text stringlengths 6 976k | token_count float64 677 677 | cluster_id int64 1 1 |
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Math 009A, also known as Introduction to Calculus, is a common course that many majors have to take. It is usually reserved the people in BCOE who are engineering majors because they need the course the most. Though, people who are in Biology and Economics all need this course. Math is a fundamentally important subject and many majors are all fighting for this course
Derivatives and Limits
Calculus is where many people believe math starts to become difficult. After all, there can be over 10 steps involved just for doing 1 problem. One silly mistake in those countless steps and you can get the wrong answer. These concepts are not actually easy to grasp as they are unheard of. With a mixture of complex algebra and new concepts, calculus creates numerous struggles among students. Plus, this is not even applied calculus, so professors are simply teaching in theory. Students cannot physically grasp onto to these concepts as they are intangible.
Showing Work
Some professors can be extremely picky at how work is shown. Notation for limits and derivatives have to precise and accurate. You even have to rewrite the entire limit equation at the end as some professors can be very picky. Since the class size is so large, you won't always have the opportunity to show how your work should be done. Although, there are some laid back professors who do not care about how you show your work.
Biweekly Quizzes
The one thing that many students hate about math are the constant quizzes. There is almost a quiz each time you meet up with your discussion class. Some classes even give out quizzes each week for 10 whole weeks. Other classes give out no quizzes. Most classes give out five classes in total. That means there is a quiz almost every other week you meet up with your discussion class.
Finals and Midterm
The math classes here place a heavy weight on the midterm and final. The midterm is usually 30% of your overall grade and the final is 45%. That means those two tests added together make up 75% of your total grade. In general, that makes up your total grade. Doing badly in one of those exams could risk your A. Also be aware that the class may not be curved.
Online Homework
There is homework assigned after each lecture and due at the next week. Your friends will not have the same problems as the computer generates different problems. This makes students understand the concept and avoid copying homework.
Math 9A is a mandatory course for many majors. The difficulty can vary depending on your professor. Though, students inevitably have to take this class. As long as you put in the time and effort, you should do fine. | 677.169 | 1 |
09 Jun
The study and mastery of mathematical concepts and skills is a highly sequential learning process | 677.169 | 1 |
This book contains answers to all the exercises in Science Year 4, which goes beyond the requirements of the National Curriculum for Year 4 pupils (aged 8 and above) preparing for Common Entrance and other independent school entrance exams at 11+.
Essential Mathematical Methods 1&2 has been specifically designed for students undertaking Mathematical Methods Units 1&2 and intending to proceed to Specialist Mathematics and Mathematical Methods 3&4, or just Mathematical Methods 3&4. The material is chosen from the revised Mathematics Study Design and supplementary material has been added to enable a thorough preparation for these Year 12 courses. The new book also features questions that require the use of a graphics calculator in certain exercises. The CD-ROM includes chapter sections that contain theory, worked examples and exercises linked to answers. | 677.169 | 1 |
The Standard Model is the foundation of modern particle and high energy physics. This book explains the mathematical background behind the Standard Model, translating ideas from physics into a mathematical language and vice versa.
This textbook gives a systematized and compact summary, providing the most essential types of modern models for languages and computation together with their properties and applications. Most of these models properly reflect and formalize current computational methods, based on parallelism, distribution and cooperation covered in this book. As a result, it allows the user to develop, study, and improve these methods very effectively.
The book has many important features which make it suitable for both undergraduate and postgraduate students in various branches of engineering and general and applied sciences. The important topics interrelating Mathematics & Computer Science are also covered briefly. The book is useful to readers with a wide range of backgrounds including Mathematics, Computer Science/Computer Applications and Operational Research. | 677.169 | 1 |
CFX-9850Gb PLUS
(With Built In Program Library for EA100 Experiments)
COLOR GRAPHING CALCULATOR
The CFX-9850Gb PLUS
Color Graphing calculator shows three distinctive colors that can be used
in a variety of ways. This new calculator makes implementing the NCTM standards
in your classroom easier than ever and is affordable at about $84 (Educator
Price) Casio's Icon Menu makes it easy to find the functions you need.
Six function keys select on screen Soft Keys that make the
CFX-9850GbPLUS easy to use because all options are displayed on the
screen at one time.
The CFX-9850Gb PLUS Television Interface Unit
Allows direct connection from the CFX-9850Gb+ color graphing calculator teaching
unit to any television in crisp, clear color. This stand alone unit
is the OH-9850 PLUS
Dual Screen capability shows a two graphs, a graph and a zoom or a graph
and table on the same display.
A new CONICS feature draws and analyzes standard Conic Sections (Ellipses,
Circles, Hyperbolas and Parabolas). Students can trace along the curves,
or solve to find points and values of interest like Foci, Asymptotes, Directrix,
Tangent, Slope and more. Using the Background feature, students can compare
Conics with the graphs of functions drawn in other calculator modes.
The CFX-9850Gb PLUS powerful statistics packages has the ability to
build and manipulate up to 36 lists of data. A wide variety of statistical
calculations are easily completed on each data list with the touch of a single
button. With the PLUS, you can master list-based statistical calculations
with both single and double variables. It also performs linear, median-median,
quadratic, polynomial, cubic, quartic, logarithmic, exponential, power, logistic,
and sinusoidal regressions. The advance statistics include 9 different hypothesis
testing and six confidence interval functions.
The powerful dynamic graphing feature allows the opportunity to dynamically
see a graph unfold before your eyes. Enter your own equations or use the
built in equations. For example, if you use y = mx + b and want to see what
happens to the line as the slope is altered from -4 to 4, you set m as your
dynamic variable with a range of -4 to 4. Then set your y-intercept (b) to
any desired value. Now sit back and watch to show. Great for discovery learning!
You have to see this live!
The EA-100 Data Analysis System will be able to measure and collect data
for temperature, light, motion, sound, force, and pH. Data is collected and
stored in real time by sensitive, veneer compatible probes. The data can
then be transferred directly into the CFX-9850G or CFX-9850Ga PLUS
color graphing calculator, which automatically creates applicable graphs
and tables for the collected data. The 9850GB+ also has a built
in library of programs to use for the EA100.
CFX 9850G Specifications
8 line by 21 character, 3 color display
Standard Calculations with real and complex numbers
Advanced functions available through on- screen, soft
key menus
15 digit calculation accuracy
Up to 20 functions can be stored, graphed, and analyzed
simultaneously | 677.169 | 1 |
Essential Mathematics and Statistics for Science, 2nd Edition
This book is a completely revised and updated version of
this invaluable text which allows science students to extend
necessary skills and techniques, with the topics being developed
through examples in science which are easily understood by students
from a range of disciplines. The introductory approach eases
students into the subject, progressing to cover topics relevant to
first and second year study and support data analysis for final
year projects.
The revision of the material in the book has been matched, on
the accompanying website, with the extensive use of video,
providing worked answers to over 200 questions in the book plus
additional tutorial support. The second edition has also improved
the learning approach for key topic areas to make it even more
accessible and user-friendly, making it a perfect resource for
students of all abilities. The expanding website provides a wide
range of support material, providing a study environment within
which students can develop their independent learning skills, in
addition to providing resources that can be used by tutors for
integration into other science-based programmes.
Resource Site Click here to find links to:
Over 200 videos showing step-by-step workings of problems in the book
Additional materials including related topic areas, applications, and tutorials on Excel and Minitab
Interactive multiple-choice questions for self-testing, with step-by-step video feedback for any wrong answers | 677.169 | 1 |
Description
Math Essentials, Middle School Level gives middle school math teachers the tools they need to help prepare all types of students (including gifted and learning disabled) for mathematics testing and the National Council of Teachers of Mathematics (NCTM) standards. Math Essentials highlights Dr. Thompson's proven approach by incorporating manipulatives, diagrams, and independent practice. This dynamic book covers thirty key objectives arranged in four sections. Each objective includes three activities (two developmental lessons and one independent practice) and a list of commonly made errors related to the objective. The book's activities are designed to be flexible and can be used as a connected set or taught separately, depending on the learning needs of your students. Most activities and problems also include a worksheet and an answer key and each of the four sections contains a practice test with an answer key.
Buy Both and Save 25%!
This item: Math Essentials, Middle School Level: Lessons and Activities for Test Preparation
About the Author
Frances McBroom Thompson, Ph.D. has taught mathematics at the junior and senior high school levels and has served as a K-12 mathematics specialist. She holds a B.S. in mathematics education from Abilene Christian University (Texas), a master's degree in mathematics from the University of Texas in Austin, and a doctoral degree in mathematics education from the University of Georgia at Athens. Dr. Thompson has published numerous articles and conducts workshops for teachers at the elementary and secondary levels.
5. Generate the formulas for the circumference and the area of a circle; apply the formulas to solve word problems.
6. Generate and apply the area formula for a parallelogram (including rectangles); extend to the area of a triangle.
7. Generate and apply the area formula for a trapezoid.
8. Apply nets and concrete models to find total or partial surface areas of prisms and cylinders.
9. Find the volume of a right rectangular prism, or find a missing dimension of the prism; find the new volume when the dimensions of a prism are changed proportionally.
Practice Test.
Section 4: Graphing, Statistics, and Probability.
Objectives.
1. Locate and name points using ordered pairs of rational numbers or integers on a Cartesian coordinate plane.
2. Construct and interpret circle graphs.
3. Compare different numerical or graphical models for the same data, including histograms, circle graphs, stem-and-leaf plots, box plots, and scatter plots; compare two sets of data by comparing their graphs of similar type.
4. Find the mean of a given set of data, using different representations such as tables or bar graphs.
5. Find the probability of a simple event and its complement.
6. Find the probability of a compound event (dependent or independent). | 677.169 | 1 |
Curriculum Frameworks established by the Massachusetts Department of
Education. Students will use mathematical standards for problem solving,
communicating, reasoning and making connections to real world situations.
Mathematical terms and symbols will be reviewed and is a major part of the
course
• Making estimation will determine the reasonableness of an answer
• We use number sense and operations that relate decimals and fractions
• Major focus is on the relationships between decimals, fractions, ratios,
percent and proportions
• Patterns relationship to algebraic thinking is encouraged
• Data will be collected and organizing into charts and graphs that relate to
everyday life situations
• Appreciation of geometry and measurement as seen in our world is explored
• Statistics and probability as it relates to games and choices is practiced
• Problem solving strategies will be formulated to attack a wide variety of
situations
Ratios and Proportional Relationships Analyze proportional relationships and use them to solve real-world and mathematical problems. The Number System Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers. Expressions and Equations Use properties of operations to generate equivalent expressions.
Solve real-life and mathematical problems using numerical and algebraic expressions and equations. Geometry Draw, construct and describe geometrical figures and describe the relationships between them.
Solve real-life and mathematical problems involving angle measure, area, surface area, and volume. Statistics and Probability Use random sampling to draw inferences about a population.
Draw informal comparative inferences about two populations.
Investigate chance processes and develop, use, and evaluate probability models.
8th Grade Math
The Number System Know that there are numbers that are not rational, and approximate them by rational numbers. Expressions and Equations Work with radicals and integer exponents.
Understand the connections between proportional relationships, lines, and linear equations.
Analyze and solve linear equations and pairs of simultaneous linear equations. Functions Define, evaluate, and compare functions.
Use functions to model relationships between quantities. Geometry Understand congruence and similarity using physical models, transparencies, or geometry software.
Understand and apply the Pythagorean Theorem.
Solve real-world and mathematical problems involving volume of cylinders, cones and spheres. Statistics and Probability Investigate patterns of association in bivariate data | 677.169 | 1 |
In some programs, all it will require to go an exam is take note using, memorization, and recall. On the other hand, exceeding in a math class usually takes a distinct variety of effort and hard work. You can not simply just demonstrate up for your lecture and look at your teacher "talk" about algebra and . You find out it by executing: being attentive in school, actively researching, and resolving math complications trigonometry problems – regardless if your instructor has not assigned you any. Should you end up battling to perform perfectly within your math class, then take a look at very best site for solving math challenges to learn the way you may become a far better math scholar.
Affordable math specialists on the web
Math programs comply with a normal progression – every one builds on the knowledge you have obtained and mastered from the previous program. In case you are finding it rough to observe new ideas at school, pull out your outdated math notes and review prior materials to refresh oneself. Make sure that you fulfill the conditions ahead of signing up for any class.
Critique Notes The Night Prior to Course
Despise any time a teacher phone calls on you and you've overlooked how to address a specific issue? Avoid this moment by examining your math notes. This could make it easier to establish which concepts or questions you'd wish to go above in school the subsequent working day.
The considered executing research every night may seem aggravating, however, if you would like to achieve , it is actually essential that you continuously exercise and learn the problem-solving methods. Make use of your textbook or on the net guides to work as a result of top rated math troubles on a weekly foundation – even though you have no research assigned.
Make use of the Health supplements That come with Your Textbook
Textbook publishers have enriched fashionable publications with further materials (which include CD-ROMs or on the web modules) that will be used to enable students acquire extra exercise in . Many of these supplies can also contain a solution or clarification guidebook, which could make it easier to with functioning by way of math troubles by yourself.
Go through Ahead To stay In advance
If you'd like to reduce your in-class workload or maybe the time you devote on homework, make use of your spare time soon after university or to the weekends to study forward for the chapters and ideas which will be covered the subsequent time you happen to be at school.
Assessment Old Assessments and Classroom Examples
The get the job done you are doing in class, for research, and on quizzes can provide clues to what your midterm or ultimate exam will glimpse like. Make use of your old exams and classwork to create a personalized research information for your upcoming exam. Search for the way your trainer frames issues – this really is most likely how they're going to look on your own check.
Learn how to Work From the Clock
This is a preferred study idea for individuals having timed tests; primarily standardized assessments. For those who have only forty minutes for your 100-point take a look at, you'll be able to optimally commit four minutes on every 10-point query. Get facts regarding how very long the test are going to be and which forms of issues are going to be on it. Then approach to assault the better issues very first, leaving you ample time for you to shell out around the much more tough types.
Maximize your Resources to get math research help
If you're owning a hard time comprehension concepts in school, then you'll want to get aid outside of course. Talk to your friends to create a examine team and stop by your instructor's business office hrs to go over rough complications one-on-one. Go to study and critique classes once your instructor announces them, or retain the services of a non-public tutor if you need a single.
Communicate To On your own
If you are reviewing difficulties for an examination, attempt to elucidate out loud what technique and strategies you accustomed to get the remedies. These verbal declarations will occur in handy in the course of a examination once you must remember the ways you need to take to locate a solution. Get additional practice by seeking this tactic having a pal.
Use Analyze Guides For Additional Practice
Are your textbook or class notes not aiding you fully grasp what you ought to be mastering in school? Use examine guides for standardized tests, including the ACT, SAT, or DSST, to brush up on outdated product, or . Research guides typically come outfitted with comprehensive explanations of ways to remedy a sample dilemma, , therefore you can generally find where may be the much better get mathtroubles. | 677.169 | 1 |
A Portrait of Linear Algebra
A Portrait of Linear Algebra provides students with a unified, elegant, modern, and comprehensive introduction to linear algebra that emphasizes the reading, understanding, and writing of proofs, while giving them advice on how to master these skills.
Written in a student-friendly style, with precisely stated definitions and theorems, the NEW Third Edition of A Portrait of Linear Algebra by Jude Thaddeus Socrates:
Features more than 500 additional exercises from the previous edition, including basic computations, assisted computations, true or false questions, mini-projects, and multi-step proofs broken down with hints for the student;
Presents a thorough introduction to basic logic, set theory, axioms, theorems, and methods of proof;
Includes topics not usually seen in an introductory book, such as the exponential of a matrix, the intersection of two subspaces, the pre-image of a subspace, cosets, quotient spaces, and the Isomorphism Theorems of Emmy Noether, a complete proof of Schur's Lemma and the Spectral Theorems, The Fundamental Theorem of Linear Algebra, the Singular Value Decomposition, and simultaneous diagonalization of commuting normal matrices, providing enough material for two full semesters. | 677.169 | 1 |
Objective: Calculus
is often a prerequisite course to linear algebra. As such, it is beneficial
for students to see connections between concepts from linear algebra and
more familiar topics from Calculus. This demo allows the instructor to
tie together the concepts of dot product and a function—we use dot products
to define a function that can be graphed in the plane. The ideas of orthogonality
and parallel vectors can then be connected to the concepts from calculus
that are important in the study of curve sketching.
Level: Any first course
in linear algebra, vector calculus, or any course that includes a study
of dot products.
Platform: This
demo can be used in a browser by instructors for a classroom demonstration
of the topic and by students to carry out an investigation of the topic.
A MATLAB m-file, Mathematica notebook, and Mathcad worksheet may
be downloaded to provide routines for interactive experimentation beyond
the examples included as animations in this browser file.
Instructor's Notes:
Fix some vector .
Plot the dot product of the vector as
t varies from
This dot product operation yields the function
which traces out a curve as t varies from .
For fixed vector we
obtain the graph in Figure 1.
Figure 1.
Rather than look at the final curve as shown in Figure
1, it is helpful to see the curve generated by dot product computations.
To do this, we display the vectoras
a clock hand moving about the origin. To simulate this action, we
select a set of evenly spaced points between .
As the clock hand moves, each of the ordered pairs (t, f(t)) is then shown
on a separate graph as a red dot. The dots are connected by line
segments to obtain the graph in Figure 1. This generation is shown
in the following animation.
Begin by studying the values of f(t) for the
vector
as the animation above progresses. We want to investigate the relationship
between the dot product of the vector v and the set of vectors w(t) = .
This relationship is shown geometrically in the graph of f(t). In
particular, we want to answer the following questions:1. What is the geometric relationship
between
v and w(t) when f(t) is a maximum? 2. What is the geometric relationship
between
v and w(t) when f(t) is a minimum? 3. What is the geometric relationship
between
v and w(t) when f(t) = 0?Use the animation above to investigate the
connection between v and w(t) for the stated properties of f(t).
As an aid, by clicking on the box below
you may initiate an avi-file for the animation. You can stop the
execution of the avi-file at any point by clicking the mouse. Restart
by clicking the mouse again. (WARNING: This file is quite large
and takes a long time to load. If you have a Windows computer, you
may save the avi file to your local computer by right-clicking on the picture
below and "Save Target as..". If you have a Macintosh computer (or
wish to have a Quicktime 3.0 or higher version), click here.
If an instructor has used
the preceding material along with class discussion, the fundamental ideas
have been discussed. Optionally, the instructor can continue to use
the material below in class, or assign students the material to be used
as part of an individual or group investigation that leads to the
answers to the three questions stated below.
Further Discussion or
Student Investigation
As you change vector v
= the
graph of f(t) will change. For example, Figure 2 shows f(t) when
v
= and
Figure 3 shows f(t) when v = .
Click in the boxes below to see the animations
that generate Figure 2 and Figure 3. As you view the animations,
use these choices of the vector v to answer the three questions
posed above.
AVI (Windows) and MOV (Quicktime) movies
may be downloaded by clicking on the links below. Note that the files
are fairly large, so download time may be slow.
For the three cases investigated so far,
complete the following statements.
1. f(t)
has its maximum value when vectors v and w(t) are ______________. 2. f(t)
has its minimum value when vectors v and w(t) are _______________. 3. f(t)
= 0 when vectors v and w(t) are _____________________________.
Use the answers to the preceding statements
to answer the following. Let .
Then
1. For what
value of t is f(t) a maximum?___________________ 2. For
what value of t is f(t) a minimum?___________________ 3. For
what value of t is f(t) = 0?_________________________
By inspecting Figures 1-3 and keeping in
mind the vector v, conjecture the maximum and minimum values of f(t). (Hint: Consider the
graphs that would be generated if v =
or v = .
How are the maximum and minimum values related to vector v in these
special cases?)
Calculus Connection (Optional)
The function
is a linear combination of cos(t) and sin(t). Use max-min techniques
from calculus to determine the location and values of the maximum and minimums
in the interval .
and is included in Demos with
Positive Impact with their permission. A version of this demo
appears in their book Interactive Linear Algebra, A Laboratory Course
Using Mathcad, Springer-Verlag New York, 1996.
Interactive Files:
MATLAB:
MATLAB file dproddemo can
be downloaded and was created by David R. Hill. To see a sample screen
of the MATLAB routine click on the following box.
Mathematica:
Mathematica notebook dproddemo.nb
for the above demo/investigation can be downloaded. It was created
by Elizabeth G. Carver and Lila F. Roberts.
Mathcad:
Mathcad worksheet dproddemo.mcd for
the above demo/investigation was adapted from Interactive
Linear Algebra, A Laboratory Course Using Mathcad (Porter/Hill) by
Lila F. Roberts. | 677.169 | 1 |
This book is designed primarily for undergraduates in mathematics, engineering, and the physical sciences. Rather than concentrating on technical skills, it focuses on a deeper understanding of the subject by providing many unusual and challenging examples. The basic topics of vector geometry, differentiation and integration in several variables are explored. It also provides numerous computer illustrations and tutorials using MATLAB® and Maple®, that bridge the gap between analysis and computation.
Features:
·Includes numerous computer illustrations and tutorials using MATLAB® and Maple® ·Covers the major topics of vector geometry, differentiation, and integration in several variables ·Instructors' ancillaries available upon adoption
"synopsis" may belong to another edition of this title.
About the Author:
David A. Santos (late) held a PhD from the University of Michigan and was an instructor at the Community College of Philadelphia.
Book Description Mercury Learning Information, United States, 2015. Mixed media product. Book Condition: New. Language: English . This book usually ship within 10-15 business days and we will endeavor to dispatch orders quicker than this where possible. BTE 2014. Hardcover. Book Condition: New. Hardcover. This title is designed for the undergraduate course in multivariable and vector calculus. Rather than concentrating on technical skills, it focuses on a deeper understanding of the subject.Shipping may be from multiple locations in the US or from the UK, depending on stock availability. 450 pages. 1.043. Bookseller Inventory # 9781936420285 | 677.169 | 1 |
About this product
Description
For Math and Numeracy Aficionados from All Walks of Life! Fine-tune your numerical mindset with a quantitative review that serves as a tool for perceiving probability in a new way. Whether you're a high school student, college student, or a test-prep candidate, this book's wealth of explanations and insights makes it a perfect learning companion. Enjoy the benefits of your own short course in probability: *Be able to think conceptually by understanding how key problems fit within the main topics of probability, permutations, combinations, and enumerations. *Master basic probability using a simple flowchart to identify the correct formulas. *Understand when to add probabilities and when to multiply probabilities. *Be able to distinguish between events that are independent versus t independent and events that are mutually exclusive versus t mutually exclusive. *Grasp key differences between permutations and combinations and look for key words such as arrangements or selections to indicate the correct problem type. *Solve tricky permutation problems that involve repeated letters or numbers. *Approach probability problems with a newfound confidence and competency. This book is focused on those thinking skills that are essential for mastering basic probability. Such thinking skills make it much more likely that a person will be able to understand the how and why of problem solving, approach the subject in a conceptual way, and grasp those key principles that act as themes to bind related problems. These skills combine the science of math with the art of numbers. Author's bio: Brandon Royal (CPA, MBA) is an award-winning writer whose educational authorship includes The Little Green Math Book, The Little Blue Reasoning Book, The Little Red Writing Book, and The Little Gold Grammar Book, During his tenure working in Hong Kong for US-based Kaplan Educational Centers -- a Washington Post subsidiary and the largest test-preparation organization in the world -- Brandon honed his theories of teaching and education and developed a set of key learning principles to help define the basics of writing, grammar, math, and reasoning. A Canadian by birth and graduate of the University of Chicago's Booth School of Business, his interest in writing began after completing writing courses at Harvard University. Since then he has authored a dozen books and reviews of his books have appeared in Time Asia magazine, Publishers Weekly, Library Journal of America, Midwest Book Review, The Asian Review of Books, Choice Reviews Online, Asia Times Online, and About.com. Brandon is a five-time winner of the International Book Awards, a five-time gold medalist at the President's Book Awards, as well as a winner of the Global eBook Awards, the USA Book News Best Book Awards, and recipient of the 2011 Educational Book of the Year award as presented by the Book Publishers Association of Alberta. To get started in probability theory, all you need are a few basic principles. Here they are, clear and uncluttered, in a short, simple book that comes as a welcome breath of fresh air. --Dr. Ian Steward, author of 17 Equations That Changed the World and the Cabinet of Mathematical Curiosities | 677.169 | 1 |
Pre algebra
Pre-algebra is a common name for a course in middle school math.The official online store of Demme Learning, the authors of Math-U-See, Spelling You See, and Building Faith Families.In this area we build the foundation of Algebra as we study the topic of Pre-Algebra.Homeschooling through high school just got a whole lot easier.
Free Pre-Algebra Math Worksheets | KidSmart Education
Pre-Algebra Tutors | Get Help from SchoolTutoring Academy
Students, teachers, parents, and everyone can find solutions to their math.
Pre-Algebra (Grade 8) - Glencoe/McGraw-Hill
Prealgebra Upgrade On-Line Course
Free Algebra PDF Worksheets - Algebra for Children
These dynamically created Algebra Worksheets cover Pre-Algebra, Algebra 1, and Algebra 2 topics that are suitable for students in the 5th through 8th Grades.Math Goodies is a free math help portal for students, teachers, and parents.
In the United States, pre-algebra is usually taught in the 7th grade The objective of it is to.
Prealgebra / Edition 5 by Elayn Martin-Gay | 2900132319514
Martin-Gay, Prealgebra, 6th Edition - Pearson Higher Ed
Really clear math lessons (pre-algebra, algebra, precalculus), cool math games.These are times of rapid change, and this study program is changing with those.
One-step equations word problem: super yoga (1 of 2) One-step equations word problem: super yoga (2 of 2) Modeling with one-step equations Modeling with one-step equations Practice One-step equations review Inequalities: Greater than and less than basics Equality is usually a good thing, but the world is not a perfect place.Let us throw some explanations, examples, and practice problems at your problem.Pre-Algebra. Algebra 2. Algebra. Algebra is great fun - you get to solve puzzles. | 677.169 | 1 |
Synopsis
This eBook deals with eight different methods of solving vectors that one might come across in statics problems. Some students may find the method used in their course text to be too complicated, or that it may not be described very well. The purpose of this eBook is to empower the student to have choices as to how they might tackle a particular problem, or become familiar with the different methods to further help them understand the concept. Many students tend to study the night before the exam, so this eBook is meant to be short and provide a fast informative read for those students who need fast answers. Many students try the internet, or YouTube only to find that problems are solved in many formats with different symbols as snippets to a particular solution. One method can be used to verify the accuracy of another method, or just check if the solution makes sense | 677.169 | 1 |
Hi, so I'll be starting a physics degree this Oct, and I'm self studying maths a level before I begin(mature student). I'm wondering whether it would be better to, once I've finished with the basic core modules of c1-4, to go as far as possibly with further 1, 2, and 3, or to do m1 and m2, and then possibly if I have time start on fp1?
Just wondering how useful mechanics at a level is for the first year of a physics degree, as in, whether the info is covered or not. | 677.169 | 1 |
Develop strategies to simplify mathematics through "Leveled Texts for Mathematics: Geometry"Highlighting geometry, this resource provides the know-how to use leveled texts to differentiate instruction in mathematics. A total of 15 different topics are featured in and the high-interest text is written at four different reading levels with matching visuals. Practice problems are provided to reinforce what is taught in the passage. The included Teacher Resource CD features a modifiable version of each passage in text format and full-color versions of the texts and image files. This resource is correlated to the Common Core State Standards. 144 pp.
Grade: 3 - 12 / Ages 8 - 18
Language: English
Author Name: Lori Barker
Publisher: Shell Education
Publishing Date/Year: June 1, 2011
No. of Pages: 144
ISBN -10: 1425807178
ISBN -13: 978-1425807177
Series: Leveled Texts for Mathematics
Edition: 1
CD included
Dimensions: 11"H x 8.4"W x 0.4"D
A great resources for integrating Science, Technology, Engineering, Arts and Math (STEAM) into your classroom
Specifications
Audience : Students
Author Name : Lori Barker
Contents : Leveled Texts for Mathematics - Geometry
Education Subject Matter : Math
ISBN : 9781425807177
Language Options : English
Length in Inches : 11
Media Format : Printed Book
Number of Pages : 144
Publisher : Shell Education
Quantity : 1
School Grade : Multi-Grade
Subject and Theme : Mathematics Book
Thickness in Inches : 0.4
Title : Leveled Texts for Mathematics - Geometry
True Color : Multicolor
Type of Book Covers : Paperback
Type of Books : Book
Width in Inches : 8.4
Age Set : 8 - 18 Years
Weight : 1.00 lbs. per Each
Brand : Shell Education | 677.169 | 1 |
Elementary Set Theory: Pt. 1 by Kam-Tim Leung, Doris Lai-Chue Chen
February 28, 2017 @ 12:04 am
By Kam-Tim Leung, Doris Lai-Chue Chen
This e-book presents scholars of arithmetic with the minimal volume of information in good judgment and set conception wanted for a ecocnomic continuation in their experiences. there's a bankruptcy on assertion calculus, via 8 chapters on set thought.
How do you draw a directly line? How do you establish if a circle is basically around? those could sound like uncomplicated or maybe trivial mathematical difficulties, yet to an engineer the solutions can suggest the variation among luck and failure. How around Is Your Circle? invitations readers to discover a number of the comparable basic questions that operating engineers care for each day--it's demanding, hands-on, and enjoyable.
This e-book, designed for complex graduate scholars and post-graduate researchers, introduces Lie algebras and a few in their purposes to the spectroscopy of molecules, atoms, nuclei and hadrons. The ebook includes many examples that aid to explain the summary algebraic definitions. It presents a precis of many formulation of useful curiosity, comparable to the eigenvalues of Casimir operators and the scale of the representations of all classical Lie algebras.
This finished, best-selling textual content makes a speciality of the research of many various geometries -- instead of a unmarried geometry -- and is carefully glossy in its procedure. every one bankruptcy is largely a quick path on one element of contemporary geometry, together with finite geometries, the geometry of alterations, convexity, complicated Euclidian geometry, inversion, projective geometry, geometric features of topology, and non-Euclidean geometries.
3. 4. empty set. The unique set which contains no element is called the The empty set is also called the void set or the null sety and is denoted throughout this book by the symbol 0 . Any set which is not the empty set is called a non-empty set. Clearly the only subset of 0 is 0 itself. Moreover, 0 is also characterized by the property that 0 is a subset of any set. This means that (i) for any set A, 0 c At and (ii) if a set B is such that B c A for every set Ay then B = 0 . Since xe0 is always false, the conditional xe0 -* xeA for every set A is always true.
Cz D. -related to br) for (ayb)eGy where R = (A,B,G). D. Inverses and compositions Intersection and union of two relations Rx — (AyB,Gx) and R2 = (AyByG2) having the same set of departure and the same set of destination are defined as follows: RX()R2 = (AyByGx n G2) # ! U R2 = (AyByG^ u G2) But we shall not discuss these further, as they are seldom used in mathematics. The more important operations on relations are the forming of inverses and compositionSy which we introduce now. 8. If R = (AyB,G) is a relation, then the inverse relation of the relation R is the relation R1 = (ByAyG~l) where G-i = {(bta)eB x A:(ayb)eG} Notice that the set of departure of R is the set of destination of R'1 and that the set of destination of R is the set of departure of R1. | 677.169 | 1 |
Basic Math - PrealBasic Math-Prealgebra Course Menu
This course was designed for students who need to build skills in Basic Math and Prealgebra. The course covers all the essential topics needed to be successful in future Algebra courses. Topics include: basic operations with real numbers, fractions, decimals, exponents, order of operation, conversion of units, percents, radicals, basic operations of Prealgebra, linear equations, mathematical modeling, data interpretation, area, perimeter and volume of geometric figures.
Prerequisite: None
UC Approved: UC approval is not required at this level.
Meets Common Core Requirements: Yes | 677.169 | 1 |
Course Description
Email: asimser@lowvilleacademy.org
Course Description
This course is designed to prepare students for first year algebra. Students are introduced to working with radicals and integer exponents, proportional relationships, lines, linear equations, pairs of simultaneous linear equations, functions, the Pythagorean Theorem, transformations, and congruence/similarity of shapes. A State Assessment to evaluate each students knowledge in this level of mathematics will be administered in April.
Class Expectations
· RESPECT yourself and each other.
· Be on time
· Be Prepared
· Be Ready to Learn
· Bring a Pencil and binder each day
Class Materials
· Student Planner
· 3-ring binder (2 inch)
· Folder
· Pencils
Class Procedure
· Before you enter the class:
o Make sure you have all materials needed for class time
o Have a PENCIL
· When you enter the classroom:
o Pick up classroom materials located on counter in back of room
o Read and fill in the blanks for the focus question of the day
o Get homework out to be turned in on day that it is due on (Thursdays)
Classroom Behavior Plan
· Strike #1: Verbal Warning
· Strike #2: Seat Change
· Strike #3: Removal from class/Discipline referral
Note to parents: In order to keep you informed of important events and upcoming tests, I would like to generate an email list. If you are interested in receiving class updates, please send me an email at asimser@lowvilleacademy.org and I will add you to the class list.
Students can redo/retake any homework assignment or quiz that they would like to get a better grade on.
Corrections can be completed on Unit Tests to get their grade half way to 100%.
o Quizzes are administered after each topic is completed and worth 30 points each
o Unit Assessments are administered at the end of a Unit consisting of about 3 topics and are worth 100 points.
o All tests are announced well ahead of time to allow time for studying and review will be done in class for each Assessment.
· Project
o Project will be given a point value based on the length and amount of time required on the project
· Extra Credit
o Extra Credit will be provided throughout the marking periods and students will be awarded bonus points upon completion.
Absentee Responsibilities
Fill in the lesson notes and complete practice set problems in your math packet by viewing a peers work or the notes binder in the back of the classroom to get caught up on the days you were absent. Homework is due the day that you return to classes if you miss a Thursday class. | 677.169 | 1 |
This dictionary includes explanations of over 200 mathematical words and phrases. Other features include: multiplication tables; table of squares and cubes; frequently-used fractions, decimals and percentages; metric and imperial units; simple coordinate graphs; angle and circle rules | 677.169 | 1 |
matlab for beginners and experienced users hunt lipsman & rosenberg
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A Guide to MATLABThis book is a short, focused introduction to MATLAB, a comprehen-sive software system for mathematics and technical computing. It willbe useful to both beginning and experienced users. It contains conciseexplanations of essential MATLAB commands, as well as easily under-stood instructions for using MATLAB's programming features, graphi-cal capabilities, and desktop interface. It also includes an introductionto SIMULINK, a companion to MATLAB for system simulation. Written for MATLAB 6, this book can also be used with earlier (andlater) versions of MATLAB. This book contains worked-out examplesof applications of MATLAB to interesting problems in mathematics,engineering, economics, and physics. In addition, it contains explicitinstructions for using MATLAB's Microsoft Word interface to producepolished, integrated, interactive documents for reports, presentations,or online publishing. This book explains everything you need to know to begin usingMATLAB to do all these things and more. Intermediate and advancedusers will find useful information here, especially if they are makingthe switch to MATLAB 6 from an earlier version.Brian R. Hunt is an Associate Professor of Mathematics at the Univer-sity of Maryland. Professor Hunt has coauthored four books on math-ematical software and more than 30 journal articles. He is currentlyinvolved in research on dynamical systems and fractal geometry.Ronald L. Lipsman is a Professor of Mathematics and Associate Deanof the College of Computer, Mathematical, and Physical Sciences at theUniversity of Maryland. Professor Lipsman has coauthored five bookson mathematical software and more than 70 research articles. ProfessorLipsman was the recipient of both the NATO and Fulbright Fellowships.Jonathan M. Rosenberg is a Professor of Mathematics at the Univer-sity of Maryland. Professor Rosenberg is the author of two books onmathematics (one of them coauthored by R. Lipsman and K. Coombes)and the coeditor of Novikov Conjectures, Index Theorems, and Rigidity,a two-volume set from the London Mathematical Society Lecture NoteSeries (Cambridge University Press, 1995).
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PrefaceMATLAB is an integrated technical computing environment that combinesnumeric computation, advanced graphics and visualization, and a high-level programming language. – statement encapsulates the view of The MathWorks, Inc., the developer ofMATLAB . MATLAB 6 is an ambitious program. It contains hundreds of com-mands to do mathematics. You can use it to graph functions, solve equations,perform statistical tests, and do much more. It is a high-level programminglanguage that can communicate with its cousins, e.g., FORTRAN and C. Youcan produce sound and animate graphics. You can do simulations and mod-eling (especially if you have access not just to basic MATLAB but also to itsaccessory SIMULINK ). You can prepare materials for export to the WorldWide Web. In addition, you can use MATLAB, in conjunction with the wordprocessing and desktop publishing features of Microsoft Word , to combinemathematical computations with text and graphics to produce a polished, in-tegrated, and interactive document. A program this sophisticated contains many features and options. Thereare literally hundreds of useful commands at your disposal. The MATLABhelp documentation contains thousands of entries. The standard references,whether the MathWorks User's Guide for the product, or any of our com-petitors, contain myriad tables describing an endless stream of commands,options, and features that the user might be expected to learn or access. MATLAB is more than a fancy calculator; it is an extremely useful andversatile tool. Even if you only know a little about MATLAB, you can use itto accomplish wonderful things. The hard part, however, is figuring out whichof the hundreds of commands, scores of help pages, and thousands of items ofdocumentation you need to look at to start using it quickly and effectively. That's where we come in. xiii
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xiv PrefaceWhy We Wrote This Book The goal of this book is to get you started using MATLAB successfully and quickly. We point out the parts of MATLAB you need to know without over- whelming you with details. We help you avoid the rough spots. We give you examples of real uses of MATLAB that you can refer to when you're doing your own work. And we provide a handy reference to the most useful features of MATLAB. When you're finished reading this book, you will be able to use MATLAB effectively. You'll also be ready to explore more of MATLAB on your own. You might not be a MATLAB expert when you finish this book, but you will be prepared to become one — if that's what you want. We figure you're probably more interested in being an expert at your own specialty, whether that's finance, physics, psychology, or engineering. You want to use MATLAB the way we do, as a tool. This book is designed to help you become a proficient MATLAB user as quickly as possible, so you can get on with the business at hand.Who Should Read This Book This book will be useful to complete novices, occasional users who want to sharpen their skills, intermediate or experienced users who want to learn about the new features of MATLAB 6 or who want to learn how to use SIMULINK, and even experts who want to find out whether we know any- thing they don't. You can read through this guide to learn MATLAB on your own. If your employer (or your professor) has plopped you in front of a computer with MATLAB and told you to learn how to use it, then you'll find the book par- ticularly useful. If you are teaching or taking a course in which you want to use MATLAB as a tool to explore another subject — whether in mathematics, science, engineering, business, or statistics — this book will make a perfect supplement. As mentioned, we wrote this guide for use with MATLAB 6. If you plan to continue using MATLAB 5, however, you can still profit from this book. Virtually all of the material on MATLAB commands in this book applies to both versions. Only a small amount of material on the MATLAB interface, found mainly in Chapters 1, 3, and 8, is exclusive to MATLAB 6.
