this group are independent (x=y+2z) so the rank is 2
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Vectors: Summary Definitions you should now learn: –Vector, matrix, null vector, scalar, vector product, linear combination, linear dependence, linear independence, rank 4 skills you should be able to do: –Add two nx1 or 1xn vectors –Multiply a vector by a scalar and do the inner (dot) product –Depict 2 dimensional vectors and their addition in a diagram –Find if a group of vectors are linearly dependent or not.
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Matrices 1.Introduction Recall that matrices are tables where the order of columns and rows matters –E.g. marks in a course test for each question and each student AndyAhmadAnka Qn Qn Qn
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In general any system of m equations with n variables (x1, x2,...xn) Can be written in matrix form Ax = b where
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Dimensions of a matrix are always defined by the number of rows followed by the number of columns So is a matrix with m rows and n columns. Example So A is a 2 x 5 matrix(2 rows, 5 columns) B is 5x2 matrix Definitions
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Can think of vectors as special cases of matrices Hence a row vector is a 1 x n matrix B is a 1 by 3 row vector Similarly a column vector is an n x 1 matrix So C is a 3 x 1 column vector ( note the content is the same as B. This means that you can store the same information in different ways) Definitions
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Sometimes we wish to refer to individual elements in a matrix E.g. the number in the third row, second column. We use the notation a ij (or b ij etc.) to indicate the appropriate element i refers to the row j refers to the column Example So a 13 = 3 and a 21 = 7 If What is b 21 ? Definitions
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The null matrix or zero matrix is a matrix consisting entirely of zeros A square matrix is one where the number of rows equals the number of columns i.e. nxn Eg. For a square matrix, the main (or leading) diagonal is all the elements a ij where i = j So in the above the main diagonal is (2 1) Definitions
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The identity matrix is a square matrix consisting of zeros except for the leading diagonal which consists of 1s: We write I for the identity matrix If we wish to identify its size (number of rows or columns) we write I n Definitions
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A diagonal matrix is a square matrix with a ik = 0 whenever i ≠ k And so consists of zeros everywhere except for the main diagonal which consists of non-zero numbers (so the identity matrix is a special case of a diagonal matrix since it has just ones along the main diagonal) Definitions
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Trace of a matrix is the sum of the elements on the main (leading) diagonal of any square matrix So tr(A) = = -4
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1.What are the dimensions of A? 2.What is a 21 ? 3.Is B a square matrix? 4.What is the largest element on the main diagonal of B? 5.What is the value of the largest element on the main diagonal of B? Mini quiz
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The transpose of a matrix A is obtained by by turning rows into columns and vice versa swapping a ij for a ji for all i and j We write the transpose as A' or A T ( A "prime") A symmetric matrix is one where A' = A A positive matrix is one where all of the elements are strictly positive A non-negative matrix is one where all of the elements are either positive or zero More Definitions
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Useful Properties of Transposes 1.(A')' = A -The transpose of a transpose is the original matrix 2. (A + B)' = A' + B' -The transpose of a sum is the sum of the transposes 3. (AB)' = B'A' -The transpose of a product is the product of the transposes in reverse order Eg. Given Find (AB)'and B'A'
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A negative matrix is one where none of the elements are positive A strictly negative matrix is one where all of the elements are strictly negative C is strictly positive and symmetric; B is negative; A is neither positive nor negative. More Definitions
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Adding matrices -add each element from the corresponding place in the matrices. i.e. if A and B are m x n matrices, then A+B is the m x n matrix where c ij = a ij +b ij for i = 1,..,m and j = 1,…,n. You can only add two matrices if they have the same dimensions. e.g. you cannot add A and B Some rules
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Multiplying by a scalar When you multiply by a scalar (e.g. 3, 23.1 or -2), then you multiply each element of the matrix by that scalar Example 1: what is 4A if Example 2: what is xB if Matrix Multiplication
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In general multiplication of 2 or more matrices has some special rules 1.The first rule is that the order of multiplication matters. In general AxB (or AB) is not the same as BA (so this is very different to multiplying numbers where the order doesn't matter – e.g. 3x4 = 4x3 = 12 ) Asides: Addition, multiplication, matrix multiplication etc. are examples of operators An operator is said to be commutative if x operator y = y operator x for any x and y (the order of multiplication does not matter) Addition is commutative: x+y = y+x; multiplication is commutative; subtraction is not commutative (2-1 ≠ 1-2) Matrix multiplication is commutative but matrix multiplication is not
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2. The second rule is that you can only multiply two matrices if they are conformable Two matrices are conformable if the number of columns for the first matrix is the same as the number of rows for the second matrix If the matrices are not conformable they cannot be multiplied. Example 1: does AB exist? –Answer: A is a 2x4 matrix. B is a 3x3. –So A has 4 columns and B has 3 rows. –Therefore AB does not exist. A and B are not conformable. Multiplying two matrices
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Example 2: does AB exist? –Answer: A is a 2x5 matrix. B is a 5x2. So A has 5 columns and B has 5 rows. A and B are conformable. –Therefore AB exists. –How to find it? Multiplying two matrices
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Finding AB –A and B are conformable. So C = AB exists and will be a 2 x 2 matrix (no. of rows of A by no. Columns of B) To calculate it: i.To get the first element on the first row of C take the first row of A and multiply each element in turn against its corresponding element in the first column of B. Add the result. –Example: c 11 = (4x4) + (-1x-1) + (3x3) + (3x3) + (0x1) = 35 ii.To get the remaining elements in the first row: repeat this procedure with the first row of A multiplying each column of B in turn. Multiplying two matrices
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So top left hand element is And top right hand element is Multiplying two matrices
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......and so on to give Multiplying two matrices
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Suppose A is an mxn matrix and and B is an nxr matrix with typical elements a ik and b kj respectively then AB =C where element c ij is : Note that the result is an mxr matrix Multiplying two matrices - formally
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Can you multiply the following matrices? If so, what is the dimension of the result? 1.BA 2.BC 3.AA ' 4.A ' A 5.CC Quiz.
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Answers Finding CC (sometimes written C 2 ).
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Recall: 1.A square matrix has the same number of rows and columns – it's nxn 2.The identity matrix is a square matrix with 1s in the leading diagonal and 0s everywhere else. E.g. Multiplying square matrices by the identity matrix
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The usefulness of the identity matrix is similar to that of the number 1 in number algebra Since IA = AI = A - if multiply a matrix by the identity matrix the product is the original matrix Eg (leave it to you to show AI=A)
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1.If A is a square matrix then IA = AI = A NB. This only applies to square matrices A general result for square matrices
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This result can be useful sometimes to help solve matrix algebra Since if AI = A Then AIB = (AI) B = A (BI) = AB -The inclusion of the identity matrix does not affect the matrix product result (since like multiplying by "1" (will see example of this in econometrics EC5040) Also note that -An identity matrix squared is equal itself Any matrix with this property AA = A Is said to be idempotent
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Idea: In standard multiplication every number has an inverse (except maybe zero unless you count infinity) The inverse of 3 is 1/3; the inverse of 27 is 1/27, the inverse of -1.1 = -1/1.1 Also a number times its inverse equals 1: x(1/x) = 1 and the inverse times the number equals 1: (1/x)x = 1 and the inverse of the inverse is the original number 1/(1/x) = x Inverses for (square) matrices
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The rules for the inverse of a matrix are similar (but not identical) If A is an nxn matrix then the inverse of A, written A -1, is an nxn matrix such that: 1.AA -1 = I 2.A -1 A = I Notes: 1.This means that A is the inverse of A -1 2.But... 3.A -1 may not always exist Inverses for (square) matrices
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1.Supposeand Then So given AA -1 =Iit must be that in this case B=A -1 An Inverse matrix example
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In general need to introduce some more terminology before can invert a matrix 1.Only square matrices can be inverted. Not all square matrices can be inverted however 2.A matrix that can be inverted is said to be nonsingular (so squareness is a necessary but not sufficient condition to invert) 3. The sufficient condition is that the columns (or rows since it is square) be linearly dependent - think of this as being separate equations so the equations must be independent (n equations and n unknowns) if a solution is to be found
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Eg So that the 1 st row of A is twice that of the 2 nd row and there is linear dependence One equation is redundant (no extra information) and the system reduces to a single equation with 2 unknowns So no unique solution for x 1 and x 2 exists
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Rank of a matrix The idea of vector rank can be easily extended to a matrix The rank of a matrix is the maximum number of linearly independent rows or columns If the matrix is square the maximum number of independent rows must be the same as the maximum number of independent columns If the matrix is not square then the rank is equal to the smaller of the maximum number of rows or columns, ρ<=min(rows, cols) If a matrix of order n is also of rank n, the matrix is said to be of full rank Important: Only full rank matrices can be inverted Matrix ranks are closely linked to the concept of determinants
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Let A be an nxn matrix then the determinant of A is a unique number (scalar), defined as: (1) Notes: In each term there are three components: 1.(-1) 1+j 2.a 1j 3.Det(A 1j ) 4.What does this mean? Start with a 2 x 2 matrix which gives a single number (scalar) as the answer – as do all determinants Can you see how this relates to equation (1) ? Determinants
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Eg What is the determinant of So matrices that are not full rank – have linear dependent rows/columns - have zero determinants (will come back to this) and are singular Determinants
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The determinant of a matrix is defined iteratively 1.An nxn is calculated as the sum of terms involving the determinants of nx(n-1)x(n-1) (ie n!) matrices 2.Each (n-1)x(n-1) matrix determinant is the sum of terms involving n-1 determinants of (n-2)x(n-2) matrices and so on 3.Since we know how to calculate the determinant of a 2x2 matrix we can always use this definition to find the determinant of an nxn matrix 4.In practice we shall not go above 3x3 matrices (unless using a computer program) but we need to know the general formula for an inverse Determinants.
