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Upon completion of the course the student is able to: 1. Write logical mathematical argumentation. 2. Use the key concepts and techniques from algebra. 3. Analyze the properties of functions of one variable. 4. Solve optimization problems for functions of one variable. 5. Differentiate functions of several variables. 6. Use the basic rules of integral calculus. 7. Understand and apply the basics of matrix algebra and vector algebra.
Omschrijving
The objective of this course is to learn the mathematical tools necessary to understand topics covered within the minor Finance. The course begins with topics in algebra and solving equations. Students learn the techniques of differentiation and integration and learn to work with interest rates and present values. The course also aims to make students sufficiently familiar with matrix algebra and calculus as preparation for the course Econometrics for Minor Finance. | 677.169 | 1 |
Top 50 Math Skills for GED Success is a fast course for the GED Mathematics Test!
The text is designed to present the top 50 math skills as a review and practice for students. Each skill is targeted in a single lesson and contains instruction and practice. The text contains a pre-test and a post-test to assess student's skills deficiencies and areas of strength.
Pretest
The
Lesson Structure
Each
Posttest
The posttest in Top 50 Math Skills for GED Success is formatted just like the GED Test, and is designed to assess student's
Computation Review Section
Students can use the Computation Review section for a quick review of the basic math skills important for GED success. Each math skill is presented with an explanation, examples, and brief practice problems.
Other Features
Top 50 Math Skills for GED Success contains a detailed Annotated Answer Key, which not only shows the correct answer for each problem in the text but a step-by-step illustration of how the answer was derived.
Casio fx-260 instructions are included with an illustration of the calculator face that shows which functions are used on the GED Test. Instructions on using the basic functions and step-by-step examples using the calculator for certain functions are included.
An
Top Education, United States, 2004. Paperback. Estado de conservación: New. Annotated edition. Language: English . This book usually ship within 10-15 business days and we will endeavor to dispatch orders quicker than this where possible. Brand New Book. Top 50 Math Skills for GED Success is a fast course for the GED Mathematics Test! The text is designed to present the top 50 math skills as a review and practice for students. Each skill is targeted in a single lesson and contains instruction and practice. The text contains a pre-test and a post-test to assess student s skills deficiencies and areas of strength. Pretest The Lesson Structure Each Posttest The posttest in Top 50 Math Skills for GED Success is formatted just like the GED Test, and is designed to assess student s Computation Review Section Students can use the Computation Review section for a quick review of the basic math skills important for GED success. Each math skill is presented with an explanation, examples, and brief practice problems. Other Features Top 50 Math Skills for GED Success contains a detailed Annotated Answer Key, which not only shows the correct answer for each problem in the text but a step-by-step illustration of how the answer was derived. Casio fx-260 instructions are included with an illustration of the calculator face that shows which functions are used on the GED Test. Instructions on using the basic functions and step-by-step examples using the calculator for certain functions are included. An Top Nº de ref. de la librería BTE9780072973839
Descripción 2004. Paperback. Estado de conservación: New. Paperback. "Top 50 Math Skills for GED Success" is a fast course for the GED Mathematics TestThe text is designed to present the top 50 math skills as a review and practice for.Shipping may be from our Sydney, NSW warehouse or from our UK or US warehouse, depending on stock availability. 0.363. Nº de ref. de la librería 9780072973839 | 677.169 | 1 |
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Written as much for teachers as it was for students, this 61 page book explains how to use the TI-83/TI-84 Plus with step by step tutorials on many of the common topics covered in Secondary Mathematics courses.
The sections can be printed off separately allowing teachers to use these tutorials in parallel with the math topics covered in the classroom.
Each of the 19 sections contain specific examples and work through them showing the proper syntax and graphics as they apply to the TI-83/TI-84 Plus.
The topics covered include Functions, Polynomial Functions, Quadratic Relations and Systems of Equations, Exponential and Logarithmic Functions, and Sequences and Series.
There is also an introduction to programming on the TI stepping students through how to create two simple but useful programs.
A table of contents and a helpful hints section are also provided in the book. | 677.169 | 1 |
"A pleasant arithmetic pageant" tells the tale of the Indiana collage arithmetic pageant (ICMC) through featuring the issues, options, and result of the 1st 35 years of the ICMC. The ICMC was once geared up in response to the Putnam examination - its difficulties have been to be extra consultant of the undergraduate curriculum, and scholars may well paintings on them in groups.
Originally participation used to be initially limited to the small, deepest faculties and universities of the country, yet used to be later spread out to scholars from all the faculties in Indiana. the contest was once quick nicknamed the "Friendly" pageant due to its concentrate on fixing mathematical difficulties, which introduced college and scholars jointly, instead of at the aggressive nature of profitable. equipped by means of 12 months, the issues and recommendations during this quantity current an exceptional archive of data approximately what has been anticipated of an undergraduate arithmetic significant during the last 35 years. With greater than 245 difficulties and strategies, the booklet can be a needs to purchase for school and scholars attracted to problem-solving.
Bob Miller's humor-laced, step by step studying assistance make even the main tough math difficulties regimen. in keeping with greater than 28 years of educating and scholar suggestions, his easy-to-grasp innovations supply scholars much-needed self belief.
This publication introduces the math that helps complex machine programming and the research of algorithms. the first goal of its recognized authors is to supply a great and correct base of mathematical abilities - the abilities had to resolve advanced difficulties, to judge horrendous sums, and to find refined styles in info.
This article addresses the necessity for a brand new arithmetic textual content for careers utilizing electronic know-how. the cloth is delivered to lifestyles via a number of purposes together with the maths of reveal and printer monitors. The direction, which covers binary mathematics to Boolean algebra, is rising through the kingdom and should fill a necessity at your tuition.
Observe: this can be a STAND by myself ebook. entry CODE isn't incorporated WITH THIS publication utilizing and figuring out arithmetic: A Quantitative Reasoning process prepares scholars for the math they are going to come across in university classes, their destiny occupation, and existence mostly. Its quantitative reasoning procedure is helping scholars to construct the talents had to comprehend significant concerns in daily life, and compels scholars to obtain the problem-solving instruments that they are going to have to imagine significantly approximately quantitative concerns in modern society.
S1966-5 resolution 1: (In what follows, "sequence" refers to an ascending series of optimistic integers as within the challenge. ) permit S(n, okay) be the variety of assorted sequences of size ok within which each quantity from 1 to n happens once or more. Then S(n, okay) is just the variety of how one can partition a ok k−1 . enable point series into n non-empty subsets. as a result, S(n, okay) = n−1 T (n, ok, r) be the variety of various sequences of size okay within which purely r (out of n) specific numbers happen, then we've T (n, okay, r) = n S(r, okay) = r n r eventually, we see that the answer is n n T (n, ok, r) = r=1 r=1 n r ok −1 r−1 = k−1 . r−1 n+k−1 okay the place the latter equality is an invocation of the Vandermonde convolution. resolution 2: Make n + okay − 1 blanks and fill in ok of them with x's. For this type of association of x's, outline ai = 1 + the variety of clean to the left of the ith x 54 recommendations for every i from 1 to okay. this provides a 1-to-1 correspondence among the sequences we're attempting to count number and the methods of placing ok x's in the various n+k −1 blanks. however the variety of methods of doing the latter is clearly n+k−1 . ok glance less than Enumeration within the Index for comparable difficulties. S1966-6 √ resolution 1: allow f(x) = ax2 + b. we wish to convey that f(x) is a contraction, i. e. ∃ zero < C < 1 s. t. |f(x)−f(y)| ≤ C|x −y| for all x, y ∈ R. as a consequence, the series x1 = c, xn+1 = f(xn ) converges to the original fastened aspect of f. |f(x) − f(y)| = | ax2 + b − ay2 + b| √ = a | x2 + b/a − y2 + b/a| √ ≤ a |x − y|. √ to work out that | x2 + c − y2 + c| ≤ |x − y| for any c ≥ zero, Multiply and divide through the conjugate to get | x2 + c − |x2 − y2 | y2 + c| = √ | x2 + c + y2 + c| |x − y||x + y| = √ . | x2 + c + y2 + c| considering | x2 + c + y2 + c| ≥ |x| + |y| ≥ |x + y|, the inequality follows. √ because zero < a < 1, f is a contraction. The √ restrict of the series {xn } is just the fastened aspect of f. fixing x = ax2 + b yields limn→∞ xn = b/(1 − a). answer 2: After computing the 1st numerous phrases of the series, we see that xn+1 = = an c2 + b(1 + a + a2 + · · · + an−1 ) a n c2 + b 1 − an . 1−a 55 examination #2–1967 considering the fact that zero < a < 1, we could take the restrict giving b 1−0 = . 1−a 1−a glance less than restrict overview within the Index for related difficulties. lim (xn+1 ) = n→∞ zero · c2 + b S1966-7 suppose that the set {Ik } is minimum within the feel that not one of the durations is a formal subset of the union of a few others. Then we will order the durations by means of the worth in their left-endpoint, in order that Ii starts off to the left of Ii+1 for all i. Then we now have Ij and Ij+2 are disjoint for every j = 1, . . . , n − 2; differently Ij+1 ⊂ Ij Ij+2 , or Ij+2 ⊂ Ij+1 , both of which violate the minimality assumption. hence, the set of durations with peculiar subscripts involves jointly disjoint units, as does the set of periods with even subscripts. a minimum of this kind of units covers 0.5 of I, because their union covers all of I. S1966-8 decide on someone P . If P is a pal of okay other folks, then P is a stranger of five − ok people, and one among ok or five − okay has to be at the least three. | 677.169 | 1 |
Mathematics 12 is an academic high school mathematics
course that is offered over one semester. Students who select Mathematics 12
should have a solid understanding of the Mathematics 11curriculum. Course content
includes: finance, polynomial, exponential and sinusoidal functions, Set
theory, counting and probability, logical puzzles and games. Students will
also complete a math research project.
Relations and
Functions 35% RF01 Students will be expected to represent data, using polynomial
functions (of degree ≤ 3), to solve problems
RF02 Students will be expected to represent data, using exponential and
logarithmic functions, to solve problems.
RF03 Students will be expected to represent data, using sinusoidal functions
, to solve problems.
Finance 20% FM01 Students will be expected to solve problems that involve compound
interest in financial decision making.
FM02 Students will be expected to analyze costs and benefits of renting,
leasing and buying.
FM03 Students will be expected to analyze an investment portfolio in terms
of interest rate, rate of return, and total return.
Set Theory 15% LR02 Students will be expected to solve problems that involve the
application of set theory.
LR03 Students will be expected to solve problems that involve conditional
statements.
Counting and
Probability 20% P01 Students will be expected to interpret and assess the validity of
odds and probability statements.
P02 Students will be expected to solve problems that involve the probability
of mutually exclusive and nonmutually exclusive events.
P03 Students will be expected to solve problems that involve the probability
of two events.
P04 Students will be expected to solve problems that involve the fundamental
counting principle.
P05 Students will be expected to solve problems that involve permutations.
P06 Students will be expected to solve problems that involve combinations.
Math Research
Project / Logical Puzzles and Games 10% MRP01 Students will be expected to research and give a presentation on a
topic that involves the application of mathematics.
LR01 Students will be expected to analyze puzzles and games that involve
numerical and logical reasoning, using problem-solving strategies.
ASSESSMENT: Exam: The exam will be based on the entire year and is worth 20% of the
final grade. The exam will be written in June. Tests: Tests are major
assessments and will be announced several days in advance. Expect tests to
be cumulative throughout the year. Quizzes: Quizzes provide
opportunities to assess ongoing progress and give indications as to whether
extra help is needed. Quizzes are generally short and will be announced in
advance. Assignments: Assignments will be
given and graded on a regular basis. They may be in class or take home,
individual or group effort, as indicated by your teacher. Submissions will
not be accepted after the corrected assignments have been handed back to
students. Homework Probes: Homework probes
are unannounced short evaluations based on recent work. Students must be
sure to keep up with their work on a daily basis and come in for extra help
when needed.
Note: Marks throughout the semester will adhere to
these proportions:
Tests and
Quizzes 50%
Assignments and Homework Probes 50%
** Absence from an assessment will require a written
excuse from a parent or guardian upon your return to class. Failure to do so
will result in a mark of zero for that assessment. You will be expected to
make up missed tests. If you are absent from class, it is your
responsibility to catch up on any missed work. Check the class website, ask
classmates for notes, and see your teacher to get any handouts. Any
assignments due during your absence will be due at the beginning of class on
the day you return (provided a note excusing your absence has been presented
to your teacher). Graphing calculators will be provided for use during class
and on tests. | 677.169 | 1 |
NCERT Maths Class 10 solutions | DronStudy.com
NCERT books cowl the basics and basics on all topics of Physics, Chemistry and Biology
NCERT books are written and published through extraordinarily educated and honoured persons and Establishments as soon as an awesome analysis
NCERT books provide an explanation for all of the ideas in a very smooth and lucid manner. Thus, keeping in mind the vast importance and significance of the NCERT Textbooks for a student, DronStudy consultants have ready the chapter wise cbse sample papers for class 10.
Learning at DronStudy.com is an advantageous proposition for you. You are tutored for NCERT solutions for class 10 maths and the complete class 9 syllabus at home through the internet, in real-time, via highly interactive sessions.
Download Maths Class 10 NCERT Solutions in PDF format free. We are also providing help in solving holiday homework, if you are facing problem in doing holiday homework, upload in our website and get the solution with a week. | 677.169 | 1 |
Mathematics and the Real World: The Remarkable Role of Evolution in the Making of Mathematics
During this obtainable and illuminating research of ways the technological know-how of arithmetic constructed, a veteran math researcher and educator seems to be on the ways that our evolutionary make-up is either a aid and a difficulty to the examine of math.
Artstein chronicles the invention of significant mathematical connections among arithmetic and the true global from precedent days to the current. the writer then describes the various modern purposes of mathematics—in chance idea, within the examine of human habit, and together with desktops, which provide arithmetic unheard of power.
The writer concludes with an insightful dialogue of why arithmetic, for many humans, is so difficult. He argues that the rigorous logical constitution of math is going opposed to the grain of our predisposed methods of considering as formed via evolution, most likely as the expertise had to deal with logical arithmetic gave the human race as an entire no evolutionary virtue. With this in brain, he deals how one can triumph over those innate impediments within the educating of math.
Bob Miller's humor-laced, step by step studying assistance make even the main tricky math difficulties regimen. in line with greater than 28 years of educating and scholar suggestions, his easy-to-grasp innovations provide scholars much-needed self assurance.
This booklet introduces the math that helps complex laptop programming and the research of algorithms. the first goal of its recognized authors is to supply a superb and appropriate base of mathematical talents - the abilities had to clear up advanced difficulties, to guage horrendous sums, and to find refined styles in info.
This article addresses the necessity for a brand new arithmetic textual content for careers utilizing electronic know-how. the fabric is dropped at lifestyles via numerous purposes together with the math of reveal and printer monitors. The path, which covers binary mathematics to Boolean algebra, is rising in the course of the kingdom and will fill a necessity at your institution.
Notice: it is a STAND on my own publication. entry CODE isn't incorporated WITH THIS e-book utilizing and realizing arithmetic: A Quantitative Reasoning procedure prepares scholars for the maths they're going to stumble upon in university classes, their destiny profession, and existence often. Its quantitative reasoning method is helping scholars to construct the talents had to comprehend significant concerns in way of life, and compels scholars to procure the problem-solving instruments that they're going to have to imagine severely approximately quantitative matters in modern society.
This type of basic rule was once very important to justify its makes use of for occasions that seem in nature. the inability of interdependence in nature among the random occasions will be justified, however it is tougher to justify the belief that the random features are equivalent for all occasions. The paintings of the Russian mathematicians lowered the distance among the mathematical theorem and its attainable functions, and therefore the crucial restrict theorem, including different restrict theorems, grew to become the norm for some of the makes use of of records. the most use of restrict theorems is to estimate statistical values similar to the common, the dispersion, etc, of knowledge that experience random inaccuracy. it truly is usually tough to evaluate even if the mistakes are random or now not. no matter if the blunders are random, although, soaking up and knowing the procedure required for using the math that built contains problems which may themselves reason error. back, the problems derive from the way in which our instinct pertains to information. we are going to point out such problems. we're uncovered to large numbers of statistical surveys in our daily lives. whilst the result of a survey are released, it is often performed within the following shape: The survey came upon that (say) forty seven percentage of the citizens intend to vote for a selected candidate, with a survey mistakes of plus or minus 2 percentage. but simply a part of the stipulations and reservations in regards to the effects are provided within the survey file. In impression, the proper end from the survey will be that there's a ninety five percentage likelihood that the percentage of electorate who intend to aid that individual candidate is among forty seven percentage plus 2 percentage and forty seven percentage minus 2 percentage. The ninety five percentage certain is sort of basic in perform in data. The survey may be devised such that the opportunity that the review derived from the survey is true is ninety nine percentage (it will then price extra to hold out the survey), or the other quantity lower than 100. The of ninety five or ninety nine percentage isn't really released. Why now not? the truth that the declared limits of the implications, that's, that the ensuing period of forty five to forty nine percentage, applies with just a ninety five percentage chance may have nice significance. the explanation would appear to be the trouble in soaking up quantifications. there's one other element of statistical samples that's obscure. allow us to say that we're informed survey of 5 hundred humans chosen at random in Israel is enough to make sure a end result with a plus or minus 2 percentage survey mistakes (with ninety five percentage accuracy). The inhabitants of Israel is set 8 million. What dimension pattern will be required to accomplish that point of self assurance within the usa with 320 million population? would it not have to be 40 occasions the pattern measurement in Israel? most folks requested that question might resolution intuitively a lot higher pattern is required within the usa than in Israel. the proper solution is that an analogous dimension pattern is needed in either situations. the dimensions of the sampled inhabitants impacts in simple terms the trouble of choosing the pattern randomly. | 677.169 | 1 |
as we
a test or quiz lately? Avon View provides one parent-teacher opportunity per semester
as well as frequent subject reports.
Please watch for those. Parents can also gain access to students' grades online with
PowerSchool.
Contact the office for details. Feel free to contact us by e- mail with questions,
concerns or for updates, and see the class website -
It is ultimately the responsibility of the student to fulfill all of the
requirements (i.e. homework, proper attendance, studying, extra help etc.) to ensure
success
the scheduled/due date, actions taken by the teacher may include an alternate
assignment, an alternate
discretion of the teacher.
If you miss an assessment as a result of truancy, a mark of zero will be
assigned.
You are still expected to complete the missed work so that you still receive the
benefit of
address: [email protected]
Mathematics 11 and study habits
Read carefully and deliberately
In math you must read slowly, absorbing each word.
It is sometimes necessary to read something several times before it makes sense to
you.
In math each word or symbol is important because there are many thoughts condensed
into a few statements.
Think with pencil and paper
Test out the ideas on paper that the textbook authors are discussing.
Work out examples for yourself and then check with the book.
Be independent
It takes exercise to become strong.
You cannot improve your strength through someone else's exercise.
So practice, practice, practice but ask for help when you need it.
Listen and write notes in class
You must pay attention to the discussion in order to understand what is going
on.
A good set of notes will be an excellent source to study from for your exam.
Take time to do your work and do it on time
Learning mathematics is not an activity for the intellectually lazy.
It requires a strong steady effort.
With a new concept being introduced each day it is very easy to fall behind quickly
if you do not keep up with the pace of the class.
Be neat and accurate
Notes should be kept in order in a 3 ring binder.
Dates and headings on your notes will help you find things easily.
Write neatly so you can read your notes.
Copy steps from the board accurately so that you are not learning incorrect
procedures.
Persevere
Stick with it.
There will be times of confusion and frustration when nothing seems to work but
there will always be an answer. | 677.169 | 1 |
Geometer's Sketchpad
Sketchpad is a program for PC and Mac that gives students a tangible, visual way to learn mathematics that increases their engagement, understanding, and achievement. Sketchpad also has uses in elementary or middle school math, algebra, precalculus, and calculus. | 677.169 | 1 |
Measure Theory
Duration
Four Weeks
Lecturer
Marty Ross (Melbourne)
Consultation Hours:
Tuesdays 16–17h Carslaw Room 821
Assessment
50% problems assigned in lectures and 50% take home exam.
Assumed Knowledge
We will assume familiarity with the fundamental concepts of analysis in
Euclidean Space (infimum and supremum, open and closed sets, continuity, completeness and
compactness, countability). Some corresponding familiarity with these notions in metric spaces
would be helpful but will not be assumed; familiarity with these notions in topological spaces would
be just peachy.
Background Material
Lecture notes summarising the relevant background on sets and real analysis
(PDF) are available. Some (but definitely not all) of this material will be reviewed along the way,
particularly the material on metric spaces and topological spaces. Before the summer school begins,
you should definitely take a good look at the background notes and, if need be, browse through a
real analysis text or two.
Course Outline
Measure theory is the modern theory of integration, the method of assigning a "size"
to subsets of a universal set. It is more general, more powerful and more beautiful (though
also more technical) than the classical theory of Riemann integration. The course will be a
reasonably standard introduction to measure theory, with some emphasis upon geometric aspects.
We will cover most (but definitely not all) of the topics listed below, subject to time and
taste: | 677.169 | 1 |
Re: Linear Algebra Homework help!! Welcome to PF; It helps to give it a shot and tell us about it. You should have some examples of this sort of thing in your notes. Linear Algebra Homework Help. Basic linear algebra is a department of mathematics that targets solving systems of linear equations. The class will typically evolve to. Pre-Algebra, Algebra I, Algebra II, Geometry: homework help by free math tutors, solvers, lessons. Each section has solvers (calculators), lessons, and a place where. The following 5 questions have to do with Vector Space.They are based upon information from chapter 3 of Linear Algebra with Applications, 8th edition, Steven J Leon. Linear algebra homework (if xxxx then xx xxxxx a fourth vector xxxx cannot be rewritten as a linear combo of xxx xxxxx xxxxx xxx then xxx. Algebra homework help.
Linear algebra homework help
Linear algebra homework (if xxxx then xx xxxxx a fourth vector xxxx cannot be rewritten as a linear combo of xxx xxxxx xxxxx xxx then xxx. Algebra homework help. Professional Linear Algebra Homework Help Online. Linear Algebra assignments are some of the most complicated problems developed in Math, and we understand how. Re: Linear Algebra Homework help!! Welcome to PF; It helps to give it a shot and tell us about it. You should have some examples of this sort of thing in your notes. Get online tutoring and college homework help for Linear Algebra. We have a full team of professional Linear Algebra tutors ready to help you today. The following 5 questions have to do with Vector Space.They are based upon information from chapter 3 of Linear Algebra with Applications, 8th edition, Steven J Leon.
Linear Algebra Assignment Help Service– Timely and Expert Assistance. Linear Algebra is a vital component of mathematics and it deals with linear functions, linear. Linear Algebra Homework Help. Basic linear algebra is a department of mathematics that targets solving systems of linear equations. The class will typically evolve to. Professional Linear Algebra Homework Help Online. Linear Algebra assignments are some of the most complicated problems developed in Math, and we understand how. Pre-Algebra, Algebra I, Algebra II, Geometry: homework help by free math tutors, solvers, lessons. Each section has solvers (calculators), lessons, and a place where.
We are always 24/7 available to render our highly rated services and you can reach us at any time you need urgent linear algebra homework assistance. Linear Algebra Homework Help : If you are a Linear Algebra Homework student and seeking help in Linear Algebra Homework, then here is most reliable, precise and 100 %. Popular Linear algebra Textbooks See all Linear algebra textbooks up to: 750 gold Linear Algebra and Its Applications, 5th Edition up to. Get online tutoring and college homework help for Linear Algebra. We have a full team of professional Linear Algebra tutors ready to help you today. Linear Algebra Assignment Help. Any student who has chosen the field of science or engineering has to have a command over linear algebra. This topic of mathematics.
Get homework help at HomeworkMarket.com is an on-line marketplace for homework assistance and tutoring. You can ask homework questions.
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Popular Linear algebra Textbooks See all Linear algebra textbooks up to: 750 gold Linear Algebra and Its Applications, 5th Edition up to.
Linear Algebra. Linear Algebra is a branch of algebra, a sub discipline of mathematics, which takes into account solutions of systems of linear equations as well as.
Linear Algebra Assignment Help Service– Timely and Expert Assistance. Linear Algebra is a vital component of mathematics and it deals with linear functions, linear. Get homework help at HomeworkMarket.com is an on-line marketplace for homework assistance and tutoring. You can ask homework questions. Need linear algebra homework help? Our linear algebra tutors are available 24/7. Ask a linear algebra question now. | 677.169 | 1 |
If
Appropriate for students in grades 6 through 9, Practice Makes Perfect: Pre-Algebra gives your child the tools to master:
Integers
Rational numbers
Patterns
Equations
Graphing
Functions
And more
"synopsis" may belong to another edition of this title.
About the Author:
Erin Muschla-Berry is a middle school math teacher in New Jersey and has written extensively for teachers, including Practice Makes Perfect: Fractions, Decimals, and Percents. | 677.169 | 1 |
Free Math Program for K-12 & College Level
Khan Academy offers two things: A "free, adaptive math program" and over 800 how-to videos on almost any math concept you can think of.
About the math program: Our children are doing the math program this year and really prefer it over our past method of using a textbook math program. Khan Academy's program "learns" how much you know after a couple hours of working on it. Soon, it will have you up to the math level you should be on, and with the aid of the videos, you can keep learning math concepts, building on your knowledge.
Khan Academy also has "over 800 videos covering everything from basic arithmetic and algebra to differential equations, physics, and finance," including many videos on the current economic crisis.
About the professor (the following text is taken from the Khan Academy website) :
Salman Khan (Sal) founded the Khan Academy with the hope of using technology to foster new learning models. He is currently an investment professional in Palo Alto, CA and has held positions in venture capital, product management, and engineering. Sal received his MBA from Harvard Business School. He also holds a Masters in electrical engineering and computer science, a BS in electrical engineering and computer science, and a BS in mathematics from the Massachusetts Institute of Technology.
Testimonials: Many testimonials are posted on the Khan Academy site, including these:
"I am a high school Math/Physics Teacher working in Toronto Ontario. I have been referring students to your videos for the past year and have referenced many of them on my website. Those that have used it find it very useful, and have commented that you explain content better than I do…"
"…Your videos have become very popular in my school, and my grades have sky-rocketed ever since I became accustomed to your methods of teaching..."
"I've never been very successful with algebra, I know this is pretty basic, but I have almost no comprehension of mathematics. I was honestly going to be kicked out of school for being non-productive, I was ready to be thrown out, but now I get this. Thank you so much…"
Linda Barger, who emailed me about Khan Academy said her son "loves the site." Thank you, Linda, for this high quality, 100% free, educational find!
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Comments
Thanks so much! I have been looking for a free no nonsense math program. Love it…especially the way it tracks your progress and only moves you forward when your ready. I am doing right along with the kiddos tho brush up on my skills.
Lesa, I'm not sure of the exact definition of "curriculum." The program
does cover all math concepts needed for at least through college level. If
you create an account, after logging in, you can click on a "Knowledge
Map." That map shows how the program works, by showing you what you need to
accomplish (eg. addition) before moving onto another concept (eg.
fractions).
My 6th grader has been struggling with various concepts in math. She has declared Dr. Khan the greatest math teacher EVER!! Somehow, his explanations clarify the point, and his humor takes all the fear out of math. In fact, she usually ends a session by going to a video of something wildly difficult (like some topic in calculus) just to watch him work out the problems. I give him an A +++++++…
At what age would you start this program. My daugher will be in 1st grade next year and I don't think this has enough instruction at the younger levels. Like telling time and some of the other simple math lessons. I can't wait to use this later on.
Right now, we use Khan Academy for our 5th grader on up. You're right, it doesn't seem to have enough for the younger grades. Our 3rd grader doesn't get much out of it. I think I'll start her next year, in 4th. It's totally up to the child, though! Our eldest son would have probably been ready much younger than our 3rd grader is. | 677.169 | 1 |
A modern treatment of the essential topics of college algebra and trigonometry
(a.k.a, Precalculus). Topics include the real number system, functions and graphs,
polynomial and rational functions, trigonometric functions, exponential and logarithmic
functions. This course involves work using a graphics calculator.
Please note:
Students who take MATH 119 must make a grade of C or higher in order to
enroll in MATH 131 (Analytic Geometry and Calculus I).
A student who passes MATH 119 and subsequently changes to a major that does
not require MATH 131 may substitute MATH 119 for MATH 104.
Students may not receive credit for both MATH 104 and MATH 119 in meeting
their core curriculum mathematics requirement.
In addtion to the textbook, we will by using Pearson's MyMathLab for homework. You will need
to register for this course at
The course ID for this section of MATH 119 is moore04246.
Calculator
Recommend TI-83 Plus, TI-84, or TI-89
Grading
The final grade for the course is based on 7 grades as follows:
Five assigned in-class tests. Each test counts as a separate gradeHomework will be assigned from the textbook, with corresponding assignments
on MyMathLab from the same section of the textbook. Daily quizzes will come
directly from the homework problems assigned from the textbook. Students
may replace any quiz grade with a grade from MyMathLab on the corresponding
assigment. Please note that homework assignments are
excluded from this policy. You are encouraged to get help on assigned
homework problems when you need it.
Class attendance and participation can influence borderline grades.
A total of twelve, moving around, or leaving the
room during class without explicit permission. These actions are disturbing
to other | 677.169 | 1 |
Mathematics Illuminated by MacGregor Campbell
Description: Mathematics Illuminated is a thirteen-part series for adult learners and high school teachers. The series explores major themes in the field of mathematics, from humankind's earliest study of prime numbers, to the cutting-edge mathematics used to reveal the shape of the universe.
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Six Septembers: Mathematics for the Humanist by Patrick Juola, Stephen Ramsay - Zea Books Scholars of all stripes are turning their attention to materials that represent enormous opportunities for the future of humanistic inquiry. The purpose of the book is to impart the concepts that underlie the mathematics they are likely to encounter. (651 views)
Mathematics for Engineers by William Neville Rose - Chapman These two volumes form a most comprehensive and practical treatise on the subject. They show the direct bearing of all principles to engineering practice, and will prove a valuable reference work embracing all the mathematics needed by engineers. (6621 views)
Higher Mathematics for Students of Chemistry and Physics by Joseph William Mellor - Longmans, Green Long a standard textbook for graduate use in both Britain and America, this 1902 classic of modern mathematics remains a lucid, if advanced introduction to higher mathematics as used in advanced chemistry and physics courses. (2788 views) | 677.169 | 1 |
SOS Math Tutoring Website
This site, aimed at students in grades 9 and up, provides an abundance of examples for algebra, matrices, calculus, and everything in between, without being overwhelming — kids can choose to either display additional examples if needed or keep them hidden. While SOS Math touched on each of our three sample math questions, the lessons provided only a brief overview, which may not be enough for a student who needs more explanation. But teens who are struggling with a concept will appreciate this site's step-by-step solutions (as opposed to just providing the answer for its sample problems). SOS Math also offers excellent practice quizzes to let students learn by trial and error. Take note: Finding a topic (particularly the quizzes) is easier via the search engine than by basic navigation. sosmath.com. | 677.169 | 1 |
CSE 291, Additive Combinatorics and its Applications, Winter 2014
Additive combinatorics is an emerging area in the intersection of combinatorics, number theory, harmonic analysis and ergodic theory. It studies the structure of "approximate algebraic" objects, such as subsets of groups or fields, and their behavior under algebraic operations, such as addition or multiplication. While originally, the development of this field was motivated by problems in number theory, in recent years deep connections between additive combinatorics and several fundamental problems in theoretical computer science have been discovered. In this course, we will cover basic topics in additive combinatorics, discuss applications to theoretical computer science, and the many open problems that remain. There are no specific prerequisites beyond a certain level of mathematical maturity.
The course material will be broadly based on the survey Additive Combinatorics and its Applications in Theoretical Computer Science, as well as some more advanced topics, based on audience interest and time permitting. Topics which will be covered include: structure of set addition, arithmetic progressions in sets, sum-product theorems and higher-order Fourier analysis. We will discuss applications of these to several areas in theoretical computer science, including: property testing, explicit constructions of pseudo-random objects, and hardness results in complexity. | 677.169 | 1 |
Explorations in College Algebra, 5th Edition
Explorations in College Algebra, 5th Edition is designed to make algebra interesting and relevant to the student. The text adopts a problem-solving approach that motivates readers to grasp abstract ideas by solving real-world problems. The problems lie on a continuum from basic algebraic drills to open-ended, non-routine questions. The focus is shifted from learning a set of discrete mathematical rules to exploring how algebra is used in the social, physical, and life sciences. The goal of Explorations in College Algebra, 5th Edition is to prepare students for future advanced mathematics or other quantitatively based courses, while encouraging them to appreciate and use the power of algebra in answering questions about the world around us. Access to WileyPLUS sold separately.
Explore & Extend: are a new feature in this edition.
These "mini exploration" problems provide students with ideas for
going deeper into topics or previewing new concepts. They can be
found in every section.
Accumulation of tools: The authors added a
transformation to each chapter that covers functions. The intention
of these changes is to progressively accumulate (about one per
chapter) the tools for transforming functions. This is a way to
discuss properties of functions using function notation and to use
the transformations as new functions are introduced.
Chapter 8 reconfigured: Ch. 8 has been split into two
chapters: New Ch 8 covers quadratics and polynomials, while Ch. 9
is titled "Creating New Functions from Old."
Data: Data are updated throughout.
Extended Explorations: The two Extended Explorations
have been integrated into the text as sections in relevant
chapters.
NEW WileyPLUS Math Enhancements: a blend of conceptual
and skills, the new Enhancements feature Graphing,
"Intelligent Tutoring", Simplified Answer Recognition,
Improvements to Show Work Whiteboard, improved correction
process and computerized test back improvements
With 3,000+ questions in total, the Technology Update features
over 700 new questions, many conceptual in nature, including the
Check Your Understanding, Algebra Aerobics, and Explore and Extend
questions from the text
Algebra developed from real-world applications: These materials are based on problems using actual data drawn from a wide variety of sources including: the U.S. Census, medical texts, the Educational Testing Services, the U.S. Olympic Committee, and the Center for Disease Control.
Flexibility of use: The materials are designed for flexibility of use and offer multiple options for adapting them to a wide range of skill levels and departmental needs. The text is currently used in both small and large classes, two and four-year institutions, and taught with technology (graphing calculators and/or computers) or without. Many optional special features are described in following points.
Many opportunities for students to practice: Each section includes Algebra Aerobics, which are intended to help the student practice the skills they just learned. At the end of each chapter the Check Your Understanding and Review Summary sections can help students review the major ideas of the chapter.
Actively involved students: The text advocates the active engagement of students in class discussions and teamwork. The Something To Think About questions, open-ended exercises, and the Explorations are tools for stimulating student thought. The Explorations are an opportunity for students to work collaboratively or on their own to synthesize information from class lectures, the text, and the readings, and most importantly from their own discoveries.
Emphasis on verbal and written communication: This text encourages students to verbalize their ideas in small group and class discussions. Suggestions for writing "60-second summaries" are included in the first chapter, and many of the assignments require students to describe their observations in writing. Throughout the text there references to wide a variety of essays, articles, and reports included either in the Anthology of Readings (in the appendix of the text) or on the book companion website at: Many of the Explorations conclude with group presentations to the class.
Technology integrated throughout: While the materials promote the use of technology and include many explorations and exercises using graphing calculators and computers, there are no specific technology requirements. Some schools use graphing calculators only, others use just computers, and some use a combination of both. This flexible approach allows Explorations in College Algebra, 4/e to meet the needs of many varying courses.
Student Solutions: Step-by-step solutions to selected problems are provided at the end of the book | 677.169 | 1 |
I'm a high school graduate (2016, took a gap year) and I've been lurking on HN for almost a year in my free time. All the folks on here have really piqued my interest for math (I hear terms like category theory and abstract algebra being thrown around) and CS theory. If there's anything I'm thankful for from this community it's this thing. However I cannot bring myself to tackle such topics(because I feel that I'm not armed enough to learn them). How do you think I can overcome that?
I would first try to learn how to do proofs. I did no math since high school, then started again a few years ago just for fun . All higher level math (upper division and graduate school) is based on being able to read and write proofs. However, you don't need anything above high school algebra to learn proofs, so you don't have to wait, you can just get started now!
My favorite book, that I strongly recommend despite the high price of around $100 in the US is "Mathematical Proofs" by Chartrand.
You can get an international copy off eBay for around $45.
If you're weak on basic algebra etc, then you should instead start with "engineering mathematics" by Stroud, which has a foundations section that I started with several years ago when I started relearning math. It's designed for self-study.
I actually did find it helpful to do classes, I found most of the lower division math classes available online (i.e. calculus 1,2,3 and linear algebra). Sometimes, it helps to have deadlines, exams etc :)
Btw, if anyone out there already has a non math degree, but wants to study upper division and graduate level math formally, it turns out the way that is usually done on the US is to apply to a Math Masters program for "conditional admission" to the masters programs. They admit you, and then you do the upper division undergrad courses first, then move onto the masters programs. It's also possible to sign up for one-off classes at various universities via some kind of "open university" program, which is much easier to get into than formal admission to a degree course- I'm actually starting an Analysis course and a Linear algebra course at Berkeley tomorrow, as part of their "summer session", and you basically just sign up, pay your money, and turn up :)
Feel free to get in touch if anyone has any questions (email in profile)
If you have an interest in these topics, you can learn them. If you spend enough time doing something, you will learn it. Everyone has a different number for how long it will take, but depending on your "intelligence" skill level you will eventually grasp the subject.
This feeling that you are not armed for the subject is because there is a lot of dependent information between what you know and subjects like category theory and abstract algebra. Since you just got outta high school, you still have a lot to learn between where you are and where you want to be. Do not let that dissuade you tho, you can learn it, just gotta start.
Both MIT[1] and Stanford[2] have category theory as a graduate level course. I was not a math major but I assume that means you're like 4+ years away from learning this on the college track. Now, do not take that as a personal endorsement for going to college, you do you.
But, you are on hacker news, so I assume you want to learn, Well here is the MIT undergrad pure math major class requirements[3]. Its a good place to start learning an undergrad amount of math, the internet has resources everywhere to learn this stuff, it just takes time. Lots and lots of time.
One more tip, there is a trade-off between how hard something is to learn and how quickly you can learn it [4]. Do not over exert yourself too far in the difficult to learn direction, because you will become frustrated. Try and find a spot that is still fun, but not too fun, because then you are not maximizing your learning potential, assuming that is your goal. Learning how to learn can be very helpful, maximize your gains.
Also shout out to Numberphile on Youtube [5]. If you like math, you will like the channel.
Steal a copy of a textbook on libgen then read it. Try the exercises, if you can't do them then find out what you need to learn. This certainly works for physics (Obviously don't start with a graduate QED textbook).
What do you want to do with this mathematical knowledge you want to acquire? Learning for the sake of learning is fine, but, like programming and many other big topics, it can be much easier if you have specific goals and motivations.
Personally, I only started to enjoy math when I started hanging out with PhD students (in engineering as I was an engineer). They showed me what you can do with upper level math and that motivated me to learn it. I discovered that most math isn't like high school at all and is way cooler than I imagined. | 677.169 | 1 |
Students will develop a thorough understanding of the principal tools of calculus--how the tools are designed and how they are used to analyze the behavior of functions.
By modeling and solving a variety of problems, including some with real-world applications, students will acquire problem-solving skills and strategies--such as breaking complex problems into simpler subproblems. They will also come to understand how theoretical results and concepts can be developed and then used for problem solving as well as further investigation.
Students will gain an appreciation for the mathematical process, which consists of pattern recognition, pattern description, and pattern explanation (proof).
Students will develop a sense of the importance of criteria, such as clarity, efficiency, and elegance, by which competing solutions to problems might be judged. | 677.169 | 1 |
Foundations of Algorithms, Fourth Edition offers a well-balanced presentation of algorithm design, complexity analysis of algorithms, and computational complexity. The volume is accessible to mainstream computer science students who have a background in college algebra and discrete structures. To support their approach, the authors present mathematical concepts using standard English and a simpler notation than is found in most texts. A review of essential mathematical concepts is presented in three appendices. The authors also reinforce the explanations with numerous concrete examples to help students grasp theoretical concepts.
In short, I like this book. The contents are well-organized. Graphs, tables and examples are effectively used as well. Each section has its specific example, which helps beginners easily grasp its concept.
Using this book for my algorithms analysis class and while the algorithms in the book are efficient the text itself is lacking without a supplemental book. Even my instructor says that without the supplemental book some of the problems in the book are not really doable.
This book is truly something. Everything is explained in great amount of detail and several examples of the different techniques used for algorithms are really well explained, and the best part is that you can actually understand it. While everything they say is demonstrated, they dont use as much mathematical notation as other books, instead, they take their time to explain it with words and examples (yet they do use just the right amount of math notation to prove everything they say).
The pseudocode is c++like and that helped me understand everything so much better. They explain complexity, divide and conquer, dynamic programming, greedy algotirithms, and so much more in a way thats easy to understand and well documented. They also include several books after every chapter where you can expand what you have learned. They also include a good amount of pictures.
All in all this is the only book which actually helped me understand prim, kruskal, and dijkstra in a way I could actually code it myself. So I am extremely thankful for that. Also, beware that I found one typo in prim which changed everything. :) | 677.169 | 1 |
Book Description
BestBook Details
Amazon Sales Rank: #1128490 in Books
Published on: 1996-09-08
Original language: English
Number of items: 1
Dimensions: 9.00" h x 1.00" w x 7.50" l, 1.15 pounds
Binding: Paperback
332 pages
Editorial Reviews
From the Publisher Best-selling author Delores Etter provides an up-to-date introduction to MATLAB. Presenting a consistent five-step problem-solving methodology, Etter describes the computational and visualization capabilities of MATLAB and illustrates the problem solving process through a variety of engineering examples and applications.
From the Back Cover BestCustomer Reviews
Most helpful customer reviews
4 of 4 people found the following review helpful. Good text, but the typos in problem sets and examples make it more frustrating than helpful. By smiles This book is required for my C++ class. Although it covers pertinent topics, and the text describes how to solve problems well, the examples are poorly written. They contain many errors, and don't explain why they are using certain functions. Not only does someone who is just learning C++ have to figure out where the errors are, but they have to figure out why they are using a certain code. Not only do the examples have errors, but the problem sets have all sorts of typos. A general example: it will tell the student to alter the program they wrote for problem x on page y, and said problem doesn't exist. This results in searching the text for the proper problem. This causes errors in problems assigned by my professor, studying, and trying to find an example in class. My husband holds a degree in computer science, and the book even annoys him. At least buying it through Amazon saved me a lot of money compared to my school's bookstore, but still, I wish I didn't have to buy this for class. I have a library of more helpful books, and usually resort to them, or asking "the Internet" questions to clarify the text.
5 of 5 people found the following review helpful. Clear presentation - an excellent textbook By Lauren Elizabeth The layout of this text is very helpful for keeping the material straight. It is written with clarity. The real-world sample problems are interesting. I have just one concern, which kept me from giving it 5 stars: there are errors in some of the answers to practice problems. I don't see why these would not have all been carefully checked. (For reference, I have the international edition.) Nonetheless, a great text!
0 of 0 people found the following review helpful. Avoid, disorganized pseudointellectual obfuscation layer. By Mr. Lobotomy Avoid. Rent, do not buy, if you need it for class.
Introductory class text for freshman programming. Luckily, I was able to spend another $50 and get four used books that are doing the job of getting me through class. Buy the Kernigan and Ritchie C text, also bought James Coplien's C++ which are well organized and readable.
Found one helpful insight in the book where Etter says a function call may be changed from pass by value to pass by reference by adding the & operator. That works perfectly.
Pseudointellectual. Thank God for Amazon and a place to register my complaint. She has a brute force sort with variables like (int j=k+1; j I'm typing in her OOP class example and once I understand what is supposed to be happening, I'm getting the feeling I will want to rewrite this entirely. No understanding of, or structure of style idioms is conveyed. I would like a book recommendation that will do this for me.
She uses an example problem from systems analysis which is a 3rd year EE course, damped responses. A formula is provided as a black box to just type in. Having had systems analysis, I have no idea what the student was supposed to get out of the way it was presented. This is an introductory text, though system response damping could have been introduced in an insightful way instead of as a black box math formula. I'm guessing the first year student would see this and feel like they were rowed out to the deep water and tossed overboard. That's how I felt looking at it and I took a semester class in that
Book covers introductory material by dispersing it in different chapters. Struct is not in the index. Pointers could have been chpt 2 or 3 and not 9. I'm looking for where she says a pointer is a data type that hold a memory address but i cannot find | 677.169 | 1 |
All Sequenceshis is a set of 16 task cards involving explicit formulas for geometric and arithmetic sequences.
Questions involve:
Finding the nth term of geometric and arithmetic sequences.
Solving word problems.
Determining the basic graph of each type of sequence
Writing explicit formulas for geometric and arithmetic sequences | 677.169 | 1 |
Mathematics 2
Objectives
This subject provides students with the skills to succeed in university courses that require a high level of analytical and logical thought, such as science, engineering or actuarial studies. This subject also ensures that students have the language skills to enable them to ask mathematical questions necessary for them to undertake further studies in mathematics in English. | 677.169 | 1 |
Factoring Trinomials Level 2 Card Match
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Factoring Polynomials LEVEL 2
Students will FACTOR POLYNOMIALS in this 26 problem resource. Level 2 of my Factoring Polynomials series is for students who have successfully grasped the introduction to factoring polynomials in Factoring Polynomials LEVEL 1.