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Preface xvHow This Book Is Organized In writing, we drew on our experience to provide important information as quickly as possible. The book contains a short, focused introduction to MATLAB. It contains practice problems (with complete solutions) so you can test your knowledge. There are several illuminating sample projects that show you how MATLAB can be used in real-world applications, and there is an en- tire chapter on troubleshooting. The core of this book consists of about 75 pages: Chapters 1–4 and the begin- ning of Chapter 5. Read that much and you'll have a good grasp of the funda- mentals of MATLAB. Read the rest — the remainder of the Graphics chapter as well as the chapters on M-Books, Programming, SIMULINK and GUIs, Ap- plications, MATLAB and the Internet, Troubleshooting, and the Glossary — and you'll know enough to do a great deal with MATLAB. Here is a detailed summary of the contents of the book. Chapter 1, Getting Started, describes how to start MATLAB on different platforms. It tells you how to enter commands, how to access online help, how to recognize the various MATLAB windows you will encounter, and how to exit the application. Chapter 2, MATLAB Basics, shows you how to do elementary mathe- matics using MATLAB. This chapter contains the most essential MATLAB commands. Chapter 3, Interacting with MATLAB, contains an introduction to the MATLAB Desktop interface. This chapter will introduce you to the basic window features of the application, to the small program files (M-files) that you will use to make most effective use of the software, and to a simple method (diary files) of documenting your MATLAB sessions. After completing this chapter, you'll have a better appreciation of the breadth described in the quote that opens this preface. Practice Set A, Algebra and Arithmetic, contains some simple problems for practicing your newly acquired MATLAB skills. Solutions are presented at the end of the book. Chapter 4, Beyond the Basics, contains an explanation of the finer points that are essential for using MATLAB effectively. Chapter 5, MATLAB Graphics, contains a more detailed look at many of the MATLAB commands for producing graphics. Practice Set B, Calculus, Graphics, and Linear Algebra, gives you another chance to practice what you've just learned. As before, solutions are provided at the end of the book.
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xvi Preface Chapter 6, M-Books, contains an introduction to the word processing and desktop publishing features available when you combine MATLAB with Microsoft Word. Chapter 7, MATLAB Programming, introduces you to the programming features of MATLAB. This chapter is designed to be useful both to the novice programmer and to the experienced FORTRAN or C programmer. Chapter 8, SIMULINK and GUIs, consists of two parts. The first part de- scribes the MATLAB companion software SIMULINK, a graphically oriented package for modeling, simulating, and analyzing dynamical systems. Many of the calculations that can be done with MATLAB can be done equally well with SIMULINK. If you don't have access to SIMULINK, skip this part of Chapter 8. The second part contains an introduction to the construction and deployment of graphical user interfaces, that is, GUIs, using MATLAB. Chapter 9, Applications, contains examples, from many different fields, of solutions of real-world problems using MATLAB and/or SIMULINK. Practice Set C, Developing Your MATLAB Skills, contains practice problems whose solutions use the methods and techniques you learned in Chapters 6–9. Chapter 10, MATLAB and the Internet, gives tips on how to post MATLAB output on the Web. Chapter 11, Troubleshooting, is the place to turn when anything goes wrong. Many common problems can be resolved by reading (and rereading) the advice in this chapter. Next, we have Solutions to the Practice Sets, which contains solutions to all the problems from the three practice sets. The Glossary contains short de- scriptions (with examples) of many MATLAB commands and objects. Though not a complete reference, it is a handy guide to the most important features of MATLAB. Finally, there is a complete Index.Conventions Used in This Book We use distinct fonts to distinguish various entities. When new terms are first introduced, they are typeset in an italic font. Output from MATLAB is typeset in a monospaced typewriter font; commands that you type for interpretation by MATLAB are indicated by a boldface version of that font. These commands and responses are often displayed on separate lines as they would be in a MATLAB session, as in the following example: >> x = sqrt(2*pi + 1) x = 2.697
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Preface xvii Selectable menu items (from the menu bars in the MATLAB Desktop, figure windows, etc.) are typeset in a boldface font. Submenu items are separated from menu items by a colon, as in File : Open.... Labels such as the names of windows and buttons are quoted, in a "regular" font. File and folder names, as well as Web addresses, are printed in a typewriter font. Finally, names of keys on your computer keyboard are set in a SMALL CAPS font. We use four special symbols throughout the book. Here they are together with their meanings.
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Paragraphs like this one contain cross-references to other parts of the book or suggestions of where you can skip ahead to another chapter. ➱ Paragraphs like this one contain important notes. Our favorite is "Save your work frequently." Pay careful attention to these paragraphs. Paragraphs like this one contain useful tips or point out features of interest in the surrounding landscape. You might not need to think carefully about them on the first reading, but they may draw your attention to some of the finer points of MATLAB if you go back to them later. Paragraphs like this discuss features of MATLAB's Symbolic Math Toolbox, used for symbolic (as opposed to numerical) calculations. If you are not using the Symbolic Math Toolbox, you can skip these sections. Incidentally, if you are a student and you have purchased the MATLAB Student Version, then the Symbolic Math Toolbox and SIMULINK are auto- matically included with your software, along with basic MATLAB. Caution: The Student Edition of MATLAB, a different product, does not come with SIMULINK.About the Authors We are mathematics professors at the University of Maryland, College Park. We have used MATLAB in our research, in our mathematics courses, for pre- sentations and demonstrations, for production of graphics for books and for the Web, and even to help our kids do their homework. We hope that you'll find MATLAB as useful as we do and that this book will help you learn to use it quickly and effectively. Finally, we would like to thank our editor, Alan Harvey, for his personal attention and helpful suggestions.
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Chapter 1 Getting Started In this chapter, we will introduce you to the tools you need to begin using MATLAB effectively. These include: some relevant information on computer platforms and software versions; installation and location protocols; how to launch the program, enter commands, use online help, and recover from hang- ups; a roster of MATLAB's various windows; and finally, how to quit the soft- ware. We know you are anxious to get started using MATLAB, so we will keep this chapter brief. After you complete it, you can go immediately to Chapter 2 to find concrete and simple instructions for the use of MATLAB. We describe the MATLAB interface more elaborately in Chapter 3.Platforms and Versions It is likely that you will run MATLAB on a PC (running Windows or Linux) or on some form of UNIX operating system. (The developers of MATLAB, The MathWorks, Inc., are no longer supporting Macintosh. Earlier versions of MATLAB were available for Macintosh; if you are running one of those, you should find that our instructions for Windows platforms will suffice for your needs.) Unlike previous versions of MATLAB, version 6 looks virtually identi- cal on Windows and UNIX platforms. For definitiveness, we shall assume the reader is using a PC in a Windows environment. In those very few instances where our instructions must be tailored differently for Linux or UNIX users, we shall point it out clearly. ➱ We use the word Windows to refer to all flavors of the Windows operating system, that is, Windows 95, Windows 98, Windows 2000, Windows Millennium Edition, and Windows NT. 1
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2 Chapter 1: Getting Started This book is written to be compatible with the current version of MATLAB, namely version 6 (also known as Release 12). The vast majority of the MATLAB commands we describe, as well as many features of the MATLAB interface (M-files, diary files, M-books, etc.), are valid for version 5.3 (Release 11), and even earlier versions in some cases. We also note that the differences between the Professional Version and the Student Version (not the Student Edition) of MATLAB are rather minor and virtually unnoticeable to the new, or even mid-level, user. Again, in the few instances where we describe a MATLAB feature that is only available in the Professional Version, we highlight that fact clearly.Installation and Location If you intend to run MATLAB on a PC, especially the Student Version, it is quite possible that you will have to install it yourself. You can easily accomplish this using the product CDs. Follow the installation instructions as you would with any new software you install. At some point in the installation you may be asked which toolboxes you wish to include in your installation. Unless you have severe space limitations, we suggest that you install any that seem of interest to you or that you think you might use at some point in the future. We ask only that you be sure to include the Symbolic Math Toolbox among those you install. If possible, we also encourage you to install SIMULINK, which is described in Chapter 8. Depending on your version you may also be asked whether you want to specify certain directory (i.e., folder) locations associated with Microsoft Word. If you do, you will be able to run the M-book interface that is described in Chapter 6. If your computer has Microsoft Word, we strongly urge you to include these directory locations during installation. If you allow the default settings during installation, then MATLAB will likely be found in a directory with a name such as matlabR12 or matlab SR12 or MATLAB — you may have to hunt around to find it. The subdirectory binwin32, or perhaps the subdirectory bin, will contain the executable file matlab.exe that runs the program, while the current working directory will probably be set to matlabR12work.Starting MATLAB You start MATLAB as you would any other software application. On a PC you access it via the Start menu, in Programs under a folder such as MatlabR12
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Typing in the Command Window 3 or Student MATLAB. Alternatively, you may have an icon set up that enables you to start MATLAB with a simple double-click. On a UNIX machine, gen- erally you need only type matlab in a terminal window, though you may first have to find the matlab/bin directory and add it to your path. Or you may have an icon or a special button on your desktop that achieves the task. ➱ On UNIX systems, you should not attempt to run MATLAB in the background by typing matlab . This will fail in either the current or older versions. However you start MATLAB, you will briefly see a window that displays the MATLAB logo as well as some MATLAB product information, and then a MATLAB Desktop window will launch. That window will contain a title bar, a menu bar, a tool bar, and five embedded windows, two of which are hidden. The largest and most important window is the Command Window on the right. We will go into more detail in Chapter 3 on the use and manipulation of the other four windows: the Launch Pad, the Workspace browser, the Command History window, and the Current Directory browser. For now we concentrate on the Command Window to get you started issuing MATLAB commands as quickly as possible. At the top of the Command Window, you may see some general information about MATLAB, perhaps some special instructions for getting started or accessing help, but most important of all, a line that contains a prompt. The prompt will likely be a double caret ( or ). If the Command Window is "active", its title bar will be dark, and the prompt will be followed by a cursor (a vertical line or box, usually blinking). That is the place where you will enter your MATLAB commands (see Chapter 2). If the Command Window is not active, just click in it anywhere. Figure 1-1 contains an example of a newly launched MATLAB Desktop. ➱ In older versions of MATLAB, for example 5.3, there is no integrated Desktop. Only the Command Window appears when you launch the application. (On UNIX systems, the terminal window from which you invoke MATLAB becomes the Command Window.) Commands that we instruct you to enter in the Command Window inside the Desktop for version 6 can be entered directly into the Command Window in version 5.3 and older versions.Typing in the Command Window Click in the Command Window to make it active. When a window becomes active, its titlebar darkens. It is also likely that your cursor will change from
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4 Chapter 1: Getting Started Figure 1-1: A MATLAB Desktop. outline form to solid, or from light to dark, or it may simply appear. Now you can begin entering commands. Try typing 1+1; then press ENTER or RETURN. Next try factor(123456789), and finally sin(10). Your MATLAB Desktop should look like Figure 1-2.Online Help MATLAB has an extensive online help mechanism. In fact, using only this book and the online help, you should be able to become quite proficient with MATLAB. You can access the online help in one of several ways. Typing help at the command prompt will reveal a long list of topics on which help is available. Just to illustrate, try typing help general. Now you see a long list of "general purpose" MATLAB commands. Finally, try help solve to learn about the command solve. In every instance above, more information than your screen can hold will scroll by. See the Online Help section in Chapter 2 for instructions to deal with this. There is a much more user-friendly way to access the online help, namely via the MATLAB Help Browser. You can activate it in several ways; for example, typing either helpwin or helpdesk at the command prompt brings it up.
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Interrupting Calculations 5 Figure 1-2: Some Simple Commands. Alternatively, it is available through the menu bar under either View or Help. Finally, the question mark button on the tool bar will also invoke the Help Browser. We will go into more detail on its features in Chapter 2 — suffice it to say that as in any hypertext browser, you can, by clicking, browse through a host of command and interface information. Figure 1-3 depicts the MATLAB Help Browser. ➱ If you are working with MATLAB version 5.3 or earlier, then typing help, help general, or help solve at the command prompt will work as indicated above. But the entries helpwin or helpdesk call up more primitive, although still quite useful, forms of help windows than the robust Help Browser available with version 6. If you are patient, and not overly anxious to get to Chapter 2, you can type demo to try out MATLAB's demonstration program for beginners.Interrupting Calculations If MATLAB is hung up in a calculation, or is just taking too long to perform an operation, you can usually abort it by typing CTRL+C (that is, hold down the key labeled CTRL, or CONTROL, and press C).
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6 Chapter 1: Getting Started Figure 1-3: The MATLAB Help Browser.MATLAB Windows We have already described the MATLAB Command Window and the Help Browser, and have mentioned in passing the Command History window, Cur- rent Directory browser, Workspace browser, and Launch Pad. These, and seve- ral other windows you will encounter as you work with MATLAB, will allow you to: control files and folders that you and MATLAB will need to access; write and edit the small MATLAB programs (that is, M-files) that you will utilize to run MATLAB most effectively; keep track of the variables and functions that you define as you use MATLAB; and design graphical models to solve prob- lems and simulate processes. Some of these windows launch separately, and some are embedded in the Desktop. You can dock some of those that launch separately inside the Desktop (through the View:Dock menu button), or you can separate windows inside your MATLAB Desktop out to your computer desktop by clicking on the curved arrow in the upper right. These features are described more thoroughly in Chapter 3. For now, we want to call your attention to the other main type of window you will en- counter; namely graphics windows. Many of the commands you issue will generate graphics or pictures. These will appear in a separate window. MAT- LAB documentation refers to these as figure windows. In this book, we shall
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Ending a Session 7 also call them graphics windows. In Chapter 5, we will teach you how to gen- erate and manipulate MATLAB graphics windows most effectively.
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See Figure 2-1 in Chapter 2 for a simple example of a graphics window. ➱ Graphics windows cannot be embedded into the MATLAB Desktop.Ending a Session The simplest way to conclude a MATLAB session is to type quit at the prompt. You can also click on the special symbol that closes your windows (usually an × in the upper left- or right-hand corner). Either of these may or may not close all the other MATLAB windows (which we talked about in the last section) that are open. You may have to close them separately. Indeed, it is our experience that leaving MATLAB-generated windows around after closing the MATLAB Desktop may be hazardous to your operating system. Still another way to exit is to use the Exit MATLAB option from the File menu of the Desktop. Before you exit MATLAB, you should be sure to save any variables, print any graphics or other files you need, and in general clean up after yourself. Some strategies for doing so are addressed in Chapter 3.
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Chapter 2 MATLAB Basics In this chapter, you will start learning how to use MATLAB to do mathematics. You should read this chapter at your computer, with MATLAB running. Try the commands in a MATLAB Command Window as you go along. Feel free to experiment with variants of the examples we present; the best way to find out how MATLAB responds to a command is to try it.
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For further practice, you can work the problems in Practice Set A. The Glossary contains a synopsis of many MATLAB operators, constants, functions, commands, and programming instructions.Input and Output You input commands to MATLAB in the MATLAB Command Window. MAT- LAB returns output in two ways: Typically, text or numerical output is re- turned in the same Command Window, but graphical output appears in a separate graphics window. A sample screen, with both a MATLAB Desktop and a graphics window, labeled Figure No. 1, is shown in Figure 2–1. To generate this screen on your computer, first type 1/2 + 1/3. Then type ezplot('xˆ3 - x'). While MATLAB is working, it may display a "wait" symbol — for example, an hourglass appears on many operating systems. Or it may give no visual evidence until it is finished with its calculation.Arithmetic As we have just seen, you can use MATLAB to do arithmetic as you would a calculator. You can use "+" to add, "-" to subtract, "*" to multiply, "/" to divide, 8
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Arithmetic 9 Figure 2-1: MATLAB Output.and "ˆ" to exponentiate. For example, 3ˆ2 - (5 + 4)/2 + 6*3 ans = 22.5000 MATLAB prints the answer and assigns the value to a variable called ans.If you want to perform further calculations with the answer, you can use thevariable ans rather than retype the answer. For example, you can computethe sum of the square and the square root of the previous answer as follows: ansˆ2 + sqrt(ans) ans = 510.9934 Observe that MATLAB assigns a new value to ans with each calculation.To do more complex calculations, you can assign computed values to variablesof your choosing. For example, u = cos(10) u = -0.8391
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10 Chapter 2: MATLAB Basics v = sin(10) v = -0.5440 uˆ2 + vˆ2 ans = 1 MATLAB uses double-precision floating point arithmetic, which is accurate to approximately 15 digits; however, MATLAB displays only 5 digits by default. To display more digits, type format long. Then all subsequent numerical output will have 15 digits displayed. Type format short to return to 5-digit display. MATLAB differs from a calculator in that it can do exact arithmetic. For example, it can add the fractions 1/2 and 1/3 symbolically to obtain the correct fraction 5/6. We discuss how to do this in the section Symbolic Expressions, Variable Precision, and Exact Arithmetic on the next page.Algebra Using MATLAB's Symbolic Math Toolbox, you can carry out algebraic or symbolic calculations such as factoring polynomials or solving algebraic equations. Type help symbolic to make sure that the Symbolic Math Tool- box is installed on your system. To perform symbolic computations, you must use syms to declare the vari- ables you plan to use to be symbolic variables. Consider the following series of commands: syms x y (x - y)*(x - y)ˆ2 ans = (x-y)^3 expand(ans)
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Algebra 11 ans = x^3-3*x^2*y+3*x*y^2-y^3 factor(ans) ans = (x-y)^3 Notice that symbolic output is left-justified, while numeric output is indented. This feature is often useful in distinguishing symbolic output from numerical output. Although MATLAB makes minor simplifications to the expressions you type, it does not make major changes unless you tell it to. The command ex- pand told MATLAB to multiply out the expression, and factor forced MAT- LAB to restore it to factored form. MATLAB has a command called simplify, which you can sometimes use to express a formula as simply as possible. For example, simplify((xˆ3 - yˆ3)/(x - y)) ans = x^2+x*y+y^2 MATLAB has a more robust command, called simple, that sometimes does a better job than simplify. Try both commands on the trigonometric expression sin(x)*cos(y) + cos(x)*sin(y) to compare — you'll have to read the online help for simple to completely understand the answer.Symbolic Expressions, Variable Precision, and Exact Arithmetic As we have noted, MATLAB uses floating point arithmetic for its calculations. Using the Symbolic Math Toolbox, you can also do exact arithmetic with sym- bolic expressions. Consider the following example: cos(pi/2) ans = 6.1232e-17 The answer is written in floating point format and means 6.1232 × 10−17 . However, we know that cos(π/2) is really equal to 0. The inaccuracy is due to the fact that typing pi in MATLAB gives an approximation to π accurate
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12 Chapter 2: MATLAB Basics to about 15 digits, not its exact value. To compute an exact answer, instead of an approximate answer, we must create an exact symbolic representation of π/2 by typing sym('pi/2'). Now let's take the cosine of the symbolic representation of π/2: cos(sym('pi/2')) ans = 0 This is the expected answer. The quotes around pi/2 in sym('pi/2') create a string consisting of the characters pi/2 and prevent MATLAB from evaluating pi/2 as a floating point number. The command sym converts the string to a symbolic expression. The commands sym and syms are closely related. In fact, syms x is equiv- alent to x = sym('x'). The command syms has a lasting effect on its argu- ment (it declares it to be symbolic from now on), while sym has only a tempo- rary effect unless you assign the output to a variable, as in x = sym('x'). Here is how to add 1/2 and 1/3 symbolically: sym('1/2') + sym('1/3') ans = 5/6 Finally, you can also do variable-precision arithmetic with vpa. For example, √ to print 50 digits of 2, type vpa('sqrt(2)', 50) ans = 1.4142135623730950488016887242096980785696718753769 ➱ You should be wary of using sym or vpa on an expression that MATLAB must evaluate before applying variable-precision arithmetic. To illustrate, enter the expressions 3ˆ45, vpa(3ˆ45), and vpa('3ˆ45'). The first gives a floating point approximation to the answer, the second — because MATLAB only carries 16-digit precision in its floating point evaluation of the exponentiation — gives an answer that is correct only in its first 16 digits, and the third gives the exact answer.
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See the section Symbolic and Floating Point Numbers in Chapter 4 for details about how MATLAB converts between symbolic and floating point numbers.
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Managing Variables 13Managing Variables We have now encountered three different classes of MATLAB data: floating point numbers, strings, and symbolic expressions. In a long MATLAB session it may be hard to remember the names and classes of all the variables you have defined. You can type whos to see a summary of the names and types of your currently defined variables. Here's the output of whos for the MATLAB session displayed in this chapter: whos Name Size Bytes Class ans 1x1 226 sym object u 1x1 8 double array v 1x1 8 double array x 1x1 126 sym object y 1x1 126 sym object Grand total is 58 elements using 494 bytes We see that there are currently five assigned variables in our MATLAB session. Three are of class "sym object"; that is, they are symbolic objects. The variables x and y are symbolic because we declared them to be so using syms, and ans is symbolic because it is the output of the last command we executed, which involved a symbolic expression. The other two variables, u and v, are of class "double array". That means that they are arrays of double-precision numbers; in this case the arrays are of size 1 × 1 (that is, scalars). The "Bytes" column shows how much computer memory is allocated to each variable. Try assigning u = pi, v = 'pi', and w = sym('pi'), and then type whos to see how the different data types are described. The command whos shows information about all defined variables, but it does not show the values of the variables. To see the value of a variable, simply type the name of the variable and press ENTER or RETURN. MATLAB commands expect particular classes of data as input, and it is important to know what class of data is expected by a given command; the help text for a command usually indicates the class or classes of input it expects. The wrong class of input usually produces an error message or unexpected output. For example, type sin('pi') to see how unexpected output can result from supplying a string to a function that isn't designed to accept strings. To clear all defined variables, type clear or clear all. You can also type, for example, clear x y to clear only x and y. You should generally clear variables before starting a new calculation. Otherwise values from a previous calculation can creep into the new
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14 Chapter 2: MATLAB Basics Figure 2-2: Desktop with the Workspace Browser. calculation by accident. Finally, we observe that the Workspace browser pre- sents a graphical alternative to whos. You can activate it by clicking on the Workspace tab, by typing workspace at the command prompt, or through the View item on the menu bar. Figure 2-2 depicts a Desktop in which the Command Window and the Workspace browser contain the same information as displayed above.Errors in Input If you make an error in an input line, MATLAB will beep and print an error message. For example, here's what happens when you try to evaluate 3uˆ2: 3uˆ2 ??? 3u^2 | Error: Missing operator, comma, or semicolon. The error is a missing multiplication operator *. The correct input would be 3*uˆ2. Note that MATLAB places a marker (a vertical line segment) at the place where it thinks the error might be; however, the actual error may have occurred earlier or later in the expression.
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Online Help 15 ➱ Missing multiplication operators and parentheses are among the most common errors. You can edit an input line by using the UP-ARROW key to redisplay the pre- vious command, editing the command using the LEFT- and RIGHT-ARROW keys, and then pressing RETURN or ENTER. The UP- and DOWN-ARROW keys allow you to scroll back and forth through all the commands you've typed in a MATLAB session, and are very useful when you want to correct, modify, or reenter a previous command.Online Help There are several ways to get online help in MATLAB. To get help on a particu- lar command, enter help followed by the name of the command. For example, help solve will display documentation for solve. Unless you have a large monitor, the output of help solve will not fit in your MATLAB command window, and the beginning of the documentation will scroll quickly past the top of the screen. You can force MATLAB to display information one screen- ful at a time by typing more on. You press the space bar to display the next screenful, or ENTER to display the next line; type help more for details. Typing more on affects all subsequent commands, until you type more off. The command lookfor searches the first line of every MATLAB help file for a specified string (use lookfor -all to search all lines). For example, if you wanted to see a list of all MATLAB commands that contain the word "factor" as part of the command name or brief description, then you would type lookfor factor. If the command you are looking for appears in the list, then you can use help on that command to learn more about it. The most robust online help in MATLAB 6 is provided through the vastly improved Help Browser. The Help Browser can be invoked in several ways: by typing helpdesk at the command prompt, under the View item in the menu bar, or through the question mark button on the tool bar. Upon its launch you will see a window with two panes: the first, called the Help Navigator, used to find documentation; and the second, called the display pane, for viewing documentation. The display pane works much like a normal web browser. It has an address window, buttons for moving forward and backward (among the windows you have visited), live links for moving around in the documentation, the capability of storing favorite sites, and other such tools. You use the Help Navigator to locate the documentation that you will ex- plore in the display pane. The Help Navigator has four tabs that allow you to
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16 Chapter 2: MATLAB Basics arrange your search for documentation in different ways. The first is the Con- tents tab that displays a tree view of all the documentation topics available. The extent of that tree will be determined by how much you (or your system administrator) included in the original MATLAB installation (how many tool- boxes, etc.). The second tab is an Index that displays all the documentation available in index format. It responds to your key entry of likely items you want to investigate in the usual alphabetic reaction mode. The third tab pro- vides the Search mechanism. You type in what you seek, either a function or some other descriptive term, and the search engine locates corresponding documentation that pertains to your entry. Finally, the fourth tab is a roster of your Favorites. Clicking on an item that appears in any of these tabs brings up the corresponding documentation in the display pane. The Help Browser has an excellent tutorial describing its own operation. To view it, open the Browser; if the display pane is not displaying the "Begin Here" page, then click on it in the Contents tab; scroll down to the "Using the Help Browser" link and click on it. The Help Browser is a powerful and easy-to-use aid in finding the information you need on various components of MATLAB. Like any such tool, the more you use it, the more adept you become at its use. If you type helpwin to launch the Help Browser, the display pane will contain the same roster that you see as the result of typing help at the command prompt, but the entries will be links.Variables and Assignments In MATLAB, you use the equal sign to assign values to a variable. For instance, x = 7 x = 7 will give the variable x the value 7 from now on. Henceforth, whenever MAT- LAB sees the letter x, it will substitute the value 7. For example, if y has been defined as a symbolic variable, then xˆ2 - 2*x*y + y ans = 49-13*y
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Solving Equations 17 ➱ To clear the value of the variable x, type clear x. You can make very general assignments for symbolic variables and then manipulate them. For example, clear x; syms x y z = xˆ2 - 2*x*y + y z = x^2-2*x*y+y 5*y*z ans = 5*y*(x^2-2*x*y+y) A variable name or function name can be any string of letters, digits, and underscores, provided it begins with a letter (punctuation marks are not al- lowed). MATLAB distinguishes between uppercase and lowercase letters. You should choose distinctive names that are easy for you to remember, generally using lowercase letters. For example, you might use cubicsol as the name of the solution of a cubic equation. ➱ A common source of puzzling errors is the inadvertent reuse of previously defined variables. MATLAB never forgets your definitions unless instructed to do so. You can check on the current value of a variable by simply typing its name.Solving Equations You can solve equations involving variables with solve or fzero. For exam- ple, to find the solutions of the quadratic equation x 2 − 2x − 4 = 0, type solve('xˆ2 - 2*x - 4 = 0') ans = [ 5^(1/2)+1] [ 1-5^(1/2)] Note that the equation to be solved is specified as a string; that is, it is sur- rounded by single quotes. The answer consists of the exact (symbolic) solutions
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18 Chapter 2: MATLAB Basics √ 1 ± 5. To get numerical solutions, type double(ans), or vpa(ans) to dis- play more digits. The command solve can solve higher-degree polynomial equations, as well as many other types of equations. It can also solve equations involving more than one variable. If there are fewer equations than variables, you should spec- ify (as strings) which variable(s) to solve for. For example, type solve('2*x - log(y) = 1', 'y') to solve 2x − log y = 1 for y in terms of x. You can specify more than one equation as well. For example, [x, y] = solve('xˆ2 - y = 2', 'y - 2*x = 5') x = [ 1+2*2^(1/2)] [ 1-2*2^(1/2)] y = [ 7+4*2^(1/2)] [ 7-4*2^(1/2)] This system of equations has two solutions. MATLAB reports the solution by giving the two x values and the two y values for those solutions. Thus the first solution consists of the first value of x together with the first value of y. You can extract these values by typing x(1) and y(1): x(1) ans = 1+2*2^(1/2) y(1) ans = 7+4*2^(1/2) The second solution can be extracted with x(2) and y(2). Note that in the preceding solve command, we assigned the output to the vector [x, y]. If you use solve on a system of equations without assigning the output to a vector, then MATLAB does not automatically display the values of the solution: sol = solve('xˆ2 - y = 2', 'y - 2*x = 5')
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Solving Equations 19 sol = x: [2x1 sym] y: [2x1 sym] To see the vectors of x and y values of the solution, type sol.x and sol.y. To see the individual values, type sol.x(1), sol.y(1), etc. Some equations cannot be solved symbolically, and in these cases solve tries to find a numerical answer. For example, solve('sin(x) = 2 - x') ans = 1.1060601577062719106167372970301 Sometimes there is more than one solution, and you may not get what you expected. For example, solve('exp(-x) = sin(x)') ans = -2.0127756629315111633360706990971 +2.7030745115909622139316148044265*i The answer √ a complex number; the i at the end of the answer stands for is the number −1. Though it is a valid solution of the equation, there are also real number solutions. In fact, the graphs of exp(−x) and sin(x) are shown in Figure 2-3; each intersection of the two curves represents a solution of the equation e−x = sin(x). You can numerically find the solutions shown on the graph with fzero, which looks for a zero of a given function near a specified value of x. A solution of the equation e−x = sin(x) is a zero of the function e−x − sin(x), so to find the solution near x = 0.5 type fzero(inline('exp(-x) - sin(x)'), 0.5) ans = 0.5885 Replace 0.5 with 3 to find the next solution, and so forth.
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In the example above, the command inline, which we will discuss further in the section User-Defined Functions below, converts its string argument to a
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20 Chapter 2: MATLAB Basics exp(-x) and sin(x) 1 0.5 0 -0.5 -1 0 1 2 3 4 5 6 7 8 9 10 x Figure 2-3 function data class. This is the type of input fzero expects as its first argument. In current versions of MATLAB, fzero also accepts a string expression with independent variable x, so that we could have run the command above without using inline, but this feature is no longer documented in the help text for fzero and may be removed in future versions.Vectors and Matrices MATLAB was written originally to allow mathematicians, scientists, and engineers to handle the mechanics of linear algebra — that is, vectors and matrices — as effortlessly as possible. In this section we introduce these concepts.
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Vectors and Matrices 21Vectors A vector is an ordered list of numbers. You can enter a vector of any length in MATLAB by typing a list of numbers, separated by commas or spaces, inside square brackets. For example, Z = [2,4,6,8] Z = 2 4 6 8 Y = [4 -3 5 -2 8 1] Y = 4 -3 5 -2 8 1 Suppose you want to create a vector of values running from 1 to 9. Here's how to do it without typing each number: X = 1:9 X = 1 2 3 4 5 6 7 8 9 The notation 1:9 is used to represent a vector of numbers running from 1 to 9 in increments of 1. The increment can be specified as the second of three arguments: X = 0:2:10 X = 0 2 4 6 8 10 You can also use fractional or negative increments, for example, 0:0.1:1 or 100:-1:0. The elements of the vector X can be extracted as X(1), X(2), etc. For ex- ample, X(3) ans = 4
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22 Chapter 2: MATLAB Basics To change the vector X from a row vector to a column vector, put a prime (') after X: X' ans = 0 2 4 6 8 10 You can perform mathematical operations on vectors. For example, to square the elements of the vector X, type X.ˆ2 ans = 0 4 16 36 64 100 The period in this expression is very important; it says that the numbers in X should be squared individually, or element-by-element. Typing Xˆ2 would tell MATLAB to use matrix multiplication to multiply X by itself and would produce an error message in this case. (We discuss matrices below and in Chapter 4.) Similarly, you must type .* or ./ if you want to multiply or di- vide vectors element-by-element. For example, to multiply the elements of the vector X by the corresponding elements of the vector Y, type X.*Y ans = 0 -6 20 -12 64 10 Most MATLAB operations are, by default, performed element-by-element. For example, you do not type a period for addition and subtraction, and you can type exp(X) to get the exponential of each number in X (the matrix ex- ponential function is expm). One of the strengths of MATLAB is its ability to efficiently perform operations on vectors.
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Vectors and Matrices 23Matrices A matrix is a rectangular array of numbers. Row and column vectors, which we discussed above, are examples of matrices. Consider the 3 × 4 matrix 1 2 3 4 A = 5 6 7 8 . 9 10 11 12 It can be entered in MATLAB with the command A = [1, 2, 3, 4; 5, 6, 7, 8; 9, 10, 11, 12] A = 1 2 3 4 5 6 7 8 9 10 11 12 Note that the matrix elements in any row are separated by commas, and the rows are separated by semicolons. The elements in a row can also be separated by spaces. If two matrices A and B are the same size, their (element-by-element) sum is obtained by typing A + B. You can also add a scalar (a single number) to a matrix; A + c adds c to each element in A. Likewise, A - B represents the difference of A and B, and A - c subtracts the number c from each element of A. If A and B are multiplicatively compatible (that is, if A is n × m and B is m × ), then their product A*B is n × . Recall that the element of A*B in the ith row and jth column is the sum of the products of the elements from the ith row of A times the elements from the jth column of B, that is, m (A ∗ B)i j = AikBkj , 1 ≤ i ≤ n, 1 ≤ j ≤ . k=1 The product of a number c and the matrix A is given by c*A, and A' represents the conjugate transpose of A. (For more information, see the online help for ctranspose and transpose.) A simple illustration is given by the matrix product of the 3 × 4 matrix A above by the 4 × 1 column vector Z': A*Z' ans = 60 140 220
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24 Chapter 2: MATLAB Basics The result is a 3 × 1 matrix, in other words, a column vector.
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MATLAB has many commands for manipulating matrices. You can read about them in the section More about Matrices in Chapter 4 and in the online help; some of them are illustrated in the section Linear Economic Models in Chapter 9.Suppressing Output Typing a semicolon at the end of an input line suppresses printing of the output of the MATLAB command. The semicolon should generally be used when defining large vectors or matrices (such as X = -1:0.1:2;). It can also be used in any other situation where the MATLAB output need not be displayed.Functions In MATLAB you will use both built-in functions as well as functions that you create yourself.Built-in Functions MATLAB has many built-in functions. These include sqrt, cos, sin, tan, log, exp, and atan (for arctan) as well as more specialized mathematical functions such as gamma, erf, and besselj. MATLAB also has several√ built- in constants, including pi (the number π ), i (the complex number i = −1), and Inf (∞). Here are some examples: log(exp(3)) ans = 3 The function log is the natural logarithm, called "ln" in many texts. Now consider sin(2*pi/3) ans = 0.8660
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Functions 25 To get an exact answer, you need to use a symbolic argument: sin(sym('2*pi/3')) ans = 1/2*3^(1/2)User-Defined Functions In this section we will show how to use inline to define your own functions. Here's how to define the polynomial function f (x) = x 2 + x + 1: f = inline('xˆ2 + x + 1', 'x') f = Inline function: f(x) = x^2 + x + 1 The first argument to inline is a string containing the expression defining the function. The second argument is a string specifying the independent variable.
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The second argument to inline can be omitted, in which case MATLAB will "guess" what it should be, using the rules about "Default Variables" to be discussed later at the end of Chapter 4. Once the function is defined, you can evaluate it: f(4) ans = 21 MATLAB functions can operate on vectors as well as scalars. To make an inline function that can act on vectors, we use MATLAB's vectorize function. Here is the vectorized version of f (x) = x 2 + x + 1: f1 = inline(vectorize('xˆ2 + x + 1'), 'x') f1 = Inline function: f1(x) = x.^2 + x + 1
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26 Chapter 2: MATLAB Basics Note that ^ has been replaced by .^. Now you can evaluate f1 on a vector: f1(1:5) ans = 3 7 13 21 31 You can plot f1, using MATLAB graphics, in several ways that we will explore in the next section. We conclude this section by remarking that one can also define functions of two or more variables: g = inline('uˆ2 + vˆ2', 'u', 'v') g = Inline function: g(u,v) = u^2+v^2Graphics In this section, we introduce MATLAB's two basic plotting commands and show how to use them.Graphing with ezplot The simplest way to graph a function of one variable is with ezplot, which expects a string or a symbolic expression representing the function to be plot- ted. For example, to graph x 2 + x + 1 on the interval −2 to 2 (using the string form of ezplot), type ezplot('xˆ2 + x + 1', [-2 2]) The plot will appear on the screen in a new window labeled "Figure No. 1". We mentioned that ezplot accepts either a string argument or a symbolic expression. Using a symbolic expression, you can produce the plot in Figure 2-4 with the following input: syms x ezplot(xˆ2 + x + 1, [-2 2]) Graphs can be misleading if you do not pay attention to the axes. For example, the input ezplot(xˆ2 + x + 3, [-2 2]) produces a graph
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Graphics 27 2 x +x+1 7 6 5 4 3 2 1 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 x Figure 2-4 that looks identical to the previous one, except that the vertical axis has different tick marks (and MATLAB assigns the graph a different title).Modifying Graphs You can modify a graph in a number of ways. You can change the title above the graph in Figure 2-4 by typing (in the Command Window, not the figure window) title 'A Parabola' You can add a label on the horizontal axis with xlabel or change the label on the vertical axis with ylabel. Also, you can change the horizontal and vertical ranges of the graph with axis. For example, to confine the vertical range to the interval from 1 to 4, type axis([-2 2 1 4]) The first two numbers are the range of the horizontal axis; both ranges must
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28 Chapter 2: MATLAB Basics be included, even if only one is changed. We'll examine more options for ma- nipulating graphs in Chapter 5. To close the graphics window select File : Close from its menu bar, type close in the Command Window, or kill the window the way you would close any other window on your computer screen.Graphing with plot The command plot works on vectors of numerical data. The basic syntax is plot(X, Y) where X and Y are vectors of the same length. For example, X = [1 2 3]; Y = [4 6 5]; plot(X, Y) The command plot(X, Y) considers the vectors X and Y to be lists of the x and y coordinates of successive points on a graph and joins the points with line segments. So, in Figure 2-5, MATLAB connects (1, 4) to (2, 6) to (3, 5). 6 5.8 5.6 5.4 5.2 5 4.8 4.6 4.4 4.2 4 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 Figure 2-5
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Graphics 29 To plot x 2 + x + 1 on the interval from −2 to 2 we first make a list X ofx values, and then type plot(X, X.ˆ2 + X + 1). We need to use enoughx values to ensure that the resulting graph drawn by "connecting the dots"looks smooth. We'll use an increment of 0.1. Thus a recipe for graphing theparabola is X = -2:0.1:2; plot(X, X.ˆ2 + X + 1)The result appears in Figure 2-6. Note that we used a semicolon to suppressprinting of the 41-element vector X. Note also that the command plot(X, f1(X))would produce the same results (f1 is defined earlier in the section User-Defined Functions). 7 6 5 4 3 2 1 0 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Figure 2-6
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We describe more of MATLAB's graphics commands in Chapter 5. For now, we content ourselves with demonstrating how to plot a pair of expressions on the same graph.Plotting Multiple Curves Each time you execute a plotting command, MATLAB erases the old plot and draws a new one. If you want to overlay two or more plots, type hold on. This command instructs MATLAB to retain the old graphics and draw any new graphics on top of the old. It remains in effect until you type hold off. Here's an example using ezplot: ezplot('exp(-x)', [0 10]) hold on ezplot('sin(x)', [0 10]) hold off title 'exp(-x) and sin(x)' The result is shown in Figure 2-3 earlier in this chapter. The commands hold on and hold off work with all graphics commands. With plot, you can plot multiple curves directly. For example, X = 0:0.1:10; plot(X, exp(-X), X, sin(X)) Note that the vector of x coordinates must be specified once for each function being plotted.