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General properties of determinants 1)If B = A', then det. B = det. A 2)If B is constructed from A by swapping two rows, then det. B = -det. A 3)If B is constructed from A by swapping two columns, then det. B = -det. A 4)If B is constructed from A by multiplying one row (or column) by a constant, c, then det. B = c det. A 5)If B is constructed from A by adding a multiple of one row to another, then det. B = det. B
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Determinants of triangular matrices Are examples of – respectively – an upper triangular and a lower triangular matrix (zeros below or above the main diagonal) The determinant of either an upper or lower triangular matrix is equal to the product of the elements on the main diagonal Eg det.A = 1(24-0) - 2(0) + 3(0) = 24 = 1*4*6
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For a 3 x 3 matrix, using Question: What is the determinant of Determinants
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Method 1: Laplace expansion of an n x n matrix. Can generalise this rule for the determinant of any n by n matrix As part of this method, you need to know the following: 1.Minor M 2.Co-factor C (which are also essential to invert a matrix)
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There is a minor M ij for each element a ij in the square matrix. 1.To find it construct a new matrix by deleting the row i and deleting the column j. 2.Then find the determinant of what's left 3.E.g. M 11 4.Example M 12 nxn Matrix inversion - minors 123 A= Delete row and column
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The cofactor is C ij is a minor with a pre-assigned algebraic sign given to it 1.For each element a ij, work out the minor 2.Then multiply it by (-1) i+j 3.In simpler language: if i+j is even then C ij = M ij 4.If i+j is odd, then C ij = -M ij 5.The co-factor matrix is just 6.The adjoint matrix is C' – i.e. the transpose of C. N xn matrix inversion: Co-factor and adjoint matrices
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1.The inverse matrix, A -1 is just So i) find the determinant – if it is non-zero, the matrix is non-singular so its inverse exists ii) Find the cofactors of all the elements of A and arrange them in the cofactor matrix iii) Transpose this matrix to get the adjoint matrix iv) Divide the adjoint matrix by the determinant to get the inverse Co-factor, adjoint matrices and the inverse matrix
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1.Iffind A -1 Use the formula First find the determinant which is non-zero so can continue Now find matrix of cofactors, which in the 2 x 2 case is a set of 1 X 1 determinants Example 1 (2 x 2 matrix)
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Now transpose the matrix of cofactors to get the adjoint matrix Now using the formula above which is non-zero so can continue then NB. Always check that the answer is right by looking if AA -1 = I Example 1 (2 x 2 matrix)
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While Example 2: 3 x 3 matrix
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Example continued
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1.In each case find the matrix of minors 2.Find the determinant 3.Find the inverse and check it. Quiz
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The termis called the determinant of A, often written det.(A). Vertical lines surrounding the original matrix entries also means 'determinant of A'. The matrix part of the solution is called the 'adjoint of A' written adj. A. The elements of the adj.A are called co-factors. So, Some more jargon
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Summary 11 Definitions you should now memorise: Matrix dimensions, null matrix, identity matrix, transpose, symmetric matrix, square matrix, leading diagonal, nonnegative, positive, nonpositive, and negative matrices. 5 skills you should be able to do: –Add two nxm matrices –Multiply a matrix by a scalar –Transpose a matrix. –Identify the element a ij in any matrix –Understand and have practised matrix multiplication
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For home study: Method - the Gaussian approach. In a row operation multiples of one row are added or subtracted from multiples of another row to produce a new matrix. Example: transform A by replacing row 1 by the sum of row 1 and row 2: A row operation can be represented by matrix multiplication. A' = BA where
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Method 2: the Gaussian method. Suppose, by a series of m row operations we transform A into I, the identity matrix. Let B i indicate the series of matrices in this sequence of m row operations: B m B m-1 …B 1 A = I. Let C = B m B m-1 …B 1 so that CA = I. It follows that, by the definition of the inverse that C = A -1. Since C = CI we can find C by taking the row operations conducted on A and conducting them in parallel on I. Method. Begin with the extended matrix [A I]: Carry out row operations on the extended matrix until it has the form [I C] C = A -1
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1.Use the Gaussian method to find the inverse and check it works. Quiz
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5. An odd example. Joan wishes to own and consume exactly one latte and one muffin. Can she buy to achieve her goal? Answer: She buys 1/2 of latte lover combo and 1/6 of muffintopia combo. This mix of vectors is called a linear combination. More formally, if x and y are n x 1 vectors and a and b are scalars, ax + by is a linear combination. Joan's purchase is (1/2)x + (1/6)y
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The example again Can Joan construct any combination of latte and muffin out of x and y? Any combination: So formally, is there an a and a b such that: Or
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10. Vector Space The set of vectors generated by the various linear combinations of 2 vectors is called a 2-dimensional vector space Consider a space with n dimensions It follows that a single nx1 vector is a point in that space A group of nx1 vectors is said to form a basis for the space if any point in that space can be represented as a linear combination of the vectors in the group. In the example,formed a basis for 2 dimensional space. These vectors are also said to span 2 dimensional space In other words if a group of vectors form a basis for an n-dimensional space that's the same as saying that they span the n-dimensional space. | 677.169 | 1 |
What's Different
We're asked that question all the time. To answer that, we should start by pointing out what's the same about iLearn and other programs.
We cover all the content students need in a developmental math sequence.
We provide extensive practice with immediate feedback.
Students must meet rigorous mastery standards.
We provide extensive diagnostics to determine needs. (only one other program does this)
We have extensive reports that make it easy for the instructor to manage the process.
So what's different?
We do something no one else has even conceived of. We make it much easier for students to learn. In the process, we transform the learning process from one filled with frustration and resistance to one filled with success and motivation. This is not an empty claim. It's supported by data.
Error Prevention vs. Error Correction
The key to making it easier to learn is reducing the rate of errors. This may seem trivially obvious, but actually doing it is an entirely different matter. Errors are the enemy of success. In fact, failure is just the accumulation of too many errors. Errors are not "neutral events." They have two very powerful negative impacts:
they slow down learning and
they frustrate, demoralize, and demotivate students.
Other products try to correct errors, but do nothing to help the student avoid errors. Others tout the fact that they use an approach called "error correction." This means they wait for errors to occur, then provide what they call "corrective feedback." However, there is little evidence that this process actually helps to "correct" errors. If it does help, it comes at a very high cost. The problem is that once an error occurs, the damage is done. Students usually "tune out" at that point. Errors continue at a high rate, even when the "feedback" is "specific to the error."
The iLearn Math approach is very different. It's called Optimized and it's based on the use of multiple "error prevention" strategies built into the design of the instructional process. We anticipate the kinds of errors students make, and then build an instructional process that prevents students from developing these erroneous thought patterns. The result is that students make dramatically fewer errors as they learn the math content.
The Negative Impact of Errors in Current Practice
There are several deficiencies in the other programs that have a detrimental impact. First, every other product lets students start working "practice" or "homework" problems without adequate preparation. The result is that students make errors at a very high rate. This "trial and error" approach results in far more errors than anything else.
The frustration students feel from this process motivates them in the wrong way to do the wrong things. A high percentage of students focus on trying to "beat the system," by what would be called "cheating" if it occurred in class. Mimicking answers from examples given as they work a problem is a well-known "gimmick" to "beat the system." Finding creative ways to copy answers is another. Students invest time and energy trying to find ways not to have to learn . They'd rather try just about anything to "meet the requirements" without having to learn anything. The result of this situation is consistently reported by those who use these other products. Students meet the requirements in the program, but cannot pass the mastery test given by the instructor. A high percentage of students then take the mastery test over and over again, trying to pass. Many instructors report students taking a test as many as 15-20 times before they eventually pass. This is not evidence of real learning, it's about "meeting the requirement." There is little likelihood that students understand what they have "learned." What is overlooked is that students consistently have a very high rate of errors. That's why they fail. That's also the main reason why they avoid trying to learn.
This counterproductive mindset is widespread but to date, no other software provider has provided an adequate solution . If you use one of these programs, it's probably a good idea to ask the provider to give you data on the overall percent correct for students during practice and test activities. The data will probably surprise you - if you can get them.
The good news is that this counterproductive mindset is not inevitable . It can be changed. The real difference between iLearn Math and these other programs is that we take a completely different approach to the learning process that changes this scenario. Students become much more focused on productive activity than unproductive activity . Instructors see the powerful impact in working with students.
Optimizing Learning
A key element of this approach to changing students' focus is that we do far more to make sure every student first understands the math that is taught. Then, and only then, do we allow them to start solving practice problems. We require that students complete instruction and demonstrate understanding of the concepts. They simply can't avoid this requirement because of the way the program is designed.
When we deliver instruction, we don't do it with "talking head" videos of lectures, and we don't have students read text on screen. We use state-of-the art multimedia designs that are built on rigorous scientific research on how to communicate information effectively and efficiently to students. They combine graphics, animation and narration to illustrate and represent math concepts in a way that makes them more engaging and much easier to understand. This delivery format takes advantage of the latest knowledge about facilitating cognitive processing of information.
So, in summary, what's different about iLearn Math? In short, it's the way we transform the learning experience. We make it easier for students to learn.
The expertise of iLearn is in designing and delivering instruction in ways that deliver on this claim.