Level 2 includes trinomials with prime "a" and "c" values of 2, 3 and 5. The Factoring Polynomials Series each include 3 formats: Card Match, Around the Room and Whole Class Instruction to accommodate various teaching and learning styles.
************************************************************************* Marie.DLR@AlgebraAccents.com. This resource may not be uploaded to the internet in any form, including classroom/personal websites or network drives. | 677.169 | 1 |
introduction to numerical methods for students in engineering. It covers solution of equations, interpolation and data fitting, solution of differential equations, eigenvalue problems and optimisation. The algorithms are implemented in Python 3, a high-level programming language that rivals MATLABŪ in readability and ease of use. All methods include programs showing how the computer code is utilised in the solution of problems. The book is based on Numerical Methods in Engineering with Python, which used Python 2. This new edition demonstrates the use of Python 3 and includes an introduction to the Python plotting package Matplotlib. This comprehensive book is enhanced by the addition of numerous examples and problems throughout. | 677.169 | 1 |
Quadratics MATHO
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MATHO is a excellent way to review!
Students enjoy it and so do teachers because it is easy to use, fun, and serves as an effective review.
This MATHO game includes the following topics:
- Will the parabola open up or down?
- Find the vertex.
- Find the roots/solutions/x-intercepts.
- Find the y-intercept.
- Identify simple parent graph shifts and transformations.
- Find the discriminant. | 677.169 | 1 |
Objectives This chapter will provide students with specific strategies to be successful in math and science courses. Part Eight, Additional Learning Skills.
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Presentation on theme: "Objectives This chapter will provide students with specific strategies to be successful in math and science courses. Part Eight, Additional Learning Skills."— Presentation transcript:
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Objectives This chapter will provide students with specific strategies to be successful in math and science courses. Part Eight, Additional Learning Skills Studying Math and Science
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There is no way to pass mathematics and science courses without doing a great deal of hard work. Mathematics and science courses demand: Excellent attendance Complete notes Extensive homework Intensive study sessions. Doing well in these classes is possible. You must be aware of the adjustments you should make when you switch from mathematics and science form your less technical subjects.
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In mathematics and science courses: Knowledge is cumulative. Each fact or formula you learn must rest on a basic structure of all you have learned before. Great emphasis is placed on specialized vocabulary, rules, and formulas. Mathematics and science deal in precision. You need to understand specific rules, formulas, and vocabulary terms Special emphasis is placed on homework. You may be given numerical problems to complete. Their purpose is to give you practice in the kinds of material you will find on tests.
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In mathematics and science courses: Taking clear notes is crucial. You will often be copying problems, diagrams, formulas, and definitions from the blackboard. You will also be trying to follow your instructors train of thought as he or she explains how a problem is solved or how a process works. Be sure to include in your notes any information provided that helps you see the connections between steps or the relationship of one fact to another. As soon as possible after class, clarify and expand your notes while the material is still clear in your mind.
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In mathematics and science courses: Patient, slow reading is required. The information in math and science texts is densely packed. Texts are filled with special terms that are often unfamiliar. Blocks of text are interspersed with numerical formulas, problems to solve, charts, diagrams, and drawings. Such textbooks cannot be read quickly, so you must keep up with assigned reading. Math and science texts are usually organized very clearly. They also include glossaries of terms and concise reviews at the ends of chapters.
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Read slowly. Dont skip over unfamiliar terms; take the time to look them up. Spend time with each sample problem. Study visual material accompanying the written explanations. Be organized, persistent, and willing to work, and you will succeed. | 677.169 | 1 |
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Students work in groups of three or four to compare and contrast the three methods of solving systems: elimination, substitution and graphing. Four different sets of systems including a word problem. This is a work in progress...please email any needed corrections to kimberlybwood@windstream.net | 677.169 | 1 |
Part One: Connecting Mathematics with Work and Life | High School Mathematics at Work: Essays and Examples for the Education of All Students | The National Academies Press
Part One— Connecting Mathematics with Work and Life
Page 8This page in the original is blank.
Page 9Overview
Mathematics is the key to opportunity. No
longer just the language of science, mathematics now contributes in
direct and fundamental ways to business, finance, health, and defense.
For students, it opens doors to careers. For citizens, it enables
informed decisions. For nations, it provides knowledge to compete in a
technological community. To participate fully in the world of the
future, America must tap the power of mathematics. (NRC, 1989, p. 1)
The above statement remains true today, although
it was written almost ten years ago in the Mathematical Sciences
Education Board's (MSEB) report Everybody Counts (NRC, 1989).
In envisioning a future in which all students will be afforded such
opportunities, the MSEB acknowledges the crucial role played by formulae
and algorithms, and suggests that algorithmic skills are more flexible,
powerful, and enduring when they come from a place of meaning and
understanding. This volume takes as a premise that all students can
develop mathematical understanding by working with mathematical tasks
from workplace and everyday contexts. The essays in this report
provide some rationale for this premise and discuss some of the issues
and questions that follow. The tasks in this report illuminate some of
the possibilities provided by the workplace and everyday life.
Page 10Contexts from within mathematics also can be
powerful sites for the development of mathematical understanding, as
professional and amateur mathematicians will attest. There are many good
sources of compelling problems from within mathematics, and a broad
mathematics education will include experience with problems from
contexts both within and outside mathematics. The inclusion of tasks in
this volume is intended to highlight particularly compelling problems
whose context lies outside of mathematics, not to suggest a curriculum.
The operative word in the above premise is "can."
The understandings that students develop from any encounter with
mathematics depend not only on the context, but also on the students'
prior experience and skills, their ways of thinking, their engagement
with the task, the environment in which they explore the task—including
the teacher, the students, and the tools—the kinds of interactions that
occur in that environment, and the system of internal and external
incentives that might be associated with the activity. Teaching and
learning are complex activities that depend upon evolving and rarely
articulated interrelationships among teachers, students, materials, and
ideas. No prescription for their improvement can be simple.
This volume may be beneficially seen as a rearticulation and elaboration of a principle put forward in Reshaping School Mathematics:
Principle 3: Relevant Applications Should be an Integral Part of the Curriculum.
Students need to experience mathematical ideas
in the context in which they naturally arise—from simple counting and
measurement to applications in business and science. Calculators and
computers make it possible now to introduce realistic applications
throughout the curriculum.
The significant criterion for the suitability
of an application is whether it has the potential to engage students'
interests and stimulate their mathematical thinking. (NRC, 1990, p. 38)
Mathematical problems can serve as a source of
motivation for students if the problems engage students' interests and
aspirations. Mathematical problems also can serve as sources of meaning
and understanding if the problems stimulate students' thinking. Of
course, a mathematical task that is meaningful to a student will provide
more motivation than a task that does not make sense. The rationale
behind the criterion above is that both meaning and motivation are
required. The motivational benefits that can be provided by workplace
and everyday problems are worth mentioning, for although some students
are aware that certain mathematics courses are necessary in order to
gain entry into particular career paths, many students are unaware of
how particular topics or problem-solving approaches will have relevance
in any workplace. The power of using workplace and everyday problems to
teach mathematics lies not so much in motivation, however, for no con-
Page 11text by itself will motivate all students. The real power is in connecting to students' thinking.
There is growing evidence in the literature that
problem-centered approaches—including mathematical contexts, "real
world" contexts, or both—can promote learning of both skills and
concepts. In one comparative study, for example, with a high school
curriculum that included rich applied problem situations, students
scored somewhat better than comparison students on algebraic procedures
and significantly better on conceptual and problem-solving tasks (Schoen
& Ziebarth, 1998). This finding was further verified through
task-based interviews. Studies that show superior performance of
students in problem-centered classrooms are not limited to high schools.
Wood and Sellers (1996), for example, found similar results with second
and third graders.
Research with adult learners seems to indicate
that "variation of contexts (as well as the whole task approach) tends
to encourage the development of general understanding in a way which
concentrating on repeated routine applications of algorithms does not
and cannot" (Strässer, Barr, Evans, & Wolf, 1991, p. 163). This
conclusion is consistent with the notion that using a variety of
contexts can increase the chance that students can show what they know.
By increasing the number of potential links to the diverse knowledge and
experience of the students, more students have opportunities to excel,
which is to say that the above premise can promote equity in mathematics
education.
There is also evidence that learning mathematics
through applications can lead to exceptional achievement. For example,
with a curriculum that emphasizes modeling and applications, high school
students at the North Carolina School of Science and Mathematics have
repeatedly submitted winning papers in the annual college competition,
Mathematical Contest in Modeling (Cronin, 1988; Miller, 1995).
The relationships among teachers, students,
curricular materials, and pedagogical approaches are complex.
Nonetheless, the literature does supports the premise that workplace and
everyday problems can enhance mathematical learning, and
suggests that if students engage in mathematical thinking, they will be
afforded opportunities for building connections, and therefore meaning
and understanding.
In the opening essay, Dale Parnell argues that
traditional teaching has been missing opportunities for connections:
between subject-matter and context, between academic and vocational
education, between school and life, between knowledge and application,
and between subject-matter disciplines. He suggests that teaching must
change if more students are to learn mathematics. The question, then, is
how to exploit opportunities for connections between high school
mathematics and the workplace and everyday life.
Rol Fessenden shows by example the importance of
mathematics in business, specifically in making marketing decisions. His
essay opens with a dialogue among employees of a company that intends
to expand its business into
Page 12Japan, and then goes on to point out many of the
uses of mathematics, data collection, analysis, and non-mathematical
judgment that are required in making such business decisions.
In his essay, Thomas Bailey suggests that
vocational and academic education both might benefit from integration,
and cites several trends to support this suggestion: change and
uncertainty in the workplace, an increased need for workers to
understand the conceptual foundations of key academic subjects, and a
trend in pedagogy toward collaborative, open-ended projects.
Further-more, he observes that School-to-Work experiences, first
intended for students who were not planning to attend a four-year
college, are increasingly being seen as useful in preparing students for
such colleges. He discusses several such programs that use work-related
applications to teach academic skills and to prepare students for
college. Integration of academic and vocational education, he argues,
can serve the dual goals of "grounding academic standards in the
realistic context of workplace requirements and introducing a broader
view of the potential usefulness of academic skills even for entry level
workers."
Noting the importance and utility of mathematics
for jobs in science, health, and business, Jean Taylor argues for
continued emphasis in high school of topics such as algebra, estimation,
and trigonometry. She suggests that workplace and everyday problems can
be useful ways of teaching these ideas for all students.
There are too many different kinds of workplaces
to represent even most of them in the classrooms. Furthermore, solving
mathematics problems from some workplace contexts requires more
contextual knowledge than is reasonable when the goal is to learn
mathematics. (Solving some other workplace problems requires more
mathematical knowledge than is reasonable in high school.) Thus,
contexts must be chosen carefully for their opportunities for sense
making. But for students who have knowledge of a workplace, there are
opportunities for mathematical connections as well. In their essay,
Daniel Chazan and Sandra Callis Bethell describe an approach that
creates such opportunities for students in an algebra course for 10th
through 12th graders, many of whom carried with them a "heavy burden of
negative experiences" about mathematics. Because the traditional Algebra
I curriculum had been extremely unsuccessful with these students,
Chazan and Bethell chose to do something different. One goal was to help
students see mathematics in the world around them. With the help of
community sponsors, Chazen and Bethell asked students to look for
mathematics in the workplace and then describe that mathematics and its
applications to their classmates.
The tasks in Part One complement the points made in the essays by making direct connections to the workplace and everyday life. Emergency Calls
(p. 42) illustrates some possibilities for data analysis and
representation by discussing the response times of two ambulance
companies. Back-of-the-Envelope Estimates (p. 45) shows how quick, rough estimates and calculations
Page 13are useful for making business decisions. Scheduling Elevators
(p. 49) shows how a few simplifying assumptions and some careful
reasoning can be brought together to understand the difficult problem of
optimally scheduling elevators in a large office building. Finally, in
the context of a discussion with a client of an energy consulting firm, Heating-Degree-Days (p. 54) illuminates the mathematics behind a common model of energy consumption in home heating.
References
Cronin, T. P. (1988). High school students win "college" competition. Consortium: The Newsletter of the Consortium for Mathematics and Its Applications,26, 3, 12.
Miller, D. E. (1995). North Carolina sweeps MCM '94.SIAM News,28 (2).
National Research Council. (1989). Everybody counts: A report to the nation on the future of mathematics education. Washington, DC: National Academy Press.
National Research Council. (1990). Reshaping school mathematics: A philosophy and framework for curriculum. Washington, DC: National Academy Press.1— Mathematics as a Gateway to Student Success
DALE PARNELL
Oregon State University
The study of mathematics stands, in many ways, as a
gateway to student success in education. This is becoming particularly
true as our society moves inexorably into the technological age.
Therefore, it is vital that more students develop higher levels of
competency in mathematics.1
The standards and expectations for students must
be high, but that is only half of the equation. The more important half
is the development of teaching techniques and methods that will help all
students (rather than just some students) reach those higher
expectations and standards. This will require some changes in how
mathematics is taught.
Effective education must give clear focus to
connecting real life context with subject-matter content for the
student, and this requires a more ''connected" mathematics program. In
many of today's classrooms, especially in secondary school and college,
teaching is a matter of putting students in classrooms marked "English,"
"history," or "mathematics," and then attempting to fill their heads
with facts through lectures, textbooks, and the like. Aside from an
occasional lab, workbook, or "story problem," the element of contextual
teaching and learning is absent, and little attempt is made to connect
what students are learning with the world in which they will be expected
to work and spend their lives. Often the frag-
Page 15mented information offered to students is of little use or application except to pass a test.
What we do in most traditional classrooms is
require students to commit bits of knowledge to memory in isolation from
any practical application—to simply take our word that they "might need
it later." For many students, "later" never arrives. This might well be
called the freezer approach to teaching and learning. In effect, we are
handing out information to our students and saying, "Just put this in
your mental freezer; you can thaw it out later should you need it." With
the exception of a minority of students who do well in mastering
abstractions with little contextual experience, students aren't buying
that offer. The neglected majority of students see little personal
meaning in what they are asked to learn, and they just don't learn it.
I recently had occasion to interview 75 students
representing seven different high schools in the Northwest. In nearly
all cases, the students were juniors identified as vocational or general
education students. The comment of one student stands out as
representative of what most of these students told me in one way or
another: "I know it's up to me to get an education, but a lot of times
school is just so dull and boring. … You go to this class, go to that
class, study a little of this and a little of that, and nothing
connects. … I would like to really understand and know the application
for what I am learning." Time and again, students were asking, "Why do I
have to learn this?" with few sensible answers coming from the
teachers.
My own long experience as a community college
president confirms the thoughts of these students. In most community
colleges today, one-third to one-half of the entering students are
enrolled in developmental (remedial) education, trying to make up for
what they did not learn in earlier education experiences. A large
majority of these students come to the community college with limited
mathematical skills and abilities that hardly go beyond adding,
subtracting, and multiplying with whole numbers. In addition, the need
for remediation is also experienced, in varying degrees, at four-year
colleges and universities.
What is the greatest sin committed in the teaching
of mathematics today? It is the failure to help students use the
magnificent power of the brain to make connections between the
following:
subject-matter content and the context of use;
academic and vocational education;
school and other life experiences;
knowledge and application of knowledge; and
one subject-matter discipline and another.
Why is such failure so critical? Because
understanding the idea of making the connection between subject-matter
content and the context of application
Page 16is what students, at all levels of education,
desperately require to survive and succeed in our high-speed,
high-challenge, rapidly changing world.
Educational policy makers and leaders can issue
reams of position papers on longer school days and years, site-based
management, more achievement tests and better assessment practices, and
other "hot" topics of the moment, but such papers alone will not make
the crucial difference in what students know and can do. The difference
will be made when classroom teachers begin to connect learning with
real-life experiences in new, applied ways, and when education reformers
begin to focus upon learning for meaning.
A student may memorize formulas for determining
surface area and measuring angles and use those formulas correctly on a
test, thereby achieving the behavioral objectives set by the teacher.
But when confronted with the need to construct a building or repair a
car, the same student may well be left at sea because he or she hasn't
made the connection between the formulas and their real-life
application. When students are asked to consider the Pythagorean
Theorem, why not make the lesson active, where students actually lay out
the foundation for a small building like a storage shed?
What a difference mathematics instruction could
make for students if it were to stress the context of application—as
well as the content of knowledge—using the problem-solving model over
the freezer model. Teaching conducted upon the connected model would
help more students learn with their thinking brain, as well as with
their memory brain, developing the competencies and tools they need to
survive and succeed in our complex, interconnected society.
One step toward this goal is to develop
mathematical tasks that integrate subject-matter content with the
context of application and that are aimed at preparing individuals for
the world of work as well as for post-secondary education. Since many
mathematics teachers have had limited workplace experience, they need
many good examples of how knowledge of mathematics can be applied to
real life situations. The trick in developing mathematical tasks for use
in classrooms will be to keep the tasks connected to real life
situations that the student will recognize. The tasks should not be just
a contrived exercise but should stay as close to solving common
problems as possible.
As an example, why not ask students to compute the
cost of 12 years of schooling in a public school? It is a sad irony
that after 12 years of schooling most students who attend the public
schools have no idea of the cost of their schooling or how their
education was financed. No wonder that some public schools have
difficulty gaining financial support! The individuals being served by
the schools have never been exposed to the real life context of who pays
for the schools and why. Somewhere along the line in the teaching of
mathematics, this real life learning opportunity has been missed, along
with many other similar contextual examples.
The mathematical tasks in High School Mathematics at Work provide students (and teachers) with a plethora of real life mathematics problems and
Page 17challenges to be faced in everyday life and
work. The challenge for teachers will be to develop these tasks so they
relate as close as possible to where students live and work every day.
References
Note
1.
For further discussion of these issues, see Parnell (1985, 1995).
DALE PARNELL is
Professor Emeritus of the School of Education at Oregon State
University. He has served as a University Professor, College President,
and for ten years as the President and Chief Executive Officer of the
American Association of Community Colleges. He has served as a
consultant to the National Science Foundation and has served on many
national commissions, such as the Secretary of Labor's Commission on
Achieving Necessary Skills (SCANS). He is the author of the book The Neglected Majority which provided the foundation for the federally-funded Tech Prep Associate Degree Program.
Page 182— Market Launch
ROL FESSENDEN
L. L. Bean, Inc.
"OK, the agenda of the meeting is to review the
status of our launch into Japan. You can see the topics and presenters
on the list in front of you. Gregg, can you kick it off with a strategy
review?"
"Happy to, Bob. We have assessed the
possibilities, costs, and return on investment of opening up both store
and catalog businesses in other countries. Early research has shown that
both Japan and Germany are good candidates. Specifically, data show
high preference for good quality merchandise, and a higher-than-average
propensity for an active outdoor lifestyle in both countries. Education,
age, and income data are quite different from our target market in the
U.S., but we do not believe that will be relevant because the cultures
are so different. In addition, the Japanese data show that they have a
high preference for things American, and, as you know, we are a classic
American company. Name recognition for our company is 14%, far higher
than any of our American competition in Japan. European competitors are
virtually unrecognized, and other Far Eastern competitors are perceived
to be of lower quality than us. The data on these issues are quite
clear.
"Nevertheless, you must understand that there is a
lot of judgment involved in the decision to focus on Japan. The
analyses are limited because the cultures are different and we expect
different behavioral drivers. Also,
Page 19
Suggested Citation: "Part One: Connecting Mathematics with Work and Life." National Research Council. 1998. High School Mathematics at Work: Essays and Examples for the Education of All Students. Washington, DC: The National Academies Press. doi: 10.17226/5777.
×
much of the data we need in Japan are simply not
available because the Japanese marketplace is less well developed than
in the U.S. Drivers' license data, income data, lifestyle data, are all
commonplace here and unavailable there. There is little prior
penetration in either country by American retailers, so there is no
experience we can draw upon. We have all heard how difficult it will be
to open up sales operations in Japan, but recent sales trends among
computer sellers and auto parts sales hint at an easing of the
difficulties.
"The plan is to open three stores a year, 5,000
square feet each. We expect to do $700/square foot, which is more than
double the experience of American retailers in the U.S. but 45% less
than our stores. In addition, pricing will be 20% higher to offset the
cost of land and buildings. Asset costs are approximately twice their
rate in the U.S., but labor is slightly less. Benefits are more
thoroughly covered by the government. Of course, there is a lot of
uncertainty in the sales volumes we are planning. The pricing will cover
some of the uncertainty but is still less than comparable quality goods
already being offered in Japan.
"Let me shift over to the competition and tell you
what we have learned. We have established long-term relationships with
500 to 1000 families in each country. This is comparable to our practice
in the U.S. These families do not know they are working specifically
with our company, as this would skew their reporting. They keep us
appraised of their catalog and shopping experiences, regardless of the
company they purchase from. The sample size is large enough to be
significant, but, of course, you have to be careful about small
differences.
"All the families receive our catalog and catalogs
from several of our competitors. They match the lifestyle, income, and
education demographic profiles of the people we want to have as
customers. They are experienced catalog shoppers, and this will skew
their feedback as compared to new catalog shoppers.
"One competitor is sending one 100-page catalog
per quarter. The product line is quite narrow—200 products out of a
domestic line of 3,000. They have selected items that are not likely to
pose fit problems: primarily outerwear and knit shirts, not many pants,
mostly men's goods, not women's. Their catalog copy is in Kanji, but the
style is a bit stilted we are told, probably because it was written in
English and translated, but we need to test this hypothesis. By
contrast, we have simply mailed them the same catalog we use in the
U.S., even written in English.
"Customer feedback has been quite clear. They
prefer our broader assortment by a ratio of 3:1, even though they don't
buy most of the products. As the competitors figured, sales are focused
on outerwear and knits, but we are getting more sales, apparently
because they like looking at the catalog and spend more time with it.
Again, we need further testing. Another hypothesis is that our brand
name is simply better known.
"Interestingly, they prefer our English-language
version because they find it more of an adventure to read the catalog in
another language. This is probably
Page 20a built-in bias of our sampling technique
because we specifically selected people who speak English. We do not
expect this trend to hold in a general mailing.
"The English language causes an 8% error rate in
orders, but orders are 25% larger, and 4% more frequent. If we can get
them to order by phone, we can correct the errors immediately during the
call.
"The broader assortment, as I mentioned, is
resulting in a significantly higher propensity to order, more units per
order, and the same average unit cost. Of course, paper and postage
costs increase as a consequence of the larger format catalog. On the
other hand, there are production efficiencies from using the same
version as the domestic catalog. Net impact, even factoring in the error
rate, is a significant sales increase. On the other hand, most of the
time, the errors cause us to ship the wrong item which then needs to be
mailed back at our expense, creating an impression in the customers that
we are not well organized even though the original error was theirs.
"Final point: The larger catalog is being kept by
the customer an average of 70 days, while the smaller format is only
kept on average for 40 days. Assuming—we need to test this—that the
length of time they keep the catalog is proportional to sales volumes,
this is good news. We need to assess the overall impact carefully, but
it appears that there is a significant population for which an
English-language version would be very profitable."
"Thanks, Gregg, good update. Jennifer, what do you have on customer research?"
"Bob, there's far more that we need to know than
we have been able to find out. We have learned that Japan is very
fad-driven in apparel tastes and fascinated by American goods. We expect
sales initially to sky-rocket, then drop like a stone. Later on, demand
will level out at a profitable level. The graphs on page 3 [Figure 2-1]
show demand by week for 104 weeks, and we have assessed several
scenarios. They all show a good underlying business, but the uncertainty
is in the initial take-off. The best data are based on the Italian
fashion boom which Japan experienced in the late 80s. It is not strictly
analogous because it revolved around dress apparel instead of our
casual and weekend wear. It is, however, the best information available.
FIGURE 2-1: Sales projections by week, Scenario A
Page 21 2-2: Size distributions, U.S. vs. Japan
"Our effectiveness in positioning inventory for
that initial surge will be critical to our long-term success. There are
excellent data—supplied by MITI, I might add—that show that Japanese
customers can be intensely loyal to companies that meet their high
service expectations. That is why we prepared several scenarios. Of
course, if we position inventory for the high scenario, and we
experience the low one, we will experience a significant loss due to
liquidations. We are still analyzing the long-term impact, however. It
may still be worthwhile to take the risk if the 2-year ROI1 is sufficient.
"We have solid information on their size scales [Figure 2-2].
Seventy percent are small and medium. By comparison, 70% of Americans
are large and extra large. This will be a challenge to manage but will
save a few bucks on fabric.
"We also know their color preferences, and they
are very different than Americans. Our domestic customers are very
diverse in their tastes, but 80% of Japanese customers will buy one or
two colors out of an offering of 15. We are still researching color
choices, but it varies greatly for pants versus shirts, and for men
versus women. We are confident we can find patterns, but we also know
that it is easy to guess wrong in that market. If we guess wrong, the
liquidation costs will be very high.
"Bad news on the order-taking front, however. They don't like to order by phone. …"
Analysis
In this very brief exchange among
decision-makers we observe the use of many critically important skills
that were originally learned in public schools. Perhaps the most
important is one not often mentioned, and that is the ability to convert
an important business question into an appropriate mathematical one, to
solve the mathematical problem, and then to explain the implications of
the solution for the original business problem. This ability to inhabit
simultaneously the business world and the mathematical world, to
translate between the two, and, as a consequence, to bring clarity to
complex, real-world issues is of extraordinary importance.
In addition, the participants in this conversation
understood and interpreted graphs and tables, computed, approximated,
estimated, interpolated, extrapolated, used probabilistic concepts to
draw conclusions, generalized from
Page 22small samples to large populations, identified
the limits of their analyses, discovered relationships, recognized and
used variables and functions, analyzed and compared data sets, and
created and interpreted models. Another very important aspect of their
work was that they identified additional questions, and they suggested
ways to shed light on those questions through additional analysis.
There were two broad issues in this conversation
that required mathematical perspectives. The first was to develop as
rigorous and cost effective a data collection and analysis process as
was practical. It involved perhaps 10 different analysts who attacked
the problem from different viewpoints. The process also required
integration of the mathematical learnings of all 10 analysts and
translation of the results into business language that could be
understood by non-mathematicians.
The second broad issue was to understand from the
perspective of the decision-makers who were listening to the
presentation which results were most reliable, which were subject to
reinterpretation, which were actually judgments not supported by
appropriate analysis, and which were hypotheses that truly required more
research. In addition, these business people would likely identify
synergies in the research that were not contemplated by the analysts.
These synergies need to be analyzed to determine if—mathematically—they
were real. The most obvious one was where the inventory analysts said
that the customers don't like to use the phone to place orders. This is
bad news for the sales analysts who are counting on phone data
collection to correct errors caused by language problems. Of course, we
need more information to know the magnitude—or even the existance—of the
problem.
In brief, the analyses that preceded the dialogue might each be considered a mathematical task in the business world:
A cost analysis of store operations and catalogs was conducted using data from existing American and possibly other operations.
Customer preferences research was analyzed to
determine preferences in quality and life-style. The data collection
itself could not be carried out by a high school graduate without
guidance, but 80% of the analysis could.
Cultural differences were recognized as a causes
of analytical error. Careful analysis required judgment. In addition,
sources of data were identified in the U.S., and comparable sources were
found lacking in Japan. A search was conducted for other comparable
retail experience, but none was found. On the other hand, sales data
from car parts and computers were assessed for relevance.
Rates of change are important in understanding how Japanese and American stores differ. Sales per square foot, price increases,
Page 23asset costs, labor
costs and so forth were compared to American standards to determine
whether a store based in Japan would be a viable business.
"Nielsen" style ratings of 1000 families were
used to collect data. Sample size and error estimates were mentioned.
Key drivers of behavior (lifestyle, income, education) were mentioned,
but this list may not be complete. What needs to be known about these
families to predict their buying behavior? What does "lifestyle"
include? How would we quantify some of these variables?
A hypothesis was presented that catalog size and
product diversity drive higher sales. What do we need to know to assess
the validity of this hypothesis? Another hypothesis was presented about
the quality of the translation. What was the evidence for this
hypothesis? Is this a mathematical question? Sales may also be
proportional to the amount of time a potential customer retains the
catalog. How could one ascertain this?
Despite the abundance of data, much uncertainty
remains about what to expect from sales over the first two years.
Analysis could be conducted with the data about the possible inventory
consequences of choosing the wrong scenario.
One might wonder about the uncertainty in size
scales. What is so difficult about identifying the colors that Japanese
people prefer? Can these preferences be predicted? Will this increase
the complexity of the inventory management task?
Can we predict how many people will not use phones? What do they use instead?
As seen through a mathematical lens, the
business world can be a rich, complex, and essentially limitless source
of fascinating questions.
Note
1.
Return on investment.
ROL FESSENDEN is
Vice-President of Inventory Planning and Control at L. L. Bean, Inc. He
is also Co-Principal Investigator and Vice-Chair of Maine's State
Systemic Initiative and Chair of the Strategic Planning Committee. He
has previously served on the Mathematical Science Education Board, and
on the National Alliance for State Science and Mathematics Coalitions
(NASSMC).
Page 243— Integrating Vocational and Academic Education
THOMAS BAILEY
Columbia University
In high school education, preparation for work
immediately after high school and preparation for post-secondary
education have traditionally been viewed as incompatible. Work-bound
high-school students end up in vocational education tracks, where
courses usually emphasize specific skills with little attention to
underlying theoretical and conceptual foundations.1
College-bound students proceed through traditional academic
discipline-based courses, where they learn English, history, science,
mathematics, and foreign languages, with only weak and often contrived
references to applications of these skills in the workplace or in the
community outside the school. To be sure, many vocational teachers do
teach underlying concepts, and many academic teachers motivate their
lessons with examples and references to the world outside the classroom.
But these enrichments are mostly frills, not central to either the
content or pedagogy of secondary school education.
Rethinking Vocational and Academic Education
Educational thinking in the United States has
traditionally placed priority on college preparation. Thus the distinct
track of vocational education has been seen as an option for those
students who are deemed not capable of success in the more desirable
academic track. As vocational programs acquired a reputation
Page 25as a ''dumping ground," a strong background in
vocational courses (especially if they reduced credits in the core
academic courses) has been viewed as a threat to the college aspirations
of secondary school students.
This notion was further reinforced by the very influential 1983 report entitled A Nation at Risk
(National Commission on Excellence in Education, 1983), which
excoriated the U.S. educational system for moving away from an emphasis
on core academic subjects that, according to the report, had been the
basis of a previously successful American education system. Vocational
courses were seen as diverting high school students from core academic
activities. Despite the dubious empirical foundation of the report's
conclusions, subsequent reforms in most states increased the number of
academic courses required for graduation and reduced opportunities for
students to take vocational courses.
The distinction between vocational students and
college-bound students has always had a conceptual flaw. The large
majority of students who go to four-year colleges are motivated, at
least to a significant extent, by vocational objectives. In 1994, almost
247,000 bachelors degrees were conferred in business administration.
That was only 30,000 less than the total number (277,500) of 1994
bachelor degree conferred in English, mathematics, philosophy, religion,
physical sciences and science technologies, biological and life
sciences, social sciences, and history combined. Furthermore,
these "academic" fields are also vocational since many students who
graduate with these degrees intend to make their living working in those
fields.
Several recent economic, technological, and
educational trends challenge this sharp distinction between preparation
for college and for immediate post-high-school work, or, more
specifically, challenge the notion that students planning to work after
high school have little need for academic skills while college-bound
students are best served by an abstract education with only tenuous
contact with the world of work:
First, many employers and
analysts are arguing that, due to changes in the nature of work,
traditional approaches to teaching vocational skills may not be
effective in the future. Given the increasing pace of change and
uncertainty in the workplace, young people will be better prepared, even
for entry level positions and certainly for subsequent positions, if
they have an underlying understanding of the scientific, mathematical,
social, and even cultural aspects of the work that they will do. This
has led to a growing emphasis on integrating academic and vocational
education.2
Views about teaching and pedagogy have increasingly
moved toward a more open and collaborative "student-centered" or
"constructivist" teaching style that puts a great deal of emphasis on
having students work together on complex, open-ended projects. This
reform strategy is now widely implemented through the efforts of
organizations such as the Coalition of Essential Schools, the National
Center for Restructuring Education, Schools, and Teaching at
Page 26Teachers College, and
the Center for Education Research at the University of Wisconsin at
Madison. Advocates of this approach have not had much interaction with
vocational educators and have certainly not advocated any emphasis on
directly preparing high school students for work. Nevertheless, the
approach fits well with a reformed education that integrates vocational
and academic skills through authentic applications. Such applications
offer opportunities to explore and combine mathematical, scientific,
historical, literary, sociological, economic, and cultural issues.
In a related trend, the federal School-to-Work
Opportunities Act of 1994 defines an educational strategy that combines
constructivist pedagogical reforms with guided experiences in the
workplace or other non-work settings. At its best, school-to-work could
further integrate academic and vocational learning through appropriately
designed experiences at work.
The integration of vocational and academic education
and the initiatives funded by the School-to-Work Opportunities Act were
originally seen as strategies for preparing students for work after high
school or community college. Some educators and policy makers are
becoming convinced that these approaches can also be effective for
teaching academic skills and preparing students for four-year college.
Teaching academic skills in the context of realistic and complex
applications from the workplace and community can provide motivational
benefits and may impart a deeper understanding of the material by
showing students how the academic skills are actually used. Retention
may also be enhanced by giving students a chance to apply the knowledge
that they often learn only in the abstract.3
During the last twenty years, the real wages of high
school graduates have fallen and the gap between the wages earned by
high school and college graduates has grown significantly. Adults with
no education beyond high school have very little chance of earning
enough money to support a family with a moderate lifestyle.4
Given these wage trends, it seems appropriate and just that every high
school student at least be prepared for college, even if some choose to
work immediately after high school.
Innovative Examples
There are many examples of programs that use
work-related applications both to teach academic skills and to prepare
students for college. One approach is to organize high school programs
around broad industrial or occupational areas, such as health,
agriculture, hospitality, manufacturing, transportation, or the arts.
These broad areas offer many opportunities for wide-ranging curricula in
all academic disciplines. They also offer opportunities for
collaborative work among teachers from different disciplines. Specific
skills can still be taught in this format but in such a way as to
motivate broader academic and theoretical themes. Innovative programs
can now be found in many vocational
Page 27high schools in large cities, such as Aviation
High School in New York City and the High School of Agricultural Science
and Technology in Chicago. Other schools have organized
schools-within-schools based on broad industry areas.
Agriculturally based activities, such as 4H and
Future Farmers of America, have for many years used the farm setting and
students' interest in farming to teach a variety of skills. It takes
only a little imagination to think of how to use the social, economic,
and scientific bases of agriculture to motivate and illustrate skills
and knowledge from all of the academic disciplines. Many schools are now
using internships and projects based on local business activities as
teaching tools. One example among many is the integrated program offered
by the Thomas Jefferson High School for Science and Technology in
Virginia, linking biology, English, and technology through an
environmental issues forum. Students work as partners with resource
managers at the Mason Neck National Wildlife Refuge and the Mason Neck
State Park to collect data and monitor the daily activities of various
species that inhabit the region. They search current literature to
establish a hypothesis related to a real world problem, design an
experiment to test their hypothesis, run the experiment, collect and
analyze data, draw conclusions, and produce a written document that
communicates the results of the experiment. The students are even
responsible for determining what information and resources are needed
and how to access them. Student projects have included making plans for
public education programs dealing with environmental matters, finding
solutions to problems caused by encroaching land development, and making
suggestions for how to handle the overabundance of deer in the region.
These examples suggest the potential that a more
integrated education could have for all students. Thus continuing to
maintain a sharp distinction between vocational and academic instruction
in high school does not serve the interests of many of those students
headed for four-year or two-year college or of those who expect to work
after high school. Work-bound students will be better prepared for work
if they have stronger academic skills, and a high-quality curriculum
that integrates school-based learning into work and community
applications is an effective way to teach academic skills for many
students.
Despite the many examples of innovative
initiatives that suggest the potential for an integrated view, the
legacy of the duality between vocational and academic education and the
low status of work-related studies in high school continue to influence
education and education reform. In general, programs that deviate from
traditional college-prep organization and format are still viewed with
suspicion by parents and teachers focused on four-year college. Indeed,
college admissions practices still very much favor the traditional
approaches. Interdisciplinary courses, "applied" courses, internships,
and other types of work experience that characterize the school-to-work
strategy or programs that integrate academic and vocational education
often do not fit well into college admissions requirements.
Page 28Joining Work and Learning
What implications does this have for the
mathematics standards developed by the National Council of Teachers of
Mathematics (NCTM)? The general principle should be to try to design
standards that challenge rather than reinforce the distinction between
vocational and academic instruction. Academic teachers of mathematics
and those working to set academic standards need to continue to try to
understand the use of mathematics in the workplace and in everyday life.
Such understandings would offer insights that could suggest reform of
the traditional curriculum, but they would also provide a better
foundation for teaching mathematics using realistic applications. The
examples in this volume are particularly instructive because they
suggest the importance of problem solving, logic, and imagination and
show that these are all important parts of mathematical applications in
realistic work settings. But these are only a beginning.
In order to develop this approach, it would be
helpful if the NCTM standards writers worked closely with groups that
are setting industry standards.5 This would allow both groups to develop a deeper understanding of the mathematics content of work.
The NCTM's Curriculum Standards for Grades 9-12
include both core standards for all students and additional standards
for "college-intending" students. The argument presented in this essay
suggests that the NCTM should dispense with the distinction between
college intending and non-college intending students. Most of the
additional standards, those intended only for the "college intending"
students, provide background that is necessary or beneficial for the
calculus sequence. A re-evaluation of the role of calculus in the high
school curriculum may be appropriate, but calculus should not serve as a
wedge to separate college-bound from non-college-bound students.
Clearly, some high school students will take calculus, although many
college-bound students will not take calculus either in high school or
in college. Thus in practice, calculus is not a characteristic that
distinguishes between those who are or are not headed for college.
Perhaps standards for a variety of options beyond the core might be
offered. Mathematics standards should be set to encourage stronger
skills for all students and to illustrate the power and usefulness of
mathematics in many settings. They should not be used to
institutionalize dubious distinctions between groups of students.
References
Bailey, T. & Merritt, D. (1997).School-to-work for the collegebound. Berkeley, CA: National Center for Research in Vocational Education.
National Commission on Excellence in Education. (1983). A nation at risk: The imperative for educational reform. Washington, DC: Author.
Page 29Notes
1.
Vocational education has been shaped by
federal legislation since the first vocational education act was passed
in 1917. According to the current legislation, the Carl D. Perkins
Vocational and Technical Education Act of 1990, vocational students are
those not headed for a baccalaureate degree, so they include both
students expecting to work immediately after high school as well as
those expecting to go to a community college.
2.
This point of view underlies the reforms
articulated in the 1990 reauthorization of the Carl Perkins Vocational
and Technical Education Act (VATEA). VATEA also promoted a program,
dubbed "tech-prep," that established formal articulations between
secondary school and community college curricula.
3.
This argument is reviewed in Bailey &
Merritt (1997). For an argument about how education may be organized
around broad work themes can enhance learning in mathematics see
Hoachlander (1997).
4.
These wage data are reviewed in Levy & Murnane (1992).
5.
The Goals 2000: Educate America Act, for
example, established the National Skill Standards Board in 1994 to serve
as a catalyst in the development of a voluntary national system of
skills standards, assessments, and certifications for business and
industry.
THOMAS BAILEY
is an Associate Professor of Economics Education at Teachers College,
Columbia University. He is also Director of the Institute on Education
and the Economy and Director of the Community College Research Center,
both at Teachers College. He is also on the board of the National Center
for Research in Vocational Education.
Page 304— The Importance of Workplace and Everyday Mathematics
JEAN E. TAYLOR
Rutgers University
For decades our industrial society has been based
on fossil fuels. In today's knowledge-based society, mathematics is the
energy that drives the system. In the words of the new WQED television
series, Life by the Numbers, to create knowledge we "burn
mathematics." Mathematics is more than a fixed tool applied in known
ways. New mathematical techniques and analyses and even conceptual
frameworks are continually required in economics, in finance, in
materials science, in physics, in biology, in medicine.
Just as all scientific and health-service careers
are mathematically based, so are many others. Interaction with computers
has become a part of more and more jobs, and good analytical skills
enhance computer use and troubleshooting. In addition, virtually all
levels of management and many support positions in business and industry
require some mathematical understanding, including an ability to read
graphs and interpret other information presented visually, to use
estimation effectively, and to apply mathematical reasoning.
What Should Students Learn for Today's World?
Education in mathematics and the ability to
communicate its predictions is more important than ever for moving from
low-paying jobs into better-paying ones. For example, my local paper, The Times of Trenton, had a section "Focus
Page 31on Careers" on October 5, 1997 in which the
majority of the ads were for high technology careers (many more than for
sales and marketing, for example).
But precisely what mathematics should students
learn in school? Mathematicians and mathematics educators have been
discussing this question for decades. This essay presents some thoughts
about three areas of mathematics—estimation, trigonometry, and
algebra—and then some thoughts about teaching and learning.
Estimation is one of the harder skills for
students to learn, even if they experience relatively little difficulty
with other aspects of mathematics. Many students think of mathematics as
a set of precise rules yielding exact answers and are uncomfortable
with the idea of imprecise answers, especially when the degree of
precision in the estimate depends on the context and is not itself given
by a rule. Yet it is very important to be able to get an approximate
sense of the size an answer should be, as a way to get a rough check on
the accuracy of a calculation (I've personally used it in stores to
detect that I've been charged twice for the same item, as well as often
in my own mathematical work), a feasibility estimate, or as an
estimation for tips.
Trigonometry plays a significant role in the
sciences and can help us understand phenomena in everyday life. Often
introduced as a study of triangle measurement, trigonometry may be used
for surveying and for determining heights of trees, but its utility
extends vastly beyond these triangular applications. Students can
experience the power of mathematics by using sine and cosine to model
periodic phenomena such as going around and around a circle, going in
and out with tides, monitoring temperature or smog components changing
on a 24-hour cycle, or the cycling of predator-prey populations.
No educator argues the importance of algebra for
students aiming for mathematically-based careers because of the
foundation it provides for the more specialized education they will need
later. Yet, algebra is also important for those students who do not
currently aspire to mathematics-based careers, in part because a lack of
algebraic skills puts an upper bound on the types of careers to which a
student can aspire. Former civil rights leader Robert Moses makes a
good case for every student learning algebra, as a means of empowering
students and providing goals, skills, and opportunities. The same idea
was applied to learning calculus in the movie Stand and Deliver. How, then, can we help all students learn algebra?
For me personally, the impetus to learn algebra
was at least in part to learn methods of solution for puzzles. Suppose
you have 39 jars on three shelves. There are twice as many jars on the
second shelf as the first, and four more jars on the third shelf than on
the second shelf. How many jars are there on each shelf? Such problems
are not important by themselves, but if they show the students the power
of an idea by enabling them to solve puzzles that they'd like to solve,
then they have value. We can't expect such problems to interest all
students. How then can we reach more students?
Page 32Workplace and Everyday Settings as a Way of Making Sense
One of the common tools in business and industry
for investigating mathematical issues is the spreadsheet, which is
closely related to algebra. Writing a rule to combine the elements of
certain cells to produce the quantity that goes into another cell is
doing algebra, although the variables names are cell names rather than x or y. Therefore, setting up spreadsheet analyses requires some of the thinking that algebra requires.
By exploring mathematics via tasks which come from
workplace and everyday settings, and with the aid of common tools like
spreadsheets, students are more likely to see the relevance of the
mathematics and are more likely to learn it in ways that are personally
meaningful than when it is presented abstractly and applied later only
if time permits. Thus, this essay argues that workplace and everyday
tasks should be used for teaching mathematics and, in particular, for
teaching algebra. It would be a mistake, however, to rely exclusively on
such tasks, just as it would be a mistake to teach only spreadsheets in
place of algebra.
Communicating the results of an analysis is a
fundamental part of any use of mathematics on a job. There is a growing
emphasis in the workplace on group work and on the skills of
communicating ideas to colleagues and clients. But communicating
mathematical ideas is also a powerful tool for learning, for it requires
the student to sharpen often fuzzy ideas.
Some of the tasks in this volume can provide the
kinds of opportunities I am talking about. Another problem, with clear
connections to the real world, is the following, taken from the book
entitled Consider a Spherical Cow: A Course in Environmental Problem Solving,
by John Harte (1988). The question posed is: How does biomagnification
of a trace substance occur? For example, how do pesticides accumulate in
the food chain, becoming concentrated in predators such as condors?
Specifically, identify the critical ecological and chemical parameters
determining bioconcentrations in a food chain, and in terms of these
parameters, derive a formula for the concentration of a trace substance
in each link of a food chain. This task can be undertaken at several
different levels. The analysis in Harte's book is at a fairly high
level, although it still involves only algebra as a mathematical tool.
The task could be undertaken at a more simple level or, on the other
hand, it could be elaborated upon as suggested by further exercises
given in that book. And the students could then present the results of
their analyses to each other as well as the teacher, in oral or written
form.
Concepts or Procedures?