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Chapter 3 Interacting with MATLAB In this chapter we describe an effective procedure for working with MATLAB, and for preparing and presenting the results of a MATLAB session. In parti- cular we will discuss some features of the MATLAB interface and the use of script M-files, function M-files, and diary files. We also give some simple hints for debugging your M-files.The MATLAB Interface MATLAB 6 has a new interface called the MATLAB Desktop. Embedded inside it is the Command Window that we described in Chapter 2. If you are using MATLAB 5, then you will only see the Command Window. In that case you should skip the next subsection and proceed directly to the Menu and Tool Bars subsection below.The Desktop By default, the MATLAB Desktop (Figure 1-1 in Chapter 1) contains five windows inside it, the Command Window on the right, the Launch Pad and the Workspace browser in the upper left, and the Command History window and Current Directory browser in the lower left. Note that there are tabs for alternating between the Launch Pad and the Workspace browser, or between the Command History window and Current Directory browser. Which of the five windows are currently visible can be adjusted with the View : Desktop Layout menu at the top of the Desktop. (For example, with the Simple option, you see only the Command History and Command Window, side-by-side.) The sizes of the windows can be adjusted by dragging their edges with the mouse. 31
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32 Chapter 3: Interacting with MATLAB The Command Window is where you type the commands and instructions that cause MATLAB to evaluate, compute, draw, and perform all the other wonderful magic that we describe in this book. The Command History window contains a running history of the commands that you type into the Command Window. It is useful in two ways. First, it lets you see at a quick glance a record of the commands that you have entered previously. Second, it can save you some typing time. If you click on an entry in the Command History with the right mouse button, it becomes highlighted and a menu of options appears. You can, for example, select Copy, then click with the right mouse button in the Command Window and select Paste, whereupon the command you selected will appear at the command prompt and be ready for execution or editing. There are many other options that you can learn by experimenting; for instance, if you double-click on an entry in the Command History then it will be executed immediately in the Command Window. The Launch Pad window is basically a series of shortcuts that enable you to access various features of the MATLAB software with a double-click. You can use it to start SIMULINK, run demos of various toolboxes, use MATLAB web tools, open the Help Browser, and more. We recommend that you experiment with the entries in the Launch Pad to gain familiarity with its features. The Workspace browser and Current Directory browser will be described in separate subsections below. Each of the five windows in the Desktop contains two small buttons in the upper right corner. The × allows you to close the window, while the curved arrow will "undock" the window from the Desktop (you can return it to the Desktop by selecting Dock from the View menu of the undocked window). You can also customize which windows appear inside the Desktop using its View menu. While the Desktop provides some new features and a common interface for both the Windows and UNIX versions of MATLAB 6, it may also run more slowly than the MATLAB 5 Command Window interface, especially on older computers. You can run MATLAB 6 with the old interface by starting the program with the command matlab /nodesktop on a Windows system or matlab -nodesktop on a UNIX system. If you are a Windows user, you probably start MATLAB by double-clicking on an icon. If so, you can create an icon to start MATLAB without the Desktop feature as follows. First, click the right mouse button on the MATLAB icon and select Create Shortcut. A new, nearly identical icon will appear on your screen (possibly behind a window — you may need to hunt for it). Next, click the right mouse button on the new icon, and select Properties. In the panel that pops up, select the
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The MATLAB Interface 33 Shortcut tab, and in the "Target" box, add to the end of the executable file name a space followed by /nodesktop. (Notice that you can also change the default working directory in the "Start in" box.) Click OK, and your new icon is all set; you may want to rename it by clicking on it again with the right mouse button, selecting Rename, and typing the new name.Menu and Tool Bars The MATLAB Desktop includes a menu bar and a tool bar; the tool bar contains buttons that give quick access to some of the items you can select through the menu bar. On a Windows system, the MATLAB 5 Command Window has a menu bar and tool bar that are similar, but not identical, to those of MATLAB 6. For example, its menus are arranged differently and its tool bar has buttons that open the Workspace browser and Path Browser, described below. When referring to menu and tool bar items below, we will describe the MATLAB 6 Desktop interface. ➱ Many of the menu selections and tool bar buttons cause a new window to appear on your screen. If you are using a UNIX system, keep in mind the following caveats as you read the rest of this chapter. First, some of the pop-up windows that we describe are available on some UNIX systems but unavailable on others, depending (for instance) on the operating system. Second, we will often describe how to use both the command line and the menu and tool bars to perform certain tasks, though only the command line is available on some UNIX systems.The Workspace In Chapter 2, we introduced the commands clear and whos, which can be used to keep track of the variables you have defined in your MATLAB session. The complete collection of defined variables is referred to as the Workspace, which you can view using the Workspace browser. You can make the browser appear by typing workspace or, in the default layout of the MATLAB Desktop, by clicking on the Workspace tab in the Launch Pad window (in a MATLAB 5 Command Window select File:Show Workspace instead). The Workspace browser contains a list of the current variables and their sizes (but not their values). If you double-click on a variable, its contents will appear in a new window called the Array Editor, which you can use to edit individual entries in a vector or matrix. (The command openvar also will open the Array Editor.)
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34 Chapter 3: Interacting with MATLAB You can remove a variable from the Workspace by selecting it in the Workspace browser and choosing Edit:Delete. If you need to interrupt a session and don't want to be forced to recompute everything later, then you can save the current Workspace with save. For example, typing save myfile saves the values of all currently defined vari- ables in a file called myfile.mat. To save only the values of the variables X and Y, type save myfile X Y When you start a new session and want to recover the values of those variables, use load. For example, typing load myfile restores the values of all the variables stored in the file myfile.mat.The Working Directory New files you create from within MATLAB will be stored in your current working directory. You may want to change this directory from its default location, or you may want to maintain different working directories for dif- ferent projects. To create a new working directory you must use the standard procedure for creating a directory in your operating system. Then you can make this directory your current working directory in MATLAB by using cd, or by selecting this directory in the "Current Directory" box on the Desktop tool bar. For example, on a Windows computer, you could create a directory called C:ProjectA. Then in MATLAB you would type cd C:ProjectA to make it your current working directory. You will then be able to read and write files in this directory in your current MATLAB session. If you only need to be able to read files from a certain directory, an alterna- tive to making it your working directory is to add it to the path of directories that MATLAB searches to find files. The current working directory and the directories in your path are the only places MATLAB searches for files, unless you explicitly type the directory name as part of the file name. To add the directory C:ProjectA to your path, type addpath C:ProjectA When you add a directory to the path, the files it contains remain available for the rest of your session regardless of whether you subsequently add another
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The MATLAB Interface 35 directory to the path or change the working directory. The potential disadvan- tage of this approach is that you must be careful when naming files. When MATLAB searches for files, it uses the first file with the correct name that it finds in the path list, starting with the current working directory. If you use the same name for different files in different directories in your path, you can run into problems. You can also control the MATLAB search path from the Path Browser. To open the Path Browser, type editpath or pathtool, or select File:Set Path.... The Path Browser consists of a panel, with a list of directories in the current path, and several buttons. To add a directory to the path list, click on Add Folder... or Add with Subfolders..., depending on whether or not you want subdirectories to be included as well. To remove a directory, click on Remove. The buttons Move Up and Move Down can be used to reorder the directories in the path. Note that you can use the Current Directory browser to examine the files in the working directory, and even to create subdirectories, move M-files around, etc. The information displayed in the main areas of the Path Browser can also be obtained from the command line. To see the current working directory, type pwd. To list the files in the working directory type either ls or dir. To see the current path list that MATLAB will search for files, type path. If you have many toolboxes installed, path searches can be slow, especially with lookfor. Removing the toolboxes you are not currently using from the MATLAB path is one way to speed up execution.Using the Command Window We have already described in Chapters 1 and 2 how to enter commands in the MATLAB Command Window. We continue that description here, presenting an example that will serve as an introduction to our discussion of M-files. Suppose you want to calculate the values of sin(0.1)/0.1, sin(0.01)/0.01, and sin(0.001)/0.001 to 15 digits. Such a simple problem can be worked directly in the Command Window. Here is a typical first try at a solution, together with the response that MATLAB displays in the Command Window: x = [0.1, 0.01, 0.001]; y = sin(x)./x
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36 Chapter 3: Interacting with MATLAB y = 0.9983 1.0000 1.0000 After completing a calculation, you will often realize that the result is not what you intended. The commands above displayed only 5 digits, not 15. To display 15 digits, you need to type the command format long and then repeat the line that defines y. In this case you could simply retype the latter line, but in general retyping is time consuming and error prone, especially for complicated problems. How can you modify a sequence of commands without retyping them? For simple problems, you can take advantage of the command history fea- ture of MATLAB. Use the UP- and DOWN-ARROW keys to scroll through the list of commands that you have used recently. When you locate the correct com- mand line, you can use the LEFT- and RIGHT-ARROW keys to move around in the command line, deleting and inserting changes as necessary, and then press the ENTER key to tell MATLAB to evaluate the modified command. You can also copy and paste previous command lines from the Command Window, or in the MATLAB 6 Desktop from the Command History window as described earlier in this chapter. For more complicated problems, however, it is better to use M-files.M-Files For complicated problems, the simple editing tools provided by the Command Window and its history mechanism are insufficient. A much better approach is to create an M-file. There are two different kinds of M-files: script M-files and function M-files. We shall illustrate the use of both types of M-files as we present different solutions to the problem described above. M-files are ordinary text files containing MATLAB commands. You can cre- ate and modify them using any text editor or word processor that is capable of saving files as plain ASCII text. (Such text editors include notepad in Win- dows or emacs, textedit, and vi in UNIX.) More conveniently, you can use the built-in Editor/Debugger, which you can start by typing edit, either by itself (to edit a new file) or followed by the name of an existing M-file in the current working directory. You can also use the File menu or the two leftmost buttons on the tool bar to start the Editor/Debugger, either to create a new file or to open an existing file. Double-clicking on an M-file in the Current Directory browser will also open it in the Editor/Debugger.
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M-Files 37Script M-Files We now show how to construct a script M-file to solve the mathematical prob- lem described earlier. Create a file containing the following lines: format long x = [0.1, 0.01, 0.001]; y = sin(x)./x We will assume that you have saved this file with the name task1.m in your working directory, or in some directory on your path. You can name the file any way you like (subject to the usual naming restrictions on your operating system), but the ".m" suffix is mandatory. You can tell MATLAB to run (or execute) this script by typing task1 in the Command Window. (You must not type the ".m" extension here; MATLAB automatically adds it when searching for M-files.) The output — but not the commands that produce them — will be displayed in the Command Window. Now the sequence of commands can easily be changed by modifying the M-file task1.m. For example, if you also wish to calculate sin(0.0001)/0.0001, you can modify the M-file to read format long x = [0.1, 0.01, 0.001, 0.0001]; y = sin(x)./x and then run the modified script by typing task1. Be sure to save your changes to task1.m first; otherwise, MATLAB will not recognize them. Any variables that are set by the running of a script M-file will persist exactly as if you had typed them into the Command Window directly. For example, the program above will cause all future numerical output to be displayed with 15 digits. To revert to 5-digit format, you would have to type format short. Echoing Commands. As mentioned above, the commands in a script M-file will not automatically be displayed in the Command Window. If you want the commands to be displayed along with the results, use echo: echo on format long x = [0.1, 0.01, 0.001]; y = sin(x)./x echo off
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38 Chapter 3: Interacting with MATLAB Adding Comments. It is worthwhile to include comments in a lengthly script M-file. These comments might explain what is being done in the calculation, or they might interpret the results of the calculation. Any line in a script M-file that begins with a percent sign is treated as a comment and is not executed by MATLAB. Here is our new version of task1.m with a few comments added: echo on % Turn on 15 digit display format long x = [0.1, 0.01, 0.001]; y = sin(x)./x % These values illustrate the fact that the limit of % sin(x)/x as x approaches 0 is 1. echo off When adding comments to a script M-file, remember to put a percent sign at the beginning of each line. This is particularly important if your editor starts a new line automatically while you are typing a comment. If you use echo on in a script M-file, then MATLAB will also echo the comments, so they will appear in the Command Window. Structuring Script M-Files. For the results of a script M-file to be reproducible, the script should be self-contained, unaffected by other variables that you might have defined elsewhere in the MATLAB session, and uncorrupted by leftover graphics. With this in mind, you can type the line clear all at the beginning of the script, to ensure that previous definitions of variables do not affect the results. You can also include the close all command at the beginning of a script M-file that creates graphics, to close all graphics windows and start with a clean slate. Here is our example of a complete, careful, commented solution to the problem described above: % Remove old variable definitions clear all % Remove old graphics windows close all % Display the command lines in the command window echo on % Turn on 15 digit display format long | 677.169 | 1 |
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Mini Book for Pre-Algebra! Students will explore slopes of lines using the slope formula and rise over run "slope triangles." Writing in math class helps students to retain and understand new concepts. This is an engaging way for students to retain concepts about slopes: positive, negative, zero, and undefined. This Christmas mini book is a great addition to students' interactive notebooks!
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SimilarThe PWN the SAT Math Guide was created to help ambitious, highly motivated kids maximize their SAT math scores. Do you crave a higher score? Are you willing to do a little hard work to achieve it? Good. I knew I liked you.
Read this book from beginning to end, with a pencil in hand and a calculator and an Official SAT Study Guide by your side. When you're done, you'll be able to approach the SAT with confidence—very few questions will surprise you, and even fewer will be able to withstand your withering attacks.
Stand tall, intrepid student. Destiny awaits.
Updated for the New SAT
This new edition of the Math Guide has been updated, rather painstakingly, to reflect the realities of the new SAT coming March 2016. This book was not rushed to market to take advantage of interest in the new exam. I took my time, and hopefully I got it right.
Chapters are broken into five major sections: Techniques, Heart of Algebra, Passport to Advanced Math, Problem Solving and Data Analysis, and Additional Topics in Math. Each chapter concludes with a reference list of similar questions from official practice tests.
Practice questions are designated as either "Calculator" or "No calculator." Students will be forbidden from using their calculators for one whole section of the new SAT.
Emphasis is placed on nimbleness—the ability to approach problems in multiple ways to find the one that works best. Calculator solutions and shortcuts are provided where appropriate.
Join me online
Readers of this book are encouraged to register as Math Guide Owners at the PWN the SAT website. There will be video solutions and other bonus content there. Signing up there will also give me a way to get in touch with you if I make book updates. See details at This comprehensive quick and easy-to-use supplement complements any fifth grade math curriculum. The reproducible activities review twenty-four essential math skills and concepts in only ten minutes each day during a four-day period. On the fifth day, a 20-minute ten problem assessment is provided. The exercises in this book cover a 40-week period and are designed on a continuous spiral so concepts are repeated weekly. Concepts include place value, geometry, fractions, decimals, patterns, time and money, measurement, graphs and tables, estimation, problem solving, word problems, probability, and more. It also includes test-taking tips, skills and concepts charts, scope and sequence charts, and an answer key.
Helpful advice for teaching Common Core Math Standards to middle-school students
The new Common Core State Standards for Mathematics have been formulated to provide students with instruction that will help them acquire a thorough knowledge of math at their grade level, which will in turn enable them to move on to higher mathematics with competence and confidence. Hands-on Activities for Teaching the Common Core Math Standards is designed to help teachers instruct their students so that they will better understand and apply the skills outlined in the Standards.
This important resource also gives teachers a wealth of tools and activities that can encourage students to think critically, use mathematical reasoning, and employ various problem-solving strategies.
Filled with activities that will help students gain an understanding of math concepts and skills correlated to the Common Core State Math Standards Offers guidance for helping students apply their understanding of math concepts and skills, develop proficiency in calculations, and learn to think abstractly Describes ways to get students to collaborate with other students, utilize technology, communicate ideas about math both orally and in writing, and gain an appreciation of the significance of mathematics to real life
This practical and easy-to-use resource will help teachers give students the foundation they need for success in higher mathematics ReveSingapore Math creates a deep understanding of each key math concept, includes an introduction explaining the Singapore Math method, is a direct complement to the current textbooks used in Singapore, and includes step-by-step solutions in the answer key. Singapore Math, for students in grades 2 to 5, provides math practice while developing analytical and problem-solving skills. This series is correlated to Singapore Math textbooks and creates a deep understanding of each key math concept. Learning objectives are provided to identify what students should know after completing each unit, and assessments are included to ensure that learners obtain a thorough understanding of mathematical concepts. Perfect as a supplement to classroom work, these workbooks will boost confidence in problem-solving and critical-thinking skillsEvery day, your child encounters math in many different situations. The activities in Creative Kids Math make learning math fun while also challenging your child to use math skills in different subject areas. The activities include stories, games, science experiments, and crafts! assessment through fifth grade to help ensure that children master fractions, decimals, and percentages this clever guide, young readers previously daunted by math will discover they're better at it than they thought.
With clear and accessible examples, How to Be a Math Genius explores the math brain and demonstrates to readers that they use math skills all the time-they just don't know it yet. Explaining fascinating ideas in a simple and fun way, the book is also packed with activities and puzzles, compelling stories of math geniuses, irresistible facts and stats, and more, making the dreaded subject of math both engaging and relevant.
For decades teachers and parents have accepted the judgment that some students just aren't good at math. John Mighton-the founder of a revolutionary math program designed to help failing math students-feels that not only is this wrong, but that it has become a self-fulfilling prophecy.
A pioneering educator, Mighton realized several years ago that children were failing math because they had come to believe they were not good at it. Once students lost confidence in their math skills and fell behind, it was very difficult for them to catch up, particularly in the classroom. He knew this from experience, because he had once failed math himself.
Using the premise that anyone can learn math and anyone can teach it, Mighton's unique teaching method isolates and describes concepts so clearly that students of all skill levels can understand them. Rather than fearing failure, students learn from and build on their own successes and gain the confidence and self-esteem they need to be inspired to learn. Mighton's methods, set forth in The Myth of Ability and implemented in hundreds of Canadian schools, have had astonishing results: Not only have they helped children overcome their fear of math, but the resulting confidence has led to improved reading and motor skills as well.
The Myth of Ability will transform the way teachers and parents look at the teaching of mathematics and, by extension, the entire process of education theMath Activities Homework Helper provides children in second grade with extra help in learning important math skills. Packed full of fun-to-do activities and appealing art, children will have fun completing the reproducible pages while learning math one- and two-digit addition and subtraction, problem solvingThe 100+ Series, Algebra, offers in-depth practice and review for challenging middle school math topics such as radicals and exponents; factoring; and solving and graphing equationsThe 100+ Series, Algebra II, offers in-depth practice and review for challenging middle school math topics such as factoring and polynomials; quadratic equations; and trigonometric functionsAn uncommon guide for accomplishing more every day by engaging the unique skill of forgetting, from the creator of the award-winning memory training system Brainetics
Is it possible that the answer to becoming a more efficient and effective thinker is learning how to forget? Yes! Mike Byster will show you how mastering this extraordinary technique—forgetting unnecessary information, sifting through brain clutter, and focusing on only important nuggets of data—will change the quality of your work and life balance forever.
Using Byster's exclusive quizzes and games, you'll develop the critical skills to become more successful in all that you do, each and every dayIntegrate TI Graphing Calculator technology into your algebra instruction with this award-winning resource book. Perfect for grades 6-12, this resource includes lessons, problem-solving practice, and step-by-step instructions for using graphing calculator technology. 238pp plus Teacher Resource CD with PDF files of the tables, templates, activity sheets, and student guides for TI-83/84 Plus Family and TI-73 Explorer(tm). This resource is correlated to the Common Core State Standards, is aligned to the interdisciplinary themes from the Partnership for 21st Century Skills, and supports core concepts of STEM instructionSpectrum Test Prep Grade 7 includes strategy-based activities for language arts and math, test tips to help answer questions, and critical thinking and reasoning. The Spectrum Test Prep series for grades 1 to 8 was developed by experts in education and was created to help students improve and strengthen their test-taking skills. The activities in each book not only feature essential practice in reading, math, and language arts test areas, but also prepare students to take standardized tests. Students learn how to follow directions, understand different test formats, use effective strategies to avoid common mistakes, and budget their time wisely. Step-by-step solutions in the answer key are included. These comprehensive workbooks are an excellent resource for developing skills for assessment success. Spectrum, the best-selling workbook series, is proud to provide quality educational materials that support your students' learning achievement and success.
Spectrum(R) Grade Specific for Grade 6 includes focused practice for reading, language arts, and math mastery. Skills include grammar and usage, parts of speech and sentence types, vocabulary acquisition and usage, multiplying and dividing fractions and decimals, equations and inequalities, problem solving in the coordinate plane, probability and statistics, and ratios, rates, and percents. Spectrum Grade Specific workbooks contain focused practice for language arts mastery. Each book also includes a writer's guide. Step-by-step instructions help children with planning, drafting, revising, proofreading, and sharing writing. The math activities build the skills that children need for math achievement and success. Children in grades 1 to 6 will find lessons and exercises that help them progress through increasingly difficult subject matter. Aligned to current state standards, Spectrum is your child's path to language arts and math mastery.
Mastering Basic Skills(R) Third Grade includes comprehensive content essential to third graders. Topics include reading comprehension, phonics, grammar, writing, dictionary skills, math, time and moneyMake math matter to students in grades 5 and up using Math Logic! This 80-page book includes logic problems at three skill levels. Each nonroutine problem includes the situation, variables involved, and clues that help students work through the problem. The logic problems meet NCTM standards for reasoning, proof, and problem solvingReinforce your second grader's essential skills with the Complete Book of Grade 2. With the colorful lessons in this workbook, your child will strengthen skills that include prefixes and suffixes, word relationships, vowel sounds, and multiplicationUse children's literature as a springboard to successful mathematical literacy. This book contains summaries of books, each related to the NCTM Standards, that will help children gain familiarity with and an understanding of mathematical concepts. Each chapter has classroom-tested activities and a bibliography of additional books to further expand student learning.
In Interactive Notebooks: Math for first grade, students will complete hands-on activities about place value, addition and subtraction, word problems, time, nonstandard measurement, shape attributesNo author has gone as far as Doerfler in covering methods of mental calculation beyond simple arithmetic. Even if you have no interest in competing with computers you'll learn a great deal about number theory and the art of efficient computer programming. —Martin Gardner
A fun, easy-to-implement collection of activities that give elementary and middle-school students a real understanding of key math concepts
Math is a difficult and abstract subject for many students, yet teachers need to make sure their students comprehend basic math concepts. This engaging activity book is a resource teachers can use to give students concrete understanding of the math behind the questions on most standardized tests, and includes information that will give students a firm grounding to work with more advanced math concepts.
Contains over 100 activities that address topics like number sense, geometry, computation, problem solving, and logical thinking. Includes projects and activities that are correlated to National Math Education Standards Activities are presented in order of difficulty and address different learning styles
Math Wise! is a key resource for teachers who want to teach their students the fundamentals that drive math problems.
New Syllabus Mathematics (NSM) is a series of textbooks specially designed to provide valuable learning experiences to engage the hearts and minds of students sitting for the GCE O-level examination in Mathematics. Included in the textbooks are Investigation, Class Discussion, Thinking Time, Journal Writing, Performance Task and Problems in Real-World Contexts to support the teaching and learning of Mathematics.
Every chapter begins with a chapter opener which motivates students in learning the topic. Interesting stories about Mathematicians, real-life examples and applications are used to arouse students' interest and curiosity so that they can appreciate the beauty of Mathematics in their surroundings.
The use of ICT helps students to visualise and manipulate mathematical objects more easily, thus making the learning of Mathematics more interactive. Ready-to-use interactive ICT templates are available at Brimming with fun and educational games and activities, the Magical Math series provides everything you need to know to become a master of mathematics! In each of these books, Lynette Long uses her won unique style to help you truly understand mathematical concepts with common objects such as playing cards, dice, coins, and every mathematician's basic tools: paper and pencil.
Inside Wacky Word Problems, you'll discover how to decode many different types of word problems-from counting, logic, and percentage problems to distance, algebra, geometry, and graphing problems-in order to solve real-world dilemmas. While you play exciting games like Measurement Jeopardy and Percentage War, you'll learn how to identify word cues, develop reasoning skill,s and spot key formulas that will help you solve any problem with ease. You'll also boost your math skills as you enter into crazy contests with your friends, create mystery word problems, and play word-problem charades-and have a great time doing it!
So why wait? Jump right in and find out how easy it is to become a word-problem master!
Also available in this series: Dazzling Division, Delightful Decimals and Perfect Percents, Fabulous Fractions, Groovy Geometry, Marvelous Multiplication, and Measurement Mania, all from WileyYour Total Solution for Kindergarten will delight young children with activities that teach position words, letter recognition, vowel sounds, making predictions, numbers 0 to 20, sequencing, opposites, graphing, telling time, and more. Your Total Solution provides lots of fun-to-do practice in math, reading, and language skills for children in prekindergarten to second grade. Colorful pages teach numbers, counting, sorting, sequencing, shapes, patterns, measurement, letters and sounds, basic concepts, early writing skills, vocabulary, and more. Loaded with short, engaging activities, these handy workbooks are a parent's total solution for supporting learning at home during the important early yearsHello Hi-Lo: Readers Theatre Math offers a proven way for teachers to build reading fluency in their classrooms, even if students do not all read at the same level. The book offers 15 readers theatre plays, each of which includes various levels of readability within one script. The plays thus provide lower-level readers the opportunity to follow along with accelerated reading parts, building confidence by reading aloud at their own levels.
Each play includes speakers for three different grade levels of readability: 4th grade and lower, 5th-6th grade, and 7th-8th grade. Concepts are presented so that they are fluid among these grade levels. Taking a cross-curricular approach, the scripts reinforce key math concepts and standards-based math skills taught in the middle grades, such as order of operations, fractions, inequalities, positive and negative numbers, and graphing on a coordinate plane. Each play includes extended activities that will help the teacher incorporate the math concept into the classroom.
Mathematics is often thought of as the coldest expression of pure reason. But few subjects provoke hotter emotions--and inspire more love and hatred--than mathematics. And although math is frequently idealized as floating above the messiness of human life, its story is nothing if not human; often, it is all too human. Loving and Hating Mathematics is about the hidden human, emotional, and social forces that shape mathematics and affect the experiences of students and mathematicians. Written in a lively, accessible style, and filled with gripping stories and anecdotes, Loving and Hating Mathematics brings home the intense pleasures and pains of mathematical life.
These stories challenge many myths, including the notions that mathematics is a solitary pursuit and a "young man's game," the belief that mathematicians are emotionally different from other people, and even the idea that to be a great mathematician it helps to be a little bit crazy. Reuben Hersh and Vera John-Steiner tell stories of lives in math from their very beginnings through old age, including accounts of teaching and mentoring, friendships and rivalries, love affairs and marriages, and the experiences of women and minorities in a field that has traditionally been unfriendly to both. Included here are also stories of people for whom mathematics has been an immense solace during times of crisis, war, and even imprisonment--as well as of those rare individuals driven to insanity and even murder by an obsession with math.
This is a book for anyone who wants to understand why the most rational of human endeavors is at the same time one of the most emotional.
In the second book in the Uncomplicating Mathematics Series, professional developer Marian Small shows teachers how to uncomplicate the teaching of algebra by focusing on the most important ideas that students need to grasp. Organized by grade level around the Common Core State Standards for Mathematics, Small shares approaches that will lead to a deeper and richer understanding of algebra for both teachers and students. The book opens with a clear discussion of algebraic thinking and current requirements for algebraic understanding within standards-based learning environments. The book then launches with Kindergarten, where the first relevant standard is found in the operations and algebraic thinking domain, and ends with Grade 8, where the focus is on working with linear equations and functions. In each section the relevant standard is presented, followed by a discussion of important underlying ideas associated with that standard, as well as thoughtful, concept-based questions that can be used for classroom instruction, practice, or assessment. Underlying ideas include: Background to the mathematics of each relevant standard. Suggestions for appropriate representations for specific mathematical ideas. Suggestions for explaining ideas to students. Cautions about misconceptions or situations to avoid.
The Common Core State Standards for Mathematics challenges students to become mathematical thinkers, not just mathematical "doers." This resource will be invaluable for pre- and inservice teachers as they prepare themselves to understand and teach algebra with a deep level of understanding.
"Uncomplicating Algebra is an excellent resource for teachers responsible for the mathematical education of K–8 students. It is also a valuable tool for the training of preservice teachers of elementary and middle school mathematics." —Carole Greenes, associate vice provost for STEM education, director of the Practice Research and Innovation in Mathematics Education (PRIME) Center, professor of mathematics education, Arizona State University
"The current climate in North America places a major emphasis on standards, including the Common Core State Standards for Mathematics in the U.S. In many cases, teachers are being asked to teach content with which they themselves struggle. In this book, Dr. Small masterfully breaks down the big ideas of algebraic thinking to assist teachers, math coaches, and preservice teachers—helping them to deepen their own understanding of the mathematics they teach. She describes common error patterns and examines algebraic reasoning from a developmental viewpoint, connecting the dots from kindergarten through grade 8. The book is clearly written, loaded with specific examples, and very timely. I recommend it strongly as a 'must-read' for all who are seeking to broaden their understanding of algebra and how to effectively teach this important content area to children." —Daniel J. Brahier, director, Science and Math Education in ACTION, professor of mathematics education, School of Teaching and Learning, Bowling Green State University fifth grade to help ensure that children master necessary mathA barber in Chicago says he'd rather cut the hair of ten red-headed men than the hair of one brown-haired man. Can you guess why? Ask Professor Picanumba, a master of riddles who carries dozens of surefire tricks up his sleeve. He'll show you how to astonish your friends and family by predicting the answers to 88 word and number challenges. These tricks require only simple props—a deck of cards or a couple of pairs of dice, a calculator, and a pencil and paper. With or without an audience, these foolproof feats of mental magic offer hours of amusement. Solutions appear at the end, with 64 illustrations in between. Author Martin Gardner has written more than 70 books on subjects from science and math to poetry and religion. Well known for the mathematical games that appeared in Scientific American for decades and for his "Trick of the Month" column in Physics Teacher magazine, Gardner has had a lifelong passion for magic tricks and puzzles.
Is this poetry? Math? A brainteaser? Yes! It's all that and more. The poet J. Patrick Lewis has reimagined classic poems—such as Edgar Allan Poe's "The Raven" and Langston Hughes's "April Rain Song"—and added a dash of math. Between the silly parodies and the wonderfully wacky art, kids will have so much fun figuring out the puzzles, they won't guess they're learning! Answers appear unobtrusively on each page, and engaging information about the original poets is included. Math games and concepts, poetry and poet biographies—it's all so cleverly put together. This funny book is a treat for fans of words and numbers alikeImprove students' math skills in the classroom while also providing a way to continue the learning process at home. Weekly Practice: Math for grade 4 allows you to reinforce math topics at school and at home by offering 40 weeks of standards-based activities and skill review. The unique layout and engaging exercises keep students interested as they build concept knowledge and essential skills. Reproducible at-home activities and flash cards are also included to encourage the home-to-school connection that's essential for student success.
Weekly Practice is the perfect time-saving resource for creating standards-aligned homework packets and keeping students' skills sharp all year long. The Weekly Practice series for kindergarten to grade 5 provides 40 weeks of comprehensive skill review. Each 192-page supplemental workbook focuses on critical skills and concepts that meet the standards for language arts or math. Designed to help students achieve subject mastery, each book includes four days of practice activities, weekly off-the-page activities, Common Core State Standards alignment matrix, flash cards, and an answer key. Weekly Practice offers an effortless way to integrate language arts or math practice into daily classroom instruction.
Kelley Wingate's Math Practice for second place value, more complex addition and subtraction, standard measurement, analyzing shapes, and a comprehensive selection of other second | 677.169 | 1 |
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Algebra 2 Music & Lyrics DVD
MathOdes: Etching Math in Memory is an artistic fun and creative math teaching and study aid designed to help students remember math concepts and formulas.
In the form of poetry and music, the formulas and rules focus on Algebra 2. Each "ode" details a particular math concept such as logarithmic functions, matrices, and conics. This DVD provides the music and visual lyrics in time with the music for each song which can be projected in a classroom session or used on individual computing devices and TV's for a fun and effective personal study experience!
Solving Polynomials
Asymptotes
Quadratic Inequalities
Exponential Functions
Logarithmic Functions
Systems of Linear Equations
Matrix Operations
Determinants and Cramer's Rule
Partial Fractions
Conics – Parabola
Conics – Hyperbola
Conics – Ellipse & Circle
Function Transformations
Composition & Inverse Functions
Sequence, Series, & Notation
Hover over the thumbnails to view the image or select the thumbnail to see the full size product image. | 677.169 | 1 |
Evaluating Algebraic Expressions Smart Notebook
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This Smart Notebook can be used to teach students how to evaluate algebraic expressions by substituting a value for a variable.
Important vocabulary words are also defined in this notebook as well as practice problems.
You can only download this file if you have Smart Notebook Software. | 677.169 | 1 |
"Setbacks are short-term and changeable.
Setbacks and failure linger only if you
allow them to."
- Jay Bilas' book "Toughness"-
About Mr. Crowley:
I have been teaching math at Wamogo since August 2001. This year I will be teaching Algebra 2, both Honors and CP and one section of Geometry. I have been the Girls Basketball Team's Head Coach since 2000. I am married and have two sons.
About the classes:
Each class is based on a rigorous approach to Algebra 2 and Geometry with an emphasis on tools and strategies. Analytical and logical thinking and problem solving will be an emphasis throughout the year. Most students know how to do things in math, I want them to understand why they do them as well. Each student will be asked to communicate their knowledge in written and spoken forms. Group work will be utilized to encourage communication among students and to help build understanding. Common Core will be a guiding influence throughout each course.
Important dates for classes:
All Classes will have a FINAL EXAM!!
MP 2 on line Progress Reports: 12/14/17 MP 2Grades for Mr. Crowley close on: 1/26/18 You can check your progress anytime on Power School!
Please remember you can check your grade anytime using Power School.
Calculators:
During class we will use TI-83 or TI-83+ graphing calculators. I have a classroom set of TI-83 calculators for use during class. Assignments will not require students to use a graphing calculator outside the classroom...however having your own graphing calculator is recommended for use in and out of class. | 677.169 | 1 |
Mathematica is a formidable commercial
system for symbolic mathematics and
graphics. Most of the system is written
in the Mathematica language, a powerful
hybrid interpreted language for expressing
mathematical formulae and procedures.
The end user also employs the Mathematica
language to perform describe the problems
they wish Mathematica to solve to computations
they want it to undertake.
The language has a very broad complement
of features:
Data elements in Mathematica are
strongly typed, but the language system
performs many types of automatic
conversion, especially on numbers.
All numbers in Mathematica are
unlimited precision: integers, reals,
rationals, and complex. Programmers
can also define their own data types,
after a fashion.
[Note: in a sense, Mathematica has only
one non-primitive data type: the "basic form." All
aggregate data and symbolic expressions in Mathematica are internally stored
as a head and a body, where the head defines the
data type. Programmers can create their
own head types, and do, and in that sense
they are defining new data types.]
For many users, the power of the Mathematica
system is its immense library of pre-defined
functions. These functions fall into
several broad groups, some of which are: arithmetic,
algebra, calculus, graphics, I/O,
programming, and the GUI interface.
As an interpreter, Mathematica can
easily import new functions and
package dynamically. Several hundred
such packages have been written for
the system, in academia and industry.
There is only one Mathematica, now in
its third major release. The system
is available commercially for Windows,
Macintosh, and most Unix platforms.
Programming with Mathematica has all the
ease and pleasure of using
the best interpreters. Availability
of procedural, functional, and rule-based
programming models allow the user to
experiment and choose the most suitable
style for their background and problem
domain. Of course, choice of style can
have very serious implications for
performance -- the mathematical core is
far faster at performing some kinds of
operations than others.
Like Hope, Mathematica allows the
programmer to set up very complex
pattern matching rules to control how
functions are applied during computation.
This capabilities is very powerful;
users have coded up significant areas of
mathematical research in the language.
The rivalry between Mathematica and its
leading competitor Maple sometimes gets
bitter. Both have their strengths, but
in the area of the programming language,
Mathematica is somewhat more powerful. | 677.169 | 1 |
Certificate of Higher Education in Mathematical Sciences
Mathematics and statistics are fascinating subjects and highly relevant to contemporary life. Their tools and techniques are essential for solving problems and making decisions – and for many exciting careers from IT to engineering. This certificate is ideal if you need some underpinning mathematics for other studies or at work; as a first step towards higher level qualification; or if you're simply curious and want to take your interest a bit further.
Key features of the course
Introduces pure mathematics, applied mathematics and statistics
Builds your confidence working with abstract ideas; modelling real world problems and using mathematical and statistical software
Offers a choice of two start points to suit your level of mathematical knowledge
Mathematics is a linear subject – it's important to have a good understanding of the basics before moving on to more advanced topics. You'll begin with an introduction to key mathematical ideas, ideal if you're not confident with algebra and trigonometry; if you haven't previously studied mathematics to an advanced level; or if you haven't studied mathematics for some time and need to refresh your skills.
If you're confident about studying mathematics at university level and, in particular, have a good understanding of algebra and trigonometry, you can start your studies at a higher level and at a faster pace. The intensive start skips the first module and instead allows you to choose a module from a selection of other modules to complete Stage 1. Am I ready? will help you decide which start best suits your level of mathematics skills and knowledge.
To allow mobility between qualifications, Stage 1 is common to the BSc (Honours) Mathematics, BSc (Honours) Mathematics and Statistics, BSc (Honours) Mathematics and its Learning, Diploma of Higher Education in Mathematical Sciences and Certificate of Higher Education in Mathematical Sciences.
The modules quoted in this description are currently available for study. However, as we review the curriculum on a regular basis, the exact selection may change over time.