"With iLearn Math, my students now see me as a partner in their success instead of an adversary." -VK Bussen, Instructor, Clovis Community College | 677.169 | 1 |
Math Tutorials
We are continuously adding new math tutorials. The main goal is to present different math topics in a clear, step-by-step fashion, in order to help our visitors to gain understanding and adquire familiary with the topic in question. | 677.169 | 1 |
An understanding of functions of a complex variable, together with the importance of their applications, form an essential part of the study of mathematics. Complex Variables and their Applications assumes as little background knowledge of the reader as is practically possible, a sound knowledge of calculus and basic real analysis being the only essential pre-requisites.
With an emphasis on clear and careful explanation, the book covers all the essential topics covered in a first course on Complex Variables, such as differentiation, integration and applications, Laurent series, residue theory and applications, and elementary conformal mappings. The reader is also introduced to the Schwarz-Christoffel transformation, Dirchlet problems, harmonic functions, analytic continuation, infinite products, asymptotic series and elliptic functions. Applications of complex variable theory to linear ordinary differential equations and integral transforms are also included.
Complex Variables and their Applications is an ideal textbook and resource for second and final year students of mathematics, engineering and physics.
REVIEWS for Complex Variables | 677.169 | 1 |
The true power of vectors has never been exploited, for over a century, mathematicians, engineers, scientists, and more recently programmers, have been using vectors to solve an extraordinary range of problems. However, today, we can discover the true potential of oriented, lines, planes and volumes in the form of geometric algebra. As such geometric elements are central to the world of computer games and computer animation, geometric algebra offers programmers new ways of solving old problems.
John Vince (best-selling author of a number of books including Geometry for Computer Graphics, Vector Analysis for Computer Graphics and Geometric Algebra for Computer Graphics) provides new insights into geometric algebra and its application to computer games and animation.
The first two chapters review the products for real, complex and quaternion structures, and any non-commutative qualities that they possess. Chapter three reviews the familiar scalar and vector products and introduces the idea of 'dyadics', which provide a useful mechanism for describing the features of geometric algebra. Chapter four introduces the geometric product and defines the inner and outer products, which are employed in the following chapter on geometric algebra. Chapters six and seven cover all the 2D and 3D products between scalars, vectors, bivectors and trivectors. Chapter eight shows how geometric algebra brings new insights into reflections and rotations, especially in 3D. Finally, chapter nine explores a wide range of 2D and 3D geometric problems followed by a concluding tenth chapter.
Filled with lots of clear examples, full-colour illustrations and tables, this compact book provides an excellent introduction to geometric algebra for practitioners in computer games and animation.
"Geometric algebra (GA), a truly fascinating area of mathematics, provides a powerful, unified language of exceptional clarity and generality to describe one-, two-, three-, and higher-dimensional geometries. … This book's outstanding feature is the use of tables and colors to develop some arithmetical details. … The book is better suited for self-study than for the classroom. I recommend it for upper-level undergraduates, graduate students, teachers, researchers, and technical libraries." (Edgar R. Chavez, ACM Computing Reviews, December, 2009)
"In the current volume, the author simplifies the presentation based on some of his new ideas on the subject. … The volume is self-contained and can be used by students and computer graphics professionals. … a good course in linear algebra and some mathematical maturity. Summing Up: Recommended. Computer graphics, computer animation, and computer games collections for upper-division undergraduates, graduate students, and professionals." (D. Z. Spicer, Choice, Vol. 47 (7), March, 2010)
"Geometric algebra is a topic of current interest in mathematical research and in applications in physics, engineering, and computer science … . the book is directed to a computer programming audience. … this accessible, introductory book may convince some computer graphics programmers of the usefulness of geometric algebra … ." (Adam Coffman, Mathematical Reviews, Issue 2011 i)
"The book's true value lies in describing important geometric transformations like reflection and rotation in a systematic way, and in listing many geometric primitives … . For people working in computer graphics or in game design, these topics could be of considerable value, and they certainly justify the book's title." (Rolf Klein, Zentralblatt MATH, Vol. 1226, 2012) | 677.169 | 1 |
AP Calculus First Day Of School Homework
272 Downloads
PDF (Acrobat) Document File
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0.27 MB | 4 pages
PRODUCT DESCRIPTION
In order to set the stage and standard of my AP Calculus AB class, I use this 4-page assignment for homework in order to give me a sense of where my students are at mathematically. This can be used as a group activity in class or as a homework assignment. Concepts include: Function Notation, Position Functions, Graphs of Foundation Functions, Domain and Range, Function Arithmetic, Composite Functions, and Inverse Functions. Students are asked to explain their reasoning to get them prepared for the rigor and expectations of the AP Exam.
NOTE: Look for the Answer Key in my store titled "AP Calculus First Day Of School Homework SOLUTIONS | 677.169 | 1 |
MCP Mathematics Level E Student Edition 2005c
(MCP44CP Mathematics promotes mathematical success for all students, especially those who struggle with their core math program. This trusted, targeted program uses a traditional drill and practice format with a predictable, easy-to-use lesson format. MCP Math is flexible and adaptable to fit a variety of intervention settings including after school, summer school, and additional math instruction during the regular school day.By teaching with MCP Math, you can: Provide targeted intervention through a complete alternative program to core math textbooks. Help students learn and retain new concepts and skills with extensive practice. Prepare students at a wide range of ability levels for success on standardized tests of math proficiency. | 677.169 | 1 |
PDF (Acrobat) Document File
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0.4 MB | 2 pages
PRODUCT DESCRIPTION
Your students will beg for more after they do this 24-question self-checking circuit on order of operations. Includes evaluating algebraic expressions. Written for an algebra one level, but can be used for pre-algebra, college algebra, algebra two review, etc. Intended for use without calculator, but calculator use may be allowed. May be turned into a scavenger hunt or task cards | 677.169 | 1 |
NCERT Solutions For Class 8 Maths
Byju's NCERT solutions for class 8 maths will help students find the right answers from their maths textbooks. Byju's solutions and study material to NCERT textbooks for class 8 maths will help students perform better in their exams and also understand the concepts clearly.
These textbooks can be very helpful for students in achieving maximum marks in their examinations. Student having trouble finding solutions to difficult math problem can refer to the NCERT maths solutions pdf's which we also provide. Students just need to cover these books and solve the questions and exercises given in each chapter. Students can be assured of positive results. | 677.169 | 1 |
Duration
Publisher
Video
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Certification
Minimum Grade/Class Level
Course Description
ALISON.com's free online Diploma in Mathematics course gives you comprehensive knowledge and understanding of key subjects in mathematics. This course covers calculus, geometry, algebra, trigonometry, functions, vectors, data distributions, probability and probability and statistics. Math qualifications are in great demand from employers and this math course will greatly enhance your career prospectsALISON.com's math course will greatly enhance your skills, giving you a greater understanding of core mathematics components such as geometry, trigonometry, calculus and more as well as expanding your knowledge base in areas such as chance, data distributions, statistics, probability, correlations and regression.
You will learn about using binomial expansions for problem solving and will understand the relationship between the graphs of functions and their anti-derivatives.
You will be able to confidently create graphs and make advanced calculations such as straight-line calculations, kinematics, motion, vectors, algebra, binomial expressions, and quadratic functions.
Claire GrossmanUnited Kingdom
it was great and at last I've passed with 84% i'm so chuffed I will get the Diploma framed and put it on the wall'
Not bad for somebody whose had 2 strokes 2015-02-17 12:02:55
Cain A MupasiUnited Kingdom
Very well put together. Just needs more examples to try out and then the right answers to check that I have done them right. 2015-02-05 00:02:43 | 677.169 | 1 |
Polynomials Review Activity
233 Downloads
Word Document File
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0.02 MB | 1 pages
PRODUCT DESCRIPTION
This was used right before a test on polynomials. It covers graphing, writing the equation based on a graph, describing end behavior, factoring, and using graphing calculators.
Each student gets a worksheet. They are creating the 5 problems on this page, but they must follow the instructions. The instructions in each box are directed toward the student while they are creating the questions.
Then have the students pass their worksheet to the person on their right (or left). That student writes their name in the #1 box and solves the problem.
Have them pass their paper again. That student writes their name in the #2 box and solves the problems.
Continue until all problems have been solved | 677.169 | 1 |
Maths Quest 12
Maths Quest 12
Maths Quest 12 course guides to prepare for your mathematics exam! Maths Quest 12 includes student workbooks, worksheets, teacher's guides and online components. Jacaranda also offers a series of StudyON guides designed to prepare year 12 students to take VCE exams.
Maths Quest 12 includes supporting calculator companions containing comprehensive step-by-step CAS calculator instructions, fully integrated into worked examples for the Casio ClassPad and TI-Nspire CAS calculators. The series also includes studyON: our revolutionary online study, revision and exam practice tool for students and studyON Teacher Editon: an innovative aid for teachers to monitor students' progress and pinpoint strengths and weaknesses for improved exam results.
StudyON Maths Quest 12 Further Mathematics 4E Value Packs
StudyON provides students with practice exam questions taken from official VCE exams, there are over 800 official questions to practice on. As students complete the activities, there is automatic marking with instant feedback. The learning activities are visual and interactive, including videos and animations, helping students retain more information. StudyON also offers value packs that are specialized according to the calculator needed to complete the exercises. Further Mathematics 4E Casio Classpad Calculator Companion and Further Mathematics 4E TI-Nspire Calculator Companion are the most popular value packs that serve as companions to Jacaranda's Quest 12 materials.
Maths Quest 12 Mathematical Methods CAS 2E
Calculator companion guides contain comprehensive step-by-step calculator instructions, worked examples that are fully integrated and electronic tutorials for select worked examples in each chapter. The books use full colour and offer helpful photographs and graphics. Each chapter offers a comprehensive chapter reviews with sample exam questions for practice.