When teaching mathematics, it is easy to spend
so much time and energy focusing on the procedures that the concepts
receive little if any attention. When teaching algebra, students often
learn the procedures for using the quadratic formula or for solving
simultaneous equations without thinking of intersections of curves and
lines and without being able to apply the procedures in unfamiliar
settings. Even
Page 33when concentrating on word problems, students
often learn the procedures for solving "coin problems" and "train
problems" but don't see the larger algebraic context. The formulas and
procedures are important, but are not enough.
When using workplace and everyday tasks for
teaching mathematics, we must avoid falling into the same trap of
focusing on the procedures at the expense of the concepts. Avoiding the
trap is not easy, however, because just like many tasks in school
algebra, mathematically based workplace tasks often have standard
procedures that can be used without an understanding of the underlying
mathematics. To change a procedure to accommodate a changing business
climate, to respond to changes in the tax laws, or to apply or modify a
procedure to accommodate a similar situation, however, requires an
understanding of the mathematical ideas behind the procedures. In
particular, a student should be able to modify the procedures for
assessing energy usage for heating (as in Heating-Degree-Days, p. 54) in order to assess energy usage for cooling in the summer.
To prepare our students to make such modifications
on their own, it is important to focus on the concepts as well as the
procedures. Workplace and everyday tasks can provide opportunities for
students to attach meaning to the mathematical calculations and
procedures. If a student initially solves a problem without algebra,
then the thinking that went into his or her solution can help him or her
make sense out of algebraic approaches that are later presented by the
teacher or by other students. Such an approach is especially appropriate
for teaching algebra, because our teaching of algebra needs to reach
more students (too often it is seen by students as meaningless symbol
manipulation) and because algebraic thinking is increasingly important
in the workplace.
An Example: The Student/Professor Problem
To illustrate the complexity of learning algebra
meaningfully, consider the following problem from a study by Clement,
Lockhead, & Monk (1981):
Write an equation for the following statement: "There are six times as many students as professors at this university." Use S for the number of students and P for the number of professors. (p. 288)
The authors found that of 47 nonscience majors
taking college algebra, 57% got it wrong. What is more surprising,
however, is that of 150 calculus-level students, 37% missed the problem.
A first reaction to the most common wrong answer, 6S = P,
is that the students simply translated the words of the problems into
mathematical symbols without thinking more deeply about the situation or
the variables. (The authors note that some textbooks instruct students
to use such translation.)
By analyzing transcripts of interviews with
students, the authors found this approach and another (faulty) approach,
as well. These students often drew a diagram showing six students and
one professor. (Note that we often instruct students to draw diagrams
when solving word problems.) Reasoning
Page 34from the diagram, and regarding S and P as units, the student may write 6S = P,
just as we would correctly write 12 in. = 1 ft. Such reasoning is quite
sensible, though it misses the fundamental intent in the problem
statement that S is to represent the number of students, not a student.
Thus, two common suggestions for
students—word-for-word translation and drawing a diagram—can lead to an
incorrect answer to this apparently simple problem, if the students do
not more deeply contemplate what the variables are intended to
represent. The authors found that students who wrote and could explain
the correct answer, S = 6P, drew upon a richer understanding of what the equation and the variables represent.
Clearly, then, we must encourage students to
contemplate the meanings of variables. Yet, part of the power and
efficiency of algebra is precisely that one can manipulate symbols
independently of what they mean and then draw meaning out of the
conclusions to which the symbolic manipulations lead. Thus, stable,
long-term learning of algebraic thinking requires both mastery of
procedures and also deeper analytical thinking.
So it is education in mathematical thinking, in
applying mathematical computation, in assessing whether an answer is
reasonable, and in communicating the results that is essential. Teaching
mathematics via workplace and everyday problems is an approach that can
make mathematics more meaningful for all students. It is important,
however, to go beyond the specific details of a task in order to teach
mathematical ideas. While this approach is particularly crucial for
those students intending to pursue careers in the mathematical sciences,
it will also lead to deeper mathematical understanding for all
students.
JEAN E. TAYLOR is
Professor of Mathematics at Rutgers, the State University of New Jersey.
She is currently a member of the Board of Directors of the American
Association for the Advancement of Science and formerly chaired its
Section A Nominating Committee. She has served as Vice President and as a
Member-at-Large of the Council of the American Mathematical Society,
and served on its Executive Committee and its Nominating Committee. She
has also been a member of the Joint Policy Board for Mathematics, and a
member of the Board of Advisors to The Geometry Forum (now The
Mathematics Forum) and to the WQED television series, Life by the Numbers.
Page 355— Working with Algebra
DANIEL CHAZAN
Michigan State University
SANDRA CALLIS BETHELL
Holt High School
Teaching a mathematics class in which few of the
students have demonstrated success is a difficult assignment. Many
teachers avoid such assignments, when possible. On the one hand, high
school mathematics teachers, like Bertrand Russell, might love
mathematics and believe something like the following:
Mathematics, rightly viewed, possesses not
only truth, but supreme beauty—a beauty cold and austere, like that of
sculpture, without appeal to any part of our weaker nature, without the
gorgeous trappings of painting or music, yet sublimely pure, and capable
of a stern perfection such as only the greatest art can show. … Remote
from human passions, remote even from the pitiful facts of nature, the
generations have gradually created an ordered cosmos, where pure thought
can dwell as in its nature home, and where one, at least, of our nobler
impulses can escape from the dreary exile of the natural world.
(Russell, 1910, p. 73)
But, on the other hand, students may not have
the luxury, in their circumstances, of appreciating this beauty. Many of
them may not see themselves as thinkers because contemplation would
take them away from their primary
Page 36focus: how to get by in a world that was not created for them. Instead, like Jamaica Kincaid, they may be asking:
What makes the world turn against me and all
who look like me? I won nothing, I survey nothing, when I ask this
question, the luxury of an answer that will fill volumes does not
stretch out before me. When I ask this question, my voice is filled with
despair. (Kincaid, 1996, pp. 131-132)
Our Teaching and Issues it Raised
During the 1991-92 and 1992-93 school years, we
(a high school teacher and a university teacher educator) team taught a
lower track Algebra I class for 10th through 12th grade students.1
Most of our students had failed mathematics before, and many needed to
pass Algebra I in order to complete their high school mathematics
requirement for graduation. For our students, mathematics had become a
charged subject; it carried a heavy burden of negative experiences. Many
of our students were convinced that neither they nor their peers could
be successful in mathematics.
Few of our students did well in other academic
subjects, and few were headed on to two- or four-year colleges. But the
students differed in their affiliation with the high school. Some,
called ''preppies" or "jocks" by others, were active participants in the
school's activities. Others, "smokers" or "stoners," were rebelling to
differing degrees against school and more broadly against society. There
were strong tensions between members of these groups.2
Teaching in this setting gives added importance
and urgency to the typical questions of curriculum and motivation common
to most algebra classes. In our teaching, we explored questions such as
the following:
What is it that we really want
high school students, especially those who are not college-intending, to
study in algebra and why?
What is the role of algebra's manipulative skills
in a world with graphing calculators and computers? How do the
manipulative skills taught in the traditional curriculum give students a
new perspective on, and insight into, our world?
If our teaching efforts depend on students'
investment in learning, on what grounds can we appeal to them,
implicitly or explicitly, for energy and effort? In a tracked,
compulsory setting, how can we help students, with broad interests and
talents and many of whom are not college-intending, see value in a
shared exploration of algebra?
An Approach to School Algebra
As a result of thinking about these questions,
in our teaching we wanted to avoid being in the position of exhorting
students to appreciate the beauty or utility of algebra. Our students
were frankly skeptical of arguments based on
Page 37utility. They saw few people in their community
using algebra. We had also lost faith in the power of extrinsic rewards
and punishments, like failing grades. Many of our students were
skeptical of the power of the high school diploma to alter fundamentally
their life circumstances. We wanted students to find the mathematical
objects we were discussing in the world around them and thus learn to
value the perspective that this mathematics might give them on their
world.
To help us in this task, we found it useful to
take what we call a "relationships between quantities" approach to
school algebra. In this approach, the fundamental mathematical objects
of study in school algebra are functions that can be represented by
inputs and outputs listed in tables or sketched or plotted on graphs, as
well as calculation procedures that can be written with algebraic
symbols.3
Stimulated, in part, by the following quote from August Comte, we
viewed these functions as mathematical representations of theories
people have developed for explaining relationships between quantities.
In the light of previous experience, we must
acknowledge the impossibility of determining, by direct measurement,
most of the heights and distances we should like to know. It is this
general fact which makes the science of mathematics necessary. For in
renouncing the hope, in almost every case, of measuring great heights or
distances directly, the human mind has had to attempt to determine them
indirectly, and it is thus that philosophers were led to invent
mathematics. (Quoted in Serres, 1982, p. 85)
The "Sponsor" Project
Using this approach to the concept of function,
during the 1992-93 school year, we designed a year-long project for our
students. The project asked pairs of students to find the mathematical
objects we were studying in the workplace of a community sponsor.
Students visited the sponsor's workplace four times during the
year—three after-school visits and one day-long excused absence from
school. In these visits, the students came to know the workplace and
learned about the sponsor's work. We then asked students to write a
report describing the sponsor's workplace and answering questions about
the nature of the mathematical activity embedded in the workplace. The
questions are organized in Table 5-1.
Using These Questions
In order to determine how the interviews could
be structured and to provide students with a model, we chose to
interview Sandra's husband, John Bethell, who is a coatings inspector
for an engineering firm. When asked about his job, John responded, "I
argue for a living." He went on to describe his daily work inspecting
contractors painting water towers. Since most municipalities contract
with the lowest bidder when a water tower needs to be painted, they will
often hire an engineering firm to make sure that the contractor works
according to specification. Since the contractor has made a low bid,
there are strong
Page 38 5-1: Questions to ask in the workplace
QUANTITIES: MEASUREDORCOUNTEDVERSUSCOMPUTED
What quantities are measured or counted by the people you interview?
What kinds of tools are used to measure or count?
Why is it important to measure or count these quantities?
What quantities do they compute or calculate?
What kinds of tools are used to do the computing?
Why is it important to compute these quantities?
COMPUTINGQUANTITIES
When a quantity is computed, what information is needed and then what computations are done to get the desired result?
Are there ever different ways to compute the same thing?
REPRESENTINGQUANTITIESANDRELATIONSHIPSBETWEENQUANTITIES
How are quantities kept track of or represented in this line of work?
Collect examples of graphs, charts, tables, etc. that are used in the business.
How is information presented to clients or to others who work in the business?
COMPARISONS
What kinds of comparisons are made with computed quantities?
Why are these comparisons important to do?
What set of actions are set into motion as a result of interpretation of the computations?
financial incentives for the contractor to compromise on quality in order to make a profit.
In his work John does different kinds of
inspections. For example, he has a magnetic instrument to check the
thickness of the paint once it has been applied to the tower. When it
gives a "thin" reading, contractors often question the technology. To
argue for the reading, John uses the surface area of the tank, the
number of paint cans used, the volume of paint in the can, and an
understanding of the percentage of this volume that evaporates to
calculate the average thickness of the dry coating. Other examples from
his workplace involve the use of tables and measuring instruments of
different kinds.
Page 39Some Examples of Students' Work
When school started, students began working on
their projects. Although many of the sponsors initially indicated that
there were no mathematical dimensions to their work, students often were
able to show sponsors places where the mathematics we were studying was
to be found. For example, Jackie worked with a crop and soil scientist.
She was intrigued by the way in which measurement of weight is used to
count seeds. First, her sponsor would weigh a test batch of 100 seeds to
generate a benchmark weight. Then, instead of counting a large number
of seeds, the scientist would weigh an amount of seeds and compute the
number of seeds such a weight would contain.
Rebecca worked with a carpeting contractor who, in
estimating costs, read the dimensions of rectangular rooms off an
architect's blueprint, multiplied to find the area of the room in square
feet (doing conversions where necessary), then multiplied by a cost per
square foot (which depended on the type of carpet) to compute the cost
of the carpet. The purpose of these estimates was to prepare a bid for
the architect where the bid had to be as low as possible without making
the job unprofitable. Rebecca used a chart (Table 5-2) to explain this procedure to the class.
Joe and Mick, also working in construction, found
out that in laying pipes, there is a "one by one" rule of thumb. When
digging a trench for the placement of the pipe, the non-parallel sides
of the trapezoidal cross section must have a slope of 1 foot down for
every one foot across. This ratio guarantees that the dirt in the hole
will not slide down on itself. Thus, if at the bottom of the hole, the
trapezoid must have a certain width in order to fit the pipe, then on
ground level the hole must be this width plus twice the depth of the
hole. Knowing in advance how wide the hole must be avoids lengthy and
costly trial and error.
Other students found that functions were often
embedded in cultural artifacts found in the workplace. For example, a
student who visited a doctor's office brought in an instrument for
predicting the due dates of pregnant women, as well as providing
information about average fetal weight and length (Figure 5-1).
TABLE 5-2: Cost of carpet worksheet
INPUTS
OUTPUT
LENGTH
WIDTH
AREA OF THE ROOM
COST FOR CARPETING ROOM
10
35
20
25
15
30
Page 40 5-1: Pregnancy wheel
Conclusion
While the complexities of organizing this sort
of project should not be minimized—arranging sponsors, securing parental
permission, and meeting administrators and parent concerns about the
requirement of off-campus, after-school work—we remain intrigued by the
potential of such projects for helping students see mathematics in the
world around them. The notions of identifying central mathematical
objects for a course and then developing ways of identifying those
objects in students' experience seems like an important alternative to
the use of application-based materials written by developers whose lives
and social worlds may be quite different from those of students.
References
Eckert, P. (1989). Jocks and burnouts: Social categories and identity in the high school. New York: Teachers College Press.
Page 41Notes
1.
For other details, see Chazan (1996).
2.
For more detail on high school students' social groups, see Eckert (1989).
3.
Our ideas have been greatly influenced by
Schwartz & Yerushalmy (1992) and Yerushalmy & Schwartz (1993)
and are in the same spirit as the approach taken by Fey, Heid, et al.
(1995), Kieran, Boileau, & Garancon (1996), Nemirovsky (1996), and
Thompson (1993).
DANIEL CHAZAN
is an Associate Professor of Teacher Education at Michigan State
University. To assist his research in mathematics teaching and learning,
he has taught algebra at the high school level. His interests include
teaching mathematics by examining student ideas, using computers to
support student exploration, and the potential for the history and
philosophy of mathematics to inform teaching.
SANDRA CALLIS BETHELL
has taught mathematics and Spanish at Holt High School for 10 years.
She has also completed graduate work at Michigan State University and
Western Michigan University. She has interest in mathematics reform,
particularly in meeting the needs of diverse learners in algebra
courses.
Page 42Emergency Calls
Task
A city is served by two different ambulance
companies. City logs record the date, the time of the call, the
ambulance company, and the response time for each 911 call (Table 1).
Analyze these data and write a report to the City Council (with
supporting charts and graphs) advising it on which ambulance company the
911 operators should choose to dispatch for calls from this region.
TABLE 1: Ambulance dispatch log sheet, May 1–30
DATE OF CALL
TIME OF CALL
COMPANY NAME
RESPONSE TIME IN MINUTES
DATE OF CALL
TIME OF CALL
COMPANY NAME
RESPONSE TIME IN MINUTES
1
7:12 AM
Metro
11
12
8:30 PM
Arrow
8
1
7:43 PM
Metro
11
15
1:03 AM
Metro
12
2
10:02 PM
Arrow
7
15
6:40 AM
Arrow
17
2
12:22 PM
Metro
12
15
3:25 PM
Metro
15
3
5:30 AM
Arrow
17
16
4:15 AM
Metro
7
3
6:18 PM
Arrow
6
16
8:41 AM
Arrow
19
4
6:25 AM
Arrow
16
18
2:39 PM
Arrow
10
5
8:56 PM
Metro
10
18
3:44 PM
Metro
14
6
4:59 PM
Metro
14
19
6:33 AM
Metro
6
7
2:20 AM
Arrow
11
22
7:25 AM
Arrow
17
7
12:41 PM
Arrow
8
22
4:20 PM
Metro
19
7
2:29 PM
Metro
11
24
4:21 PM
Arrow
9
8
8:14 AM
Metro
8
25
8:07 AM
Arrow
15
8
6:23 PM
Metro
16
25
5:02 PM
Arrow
7
9
6:47 AM
Metro
9
26
10:51 AM
Metro
9
9
7:15 AM
Arrow
16
26
5:11 PM
Metro
18
9
6:10 PM
Arrow
8
27
4:16 AM
Arrow
10
10
5:37 PM
Metro
16
29
8:59 AM
Metro
11
10
9:37 PM
Metro
11
30
11:09 AM
Arrow
7
11
10:11 AM
Metro
8
30
9:15 PM
Arrow
8
11
11:45 AM
Metro
10
30
11:15 PM
Metro
8
Page 43Commentary
This problem confronts the student with a
realistic situation and a body of data regarding two ambulance
companies' response times to emergency calls. The data the student is
provided are typically "messy"—just a log of calls and response times,
ordered chronologically. The question is how to make sense of them.
Finding patterns in data such as these requires a productive mixture of
mathematics common sense, and intellectual detective work. It's the kind
of reasoning that students should be able to do—the kind of reasoning
that will pay off in the real world.
Mathematical Analysis
In this case, a numerical analysis is not
especially informative. On average, the companies are about the same:
Arrow has a mean response time of 11.4 minutes compared to 11.6 minutes
for Metro. The spread of the data is also not very helpful. The ranges
of their distributions are exactly the same: from 6 minutes to 19
minutes. The standard deviation of Arrow's response time is a little
longer—4.3 minutes versus 3.4 minutes for Metro—indicating that Arrow's
response times fluctuate a bit more.
Graphs of the response times (Figures 1 and 2)
reveal interesting features. Both companies, especially Arrow, seem to
have bimodal distributions, which is to say that there are two clusters
of data without much data in between.
FIGURE 1: Distribution of Arrow's response times
FIGURE 2: Distribution of Metro's response times
Page 44 distributions for both companies suggest
that there are some other factors at work. Might a particular driver be
the problem? Might the slow response times for either company be on
particular days of the week or at particular times of day? Graphs of the
response time versus the time of day (Figures 3 and 4) shed some light on these questions.
FIGURE 3: Arrow response times by time of day
FIGURE 4: Metro response times by time of day
These graphs show that Arrow's response times
were fast except between 5:30 AM and 9:00 AM, when they were about 9
minutes slower on average. Similarly, Metro's response times were fast
except between about 3:30 PM and 6:30 PM, when they were about 5 minutes
slower. Perhaps the locations of the companies make Arrow more
susceptible to the morning rush hour and Metro more susceptible to the
afternoon rush hour. On the other hand, the employees on Arrow's morning
shift or Metro's afternoon shift may not be efficient. To avoid slow
responses, one could recommend to the City Council that Metro be called
during the morning and that Arrow be called during the afternoon. A
little detective work into the sources of the differences between the
companies may yield a better recommendation.
Extensions
Comparisons may be drawn between two samples in
various contexts—response times for various services (taxis,
computer-help desks, 24-hour hot lines at automobile manufacturers)
being one class among many. Depending upon the circumstances, the data
may tell very different stories. Even in the situation above, if the
second pair of graphs hadn't offered such clear explanations, one might
have argued that although the response times for Arrow were better on
average the spread was larger, thus making their "extremes" more risky.
The fundamental idea is using various analysis and representation
techniques to make sense of data when the important factors are not
necessarily known ahead of time.
Page 45Back-of-the-Envelope Estimates
Task
Practice "back-of-the-envelope" estimates based
on rough approximations that can be derived from common sense or
everyday observations. Examples:
Consider a public high school
mathematics teacher who feels that students should work five nights a
week, averaging about 35 minutes a night, doing focused on-task work and
who intends to grade all homework with comments and corrections. What
is a reasonable number of hours per week that such a teacher should
allocate for grading homework?
How much paper does The New York Times
use in a week? A paper company that wishes to make a bid to become their
sole supplier needs to know whether they have enough current capacity.
If the company were to store a two-week supply of newspaper, will their
empty 14,000 square foot warehouse be big enough?
Commentary
Some 50 years ago, physicist Enrico Fermi asked
his students at the University of Chicago, "How many piano tuners are
there in Chicago?" By asking such questions, Fermi wanted his students
to make estimates that involved rough approximations so that their goal
would be not precision but the order of magnitude of their result. Thus,
many people today call these kinds of questions "Fermi questions."
These generally rough calculations often require little more than common
sense, everyday observations, and a scrap of paper, such as the back of
a used envelope.
Scientists and mathematicians use the idea of order of magnitude,
usually expressed as the closest power of ten, to give a rough sense of
the size of a quantity. In everyday conversation, people use a similar
idea when they talk about "being in the right ballpark." For example, a
full-time job at minimum wage yields an annual income on the order of
magnitude of $10,000 or 104 dollars. Some corporate
executives and professional athletes make annual salaries on the order
of magnitude of $10,000,000 or 107 dollars. To say that these salaries differ by a factor of 1000 or 103,
one can say that they differ by three orders of magnitude. Such a lack
of precision might seem unscientific or unmathematical, but such
approximations are quite useful in determining whether a more precise
measurement is feasible or necessary, what sort of action might be
required, or whether the result of a calculation is "in the right
ballpark." In choosing a strategy to protect an endangered species, for
example, scientists plan differently if there are 500 animals remaining
than if there are 5,000. On the other hand, determining whether 5,200 or
6,300 is a better estimate is not necessary, as the strategies will
probably be the same.
Careful reasoning with everyday observations can
usually produce Fermi estimates that are within an order of magnitude of
the exact answer (if there is one). Fermi estimates encourage students
to reason creatively with approximate quantities and uncertain
information. Experiences with such a process can help an individual
function in daily life to determine the reasonableness of numerical
calculations, of situations or ideas in the workplace, or of a proposed
tax cut. A quick estimate of some revenue- or profit-enhancing scheme
may show that the idea is comparable to suggesting that General Motors
enter the summer sidewalk lemonade market in your neighborhood. A quick
estimate could encourage further investigation or provide the rationale
to dismiss the idea.
Page 46Almost any numerical claim may be treated as a
Fermi question when the problem solver does not have access to all
necessary background information. In such a situation, one may make
rough guesses about relevant numbers, do a few calculations, and then
produce estimates.
Mathematical Analysis
The examples are solved separately below.
Grading Homework
Although many component factors vary greatly
from teacher to teacher or even from week to week, rough calculations
are not hard to make. Some important factors to consider for the teacher
are: how many classes he or she teaches, how many students are in each
of the classes, how much experience has the teacher had in general and
has the teacher previously taught the classes, and certainly, as part of
teaching style, the kind of homework the teacher assigns, not to
mention the teacher's efficiency in grading.
Suppose the teacher has 5 classes averaging 25
students per class. Because the teacher plans to write corrections and
comments, assume that the students' papers contain more than a list of
answers—they show some student work and, perhaps, explain some of the
solutions. Grading such papers might take as long as 10 minutes each, or
perhaps even longer. Assuming that the teacher can grade them as
quickly as 3 minutes each, on average, the teacher's grading time is:
This is an impressively large number, especially
for a teacher who already spends almost 25 hours/week in class, some
additional time in preparation, and some time meeting with individual
students. Is it reasonable to expect teachers to put in that kind of
time? What compromises or other changes might the teacher make to reduce
the amount of time? The calculation above offers four possibilities:
Reduce the time spent on each homework paper, reduce the number of
students per class, reduce the number of classes taught each day, or
reduce the number of days per week that homework will be collected. If
the teacher decides to spend at most 2 hours grading each night, what is
the total number of students for which the teacher should have
responsibility? This calculation is a partial reverse of the one above:
If the teacher still has 5 classes, that would mean 8 students per class!
The New York Times
Answering this question requires two preliminary estimates: the circulation of The New York Times and the size of the newspaper. The answers will probably be different on Sundays. Though The New York Times
is a national newspaper, the number of subscribers outside the New York
metropolitan area is probably small compared to the number inside. The
population of the New York metropolitan area is roughly ten million
people. Since most families buy at most one copy, and not all families
buy The New York Times, the circulation might be about 1
million newspapers each day. (A circulation of 500,000 seems too small
and 2 million seems too big.) The Sunday and weekday editions probably
have different
Page 47circulations, but assume that they are the same
since they probably differ by less than a factor of two—much less than
an order of magnitude. When folded, a weekday edition of the paper
measures about 1/2 inch thick, a little more than 1 foot long, and about
1 foot wide. A Sunday edition of the paper is the same width and
length, but perhaps 2 inches thick. For a week, then, the papers would
stack 6 × 1/2 + 2 = 5 inches thick, for a total volume of about 1 ft × 1
ft × 5/12 ft = 0.5 ft3.
The whole circulation, then, would require about
1/2 million cubic feet of paper per week, or about 1 million cubic feet
for a two-week supply.
Is the company's warehouse big enough? The paper
will come on rolls, but to make the estimates easy, assume it is
stacked. If it were stacked 10 feet high, the supply would require
100,000 square feet of floor space. The company's 14,000 square foot
storage facility will probably not be big enough as its size differs by
almost an order of magnitude from the estimate. The circulation estimate
and the size of the newspaper estimate should each be within a factor
of 2, implying that the 100,000 square foot estimate is off by at most a
factor of 4—less than an order of magnitude.
How big a warehouse is needed? An acre is 43,560
square feet so about two acres of land is needed. Alternatively, a
warehouse measuring 300 ft × 300 ft (the length of a football field in
both directions) would contain 90,000 square feet of floor space, giving
a rough idea of the size.
Extensions
After gaining some experience with these types
of problems, students can be encouraged to pay close attention to the
units and to be ready to make and support claims about the accuracy of
their estimates. Paying attention to units and including units as
algebraic quantities in calculations is a common technique in
engineering and the sciences. Reasoning about a formula by paying
attention only to the units is called dimensional analysis.
Sometimes, rather than a single estimate, it is
helpful to make estimates of upper and lower bounds. Such an approach
reinforces the idea that an exact answer is not the goal. In many
situations, students could first estimate upper and lower bounds, and
then collect some real data to determine whether the answer lies between
those bounds. In the traditional game of guessing the number of jelly
beans in a jar, for example, all students should be able to estimate
within an order of magnitude, or perhaps within a factor of two. Making
the closest guess, however, involves some chance.
Fermi questions are useful outside the workplace. Some Fermi questions have political ramifications:
How many miles of streets are in
your city or town? The police chief is considering increasing police
presence so that every street is patrolled by car at least once every 4
hours.
When will your town fill up its landfill? Is this
a very urgent matter for the town's waste management personnel to
assess in depth?
In his 1997 State of the Union address, President
Clinton renewed his call for a tax deduction of up to $10,000 for the
cost of college tuition. He estimates that 16.5 million students stand
to benefit. Is this a reasonable estimate of the number who might take
advantage of the tax deduction? How much will the deduction cost in lost
federal revenue?
Page 48Creating Fermi problems is easy. Simply ask
quantitative questions for which there is no practical way to determine
exact values. Students could be encouraged to make up their own.
Examples are: ''How many oak trees are there in Illinois?" or "How many
people in the U.S. ate chicken for dinner last night?" "If all the
people in the world were to jump in the ocean, how much would it raise
the water level?" Give students the opportunity to develop their own
Fermi problems and to share them with each other. It can stimulate some
real mathematical thinking.
Page 49Scheduling Elevators
Task
In some buildings, all of the elevators can
travel to all of the floors, while in others the elevators are
restricted to stopping only on certain floors. What is the advantage of
having elevators that travel only to certain floors? When is this worth
instituting?
Commentary
Scheduling elevators is a common example of an
optimization problem that has applications in all aspects of business
and industry. Optimal scheduling in general not only can save time and
money, but it can contribute to safety (e.g., in the airline industry).
The elevator problem further illustrates an important feature of many
economic and political arguments—the dilemma of trying simultaneously to
optimize several different needs.
Politicians often promise policies that will be
the least expensive, save the most lives, and be best for the
environment. Think of flood control or occupational safety rules, for
example. When we are lucky, we can perhaps find a strategy of least
cost, a strategy that saves the most lives, or a strategy that damages
the environment least. But these might not be the same strategies:
generally one cannot simultaneously satisfy two or more independent
optimization conditions. This is an important message for students to
learn, in order to become better educated and more critical consumers
and citizens.
In the elevator problem, customer satisfaction can
be emphasized by minimizing the average elevator time (waiting plus
riding) for employees in an office building. Minimizing wait-time during
rush hours means delivering many people quickly, which might be
accomplished by filling the elevators and making few stops. During
off-peak hours, however, minimizing wait-time means maximizing the
availability of the elevators. There is no reason to believe that these
two goals will yield the same strategy. Finding the best strategy for
each is a mathematical problem; choosing one of the two strategies or a
compromise strategy is a management decision, not a mathematical
deduction.
This example serves to introduce a complex topic
whose analysis is well within the range of high school students. Though
the calculations require little more than arithmetic, the task puts a
premium on the creation of reasonable alternative strategies. Students
should recognize that some configurations (e.g., all but one elevator
going to the top floor and the one going to all the others) do not merit
consideration, while others are plausible. A systematic evaluation of
all possible configurations is usually required to find the optimal
solution. Such a systematic search of the possible solution space is
important in many modeling situations where a formal optimal strategy is
not known. Creating and evaluating reasonable strategies for the
elevators is quite appropriate for high school student mathematics and
lends itself well to thoughtful group effort. How do you invent new
strategies? How do you know that you have considered all plausible
strategies? These are mathematical questions, and they are especially
amenable to group discussion.
Students should be able to use the techniques
first developed in solving a simple case with only a few stories and a
few elevators to address more realistic situations (e.g., 50 stories,
five elevators). Using the results of a similar but simpler problem to
model a more complicated problem is an important way to reason in
mathematics. Students
Page 50need to determine what data and variables are
relevant. Start by establishing the kind of building—a hotel, an office
building, an apartment building? How many people are on the different
floors? What are their normal destinations (e.g., primarily the ground
floor or, perhaps, a roof-top restaurant). What happens during rush
hours?
To be successful at the elevator task, students
must first develop a mathematical model of the problem. The model might
be a graphical representation for each elevator, with time on the
horizontal axis and the floors represented on the vertical axis, or a
tabular representation indicating the time spent on each floor. Students
must identify the pertinent variables and make simplifying assumptions
about which of the possible floors an elevator will visit.
Mathematical Analysis
This section works through some of the details
in a particularly simple case. Consider an office building with six
occupied floors, employing 240 people, and a ground floor that is not
used for business. Suppose there are three elevators, each of which can
hold 10 people. Further suppose that each elevator takes approximately
25 seconds to fill on the ground floor, then takes 5 seconds to move
between floors and 15 seconds to open and close at each floor on which
it stops.
Scenario One
What happens in the morning when everyone
arrives for work? Assume that everyone arrives at approximately the same
time and enters the elevators on the ground floor. If all elevators go
to all floors and if the 240 people are evenly divided among all three
elevators, each elevator will have to make 8 trips of 10 people each.
When considering a single trip of one elevator,
assume for simplicity that 10 people get on the elevator at the ground
floor and that it stops at each floor on the way up, because there may
be an occupant heading to each floor. Adding 5 seconds to move to each
floor and 15 seconds to stop yields 20 seconds for each of the six
floors. On the way down, since no one is being picked up or let off, the
elevator does not stop, taking 5 seconds for each of six floors for a
total of 30 seconds. This round-trip is represented in Table 1.
TABLE 1: Elevator round-trip time, Scenario one
TIME (SEC)
Ground Floor
25
Floor 1
20
Floor 2
20
Floor 3
20
Floor 4
20
Floor 5
20
Floor 6
20
Return
30
ROUND-TRIP
175
Since each elevator makes 8 trips, the total time will be 1,400 seconds or 23 minutes, 20 seconds.
Scenario Two
Now suppose that one elevator serves floors 1–3
and, because of the longer trip, two elevators are assigned to floors
4–6. The elevators serving the top
Page 51 2: Elevator round-trip times, Scenario two
ELEVATOR A
ELEVATORS B & C
Stop Time
STOP TIME
Ground Floor
25
25
Floor 1
1
20
5
Floor 2
2
20
5
Floor 3
3
20
5
Floor 4
0
4
20
Floor 5
0
5
20
Floor 6
0
6
20
Return
15
30
ROUND-TRIP
100
130
floors will save 15 seconds for each of floors
1–3 by not stopping. The elevator serving the bottom floors will save 20
seconds for each of the top floors and will save time on the return
trip as well. The times for these trips are shown in Table 2.
Assuming the employees are evenly distributed
among the floors (40 people per floor), elevator A will transport 120
people, requiring 12 trips, and elevators B and C will transport 120
people, requiring 6 trips each. These trips will take 1200 seconds (20
minutes) for elevator A and 780 seconds (13 minutes) for elevators B and
C, resulting in a small time savings (about 3 minutes) over the first
scenario. Because elevators B and C are finished so much sooner than
elevator A, there is likely a more efficient solution.
Scenario Three
The two round-trip times in Table 2
do not differ by much because the elevators move quickly between floors
but stop at floors relatively slowly. This observation suggests that a
more efficient arrangement might be to assign each elevator to a pair of
floors. The times for such a scenario are listed in Table 3.
Again assuming 40 employees per floor, each
elevator will deliver 80 people, requiring 8 trips, taking at most a
total of 920 seconds. Thus this assignment of elevators results in a
time savings of almost 35% when compared with the 1400 seconds it would
take to deliver all employees via unassigned elevators.
TABLE 3: Elevator round-trip times, Scenario three
ELEVATOR A
ELEVATOR B
ELEVATOR C
STOP TIME
STOP TIME
STOP TIME
Ground Floor
25
25
25
Floor 1
1
20
5
5
Floor 2
2
20
5
5
Floor 3
0
3
20
5
Floor 4
0
4
20
5
Floor 5
0
0
5
20
Floor 6
0
0
6
20
Return
10
20
30
ROUND-TRIP
75
95
115
Page 52Perhaps this is the optimal solution. If so, then the above analysis of this simple case suggests two hypotheses:
The optimal solution assigns each floor to a single elevator.
If the time for stopping is sufficiently larger than
the time for moving between floors, each elevator should serve the same
number of floors.
Mathematically, one could try to show that
this solution is optimal by trying all possible elevator assignments or
by carefully reasoning, perhaps by showing that the above hypotheses are
correct. Practically, however, it doesn't matter because this solution
considers only the morning rush hour and ignores periods of low use.
The assignment is clearly not optimal during
periods of low use, and much of the inefficiency is related to the first
hypothesis for rush hour optimization: that each floor is served by a
single elevator. With this condition, if an employee on floor 6 arrives
at the ground floor just after elevator C has departed, for example, she
or he will have to wait nearly two minutes for elevator C to return,
even if elevators A and B are idle. There are other inefficiencies that
are not considered by focusing on the rush hour. Because each floor is
served by a single elevator, an employee who wishes to travel from floor
3 to floor 6, for example, must go via the ground floor and switch
elevators. Most employees would prefer more flexibility than a single
elevator serving each floor.
At times when the elevators are not all busy,
unassigned elevators will provide the quickest response and the greatest
flexibility.
Because this optimal solution conflicts with
the optimal rush hour solution, some compromise is necessary. In this
simple case, perhaps elevator A could serve all floors, elevator B could
serve floors 1-3, and elevator C could serve floors 4-6.
The second hypothesis, above, deserves some further thought. The efficiency of the rush hour solution Table 3
is due in part to the even division of employees among the floors. If
employees were unevenly distributed with, say, 120 of the 240 people
working on the top two floors, then elevator C would need to make 12
trips, taking a total of 1380 seconds, resulting in almost no benefit
over unassigned elevators. Thus, an efficient solution in an actual
building must take into account the distribution of the employees among
the floors.
Because the stopping time on each floor is
three times as large as the traveling time between floors (15 seconds
versus 5 seconds), this solution effectively ignores the traveling time
by assigning the same number of employees to each elevator. For taller
buildings, the traveling time will become more significant. In those
cases fewer employees should be assigned to the elevators that serve the
upper floors than are assigned to the elevators that serve the lower
floors.
Extensions
The problem can be made more challenging by
altering the number of elevators, the number of floors, and the number
of individuals working on each floor. The rate of movement of elevators
can be determined by observing buildings in the local area. Some
elevators move more quickly than others. Entrance and exit times could
also be measured by students collecting
Page 53data on local elevators. In a similar manner, the number of workers, elevators, and floors could be taken from local contexts.
A related question is, where should the elevators
go when not in use? Is it best for them to return to the ground floor?
Should they remain where they were last sent? Should they distribute
themselves evenly among the floors? Or should they go to floors of
anticipated heavy traffic? The answers will depend on the nature of the
building and the time of day. Without analysis, it will not be at all
clear which strategy is best under specific conditions. In some
buildings, the elevators are controlled by computer programs that
"learn" and then anticipate the traffic patterns in the building.
A different example that students can easily
explore in detail is the problem of situating a fire station or an
emergency room in a city. Here the key issue concerns travel times to
the region being served, with conflicting optimization goals: average
time vs. maximum time. A location that minimizes the maximum time of
response may not produce the least average time of response. Commuters
often face similar choices in selecting routes to work. They may want to
minimize the average time, the maximum time, or perhaps the variance,
so that their departure and arrival times are more predictable.
Most of the optimization conditions discussed so
far have been expressed in units of time. Sometimes, however, two
optimization conditions yield strategies whose outcomes are expressed in
different (and sometimes incompatible) units of measurement. In many
public policy issues (e.g., health insurance) the units are lives and
money. For environmental issues, sometimes the units themselves are
difficult to identify (e.g., quality of life).
When one of the units is money, it is easy to find
expensive strategies but impossible to find ones that have virtually no
cost. In some situations, such as airline safety, which balances lives
versus dollars, there is no strategy that minimize lives lost (since
additional dollars always produce slight increases in safety), and the
strategy that minimizes dollars will be at $0. Clearly some compromise
is necessary. Working with models of different solutions can help
students understand the consequences of some of the compromises.
Page 54Heating-Degree-Days
Task
An energy consulting firm that recommends and
installs insulation and similar energy saving devices has received a
complaint from a customer. Last summer she paid $540 to insulate her
attic on the prediction that it would save 10% on her natural gas bills.
Her gas bills have been higher than the previous winter, however, and
now she wants a refund on the cost of the insulation. She admits that
this winter has been colder than the last, but she had expected still to
see some savings.
The facts: This winter the customer has used 1,102
therms, whereas last winter she used only 1,054 therms. This winter has
been colder: 5,101 heating-degree-days this winter compared to 4,201
heating-degree-days last winter. (See explanation below.) How does a
representative of the energy consulting firm explain to this customer
that the accumulated heating-degree-days measure how much colder this
winter has been, and then explain how to calculate her anticipated
versus her actual savings.
Commentary
Explaining the mathematics behind a situation
can be challenging and requires a real knowledge of the context, the
procedures, and the underlying mathematical concepts. Such communication
of mathematical ideas is a powerful learning device for students of
mathematics as well as an important skill for the workplace. Though the
procedure for this problem involves only proportions, a thorough
explanation of the mathematics behind the procedure requires
understanding of linear modeling and related algebraic reasoning,
accumulation and other precursors of calculus, as well as an
understanding of energy usage in home heating.
Mathematical Analysis
The customer seems to understand that a straight
comparison of gas usage does not take into account the added costs of
colder weather, which can be significant. But before calculating any
anticipated or actual savings, the customer needs some understanding of
heating-degree-days. For many years, weather services and oil and gas
companies have been using heating-degree-days to explain and predict
energy usage and to measure energy savings of insulation and other
devices. Similar degree-day units are also used in studying insect
populations and crop growth. The concept provides a simple measure of
the accumulated amount of cold or warm weather over time. In the
discussion that follows, all temperatures are given in degrees
Fahrenheit, although the process is equally workable using degrees
Celsius.
Suppose, for example, that the minimum temperature
in a city on a given day is 52 degrees and the maximum temperature is
64 degrees. The average temperature for the day is then taken to be 58
degrees. Subtracting that result from 65 degrees (the cutoff point for
heating), yields 7 heating-degree-days for the day. By recording high
and low temperatures and computing their average each day,
heating-degree-days can be accumulated over the course of a month, a
winter, or any period of time as a measure of the coldness of that
period.
Over five consecutive days, for example, if the
average temperatures were 58, 50, 60, 67, and 56 degrees Fahrenheit, the
calculation yields 7, 15, 5, 0, and 9 heating-degree-days respectively,
for a total accumulation of 36 heating-degree-days for the five days.
Note that the fourth day contributes 0 heating-degree-days to the total
because the temperature was above 65 degrees.
Page 55 relationship between average temperatures and heating-degree-days is represented graphically in Figure 1.
The average temperatures are shown along the solid line graph. The area
of each shaded rectangle represents the number of heating-degree-days
for that day, because the width of each rectangle is one day and the
height of each rectangle is the number of degrees below 65 degrees. Over
time, the sum of the areas of the rectangles represents the number of
heating-degree-days accumulated during the period. (Teachers of calculus
will recognize connections between these ideas and integral calculus.)
The statement that accumulated heating-degree-days
should be proportional to gas or heating oil usage is based primarily
on two assumptions: first, on a day for which the average temperature is
above 65 degrees, no heating should be required, and therefore there
should be no gas or heating oil usage; second, a day for which the
average temperature is 25 degrees (40 heating-degree-days) should
require twice as much heating as a day for which the average temperature
is 45
FIGURE 1: Daily heating-degree-days
degrees (20 heating-degree-days) because there is twice the temperature difference from the 65 degree cutoff.
The first assumption is reasonable because most
people would not turn on their heat if the temperature outside is above
65 degrees. The second assumption is consistent with Newton's law of
cooling, which states that the rate at which an object cools is
proportional to the difference in temperature between the object and its
environment. That is, a house which is 40 degrees warmer than its
environment will cool at twice the rate (and therefore consume energy at
twice the rate to keep warm) of a house which is 20 degrees warmer than
its environment.
The customer who accepts the heating-degree-day
model as a measure of energy usage can compare this winter's usage with
that of last winter. Because 5,101/4,201 = 1.21, this winter has been
21% colder than last winter, and therefore each house should require 21%
more heat than last winter. If this customer hadn't installed the
insulation, she would have required 21% more heat than last year, or
about 1,275 therms. Instead, she has required only 5% more heat
(1,102/1,054 = 1.05), yielding a savings of 14% off what would have been
required (1,102/1,275 = .86).
Another approach to this would be to note that
last year the customer used 1,054 therms/4,201 heating-degree-days =
.251 therms/heating-degree-day, whereas this year she has used 1,102
therms/5,101 heating-degree-days = .216 therms/heating-degree-day, a
savings of 14%, as before.
Page 56Extensions
How good is the heating-degree-day model in
predicting energy usage? In a home that has a thermometer and a gas
meter or a gauge on a tank, students could record daily data for gas
usage and high and low temperature to test the accuracy of the model.
Data collection would require only a few minutes per day for students
using an electronic indoor/outdoor thermometer that tracks high and low
temperatures. Of course, gas used for cooking and heating water needs to
be taken into account. For homes in which the gas tank has no gauge or
doesn't provide accurate enough data, a similar experiment could be
performed relating accumulated heating-degree-days to gas or oil usage
between fill-ups.
It turns out that in well-sealed modern houses,
the cutoff temperature for heating can be lower than 65 degrees
(sometimes as low as 55 degrees) because of heat generated by light
bulbs, appliances, cooking, people, and pets. At temperatures
sufficiently below the cutoff, linearity turns out to be a good
assumption. Linear regression on the daily usage data (collected as
suggested above) ought to find an equation something like U = -.251(T - 65), where T is the average temperature and U
is the gas usage. Note that the slope, -.251, is the gas usage per
heating-degree-day, and 65 is the cutoff. Note also that the
accumulation of heating-degree-days takes a linear equation and turns it
into a proportion. There are some important data analysis issues that
could be addressed by such an investigation. It is sometimes dangerous,
for example, to assume linearity with only a few data points, yet this
widely used model essentially assumes linearity from only one data
point, the other point having coordinates of 65 degrees, 0 gas usage.
Over what range of temperatures, if any, is this a
reasonable assumption? Is the standard method of computing average
temperature a good method? If, for example, a day is mostly near 20
degrees but warms up to 50 degrees for a short time in the afternoon, is
35 heating-degree-days a good measure of the heating required that day?
Computing averages of functions over time is a standard problem that
can be solved with integral calculus. With knowledge of typical and
extreme rates of temperature change, this could become a calculus
problem or a problem for approximate solution by graphical methods
without calculus, providing background experience for some of the
important ideas in calculus.
Students could also investigate actual savings
after insulating a home in their school district. A customer might
typically see 8-10% savings for insulating roofs, although if the house
is framed so that the walls act like chimneys, ducting air from the
house and the basement into the attic, there might be very little
savings. Eliminating significant leaks, on the other hand, can yield
savings of as much as 25%.
Some U.S. Department of Energy studies discuss the
relationship between heating-degree-days and performance and find the
cutoff temperature to be lower in some modern houses. State energy
offices also have useful documents.