Accessibility
We make all our qualifications as accessible as possible and have a comprehensive range of services to support all our students. The Certificate of Higher Education in Mathematical Sciences uses a variety of study materials and has the following elements:
using mathematical and scientific expressions, notations and associated techniques
using and producing diagrams and/or screenshots
studying a mixture of printed and online material – online learning resources may include websites, audio/video media clips, and interactive activities such as online quizzes
in some modules undertaking small amounts of practical work
using specialist mathematical or statistical software.
If you feel you may need additional support with any of the elements above, visit our disability page to find more about what we offer. Please contact us as soon as possible to discuss your individual requirements, so we can put arrangements in place before you start.
Learning outcomes, teaching and assessment
This qualification develops your learning in four main areas:
Knowledge and understanding
Cognitive skills
Practical and professional skills
Key skills
The level and depth of your learning gradually increases as you work through the qualification. You'll be supported throughout by the OU's unique style of teaching and assessment – which includes a personal tutor to guide and comment on your work; top quality course texts; e-learning resources like podcasts, interactive media and online materials; tutorial groups and community forums.
Credit transfer
If you have already studied at university level, you may be able to count it towards your Open University qualification – which could save you time and money by reducing the number of modules you need to study. At the OU we call this credit transfer.
It's not just university study that can be considered, you can also transfer study from a wide range of professional or vocational qualifications such as HNCs and HNDs.
You should apply for credit transfer before you register, at least 4 weeks before the registration closing date. We will need to know what you studied, where and when and you will need to provide evidence of your previous study.
For more details of when you will need to apply by and to download an application form, visit our Credit Transfer website.
On completion
On completion of this undergraduate course, we'll award you the Certificate of Higher Education in Mathematical Sciences.
Recognition in your country
If you intend to use your Open University qualifications to seek work or undertake further study outside the UK, we recommend checking whether your intended qualification will meet local requirements for your chosen career. Read recognition in my country.
Regulations
As a student of The Open University, you should be aware of the content of the qualification-specific regulations below and the academic regulations that are available on our Essential Documents website.
If you can answer most of the questions, or could do them with a quick reminder (because the topics are familiar to you but you can't quite remember the details), then you are ready.
If you achieved a very high score, our intensive start may be an option for you. First complete our intensive start Are you ready? quiz to check if you have the necessary experience and confidence with mathematics.
If you would like more advice and guidance about where to start, please visit our MathsChoices website.
How much time do I need?
Most of our students study part time, completing 60 credits a year.
This will usually mean studying for 16–18 hours a week.
Counting previous study
You could save time and money by reducing the number of modules you need to study towards this qualification if you have:
already studied at university level (even if you didn't finish your studies)
other professional or vocational qualifications such as HNCs and HNDs.
How much will it cost in England?
We believe cost shouldn't be a barrier to achieving your potential. That's why we work hard to keep the cost of study as low as possible and have a wide range of flexible ways to pay to help spread, or even reduce, the cost.
Fees are paid on a module-by-module basis – you won't have to pay for the whole of your qualification up front.
A qualification comprises a series of modules, each with an individual fee. Added together, they give you the total cost.
If, like most OU students, you study part time at a rate of 60 credits a year, you'll take six years to complete an honours degree.
Our current fee for 60 credits is £2,864*.
Our current fee for 120 credits – which is equivalent to a year's full-time study – is £5,728*.
At current prices, the total cost of your qualification would be £5,728*.
.
*The fee information provided here is valid for modules starting before 31 July 2018. Fees normally increase annually in line with inflation and the University's strategic approach to fees.
Ways to pay for your qualification and other support
We know there's a lot to think about when choosing to study, not least how you can pay. That's why we offer a wide range of flexible payment and funding options to help make study more affordable. Options include Part-Time Tuition Fee Loans (also known as student loans), monthly payment plans and employer sponsorship.
We're confident we can help you find an option that's right for you.
Just answer these simple questions to find out more about the options available to you for courses starting before 31 July 2018.
To find out what funding options are available you need to tell us:
how many credits you want to study
if you already hold a degree
if your household is in receipt of benefits
about your household income
if you are employed
if you are a member of the British forces overseas
How many credits are you planning to study per year?
Credits
You will need [xxx] credits to complete this qualification.
30 credits per year means you will study 8 - 9 hours per week
60 credits per year means you will study 16 - 18 hours per week
Do you already hold a degree?
Yes, I already hold a degree
No, I do not hold a degree
Was your previous degree in the same subject you wish to study now?
Yes, it is in the same subject
No, it is in a different subject
Was it achieved in the last 5 years?
Yes, it was
No, it wasn't
Are you employed?
Yes, I'm employed
No, I'm not employed
Are you a member of British Forces Posted Overseas?
British Forces
If you have a BFPO address, you are only eligible for UK course fees if you are a currently serving member of the British armed forces, and you're temporarily and unavoidably working abroad. Other students using BFPO addresses should contact us on +44 (0)300 303 5303 for UK fee eligibility to be assessed.
Yes, I am a member of British Forces Posted Overseas
No, I am not a member of British Forces Posted Overseas
* The fee and funding information provided here is valid for courses starting before 31 July 2018. Fees normally increase in line with inflation and the University's strategic approach to fees.
Skills for career development
This certificate course provides skills and knowledge required for jobs in a wide range of fields. It will particularly enhance the following transferable and highly valued skills:
communicating mathematical ideas clearly and succinctly
explaining mathematical ideas to others
understanding complex mathematical texts
working with abstract concepts
thinking logically
expressing problems in mathematical language
constructing logical arguments
finding solutions to problems
interpreting mathematical results in real-world terms
using relevant professional software.
Career relevance
Career areas directly related to mathematics include:
banking
bioinformatics
economics
insurance
investments
market research
pensions
quantitative analysis/risk analysis
retail
stockbroking/trading
tax.
Other careers
Mathematical knowledge is much sought after by a wide variety of employers in fields such as education, engineering, business, finance, and accountancy. A qualification in mathematics also offers opportunities for self-employment – as a financial adviser, for example.
In addition to improving your career prospects, studying with the OU is an enriching experience that broadens your horizons, develops your knowledge, builds your confidence and enhances your life skills.
Exploring your options
Once you register with us (and for up to three years after you finish your studies), you'll have full access to our careers service for a wide range of information and advice – including online forums, website, interview simulation, vacancy service as well as the option to email or speak to a careers adviser. Some areas of the careers service website are available for you to see now, including help with looking for and applying for jobs. You can also read more general information about how OU study enhances your career.
In the meantime if you want to do some research around this qualification and where it might take you, we've put together a list of relevant job titles as a starting point (note that some careers may require further study, training and/or work experience):
actuary
aeronautical engineer
auditor
chartered accountant
data scientist
financial risk analysist
investment analyst
lecturer
management consultant
meteorologist
operational researcher
pensions administrator
secondary school teacher
statistician
systems developer.
Want to see more jobs? Use the career explorer for job ideas from the National Careers Service, PlanIT Plus in Scotland and Prospects across all nations. You can also visit GradIreland for the Republic of Ireland. | 677.169 | 1 |
Dr. Sarah's Survey Assignment
Work on the answers to the following questions
in a word processor and save the file often
as yourfirstnamelab1.doc (ie something like
drsarahlab1.doc) from Word.
Note: grammar and spelling are important.
You are probably taking this course
as part of a general education requirement. What skills central
to your intellectual development do you think society expects you to take
away from a college level mathematics course such as Math 1010?
What do you think mathematics is (ie use your own ideas
to define mathematics)?
Name a mathematical experience you had that was
rewarding for you.
Think back to the first mathematical experience you had that
was destructive to your mathematical self-confidence. (We have
all had such experiences - even Dr. Sarah!)
Describe this experience.
In the last question, you explored a negative
mathematical experience.
If you could rewrite your role in that experience in order to prevent it
from being destructive to your mathematical self-confidence
what would you change about your behavior and reactions?
When you are finished
follow directions on the computer information sheet
to enter WebCT and compose a WebCT
message to Dr. Sarah with your file attached to the forum containing you
and her.
Note that Dr. Sarah can read Microsoft Word attachments
Save the file as
yourfirstnamelab1.doc (ie something like drsarahlab1.doc) from Word),
but if you are working in a different
word processor such as WordPerfect or Works,
then you must first save the file in a different format.
For these word processors, use Save as... and choose
rich text format and then save the file as yourfirstnamelab1.rtf
(ie something like drsarahlab1.rtf)
before attaching.
Begin working on Are
The Simpsons 2-D or 3-D? by going through the given links.
We will watch the segment together as a class, but you should go through
the links on your own. You will answer questions 1 and 2 for homework.
Homework for Tuesday and Thursday
Tues
Get a scientific calculator (with a
yx, xy, or ^ key on it),
a 3-ring binder notebook, and a hole puncher.
Get the How Do You Know? book from the Mat 1010 section of the
bookstore (open 8am-6pm this week).
Get the Heart of Math book by showing your syllabus upstairs in the
IDS section of the bookstore.
Work on lab 1 due Thursday.
Bring your calculator, 3-ring binder notebook, hole puncher and
all handouts I give you to classes and labs until otherwise notified.
Thur
**Lab 1**
due at the start of class (no lates allowed)
At the beginning of class on Thursday, you will turn in the
completed
perspective drawing worksheet
sheet and your response to Are
The Simpsons 2-D or 3-D?.
In addition, your survey
is due
as a Microsoft Word attachment posting on WebCT.
I'm happy to help or answer questions in office hours. | 677.169 | 1 |
Understanding Calculus is a complete online introductory book that focuses on concepts. Integrated throughout the e-book are many engineering applications aimed at developing the student's scientific approach towards problem solving.
Understanding Calculus is a complete online introductory book that focuses on concepts. Integrated throughout the e-book are many engineering applications aimed at developing the student's scientific approach towards problem solving.
Faraz Hussain wrote:The purpose of this book is to present mathematics as the science of deductive reasoning and not as the art of manipulation. Unfortunately, many students feel mathematics is incomprehensible and is riddled with complex and abstract jargon. My goal is to impose a lasting understanding of and appreciation for calculus on the student.
Unfortunately, students are rarely given any example of practical applications. The curriculum's idea of exercises is nothing more than sheer number-crunching and manipulation of variables. The entire underlying principle of order and beauty upon which calculus is based, is neglected. The problems never call upon the student's ability to think logically. Rather, they require no more than time and persistence, to allow the trial and error method to succeed.
Faraz Hussain wrote:It is all these shortcomings I set out to correct in writing my book. My book is intended to give the student a real for and understanding of what Calculus is truly about. Only by explaining where something has come from will I be able to show where it is going. It does not take more intelligence than that of a parrot to be able to go through a list of theorems and equations; but only when one understands their origins can one correctly and confidently apply them in the real world.
I assume absolutely nothing and neither do I take anything for granted. Each chapter has a definite beginning, followed by logical, elegant and clear proofs, concluded with a brief summary that ties everything together. My only reference has been reason and if you could find just three lines that remotely resemble the lines from any other book, I would feel greatly distressed. | 677.169 | 1 |
A First Course in Modular Forms
1st ed. 2005, Corr. 4th printing 2016
Description - A First Course in Modular Forms by Fred Diamond
This book introduces the theory of modular forms with an eye toward the Modularity Theorem:All rational elliptic curves arise from modular forms.The topics covered include:- elliptic curves as complex tori and as algebraic curves- modular curves as Riemann surfaces and as algebraic curves- Hecke operators and Atkin-Lehner theory- Hecke eigenforms and their arithmetic properties- the Jacobians of modular curves and the Abelian varieties associated to Hecke eigenforms- elliptic and modular curves modulo p and the Eichler-Shimura Relation- the Galois representations associated to elliptic curves and to Hecke eigenformsAs it presents these ideas, the book states the Modularity Theorem in various forms, relating them to each other and touching on their applications to number theory.A First Course in Modular Forms is written for beginning graduate students and advanced undergraduates. It does not require background in algebraic number theory or algebraic geometry, and it contains exercises throughout. | 677.169 | 1 |
Courses undergraduate courses apma 0090 introduction to modeling topics of applied mathematics introduced in the context of practical applications where . List of the new elected members to the european academy of sciences. Retrouvez toutes les discotheque marseille et se retrouver dans les plus grandes soirees en discotheque a marseille | 677.169 | 1 |
lessons that introduce the concept of shading regions on a 2D plane to help solve linear programming examples in the Decision 1 module of A level maths. The first lesson is simple shading out of regions and then the students are introduced to the concept of a linear programming problem, maximising profit. All powerpoints are fully animated to show shading and the solutions sets, all resources are printed so that students can draw straight on the graphs.
A series of lesson sheets that take you through the whole of the Edexcel AS Further Decision Maths 1 course. They are intended to be used in the order they are numbered. When used in conjunction with exercises on the topics these deliver the entire content of the course.
These resources are designed to aid the teaching and learning of using a graphical method to solve linear programming problems.
The first resource introduces the idea of representing inequalities on graphs and finding the point(s) that maximise a given objective function. There are also some examples that require integer solutions so the optimal point is not at a vertex of the feasible region.
The second resource provides practice of solving problems with a provided graph - these are examination style questions and involve considering how changes to the objective function may change the optimal point(s).
The third resource has 2 example questions in context where the students must use a description of a problem to formulate the objective function and the non-trivial constraints, and then go on to solve the problem graphically.
Grids are provided for all graphs and solutions are included for all questions.
A couple of lessons clearly explaining the difficult Linear Programming questions in the iGCSE. Animation shows step by step method. Use the worksheets alongside the PowerPoint. This resource assumes that students already know how to use inequalities to shade regions.
SMART notebook file guiding through Decision 1 Chapter 6 Linear Programming.
Based on Edexcel Pearson.
Thank you to another tes author who designed the starter inequalities questions. (Can't remember username, but will credit you if you let me know who you are!)
PowerPoints written so that a teacher can learn from the PowerPoints and teach from them. Students can use the PowerPoints to learn independently too. Many colleagues have used my lessons on Decision 1 without a strong understanding of Decision themselves.
Includes all chapters: Algorithms, Networks, Route Inspection, Critical paths, Linear Programming and Matchings
Includes all worksheets, answers and assessments the Decision Mathematics module for A level | 677.169 | 1 |
Immersive Linear Algebra by J. Strom, K. Astrom, T. Akenine-Moller - immersivemath This is a linear algebra book built around interactive illustrations. Each chapter starts with an intuitive concrete example that practically shows how the math works using interactive illustrations. After that, the more formal math is introduced. (310 views)
A First Course in Linear Algebra by Ken Kuttler - Lyryx The book presents an introduction to the fascinating subject of linear algebra. It is designed as a course in linear algebra for students who have a reasonable understanding of basic algebra. Major topics of linear algebra are presented in detail. (1019 views) | 677.169 | 1 |
Winning At Math Paul Nolting Ph.D. Chapter 3
Transcription
1 Winning At Math Paul Nolting Ph.D Chapter 3
2 Definition Of Math Anxiety Mathophobia and mathemaphobia are both extreme emotional and/ or physical reaction for low confidence and high anxiety. Math anxiety means a state of panic. One third of students had been counseled for student math anxiety at major universities. Students that had a poor high school background and low placement scored could be considered to have math anxiety.
3 Types Of Math Anxiety Math Test anxiety: Involves anticipation, completion and feedback of math tests. Numerical Anxiety: Includes students who are trying to figure out everyday situations for example figuring out the amount for a tip, thinking about mathematics, studying homework ect. Abstraction Anxiety: Is variables and mathematical concepts used to solve equations. Also to not usually have problems with numbers but once learning algebra, they develop Test and Abstraction Anxiety.
4 The Causes of Math Anxiety Anxiety can begin as early as elementary school where you where last in class or got the answer wrong. Or even coming last in a math races are examples of a negative math experience. Humiliation, is a personal view of put downs and are another way setting your self up and lead to a math anxiety and could use a positive math experience to help with this anxiety. Being embarrassed by family members can also cause math anxiety and sometimes led to serious trauma. A good example of anxiety was a student that completed her BS degree and returned to college that require to take math and a placement test and said, I can t do math and I will have to wait a few days to get psychologically ready to take the math test. She indicated her old feelings of not being able to do math rushed her and almost had an anxiety attack. Writing out the incidents could help start to better your anxiety and to write down the positive experiences, when you received a good grade remembering might influence you to do better and think positive.
5 How Math Anxiety Affects Learning Does home work affect your anxiety? Does it cause you to difficulty start or complete homework? Then does it remind them of their pervious math failure? Approach avoidance is procrastination to pull off tackling their homework, then to be prepared to see test day happen after procrastinations leads these effects of your study learning skills. Students are then usually afraid to speak out and ask questions. Key Points: 1.) Make an appointment to talk to math teacher. 2.) Before class, ask the instructor to work on a homework problem. So it is not hard to answer questions. 3.) Prepare one question from homework problem and ask with in the first 15 minutes of class. 4.) Ask a question you know already. 5.) Use to send questions The instructors job is to be asked questions. Don t avoid study groups it helps gain positive attitude It is like asking a person with aquaohobia enjoys yourself and not get wet. So do your study group and supplemental instruction and just listen. Don t live in the past with fear. The next step is understanding anxiety affect demonstration of math knowledge so be prepared.
6 How To Recognize Test Anxiety Anxiety + High Ability = Improvements Anxiety + Low or Average Ability= No Improvements The defining test anxiety Emotional habit to either a single experience or a recurring experience of high tolerant experience. Educational anxiety experiences show during proof of test and indicating process work. These are learning responses and can be undone The cause of Test Anxiety Identify the loss 1.)Test anxiety can be a learning behavior resulting from the expectations of parents, teachers or other significant people in the students life. 2.)Test anxiety can be caused by the association between grades and a student s personal worth. 3.) Test anxiety develops form fear of alienating parents, family or friends due to poor grades. 4.) Test anxiety can stem from a feeling of lack of control and an inability to change one s life situation. 5.) Test anxiety can be caused by a student being embarrassed by the teacher or other students when trying to do math problems. 6.) Test anxiety can be caused by times tests and the fear of not finishing the test even if the student can do all of the problems. 7.) Test anxiety can be caused by being put in math courses that are above the students level of competence. 8.) Students leaving the room before the test time is up.
7 Different Types Of Test Anxiety Emotional anxiety causes: upset stomach, nausea, sweaty palms, rapid heartbeat etc. Worry anxiety cause students to think about failing the test, which can happen either before or after the test These feelings and physical inconveniences can affect your concentration, your testing speed, and it can cause you to completely draw a blank
8 The Effects of Anxiety On Learning & Testing Anxiety can interfere with how fast people process information & decrease the amount of information they can hold for a short period of time Math anxiety temporarily disrupts mental processing in working memory that causes poorer math achievement Test anxiety gets so bad that students would rather leave early and receive a lower grade than to stay and complete the assignment
9 The 12 Myths About Test Anxiety 1.)Students are born with test anxiety 2.)Text anxiety is a mental illness 3.)Test anxiety cannot be reduced 4.)Any level of test anxiety is bad 5.)All students who are not prepared will have test anxiety 6.)Students with test anxiety cannot learn math 7.)Students who are well prepared will not have test anxiety 8.)Very intelligent students and students taking high-level courses, such as calculus, do not have test anxiety 9.)Attending class and doing my homework should reduce all of test anxiety 10.)Being told to relax during a test will make you relaxed 11.)Doing nothing about test anxiety will make it go away 12.)Reducing test anxiety will guarantee better grades
10 How to Reduce Math/ Test Anxiety To reduce math anxiety and math test anxiety you need to understand both the relaxation response and how negative self talk undermines your abilities. The relaxation response is any technique or procedure that helps you become relaxed and will take the place of anxiety response
11 Short-Term Relaxation Techniques The Tensing and Differential relaxation method helps you relax by tensing and relaxing your muscles all at once. Follow these procedures: 1.Put your feet flat on the floor 2.With your hands, grab underneath the chair 3.Push down with your feet and pull up on your chair at the same time for about five seconds 4.Relax for five to ten seconds 5. Repeat the procedure two to three times 6.Relax all your muscles except the ones that are actually used to take the test
12 The Palming Method The palming method is a visualization procedure used to reduce test anxiety. While you are at your desk before or during a test try theses methods to reduce anxiety Close and cover your eyes using the center of your palms of your hands Visualize a relaxing scene for one or two minutes Prevent your hands from touching your eyes by resting them on your cheekbones
13 Managing Self-Talk Self talk is like a telegraphic message where one or two words can bring up many different thoughts & feelings Self talk can increase or decrease test anxiety, it just depends on what you tell yourself
14 Self-Talk Habits Types The Worrier-always asks What if I fail the test The Victim-believes things are hopeless say s no way will I ever pass math The Critic-Puts themselves down by saying I cannot reduce my test anxiety and will fail The Perfectionist- Who says I must make an A or I am a failure
15 Negative/Positive Self-Talk No matter what I do I will not pass this course I know that, with hard work, I will pass math I am no good in math, so why should I try? I failed the course last semester, but I can now use my math study skills to pass this course I am going to fail this test and never graduate I went blank on the last test, but I now know how to reduce my test anxiety
16 Thought-Stopping Technique Making a loud noise such as slapping a desk can effectively interrupt the negative thoughts To stop negative thoughts in a classroom is to shout at yourself, stop thinking about or think positive thoughts The way stop-thinking works is by interrupting the worry response before it can create the type of anxiety that gets out of control. During the interruptions you gain control and replace the negative self-talk with positive responses
THE FIVE CAUSES OF TEST ANXIETY: Test anxiety is a learned behavior. The association of grades and personal worth causes test anxiety. Test anxiety can come from a feeling of a lack of control. Test anxiety
Test Anxiety Inventory Read each statement carefully. If the statement reflects your experience in taking a test, place a check ( ) before that statement. Check as many statements as apply to you. Check
Cooper Counseling, LLC 251 Woodford St Portland, ME 04103 (207) 773-2828(p) (207) 761-8150(f) Psychological Assessment Intake Form This form has been designed to ask questions about your history and currentTest Taking and Test Anxiety University of St. Thomas Counseling and Disability Services Tutorial Services Center 2 Test Anxiety Checklist I worry about failing tests. I do not sleep well the night before
How to Use the Math Study Skills Workbook Houghton Mifflin Company. All rights reserved. Use in the Math Classroom Math classroom time directly influences how students shape their studying time outside
1 Running head: STUDY GUIDE Test Anxiety in the Classroom How Educators Can Help Their Students Iska Harriott University of Pittsburgh 2 Definition Test Anxiety usually occurs when a student has to take
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How to Curb Anxiety and Panic Attacks Using CBT Techniques 1 How to Stop Having Panic Attacks Using CBT Techniques Table of Contents. Introduction 2 Worry Wart 3 What is CBT? 4 The Big Difference 5 Why | 677.169 | 1 |
Mathematics
The mathematics curriculum at Key Stage 3 is loosely based on the English national curriculum. We have adapted it to fit our students' needs. As a guide, we aim to cover up to level 6 in Year 7, level 7 in Year 8 and level 8 in year 9.
We are now increasingly starting to deliver our curriculum by means of tasks that encourage the use of Higher Order Thinking Skills. A useful website with more information on this is
Year 7
Term 1
Term 2
Term 3
Working with decimals & integers
Linear expressions & graphs
Rounding Powers
Sequences & simplifying expressions
Reflections & rotations
Solving equations & conversion graphs
Length and area
Collecting & representing data
Geometrical Constructions
Fractions, decimals & percentages
Fractions & ratios
Probability & Averages
Transformations
Volume & Surface area
Statistics Project
Year 8
Term 1
Term 2
Term 3
Multiples, factors, primes & powers
Linear & quadratic graphs
Solving equations & inequalities
Angles in polygons
Decimals & fractions
Collection & presentation of data
Calculating probabilities
Transformations & tessellations
Geometrical Constructions
Fractions, decimals & percentages
Statistical charts
Algebraic manipulation
Perimeters, areas & volumes
Year 9
Term 1
Term 2
Term 3
Researching the golden ratio
Areas & volumes
Transformations
Sequences
Standard form
Right angled triangle trigonometry
Linear graphs
Indices, surds
Algebraic manipulation
Fractions, decimals, percentage & ratio
Graphs of quadratics & cubics
Sectors, arcs & cylinders
Equations, formulae and identities
Calculating probabilities
Simultaneous equations
Construction and Loci
Pythagoras' theorem
Data collection & representation
Key Stage 4
In Key Stage 4 this year we are following a new IGCSE syllabus, the Cambridge International Mathematics course. This course builds on the work done in KS3 as it incorporates investigational and mathematical modelling skills and also prepares students very well for the IB programme. A graphical display calculator is required for this course and this facilitates an enquiry based approach to learning and encourages students to use technology to model mathematics. There are two levels of examination for this course: Extended (grades A* - C) and Core (grades C and below).
Year 10
IGCSE
Term 1
Term 2
Term 3
Number
Trigonometry 1
Geometry
Algebra 1
Mensuration
Algebra 2
Functions & Graphs
Trigonometry 2
Year 11
IGCSE
Term 1
Term 2
Term 3
Sets & Vectors
Probability
Transformations
Revisions & Past Paper Practice
Statistics
Key Stage 5 - IB
The IB has three levels of mathematics:
Mathematics HL
Mathematics SL
Mathematics Studies
Mathematics HL
This course provides for students with a strong background in mathematics who are competent in a range of analytical and technical skills. The majority of these students will be expecting to include mathematics as a major component of their university studies. Others may take this subject because they have a strong interest in mathematics and enjoy meeting its challenges and engaging with its problems.
Summary of the course
Core Content: Students must study all of the following
1. Algebra
2. Functions and Equations
3. Circular functions and trigonometry
4. Vectors
5. Statistics and probability
6. Calculus
Option: Students must study one of the following
7. Statistics and probability
8. Sets, relations and groups
9. Calculus
10. Discrete mathematics
Mathematical Exploration
Assessment
The topics above are assessed by external examinations (three A or above in their IGCSE extended examination for mathematics or its equivalent.
Mathematics SL
This course provides for students with a strong background in mathematics, and whose interests lie in the fields where mathematical skills and techniques are likely to be needed. It is a course that provides the students with skills needed to cope with the demands of a technological society. Emphasis is placed on the application of mathematics to real-life situations.
Summary of the course
1. Algebra
2. Functions and Equations
3. Circular functions and trigonometry
4. Vectors
5. Statistics and probability
6. Calculus
Mathematical Exploration
Assessment
The topics above are assessed by external examinations (two B or above in their IGCSE extended examination for mathematics or its equivalent.
Mathematical Studies
Mathematical studies is designed to build confidence and encourage an appreciation of mathematics in students who do not need mathematics in their future studies. Mathematical studies concentrates on mathematics that can be applied to other curriculum subjects, to common general world occurrences and to topics that relate to home, work and leisure situations. Emphasis is placed on the application of mathematics based on real-life situations. It encourages students to become critical thinkers and life long learners.
The course includes a project - a piece of written work based on personal research, guided and supervised by the teacher. In this students undertake an investigation of a mathematical nature in the context of another subject in the curriculum, or a hobby or interest of their choice, using skills learned before and during the mathematical studies course. Projects may take the form of mathematical modelling, investigations, applications or statistical surveys.
Students most likely to select this subject are those whose main interests lie outside the field of mathematics, and for many mathematical studies students this will be their last formal mathematics course.
We internally assess students by means of termly topic tests, end of year exams, and mini projects. | 677.169 | 1 |
Difficulty of undergraduate homework problems
I am an advanced undergraduate student, and I still remember the days when I could open my mathematics textbooks, read through a few sample problems, and then blaze my way through the exercises with little or no help.
Come university, and my freshman year, and things changed dramatically. I could still more or less blaze through the Resnick and Halliday exercises, but I was beginning to see that textbooks such as Griffiths, Purcell, Marion and Thornton (I mean, sophomore and beyond) contain exercises I could never solve on my own. In the majority of the cases, I've had to rely on solution manuals to work my way through the problems. It's not like I don't understand the solutions - once I understand the solutions, I can again blaze my way through similar calculations, but for some reason, every problem is different and poses new challenges.
I have been wondering if I have reached the pinnacle of my academic ability. Are there really students who could simply take a look at those problems and figure out the answers instantly?
Thus far, all the top-notch undergraduate students I've encountered get stuck in the exercises, but what separates them from the rest is that they've actually been able to work their way through textbooks so that they've struggled with these exercises, unlike the majority of the students who don't solve all the exercises in the textbook. Therefore, when these top-notch students encounter similar exercises in future courses, they can actually blaze through the calculations, seemingly giving the impression that they've produced the solutions out of nowhere. That's why I think it's so very important to read through textbooks and solve all the exercises because the techniques you learn becomes indispensible in future courses.
I think that's what separates the top-notch students from the rest - Hard work and perseverence! | 677.169 | 1 |
Content Vocabulary: (Be able to define)
Binomial – an algebraic expression of the sum or the difference of two terms
Trinomial – an algebraic expression of the sum or the difference of three terms
Expression – A mathematical phrase involving at least one variable and sometimes numbers and operation symbols.
Function – a special relationship where each input has a single output
Composite function – a function obtained from two given functions, where the range of one function is contained in the domain of the second function, by assigning to an element in the domain of the first function that element in the range of the second function whose inverse image is the image of the element.
Pascal's Triangle – an arrangement of the values of n are in a triangular pattern where each row corresponds to a value of n
Polynomial – a mathematical expression involving a sum of nonnegative integer powers in one or more variables multiplied by coefficients.
Inverse – a function that "reverses" another function
Essential Questions: (Be able to answer)
What is a polynomial?
What characteristics are used to describe polynomials?
How do you add polynomials?
How do you subtract polynomials?
How do you multiply polynomials?
What is Pascal's Triangle and how does it apply to polynomials?
How is composition of functions done?
What is an inverse?
How do you prove two functions are inverses of each other?
Objectives: (Be able to do)
Use the definition of a polynomial to identify polynomials
Interpret the structure and parts of a polynomial expression including terms, factors, and coefficients | 677.169 | 1 |
This book is intended for undergraduate and first-year graduate courses of all engineering streams. It provides the basic concepts of analysis and design of discrete-time systems. It covers the theory and applications of digital control systems, assuming knowledge of matrix algebra, difference equations, Laplace/Z-transforms, and the basic principles of continuous-data control systems. Although the chapters are written in the digital domain, the principle is the same for both the continuous-time control and the discrete-time control. Introduces special topics on robust and H-infinity control Contains a unified presentation of the concepts of digital control theory Includes numerous illustrations and solved examples Provides the importance of MATLAB for computation Comprises additional questions and objective-type questions on control systems Chapter 1: Introduction Chapter 2: Z-Transforms and Inverse Z-Transforms Chapter 3: Pulse Transfer Function and Steady-State Errors Chapter 4: Stability Analysis Chapter 5: State Space Analysis Chapter 6: Feedback Controllers and Lyapunov Stability Analysis Chapter 7: Design of Control System Chapter 8: Robust Control Appendix A: MATLAB Programs Appendix B: Additional Questions Appendix C: Objective Questions Bibliography Index Printed Pages: 642. Bookseller Inventory # 73067 | 677.169 | 1 |
歡迎光臨we3tp7在痞客邦的小天地
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- Join a website and go through both online courses. You will fondness it! There are online communities where on earth grouping cover their pure mathematics complications and support each opposite. There are a lot of websites subject matter on the loose online math module. Join them. Have some fun.
Even if you are a contestant of one the abundant popular with interactive pure mathematics communities you inactive have need of to den and perform a lot. You can harvester perusing and practising next to synergistic module which is fun. Interactive courses, Math labs, Algebra Tutorials and Books are severe but what's genuinely importand and influential is to do whatever school assignment.
Never forget that the secretive is review. It's astir perception the cause trailing the concepts and the way. By clip you will discovery it a lot easier to track the method. I aspiration you the unexcelled of circumstances and optimism you'll get the next pure mathematics personality. | 677.169 | 1 |
You are permitted to use a ScientificCalculator Students take the Mathematics Placement Test For various purposes Some are attempting toshow proficiency of basic skills developmental skills levels MAT 090 or MAT 095 and othersare attempting to demonstrate a skill level needed For College Algebra MAT 107 The practicetest below demonstrates the types of questions that appear on the actual testThe t...
Geometry Welcome to GeometryGeometry requires more thought and creativity than any other math courseyou may have taken before It also requires a lot of patience and a lot of studying If you are willingto work extremely hard you will succeedMaterials needed For class every day areCovered textbook3-Ring Binder with sections For Notes Classwork Homework Tests QuizzesHole-punched lined paper and grap...
Orion TI-36X Talking ScientificCalculator Leaflet Introducing ORION TI-36XThe World s Most Advanced and AffordableTalking Scientific CalculatorSpecially designed by Orbit Research based on thepopular TI-36X Solar Calculator from TexasInstrumentsPerfect For students of junior HighSchool algebra tocollege calculusEasy For teachers to help with - LCD display andfunctionality are identical to the st...
CITY CHRISTIAN HighSchoolSchool SUPPLIES 2013-2014For the beginning of the School year and throughout the School yearAll HighSchool classesPhysical ScienceAccess to a computer with word processing and power4 6 black white board markers to be turned in to thepoint programsteacher the first weekPrinter For printing papers that are compatible withMicrosoft Office 2007USB drive For storing document...
Woodbury Junior – Senior HighSchool - School Supplies List Woodbury Junior Senior HighSchool - School Supplies List 2011-20126th Grade Supplies List 7th 8th Grade Senior HighSchool 9-12MATERIALS SUPPLIED BY TEACHERS TO MATERIALS SUPPLIED BY TEACHERS TO Below is a short list of supplies recommendedSTUDENTS ON THE OPENING DAYS OF School STUDENTS ON THE OPENING DAY OF SchoolFor parents to purch...
August 1997 June 2007Dear parents of students entering Grade 7 in September 2007We would like to welcome you and your son to Middle School at Selwyn House The beginning of the newyear requires the purchase of School supplies You will find below the list For your son These items will needto be replenished periodically We suggest that you purchase duplicates of some items pencils glue erasersand kee...
UPPER ELEMENTARY School A student s success in School is greatly enhanced when he she is organized and has the proper tools andmaterials This list will help parents know what the student will be using throughout the School yearPlease note -We will be making a concerted effort to help students develop their organizational and studyskills This can only begin when the student has the necessary materi...
Dublin Middle School Sixth Grade Supply ListEvery sixth grader is expected to bring the following to all classesOne box of tissues to be turned into homeroom teacher on open house night or first day of schoolOne folder to be used as a homework folder For ALL classesMetric Standard RulerPencil pouch with the following pencils colored pencils markers glue sticks highlighters scissorsLanguage Arts1 B...
Middle School Students Supplies Lists For 2012 13 Dear Parents of Middle School StudentsThe majority of our middle School students School supplies are now part of the tablet computer i edigital binders digital binder paper writing stylus etc This significantly decreases the number ofsupplies that each student will need on the first day of SchoolFor each subject Therefore it is bestto wait until t...
Canwood Community School Canwood Community School2013-2014 School Supply ListsKindergarten Mrs HowatYour child will need the following items marked with their name Velcro shoes For gym a back packa lunch kit and a box of KleenexMrs Howat provides all other supplies and each student pays 15 00Grade 1 2 Mr SchwehrMr Schwehr provides all the supplies and each students pays 30 00Student FeesAcademic F...
Lesmahagow HighSchool Electronics S3 Physics Summary Notes Exercise Lesson 1 Electronic componentsThis is the symbol For a This device can only be on or off so it is calleda device The energy change that takes place iselectricalThis is the symbol For a This device can be a range of valuesbetween on and off so it is called an device The energychange that takes place isThis is the symbol For a swit...
DRACUT HighSchool Dracut Senior High School1540 Lakeview AvenueDracut MA 01826PRECALCULUS TRIGONOMETRY 332COURSE DESCRIPTION This course provides a comprehensive examination of all traditionalPrecalculus Trigonometry topics including the concepts and operations that are prerequisite to thesuccessful study of Honors Calculus in the next academic year Prerequisites B average or higherin both Algebr...
Springwood Middle School SUPPLY LIST 2002 / 2003 Grade 6 or 7 Supply List 2014NOTE Clearly mark student s name on all items including clothing excluding pencils and erasersBasic supplies and equipment For September this is a starter list Some items may need to bereplenished during the year Teachers may request specific supplies in addition to this list Newsupplies are not re...
2009 PhysicsBowl Results 2009 PhysicsBowl ResultsDear Physics TeacherThank you For having your students participate in this year s AAPT PHYSICSBOWL contestThis year there were almost 4500 students participating from approximately 225 schools across the United States andCanada as well as a School in China Enclosed are lists of the first and second place schools in each division and the firstand sec... | 677.169 | 1 |
International Baccalaureate Math (Higher Level) – Year One
The International Baccalaureate describes this as a two-year course that caters for students with a good background in mathematics who are competent in a range of analytical and technical skills. The majority of these students will be expecting to include mathematics as a major component of their university studies, either as a subject in its own right or within courses such as physics, engineering and technology. Others may take this subject because they have a strong interest in mathematics and enjoy meeting its challenges and engaging with its problems.
The first year of IB Math HL includes a review of all topics necessary to prepare for studying Caclulus as well as some additional mathematical topics which will be useful at the college level. The former category includes algebraic functions, exponents, logarithms, and trigonometry. The latter category includes vectors, sequences, probability, statistics, complex numbers and mathematical induction. | 677.169 | 1 |
Hello Math experts ! I am a beginner at simplifying radicals. I seem to understand the lectures in the class well, but when I begin to solve the questions at home myself, I commit mistakes. Does anyone know of any resource where I can get my solutions checked before submitting them for grading? Or any resource where I can get to see a step by step answer ?
Being a professor , this is a comment I usually hear from students. simplifying radicals is not one of the most popular topics amongst kids. I never encourage my students to get ready made solutions from the internet , however I do encourage them to use Algebrator. I have developed a liking for this tool over time. It helps the children learn math in an easy to understand way.
system of equations, binomial formula and relations were a nightmare for me until I found Algebrator, which is truly the best math program that I have come across. I have used it frequently through several algebra classes – Algebra 1, College Algebra and College Algebra. Just typing in the math problem and clicking on Solve, Algebrator generates step-by-step solution to the problem, and my algebra homework would be ready. I highly recommend the program. | 677.169 | 1 |
Anyone intending to tackle both the Linear Algebra Insight and the Intro Analysis Insight, will probably notice that there is some overlap between the two. Micromass was kind enough to provide an efficient way to navigate through them, which he gave permission to repost here:
"So if you're doing both of them, then I would recommend:
Do Bloch Analysis and MacDonald in parallel.
Then after Bloch do Hubbard, and after MacDonald do Axler.
This way you'll get everything without too much repetition. MacDonald will teach you the basics of LA (vector spaces, linear transformations), but will also do geometric algebra. Hubbard will repeat the basics but not from a point of view of analysis. And Axler will do things in the most rigorous light. Avoiding determinants in Axler is not a problem since Hubbard and MacDonald cover those. What do you think? It is possible to do Treil instead of Axler if you prefer Treil, but it's really up to you."