Maths Quest 12 Specialist Mathematics
StudyON in Specialist Mathematics. This product has been specially designed to be used with VCE Mathematics Units 3 & 4. The book includes a CD-ROM with examples of worked exercises, real life examples and detailed explanations. The Specialist Mathematics workbook chapters are: Chapter 1: Coordinate geometry; Chapter 2: Circular (trigonometric) functions; Chapter 3: Complex numbers; Chapter 4: Representations of relations and regions in the complex plane; Chapter 5: Differential calculus; Chapter 6: Integral calculus; Chapter 7: Differential equations; Chapter 8: Kinematics; Chapter 9: Vectors; Chapter 10: Vector calculus and Chapter 11: Mechanics. Jacaranda offers a curriculum targeted to VCE Mathematics Units 3 & 4, with these guides students can quickly discover their areas of strength and weaknesses and spend time working only in the areas where they need improvement before exams | 677.169 | 1 |
Methods of Proof in MathematicsMethods of Proof in Mathematics
Course Description
This course is an introduction to abstract mathematics, with an emphasis on the techniques of mathematical proof (direct, contradiction, conditional, contraposition). Topics to be covered include logic, set theory, relations, functions and cardinality. Prerequisite: Calculus lll
Note: This course is proof based. All homework assignments, exams and the final are graded by the instructor.
For More Information, Select Course Type Below:384.65 and a second payment 30 days later of $315 | 677.169 | 1 |
Relations and Functions
Presentation (Powerpoint) File
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9.19 MB | 13 pages
PRODUCT DESCRIPTION
This powerpoint introduces students to the idea of relations, beginning with some examples, basic definitions and terminology. Next comes an introduction/review of the ways in which functions can be represented; ordered pairs, mappings, and coordinate graphs. Following this, we examine how a two-variable equation represents a relation. Next, the concept of a function is introduced, starting with a basic definition followed by a video discussing a few examples of whether or not a given relation is a function. Finally, students are introduced to vertical line test as a way to check whether a graph represents a function | 677.169 | 1 |
Download Free Hands On Start To Wolfram Mathematica Book in PDF and EPUB Free Download. You can read online Hands On Start To Wolfram Mathematica and write the review.
For more than 25 years, Mathematica has been the principal computation environment for millions of innovators, educators, students, and others around the world. This book is an introduction to Mathematica. The goal is to provide a hands-on experience introducing the breadth of Mathematica, with a focus on ease of use. Readers get detailed instruction with examples for interactive learning and end-of-chapter exercises. Each chapter also contains authors tips from their combined 50+ years of Mathematica use.
With over a million users around the world, the Mathematica software system created by Stephen Wolfram has defined the direction of technical computing for the past decade. The enhanced text and hypertext processing and state-of-the-art numerical computation features will ensure that Mathematica 4, takes scientific computing into the next century.The Mathematica Book continues to be the definitive reference guide to this revolutionary software package and is released in this new edition to coincide with the release of the new version of Mathematica.The Mathematica Book is a must-have purchase for anyone who wants to understand the opportunities in science, technology, business, and education made possible by Mathematica 4. This encompasses a broad audience of scientists and mathematicians; engineers; computer professionals; financial analysts; medical researchers; and students at high-school, college, and graduate levels.Written by the creator of the system, The Mathematica Book includes both a tutorial introduction and complete reference information, and contains comprehensive description of how to take advantage of Mathematica's ability to solve myriad technical computing problems and its powerful graphical and typesetting capabilities.New to this version:* Major efficiency enhancements in handling large volumes of numerical data.* Internal packed array technology to make repetitive operations on large numerical datasets radically more efficient in speed and memory.*Improved algebraic computation facilities, including support for assumptions within Simplify, and related functions, and specification of domains for variables, as well as full support of symbolic Laplace, Fourier, and Z transforms.* Additional Mathematica functions, including Dirac Delta, Stuve, Harmonic numbers, etc.* Enhanced graphics and sound capabilities, including faster graphic generation and additional format support for graphics and sound.* Full-function spell checking including special technical dictionaries.
The Wolfram Language represents a major advance in programming languages that makes leading-edge computation accessible to everyone. Unique in its approach of building in vast knowledge and automation, the Wolfram Language scales from a single line of easy-to-understand interactive code to million-line production systems. This book provides an elementary introduction to the Wolfram Language and modern computational thinking. It assumes no prior knowledge of programming, and is suitable for both technical and non-technical college and high-school students, as well as anyone with an interest in the latest technology and its practical application.
Adapted from Stephen Wolfram's definitive work Mathematica: A System for Doing Mathematics by Computer, 2nd Ed., this is the beginning student's ideal road map and guidebook to Mathematica. This adaptation addresses the student's need for more concise and accessible information. Beck has trimmed to book to half its original size, focusing on the functions and topics likely to be encountered by students.
Starting with an introduction to the numerous features of Mathematica®, this book continues with more complex material. It provides the reader with lots of examples and illustrations of how the benefits of Mathematica® can be used. Composed of eleven chapters, it includes the following: A chapter on several sorting algorithms Functions (planar and solid) with many interesting examples Ordinary differential equations Advantages of Mathematica® dealing with the Pi number The power of Mathematica® working with optimal control problems Introduction to Mathematica® with Applications will appeal to researchers, professors and students requiring a computational tool.
Learn Raspberry Pi 2 with Linux and Windows 10 will tell you everything you need to know about working with Raspberry Pi 2 so you can get started doing amazing things. You'll learn how to set up your new Raspberry Pi 2 with a monitor, keyboard and mouse, and how to install both Linux and Windows on your new Pi 2. Linux has always been a great fit for the Pi, but it can be a steep learning curve if you've never used it before. With this book, you'll see how easy it is to install Linux and learn how to work with it, including how to become a Linux command line pro. You'll learn that what might seem unfamiliar in Linux is actually very familiar. And now that Raspberry Pi also supports Windows 10, a chapter is devoted to setting up Windows 10 for the Internet of Things on a Raspberry Pi. Finally, you'll learn how to create these Raspberry Pi projects with Linux: Making a Pi web server: run LAMP on your own network Making your Pi wireless: remove all the cables and retain all the functionality Making a Raspber ry Pi-based security cam and messenger service Making a Pi media center: stream videos and music from your Pi | 677.169 | 1 |
Translation Discovery
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0.07 MB | 2 pages
PRODUCT DESCRIPTION
This discovery activity allows students to construct their own knowledge with translations of functions, namely quadratic and cubic functions. This is a guided activity best coupled within a cooperative learning | 677.169 | 1 |
PreMBA Analytical Primer by Regina Trevino
Book Description
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 | 677.169 | 1 |
Intermediate Algebra:
Intermediate Algebra is a branch of mathematics that substitutes letters for numbers and uses simplification techniques to solve equations.
Algebraic equations:
A scale, what is done on one side of the scale with a number is also done to the other side of the scale.
Variables:
Variables are symbols that are usually defined to denote any member of a set of objects.
Functions:
Functions are relations defined in such a way that for each element of a set there is a unique element of another set that is associated it with.
VANG: VANG stands for Verbal Analytical/Algebraic Numerical Graphical
Rapid Study Kit for "Title":
Flash Movie
Flash Game
Flash Card
Core Concept Tutorial
Problem Solving Drill
Review Cheat Sheet
"Title" Tutorial Summary :
Intermediate Algebra is a course that involves relations and its use. Intermediate Algebra along with Elementary Algebra course provides a solid foundation to higher mathematics course such as College Algebra, Pre-Calculus and Calculus. To be proficient in mathematics, we highly recommend you to examine the examples and properties discussed in this course and ponder any of the subtleties encountered in any mathematics problems.
This tutorial gives an overview of what to look forward to in this Inermediate Algebra course. This introductory tutorial summarized the most important principles encountered in an Intermediate Algebra course. It is a strong tool to use for anyone who is unfamiliar or is learning Inermediate Algebra for the very first time. The meaning of VANG is discussed in this tutorial.
Tutorial Features:
Specific Tutorial Features:
• Definition of Inermediate Algebra and its relationship with other mathematics courses is discussed in this tutorial.
• The use of relations and its definition is shown here with the use of step by step examples.
Series Features:
• Concept map showing inter-connections of concepts introduced.
• Definition slides introduce terms as they are needed.
• Examples given throughout to illustrate how the concepts apply.
• A concise summary is given at the conclusion of the tutorial.
"Title" Topic List:
Getting to Know Intermediate Algebra What is Intermediate Algebra? Intermediate Algebra and relations The construction of relations and examples Intermediate Algebra and Other Mathematics Courses Differences and similarities between Inermediate Algebra and other mathematics courses Intermediate Algebra Tips VANG | 677.169 | 1 |
Synopses & Reviews
Publisher Comments
This ingenious, user-friendly introduction to calculus recounts adventures that take place in the mythical land of Carmorra. As the story's narrator meets Carmorra's citizens, they confront a series of practical problems, and their method of working out solutions employs calculus. As readers follow their adventures, they are introduced to calculating derivatives; finding maximum and minimum points with derivatives; determining derivatives of trigonometric functions; discovering and using integrals; working with logarithms, exponential functions, vectors, and Taylor series; using differential equations; and much more. This introduction to calculus presents exercises at the end of each chapter and gives their answers at the back of the book. Step-by-step worksheets with answers are included in the chapters. Computers are used for numerical integration and other tasks. The book also includes graphs, charts, and whimsical line illustrations. Barron's Easy Way books introduce a variety of academic and practical subjects to students and general readers in clear, understandable language. Ideal as self-teaching manuals for readers interested in learning a new career-related skill, these books have also found widespread classroom use as supplementary texts and brush-up test-preparation guides. Subject heads and key phrases that need to be learned are set in a second color.