What is the relationship between
heating-degree-days computed using degrees Fahrenheit, as above, and
heating-degree-days computed using degrees Celsius? Showing that the
proper conversion is a direct proportion and not the standard
Fahrenheit-Celsius conversion formula requires some careful and
sophisticated mathematical thinking. | 677.169 | 1 |
Once again keeping a keen ear to the needs of the evolving calculus community, Stewart created this text at the suggestion and with the collaboration of professors in the mathematics department at Texas A&M University. With an early introduction to vectors and vector functions, the approach is ideal for engineering students who use vectors early in their curriculum. Stewart begins by introducing vectors in Chapter 1, along with their basic operations, such as addition, scalar multiplication, and dot product. The definition of vector functions and parametric curves is given at the end of Chapter 1 using a two-dimensional trajectory of a projectile as motivation. Limits, derivatives, and integrals of vector functions are interwoven throughout the subsequent chapters.
As with the other texts in his Calculus series, in Early Vectors Stewart makes us of heuristic examples to reveal calculus to students. His examples stand out because they are not just models for problem solving or a means of demonstrating techniques - they also encourage students to develop an analytic view of the subject. This heuristic or discovery approach in the examples give students an intuitive feeling for analysis.
Meet the Authors
James Stewart,
McMaster UniversityFeatures
Stewart makes use of heuristic examples to reveal calculus to students. His examples stand out because they are not just models for problem solving or a means of demonstrating techniques, they also encourage students to develop an analytic view of the subject. This heuristic or discovery approach in the examples gives students an intuitive feeling for calculus.
Vectors are introduced in Chapter 1, along with their basic operations, such as addition, scalar multiplication, and dot product. The definition of vector functions is given at the end of Chapter 1 using a two-dimensional trajectory of a projectile as motivation.
Chapters 2 through 6 are modifications of Chapters 1 through 5 of CALCULUS: EARLY TRANSCENDENTALS, Third Edition.
In Chapter 2, on Limits, the usual definitions of limits and continuity are unchanged; however the end of these sections contain the necessary modifications to accommodate vector functions.
Chapter 3, on Derivatives, contains two sections on the derivative of a vector function and parameterized curve.
Chapters 4 and 5 (inverse functions and curve sketching) are the same as Chapters 3 and 4 of CALCULUS: EARLY TRANSCENDENTALS, Third Edition with the addition of a few vector-valued examples and exercises.
This new text is a version of Stewart's renowned CALCULUS series designed to meet the needs of engineering students who use vectors early in the course. | 677.169 | 1 |
How to Read and Do Proofs an Introduction to Math Ematical Thought Processes 6E(Paperback)
Synopsis
This text makes a great supplement and provides a systematic approach for teaching undergraduate and graduate students how to read, understand, think about, and do proofs. The approach is to categorize, identify, and explain (at the student's level) the various techniques that are used repeatedly in all proofs, regardless of the subject in which the proofs arise. How to Read and Do Proofs also explains when each technique is likely to be used, based on certain key words that appear in the problem under consideration. Doing so enables students to choose a technique consciously, based on the form of the problem | 677.169 | 1 |
Mathematics
A Basic Introduction
The book covers all the basic areas of mathematics including calculating, fractions, decimals, percentages, measuring, graphs and formulae. Each chapter includes not only an explanation of the knowledge and skills you need, but also worked examples and test questions. | 677.169 | 1 |
Trigonometry, 3rd Edition
The third edition of Cynthia Young's TrigonometryThe seamless integration of Cynthia Young's Trigonometry 3rd edition with WileyPLUS, a research-based, online environment for effective teaching and learning, continues Young's vision of building student confidence in mathematics because it takes the guesswork out of studying by providing them with a clear roadmap: what to do, how to do it, and whether they did it right.
Clear, Concise and Inviting
Writing. The author's engaging and
clear presentation is presented in a layout that is designed to
reduce math anxiety in students.
Author Lecture Videos. For
review at home or for a class missed, Cynthia Young has recorded
over 300 instructional videos including worked examples and "Your
Turn" problems from the text. These videos can
be found in WileyPLUS and are denoted in the text with an
icon.
WileyPLUS, accompanied with
Young,Trigonometry3rd
edition, provides a research based, online
environment for effective teaching and learning. WileyPLUS
builds students' confidence because it takes the guesswork
out of studying by providing students with a clear roadmap: what to
do, how to do it, if they did it right.
Six Different Types of Exercises. Every chapter has skill, application, catch the mistake,
challenge, conceptual, and technology exercises. The
exercises gradually increase in difficulty and vary skill and
conceptual emphasis.
Correct vs. Incorrect. In
addition to standard examples, some problems are worked both
correctly and incorrectly to highlight common errors students
make. Counter examples, like these, are often an effective
learning approach for many students.
Catch the Mistake. In every
section, 'Catch the Mistake' exercises put the students
in the role of the instructor grading homework which increases the
depth of understanding and reinforces what they have
learned.
Your Turn. Students are often
asked to work a problem immediately following an example to
reinforce and check their understanding. This helps them
build confidence as they progress in the chapter. These are ideal
for in-class activity and preparing the student to work homework
later.
Parallel Words and Math. This
text reverses the common presentation of examples by placing the
explanation in words on the left and the mathematics in
parallel on the right. This makes it easier for
students to read through examples as the material flows more
naturally and as commonly presented in lecture.
Modeling Our World.Found in every
chapter, these projects engage students by using real world data to
model mathematical applications found in everyday life.
Chapter Cumulative Test.Included at
the end of each chapter to assess and improve students' retention
of material.
Authored by Cynthia Young, the manual provides practical advice on teaching with the text.
PowerPoint Slides
For each section of the book, a corresponding set of lecture notes and worked out examples are presented as PowerPoint slides, available on the Book Companion Site and WileyPLUS.
Computerized/Printed Test Bank
Contains approximately 900 questions and answers from every section of the text
Annotated Instructor's Edition
Displays answers to all exercise questions (which can be found in the back of the book) and provides additional classroom examples within the standard difficulty range of the in-text exercises, as well as challenge problems to assess your students mastery of the material. | 677.169 | 1 |
Teaching Math 5H: UNIT 3 Differentiation, Quarter I, Week 5
We started UNIT 3 about Differentiation! We derived the Product and Quotient Rules. We did all 6 Trig Rules. We started talking about the Chain Rule.
Quarter I Week 3: 9/23-9/27
We finished UNIT 2 with a test! Our last topic concerned the existence of the derivative. BTW, we start with UNIT 2 since UNIT 1 is just a review of PreCalculus. We did a bit of that in 106 and 107 on Conics and Polar Notation.
Quarter I Week 2: 9/16-9/20
We are almost finished with UNIT 2: Continuity and Differentiablility! We demostrated the Power Rule using the Definition of the Derivative as the limit of the Difference Quotient. We even started the Trig Rules!
Quarter I Week 1: 9/9-9/13
AP Calculus BC started with a preCalc review in the form of the topic of Conic Sections! This is a great topic to review Cartesian and Polar Coordinates, as well as some algebra, trig and TI89 usage!
YouTube Wednesdays this month are about Admiral Grace Murray Hopper! The first video is linked above as it's not on YouTube anymore, CBS had me remove it.... I know Fridays aren't YouTube Wednesdays, but I had to show the following filks about continuity and differentiability!
ScreenCasts and SmartNotes and Code, oh my:
Below, you will usually find ScreenCasts from this week. SmartNotes are from a previous year's TI89 based course. We don't code much in this class except for programming the TI89. | 677.169 | 1 |
Pre-requisites
Restrictions
Overview
This module will give an introduction to nonlinear ordinary differential equations and difference equations. Such ordinary differential equations and difference equations have a variety of applications such as Mathematical Biology and Ecology. The emphasis will be on developing an understanding of ordinary differential equations and difference equations and using analytical and computational techniques to analyse them. Topics include: phase plane, equilibria and stability analysis; periodic solutions and limit cycles; Poincare-Bendixson theorem; dynamics of difference equations: cobwebs, equilibria, stability and periodic solutions; the discrete logistic model and chaos. The material is chosen so as to demonstrate the range of modern analytical and computational techniques available for solving nonlinear ordinary differential equations and difference equations and to illustrate the many different applications which are modelled by such equations. A range of Mathematical tools are drawn together to study the nonlinear equations, including computation through the use of MAPLE.
Learning outcomes
On successfully completing the level 6 module students will be able to:
1 demonstrate systematic understanding of key aspects of introductory nonlinear systems;
2 demonstrate the capability to deploy established approaches accurately to analyse and solve problems using a reasonable level of skill in calculation and manipulation of the material in the following areas: equilibra for both nonlinear differential and difference equations and their stability, phase portraits, the existence of limit cycles;
3 apply key aspects of nonlinear systems in well-defined contexts, showing judgement in the selection and application of tools and techniques;
4 show judgement in the selection and application of Maple.
The intended generic learning outcomes.
On successfully completing the level 6 module students will be able competent use of information technology skills such online resources (Moodle), internet communication;
7 communicate technical material competently;
8 demonstrate an increased level of skill in numeracy and computation;
9 demonstrate the acquisition of the study skills needed for continuing professional development.
University of Kent makes every effort to ensure that module information is accurate for the relevant academic session and to provide educational services as described. However, courses, services and other matters may be subject to change. Please read our full disclaimer. | 677.169 | 1 |
Educational Aims
This course aims to provide the student with an understanding of functions, limits, and series, and a knowledge of the basic techniques of differentiation and integration. The purpose of this course is to study functions of a single real variable. Some of the topics will be familiar, others will be studied more thoroughly in subsequent courses.
The module begins by introducing examples of functions and their graphs, and techniques for building new functions from old. We consider rational functions and the exponential function. We then consider the notion of a limit, sequences and series and then introduce the main tools of calculus. The derivative measures the rate of change of a function and the integral measures the area under the graph of a function. The rules for calculating derivatives are obtained form the definition of the derivative as a rate of change. Taylor series are calculated for functions such as sin, cos and the hyperbolic functions. We then introduce the integral and review techniques for calculating integrals. We learn how to add, multiply and divide polynomials and introduce rational functions and their partial fractions. Rational functions are important in calculations, and we learn how to integrate rational functions systematically. The exponential function is defined by means of a power series which is subsequently extended to the complex exponential function of an imaginary variable, so that students understand the connection between analysis, trigonometry and geometry. The trigonometric and hyperbolic functions are introduced in parallel with analogous power series, so that students understand the role of functional identities. Such functional identities are later used to simplify integrals and to parameterize geometrical curves.
Outline Syllabus
Arithmetic of complex numbers;
Polynomials;
Rational functions and partial fractions;
Exponential and hyperbolic functions;
Compositions and inverses;
Induction;
Sequences and limits;
Differentiation;
Product and Chain rules;
Maxima and minima;
Taylor series;
Complex exponentials and trigonometric functions;
Definite integral as areas;
Fundamental theorem of calculus;
Integration by parts and by substitution;
Assessment Proportions
Coursework: 50%
Exam: 50%
MATH102: Further Calculus
Terms Taught: Michaelmas Term Only
US Credits: 2 semester credits
ECTS Credits: 4 ECTS
Pre-requisites: A year of general mathematics, including calculus.
Course Description
The course covers: improper integrals; integration over infinite ranges; Simpson's rule; functions of two or more real variables; partial derivatives; curves in the plane; implicit functions; the chain rule for differentiating along a curve; stationary points for functions of two real variables; double and repeated integrals; Cavalieri's slicing principle; volumes.
Educational Aims
The first part of this course extends ideas of MATH101 from functions of a single real variable to functions of two real variables. The notions of differentiation and integration are extended from functions defined on a line to functions defined on the plane. Partial derivatives help us to understand surfaces, while repeated integrals enable us to calculate volumes.
Outline Syllabus
Complex polynomials and complex roots;
Integration of rational functions;
Improper integrals
Integration over infinite ranges;
Simpson's rule;
Functions of two or more real variables;
Partial derivatives;
Curves in the plane;
Implicit functions;
The chain rule for differentiating along a curve;
Stationary points for functions of two real variables;
Double and repeated integrals;
Cavalieri's slicing principle;
Volumes
Assessment Proportions
Coursework: 50%
Exam: 50%
MATH103: Probability
Terms Taught: Lent and Summer Terms Only
US Credits: 2 Semester credits
ECTS Credits: 4 ECTS credits
Pre-requisites: A year of general mathematics
Course Description
Probability theory is the study of chance phenomena; the concepts of probability are fundamental to the study of statistics. The course will emphasise the role of probability models which characterise the outcomes of different types of experiment that involve a chance or random component. The course will cover the ideas associated with the axioms of probability, conditional probability, independence, discrete random variables and their distributions, expectation and probability models. No previous exposure to the subject will be assumed.
Educational Aims
To provide an introduction to probability theory for discrete distributions.
To introduce students to some simple combinatorics, set theory and the axioms of probability.
To make students aware of the different probability models used to model varied practical situations.
Outline Syllabus
The axioms of probability
Conditional probability
Independence
Discrete Random variables
Expectation, mean and variance.
The binomial, Poisson and geometric distributions
Assessment Proportions
Coursework: 50%
Exam: 50%
MATH104: Statistics
Terms Taught: Lent and Summer Terms Only
US Credits: 2 Semester credits
ECTS Credits: 4 ECTS credits
Pre-requisites: A year of general mathematics
Course Description
Educational Aims
The module aims to enable students to achieve a solid understanding of the broad role that statistical thinking plays in addressing scientific problems in which the recorded information is subject to systematic and random variations. Specifically, by the end of the module, students should be able to select and formulate appropriate probability models, to implement the associated statistical techniques, and to draw clear and informative statistical conclusions for a range of simple scientific problems.
The module starts with the description of examples of scientific investigations in which specific questions are of interest, but they are not straightforward to answer, as the available data are subject to systematic and random variations. A range of exploratory data analysis methods for gaining insight into the sources of variations will be introduced. Then a general strategy for the statistical treatment of such problems will be developed, involving aspects of modelling, investigation, and conclusions.
Outline Syllabus
Data collection and summary
Modelling discrete data
Continuous distributions
Modelling continuous data
Assessment Proportions
Coursework: 50%
Exam: 50%
MATH105: Linear Algebra
Terms Taught: Lent and Summer Terms Only
US Credits: 2 semester credits
ECTS Credits: 4 ECTS credits
Pre-requisites:
A year of general mathematics, including matrices
Course Description
Educational Aims
The specific aim of this module is to introduce the notion of matrices and their basic uses, mainly in algebra. The main goals are to learn how the algorithm of elementary row and column operations is used to solve systems of linear equations, the concept and use of determinant, and the notion of a linear transformation of the Euclidean space. The course also aims at defining the main concepts underlying linear transformations, namely singularity, the characteristic equation and the Eigen spaces.
Outline Syllabus
Matrices: addition and multiplication, transpose and inverse.
Simultaneous linear equations
Reduction to echelon form by elementary row operations
Elementary matrices
Determinants: expansions about a row or column
Elementary row and column operations on determinants
Properties of determinants
Linear transformations of Euclidean space
The matrix of a linear transformation
Non-singular linear transformations
Eigenvectors and eigenvalues
The characteristic equation
Assessment Proportions
Coursework: 50%
Exam: 50%
MATH111: Numbers and Relations
Terms Taught: Michaelmas Term Only
US Credits: 2 semester credits
ECTS Credits: 4 ECTS
Pre-requisites: A year of general mathematic.
Course Description
The course covers: truth tables; methods of proof; lowest common multiples; prime numbers; the Fundamental Theorem of Arithmetic; the existence of infinitely many prime numbers; applications of prime factorisation; solving congruences; the Chinese Remainder Theorem; equivalence relations; constructions of number systems; the division algorithm; highest common factors; the Euclidean algorithm.
Educational Aims
This course aims to:
introduce students to mathematical proofs;
to state and prove fundamental results in number theory;
to generalize the notion of congruence to that of an equivalence relation and explain its usefulness;
to generalize the notion of a highest common factor from pairs of integers to pairs of real polynomials.
Outline Syllabus
Number theory: division with remainder; highest common factors and the Euclidean algorithm; lowest common multiples; prime numbers; the Fundamental Theorem of Arithmetic and the existence of infinitely many prime numbers; applications of prime factorization.
Course Description
Educational Aims
The module formulates the language in which students can describe certain mathematical models or enumerative problems in precise mathematical terms, especially using set terminology. Students can then formulate solve such problems using combinatorial proofs. It also equips students with certain tools such as generating functions and graphs which are helpful in solving combinatorial problems.
Outline Syllabus
Introduction to set notation;
Manipulation of sets; inclusion, intersection, union, complements;
Inclusion-exclusion;
Countability;
Functions and composition;
Injectivity, surjectivity and bijectivity;
Invertibility of functions;
Selecting and counting elements from finite sets;
The pigeonhole principle;
Generating functions;
Recurrence relations;
Graphs and trees;
Isomorphism, planarity, traversing and colouring of graphs
Assessment Proportions
Coursework: 20%
Exam: 50%
Project: 5%
Test: 25%
MATH113: Convergence and Continuity
Terms Taught: Lent / Summer Terms Only
US Credits: 2 semester credits
ECTS Credits: 4 ECTS
Pre-requisites: A year of general mathematics, including basic calculus.
Course Description
This course is an introduction to the foundations of Real Analysis, including suprema and infima of real numbers, limits of sequences, convergence and continuity of functions.
Educational Aims
You will learn will learn: the structure of the real number system and the notions of supremum and infimum for sets of real numbers, the mathematical notion of sequences, subsequences, boundedness, limit points, and convergence. The mathematical notion of continuity and related properties of functions. How to understand mathematical notation and how to read and write proofs related to the above topics. How to provide examples and counter-examples to mathematical definitions and statements regarding the above topics.
Assessment Proportions
MATH114: Integration and Differentiation
Pre-requisites: A year of general mathematics, including basic calculus
Course Description
The course covers: Integration, series and convergence tests, and differentiation.
Educational Aims
In this module the student will learn: the concept of integration of continuous functions, the notion of series and convergence of series and convergence tests, the relation of series to sequences and to integrals, the concept of differentiability of functions, and its relation to continuity and integration. How to understand mathematical notation and how to read and write proofs related to the above topics. How to provide examples and counter-examples to mathematical definitions and statements regarding the above topics
Course Description
Educational Aims
A vast number of naturally occurring phenomena are modelled by differential equations, for which solutions are required to explain these phenomena. This course, which should be particularly useful for science students, sets about obtaining solutions to a number of standard types of differential equations.
Assessment Proportions
MATH141: Introductory Engineering Mathematics
Terms Taught: Michaelmas Term Only
US Credits: 2 semester credits
ECTS Credits: 4 ECTS
Pre-requisites: High school mathematics.
Course Description
Manipulation of algebraic expressions and equations; partial fractions; complex numbers, including Cartesian and exponential forms, and use of de Moivre's theorem; common series and the binomial theorem; square matrices, including inverses and determinants; vectors, including scalar and vector products; complex numbers and vectors to represent physical quantities and harmonically varying quantities.
Educational Aims
To provide students with knowledge and understanding of a range of mathematical techniques appropriate to engineering. The course will enable students to find algebraic, numerical or graphical solutions to mathematical problems. In particular the course will concentrate on learning to use mathematics as a tool and will include many worked examples.
Assessment Proportions
Coursework: 60%
Exam: 40%
MATH142: Calculus
Terms Taught: Michaelmas Term Only
US Credits: 2 semester credits
ECTS Credits: 4 ECTS
Pre-requisites: High school mathematics.
Course Description
The course covers: limit of a function of a real variable, the derivative rules of differentiation, parametric, implicit and logarithmic differentiation; integration as anti derivative, definite integrals, integration by parts and substitution; applications of integration; numerical integration (trapezium and Simpson's rule). Also the Newton-Raphson method for finding roots, Taylor and Maclaurin series to estimate definite integrals and L'Hôpital's ruleThe limit of a function of a real variable; the derivative as a limit.
The graphical interpretation of the derivative.
Rules of differentiation: linearity, the product rule and the chain rule.
Stationary points and their classification.
Parametric, implicit and logarithmic differentiation.
Higher-order derivatives.
Definite integrals; integration as antiderivative.
Integration by parts and by substitution.
Applications of integration to calculations of quantities such as: arc length; the area and centroid of a plane region; the surface area, volume and centre of mass of a surface of revolution; RMS. valuesPeriodic functions. Fourier series and their properties; Parseval's theorem.
Transformation matrices. The solution of simultaneous linear equations using row reduction and using Gaussian elimination with partial pivoting; inherent and induced instabilities. The rank of a matrix. Eigenvalues and eigenvectors.
The Laplace transform. The solution of differential equations using Laplace transforms.
Partial differentiation. The stationary points of a function of two variables. First and second-order Taylor polynomials in two variables.
The solution of equations iteratively using the one-point, bisection and false-position methods. The Jacobi and Gauss-Seidel iterative methods of solution for sets of simultaneous linear equations.
Assessment Proportions
Coursework: 60%
Exam: 40%
MATH210: Real Analysis
Course Description
This course gives a rigorous series and applications.
Educational Aims
The notion of a limit underlies a whole range of concepts that are really basic in mathematics, including sums of infinite series, continuity, differentiation and integration. After the more informal treatment in the first year, our aim now is to develop a really precise understanding of these notions and to provide fully watertight proofs of the theorems involving them. We also show how the theorems apply to give useful facts about specific functions such as exp, log, sin, cos, including some integrals and other unexpected identities.
Definition of the Riemann integral. The fundamental theorem of calculus.
Inequalities for integrals; application to the estimation of discrete sums; the series for tan ^-1 x and log (1+ x); Euler's constant.
Infinite products.
Sequences and series of functions: uniform convergence.
Fourier series: examples, convergence theorems and applications.
Assessment Proportions
Coursework: 15%
Exam: 85%
MATH211: Introductory Real Analysis
Terms Taught: Michaelmas Term Only
US Credits: 3 semester credits
ECTS Credits: 6 ECTS
Pre-requisites: Mathematics minors or theoretical physics major.
Course Description
This course gives an series and applications.
Educational Aims
To introduce the basic concept of a limit, together with the derived concepts of convergent series, continuous functions and differentiation. To present the most important results connected with these concepts.
Educational Aims
The purpose of this course is to give an introduction to the theory of functions of a single complex variable together with some basic applications. The treatment will be analytical, and develops ideas from calculus and real analysis. The first part of the course reviews complex numbers, and presents complex series and the complex derivative in a style similar to calculus. The course then introduces integrals along curves and develops complex function theory from Cauchy's Theorem for a triangle, which is proved by a bisection argument. The results of function theory are used to evaluate some definite integrals. The course aims to strengthen students' understanding of the geometry of the plane by analysis on starlike regions and by discussion of the Möbius group of transformations. The Cauchy-Riemann equations emphasize the link between real and complex analysis; the maximum modulus theorem for harmonic functions is obtained in this way.
Outline Syllabus
The Argand diagram: polar form for complex numbers.
Convergence: Cauchy's criterion; uniform convergence and the M-test.
Continuity and differentiability of complex functions; rational functions; differentiability of power series; the exponential function as a power series.
Line integrals and contours; fundamental theorem of calculus; Cauchy's theorem for a triangle; Cauchy's formula for a disc.
Educational Aims
The course begins by introducing the idea of a vector space, showing how it grows naturally out of the ideas developed for studying vectors in two and three dimensions. It shows how the abstraction has a number of advantages, including making precise and underlying assumptions, suggesting concepts of utility in areas where their significance might not otherwise have been recognised, and simplifying matters by trimming away superfluous information. It then goes on to consider the structure-preserving maps between vector spaces and shows how the study of these leads to a deeper understanding of matrices and important matrix questions. The next section is concerned with the effect of changing bases on the matrix representing one of these maps, and examines how we can choose bases so that this matrix is as simple as possible. The final section takes up the theme of scalar products to examine vector spaces in which the concepts of length and angle can be studied, leading to a means of diagonalising real symmetric matrices using geometrically interesting changes of basis.
MATH225: Abstract Algebra
Course Description
The course covers: an introduction to the basic notions of abstract algebra, covering: binary operations; the definition of a group and examples; the cancellation law; subgroups; cyclic groups and the order of an element; co-sets and Lagrange's theorem; homomorphisms; normal subgroups; quotient groups; the fundamental isomorphism theorem for groups; groups of permutations; the definition of a ring and examples; subrings and ideals; quotient rings; the fundamental isomorphism theorem for rings; integral domains; fields; polynomial rings; principal ideal domains.
Educational Aims
The aim of this module is to introduce students to the basics of the theory of groups and rings. The first part of the module emphasizes finite groups which can be considered in terms of their Cayley tables. The module stresses the importance of equivalence relations, which occur at several points. In the second part of the module fundamental concepts of ring theory are introduced. General results and examples are presented, before some important special classes of commutative rings such as integral domains, fields and principal ideal domains are considered in greater detail.
The most important results are:
Lagrange's theorem;
The fundamental isomorphism theorem for groups;
The fundamental isomorphism theorem for rings;
Ideal structure in the ring of integers and the polynomial ring over a field
Outline Syllabus
Binary operations; definition of a group and examples; elementary properties; equivalence relations and modular arithmetic; further examples.
Subgroups; cyclic groups and order of elements; cosets; Lagrange's theorem and applications.
Educational Aims
This course gives a formal introduction to probability and random variables. In the first half we introduce methods for dealing with continuous random variables, building on the work in MATH104 Probability for discrete distributions. We shall use many examples from a variety of statistical applications to illustrate the theoretical ideas.
The second half aims to extend knowledge of probability and distribution theory so that the student should become competent in manipulating functions of one or more random variables, develop probability models for more realistic problems, and discover how distributions that are important in statistical inference are interlinked.
MATH313 characteristic function is used to study the distributions of sums of independent variables. The results are illustrated in applications to random walks and to statistical physics.
The aim is to teach probability theory within a general axiomatic approach which emphasizes rigour and mathematical analysis. This module links second year analysis courses with statistics courses. The course also shows how one can describe various probabilistic models within this framework and particularly emphasizes the standard distributions such as the Poisson and Gaussian. A significant application is to ideal gases.
MATH314To develop a rigorous approach to the notions of length and area by expressing these analytically
To provide a deeper understanding of the concept of a real number
To introduce the Lebesgue integral as a tool, enabling students to study advanced topics in mathematical analysis and its applications.
Outline Syllabus
Lebesgue's definition of the integral. Integral of a step function. Subsets of the real line; open sets and countable sets. Measure of an open set. Measurable sets and null sets.
Inversion formula for the Fourier cosine transform: Properties of the Fourier transform; Plancherel's formula; Applications of the Fourier transform.
Assessment Proportions
Exam: 90%
Coursework: 10%
MATH317: Hilbert Space
Terms Taught: Michaelmas Term Only.
US Credits: 4 Semester Credits.
ECTS Credits: 8 ECTS
Course Description
A Hilbert space is a linear space with the additional feature of an 'inner product', which allows us to extend the notions of distance and angle to much more general settings than ordinary geometrical space, for example, to infinite-dimensional spaces of functions. The concepts of linear algebra and analysis combine to produce a powerful theory. The notion of orthogonality is applied to best approximations, bases and linear operators.
Educational Aims
This course introduces the student to an area of Mathematics in which the concepts of linear algebra, analysis and geometry are harnessed together. It is shown how this leads to powerful and elegant generalizations of earlier results, many of which are fundamental to modern applications of analysis.
MATH318: Differential Equations
Course Description
Differential equations arise throughout the applications of mathematics and consequently the study of them has always been recognised as a fundamental branch of the subject. This course aims to give a systematic introduction to the topic, striking a balance between methods for finding solutions of particular types of equations and theoretical results about the existence, uniqueness and nature of solutions.
Educational Aims
On successful completion of this module students will be able to:
solve first order and elementary types of second order linear ordinary differential equations;
use several general methods for second order linear equations and find series solutions;
understand Wronskians, the uniqueness theorem and Sturm's separation and comparison theorems for zeros of solutions;
understand Sturm-Liouville systems and solve certain examples;
solve constant coefficient first order systems of equations and sketch solution curves for 2x2 systems;
Educational Aims
The aim of this module is to build on the theory of groups as introduced in the 2nd year module MATH225:
Groups and Rings. Emphasis will be given to finite groups. The most important results covered will be as follows:
The classification of finite abelian groups.
The orbit-stabilizer theorem.
The Jordan-Holder theorem.
The classification and symmetry groups of the Platonic solids.
Sylow's theorems.
We shall first consider a way of comparing the elements of a group and show how a group may be built up from smaller components using 'direct products'. Next we shall treat situations in which a group 'acts' on a set by permuting its elements; after identifying the five Platonic solids, we shall use group actions to determine their symmetry groups. Finally we shall prove some interesting and important results, known as the 'Sylow theorems', relating to subgroups of certain orders.
To continue developing students algebraic understanding and their ability to reason from stated axioms and definitions, and introduce them to the use of algebraic ideas in the study of symmetry and geometry.
MATH322: Rings Fields and Polynomials
Course Description
The course introduces two important classes of commutative rings: Euclidean domains and unique factorisation domains, and then studies the properties of such rings, especially factorisation of polynomials over them. Topics include: Euclidean domains (ED), every ED is a principal ideal domain (PID), highest common factors, irreducible elements and prime elements, unique factorization domains (UFD), every PID is a UFD, primitive polynomials, Gauss' Lemma, Eisenstein's Irreducibility Criterion.
Educational Aims
This is a theory-based course whose aim is to study factorization in integral domains, in particular in polynomial rings, and to present some applications of this theory. It is intended to build upon the material encountered in the second-year courses in Rings and Linear Algebra, and to provide an introduction to the fourth year Galois Theory course.
The course presents students with a hierarchy of properties that certain integral domains possess. The weakest of these properties leads to a factorization theory which is analogous to the prime factorization of integers. The aim of the course is to study these properties and their relationship, to determine which integral domains possess them, and in the positive case to understand how the "prime factorizations" mentioned above can be found in practice.
Factorization of polynomials over an integral domain: Gauss's Lemma and Gauss's Factorization Theorem; examples and applications; the polynomial ring over a unique factorization domain is a unique factorization domain.
The field of fractions of an integral domain; the characteristic of a unital ring, finite fields.
Assessment Proportions
Coursework: 10%
Exam: 90%
MATH323: Elliptic Curves
Course Description
This module gives you a solid foundation in the basics of algebraic geometry. You will explore how curves can be described by algebraic equations, and learn how to understand and use abstract groups in dealing with geometrical objects (curves). You will also gain an understanding of the notions and the main results pertaining to elliptic curves, and the way that algebra and geometry are linked via polynomial equations. Finally you'll learn to perform algebraic computations with elliptic curves.
Educational Aims
This course is an introduction to elliptic curves, and hence to algebraic geometry. It also presents applications and results of the theory of elliptic curves. The course also provides a useful link between concepts from algebra and geometry .
The student should learn the basics of algebraic geometry; understand and use abstract groups in dealing with geometrical objects (curves), know the notions and the main results pertaining to elliptic curves.
Outline Syllabus
Algebraic curves: Parametrisation of the projective line, Rational points on curves of degree 2 and 3.
Intersection multiplicity and singularity, Bézout's theorem.MATH325: Representation Theory of Finite Groups
Course Description
The primary aim of this course is to provide an introduction to representation theory. The main part of the course treats the ordinary representations of finite groups, for which two traditional approaches are taken: representations via coursesEducational Aims
At the end of the course the students should be able to demonstrate subject specific knowledge, understanding and skills and have the ability to:
understand the basics of representation theory -understand the concept of CG-module, the use of matrix groups in the study of the representations of an abstract finite group, and the correspondence between group representations and CG-modules.
know and be able to apply the main results such as Schur's lemma and Maschke's theorem.
know how to find the irreducible representations of finite abelian groups.
Outline Syllabus
the kernel and the image of an R-homomorphism; isomorphism theorems for R-modules.
Representations of Finite groups; correspondence between \CG-modules and the representations of a Finite group G.
Maschke's theorem and complete reducibility.
\CG-homomorphisms; Schur's lemma; spaces of \CG-homomorphisms and their dimensions.
The representations of Finite abelian groups.
The group algebra \CG of a Finite group G and \CG-modules; composition factors; the regular representation of a Finite group, and the decomposition of the group algebra.
Assessment Proportions
Coursework: 10%
Exam: 90%
MATH326: Graph Theory
Terms Taught: Michaelmas Term Only.
US Credits: 4 Semester Credits
ECTS Credits: 8 ECTS Credits.
Pre-requisites: Mathematics Majors or appropriate College Mathematics
Course Description
The course aims to introduce students to graph theory, a substantial modern area of mathematics with many applications in pure and applied sciences. Graph theory originated in Euler's 1736 paper on the seven bridges of Konigsberg. Concepts are introduced alongside examples and, where possible, discussion of how the concepts can be applied in real world situations.
Educational Aims
Graph theory is the cornerstone of discrete mathematics. This course will introduce a range of fundamental topics in graph theory. Graphs are mathematical structures used to model pairwise relations between objects, for example networks of communications. Students will develop an appreciation for a range of discrete mathematical techniques. Emphasis will be placed on topics linking graph theory to linear algebra and to topology.
Assessment Proportions
Exam 90%
Coursework 10%
MATH327: Combinatorics
Course Description
Historically, combinatorics has its roots in mathematical recreations, puzzles and games. However, many problems in combinatorics that were previously studied simply for amusement or for their aesthetic appeal, are today of great importance in many areas of pure and applied sciences. For example, applications of combinatorics include topics as diverse as codes, circuit design, molecular assembly and drug design, and algorithm complexity. Moreover, in pure mathematics, combinatorial problems and methods frequently arise in areas such as algebra, probability, topology and geometry. It has thus become essential for workers in many scientific fields to have some familiarity with this subject.
Educational Aims
The course provides an introduction to some of the key concepts and methods in combinatorics and graph theory, the cornerstones of discrete mathematics.
In particular, the course will introduce students to various important counting methods and counting coefficients, to some key concepts, methods and algorithms in graph theory and to the theory of combinatorial designs.
Some applications of the results and methods will also briefly be discussed.
The course aims to introduce students to discrete mathematics, a fundamental part of mathematics with many applications in computer science and other pure and applied sciences.
MATH328: Number Theory
Course Description
The course gives an all round introduction to the concepts, results and methods of number theory, including topics such as the following: Greatest common divisors, congruence, prime numbers, arithmetic functions, the divisor and phi functions, the Euler-Fermat theorem and applications to coding, quadratic residues, Dirichlet series, convolutions, the Möbius function, sums of two squares, partial sums of arithmetic functions. Connections between these topics will be emphasised, and results will be illustrated by numerical examples.
MATH329: Geometry of Curves and Surfaces
Course Description
This course is an introduction to the study of smooth curves and surfaces in three-dimensional space. Various geometrical properties of these objects will be encountered; some familiar ones, such as length and area, and some less familiar, such as torsion and curvature. The meaning of these quantities will be explored and their values will be calculated for a variety of examples, applying techniques from calculus and linear algebra.
Educational Aims
To provide an introduction to the differential geometry of curves and surfaces in three-dimensional space.
To allow the student to appreciate geometric concepts, by showing how ideas from calculus can be used to compute familiar and novel quantities which allow the description of three-dimensional curves and surfaces
MATH330: Likelihood Inference
Course Description
ThisEducational Aims
TheThis course aims for students:
to gain skills in problem solving and critical thinking.
to appreciate the importance of communicating technical ideas at an appropriate level.
to appreciate the importance of making evidence-based decisions.
Outline Syllabus
The likelihood function and maximum likelihood estimation.
Comparing estimators and the Cramér-Rao lower bound.
Multiparameter problems and examples.
Properties of the likelihood.
Asymptotics: the distribution of maximum likelihood estimators
The multivariate normal distribution.
Model selection and the generalised likelihood test.
Graphical and geometrical interpretations.
Use of the profile likelihood for dealing with nuisance parameters.
Parameter functions.
Substantial applications of the above topics e.g. to ANOVA testing and extensions outside the Gaussian family.
Assessment Proportions
Coursework: 10%
Exam: 90%
MATH331: Bayesian Inference
Terms Taught: Michaelmas Term Only.
US Credits: 4 Semester Credits.
ECTS Credits: 8 ECTS
Pre-requisites:
Equivalent of MATH 330 Likelihood Inference;
Mathematics Majors or appropriate College Mathematics.
Course Description
This course aims to introduce the Bayesian view of statistics, stressing its philosophical contrasts with classical statistics, its facility for including information other than 'the data' into an analysis, its coherent approach to inference and its decision theoretic foundations.
Educational Aims
At the end of the course the students should be able to demonstrate subject specific knowledge,understanding and skills and have the ability to:
To be able to resolve some well-known discrete paradoxes using a Bayesian formulation
To understand the advantages that a prior brings to statistical analysis, appreciate some of the difficulties it creates and show how these difficulties are resolved.
To recognise distributions from the exponential family, derive their conjugate priors, the posterior distributions, the marginal likelihoods and the predictive distributions.
To appreciate the role of each of the above distributions in integrating statistical reasoning and to simulate or calculate them using the language R from simple datasets.
To be familiar with the range of loss functions, to select an appropriate loss function for a given problem and show how the minimisation of the expected loss leads to a rational action
To be able to carry out a variety of strategies for dealing with multi-parameter problems using directed graphs and the language R
To be able to contrast the Bayesian and classical approaches toward making inferences, choosing models and predicting
Outline Syllabus
Bayesian updating of belief
The formulation of a prior belief for a Bayesian analysis
Bayesian decision theory and the role of the utility function in Bayesian estimation
The predictive and marginal distributions for model checking and selection and model selection
Multi-parameter models
Comparison with likelihood methods
The asymptotic approximation for multi-parameter and non-conjugate models
The use of all the above in substantive applications
Assessment Proportions
Coursework: 10%
Exam: 90%
MATH332: Stochastic Processes
Terms Taught: Lent/Summer Terms Only.
US Credits: 4 Semester Credits.
ECTS Credits: 8 ECTS
Pre-requisites: Equivalent of MATH 230 Probability. you should be able to use conditioning arguments to calculate probabilities and expectations of random variables for stochastic processes; to calculate the distribution of a Markov process at different time pointsAssessment Proportions
MATH333: Statistical Models
Terms Taught: Lent / Summer Terms Only.
US Credits: 4 Semester Credits.
ECTS Credits: 8 ECTS
Pre-requisites: Equivalent of MATH 330 Likelihood Inference.
Course Description
The course introduces a class of well known statistical models for regression problems. The class includes linear regression for normal data, generalised linear models for non-normal data. By the end of the course you should be able to formulate sensible models for different sets of data, taking account of the constraints on the data, and to explore and analyse the data using R.
Educational Aims
To understand the theoretical basis of generalized linear models and to apply to a diverse range of practical problems.
To understand the effect of censoring in the statistical analyses and to use appropriate statistical techniques for lifetime data.
To relate modern statistical models and methods to real life situations and use relevant computer software for statistical analysis.
* The R language is required for the laboratory sessions of MATH333
Single334: Time series analysis
Terms Taught: Lent / Summer Terms Only.
US Credits: 4 Semester Credits.
ECTS Credits: 8 ECTS
Pre-requisites: Equivalent of MATH 331 Bayesian Inference.
Course Description
This course aims to provide an introduction to recent developments in statistics. This may include statistical methods for analysing time series, multivariate data with emphasis on the financial applications, change-point analysis and stochastic volatility models.
Educational Aims
To provide an introduction to recent developments in statistics. This may include statistical methods for analysing time series, multivariate data with emphasis on the financial applications, change point analysis and stochastic volatility models.
To allow the student to appreciate statistical methods and data analysis concepts as well as the use of the statistical software R.
Outline Syllabus
Topics taken from the following: Time Series, Volatility Modelling, Multivariate Analysis, Change Point Methods. The choice of topics and level of depth will reflect the lecturer's personal research interests.
Change Point: Methods: general introduction to change point problems; CUSUM; likelihood based approaches; penalised likelihood; current trends in change point methods.
* The R language is required for the laboratory sessions of MATH334 Single335: Medical Statistics: study design and data analysis
Course Description
This course aims to understand the conceptual and theoretical basis of health investigations including measures of disease, study design, causality, confounding and measures of disease-exposure association (mortality and morbidity). The course develops a firm understanding of key analytical methods and procedures used in studies of disease aetiology, disease screening/diagnosis and clinical trials. It includes the understanding of the effect of censoring in the statistical analyses and the use of appropriate statistical techniques for time to event data.
Educational Aims
To understand the conceptual and theoretical basis of health investigations including measures of disease, study design, causality, confounding and measures of disease-exposure association (mortality and morbidity). Firm understanding of key analytical methods and procedures used in studies of disease aetiology. disease screening/diagnosis and clinical trials including the impact of key study designs upon inference.
To understand the effect of censoring in the statistical analyses and to use appropriate statistical techniques for time to event data.
To relate statistical models and methods to address 'real' health research questions and to use relevant computer software for statistical analysis. To critique published articles related to health research and discuss in context.
Clinical Trials: intervention studies and causality, measurements and end points, bias and replication, treatment allocation: randomisation and control, group comparisons: t-tests, use of baseline measurements.
Assessment Proportions
MATH336: Multivariate Statistics in Machine Learning
Pre-requisites: Students should be familiar with multivariate random variables and matrix algebra
Course Description
This module aims to introduce students to some of the theoretical and practical elements of machine learning and multivariate statistics. The specific focus will be on, multivariate data representation/visualisation,feature extraction and dimensionality reduction, e.g. through Principal Component Analysis (PCA),multivariate data classification using e.g. discriminant analysis and Support Vector Machines (SVMs).Apart from learning the theoretical aspects of the above methods, the students will also learn how to apply them in practice using R.
Educational Aims
On successful completion of this module students will be able to:
Appreciate the need for multivariate statistical analysis
Represent and visualise high-dimensional data
Understand the concept of dimensionality reduction and its use in feature extraction from high-dimensional data
Assessment Proportions
MATH345: Financial Mathematics
Terms Taught: Lent and Summer Term Only
US Credits: 4 Semester Credits
ECTS Credits: 8 ECTS
Pre-requisites: Mathematics Majors or appropriate College Mathematics
Course Description
This module will give a simple introduction to mathematical finance. This includes some financial terminology and the study of European and American option pricing with respect to different models. More precisely, we consider two discrete models, the binomial Model and finite market model, and one continuous model, the Black Scholes model. We also introduce some probabilistic terminology, which is required to study the properties of these models. This includes martingales, stopping times and an overview over the mathematical theory of Brownian motion, including stochastic integration with respect to Brownian motion.
MATH361: Mathematical Education
Course Description
The aims of this course are to provide you with opportunities to gain insights into a range of issues relating to mathematics education in general, and consider theories about teaching and learning in particular. This will be achieved in practical, activity based workshops and in seminars.
Educational Aims
The aims of this course are as follows:
To enable students to reflect on mathematics, in particular on its history, application and significance in human culture;
To foster in students a deeper understanding of philosophies on the learning of mathematics and their application to mathematics education;
To consider the position of mathematics within the curriculum;
To consider the purposes of assessment, and the advantages and disadvantages of various forms there of.
Outline Syllabus
Mathematics as a problem solving discipline;
using ICT to do, learn and teach mathematics;
different ways that children learn mathematics;
different styles and strategies a mathematics teacher might use to enhance learning;
significant developments and government initiated publications since 1980, e.g. the national Curriculum (DCSF 2008);
assessing children's mathematical achievements.
Assessment Proportions
Coursework: 100%
MATH411: Operator Theory
Terms Taught: Lent / Summer Terms Only.
US Credits: 4 Semester Credits.
ECTS Credits: 8 ECTS Credits.
Pre-requisites: Equivalent of MATH 317 Hilbert Space.
Course Description
The aims of this course are to build on the theory of Hilbert spaces, to analyse a variety of Hilbert space operators, together with spectral theory for self adjoint operators, and basic integral operators.
Educational Aims
At the end of the module students should be able to:
define and give examples of various types of bounded linear operator on Hilbert space;
Assessment Proportions
MATH412: Topology and Fractals
Terms Taught: Michaelmas Term Only.
US Credits: 4 Semester Credits.
ECTS Credits: 8 ECTS
Pre-requisites: Equivalent of MATH 210 Real Analysis.
Course Description
The aims of this course are to understand the definitions and the basic properties (such as closedness, connectedness, fractal dimension) of a variety of fractals. These include, the Cantor set and Cantor dust, the Sierpinski sieve, the von Koch (snowflake) curve, number system fractals, fractals defined by iterated function systems
Educational Aims
The course gives an introduction to topology in the Euclidean spaces Rn, embracing fundamental terms such as connectedness, total disconnectedness and compactness.
These notions are illustrated with fractal sets and the basic theory of these sets is developed, including their generation by iterated function systems and identification of fractional dimensions.
To develop knowledge and skills in analytical arguments in set theory and in analysis. To analyse diverse fractal sets rigorously.