One thing about the Macdonald book is how surprisingly small it is (204 pages) for the amount of content it seems to cover. This is mainly for 2 reasons: (1) it handles worked exercises in a cool way and (2) he doesn't devote space to learning what he calls algorithms (e.g. the mechanistic cookbook recipe for row reduction, etc.)
Regarding worked exercises, the trick is he has you do them! Almost every page has a couple of small exercises that relate to the text you just read. They really make you engage with the content as you go in a neat way that I haven't seen before. Sometimes you'll want a little scratch pad and a pencil to work it out and other times it'll be something simple that you can work out in your head like "what happens if you set t = 0 or 1?" and then you have an aha moment as you realize it simplifies to something you've seen before. This is quite rewarding as opposed to being given the same information in a paragraph.
Regarding algorithms, an example is matrix inversion - he goes through the concept and applications of it, thereafter using it throughout the book but he does not devote space to building up the detailed recipe for mechanistically computing one by hand. Same goes for row reduction, determinants, eigenstuff, etc. In the Preface he argues that the recipes are not needed for theoretical development, and no one solves them by hand anymore anyways except as exercises in Linear Algebra textbooks.Don't worry, Shilov is an excellent book. You really can't go wrong with it. I personally wouldn't recommend it because it does determinants in the beginning, which I find a fairly unintuitive and perhaps too abstract approach. Also, it never really says what a determinant is geometrically (as far as I recall). But if you like it, it's a nice book. | 677.169 | 1 |
College Algebra Advice
Showing 1 to 1 of 1
I would recommend this course to someone because math is necessary for life. This course requires critical thinking about a variety of different problems. It is also essential for almost all majors in order to complete their schooling.
Course highlights:
A major highlight of this course was that it helped me to pass my Praxis Core I Math test. Education majors are required to take the Praxis Core I, which is composed of Math, Reading and Writing, I immediately passed the Reading and Writing portion but failed the Math portion by two points the first time I took it. This course helped me to take a step back and reevaluate math problems. I went back and took the Math portion of the test half way through this course and passed with flying colors.
Hours per week:
6-8 hours
Advice for students:
Make sure you do every homework assignment given out. The teacher does not always collect every assignment, but in the long run doing the work will make the tests that much easier for you.
Course Term:Spring 2016
Professor:Schmahl
Course Required?Yes
Course Tags:Great Intro to the SubjectMany Small AssignmentsParticipation Counts | 677.169 | 1 |
Eyes on Math is a different source that indicates the best way to use photographs to stimulate mathematical educating conversations round K-8 math concepts.
Includes greater than a hundred and twenty full-colour pix and images that illustrate mathematical topics
Each picture is supported with:
- a quick mathematical history and context
- inquiries to use with scholars to guide the academic conversation
- anticipated solutions for every question
- reasons for why every one query is important
- Follow-up extensions to solidify and determine pupil knowing won via discussion
Images could be downloaded for projection onto interactive whiteboards or screens
Provides new methods for lecturers to explain ideas that scholars locate difficult
Invaluable for lecturers operating with scholars with decrease interpreting skill, together with ELL and designated schooling scholars.
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Additional resources for A Bernstein property of solutions to a class of prescribed affine mean curvature equations
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Quot; There was to suppose that our author good reason is a diligent and that in conjunction student of the writings of Jacob Behmen with a relative, Dr. Newton, he was busily engaged, for several ; months in the earlier part tincture. " Great of Alchymist," scribes the character of quest of the philosopher s however, very imperfectly de life, in Behmen, whose researches into things material and things spiritual, things human and things divine, aiford the strongest evidence of a great and original mind.
February, 3. one Mr. Newton I ; : other very learned books and tracts, he s written one upon the mathe matical principles of philosophy, which has given him a mighty name, he having received, especially from Scotland, abundance of congratulatory letters for the same but of all the books he ever ; wrote, there was one of colours and light, established upon thou sands of experiments which he had been twenty years of making, and which had cost him many hundreds of pounds. This book which he vaiued so much, and which was so much talked of, had the ill luck to perish, and be utterly lost just when the learned author was almost at pitting a conclusion at the same, after this In a winter s morning, leaving it among his other papers manner : on his study table while he went to chapel, the candle, which he had unfortunately left burning there, too, catched hold by some means of other papers, and they fired the aforesaid book, and ut consumed it and several other valuable writings arid which most wonderful did no further mischief.
In Newton retained his Professorship at Cambridge till 1703. But he had, on receiving the appointment of Master of the Mint, in 1699, made Mr. Whiston his deputy, with all the emoluments of the office and, on finally resigning, procured his nomination to ; the vacant Chair. John Bernouilli proposed to the most distin of Europe two problems for solution. mathematicians guished Leibnitz, admiring the beauty of one of them, requested the time In January 1697, for solving it to be extended to twelve months twice the period readily granted. | 677.169 | 1 |
This Textbooks New, Books~~Mathematics~~Applied, Elementary-Analysis-Through-Examples-and-Exercises~~John-Schmeelk, 999999999, Elementary Analysis through Examples and Exercises, John Schmeelk, Arpad Takaci, Djurdjica Takaci, 079233597X, Springer Netherlands, , , , , Springer Netherlands
It. It seems to be a constant ingredient to foster deeper mathematical understanding. To a talented mathematical student, many elementary concepts seem clear on their first encounter. However, it is the belief of the authors, this understanding can be deepened with a guided set of exercises leading from the so called "elementary" to the somewhat more "advanced" form. Insight is instilled into the material which can be drawn upon and implemented in later development. The first year graduate student attempting to enter into a research environment begins to search for some original unsolved area within the mathematical literature. It is hard for the student to imagine that in many circumstances the advanced mathematical formulations of sophisticated problems require attacks that draw upon, what might be termed elementary techniques. However, if a student has been guided through a serious repertoire of examples and exercises, he/she should certainly see connections whenever they are encountered. Books, Science and Geography~~Mathematics~~Calculus & Mathematical Analysis, Elementary Analysis Through Examples And Exercises~~Book~~9780792335979~~John Schmeelk, , , , , , , , , ,, [PU: Kluwer Academic Publishers] Elementary Analysis Through Examples and Exercises Schmeelk, John / Schmeelk, J. / Takaci, Djurdjica, Springer
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Extra Lessons In Mathematics Book 5
Extra Lessons in Mathematics Book 5
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Author: Keshmani Dhaniram-Seebaransingh Number of Pages: 199ISBN: 9789796461127Publisher: Charran Publishing HousePublication date: 2004; reprinted 2014 The primary objective of any Mathematics text should be to develop the conceptual understanding of the topics in the syllabus - to develop the child's problem solving skills and to ensure that after completing the exercises, the outcome is a pupil who can use the skills learnt in the external environment whilst deriving pleasure in learning Mathematics. | 677.169 | 1 |
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Here's a real life CSI application of algebra for your students....
There has been a suspicious death. Using their knowledge of formulae and quadratic equations, students have to gather evidence from the crime scene and mortuary to determine (1) who was with the victim moments before death and (2) whether the death was a tragic accident or cold-blooded murder.
At the Crime Scene - The victim died after falling from a balcony. Students have to collect information on the distance of the body from the building to determine whether it was an accidental fall or murder.
At the Mortuary - Students must match a bite mark left on the body with teeth impressions taken from three suspects to find out who was with the victim when he fell.
The preview shows some of the pages from the resource. Please note, it does include an answer page. | 677.169 | 1 |
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All students take the same mathematics courses for grades 8 and 9. In grade 10 students can choose between: Foundations of Mathematics 10 and Apprenticeship and Workplace Mathematics 10. In grade 11 and 12, depending on their planned post-secondary education path, students have the option of choosing between: Foundations of Mathematics 11/12, Apprenticeship and Workplace Mathematics 11, and Pre-Calculus 11/12. Students must take a minimum of one mathematics course at the grade 11 level for graduation and depending on post-secondary education plans, students will need to take more.
Secondary mathematics at LFAS starts off by building upon the mathematics skills that students have already acquired and working to develop them further each year, broadening their abilities to apply and make connections with various problems. Throughout their years in Mathematics at LFAS, students will work on developing their mathematics skills through practice, projects, life application, problem solving, collaboration and learning transferable | 677.169 | 1 |
Prepare, Practice, Review The Sullivan's time-tested approach focuses students on the fundamental skills they need
for the course: preparing for class, practicing with homework, and reviewing the concepts. The Enhanced with Graphing Utilities Series has evolved to meet today's course ...
Represents mathematics as it appears in life, providing understandable, realistic applications consistent with the abilities
of any reader. This book develops trigonometric functions using a right triangle approach and progresses to the unit circle approach. Graphing techniques are emphasized, including ...
For courses in college algebra. Prepare, Practice, Review The Sullivan's time-tested approach focuses students on
the fundamental skills they need for the course: preparing for class, practicing with homework, and reviewing the concepts. The Enhanced with Graphing Utilities Series has ...
Using authentic data to make math meaningful to students, Jay Lehmann's algebra series uses a
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УСЛОВИЯ ИСПОЛЬЗОВАНИЯ Для работы голосового калькулятора необходимо чтобы на вашем устройстве были установлены стандартные голосовые фукнкции на русском языке. VOICE CALCULATOR The application is designed to perform calculations of arithmetic expressions of any complexity. Realized the possibility of using standard functions and grouping expressions in parentheses. The interaction with the application by voice in Russian. Trigonometric functions take parameters in radians.
Sample queries 1. Two added seven. 2. Forty-five multiplied by fifty-six. 3. One divided into twenty-five plus one divided by thirty-five all squared. 4. One plus sinus opening of pi divided by three plus the number of PI divided into four close to the cube? 5. The square root of two point fifty-eight in the box? 6. Sixty-plus degrees open tangent of zero point five open close close? 7. ((2.05 * COS (constant pi / 2) + SIN (0.5)) / 2? 8. E constant squared plus the constant E in a cube all in half? 8. The square root of the constant E in the square in half?
To group expressions in complex mathematical calculations and to set the parameters in mathematical functions should be pronounced the name of parentheses: - bracket opens - brace closes - open bracket - Close parenthesis or easier and preferable to say: - open from or - close Also for the bracketed arithmetic sequence from the beginning of the expression and to its current location, you can use the phrase: - all this - and that the result will be - and what happens as a result of - and what happens - everything that happens as a result - all that will - and it's all - all - all this - all - result - the result - what happens - what happened
For simple single expression with features you can not pronounce the name of the closing bracket at the end, for example: - Sine of pi in half - the sine of half the number of PI - the square root of three
For the summation of the numbers start with the words of the phrase: - sum - the amount of - a plus and then the first or the next number. When you enter the number of the next program repeats it. To cancel the repetition of numbers need to apply one of the commands: - do not speak - do not say - silently - be silent
To get the result the amount necessary to say one of the following phrases: - how many - a total of - say the amount - as the result of - how many total - as happened To calculate the percentage of the arithmetic expression is necessary to say: - NNN percent of the MMM Where NNN is the size of the desired percentage, and MMM any number or an arithmetic expression in parentheses FEATURE LIST: - MULTIPLY - section or share - added - PLUS or ADD - MINUS or subtract - Popol - in the square - cubed - To the extent | 677.169 | 1 |
A-APR.B | Understand the relationship between zeros and factors of polynomials
A-APR.D | Rewrite rational expressions
A-REI.A | Understand solving equations as a process of reasoning and explain the reasoning
A-REI.B | Solve equations and inequalities in one variable: Solve quadratic equations in one variable
A-REI.D | Represent and solve equations and inequalities graphically
A-SSE.A | Interpret the structure of expressions
N-RN.A | Extend the properties of exponents to rational exponents
N-CN.A | Perform arithmetic operations with complex numbers
N-CN.C | Use complex numbers in polynomial identities and equations
Welcome to the UnboundEd Mathematics Guide series! These guides are designed to explain what new, high standards for mathematics say about what students should learn in each grade, and what they mean for curriculum and instruction. This guide, the first for Algebra II, includes two parts. The first part gives a "tour" of the standards focused on reasoning about polynomial (and quadratic), radical and rational equations using freely available online resources that you can use or adapt for your class. It then explains how reasoning about equations relates to other mathematical content in Algebra II, especially graphing, and how to use understandings from prior grades to support students who enter Algebra II with gaps in their learning.
Part 1: What do the standards say?
Algebra II is full of rich content, much of which is important for success in higher mathematics and in a range of careers. So why should this series, and probably your year, begin with reasoning about equations? Like Algebra I and Grade 6 even earlier, Algebra II represents a crucial inflection point in students' learning. In Grade 6, we saw the culmination of fractions and the launching of ratios and proportional relationships. In Algebra I, we saw the culmination of reasoning about linear equations and the maturation of reasoning about quadratic equations. In Algebra II, and in this guide in particular, we see the culmination of reasoning about quadratic equations and the launching of reasoning about higher-order polynomial equations, radical equations and rational equations. Reasoning about equations, therefore, is a good way for students to "ease into" Algebra II with material that relates strongly to what they already know. At the same time, the core idea of reasoning (starting with a claim and making a precise argument to support that claim using a series of logical steps supported by detail) is used again and again in other course work—like literary analysis, social studies, science and later mathematics—and in many careers, so it's good that students continue to build expertise with it.
Another reason to focus on reasoning with equations is that five of the nine clusters covered in this guide (A-SSE.A, A-APR.B, A-REI.A, A-REI.D, and N-RN.A) are also recognized as "major" by PARCC's Model Content Frameworks, meaning they deserve a significant amount of attention over the course of the school year.1 (It's generally a good idea to prioritize major standards within the year to make sure they get the attention they deserve.)
The high school standards are organized into five "categories," and within each category are a number of "domains." The standards involving reasoning about polynomial and quadratic, radical, rational equations are spread across three domains in the Algebra category—"Seeing Structure in Expressions" (A-SSE), "Arithmetic with Polynomials and Rational Expressions" (A-APR) and "Reasoning with Equations and Inequalities" (A-REI). We also have two domains from the Number & Quantity category ("The Real Number System" and "The Complex Number System," or N-RN and N-CN, respectively) that will impact work with radical equations and quadratic equations in particular. Before we get started with the content in these standards, let's pause and take a look at the standards themselves. As you read, think about:
Where do these standards emphasize conceptual understanding of important ideas?
Where do these standards include opportunities to develop key procedural skills?
A-APR.B | Understand the relationship between zeros and factors of polynomials
A-APR.B.2
Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x − a is p(a), so p(a) = 0 if and only if (x − a) is a factor of p(x).
A-APR.B.3
Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.
A-APR.D | Rewrite rational expressions
A-APR.D.6
Rewrite simple rational expressions in different forms: write a(x)/
A-REI.A | Understand solving equations as a process of reasoning and explain the reasoning
A-REI.A.1
Explain each step in solving a simple [polynomial, quadratic, rational, and radical] equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
A-REI.B | Solve equations and inequalities in one variable: Solve quadratic equations in one variable
A-REI.B.4.B
Solve quadratic equations by inspection,
A-REI.D | Represent and solve equations and inequalities graphically
A-REI.D.11
Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equations
A-SSE.A | Interpret the structure of expressions
A-SSE.A.2
Use the structure of an expression to identify ways to rewrite it.
N-RN.A | Extend the properties of exponents to rational exponents
N-RN.A.1
Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents.
N-RN.A.2
Rewrite expressions involving radicals and rational exponents using the properties of exponents.
N-CN.A | Perform arithmetic operations with complex numbers
N-CN.A.1
Know there is a complex number i such that i2= −1, and every complex number has the form a + bi with a and b real.
N-CN.C | Use complex numbers in polynomial identities and equations
N-CN.C.7
Solve quadratic equations with real coefficients that have complex solutions.
The order of the standards doesn't indicate the order in which they have to be taught. Standards are only a set of expectations for what students should know and be able to do by the end of each year; they don't prescribe an exact sequence or curriculum. The high school standards can be sequenced in a variety of ways that result in a coherent experience for students.2
The importance of coherence
Historically, Algebra II has been organized more like a checklist than a coherent inflection point in students' mathematical journey. In this guide, we suggest that working with a variety of different equations is not an end unto itself, but rather a way for students to understand and use mathematical reasoning. The standards above (and their associated clusters, domains and categories) all relate to the idea of reasoning about equations. We begin with quadratic equations, which completes a progression from Algebra I. Then we turn to higher order polynomial equations, a natural extension from quadratic equations. Next, we consider rational equations which follow from the long division necessary to solve polynomial equations. We finish with radical equations, where we study in more detail the reversibility and irreversibility of certain reasoning, which can lead to extraneous solutions.
Reasoning about quadratic equations and the structure of the number system
We begin our reasoning work by considering our well-worn friend, the quadratic equation. As with all equations, students should treat solving equations a process of reasoning, transforming one equation into another equation with the same solutions and justifying their thinking at each step. (A-REI.A.1) The idea is to make equation-solving a conceptual undertaking, focusing on why the process works while learning how to complete the necessary calculations. If students only learn algorithmic steps, they run the risk of forgetting why their methods work or making up invalid moves.
While the methods used in Algebra I to reason through to solutions—factoring, completing the square, and the quadratic formula—continue to be helpful, the kinds of quadratic equations that have been solvable to this point have been constrained to those that exist within the set of real numbers (x2 − 3x − 12 = 0, for example). However, there are quadratic equations (like x2 + 6x + 10 = 0) that can be solved, but only by working within a superset of the real numbers: the complex numbers. (A-REI.B.4, N-CN.C.7) In other words, by considering the structure of number systems, and by working within a larger system, we are able to solve more problems. This is particularly helpful with engineering applications and modeling certain kinds of physics problems.
In Algebra II, students are introduced to complex numbers in service of solving quadratic equations that otherwise would be unsolvable. The introduction of the new idea of complex numbers to a problem type with which they are already familiar is a good way to link students' prior knowledge of quadratics to new learning. The example below is taken from a lesson that does just this. The lesson begins by positing an "unsolvable" equation (x2 + 1 = 0), and this is the conclusion of the resulting discussion. In the process, they come to see that the equation is only unsolvable over the real numbers, but has a solution within the complex numbers. (It's definitely worth viewing the whole lesson plan to see how imaginary numbers are derived from real ones via rotations of the number line, but for now we'll focus on the moment in which complex numbers emerge.)
Algebra II, Module 1, Lesson 37: Discussion
When we perform two 90° rotations, it is the same as performing a 180° rotation, so multiplying by twice results in the same rotation as multiplying by . Since two rotations by 90° is the same as a single rotation by 180°, two rotations by 90° is equivalent to multiplication by twice, and one rotation by 180° is equivalent to multiplication by , we have
for any real number ; thus,
Why might this new number be useful?
Recall from the Opening Exercise that there are no real solutions to the equation
However, this new number is a solution.
In fact, "solving" the equation , we get
=
= or =
However, because we know from above that , and , we have two solutions to the quadratic equation , which are and .
These results suggests that . That seems a little weird, but this new imagined number already appears to solve problems we could not solve before.
This lesson (as with many in Algebra II) relies on nuanced reasoning to derive new ideas from old ones. The process starts with a few assumptions: that the square of a number is always positive, based on the properties of integers and exponents, and that every number has two square roots (positive and negative). If you believe your students might struggle to articulate these key ideas, it might help to begin the lesson with a series of simple review problems, similar to those they first encountered in Grades 6-8. You might, for example, have them evaluate several pairs of integer expressions, such as 32 and (−3)2 and ask them what they notice. And you might have them solve a few equations, such as x2 = 25, and have students explain why there are two solutions. Having these prerequisite ideas at the forefront will help students make sense of the reasoning involved with the rest of the lesson, and can help them answer more advanced questions. For example, can there be a square root of −1? Why or why not? What if there were a number that allowed us to solve x2 = −1? What would the properties of that number need to be?
Working with complex numbers
Once students see how complex numbers arise, they should have opportunities to understand the structure of complex numbers, discovering and articulating patterns along the way. (N-CN.A.1, N-CN.A.2) However, consider spending time judiciously on this topic, as precious instructional time in Algebra II is best spent on the major work of the course. With complex numbers in their toolboxes, students have the tools necessary to reason about the solutions to any quadratic equation. And because reasoning about equations is the main goal here, the following task helps give students an opportunity to apply previous approaches in new ways:
N-CN, A- REI Completing the Square
Renee reasons as follows to solve the equation
First I will rewrite this as a square plus some number.
+
Now I can subtract from both sides of the equation
= -
But I can't take the square root of a negative number so I can't solve this equation.
Show how Renee might have continued to find the complex solutions of .
What's nice about this task is both its connection back to a well-known approach from Algebra I—completing the square—and, its connection to the new idea, complex numbers. Notice also the transparency of the reasoning in Renee's example, and how it shows her thinking and her work, including the moment she could no longer solve the problem. Another approach to building reasoning with quadratic equations is to introduce the discriminant, which allows students to reason first about the nature of the solutions that they are looking for, before they start looking.
Reasoning about polynomial equations
Having examined quadratic equations, let's look at equations of a higher degree, namely polynomial equations. Students learned about polynomial arithmetic and solving quadratic equations in Algebra I, and extending that learning to higher order polynomials. Note, though, that we are not just "doing more work with polynomials." Rather, we are deepening, extending and becoming more expert in reasoning with equations. The goal, as indicated by standard A-REI.A.1, is to "explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method." In other words, rather than focusing solely on the procedures involved, the standards describe that students should understand why the process of solving an equation works, and what each step in that process means. (A-REI.A.1) Throughout this guide, we are going to return to these ideas repeatedly. Reasoning, as the standards tell us, is about solving with intent and purpose, justifying solution steps, and taking steps that are logically connected.
In the past, reasoning through solutions for polynomial equations has often been taught as rote procedure, with little or no connection to learning from prior grades: Think FOIL or Synthetic Division. As a result, students were often able to perform the procedures to add, subtract, multiply and divide polynomials, but lacked any conceptual understanding of why those procedures worked, or even when they should be used in a solution process. These methods also had limited utility. FOIL, for example, is a method that works to find the product of (2x + 5) and (3x − 8), but what about multiplying (2x + 5) by (x2 − 3x + 10)? When framed within the context of structures that students already know, however, reasoning about polynomial equations—and division of polynomials in particular—makes sense conceptually and is easier for students to retain. Moreover, students are also able to solve a greater variety of problems.
The Remainder Theorem and polynomial division
In Algebra II, reasoning about polynomial equations such as x3 − 3x2 − x = −3 depends greatly on a student's ability to use both the Remainder Theorem and polynomial division. These can seem like complicated ideas, but both have their structural roots in the division of integers. In Grade 6, students are expected to master the standard algorithm for division after years of coming to understand the concept of division using strategies based on place value, the properties of operations, and the relationship between multiplication and division. (6.NS.B.2) Later, in Algebra I, students learn that polynomials have a structure similar to the integers. As with integers, polynomial division uses the same structure, and they can also be divided using "long division." In the elementary grades, students learn that if one number divided by another leads to no remainder (8 ÷ 4 = 2), then the divisor (4) is a factor of the dividend (8). In Algebra II, students extend this idea to polynomials using the Remainder Theorem. (A-APR.B.2) The Remainder Theorem essentially states that when a polynomial is divided by another polynomial, and the remainder is zero, the divisor is a factor of the dividend. The example below shows the connection between whole number division and polynomial division.
Algebra II, Module 1, Lesson 4: Example 1
If , then the division ÷ can be represented using polynomial division.
The quotient is .
The completed board work for this example should look something like this:
Notice that in both cases, the same algorithm is used; this is a key connection point that can really help students make sense of polynomial division. Also, the numbers in the first division match the coefficients in the second. This is also a nice scaffold for students making the shift from whole number to polynomial division. Essentially, we can simplify the problem to understand the kinds of coefficients we should have in the answer. Finally, students should see that since both answers contain no remainders, the divisor is a factor of the dividend.
Why not just use synthetic division?
As with the rest of the standards, fluency in polynomial division should follow conceptual understanding of why the process works. While synthetic division can be a useful shortcut, it doesn't clearly relate to what students have learned before, and is best reserved for a fourth-year course (if it is to be introduced at all) due to its relative abstraction. Connections between long division of integers and long division of polynomials builds understanding, while synthetic division, offered too soon, can become a distraction.
Solving using the Remainder Theorem and long division
Once students understand polynomial long division and the relationship between remainders and factors with polynomials, you can shift from a conversation about the structure of division to reasoning about equations. Briefly, the goal is for students to find the values of x that make both sides of an equation equal 0, just as they did with quadratics. (A-APR.B.3) And, just as they did with quadratic equations, students are looking for factors. The nice thing about the Remainder Theorem and long division is that it allows students to get a foothold into the solution of certain higher-order polynomials by reducing their degree through long division. (Equations used in lessons on this idea will, of course, need to be amenable to long division.) Once the degree is reduced to 2 (a quadratic), they have plenty of tools to finish solving. In later courses, students will learn about the Rational Roots Theorem, which guides them in choosing a potential root to start with. For now, though, we suggest either looking at parts of a graph to get started, or to start by checking a given potential candidate and then finding others.
Let's go back to our example above: x3 − 3x2 − x = −3. As we did with quadratics, we start by adding 3 to both sides, in order to set the equation equal to 0: x3 − 3x2 − x + 3 = 0. From here, we begin looking for binomials that will divide into the polynomial with no remainder. If we try (x − 1), we get x2 − 2x − 3. Since there was no remainder, we know that (x − 1) is a factor. So we have (x − 1)(x2 − 2x − 3) = 0. From there, it is a simple process of factoring the remaining quadratic: (x − 3)(x + 1). Based on the factor theorem, we know that the solutions to x3 − 3x2 − x = −3 are 1, 3, and −1. The example below shows a slightly different twist:
Algebra II, Module 1, Lesson 19: Example 4
1. Consider the polynomial .
a. Find the value of so that is a factor of .
In order for to be a factor of , the remainder must be zero. Hence, since , we must have so that .
In this task, students need to reason about the nature of factor: namely, by the zero product property, if x + 1 is a factor, then x = −1 is the root, and evaluation of the function at the value of the root is 0. After that, they can put the factor to use in finding k, which requires additional application of the Remainder Theorem. Note that students seek all of the factors—not just one.
Here's another example, which is a bit more classical.
<task>
Using the Remainder Theorem and Long Division to Solve a Polynomial Equation
Suppose we have the polynomial equation and we want to determine whether is a factor (and thus, that 6 is a solution), and, if it is, what the other factors and solutions are.
The Remainder Theorem tells us two things. First, it tell us that if we substitute 6 in for , then both sides of the equation should be equal.
Let's check: . Great! Since 18 = 18, we know that 6 is a solution.
But, how do we find the others? This is the second way in which the Remainder Theorem is helpful. Since 6 is a solution, is a factor. Knowing this, we can boldly divide our original polynomial by . We can predict that our quotient is now a very tolerable quadratic, which we can solve any number of ways.
Let's start by checking: yields as a quotient, with no remainder.
As such, we are both sure that is a factor (and 6 is a solution), and we can factor quite easily.
Factoring yields , leaving us with -3 and -1 as the other two solutions, when each expression is set to 0. So, is really , so, by the zero product property, our solutions are 6, −3, and −1.
In previous courses, reasoning about the solution(s) to a third-degree polynomial may not have been manageable unless through technology. Now, with tools like the Remainder Theorem and long division, students can find all solutions. (It's important to note that A-APR.B.3 also mentions sketching graphs once we find the zeros. More on that in Part 2 of this guide.)
Factoring with higher degree polynomials: Another path to solving
The Remainder Theorem and long division are not the only tools available to students in solving problems with higher degree polynomials. In fact, students should be extending their work from Algebra I around factoring quadratics and special case polynomials to Algebra II, using structure as a tool to support problem-solving. (A-SSE.A.2) In the same way students come to see polynomial long division as based on the same structure of whole number long division, they come to see some special case higher order polynomial factorizations as based on many of the same structure as quadratic factorization, which allows for more tools in solving problems. The exercise below illustrates a way to bring this point home:
Algebra II, Module 1, Lesson 13: Opening Exercise
Factor each of the following expressions. What similarities do you notice between the examples in the left column and those on the right?
The exercise serves as a launch point for being able to utilize structure to solve problems with higher order polynomials using factoring. In particular, notice how part (f) changes a difference of two quartics into two factors, which are both quadratic in nature, and where the first quadratic is factorable as the difference of two squares.
Reasoning about rational and radical equations
After students have reasoned about polynomial equations and about constraints through quadratic equations, it's time to dive into rational and radical equations. Let's take rational equations first, starting with rational expressions. When applying the Remainder Theorem using long division, students will likely wonder how to handle situations where the remainder is not zero. Rational expressions are structured in the form a(x)/b(x), where a(x) and b(x) are polynomials. Ideally, you'll be able to help students see that rational expressions are structured like rational numbers, and can be rewritten like fractions. For example, where 1473/15 can be rewritten as 98+ 1/5 or 98 1/5, (x2 − 5x + 7)/(x − 2) can be rewritten as (x − 3) + 1/(x − 2) using long division. (A-SSE.A.2) Moreover, just as rewriting fractions helps us understand them better, rewriting and reasoning about rational expressions gives additional insight into their structure and uses. (A-APR.D.6) We can approach the rewriting through a few different methods, depending on the complexity of the rational expression. One way is through inspection: just as we know that 25/3 is 8⅓ by looking at it, we know that (x − 3)/(x −3)2 is 1/(x − 3) quickly and accurately (keeping in mind that x ≠ 3). In other words, inspection can be thought of as reasoning fluently. Here's another nice example of what we're talking about. After getting a common denominator on the right-hand side, we have which, by inspection, we can immediately see is the same as
Egyptian Fractions II (excerpted)
b. We will see how we can use identities between rational expressions to help in our understanding of Egyptian fractions. Verify the following identity for any
Another way is through long division: Some expressions take a little more work, like our example above. Lastly, the standard (A-APR.D.6) also suggests the possible use of computer algebra systems (CAS) to solve "more complicated examples." These can be useful tools for students, but only once they have a solid understanding of simpler examples.
Once students know how to work with rational expressions, they can move into work with rational equations and reason about their solutions. Remember our mantra here: It's not about more procedural gymnastics. Let's take the following task as an example.
Algebra II, Module 1, Lesson 26: Exercise 3
Solve the following equation:
Method 1: Convert both expressions to equivalent expressions with a common denominator. The common denominator is so we use the identity property of multiplication to multiply the left side by and the right side by This does not change the value of the expression on either side of the equation.
Since the denominators are equal, we can see that the numerators must be equal; thus, . Solving for gives a solution of At the outset of this example, we noted that cannot take on the value of or , but there is nothing preventing from taking on the value Thus, we have found a solution. We can check our work. Substituting into gives us , and substituting into gives us . Thus, when , we have ; therefore, is indeed a solution.
The example above makes wonderfully clear how reasoning looks in Algebra II. Notice how each step taken in the solution process is communicated. Notice also how students can reason that, "since the denominators are equal, we can see that the numerators must be equal: thus, 3x − 6 = 8x. Finally, notice how extraneous solutions and constraints are accounted for in defense of the solution, including the checking of the answer.
With a more straightforward example under their belts, students are ready to consider some modeling, too:
Canoe Trip
Jamie and Ralph take a canoe trip up a river for 1 mile and then return. The current in the river is 1 mile per hour. The total trip time is 2 hours and 24 minutes. Assuming that they are paddling at a constant rate throughout the trip, find the speed that Jamie and Ralph are paddling.
Suppose we let denote the speed, in miles per hour, that the canoe would travel with no current. When they are traveling against the current, Jamie and Ralph's speed will be miles per hour and when they are traveling with the current their speed will be miles per hour. The trip upstream will take hours and the trip downstream will take hours. There are of an hour in 24 minutes so the total trip lasts for hours giving us
Multiplying both sides of the equation by gives
This equation simplifies to or, after further manipulation,
We can use the quadratic formula to solve for :
We have so the two solutions are or and The second solution does not make any sense in this context as the speed cannot be negative. So Jamie and Ralph are paddling at a rate of miles per hour. Going upstream, the trip takes longer against the current and going with the current the trip is shorter.
There are a number of noteworthy aspects to this problem. First, it works with a "simple rational equation," as described in A-REI.A.2. Second, it works with a mainstay concept in mathematics, namely that the distance an object travels is proportional to its rate and the time it travels (d = rt). This should be a fairly easy one for students to remember, but is also a quick review if they've forgotten. Third, the resulting equation is our old friend—a quadratic equation. Finally, and perhaps most interesting, is the task offers another opportunity to discuss constraints. This time, it's not about the number set, but rather context and which answer is right given the context. The constraint of the context yields an extraneous solution: −2/3.
While long division is not needed here, other reasoning processes about equations are. Students use properties of equality (e.g. multiplying both sides of the equation by the same value). They also use properties of operations with fractions (i.e., finding a common denominator) learned in working with uncommon denominators in elementary school and proportional relationships in middle school to simplify the rational expressions.
From the perspective of coherence across grades, we again see Algebra II, and reasoning about rational equations in particular, as both a culmination of prior work and launching of new ideas. Radical equations follow the same suit. Students should begin by making connections between radical equations and the study of rational exponents. (N-RN.A.1, N-RN.A.2) They should see, as the standard states, that "the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents." The lesson below shows an interesting and fairly straightforward way of introducing this idea using the properties of integer exponents as a launching pad. Notice how, once again, students have an opportunity to make use of structure (in this case, the structure of the properties of integer exponents) to bring coherence to a new idea. In particular, students use the product rule for integer exponents, xmxn = xm + n, to explain the meaning of non-integer exponents. For example, we can use the product rule to see that the following is true: by showing this: , and, therefore, that . This task is another example.
Algebra II, Module 3, Lesson 3: Discussion
Assume for the moment that whatever means, it satisfies our known rule for integer exponents
Working with this assumption, what is the value of ?
It would be because .
What unique positive number squares to ? That is, what is the only positive number that when multiplied by itself is equal to ?
By definition, we call the unique positive number that squares to the square root of , and we write .
Write the following statements on the board, and ask students to compare them and think about what the statements must tell them about the meaning of .
and
What do these two statements tell us about the meaning of ?
Since both statements involve multiplying a number by itself and getting , and we know that there is only one number that does that, we can conclude that .
At this point, have students confirm these results by using a calculator to approximate both and to several decimal places. In the Opening, was approximated graphically, and now it has been shown to be an irrational number.
Next, ask students to think about the meaning of using a similar line of reasoning
Assume that whatever means will satisfy
What is the value of ?
The value is because
What is the value of ?
The value is because
What appears to be the meaning of ?
Since both the exponent expression and the radical expression involve multiplying a number by itself three times and the result is equal to we know that .
The scaffolding suggestion in the right margin is a good one. Some students may have an easier time with this line of questioning if they're able to work with perfect squares and perfect cubes. For example, in the first sequence of questions, 91/2 might work even better than 21/2, since 9 has a well-known square root. Moreover, using a number with an integer square root would allow students to see that 91/2 ≠ 9(1/2), while also connecting back to the exponent rules for integer exponents. Number 2 in the following task shows a nice example rewriting such expressions:
Properties of Exponents and Radicals
Find the exact value of without using a calculator.
Justify that using the properties of exponents in at least two different ways.
This problem is nice and simple for two main reasons. First, it asks students to prove the truth of an equation, rather than rewriting for its own sake. In other words, it involves a bit of argument, which engages students in a different way than simply evaluating an expression. Second, it asks students to do the rewriting in two ways (using radicals and using rational exponents), which helps them synthesize the relationships between the two.
Once students see both the properties and notation that rational exponents provide, they can move into reasoning about radical equations. Students should see how strategies for solving a radical equation are a consequence of the properties of exponents, as well as how how the structure of radical equations often gives rise to extraneous solutions. This task offers them the opportunity to do both.
Radical Equations
Solve the following two equations by isolating the radical on one side and squaring both sides:
Be sure to check your solutions.
If we raise both sides of an equation a power, we sometimes obtain an equation which has more solutions than the original one. (Sometimes the extra solutions are called extraneous solutions.) Which of the following equations result in extraneous solutions when you raise both sides to the indicated power? Explain.
square both sides
square both sides
cube both sides
cube both sides
Create a square root equation similar to the one in part (a) that has an extraneous solution. Show the algebraic steps you would follow to look for a solution, and indicate where the extraneous solution arises.
Here, we see the use of "explain" in Part (b) to engage students more deeply in reasoning—they need to be able to tell us how extraneous solutions rise—namely, that some steps are not reversible. For example, if we assume that x is a number that satisfies situation (i), then we can quickly see that x is 25. However, that is different than saying, if x2 = 25, then x is 5 (it might be −5).3 Situation (ii) is also a great chance to dig into this with kids. Opportunities exist in Part (a) as well, should you choose to ask students to explain how they used properties of operations, rational exponents, and equality to arrive at their solutions. (Note how extraneous solutions are raised there, too.)
The role of Mathematical Practices
The Standards don't just include knowledge and skills; they also recognize the need for students to engage in certain important practices of mathematical thinking and communication. These "mathematical practices" have their own set of standards, which contain the same basic objectives for Grades K-12.4 You can read the full text of the Standards for Mathematical Practice here. You can read the full text of the Standards for Mathematical Practice here. You can read the full text of the Standards for Mathematical Practice here. (The idea is that students should cultivate the same habits of mind in increasingly sophisticated ways over the years.) But rather than being "just another thing" for teachers to incorporate into their classes, the practices are ways to help students arrive at the deep conceptual understandings required in each grade. The table below contains a few examples of how the practices might help students understand and work with equations in Algebra II.
Opportunities for Mathematical Practices:
Teacher actions:
The Remainder Theorem is based on the same basic concepts of division that students get in the elementary grades (for example, the idea that one number divided by another equals a quotient plus a remainder). Therefore, students can better understand the Remainder Theorem when they relate it to division with integers. In this way, they make use of structures they already know to understand new ideas. (MP.7)
Throughout work with polynomials, introduce concepts as "an extension of a structure we already know." When examining division of polynomials, for example, you can show students examples of integer long division next to examples of polynomial long division, and ask them to draw connections between the two methods (see here for an example).
Each of the equation types in this guide, which are mainstays of Algebra II, have their own unique structures for students to discover and use. (MP.7) In particular, quadratic structures offer many opportunities to rewrite expressions in the process of solving equations. Radical and rational equations also offer rich opportunities to consider structure.
For example, students can view rational expressions in light of what they know about division and fractions from previous grades. You might begin a lesson on rewriting rational expressions by linking to what students learned earlier in the course about polynomial long division. (A-APR.D.6)
Equations derived from a real-world context are abstractions of relationships between quantities. When students solve equations and interpret their solutions in terms of that context, they reason abstractly and quantitatively. (MP.2) They also attend to precision (MP.6) when they use precise reasoning in their solutions, understanding the assumptions and limits involved in that reasoning.
For example, students should realize that every solution process begins with the assumption that an equation has a solution. (A-REI.A.1) In the case of quadratic equations, they'll discover that there may not be a solution over the real numbers. When no solution exists, they'll have the chance to abstractly reason about larger number sets.