Synopsis
(back cover)
ALL THE ESSENTIALS IN ONE CLEAR VOLUME
by Douglas Downing, Ph.D.
A user-friendly introduction to calculus in the form of a novel
Exercises with answers at the end of each chapter
Step-by-step worksheets
Graphs, charts, and line illustrations
Calculating derivatives
Finding maximum and minimum points with derivatives
Determining derivatives of trigonometric functions
Exponential functions, vectors, and Taylor series
Using differential equations
Much more
Synopsis
(back cover)
Calculus makes sense when you approach them the E-Z way! Open this book for a clear, concise, step-by-step review of:
Derivatives
Natural logarithms
Exponential functions
Integrals and much more
Measure your progress as you gain a command of Calculus:
Develop your understanding of calculus and attain higher grades
Take each chapters quizzes and check your answers
And discover that learning Calculus can be E-Z!
Synopsis
The author of this imaginative self-teaching book tells an entertaining story about travels in the fictional land of Carmorra. In the process he introduces a series of problems and solves them by applying principles of calculus. Readers are introduced to derivatives, natural logarithms, exponential functions, differential equations, and much more. Skill-building exercises are presented at the end of every chapter. Books in Barron's new E-Z series are enhanced and updated editions of Barron's older, highly popular Easy Way books. New cover designs reflect the brand-new interior layouts, which feature extensive two-color treatment, a fresh, modern typeface, and more graphic material than ever. Charts, graphs, diagrams, line illustrations, and where appropriate, amusing cartoons help make learning E-Z in a variety of subjects. Barron's E-Z books are self-teaching manuals focused to improve students' grades in skill levels that range between senior high school and college-101 standards.
About the Author
Douglas Downing earned B.S. and Ph.D. degrees from Yale, and has taught economics at Seattle Pacific University since 1983. Calculus the Easy Way is part of a trilogy with Algebra the Easy Way and Trigonometry the Easy Way, all written be Douglas Downing and available from Barron's. | 677.169 | 1 |
in a traditional course without a textbook. Purpose:Core Curriculum ALEKS Course:Basic Math
David C. Edwards, Instructor After implementing ALEKS in my math courses, I have seen a marked improvement in my students' learning. Students love ALEKS and my time is much better used now. Instead of shuffling papers and grading, I spend my time essentially tutoring each student one on one. ALEKS has been a huge benefit for improving the way students feel about their math confidence.
Scenario
Was ALEKS used in your course with all students or with targeted students? All students.
Number of students who used ALEKS for the course and term:
Number of sections:1 Number of students per ALEKS section:28 Total students enrolled in this ALEKS course:28
How do you structure your course periods with ALEKS? The students work independently on their pies, while I am available for individual questions.
How often are students encouraged or required to use ALEKS?
Days per week:4 Hours per week:5 Hours per term:55
Implementation
Please describe how you implement ALEKS into your course curriculum. Students are required to be in class and work on ALEKS. After completing the Basic Math pie, students are then able to continue on to higher math courses at their own speed. Many students learn much more than what is required for the class.
Do you cover ALEKS topics in a particular order? No. Students choose the order in which they want to cover topics homework assignments into your course? For homework, students must work at their own pace to complete the ALEKS pie.
How do you incorporate ALEKS scheduled Assessments into your course? An ALEKS assessment is used for the final exam.
How do you modify your regular instructional approach as a result of ALEKS? Students are encouraged to work at their own pace. There are minimum requirements that need to be met, however, many students go above and beyond the requirements of the course.
How do you use information from ALEKS to focus your teaching? I use the time logged feature in ALEKS to check the progress made by students. If a student is not progressing well through their pie, I offer individual help.
Grading
What percent of a student's grade does ALEKS make up? 80 percent.
How do you incorporate ALEKS into your grading system? ALEKS comprises 100 percent of the final grade with the following breakdown: 25 percent from attendance; 25 percent from time logged in ALEKS; 25 percent from completing the Basic Math pie; and 25 percent from the final exam.
Is ALEKS assigned to your students as all or part of their homework responsibilities? ALEKS makes up 100 percent of the students' homework responsibilities.
Do you require students to make regular amounts of progress in ALEKS? Students have to finish the pie by the end of the quarter. I tell them to figure out how far they need to go after their initial assessment and to divide that by ten. In each of ten weeks, students should progress that far to reach their goal.
How do you track student progress in ALEKS? I look at the amount of topics completed by each student and the number of topics they learn per hour.
Do you notice that students who spend more time in ALEKS perform better in the course than those students who spend less time in ALEKS? I am sold on ALEKS. Students who have hated math their whole lives now love math in this course.
Would you attribute any improved student performance in the course to ALEKS methodology or to some other factor? Slower students don't feel ashamed to ask questions and they get lots of individual help. If they don't understand something foundational, they can stick with it until they get it rather than being rushed to keep up with the herd. Furthermore, faster students can learn as much math as they want, in some cases, above and beyond the course requirement, and not get bogged down listening to things they already know.
In general, how do the students feel about their progress in ALEKS? ALEKS has been a huge benefit for improving the way students feel about their math confidence.
Best Practices
What challenges did you encounter when first implementing ALEKS and how can other instructors avoid these pitfalls? I discovered too late that ALEKS is available for purchase online and in the bookstore, which is very helpful for students who must purchase their materials in a certain way depending on their funding source.
What will you do differently when you implement ALEKS in your future course(s)? I originally waited for students to ask me for help when they got stuck. However, I now initiate questions so the students are more comfortable with the program.
Which ALEKS course product(s) have you used in the past? Basic Math, Intermediate Algebra, College Algebra with Trigonometry, College Algebra, Prep for College Algebra
What best practices would you like to share with other instructors who are implementing ALEKS? The best thing to do is remove the boredom and embarrassment from math. ALEKS gives the students the confidence needed to start learning and enjoying math.
Additional Insights: Students love ALEKS and my time is much better used now. Instead of shuffling papers and grading, I spend my time essentially tutoring each student one on one. | 677.169 | 1 |
App Info. Find out more information about subscription plans to app.ibmaths.com or about a free trial for a school, teacher, or student. IB Math Studies (Standard Level) is designed specifically for out-of-field instructors or teachers new to the IB curriculum. This course is designed for students who require a basic college preparatory course in mathematics. Coursework. Algebra 2, Math Studies 1 or Mathematics SL 1. IB. For the Math Studies IA the student is to pick 2 variables they believe are related to each other in. International Baccalaureate A detailed guide by criteria. Online Homework Help For Math Ib Maths Studies Coursework Mark Scheme How To Write An Abstract For A Biology Research Paper. Triangular Slave Trade Essay. Math Coursework. The IB exam for Math. For the IB Math Methods HL students must complete math portfolio assignments along with a math studies project. Ib Math Studies Coursework Ideas Literature Review Of India Fmcg Writing A Thesis Statement For A History Research Paper Literature Review On Employee Attendance.
Ib math studies coursework
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This is the British International School Phuket's IB maths exploration (IA) page. This list is for SL and HL students – if you are doing a Maths Studies IA then.
How to score level 7 in Math IA/ Exploration in 2 hours?. How to score a L7 for IB Math - Duration:. Maths Studies - Duration:. Ib Math Research Paper Literature Review Web Development. How Democratic Was Andrew Jackson Dbq Essay. Ocr Media Studies A Level Coursework. Research. For my Ib maths coursework for one section i have to find the interquartile range. im comparing head size and how many words you can remember. Mathematical studies SL guide International Baccalaureate, Baccalauréat International and Bachillerato Internacional are. of coursework assessed by teachers. Ib English Coursework Help A description of a good coursework example that students can follow and apply to minimize the potential of errors while writing their. Xoxxrf - f.g496courseworkideas.writeessayhelpkrok.tech. IB SL Math vs IB SL Math Studies?. IB HL Bio IB SL math or math studies?. I'm making an inference that IB History would have a large amount of coursework. | 677.169 | 1 |
Pre-collegiate Mathematics
Currently, I plan on being a math or physics major in college. Under the assumption that I'll be accepted into a top university, - not necessarily likely - what mathematical concepts should I have mastered, be familiar with, etc. before enrolling in college? | 677.169 | 1 |
Software tag 'mathPTC Mathcad Express is free-for-life engineeringmath software. You get unlimited use of the most popular capabilities in PTC Mathcad allowing you to solve, document, share and reuse vital calculationsCalculator Prompter is a math expression calculator. Calculator Prompter has a built-in error recognition system that helps you get correct results. With Calculator Prompter you can enter the whole expression, including brackets, and operators. | 677.169 | 1 |
The General Certificate of Secondary Education (GCSE) is an academically rigorous, internationally recognised qualification (by Commonwealth countries with education. Arrow_back Back to Home Free Maths Revision and Help for Students. Whatever your age or ability (I bet you are a lot better than you think you are), if you are. This presentation goes through how to use Spearman's Rank for a piece of Geography coursework. It does this through a worked example, stage by stage. Each Unplugged activity is available to download in PDF format, with full instructions and worksheets. Background sections explain the significance of each activity. Each Unplugged activity is available to download in PDF format, with full instructions and worksheets. Background sections explain the significance of each activity. Cambridge GCSE Computing Online - the home of Computer Science teaching and assessment with OCR and Cambridge University Press in association with Rasberry Pi. Cambridge GCSE Computing Online - the home of Computer Science teaching and assessment with OCR and Cambridge University Press in association with Rasberry Pi. Objective: To develop our general Geographical knowledge of Brazil & FIFA by completing a Mapping from Memory activity. Task 1 - Where is Brazil. Curriculum Information. Subject. Maths. Qualification. GCSE. Exam Board. Edexcel. Course Leader. Mrs A Cooper. Course Summary. Students will follow a three year.