MATH413On successful completion of this module students will be able to:
understand the abstract notion of a probability space;
deduce properties of measure from the axioms;
verify that certain discrete probability spaces satisfy the axioms;
verify that certain collections of sets are sigma-algebras;
understand the abstract notion of a random variable;
derive fundamental properties of the cumulative distribution function from the axioms of measure;
understand the basic properties of probability density functions;
find the probability density function of the x2 random variable using the method of distribution functions;
prove basic properties of the expectation of simple random variables;
prove basic properties of expectation and variance;
derive Chebyshev's inequality from the axioms;
calculate the expectation of certain functions of continuous random variables;
understand the technical definition of independence;
understand different probabilistic notions of convergence;
prove comparative strengths of probabilistic modes of convergence;
prove the weak law of large numbers and apply it to simple examples;
prove the Borel-Cantelli Lemmas, and apply them to simple examples;
calculate the characteristic functions of some random variables, and understand its properties;
MATH414On successful completion of this module students will be able to:
understand the basic concepts of Lebesgue's definition of the integral;
appreciate Dirichlet's comb function;
prove basic properties of countable sets;
understand Archimedes' axiom and Cantor's uncountability theorem;
prove the structure theorem for open sets;
prove covering lemmas for open sets;
understand the statement (not proof) of Heine-Borel theorem;
understand the concept and prove basic properties of outer measure;
understand inner measure;
prove Lebesgue's theorem on countable additivity of measure;
understand the notion of a measurable function, and prove continuous functions are measurable;
define the integral of a bounded measurable function;
extend the definition of the integral to unbounded functions and understand its properties;
MATH417: Hilbert Space
Course Description
This course introduces you to an area of Mathematics in which the concepts of linear algebra, analysis and geometry are harnessed together. It is shown how this leads to powerful and elegant generalisations of earlier results, many of which are fundamental to modern applications of analysis.
Educational Aims
On successful completion of this module students will be able to...
define the notions norm and inner product;
define and give examples of equivalent and inequivalent norms;
evaluation norms of linear operators and functionals;
understand and apply the Cauchy-Schwarz inequality;
use orthogonality to find best approximations in spaces of sequences or functions;
understand the existence proof for orthonormal basses, and their application to isomorphisms and Parseval's identity;
understand the existence proof for closest points and the applications to orthogonal projections and the Riesz representation theorem;
define the adjoint of an operator and understand the proof of the spectral theorem for self adjoint operators in finite dimensions.
MATH423: Elliptic Curves
Course Description
This course is an introduction to elliptic curves, and hence to algebraic geometry. It also presents applications and results of the theory of elliptic curves. The course also provides a useful link between concepts from algebra and geometry, and introduces in a specific context some ideas that have motivated the development of modern algebraic geometry.
Educational Aims
At the end of the course the students should be able to demonstrate subject specific knowledge, understanding and skills and have the ability to use:Assessment Proportions
MATH424: Galois Theory
Terms Taught: Lent/Summer Terms Only
US Credits: 4 Semester Credits
ECTS Credits: 8 ECTS
Pre-requisites: Mathematics Majors or appropriate College Mathematics
Course Description
With this module students will apply Galois theory in a number of important situations, including deciding when a polynomial equation is solvable by radicals, and which regular n-gons are constructible with ruler and compasses;
Students will also appreciate the historical development of Galois Theory and its influence on the subsequent development of pure mathematics as a whole.
Solvability by radicals: radical extensions, cyclic and abelian extensions, Galois's criterion for solvability by radicals, the field of symmetric rational expressions, solution of general quadratics, cubics and quartics, rational equations which are not solvable by radicals.
Assessment Proportions
Assessment will be through:
Weekly coursework, aimed at testing and consolidating understanding of the basic elements of the course;
An examination in the Summer which assesses more fully the students' understanding and summative knowledge of the topics
Course Description
This course sets out to give an appreciation of the significance of the Fundamental Theorem of Galois Theory through a consideration of its historical context, and through a number of important applications, including the solvability of polynomial equations.
Educational Aims
The primary aim of this course is to provide an introduction to representation theory.
The main part of the course treats the ordinary representations of finite groups, for which two traditional approaches are taken: representations via modulesThe student should learn the basics of representation theory. In particular, the concept of ordinary group representations and the pertaining important results.
Outline Syllabus
R-modules and R-homomorphisms for a unital ring R: submodules, direct sums and quotient modules; the kernel and the image of an R-homomorphism; isomorphism theorems for R-modules.
Representations of finite groups; correspondence between CG-modules and the representations of a finite group G.
Maschke's theorem and complete reducibility.
CG-homomorphisms; Schur's lemma; spaces of CG-homomorphisms and their dimensions.
The representations of finite abelian groups.
The group algebra of a finite group G and CG-modules; composition factors; the regular representation of a finite group, and the decomposition of the group algebra.
Assessment Proportions
Coursework: 10%
Exam: 90%
MATH426: Lie Groups and Lie Algebras
Course Description
Lie groups and Lie algebras form an indispensable part of the toolkit of any pure and applied mathematician; they are also widely used in theoretical physics and even chemistry. It is, therefore, important that students have at least some initial exposure to this fundamental and exciting area. This course will allow the students to appreciate the subtle and pervasive interplay between algebra and geometry and appreciate the unified nature of mathematics. The abstract nature of the course will give students a taste of modern research in pure mathematics.
Educational Aims
At the end of the course the students will gain understanding of the structure theory of Lie algebras, manifolds and Lie groups. They will also gain basic knowledge of representations of Lie algebras. They will be able to demonstrate subject specific knowledge, understanding and skills and have the ability to:
Construct a Lie algebra associated with a given Lie group.
Integrate a Lie algebra to a Lie group, in the nilpotent and simply connected cases.
For a given homomorphism of Lie groups construct homomorphisms between their Lie algebras.
Recover a homomorphism between two Lie groups from the tangent homomorphism of their Lie algebras.
Analyse various examples of Lie groups and Lie algebras, particularly those associated with matrices and vector fields on manifolds.
MATH432: Stochastic Processes the you should be able to use conditioning arguments and the reflection principle to calculate probabilities and expectations of random variables for stochastic processes; to calculate the distribution of a Markov Process at different time points and to calculate expected hitting times:MATH440: Stochastic Calculus for Finance
Course Description
Stochastic processes are widely used to model uncertainty in areas ranging from finance to the physical sciences. This module aims to show how stochastic calculus can be used to formulate and solve problems arising from these areas of application.
Educational Aims
Stochastic Calculus is a theory that enables the calculation of integrals with respect to stochastic processes. This module begins with the study of discrete-time stochastic processes, in particular defining key concepts such as martingales and stopping times. This leads on to the exploration of continous-time processes, in particular Brownian motion. The final section covers integration with respect to Brownian motion, and the derivation of Ito's formula, a stochastic analogue of the chain rule. This allows the definition and solution of stochastic differential equations (SDEs), the stochastic analogue to ordinary differential equations (ODEs).
Assessment Proportions
Exam 70%
Project 20%
Coursework 10%
MATH451: Likelihood Inference
Terms Taught: Michaelmas Term Only
US Credits: 4 Semester Credits.
ECTS Credits: 8 ECTS
Pre-requisites: Equivalent of MATH 235 (excl. MATH 331).
Course Description
Statistical theory is the theory of the extraction of information about the unknown parameters of an underlying probability model from observed data. This underpins all practical statistical applications, such as those considered in later MSc courses. This course considers how the likelihood function, the probability of the observed data viewed as a function of unknown parameters, can be used to make inference about those parameters. This inference includes both estimates of the values of these parameters, and measures of the uncertainty surrounding these estimates. We consider multi-parameter models, and models which do not assume the data are independent and identically distributed. We also cover basic computational aspects of likelihood inference that are required in many practical applications.
Educational Aims
On successful completion of this module students will be able to:
appreciate how information about the unknown parameters is obtained and summarized via the likelihood function;
be able to calculate the likelihood function for some statistical models which do not assume independent identically distributed data;
be able to evaluate point estimates and make statements about the variability of these estimates;
understand about the inter-relationships between parameters, and the concept of orthogonality;
be able to perform hypothesis tests using the generalised likelihood ratio statistic;
use computational methods to calculate maximum likelihood estimates.
Outline Syllabus
The course presents the key tools for statistical inference, stressing the fundamental role of the likelihood function.
It will cover:
Definition of the likelihood function for multi-parameter models, and how it is used to calculate point estimates (maximum likelihood estimates).
Asymptotic distribution of the maximum likelihood estimator, and the profile deviance, and how these are used to quantify uncertainty in estimates.
Inter-relationships between parameters, and the definition and use of orthogonality.
Generalised Likelihood Ratio Statistics, and their use for hypothesis tests.
Calculating likelihood functions for non-iid data.
Simple use of computational methods to calculate maximum likelihood estimates.
Assessment Proportions
Coursework: 50%
Exam: 50%
MATH452: Generalised Linear Models
Terms Taught: Michaelmas Term Only
US Credits: 4 Semester Credits
ECTS Credits: 8 ECTS
Pre-requisites: Equivalent of MATH 451 or MATH 331 (excl. MATH 333).
Course Description
Generalised linear models are now one of the most frequently used statistical tools of the applied statistician. They extend the ideas of regression analysis to a wider class of problems that involves exploring the relationship between a response and one or more explanatory variables. In this course we aim to discuss applications of the generalised linear models to diverse range of practical problems involving data from the area of biology, social sciences and time series to name a few and to explore the theoretical basis of these models.
Educational Aims
On successful completion of this module students will be able:
To learn techniques for formulating sensible models for set of data that enables to answer question such as how the probability of success of a particular treatment will depend on the patient's age, weight, blood pressure and so on.
To introduce a large family of models, called the generalised linear models (GLMs), that includes the standard linear regression model as a special case and to discuss the theoretical properties of these models.
To learn a common algorithm called iteratively reweighted least squares algorithm for the estimation of parameters.
To fit and check these models with the statistical package R; produce confidence intervals and tests corresponding to questions of interest; and state conclusions in everyday language.
Outline Syllabus
The course introduces a large family of models, the generalised linear models (GLMs), that includes the standard linear regression model as a special case. By the end of the course students should be able to formulate sensible models for sets of data, taking account of the motivation for data collection; fit and check these models in the statistical package R; produce confidence intervals and tests corresponding to questions of interest; and state conclusions in everyday language.
Assessment Proportions
Coursework: 50%
Exam: 50%
MATH453: Bayesian Inference
Terms Taught: Michaelmas Term Only
US Credits: 4 Semester Credits.
ECTS Credits: 8 ECTS
Pre-requisites: Equivalent of MATH 451 or MATH 331.
Course Description
This course aims to introduce the Bayesian view of statistics, stressing its philosophical contrasts with classical statistics, its facility for including information other than the data into the analysis and its decision theoretic foundations. By the end of the course you should be able to formulate an appropriate prior to a variety of problems, calculate, simulate from and interpret the posterior distributions, carry out Bayesian model selection using the marginal likelihood and conduct model checking using the predictive distribution. In addition you should be able to formulate and carry out a Bayesian analysis from a decision theoretical point of view upon consideration of an appropriate loss or utility function.
Educational Aims
On successful completion of this module students will be able to:
define the components of a glm;
express standard models (normal, Poisson, . . . ) in glm form;
derive relationships between mean and variance and parameters of an exponential family distribution;
specify design matrices for given problems;
define and interpret model deviance and degrees of freedom;
use model deviances to assist in model selection;
Outline Syllabus
Inference by updating belief.
The ingredients of Bayesian inference: The prior, the likelihood, the posterior, the predictive and the marginal distributions.
Methods for formulating the prior (depending on the research question).
Conjugate priors for single parameter models
Multiple regression
Sampling from the posterior and the predictive distribution in the case of conjugate priors.
For non-conjugate priors, sampling from the posterior using an approximation and then correcting for the approximation using importance sampling-resampling methods.
Methods for summarising the posterior distribution
Model checking using the predictive and model selection using the marginal likelihood
Course Description
To introduce you to Markov chain Monte Carlo (MCMC) algorithms and their applications in Bayesian statistics; and to introduce Bayesian hierarchical modeling and its Bayesian implementation using MCMC.
Educational Aims
This course introduces the use of Markov chain Monte Carlo methods as a powerful technique for performing Bayesian inference on complex stochastic models. The first part of the course introduces in detail the necessary concepts and theory for finite state-space Markov chains; analogous concepts and theory for continuous state-space Markov chains are then introduced heuristically. The second part of the course introduces the Metropolis-Hastings algorithm as an algorithm for sampling from a distribution known up to a constant of proportionality, and the third (and largest) part applies this to Bayesian inference and introduces the Gibbs sampler. The two most common Metropolis-Hastings algorithms (the random walk and the independence sampler) will be examined in detail, as will the Gibbs sampler. Examples will include hierarchical models, random effects models, and mixture models.
Assessment Proportions
MATH463: Clinical Trials
Terms Taught: Lent / Summer Terms Only
US Credits: 4 Semester Credits
ECTS Credits: 8 ECTS
Pre-requisites: Equivalent of MATH 235 and MATH 390.
Course Description
This course aims to introduce you to aspects of statistics, which are important in design and analysis of clinical trials. By the end of the course you should understand the basic elements of clinical trials, be able to recognise and use principles of good study design, and be able to analyse and interpret study results to make correct scientific inferences.
Educational Aims
On successful completion of this module students will be able to:
understand the basic elements of clinical trials,
be able to recognise and use principles of good study design,
be able to analyse and interpret study results to make correct scientific inferences
Assessment Proportions
MATH464: Principles of Epidemiology
Terms Taught: This module is an intensive one week course taught in Lent Term only.
US Credits: 2 Semester Credits.
ECTS Credits: 4 ECTS
Pre-requisites: Equivalent of MATH 235 and MATH 390.
Course Description
Epidemiology is the study of the distribution and determinants of disease in human populations. This course provides an introduction to the principles and statistical methods of epidemiology. Various concepts and strategies used in epidemiological studies are examined. Most inference will be likelihood based, although the emphasis is on conceptual considerations.
Educational Aims
On successful completion of this module students will be able to:
Calculate appropriate measures of disease incidence and mortality.
Understand the key statistical issues in the design of surveys, case-control studies, cohort studies and RCTs..
Be able to discuss and address strategies for dealing with bias and confounding
Evaluate diagnostic and screening tests in terms of design and analysis issues.
Outline Syllabus
Fundamental measures of disease: incidence, prevalence, risk, etc.
Indices of morbidity and mortality
Epidemiologic concepts: transmission of disease, investigation of an epidemic, vaccination
Assessment Proportions
MATH466: Longitudinal Data Analysis
Terms Taught: This module is an intensive one week course taught in Lent Term only.
US Credits: 4 Semester Credits.
ECTS Credits: 8 ECTS
Pre-requisites: Equivalent of MATH 451 or MATH 331.
Course Description
The specific aim of this course is to teach you a modern approach to the analysis of longitudinal data. Upon completion of this course you should have acquired, from lectures and practical classes, the ability to build statistical models for unseen sets of longitudinal data, and to draw valid conclusions from their models.
Educational Aims
Upon completion of this course the students should have acquired, from lectures and practical classes, the ability to build statistical models for longitudinal data, and to draw valid conclusions from their models.
Outline Syllabus
What are Longitudinal data?
Exploratory and simple analysis strategies
The independence working assumption
Normal linear model with correlated errors
Linear mixed effects models
Generalised estimating equations
Dealing with dropout.
Assessment Proportions
Coursework: 50%
Exam: 50%
MATH482: Assessing Financial Risk: Extreme Value Methods
Course Description
Assessment of financial risk requires accurate estimates of the probability of rare events. For example, in the next day of trading what is the risk of a share portfolio losing half of its value, or equivalently what is the value of the portfolio at risk of being lost with a specified probability? Estimating the probability of such 'extreme' events is challenging, as by nature they are sufficiently rare that there is little direct empirical evidence on which to base inference. Instead we have to extrapolate based on the past frequency of the occurrence of less extreme events. This course aims to develop the asymptotic theory and associated techniques from Extreme Value Theory which give a sound mathematical basis to such extrapolation, and shows practically how it can be used to give accurate assessments of financial risk in a wide range of scenarios. Analysis of financial data within the computer package R will be covered.
Educational Aims
On successful completion of this module students will be able to:
understand the need for special models to describe the extreme values of a financial times series
fit appropriate extreme value models to data which are maxima or threshold exceedances
use extreme value models to evaluate value and risk and expected short fall | 677.169 | 1 |
About this item
Comments:ALTERNATE EDITION: This is an international edition textbook (same content, just cheaper!!). Will show signs of use, and may contain writing, underlining, &/or highlighting within. May not contain supplementary materials. Second day shipping available. Ships same or next day.This is the U.S. student edition as pictured unless otherwise stated.
Product details
ISBN-13: 9780321390530
ISBN: 0321390539
Edition: 2
Publication Date: 2007
Publisher: Pearson Addison-Wesley
AUTHOR
Chartrand, Gary, Polimeni, Albert D., Zhang, Ping
SUMMARY
Mathematical Proofs: A Transition to Advanced Mathematics, 2/e,prepares students for the more abstract mathematics courses that follow calculus. This text introduces students to proof techniques and writing proofs of their own. As such, it is an introduction to the mathematics enterprise, providing solid introductions to relations, functions, and cardinalities of sets.KEY TOPICS: Communicating Mathematics, Sets, Logic, Direct Proof and Proof by Contrapositive, More on Direct Proof and Proof by Contrapositive, Existence and Proof by Contradiction, Mathematical Induction, Prove or Disprove, Equivalence Relations, Functions, Cardinalities of Sets, Proofs in Number Theory, Proofs in Calculus, Proofs in Group Theory.MARKET: For all readers interested in advanced mathematics and logic.Chartrand, Gary is the author of 'Mathematical Proofs A Transition to Advanced Mathematics', published 2007 under ISBN 9780321390530 and ISBN 03213905 | 677.169 | 1 |
Quadratics Notes/Practice Packet
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22 pages of quadratics!! I created a workbook to guide me and my students through quadratics. Included are guided notes on: Vertex form, factored form, standard form, graphing, complex, imaginary numbers, quadratic formula, and rate of change. There are vocabulary pages for students to write definitions, and after each note page are practice problems for the students to try. There is a review page at the end that can be used as a worksheet or quiz. It is a word doc, so you can edit as you like. | 677.169 | 1 |
College Algebra, 1st Edition
Axler's College Algebra focuses on depth, not breadth of topics by exploring necessary topics in greater detail. Students will benefit from the straightforward definitions and plentiful examples of complex concepts.
The Student Solutions Manual is integrated at the end of every section. The proximity of the solutions encourages students to go back and read the main text as they are working through the problems and exercises. The inclusion of the manual also saves students money.
Axler's College Algebra is available with WileyPLUS; an innovative, research-based, online environment for effective teaching and learning.
Depth, Not Breadth: Topics have been carefully selected to get at the heart of algebraic weakness by narrowing down to key sets of skills which are regularly revisited from varied perspectives.
Exercises and Problems: The difference between an exercise and a problem is that each exercise has a unique correct answer that is a mathematical object such as a number or a function, while the solutions to problems consist of explanations or examples. The solutions to the odd-numbered exercises appear directly behind the relevant section.
Variety: Exercises and problems in this book vary greatly in difficulty and purpose. Some exercises and problems are designed to hone algebraic manipulation skills; other exercises and problems are designed to push students to genuine understanding. Applications are written to reflect real scenarios, not artificial examples.
Integrated Student's Solutions Manual: The solutions manual encourages students to read the main text and students will save money by not having to purchase a separate solutions manual.
Designed To Be Read: The writing style and layout are meant to induce students to read and understand the material. Explanations are more plentiful than typically found in College Algebra books, with examples of concepts making the ideas concrete whenever possible.
Calculator Problems: A symbol appears next to problems that require a calculator; some exercises and problems are designed to make students realize that by understanding the material, they can overcome the limitations of calculators | 677.169 | 1 |
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However, calculus is about calculating and with the advent of computers it has become possible to attack non-linear problems. For more information on tiering, contact the Center for Gifted Studies and Talent Development, Ball State University (BSU) Earlier in item three the "new syllabus" suggested students should be responsible for "taking out a common monomial factor and factoring the difference of two squares.
The specificity of the objectives or learning targets also makes it easier for teachers to create assessment tests of high validity and high reliability. Seeley points to an alternative and equally unfortunate scenario: "In other districts, compassionate teachers concerned about the potential for students failing .. or teachers pushed by administrators concerned about high failure rates, will see to it that algebra becomes accessible to their students by watering down the content of their course " Neither scenario will have a positive impact.
Student performance requires students to act and do. Project MATHEMATICS!��videos explore basic topics in high school mathematics �in ways that cannot be done at the chalkboard or in a textbook. In a vertically-sequenced course of study, it is quite easy for a student to encounter new material that assumes prerequisite knowledge and skills that were previously covered, but that were not understood and/or learned very well by that student, or that have been forgotten.
Holistic Numerical Methods licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License. Montessori was a major proponent of tactile learning. Just one simple example: When working with the equation 7x +15=8x+14, he said, "Since 8 is bigger than 7, if I move the 7x to the right, it will become negative, and if I combine the x's, I'll have a positive x [8x-7x=1x] instead of a negative x [7x-8x=-1x]. Note that all standards are important and are eligible for inclusion on the large scale assessment to be administered during the 2014-15 school year.
Curriculum-Based Measurement ( CBM ) is an approach for assessing the growth of students in basic skills that originated uniquely in special education. Third, group incentive structures were associated with higher achievement if the performance of individual students was accounted for and was reflected in the group rewards. In the second of this newsletter series, I write about Joy in Learning and Playing Games.
In this ground-breaking handbook, Richele Baburina reveals what every parent-teacher wants to know about Charlotte's approach to teaching math. If you teach in a remote school or on certain islands you may get an additional allowance: £1,928 for a distance island and £1,320 or £2,475 for a remote school. When my husband was talking to my 2nd grader about "carrying the one," he had no idea what he was talking about. Cramming for tests (memorize, regurgitate, and forget) is a common study skill!
We all understand the idea of a native language speaker of a natural language. Maria Andersen of Muskegon Community College has a great overview of common instructional practices in mathematics. Step 7: Solve the problem completely by writing the answer in full sentences. State University of New York at Buffalo, Buffalo, NY 14260 In reality, no one can teach mathematics. There is truly something for everyone! presentations for free. Jennifer Adams, EdD (Harvard University). Comparative and international education; Poverty and education; Child welfare; Educational policy.
To each his own, but I am sure that the time they spent on trying to learn math this way was not wasted anyway: their visual pathway is definitely a lot more developed and they've spent a lot of quality (and fun!) time with their parents – what can be more rewarding? Prerequisites: none. (Cross-listed with EDS 30.) Revisit students' learning difficulties in mathematics in more depth to prepare students to make meaningful observations of how K–12 teachers deal with these difficulties.
The emphasis is clearly on the acquiring of information or procedural skills. Multiplication is an operation that requires you to add another number to itself a certain number of times as indicated in the multiplication equation. They also need to be able to apply these concepts in active mathematical problem solving in order to be more confident with their mathematical skills. [3] Develop the mathematical skills.
How are the Common Core State Standards changing teaching, learning, and assessment in K-12 math education? While you watch a child do math homework, you probably won't recognize the methods they're using. We train trainers for teaching class LKG- 1 (4 - 7 years), Class II - V (8 years to 11 years), Class VII - X (12 years to 15 years) and Advance Levels for teaching students preparing for for competitive examination. There is review is included in each lesson, but in very reasonable amounts, 6-9 problems. | 677.169 | 1 |
Teachers and tutors may have their own personal philosophies, but there is one thing that they all agree on - every student should be attempting the questions from the practice SATs given in The Official SAT Study Guide. The SAT Prep Official Study Guide Math Companion contains solutions to every math question from each of the ten SATs in the 2nd edition of the Official SAT Study Guide. Access will also be given to solutions to the questions in the Official DVD Test. As usual, Dr. Warner gives simple, efficient, in-depth solutions to each of these problems and most problems are solved using several different methods; the quickest way to solve each of these problems is included - a feature ideal for those students going for a perfect 800 or near perfect score! Using this book you will learn to solve the SAT math problems from the Official SAT Study Guide in clever and efficient ways that will have you spending less time on each problem, and answering even the most difficult of these questions with ease. All definitions and concepts are reviewed in detail as they come up with each questionThis book contains everything you need to know in order to achieve your full potential on the math SAT. The first part of the book contains strategies to be used when taking practice tests published by the College Board in The Official SAT Study Guide, Second Edition. These include setting a target score, guessing and skipping rules, problem solving techniques, and detailed instructions for taking a cycle of practice tests. The second part of the book contains a review of the math topics that are on the test. For each math topic there is a lesson, homework problems in multiple choice format, and answer explanations. The book also contains cross-references between math topics and problems in the official guide. Written by an active math tutor, the material in this book has been used by more than 7,000 students and has been field tested over and over. | 677.169 | 1 |
Advances in mathematical economics. by S. Kusuoka, A. Yamazaki
A lot of financial difficulties can formulated as restricted optimizations and equilibration in their ideas. a number of mathematical theories were delivering economists with imperative machineries for those difficulties bobbing up in financial conception. Conversely, mathematicians were inspired by means of a variety of mathematical problems raised through monetary theories. The sequence is designed to collect these mathematicians who have been heavily attracted to getting new demanding stimuli from monetary theories with these economists who're looking for potent mathematical instruments for his or her researchers.
The highly-acclaimed MEI sequence of textual content books, helping OCR's MEI based arithmetic specification, totally fit the necessities of the necessities, and are reknowned for his or her pupil pleasant strategy.
An entire consultant to the mathematical instruments and methods used to unravel difficulties in physics, with a brand new artwork application, and references for utilizing Numerical Recipes and Mathematica.
Ziffel bought 3 cows and 12 sheep for $2400. If all the cows were the same price and all the sheep were another price, find the price charged for a cow or for a sheep. Let x = price of a cow Let y = price of a sheep Then Farmer Greenjeans' equation would be: 4 x + 6 y = 1700 Mr. Ziffel's equation would be: 3 x + 12y = 2400 To solve by addition-subtraction: Multiply the first equation by −2 : −2(4 x + 6 y = 1700) Keep the other equation the same: (3 x + 12y = 2400) By doing this, the equations can be added to each other to eliminate one variable and solve for the other variable. | 677.169 | 1 |
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This is a book full of ideas for introducing real world problems into mathematics classrooms and assisting teachers and students to benefit from the experience. Taken as a whole these contributions provide a rich resource for mathematics teachers and their students that is readily available in a single volume. Nowadays there is a universal emphasis on teaching for understanding, motivating students to learn mathematics and using real world problems to improve the mathematics experience of school students. However, using real world problems in mathematics classrooms places extra demands on teachers in terms of extra-mathematical knowledge e.g. knowledge of the area of applications, and pedagogical knowledge. Care must also be taken to avoid overly complex situations and applications. Papers in this collection offer a practical perspective on these issues, and more. While many papers offer specific well worked out lesson type ideas, others concentrate on the teacher knowledge needed to introduce real world applications of mathematics into the classroom. We are confident that mathematics teachers who read the book will find a myriad of ways to introduce the material into their classrooms whether in ways suggested by the contributing authors or in their own ways, perhaps through mini-projects or extended projects or practical sessions or enquiry based learning. We are happy if they do! This book is written for mathematics classroom teachers and their students, mathematics teacher educators, and mathematics teachers in training at pre-service and in-service phases of their careerser | 677.169 | 1 |
Author Biography
Table of Contents
CYCLE 1
Part 1: Gear Up
1.1 Focus Problem, Step 1: Understand the Problem
Exposure to an open-ended problem on a medical error
Previews the use of rates and units and Polya's problem solving process
1.2 Getting Started: Syllabus
Group activity to learn course policies
Develops critical reading skills
1.3 Getting Started: Skills
Venn diagrams are used to explore prerequisite knowledge
Reviews prerequisite skills
Develops Venn diagrams
1.4 Getting Started: Groups
Personality test to improve group dynamics
Develops knowledge of the Cartesian coordinate system
Develops the ability to plot and read ordered pairs
Exposes students to the term variable
1.5 A Tale of Two Numbers(FP)
Exploration of ratios and rates used in daily life
Previews scaling fractions
Develops ratio and rates concepts including notation
1.6 Part and Whole
Conceptual review of fraction operations
Reviews fraction concepts and operations
Develops the skill of drawing a picture
1.7 The Elusive A in Math
Assess traits necessary to success in mathematics
Develops pie and bar graphs
Connects equivalent fractions and scaling
Previews working with axes and increments on them
1.8 Two by Two
Visualizing situations with scatterplots
Develops scatterplots
Applies pie graphs and plotting points
1.9 Multiply or Divide? (FP)
Daily situations that involve unit conversions by multiplying or dividing
Develops unit conversions
Applies student success knowledge from 1.7
Part 1 Recap
Part 2: Shift Gears
1.10 Focus Problem, Step 2: Devise a Plan
Revisit focus problem and develop a plan to solve it
Applies knowledge gained to date
1.11 Higher or Lower? (FP)
Compare two pay structures with graphs, tables, and Excel
Previews integers, equations, like terms, and the commutative property
Develops percent calcuations, the concept of a function, and generalizing a calculation | 677.169 | 1 |
This presentation will look at using the Wolfram Demonstrations and Wolfram|Alpha Widgets to augment the classroom. These tools, which provide an interactive environment for mathematical experimentation, are freely available and provide a rich set of capabilities. Those capabilities include rapid deployment, real-time updates, and symbolic computation. Further, there are thousands of freely available interactive examples ready to deploy in undergraduate instruction. Finally, examples of combining these tools with other freely-available are presented. Focus is given to applying these tools in the introductory college mathematics courses along with discussion of the purely online mathematics education environment. | 677.169 | 1 |
Description - Essentials of Mathematics by Margie Hale
Essentials of Mathematics is designed as both a textbook and outside reading for college students who want to prepare themselves for mathematics courses beyond the first-year level. The mathematical content includes logic, set theory and a theoretical development of the number systems, giving students practice at proving mathematical statements. There are no answers in the book, but a separate manual provides instructor support. The book makes an excellent reference for students beginning to take courses in which proofs play a major role. In addition to the course material, there are narratives on the nature of mathematics and the mathematics profession. These sections can be read without help or guidance. | 677.169 | 1 |
Functions - MCR3U
Description de cours
This course introduces the mathematical concept of the function by extending your experiences with linear and quadratic relations. You will investigate properties of discrete and continuous functions, including trigonometric and exponential functions; represent functions numerically, algebraically, and graphically; solve problems involving applications of functions; investigate inverse functions; and develop facility in determining equivalent algebraic expressions. You will reason mathematically and communicate your thinking as you solve multi-step problems.
Le matériel supplémentaire
In mathematics, the word "function" refers to the idea that one quantity depends on another. The concept of a function is evident in many daily situations. For example, the time it takes for a vehicle to travel 10 km depends on its speed; the length of a tree's shadow depends on the time of day; the amount of an investment depends on the rate of interest earned.
In previous math courses, you studied linear functions and quadratic functions. You learned about the shape and features of the graphs of these functions in relation to their corresponding equations. In this course, you will extend your knowledge of linear and quadratic functions, as well as investigate and learn about other types of functions. Before you do that, you will need to develop a set of tools that can be used to explore and identify key characteristics of any type of function.
In the Key Questions for this unit you'll practise simplifying polynomial expressions by using inverse operations. You'll distinguish between relations and functions, and you'll determine if the inverse of a function is also a function. Finally, you'll identify the parameters of both linear and quadratic functions and use them to sketch the functions and solve problems.
Unit 2: Exponential Functions
In this unit, you will learn about a new type of function called an exponential function. This type of function arises from a variety of real-world applications, such as the radioactive decay of a chemical substance, population growth, and compound interest earned on investments and annuities.
Exponential functions are represented by powers that have a numerical base and a variable exponent. Before you can solve problems involving exponential functions, you will have to review and extend skills with powers, numerical bases, and exponents.
In the Key Questions for this unit you'll identify exponential functions based on their characteristics, graph them, and solve problems involving them.
Unit 3: Trigonometric Functions
This unit is different from the other units in this course because it deals specifically with functions that are connected to the properties of triangles. Trigonometry, which is the study of the relationships between the angles and sides of triangles, has many real-world applications, as you will see. This unit begins by reviewing familiar concepts from Grade 10 and then extending them. Real-world situations that are periodic, such as tides and temperature changes, can be modelled by trigonometric functions and their transformations. Connections to the transformations investigated in Units 1 and 2 will be made.
In the Key Questions for this unit you'll solve problems involving the six trigonometric ratios, the sine law, and the cosine law. You'll also prove trigonometric identities, and pose and solve problems regarding the characteristics of sinusoidal functions.
Unit 4: Discrete Functions
In Unit 4, you will explore the concepts of sequences and series. You will investigate the patterns found in the Fibonacci sequence and Pascal's triangle. This will lay the groundwork for your study of geometric and arithmetic sequences. In Lesson 18, you will examine series and solve real-world problems that relate to series. The remainder of the unit covers topics that someday could provide you with a big payoff: annuities, simple interest, and compound interest.
In the Key Questions for this unit you'll represent sequences and series in a variety of ways and make connections between them. You'll find the sums of arithmetic and geometric series, and apply what you've learned to practical examples involving compound interest and annuities. | 677.169 | 1 |
Foundations for Algebra Review
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This review can be used for the beginning of the year in an Algebra classroom. The review consists of algebraic expressions, simplifying and evaluating expressions, real numbers, estimating square roots, properties, equivalent expressions, ordered pairs, and solutions to equations. There are multiple examples in the word document that will allow for the students to review this material. This review can also be used as a pre-test to see what the students remember from their pre-algebra course. | 677.169 | 1 |
Bach/Leitner's progressive text lays a solid foundation for elementary algebra that carefully addresses student needs. The authors' clear, non-intimidating, and humorous style reassures math-anxious readers. Unlike workbook-format Prealgebra texts that stress competence at procedures, this text emphasizes understanding and mastery through careful step-by-step explanations that strengthen students' long-term abilities to conceptualize and solve problems. The text's innovative sequencing builds students' confidence with arithmetic operations early on before extending the basic concepts to algebraic expressions and equations. The authors' unusually thorough introduction to variables eases students through the crucial transition from working with numbers. Throughout the text, interesting applied examples and exercises and math-appreciation features highlight key concepts at work in a wide variety of real-world | 677.169 | 1 |
Description of the book "Usamos Matematicas en la Fiesta del Salon":
Key features:
- Native language instruction in a core content area
- Topics correlated to the first grade math curriculum
- Text reviewed by a math curriculum consultant and a reading consultant
- Engaging full-color photographs and illustrations that support the text and aid comprehension
- Examples that relate math to real-world situations
Special Features:
- Photographs that support the text and build math skills
- Glossary
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Unit 1: Analytical Methods for Engineers
Unit 1:
Analytical
Methods for Engineers
This unit will provide the analytical knowledge and techniques needed to
carry out a range of engineering tasks and will provide a base for further
study of engineering mathematics.
• Unit abstract
This unit enables learners to develop previous mathematical knowledge
obtained at school or college and use fundamental algebra, trigonometry,
calculus, statistics and probability for the analysis, modelling and solution
of realistic engineering problems.
Learning outcome 1 looks at algebraic methods, including polynomial
division, exponential, trigonometric and hyperbolic functions, arithmetic and
geometric progressions in an engineering context and expressing variables as
power series.
The second learning outcome will develop learners' understanding of
sinusoidal functions in an engineering concept such as AC waveforms, together
with the use of trigonometric identities.
The calculus is introduced in learning outcome 3, both differentiation
and integration with rules and various applications.
Finally, learning outcome 4 should extend learners' knowledge of
statistics and probability by looking at tabular and graphical representation
of data; measures of mean, median, mode and standard deviation; the use of
linear regression in engineering situations, probability and the Normal
distribution.
• Learning outcomes
On successful completion of this unit a
learner will:
1Be able to
analyse and model engineering situations and solve problems using algebraic
methods
2Be able to
analyse and model engineering situations and solve problems using trigonometric
methods
3Be able to analyse and model engineering
situations and solve problems using calculus
4Be able to
analyse and model engineering situations and solve problems using statistics
and probability.
Unit content
1Be able
to analyse and model engineering situations and solve problems using algebraic
methods
Arithmetic
and geometric: notation
for sequences; arithmetic and geometric progressions;the limit of a
sequence; sigma notation; the sum of a series; arithmetic and geometric series;
Pascal's triangle and the binomial theorem
Power series: expressing variables as power series functions and use series to findapproximate values eg exponential series, Maclaurin's series, binomial
series
2Be able
to analyse and model engineering situations and solve problems using
trigonometric methods
Trigonometric
identities: relationship between
trigonometric and hyperbolic identities; doubleangle and compound angle
formulae and the conversion of products to sums and differences; use of
trigonometric identities to solve trigonometric equations and simplify
trigonometric expressions
3Be able
to analyse and model engineering situations and solve problems using calculus
Calculus: the concept of the limit and continuity;
definition of the derivative; derivatives ofstandard functions; notion
of the derivative and rates of change; differentiation of functions using the
product, quotient and function of a function rules; integral calculus as the
calculation of area and the inverse of differentiation; the indefinite integral
and the constant of integration; standard integrals and the application of
algebraic and trigonometric functions for their solution; the definite integral
and area under curves
Further differentiation: second order and higher derivatives; logarithmic
differentiation;differentiation of inverse trigonometric functions;
differential coefficients of inverse hyperbolic functions
Further
integration:
integration by parts; integration by substitution; integration using partialfractions
Probability distributions: discrete and continuous distributions,
introduction to the binomial,Poisson and normal distributions; use of
the normal distribution to estimate confidence intervals and use of these
confidence intervals to estimate the reliability and quality of appropriate
engineering components and systems
Learning outcomes and assessment criteria
Learning outcomes
Assessment criteria for pass
On successful completion of
The learner can:
this unit a learner will:
LO1 Be able to analyse and model
1.1
determine the quotient and remainder for
algebraic
engineering
situations and
fractions and reduce algebraic fractions to
partial
solve
problems using
fractions
algebraic
methods
1.2
solve engineering problems that involve the
use and
solution of exponential, trigonometric and
hyperbolic
functions and equations
1.3
solve scientific problems that involve
arithmetic and
geometric series
1.4
use power series methods to determine
estimates of
engineering variables expressed in power
series
form
LO2 Be able to analyse and model
2.1
use trigonometric functions to solve
engineering
engineering
situations and
problems
solve
problems using
2.2
use sinusoidal functions and radian measure
to solve
trigonometric
methods
engineering problems
2.3
use trigonometric and hyperbolic identities
to solve
trigonometric equations and to simplify
trigonometric expressions
LO3 Be able to analyse and model
3.1
differentiate algebraic and trigonometric
functions
engineering
situations and
using the product, quotient and function of
function
solve problems using calculus
rules
3.2
determine higher order derivatives for
algebraic,
logarithmic, inverse trigonometric and
inverse
hyperbolic functions
3.3
integrate functions using the rules, by
parts, by
substitution and partial fractions
3.4
analyse engineering situations and solve
engineering
problems using calculus
LO4 Be able to analyse and model
4.1
represent engineering data in tabular and
graphical
engineering
situations and
form
solve
problems using
4.2
determine measures of central tendency and
statistics
and probability
dispersion
4.3
apply linear regression and product moment
correlation to a variety of engineering
situations
4.4
use the normal distribution and confidence
intervals
for estimating reliability and quality of
engineering
components and systems.
Guidance
Links
This unit
can be linked with the core units and other principles and applications units
within the programme. It will also form the underpinning knowledge for the
study of further mathematical units such as Unit 35: Further Analytical
Methods for Engineers, Unit 59: Advanced Mathematicsfor Engineering.
Entry
requirements for this unit are at the discretion of the centre. However, it is
strongly advised that learners should have completed the BTEC National unit Mathematics
for EngineeringTechnicians or equivalent. Learners who have not
attained this standard will require appropriatebridging studies.
Essential requirements
There are no essential resources for this unit.
Employer engagement and vocational contexts
The delivery
of this unit will benefit from centres establishing strong links with employers
willing to contribute to the delivery of teaching, work-based placements and/or
detailed case study materials. | 677.169 | 1 |
In this short "remember when," Jackie reflects on her experience with hair pressing and the life-changing experience of hot grease and smoking hot pressing combs; A humorous short-story, for those of you who recall the red coils of the hot plates, the silver-top can of grease, and grabbing your ears. More info →
Growing up Jackie was full of questions, most of which she gained the answers to through a good dose of home-grown wisdom from the grandparents who raised her. In her down-to-earth Southern style Jackie weaves true tales of learning how to decipher the code of "grown-folks" speech, the proper way to behave in a traditional Baptist church, as well as the pleasure of eating fresh cornbread, and so many more. These heart-warming and often humorous stories reveal the meaningful experiences that shaped her life, teaching her the importance of family and the power of love. More info →
Book 1 of the Strategies For Your Math Success Series utilizes visualization techniques and strategies to assist you with changing your "pre-determined" ideas about your math success. Leave those predetermined outcomes at the door. Let go of the "I just want to pass this class" mentality and raise your expectations. Everyone starts with an A, including you! More info →
(Book 2) The 5 components of R.U.D.A.R™ are Read, Understand, Do, Apply, Retain. Each is vital and fundamental to your overall math success. You begin the semester with an excellent grade (visit A New Math Attitude). What are you going to do to keep it? RUDAR can help you accomplish this task by going over some basics you need to remember. Take the time to review the fundamentals. Our awareness of the small yet consequential details that we tend to ignore, avoid, or appear obvious can make a tremendous difference in our math success.
Your goal is math success. It all begins with R.U.D.A.R™!
More info →
(Book 3) Say the words, "word problems" and you will see the majority of students cringe. Along with fractions, many students find the idea of word problems tedious and unnecessary. However, quite the contrary. We use words every day to verbalize our needs, wants, and desires. We complete word problems every day. Yet, it seems that once those same complex problems that we work daily are transferred to the pages of a mathematics book, homework assignment, or exam they become "undoable". With a well-organized and consistent strategy, you will find the problems are doable. They require the critical thinking and problem-solving skills which you possess. Using the 7 Easy Moves Method, you will take an "I did it in my head" strategy and transform it into a format that reflects true understanding and application of mathematical concepts. More info →
: (Book 4) Homework contributes to your math success. Homework provides opportunities to get a better handle on the concepts and ideas of a particular topic. Homework gives you the chance to learn, to reinforce and to review the subject material without the stress of an exam environment. If graded, successful completion of homework can contribute to your over-all class grade. These ten tips can be your guide in completing and submitting successful homework. More info →
(Book 5) Written for you, the student, this ebook is designed to introduce and reinforce , KCM, three basic steps to understanding the most introductory (Movies' Theorem™) to the most advanced theorem, property or rule. More info → | 677.169 | 1 |
Math Mammoth Ratios, Proportions & Problem Solving is a worktext that concentrates, first of all, on two important concepts: ratios and proportions, and then on problem solving. The book suits... More > best grade levels 6 and 7.
The book starts out with the basic concepts of ratio, rate, and unit rate. The lessons on equivalent rates allow students to solve a variety of problems involving ratios and rates. We connect the tables of equivalent rates with ordered pairs, use equations (such as y = 3x) to describe these tables, and plot the ordered pairs. The lessons about proportions show how to solve proportions using cross-multiplying and how to set up proportions the correct way.
The concept of direct variation is introduced in the lesson Proportional Relationships.
The lessons Scaling Figures, Floor Plans, and Maps give useful applications and more practice to master the concepts of proportions.
Lastly, we study various kinds of word problems involving ratios and use a bar model to solve these problems.< Less
Math on the Fly presents its simple and easy-to-understand ebook series to the public!
Dozens of examples with step-by-step explanations, along with a wide variety of practice problems, sets this... More > book series apart from others in its industry.
Written in a language that children and adults alike can learn from, this book covers everything you wanted to know about ratios. Topics include rates and unit rates, equivalent ratios, groupings, combining ratios, similar figures, solving proportions, setting up proportions to solve word problems, and much more!
The author, an Electrical Engineering graduate, relies on his years of experience as a professional math tutor to show that math fundamentals can be mastered and learned at any age.
Make Math on the Fly your number one source for math knowledge. Simple and easy, like it should be!< Less
Math need not be a lonely endeavor! This is a collection of 116 mathematics problems designed especially for groups. The problems focus on proportional reasoning, spatial visualization, and learning... More > to generalize from patterns -- central pillars of any math curriculum.
The book includes help for the teacher, a topics grid, and connections to several high-quality middle-grades math programs.< Less
Descriptive and Inferential Statistics, is a book that is intended for university students of any college. You'll find theory as summaries, and exercises solved, on the following topics: Descriptive... More > Statistics, Confidence Intervals and Test Hypothesis for means, proportions and variances for one sample, Chi Square Test, Test Hypothesis for means, proportions and variances, for two or more samples, and Regression line. Statistical software such as SPSS, Minitab, programs have been used in the resolution of problems and in some cases have been resolved by using the Excel and also manually end result will be a complete and total reversal of focus that will bring about a new human with entirely new and radical ways of achieving goals and solving problems. But we cannot do that while the pestilence prevents us and civilization from entering the promised land. Learn how to become powerful and unstoppable by having the right knowledge. We unknowingly wield immense power. We must recognize the fantastic power we possess and learn to use it constructively result will be a complete and total reversal of focus that will bring about a new human with entirely new and radical ways of achieving goals and solving problems. Learn how to become powerful and unstoppable by having the right knowledge. We unknowingly wield immense power. We must recognize the fantastic power we possess and learn to use it constructively. But we cannot do that while the pestilence prevents us and civilization from entering the promised land | 677.169 | 1 |
Precalculus: A Unit Circle Approach, Books a la Carte Edition
Ratti and McWaters teach the way you teach. Included are relevant and interesting applications; clear, helpful examples; and lots and lots of exercises--all the tools that you and your students need to succeed. This title offers a faster paced alternative and includes more rigorous topics ideal for students heading for calculus.
J.S.Ratti has been teaching mathematics at all levels for over 35 years. He is currently a full professor of mathematics and director of the "Center for Mathematical Services" at the University of South Florida. Professor Ratti is the author of numerous research papers in analysis, graph theory, and probability. He has won several awards for excellence in undergraduate teaching at University of South Florida and known as the coauthor of a successful finite mathematics textbook.