Part 2: Reasoning about graphical solutions to equations
So far in this guide, we've focused almost exclusively on symbolic solutions to equations. But graphical methods for reasoning about solutions to equations (A-REI.D.11) can also be powerful. Not only does graphing provide a more concrete representation for students to consider, but it also provides additional tools to support reasoning. While this guide has addressed the symbolic and graphical approaches separately, you are encouraged to teach them simultaneously, so as to maximize their mutually reinforcing attributes.
If your instruction integrates equations and related functions (for example, you teach a unit involving polynomial equations and polynomial functions), your students are well-positioned to see the connections between symbolic and graphical approaches to solving equations. (F-IF.C.7.C) More broadly, they'll have the opportunity to see and explain the relationship between equations and functions. However, if your plan for the year touches first on various types of equations, and then turns to functions much later in the year, you may have to help them tie these two threads together. In either case, students will benefit from questions that highlight the commonalities between different-looking methods. The following example helps highlight the connections:
Solving a Simple Cubic Equation
Find all the values of for which the equation is true.
Use graphing technology to graph ƒ Explain where you can see the answers from part (a) in this graph, and why.
Someone attempts to solve by dividing both sides by yielding and going from there. Does this approach work? Why or why not?
This task explicitly draws a connection between symbolic and graphical methods: Students solve one way, then the other way, and describe the relationship between the two solutions. They also use the two solutions to discover an important takeaway regarding the reasoning involved with solving equations.The use of graphical methods to solve equations become particularly helpful when students begin modeling real-world situations with mathematics. Here are two examples—one using polynomial expressions and equations and one about rational expressions and equations—that show the power of reasoning graphically about solutions with or without tools.
College Fund
When Marcus started high school, his grandmother opened a college savings account. On the first day of each school year she deposited money into the account: $1000 in his freshmen year, $600 in his sophomore year, $1100 in his junior year and $900 in his senior year. The account earns interest of at the end of each year. When Marcus starts college after four years, he gets the balance of the savings account plus an extra $500.
If is the annual interest rate of the bank account, the at the end of the year the balance in the account is multiplied by a growth factor of Find an expression for the total amount of money Marcus receives from his grandmother as a function of this annual growth factor
Suppose that altogether he receives $4400 from his grandmother. Use appropriate technology to find the interest rate that the bank account earned.
How much total interest did the bank account earn over the four years?
Suppose the bank account had been opened when Marcus started Kindergarten. Describe how the expression for the amount of money at the start of college would change. Give an example of what it might look like.
Though the task does not explicitly require students to create the graph of the function, part b offers an opportunity to highlight using a graph to reason about the solutions (a computer algebra system or graphing calculator may be a useful tool here). By examining the graph, students get a better understanding of the various aspects of meaning of the variables and their relationships in the function.
Ideal Gas Law
A certain number of Xenon gas molecules are placed in a container at room temperature. If is the volume of the container and is the pressure exerted on the container by the Xenon molecules, a model predicts that
for all Here the units for volume are liters and the units for pressure are atmospheres.
Sketch a graph of
Using the graph, approximate the volume for which the pressure is 10 atmospheres.
Note that the directions call for students to "sketch" a graph of P in part a. Certainly, students could be asked to solve the equation by hand, but by sketching a graph, they have a ready-made representation of the function from which they can find V when P(V) is 10. The helpfulness of the graph becomes more apparent when we attempt to solve the problem using algebraic techniques only.
In addition to reasoning about solutions by graphing, A-APR.B.3 suggests that solutions can be helpful in graphing. Namely, if we know the solutions—or zeros—to a polynomial equation, we can also sketch a graph of it. When graphing by hand in an Algebra II course, students graph the zeros, and then by find additional pairs of points through evaluations at chosen x-values. In most cases, tasks should provide fairly simple graphs when hand graphing is expected. Here's a great example that ties all of the work with polynomials together.
Graphing from Factors III
Mike is trying to sketch a graph of the polynomial
He notices that the coefficients of add up to zero and says
This means that 1 is a root of and I can use this to help factor and produce the graph.
Students can find the first factor by reasoning about the nature of the function and its coefficients (if all the coefficients add to zero, then 1 is a root because evaluating the function at 1 is the same as working with only the coefficients). Once they find a root of 1, long division (the Remainder Theorem) can be used to find the other factors. Students can also see where the function is negative. As the commentary to this task mentions, "to give a negative output, exactly one of the three factors (or all three factors) has to be negative, giving x < −3 or −2 < x < 1."
Part 3: Where does reasoning with equations come from?
There's quite a bit of important content packed into an Algebra II course, but these standards are intended as a capstone of the learning that occurs in Grades K-8. If your students have been following a strong, standards-aligned program for several years, you might be reaping the benefits of their experience as you read this. But high school teachers often find themselves in a challenging position, as it can be tough to ascertain what students learned in previous grades. And sometimes students have real gaps in their learning that need to be filled before high school work can begin. In this section, we'll consider a few questions you might have as you're planning. First, what exactly were students supposed to learn in the middle school grades that supports their work with equations in Algebra II? How do I leverage what they already know to make high school concepts accessible? And if some of my students are behind, how can I meet their needs without sacrificing focus on high school content?
Algebra I: Solving quadratic equations
In Grade 8 (8.EE.A.2) and in Algebra I, students begin to understand solving equations as a process of reasoning, developing successive equations with the same solutions by applying properties of operations. (A-REI.A.1) Most of their early work in this arena involves linear equations, with quadratics emerging later in the year. In Algebra I, quadratics are limited to equations with real solutions, (A-REI.B.4.B) and the methods used to solve quadratic equations are generally factoring, completing the square, and in some cases, the quadratic formula. (While some students may not be comfortable using the quadratic formula to solve equations, they should at least be aware of how it's derived and why it's useful. (A-REI.B.4)) The progression of problems below illustrates the development of equation-solving throughout Algebra I and II.
➔ This problem, taken from an early lesson on the reasoning involved in equation solving, asks students to consider how the properties of equality can be used to obtain equations with the same solutions. This conceptual foundation allows students to understand why the algebraic procedures for solving linear and quadratic equations work, and prepares them to see the limitations of these procedures when working with rational and radical equations in Algebra II.
➔ This equation, typical of quadratics in Algebra I, has real number solutions and is solvable by factoring. Other equations might lend themselves to solution by completing the square, which should be viewed as an extension of factoring.
➔ Equations like this one, which is similar to others in this guide, expand upon the work students do in Algebra I, requiring them to use the Quadratic Formula and understand the complex number system in order to make sense of previouslyunsolvable equations.
Grade 8 and Algebra I: Exponents and radicals
Starting in Grade 8, students should be able to describe the properties of integer exponents (8.EE.A.1) and use square and cube roots to solve problems. (8.EE.A.2) In the course of solving these types of problems, they should also come to understand that there are numbers which can't be represented in rational form, (8.NS.A.1) and should recognize common irrational numbers (e.g. √2). In Algebra I, students encounter exponents and radicals in the context of equation solving and performing operations with polynomials. Completing the square, for example, relies on an understanding of the relationship between a square and a square root. Moreover, the solutions to equations for which completing the square is useful may well be irrational, so students should also be able to explain the relationship between rational and irrational numbers under the operations of addition and multiplication. (N-RN.B.3) Another progression of problems shows how exponents and radicals evolve over the years.
➔ With problems like these, students begin to understand how the structure of exponential expressions gives rise to their properties. It's important that students take the time to rewrite expressions and explain these properties as they learn them, rather than memorizing a set of rules.
➔ In Algebra I, solving equations like this one requires students to apply their basic understandings of exponents and radicals. Along with just being able to solve, students should also be able to explain why both solutions, which have a rational and an irrational component, are irrational.
➔ Equations like these require students to utilize the properties of exponents to a greater extent than ever before, and to understand why squaring a number in the solution process often entails extraneous solutions.
Suggestions for students who are behind
If you know your students don't have a solid grasp of the pre-requisites to the ideas named above (or haven't encountered them at all), what can you do? It's not practical (or even desirable) to re-teach everything students should have learned in Grades 7, 8 and Algebra I, so the focus needs to be on grade-level standards. At the same time, there are strategic ways of wrapping up "unfinished learning" from prior grades and honing essential competencies within instruction focused on the content above. Here are a few ideas for adapting your instruction to bridge the gaps.
If a significant number of students don't understand the notion of solutions to quadratic equations, you could plan a lesson or two on that idea before starting work with solving quadratics over the complex numbers. This lesson, which introduces the Zero Product Property and includes work with visual models, could be a good place to start. And if you think an entire lesson is too much, but your students could still use some review, you could use 2-3 quadratic equation problems as "warm-ups" to start your first few lessons.
If a significant number of students don't understand fractions as division, you could likewise plan a lesson or two on that before introducing polynomial division. (This Grade 5 lesson may be a good starting place.) Again, if you think that an entire lesson is too much, you could use some problems involving this idea as warm-ups for a few lessons.
If a significant number of students don't understand the concept of constraints on solutions to radical and rational equations you can use a few warm-ups to review the existence of two solutions to square roots of positive integers. (This Grade 8 lesson has some problems that might come in handy.)
For students who lack fluency with important operations—including operations with fractions, decimals and applications of properties—consider incorporating drills involving these operations into your weekly routine. (Drills are more common in the elementary and middle grades, but with a little convincing, even high school students will engage in these sorts of activities.) These could be as simple as a set of ten problems on a certain focus skill (page 22 of this Grade 6 lesson has an example) or as involved as a timed "sprint" exercise (pages 26-29 of this Grade 6 lesson show how these might look).
If you've just finished this entire guide, congratulations! Hopefully it's been informative, and you can return to it as a reference when planning lessons, creating units, or evaluating instructional materials. For more guides in this series, please visit our Enhance Instruction page. For more ideas of how you might use these guides in your daily practice, please visit our Frequently Asked Questions page. And if you're interested in learning more about Algebra II, don't forget these resources:
Endnotes
[1]
[4] You can read the full text of the Standards for Mathematical Practice here.
FAQs
1. What is a Content Guide?
In our work with high academic standards, we often hear educators ask, "What does standards-aligned instruction look like?" Our Content Guides aim to answer this question by providing an in-depth look at one or a few clusters of math standards at a time. The Content Guides are grade-level and content area-specific, and there are guides for each grade or course, from Kindergarten to Algebra II. If you want to learn more about teaching Ratios and Proportional Relationships in Grade 6, for example,our associated Content Guide will give you a comprehensive but accessible explanation about these standards, multiple Open Educational Resource (OER) examples that are aligned to the standards, and concrete suggestions to support the teaching of Grade 6 ratios and proportional reasoning.
Our goal in creating the Content Guides has been to provide busy teachers with a practical and easy-to-read resource on what the grade-level math standards are saying, along with examples of instructional materials that support conceptual understanding, problem-solving, and procedural skill and fluency for students.
It's important to note that content guides are not meant to serve as a curriculum (or any kind of student-facing document), a guide or source material for test-preparation activities, or any kind of teacher evaluation tool.
2. What's in a Content Guide?
Each Content Guide is focused on a specific group of standards. Most Content Guides follow the same three-part structure:
Part 1 makes clear the student skills and understandings described by this group of standards. This section illustrates the standards using multiple student tasks from freely available online sources. Teachers can use or adapt these tasks for their students.
Part 2 explains how this group of standards is connected to other standards in the same grade. We highlight how these connections have implications for planning and teaching, and how this within-grade coherence can increase access for students. Part 2 also includes multiple student tasks from freely available online sources.
Part 3 traces selected progressions of learning leading to grade-level content discussed in the specific Content Guide. This discussion segues into a series of concrete and practical suggestions for how teachers can leverage the progressions to teach students who may not be prepared for grade-level mathematics. Finally, Part 3 traces the progression to content in higher grades.
3. How can I use the Content Guides?
Teachers who have read our Content Guides say they see benefits for all educators. Here are some suggestions for how different educators might use them.
Teachers can use the Mathematics Content Guides to:
Increase or refresh their knowledge of the standards and the expectations for what students should know by the end of the year.
Adapt lessons and units using appropriate pre-requisites to support students who are behind grade-level.
Gain access to the best available OER for math to use for introducing and/or reinforcing concepts
Ensure their curriculum and/or units:
Focus on the major work of the grade and the appropriate depth of each standard.
Target the appropriate aspects of rigor—procedural skill and fluency, modeling and application, and conceptual understanding described by the standards.
Help students make coherent connections within and across grades.
Create or revise their lessons and questioning to focus on important concepts in the standards.
Instructional coaches and school leaders can use the Mathematics Content Guides to:
Refresh or increase their knowledge of the standards and the expectations for what students should know by the end of the year.
Develop and communicate consistent expectations for lesson planning and instruction aligned to the standards.
Provide a reference when planning and/or discussing instruction with teachers.
Gain insight into what instruction and student work should look like in order to meet the demands of the standards.
Develop and design content and standards-driven professional development sessions/workshops.
Develop and/or revise school improvement plans in order to support and incorporate content and practice-based teaching and learning.
4. Why the Content Guides?
The transition to higher standards has led teachers all over the country to make significant changes in their planning and instruction, but only one-third of teachers feel they are prepared to help their students pass the more rigorous standards-aligned assessments (Kane et.al., 2016). This is to be expected because the new high standards are a significant departure from prior standards. The standards require a deeper level of understanding of the math content they teach; a different progression of what students need to learn by which grade; as well as different pedagogy that emphasizes student conceptual understanding, problem solving and procedural fluency in equal intensity.
The support for teachers to bring high standards to their classrooms, however, has lagged behind. Research shows that teacher training in the U.S. is currently insufficient in preparing teachers to teach the demanding new standards (Center for Research in Mathematics and Science Education, 2010). And though some resources exist that "unpack" the standards, few, if any, explain and illustrate the standards. "Unpacking" the standards one by one can also result in a disjointed presentation that neglects the structure and coherence of the standards. In creating the Content Guides, we aimed to provide busy teachers with a practical, easy-to-read resource on their grade-specific standards and how to help all students learn them. There is ample empirical evidence that when teachers have both strong knowledge of the math content that they teach, and the pedagogical knowledge to help students master that content knowledge, their students learn more (Baumert et. al., 2010; Hill, Rowan and Ball, 2005; Rockoff et. al., 2008). With the Content Guides in hand, we hope that teachers will find more success in helping their students make progress toward college- and career-readiness.
5. What is the relationship between the Content Guides and the Progressions?
The Progressions documents describe the grade-to-grade development of understanding of mathematics. These were informed by research on children's cognitive development as well as the logical structure of mathematics. The Progressions explain why standards are sequenced the way they are. The Content Guides often highlight key ideas from the Progressions, but do not add new standards or change the expectations of what students should know and be able to do; they aim to explain and illustrate a group of standards at a time using freely available online sources. While the OER tasks and lessons in the Content Guides are one way to meet the grade-level standards, they are not the only means for doing so.
6. How were the resources selected?
We selected sample tasks and lessons from freely available online sources such as EngageNY, Illustrative Mathematics and Student Achievement Partners to illustrate the Standards. These sources are chosen because they are fully aligned to the new high standards based on national review of K-12 curricula or are created by organizations led by the writers of the new high standards. In addition, because they are open educational resources (OER), they are freely accessible for all uses. All UnboundEd materials are also OER, as part of our commitment to make high-quality, highly aligned content available to all educators.
7. Why are the Content Guides only about a few standards? Where are the rest of the standards?
Each Content Guide addresses a subset of the standards for the grade. The standards addressed in the first set of Content Guides for each grade usually address high-priority content; these standards are also often a good choice for teaching at the beginning of the year. More information about the selection of standards can be found in the introduction to each Content Guide. Over time, we will develop additional Content Guides for each grade and update existing ones. We plan to have four Content Guides for each grade or course, from Kindergarten to Algebra II. The guides will be published in waves, with each wave consisting of one guide for each grade. We plan to release a second set of Content Guides for each grade by the end of the 2016-17 school year.
8. How do I stay informed about new Content Guides?
If you would like to receive updates on content and events from UnboundEd, including new Content Guides, please sign up for UnboundEd announcements here.
Understand the relationship between zeros and factors of polynomials
Rewrite rational expressions
Understand solving equations as a process of reasoning and explain the reasoning
Solve equations and inequalities in one variable
Represent and solve equations and inequalities graphically
Interpret the structure of expressions
Extend the properties of exponents to rational exponents.
Perform arithmetic operations with complex numbers.
Use complex numbers in polynomial identities and equations.
Interpret the structure of expressions
Understand the relationship between zeros and factors of polynomials
Understand solving equations as a process of reasoning and explain the reasoning
Represent and solve equations and inequalities graphically
Extend the properties of exponents to rational exponentsRewrite rational expressionsSolve equations and inequalities in one variableRepresent and solve equations and inequalities graphicallyInterpret the structure of expressions
Use the structure of an expression to identify ways to rewrite it.
Extend the properties of exponents to rational exponentsPerform arithmetic operations with complex numbers.
Know there is a complex number i such that i² = -1, and every complex number has the form a + bi with a and b real.
Use complex numbers in polynomial identities and equations.
Solve quadratic equations with real coefficients that have complex solutions in one variable.
Solve quadratic equations with real coefficients that have complex solutions.
Know there is a complex number i such that i² = -1, and every complex number has the form a + bi with a and b real.
Use the relation i² = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbersFluently divide multi-digit numbers using the standard algorithmIdentify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.
Use the structure of an expression to identify ways to rewrite it.
Use the structure of an expression to identify ways to rewrite itSolve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may ariseLook for and make use of structure.
Look for and make use of structureReason abstractly and quantitatively.
Attend to precision.
Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomialSolve quadratic equations in one variable.
Know and apply the properties of integer exponents to generate equivalent numerical expressionsKnow that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.
Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.
Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
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Presents information about the American Mathematical Society (AMS), an organization created to further mathematical research and scholarship. Includes details about member activities, government affairs and education, publications and research tools, employment and careers, meetings and conferences, and other informationThis book is a compendium of mathematical formulas, tables, and graphs. It contains a table of analytical integrals, differential equations, and numerical series; and includes tables of trigonometric and hyperbolic functions, tables for numerical integration, rules for differentiation and integration, and techniques for point interpolation and function approximation. Additionally, it devotes a entire section to mathematical and physical constants as fractions and powers of Pi, e, and prime numbers; and discusses statistics by presenting combinatorial analysis and probability functions. | 677.169 | 1 |
2 Preface Dear student; Welcome to GAGE college distance education center. GAGE College is one of a pioneer private higher education institution with commendable achievements in teaching-learning, research and outreach activities since 1995E.C. Since its establishment, GC has been working hard to contribute its own part for the country s development effort. Distance education is emerging as a vigorous educational alternative in nearly every corner of the world. Many developed and developing nations now use it as a potent tool for the development of human resources. GAGE acronym represents Great Achievement with Good Education. Vision: To be leading centers of learning and research and to be the fountain of new ideas and innovators in Business, technology and science globally. Mission: Provide quality higher education at all levels with affordable price including through e- learning distance education modes so as to produce competent professionals who can support the development endeavors of the country. o Providing Accounting research and consultancy services to the business community. The major goals of the Department are: o Producing Accountants & Financial Managers who are capable of doing overall activities of Accountants in private, public, non-governmental and governmental organizations. 2
6 UNIT 1: LINEAR EQUATIONS AND THEIR INTERPRETATIVE APPLICATIONS IN BUSINESS Contents 1.0 Aims and Objectives 1.1 Introduction 1.2 Linear equations Developing equation of a line Special formats 1.3 Application of linear equations Linear cost-output relationships Break-even analysis 1.4 Model examination questions 1.0 objectives After reading the chapter students must be able to: define algebraic expression, equation & linear equation explain the different ways of formulating or developing equations of a line understand the breakeven point and its application define the cost output relation ship explain the different cost elements 1.1 INTRODUCTION Mathematics, old and newly created, coupled with innovative applications of the rapidly evolving electronic computer and directed toward management problems, resulted in a new field of study called quantitative methods, which has become part of the curriculum of colleges of business. The importance of quantitative approaches to management problems is now widely accepted and a course in mathematics, with management applications is included in the core of subjects studied by almost all management students. This manual develops mathematics in the applied context required for an understanding of the quantitative approach to management problems. 6
7 1.2 Linear Equations Equation: - A mathematical statement which indicates two algebraic expressions are equal Example: Y = 2X + 3 Algebraic expressions: - A mathematical statement indicating that numerical quantities Example: X + 2 are linked by mathematical operations. Linear equations: - are equations with a variable & a constant with degree one. - Are equations whose terms (the parts separated by +, -, = signs) - Are a constant, or a constant times one variable to the first power Example: 2X 3Y = 7 - the degree (the power) of the variables is 1 - the constant or the fixed value is 7 - the terms of the equation are 2X and 3Y separated by sign However 2X + 3XY = 7 isn t a linear equation, because 3XY is a constant times the product of 2 variables. * No X 2 terms, No X/Y terms, and no XY terms are allowed. - Linear equations are equations whose slope is constant throughout the line. - The general notion of a linear equation is expressed in a form Y = mx + b where m = slope, b = the Y- intercept, Y = dependent variable and X = independent variable. If Y represents Total Cost, the cost is increased by the rate of the amount of the slope m. Slope (m) = Y X rise fall run Y X 2 2 Y1 X 1 if X 1 X 2 Slope measures the steepness of a line. The larger the slope the more steep (steeper) the line is, both in value and in absolute value. 7
8 Y Y m = undefined +ive slope -ve slope X m = 0 - A line that is parallel to the X-axis is the gentlest of all lines i.e. m = 0 - A line that is parallel to the Y-axis is the steepest of all lines i.e. m = undefined or infinite. The slope of a line is defined as the change-taking place along the vertical axis relative to the corresponding change taking place along the horizontal axis, or the change in the value of Y relative to a one-unit change in the value of X Developing equation of a line There are at least three ways of developing the equation of a line. These are: 1. The slope-intercept form 2. The slope-point form 3. Two-point form 1. The slope-intercept form This way of developing the equation of a line involves the use of the slope & the intercept to formulate the equation. Often the slope & the Y-intercept for a specific linear function are obtained directly from the description of the situation we wish to model. Example # 1 Given Slope = 10 Y-intercept = +20, then Slope-intercept form: the equation of a line with slope = m and Y-intercept b is Y = mx + b Y = 10X + 20 X 8
9 Interpretative Exercises #2 Suppose the Fixed cost (setup cost) for producing product X be br After setup it costs br. 10 per X produced. If the total cost is represented by Y: 1. Write the equation of this relationship in slope-intercept form. 2. State the slope of the line & interpret the number 3. State the Y-intercept of the line & interpret the number #3. A sales man has a fixed salary of br. 200 a week In addition; he receives a sales commission that is 20% of his total volume of sales. State the relationship between the sales man s total weekly salary & his sales for the week. Answer Y = 0.20X The slope point form The equation of a non-vertical line, L, of slope, m, that passes through the point (X 1, Y 1 ) is : defined by the formula Y Y 1 = m (X X 1 ) Y Y 1 = m (X X 1 ) Example #1 Y 2 = 4 (X 1) Given, slop = 4 and Y 2 = 4X - 4 Point = (1, 2) Y = 4X 2 #2 A sales man earns a weekly basic salary plus a sales commission of 20% of his total sales. When his total weekly sales total br. 1000, his total salary for the week is 400. derive the formula describing the relationship between total salary and sales. Answer Y = 0.2X #3 If the relationship between Total Cost and the number of units made is linear, & if costs increases by br for each additional unit made, and if the Total Cost of 10 units is br Find the equation of the relationship between Total Cost (Y) & number of units made (X) Answer: Y = 7X Two-point form Two points completely determine a straight line & of course, they determine the slope of the line. Hence we can first compute the slope, then use this value of m together with 9
10 either point in the point-slope form Y Y 1 = m (X X 1 ) to generate the equation of a line. By having two coordinate of a line we can determine the equation of the line. Y2 Y1 Two point form of linear equation: (Y Y 1 ) = X X 1 X X 2 1 Example #1 given (1, 10) & (6, 0) First slope = , then Y Y 1 = m (X X 1 ) Y 10 = -2 (X 1) Y 10 = -2X + 2 Y = -2X + 12 #2 A salesman has a basic salary &, in addition, receives a commission which is a fixed percentage of his sales volume. When his weekly sales are Br. 1000, his total salary is br When his weekly sales are , his total salary is br Determine his basic salary & his commission percentage & express the relationship between sales & salary in equation form. Answer: Y = 0.2X #3 A printer costs a price of birr 1,400 for printing 100 copies of a report & br for printing 500 copies. Assuming a linear relationship what would be the price for printing 300 copies? Answer: Y = 4X Cost = 4.0 (300) = br Special formats a) Horizontal & vertical lines When the equation of a line is to be determined from two given points, it is a good idea to compare corresponding coordinates because if the Y values are the same the line is horizontal & if the X values are the same the line is vertical Example: 1 Given the points (3, 6) & (8, 6) the line through them is horizontal because both Y-coordinates are the same i.e. 6 10
11 The equation of the line becomes Y = 6,which is different from the form Y = mx + b If the X-coordinates of the two different points are equal Example (5, 2) & (5, 12) the line through them is vertical, & its equations is X = 5 i.e. X is equal to a constant. If we proceed to apply the point slope procedure, we would obtain. Slope (M) = equation is: X = constant 2 0 & if m = b) Parallel & perpendicular lines ( and ) infinite the line is vertical & the form of the Two lines are parallel if the two lines have the same slope, & two lines are perpendicular if the product of their slope is 1 or the slope of one is the negative reciprocal of the slope of the other. However, for vertical & horizontal lines. (They are perpendicular to each other), this rule of M 1 (the first slope) times M 2 (the second slope) equals 1 doesn t hold true. i.e. M 1 x M 2-1 Example: Y = 2X 10 & Y = 2X + 14 are parallel because their slope are equal i.e. 2 Y = 3/2X + 10 & Y = -2/3X are perpendicular to each other because the 3 2 multiplication result of the two slope are 1 i.e. x c) Lines through the origin Any equation in the variables X & Y that has no constant term other than zero will have a graph that passes through the origin. Or, a line which passes through the origin has an X- intercept of (0, 0) i.e. both X and Y intercepts are zero. 11
13 5. As production increases, Total Variable Cost increases at the same rate and Marginal cost is equal with Unit Variable Cost (MC = VC) only in linear equations. 6. As production increases TC increases by the rate equal to the AVC = MC (average cost equal to marginal cost) 7. AVC is the same through out any level of production, however Average Fixed Cost (AFC) decreases when Quantity increases & ultimately ATC decreases when Q increases because of the effect of the decrease in AFC. 8. As Quantity increases TR increases at a rate of P. and average revenue remains constant. AR = TR P. Q Q Q = P AR = P in linear functions Breakeven Analysis Break-even point is the point at which there is no loss or profit to the company. It can be expressed as either in terms of production quantity or revenue level depending on how the company states its cost equation. Manufacturing companies usually state their cost equation in terms of quantity (because they produce and sell) where as retail business state their cost equation in terms of revenue (because they purchase and sell) Case 1: Manufacturing Companies Consider a Company with equation TC = VC + FC / Total cost = Variable cost + Fixed cost TR = PQ/ Total Revenue = Price x Quantity At Break-even point, TR = TC i.e TR TC = 0 PQ = VC + FC PQ VC.Q = FC Q (P VC) = FC where Qe = Breakeven Quantity FC = Fixed cost P = unit selling price 13
19 0.8X = X FC 1 m or FC 1 m where m VC P TVC TR -0.2X = X = 60,000 br. When the co. receives br. 60,000 as sales revenue, there will be no loss or profit. The Breakeven revenue (BER = FC 1 m ) method is useful, because we can use a single formula for different goods so far as the company uses the same amount of profit margin for all goods. However, in Breakeven quantity method or BEQ = possible and hence we have to use different formula for different items. Example #1 FC P V it is not It is estimated that sales in the coming period will be br & that FC will be br & variable costs br. 3600, develop the total cost equation & the breakeven revenue Answer: Y = X = 0.6X Where Y = Total Cost X = Total revenue BER = Xe = 2500br At the sales volume of br. 2500, the company breaks even. * When the breakeven revenue equation is for more than one item it is impossible to find the breakeven quantity. It is only possible for one item by Qe = Xe/P Where Xe = Break even revenue P = selling price Qe = Breakeven quantity To change the breakeven revenue equation in to Breakeven quantity. We have to multiple price by the coefficient of X. likewise, to change in to breakeven revenue from Break even quantity, we have to divide the unit VC by price. 1.4 Model examination questions 1. XYZ company s cost function for the next four months is C = 500, Q 19
20 a) Find the BE dollar volume of sales if the selling price is br. 6 / unit b) What would be the company s cost if it decides to shutdown operations for the next four months c) If, because of strike, the most the company can produce is br. 100,000 units, should it shutdown? Why or why not? 2. In its first year, Abol Buna Co had the following experience Sales = 25,000 units Selling price = br. 100 TVC = br. 1,500,000 TFC = br. 350,000 Required: 1. Develop Revenue, cost & profit functions for the co. in terms of quantity. 2. Find the Breakeven point in terms of quantity 3. Convert the cost equation in terms of quantity in to a cost equation in terms of revenue 4. Find the Breakeven revenue 5. If profit had been br. 500,000 what would have been the sales volume (revenue) & the quantity of sales 6. What would have been the profit if sales are br. 2,000, A small home business set up with an investment of $ 10,000 for equipment. The business manufactures a product at a cost of br per unit. If the product sales for Br per unit how many units must be sold before the business breaks even? 4. A retail co. plans to work on a margin of 44% of retail price & to incur other Variable Cost of 4%. If is expected Fixed cost of Br. 20,000. i. Find the equation relating Total Cost to sales ii. Find the profit if sales are Br. 60,000 iii. Find the breakeven revenue iv. If profit is Br. 15,000 what should be the revenue level? v. If you have any one item at a price of Br. 15/unit how do you convert the cost equation in terms of revenue in to a cost equation in terms of quantity? 20
21 UNIT 2 MATRIX ALGEBRA AND ITS APPLICATION Contents 2.0 Aims and Objectives 2.1 Introduction 2.2 Matrix algebra Types of matrices Matrix operation The multiplicative inverse of a matrix 2.3 Matrix Application Solving systems of linear equations Word problems Markov Chains 2.0 AIMS AND OBJECTIVES After reading this chapter students will be able to: explain what a matrix is define the different types of matrices perform matrix operations find inverse of a square matrix explain how to solve a systems of linear equations solve word problems applying matrices understand the concept of Markov chain 2.1 INTRODUCTION Brevity in mathematical statements is achieved through the use of symbols. The price paid for brevity, of course, is the effort spent in learning the meaning of the symbol. In this unit we shall learn the symbols for matrices, and apply them in the statement and solution of input-output problems and other problem involving linear systems 21
22 2.2 MATRIX ALGEBRA Algebra is a part of mathematics, which deals with operations (+, -, x, ). A matrix is a rectangular array of real numbers arranged in m rows & n columns. It is symbolized by a bold face capital letter enclosed by a bracket or parentheses. eg. A a a a m1 a a a m2 a 1n a a 2n mn in which a jj are real numbers Each number appearing in the array is said to be an element or component of the matrix. Element of a matrix are designated using a lower case form of the same letter used to symbolize the matrix itself. These letters are subscripted as a ij, to give the row & column location of the element with in the array. The first subscript always refers to the raw location of the element; the second subscript always refers to its column location. Thus, component a ij is the component located at the intersection of the i th raw and j th column. The number of rows (m) & the number of columns (n) of the array give its order or its dimension, M x n (reads M by n ) Eg. The following are examples of matrices 1 7 element a 12 = 7 A = 5 3 this is 3 x 2 matrix a 21 = a 32 = 2 X = This is a 4 x 4 matrix Element X 44 = X 34 = X 42 = 8 X 32 = 7 22
23 2.2.1 Types of Matrices There are deferent types of matrices. These are 1. Vector matrix is a matrix, which consists of just one row or just one column. It is an m x 1 or 1 x n matrix. 1.1 Row vector is a 1 x n matrix i.e. a matrix with 1 row eg. W = x Column vector: is an m x 1 matrix i.e. a matrix with one column only eg. 0 Z = x 1 2. Square matrix: - a matrix that has the same number of rows & columns. It is also called n-th order matrix eg. 2 x 2, 3 x 3, n x n X = x 2 3. Null or zero matrix: - is a matrix that has zero for every entry. It s generally denoted by Om x n eg. Y = Identity (unit) matrix: - a square matrix in which all of the primary diagonal entries are ones & all of the off diagonal entries are zeros. Its denoted by I. eg. I 2 = x 2 I 2 = x 4 N.B. Each identity matrix is a square matrix * Primary diagonal represents: a 11, a 22, a 33, a a nn entries element A x I = A & I x A = A that is, the product of any given matrix & the identity matrix is the given matrix itself. Thus, the identity matrix behaves in a matrix multiplication like number 1 in an ordinary arithmetic. 23
24 5. Scalar matrix: - is a square matrix where elements on the primary diagonal are the same. An identity matrix is a scalar matrix but a scalar matrix may not be an identity matrix Matrix operations (Addition, Subtraction, Multiplication) Matrix Addition/ Subtraction Two matrices of the same dimension are said to be CONFORMABLE FOR ADDITION. Adding corresponding elements from the two matrices & entering the result in the same raw-column position of a new matrix perform the addition. If A & B are two matrices, each of site m x n, then the sum of A & B is the m x n matrix C whose elements are: C ij = a ij +b ij for i = 1, m C 11 = a 11 + b 11 j = 1, n C 22 = a 22 + b 22 C 12 = a 12 + b 12 etc eg = 10 6 eg These two matrices aren t = conformable for addition because they aren t of the same dimension. Laws of matrix addition The operation of adding two matrices that are conformable for addition has these two basic properties. 1. A + B = B + A The commutative law of matrix addition 2. (A + B) + C = A + (B + C) the associative law of matrix addition 24
25 The laws of matrix addition are applicable to laws of matrix subtraction, given that the two matrices are conformable for subtraction A B = A + (-B) eg.a= 1 2 B = A B = 1 1 Matrix Multiplication 1-1 a) By a constant (scalar multiplication) A matrix can be multiplied by a constant by multiplying each component in the matrix by a constant. The result is a new matrix of the same dimension as the original matrix. If K is any real number & A is an M x n matrix, then the product KA is defined to be the matrix whose components are given by K times the corresponding component of A; i.e. KA = K aij (m x n) eg. If X = 6 5 7, then 2X = (2 x 6) (2 x 5) (2 x 7) 2X = Laws of scalar multiplication The operation of multiplying a matrix by a constant (a scalar) has the following basic properties. If X & Y are real numbers & A & B are m x n matrices, conformable for addition, then 1. XA = AX 3. X (A + B) = XA + XB 2. (X + Y) A = XA + YA 4. X (YA) = XY (A) Laws of scalar multiplication eg. Given matrices A & B and two real numbers X & Y 25
27 Then add the result XA with YA XA + YA = = Therefore it is true that (X + Y) A is equal with XA + YA 3) X (A + B) = XA + XB Proof: X (A + B) means add matrix A and B first and then multiply the result by a constant X Given constant number X = 2 matrices A and B then the result of X (A + B) will be A = B = A + B = X (A + B) = = XA + XB means multiply matrices A and B by a constant number X independently then add the results XA = XB = XA = XB=
28 Then Add XA with XB i.e. XA + XB = = Therefore it is true that X (A + B) is equivalent with XA + XB 4) X (YA) = XY (A) Proof: X (YA) means multiply the second constant number Y with matrix A first and then multiply the first constant number X with the result. YA = X(YA) = = = XY (A) means multiply the two constant real numbers X and Y first and multiply the result by matrix A. XY = 2 X 4 XY (A) = = = Therefore it is also true that X (YA) = XY (A) of columns in B 28
29 b ) Matrix by matrix multiplication If A & B are two matrices, the product AB is defined if and only if the number of Columns in A is equal to the number of rows in B, i.e. if A is an m x n matrix, B should be an n x b. If this requirement is met., A is said to be conformable to B for multiplication. The matrix resulting from the multiplication has dimension equivalent to the number of rows in A & the number columns in B If A is a matrix of dimension n x m (which has m columns) & B is a matrix of dimension p x q (which has p rows) and if m and p aren t the same product A.B is not defined. That is, multiplication of matrices is possible only if the number of columns of the first equals the number of rows of the second. If A is of dimension n x m & if B is of dimension m x p, then the product A.B is of dimension n x p A B Dimension Dimension n x m m x p Must be the same Dimension of A.B n x p eg. A = B = x x 2 AB = (2x 1) + (3 x 0) + (4 x 5) (2 x 7) + (3 x 8) + (4 x 1) = 18 = 42 (6x 1) + (9 x 0) + (7 x 5) (6 x 7) + (9 x 8) + (7 x 1) = 29 =