Gcse maths coursework data
Free statistics coursework papers, essays, and research papers. A collection of fantastic teaching resource websites that every maths teacher must have in their bookmarks: 1. TES Connect. Thousands of maths teaching resources. A web server may return an HTTP 403 Forbidden status in response to a request from a client for a web page, or it may indicate that the server can be reached and. The General Certificate of Secondary Education (GCSE) is an academically rigorous, internationally recognised qualification (by Commonwealth countries with education. We've gathered the best GCSE revision tips for students who want to achieve top results in their exams. Click here to see how you can improve your results.
Volume - cuboids, prisms,spheres,cones and frustrums,examples,worksheets,interactive pages from GCSE Maths Tutor. We provide excellent essay writing service 24/7. Enjoy proficient essay writing and custom writing services provided by professional academic writers. Since the abolition of maths coursework, there is no formal requirement to carry out investigations with your students. However, it is only through investigative work. We provide excellent essay writing service 24/7. Enjoy proficient essay writing and custom writing services provided by professional academic writers. Free statistics coursework papers, essays, and research papers.
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A web server may return an HTTP 403 Forbidden status in response to a request from a client for a web page, or it may indicate that the server can be reached and. A comparison of the time at which students are taught data handling skills in maths compared to when they are expected to use these skills in science. Plus a. Looking at the diagram above: Channel 1 contains two stages between producer and consumer - a wholesaler and a retailer. A wholesaler typically buys and stores large. Home; Curriculum. Key Stage 3. Core Subject Homework Timetable; Year 7 Assessment Descriptors. End of Year Assessment; Food Technology Recipe Booklets; Key Stage 4 GCSE Maths Revision. Lower case sigma means 'standard deviation'. Capital sigma means 'the sum of'. x bar means 'the mean' The standard deviation.
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Science GCSE from 2016. Here are the main points: Totally exam-based. There is no practical exam, although you have to do certain practicals in school.
A comparison of the time at which students are taught data handling skills in maths compared to when they are expected to use these skills in science. Plus a.
Arrow_back Back to Home Free Maths Revision and Help for Students. Whatever your age or ability (I bet you are a lot better than you think you are), if you are. If you're after some help in preparing for your ICT exams, you've come to the right place! There is plenty to help you here: theory notes, practical tips and exam tricks. Since the abolition of maths coursework, there is no formal requirement to carry out investigations with your students. However, it is only through investigative work. GCSE Maths Revision. Lower case sigma means 'standard deviation'. Capital sigma means 'the sum of'. x bar means 'the mean' The standard deviation. A-level and IGCSE/GCSE lessons and courses taught by professional tutors: English | Maths | Econ | Physics | Chemistry | Biology. Welcome lounge and forum help Welcome to TES Community. If this is your first visit don't forget to read the how to guide. Submit your first post here and let. | 677.169 | 1 |
MYTUTOR SUBJECT ANSWERS
Why bother with learning calculus?
Calculus is the maths of change. This sounds familiar;
In mechanics changing distances, velocities etc...
In motor sport Integration would be the sum of momentum over time to find the kinetic energy. This could help with working out how to optimise energy. Differentiation would be maximum and minimum points in the laps of variables
In economics integration could be the sum of your savings. Differentiation could be a maximum or a minimum of a stock market. Then buy at a low to sell for profit and knowing when the markets about to drop from a maximum. Game theory revolutionised this though.
The most beautiful thing about calculus is that it can take you to space. To get to the moon first you have to accelerate very quickly to be higher than the escape velocity (first change). The mass of the rocket decreases as the fuel is burnt at a rate (second change). The rocket then slower down after escaping (third change). The gravitation pull from the Earth decreases and the Moon pull increases (fourth and fifth change). Opposite for the return journey.
The gravitational change can be demonstrated for a particle with F=ma in quite a nice way.
Still stuck? Get one-to-one help from a personally interviewed subject specialist
257 | 677.169 | 1 |
Peer Review
Ratings
Overall Numeric Rating:
Students are provided with exponential expressions that are categorized as Easier, Middle, Difficult, or Random. Students use the exponent rules to simplify the expressions. Easy problems require the use of one exponent rule. Difficult problems may require the use of up to four rules. Students can check their answer or ask for a hint if they are unable to simplify on their own.
Type of Material:
This is an interactive Java applet.
Recommended Uses:
This material should be used as re-enforcement for the exponent rules.
Technical Requirements:
A JAVA 2 plug-in is required.
Identify Major Learning Goals:
The student will learn by practicing the following exponent rules: the product rule, the quotient rule, the power rule, and the zero exponent rule.
Target Student Population:
Intermediate to College Algebra Students
Prerequisite Knowledge or Skills:
Students should understand what an exponent is and have an understanding of the four rules mentioned above.
Evaluation and Observation
Content Quality
Rating:
Strengths:
Students can pick the desired skill level they feel comfortable with. Hints are given when students click on the hint button. Very easy instructions to follow.
Concerns:
Although the quotient rule is used, most students are told a problem is not considered simplified if it contains a negative exponent. Instructions should be given to students explaining that they will have negative exponents in their answers and therefore, may not be considered completely simplified.
Potential Effectiveness as a Teaching Tool
Rating:
Strengths:
This is a great practice tool that reinforces the exponent rules. Students are able to decide if they should add, subtract or multiply the powers by using the rules. Feedback is given instantaneously and helpful hints are given when the student is not successful.
Concerns:
Again, students may be confused that they will have problems that have negative exponents in their final answer.
Ease of Use for Both Students and Faculty
Rating:
Strengths:
The program is easy to follow. The instructions are clear and precise.
Concerns:
none | 677.169 | 1 |
Product Description:
In Euclidean geometry, constructions are made with ruler and compass. Projective geometry is simpler: its constructions require only a ruler. In projective geometry one never measures anything, instead, one relates one set of points to another by a projectivity. The first two chapters of this book introduce the important concepts of the subject and provide the logical foundations. The third and fourth chapters introduce the famous theorems of Desargues and Pappus. Chapters 5 and 6 make use of projectivities on a line and plane, respectively. The next three chapters develop a self-contained account of von Staudt's approach to the theory of conics. The modern approach used in that development is exploited in Chapter 10, which deals with the simplest finite geometry that is rich enough to illustrate all the theorems nontrivially. The concluding chapters show the connections among projective, Euclidean, and analytic geometry.
REVIEWS for Projective Geometry | 677.169 | 1 |
Now anyone with an interest in basic, practical trigonometry can master it -- without formal training, unlimited time, or a genius IQ. In Trigonometry Demystified, best-selling author Stan Gibilisco provides a fun, effective, and totally painless way to learn the fundamentals and general concepts of trigonometry.
With Trigonometry Demystified you master the subject one simple step at a time -- at your own speed. Unlike most books on trigonometry, this book uses prose and illustrations to describe the concepts where others leave you pondering abstract symbology. This unique self-teaching guide offers questions at the end of each chapter and section to pinpoint weaknesses, and a 100-question final exam to reinforce the entire book.
Simple enough for beginners but challenging enough for professional enrichment, Trigonometry Demystified is your direct route to learning or brushing up on trigonometry | 677.169 | 1 |
Conic Section Card Sort (Precalculus)
PDF (Acrobat) Document File
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0.52 MB | 24 pages
PRODUCT DESCRIPTION
Students will match the graphs of Conic Sections for circle, parabola (opening up, down, left, right), ellipse, hyperbola (opening up/down, left/right),and truncated parabola with the rectangular equations and parametric equations. Domain and Range are noted on the cards to reinforce equation requirements and characteristics of graphs.
Follow-up Activity (4 different versions) allows students to demonstrate what they should have learned from the card match may be used as a quiz or collaborative activity | 677.169 | 1 |
Engineering Mathematics Through Applications (2nd Revised edition)
Description
This text teaches maths in a step-by-step fashion - ideal for students on first-year engineering and pre-degree courses. - Hundreds of examples and exercises, the majority set in an applied engineering context so that you immediately see the purpose of what you are learning - Introductory chapter revises indices, fractions, decimals, percentages and ratios - Fully worked solutions to every problem on the companion website at plus searchable glossary, e-index, extra exercises, extra content and more!
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About Author
Kuldeep Singh is Senior Lecturer in Mathematics at the School of Physics, Astronomy and Mathematics at the University of Hertfordshire, UK. He teaches mathematics to a wide range of engineering and science students. | 677.169 | 1 |
In Calculus Revisited Part 1, Professor Gross discussed the
calculus of a single real variable in which the domain of a function was a subset of the real numbers. Geometrically
speaking, the domain of a function was a subset of the x-axis. In this block he generalizes the domain as being a subset of
either the two-dimensional xy-plane and/or the three-dimensional xyz-space. In the language of vectors, in this block a function
maps 2 and 3 dimensional vectors into the set of real numbers. He then uses these functions to show how we compute the velocity
and acceleration of an object moving in space.
In this block
Professor Gross uses the "game of mathematics" concept to develop an analytical way to extend the domain of a function
to beyond 3 dimensions. In particular, he shows how by using vector arithmetic, the rules of arithmetic that were used in
developing the calculus of a single variable turn out to be the same that we use to develop the calculus of several variables.
This leads to a discussion of how we replace the concept of slope in the 2 and 3-dimensional calculus by such concepts as
the directional derivative when dealing with more than 3 dimensions.