Marcus McWaters is currently the chair of the Mathematics Department at the University of South Florida, a position he has held for the last eleven years. Since receiving his PhD in mathematics from the University of Florida, he has taught all levels of undergraduate and graduate courses, with class sizes ranging from 3 to 250. As chair, he has worked intensively to structure a course delivery system for lower level courses that would improve the low retention rate these courses experience across the country. When not involved with mathematics or administrative activity, he enjoys playing racquetball, spending time with his two daughters, and traveling the world with his wife. | 677.169 | 1 |
Guide to Abstract Algebra is a comprehensive and accessible text covering the basic topics of an introductory abstract algebra course. New concepts are introduced gradually and illustrated by a variety of worked examples.
New features in this second edition are: · Two new chapters on Number Systems and Polynomials · Proofs by induction are introduced through a new section on sequences and recurrence relations · Fully updated to reflect the needs of today's first year undergraduate students
This book is ideal for first year undergraduate courses in Mathematics or Computer Science.
"synopsis" may belong to another edition of this title.
About the Author:
CAROL WHITEHEAD is a Lecturer in Mathematics at Goldsmiths' College, University of London214
Book Description Paperback. Book Condition: New. New, Softcover International Edition, Printed in Black and White, Different ISBN, Same Content As US edition, Book Cover may be Different, in English Language. Bookseller Inventory # 17968 | 677.169 | 1 |
Crossroads In The History Of Mathematics And Mathematics Education
Covers topics such as the development of Calculus through the actuarial sciences and map making, logarithms, the people and practices behind real world mathematics, and fruitful ways in which the history of mathematics informs mathematics education. The book is aimed at undergraduate mathematics majors and for use in mathematics education courses aimed at teachers.
The interaction of the history of mathematics and mathematics education has long been construed as an esoteric area of inquiry. Much of the research done in this realm has been under the auspices of the history and pedagogy of mathematics group. However there is little systematisation or consolidation of the existing literature aimed at undergraduate mathematics education, particularly in the teaching and learning of the history of mathematics and other undergraduate topics. In this monograph, the chapters cover topics such as the development of Calculus through the actuarial sciences and map making, logarithms, the people and practices behind real world mathematics, and fruitful ways in which the history of mathematics informs mathematics education. The book is meant to serve as a source of enrichment for undergraduate mathematics majors and for mathematics education courses aimed at teachers.
This website requires cookies to provide all of its features. For more information on what data is contained in the cookies, please see our Privacy Policy page. To accept cookies from this site, please click the Allow button below. | 677.169 | 1 |
EEC 510: Linear Systems
9/21/2015
Lecture 7: Linear vector and space (continued)
1. Dimension and basis
Dimension: The maximum number of linearly independent vectors in a linear space (X, F)
is called the dimension of the space.
- In two dimensional real
EEC 510: Linear Systems
9/14/2015
Lecture 6: Linear Vector and Space
1. Vectors
Vectors are interpreted as a magnitude and direction. (Defined physically)
- They are often restricted to two or three dimensions of physical space.
- They often have physical
EEC510: Linear Systems
8/26/2015
Lecture 2: Modeling of Physical Systems and
Introduction to State Equations
1. Modeling of physical systems
Clearly understand the purpose of this system
Determine the energy/power transfer characteristics of the system
EEC 510: Linear Systems
8/31/2015
Lecture 3: State Equation Modeling
1. Examples of state equation modeling.
Example 1: A mechanical system is shown as below. We take external force F as input and
displacements x1 and x2 as outputs. Please derive state eq
EEC 510: Homework 2
Due on Sept. 28, 2016
This homework is classified into two separate assignments. One is written assignment,
the other is Matlab assignment. For written assignment, you have to solve every problem
using hand calculation. For Matlab assi
EEC 510: Homework 1
Fall, 2016
Due on Sep. 14, 2016
You are encouraged to discuss with others to work out the homework problems. However, you
have to turn in your own work. Identical homework will receive a grade of zero. The problems
should be in order a
EEC510: Linear Systems
8/31/2016
Lecture 2: Introduction to State Equations
1. Introduction to state equations
Definition
- A collection of first-order differential equations
- Representing the same information as the original high-order differential equ
EEC 510: Linear Systems
9/14/2016
Lecture 5: Linear Vector and Space
1. Vectors
Vectors are interpreted as a magnitude and direction. (Defined physically)
- They are often restricted to two or three dimensions of physical space.
- They often have physical | 677.169 | 1 |
VIDEO
It doesn't matter what part of math you study, there will always be pages in a textbook that take a solid day or two to really understand. i guess it could be slightly easier for someone to study a subject & then study a subject that is relatively close to it.
Like some sort of algebraist might not have as much trouble working on some other kind of algebra because of their background. It would probably be harder for an analyst to start working on graph theory because they don't have a lot to do with each other.
Pure mathematics is more like art. Pure mathematicians work on building a foundation for a theory. One nice feature about pure mathematics is that it is free from argument. When a mathematician makes a discovery there is no opposition, as in science. And his theory stands the test of time, unlike science where one law is shown to be wrong in special cases. But once a foundation is build (like complex analysis) applied mathematicians take its result and use it to solve important problems.
Pure math is much more difficult. Classes in applied math consist of memorizing the steps to solve problems. However, classes in pure math involve proofs, which implies a good understanding of the subject matter is required. In pure math you need to justify everything you do. Which can sometimes make a simple argument long and complicated. It is easier for someone in pure math to learn applied math rather than someone in applied math to learn pure math.
The problems relating to geometry cover mostly triangles and circles. Even though polygons also are covered, the emphasis on polygons is not as much as on triangles and circles.
An angle of 90⁰ is a right angle; an angle less than 90⁰ is an acute angle; an angle between 90⁰ and 180⁰ is an obtuse angle; and angle between 180⁰ and 360⁰ is a reflex angle. The sum of all angles on one side of a straight line AB at a point O by any number of lines joining the line AB at O is 180⁰. When any number of straight lines join at a point, the sum of all the angles around that point is 360⁰. Two angles whose sum is 90⁰ are said to be complementary to each other and two angles whose sum is 180⁰ are said to be supplementary angles.
When two straight lines intersect, vertically opposite angles are equal. In the figure given alongside,
ii.Division Method : Divide the larger number by smaller one. Now divide the divisor by remainder. Repeat the process of dividing preceding number last obtained till zero is obtained as number. The last divisor is HCF.
Least common multiple[LCM] : The least number which is divisible by each one of given numbers is LCM.
There are two methods for this:
i.Factorization method : Resolve each one into product of prime factors. Then LCM is product of highest powers of all factors.
When a straight line XY cuts two parallel line PQ and RS [as shown in figure], the following are the relationships between various angles that are formed. [M and N are the points of intersection of XY with PQ and RS respectively]. | 677.169 | 1 |
PRECALCULUS: REAL MATHEMATICS, REAL PEOPLE, 6th Edition, is an ideal student and instructor resource for courses that require the use of a graphing calculator. The quality and quantity of the exercises, combined with interesting applications and innovative resources, make teaching easier and help students succeed. Retaining the series' emphasis on student support, selected examples throughout the text include notations directing students to previous sections to review concepts and skills needed to master the material at hand. The book also achieves accessibility through careful writing and design--including examples with detailed solutions that begin and end on the same page, which maximizes readability. Similarly, side-by-side solutions show algebraic, graphical, and numerical representations of the mathematics and support a variety of learning styles. Reflecting its new subtitle, this significant revision focuses more than ever on showing students the relevance of mathematics in their lives and future careers. Important Notice: Media content referenced within the product description or the product text may not be available in the ebook version. | 677.169 | 1 |
Essential math skills for engineers, Paul
Автор: Paul Название: Essential math skills for engineers (Существенные математические навыки для инженеров) Издательство:Wiley Классификация: Прикладная математика ISBN: 0470405023 ISBN-13(EAN): 9780470405024 ISBN: 0-470-40502-3 ISBN-13(EAN): 978-0-470-40502-4 Обложка/Формат: Paperback Страницы: 232 Вес: 0.37 кг. Дата издания: 28.04.2009 Серия: Numerical Methods & Algorithms Язык: ENG Иллюстрации: Illustrations Размер: 23.11 x 15.49 x 1.78 book provides the mathematical basics for any student of Engineering and Physics in either beginning or senior levels to efficiently relearn or brush up on the math skills required of them at all levels of study such as: trigonometry, and differential and integral calculus. This book is distinguished by the concentration of applied engineering mathematics most relevant to every-day engineering practice, as opposed to the numerous textbooks that are not suitable for extraction of the most relevant mathematical tools and concepts. Review of Essential Math Skills is based on the idea that in order to be a good engineer, one must also be a very good applied mathematician. If students have poor mathematical skills, they cannot hope to understand the highly technical analysis of engineering systems that they are supposed to be designing. This book offers the solid and basic math skills that all engineers and students of engineering need to have at their fingertips for immediate and day-to-day use. Some of the five topics covered include: solution of simultaneous, linear, algebraic equations, solution of linear, constant coefficient ordinary differential equations, solution of linear, constant coefficient difference equations, solution of linear, and constant coefficient partial differential equations. Описание: Just the math skills you need to excel in the study or practice of engineering Good math skills are indispensable for all engineers regardless of their specialty, yet only a relatively small portion of the math that engineering students study in college mathematics courses is used on a frequent basis in the study or practice of engineering.
Описание: Essential Mathematical Biology is a self-contained introduction to the fast-growing field of mathematical biology. Written for students with a mathematical background, it sets the subject in its historical context and then guides the reader towards questions of current research interest, providing a comprehensive overview of the field and a solid foundation for interdisciplinary research in the biological sciences. A broad range of topics is covered including: Population dynamics, Infectious diseases, Population genetics and evolution, Dispersal, Molecular and cellular biology, Pattern formation, and Cancer modelling.
Описание: This introduction to complex variable methods begins by carefully defining complex numbers and analytic functions, and proceeds to give accounts of complex integration, Taylor series, singularities, residues and mappings. Both algebraic and geometric tools are employed to provide the greatest understanding, with many diagrams illustrating the concepts introduced. The emphasis is laid on understanding the use of methods, rather than on rigorous proofs. Throughout the text, many of the important theoretical results in complex function theory are followed by relevant and vivid examples in physical sciences. This second edition now contains 350 stimulating exercises of high quality, with solutions given to many of them. Material has been updated and additional proofs on some of the important theorems in complex function theory are now included, e.g. the Weierstrass–Casorati theorem. The book is highly suitable for students wishing to learn the elements of complex analysis in an applied context.
Описание: Provides an introduction to applied probability and statistics for engineering or science majors. This book emphasizes the manner in which probability yields insight into statistical problems; ultimately resulting in an intuitive understanding of the statistical procedures most often used by practicing engineers and scientists.
Описание: Aimed at graduates and advanced undergraduates, this book presents methods of asymptotics and perturbation theory for obtaining solutions to differential and difference equations. They allow one to analyze physics and engineering problems that may not be solvable in closed form and for which brute-force numerical methods may not provide solutions. | 677.169 | 1 |
Tuesday, August 8, 2017
No-Nonsense Algebra from Math Essentials: A TOS Crew Review
My girls are getting older and I'm always on the lookout for where we are heading with math in the higher grades. I've had my eye on Math Essentials for a while now because of recommendations, incredibly low price, and the availability of video lessons if desired. I was super excited to have the opportunity to review No-Nonsense Algebra from Math Essentials.
I decided to have my middle daughter, Beth, give this a try. She is beginning 5th grade, but has always found math to be both enjoyable and fairly easy. To be honest, she's never been challenged really. My husband and I decided it would be good for her to give this a try and see how she did. I also have been working hard to brush up on my own mathematics skills. It isn't my strength, but this does not need to mean it is a weakness. I'll touch on my own experience at the end of the review.
I love the title. No-Nonsense is absolutely the best description for this course! There are no frills or excessive practice pages. Honestly, there isn't a need with the direct way the course approaches math is beautiful. I struggled to excel in mathematics once I passed the elementary levels and I can't help but think a more stream-lined approach may have helped me. The instructional videos are clear and easy to follow. They are also fairly short, so if a student needs to watch a video again it will not feel it is taking an excessive amount of time.
As I said before Beth is entering into 5th grade, but has never found math challenging. She just turned 10 and almost always makes a perfect score on every math assignment I give her. I honestly rarely even have to "teach" a concept to her in the traditional way because she intuitively understands mathematics concepts and reading the instructional information is usually enough. Because of these things which are consistently true to her nature I was pretty certain she would do well with this Algebra course. I was not disappointed!
Beth thoroughly enjoyed doing her math on the computer. I was a little surprised because while she enjoys the screen time we give her, she's never been one to ask for computer-based learning. She told me she liked the lesson because it was easy to understand. Because Beth finds math easy, it is never a subject she complains over completing. What was really neat to see was that she was even more eager to complete math lessons than usual and this is really high praise indeed!
Our standard course for mathematics is to use graphing paper for written assignments. This math course is non-consumable (My favorite kind!!!!) so students will need to copy the problems onto their own paper. This was excellent for Beth. Our regular curriculum does not function this way and I found this to be a skill in which she was lacking! For the first time in a very long time Beth missed several answers. Now this may not be cause for celebration for most people, but it was for me. Finallymy smart little girl was being challenged! What I discovered was all but one of the mistakes was due to a transcription error. This was only in the first lesson, but she was learning about the concept of negative numbers which was completely new to her. She comprehended the lesson completely, but the transcription errors were her downfall. This was an incredible learning experience for her I had not anticipated.
Since Beth is young compared to most students beginning an algebra course, I had her take more than one day for each lesson. This was an excellent pace for her. The first day she watched the video and completed half of the written work. I would check the work. If there were multiple errors the second day she would watch the lesson again if needed and correct her errors. If there were no errors we moved forward and finish the rest of the written work. This gave her between 8 and 10 problems a day which worked very well.
Beth has thoroughly enjoyed this program. I too have enjoyed working through it, although at a slightly faster pace. The concepts were more familiar for me. I did not feel the need to watch the videos as the instruction in the book were adequate for review. Because this is review I felt very comfortable working on one lesson a day and did so rather quickly. If I were not already stretching my time as far as possible I would have been able to complete two lessons in a day. For anyone needing to brush up on higher mathematics prior to testing for college entry this would be a good course with which to begin.
I highly, highly recommend this math course. In fact, it has been such a great fit for our style of learning that I fully intend to explore the other courses available from Math Essentials. | 677.169 | 1 |
Math For College Readiness Flvs Answers - fasera.herokuapp.com
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A computer program for the learning of algebra: description and first experiment
A computer program for the learning of algebra: description and first experiment - Download this document for free, or read online. Document in PDF available to download.
1 Leibniz - IMAG - Laboratoire Leibniz 2 ERES - ERES
Abstract : We present APLUSIX, a computer system that helps students to learn algebra, available at http:-aplusix.imag.fr. APLUSIX contains an advanced 2D editor of algebraic expressions that allows students to make the calculations she wants, like in a paper-pencil context. The system verifies the student-s calculations, by calculating the equivalence between two consecutives expressions, and shows the result to the student. It provides information concerning the progress of the resolution. Furthermore, the students can use commands like Reduce, Expand to ask the system to make some particular calculations. So APLUSIX combines features of a Microworld and of a CAS Computer Algebra System for education. This system has a lot of parameters allowing the teacher to choose the way it works. In particular, the teacher may active or not the verification of the equivalence between calculation steps, the progress indicators and the commands, to suit its own teaching strategy. We present also an experiment of the APLUSIX system realised in September 2002 with ten classes of 15 to 17 years old students grade 9, 10, 11 on the following tasks: expand or factor polynomial expressions, solve equations, inequalities or systems of linear equations. We describe the influence of some parameters on the students- behaviour and statistics concerning the work of the ten classes. We have observed and analysed the resolutions of the students. The analysis was made by replaying their actions they are recorded by the system. We present some results of this analysis. Last, we summarize an inquiry among the teachers who participated to the experiment
* Images in this website could be protected under copyright laws. They are picked automatically from bing. Please use the contact form to claim a copyright infriction and the item will be removed immediatelly. | 677.169 | 1 |
Contemporary College Algebra and Trigonometry: A Graphing Approach (with CD-ROM, Make the Grade, and InfoTrac)
Intended to provide a flexible approach to the college algebra and trigonometry curriculum that emphasizes real-world applications, this text integrates technology into the presentation without making it an end in itself, and is suitable for a variety of audiences. Mathematical concepts are presented in an informal manner that stresses meaningful motivation, careful explanations, and numerous examples, with an ongoing focus on real-world problem solving. Pedagogical elements including chapter opening applications, graphing explorations, technology tips, calculator investigations, and discovery projects are some of the tools students will use to master the material and begin applying the mathematics to solve real-world problems. CONTEMPORARY COLLEGE ALGEBRA AND TRIGONOMETRY includes a full review of basic algebra in Chapter 0 and full coverage of trigonometry to prepare students for the standard science/engineering calculus sequence. (The companion volume, CONTEMPORARY COLLEGE ALGEBRA includes all of the non-trigonometry topics, covered in sufficient detail to prepare a student for a business/social science calculus course.) Those who are familiar with the author's CONTEMPORARY PRECALCULUS should note that this book covers topics in a different order, and with a slower, gentle approach. Also, more drill exercises are included.
"synopsis" may belong to another edition of this title.
About the Author:
Thomas W. Hungerford received his M.S. and Ph.D. from the University of Chicago. He has taught at the University of Washington and at Cleveland State University, and is now at St. Louis University. His research fields are algebra and mathematics education. He is the author of many notable books for undergraduate and graduate level courses. In addition to ABSTRACT ALGEBRA: AN INTRODUCTION, these include: ALGEBRA (Springer, Graduate Texts in Mathematics, #73. 1974); MATHEMATICS WITH APPLICATIONS, Tenth Edition (Pearson, 2011; with M. Lial and J. Holcomb); and CONTEMPORARY PRECALCULUS, Fifth Edition (Cengage, 2009; with D. Shaw).
Review:
"It is obvious that the author has spent a great deal of time to incorporate technology in an appropriate way. I particularly like the way in which he asks the student to first check graphically the trig identity(s) he/she is trying to prove." "[The] Technology Investigations really help students to have a better and more meaningful understanding of the subject matter. They also help students to have a deeper conceptual understanding. Both the presentation of the materials and the problem sets are excellent. The text contains outstanding applied problems. Additionally, it contains well thought out "Discovery Projects."
"The use of the calculator investigations/explorations within the text does a very good job of illustrating the algebraic properties or solving techniques that are being discussed. The directions are easy to follow and it is clear as to what the student is actually trying to explore." "I found the text easy to read, there were plenty of examples for the students to use to help guide them with the homework, and the page layouts were neither cluttered nor confusing."
Book Description Brooks Cole, 20007672
Book Description Brooks Cole32417-2-4 | 677.169 | 1 |
Competition to win the efficient pro forma Mathematics
Math in Focus® and Marshall Cavendish® are registered trademarks of Times Publishing Limited. Or, do you need to start with some very basic information in order for people to understand your work? The emphasis of this program is on methods of discovery and communication of knowledge, not the mastery of established facts. As other sciences develop, problems which require the use of these tools are numerous and pressing. DB had this notion of "alphabet soups", which I liked, referring to all these acronyms (MKT, PCK, etc.), and the question was raised: have we done ourselves any good by doing this?
Pages: 0
Publisher: China Press (January 1, 2000)
ISBN: 7560274145
Mathematics Unbound: The Evolution of an International Mathematical Research Community, 1800--1945
Numerical Solutions of Nonlinear Differential Equations: Publication No 17) Proceedings Advanced Symposium, Mathematics Research Center, United States Army, University Wisconsin, May 1966)
For me, the Arnold Trick was becoming a favorite tool. Several years later, a fairly eminent abelian group theorist who should have known better referred to my almost completely decomposable paper as "historic." In an effort to acquaint you with our academic programs and areas of ongoing research, Jefferson College of Graduate Studies sponsors summer research positions that provide opportunities for upper-level undergraduate students (those who have completed their junior year), to work in the laboratories of active faculty researchers Measure for Measure quartzrecordings.com. The main idea in these papers is often to consider the zero set as a measure and then use harmonic analysis, related to an algebraic curve, the so-called characteristic curve of the equation Introduction to Metric and download pdf wypozyczsobiebusa.pl. Pacing guides indicate how to spread instruction across multiple days. 5-minute Warm Ups get students ready for the lesson. Technology resources enhance instruction and aid in planning. Differentiated Instruction and ELL Vocabulary Highlights help you reach all students ref.: Building Regression Models in read online Building Regression Models in Social. Whether utilizing print and/or online formats, students can develop deeper understanding of mathematics with Motivation Math. Hiebert and Wearne (1992; 1993; 1996) reported that a critical attribute in regards to student learning in mathematics, is the nature of the learning task in which to engage students. Students need mental engagement in challenging and worthwhile mathematical tasks that emphasize the conceptual aspects of the topic and promote the formation of mathematical connections The latest national and provincial research and solution papers in mathematics college entrance examination coaching assessment(Chinese Edition) Multidisciplinary REU Program: Research, Education and Training in Computational Mathematics and Nonlinear Dynamics of Biological, Bio-inspired and Engineering Systems The Department of Mathematical Sciences at GMU will host a multidisciplinary undergraduate research program in computational mathematics and nonlinear dynamics of biological, bio-inspired and engineering systems, from June 3 to Aug 2, 2013 , source: Journal for Research in Mathematics Education Vol. 30, No. 5, November 1999 Journal for Research in Mathematics.
Mat Prentice Hall Mathematics Research Reports: Studies Supporting the Efficacy of the "Prentice Hall Mathematics" Program -- Grades 6 - 12 It is rather quiet, as the students are all taking their 5th "MOP test" in the last two weeks, and my fellow instructors and graders are either proctoring the tests or preparing more material to teach these brilliant kids Shadow prices, duality and download pdf Shadow prices, duality and Green's. This game was played at a "Discovery Day" organized by Big Brother Mouse in Laos. Computer-based math an approach based around use of mathematical software as the primary tool of computation. Conventional approach: the gradual and systematic guiding through the hierarchy of mathematical notions, ideas and techniques. Starts with arithmetic and is followed by Euclidean geometry and elementary algebra taught concurrently Contributions to Nonlinear Partial Differential Equations: v. 2: Conference Proceedings (Pitman Research Notes in Mathematics Series)
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Copies may not be duplicated for commercial purposes. Unauthorized posting of RAND PDFs to a non-RAND Web site is prohibited The Theory and Practice of download pdf You don't want to do any proving here, just tell the reader what questions you plan to discuss in your paper. Take the reader on a trip through your research project. Using your log book as your guide, start with your initial explorations and conjectures Reconstructing School Mathematics: Problems with Problems and the Real World Mathematical research papers - #1 reliable and professional academic writing aid. Forget about those sleepless nights working on your coursework with our writing ... Mathematical Research Paper Advice for amateur mathematicians on writing and publishing papers Theres no reason why amateurs cant make worthwhile research ...mathematical research paper Numerical Analysis and read epub projectbaseline.org. Best practices provide tips and opportunities to incorporate manipulatives into instruction. Chapter at a Glance in the Teacher's Edition makes it easy to prepare for lessons. Multi-day lessons allow enough time for students to reach mastery. A skills trace highlights previous and future connections , cited: 6 Pages To Open Your Eyes To download here download here. Amazingly enough, my summer job serving coffee and eggs to grouchy folks in Boston led me to an interesting combinatorics problem that I am going to talk to you about today." A short joke related to your topic can be an engaging way to start your speech , cited: Mathematics, Game Theory and Algebra Compendium (Mathematics Research Developments) The program provides students who plan to pursue a Ph. D. degree and enter academic careers with the tools needed to facilitate the application, admission, and enrollment process for graduate school , source: Introduction to MATLAB 7 for download for free quartzrecordings.com. Item and rating scale analyses indicate that the items and the six-category scale perform as intended. These indicators suggest that the SETS instrument may be appropriate for measuring preservice teacher levels of self-efficacy to teach statistics Nonlinear Elasticity: read online
Fluently add and subtract within 20 using mental strategies.2 By end of Grade 2, know from memory all sums of two one-digit numbers. 3. Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division. By the end of Grade 3, know from memory all products of two one-digit numbers. "Finding single-digit products and related quotients is a required fluency for grade 3 Shape-Preserving Approximation by Real and Complex Polynomials Shape-Preserving Approximation by Real. CCSE operates a group of scholarly journals covering the natural sciences, social sciences, and engineering. The editorial board welcomes you to join us as a reviewer. Status: Part time Working language: English Participation in the peer-review process is absolutely essential to the success and reputation of the journal. Reviewers and editors determine which works are of quality and significance ref.: On the waiting time in the read for free On the waiting time in the queuing. Teachers can easily individualize instruction to meet RtI Tiers 1, 2, and 3. Math in Focus uses the bar modeling method as a problem-solving tool that is taught explicitly beginning in Grade 2. Students become familiar with this systematic way to translate complex word problems into mathematical equations and avoid the common issue of not knowing where to start. Word problems grow in complexity from one step to two step to multistep, which enhances students' ability to think critically in a systematic process Handbook of Design Research Methods in Education: Innovations in Science, Technology, Engineering, and Mathematics Learning and Teaching If you include photographs or drawings, be sure to write captions that explain what the reader is looking at Twenty-Four Rembrandt's download for free Twenty-Four Rembrandt's Paintings. A minimum overall GPA of 3.0 is required with upward trends in grades being preferable. The program is open to students with or without prior research experience, and students who have not participated in SUPERB previously. The University of California Santa Cruz (UCSC) Summer Undergraduate Research Fellowship in Information Technology (SURF-IT) offers opportunities for both UCSC students and non-UCSC students , cited: The response of a nuclear read epub You can imagine that three of the ten seats would be introduced by three of the customers. In general, given n seats and c customers, we remove c-1 chairs and select the seats for the c customers Descriptive And Normative Approaches To Human Behavior (Advanced Series on Mathematical Psychology) read pdf. Emily Putnam (2001): "The REU at OSU is a wonderful, intellectually stimulating program, providing great exposure to higher math. Come and have fun for your summer working in areas you won't see in your regular program." Jennifer Kimble (2001): "The REU at Oregon State is definitely worth your time 60 Addition Worksheets with read online read online. It was a sort of stupid little obvious proof, in fact, but because the problem was so well known, I thought that I could probably get away with publishing this special case. Then over the next few days I started thinking that maybe the endomorphism ring didn't have to be so special after all. I could use a fairly standard trick in non-commutative ring theory (a subject which at at this point was seen as fairly far removed from abelian group theory) to extend the result a little further ref.: School effects on educational achievement in mathematics and science, 1985-86: National Assessment of Educational Progress (Research and development report) read for free. | 677.169 | 1 |
Mathematics
The study of Mathematics is an integral part of a student's education
and students must now be successful in Mathematics in order to qualify to graduate from high school.
Mathematics
Comet Bay College differentiates its Mathematics courses so that all students have the opportunity to learn at a level which best suits their ability and can achieve success at that level.
In this way, students can be accelerated or remediated according to need. This is illustrated in the addition of a pathway grade on their reports, which supplements the learning area grade. This is particularly useful in Year 10, as it is used to determine if a student is eligible for specific senior school courses.
The Year 11-12 courses are differentiated into Methods and Applications for those students who wish to sit the university entrance exams; and into Essentials for those students who wish to attend other tertiary education or enter the workforce at the completion of their schooling.
Homework is an essential element of studying Maths, just as a sportsman needs to practice skills in order to be proficient in a particular sporting field. In Years 7-10, there is a comprehensive weekly homework schedule which is supplemented by nightly skills work at the discretion of the teacher.
The Mathematics teachers at Comet Bay College also offer free support sessions for students on set nights, according to age and ability. Your child is encouraged to attend in order to get the personal help that he or she needs in order to improve in Mathematics | 677.169 | 1 |
Mathematics
Mathematics is one of the great achievements of the human mind. The mathematical way of thinking empowers the human intellect, enhances critical thinking, and facilitates rational decision-making. This has been recognized since classical times, and for this reason mathematics has become part of a great intellectual tradition, which includes the philosophical writing of Plato and Descartes and the four subjects of the medieval quadrivium. Representing ideas in a symbolic manner and analyzing arguments with the help of logic are, first and foremost, mathematical exercises.
Applications of mathematics pervade today's culture. Policy makers and ordinary citizens increasingly confront issues in science and technology which are formulated in mathematical language. However, mathematics far exceeds this historical scope. For example, mathematics is used to analyze personal finances, formulate government fiscal policy, justify data-based decisions, encode and protect information, provide a unified understanding of the forces of nature, and manage the treatment of disease. Understanding the scope and power of mathematics enables graduates to better make informed decisions as citizens and as potential leaders of the country and of the world.
Goals and Perspectives
Following are specific learning goals for courses which satisfy the University mathematics requirement, and a few comments about these courses.
1. The main goal is to provide students with experience in the mathematical way of thinking, especially insofar as this way of thinking fosters the development of disciplined habits of mind and enhances the power of the intellect. Students will learn deductive reasoning in problem solving and the inductive process in drawing conclusions from mathematical analysis.
2. Students will learn to read and understand mathematical symbols and formulas, and to be able to express their thoughts by using symbols and equations. They will demonstrate the ability to use mathematical formulas to express clear and precise relationships between the variables involved. It is sometimes said that the best way to clarify a thought is to write it as an equation. Students will work as active learners to understand the language of mathematics.
3. The courses will stress conceptual learning along with technique. Students will learn that there is a commonality in the world of mathematics by seeing fundamental concepts in different settings. For instance, in calculus they will learn that notions like velocity, acceleration, and marginal profit are all particular manifestations of the general concept of derivative. Students will learn how mathematics can be used to abstract key features of our world and reason about these features in a general context.
4. Students will learn mathematical techniques and methods by using algebraic or analytic manipulations to produce explicit solutions to computational problems. Students will learn mathematics by doing mathematics.
5. Students will develop modeling skills. These skills include describing the situation under consideration clearly, translating appropriate aspects into equations using suitable variables, symbols, and mathematical concepts, and interpreting possible mathematical solutions in terms of the original process. Students will come to appreciate that since mathematical models only approximate real world situations, they sometimes require adjustment or improvement.
6. Students will learn that mathematics informs solutions to a wide variety of problems in modern society. Problem areas include but are not limited to the environment, the economy, health care, and politics.
While it is important that there be a variety of mathematics offerings which satisfy the mathematics requirement, we expect that these will include a few specific subjects. We mention three in particular: calculus for its central place in the western intellectual tradition and its importance in modeling the physical world; statistics, because it has become an essential tool for engaging the vast amounts of data that accompany many problems in business, engineering, science, and social science; logic, since it is the foundation of rational discourse. | 677.169 | 1 |
Having specialized online calculator:
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Logarithmic and Trigonometric Tables to Five Places with
Special table: shows each trig function evaluated for special angles, like 30, 45, and 60 degrees. Avatar: The Last Airbender: During one of Aang's pre-invasion nightmares, he must ... take a math test he didn't study for! The most downloaded graphing calculator for the iPhone/iPod is now available for the iPad. Most calculators can display up to eight figures, although some of the more expensive ones may have ten or even twelve. Height of transmission tower can be calculated as follows: Question: 8 - A statue, 1.6 m tall, stands on the top of a pedestal.
It refers also to math packages with an emphasis on MAPLE and a disk comes with the package, which I have ignored. Note that coding theory is different from cryptography. An Introduction to Information Theory: Symbols, Signals and Noise. Dover. 1980. 0486240614 A very good introduction by a major contributor seems to be out of print (Dover, where are you?!): For an introduction to coding theory, look at books on abstract algebra that do applications such as Childs or Lidl and Pilz ref.: Algebra and Trigonometry: read epub Algebra and Trigonometry: Graphs and. Feel free to copy-and-paste anything you find useful here. All we ask is that you link back to this site. Simplify: √50 - √18√ is the square root sign, copy and paste it for your answer Evaluate the function at each specified value of the independent variable and simply: f(x) = x2 + 1 (a) f(t2) (b) f(t + 1)remember: just put x3 for x3(place commas between multiple answers) in order Suppose z varies directly as the square of x and inversely as y Plane Trigonometry - Primary download here Find hypotenuse length BA to the nearest hundredth. Set up the diagram and the formula in the same manner as was done in Example 1. You should arrive at the drawing and the formula shown here. Hint: If you are having a problem solving the equation algebraically, remember that when x is on the bottom, you must divide to arrive at your answer. The division is always "divide BY the trig value decimal" , source: Trigonometry: Student's Solutions Manual, 8th Edition Trigonometry: Student's Solutions. They should use a graphing calculator to visualize how these function graphs interlace High School Subjects Self read for free subtractionrecords.com. In other words, the angles are opposite from the side of the same letter Logarithmic and trigonometric tables Logarithmic and trigonometric tables.
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See problem #6 from the practice problems Know the reciprocal identities and the cofunction identities. Be able to use these identities to find trigonometric values. See problem #7-8 from the practice problems Be able to convert angle measures in degrees to radians and vice versa download. A gapped worksheet which allows calculation of the exact trigonometric ratios for certain angles for sine, cosine and tangent. Three helpful worksheets which support the derivation of the exact values for sin, cos and tan of key angles. You will need to register for a TES account to access this resource, this is free of charge Logarithmic and trigonometric tables, by Edward A. Bowser Find the gcf on a TI 83 Calculator, complex variable simultaneous equations matlab, radicals in quadratic equations, texas financial calculator cube root calculation, Find each sum or difference , cited: Student's Resource for Hecht's Physics: Algebra and Trigonometry Student's Resource for Hecht's Physics:. Take a piece of cardboard ABeD, Fig. 30, and at a point 0 in it fix a needle ON at any angle. At any point P on the needle stick another needle PO into the board, and perpendicular to it download. The Trigonometrical Ratios 59 triangle with the usual construction (Fig. 51) as follows: In continuation of section 48 we note that: or the tangent of an angle is equal to the cotangent of its complement Delta Instruction Manual read epub. The trigonometric functions can be defined in other ways besides the geometrical definitions above, using tools from calculus and infinite series , source: Modern algebra and download online Modern algebra and trigonometry:. For example, from the table above we see that This equivalence is called an identity. If we had an equation with sec x in it, we could replace sec x with one over cos x if that helps us reach our goals , source: Surveying and Navigation, With a Preliminary Treatise on Trigonometry and Mensuration subtractionrecords.com. The reason I believe the way it did after the inadvertent space vote with. Every year during early kind of weird having border with Canada yet right in epub.
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About this item
Comments: Fast Shipping! Ships within 1 business day. Arrives within 3-8 business days. May contain minor highlighting/markings or minor1296429
ISBN: 0321296427
Edition: 1
Publication Date: 2007
Publisher: Addison-Wesley Longman, Incorporated
AUTHOR
Ratti, Jogindar, McWaters, Marcus M.
SUMMARY
A guide for college professors and students on college algebra and trigonometry, this text contains clear, helpful examples and many practice exercises.Ratti, Jogindar is the author of 'College Algebra and Trigonometry College Algebra And Trigonometry', published 2007 under ISBN 9780321296429 and ISBN 03212964 | 677.169 | 1 |
Rattan and Klingbeil's Introductory Mathematics for Engineering Applications is designed to help improve engineering student success through application-driven, just-in-time engineering math instruction. Intended to be taught by engineering faculty rather than math faculty, the text emphasizes using math to solve engineering problems instead of focusing on derivations and theory
This textbook provides a step-by-step approach to numerical methods in engineering modelling. The authors provide a consistent treatment of the topic, from the ground up, to reinforce for students that numerical methods are a set of mathematical modelling tools which allow engineers to represent real-world systems and compute features of these systems with a predictable error rate. | 677.169 | 1 |
This article talks about the important concepts, formulae and previous year questions related to Chapter Permutations and Combinations for JEE Main and JEE Advanced Examination 2018. This can be used as a quick revision note before JEE Examination 2018.
In this article you will get complete information about all the topics of Chapter Sequence and Series like arithmetic progression, geometric progression, harmonic progression etc. About 2-4 questions are being asked from this chapter. This can be used as quick revision notes before the JEE Examination 2018.
In this article you will get to Online test for JEE Main, JEE Advanced, UPSEE, WBJEE and other engineering entrance examinations that will help the students in their preparation. These tests are free of cost and will useful in performance and inculcating knowledgeGet important formulae from unit Integral Calculus for quick revision. These formulae are very useful during competitive examination. This revision notes includes chapters – Indefinite Integral, Definite Integral.
Find Chapter Notes for IIT JEE, UPSEE and WBJEE engineering entrance examination. These chapter notes will be very useful during entrance examination as they will provide complete overview of the chapter.
In this article we are going to talk about circular permutation. This is one of the most important topics of Permutation. There are two important formulae in this topic. First, the number of circular permutations of n different things taken all at a time is (n - 1)! when clockwise and anti clockwise direction matters. Read this article to understand these formulae and other concepts.
In this article find quick revision notes of unit Probability and the Binomial Theorem. This revision notes is very important for UPSEE, WBJEE and other state level engineering entrance examination as direct question based on formulae is asked in thisOnline Tests play an important role in self assessing and understanding the level of questions asked in the examination. Find free online test of chapter Trigonometric Functions of Mathematics for JEE MAIN, JEE Advanced, UPSEEE, WBJEE and other engineering entranceFind Mathematics Online Test of the Chapter Inverse Circular Function in this article. This chapter is very important for all engineering entrance examinations. About 3-5 questions are being asked from this chapter. | 677.169 | 1 |
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Math Technology for Visually Impaired Students
By Susan Osterhaus, Secondary Mathematics Teacher, TSBVI
The use of scientific graphing calculators is now a mainstay of secondary math classrooms, and they must be provided to students for the End-of-Course Examination for Algebra I. When the new statewide assessment (TAKS) begins in Spring 2003, the Algebra I exam will be eliminated, and it is expected that the TAKS 9th, 10th and 11th Grade (Exit Level) Mathematics will require the use of a scientific graphing calculator. Initially, calculators will not be permitted on the 8th Grade math TAKS. However, their use may be included in the near future. Each student must also have access to a graphing calculator for routine classwork and practice. Visually impaired students must meet the same requirements as their peers, and Teachers of the Visually Impaired (TVIs) must be prepared to teach their students the skills needed to be successful in mathematics courses. A review of the technology currently available may help TVIs determine the most appropriate ways to teach their students and prepare them for statewide assessments.
The Administrator's Manual of the 2001 End-of-Course Test states that each student must have access to a graphing calculator during the testing. Students may use any graphing calculator except those with typewriter-style keypads (known as QWERTY) or those that include a computer algebra system. Students may also use any four-function or scientific calculator on the test, but hand-held minicomputers or laptop computers may not be used. TEA usually accommodates the needs of special education students on an individual basis. If a visually impaired student uses, or wishes to use, a piece of technology not currently approved by TEA, the student's TVI should contact the Accommodations Task Force at TEA, which may give permission to use it on statewide assessments. For further information, the TVI might also call the Student Assessment Division of TEA, at (512) 463-9536. Remember, the device must be an accommodation that the student routinely uses in class.
Large Display Scientific/Graphing Calculators
Texas Instruments (TI) makes a ViewScreen package for several TI calculators including the TI-82 and TI-83. It has worked well with some low vision students. They use a ViewScreen calculator connected to a ViewScreen LCD display panel placed on a light box. Some students prefer using their calculator on the newer color CCTVs. This technology is easily available and has been approved by TEA. The ViewScreen package is already used by math teachers on their overhead projectors, so they should have no problem training visually impaired students with this technology.
The VisAble is the only large display scientific calculator made as a one-piece portable unit, and is an alternative for low vision students who are unable to use one of the TI solutions. However, it does not have graphing capabilities. Betacom Corporation manufactures it. Although most general education math teachers will be unfamiliar with the VisAble, the various functions are easily identifiable, and a willing math teacher should have little difficulty orienting the visually impaired student to the VisAble. This technology is more expensive than some, but it does meet with TEA's approval.
TI-83 Trainer (Current price: $49.50)
Professor Goldstein's TI-83 Trainer is an affordable computer software program with complete TI-83 calculator emulation. When installed on a laptop, the student has a very portable device. The addition of magnification software provides even better accessibility. If the math teacher has selected the TI-83 as the class calculator of choice, the TI-83 Trainer is an excellent option for the low vision student. The math teacher should have little difficulty orienting the visually impaired student. However, this option has not been approved by TEA at the present time.
Scientific Notebook (SN) is another software package. When installed on a laptop, the student has a very portable device, which is more than just a graphing scientific calculator. SN is also a math/text processor, so the student can do all assignments, calculations, and graphs in one document directly on the laptop. It has onscreen magnification up to 400%, or additional magnification software may be used. In addition, two large print fonts are available from MAVIS at NMSU, which will allow further onscreen magnification and large print hard copies. Metroplex Voice Computing is even working on voice recognition software to make it accessible to the totally blind. With the right techniques, it is also possible for a blind student to work with matrices using Scientific Notebook and a screen reader to solve systems of equations and find regression lines. Furthermore, math teachers can enter all their worksheets, tests, etc. on SN, and the teacher of the visually impaired can easily translate them into Nemeth code. Many general education math teachers are just now discovering SN and seem quite excited about its potential. Although it has not been approved by TEA at the present time, most math teachers should find it to be affordable and user-friendly.
Braille Scientific Calculator
The Leo is the only stand-alone braille-display scientific calculator, and is an alternative for the deafblind student who does not use a notetaker with braille display. However, it does not have graphing capabilities. Robotron Sensory Tools manufactures it. Most math teachers will need training before they can assist the student with this technology. The cost may be prohibitive for most, but it does meet with TEA's approval.
Talking Scientific/Graphing Calculators
Certain low vision students may prefer a stand-alone talking scientific calculator, and although there are many such calculators on the market today, the ORION TI-34 from Orbit Research is currently the most affordable and user-friendly. It is also approved by TEA. While it does not have graphing capabilities, it is easily accessible by totally blind students (unlike the TIs and the VisAble), and features a built-in learning mode. The ORION's LCD display and functionality are identical to the TI-34, so math teachers should feel very comfortable orienting the visually impaired student.
Graph-It is a tactile scientific graphing calculator program for Blazie Engineering Note-Takers. Graph-It PC is designed for use with IBM compatible PCs. Both are available from Freedom Scientific. The student can type in an equation and produce a tactile graphic on most embossers. An audio representation of the graph can also be played through the speaker for a quick, sound-picture of the graph. The software is quite limited, however, and the tactile graphics and audio graph lack precision. The note-takers also include a built-in scientific calculator. Although this combination is not the most user-friendly or time efficient scientific graphing calculator solution, it may be the only option for a deafblind student. Most math teachers will need training before they can assist a visually impaired student with Graph-It, and this is not a TEA-approved solution.
The Accessible Graphing Calculator (AGC) from ViewPlus Software, Inc. is a self-voicing graphing scientific calculator software program. Unlike a hand-held calculator, it displays results through speech and sounds, as well as visually presenting numbers and graphs. This program is intended to have capabilities comparable to a full-featured hand-held scientific and statistical graphing calculator. The AGC is truly accessible for all students, and could be used for the entire class. The onscreen graphics are easily seen by a low vision student via an enlargement feature, and the graph can be listened to by using the sophisticated audio wave feature. Print copies can be made with any standard printer using a variety of fonts, including braille. The print copies with braille fonts can be copied onto swell paper and run through a tactile imaging machine. One of the best ways to use the AGC is with a TIGER Braille/graphics embosser from ViewPlus Technologies, Inc., but the TIGER is rather expensive. Although considerable time is typically needed for training a blind student to use the ACG totally independently, the math teacher is usually able to assist the student because it is so user-friendly for the sighted individual. The AGC cannot do matrices or parent functions, but the various functions it will perform are quite impressive. (The vendor plans to continue upgrading the software, including working with matrices.) It has not been approved by TEA at the present time.
Conclusion
Teachers of the visually impaired must make many decisions about appropriate programming and technology for their students. They must not only be aware of the different kinds of technology that are available, but also be able to teach their students how to use them. For TEA to approve the use of a piece of technology, it should be routinely used by a student to complete assignments at school. The student should (a) have access to a graphing calculator, (b) know how to use a graphing calculator, and (c) use it to practice routine class work at the same time as his or her classmates.
TSBVI is committed to being a resource for visually impaired students, their teachers, and their families. It sponsors a variety of workshops and training opportunities, and maintains a math website at The website not only provides information about appropriate materials, tools and technology, but also offers specific suggestions for collaboration between teachers of the visually impaired and general education math teachers. In addition, the Special Programs Department offers a three-week summer class, Adaptive Tools and Technology for Accessible Mathematics (ATTAM), and a one-week ATTAM class during the regular school year. They also offer several one-week sessions of individualized instruction on specific IEP objectives throughout the year, which might include math and/or math technology goals. Finally, the Comprehensive Programs Department offers Algebra I, Geometry, and Algebra II in two-period blocks during the academic school year. The two-period block allows additional time for learning the necessary Texas Essential Knowledge and Skills (TEKS), perfecting the Nemeth code, and training in the use of adaptive math tools and technology.