31 iii. (A + B) C = AC + BC Distributive property # 2 on the other hand, the commutative law of multiplication doesn t apply to matrix multiplication. For any two real numbers X & Y, the product XY is always identical to the product YX. But for two matrices A & B, it is not generally true that AB equals BA. (in the product AB, we say that B is pre multiplied by A & that A is post multiplied by B.) # 3 In many instances for two matrices, A & B, the product AB may be defined while the product BA is not defined or vice versa. In some special cases, AB does equal BA. In such special cases A & B are said to be Commute. A = 1 1 B = 2 2 AB = x 2 2 x 2 BA = 4 4 # 4 Another un usual property of matrix multiplication is that the product of two matrices can be zero even though neither of the two matrices themselves is zero: we can t conclude from the result AB = 0 that at least one of the matrices A or B is a zero matrix A = B = AB = # 5 Also we can t, in matrix algebra, necessarily conclude from the result ab = AC that B= C even if A 0. Thus the cancellation law doesn t hold, in general, in matrix multiplication eg. A = 1 3 B = 4 3 C =
32 AB = AC = but B C The multiplicative inverse of a matrix If A is a square matrix of order n, then a square matrix of its inverse (A -1 ) of the same order n is said to be the inverse of A, if and only if A x A -1 = I = A -1 x A Two square matrices are inverse of each other, if their product is the identity matrix. AA -1 = A -1 A = I Not all matrices have an inverse. In order for a matrix to have an inverse, the matrix must, first of all, be a square matrix. Still not all square matrices have inverse. If a matrix has an inverse, it is said to be INVERTIBLE OR NON-SINGULAR. A matrix that doesn t have an inverse is said to be singular. An invertible matrix will have only one inverse; that is, if a matrix does have an inverse, that inverse will be unique. Note: i. Inverse of a matrix is defined only for square matrices ii. If B is an inverse of A, then A is also an inverse of B iii. Inverse of a matrix is unique iv. If matrix A has an inverse, A is said to be invertible & not all. Square matrices are invertible. Finding the inverse of a matrix Lets begin by considering a tabular format where the square matrix A is AUGMENTED with an identity matrix of the same order as A / I i.e. the two matrices separated by a vertical line Now if the inverse matrix A -1 were known, we could multiply the matrices on each side of the vertical line by A -1 as AA -1 / A -1 I Then because AA -1 = I & A -1 I = A -1, we would have I / A -1. We don t follow this procedure, because the inverse is not known at this juncture, we are trying to determine 32
33 the inverse. We instead employ a set of permissible row operations on the augmented matrix A / I to transform A on the left of the vertical line in to an identity matrix (I). As the identity matrix is formed on the left of the vertical line, the inverse of A is formed on the right side. The allowable manipulations are called Elementary raw operations. ELEMENTARY ROW OPERATIONS: are operations permitted on the rows of a matrix. In a matrix Algebra there are three types of row operations Type 1: Any pair of rows in a matrix may be interchanged / Exchange operations Type 2: a row can be multiplied by any non-zero real number / Multiple operation Type 3: a multiple of any row can be added to any other row. / Add A-multiple operation In short the operation can be expressed as 1. Interchanging rows 2. The multiplication of any row by a non-zero number. 3. The addition / subtraction of (a multiple of) one row to /from another row eg.1. A = B = interchanging rows 2. A = B = Multiplying the first rows by 2 3. A = B = multiplying the 1 st row by 2 & add to the 2 nd row. This case there is no charge to the first row. Theorem on row operations A row operation performed on product of two matrices is equivalent to row operation performed on the pre factor matrix. 33
35 (-3R 1 + R 2 ) i.e. No change to R Multiply R 2 by 1 = (-R 2 ) Multiply R 2 by 1 & add to R 1 Ones first: try to set ones first in a column and then zeros of the same column. Goes from left to right Therefore inverse of A is A -1 = Zeros first method A = Find inverse of A Augmentation R 2 + R R 1 + R 2 Therefore, inverse of A i.e. A -1 = Exercise: Find the inverse for the following matrices (if exist) 1. A = A -1 = 1/
36 2. B = B -1 = What do you conclude from question 2 and 3? C = D = Answers for exercises 1) Finding inverse of matrix A next change the remaining number Uses ones first method. First augment the matrix with The same dimension identity matrix i.e row number i.e. 1R 2 + R 1 With in the same column into zero. the appropriate operation is multiply row 2 by 1 and add the result to Then our objective is trying to change the given matrix into identify format by applying elementary row operations exchange row 1 with row 2 now proceed to the third column and change the column into its required form. First change the primary diagonal entries into one. By multiplying the third row by 1/3 36
37 Next multiply Row 1 by 2 and add the i.e. 1/3 R 3 result to Row 2 i.e.2r 1 + R /3 2/ then change the remaining numbers Now proceed to the 2 nd column and change the primary diagonal entry into zero. multiply Row 3 by 4 and add into positive one by applying elementary the result to raw 2 row operation. The best operation is is exchanging row 2 with row /3 8/ /3 2/ now multiply the third row by 4 add the result to row 1 i.e. 4R 3 + R /3 5/ /3 8/ /3 2/3 0 Therefore the resulting matrix, that is a matrix consisting of the elements at the right side is assumed to be inverse of matrix A i.e. A -1 A -1 = -4/3 5/3 1-4/3 8/3 1 1/3 2/3 0 2) Inverse of matrix B is B -1 = ) Inverse of matrix C is C -1 =
38 4) We can conclude (observe) that matrix B and C are inverse to each other. 5) Matrix d doesn t have an inverse because it is not a square matrix. 2.3 MATRIX APPLICATIONS Solving Systems of Linear Equations I. n by n systems Systems of linear equations can be solved using different methods. Some are: i. Estimation method for two (2) variable problems (equation) ii. Matrix method - Inverse method - Gaussian method Inverse method: Steps 1. Change the system of linear equation into matrix form. The result will be 3 different matrices constructed using coefficient of the variables, unknown values and right hand side (constant) values 2. Find the inverse of the coefficient matrix 3. Multiply the inverse of coefficient matrix with the vector of constant, and the resulting values are the values of the unknown matrix. eg. 2X + 3Y = 4 Given this system of linear equation applying X + 2Y = 2 inverse method we can find the unknown values. Step 1. Change it into matrix form - Using coefficient construct one matrix i.e. coefficient matrix 1 3 = Coefficient matrix Using the unknown variables construct unknown matrix & it is a column vector (a matrix which has one column) 38
39 X Y = vector of unknown -Using the constant values again construct vector of constant 4 = vector of constant 2 Step 2. Find inverse of the coefficient matrix Now we are familiar how to find an inverse for any square matrix. Assuming once first method find the inverse for matrix Its inverse become Step 3. Multiply the coefficient inverse with the vector of constant = Therefore the resulting matrix that is 2 is 0 the value for the unknown variables i.e. X = 2 Y 0 Then X = 2 and Y = 0 that is unique solution * The logic is this given three matrices, coefficient matrix, unknown matrix and vector of constant in the following order. AX = B A = coefficient matrix Given this we can apply different X = vector of unknown Operations, say multiply both sides B = vector of constant Of the expression by A -1 39
40 A -1 AX = A -1 B IX = A -1 B X = A -1 B this implies that multiplying inverse of the coefficient matrix will gives us the value of the unknown matrix Limitations of inverse method - It is only used whenever the coefficient matrix is square matrix - In addition to apply the method the coefficient matrix needs to have an inverse - It doesn t differentiate between no solution and infinite solution cases. Gausian method It is developed by a mathematician Karl F. Gauss ( ). It helps to solve systems of linear equations with different solution approaches i.e. unique solution, No solution and infinite solution cases. n by n systems Step: 1. Change the system of linear equation into a matrix form 2. Augument the coefficient matrix with the vector of constant. 3. Change the coefficient matrix into identity form by applying elementary row operation and apply the same on the vector of constant. 4. The resulting values of the vector of constant will be the solution or the value of the unknown Example: 2X + 3Y = 4 X + 2Y = 2 Step 1. Change it into matrix form 2 3 X = Y 2 Coefficient unknown vector of Matrix matrix constant 40
41 Step: 2. Augumentation Step: 3. Change the coefficient matrix into identity form by applying elementary row operation (use ones first method) Change first the primary diagonal entry from the first row into positive one. Possible operation is exchange row one with row two Next change the remaining numbers in the first column into zero, this case number 2 Now multiply the 1 st row by 2 & add the result to row Then proceed to column 2 and change the primary diagonal entry i.e. 1 into 1 Multiply the 2 nd row by 1 (-1R 2 ) Now change the remaining number with in the same column (column 2) into zero i.e. number 2 Multiply 2 nd row by 2 and add the result to the 1 st row Therefore X = 2 and Y = 0 Example 2. X + Y = 2 2X + 2Y = 4 41
42 Step X = Y 4 Step Multiply Row-1 by 2 & add the result to raw-1 (-2R 1 + R 2 ) The next step is changing the primary diagonal entry in the 2 nd row to 1. But there is no possible operation that can enable you to change it in to number 1 Therefore the implication is that you can t go further but we can observe something from the result. And it is implying an infinite solution case Example 3. X + Y = 5 X + Y = 9 Step X = Y 9 Step Change the encircled number above in to zero Multiply the first row by 1 & add the result to the 2 nd row = 4 no solution 42
43 There is no possible operation that we can apply in order to change the primary diagonal entry in the 2 nd column without affecting the first column structure. Therefore stop there, but here we can observe something i.e. it is no solution case. Therefore, Gaussian method makes a distinction between No solution & infinite solution. Unlike the inverse method. * Summarizing our results for solving an n by n system, we start with the matrix. (A/B), & attempt to transform it into the matrix (I/C) one of the three things will result. 1. an n by n matrix with the unique solution. eg A row that is all zeros except in the constant column, indicating that there are no solutions, eg A matrix in a form different from (1) & (2), indicating that there are an unlimited number of solutions. Note that for an n by n system, this case occurs when there is a row with all zeros, including the constant column. Eg Reference Exercise 1. X + 2Y 3Z = X + Y + Z = 4 3. X + Y + Z = 4 3X + 2Y + Z = 1 5X Y + 7Z = 25 5X Y + 7Z =20 2X + Y - 5Z = 11 2X Y + 3Z = 8 X Y + 3Z = 8 43
49 2.3.2 Word problems Steps 1. Represent one of the unknown quantities by a letter usually X & express other unknown quantities if there is any in terms of the same letter like X 1 X 2 etc 2. Translate the quantities from the statement of the problem in to algebraic form & set up an equation 3. Solve the equation (s) for the unknown that is represented by the letter & find other unknowns from the solution 4.check the findings according to the statement in the problem Example: 1) A Manufacturing firm which manufactures office furniture finds that it has the following variable costs per unit in dollar/unit Desks Chairs Tables Cabinet Material Labor Overhead Assume that an order of 5 desks, 6 chairs, &4 tables & 12 cabinets has just been received. What is the total material, labor & overhead costs associated with the production of ordered items? Answer: Material cost = $ 750 Labor cost = $ 918 Overhead cost = $ Kebede carpet co. has an inventory of 1,500 square yards of wool & 1,800 square yards of nylon to manufacture carpeting. Two grades of carpeting are produced. Each roll of superior grade carpeting requires 20 sq. yards of wool & 40sq. yards of nylon. Each roll of quality-grade carpeting requires 30 square yards of wool & 30 square yard of 49
50 nylon. If Kebede would like to use all the material in inventory, how many rolls of superior & how may rolls of quality carpeting should be manufactured? 15 & Getahun invested a total of br in three different saving accounts. The accounts paid simple interest at an annual rate of 8%, 9% & 7.5% respectively. Total interest earned for the year was br The amount in the 9% account was twice the amount invested in the 7.5% account. How much did Getahun invest in each account? 1000, 6000, A certain manufacturer produces two product P & q. Each unit of product P requires (in its production) 20 units of row material A & 10 units of row material B. each unit of product of requires 30 units of raw material A & 50 units of raw maternal B. there is a limited supply of 1200 units of raw material A & 950 units of raw material B. How many units of P & Q can be produced if we want to exhaust the supply of raw materials? Answer: 45 units of P and 10 units of Q 5. Attendance records indicate that 80,000 South Koreans attended the 2002 world cup at its opening ceremony. Total ticket receipts were Birr 3,500,000. Admission prices were Birr 37.5 for the second-class and Birr for the first class. Determine the number of South Koreans who attended the football game at first class and second class. Solutions / word problems 1) D Ch T Cb Mt Desk Lab Chair FOH Tables 12 Cabinet Material cost = (50 x 5) + (20 x 6) + (5 x 4) + (25 x 12) = 730 br. Labor cost = (30 x 5) + (15 x 6) + (12 x 4) + (15 x 12) = $
Chapter Two: Matrix Algebra and its Applications 1 CHAPTER TWO Matrix Algebra and its applications Algebra - is a part of mathematics that deals with operations (+, -, x ). Matrix is A RECTANGULAR ARRAY
Ordered Pairs Graphing Lines and Linear Inequalities, Solving System of Linear Equations Peter Lo All equations in two variables, such as y = mx + c, is satisfied only if we find a value of x and a valueCHAPTER 13 SECTION 13-1 Geometry and Algebra The Distance Formula COORDINATE PLANE consists of two perpendicular number lines, dividing the plane into four regions called quadrants X-AXIS - the horizontal
1 Systems Of Linear Equations and Matrices 1.1 Systems Of Linear Equations In this section you ll learn what Systems Of Linear Equations are and how to solve them. Remember that equations of the form a
Matrices define matrix We will use matrices to help us solve systems of equations. A matrix is a rectangular array of numbers enclosed in parentheses or brackets. In linear algebra, matrices are important.5 Elementary Matrices and a Method for Finding the Inverse Definition A n n matrix is called an elementary matrix if it can be obtained from I n by performing a single elementary row operation Reminder:
Algebra Review Numbers FRACTIONS Addition and Subtraction i To add or subtract fractions with the same denominator, add or subtract the numerators and keep the same denominator ii To add or subtract fractionsPOL502: Linear Algebra Kosuke Imai Department of Politics, Princeton University December 12, 2005 1 Matrix and System of Linear Equations Definition 1 A m n matrix A is a rectangular array of numbers withChapter 3 Vocabulary equivalent - Equations with the same solutions as the original equation are called. formula - An algebraic equation that relates two or more real-life quantities. unit rate - A rateSection 1: Linear Algebra ECO4112F 2011 Linear (matrix) algebra is a very useful tool in mathematical modelling as it allows us to deal with (among other things) large systems of equations, with relative.1 Functions and Function Notation The definition of a function MAT 111 Summary of Key Points for Section 1.1 A function is a rule which takes certain numbers as inputs and assigns to each input number
. Matrix Inverses Question : What is a matrix inverse? Question : How do you find a matrix inverse? For almost every real number, there is another number such that their product is equal to one. For instance,
1 Introduction to Matrices In this section, important definitions and results from matrix algebra that are useful in regression analysis are introduced. While all statements below regarding the columns
Understanding Basic Calculus S.K. Chung Dedicated to all the people who have helped me in my life. i Preface This book is a revised and expanded version of the lecture notes for Basic Calculus and otherShort-Run Production and Costs The purpose of this section is to discuss the underlying work of firms in the short-run the production of goods and services. Why is understanding production important toAppendix A Introduction to Matrix Algebra I Today we will begin the course with a discussion of matrix algebra. Why are we studying this? We will use matrix algebra to derive the linear regression model
Math 1050 2 ~ Final Exam Review Guide* *This is only a guide, for your benefit, and it in no way replaces class notes, homework, or studying General Tips for Studying: 1. Review this guide, class notes,
Helpsheet Giblin Eunson Library ATRIX ALGEBRA Use this sheet to help you: Understand the basic concepts and definitions of matrix algebra Express a set of linear equations in matrix notation Evaluate determinants
Basic Linear Algebra In this chapter, we study the topics in linear algebra that will be needed in the rest of the book. We begin by discussing the building blocks of linear algebra: matrices and vectors.
1 P a g e Mathematics Notes for Class 12 chapter 3. Matrices A matrix is a rectangular arrangement of numbers (real or complex) which may be represented as matrix is enclosed by [ ] or ( ) or Compact form
To the applicant: The following information will help you review math that is included in the Paraprofessional written examination for the Conejo Valley Unified School District. The Education Code requires
Math 141 Linear systems and Matrices Determine whether each system of linear equations has (a one and only one solution, (b infinitely many solutions, or (c no solution. Find all solutions whenever they
SYSTEMS OF EQUATIONS AND MATRICES WITH THE TI-89 by Joseph Collison Copyright 2000 by Joseph Collison All rights reserved Reproduction or translation of any part of this work beyond that permitted by Sections
Math 111 University of Washington Graphical Approaches to Rates of Change 1 1 Introduction to Speed and Rates of Change Speed is an example of a rate of change. You may be familiar with the formula speed
MA 15910, Lesson 8 notes Algebra part: Sections 3. and 3.3 Calculus part: Section 1.1 Slope: Definition: The slope of a line is the ratio of the change in y to the change in x (ratio of vertical changeVocabulary & Definitions Algebra 1 Midterm Project 2014 15 Associative Property When you are only adding or only multiplying, you can group any of the numbers together without changing the value of the0 CHAPTER. SYSTEMS OF LINEAR EQUATIONS AND MATRICES. Matrices and Matrix Operations.. De nitions and Notation Matrices are yet another mathematical object. Learning about matrices means learning what they
LECTURE 5 Solving Systems of Linear Equations Recall that we introduced the notion of matrices as a way of standardizing the expression of systems of linear equations In today s lecture I shall show howGaussian Elimination We list the basic steps of Gaussian Elimination, a method to solve a system of linear equations. Except for certain special cases, Gaussian Elimination is still state of the art. After
Contents 1 Review of two equations in two unknowns 1.1 The "standard" method for finding the solution 1.2 The geometric method of finding the solution 2 Some equations for which the "standard" method doesn't
. Systems of Linear Equations Question : What is a system of linear equations? Question : Where do systems of equations come from? In Chapter, we looked at several applications of linear functions. One
Chapter 5 Vector Spaces and Subspaces 5. The Column Space of a Matrix To a newcomer, matrix calculations involve a lot of numbers. To you, they involve vectors. The columns of Av and AB are linear combinations
CHAPTER 1 Linear Equations 1.1. Lines The rectangular coordinate system is also called the Cartesian plane. It is formed by two real number lines, the horizontal axis or x-axis, and the vertical axis or
21-105 Pre-Calculus Notes for Week # 3 Higher-Degree Polynomial Equations. At this point we have seen complete methods for solving linear and quadratic equations. For higher-degree equations, the questionFM 5001 NOTES 9/19/2012 1. Quiz Problem 1. Use the prediction method to find the derivative of the function (cos(x)) 2 at x = π/3. You are allowed to use the fact that the derivative of cos(x) is sin(x), | 677.169 | 1 |
MATH Algebra Advice
Showing 1 to 3 of 3
its a basic math course. You will have to take it, so study hard and you should be fine. Also ask for help when you need it. Don't fall behind.
Course highlights:
pie day was alot of fun. we got eat a bunch of pie. this is not part of the course, it all depends on the teacher you have. so hope you get a fun one.
Hours per week:
0-2 hours
Advice for students:
Study every day for atleast a our. unless you are good at math ask for help. do not be afraid to ask for help. once you fall behind it is a pain to catch up and get good grades.
Course Term:Fall 2014
Professor:Ms. Petras
Course Required?Yes
Course Tags:Math-heavyMany Small AssignmentsA Few Big Assignments
Oct 07, 2016
| Would highly recommend.
Not too easy. Not too difficult.
Course Overview:
If you want to be smart at math, all you need to do is know algebra ( algebra 1 and 2). Algebra is one of the fundamentals; it is the root of Math. Even in the SATs, a lot of the questions are algebra problems, so if you are someone who's going to take the SAT soon or near in the future, you have to know a lot of algebra in order to get a high score in the math section of the test.
Course highlights:
Algebra helped me a lot, more than ever! One thing that it did help me with is when I took the SAT. If you're good at algebra, you won't have no problems whatsoever answering the questions on the math section on the SAT exam.
Hours per week:
3-5 hours
Advice for students:
Every new lesson you take or learn in class is important. Study everyday and review everything you have learned in class, so that you won't forget. It's all about reviewing the things you learn everyday.
Course Term:Fall 2016
Professor:Mada
Course Required?Yes
Course Tags:Math-heavyBackground Knowledge Expected
Dec 20, 2015
| Would highly recommend.
Pretty easy, overall.
Course Overview:
It's fun, and Mr. Mada is really good at hat he does!
Course highlights:
Gained a lot of useful information and math tricks that no other teachers had taught me. | 677.169 | 1 |
This thorough overview of the major computer algebra (symbolic mathematical) systems compares and contrasts their strengths and weaknesses, and gives tutorial information for using these systems in various ways.
The Quick and Effective Way to Learn Algebra! The fastest way to learn algebra is to build a solid foundation in the basics. Inside this book you won't find a lot of endless drills. Instead, you get an original, step-by-step approach to learning algebra. In your first steps, you are introduced to essential concepts―allowing you to grasp the subject almost immediately. You will gradually progress to more challenging skills. | 677.169 | 1 |
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The GCSE Maths Revision has seven practice opportunities ensure the best results on exam
This Maths all-in-one revision and practice book contains clear, accessible explanations of GCSE content. There are lots of practice opportunities for each topic throughout the book, suitable for the new Edexcel, AQA, OCR and WJEC Eduqas GCSE courses. There are clear and concise revision notes for every topic covered in the curriculum. Check your newly gathered knowledge by doing the quick tests provided in this book. Each topic ends with a few practice questions. | 677.169 | 1 |
Friday, September 7, 2012
Teaching 5HBC: UNIT 2 Conics and Limits!
We started with a quick review of preCalculus topics. We did this by investigating how to graph conic sections. This lead to fruit full discussion about algebraic techniques such as completing the square, trigonometric techniques such as polar notation and reviewing the use of a Graphing Calculator. | 677.169 | 1 |
MTB 1103 - Business Mathematics
3 Credit Hours (Fall)
This course presents basic principles used to solve everyday business problems, including a review of basic skills and business terminology. Topics in the course include the following: base, rate and percentage, trade and cash discounts, wage and salary administration, insurance (fire and automobile), depreciation and business profits, distribution of corporate dividends, simple interest and bank discount, and buying and selling of corporation bonds and stocks. | 677.169 | 1 |
Archived Semesters
Class Schedule
A problem solving approach to mathematics as it applies to real-life situations. Development, use and communication of mathematical concepts and applications that relate to measurement, percentage, practical geometry, statistics, finance, and unit conversions. Prerequisites: An appropriate mathematics placement score, OR a grade of "C" or better for MAT090, or MAT091, or MAT092, OR (an appropriate diagnostic score, or a grade of "C" or better in each of the following courses: MAT055, MAT056, and MAT057).
Notes: Class 29304Notes: Class 23456 | 677.169 | 1 |
Month: June 2015What an honor it is to be named the valedictorian of the first graduating class of Wilson Hill Academy. I still remember that first day of online Latin class the summer before I officially entered high school. Little did I know then that I was about to embark on a journey, not always smooth and easy, but always incredibly rich and rewarding. My online classical Christian education has taught me to count learning as a
On what rock should your mathematical house be built? In Matthew 7:24-27, the wise man built his house upon the rock. In all branches of mathematics, knowledge is built on previous learning, so for the higher mathematics courses, what is that secure foundation? Many suppose, erroneously, that it is the work done with arithmetic in the early grades. While manipulation of numbers cannot be discounted in importance, the real bedrock is algebraic studies, specifically Algebra | 677.169 | 1 |
Description
This book is a readable introduction to linear algebra, starting at an elementary level. It is intended to be useful both for students of pure mathematics who may subsequently pursue more advanced study in this area and for students who require linear algebra and its applications in other subjects. Throughout, stress is placed on applications of the subject in preference to more theoretical aspects. The book has worked examples on every left-hand page concurrently with the text on the right-hand page, which allows the reader to follow the text uninterrupted. The book is intended to be worked through and learned from, and contains numerous exercises with solutions.show more
Review quote
'The book is clearly directed at students who have had little or no experience of solving linear equations at school. Moreover, it is intended for people who do not find learning mathematics easy. There are no short cuts, or bribes in the form of interesting applications, but the student who is motivated to learn about systems of linear equations, and willing to work through the exercises, will have an excellent opportunity to acquire proficiency and understanding.' The Times Higher Education Supplementshow more | 677.169 | 1 |
Powerful, four-line scientific calculator for high school math and science exploration. Designed with unique features to
allow you to enter more than one calculation compare results and explore patterns all on the same screen. Enter and view
calculations in common Math Notation via the MATHPRINT Mode including stacked fractions exponents exact square roots and
more. Quickly view fractions and decimals in alternate forms by using the Toggle Key. Scroll through previous entries
and investigate critical patterns as well as viewing and pasting into a new calculation. Explore an x y table of values
for a given function automatically or by entering specific x values. Power Source(s): Battery Solar Display Notation:
Numeric Number of Display Digits: N/A Display Characters x Display Lines: 16 x 4.Unit of Measure : Each | 677.169 | 1 |
Basic Concepts of Mathematics
Publisher: The Trillia Group2007 ISBN/ASIN: 1931705003 Number of pages: 208
Description: The book will help students complete the transition from purely manipulative to rigorous mathematics. It covers basic set theory, induction, quantifiers, functions and relations, equivalence relations, properties of the real numbers, fields, and basic properties of n-dimensional Euclidean spaces | 677.169 | 1 |
math introduction frustrated | 677.169 | 1 |
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Data Management is a skill needed for not only upper level math, but for many professions as well. Provide your students with a firm understanding of how graphing and probability studies work with these reproducible worksheets. Students will analyze and graph information, practicing the various forms and styles of graphing, including a master project of using $15,000.00 dollars to "set up" a new, inherited house. 64 classroom reproducible pages, softcover; answer key included. Grades 7-8. | 677.169 | 1 |
The Quick and Effective Way to Learn Algebra! The fastest way to learn algebra is to build a solid foundation in the basics. Inside this book you won't find a lot of endless drills. Instead, you get an original, step-by-step approach to learning algebra. In your first steps, you are introduced to essential concepts―allowing you to grasp the subject almost immediately. You will gradually progress to more challenging skills. | 677.169 | 1 |
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A quick review on Teaching Textbooks.
First let me say their name is beyond boring… but they went to MIT… brilliant in math… lacking in the loquaciousness… we will forgive them..
Up to this point, we have used their Math 7, Pre-algebra, Algebra 1, and Geometry. I am waiting for Math 3 to come out so I can begin it with my youngest child. I have one child who just "gets" math…. the other, is like her mama…. and struggles with math over all. Both students enjoyed these books. My oldest, ( and the one who "gets" it) was able to come from public school 6th grade math, and then a year of no math, straight into the Algebra 1. I don't mean to say that this Algebra course is that easy, because I do believe it to be academic. I feel he had such success because the concepts were made clear, in every lesson. I tried to do Pre-Algebra that same year with my math struggler, and it was melt-downs and tears. So we went and printed out the placement tests they provide, and moved her to Math 7 for a wonderful tear-free year. I do like the fact that my children can work independently with their math curriculum, and come to me with questions vs. me standing there reading out of a teachers manual and explaining all the concepts. I think that when you home school multiple grades, you need to take the opportunity to let your children work alone from time to time so that you are available to help your other students. These books do a very good job of explaining the concept, and they provide either CD-ROM lectures, as well as the lecture written in the book. It is a nice tool for those who are more auditory in their learning process.
I learned things myself in Algebra and 7th grade math, that I was never taught. I feel like it filled the holes that were there from my own math education in "traditional" school. I have read dissenting opinions on this curriculum, and I do know if it is "behind" or not; I do not hold a degree in Math. To me, some of the word problems tend to be repetitive within a chapter, but it does the job of cementing the concept. It is expensive, however, I do plan to use the books with all of my 4 of my children, making the cost more like 50.00 a kid. I have noticed that re-sale value on this curriculum remains fairly high, should you choose that option to recuperate some of your expenses.
If you have a math genius, maybe this isn't your curriculum… But if understanding processes, and actually enjoying math is your goal…. these are the books for you. | 677.169 | 1 |
Enterprex SR-55 Jr. Sliderule and Electric Sliderule
The Enterprex SR-55 Jr. Sliderule and Electric Sliderule
is
an arithmetic
calculator with unknown digits precision
and
algebraic logic.
It has
12 functions, 30 keys
and
a VFD (vacuum fluorescent) display. The power source is
2xAA batteries. | 677.169 | 1 |
Books by O.P. Gupta
Hello folks. CBSE, New Delhi conducts the annual examinations in the months of March and April (usually) every year for class XII. For the purpose of reference, CBSE issues Sample Papers. Here we are providing you Sample Papers of Maths for the purpose of practice. These Sample Papers are based on the Latest Guidelines of CBSE, New Delhi. Solutions, Hints and/or Answers of the questions in Sample Papers have been added too NCERT publishes two Textbooks for Maths, Part I and Part II for the students of class XII. Maths Part I of NCERT covers six chapters whereas Maths Part II of NCERT includes seven chapters; in total we have to study 13 chapters. Followings are the List Of Important Formulae based on all the Topics Of NCERT Textbooks of class 12. The presentation of each topic has been done in an illustrative format. Covering all the aspects of the concerned topic, these Formulae Guides will be quite helpful for all | 677.169 | 1 |
Grade 7 Math Prep
The course is a rigorous three-week blended learning program designed to both review skills and content, and to push students forward conceptually to prepare for Grade 7 mathematics. This guided program focuses on a weekly topic with skill review, practice and assessments as benchmarks for mastery of skill. Students will be presented material in a variety of ways and will be able to sharpen both their analytical and problem solving skills. Designed to spotlight and strengthen the areas where students have the most trouble in Grade 7, topics covered will include: linear equations, positive and negative integer operations, and exponent rules. While this course is designed to be completed in its entirety online, in person sessions will be given each Monday of the course from 3:30pm to 4:30pm on campus. These sessions will be available live online through Google Hangouts and will be saved for later viewing if students aren't available during that time.
Instructor
Peter Smith
Peter Smith is the Director of Online Education and Blended Learning at Worcester Academy. He also teaches Pre-Algebra. He has 12 years of experience working in independent schools. He has his B.S. in Computer Science from Allegheny College and an M.A. in Computing in Education from The Teacher's College at Columbia University. He earned his STEM Leadership Certificate through NASA and Columbia University. He has presented at ISTE, the Lausanne Laptop Institute, NEALS, AISNE, and at other independent schools and online webinars. | 677.169 | 1 |
Kaplan
Intensive Practice Build the skills you need to master every math concept and every math question type.
Targeted Review Prepare with comprehensive, step-by-step training in the four major content areas: arithmetic, algebra, word problems, and geometry. | 677.169 | 1 |
Vidcode Standards Alignment
The Vidcode curriculum is aligned to Common Core Standards for Mathematics, making it possible to integrate Vidcode smoothly into your classroom. Vidcode also satisfies several of the CSTA (Computer Science Teacher's Association) standards.
COMMON CORE MATHematical Content Standards
N-Q - Reason quantitatively and use units to solve problems.
5.OA.1 - Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.
6.NS.5 - UnderstandA.REI.1 - Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. | 677.169 | 1 |
Y11 Options - PowerPoint PPT Presentation
. Y11 OptionsMathematics . What will we do in Mathematics in year 11? A further year of GCSE study to re-sit Mathematics A recognised qualification in Additional Mathematics A recognised qualification about dealing with money A recognised qualification in another subject areaA taster course11 Options' - rico | 677.169 | 1 |
1. The complex number system includes real numbers and imaginary numbers
Evidence Outcomes
21st Century Skill and Readiness Competencies
Students Can:
Extend the properties of exponents to rational exponents. (CCSS: N-RN)
Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents.1 (CCSS: N-RN.1)
Use square root and cube root symbols to represent solutions to equations of the form \(x^2=p\) and \(x^3=p\), where \(p\) is a positive rational number. (CCSS: 8.EE.2)
Evaluate square roots of small perfect squares and cube roots of small perfect cubes.4 (CCSS: 8.EE.2)
Use numbers expressed in the form of a single digit times a whole-number power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other.5 (CCSS: 8.EE.3)
Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. (CCSS: 8.EE.4)
Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities.6 (CCSS: 8.EE.4)
Interpret scientific notation that has been generated by technology. (CCSS: 8.EE.4)
Inquiry Questions:
Why are real numbers represented by a number line and why are the integers represented by points on the number line?
Why is there no real number closest to zero?
What is the difference between rational and irrational numbers?
Relevance & Application:
Irrational numbers have applications in geometry such as the length of a diagonal of a one by one square, the height of an equilateral triangle, or the area of a circle.
Different representations of real numbers are used in contexts such as measurement (metric and customary units), business (profits, network down time, productivity), and community (voting rates, population density).
Technologies such as calculators and computers enable people to order and convert easily among fractions, decimals, and percents.
Nature Of:
Mathematics provides a precise language to describe objects and events and the relationships among them.
Mathematicians reason abstractly and quantitatively. (MP)
Mathematicians use appropriate tools strategically. (MP)
Mathematicians attend to precision. (MP)
1 Know that numbers that are not rational are called irrational. (CCSS: 8.NS.1)
2 e.g., \(\pi^2\). (CCSS: 8.NS.2)
For example, by truncating the decimal expansion of \(\sqrt{2}\), show that \(\sqrt{2}\) is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations. (CCSS: 8.NS.2)
5 For example, estimate the population of the United States as 3 times \(10^8\) and the population of the world as 7 times \(10^9\), and determine that the world population is more than 20 times larger. (CCSS: 8.EE.3)
6 e.g., use millimeters per year for seafloor spreading. (CCSS: 8.EE.4)
3. In the real number system, rational numbers have a unique location on the number line and in space
Evidence Outcomes
21st Century Skill and Readiness Competencies
Students Can:
Explain why positive and negative numbers are used together to describe quantities having opposite directions or values.11 (CCSS: 6.NS.5)
Use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation. (CCSS: 6.NS.5)
Use number line diagrams and coordinate axes to represent points on the line and in the plane with negative number coordinates.12 (CCSS: 6.NS.6)
Describe a rational number as a point on the number line. (CCSS: 6.NS.6)
Use opposite signs of numbers to indicate locations on opposite sides of 0 on the number line. (CCSS: 6.NS.6a)
Identify that the opposite of the opposite of a number is the number itself.13 (CCSS: 6.NS.6a)
Explain when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes. (CCSS: 6.NS.6b)
Find and position integers and other rational numbers on a horizontal or vertical number line diagram. (CCSS: 6.NS.6c)
Find and position pairs of integers and other rational numbers on a coordinate plane. (CCSS: 6.NS.6c)
Order and find absolute value of rational numbers. (CCSS: 6.NS.7)
Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram.14 (CCSS: 6.NS.7a)
Write, interpret, and explain statements of order for rational numbers in real-world contexts.15 (CCSS: 6.NS.7b)
Define the absolute value of a rational number as its distance from 0 on the number line and interpret absolute value as magnitude for a positive or negative quantity in a real-world situation.16 (CCSS: 6.NS.7c)
Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane including the use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate. (CCSS: 6.NS.8)
Inquiry Questions:
Why are there negative numbers?
How do we compare and contrast numbers?
Are there more rational numbers than integers?
Relevance & Application:
Communication and collaboration with others is more efficient and accurate using rational numbers. For example, negotiating the price of an automobile, sharing results of a scientific experiment with the public, and planning a party with friends.
Negative numbers can be used to represent quantities less than zero or quantities with an associated direction such as debt, elevations below sea level, low temperatures, moving backward in time, or an object slowing down
Nature Of:
Mathematicians use their understanding of relationships among numbers and the rules of number systems to create models of a wide variety of situations.
1. The decimal number system describes place value patterns and relationships that are repeated in large and small numbers and forms the foundation for efficient algorithms
Evidence Outcomes
21st Century Skill and Readiness Competencies
Students Can:
Explain that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left. (CCSS: 5.NBT.1)
Explain patterns in the number of zeros of the product when multiplying a number by powers of 10. (CCSS: 5.NBT.2)
Explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. (CCSS: 5.NBT.2)
Use whole-number exponents to denote powers of 10. (CCSS: 5.NBT.2)
Read, write, and compare decimals to thousandths. (CCSS: 5.NBT.3)
Read and write decimals to thousandths using base-ten numerals, number names, and expanded form.1 (CCSS: 5.NBT.3a)
Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons. (CCSS: 5.NBT.3b)
Use place value understanding to round decimals to any place. (CCSS: 5.NBT.4)
Convert like measurement units within a given measurement system. (CCSS: 5.MD)
Convert among different-sized standard measurement units within a given measurement system.2 (CCSS: 5.MD.1)
4. The concepts of multiplication and division can be applied to multiply and divide fractions (CCSS: 5.NF)
Evidence Outcomes
21st Century Skill and Readiness Competencies
Students Can:
Interpret a fraction as division of the numerator by the denominator (a/b = a ๗ b). (CCSS: 5.NF.3)
Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers.8 (CCSS: 5.NF.3)
Interpret the product (a/b) ื q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a ื q ๗ b. 9 In general, (a/b) ื (c/d) = ac/bd. (CCSS: 5.NF.4a)
Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. (CCSS: 5.NF.4b)
How do operations with fractional numbers compare to operations with whole numbers?
Relevance & Application:
Rational numbers are used extensively in measurement tasks such as home remodeling, clothes alteration, graphic design, and engineering.
Situations from daily life can be modeled using operations with fractions, decimals, and percents such as determining the quantity of paint to buy or the number of pizzas to order for a large group.
Rational numbers are used to represent data and probability such as getting a certain color of gumball out of a machine, the probability that a batter will hit a home run, or the percent of a mountain covered in forestMathematicians make sense of problems and persevere in solving them. (MP)
Mathematicians model with mathematics. (MP)
Mathematicians look for and express regularity in repeated reasoning. (MP)
8 (CCSS: 5.NF.3)
9 For example, use a visual fraction model to show (2/3) ื 4 = 8/3, and create a story context for this equation. Do the same with (2/3) ื (4/5) = 8/15. (CCSS: 5.NF.4a)
10 Explain why multiplying a given number by a fraction greater than 1 results in a product greater than the given number. (CCSS: 5.NF.5b) Explain why multiplying a given number by a fraction less than 1 results in a product smaller than the given number (CCSS: 5.NF.5b)
11 e.g., by using visual fraction models or equations to represent the problem. (CCSS: 5.NF.6)
12 For example, create a story context for (1/3) ๗ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ๗ 4 = 1/12 because (1/12) ื 4 = 1/3. (CCSS: 5.NF.7a)
13 For example, create a story context for 4 ๗ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ๗ (1/5) = 20 because 20 ื (1/5) = 4. (CCSS: 5.NF.7b)
14 (CCSS: 5.NF.7c)
Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100.1 (CCSS: 4.NF.5)
Use decimal notation for fractions with denominators 10 or 100.2 (CCSS: 4.NF.6)
Compare two decimals to hundredths by reasoning about their size.3 (CCSS: 4.NF.7)
Inquiry Questions:
Why isn't there a "oneths" place in decimal fractions?
How can a number with greater decimal digits be less than one with fewer decimal digits?
Is there a decimal closest to one? Why?
Relevance & Application:
Decimal place value is the basis of the monetary system and provides information about how much items cost, how much change should be returned, or the amount of savings that has accumulated.
Knowledge and use of place value for large numbers provides context for population, distance between cities or landmarks, and attendance at events2 For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram. (CCSS: 4.NF.6)
3 Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model. (CCSS: 4.NF.7)
1. The whole number system describes place value relationships through 1,000 and forms the foundation for efficient algorithms
Evidence Outcomes
21st Century Skill and Readiness Competencies
Students Can:
Use place value to read, write, count, compare, and represent numbers. (CCSS: 2.NBT)
Represent the digits of a three-digit number as hundreds, tens, and ones.1 (CCSS: 2.NBT.1)
Count within 1000. (CCSS: 2.NBT.2)
Skip-count by 5s, 10s, and 100s. (CCSS: 2.NBT.2)
Read and write numbers to 1000 using base-ten numerals, number names, and expanded form. (CCSS: 2.NBT.3)
Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons. (CCSS: 2.NBT.4)
Use place value understanding and properties of operations to add and subtract. (CCSS: 2.NBT)
Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction. (CCSS: 2.NBT.5)
Add up to four two-digit numbers using strategies based on place value and properties of operations. (CCSS: 2.NBT.6)
Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method.2 (CCSS: 2.NBT.7)
Mentally add 10 or 100 to a given number 100–900, and mentally subtract 10 or 100 from a given number 100–900. (CCSS: 2.NBT.8)
Explain why addition and subtraction strategies work, using place value and the properties of operations. (CCSS: 2.NBT.9)
Inquiry Questions:
How big is 1,000?