Block 4 extends the concept of inverse functions to
the case where y = f(x) with y = (y1, y2, ..., yn) and x = (x1, x2, ..., xn). In more user-friendly
terms this block asks us to determine when and how the system of equations that expresses y1, y2, ... and yn as
functions of x1, x2, and xncan be "inverted" to express x1, x2, ... and xn as functions of y1,
y2, ... and yn. This motivates the study of matrix algebra since the process of inverting an n x n square
matrix is used to show how we decide whether a function f(x1,
x2, ..., xn) has an inverse and how we find the inverse function if it exists | 677.169 | 1 |
Resource Added!
Type:
Graphic Organizer/Worksheet, Other
Description:
Subjects:
Mathematics > General
Education > General
Education Levels:
Grade 9
Grade 10
Grade 11
Grade 12
Keywords:
Math NCSSM Trigonometry North Carolina School of Science and Mathematics optimization07-28.
Component Ratings:
Technical Completeness: 3 Content Accuracy: 3 Appropriate Pedagogy: 3
Reviewer Comments:
This fairly irresistible exercise tells a long story about two students on an adventure holiday in New York, visiting local tourist places and solving mysteries involving trig. It's lots of fun and presents some interesting problems for the students to solve.
Not Rated Yet. | 677.169 | 1 |
Be sure that you have an application to open this file type before downloading and/or purchasing.
PRODUCT DESCRIPTION
In this video I work through the practice problems in Unit 1 of Integrated Math 2 in the Common Core. Students review rational exponents, properties of rational and irrational numbers, complex numbers, and polynomial operations. Practice Worksheet and Key included. This is a great hour of online tutoring for only $9.99 | 677.169 | 1 |
From the beginning of time man has hated math. Fortunately, that was a long time ago and we have Harold Jacobs now. Harold Jacobs' premise is that everyone is a born mathemetician, and that, provided the proper instruction, anyone can do well in math. His texts appeal to students at all stages of mathematical competence due to their literate presentation of material, engaging illustrations and comics (including Peanuts, B.C., and the New Yorker), and comprehensive problem sets designed to reinforce and contextualize key concepts. Jacobs promotes creativity in learning, and his books are creative and even exciting.
The Elementary Algebra text has frequently been compared to the Saxon math curriculum for high school students. Both are incremental in approach (introducing concepts a little at a time and reintroducing them later as parts of larger wholes) and both emphasize constant review so that building-block principles aren't forgotten as students progress into more complex territory. However, Jacobs' text is far more engaging with its use of meaningful illustrations and references to culture, literature and art. It is also better written; whereas Saxon tends to write in a breathless, choppy style, Jacobs' prose is clear and to-the-point.
Geometry follows the same general pattern of Elementary Algebra, but offers more supplementary information. The author describes it as a "museum in a book" because of all the historical and cultural references to the various uses of geometry. It's also more colorfully illustrated than the other books, with photographs of famous artwork and architecture illustrating different concepts. There is more information in the teacher's manual than in either of the others, though full step by step solutions are not included. We don't think of this as a huge problem, since the point of geometrical proofs is proving them, and this can be done in many different ways.
Elementary Algebra and Geometry were meant for high schoolers (grades 9 and 10), although older students have found them useful, especially those who've struggled with math. A third text, Mathematics: A Human Endeavor (subtitled A Book for Those Who Think They Don't Like the Subject) is aimed at a less focused audience. While it can be used as a follow-up to the other two books it can also be used for remedial college students who need to be re-introduced to general math concepts. Some parents have used A Human Endeavor for earlier levels, but we probably wouldn't recommend that. The three books form a loose trilogy of some of the finest math textbooks available.
How Do These Work?
Each level consists of a student text, teacher's guide, and book of test masters. (Overhead transparencies are also available on CD-ROM for the classroom setting, but due to their cost we don't currently stock them.) The teacher's guides contain all the answers to the problem sets as well as suggestions for presenting the material, although Jacobs encourages teachers to find their own ways of teaching and to be creative.
The books are broken up into chapters focused on specific topics. These chapters are further broken down into lessons containing text which introduce and explain the topic and include a series of problem sets. Algebra has four sets: the first is review; Sets 2 and 3 relate to the lesson; and Set 4 is a special math or logic puzzle for advanced students who need more challenge. Geometry only has 3 sets: only Set 2 covers the specific lesson. Answers to Set 2 are always included in the back of the student text; answers to the other sets are in the teacher's guide.
In some instances, the text leaves it up to the student to understand aspects of a principle through intuition. Jacobs heavily encourages deductive reasoning, and presents several opportunities to put it to use. While this may seem daunting to the beginner, the course is designed to help students improve their reasoning and logic skills, thus preparing them to encounter such problems. To the uninitiated, however, there will appear to be certain gaps in some of the lessons.
Elementary Algebra aims at student competency, providing a solid framework of skills and concepts from which to move on to more advanced math. Geometry is Euclidean in approach and nearly exhaustive in scope—Jacobs criticizes other curriculums (like Saxon) for failing to devote an entire text to geometry. Mathematics: A Human Endeavor is much broader, providing more basic information and filling in gaps. Together they provide an excellent, consistent overview of high school math.
Jacobs designed his books for use in a classroom setting. Teacher involvement and instruction is expected, and so too is a measure of teacher competency in the subject matter; unlike other curricula, the teacher's guides are not exhaustive references. However, the student texts are comprehensive enough that students can work through them on their own. Solo work might not yield as full an understanding as teacher-led efforts, but there is enough content to at least generate competency if not mastery.
Our Honest Opinion:
There really aren't much better math curriculums for upper-level work than Harold Jacobs'. The texts are engaging but not overly visual—there is plenty of content, and the pictures and cartoons are often used as illustrations for key ideas. Topics are easily referenced later on due to the logical structure of the books and obvious chapter and lesson breaks.
The student texts are excellent. Concise and witty, they present complex and difficult ideas in a way most students can understand. The teacher's guides from Freeman (the publisher) leave more to be desired: they offer minimal support for teachers, only rudimentary lesson plans, and no solutions manual. Happily, Dr. Callahan and his daughter Cassidy Cash have come to the rescue with video supplements, additional teaching aids and a solutions manual for Elementary Algebra. This helps us to highly recommend these books for high school (home school) students.
If you're looking for a curriculum that will make your kids sick of math this probably isn't the one you want. Many parents report that, after switching to Harold Jacobs, their kids did a lot better in math and started to actually enjoy it. Even professional mathematicians and math teachers offer glowing reviews, and as far as we can tell this is definitely a curriculum that deserves high marks. | 677.169 | 1 |
Calculus
Product Description:
Hecht brings to bear the perspective of both historical concepts and contemporary physics. While the text covers the standard range of material from kinematics to quantum physics, Hecht has carefully limited the math required to basic calculus and very basic vector analysis. He omits obscure, high-level topics while focusing on helping students understand the fundamental concepts of modern-day physics. Calculus and vector analysis are both painstakingly developed as tools, and then used only insofar as they illuminate the physics. Hecht deliberately paces comfortably, justifies where each topic is going, stops to take stock of where the students have been, and points out the marvelous unity of the discourse. Informed by a 20th century perspective and a commitment to providing a conceptual overview of the discipline, Hecht's CALCULUS 2/e keeps students involved and focused.
REVIEWS for Physics | 677.169 | 1 |
Find a HedwigThomas and received an A in the course. Linear Algebra is the study of matrices and their properties. The applications for linear algebra are far reaching whether you want to continue studying advanced algebra or computer science | 677.169 | 1 |
Compressed Zip File
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3.31 MB | 21 pages
PRODUCT DESCRIPTION
Calculus : Integration by Substitution is a review and how-to guide to help students solve problems involving integrals that can be solved using u-Substitution. Includes a handout that discusses concepts informally along with solved examples, with 20 homework problems for the student. A separate handout includes solutions to all the homework problems.
This handout and homework problems are designed for use with AP Calculus BC and College level Calculus 2.
Using an informal tone, this handout is meant to supplement the students ordinary textbook. Concepts are discussed in an easy going fashion with the focus on how to solve problems. Problems are solved in explicit detail. The purpose of the handout is to show students how to solve problems and grasp calculus concepts in an informal fashion. It can be used to supplement the textbook and for tutorial sessions or for self-learning by the student.
Topics covered include:
- Review of the power rule for integration
- Review of other basic types of integrals
- Review of integration of trig functions (optional)
- Overview of the substitution technique
- 4 solved examples to illustrate how to apply the substitution technique
- 20 homework problems
Complete solutions to the homework problems are provided in a separate handout | 677.169 | 1 |
Math Start
Program Overview
MATH Start is an intensive 8-week program for incoming CUNY students who want to increase their math proficiency before starting credit classes. The program enrolls students who have not passed both math sections of the CUNY Assessment Test (Pre-Algebra and Algebra).
The pre-college math instruction focuses on complex topics in algebra and helps students maximize their understanding through in-depth study of core math concepts in an interactive, supportive learning environment. In addition, students attend a weekly college success seminar to help them develop their academic identity and learn about college structures and campus resources.
Students pay only $35 for Math Start and receive free MetroCards to attend.**
Math Start helps students:
Eliminate or reduce remedial math needs
Prepare for success in college math coursework
Provides up to two opportunities to retake the math CUNY Assessment Test
Integrates advisement in preparation for campus life
Eligibility Requirements
Demonstrate need for skills development in math (only) based on your CUNY Assessment Test results. Specifically, you must have received non-passing scores in both Math 1 (arithmetic/ pre-algebra) and Math 2 (elementary algebra).