If you have questions about teaching math to students with visual impairments, contact Susan Osterhaus at or (512) 206-9305 | 677.169 | 1 |
Tapping into Mathematics with the TI-80 Graphics Calculator
Not only does this book teach the basics of mathematics and calculator skills, but also exploits the graphics calculator as a learning aid to develop the student's understanding of mathematical principles, explaining the facilities of the calculator and showing when and how to apply it. This approach is effective and far more exciting for the students.
Book Description Addison-Wesley / Open University, 1997. Soft Cover. Book Condition: Very Good. No Jacket. First Edition. 320pp with index, illustrated throughout with black and white diagrams. Slight edgewear and creasing to corners, otherwise very good copy with no marks or inscriptions. Size: 4to - over 9¾" - 12" tall. Bookseller Inventory # 0038981993748 | 677.169 | 1 |
Algebra 1: Linear and Exponential Model Task Cards with QR Codes
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This set of 16 task cards is made up of 4 sets of 4 cards on Linear and Exponential Models. Each set of exercises has a different focus about linear and exponential models:
Set 1. Distinguish Between Linear and Exponential Situations
Set 2. Construct Linear, Quadratic, and Exponential Models
Set 3. Compare Linear, Exponential, and Quadratic Data
Set 4. Interpret expressions for functions in terms of the situation they model.
After students record their solution to a question on their recording sheet, they can then use a QR reader to scan the QR code. The student will be shown a complete solution to the question, not just an answer. Students can then compare their solution to the one they are shown. This will help students see where their thinking went wrong. | 677.169 | 1 |
Excursions in Modern Mathematics (7th Edition)
Excursions in Modern Mathematics, Seventh Edition, shows readers that math is a lively, interesting, useful, and surprisingly rich subject. With a new chapter on financial math and an improved supplements package, this book helps students appreciate that math is more than just a set of classroom theories: math can enrich the life of any one who appreciates and knows how to use it.
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About the Author:
Peter Tannenbaum has bachelor's degrees in Mathematics and Political Science and a Ph. D. in Mathematics, all from the University of California, Santa Barbara. He has held faculty positions at the University of Arizona, Universidad Simon Bolivar (Venezuela), and is currently professor of mathematics at the California State University, Fresno. His current research interests are in the interface between mathematics, politics and behavioral economics. He is also involved in mathematics curriculum reform and teacher preparation. His hobbies are travel, foreign languages and sports. He is married to Sally Tannenbaum, a professor of communication at CSU Fresno, and is the father of three (twin sons and a daughter).
Book Description Pearson. Hardcover. Book Condition: New. 0321568036137 | 677.169 | 1 |
This second edition has been fully updated to bring it in to line with the latest University of Cambridge International ExaminationsIGCSE syllabus (2006), and is ideal for students following the Extended Curriculum (up to Grade A). Follows the order of the syllabus: Number, Algebra, Shape and Space, Statistics and Probability Most chapters contain a Core Section, which revises material already studied, plus an Extended Section, which introduces the new material International contexts are used to aid understanding and ensure this text is relevant to your students. Brand new ICT sections included, with ideas for ICT activities for each chapter Assessment: each Core Section has two short-answer Assessments for students; the Extended Sections have two short-answer and two structured-question Assessments, corresponding to Papers 1 to 4 of the IGCSE examination Worked examples and plenty of exercises throughout for rigorous drill and practice Full solutions and an index are included
"synopsis" may belong to another edition of this title.
About the Author:
The authors, Terry Wall and Ric Pimental, have also written Core Mathematics for IGCSE 2nd edition".EVIV##12751
Book Description Trans-Atlantic Publications, Inc. PAPERBACK. Book Condition: New. 034090813008130-ABA
Book Description Hodder41 ? Follows the order of the syllabus: Number, Algebra, Shape and Space, Statistics and Probability ? Most chapters contain a Core Section, which revises material already studied, plus an Extended Section, which introduces the new material ? International contexts are used to aid understanding and ensure this text is relevant to your students. ? Brand new ICT sections included, with ideas for ICT activities for each chapter ? Assessment: each Core Section has two short-answer Assessments for students; the Extended Sections have two short-answer and two structured-question Assessments, corresponding to Papers 1 to 4 of the IGCSE examination ? Worked examples and plenty of exercises throughout for rigorous drill and practice ? Full solutions and an index are included New edition of a best-selling textbook ? Fully updated to reflect changes to the University of Cambridge International Examinations IGCSE specification ? Included in the CIE recommended resources list ?Contains material on how to integrate ICT fully into the teaching of IGCSE Mathematics Printed Pages: 488. Bookseller Inventory # 86951 • Follows the order of the syllabus: Number, Algebra, Shape and Space, Statistics and Probability • Most chapters contain a Core Section, which revises material already studied, plus an Extended Section, which introduces the new material • International contexts are used to aid understanding and ensure this text is relevant to your students. • Brand new ICT sections included, with ideas for ICT activities for each chapter • Assessment: each Core Section has two short-answer Assessments for students; the Extended Sections have two short-answer and two structured-question Assessments, corresponding to Papers 1 to 4 of the IGCSE examination • Worked examples and plenty of exercises throughout for rigorous drill and practice • Full solutions and an index are included New edition of a best-selling textbook • Fully updated to reflect changes to the University of Cambridge International Examinations IGCSE specification • Included in the CIE recommended resources list •Contains material on how to integrate ICT fully into the teaching of IGCSE Mathematics Printed Pages: 488. Bookseller Inventory # 86951 | 677.169 | 1 |
General AAS
Maths for Chemists Volume II: Power Series, Complex Numbers and Linear Algebra builds on the foundations laid in Volume I, and goes on to develop more advanced material. The topics covered include: power series, which are used to formulate alternative representations of functions and are important in model building in chemistry; complex numbers and complex functions, which appear in quantum chemistry, spectroscopy and crystallography; matrices and determinants used in the solution of sets of simultaneous linear equations and in the representation of geometrical transformations used to describe molecular symmetry characteristics; and vectors which allow the description of directional properties of molecules.
Pat McKeague's passion and dedication to teaching mathematics and his ongoing participation in mathematical organizations provides the most current and reliable textbook for both instructors and students. Pat McKeague's main goal is to write a textbook that is user-friendly. Students are able to develop a thorough understanding of the concepts essential to their success in mathematics because of his attention to detail, exceptional writing style, and organization of mathematical concepts. The Fifth Edition of Intermediate Algebra: Concepts and Graphs is another extraordinary textbook with exceptional clarity and accessibility.
Students who approach math with trepidation will find that Intermediate Algebra, Second Edition, builds competence and confidence. The study system, introduced at the outset and used consistently throughout the text, transforms the student experience by applying time-tested strategies to the study of mathematics. Learning strategies dovetail nicely into the overall system and build on individual learning styles by addressing students' unique strengths. The authors talk to students in their own language and walk them through the concepts, showing students both how to do the math and the reasoning behind it. Tying it all together, the use of the Algebra Pyramid as an overarching theme relates specific chapter topics to the 'big picture' of algebra. | 677.169 | 1 |
6284 Calc
By Amit Kalra
Description
6284 Calc is a tool for students to do their math work without having to memorize hundreds of formulas throughout math or science!
Use 6284 Calc when you're stuck on a math/science problem, forget a formula, or just don't feel like doing you homework because you're too cool for school! (most likely the case)
Not a student? That's cool! Use the Daily Life Math section for real life situations like shopping, giving tips, investments, and interest! (Extremely helpful during the holidays!) There's also an Extra section that contains unique stuff! | 677.169 | 1 |
There are some people that love numbers and there are some people that hate numbers. Those that love numbers usually take up careers that involve numbers and many of those that hate numbers can be seen taking up humanities as their career. But does this mean that someone that is scared of numbers cannot have a career involving numbers? Yes they can provided they put in that extra bit to grasp numbers by their napes. Intermediate Accounting 18th Edition and Elementary Statistics A Step By Step Approach 8th Edition are two books that can help people grasp numbers.
Although none of Intermediate Accounting 18th Edition and Elementary Statistics A Step By Step Approach 8th Edition directly involve working with numbers (they are not mathematics books from any angle) but they give you that channel to handle the numbers of accounts and stats better. Yes, they are not books that teach you to learn the ABC of these subjects. But they are books that propel you on your way when you have some basic knowledge of accounts and statistics. And when I say basic knowledge I mean basic knowledge. You know a bit about accounts and stats you can really make it count using these books.
Accounting and statistics could be complex subjects depending on the way you look at them. Some people are plain scared of them. You look at an accounting or a statistical problem and the fact that you have to reach a stated end result could be daunting for you. But there are textbooks that are able to remove the scare. And in this category of books fall Intermediate Accounting 18th Edition and Elementary Statistics A Step By Step Approach 8th Edition.
Some people think that you should have some love for numbers if you want to take up accounting or statistics as your career option. But this may not be true. You may not be in love with mathematics but you can fall in love with accounting and statistics. Both these subjects have this enigma around them. You see someone studying accounting or statistics and their status suddenly goes up. And even when you never liked mathematics in school you think of taking up accounting or stats in college. And as mentioned above, if you want to grasp these subjects you will scarcely find better books than Intermediate Accounting 18th Edition and Elementary Statistics A Step By Step Approach 8th Edition.
Both these books are highly popular among teachers and students and it could be a real challenge when you try to buy them from your college bookstore. It could often happen that they run out of stock or you have to wait for hours to get to the counter. But you can skirt this problem by ordering your copies of Intermediate Accounting 18th Edition and Elementary Statistics A Step By Step Approach 8th Edition online. Choose one of the better known online bookstores and they are sure to have both these books in their stock. | 677.169 | 1 |
Explained: Matrices
December 6, 2013
by Larry Hardesty
A matrix multiplication diagram.
Among the most common tools in electrical engineering and computer science are rectangular grids of numbers known as matrices. The numbers in a matrix can represent data, and they can also represent mathematical equations. In many time-sensitive engineering applications, multiplying matrices can give quick but good approximations of much more complicated calculations.
Matrices arose originally as a way to describe systems of linear equations, a type of problem familiar to anyone who took grade-school algebra. "Linear" just means that the variables in the equations don't have any exponents, so their graphs will always be straight lines.
The equation x - 2y = 0, for instance, has an infinite number of solutions for both x and y, which can be depicted as a straight line that passes through the points (0,0), (2,1), (4,2), and so on. But if you combine it with the equation x - y = 1, then there's only one solution: x = 2 and y = 1. The point (2,1) is also where the graphs of the two equations intersect.
The matrix that depicts those two equations would be a two-by-two grid of numbers: The top row would be [1 -2], and the bottom row would be [1 -1], to correspond to the coefficients of the variables in the two equations.
In a range of applications from image processing to genetic analysis, computers are often called upon to solve systems of linear equations—usually with many more than two variables. Even more frequently, they're called upon to multiply matrices.
Matrix multiplication can be thought of as solving linear equations for particular variables. Suppose, for instance, that the expressions t + 2p + 3h; 4t + 5p + 6h; and 7t + 8p + 9h describe three different mathematical operations involving temperature, pressure, and humidity measurements. They could be represented as a matrix with three rows: [1 2 3], [4 5 6], and [7 8 9].
Now suppose that, at two different times, you take temperature, pressure, and humidity readings outside your home. Those readings could be represented as a matrix as well, with the first set of readings in one column and the second in the other. Multiplying these matrices together means matching up rows from the first matrix—the one describing the equations—and columns from the second—the one representing the measurements—multiplying the corresponding terms, adding them all up, and entering the results in a new matrix. The numbers in the final matrix might, for instance, predict the trajectory of a low-pressure system.
Of course, reducing the complex dynamics of weather-system models to a system of linear equations is itself a difficult task. But that points to one of the reasons that matrices are so common in computer science: They allow computers to, in effect, do a lot of the computational heavy lifting in advance. Creating a matrix that yields useful computational results may be difficult, but performing matrix multiplication generally isn't.
One of the areas of computer science in which matrix multiplication is particularly useful is graphics, since a digital image is basically a matrix to begin with: The rows and columns of the matrix correspond to rows and columns of pixels, and the numerical entries correspond to the pixels' color values. Decoding digital video, for instance, requires matrix multiplication; earlier this year, MIT researchers were able to build one of the first chips to implement the new high-efficiency video-coding standard for ultrahigh-definition TVs, in part because of patterns they discerned in the matrices it employs.
In the same way that matrix multiplication can help process digital video, it can help process digital sound. A digital audio signal is basically a sequence of numbers, representing the variation over time of the air pressure of an acoustic audio signal. Many techniques for filtering or compressing digital audio signals, such as the Fourier transform, rely on matrix multiplication.
Another reason that matrices are so useful in computer science is that graphs are. In this context, a graph is a mathematical construct consisting of nodes, usually depicted as circles, and edges, usually depicted as lines between them. Network diagrams and family trees are familiar examples of graphs, but in computer science they're used to represent everything from operations performed during the execution of a computer program to the relationships characteristic of logistics problems.
Every graph can be represented as a matrix, however, where each column and each row represents a node, and the value at their intersection represents the strength of the connection between them (which might frequently be zero). Often, the most efficient way to analyze graphs is to convert them to matrices first, and the solutions to problems involving graphs are frequently solutions to systems of linear equations.
In the last decade, theoretical computer science has seen remarkable progress on the problem of solving graph Laplacians—the esoteric name for a calculation with hordes of familiar applications in scheduling, image processing, ...
The maximum-flow problem, or max flow, is one of the most basic problems in computer science: First solved during preparations for the Berlin airlift, it's a component of many logistical problems and a staple of introductory ...
It took only a few years for high-definition televisions to make the transition from high-priced novelty to ubiquitous commodity—and they now seem to be heading for obsolescence just as quickly. At the Consumer Electronics ...
(PhysOrg.com) -- A new analysis of number randomness in Sudoku matrices could lead to the development of more difficult and multi-dimensional Sudoku puzzles. In a recent study, mathematicians have found that the way that | 677.169 | 1 |
ISBN-10: 0883857448
ISBN-13: 9780883857441Topology is a branch of mathematics packed with intriguing concepts, fascinating geometrical objects, and ingenious methods for studying them. The authors have written this textbook to make the material accessible to undergraduate students without requiring extensive prerequisites in upper-level mathematics. The approach is to cultivate the intuitive ideas of continuity, convergence, and connectedness so students can quickly delve into knot theory, the topology of surfaces and three-dimensional manifolds, fixed points and elementary homotopy theory. The fundamental concepts of point-set topology appear at the end of the book when students can see how this level of abstraction provides a sound logical basis for the geometrical ideas that have come before. This organization exposes students to the exciting world of topology now(!) rather than later. Students using this textbook should have some exposure to the geometry of objects in higher-dimensional Euclidean spaces together with an appreciation of precise mathematical definitions and | 677.169 | 1 |
This idea is one that has
developed from years of listening to students and watching how they learn as
well as attendance at several seminars with Saxon Math publishers.Saxon is designed for the teacher to teach no more than 10 minutes.The students learn from the material and examples presented in the text
and use the teacher as a resource.Students
have reinforced this idea for years with comments like: "I just go home and
read the book.I like to ask
questions in class."
Class time will be devoted to
problem solving with word problems form a variety of sources outside of Saxon.This will provide students to exposure to a variety of types of word
problems written in different ways.Occasionally
there will be group projects that students in one or more of the books will
complete together.We will also
work on problems together that seem to challenge a large group of the students
based on the homework errors and finally individual questions.
Assignments will be made, tests
will be given weekly.Students will
bring home their test and complete it.Parents
should initial it showing they know it was completed.Students should NOT use their book on tests or have parents help.All the students' work MUST be shown and turned in with the test.
Students who need additional assistance may sign up for private 30
minutes sessions with the teacher.Each
student may have 2 of these per month at no additional fee after that the charge
will be $10 per session.
How
can I decide what level my child should take?
You can go to the Saxon
website and have your student take the placement test. If they test ready
for Algebra 1/2 or above this class is for them. We will cover Pre-Algebra
- Calculus this year using Saxon books. | 677.169 | 1 |
The ideal review for your intro to mathematical supplies a concise guide to the standard college courses in mathematical economics 710 solved problems Clear, concise explanations of all mathematical economics concepts Supplements the major bestselling textbooks in economics courses Appropriate for the following courses: Introduction to Economics, Economics, Econometrics, Microeconomics, Macroeconomics, Economics Theories, Mathematical Economics, Math for Economists, Math for Social Sciences Easily understood review of mathematical economics Supports all the major textbooks for mathematical economics coursesSchaum's Easy Outline Series When you are looking for a quick nuts-and-bolts overview, there's no series that does it better. Schaum's Easy Outline of Introduction to Mathematical Economics is a pared-down, simplified, and tightly focused version of its predecessor text offers a presentation of the mathematics required to tackle problems in economic analysis. After a review of the fundamentals of sets, numbers, and functions, it covers limits and continuity, the calculus of functions of one variable, linear algebra, multivariate calculus, and dynamics.
The ideal review for your principles of 964 solved problems Outline format supplies a concise guide to the standard college courses in economics Clear, concise explanations of all economics concepts Complements and supplements the major economics textbooks Appropriate for the following courses: Economics, Principles of Economics, Microeconomics, Macroeconomics Easily understood review of economics Supports all the major textbooks for economics coursesConfused about financial management? Problem solved. Schaum's Outline of Financial Management provides a succinct review of all financial management concepts in topics such as financial forecasting, planning and budgeting, the management of working capital, short-term financing, time value of money, risk, return, and valuation, capital budgeting, and more.
This book provides a systematic exposition of mathematical economics, presenting and surveying existing theories and showing ways in which they can be extended. One of its strongest features is that it emphasises the unifying structure of economic theory in such a way as to provide the reader with the technical tools and methodological approaches necessary for undertaking original research. The author offers explanations and discussion at an accessible and intuitive level providing illustrative examples. He begins the work at an elementary level and progessively takes the reader to the frontier of current research. This second edition brings the reader fully up to date with recent research in the fieldConfusing Textbooks? Missed Lectures? Not Enough Time'. . . . This Schaum's Outline gives you. . Practice problems with full explanations that reinforce knowledge. Coverage of the most up-to-date developments in your course field. In-depth review of practices and applications. . . Fully compatible with your classroom text, Schaum's highlights all the important facts you need to know. Use Schaum's to shorten your study time-and get your best test scores!. . Schaum's Outlines-Problem Solved..
Providing an overview of necessary computational mathematics, this book continues with a series of key economics problems using "higher mathematics". Presenting a mix of classical and contemporary economic theory, this book covers the problems of uncertainty, continuous-time dynamics, comparative statistics, and the applications of optimization methods to economics.Confusing Textbooks? Missed Lectures? Not Enough Time? Fortunately for you, theres Schaums Outlines. More than 40 million students have trusted Schaums Schaums Outline gives you Practice problems with full explanations that reinforce knowledge Coverage of the most up-to-date developments in your course field In-depth review of practices and applications Fully compatible with your classroom text, Schaums highlights all the important facts you need to know. Use Schaums to shorten your study time-and get your best test scores! SchaumsNotes de Math matiques Appliqu es has been writing in one form or another for most of life. You can find so many inspiration from Notes de Math matiques Appliqu es also informative, and entertaining. Click DOWNLOAD or Read Online button to get full Notes de Math matiques Appliqu es book for free. | 677.169 | 1 |
ELTE Summer School in Mathematics
ELTE Summer School in Mathematics
ELTE Summer School in Mathematics
6–10 June 2017
One-week summer school for undergraduate and graduate students in Mathematics, Computer Science or Sciences, organized by the Institute of Mathematics, ELTE, Budapest. If you are planning your future and want to check our graduate school that offers an English language MSc program, then this is a good opportunity to do so with a one-week long intensive experience.
Topic:
"Higher mathematics through problem solving"
Although mathematics is much more than just going through a series of drill problems - posing new questions, building new theories, applying the existing tools to other areas of science are just as important -, problem solving remains one of the most important parts of mathematics education. And it's fun, too!
A new definition, a theorem or even whole theories are best understood when we are guided to discover these results through our own efforts. Carefully chosen problems will lead us to understand these results without feeling the difficulties of reading a "dry theorem", and we will also better appreciate the conditions which give the proper setting of a mathematical statement. By getting trained in problem solving, we will also get a training for doing research.
During the summer school there will be problem solving sessions in algebra, number theory, combinatorics, geometry, analysis and probability theory, each of them concentrating on one or two special topics. Participants will get a brief introduction into the necessary notions and results and then individual work will follow, with the guidance of the lecturers and their assistants.
Mathematics education at ELTE
Mathematics education has a long tradition in Hungary. Eötvös Loránd University (ELTE) has leading mathematicians among its former students and former and current professors. Frigyes Riesz, Lipót Fejér, John von Neumann, Pál Turán, Endre Szemerédi (Abel Prize winner in 2012), László Lovász, and Pál Erdős are all ELTE alumni.
Important dates
Early bird application: 30 April 2017
Latest deadline for applications: 31 May 2017
First day of classes: 6 June 2017
Last day of classes: 10 June 2017 | 677.169 | 1 |
Unit 8: Geometric Applications of the Integral
We consider in this unit several things that can be done with the
definite integral. In unit 7, we defined
where
as a way of calculating the area under a curve. We now extend this
idea to using integrals for other geometric purposes. In each case, we
show the quantity to be calculated can be approximated by a sum of
``slices'' of one form or another. We then take a limit process to
reach infinitely many thin slices, which will be an integral.
Objectives
After completing this unit you should be able to set up integrals to
calculate areas, areas between curves, volumes of solids of
revolutions (by both disk and shell method), arc lengths, and areas of
surfaces of revolution.
Suggested Procedure
Simmons 7.1-7.6
Work many of the problems at the ends of each section. Be sure
that you are capable of setting up the integral for any of the
geometric problems presented. The ability to solve such integrals is
also important, with the caveat that you haven't been given all of the
techniques of integration yet. Still, the problems are mostly designed
so as to produce an integral you can solve. | 677.169 | 1 |
(back cover) An ideal companion to high school geometry textbooks, this volume covers all topics prescribed by the New York State Board of Regents for the new Geometry exam. For Students: Easy-to-read topic summaries Step-by-step demonstration examples Review of all required Geometry topics Hundreds of exercises with answers for practice and review Glossary of Geometry terms In-depth Regents exam preparation, including problems similar to those you'll find on the actual Regents exam For Teachers: A valuable lesson-planning aid A helpful source of practice and test questions
Includes up-to-date information on the Geometry Common Core Regents Exam. Review topics such as triangle congruence, similarity and right triangle trigonometry, parallelograms, and more. The first two actual Regents exams in Geometry are also included, with all answers thoroughly explained.
Reflecting the latest New York State curriculum change, this brand-new addition to Barron's Let's Review series covers all topics prescribed by the New York State Board of Regents for the new Integrated Algebra Regents exam, which replaces the Math A Regents exam. This book stresses rapid learning, using many step-by-step demonstration examples, helpful diagrams, enlightening "Math Fact" summaries, and graphing calculator approaches. Fourteen chapters review the following topics: sets, operations, and algebraic language; linear equations and formulas; problem solving and technology; ratios, rates, and proportions; polynomials and factoring; rational expressions and equations; radicals and right triangles; area and volume; linear equations and graphing; functions, graphs, and models; systems of linear equations and inequalities; quadratic and exponential functions; statistics and visual representations of data; and counting and probability of compound events. Exercise sections within each chapter feature a large sampling of Regents-type multiple-choice and extended response questions, with answers at the back of the book. Students will find this book helpful when they need additional explanation and practice on a troublesome topic, or when they want to review specific topics before taking a classroom test or the Regents exam. Teachers will value it as a lesson-planning aid, and as a source of classroom exercises, homework problems, and test questions."Barron's Regents Exams and Answers: Geometry" and "Barron's Let's Review Geometry," are available as a two-book set, giving buyers a savings of $2.99 as compared with the price of the books purchased separately.
(back cover) An ideal companion to high school physics textbooks, this volume covers all topics prescribed by the New York State Board of Regents for the physics exam. For Students: Practice and review questions with answer keys Subject review covers all physics topics that appear on the Regents exam Includes a full-length Regents exam with answer key For Teachers: A valuable classroom supplement to the main textbook and a lesson planning aid A helpful source of practice and test questions | 677.169 | 1 |
You'll gain access to interventions, extensions, task implementation guides, and more for this lesson.
Big Ideas:
Problems that exist within the real-world, including seemingly random bivariate data, can be modeled by various algebraic functions.
This lesson builds on students' prior work with trigonometric functions. This task focuses on the simple harmonic motion of a spring hanging vertically with a weight on the end that is stretched and released. By considering all of the parameters of this task, students will realize that a seemingly simple task requires trigonometric modeling. This builds towards their understanding of how trigonometric functions can be used in regression modeling of real-world bivariate data which comes up later in Algebra 2 and in Advanced Placement and college-level courses.
Vocabulary:
Trigonometry, Simple Harmonic Motion, Sine, Cosine, Model, Oscillation, Amplitude, Period, Phase Shift
Special Materials:
Graphing calculator or access to | 677.169 | 1 |
Here is the catalog description of Math 215: Probability, statistics,
ratios, and proportional relationships. Experimental and theoretical
probability. Collecting, analyzing, and displaying data, including
measurement data. Multiple approaches to solving problems involving
proportional relationships, with connections to number and operation,
geometry and measurement, and algebra. Understanding data in professional
contexts of teaching. Taught primarily through student activities and
investigations. Prerequisite(s): C or better in MATH 112.
The course intends to provide future K-8 (and higher) teachers with
adequate knowledge of probability and statistics in a format that can be
adapted for use in their classrooms. We will combine hands-on activities
with calculator use to explore concepts in a concrete and tangible way. A
TI-83/84 calculator is required for the course. We will not use a textbook
but there is a course pack of materials whose price is only the cost of
Xeroxing. Some materials are online at
(click on Statistics). We hope that you will find some of them suitable to
try with students in your future classes.
This
course intends to provide K-12 preservice and inservice teachers (and
others) with adequate algebraic knowledge in a format that can be
directly used in their classrooms. We will combine algebra with
geometry based on measurement of distance, providing hands-on
activities, some adaptable even for students in early elementary
grades. Calculators (TI-108 and TI-83/84) will be used. Graduate
students/practicing teachers will take Math 513 section 1 and sit in the
same university class with Math 313 section 1 undergraduates. For
materials we will use a course pack, some units from our book, Breaking
away from the Algebra and Geometry Book (2001)), and units from
our website,
For more information (on either course, on dates to register, how to
register, etc.) contact Pat Baggett, baggett@nmsu.edu, (575) 646-2039 | 677.169 | 1 |
Intermediate Algebra: Functions and Authentic Applications
9780130144980
01301449800
Marketplace
$5.21
More Prices
Summary
For courses in Intermediate Algebra that incorporate a graphing calculator. Unique and enthusiastic in its approach, Lehmann's text is a rich combination of important skills, concepts, and applications. This text captivates students' and instructors' interest as they use curve fitting to model current, compelling, and authentic situations. The curve fitting approach emphasizes concepts related to functions in a natural, substantial way and encourages students to view functions graphically, numerically, and symbolically as well as to verbally describe concepts related to functions. The examples in the test demonstrate both how to perform skills and how to investigate concepts. Students learn why they perform skills to solve problems as well as how to solve the problems. Explorations deepen students' understanding as they investigate mathematics with graphing calculator and pencil and paper activities.
Table of Contents
Preface
xiii
Linear Functions
1
(48)
Using Qualitative Graphs to Describe Situations
1
(5)
Sketching Graphs of Linear Equations
6
(9)
Slope of a line
15
(8)
Graphical, Numerical, and Symbolic Significance of Slope
23
(9)
Finding Linear Equations
32
(6)
Functions
38
(11)
Taking It to the Lab
44
(1)
Chapter Summary
45
(1)
Key Points of This Chapter
45
(1)
Review Exercises
46
(2)
Chapter Test
48
(1)
Modeling with Linear Functions
49
(40)
Using Lines to Model Data
49
(8)
Finding Equations for Linear Models
57
(8)
Function Notation and Making Predictions
65
(11)
Slope is a Rate of Change
76
(13)
Taking It to the Lab
82
(4)
Chapter Summary
86
(1)
Key Points of This Chapter
86
(1)
Review Exercises
87
(1)
Chapter Test
88
(1)
Systems of Linear Equations
89
(42)
Using Graphs to Solve Systems
89
(11)
Using Elimination and Substitution to Solve Systems
100
(7)
Using Systems to Model Data
107
(8)
Using Linear Inequalities in One Variable to Make Predictions
115
(16)
Taking It to the Lab
124
(2)
Chapter Summary
126
(1)
Key Points of This Chapter
126
(1)
Review Exercises
127
(1)
Chapter Test
128
(1)
Cumulative Review of Chapters 1--3
129
(2)
Exponential Functions
131
(52)
Properties of Exponents
131
(7)
Rational Exponents
138
(10)
Sketching Graphs of Exponential Functions
148
(10)
Finding Equations for Exponential Models
158
(6)
Using Exponential Functions to Model Data
164
(19)
Taking It to the Lab
174
(4)
Chapter Summary
178
(1)
Key Points of This Chapter
178
(1)
Review Exercises
179
(2)
Chapter Test
181
(2)
Logarithmic Functions and Inverse Functions of Linear Functions
183
(42)
Finding Inverse Functions of Linear Functions
183
(9)
Logarithmic Functions
192
(8)
Properties of Logarithms
200
(5)
Using the Power Property with Exponential Models to Make Predictions
205
(8)
More Properties of Logarithms
213
(12)
Taking It to the Lab
219
(1)
Chapter Summary
220
(1)
Key Points of This Chapter
220
(1)
Review Exercises
221
(1)
Chapter Test
222
(1)
Cumulative Review of Chapters 1--5
223
(2)
Quadratic Functions
225
(46)
Sketching Graphs of Functions in Vertex Form
225
(9)
Expanding and Factoring Quadratic Expressions
234
(7)
Factoring Quadratic Expressions
241
(6)
Using Factoring to Solve Quadratic Equations
247
(11)
Sketching Graphs of Quadratic Functions in Standard Form
258
(13)
Chapter Summary
266
(1)
Key Points of This Chapter
266
(1)
Review Exercises
267
(1)
Chapter Test
268
(3)
Using Quadratic Functions to Model Data
271
(52)
Solving Quadratic Equations by Extracting Square Roots
271
(7)
Solving Quadratic Equations by Completing the Square
278
(6)
Solving Quadratic Equations by Using the Quadratic Formula
284
(8)
Solving Systems of Three Linear Equations to Find Quadratic Functions
292
(5)
Solving Systems of Three Linear Equations to Find Quadratic Models
297
(7)
Modeling with Quadratic Functions
304
(19)
Taking It to the Lab
313
(5)
Chapter Summary
318
(1)
Key Points of This Chapter
318
(1)
Review Exercises
319
(2)
Chapter Test
321
(1)
Cumulative Review of Chapters 1--7
321
(2)
Polynomials and Rational Functions
323
(62)
Factoring Sums and Differences of Cubes and Factoring by Grouping
323
(7)
Finding the Domains of Rational Functions and Simplifying Rational Expressions | 677.169 | 1 |
This book is an integrated introduction to the mathematics of coding, that is, replacing information expressed in symbols, such as a natural language or a sequence of bits, by another message using (possibly) different symbols. There are three main reasons for doing this: economy, reliability, and security, and each is covered in detail. Only a modest mathematical background is assumed, the mathematical theory being introduced at a level that enables the basic problems to be stated carefully, but without unnecessary abstraction. Other features include: clear and careful exposition of fundamental concepts, including optimal coding, data compression, and public-key cryptography; concise but complete proofs of results; coverage of recent advances of practical interest, for example in encryption standards, authentication schemes, and elliptic curve cryptography; numerous examples and exercises, and a full solutions manual available to lecturers from
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Common Core Math 1: Linear vs. Exponential Examples
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This handout can be used as guided notes or a worksheet. It contains three scenarios: one that is linear and two that are exponential. Students are asked to give a function rule for the given scenario, name the starting value, rate of change, make a table of values and sketch a graph of the function. | 677.169 | 1 |
ISBN 13: 9781571160140
Geometry Toolkit
Build an intuitive understanding of essential geometry concepts using this exciting, interactive educational software product. The Geometry Toolkit can be used to support any geometry curriculum. This software tool creates a variety of learning environments that can be effectively used in a discovery approach to mathematics. A comprehensive instructional guide helps teachers lead their students to discover important mathematical concepts and to build a fundamental understanding of geometry. The Geometry Toolkit will make learning geometry an enjoyable experience. Features include: 6 interactive geometry learning environments, graphically illustrated tutorials, full color printing capabilities, comprehensive teacher's guide and reproducible activity sheets (PDF), and concept development activities. Mac/Win CD-ROM.
"synopsis" may belong to another edition of this title.
About the Author:
Dr. Ventura is an experienced classroom teacher and has taught elementary, secondary and college levels. He holds a doctorate in education from the University of California, and presents workshops for educators on the instructional uses of microcomputers. Dr. Ventura has designed a collection of math, science, and graphics software products. Each program is designed to be easily integrated into the classroom and provides teachers with new tools for active teaching and active learning. | 677.169 | 1 |
Math 560: Optimization
Fall Semester 2011
Basic Information
Note: this syllabus is temporary, and may change up to the first day of class.
This version posted on: 2011-08-25
The official course title is "Introduction to Optimization Theory".
However, we will try to split our time evenly between applications/modeling,
theory, and methods.
Mathematical optimization is the science of finding the best
(e.g. lowest cost) solution to a problem. Often, we also try to
prove that it is the best, or that it is within some small percentage
of the best solution. Some common examples are:
Selecting the right amounts of various foods to meet
nutritional requirements at minimum cost,
Deciding on prices to maximize income,
Manipulating statistical parameters to maximize a likelihood function, and
Scheduling staff to meet customer demand levels.
Since you have had the prerequisite courses, you are familiar with
finding the minimum or maximum of a function (like in first-semester
calculus), and finding the minimum or maximum of a function of two or three
variables (like in third-semester calculus). In this course, we will be
working with thousands of variables--real applications can use millions
of variables. Of course, we don't solve problems like that by hand.
In our course, we will learn the methods people have developed for
computers to solve such large problems.
Official Course Catalog Entry
An introduction to various aspects of optimization theory, including linear and nonlinear programming, primal dual methods, calculus of variations, optimal control theory, sensitivity analysis and numerical methods.
Prerequisites
Completion of courses in linear algebra and multivariable calculus is assumed.
Some experience using Excel, Mathematica, Maple, or Matlab/Octave/SciLab will also
be very helpful, but it is not strictly a prerequisite.
I am also happy to make appointments if you cannot come to the general
office hours. Please send me e-mail to arrange an appointment.
I am definitely unavailable during the times I teach other classes:
M/T/W/R 12:30-1:20
T/R 2:00-3:15
Many assignments in this course will be in the form of papers, which I
want to be well written. Please consult with
The Writing Center
for help in tuning up your writing.
Required materials
Our suggested (not required!) textbook is
"Operations Research: Applications and Algorithms (4th revised
edition)" by Wayne Winston (2004), retail price around $161 (yikes!)
The price is why it's suggested but not required.
My copy of the book has around 1400 pages (!). I understand that some
students like to order international copies, which may have a
different number of pages. I would be very suspicious of anything
with less than, say, 1000 pages.
I will also be asking for your feedback at the end of many class meetings, written on a 3-by-5 inch notecard. Please pick up a pack of them (at least 25 cards); this should cost about a dollar.
We will also be using software. At least one of the following, and possibly more, is required:
Microsoft Excel (2010 preferred), or other spreadsheet software like Gnumeric or OpenOffice
(MS Works spreadsheet is not powerful enough), and
GMPL/GUSEK, and
Mathematica, Maple, or Matlab/Octave/SciLab
Course Web Page
We will use the
EMU-Online system.
You are expected to keep an eye
on your scores using the system, and get extra help if your scores
indicate the need.
Building and Solving Mathematical Programming Models in Engineering and Science, by Castillo, Conejo, Pedregal, Carcia, and Alguacil
Building Intuition, edited by Chhajed and Lowe, chapter on Knapsack Problem
Course Content
Course Goals
Our primary goal is to teach you to be a good (or great!) optimizer.
To be a good optimizer, you need:
Good habits and procedures, just like a scientist, and
Knowledge of common optimization models.
We have a few secondary goals, which may be more or less applicable to your
personal situation:/Octave/SciLab.
Teach you how to communicate your math models by writing math papers and giving math presentations.
Outline/schedule
Many classes in optimization theory start with linear problems and solution methods, then proceed to nonlinear problems,
because linear problems and solution methods are simpler. Other classes start with nonlinear, because then linear
is just a special case. We will start with mostly nonlinear problems and solution methods because then you will
have a wide base of ideas to do the first (mid-semester) project. might(s)
Instead of exams, we will have two projects.
Your results for each will presentations for the second project may be dropped entirely (at the professor's discretion), previous students
Last time I taught the course, I asked my students to give advice to you, future students,
based on their experiences in my course. Here are some of the highlights:
Previous knowledge about linear algebra would be helpful in this course.
A matlab book may be helpful.
Don't wait until the last minute to do the homework.
Start the homework questions immediately.
Ask a lot of questions
Email him when you get stuck -- his responses are always very prompt and helpful.
You probably don't need the book, but if you're a math geek you're going to want it--it's good.
Look into online tutorials for Excel and Matlab/Octave/Scilab. | 677.169 | 1 |
Main Menu
Why Math?
…in all departments of knowledge, as experience proves, any one who has studied geometry is infinitely quicker of apprehension than one who has not.
– Plato, Republic (VII)
When students begin having trouble with their math, the question they ask most often is, "Why do I even need to know this? Am I ever going to use it?" Of course, we want to affirm the value of what a student is doing in a way that makes sense to them, and so we flip open to the "applications" section of their textbook. But a careful examination of those problems reveals something very startling: the math that most students learn does not apply to really real life. In order to apply high-school level algebra to real life, we need to construct problems so carefully that they no longer resemble real-life situations even in the slightest.
If George is twice Anne's age plus 10, and if George's age minus Anne's age is 21, how old are they?
A student with some algebra under his/her belt will tell you that George is 32 and Anne is 11 (you can verify that these numbers fulfill both conditions stated in the problem). But here's the thing – so what? This variety of problem is presented by many mathematics textbooks as an example of how math applies to real life, but no one has ever encountered a problem like this in real life. The student who says "but when will we use this?" deserves to be listened to. The fact is, the majority of the time one does not use sophisticated mathematics in everyday life.
Maybe we're just not getting sophisticated enough. Say that we flip open a textbook to the quadratic equation section, where we might find a problem like this:
The path of a ball thrown at an angle from a 4-meter high platform is found to follow a path described by the equation
.
How far does the ball travel before hitting the ground?
The problem here is the same as before, namely that the information you are supposed to have access to is so much more obscure than the answer you're supposed to provide. You're really telling me that it's possible to determine – down to the third decimal place – an equation describing the ball's motion, but it's not possible to just plain measure how far it travels? In real life, no one is ever presented with a question where they have access to this sort of information and are asked for this sort of answer. (The problem is more interesting when posed in a Physics course, in which case different information is given to the student and the requested response makes more sense.)
Well wait, math is everywhere. It's used in banking, in making purchases, in measuring ingredients for cooking, in measurement of land for development, and in measurement of rhythm for music. Of course it's used in real life!
Obviously, this is correct. Math really is "all around us," as the same textbooks referred to above claim. But the sort of computational skills required to tackle truly "real-life" math are not what most students spend their time absorbing. Instead, they are taught a mixture of algebraic techniques developed over thousands of years – mostly to tackle imaginary problems. A problem from a mathematics textbook of the ancient world might read something like this:
Say that we know the sum and product of two numbers. Can we determine the two numbers themselves?
Or, in a textbook for the more advanced pupil, something like this:
Inside of a circle, draw a triangle with all three sides equal so that the corners of the triangle lie on the edge of the circle. Now outside of the circle, draw a triangle with all three sides equal so that the sides of the triangle only touch – but do not go inside – the circle. What is the relationship between the size of the triangle inside the circle and the triangle outside the circle?
Do you see the difference here? The two problems just stated do not name the numbers "George" and "Anne." The triangles inside the circle and outside the circle are not two paths traveled by two different hikers. The problems are presented as exercises in thought, and never pretend to be anything other than that. In case it's not obvious yet, the whole point here is that the connection between elementary abstract math like algebra and truly "everyday" life is a farce.
Well… then why study it? To the experts at Strength in Numbers, the plain answer is that even when mathematics is not "useful" in the ordinary sense, it is always beautiful. The ability to determine unknown numbers, unknown measurements, to look for patterns and make predictions, uses the human mind in a way that everyday life doesn't allow us to. Most of the time, everyone needs to be concerned with getting food on our tables, gas in our cars, money in our bank accounts, smiles on our kids' faces. Exercising our minds gives us a vacation from the humdrum pace of everyday life! Instead of pretending to validate mathematics by turning it into a tiresome list of "applications," our expert tutors look beyond and teach the patterns that undergird math. Math hones the mind, develops our logical faculties so we can experience the world as something understandable. It is the language through which we can understand the workings of the universe. It is these things about mathematics which still cause it to be studied, and which motivate the tutors at Strength in Numbers to continue its pursuit.
In some ways, the current math curriculum serves more to confuse than to enlighten. We can't wave a magic wand and make the traditional math curriculum go away, but we can give you the tools to understand the real nature of mathematics and how to practice it successfully. And ask any teacher – stoking our natural intellectual hunger will always lead to better educational outcomes. Get in touch with us today, so we can start working together! | 677.169 | 1 |
The Web's Most Accessible Mathematics Resource
"making mathematics common knowledge"
Public Domain Materials - for the classroom, and independent study
(no installation required) (no sign-up required) (no cookies used)
motto: "If you don't know algebra, your hands are tied."
Print and Online Materials (Completely free. No need even to acknowledge source!)
For the classroom, and independent study Public Domain instructional materials, mostly for Mathematics and Language-Arts, mostly at the high-school level, but ranging from kindergarten to college. Note: Most of the items are in English, but some are in Esperanto. External links are also given. These places are not necessarily wide-open like mine, so be sure to abide by their terms of use. ***************************************************************************************
In the late 1970's I began creating public domain Mathematics articles, and sending a copy of everything to the U.S. National Science Foundation. They sent me a congratulatory ("kudos") letter some years ago.
The purpose of the collection is to provide short, standalone units of supplemental teaching and/or self-study materials. Accordingly, most of the items are only one page in length, although there are a few ambitious projects much longer. The level of the materials ranges from kindergarten to college, with the bulk being at the high school level. I considered it a priority that there be no impediment to their use, and so put all of them into the Public Domain. The materials are all in (Mercan) English, except for an infinitesimal amount in Esperanto. (The language barrier is a non-legal impediment to the use of anything in an ethnic language such as English, and removing, or, circumventing that impediment is what Esperanto is all about.)
So, in a word, these materials are "accessible" in both senses of the word: accessible legally, and accessible linguistically.
With rare or obvious exceptions, such as for the Military Alphabet, or The Chinese Magic Square, or when I'm quoting someone, or availing myself of fair use of existing literature, or providing an external link to someone else's website, all of the materials on this website are written by me (bit by bit, starting long before the World Wide Web came into being). It is not intended that anyone else put materials on this website (except for comments on my blog posts).
Most of the materials are expository in nature, of course, but at least in a few cases I offer what may be a novel perspective on a given topic. For example, the concept of a "computational formula" (as opposed to a "definitional formula") is well-known, but I was unaware of anyone's ever having extended this concept to the absolute value function, so I wrote an article on it, which you can view/download here (also available under the "Table of Contents" tab, along with everything else). Another such item is my unified treatment of square roots, which you can view/download here.
Acknowledgement: A tip of the hat to Ipernity-memberNicole Else, for mentioning (in Esperanto, by the way) the possibility of easily building a website such as this. I saw her mention of this on Wednesday, 4.Apr.2012, and jumped right in to building this site.
The amount of materials that I have to upload is a lot (more than 30 years' worth), and so I am doing the uploading little by little.
All these items (including all my blog posts herein) are completely in the Public Domain. (No rights are reserved.) You can use them, or modify them, in whole or in part, in any way whatsoever, including for commercial purposes. There is no need even to acknowledge source. Again, you may alter, translate, sell, re-transmit, publish, post, broadcast or circulate the materials, or any portion of them, in any way you wish, including in a commercial manner or in support of a commercial enterprise, without any kind of compensation to me. It is just as if you yourself had written them, except that you cannot restrict anyone else's use of these materials. (Of course, these materials are supplied strictly on an as-is basis, with no guarantee or warranty of any kind.)
To put it another way, you can copy or modify any or all of this work, without giving me any credit or notification.
Permission to Copy and Customize My Website: In fact, if you would like a copy of this entire website, I would have no problem with that, if you have the means / expertise to obtain it. My website host is Weebly, and I hereby give them permission to give a copy of this website to anyone who wants one. I envision that you would mashup your copy of this website. For example, if you are only interested in the Language Arts portion, you would perhaps delete all the Mathematics items, and then perhaps re-arrange the links/pages and add links/pages/content of your own. The result might be your own personal password-protected library, or something that you want to share with the world, possibly on a commercial basis. Whatever. Have fun.
"The most satisfying thing in life is to have been able to give a large part of one's self to others." -- Teilhard de Chardin
If you experience any blockage or other problem, please email me, or if you want to make a suggestion, with the understanding that all such submissions are in the Public Domain.