How does the position of a digit in a number affect its value?
Relevance & Application:
The ability to read and write numbers allows communication about quantities such as the cost of items, number of students in a school, or number of people in a theatre.
Place value allows people to represent large quantities. For example, 725 can be thought of as 700 + 20 + 5.
Nature Of:
Mathematicians use place value to represent many numbers with only ten digits.
2 Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds. (CCSS: 2.NBT.7)
1. The whole number system describes place value relationships within and beyond 100 and forms the foundation for efficient algorithms
Evidence Outcomes
21st Century Skill and Readiness Competencies
Students Can:
Count to 120 (CCSS: 1.NBT.1)
Count starting at any number less than 120. (CCSS: 1.NBT.1)
Within 120, read and write numerals and represent a number of objects with a written numeral. (CCSS: 1.NBT.1)
Represent and use the digits of a two-digit number. (CCSS: 1.NBT.2)
Represent the digits of a two-digit number as tens and ones.1 (CCSS: 1.NBT.2)
Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <. (CCSS: 1.NBT.3)
Compare two sets of objects, including pennies, up to at least 25 using language such as "three more or three fewer" (PFL)
Use place value and properties of operations to add and subtract. (CCSS: 1.NBT)
Add within 100, including adding a two-digit number and a one-digit number and adding a two-digit number and a multiple of ten, using concrete models or drawings, and/or the relationship between addition and subtraction. (CCSS: 1.NBT.4)
Identify coins and find the value of a collection of two coins (PFL)
Mentally find 10 more or 10 less than any two-digit number, without counting; explain the reasoning used. (CCSS: 1.NBT.5)
Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction. (CCSS: 1.NBT.6)
Relate addition and subtraction strategies to a written method and explain the reasoning used. (CCSS: 1.NBT.4 and 1.NBT.6)
Inquiry Questions:
Can numbers always be related to tens?
Why not always count by one?
Why was a place value system developed?
How does a position of a digit affect its value?
How big is 100?
Relevance & Application:
The comparison of numbers helps to communicate and to make sense of the world. (For example, if someone has two more dollars than another, gets four more points than another, or takes out three fewer forks than needed.
1. Whole numbers can be used to name, count, represent, and order quantity
Evidence Outcomes
21st Century Skill and Readiness Competencies
Students Can:
Use number names and the count sequence. (CCSS: K.CC)
Count to 100 by ones and by tens. (CCSS: K.CC.1)
Count forward beginning from a given number within the known sequence.1 (CCSS: K.CC.2)
Write numbers from 0 to 20. Represent a number of objects with a written numeral 0-20.2 (CCSS: K.CC.3)
Count to determine the number of objects. (CCSS: K.CC)
Apply the relationship between numbers and quantities and connect counting to cardinality.3 (CCSS: K.CC.4)
Count and represent objects to 20.4 (CCSS: K.CC.5)
Compare and instantly recognize numbers. (CCSS: K.CC)
Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group.5 (CCSS: K.CC.6)
Compare two numbers between 1 and 10 presented as written numerals. (CCSS: K.CC.7)
Identify small groups of objects fewer than five without counting
Inquiry Questions:
Why do we count things?
Is there a wrong way to count? Why?
How do you know when you have more or less?
What does it mean to be second and how is it different than two?
Relevance & Application:
Counting is used constantly in everyday life such as counting plates for the dinner table, people on a team, pets in the home, or trees in a yard.
Numerals are used to represent quantities.
People use numbers to communicate with others such as two more forks for the dinner table, one less sister than my friend, or six more dollars for a new toy.
Nature Of:
Mathematics involves visualization and representation of ideas.
Numbers are used to count and order both real and imaginary objects.
Mathematicians attend to precision. (MP)
Mathematicians look for and make use of structure. (MP)
1 instead of having to begin at 1. (CCSS: K.CC.2)
2 with 0 representing a count of no objects. (CCSS: K.CC.3)
3 When counting objects, say the number names in the standard order, pairing each object with one and only one number name and each number name with one and only one object. (CCSS: K.CC.4a) Understand that the last number name said tells the number of objects counted. The number of objects is the same regardless of their arrangement or the order in which they were counted. (CCSS: K.CC.4b) Understand that each successive number name refers to a quantity that is one larger. (CCSS: K.CC.4c)
4 Count to answer "how many?" questions about as many as 20 things arranged in a line, a rectangular array, or a circle, or as many as 10 things in a scattered configuration. (CCSS: K.CC.5) Given a number from 1–20, count out that many objects. (CCSS: K.CC.5) | 677.169 | 1 |
If after watching the first video you still need more instruction you can also watch these other videos about logarithms. Do the practice problems online for practice. You can also read the instruction pages in your text book. khan academy logarithms logarithm properties
This will be a good review for the final exam which is coming up in Jan.
Day 2
On your computer go to student.desmos.com and type in the following code: H5WGW and do the activities given. I will be able to see your answers on my end.
AP CALC- Semester 2
Semester 2
Day 1
Go to the college board website, or use the link below to find the free response questions for 2012 and work through the solutions. You may contact each other to discuss how to do the problems or ask questions on this blog and I will give you hints as needed. I would expect that you all could do "part a" on all 6 questions. I hope that you could get most of the middle parts. I will be very impressed with you if you get the last part to any of the questions.
Here is a link to the problems: 2012 free response questions
It would be helpful if you timed yourself on the problems. On the real test in May you will have 90 min to do 6 problems. I would suggest doing the first parts to each question first and then going back to tackle the harder parts. Have fun!
BB Day 2
Blizzard Bag day 2:
Read through the course description and all the guidelines. Do all the practice multiple choice questions beginning on page 18. The first section is no calculator and the second section you will need your calculator to solve some of the questions. Show all the work that leads to your answer and/or write down what questions you have about how to solve the problems. Please indicate which problems you needed the calculator for. We will discuss the concepts and strategies on how to get the most mult. choice points in class. | 677.169 | 1 |
Version ???: Where Are We?
After having Mr. K as my math teacher since my first semester of grade ten, it was different not having him around in math class for two months. During his absence, we had a revolving door of teachers teaching our class; we were exposed to different teaching styles.
With one teacher, we studied directly from the textbook, using the examples it provided. Although reading the textbook is good, I thought we should have used our class time doing questions from Mr. K's old slides from last year instead of reading our textbook, because I can read my textbook at home instead.
A lot of the content in chapters 2 and 3 were weighted heavily on our calculators and graphing, especially graphing the derivative. As we ventured on into learning more about the derivative, we found out that we can find the derivative of a derivative, the second derivative (and the derivative of the derivative of a derivative, the third derivative, and so on). My muddiest point in learning must have been giving an answer to a question that asks for information when given a second derivative and I have to find something about the original function, or vice-versa, because of taking the "extra steps" involved in obtaining the answer.
On the other hand, chapter 3 was about the integral. Using my knowledge of grade 11 physics, studying the integral was a breeze (compared to studying the derivative).
So to help answer Mr. K's question, "where are we?" or more specifically, "where am I in my learning?" a review of the relationships between a function and its second derivative would be nice. | 677.169 | 1 |
Book Description This series of ten books and two teachers handbooks for senior high
students presents calculus as part of real worl learning. A feature of the
series is the use of computer graphing utilities to compare graphs of
many different functions and help students to recognize patterns and form
generalizations. | 677.169 | 1 |
Mathematical Applications on the SAT
The College Board emphasizes that the Mathematics section on the new SAT is intended to test especially the mathematical knowledge that will be relevant for a broad range of careers—not only the mathy professions like accounting, statistics, or chemistry—as well as for the needs of daily life. Mathematics for the non-mathematicians, in other words.
This shift away from abstract or specialized mathematics towards mathematical applications is evident in the many questions that draw on real-world scenarios to frame their problems, from household finances and personal loans to off-the-cuff grocery-store calculations, simple estimates of productivity, and even computation of the volume of text messages sent and received (the last could be handy for figuring out your cellphone bill!). While there remain more "traditional" math problems testing, for example, parabolic graphs and systems of inequalities, these are now a shrinking portion of the material.
In a slightly more abstract way, the shift in emphasis can also be seen in the form that the questions take. Rarely do the mathematical problems that we face in the world come to us in the form of ready-made equations. On the contrary, we have to devise the mathematical equations ourselves by selecting the variables and operations that are appropriate for a particular situation. In most cases, these real-world equations will be embedded in a broader explanatory context that uses ordinary English in addition to, or instead of, the symbolic language of mathematics. This context shows us why we are using mathematics and what problem it will help us solve.
When problems that mix mathematics and ordinary language show up on tests, they are usually called "word problems." In dealing with such problems, it can be as important to determine the significance of the information you're given and the meaning of the results as to perform the algebraic or geometric calculations themselves. It's not so much math for its own sake, as math for the sake of solving a concrete problem. My goal in this post is to provide a brief introduction to some of the forms that word problems take on the SAT, and how you can start coming to grips with them.
Words
The simplest word problems (although not always the easiest) are those that are just a mixture of mathematical equations and English statements. Consider the following example:
You are given at first two mathematical equations and a chunk of text. In all word problems, the first thing you should do is read the text carefully. Treat the word problem as though it were a passage on the reading section: underline, circle or otherwise mark significant information, which will likely include explanation of variables and a statement of your task.
Here, we are first told what the three variables (b, c, x) given in the equations represent: the price per pound of beef is b, of chicken c, and the number of months after July 1st is x. We are then given the problem: "What was the price per pound of beef when it was equal to the price per pound of chicken?"
The next step in tackling these kinds of word problems is to translate the ordinary language into math. The SAT will always give you enough information on word problems to be able to change the English expressions into mathematical expressions. By substituting variables and operations for English expressions, we turn word problems into recognizable algebraic or geometric equations.
If we perform this substitution for our question, "What was the price per pound of beef when it was equal to the price per pound of chicken?", we come up with, "What was b when b = c?" The relevance of the two equations given at the beginning of the problem should now pop out. We need to set them equal to each other, determine the value of x, and then plug the value of x back into the equation for b in order to find our answer. This is simple algebra that you can easily handle.
Of course, one of the difficulties of word problems is mastering the technique of translating English into math and vice versa. You will know much of it already from your math classes, but this is where practice with SAT sample questions will help you become familiar with the language of the test. Your tutor will also be able to give you some of the basic equivalences.
Charts and Tables
Some word problems will utilize charts or tables to visually represent information. Take this example:
It is useful to adopt a modified strategy for tackling word problems when you have a chart or table at hand. In these cases, do not begin by reading through the preliminary information and by trying to decipher the chart, but instead immediately determine the task. The reason for this is that it will be difficult to evaluate the significance of the information you are provided until you know what to look for. If you know what you need from the table, you can go directly to what is relevant for your purposes. We are told here that we need to find a "reasonable approximation" of the number of earthworms 5cm under the surface of earth in the entire plot.
Once you have determined your task, go back and read the text that you skipped while going to the task. It may contain important information that will help you understand the variables and presentation of the chart, and it is never safe to ignore it. This problem is a case in point. We learn that the students marked off 10 "randomly selected," non-overlapping regions of the field measuring 1m x 1m each. Moreover, we learn that the entire plot is 10m x 10m. This information is vital, for, once we translate the English into math, it allows us to ascertain that the randomly selected regions account for only 10% of the size of the entire plot. (1m x 1m = 1m2; there are ten such plots, thus 10 x 1m2 = 10m2; whereas the plot is 10m x 10m = 100m2; finally, 100m2 /10m2 = 10.) Therefore, we cannot merely sum the values given in the chart to determine our answer, but must rather multiply that figure by 10 in order to find an approximation for the entire field. This little trick is what makes the problem challenging.
Finally, look at the chart itself, using the task to guide your eye. We are looking for a "reasonable approximation" of the number of earthworms 5cm under the surface of earth in the entire plot. We have already hinted at the answer above: since the preliminary information lets us know that the plots do not overlap and that altogether they are 10% of the total area, we need only find their sum and then multiple by 10. A rough approximation of the number of earthworms in the 10 plots is 1500. Multiplying by 10 we get 15000, which happens to be the correct answer, C.
Graphs
Lastly, word problems can also incorporate graphs of all sorts. Here is an example:
Just as for charts and tables, do not start with the graph, but instead look for the task. In this case, our task is contained in problem #15, which asks what the C-intercept in the graphs represents.
From here, we can work our way back to the graph and the other information provided by the problem. Reading the labels for our graph proves to offer almost all of the information that we need to solve our problem. We see that C is the cost renting a boat for h hours. Answer choices B and C are immediately off the table, then, since they deal with numbers of boats and hours, not cost. A and D are better, but a moment of reflection will show that the intercept here must be a single value, not a relationship. A is the correct answer then. D is a potential pitfall for those who read the problem quickly and see the problem "What does C represent in the graph?" instead of "What does the C-intercept represent in the graph?"
Summing up
Word problems can be challenging for a number of reasons, and each one demands a slightly different approach. But if you treat each one as a self-contained reading problem and master the art of translating English into math, you will also master word problems on the SAT mathematics sections. | 677.169 | 1 |
Weber, Keith
Abstract [en]
How the concept of "key idea" can be used in high school geometry. Article discusses connecting students' intuitive informal arguments with rigorous formal mathematical proofs. Includes three examples. | 677.169 | 1 |
April 1, 2016
Learning Calculus by following a simple model of learning
Many learners find it difficult to learn a subject or anything that they want to learn. The difficulties come from the fact that people have always thought that in order to learn something somebody has to teach it in the first place. Learning doesn't always come from someone else. One can learn by oneself. In fact learning happens throughout life mostly in the informal way. Life would be impossible without learning. Learning happens explicitly after birth. Babies learn to cry to get fed. This is a natural process of a simple stimulus-response conditioning. A natural stimulus is used in order to get a response. The baby cry is a natural stimulus to get a response which is food. Learning viewed this way is a change of behavior. Later comes complex stimulus-response conditioning. The complex stimulus-response conditioning is known as classical conditioning of Pavlov. In complex stimulus-response conditioning a second stimulus is introduced, which stimulus is neutral. Dog naturally salivate when they see meat but Pavlov was able to teach a dog to salivate at the sound of a bell by associating the sound of a bell to the presentation of the meat to the dog. By repeating several times the association meat with the sound of a bell the dog learns to salivate when the bell rings. This process of conditioned learning has been used by humans to live and to create different structures in society.
Learning happens whether we want it or not. In order to learn more complex things ways of learning are necessary. One cannot depend exclusively one someone else to learn as if this person isn't present learning cannot take place. A teacher doesn't force learning to take place. He facilitates and creates conditions for learning. This starts by believing that you can learn. Then you learn the study skills and habits. You need to know the theories, rules and processes in order to learn math.
Mathematics play an important role in human activities. They are used from simple everyday activities such as personal budgeting, checkbook balancing, groceries shopping to more complicated disciplines such as Economy, Science, Computers, Engineering, etc. The buildings we live in the roads we use, the computers, cellphones, tablets, televisions, etc are designed by people who know math. Calculus is an important branch of mathematics used in various disciplines taught at the college level. The notions of limits are fundamental in understanding some very important notions in Calculus such as Continuity, Derivation and Integrals. I have designed two Calculus courses for learners taking AP Calculus or who will take it. If a student plans to take Calculus as their next math course it's good to start taking them now so that it doesn't look strange to them. They are also designed for students at the high school or college level who need a remediation course. The first one is a free Introductory Calculus course. The second is a complete Calculus course at an affordable price.
The instruction process for this course is designed in the following manner:
1. Students will watch an introductory video. The videos introduce the lessons to the learner
2. There will be some readings to do. The readings expose the learners to the theories of different topics.
3. There will be some problems completely solved. Students should master the solution process of these problems.
3. They will have to solve practice problems demonstrating an understanding of the topics.
Courses in Basic Algebra, Algebra I & II, Geometry, Trigonometry, Pre-Calculus and math for adults are also available. Other face-to-face and online courses in French and English to Speakers of other Languages are available on demand. Online and face-to-face tutoring are also available in these subjects.For more information visit New Direction Education Services at If you are interested in the 2 Calculus courses, click on the link at the end of this post. If this is not for you please share the link to people who might be interested. Here is the link: Free Introductory Calculus Course. Complete Calculus course | 677.169 | 1 |
Comment: Good condition. Fast shipping with Amazon FBA!! A treasure trove of information!!! Normal shelf wear to cover. Good copy with crisp white pages. May contain minor highlights or markings. Ships very quickly with Amazon Fulfillment!!! We stand behind everything we sell Art of Problem Solving, Volume 1, is the classic problem solving textbook used by many successful MATHCOUNTS programs, and have been an important building block for students who, like the authors, performed well enough on the American Mathematics Contest series to qualify for the Math Olympiad Summer Program which trains students for the United States International Math Olympiad team. Volume 1 is appropriate for students just beginning in math contests. MATHCOUNTS and novice high school students particularly have found it invaluable. Although the Art of Problem Solving is widely used by students preparing for mathematics competitions, the book is not just a collection of tricks. The emphasis on learning and understanding methods rather than memorizing formulas enables students to solve large classes of problems beyond those presented in the book. Speaking of problems, the Art of Problem Solving, Volume 1, contains over 500 examples and exercises culled from such contests as MATHCOUNTS, the Mandelbrot Competition, the AMC tests, and ARML. Full solutions (not just answers!) are available for all the problems in the solution manual.
Editorial Reviews
Review
Your book meets a great need for students getting into serious math competitions. It's readable, understandable, and comprehensive - a nice bridge into the AIME and the Olympiads. The kids and I love the books. --Joe Holbrook, Academy for the Advancement of Math/Science/TechnologyAbout the Author
Sandor Lehoczky participated in the Math Olympiad Summer Program in 1989, and in 1990 earned the sole perfect AIME score and led the national first place team on the AHSME (now AMC 12). Richard Rusczyk is the founder of the Art of Problem Solving website. He was a national MATHCOUNTS participant in 1985, a three-time participant in the Math Olympiad Summer Program, a perfect AIME scorer in 1989, and a USA Mathematical Olympiad winner. He is author or co-author of 6 Art of Problem Solving textbooks. Lehoczky and Rusczyk were co-founders of the Mandelbrot Competition and are board members of the Art of Problem Solving Foundation.
Top customer reviews
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I believe I've reviewed this book and vol. 1 in another area but amazon doesn't agree. this book is fantastic-particularly if you need a review of "basic" concepts. We found it when looking for a review book for the college testing but I notice it continues to have its place among college textbooks.
Excellent book for my 6-th-grade kid. It covers very advanced concepts and problems in Pre-algebra (and some beyond definitely), excluding geometry part, by including many basic examples and advanced competitive math problems such as AMC and AIME.
I originally bought this book just to help me learn a few math techniques for school and math competitions, but soon realized that it was much more. Honestly, I would have to say that this book is really beneficial to almost anyone who wants to improve in mathematics. Not only are the theorems the book teaches valuable, but the problem solving skills that you learn through reading and working on problems is amazing!
I bought this book a few months back for my son. Its a good book to have. However, many topics are beyond what's covered in middle school and hence requires a really motivated student to appreciate it. If you or your kid is serious and interested about math, I would recommend it. | 677.169 | 1 |
FUNDAMENTALS OF MATHEMATICS
With the beginning of the course, I experienced certain complexities in understanding the concepts and ideas of mathematics, which made it difficult for me to carry on the course. Mathematics is not a new subject for me, as I have been familiar with it since first grade. By coping up the conceptual problem associated with mathematical equations, I managed to understand the course. The accomplishments that I have achieved during this course are the understanding and application of mathematical equations in real life. The exercises and practices enabled me to repeat the concepts and memorize how and why specific sums are solved (Setek& Gallo, 2005).
The topics or concepts that have been a struggle so far in this course mainly include number system, sets, exponents, order and operations. Throughout the course, I found the above-mentioned mathematical topics and ideas as complex in nature that hinders in developing an understanding with them. The number system was complex due to involvement of distinct nature of numbers that are written to express the equation. Similarly, solving sets had been rather easy than applying it in real life. The difficulty faced in understanding exponents, order and operation system were based on their rules. I did not focus on memorizing and comprehending the rules involved in these ideas (Dyke, Rogers,& Adams, 2011).
In order to overcome the struggle, one action step I can take is to focus on the logic behind the mathematical equation. Fundamental nature of complexity I experienced in this course was due to the fact that I did not focus on the rules, as I find it quite difficult. Therefore, by focusing on the logic of each concept, I can overcome the struggle (Schröder, 2010). | 677.169 | 1 |
HS Performance Tasks
Students will be asked to show equivalent expressions in function form. They will use those equivalent expressions to determine information about the context of the problem. Additionally, they will determine when it is best to use a certain function form for specific reasons. Finally, students will have an opportunity to write equivalent expressions.
Students will write a system of equations to solve a problem. Using those equations, students will figure out total numbers of items sold. Finally, students will use prior knowledge of percentages to calculate profits and make a recommendation for the selling price of an item.
Students will be asked to graph points to determine if a function is linear or non-linear. Once a graph is drawn, students will be asked to interpret the graph and make predictions from it. Additionally, students will be asked to use information from a table to determine characteristics of the graph.
Students will practice writing exponential functions in this task. Once their function is written, students are tasked with using the function to evaluate given certain numbers. Finally, students are asked to use their function to determine the validity of numbers.
Students will be asked to write a linear equation. They will then use that linear equation to find a point of intersection with a quadratic equation. Different forms of a quadratic will be presented and students will be asked to show their equivalency and find key information from the different forms.
Students will use trigonometry and properties of circles and triangles to find distances. Additionally, students will be able to practice with the Pythagorean theorem. Lastly, students will use a drawing to make a conclusion about the shortest distance between points.
Students will be asked to calculate the probability of simple and compound events. Probabilities will be given and students will be asked to explain if those probabilities are correct or incorrect. Finally, students will be asked to use their knowledge of probability to make a recommendation regarding a game.
Students are presented with information about the dimensions of an actual Ferris wheel. They must come up with a model for the height of a cart on the Ferris wheel at particular times. The task requires students to use a function that they create to determine the height of a person at particular times.
Students will be asked to find the volume of spheres and cylinders. Using the volumes, students will determine whether or not items will fit into containers. Finally, students will reason about the volume of other containers based on their knowledge of given containers.
Students will investigate number patterns. They will write equations or rules from those number patterns. Using the equations they created, students will determine the number of shapes in a pattern. Finally, students will create their own pattern given certain parameters. | 677.169 | 1 |
Maths and Further Maths
Overview of the Course
Those interested in Maths can study A Level Further Mathematics alongside A Level Mathematics. This course is suitable for keen and able mathematicians and is particularly suited to those who are confident that they want to pursue a career in Mathematics, Physics, Engineering, Computer Science or related disciplines.
The A Level Further Mathematics Study Programme builds on the topics covered in A Level Mathematics as well as introducing new and challenging ideas and concepts. As with A Level Mathematics there is a large emphasis on Pure Maths with the remaining exams being optional topics chosen from Pure, Statistics, Mechanics or Decision Maths. Decision Maths includes topics relevant to Computing, Business and Economics.
What are the career opportunities?
There really is no "typical job" that maths graduates progress on to. Mathematics is more of a way of thinking, or a set of tools, than a specific learned skill. However, a mathematician's logical, problem-solving and numerical skills are highly sought after in many different areas of employment.
Mathematics is an excellent qualification to have for a wide variety of higher education courses and careers. It is vital if you wish to study Mathematics or Engineering at university and is advisable for Economics, Business, Banking, Accountancy, Design, Engineering, Statistics, Computing, IT and Sciences. If you study mathematics your skills will also be welcomed for Medicine, Dentistry, Veterinary Science, Teaching, Technology and Architecture including GCSE maths grade 9-7
(A*-A).Please note: students applying for Further Mathematics must also apply for
mathematics.
How long is the course and when can I start?
The A Level is a full-time two-year programme, starting in September, with the opportunity to take AS after one year.
How is the course structured and how will I be assessed?
You will be assessed with two exams at the end of year one for the AS qualification, and three exams at the end of year two. Each exam will be externally set and contain elements from the pure and applied sections of the course.
What employability skills will I develop on this course?
Employers value mathematics graduates or those with qualifications beyond GCSE level because they assume correctly that such people can use numeracy skills and handle data effectively to demonstrate a case, that they can show initiative in problem solving and that they can work independently and as part of a team in carrying out their work.
Throughout the course, you will be encouraged to solve problems by breaking them down into smaller parts and to identify key components of solutions. Work on statistics provides the opportunity for you to become adept in producing evidence to support or argue against a point. This will include the use of technology and spreadsheets to assist in your solutions.
A qualified mathematician who is well informed and is able to work independently whilst also being able to communicate effectively and take on board the ideas of others will be an excellent asset to an employer.
English and communication skills will be continually developed as you will be involved in discussions and must be able to explain your reasoning and assumptions in a structured and coherent manner to support your calculations.
Work experience
Work experience is an essential element of your Study Programme and you will be supported to secure a five day work experience placement Enrichment includes guest speakers and trips/visits.
You will have the opportunity to take part in competitions such as the BEBRAS Challenge and the UKMT Individual and Team Challenges. For students that are more able, there is also the opportunity to take part in the Senior Kangaroo challenge.
Are there any additional costs or requirements?
A graphical calculator that has the ability to store and retrieve statistical tables of information and draw graphs will be required. | 677.169 | 1 |
Middle School Math Courses - PowerPoint PPT Presentation
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PowerPoint Slideshow about 'Middle School Math Courses' - althe wishing to take algebra in middle school do NOT need to seek acceleration between 5th and 6th grade. It is our recommendation that only students wishing to take geometry (and those willing to commit to 6 years of HS math prior to graduation) should attempt to take the 6th grade CBE.
Geometry is taught in a non-traditional manner (on-line environment) at the middle school level. Parents should be aware of this when choosing to accelerate between 5th and 6th grade
Students enrolled in the PAP track in middle school will automatically be given the chance to accelerate between 7th and 8th grade allowing them to take Algebra as 8th graders | 677.169 | 1 |
Program Description
The Summer Mathematics Program for High School Students is an intensive four-week program in number
theory for motivated high school students. Because the integers are such a
fundamental mathematical object, for thousands of years mathematicians
from all over the world have studied number-theoretic questions;
nevertheless, the subject is rich enough that it is studied today more
actively than ever. It moreover has essential contemporary applications to
cryptography, the science of sending and deciphering secret messages.
Participants will work closely throughout the day with one another and
with the program staff (the director and three graduate students),
exploring a series of challenging problem sets. Students will be asked to
experiment with the subject, formulating conjectures about what they
believe to be true, and then to justify their claims rigorously. Each
student will, with the assistance of the counselors, substantially develop
her or his ability to communicate mathematical ideas and arguments
precisely.
Each day the students and the staff will share a lunch break in which they
get to know one another better and continue more informally their
mathematical discussions. In addition to the core number theory course,
once each week a member of the Utah mathematics department will run an
afternoon program devoted to other areas of mathematics, both theoretical
and applied, so that students can catch a glimpse of the staggering range
of the contemporary mathematical landscape. | 677.169 | 1 |
This is one of those books that you can actually learn math from without trying too hard. The explanations were clear enough that I understood most of the material even though I did not do any of the exercises. I feel like I got a fairly thorough introduction to the mathematical theory of knots. The […]
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I didn't get much out of this book. I was hoping for an elementary explanation of the underlaying principles of thermodynamics, but found only a lot of confusing derivations of formulas. What's worse is that they are all in the pre-quantum-reformation form, which actually makes them more confusing. I don't think anyone needs to read […]
Read more from the Science category. If you would like to leave a comment, click here: Comment. or stay up to date with this post via RSS from your site.
This book complemented Adams' book well. Many of the concepts that I was confused about were explained with more sophistication, which actually made them easier to understand because there was no ambiguity. Although this book might be too dense for a first introduction to knot theory, it is a very well written book, and contains […]
Read more from the Mathematics category. If you would like to leave a comment, click here: Comment. or stay up to date with this post via RSS from your site.
This book is a good source of information for setting up the Linux software of an embedded system. It does not give a lot of explanation of embedded systems or using the software. Because of its focus, it is not that enjoyable to read and it isn't a very thorough introduction to embedded system design, […]
Read more from the Computers and Technology category. If you would like to leave a comment, click here: Comment. or stay up to date with this post via RSS from your site.
I think that this book is not quite an introduction. It claims that you only need a knowledge of basic quantum mechanics, but I felt lost for large portions of it. One chapter was actually worth reading: "Quantum Computation: An Introduction". This chapter explains how quantum computing algorithms work. Also of interest was page 20 […]
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Although the content was at a lower level than I was hoping for, this book did provide a relaxing and educational overview of quantum electrodynamics. It explains how light reflects based on quantum wavefunctions and how this causes irridescence. It spends a lot of time explaining the concept that the paths of particles are actually […]
Read more from the ScienceThis book is basically a translation of "The Starry Messenger", "Letters on Sunspots", "Letter to the Grand Duchess Christina", and "The Assayer" with historical explanations between them. It is really interesting to sense the opposition the Galileo faced from the Church and the legacy of Aristotle. It is also interesting to notice how science was […]
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This is a collection of true anecdotal stories from Wall Street that the author knew about. The point of the book is that pretty much all of the big winners in the stock market are the inside traders or people who are in special positions. It really portrays the stock market as a way for […]
Read more from the Economics and Finance category. If you would like to leave a comment, click here: Comment. or stay up to date with this post via RSS from your site.
This is a book on an advanced subject of physics written by a philosopher. It is surprisingly advanced in terms of physics content, I don't know how a philosopher learned this much QFT, but I still don't like it. It is funny how he spends a whole chapter talking about "primitive thisness", which is basically […]
Read more from the Science category. If you would like to leave a comment, click here: Comment. or stay up to date with this post via RSS from your site. | 677.169 | 1 |
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Synopses & Reviews
Publisher Comments
This manual, by Reva Narasimhan, provides Excel information, including step-by-step examples and sample exercises, for finite math and applied caclulus topics. No prior knowledge of Excel is necessary. In edition, this manual references a number of exercises from the Harshbarger/Reynolds text that would work well with Excel.
Synopsis
This resource provides a brief introduction to Excel and specialized, step-by-step instructions on how to use Excel to explore calculus concepts.
About the Author
Ron Harshbarger and Jim Reynolds have worked together as co-authors on MATHEMATICAL APPLICATIONS, Nineth Edition since the book's inception. They have both taught for over 20 years, at all levels of undergraduate mathematics. Harshbarger is a professor at the University of South Carolina, and Reynolds is a professor at Clarion University in Pennsylvania. | 677.169 | 1 |
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THE DEVELOPMENTAL MATH PROGRAM
Located in the Learning Resource Center, the Developmental Math Program is designed for students who desire to improve their mathematics skills in Arithmetic, Pre-algebra, Algebra, and Geometry.
Courses:
Mathematics G010/G010L – 4 Units Elementary Algebra
Prerequisites: Mathematics G008 or Mathematics Placement Assessment
This course is equivalent to a first-year high school algebra course. The topics covered include properties of real numbers, simplifying polynomial, rational, and radical expressions, and solving linear, quadratic, rational, and radical equations in one variable. The rectangular coordinate system is covered including graphing linear equations in two variables and solving systems of linear equations in two variables. Applications of mathematical concepts are incorporated throughout the course. This course is taught in a combined large lecture and laboratory format. Lecture & lab. Letter grade only. Not transferable, not degree applicable. | 677.169 | 1 |
MATH ALGEBRA 1 Advice
Showing 1 to 2 of 2
In this course I learned the most about Math in general and I would highly recommend it to every student in order to persevere in the study of math. Algebra 1 for me was the building blocks of Math.
Course highlights:
I learned a lot about polynomials, that will stay with you all the way through College. One of the main things that I learned is how to be able to break down a solve a basic algebraic equation.
Hours per week:
3-5 hours
Advice for students:
My advise to every student that takes this course is to study at least every night. Without any studying one should expect a least a B. Even if you know all the material, study for at least 10 minutes a night.
Course Term:Fall 2013
Professor:mrs. Roger
Course Required?Yes
Course Tags:Great Intro to the SubjectMany Small AssignmentsGroup Projects
Jan 19, 2017
| Would highly recommend.
Not too easy. Not too difficult.
Course Overview:
I would recommend this course to upcoming students coming from middle school and are on there way to high school.
Course highlights:
I learned about inequalities. That means that two equations that aren't equal, you use <,>.
Hours per week:
3-5 hours
Advice for students:
To study more and pay attention in class. The main key is to stay on task and you will be fine. | 677.169 | 1 |
Highlights of Calculus
Highlights of Calculus is a series of videos that introduce the fundamental concepts of calculus to both high school and college students. Renowned mathematics professor, Gilbert Strang, will guide students through a number of calculus topics to help them understand why calculus is relevant and important to understand. | 677.169 | 1 |
Modeling and Simulation in Science and Mathematics Education: Macintosh/Windows Version(Hardback)
Synopsis
The role of simulation modeling in understanding dynamic processes is now extending beyond research and university curricula to pre-college education. Computer modeling offers the promise of transforming teaching of many subjects, notably science, and is enhanced by the adoption of new standards for science education, the increasing access to scientific information through computer networks, and the availability of powerful, user-friendly modeling software. This book and its accompanying software will bring the tools and excitement of modeling to pre-college teachers, to researchers involved in curriculum development, and to software developers interested in the pre-college market | 677.169 | 1 |
A Blog Devoted to Encouraging Homeschooling Mothers
The Burts in 2013
Sunday, September 9, 2012
Schoolhouse Review Crew ~ Math 911
I take homeschooling seriously ~ really seriously. A subject of enormous importance to me is math, especially at the high-school, college preparatory level. It's nothing to mess around with, avoid, or ignore. Every single high school student, home educated or not, needs a solid four years of high school math. Umm, did I mention that I take this seriously?!!?
You've heard me say this before, and I will say it again because it bears pretty heavily on my review of Math911; my son attends Wheaton College, a Christian liberal arts college that is ranked as an Ivy League school. For my son to even be considered for acceptance as a homeschooler, he had to take the ACT, SAT and two SAT II subject tests. One of those subject tests was in college math. In the end, his hard work paid off. Not only was he accepted into Wheaton, but he was one of a handful of freshmen who did not have to spend a few hours of their first day on campus taking an extensive math placement test. So as I have been using Math911 with my three high school aged daughters, I have been considering it's value in relation to preparing homeschoolers for college.
I have tested this math program, and I can honestly say it has NOT been found wanting! Created by Professor Martin Weissman, a math teacher in his 49th year, Math911 teaches math mastery in a way that isn't discouraging. While the graphics may seem a little dated to the average teenager, my children considered it's academic value to be more important.
Here's an overview of how Math911 works. The program itself is a download, and when you visit the website ( you first access the free Standard Version (yes, I said FREE!). This is a complete Introductory Algebra course. You can also purchase the Premier Version (which is what I received). This version includes Introductory Algebra, Intermediate Algebra, College Algebra, Statistics, Trigonometry, and Pre-Calculus. It's all included in one price ($49.95 - but keep reading because there is discount information later in my review!) and it includes free lifetime updates, free tech support, access to additional math courses with separate grade reports for multiple users, and even allows mom and dad to use Math911 to refresh their math skills. Access to the Premier Version is completed by entering a few codes that will be emailed to you upon purchase. So the first step is to download the free Standard Version, try that out and see how your teens do using it, and if you choose to purchase the Premier Version then you will enter the access codes by clicking the "Upgrade" button. The program is easy to use by clicking the desktop icon, and my daughter's are able to access their individual levels on their own.
I'm sure you can tell that I appreciate that this is a quality program. I also appreciate that it's ease of use, the way it reinforces the positive and down-plays the negative, and the customer support is fast. There is a personal feel with this company, which I love. The goal truly seems to be to help young people master the scary subject of math, and Mr. Weissman's love of the subject shows in his work.
Math doesn't have to be burdensome, difficult or despised. For students who think they'll "just never get it" Math911 offers an encouraging solution. For parents who wonder if they can teach high school math well enough to help their students get into college, Math911 provides the solution they need as well. I can't say there was much of anything that I disliked about this program, and the minor one or two hiccups I found in using it were ridiculously minor as compared to adjustments I've made with curriculum during my 14-years of homeschooling.
(Disclaimer ~ I received this product free of charge for review purposes. All opinions shared here are solely my own.) | 677.169 | 1 |
Logarithmic Equations and Inequalities Study Guide
Speed Dating with Logarithms
Students solve exponential and logarithmic equations with their 'date' in this interactive and self-checking speed dating activity. Plan your 60 minutes lesson in Math or solving equations with helpful tips from Tiffany Dawdy
Exponential functions & Finance -- Episode where Fry goes to check his bank balance and because it has been years his initial balance has grown exponentially. Use to demonstrate exponential functions
A logarithm is the power in which the number raised in order to get the other number. Logarithm and exponents are opposite to each other. The logarithm is denoted by log_a (y) = x. There are many rules and properties of the logarithm.
How to Simplify Logarithmic Functions - (21 Amazing Examples!)
Logarithmic Functions Simplified - TERRIFIC video lesson that covers the properties for both common and natural logarithms, how to expand and condense a logarithm, and finally walks you through the change of base formula. Perfect for new teachers as a review if it's been awhile since you taught this topic. Excellent for high school and middle school math courses. Check it out today! #homeschooling #teaching | 677.169 | 1 |
The description of Calculus 2: Practice & Prep
Have you already taken Calculus 1 and now you are moving on to Calculus 2? Are you unsure of your potential performance, and perhaps Calculus 1 may have been a struggle for you? You are not alone, and the Varsity Tutors Calculus 2 app for Android devices is here to help.
Many students state that Calculus 1 was challenging and difficult to learn, so going on to Calculus 2 can be a frightening experience. It does not have to be like that! With a little practice and help from our Varsity Tutors Calculus 2 app for Android, you can work to improve your understanding of the subject. The app offers free Calculus 2 resources such as practice tests, diagnostic tests, flashcards, and so much more to help build your confidence in the subject. Our app for Android-powered smartphones and tablets enables you to study on your own time and pick what areas you want to cover.
You will find that most Calculus 2 courses start with a light review of what you should have learned in Calculus 1. They will then briskly move on to arc length, convergent series, exponential growth, harmonic series, Maclaurin Series, power series, radius of convergence, ratio test, surface of revolution, and Taylor Series. This is where the Varsity Tutors app becomes your best friend.
Start your Varsity Tutors' experience by taking one of our Calculus 2 diagnostic tests. The diagnostic tests determine which concepts you are understanding, and what you need to pay closer attention to. Highlighting how you performed in each area of Calculus 2 will provide you with a visual result of where your knowledge and skills are at the moment. Throughout the course and as you review the concepts of Calculus 2, take the diagnostic test to keep your study plan up-to-date with your areas of focus. Whether you make use of the Calculus 2 flashcards to help memorize key elements or words, or you are in need of practice tests to see how to build a pace strategy for an exam, the Varsity Tutors app for Calculus 2 is a great resource. | 677.169 | 1 |
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