**MetroCards for students to participate in Bronx Community College and Hostos Community Colleges' Math Start programs are made possible due to the generous support of The Carroll and Milton Petrie Foundation. | 677.169 | 1 |
In Our Math Department
At SWVGS, students have opportunities to enroll in highly differentiated math courses. An exhaustive range of courses is offered so that all students benefit from classes with an appropriate level of challenge.
All students enroll in a non-elective math course each year.
Juniors are placed in an appropriate math course based on math placement scores, previously completed math courses, and individual background and goals.
All juniors enroll in Statistics I and II (MTH 241 and 242) where they learn appropriate methods for collection, organization, analysis, interpretation and presentation of data. Students then apply these skills when designing, conducting and analyzing their independent research project.
Juniors may advance to a higher level mathematics course by completing 85% mastery of topics in the appropriate ALEKS course during the summer.
Seniors select their non-elective math course each semester based on success in their junior year math course and their career goals.
Seniors taking only one lab-based course may opt to take Analytic Geometry or Linear Algebra. These courses are offered on Tuesdays and Thursdays.
Seniors may choose to take Finite Mathematics if they prefer to take an additional math course in place of a second lab-based course or a science elective.
Students learn from instructors who LOVE mathematics. Teacher enthusiasm and extensive knowledge are obvious and inspiring to learners. | 677.169 | 1 |
Overview
Description
Walker uses an integrated "suite" of tools, worked Examples, Active Examples, and Conceptual Checkpoints, to make conceptual understanding an integral part of solving quantitative problems. The pedagogy and approach are based on over 20 years of teaching and reflect the results of physics education research.
Series
Features
"How much time do you spend introducing vectors in this course?"
Students often enter this course without the math background, including experience with vectors, to be successful. Walker devotes a full chapter (Chapter 3) to vectors to provide an additional resource for students.
"What is the problem-solving process you teach your students to use?"
Students use the worked Examples in the text as a guide for solving problems. Every worked Example in Walker is structured to provide a systematic process for solving problems:
Picture the Problem reminds students to visualize the situation, identify and label important quantities, and set up a coordinate system. This step is always accompanied by a figure and free-body diagram when appropriate.
Strategy helps students learn to analyze the problem, identify the key physical concepts, and map a plan for the solution.
Solution is presented in two-columns to help students translate the words of the problem on the left to the equations they will use to solve it on the right.
Insight points out interesting or significant features of the problem, solution process, or the result.
Practice Problem is the last part of the Example. Here, students are given the opportunity to test their understanding and skills on a problem similar to the one just worked.
"What role does conceptual understanding play in your course?"
Much like clicker questions an instructor might ask in class, Conceptual Checkpoints serve as a pause in the reading for students to check their understanding. These multiple choice, conceptual questions recognize and address common student misconceptions.
"How do you help students go from the worked Examples in the text to the end-of-chapter Problems they are assigned for homework?"
Active Examples serve as a bridge between the fully worked Examples and the end-of-chapter Problems. Students take an active role by thinking through the logic of the steps on the left and checking their answers with the answer on the right. This unique pedagogical tool prepares students to better tackle homework problems on their own.
"Do you assign and grade conceptual questions for homework?"
Conceptual Exercises are conceptual ranking task exercises in multiple-choice format. Because they are multiple-choice they can be assigned and easily graded as paper and pencil homework, in an online homework system, or in class using a personal response system.
Students taking introductory physics often wonder how the course is applicable to the challenges they will face in their lives and their careers. Students using Walker have commented that every time they ask themselves this, the book points out a relevant and interesting application of the material. Real-world and Biological applications are identified by a marginal icon. A list of Applications is available in the Preface.
Chapter Summary
Each chapter contains a Chapter Summary organized in an outline format for easy reference and study. The Third Edition Chapter Summary includes key figures from the chapter which serve as a visual reminder as students review.
Problem Solving Summary
In addition to the Chapter Summary, each chapter includes a Problem-Solving Summary. Formatted in an easy-to-reference outline, this summary lists the types of problems covered in the chapter as well as the relevant concepts and specific Examples for each.
Conceptual Questions and Exercises
Conceptual Questions are qualitative questions that allow students to test their understanding of the chapter principles. Because these are open-ended questions, they also provide a resource for in-class or small group discussion.
Conceptual Exerices are conceptual ranking task exercises in multiple-choice format. Because they are multiple-choice they can be assigned and easily graded as paper and pencil homework, in an online homework system, or in class using a personal response system.
Answers to odd-numbered Conceptual Questions and Exercises are available in the back of the book.
Integrated Problems (IP)
Integrated Problems, marked IP, contain two parts: a quantitative problem as well as a conceptual question. This format requires students to check their answer to one part against their answer to the other and promotes this kind of thinking as an important step in solving problems.
New to This Edition
Interactive Problems and Interactive Figures
A new section of Interactive Problems has been added to the homework. In these Problems, an Example, Active Example, Conceptual Checkpoint, or Figure "comes alive" as the computer animates the corresponding physical system. These Interactive Figures, marked with an icon in the text, are designed to be flexible in their application – they can be used in lecture, as a "virtual lab", or as a component of a homework assignment. They are available in the Walker OneKey cartridges, on the Companion Website, and on the Instructor's Resource Center on CD-ROM. By giving direct visual feedback to the student, they help to reinforce what is being learned and to provide an additional pathway of understanding. (see Interactive Figure p.302 Conceptual Checkpoint 10-4 and Interactive Problems on p.314 Problems 96 and 97)
Conceptual Exercises
This NEW section includes conceptual multiple-choice and ranking task exercises. Because they are multiple-choice they can be assigned and easily graded as paper and pencil homework, in an online homework system, or in class using a personal response system. (see p.199)
New and Revised Problems
Jim Walker has been widely recognized for writing interesting, real-world Problems covering a wide range of difficulty. Based on detailed reviewer feedback, 30% of the end of chapter Problems are either new or revised making the Third Edition the best collection of Problems available for algebra-based physics. Jim's personal favorites have titles in the 3/e. (see The World's Fastest Turbine p.309 Problem 24, Dinosaur Sounds p.474 Problem 86, and Cooking Doughnuts p.513 Problem 86)
On-line Homework with Problem-specific Hints and Feedback
Prentice Hall Grade Assist (PHGA) contains end of chapter Conceptual Exercises and Problems from the Third Edition. Variables for Problems are algorithmically generated and Problems are graded automatically. PHGA for the Third Edition contains Problem-specific hints and feedback based on common student misconceptions.
Table of Contents
NOTE: All chapters conclude with a Chapter Summary, Problem-Solving Summary, Conceptual Questions, Conceptual Exercises, and Problems. (Chapter 1 does not include a Problem-Solving Summary.)
About the Author(s)
James S. Walker James Walker obtained his Ph.D. in theoretical physics from the University of Washington in 1978. He subsequently served as a post-doc at the University of Pennsylvania, the Massachusetts Institute of Technology, and the University of California at San Diego before joining the physics faculty at Washington State University in 1983. Professor Walker's research interests include statistical mechanics, critical phenomena, and chaos. His many publications on the application of renormalization-group theory to systems ranging from absorbed monolayers to binary-fluid mixtures have appeared in Physical Review, Physical Review Letters, Physica, and a host of other publications. He has also participated in observations on the summit of Mauna Kea, looking for evidence of extra-solar planets.
Jim Walker likes to work with students at all levels, from judging elementary school science fairs to writing research papers
with graduate students, and has taught introductory physics for many years. His enjoyment of this course and his empathy for students have earned him a reputation as an innovative, enthusiastic, and effective teacher. Jim's educational publications include "Reappearing Phases" (Scientific American, May 1987) as well as articles in the American
Journal of Physics and The Physics Teacher. In recognition of his contributions to the teaching of physics, Jim was named Boeing Distinguished Professor of Science and Mathematics Education for 2001–2003.
When he is not writing, conducting research, teaching, or developing new classroom demonstrations and pedagogical materials, Jim enjoys amateur astronomy, bird watching, photography, juggling, unicycling, boogie boarding, and kayaking. Jim is also an avid jazz pianist and organist. He has served as ballpark organist for several Class A minor league baseball teams, including minor league affiliates of the Seattle Mariners and San Francisco Giants | 677.169 | 1 |
REI.B.3: Common Core State Standards for Mathematics
Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
MA.9-12.CCSS.Math.Content.HSA-CED.A.1: Mathematics
Create equations and inequalities in one variable and use them to solve problems.
MA.9-12.CCSS.Math.Content.HSA-REI.B.3: Mathematics
Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters10-20.
Component Ratings:
Technical Completeness: 3 Content Accuracy: 3 Appropriate Pedagogy: 2
Reviewer Comments:
This resource is a group project that is a culminating activity based on four lessons on writing and solving equations and inequalities. The activity includes a list of materials needed, instructions and an example project. Students may need some background instruction on economics (profit, loss, mark-up) The project leaves a bit of room for teacher modifications. Overall, a useful resource that a teacher can use to reinforce multiple lessons. | 677.169 | 1 |
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Calculators Are for Calculating, Mathematica Is for Calculus
Andy Dorsett
In this Wolfram Mathematica Virtual Conference 2011 course, learn different ways to use Mathematica to enhance your calculus class, such as using interactive models and connecting calculus to the real world with built-in datasets | 677.169 | 1 |
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PRODUCT DESCRIPTION
A template that students can use to summarize a math topic and create notes for reference before tests and quizzes. This will also help promote comprehension of the topic and give an indication of where the student needs more help. Given the importance of being able to verbalize and rationalize answers, this exercise will deepen the learning of the | 677.169 | 1 |
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