Although some of these materials are original essays, poems, etc., the bulk of the materials are intended as traditional highly-focused instructional materials, e.g., explanations of concepts, exercises, and tests that would be distributed (as supplemental materials, or even as primary materials) to students in a classroom setting, or self-administered by independent learners.
The notice "In the Public Domain. No rights reserved." usually appears under the title of a document, but even if it doesn't, the document is still in the Public Domain. One place this omission happens deliberately is for tests / exercises, because this notice anywhere on the test / exercise would only be distracting for the students. (Also, the notice will not usually be given for the answer keys either.) Another place this omission happens deliberately is for documents in Esperanto. If the document is in Esperanto, and is not a test or exercise (or answer key), instead, the Esperanto translation of the notice ("En la Publika Regno. Neniuj rajtoj rezervitaj.") will probably be present. But, in any case, all documents herein are in the Public Domain. Of course, once you download any of these documents, you can remove the Public Domain notice if you wish, or modify the document in any other way, as you wish.
In case you're wondering about the combination of "Mathematics" and "Language-Arts", it is because Mathematics and Language-Arts are the two irreducible requisites for success. To succeed, you must have at least a modicum of ability in one of them, and a great deal of ability in the other.
"[T]he 3-legged stool of understanding is held up by history, languages, and mathematics. Equipped with these three you can learn anything you want to learn. But if you lack any one of them you are just another ignorant peasant with dung on your boots." --Robert A. Heinlein
My college degree (University of Dallas, 1971) is in Mathematics, but for 9 years (from 2003 to 2012) I was teaching English, in China. It was not English-only (which in the trade is called "Oral English") but content-courses taught in English, and I gradually moved to where about half of my work consisted of teaching Mathematics in English (to Chinese teenagers). As I mentioned, I already had a large stock of supplemental materials for Mathematics before I went to China, but naturally I turned my hand to detailed supplemental materials about English Language and Usage after I got to China.
So, this is not only the web's most accessible mathematics resource; it is also:
The Web's Most Accessible Language-Arts Resource
Of course, websites of others that I link to are not necessarily in the Public Domain. Please understand that and respect their terms of use.
I would like to make special mention of the breakout hit of Mamikon Mnatsakanian, which he has dubbed "Visual Calculus". His insight and technique constitute the greatest advance in Calculus since the time of Isaac Newton. Here is the link to his (copyrighted) website: | 677.169 | 1 |
Working on Your Maths CourseworkMaths coursework's research
The central part of your Maths coursework will be research, necessarily involving calculations and numbers. When you work on your Maths coursework's research, make sure that nobody will interrupt or distract you (the more you are distracted, the more mistakes you will make). Maths coursework research sources will be all kinds of textbooks that have relevant information on the issue.
Maths coursework revision
Before you hand in your Math coursework draft, you need to proofread it a few times. This is easy to proofread the text, but not the calculations, so make sure you have enough time for that. Moreover, be ready to revise it one more time after your teacher finds some minor or major mistakes. This post originally appeared on
Working on Your Maths Coursework9.1 of
10
on the basis of
2926 Review. | 677.169 | 1 |
SymbMath
(an abbreviation for Symbolic Mathematics) is a symbolic calculator that can
solve symbolic math problems.
SymbMath
is a computer algebra system that can perform exact, numeric, symbolic and
graphic computation. It manipulates complicated formulas and returns answers in
terms of symbols, formulas, exact numbers, tables and graph.
SymbMath
is an expert system that is able to learn from user's input. If the user only
input one formula without writing any code, it will automatically learn many
problems related to this formula (e.g. it learns many integrals involving an
unknown function f(x) from one derivative f'(x)).
SymbMath
is a symbolic, numeric and graphics computing environment where you can set up,
run and document your calculation, draw your graph.
SymbMath uses external functions as if standard functions
since the external functions in library are auto-loaded.
SymbMath
is a programming language in which you can define conditional, case,
piecewise, recursive, multi-value functions and procedures, derivatives,
integrals and rules.
Please
read all *.TXT files before running SymbMath. Please copy-and-past examples in
the Help window to practise. The printed documents (100+ pages) is available
from author.
If
you get the SymbMath on ZIP format (e.g. sm32a.zip), you should unzip it with
parameter -d by
pkunzip
-d sm32a c:\symbmath
If
you get the SymbMath with the install file, you should install it byinstall
On
the MS-DOS prompt to run it, typeSymbMath
SymbMath
has two versions: Shareware Version A, and Advanced Version C. The Shareware
version lacks the solve(), trig (except sin(x) and cos(x)), and hyperbolic
functions, (lack 10% keywords). You cannot input these lack functions in
Shareware version.
You
must provide the photocopy of your license or license number for upgrades.
If
you send the author your payment by cheque, money order or bank draft that must
be drawn in Australia, you will get the latest version. If you sign the license
(see the LICENSE.TXT file) and send it to the author, you will be a legal user
for upgrades. If you write a paper about this software on publication, you will
get a free upgrade.
Its
two versions (Shareware and Advanced) are available from the author. The
Shareware version is available from my web sites.
The
Advanced version is copy-protected, so you must insert the original SymbMath
disk into drive A or B before you run SymbMath. By default, it is drive B. If
you use drive A, please copy (or rename) the DRIVE.A file to the SYMBMATH.DRI
file, or you edit drive(2) into drive(1) in the SYMBMATH.DRI file.
In
the following examples, a line of "IN: " means input, which you type
in the Input window, then leave the Input window by pressing <Esc>,
finally run the program by the command "Run"; while a line of
"OUT: " means output. You will see both input and output are
displayed on two lines with beginning of "IN: " and "OUT: "
in the Output window. You should not type the word "IN: ". Some
outputs may be omitted on the examples.
# is a
comment statement.
You
can split a line of command into multi-lines of command by the comma ,. The
comma without any blank space must be the last character in the line.
Note
that you should not be suprised if some functions in the following examples are
not working when their libraries are not in the default directory or missing.
SymbMath
gives the exact value of calculation when the switch numeric := off (default),
or the approximate value of numeric calculation when the switch numeric := on
or by num().
Mathematical
functions are usually not evaluated until by num() or by setting numeric := on.
SymbMath
can manipulate units as well as numbers, be used as a symbolic calculator, and
do exact computation. The range of real numbers is from -infinity to +infinity,
e.g. ln(-inf), exp(inf+pi*i), etc. SymbMath contains many algorithms for
performing numeric calculations. e.g. ln(-9), i^i, (-2.3)^(-3.2),
2^3^4^5^6^7^8^9, etc.
Note
that SymbMath usually gives a principle value if there are multi-values, but
the solve() and root() give all values.
Example:
Exact
and numeric calculations of 1/2 + 1/3.
IN:1/2+1/3#
exact calculation
OUT: 5/6
IN:num(1/2+1/3)#
numeric calculation
OUT: 0.8333333333
Evaluate
the value of the function f(x) at x=x0 by f(x0).
Example:
Evaluate
sin(x) when x=pi, x=180 degree, x=i.
IN:sin(pi), sin(180*degree)
OUT: 0, 0
IN:sin(i), num(sin(i))
OUT: sin(i), 1.175201 i
Example:
Set
the units converter from the minute to the second, then calculate numbers with
different units.
IN:minute:=60*second
IN:v:=2*meter/second
IN:t:=2*minute
IN:d0:=10*meter
IN:v*t+d0
OUT: 250 meter
Evaluate
the expression value by
subs(y,
x = x0)
Example:
Evaluate
z=x^2 when x=3 and y=4.
IN:z:=x^2#
assign x^2 to z
IN:subs(z, x = 3)# evaluate z
when x = 3
OUT: 9
IN:x:=4#
assign 4 to x
IN:z#
evaluate z
OUT: 16
Note
that after assignment of x by x:=4, x should be cleared from assignment by
clear(x) before differentiation (or integration) of the function of x.
Otherwise the x values still is 4 until new values assigned. If evaluating z by
the subs(), the variable x is automatically cleared after evaluation, i.e. the
variable x in subs() is local variable. The operation by assignment is global
while the operation by internal function is local, but operation by external
function is global. This rule also applies to other operations.
The
complex numbers, complex infinity, and most math functions with the complex
argument can be calculated.
Example
.
IN:sign(1+i), sign(-1-i), i^2
OUT: 1, -1, -1
Example:
IN:exp(inf+pi*i)
OUT: -inf
IN:ln(last)
OUT: inf + pi*i
The
built-in constants (e.g. inf, zero, discont, undefined) can be used as numbers
in calculation of expressions or functions.
Some
math functions are discontinuous at x=x0, and only have one-sided function
value. If the function f(x0) gives the discont as its function value, you can
get its one-sided function value by f(x0-zero) or f(x0+zero).
SymbMath
automatically simplifies the output expression. You can further simplify it by
using the built-in variable last in a single line again and again until you are
happy with the answer.
Expressions
can be expanded by
expand(x)
expand
:= on
expandexp
:= on
Remember
that the operation by assignment is global while operation by function is
local. So expand(x) only expands the expression x, but the switch expand := on
expands all expressions between the switch expand := on and the switch expand
:= off. Second difference betwen them is that the switch expand := on only
expands a*(b+c) and (b+c)/p, but does not expands the power (a+b)^2. The
expandexp is exp expand.
Anytime
when you find yourself using the same expression over and over, you should turn
it into a function.
Anytime
when you find yourself using the same definition over and over, you should turn
it into a library.
You
can make your defined function as if the built-in function, by saving your
definition into disk file as a library with the function name plus extension
.li as the filename. e.g. saving the factoria function as the factoria.li file
(see Section Libraries and Packages).
The
argument in the definition should be the pattern x_. Otherwise f() works only
for a specific symbolic value, e.g. x when defining f(x):=x^2. The pattern x_
should be only on the left side of the assignment.
Once
defined, functions can be used in expressions or in other function definitions:
On
the first definition by if(), when f1() is called it gives 1 if x>0, or left
unevaluated otherwise. On the second definition by the if(), when f2() is
called it gives x^2 if x>0, x if x<=0, or left unevaluated otherwise. On
the third definition by the inequality, when f3() is called, it gives 1 for
x>0, 0 for x<=0, or x>0 for symbolic value of x. On the last
definition, when f4() is called, it is evaluated for any numeric or symbolic
value of x.
Remember
that the words "then" and "else" can be replaced by comma
,.
You
cannot differentiate nor integrate the conditional function defined by if(),
but you can do the conditional functions defined by inequalities.
You
can define a function evaluated only for numbers by
f(x_)
:= if(isnumber(x) then x^2)
This
definition is different from the definition by f(x_) := x^2. On the latter,
when f() is called, it gives x^2, regardless whatever x is. On the former, when
f() is called, it gives x^2 if x is a number, or left unevaluated otherwise.
Example:
evaluate to x^2 only if x is
number, by defining a conditional function.
By
default, all variables within procedure are global, except for variables
declared by local(). The multi-statement should be grouped by block(). The
block() only outputs the result of the last statement or the second last one as
its value. The multi-line must be terminated by a comma, (not by a comma and a
blank space). Local() must be the last one in block().
Example:
define a numeric integration
procedure ninte() and calculate integral of x^2 from x=1 to x=2 by call
ninte().
You
can define transform rules. Defining rules is similar to defining functions. In
defining functions, all arguments must be simple variables, but in defining
rules, the first argument can be a complicated expression.
You
can finds real or complex limits, and discontinuity or one-sided value.
First
find the expression value by subs(y, x = x0) or the function value by f(x0)
when x = x0.
If
the result is the discont (i.e. discontinuity), then use the one-sided value
x0+zero or x0-zero to try to find the one-sided function or expression value.
For
a function f(x), you can evaluate the left- or right-sided function value,
similar to evaluate the normal function value:
f(x0-zero)
f(x0+zero)
For
an expression y, you can evaluate its one-sided expression value by
subs(y,
x = x0-zero)
subs(y,
x = x0+zero)
The
discont (discontinuity) means that the expression has a discontinuity and only
has the one-sided value at x=x0. You should use x0+zero or x0-zero to find the
one-sided value. The value of f(x0+zero) or f(x0-zero) is the right-sided or
left-sided function value as approaching x0 from positive (+inf) or negative
(-inf) direction, respectively, i.e. as x = x0+ or x = x0-.
If
the result is undefined (indeterminate forms, e.g. 0/0, inf/inf, 0*inf, and
0^0), then find its limit by
lim(y,
x = x0)
If
the limit is discont, then you can find a left- or right-sided limit when x
approaches to x0 from positive (+inf) or negative (-inf) direction at
discontinuity by
lim(y,
x = x0+zero)
lim(y,
x = x0-zero)
Example:
Evaluate
y=exp(1/x) at x=0, if the result is discontinuity, find its left-sided and
right-sided values (i.e. when x approaches 0 from positive and negative
directions).
If
you differentiate f(x) by f'(x), x must be a simple variable and f(x) must be
unevaluated.
f'(x0)
is the same as d(f(x0),x0), but different from diff(f(x), x=x0). f'(x0) first
evaluates f(x0), then differentiates the result of f(x0). But diff(f(x), x=x0)
first differentiates f(x), then replace x with x0. Note that sin'(x^6) gives
cos(x^6) as sin'(x^6) is the same as d(sin(x^6), x^6). sin'(0) gives d(0,0) as
sin(0) is evaluated to 0 before differentiation, you should use
diff(sin(x),x=0) which gives 1.
Example:
Differentiate
the expression f=sin(x^2+y^3)+cos(2*(x^2+y^3)) with respect to x, and with
respect to both x and y.
The
equations can be operated (e.g. +, -, *, /, ^, expand(), diff(), inte()). The
operation is done on both sides of the equation, as by hand. You can find roots
of a polynomial, algebraic equations, systems of equations, differential and
integral equations.
You
can get the left side of the equation by
left(left_side
= right_side)
or get the right side by
right(left_side
= right_side)
You
can assign equations to variables.
Example:
IN:eq1:= x + y = 3
IN:eq2:= x - y = 1
IN:eq1+eq2
OUT: 2 x = 4
IN:last/2
OUT: x = 2
IN:eq1-eq2
OUT: 2 y = 2
IN:last/2
OUT: y = 1
Example:
Solve
an equation sqrt(x+2*k) - sqrt(x-k) = sqrt(k), then check the solution by
substituting the root into the equation.
IN:eq1 := sqrt(x + 2*k) - sqrt(x - k) =
sqrt(k)
OUT: eq1 := sqrt(x + 2*k) -
sqrt(x - k) = sqrt(k)
IN:eq1^2
OUT: ((2*k + x)^0.5 - ((-k) +
x)^0.5)^2 = k
IN:expand(last)
OUT: 2*x + k + (-2)*(2*k +
x)^0.5*((-k) + x)^0.5 = k
IN:last-k-2*x
OUT: (-2)*(2*k + x)^0.5*((-k) +
x)^0.5 = (-2)*x
IN:last/(-2)
OUT: (2*k + x)^0.5*((-k) +
x)^0.5 = x
IN:last^2
OUT: (2*k + x)*((-k) + x) = x^2
IN:expand(last)
OUT: (-2)*k^2 + k*x + x^2 = x^2
IN:last-x^2+2*k^2
OUT: k*x = 2*k^2
IN:last/k
OUT: x = 2*k
IN:subs(eq1, x = right(last))
OUT: k^0.5 = k^0.5
You
can solve algebraic equations step by step, as above. This method is useful in
teaching, e.g. showing students how to solve equations.
solve a polynomial and systems
of linear equations on one step. It is recommended to set the switch expand:=on
when solve the complicated equations. All of the real and complex roots of the
equation will be found by solve(). The function solve() outputs a list of roots
when there are multi-roots. You can get one of roots from the list, (see
Chapter 4.9 Arrays, Lists, Vectors and Matrices).
Example:
Solve
a+b*x+x^2 = 0 for x, save the root to x.
IN:solve(a+b*x+x^2 = 0, x)#
solve or re-arrange the equation for x
OUT: x = [-b/2 + sqrt((b/2)^2 -
a),-b/2 - sqrt((b/2)^2 - a)]
IN:x := right(last)#
assign two roots to x
OUT: x := [-b/2 + sqrt((b/2)^2
- a),-b/2 - sqrt((b/2)^2 - a)]
IN:x[1]#
the first root
OUT: -b/2 + sqrt((b/2)^2 - a)
IN:x[2]#
the second root
OUT: -b/2 - sqrt((b/2)^2 - a)
Example:
Solve
x^3 + x^2 + x + 5 = 2*x + 6.
IN:num(solve(x^3+x^2+x+5 = 2*x+6, x))
OUT: x = [1, -1, -1]
The
function solve() not only solves for a simple variable x but also solves for an
unknown function, e.g. ln(x).
You
can compute partial, finite or infinite sums and products. Sums and products
can be differentiated and integrated. You construct functions like Taylor
polynomials or finite Fourier series. The procedure is the same for sums as
products so all examples will be restricted to sums.The general formats for these functions
are:
sum(expr,
x from xmin to xmax)
sum(expr,
x from xmin to xmax step dx)
prod(expr,
x from xmin to xmax)
prod(expr,
x from xmin to xmax step dx)
The
expression expr is evaluated at xmin, xmin+dx, ...up to the last entry in the series not
greater than xmax, and the resulting values are added or multiplied.The part "step dx" is optional
and defaults to 1.The values of
xmin, xmax and dx can be any real number.
Here
are some examples:
sum(j, j from 1 to 10)
for 1 + 2 + .. + 10.
sum(3^j,
j from 0 to 10 step 2)
for 1 + 3^2 + ... + 3^10.
Here
are some sample Taylor polynomials:
sum(x^j/j!,
j from 0 to n)
for exp(x).
sum((-1)^j*x^(2*j+1)/(2*j+1)!,
j from 0 to n)
for sin(x) of degree 2*n+2.
Remember,
the 3 keywords (from, to and step) can be replaced by the comma ,.
to find the Taylor series at x=0.
The argument (order) is optional and defaults to 5.
Example:
Find
the power series expansion for cos(x) at x=0.
IN:series(cos(x), x)
OUT: 1 - 1/2 x^2 + 1/24 x^4
The
series expansion of f(x) is useful for numeric calculation of f(x). If you can
provide derivative of any function of f(x) and f(0), even though f(x) is
unknown, you may be able to calculate the function value at any x, by series
expansion. Accuracy of calculation depends on the order of series expansion.
Higher order, more accuracy, but longer calculation time.
Example:
calculate
f(1), knowing f'(x)=-sin(x) and f(0)=1, where f(x) is unknown.
You
can define a list by putting its elements between two square brackets. e.g.
[1,2,3]
You
can define lists another way, with the command:
[
list(f(x), x from xmin to xmax step dx) ]
This
is similar to the sum command,but
the result is a list:
[f(xmin),
f(xmin+dx), ..., f(xmin+x*dx), ...]
which continues until the last
value of xmin + x*dx<=
xmax.
You
also can assign the list to a variable, which variable name become the list
name:
a
:= [1,2,3]# define the list of a
b
:= [f(2), g(1), h(1)]# assumes f,g,h
defined
c
:= [[1,2],3,[4,5]]# define the list of c
Lists
are another kind of value in SymbMath, and they can be assigned to variables
just like simple values. (Since variables in SymbMath language are untyped, you
can assign any value to any variable.).
If
you have assigned a list to a variable x, you can access the j-th element by
the list index x[j]. The first element of x is alwaysx[1].If the x[j] itself is a list, then its j-th
element is accessed by repeating the similar step.But you can not use the list
index unless the list is already assigned to x.
e.g.
IN:x := [[1,2],3,[4,5]]# define the x list
IN:x[1], x[2]#
take its first and 2nd element
OUT: [1, 2], 3
IN:x#
access the entire list of x
OUT: [[1, 2], 3, [4,5]]
IN:member(x, 2)#
same as x[2]
OUT: 3
An
entire sub-list of a list xcan be
accessed with the command x[j], which is the list:
Lists
can be added, subtracted, multiplied, and divided by other lists or by
constants.When two lists are
combined, they are combined term-by-term, and the combination stops when the
shortest list is exhausted.When a
scalar is combined with a list, it is combined with each element of the
list.Try:
a
:= [1,2,3]
b
:= [4,5,6]
a
+ b
a
/ b
3
* a
b
- 4
Example
4.9.2.4.1.
Two
lists are added.
IN:[a1,a2,a3] + [b1,b2,b3]
OUT: [a1 + b1, a2 + b2, a3 +
b3]
IN:last[1]
OUT: a1 + b1
If
L is a list, thenf(L) results in a
list of the values, even though f() is the differentiation or integration
function (d() or inte()).
IN:sqrt([a, b, c])
OUT: [sqrt(a), sqrt(b),
sqrt(c)]
IN:d([x, x^2, x^3], x)
OUT: [1, 2*x, 3*x^2]
If
you use a list as the value of a variable in a function, SymbMath will try to
use the list in the calculation.
You
can sum all the elements in a list x by
listsum(x)
Example:
IN:listsum([a,b,c]^2)
OUT: a^2 + b^2 + c^2
This
function takes the sum of the squares of all the elements in the list x.
You
can do other statistical operations (see Section 4.10. Statistics) on the list,
or plot the list of numeric data (see Section 5. Plot).
You
can find the length of a list (the number of elements in a list) with:
If
you want to look at a table of values for a formula, you can use the table
command:
table(f(x),
x)
table(f(x),
x from xmin to xmax)
table(f(x),
x from xmin to xmax step dx)
It
causes a table of values for f(x) to be displayed with x=xmin, xmin+dx, ...,
xmax.If xmin, xmax, and step omit,
then xmin=-5, xmax=5, and dx=1 for default. You can specify a function to be in
table(),
Example:
Make a table of x^2.
IN:table(x^2, x)
OUT:
-5,25
-4,16
-3,9
-2,4
::
::
Its
output can be written into a disk file for interfacing with other software
(e.g. the numeric computation software).
One
of the most important feature of SymbMath is its ability to deduce and expand
its knowledge. If you provide it with the necessary facts, SymbMath can solve
many problems which were unable to be solved before. The followings are several
ways in which SymbMath is able to learn from your input.
Finding
derivatives is much easier than finding integrals. Therefore, you can find the
integrals of a function from the derivative of that function.
If
you provide the derivative of a known or unknown function, SymbMath can deduce
the indefinite and definite integrals of that function. If the function is not
a simple function, you only need to provide the derivative of its simple
function. For example, you want to evaluate the integral of f(a*x+b), you only
need to provide f'(x).
If
you know a derivative of an function f(x) (where f(x) is a known or unknown
function), SymbMath can learn the integrals of that function from its
derivative.
Example:
check
SymbMath whether or not it had already known integral of f(x)
IN:inte(f(x), x)
OUT: inte(f(x), x)
IN:inte(f(x), x, 1, 2)
OUT: inte(f(x), x, 1, 2)
As
the output displayed only what was typed in the input without any computed
results, imply that SymbMath has no knowledge of the indefinite and definite
integrals of the functions in question. Now you teach SymbMath the derivative
of f(x) on the first line, and then run the program again.
IN:f'(x_) := exp(x)/x
IN:inte(f(x), x)
OUT: x*f(x) - e^x
IN:inte(f(x), x, 1, 2)
OUT: e - f(1) + 2*f(2) - e^2
As
demonstrated, you only supplied the derivative of the function, and in exchange
SymbMath logically deduced its integral.
Another
example is
IN:f'(x_) := 1/sqrt(1-x^2)
IN:inte(f(x), x)
OUT: sqrt(1 - x^2) + x*f(x)
IN:inte(k*f(a*x+b), x)
OUT: k*(sqrt(1 - (b + a*x)^2) +
(b + a*x)*f(b + a*x))/a
IN:inte(x*f(a*x^2+b), x)
OUT: sqrt(1-(a*x^2 + b)^2) +
(a*x^2 + b)*f(a*x^2 + b)
The
derivative of the function that you supplied can be another derivative or
integral.
You
supply a simple indefinite integral, and in return, SymbMath will perform the
related complicated integrals.
Example:
Check
whether SymbMath has already known the following integrals or not.
IN:inte(f(x), x)
OUT: inte(f(x), x)
IN:inte((2*f(x)+x), x)
OUT: inte((2*f(x)+x), x)
IN:inte(inte(f(x)+y), x), y)
OUT: inte(inte(f(x)+y), x), y)
Supply,
like in the previous examples, the information: integral of f(x) is f(x) - x;
then ask the indefinite integral of 2*f(x)+x, and a double indefinite integral
of 2*f(x) + x, and a double indefinite integral of respect to both x and y.
Change the first line, and then run the program again.
SymbMath
can learn complicated derivatives from a simple derivative, even though the
function to be differentiated is an unknown function, instead of standard
function.
Example
:
Differentiate
f(x^2)^6, where f(x) is an unknown function.
IN:d(f(x^2)^6, x)
OUT: 12 x f(x^2)^5 f'(x^2)
Output
is only the part derivative. f'(x^2) in the output suggest that you should
teach SymbMath f'(x_). e.g. the derivative of f(x) is another unknown function
df(x), i.e. f'(x_) = df(x), assign f'(x_) with df(x) and run it again.
The
difference between learning and programming is as follows: the learning process
of SymbMath is very similar to the way human beings learn, and that is
accomplished by knowing certain rule that can be applied to several problems.
Programming is different in the way that the programmer have to accomplish many
tasks before he can begin to solve a problem. First, the programmer defines
many subroutines for the individual integrands (e.g. f(x), f(x)+y^2, 2*f(x)+x,
x*f(x), etc.), and for individual integrals (e.g. the indefinite integral,
definite integral, the indefinite double integrals, indefinite triple
integrals, definite double integrals, definite triple integrals, etc.), second,
write many lines of program for the individual subroutines, (i.e. to tell the
computer how to calculate these integrals), third, load these subroutines,
finally, call these subroutines. That is precisely what SymbMath do not ask you
to do.
In
one word, programming means that programmers must provide step-by-step
procedures telling the computer how to solve each problems. By contrast,
learning means that you need only supply the necessary facts (usually one f'(x)
and/or one integral of f(x)), SymbMath will determine how to go about solutions
of many problems.
If
the learning is saved as a library, then you do not need to teach SymbMath
again when you run SymbMath next time.
SymbMath
is an interpreter, and runs a SymbMath program in the Input window, which is
written by any editor in the text (ASCII) file format.
SymbMath
language is a procedure language, which is executed from top to bottom in a
program, like BASIC, FORTRAN, or PACSAL. It also is an expression-oriented
language and functional language.
The
SymbMath program consists of a number of statements. The most useful statement
contains expressions, the expression includes data, and the most important data
is functions.
The
structure of SymbMath language is:
data
-> expression -> statement -> program
Note
that upper and lower case letters are different in SymbMath language, (e.g. abc
is different from ABC) until the switch lowercase := on.
In
the following examples, a line of "IN: " means input, which you type
in the Input window, then leave the Input window by <Esc>, finally run
the program by the command "Run"; while a line of "OUT:"
means output. You will see both input and output are displayed on two lines
with beginning of "IN: " and "OUT: " in the Output window.
You should not type the word "IN: ". Some outputs may be omit on the
examples.
# is a
comment statement.
You
can split a line of command into multi-lines of command by the comma ,. The
comma without any blank space must be the last character in the line.
The
data types in SymbMath language is the numbers, constants, variables,
functions, equations, arrays, array index, lists, list index, and strings. All
data can be operated. It is not necessary to declare data to be which type, as
SymbMath can recognise it.
The
range of the output real numbers is the same as input when the switch numeric
:= off, but when the switch numeric := on, it is
-inf,
-1.E300 to -1.E-300, 0, 1.E-300 to 1.E300, inf.
It
means that the number larger than 1.e300 is converted automatically to inf, the
absolute values of the number less than 1.e-300 is converted to 0, and the
number less than -1e300 is converted to -inf.
For
examples:
-------------------------------------------
NumbersType
23integer
2/3rational
0.23real
2.3E2real
2+3*icomplex
2.3+icomplex
---------------------------------------------
That
"a" and "b" are the same means a-b = 0, while that they are
different means a-b <> 0.
For
the real numbers, the upper and lower case letters E and e in exponent are the
same, e.g. 1e2 is the same as 1E2.
Notice
that the discont and undefined constants are different. If the value of an
expression at x=x0 is discont, the expression only has the one-sided value at
x=x0 and this one-sided value is evaluated by x=x0+zero or x=x0-zero. If the
value of an expression at x=x0 is undefined, the expression may be evaluated by
the function lim().
The
sequence of characters is used as the name of variables. Variable names can be
up to 128 characters long. They must begin with a letter and use only letters
and digits.SymbMath knows upper
and lower case distinctions in variable names, so AB, ab, Ab and aB are the
different variables. They are case sensitive until the switch lowercase is set
to on (i.e. lowercase := on).
Variables
can be used to store the results of calculations. Once a variable is defined,
it can be used in another formula. Having defined X as above, you could define
Y := ASIN(X). You can also redefine a variable by storing a new value in
it.If you do this, you will lose
the original value entirely.
Assign
a result to a variable, just put
<var-name>
:=expression
e.g.x := 2 + 3# assign value to x
Variables
can be used like constants in expressions.
For
example:
a := 2 + 3
b := a*4
If
an undefined variable is used in an expression, then the expression returns a
symbolic result (which may be stored in another variable).Pick an undefined variable name, say x,
and enter:
y
:= 3 + x#
formula results since x undefined
x
:= 4#
Now x is defined
y#
y returns 7, but its value is still the formula 3 + x
x
:= 7#
revalue x
y#
new value for y
Note
that in symbolic computation, the variable has not only a numeric value but
also a symbolic value.
Symbolic
values for variables are useful mostly for viewing the definitions of functions
and symbolic differentiation and integration.
Watch
out for infinite recursion here.Defining
x
:= x+3
when x has no initial value, it
will not cause an immediate problem, but any future reference to xwill result in an infinite recursion !
A
value can be assigned to the variable, by one of three methods:
(1)
the assignment :=,
(2)
the user-defined function f(),
(3)
subs(y, x = x0).
e.g.
y:=x^2
x:=2#
assignment
y
f(2)#
if f(x) has been defined, e.g. f(x_):=x^2.
subs(x^2, x = 2)#
evaluate x^2 when x = 2.
The
variable named last is the built-in as the variable last is always automatically
assigned the value of the last output result.
The
usual used independent variable is x.
By
default, |x| < inf and all variables are complex, except that variables in
inequalities are real, as usual only real numbers can be compared. e.g. x is
complex in sin(x), but y is real in y > 1.
You
can restrict the domain of a variable by assuming the variable is even, odd,
integer, real number, positive or negative (see Chapter Simplification and
Assumption).
These
are two types of functions: internal and external. The internal function is compiled
into the SymbMath system. The external function is the library written in
SymbMath language, which is automatically loaded when it is needed. (See
Chapter Library and Package). The usage of both types are the same. You can
change the property or name of the external function by modifying its library
file, or you add a new external function by creating its library file, but you
cannot change the internal function.
Different
versions of SymbMath have different number of standard mathematical functions.
The Advanced Version C has all of them. See the following table in detail for
other versions. All below standard functions, (except for random(x), n!, fac(n)
and atan2(x,y)), can be differentiated and integrated symbolically.
If a
second argument x is omitted in the functions d(y) and inte(y), they are
implicit derivatives and integrals. If f(x) is undefined, d(f(x), x) is
differentiation of f(x). These are useful in the differential and integral
equations. (see later chapters).
For
examples:
inte(inte(F,x),
y) is double integral of F with respect to both variables x and y.
d(d(y,x),t)
is the mixed derivative of y with respect to x and t.
The
keywords "from" "to" "step" "," are the
same as separators in multi-argument functions. e.g. inte(f(x), x, 0, 1) are
the same as inte(f(x), x from 0 to 1).
You
can define the new functions, which include the standard functions, calculus
functions, and algebraic operators.
Define
a new function f(x) by
f(x_)
:= x^2
and then call f(x) as the
standard functions. The function name can be any name, except for some
keywords. (for the maximum number of arguments, see ChapterSystem Limits).
Clears
a variable or function from assignment by
clear(x)#
clear x from assignment.
clear(f(x))#
clear f(x) from assignment.
clear(a>0)#
clear a>0 from assignment.
Variables
can be used in function definitions. It leads to an important difference
between functions and variables.When a variable is defined, all terms of the definition are evaluated.When a function is defined, its terms
are not evaluated; they are evaluated when the function is evaluated. That
means that if a component of the function definition is changed, that change
will be reflected the next time the function is evaluated.
A
procedure is similar to a function, but the right side of assignment in its
definition is multi statements grouped by block(). The block(a,b,c) groups
a,b,c and only returns the last argument as its value, or returns the second
last argument as its value if the last argument is local(). It is used as
grouper in definition of a procedure. All variables in block are global, except
for variables declared by local().
e.g.
f(x_):=block(p:=x^6,p,local(p))
Remember
that you can split a line of program into multi-lines program at comma ,.
An
equation is an equality of two sides linked by an equation sign =, e.g. x^2+p =
0, where the symbol = stands for an equation. Note that the symbols
"=", "==" and ":=" are different: ":="
is the assignment, "==" is the equal sign, but "=" is the
equation sign.
A
string is a sequence of characters between two quotation marks. e.g.
"1234567890". Note that 1234 is number but "1234" is
string. The number can be calculated and only has 11 of max digits, while
string cannot be calculated and has 64000 of max characters long.
Note
that the output of strings in SymbMath is without two quotation marks. This
makes text output to graph and database more readable.
Strings
can be stored in variables, concatenated, broken, lengthen, and converted to
numbers if possible.
The
expressions (i.e. expr) are made up of operators and operands. Most operator
are binary, that is, they take two operands; the rest are unitary and take only
one operand. Binary operators use the usual algebraic form, e.g. a+b.
There
are two kinds of expressions: numeric and Boolean. The numeric expression is
combination of data and algebraic operators while the Boolean expression is
combination of data and relational operators and logic operators. These two
kinds of expressions can be mixed, but the numeric expression has higher
priority than Boolean operators. x*(x>0) is different from x*x>0.
x*x>0 is the same as (x*x)>0.
Before
you can write loops, you must be able to write statements that evaluate to 1 or
0, and before you can do that, you must be able to write useful statements with
logical values. In mathematics, these are relational statements.
SymbMath
uses the logical operators:AND,
and OR.You can combine comparison
operators with them to any level of complexity. In contrast to Pascal, logical
operators in SymbMath have a lower order or precedence than the comparisons, so
a < bandc > d works as expected.The result of combining logical values
with
logical operators is another
logical value (1 or 0).Bit
operations on integers can be performed using the same operations, but result
is integers.
SymbMath
uses the "short-circuit" definition of AND and OR when the arguments
are Boolean.Here are tables that
show how AND and OR are defined:
a
AND b
--------------------------------------------------------
b10
a
110
000
------------------------------------------------------
a
OR b
--------------------------------------------------------
b10
a
111
010
------------------------------------------------------
Short-circuit
evaluation is used because often one condition must be tested before another is
meaningful.
The
result of Boolean expression with logic operators is either 1 or 0. Boolean
expression like (1 < 3 or 1 > 4) return a real value 1 or 0.Numeric expressions can replace Boolean
ones, provided they evaluate to 1 or 0.The advantage here is that you can define the step function that is 0
for x < a and 1 for x > a by entering:
A
function call activates the function specified by the function name. The
function call must have a list of actual parameters if the corresponding
function declaration contains a list of formal parameters. Each parameter takes
the place of the corresponding formal parameter. If the function is external,
the function call will automatically load the library specified by its function
name plus extension .LI when needed.
The
assignment in SymbMath language is similar to assignment in such language as
PASCAL.
An
assignment operator is:=
The
assignment statement specifies that a new value of expr2 be assigned to expr1,
and saved into memory. The form of the assignment statements is
expr1
:= expr2
You
can use assignment for storing result.
You
can assign the result of calculation or any formula to a variable with a
command like:X :=
SIN(4.2).
The
assignments are useful for long calculations.You can save yourself a lot of
recalculations by always storing the results of your calculations in your own
variables instead of leaving them in the default variable last.
You can destroy the assignment to X with
the command clear(X). If X stored a large list, you could regain a considerable
amount of memory by clearing X. Also, since a variable and a function can have
the same name, you can clear a variable p, not a function p(x).
The
assignment operator is also used in the definition of a function or procedure.
Variables
can be used in function definitions, and that leads to an important difference
between functions and variables.When a variable is defined, all terms of the definition are
evaluated.When a function is
defined, its terms are not evaluated; they are evaluated when the function is
evaluated.That means that if a
component of the function definition is changed, that change will be reflected
the next time the function is evaluated.
e.g.
IN:p:=2+3#
2+3 is evaluated at the time of assignment, p is assigned with 5.
OUT: p := 5
IN:p(x):=2+3#
2+3 is evaluated when the value of p(x) is requested,
#
p(x) is assigned with 2+3.
OUT: p(x) := 2+3
If
the left hand side of the assignment is a variable, it is the immediate
assignment (i.e. expr2 is evaluated at the time of assignment); if the left
hand side is a function, it is the delayed assignment (i.e. expr2 is evaluated
when the value of expr1 is requested).
You
can force all the components of a function to be evaluated when the function is
defined by preceding the function with the command eval():
f(x_)
:= eval(2+3)# f(x_) is
assigned with 5
Note
that not only a variable but also any expression can be assigned. e.g. x := 2,
sin(x)/cos(x) := tan(x), a>0 := 1.
if(condition
then x) gives x if condition evaluates to 1, or no output otherwise.
if(condition
then x else y) gives x if condition evaluates to 1, y if it evaluates to 0, or
no output if it evaluates to neither 1 or 0. The 2 words (then and else) can be
replaced by comma ,.
It
is useful in definition of the use-defined function to left the function
unevaluated if the argument of the function is not number. e.g. define f(x_) :=
if(isnumber(x), 1), then call f(x), f(10) gives 1, and f(a) gives f(a).
The
filename is any MS-DOS file name. If the file is not in the current directory,
the filename should include the directory. e.g.
readfile("directory\filename")
e.g.
read a file named "inte.sm":
readfile("inte.sm")
It
seems to copy the file into the user program.
After
a file is read, you can call any part of this package from a second package, as
it seems the part of the first program has already been in the second program.
you can read many files into the SymbMath program at a time. However, all names
of the variables are public and name conflicts must be avoided.
Note
that the file must be closed by the closefile() command when writing a file
with the openfile() command, but the file is automatically closed after reading
the file. There must be the end statement at the end of file for reading.
SymbMath
can read expressions from a disk file, then manipulate the expression, and
finally write the result into another disk file.
The
all above statements are simple statements. The sequence statement specifies
that its component statements are to be executed in the same sequence as they
are written. They are separated by the separators (comma ","). e.g.
A
library is a file of an external function, which filename is its function name
within 8 letters plus extension .li. e.g. the library named sin.li is a file of
the sin(x) function definition.
The
library (the *.LI file) is similar to the MS-DOS *.BAT file. You do not need to
load or read the library by any command. SymbMath automatically load the
library when it is needed. For example, when you use the sin(x) function first
time, the library sin.li will be auto-loaded. The library must be in the
default directory, otherwise the library is not loaded and the function is not
working. Only the assignments in the library can be loaded, and others in the
library will be omitted, but all of these assignments will be not evaluated
when they are loaded. You can clear the library sin.li from memory by
clear(sin(x)).
You
can have libraries (external functions) as many as your disk space available.
You should use the "one function per file" convenience.
Note
that all names of the variables in libraries and packages are public (global)
except for those declared by local() and name conflicts must be avoided.
A
package is the SymbMath program file which filename has notextension .LI. It is recommended that its
filename has the extension .SM.
A
package is similar to a library, but the package must be read by a command
readfile("filename")
The
filename can be any MS-DOS filename. It is recommended that the filename is
same function name used in your program, plus the extension .sm. e.g. inte.sm
is the filename of the integral package as the name of integral function is
inte(). If the file is not in the current directory, the filename should
include the directory. e.g.
readfile("directory\filename")
After
reading the package, you can call the commands in the package from your
program.
The
readfile() command must be in a single line anywhere.
Many
packages can be read at a time.
You
can convert a package of f.sm into a library by renaming f.sm to f.li for auto
loading, or a library f.li to a package by renaming f.li to f.sm for not auto
loading.
You
can get help for all libraries by the library Index command in the Help menu.
You first open the library index window by this command, then open a library by
selecting its library name in the library index window.
There
are many libraries and packages. The following are some of them.
When
a program is run in the Input window, SymbMath first automatically reads (or
runs) the initial package "init.sm". The commands in the
"init.sm" package seems to be the SymbMath system commands. You can
read other packages (e.g. f.sm) in the initial package "init.sm", so
the commands in the package "f.sm" seems to be in SymbMath system.
You do this by adding the readfile("f.sm") into the init.sm file:
You
can run SymbMath from another software as a engine. Anthoer software sends a text
file to SymbMath, then run SymbMath in background, get result back from
SymbMath.
Interface
with other software, (e.g. CurFit, Lotus 123) is similar to interface with the
software PlotData in the plotdata package "plotdata.sm".
After
load the file "plotdata.sm", the functions
plotdata(y,
x)
plotdata(y,
x from xmin to xmax)
plotdata(y,
x from xmin to xmax step dx)
plot a function of y by mean of
the software PlotData. The plotdata() first opens a file
"SymbMath.Out" for writing, then write the data table of the y
function into the file "SymbMath.Out", then close the file, and
finally call the software PlotData to plot. These are done automatically by
plotdata(). After it exits from PlotData, it automatically return to SymbMath.
When
SymbMath is interfaced with the software PlotData, SymbMath produces the data
table of functions, and PlotData plots from the table. So SymbMath seems to
plot the function. This interface can be used to solve equations graphically.
Example:
plot x^2 by interfacing
software PlotData.
IN:readfile("plotdata.sm")
IN:plotdata(x^2, x)
in the software PlotData, you
just select the option to read the file "SymbMath.Out" and to plot.
PlotData reads the data in the SymbMath format without any modification (and in
many data format).
On
both monochrome and color systems, you can draw lines and graphs with different
line styles.(Since the line
segments used to draw graphs are usually very short, different line styles may
not be distinguished in graphs, but they will be distinguished on long
lines.)Linestyles are indicated by
integers in the range 0..3, and are set by the command:
setlinestyle(style,u,thickness)
where style, u and thickness are
integers.
You
can set the text style by
settextstyle(font,direction,size)
where font, direction and size
are integers.
You
can add labels to your graphs by
writes(s)
You
can put alphanumeric labels anywhere on your graphic screens. They can be
horizontal or vertical, and they can be printed in various sizes. To print a
stringshorizontally on the screen with the
lower-left corner at the screen coordinates (x,y), use two commands:
moveto(x,y),
writes(s)
To write vertically bottom to top,
use two commands:
settextstyle(1,2,2),
writes(s)
If
SymbMath attempts to graph a point (x,y) which is outside the the screen
coordinate, it ignores the point and continues.No error message is generated, and even
functions which are undefined on part of the graphing domain can be graphed.
You
can get the max x and max y on your graphics screen
coordinates by
getmaxx
getmaxy
You
can get the current point(x, y) on your graphics screen coordinates by
getx
gety
You
can get the background color and foregroud color on your graphics screen by
getbkcolor
getcolor
You
can read a character from the keyboard or pause by the command:
readchar
You
can clear graph by
cleardevice
SymbMath
auto goes back the text mode at the end of run. You can force it goes back the
text mode by the command:
You
can plot a function of y = f(x) on the xy-plane by external function:
plot(f(x),x)
plot(f(x),x,xmin,xmax)
plot(f(x),x,xmin,xmax,ymin,ymax)
plot(f(x),x,xmin,xmax,ymin,ymax,color)
f(x) can be either a
functionwith bound variable x or
an expression involving x.For
example, you could graph the parabola with the command plot(x^2,x).
The
xmin and xmax are range of x-axis, the ymin and ymax are range of y-axis. The
default values are xmin=-5, xmax=5, ymin=-5, and ymax=5. The values of xmin,
xmax, ymin, ymax are real numbers, such thatxmin < xmaxandymin < ymax.Thses values
tell SymbMath that the visible screen corresponds to a portion of the xy-plane
withxmin <= x <= xmaxandymin <= y <= ymax.
The
operator plot() plots one point (x,f(x)) for each pixel on the x-axis, and
connects successive points.To omit
the connections and just plot the points, use the command:
dotplot(f(x),x)
To
plot only every 20th point, which is useful for rapidly graphing complicated
functions, use
sketch(f(x),x)
If
you want your circles and squares to look correct --that is, if you want one
vertical unit to be really the same distance as one horizontal unit--you should
select window parameters so that the horizontal axis is 1.4 times as long as
the vertical axis.
The
"DOS shell" command executes a DOS command and automatically returns
to the SymbMath system if you provide a DOS command on the command window,
otherwise it goes to the DOS shell, the control must be returned to the
resident SymbMath system with the EXIT command on the DOS shell. | 677.169 | 1 |
06188580Functions and Change: A Modeling Approach to College Algebra and Trigonometry
Intended for precalculus courses requiring a graphing calculator, Functions and Change emphasizes the application of mathematics to real problems students encounter each day. Applications from a variety of disciplines, including Astronomy, Biology, and the Social Sciences, make concepts interesting for students who have difficulty with more theoretical coverage of mathematics. In addition to these meaningful applications, the authors' easy-to-read writing style allows students to see mathematics as a descriptive problem-solving tool. An extended version of the successful Functions and Change: A Modeling Approach to College Algebra, this text includes three chapters of trigon | 677.169 | 1 |
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